PUBLIC DRAFT
Focus in High School Mathematics:
Reasoning and Sense Making
August 28, 2008
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. ii
National Council of Teachers of Mathematics
High School Curriculum Project
Writing Group
John Carter, NCTM Board Liaison
Assistant Principal for Teaching and Learning
Adlai E. Stevenson (Illinois) High School
W. Gary Martin, Chair
Professor of Mathematics Education
Auburn University
Susan Forster
Mathematics Teacher
Bismarck (North Dakota) High School
Bill McCallum
Professor of Mathematics
University of Arizona
Albert Goetz, Staff Liaison
Mathematics Teacher Editor
National Council of Teachers of Mathematics
Eric Robinson
Professor of Mathematics
Ithaca College
Roger Howe
Professor of Mathematics
Yale University
Judith Reed, Staff Liaison
Director of Research
National Council of Teachers of Mathematics
Gary Kader
Professor of Mathematical Sciences
Appalachian State University
Vincent Snipes
Professor of Mathematics Education
Interim Director, Center for Mathematics, Science,
and Technology Education
Winston-Salem State University
Henry Kepner
President, National Council of Teachers of
Mathematics
Professor, Mathematics Education
University of Wisconsin—Milwaukee
Patricia Valdez
Mathematics Department Chair and Teacher
Pajaro Valley (California) High School
Planning Group
Ruth Casey, NCTM Board Liaison (2007−2008)
PIMSER
University of Kentucky
W. Gary Martin, Chair
Professor of Mathematics Education
Auburn University
Fred L. Dillon, NCTM Board Liaison (2008−present)
Mathematics Teacher
Strongsville (Ohio) High School
Judith Reed, Staff Liaison
Director of Research
NCTM
Kaye Forgione
Senior Associate, Mathematics
Achieve, Inc.
Jennifer J. Salls, NCTM Board Liaison (2008–
present)
Mathematics Teacher
Sparks (Nevada) High School
Henry Kepner
President, National Council of Teachers of
Mathematics
Professor, Mathematics Education
University of Wisconsin—Milwaukee
Richard Schaar
Texas Instruments
Ken Krehbiel, Staff Liaison
Associate Executive Director for Communications
NCTM
Please access the on-line review form at www.nctm.org/highschooldraft.aspx by September 19, 2008.
Focus in High School Mathematics – Public Draft, August 28, 2008
p. iii
1
Contents
2
List of Examples ........................................................................................................................vi
3
Preface......................................................................................................................................vii
4
1 Reasoning and Sense Making...................................................................................................1
5
What Are Reasoning and Sense Making?.............................................................................2
6
Why Reasoning and Sense Making? ....................................................................................4
7
How Do We Include Reasoning and Sense Making in the Classroom?.................................5
8
2 Reasoning Habits .....................................................................................................................9
9
Reasoning as the Foundation of Mathematical Competence ............................................... 11
10
Statistical Reasoning.......................................................................................................... 13
11
Mathematical Modeling ..................................................................................................... 13
12
Technology to Support Reasoning and Sense Making ........................................................ 14
13
3 Curriculum............................................................................................................................. 15
14
Reasoning with Numbers and Measurements ..................................................................... 18
15
Reasonableness of answers and measurements............................................................ 19
16
Approximations and error........................................................................................... 23
17
Number systems ......................................................................................................... 25
18
Counting..................................................................................................................... 27
19
Reasoning with Algebraic Symbols ................................................................................... 27
20
Meaningful use of symbols......................................................................................... 28
21
Mindful manipulation................................................................................................. 30
22
Reasoned solving........................................................................................................ 32
23
Connections with geometry ........................................................................................ 33
24
Linking expressions and functions.............................................................................. 35
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. iv
1
Reasoning with Functions.................................................................................................. 36
2
Multiple representations of functions.......................................................................... 37
3
Modeling using families of functions.......................................................................... 40
4
Analyzing the effects of parameters............................................................................ 45
5
Reasoning with Geometry.................................................................................................. 47
6
Conjecturing about geometric objects ......................................................................... 48
7
Construction and evaluation of geometric arguments.................................................. 52
8
Multiple geometric approaches ................................................................................... 55
9
Geometric connections and modeling ......................................................................... 57
10
Reasoning with Statistics and Probability........................................................................... 62
11
Data analysis .............................................................................................................. 64
12
Modeling randomness................................................................................................. 66
13
Connecting statistics and probability........................................................................... 69
14
Interpreting statistical studies...................................................................................... 70
15
4 Equity .................................................................................................................................... 74
16
Courses.............................................................................................................................. 75
17
Student Demographics and Course Placement.................................................................... 77
18
High Expectations.............................................................................................................. 79
19
Conclusion......................................................................................................................... 80
20
5 Coherence ............................................................................................................................. 82
21
Curriculum and Instruction ................................................................................................ 82
22
Vertical Alignment of Curriculum ..................................................................................... 83
23
Alignment with prekindergarten through grade 8 mathematics.................................... 84
24
Alignment with postsecondary education.................................................................... 85
25
Assessment ........................................................................................................................ 86
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. v
1
Conclusion......................................................................................................................... 87
2
6 Stakeholder Involvement....................................................................................................... 89
3
Students............................................................................................................................. 89
4
Parents............................................................................................................................... 90
5
Teachers ............................................................................................................................ 91
6
Administrators ................................................................................................................... 92
7
Policymakers ..................................................................................................................... 93
8
Higher Education............................................................................................................... 94
9
Curriculum Designers ........................................................................................................ 95
10
Collaboration among Stakeholders..................................................................................... 95
11
Conclusion......................................................................................................................... 96
12
References ................................................................................................................................ 98
13
14
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. vi
1
List of Examples
2
Example 1: The Surface Area of Earth ...................................................................................... 20
3
Example 2: The Illusion of Miles per Gallon............................................................................. 21
4
Example 3: Error in Approximations of π.................................................................................. 24
5
Example 4: A Geometric Representation for the Distributive Property...................................... 25
6
Example 5: Different Forms of Quadratic Expressions.............................................................. 29
7
Example 6: FOIL versus EWE .................................................................................................. 31
8
Example 7: Reasoned Solving of Equations .............................................................................. 32
9
Example 8: Completing the Square ........................................................................................... 34
10
Example 9: Finding the Next Term in the Sequence, Extreme Version...................................... 35
11
Example 10: Multiple Representations of a Function................................................................. 38
12
Example 11: Modeling a Geometric Series Using Recursive Notation....................................... 41
13
Example 12: Recursion and Loans ............................................................................................ 44
14
Example 13: The Effects of Parameters on a Graph................................................................... 46
15
Example 14: Picture This .......................................................................................................... 50
16
Example 15: Circling the Points................................................................................................ 53
17
Example 16: Taking a Spin ....................................................................................................... 55
18
Example 17: Clearing the Bridge .............................................................................................. 58
19
Example 18: Assigning Frequencies.......................................................................................... 61
20
Example 19: Meaningful Words................................................................................................ 64
21
Example 20: Random Committees ............................................................................................ 67
22
Example 21: Meaningful Words—Continued............................................................................ 70
23
Example 22: Margin of Error .................................................................................................... 72
24
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. vii
1
Preface
2
The National Council of Teachers of Mathematics (NCTM) has a long tradition of
3
providing leadership and vision to support teachers in ensuring equitable mathematics learning of
4
the highest quality for all students. In the three decades since the 1980 publication of An Agenda
5
for Action, NCTM has consistently advocated for a coherent prekindergarten through grade 12
6
mathematics curriculum focused on mathematical problem solving. This message was refined in
7
Curriculum and Evaluation Standards for School Mathematics (1989), which argued for a
8
common core of mathematics for all students, with attention to the processes of problem solving,
9
reasoning, connections, and communication. Continuing to provide leadership in 2000,
10
Principles and Standards for School Mathematics (Principles and Standards) (NCTM 2000a)
11
updated and elaborated on the 1989 recommendations in a set of five Content Standards and five
12
Process Standards to include in every school mathematics program. In 2006, the Council’s
13
Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics offered guidance on
14
how to focus the mathematical content for each grade level from prekindergarten through 8 while
15
reminding educators of the importance of consistently embedding the mathematical processes
16
throughout the content in every mathematical learning experience.
17
Realizing a parallel need for focus and coherence at the high school level, although
18
perhaps of a different kind than addressed in the Curriculum Focal Points, a 2006 NCTM task
19
force recommended the development of a framework that would guide future work regarding
20
high school mathematics. As a result, a writing group was appointed to produce the framework
21
outlined in this document. Concurrently, a planning committee was appointed to oversee the
22
larger high school curriculum project, which includes a series of follow-up publications.
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. viii
1
This document proposes that all high school mathematics programs should be focused on
2
reasoning and sense making. The audience for this document is intended to be everyone involved
3
in decisions regarding high school mathematics programs, including formal decision makers
4
within the system; people charged with implementing those decisions; and other stakeholders
5
affected by, and involved in, those decisions.
6
Chapter 1 describes what constitutes reasoning and sense making in the mathematics
7
classroom, why they should be considered as the foundation for high school mathematics, and
8
how they link with other Process Standards. Chapter 2 describes in more detail the reasoning
9
habits that students should continue to acquire throughout their high school mathematics
10
experiences. Chapter 3 demonstrates with examples how reasoning and sense making can be
11
incorporated into five overarching areas of high school mathematics—number and measurement,
12
algebraic symbols, functions, geometry, and statistics and probability. Chapter 4 focuses on what
13
it means to provide equitable opportunities for all students to engage in reasoning and sense
14
making. Chapter 5 addresses the importance of coherent expectations regarding curriculum,
15
instruction, and assessment in promoting reasoning and sense making. Finally, chapter 6 presents
16
questions to consider as stakeholders work together to improve high school mathematics
17
education.
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. 1
1
1
2
Reasoning and Sense Making
3
A high school mathematics program based on reasoning and sense making
4
will prepare students for citizenship, for the workplace, and for further study.
5
High school mathematics prepares students for possible postsecondary work and study in
6
three broad areas (cf. NCTM [2000a]):
7
1. Mathematics for life
8
2. Mathematics for the workplace
9
3. Mathematics for the scientific and technical community
10
As the demands for mathematical literacy increase, students face challenges in all three areas.
11
First, the report of the Programme for International Student Assessment (2007) suggests that
12
United States students are lagging in their ability to apply mathematics “to analyse and reason as
13
they pose, solve and interpret problems in a variety of situations” (p. 7), including as future
14
citizens. Second, globalization and the rise of technology are presenting new economic and
15
workforce challenges (Friedman 2006), and the traditional mathematics curriculum is
16
insufficient for students entering many fields (Ganter and Barker 2004). Finally, the United
17
States is in danger of losing its leadership position in science, technology, engineering, and
18
mathematics (Task Force on the Future of American Innovation 2005; Committee on Science,
19
Engineering and Public Policy 2006; Tapping America’s Potential 2008).
20
A focus on reasoning and sense making will ensure that students can accurately carry out
21
mathematical procedures but more importantly, also understand why those procedures work and
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. 2
1
how they might be used and interpreted. This understanding will expand students’ abilities to
2
apply mathematical perspectives, concepts, and tools flexibly in each of the three areas
3
mentioned above. Such a focus on reasoning and sense making will produce citizens who make
4
informed and reasoned decisions, including quantitatively sophisticated choices about their
5
personal finances, about which public policies deserve their support, and about which insurance
6
or health plans to select. It will also produce workers who can satisfy the increased mathematical
7
needs in professional areas ranging from health care to small business to graphic design
8
(American Diploma Project 2004).
9
A high school curriculum that focuses on reasoning and sense making will help to satisfy
10
the increasing demand for scientists, engineers, and mathematicians while preparing students for
11
whatever professional, vocational, or technical needs may arise. Recent studies suggest that
12
students will experience career changes multiple times during their lives (U.S. Department of
13
Labor 2006), and many of the jobs they will hold in the future do not yet exist. Mathematics is
14
increasingly essential for a wide range of careers, including finance, advertising, forensics, and
15
even sports journalism. The advent of the Internet has produced an explosion of new careers in
16
mathematics and statistics based on harnessing the huge amount of data at people’s fingertips
17
(Baker and Leak 2006).
18
What Are Reasoning and Sense Making?
19
In the most general terms, reasoning may be thought of as the process of drawing
20
conclusions on the basis of evidence or stated assumptions. Reasoning in mathematics is often
21
understood to mean formal reasoning, in which conclusions are logically deduced from
22
assumptions and definitions. However, mathematical reasoning can take many forms, ranging
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. 3
1
from informal explanation and justification to formal deduction, as well as inductive
2
observations. Reasoning often begins with explorations, conjectures at a variety of levels, false
3
starts, and partial explanations before a result is reached.
4
We define sense making as developing understanding of a situation, context, or concept
5
by connecting it with existing knowledge. In practice, reasoning and sense making are
6
intertwined across the continuum from informal observations to formal deductions, as seen in
7
figure 1.1, despite the common perception that identifies sense making with the informal end of
8
the continuum and reasoning, especially proof, with the more formal end.
9
10
11
Fig. 1.1. The relationship of reasoning and sense making
On the one hand, formal reasoning may be based on sense making in which one identifies
12
common elements across a number of observations and realizes how those common elements
13
connect with previously experienced situations. On the other hand, full appreciation of a formal
14
proof requires that students make sense of how it systematizes and explains relationships they
15
have observed rather than merely view it as a verification of the truth of a statement (Yackel and
16
Hanna 2003). As sense making develops, it increasingly incorporates more formal elements. For
17
example, students may draw on their knowledge of the effects of changes in parameters of
18
quadratic functions as they begin to make sense of how changes in parameters affect
19
trigonometric functions.
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. 4
1
We have chosen to use both terms throughout this document to emphasize the importance
2
of the two interwoven processes of reasoning and sense making. The richness of this relationship
3
is captured in the following quote from George Pólya:
4
Mathematics has two faces. Presented in finished form, mathematics appears as a purely
5
demonstrative [deductive] science, but mathematics in the making is a sort of
6
experimental science. A correctly written mathematical paper is supposed to contain
7
strict demonstrations only, but the creative work of the mathematician resembles the
8
creative work of the naturalist; observation, analogy, and conjectural generalizations, or
9
mere guesses … play an essential role in both. (Pólya 1952, p. 739)
10
Mathematical reasoning and sense making are both important outcomes of mathematics
11
instruction, as well as important means by which students come to know mathematics. In other
12
writings, Pólya (1954) describes “plausible reasoning,” the type of reasoning most often used in
13
statistics. As the term is used in this document, mathematical reasoning encompasses statistical
14
reasoning.
15
Why Reasoning and Sense Making?
16
This document emphasizes that reasoning and sense making are the foundations of the
17
NCTM Process Standards (2000a). The Process Standards—Problem Solving, Reasoning and
18
Proof, Connections, Communication, and Representation—are all manifestations of the act of
19
making sense of mathematics and of reasoning as defined above. Problem solving is impossible
20
without reasoning, and problem solving is the avenue through which students develop
21
mathematical reasoning and make sense of mathematical ideas. The communications,
22
connections, and representations chosen by a student must support reasoning and sense making,
23
and reasoning must be employed in making those decisions. Proof is a communication of formal
24
reasoning built on a foundation of sense making.
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. 5
1
At the high school level, reasoning and sense making are of particular importance, but
2
historically “reasoning” has been limited to very selected areas of the high school curriculum,
3
and sense making is in many instances not present at all. However, an emphasis on student
4
reasoning and sense making can help students organize their knowledge in ways that enhance the
5
development of number sense, algebraic fluency, functional relationships, geometric reasoning,
6
and statistical thinking, as exemplified in chapter 3. When students connect new learning with
7
their existing knowledge, they are more likely to understand and retain the new information
8
(Hiebert 2003) than when it is simply presented as a list of isolated procedures. Without such
9
conceptual understanding, “learning new topics becomes harder since there is no network of
10
previously learned concepts and skills to link a new topic to” (Kilpatrick, Swafford, and Findell
11
2001, p. 123), meaning that procedures may be forgotten as quickly as they are apparently
12
learned. A refocus on reasoning and sense making will expand access to conceptual
13
understanding and foster meaning.
14
How Do We Include Reasoning and Sense Making in the Classroom?
15
Reasoning and sense making should occur in every mathematics classroom every day. In
16
such an environment, students ask and answer such questions as “What’s going on here?” and
17
“Why do you think that?” Reasoning and sense making do need not be an extra burden for
18
teachers struggling with students who are having a difficult time just learning the procedures. On
19
the contrary, the structure that reasoning brings forms a vital support for understanding and
20
continued learning. Currently, many students have difficulty because they find mathematics
21
meaningless. Without the connections that reasoning and sense making provide, a seemingly
22
endless cycle of reteaching may result. With purposeful attention and planning, teachers can hold
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. 6
1
all students in every high school mathematics classroom accountable for personally engaging in
2
reasoning and sense making, and thus lead students to experience reasoning for themselves rather
3
than merely observe it. Moreover, technology should be used strategically throughout the high
4
school curriculum to help reach this goal.
5
What exactly do reasoning and sense making “look like” in the mathematics classroom?
6
The following example shows how reasoning and sense making can be infused into teaching a
7
formula that many students often regard as meaningless and hard to remember, the distance
8
formula. The first scenario illustrates what can happen when students are asked to recall a
9
procedure taught without understanding.
10
Teacher: Today’s lesson requires that we calculate the distance between the
11
center of a circle and a point on the circle in order to determine the circle’s radius. Who
12
remembers how to find the distance between two points?
13
Student 1: Isn’t there a formula for that?
14
Student 2: I think it’s x1 plus x2 squared, or something like that.
15
Student 1: Oh, yeah, I remember—there’s a great big square root sign, but I don’t
16
remember what goes under it.
17
Student 3: I know! It’s x1 plus x2 all over 2, isn’t it?
18
Student 4: No, that’s the midpoint formula.
19
(The discussion continues along these lines until the teacher reminds the class of
20
the formula.)
21
The next time she teaches the distance formula, the teacher engages the students in
22
solving a problem. In the following scenario, we see students reasoning about mathematics,
23
connecting what they are learning with their existing knowledge, and making sense of what the
24
distance formula means.
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Focus in High School Mathematics – Public Draft, August 28, 2008
1
p. 7
Teacher: Let’s take a look at a situation in which we need to find the distance
2
between two locations on a map. Let’s say that the school is designated as the origin of a
3
coordinate system, your house is located at two blocks west and five blocks north of
4
school, and your best friend’s house is located eight blocks east and one block south of
5
school. If the city had a system of perpendicular streets, how far would we have to drive
6
to get from your house to your friend’s house?
7
8
Student 1: Well, we would have to drive ten blocks to the east and six blocks to
the south, so I guess it would be sixteen blocks, right?
9
Teacher: Now, what if you could use a helicopter to fly straight to your friend’s
10
house? How could we find the distance “as the crow flies”? Work with your partners to
11
plot these two locations on a coordinate-axis system, and show the path you’d have to
12
drive to get to your friends house. Next, work on calculating the direct distance between
13
the houses if you could fly.
14
15
Student 1: If we use the school as the origin, then wouldn’t my house be at (–2, 5)
and my friend’s house at (8, −1)?
16
Student 2: Yeah, that sounds right. I drew the path on the streets connecting the
17
two houses, and then drew a line segment connecting the two houses. Maybe we could
18
measure the length of a block and find the distance with a ruler.
6
Your House
4
2
School
5
10
Friend's House
-2
19
20
21
Student 3: Wait a minute—you just drew a right triangle, because the streets are
perpendicular.
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Focus in High School Mathematics – Public Draft, August 28, 2008
Student 4: So that means we could use the Pythagorean theorem: 102 + 62 = c 2 , so
1
2
c = 136 .
3
Student 3: So that means the distance is between eleven and twelve blocks, since
4
121 < 136 < 144. Actually, it’s probably closer to twelve blocks, since 136 is closer to
5
144.
6
p. 8
(The teacher then extends the discussion to consider other examples and finally to
7
develop a general formula.)
8
9
By having her students approach the distance formula from a reasoning-and-sense-
10
making perspective, the teacher increases their understanding of the formula and why it is true,
11
making it more likely that they will be able to retrieve, or quickly recreate, the formula at a later
12
point in time. Restructuring the high school mathematics program around reasoning and sense
13
making enhances students’ development of both the content and process knowledge they need to
14
be successful in the continuing study of mathematics and in their lives.
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Focus in High School Mathematics – Public Draft, August 28, 2008
1
2
2
Reasoning Habits
3
Reasoning and sense making should be a part of the
4
mathematics classroom every day.
p. 9
5
As a consequence of a focus on reasoning and sense making, merely “covering”
6
mathematical topics is not enough. Students need to experience and develop the mathematical
7
reasoning habits that reflect the desired type of thinking and questioning (cf. Cuoco, Goldenberg,
8
and Mark [1996]; Driscoll [1999]; Pólya [1952, 1957]; Schoenfeld [1983]; Harel [2000]). A
9
reasoning habit is a way of thinking that becomes common in the processes of mathematical
10
inquiry and sense making. These habits should be incorporated in a natural and flexible way
11
within every high school mathematics classroom and across the high school mathematics
12
curriculum. In a well-articulated high school curriculum, students’ reasoning abilities will
13
continuously develop and expand as reasoning habits become a routine and fully expected part of
14
all mathematics classes.
15
Such fluency with mathematical reasoning habits offers student the opportunity to
16
develop increasing ability in communicating reasoning. When reasoning is interwoven with
17
sense making, and when teachers provide the necessary support and formative feedback, students
18
can be expected to demonstrate growing levels of formality in their reasoning in the classroom,
19
in their oral and written work, and in assessments throughout the high school years.
20
We offer the following list of reasoning habits. This is not intended to be a new list of
21
topics to be added to the high school curriculum, nor a list to be followed sequentially within
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Focus in High School Mathematics – Public Draft, August 28, 2008
p. 10
1
every problem situation, but rather an illustration of the types of thinking that are desirable for
2
high school students to engage in as they grow in their mathematical sophistication.
3
•
4
Analyzing a problem, for example,
–
looking for hidden structure (for example, drawing auxiliary lines in
5
geometric figures, finding equivalent forms of expressions that reveal
6
different aspects of a problem, using a different viewing window for a graph
7
that reveals different features, or looking at graphical representations of data
8
to verify normality);
9
–
looking for patterns and relationships;
10
–
considering special cases;
11
–
making connections with previous work;
12
–
making preliminary deductions and conjectures; and
13
–
determining whether a statistical solution is appropriate.
14
•
15
Initiating a strategy, for example,
–
16
selecting mathematical concepts, representations, or procedures that may be
applicable;
17
–
making purposeful use of procedures;
18
–
organizing a solution (for example, formulating an algorithm or using the
19
structure of the problem); and
20
21
–
•
making logical deductions based on current progress.
Monitoring one’s progress, for example,
22
–
reviewing a chosen strategy and evaluating progress;
23
–
analyzing the problem further if necessary; and
24
–
changing or modifying strategies used as necessary.
25
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Focus in High School Mathematics – Public Draft, August 28, 2008
1
•
Seeking and using connections, for example,
2
–
connecting seemingly different mathematical domains;
3
–
connecting seemingly different contexts; and
4
–
connecting different representations.
5
•
p. 11
Reflecting on one’s solution to a problem, for example,
6
–
interpreting a solution and how it answers the problem;
7
–
checking the reasonableness of a solution;
8
–
justifying or validating a solution, including proof;
9
–
considering other ways of thinking about the problem and how they might
10
have been more productive;
11
–
generalizing a solution to a broader class of problems; and
12
–
understanding the plausible nature of a statistical conclusion.
13
See chapter 3 for examples of how these habits can be manifested in the high school classroom,
14
with specific references to the reasoning habits listed above.
15
Reasoning as the Foundation of Mathematical Competence
16
Reasoning and sense making are inherent in developing mathematical competence, as
17
discussed in Adding It Up (Kilpatrick, Swafford, and Findell 2001); see figure 2.1. In particular,
18
procedural fluency includes learning with understanding and knowing which procedure to
19
choose, when to choose it, and for what purpose. In the absence of reasoning, students may carry
20
out procedures correctly but may also capriciously invoke incorrect or baseless rules, such as
21
“the square root of a sum is the sum of the square roots.” They come to view procedures as steps
22
they are told to do rather than a series of steps they choose to do for a specific purpose on the
23
basis of mathematical principles. Without understanding the basis of procedures in reasoning and
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sense making, they may be able to correctly perform those procedures but may think of them
2
only as a list of “tricks.” As a result, they may have difficulty selecting the procedure to use in a
3
given circumstance owing to a lack of understanding of what the procedure will accomplish.
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Authentic procedural fluency cannot be achieved unless students both master the technical skills
5
and develop the understanding needed for their continued study and use of mathematics.
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8
•
Conceptual understanding—comprehension of mathematical concepts, operations, and
relations
9
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•
Procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and
appropriately.
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•
Strategic competence—ability to formulate, represent, and solve mathematical problems
12
•
Adaptive reasoning—capacity for logical thought, reflection, explanation, and justification
13
14
•
Productive disposition—habitual inclination to see mathematics as sensible, useful, and
worthwhile, coupled with a belief in diligence and one’s own efficacy
15
16
17
18
Fig. 2.1. The strands of mathematical proficiency from Adding It Up (Kilpatrick, Swafford, and
Findell 2001)
The development of a “productive disposition” (Kilpatrick, Swafford, and Findell 2001)
19
is a high priority of high school mathematics. When students achieve this goal, they view
20
mathematics as a reasoning and sense-making enterprise. This goal can be achieved only if
21
students personally engage in mathematical reasoning and sense making as they are learning
22
mathematics content. Principles and Standards (NCTM 2000a) states, “All students are expected
23
to study mathematics each of the four years that they are enrolled in high school” (p. 288), and
24
students must experience reasoning and sense making across all four years of a high-quality high
25
school mathematics program.
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p. 13
Statistical Reasoning
Statistical reasoning involves making conclusions from, and decisions based on, data
3
from sample surveys, observational studies, or experiments. Inductive reasoning, with its active
4
use of conjectures and generalizations, lies at the heart of statistical reasoning. The need for
5
statistics arises from “the omnipresence of variability” in data (Cobb and Moore 1997, p. 801).
6
Statistics furnishes tools for summarizing the variability in data and provides strategies for
7
collecting useful data and interpreting the meaning of the data within the context of the problem.
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Both inductive and deductive reasoning are fundamental forms of mathematical reasoning, and
9
we include statistical reasoning as a particular kind of mathematical reasoning.
10
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Mathematical Modeling
The tools and reasoning processes of mathematics help us understand and operate in the
12
physical and social worlds. Mathematical modeling is a process of connecting mathematics with
13
a real-world context; figure 2.2 outlines a cycle often used to organize reasoning in mathematical
14
modeling. Mathematical modeling is essentially the axiomatization of some sliver of the real
15
world so as to deal with it mathematically. The connections between mathematics and real-world
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problems developed in mathematical modeling add value to, and provide incentive and context
17
for, studying mathematical topics. Mathematical modeling also offers opportunities to make
18
reasoned connections among different mathematical arenas, because in many situations, real-
19
world problems require a combination of mathematical tools. Example 17 in chapter 3 is a vivid
20
example of how mathematical modeling can cross a variety of mathematical strands.
21
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1. Building a mathematical model from a real-world situation (usually starting with a
simplification of the problem)
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2. Determining mathematical results within the model
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3. Interpreting the mathematical results in the real-world context
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4. Checking these results within the real-world context to determine whether the results are
feasible and adequate and, if necessary, repeating the process with a better model
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p. 14
Fig. 2.2. Four-step modeling cycle used to organize reasoning about mathematical modeling
Technology to Support Reasoning and Sense Making
Technology can be used to advance the goals of reasoning and sense making in the high
10
school mathematics classroom. It can be particularly useful in looking for patterns and
11
relationships and forming conjectures (see Examples 9, 14, and 17). Technology can allow
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students to reflect on their progress toward solving a problem rather than on carrying out
13
computations and to draw logical conclusions (see Example 9). A student’s ability to display
14
multiple representations of the same problem using technology can aid in making connections
15
(see Example 13). Technology can also be useful in generalizing a solution (see Example 22).
16
The incorporation of technology in the classroom should not overshadow the development of the
17
procedural proficiency needed by students to support continued mathematical growth. It can,
18
however, provide a lens that leads to a deeper understanding of mathematical concepts. Students
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can be challenged to take responsibility for deciding which tool might be useful in a given
20
situation when they are allowed to choose from a menu of mathematical tools that includes
21
technology. Students who have regular opportunities to discuss and reflect on how a
22
technological tool is used effectively will be less apt to use technology as a crutch.
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3
2
Curriculum
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Reasoning and sense making are integral to the experiences of all students
4
across all areas of the high school mathematics curriculum.
5
Reasoning and sense making should be pervasive across the high school mathematics
6
curriculum. Although some aspects of reasoning may be quite visible in geometry and other
7
particular areas of the curriculum, students are less likely to experience a consistent reasoning
8
approach to the content in other areas, such as developing algebraic skills. When reasoning and
9
sense making are infused throughout the high school mathematics curriculum, they provide focus
10
by strengthening connections within the discipline as well as between mathematics and the
11
world. They also help students see how new concepts connect with existing knowledge.
12
Mathematical reasoning prepares students to view mathematics as a connected whole
13
across the domains of mathematics—such as algebra, geometry, and statistics. These connections
14
are important, and they can be attained whether the curriculum is arranged in the customary U. S.
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course-based sequence (first-year algebra, geometry, second-year algebra) or in an integrated
16
manner. Students should experience mathematics as a useful tool for reasoning about the world
17
around them. Some students find the world of mathematics itself a meaningful context, but some
18
may also be motivated to study mathematics because it helps them make sense of their world
19
from varied perspectives. Contexts drawn from students’ experiences can lend meaning, insight,
20
and relevance to the mathematics they are learning.
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Whereas increasing attention to reasoning and sense making may appear to be adding yet
2
more content expectations to an overcrowded curriculum, it may indeed introduce efficiencies.
3
First, the initial part of each course that is frequently spent reteaching content from previous
4
courses may be reduced by emphasizing reasoned connections with existing knowledge, so that
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students are better able to retain what they have learned in previous courses. Second,
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emphasizing the underlying connections that promote reasoning and sense making introduces
7
coherence that allows streamlining of the curriculum. As attention turns to how those underlying
8
connections build across the high school years, less time needs to be spent addressing lists of
9
particular skills that need to be mastered.
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The following sections of this chapter illustrate how the high school mathematics
11
curriculum can be focused on broad themes that promote reasoning and sense making within five
12
specific content areas of the high school curriculum:
13
•
Reasoning with Numbers and Measurements
14
•
Reasoning with Algebraic Symbols
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•
Reasoning with Functions
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•
Reasoning with Geometry
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•
Reasoning with Statistics and Probability
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Within each content area, a number of key elements provide a broad structure for
19
thinking about how the content area can be focused on mathematical reasoning and sense
20
making. These key elements are not intended to be an exhaustive list of specific topics to be
21
addressed. Instead, they provide a lens through which to view the potential of high school
22
programs for promoting mathematical reasoning and sense making. Although some readers may
23
be looking for more-specific guidance about what topics should be covered in what order, that is
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not the intent of this document. Instead, we are advocating a very different approach to the
2
content that is taught. The task of creating a curriculum that fully achieves the goals of this
3
document will be a challenging task, one that extends well beyond what can be accomplished in
4
this document. However, all teachers in mathematics classrooms across the country can begin to
5
refocus their instruction on reasoning and sense making, thus beginning to realize some
6
immediate benefit for their students.
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Furthermore, those responsible for making decisions about the curriculum must be aware
8
of the need to provide experiences with reasoning and sense making within a broad curriculum
9
that meets the needs of a wide range of students, preparing them for future success as citizens
10
and in the workplace, as well as for careers in mathematics and science. Hard choices will be
11
need to made about the topics traditionally included in the curriculum to make room for areas
12
that have been often underemphasized, such as statistics.
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Discrete mathematics, an active branch of contemporary mathematics that is widely used
14
in business and industry, is an additional area of mathematics not directly addressed in the five
15
strands in this chapter. We argue that discrete mathematics should be an integral part of the
16
school mathematics curriculum (NCTM 2000a). As in Principles and Standards, the main topics
17
of discrete mathematics are distributed across the strands instead of receiving separate treatment,
18
as shown in figure 3.1.
19
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Attention to discrete mathematics is embedded in the following strands of Focus in High School
2
Mathematics:
3
 Counting is incorporated as a key element within Reasoning with Numbers and Measurements
4
5
and is addressed in Example 20.
 Recursion is included within the “Multiple representations of functions” key element of
6
7
Reasoning with Functions.
 Vertex-edge graphs are addressed within the last key element of Reasoning with Geometry.
8
9
10
11
Fig. 3.1. Discrete mathematics in Focus in High School Mathematics
Reasoning with Numbers and Measurements
12
Much of mathematics, particularly the domains of area and measurement, originated in
13
efforts to quantify the world. Although substantial attention is given to these areas of
14
mathematics in grades K–8, they receive much less attention in the high school years. Yet
15
number and measurement are foundational; without reasoning skills in these areas, students will
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be limited in their reasoning in other areas of mathematics.
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Key elements of reasoning and sense making with numbers and measurements include
18
the following:
19
•
Reasonableness of answers and measurements. Judging whether a given answer or
20
measurement is of the right order of magnitude, and whether it is expressed in appropriate
21
units.
22
•
Approximations and error. Realizing that all real-world measurements are approximations
23
and that unsuitably accurate values should not be used for real-world quantities; recognizing
24
the role of error in subsequent computations with measurements.
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•
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p. 19
Number systems. Understanding number-system properties deeply; extending number-system
properties to algebraic situations.
•
Counting. Recognizing when enumeration would be a productive approach to solving a
problem, then using principles and techniques of counting to find a solution.
5
We address these key elements in more detail in the following sections.
6
Reasonableness of answers and measurements
7
On the basis of their study of numbers in the elementary and middle grades, students
8
should enter high school with “number sense” related to whole numbers focused on the base-ten
9
numeration system and rational numbers. They should know that the leading (leftmost) digit of a
10
number accounts for most of the number; and that the digits to the right of the leading two or
11
three digits of a real-world number are “loose change” and may often be meaningless. For
12
example, although the U.S. Census Bureau Web site reports that the world population was
13
6,714,377,944 as of 3:11 PM on August 3, 2008, this number is only a rough estimate based on
14
data provided from nations around the world. Indeed, one can often use round numbers to do
15
approximate calculations that will yield simple but fairly accurate approximations of quantities
16
of interest. See, for example, the comments of student 3 in Example 1. This example also shows
17
the importance of thinking about the order of magnitude of one’s answer to be sure it makes
18
sense.
19
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Example 1: The Surface Area of Earth
Task
Estimate the total surface area of Earth.
In the Classroom (geometry class)
Teacher: What would you guess the total surface area of the earth is?
Student 1: Wow, I would have no idea. I’d guess either a million or maybe a billion square miles.
Student 2: Well, it is a ball, which is basically a sphere. So if we knew the radius, I guess we
could figure that out.
Teacher: OK, to help you out, I’ll tell you that its radius is 4,000 square miles.
Student 2: So then A = 4! r 2 , so we just need to plug the radius in.
Student 3: If the radius is about 4 thousand, then squaring will take us to about 16 million. 4π is a
little more than 12, and 12 × 16 = 192, so I’ll guess maybe 200 million.
Student 4: Yeah, that’s decent, but why not just use the π-button on your calculator? That’s what
I did, and I got 201,061,930.
Teacher: So which do you think is the best estimate?
Student 4: Mine, because it’s most accurate.
Student 3: But we don’t know exactly what the radius of Earth is, plus it may not be exactly
round. So I think 200,000,000 is good enough.
Key Elements of Mathematics
Reasoning with Numbers and Measurements—Reasonableness of answers and measurements;
Approximations and error
Reasoning Habits
Analyzing a problem—making preliminary deductions and conjectures
Reflecting on one’s solution—checking the reasonableness of a solution; justifying or validating
a solution
29
30
The National Mathematics Advisory Panel (2008) defines the type of number sense
31
needed for the study of algebra as “a principled understanding of place value, of how whole
32
numbers can be composed and decomposed, and of the meaning of the basic arithmetic
33
operations of addition, subtraction, multiplication, and division” (p. 27). The report goes on to
34
identify the importance of developing both conceptual and procedural knowledge of fractions for
35
further progress in mathematics and the “pervasive difficulties” (p. 28) students have. Although
36
instructional time precludes reteaching the number concepts that students should already have
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learned as they enter high school, attention to number sense can profitably be integrated into
2
instructional objectives at the high school level. For example, teachers might routinely ask
3
students to decide whether an answer to a given computation is of the right order of magnitude,
4
or how accurate an answer would be appropriate, and to then state the reasoning for their
5
judgment.
6
Additionally, “number sense” at the high school level should extend to new sets of
7
numbers and situations. Students should develop intuitions about situations involving radicals
8
and negative and fractional exponents. For example, when calculating 12 2 , a student might
9
reason that since 12 is between 9 and 16, 12 2 must be between 3 and 4. Since 12 2 = 121 !12 2 ,
3
1
10
3
1
the answer should be between 36 and 48. Thus, if a student enters “12^3/2” into his or her
3
2
11
calculator to compute 12 and gets an answer of 864, he or she should immediately realize that
12
something went terribly wrong.
13
When working with measurements, students should have a sense of what units are
14
appropriate for the solution to a problem. Example 2 shows how important the choice of units
15
can be in understanding the everyday context of fuel efficiency and costs.
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17
Example 2: The Illusion of Miles per Gallon
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Task
A teacher gives her students the following quiz taken from an article in the New York Times and
asks them to explain their reasoning (Chang 2008):
Quiz time: Which of the following would save more fuel?
a) Replacing a compact car that gets 34 miles a gallon (MPG) with a hybrid that
gets 54 MPG.
b) Replacing an SUV that gets 18 MPG with a sedan that gets 28 MPG.
c) Both changes save the same amount of fuel.
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p. 22
In the Classroom (first-year algebra)
The teacher collects the quizzes and asks two of her students to share their answers with the
class. The first student responds:
I would say the correct answer is (a). My reasoning is that the 18 to 28 MPG change
is a 56% increase, whereas the increase from 34 MPG to 54 is a 59% increase.
The second student responds:
I thought about how much gas it would take to make a 100-mile trip.
Considering the compact car:
100 miles/54 MPG = 1.85 gallons used.
100 miles/34 MPG = 2.94 gallons used.
So switching from a 34 MPG to a 54 MPG car would save 1.09 gallons of gas.
Looking at the SUV:
100 miles/28 MPG=3.57 gallons used.
100 miles/18 MPG=5.56 gallons used.
So switching from an 18 MPG car to a 28 MPG car saves 1.99 gallons of gas
every 100 miles. So you are actually saving more gas by replacing the SUV.
The teacher then asks the class to compare these two responses. After a spirited debate from
students who had chosen each of the answers, the class agreed that although the fuel efficiency
increased more for the compact car, you would actually save more gallons of gas by replacing
the SUV.
The teacher asks the class to explore the relationship of MPG with actual gas consumption. After
working in small groups for a few minutes, the teacher asks one group to show the table of
values and graph it has made, as shown below. They explain, “You save less fuel as you go up
another 5 MPG over and over. So as we saw in the quiz, MPG can be a little confusing.”
The teacher then gives the class the article from the New York Times from which the quiz was
taken to read for homework, along with some online comments from readers. She asks the
students to analyze the arguments from the article in a brief essay and to propose what unit of
measure they think will be useful for comparing the fuel consumption of cars.
Key Elements of Mathematics
Reasoning with Numbers and Measurements—Reasonableness of answers and measurements
Reasoning with Functions—Using families of functions as mathematical models
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Reasoning Habits
Initiating a strategy—selecting concepts, representations, or procedures that may be applicable;
making logical deductions based on current progress
Reflecting on one’s solution—interpreting a solution and how it answers the problem; justifying
or validating a conclusion; considering other ways of thinking about the problem
6
7
8
9
Approximations and error
Mathematics as it exists in our minds is exact, but when we want to apply it to the world,
we must learn to cope with approximation. All “real world” numbers, by which we mean
10
numbers obtained through measurement rather than counting, are subject to error. “Error” is not
11
synonymous with “mistake”; it is the unavoidable inexactness that is inherent in all real-world
12
situations. Many high school students have not grasped the significance of reporting their
13
measure to the nearest unit or subunit of measurement, depending on the measuring device.
14
Moreover, the accuracy of the measurement affects any results that are subsequently reported.
15
When scientists give results of measurements—often a mixture of direct and computed
16
measures—they are careful to specify how exactly they are known. For a real-world number,
17
students should realize that they simply do not know more than the first few digits of its “actual”
18
value, meaning that they should learn to be suspicious when they see numbers with many
19
decimal places in news reports. Most likely the places beyond the first two or three will have no
20
meaning. Students should learn that the accuracy of a result can be more precisely specified
21
using the idea of relative error. See Example 3.
22
Approximation is a crucial connection between the abstract world of pure mathematical
23
numbers (objects) and their use and interpretation in every discipline students will encounter
24
beyond the mathematics department. We need to prepare our students to connect their
25
experiences with number in the mathematics curriculum with its uses in mathematical modeling
26
and reported or predicted results in other disciplines.
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Example 3: Error in Approximations of π
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Task
Although we know that π is an irrational number, we often use such approximations as 3.14 or
22/7. How much error are we introducing by using such an approximation? Find an upper bound
for the relative error, and illustrate using a specific example.
13
value can help students to make sense of the formula for relative error,
14
15
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17
In the Classroom (second-year algebra)
Helping students move beyond the notion of error as mistake is an important part of the sensemaking process for a productive discussion of error. Students may view error as how far they are
from a correct answer. The teacher can build on this view to discus error as the difference
between an approximation (v) and “real” value (V), and absolute error as the absolute value of
the error, V ! v . Developing an understanding of relative error as a percentage of the “real”
V !v
. To better
V
understand his students’ progress in understanding relative error, a teacher assigns the task above
as an in-class assessment to be completed individually. One student’s response follows, with the
teacher’s comments in the margin:
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Key Elements of Mathematics
Reasoning with Numbers and Measurements—Approximations and error
Reasoning Habits
Analyzing a problem—making preliminary deductions and conclusions
Monitoring one’s progress—changing or modifying strategies used as necessary
Reflecting on one’s solution—checking the reasonableness of a solution
8
9
10
Number systems
Students need to have a solid background with number systems and their properties,
11
which will set a foundation for the meaningful development of algebraic understandings.
12
Example 4 illustrates how an understanding of the distributive property with multidigit
13
multiplication forms the basis for multiplication of polynomials; this point is further elaborated
14
in Example 6.
15
Example 4: A Geometric Representation for the Distributive Property
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Background
The distributive property can be nicely illustrated using a geometric representation called an area
model. This model is a powerful way to introduce students to the distributive property as it
applies to calculating products of multidigit integers. Take, for example, the product of 62 and
43. If each factor is written in expanded form, we have
(62)(43) = (60 + 2)(40 + 3) = 24(100) + 18(10) + 8(10) + 6.
One can clearly see the groups of 100, 10, and 1 represented geometrically as in the figure
below. This visual has the advantage of helping students make sense not only of the distributive
property but of perfect squares and orders of magnitude as well.
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Once the concept of using area to model multiplication of integers has been established, ideally
in grades K−8, area models can help students make sense of many processes, such as
multiplication of fractions, polynomials, and so forth.
Task
The distributive property for multiplication of variables can be modeled for such expressions
as A(B + C) . As shown in the figure below, AB + AC = A(B + C).
A
A
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AB
AC
B
C
A(B+C)
B+C
Create an area model for the product (x + 3)(x + y + 5).
In the Classroom (first-year algebra)
Students could create area models by using graph paper and pencils or algebra tiles. Two
possible area models are shown below. The first shows regions of area x 2 , xy, x, y, and 1. By
counting the number of each type of region present (i.e., one group of x 2 , three groups of y, etc.),
students can make sense of what each term in the expansion of the product represents. The
second model shows the area of each smaller rectangular region as the product of its dimensions.
This type of area model illustrates that the sum of the smaller areas (products of terms) is equal
to the area of the original factors, which can form a connection with the area addition postulate.
x
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p. 26
y
5
x
y
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x
x
x2
xy
5x
3
3
3x
3y
15
Key Elements of Mathematics
Reasoning with Numbers and Measurements—Number systems
Reasoning Habits
Analyzing a problem—making connections with previous work
Seeking and using connections—connecting seemingly different mathematical domains;
connecting different representations
29
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p. 27
Counting
2
The final area of Reasoning with Numbers and Measurements concerns using counting
3
principles and techniques, based on creating an exact enumeration of a set of objects. This
4
important topic from discrete mathematics has application in many contexts both within and
5
outside mathematics. See Example 20 for an application of counting principles to probability
6
models. Students need to recognize situations that are amenable to enumeration and to
7
understand how this approach differs from simply counting the number of cases. For example,
8
when rolling a pair of dice and summing the numbers of the topmost faces, eleven sums are
9
possible (2−12). Yet not all sums are equally likely, since five ways exist to get 6 (1 + 5, 2 + 4,
10
3 + 3, 4 + 2, and 5 + 1) but only one way to get 2 (1 + 1). Understanding the difference between
11
situations in which the order of events matters (e.g., the digits in a number) and situations in
12
which order is irrelevant (e.g., membership on a committee, as in Example 20) is also important.
13
Reasoning with Algebraic Symbols
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The algebraic notation we use today is a major accomplishment of humankind, allowing
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for the compact representation of complex calculations and problems. However, that very
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compactness can be a barrier to sense making. A basic task for teachers of algebra is to help
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students reason their way through that barrier.
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Key elements of reasoning and sense making with algebraic symbols include the
following:
•
Meaningful use of symbols. Choosing variables and constructing expressions and
equations in context; interpreting the form of expressions and equations.
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•
p. 28
Mindful manipulation. Connecting manipulation with the laws of arithmetic; anticipating
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the results of manipulations; choosing procedures purposefully in context; and picturing
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calculations mentally.
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•
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solutions in context.
•
7
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Reasoned solving. Seeing solution steps as logical deductions about equality; interpreting
Connecting algebra with geometry. Representing geometric situations algebraically and
algebraic situations geometrically; using connections in solving problems.
•
9
Linking expressions and functions. Using multiple algebraic representations to understand
functions; working with function notation.
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We address these key elements in more detail in the following sections.
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Meaningful use of symbols
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Although a long-term goal of algebraic learning is a fluid, nearly automatic facility with
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manipulating algebraic expressions that might seem to resemble what is often called “mindless
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manipulation,” this ease can be best achieved by first learning to pay close attention to
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interpreting expressions, both at a formal level and as statements about real-world situations. As
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comfort with expressions grows, constructing and interpreting them require less and less effort
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and gradually become almost subconscious, but at the outset, the reasons and justifications for
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forming and manipulating expressions should be major emphases of instruction. The true
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foundation for algebraic manipulation is close attention to meaning and structure.
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The capacity to use symbols meaningfully includes—
•
the ability to create expressions, including careful choice and careful description
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(including units of measurement) of variables to represent quantities in word problems or
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applications;
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the ability to distinguish among variables, constants, and parameters, and the
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understanding that this distinction is not a property of the letters themselves but depends
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on the sentences in which they are embedded and the context of the problem under
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consideration.
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Symbols can be used to state general properties (a + c = c + a for all a and c), as placeholders for
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representing a specific value (find the value of Q such that 3Q – 4 = 11), as parameters that affect
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the behavior of a function (what is the effect of increasing m on the graph of y = mx + b?), or as
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constants, for example, π or e. A disposition to use symbols meaningfully allows for interpreting
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the form of expressions and equations to deduce properties of the quantities or relations they
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represent, by seeing the component parts of expressions and equations not in terms of their
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placement on the page but in terms of arithmetic operations. It also allows for navigating the
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nested structure of complex expressions, for example, seeing 3 – (4 – x)2 as 3 minus something
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squared; as a function of 4 – x; and as a function of x. Example 5 gives an example of the
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information provided by different forms of a quadratic expression.
Example 5: Different Forms of Quadratic Expressions
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Task
A thrown horseshoe travels on a parabolic arc, as shown in
the figure. Its height (measured in feet) as a function of time
(measured in seconds) from the instant of release is
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1
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The expressions (a)–(d) below are equivalent. Which is most useful for finding the maximum
height of the horseshoe's path, and why is it the most useful expression?
3
16
+ 18t ! 16t 2 .
6
4
2
0.5
3
+ 18t ! 16t 2
16
1
(19 ! 16t)(16t
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(a) 1
25
(c)
(b) !16(t !
+ 1)
(d) !16(t !
1
19
1
)(t + )
16
16
9 2 100
) +
16
16
26
27
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1.5
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p. 30
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In the classroom (group reporting out in second-year algebra)
We eliminated expression (a). It tells us the starting height and starting upward speed, but the
question is not about either of those. Expressions (b) and (c) are pretty much the same, except
that the denominators in the factors have been pulled out in front in (c). One of us used (b) to
find the zeroes, and then found the midpoint. We finally decided on (d) because the term
6
!16(t !
9 2
)
16
7
is always negative or zero, so we could see that the height never goes above
8
and that it reaches this height at t =
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9
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100
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feet, or 6
1
4
feet,
seconds.
Key Elements of Mathematics
Reasoning with Algebraic Symbols—Meaningful use of symbols; Linking expressions and
functions
Reasoning Habits
Analyzing a problem—looking for hidden structure; looking for patterns and relationships
Initiating a strategy—selecting mathematical concepts, representations, or procedures that may
be applicable
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Mindful manipulation
At the outset of instruction in algebra, emphasis must be placed on understanding why
21
particular algebraic procedures and rules can be applied. Without this reasoning, students may
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resort to memorizing lists of isolated tricks or commands and not know which to choose when
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the need to use algebra arises. Truly successful algebraic manipulation is the end stage of
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practice that begins with very careful attention to the significance of each step. Without this
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initial investment in meaning, students often see the rules for manipulating expressions as rules
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for a game in which pieces are moved around on the page rather than as purposeful use of the
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nine basic rules of arithmetic. Of these, the distributive rule, which is the only rule connecting
28
the operations of addition and multiplication, is the one we must constantly appeal to when doing
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anything that involves both operations at once, which includes a wide range of manipulations:
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expanding, factoring, collecting like terms, or putting fractions over a common denominator.
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Example 6 illustrates the difference between mindless and mindful manipulation when
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multiplying polynomials.
Example 6: FOIL versus EWE
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Task
Expand:
(a) (1+ x 3 )(1+ x + x 2 )
(b) (1! x)(1+ x + x 2 )
In the Classroom (with apologies to John Bunyan)
Hopeful is a student who is accustomed to using the mnemonic FOIL (First, Outer, Inner, Last)
to expand products of two binomials, such as (1+ x)(1+ x 2 ) , applies it (mindlessly!) to the
problem (1+ x 3 )(1+ x + x 2 ) , and gets 1+ x 2 + x 3 + x 5 .
Mindful, his classmate, says, “You missed the products of the middle term, x, in the second
factor. The distributive property means we have to multiply each term of one factor with each
term of the other, and then add all the products.
So I get
(1+ x 3 )(1+ x + x 2 ) = 1!1+ 1! x + 1! x 2
+ x 3 !1+ x 3 ! x + x 3 ! x 2
= 1+ x + x 2 + x 3 + x 4 + x 5 .
“I sometimes remember this as EWE (Each with Each). Sometimes you can just multiply the
powers of x mentally. I like to visualize the steps and write as little as possible. For example,
when I apply ‘Each with Each’ to
(1! x)(1+ x + x 2 ) = 1+ x + x 2 ! x ! x 2 ! x 3 = 1! x 3 ,
I can see that the second and third terms from the multiplication by 1 are opposites of the first
and second terms from multiplication by x, so I can just write down the remaining terms.”
Key Elements of Mathematics
Reasoning with Algebraic Symbols—Meaningful use of symbols; Mindful manipulation
Reasoning Habits
Analyzing a problem—making connections with previous work
Initiating a strategy—making purposeful use of procedures; organizing a solution
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p. 32
Reasoned solving
2
Solving equations is a process of logical deduction, and the steps one can take are
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founded on reasoning about numbers. Problem solving with equations should include careful
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attention to increasingly difficult problems that span the border between arithmetic and algebra.
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Such problems can help students view algebra as a sense-making activity that extends one’s
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problem-solving skills into domains in which reasoning as done in arithmetic gets too
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complicated or cumbersome to carry out. Seeing the essential parallels between algebraic and
8
arithmetic solution methods can help students gain confidence that algebra is not something
9
totally different but simply a more powerful tool for dealing with problems that are hard to
10
approach with arithmetic by itself. In Example 7 we illustrate reasoned solving of equations.
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Worth noting is the fact that although one student used a standard algebraic approach and the
12
other used reasoning based in the concrete context, the steps in their solutions are essentially the
13
same. Examples of this sort can help students see algebra as an extension of concrete arithmetic
14
reasoning.
Example 7: Reasoned Solving of Equations
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Task
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A bar of soap on a pan of a scale balances
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How much does the bar of soap weigh? Solve the problem both with an algebraic equation and
by direct arithmetic reasoning. (Adapted from Kordemsky 1992)
3
4
of a bar of soap and
3
4
of a pound on the other pan.
In the Classroom (ninth-grade students’ solutions)
Alice: If x is the weight of the soap in pounds, then one side of the balance weighs x pounds and
23
the other weighs
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25
x = 3.
3
3
x+
4
4
, so x =
3
3
x+
4
4
. Subtracting
3
x
4
from both sides gives
1
4
3
4
x = , so
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Barbie: It’s just as easy without equations! If a bar of soap balances
3
4
2
take
3
quarter of a bar weighs
4
pounds.
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of a bar off each side. That leaves
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p. 33
3
4
of a bar and
of a bar on one side and
3
4
3
4
pounds,
lb on the other. So a
of a pound. A full bar is four quarters, and that will make 3
Teacher: Good! Can you see the connection between your two solutions?
Key Mathematical Elements
Reasoning with Algebraic Symbols—Meaningful use of symbols; Reasoned solving
Reasoning Habits
Initiating a strategy—selecting concepts, representations, or procedures; making purposeful use
of procedures
Seeking and using connections—connecting different mathematical domains; connecting
different representations
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Connections with geometry
A two-way interplay exists between algebra and geometry: such geometric
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representations as graphs or figures can cast light on algebraic expressions and equations, and
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algebraic representations can be used to deduce geometric relationships. Example 4 illustrated
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the use of an area model to multiplication of polynomials. As another example, the equivalence
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between x 2 + 10x and its factored form x(x + 10) is represented geometrically by two different
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ways of viewing the following figure:
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Students often find the process of completing the square mysterious. To represent this process in
25
a more understandable manner, we can dissect the rectangle and rearrange it in a geometrically
26
compelling way as in Example 8.
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Example 8: Completing the Square
Task
Find a way of solving the equation x 2 + 10x = 144 using an area model.
x
10
In the Classroom (guided discussion of intermediate algebra class)
Miss Landers: Can anybody see how to think of x 2 + 10x as an area?
Wally: Well, x 2 is the area of a square with side x and 10x is the area
x
10x
x2
of a rectangle with sides x and 10, so we can put the rectangle and
the square together like this. But I don’t see how this helps.
Mary Ellen: Maybe if we knew what the area of the square was, we
x+10
could just take the square root to find x.
Miss Landers: Is there a way of rearranging the figure into a square?
x
5
Wally: I know! If we cut the rectangle into two rectangles of width 5, we
could put one on each side like this! (See figure at the right.)
Mary Ellen: But it’s not a complete square, it’s missing a corner.
x
5x
x2
Miss Landers: What’s the area of the corner?
Mary Ellen: Oh, it must be 25, since the white square lines up with ends
of the rectangles, so it has side length 5. And since the gray area is
25
5x
5
144, the entire area of the big square is 144 + 25=169.
Wally: And that means the side length of the square is 13, so x + 5 = 13, which means x = 8.
Mary Ellen: Shouldn’t there be another solution since the (x + 5) is squared?
Miss Landers: Yes, there should be. Can you write down what you did algebraically?
Mary Ellen: We started with x 2 + 10x = 144 , and then we added 25 to the 144. I guess that means
we add 25 to both sides of the equation, so we get x 2 + 10x + 25 = 169 .
Wally: So to get 25, we divide the 10 by 2 to get 5, then square that to get 25.
Mary Ellen: Yeah, and then the left-hand side is a perfect square, so you get (x + 5) 2 = 169 .
Wally: The algebraic way gives you both solutions, because you get x + 5 = 13 or x + 5 = −13, so
x = 8 or x = −18, but I guess the area model can’t give you the negative solution.
Miss Landers: Good observations. The process you just did is called “completing the square”
because it always results in a perfect square. Can you see how this process might work for
other quadratic equations?
Key Mathematical Elements
Reasoning with Algebraic Symbols—Meaningful use of symbols; Connections with geometry
Reasoning Habits
Analyzing a problem—looking for hidden structure
Initiating a strategy—making logical deductions based on current progress
Seeking and using connections—connecting different representations
40
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p. 35
Linking expressions and functions
Although multiple representations of functions—symbolic, graphical, numerical, and
3
verbal—are commonly seen, the idea of multiple algebraic representations of functions is less
4
commonly made explicit. Different but equivalent ways of writing the same function can reveal
5
different properties of the function, as is illustrated in Example 5.
6
Symbolic representation shifts to a higher level when we start to use letters to stand for
7
functions and introduce function notation. The magnitude of this shift is often overlooked. High
8
school students have difficulty with extending the four basic arithmetic operations to functions
9
and also with composition of functions. Embedding these experiences in a context may enhance
10
students’ comprehension of the concepts and improve both their retention and ability to make
11
connections. See Example 9.
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Example 9: Finding the Next Term in the Sequence, Extreme Version
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Task
In earlier grades you may have seen problems that asked you to find the next term in a sequence,
such as 3, 7, 11. One possible answer is 15, assuming the sequence is generated by evaluating
f(x) = 4x – 1 at x = 1, 2, 3. But other answers are possible.
In the Classroom, with Apologies to Plato (fourth-year mathematics)
Socrates: Find the sequence generated by evaluating g(x) = x3 − 6x2 + 15x − 7 at x = 1, 2, 3.
Student: I get g(1) = 3, g(2) = 7, and g(3) = 11. It’s the same sequence: 3, 7, 11.
Socrates: What would be the next term if you used g(x) instead of f(x))?
Student: It would be g(4) = 21. That’s different from what we get using f(x).
Socrates: Can you find other polynomials that generate the sequence 3, 7, 11?
Student: No, Socrates, I am at a complete loss.
Socrates: How can two different polynomials have
the same value at x =1, 2, and 3?
Student: When I graph them on my graphing
calculator, I see that the straight line graph of
f(x) intersects the cubic graph of g(x) at three
points, corresponding to x = 1, 2, and 3.
Socrates: Can you see now how you might find
another polynomial graphically?
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Student: Perhaps I could stretch the cubic away from the straight line graph while keeping the
same intersection points.
Socrates: How would you do that algebraically?
Student: I want to stretch the difference between f(x) and g(x), which
is g(x) ! f (x) = x 3 ! 6x 2 + 11x ! 6 . I could double that, for example, and add it back on to f(x)
and get 4 x !1+ 2(x 3 ! 6x 2 + 11x ! 6) = 2x 3 !12x 2 + 26x !12 .
Socrates: Excellent. Now I want to understand what is really going on here. Is there anything
special about the polynomial x 3 ! 6x 2 + 11x ! 6 that makes this work?
Student: Not that I can see.
Socrates: Why don’t you try factoring it on your CAS?
Student: I get x 3 ! 6x 2 + 11x ! 6 = (x !1)(x ! 2)(x ! 3) . Oh, I see. When I look at the factored
form, I can see that the difference between f(x) and g(x) is zero at x = 1, 2, and 3. So I could
get many polynomials that generate the same sequence just by adding a multiple of this
polynomial to f(x).
Socrates: What is the general form of such a polynomial?
Student: It is 4 x + 1+ k(x !1)(x ! 2)(x ! 3) , where k can be any real number.
Key Mathematical Elements
Reasoning with Algebraic Symbols—Meaningful use of symbols; Expressions and functions
Reasoning Habits
Analyzing a problem—looking for patterns and relationships
Initiating a strategy—selecting concepts, representations, or procedures
Seeking and using connections—connecting different representations
25
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Reasoning with Functions
Mathematical reasoning is essential to helping students develop the concept of function
28
throughout every stage of high school mathematics. Conversely, the exploration of functions
29
gives students an opportunity to strengthen their reasoning ability, because functions themselves
30
are productive tools for reasoning. Developing procedural fluency with functions is also a
31
significant goal of high school mathematics because it will empower students to achieve the
32
greater goal of applying functions to problem-solving situations. Through these applications,
33
students will come to perceive functions as relevant mathematical tools for making sense of the
34
world around them.
35
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p. 37
Key elements of reasoning and sense making with functions include the following:
•
Multiple representations of functions. Representing functions in a variety of ways, including
3
tabular, graphic, symbolic (explicit and recursive), visual, and verbal; making decisions
4
about which representations are most helpful in problem-solving circumstances; and moving
5
flexibly among those representations.
6
•
Modeling using families of functions. Working to develop a reasonable mathematical model
7
for a particular contextual situation by applying knowledge of the characteristic behaviors of
8
different families of functions.
9
•
Analyzing the effects of parameters. Using a general representation of a function in a given
10
family (e.g., the standard form of a quadratic, f ( x) = a ( x ! h) 2 + k ), to analyze the effects of
11
varying coefficients or other parameters; converting between different forms of functions
12
(e.g., the standard form of a quadratic or the factored form) according to the requirements of
13
the problem-solving situation (e.g., finding the vertex of a quadratic or finding its zeros).
14
We address these key elements in more detail in the following sections.
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Multiple representations of functions
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Functional relationships can be represented in many ways. Common representations
17
include tables, graphs or diagrams, symbolic expressions, and verbal descriptions. Different
18
representations often provide different perspectives on the functional relationship. Using a
19
variety of representations may also serve to make functions more understandable to a wider
20
range of students than can be accomplished by working only with symbolic notation.
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An activity such as the one seen in Example 10 could be used with students entering high
22
school, who are beginning to develop the ability to express functional relationships symbolically,
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or with more advanced students who are working to improve their proficiency with algebraic
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manipulation.
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p. 38
Example 10: Multiple Representations of a Function
Task
Develop a symbolic representation of a function that produces the following ordered pairs:
In the Classroom
Students may approach this problem in several different ways, depending on their level of
mathematical experience:
1) Applying geometric reasoning, students may represent the ordered pairs by constructing a
sequence of blocks, such as that shown here:
1
When the elements in the domain of the function, x, are usedFigure
as the
row numbers and the
elements in the range, f(x), are used as the total number of blocks in the figure, one can see
that the blocks form half of a rectangle with base x and height x + 1 . The explicit definition of
this function can then be written as
1
2
19
f ( x) = x( x + 1) .
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21
22
A student who is given the opportunity to represent the function pictorially can use a simple
geometric formula to write the explicit definition of the function and then extend his or her
reasoning about symbolic representations by developing it from a familiar starting place.
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25
26
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2) Applying technology to numeric and graphical reasoning, students may enter the ordered
pairs into a graphing calculator and examine a scatterplot of the pairs to conjecture that the
relationship is quadratic because of the parabolic shape of the graph. On the basis of that
conjecture, students could use calculator-assisted regression to find the function
f ( x) = 0.5 x 2 + 0.5 x .
28
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30
3) Applying algebraic reasoning, students may choose to plot the ordered pairs or may examine
the data and observe that the function is quadratic because the second differences are
constant. Writing a system of equations of the form f ( x) = ax 2 + bx + c , substituting three
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2
1
2
ordered pairs, and solving the system would reveal that a = , b = , and c = 0 . This
strategy would require a sophisticated understanding of several mathematical topics and a
command of algebraic manipulation developed later in the high school experience.
Key Elements of Mathematics
Reasoning with Functions—Multiple representations of functions; Using families of functions as
mathematical models
Reasoning with Algebraic Symbols—Meaningful use of symbols
Reasoning with Geometry—Conjecturing about geometric objects; Geometric connections and
modeling
Reasoning Habits
Analyzing a problem—looking for hidden structure; looking for patterns and relationships;
making preliminary deductions and conjectures
Initiating a strategy—selecting concepts, representations, or procedures; making
logical deductions; using procedures with understanding
Seeking and using connections—connecting different mathematical domains; connecting
different representations
Reflecting on one’s solution—checking the reasonableness of a solution; justifying or validating
a solution; generalizing a solution
22
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Another important symbolic representation of functions is recursive notation, in which a
24
function value at stage k + 1 is defined in terms of the function value at stage k, following the
25
definition of an initial function value. High school mathematics should help students build
26
facility with both explicit and recursive definitions of functions. Contextual situations in which
27
the domain of a function is the set of natural numbers present the opportunity to help students
28
develop an understanding of sequences and determine when a discrete graph is more appropriate
29
than a continuous graph. A recursive approach, especially when supported by a calculator or an
30
electronic spreadsheet, can give students access to interesting contextual problems throughout
31
their high school experience, as shown later in Example 11.
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Modeling using families of functions
According to Curriculum Focal Points (NCTM 2006), students should enter high school
3
having had extensive experience with linear functions and some exposure to nonlinear functions.
4
High school experiences can help students build on this knowledge and perceive functions as a
5
larger algebraic structure in which different families of functions (e.g., linear, quadratic,
6
exponential, and periodic) behave in particular ways and possess unique characteristics.
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Providing opportunities to use many families of functions to model realistic data serves to
8
strengthen students’ understanding of the characteristics of various function families and
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increase their appreciation of the usefulness of mathematics. By the time they finish high school,
10
given a set of data points, students should be able to (a) decide which family of functions, among
11
those they have studied, most sensibly models the situation; (b) transform the particular function
12
to match the data provided; (c) use the model to gather the necessary information and solve a
13
problem based on the function chosen; and (d) communicate their solution in the context of the
14
problem. Throughout the high school mathematics program, students benefit from numerous
15
opportunities to discern which family of functions best fits a particular problem on the basis of
16
the context and data provided.
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Using problems that can be extended and revisited as students mature is a powerful way
18
to help students make connections between existing knowledge and new learning. Example 11
19
uses a context to which students can readily relate, and demonstrates how a problem can be
20
extended as students’ mathematical backgrounds become more sophisticated.
21
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Example 11: Modeling a Geometric Series Using Recursive Notation
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Task
A student strained her knee in an intramural volleyball game, and her doctor has prescribed an
anti-inflammatory drug to reduce the swelling. She is to take two 220-mg tablets every 8 hours
for 10 days. Her kidneys eliminate 60 percent of this drug from her body every 8 hours. Assume
that she faithfully takes the correct dosage at the prescribed regular intervals. Determine how
much of the drug is in her system after 10 days, just after she takes her last dose of medicine.
Explain what happens to the change in the amount of medicine in the body as time progresses
and why this pattern occurs. If she continued to take the drug for a year, how much of the drug
would be in her system just after she took her last dose? Explain, in mathematical terms and in
terms of body metabolism, why the long-term amount of medicine in the body is reasonable
(adapted from NCTM 2000b).
In the Classroom (variety of levels from first-year to fourth-year mathematics)
Students may examine this problem in varying degrees of depth, depending on their level of
mathematical experience:
1) Students who are near the beginning of their high school experience could produce a table of
data and generate a discrete graph of the data. The following table shows selected values of
the amount of medicine, An , in the athlete’s body just after taking n doses of medicine, with
values rounded to the nearest hundredth.
n
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An
440
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686.40
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An
733.14
733.26
733.30
733.32
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733.33
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Students may observe that the level of medicine in the body initially rises rapidly but with
time increases less rapidly. Thus, the rate of change of this function does not remain constant.
Even a beginning student could surmise from this table that the medicine level appears to
eventually stabilize, so that after about seventeen periods, the value no longer seems to
change. In terms of metabolism, students could explain that the athlete’s body is eliminating
the same amount of medication as she is taking in each dose. This observation can be
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mathematically verified by showing that 0.6(733 ) = 440 because the value of An
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approaches a constant level of 733 mg after about 17 doses.
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p. 42
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2) A problem such as this can be used to introduce students with more mathematical experience
to recursive notation.
A recursive definition for the sequence of medicine levels in the
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athlete’s body would be An = 0.40 An!1 + 440, A0 = 440 . Obtaining explicit formulas from
tables of data is often quite difficult and in some cases, impossible. A recursive approach,
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especially when 40supported
by a calculator or an electronic spreadsheet, is more intuitively
developed and gives students access to such interesting problems as this earlier in their
schooling.
3500
3) Students who are studying exponential functions might use a problem such as this to develop
their understanding of exponential decay. By examining the pattern of medicine levels,
3000 that the change of medicine level in the athlete’s body is decreasing;
students can observe
specifically, each change is 40 percent of the previous amount of change. After reasoning
about the repetitive nature of multiplying by 0.4, students could express this relationship with
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the explicit function I(n) = 440(0.4 n!1 ) , where I represents the increase in medicine level
and n represents the dose number.
2000
4) For older students with a more sophisticated mathematical background, this problem could
be revisited as a means of informally introducing students to the important mathematical
concept of limit, especially when coupled with a discrete graph of the sequence, as shown in
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the following figure.
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-500
5) Students who have more experience with reasoning and sense making may be asked to
formally justify why the level of medicine converges, thus creating a deeper understanding of
the phenomenon and why it happens. In this approach to the problem, students are asked to
use a formal proof to show that consecutive terms of the sequence will converge for large
numbers of doses. For instance, consider the recursive function for the amount A of medicine
in the athlete’s body after n doses: A(n + 1) = A(n) ! 0.6 A(n) + 440 = 0.4 A(n) + 440 . We can
define the behavior of the medicine level as a linear function C, called the transition function
governing the sequence A(n) , where C (x ) = 0.4 x + 440 . By substitution, we can write
A(n + 1) = 0.4 A(n) + 440 = C ( A(n)) . Now if the medicine concentrations truly level off, we
can let V represent the fixed value approached by A(n) after a certain number of doses, called
the fixed point of C. Doing so leads to the equation C (V ) = V , or 0.4V + 440 = V .
Solving this last equation for V gives us the value V = 733 1 , which proves that the limiting
3
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or asymptotic value of the blood concentration is 733 1 mg, as explained by the transition
3
function C. Incorporating formal proof once students have the necessary mathematical
background to understand it serves to strengthen their reasoning skills and can also ease the
transition from high school to college mathematics.
To see how the transition function guarantees the convergence, we can rewrite it to display
the difference from the fixed value. Let A(n) = V + A'(n), so that A'(n) is telling us how far the
actual amount is from the fixed value V. Substituting this in the recursion relation gives: V +
A'(n+1) = A(n + 1) = 0.4(V + A'(n)) + 440 = (0.4V + 440) + 0.4A'(n).
Since V is defined by the relation V = 0.4V + 440, we can cancel most of the terms in this
equation, and reduce to the simple recursion: A'(n + 1) = 0.4A'(n).
This shows that at each step, A' is shrunk to 40% of its previous value. Clearly after not too
many steps, A'(n) will become negligibly small, as was suggested by the numerical
calculations above.
Using problems that can be extended and revisited as students mature is a powerful way of
helping students make connections between existing knowledge and new learning.
Key Mathematical Elements
Reasoning with Functions—Multiple representations of functions; Modeling using families of
functions as mathematical models
Reasoning with Numbers and Measurements—Approximations and error
Reasoning with Algebraic Symbols—Meaningful use of symbols; Linking expressions and
functions
Reasoning Habits
Analyzing a problem—looking for patterns and relationships; making preliminary deductions
and conjectures
Initiating a strategy—selecting concepts, representations, or procedures; making purposeful use
of procedures; making logical deductions
Monitoring one’s progress—reviewing a chosen strategy
Seeking and using connections—connecting different representations
Reflecting on one’s solution—checking the reasonableness, justifying or validating
The general reasoning in this example can be extended to a range of other situations, as
shown in Example 12.
37
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Example 12: Recursion and Loans
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Task
After exploring method 5 in Example 11, a teacher asks her students to read and interpret the
following text describing how to determine the payment amount for installment loans, and to
answer a series of questions about the text.
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This implies that L '(0) = !
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In the Classroom (fourth-year mathematics)
In an installment loan, someone borrows an amount of money L and makes periodic payments
(typically monthly) to the lender until the loan is paid off. At the end of each period, interest is
charged on the unpaid amount of the loan and is added to the amount owed, and the borrower
makes a payment. Define L(n) to be the amount owed at the end of period n, where L(0) is the
amount initially borrowed. If the interest rate for a single period is r and the borrower makes a
payment P at the end of each period, then the amount owed at the end of the period (n + 1) can
be written as L(n + 1) = L(n) + rL(n) − P = (1 + r)L(n) − P.
The amounts owed in successive periods of an installment loan follow this form: the amount at
the end of period (n + 1) is obtained by applying a fixed linear function Q to the amount in
period n. In this case, Q(x) = (1 + r)x − P, and L(n + 1) = Q(L(n)) = (1 + r)L(n) − P.
We can use the fixed-point analysis to simplify the effect of Q. The fixed point of Q is
characterized by the formula Q(x) = x. Concretely, if we call the fixed point M, this means that
Q(M) = (1 + r)M − P = M, or rM − P = 0, or rM = P or M = P/r .
Next let us look at the difference between L(n) and the fixed point M, which we call L′(n). Thus,
L(n) = M + L′(n). Then Q(L(n)) = Q(M + L′(n)) = (1 + r)(M + L′(n)) − P =
(1 + r)M + (1 + r)L′(n) − P = ((1 + r)M − P) + (1 + r)L′(n) = Q(M) + (1 + r)L′(n) =
M + (1 + r)L′(n).
If we want to pay off the loan after m periods, then we will want L(m) = 0. In other words,
L(m) = Qm(L(0)) = M + (1 + r)mL′(0) = 0.
M
. Remembering that L(n) = M + L′(n) and using the formula
(1 + r ) m
above for M, we can rewrite this equation as
L(0) = M #
! (1 + r ) m # 1 "
! (1 + r ) m # 1 "
!
M
1 "
.
=
M
1
#
=
M
=
P
$
%
$
$
m %
m
m %
(1 + r ) m
& (1 + r ) '
& (1 + r ) '
& r (1 + r ) '
36
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p. 44
! (1 + r ) m "
Solving this equation for P gives P = L(0) # r # $
%.
m
' (1 + r ) & 1 (
38
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This formula gives us the amount of the monthly payment needed to pay off the original loan
amount, L(0), in m periods given interest rate r for a single period.
Answer the following questions about this analysis:
1. How does the function describing the successive periods of an installment loan compare with
the function describing the blood concentrations in successive periods when taking a drug?
2. Compare the effect of Q in the two situations. Why do we want to find a fixed point in each
situation?
3. How does the multiplier (1 + r) differ between this situation and the drug-concentration
example? How does the value of the multiplier ensure that we can eventually pay off
the loan?
4. Explain how the underlying mathematics in this situation is very similar to finding the blood
concentration of drugs, even though the two situations appear very different.
Key Mathematical Elements
Reasoning with Functions—Modeling using families of functions as mathematical models
Reasoning with Algebraic Symbols—Reasoned solving
Reasoning Habits
Analyzing a problem—making connections with previous work
Seeking and using connections—connecting different contexts
Reflecting on one’s solution—justifying or validating a solution; generalizing a solution to a
broader class of problems; interpreting a solution
24
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Analyzing the effects of parameters
Different but equivalent algebraic expressions can be used to define the same function,
27
often revealing different properties of the function. For example, writing a quadratic function in
28
the form y = ax 2 + bx + c helps us identify the y-axis intercept, whereas using the form
29
! 1 "
2
y=$
% (x # h ) + k
& 4p '
30
enables us to quickly determine the vertex of the parabolic graph and the location of its
31
focus. In Example 13, students are given the opportunity to explore the effects of changes to the
32
value of a parameter in a quadratic function.
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p. 46
Example 13: The Effects of Parameters on a Graph
Task
Using a graphing utility, investigate how the graph of y = ax2 + 2x + 1 changes as the constant a
changes. Describe what appears to be happening, and discuss why these changes occur.
In the Classroom (second-year algebra)
The students working on this problem have had experience with the slope and y-intercept of a
linear function, and are just beginning to explore the effects of parameters on the graph of a
parabola.
Students could use a graphing utility such as a graphing calculator, Java applet, or interactive
graphing software tool to explore the effects of changing the value of a. Several snapshots of the
changing function are shown in the following:
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An example of student reasoning about this group of graphs follows:
Teacher: I want you to come up with a description of what is happening to the graph of the
function as the value of a changes.
Student 1: Well, I notice that the graphs always open upward when a is positive, and downward
when a is negative.
Student 2: Could you drag it so that a = 0? I wonder what would happen then.
Student 1: Hey, now it’s a line. Maybe zero doesn’t work.
Student 2: But think about the equation—if you put zero in for a, then you get the equation of a
line, so that does make sense.
Student 3: This reminds me of the projectile formula we just worked with in physics, except the
physics a was −4.9 and the graphs were always upside down.
Student 2: So the sign of the a term must make the graph open right side up or upside down.
Student 3: See how all the graphs cross the y-axis at 1? Do you think that’s because of the 1 at
the end?
Student 2: Yeah, just think about that you are on the y-axis if x = 0, so it makes sense that the
graph will always cross the y-axis at 1 no matter what a and b are.
Student 1: It seems like the graphs are getting skinnier as our a gets bigger too. Why do you
think that is happening?
The conversation could continue like this in small groups, followed by a whole-class discussion
of the observations made by various groups.
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Key Elements of Mathematics
Reasoning with Functions—Multiple representations of functions; Effects of parameters
Reasoning with Algebraic Symbols—Meaningful use of symbols; Linking expressions and
functions
Reasoning Habits
Analyzing a problem—looking for patterns and relationships; making preliminary
deductions and conjectures
Seeking and using connections—connecting different representations
Reflecting on one’s solutions—generalizing a solution
Although high school mathematical experiences become increasingly sophisticated as
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students progress, opportunities to reason with functions and use them for modeling real-world
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situations exist at every stage of a high school student’s mathematical development.
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Unfortunately, students tend to have difficulty with function notation. By providing students with
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numerous and varied experiences with functions, we can help them internalize this sometimes
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confusing mathematical language and become better equipped for future success. Teachers can
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facilitate their students’ mathematical growth, both in reasoning ability and their understanding
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of the concept of function, which is one of the cornerstones on which a well-developed
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understanding of mathematics is built.
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Reasoning with Geometry
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Classically, the importance of high school geometry has been associated with deductive
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mathematical proof. However, geometric settings offer valuable opportunities for many
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reasoning and sense-making habits. Furthermore, studying geometry is beneficial for many other
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reasons. The ability to visualize geometric configurations, including spatial visualizations, adds
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value to anyone’s life. Many geometric ideas have useful connections with concepts in other
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mathematical domains, and geometry has important applications in careers and in everyday life.
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p. 48
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Geometric ideas are a significant part of many high-technology developments, including high-
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definition television (HDTV), global positioning systems (GPS), computer animation, computed
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axial tomography (CAT) scans, cellular telephone networks, robotics, virtual reality, and docking
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of the space shuttle. Geometric and visual reasoning often enter our daily lives. Before buying a
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new refrigerator, a consumer needs to determine whether it will fit through the door and around
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the corner. The careful geometric abstractions of the geometric ideas of a “space” and a “distance
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metric” have led to many interesting and useful mathematical results. Today, for example,
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mathematical biologists define and use a concept of distance between strands of DNA.
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Key elements of reasoning and sense making with geometry include these:
•
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Conjecturing about geometric objects. Analyzing configurations and reasoning inductively
about relationships to formulate conjectures.
•
Construction and evaluation of geometric arguments. Developing and evaluating deductive
13
arguments (both formal and informal) about figures and their properties that help make sense
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of geometric situations.
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•
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Multiple geometric approaches. Analyzing mathematical situations using transformations,
synthetic approaches, and coordinate systems.
•
Geometric connections and modeling. Using geometric ideas including spatial visualization
in other areas of mathematics, other disciplines, or in real-world situations.
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We address these key elements in more detail in the following sections.
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Conjecturing about geometric objects
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Making conjectures is a habit of mathematical reasoning that relates to all areas of
mathematics and is a fundamental activity in mathematical inquiry. Situations involving
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p. 49
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geometry are ideal for the purpose of formulating conjectures, as intriguing and often surprising
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visual or measurable geometric relationships abound.
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Geometric conjectures often arise from analyzing a planar or spatial configuration or
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from wondering whether a certain configuration can exist. Presenting situations in which
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students themselves create mathematical conjectures can activate their natural inquisitive nature
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not only about “what might be happening” (the conjecture) but “why it would be happening”
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(looking for insight, validation, or refutation.) Together with its attendant reflection, exploration,
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and explanation, the process of looking for and making conjectures gives students the
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opportunity to become immersed in these situations in ways that can foster deep understanding
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of the mathematical relationships involved, as well as the concepts and procedures that might be
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relevant in determining their validity; see Examples 14 and 15. Moreover, situations in which
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students are invited to make conjectures can contain some novel aspect or invite unique student
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perspective. Employing mathematics in new situations is a highly desirable skill in our fast-
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changing world. Further mention of this skill is made in the section titled “Geometric
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connections and modeling.”
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Using real-world contexts for mathematical analyses can be valuable for reasons other
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than the immediate utility of the results to the situation itself. The context for student conjectures
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in Example 14 is used for several reasons: (1) it provides a common, interesting, and
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recognizable situation in which students can immerse themselves for the purpose of
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mathematical analysis, (2) it offers multiple accessible methods to explore the situation for the
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purpose of creating conjectures, and (3) it fosters mathematical perspective: “mathematics is
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everywhere.”
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p. 50
1
Example 14: Picture This
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Task
Camera
Shelly wants to take an artistic photo of the
m!"#$ = 50°
P
decorated side of a building, which sits on a flat plot
of land. She has decided she wants to capture the full
width of the building (exactly) but not necessarily
Viewing
the full height of the building. She is using a camera
Angle
lens with a fixed horizontal viewing angle of 50
degrees, and she wants to take a level shot of the
building side. She has found one spot that works as
indicated by point P in the diagram. She believes
A
B
Subject
that she could stand at other places to capture the
Expanse
same horizontal expanse but from a different
perspective. Before she snaps the picture, she wants
to examine some of these other positions. Your task is to figure out the ground locations where
Shelly could stand to create a picture fitting her criteria.
Explore the situation, write down any conjectures you have regarding possible positions, and
justify any that you can. The object is to eventually create a conjecture describing the full range
of possible positions. Prepare to clearly state your conjectures as well as the thinking,
exploration, and reasoning that explain how you arrived at them.
In the Classroom
A dynamic drawing utility (with a file containing the sketch above), a hard copy of the sketch,
and a physical manipulative (a rigid angle—e.g., the expanse of a rigid compass), or a real
camera and wall could be used to facilitate exploration of possible locations. The lesson begins
with a full-class discussion of initial thoughts. After some time for individual thought or
explanation, the following dialogue ensues.
Teacher: Do you have any immediate conjectures to suggest?
Student 1: I think there is a point right in the middle where you could stand.
Teacher: What do you mean by middle?
Student 1: I mean I would stand at a point in the diagram that would be the same distance from
points A and B.
Student 2: Oh, that means the point would be somewhere on the perpendicular bisector of the
segment AB which represents the wall. This is because we have already learned that the
perpendicular bisector of the segment is the collection of points that are equidistant from
points A and B.
Teacher: How would you find such a point?
[Students engage in more work here.]
Student 1: I’d put a point on the perpendicular bisector of segment AB and
then move it back and forth until the angle exactly fit the segment.
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Teacher: Good, that gives some detail regarding how one might locate a second point. (The
teacher writes down a conjecture that there is a possible vertex location on the perpendicular
bisector of segment AB. This conjecture will be addressed in a later discussion—beyond the
confines of this example.) Are there other ideas about possible vertices?
Student 2: Based on my experience with a camera, I think there would be many points one could
stand to show the entire wall.
Many students in the class agree with the conjecture that many points would work. Another
student jumps in and suggests that that the possible positions possess symmetry. Eventually, it is
stated that reflective symmetry would be evident in the collection of possible points across the
perpendicular bisector of AB. This conjecture will be evaluated later as well.
Teacher: Now, by using either the dynamic drawing utility or the physical manipulative, plot a
collection of points that represent where the photographer might stand to snap the required
image. See if any other conjectures emerge from this exploration.
Student 4: (After some exploration) All the points seem to be
on a circular arc that goes from point A to point B.
The teacher next facilitates a discussion about stating the
conjecture abstractly; settling on “Given a line segment AB,
all vertices of angles going through points A and B and which
measure 50 degrees lie on a circular arc containing A and B.”
m!ACB = 50.00°
C
This conjecture is not resolved immediately. Rather, a set of
A
B
questions is compiled by the class for moving forward toward
a proof of this conjecture. These include the following:
1. Can we find out which circular arc we are talking about? That is, can we find the circle that
contains this arc?
2. Would every point on the arc (except points A and B) represent possible positions where the
photographer could stand?
3. Would the arc miss any possible points?
(See Example 15 along with the forthcoming topic book on Geometry for a full discussion of the
conjecture.)
Key Elements of Mathematics
Reasoning with Geometry—Conjecturing about geometric objects
Reasoning Habits
Analyzing the problem—looking for hidden structures; considering special cases; looking for
patterns and relationships; making preliminary deductions and conjectures
Initiating a strategy—selecting concepts, representations, or procedures
Monitoring one’s progress—reviewing a chosen strategy
Reflecting on one’s solution—justifying or validating a conclusion
44
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p. 52
1
Example 14 illustrates many of the points made in the discussion above it. The conjecture
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created in Example 14 is not resolved immediately. However, the context and conjecture-making
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activity set a stage for a natural discussion of several important geometric results. See Example
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15 for a description of three important results that develop in answering question 1 of Example
5
14. Example 15 makes some progress toward resolution of the conjecture. Further progress can
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be made through a subsequent development of the inscribed angle theorem.
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Making a conjecture is a first step in mathematical inquiry. Students need to follow up
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many of their conjectures with efforts to justify (or disprove) them. As in Example 14,
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conjectures may not be resolvable immediately. Intermediary results may be needed.
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Nonetheless, a natural coupling occurs between the key element of conjecturing about geometric
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objects and the key element of constructing and evaluating geometric arguments, described in the
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following section.
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Construction and evaluation of geometric arguments
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Reasoning and sense making in the high school mathematics classroom require an
understanding of —
•
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the role of empirical evidence that supports, but does not justify, a conjecture—“it
works in several cases”;
•
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the value of informal and partial explanations that lend insight into what is
happening; and
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•
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The latter includes not only the ability of students to create meaningful chains of logical
the role of logical deduction in determining mathematical certainty—proof.
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reasoning using definitions, axioms, and prior results but also the ability to read and evaluate
23
reasoning given by others. Moreover, not only is the ability to determine the validity of the
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p. 53
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deductive reasoning important, but equally important is an understanding of what a proof says
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about the mathematical ideas under consideration. See Example 15.
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Example 15: Circling the Points
Task
This task is to answer question 1 raised at the end of Example 14.
What circle contains the conjectured arc?
C
m!ACB = 50.00°
In the Classroom (geometry)
Students are divided into groups with the assignment to decide what
to do next. After a short time, the groups report on possible strategies:
A
B
Group 1: We think you need to find the center and radius of the circle.
Group 2: We think you just need to find the center of the circle. You don’t need to find the
radius because you already know at least one point on the circle—either the point P specified
in the original problem or points A or B which are part of the conjecture. You can use the
center and one of these points on the circle to draw the whole circle.
Group 3: We think you need the center. We think the center should the point on segment AB
halfway between A and B. The radius should be half the length of segment AB.
(The groups reconvene to discuss these and possibly other
thoughts. After a few more minutes the groups report.)
P
Group 4: We tested to see if the midpoint of segment AB
D
would be the center. The circle did not go through the
point P we were given. So the midpoint is not the center
in this case. But we do agree that the center (like the
A
B
midpoint) needs to be equidistant from both A and B,
since the circle needs to go through A and B. Thinking
back to what we talked about when we started Example 14, we conclude that the center of the
circle has to be on the perpendicular bisector of segment AB. But at the same time, the circle
has to go through the points A and P. So the center of the circle also has to be on the
perpendicular bisector of the segment AP. We constructed these two bisectors and found that
they intersect in one point, which we called D. That is the only point that lies on both
perpendicular bisectors. So it is the only possible point for the center of a circle.
Now, if we draw segments DP and DA, they have to have equal lengths because D is on the
perpendicular bisector of segment AP. If we then draw DB, we see that segments DA and DB
must be equal because D is on the perpendicular bisector of segment AB. So the three lengths
are equal and D really is the center of a circle that goes through the three points, A, B, and P.
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Further discussion of these arguments suggests three results: (1) Three noncollinear points in the
plane determine a unique circle. (2) (Circle construction algorithm) The center of that circle can
be found by finding the point of intersection of the perpendicular bisectors of any two sides of
the triangle with the three noncollinear points as vertices. (3) The perpendicular bisectors of the
three sides of a triangle intersect in a single point. For homework, students are asked to write
proofs of each of these statements, adding details where necessary. In addition, they are asked to
write an algorithm (i.e., well-defined sequence of steps) for constructing a circle through three
noncollinear points in the plane. Also, they are asked to check to determine whether such an arc
seems to fit the data that suggested the arclike pattern in Example 14. That is, does it appear that
headway is being made with regard to the questions in Example 14?
Key Elements
Reasoning with Geometry—Construction and evaluation of geometric arguments; Multiple
geometric approaches
Reasoning Habits
Analyzing the problem—looking for hidden structures (drawing auxiliary lines)
Initiating a strategy—selecting mathematical concepts, representations, or procedures;
organizing a solution; making logical deductions
Monitoring one’s progress—reviewing a chosen strategy
Reflecting on one’s solution—checking the reasonableness of a solution; justifying or validating
a solution
The notion of proof and its presentation are two different entities. A two-column proof is
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not more rigorous, valid, or formal than a discursive proof. In fact, one must be wary that strict
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adherence to any proof format may elevate focus on form over function, subsequently
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obstructing the creative mix of reasoning habits that should be at work. Ironically, rigid formats
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for proof may actually hinder the chance of students’ successfully understanding the
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mathematical consequences of the arguments (Herbst 2002). Although some useful guidelines
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are available, no foolproof procedure exists for proof-making. In addition to chains of reasoning,
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a mathematical proof often requires a resourcefully selected strategy, cultured intuition, and good
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judgment. Blind alleys and false starts sometimes generate insight along with the attendant
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frustration. Therein lies the challenge!
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An important point to reiterate is that such reasoning and proof are not confined to the
domain of geometry. They permeate all of mathematics.
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p. 55
Multiple geometric approaches
Geometric situations can be approached in many different ways, including synthetic
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approaches as seen in the preceding Examples 14 and 15. Coordinates can also be a useful way
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of approaching problems, as seen in the example involving the distance formula in chapter 1,
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allowing application of algebraic concepts in geometric contexts, and vice versa.
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The value of transformations in geometry has been recognized for more than thirty years
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(cf. Coxford and Usiskin [1975]; NCTM [1989, 2000a]). Geometric transformations are
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mathematical functions with special properties. Basic geometric transformations include
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rotations, translations, reflections, and dilations. Geometric transformations provide a useful way
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to understand relationships among geometric objects. Among other things, they support an
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alternative and extended way of considering congruence and similarity of objects. They are also
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an effective tool for studying symmetry. In Example 16, students are challenged to think deeply
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about rotational symmetry rather than apply what seemed to be a simple rule of merely dividing
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the number of sides of a regular polygon into 360 to find the number of degrees of rotational
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symmetry the figure has. The students are involved in multiple aspects of reasoning and sense
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making in this example.
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Example 16: Taking a Spin
Task
What regular polygons have 80-degree rotational symmetry?
In the Classroom (ninth-grade geometry)
The following scenario took place in a ninth-grade high school geometry class. Students have
been asked to answer the following problem for homework: “What regular polygons have 80degree rotational symmetry?” (Martin 1996). At the beginning of class the following day, class
members discuss this problem in small groups. One group goes to the front of the class to present
its solution.
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Student 1: We concluded that there aren’t any, since 80 does not divide into 360 evenly. When
we were looking at regular polygons last week, we saw that if you have, like, a pentagon, you
can make five triangles, and each has a 72-degree angle. So if you turn the triangle 72
degrees, it will match. But you can’t do that with 80. So it won’t work.
Teacher: What did the rest of the groups conclude?
Student 2: That’s what we thought at first. But then we thought about a 360-gon has a 1-degree
angle and so it should be able to hit every degree, so if you rotate it 80 times, it will
eventually work!
Student 1: But is that possible? 80 doesn’t divide into 360, right?
(The teacher asks the students to further discuss this issue in their small groups for a few
minutes, then pulls the class back together, asking the groups to share their observations.)
Student 3: OK, we agree that 360 will work. Like they said, it will hit every degree. [He draws a
rough sketch with many small sides.] If you rotate it once, it will match. But you can keep on
going. [He mimics doing lots of very small rotations.] And when you do this 80 times, it will
work.
Teacher: Is that really 80-degree rotational symmetry?
Student 3: We rotated 80 degrees, and it matched up with itself.
Student 4: We think that 2 degrees should work, too.
Teacher: What do you think they mean by that?
Student 4: If you rotate it forty times, it will hit 80 degrees.
Student 5: But you can’t have a 2-gon! That wouldn’t even be a polygon.
Student 4: Oh.
Student 6: It’s not a 2-gon, it has a 2-degree angle. So if you divide 2 into 360 that would be a
180-gon.
The class continues to discuss the situation and finally agrees that both regular 360-gons and
180-gons will work. Students suggest other regular polygons that they think will work, and the
teacher ends the discussion by assigning students the task of finding all regular polygons that
have 80-degree rotational symmetry. He asks them to write up their findings in a short essay that
they can share with their classmates the next day. The essay should both give all the regular
polygons that they have found and prove that they have in fact found them all.
Key Elements of Mathematics
Reasoning with Geometry—Conjecturing about geometric objects; Construction and evaluation
of geometric arguments; Multiple geometric approaches
Reasoning Habits
Analyzing the problem—considering special cases
Initiating a strategy—selecting concepts, representations, or procedures
Monitoring one’s progress—reviewing a chosen strategy; analyzing the problem further;
changing or modifying strategies used if necessary
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The idea of geometric transformations forges a link between geometry and other concepts of
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mathematics, such as the general concept of functions and the use of matrices in their
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representations.
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Geometric connections and modeling
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p. 57
The use of coordinate geometry to justify geometric properties is an important wedding
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of geometry and algebra. But geometric ideas also connect with many ideas in other
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mathematical domains. Such connections arise naturally and gainfully in mathematical modeling
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situations. In Example 17, ideas from geometry, trigonometry, algebra, functions, and
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measurement all play a role in a modeling problem that is quite simple to state and begin.
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In addition to the reasons given previously for including modeling in the high school
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mathematics curriculum, we offer several additional reasons based on Example 13. First,
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modeling gives students an opportunity to combine mathematical ideas in novel ways. Not only
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is this ability a life skill, but the ability to use knowledge effectively in new situations has been
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highlighted in several reports on the workforce and innovation (Partnership for Twenty-first
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Century Skills 2007; Secretary’s Commission on Achieving Necessary Skills 1991; APICS
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2001). Second, working to establish a mathematical model can allow students to become
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engrossed in the mathematics in ways that promote mathematical reasoning. Third, modeling can
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provide a contextual need to develop mathematical ideas as well as apply them. For example,
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creating an animation by modeling the movement of images in a plane can serve as motivation to
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introduce and study matrix theory. Fourth, modeling situations can provide points of access for
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learners with various backgrounds and skills. Many modeling examples can be continued to very
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advanced levels. For instance, Example 17 can be extended into the calculus classroom. Fifth, as
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exemplified by Example 14, some modeling contexts can serve as the basis of several lessons—
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reflecting the genuine fact that persistent, extended efforts are sometimes needed to solve
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mathematics problems.
p. 58
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Example 17: Clearing the Bridge
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Task 1
Mathematician Henry Pollak (2004) pondered
why tractor trailers often got stuck under a
certain underpass when the “maximum
clearance” was clearly labeled by a sign that
indicated the height of the bridge. The bridge
under consideration was level and located just at the base of a descending roadway as indicated
in the figure to the right. In initial tasks students are asked to construct a two-dimensional visual
model of the situation listing assumptions that they make.
In the Classroom
Upon creating a visual model, students
should easily see that if the road slopes
as indicated, one set of trailer wheels is
jacked up on the sloping part of the
road as the truck passes under the edge
of the bridge. This slope causes a
portion of the trailer to be raised higher
than it would be on a flat surface. The
real question is “how much higher?”
The model students construct likely
will include characteristics similar to
those in the diagram below. The model
should at least contain the line segment PQ—representing the “dangerous height” at which the
trailer might hit the bridge. Some detail about a student model is mentioned at this point. First,
the student model contains simplifying assumptions—which merit discussion. One of these is the
representation of the road as two straight line segments. Another is the identification of the pivot
axes of the trailer box with the wheel axels. The trailer box is represented by a rectangle, and on
a level road the trailer-box bottom is assumed to be parallel to the ground. An obvious
assumption is that the grade of the slope is less than 90 degrees. Several relevant measurement
variables are pictured: t, denoting the height of the trailer box; k, denoting the distance from the
axle (pivot axis) to the ground; l, denoting the length of the trailer; s, denoting the horizontal
distance the axle is under the bridge; w, denoting the angle representing the tip of the truck; and
g, denoting the grade of the road slope. In this iteration of the model, a further assumption is that
as the trailer tilts, a shifting load on the wheels does not distort the height k.
After students work on the construction of a model, a dynamic sketch can be given to students so
that they can explore how changes in one of these variables affects other variables including, in
particular, the length of segment PQ. Using a dynamic graphing utility can reveal some
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characteristics of these relationships and thus promote one level of profitable mathematical
analysis.
Task 2 represents another level, as it asks students to explore a relationship symbolically. This
work will combine content associated with geometry, algebra, and precalculus. (Higher levels of
analysis are possible and could involve such topics as parametric representations of functions
and topics encountered in calculus related to optimization.)
Task 2
(a) Let’s let d represent the length of segment PQ. Can you find a more precise representation
between the tilt of the trailer and the measure of the dangerous height d? In other words, by using
some or all of your measurement variables together with geometric and trigonometric concepts
and results, can you find a symbolic relationship between the tilt of the truck, as measured by w,
and the measure of the dangerous variable, d? (b) How does the relationship you found in (a)
when there is a real tilt compare with an analysis of the situation when the road is level?
In the Classroom
Group 1 presents its findings:
For part (a) we labeled some points on
the figure. Then we drew in a line segment from
point Q perpendicular to the bottom of the truck
box at point E. Then we divided segment PQ
into three parts: HQ, GH, and PG. We need the
length of each of these pieces to find the length
d. We noticed right away that (length PG) = k,
because quadrilateral PGBA is a rectangle.
We found out about (length HQ) next.
We noted that the measures of angles BHG and
QHE are equal because they are vertical angles.
So right triangles ∆ BHG and ∆QHE are
similar. The important thing is that the measure
of angle EQH must be w. Since we wanted a relationship that had (length HQ) and w in it, we
used trig because cos(angle EQH) = (length QE)/(length HQ).
We noted that the length of segment QE is t, since the truck box is a rectangle. So cos(w) ≠ 0,
and we could write
t
(length HQ) =
.
cos w
Next, we wanted a relationship that had w and (length GH) in it. We saw that
tan w = (length GH)/(length GB). Since quadrilateral PGBA is a rectangle,
(length GB) = s. So tan w = (length GH)/s. Then (length GH) = s(tan w).
t
+ s (tan w) + k .
We put these together and got d =
cos w
For part (b) we first noticed that if the road were flat, the total trailer height would be just
t
t + k. Our formula from part (a) has
+ s(tan(w)) in it instead of just t. But this is because
cos w
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p. 60
we are interested in (length GQ) instead of (length EQ). What we did next was to use the
dynamic sketch to slowly lower the grade to 0 (while keeping s and t the same). When the grade
went down to 0, we saw three things. First, we noted that segment EQ moved to fit perfectly on
t
top of segment HQ—this is why
is replaced by t in the flat road formula. Second, we
cos w
noted that segment GH disappeared—this is why the term s(tan w) disappears in the flat-road
formula. Third, we noted that w went to 0 as well. This makes sense in the problem: on a flat
road the truck should have no tilt. In fact, our reasoning in (a) still works, and we can use the
single observation about w being equal to 0 when the grade is 0 (for a flat road) in our formula
from part (a) to get
t
t
+ s (tan 0) + k = + s (0) + k = t + k.
d=
cos 0
1
So our formula for a tilting truck is a generalization of the case where the grade is 0.
Key Elements of Mathematics
Reasoning with Geometry—Geometric connections and modeling; Construction of geometric
arguments
Reasoning with Algebraic Symbols—Meaningful use of symbols; Mindful manipulation
Reasoning with Functions—Multiple representations of functions
Reasoning Habits
Analyzing the problem—looking for patterns and relationships; looking for hidden structure
Implementing a strategy—selecting concepts, representations, or procedures, making purposeful
use of procedures; making logical deductions
Monitoring one’s progress—reviewing a chosen strategy; analyzing further
Seeking and using connections—connecting different mathematical domains; connecting
different representations
Reflecting on one’s solution—checking reasonableness; justifying or validating a solution;
considering other ways of thinking; generalizing a solution
The geometry strand extends beyond two- and three-dimensional figures in Euclidean
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space to include other special configurations and visualizations. Example 18 illustrates a
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modeling context in which the most obvious geometric model of the situation—drawing circles
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of radius 5 units and looking for overlaps that represent radio interference—is not the most
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useful. Rather, a basic representation in the growing field of “graph theory”—usually considered
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a domain within discrete mathematics—is better suited to capture the relevant information in the
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modeling problem.
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Example 18: Assigning Frequencies
Task
The Federal Communications Commission (FCC)
needs to assign radio frequencies to seven new radio
stations located on the grid to the right. Such
assignments are based on several considerations
including the possibility of creating interference by
assigning the same frequency to stations that are too
close together. In this simplified situation it is assumed
that broadcasts from two stations located within 200
miles of each other will create interference if they
broadcast on the same frequency, whereas stations
more than 200 miles apart can use the same frequency
to broadcast without causing interference with each
other.
How can a vertex-edge graph be used to assign frequencies so that the fewest number of
frequencies are used and no stations interfere with each other? What would each vertex
represent? What would an edge represent? What is the fewest number of frequencies needed?
(Adapted from Hirsch et al. 2007)
In the Classroom
Students work on the task in groups. Each group seems to agree that a vertex in the graph
represents a radio station. So the graph would have seven vertices. Some groups decide to make
a graph model in which two vertices representing stations within 200 miles of each other will be
joined by an edge.
Other groups suggest two vertices will be joined with an edge if they are more than 200 miles
apart. After some discussion, the choice is made to use the first suggestion of joining vertices
with an edge if the distance between them is no more than 200 miles. The next task is to
construct the model. Doing so requires computation of the distance between each pair of stations.
Groups make these computations using the distance formula or Pythagorean theorem and a
calculator. Organization is valuable here, as a fair number of computations are involved. Some
groups might start by computing the distance from A to the remaining six stations, then the
distance from B to the remaining five, and so on. A quick determination of how many
computations are needed can help determine whether any list is missing an item. Experience with
Example 10 would be useful in this context.
Groups produce models similar to the one at the right and then work at
assigning frequencies to vertices so that no two vertices joined by a
single edge have the same frequency. They look for a method that will
use the fewest frequencies for this particular graph. One group
explained their solution this way:
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“We found that the fewest number of frequencies that could be used was 4. We reasoned this
way: First, look at the collection of vertices A through D. Each of these vertices is joined by an
edge to each of the other three in the collection. So no two vertices in the collection can have the
same frequency. That means you can have no fewer than four frequencies.
Next, suppose you assign frequency 1 to vertex A, frequency 2 to vertex B, frequency 3 to vertex C,
and frequency 4 to vertex D. You can finish the assignment by assigning frequency 1 to vertex G
(because G doesn’t interfere with A), frequency 2 to vertex F, and frequency 3 to vertex E. That
proves you don’t need any more than four frequencies. That does it!”
Key Elements of Mathematics
Reasoning with Geometry—Geometric connections and modeling; Construction of geometric
arguments
Reasoning Habits
Initiating a strategy—selecting concepts, representations, or procedures; organizing a solution;
making logical deductions
Reflecting on one’s solution—justifying or validating a conclusion
19
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Note that the vertex-edge representation of the frequency assignment problem seems to
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have facilitated the structuring of an argument that four frequencies suffice—and even a way
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(algorithm) to assign four frequencies—for this example. Finding efficient algorithms for
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analogues of the general fewest-frequency problem for larger graphs remains an active area of
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current mathematical research. Such algorithms would find uses in many contexts.
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Reasoning with Statistics and Probability
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Competence in the standards found in Principles and Standards (NCTM 2000a) depends
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on a thorough and deep understanding of the foundations of statistics, probability, and
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connections between statistics and probability. Taken together, the Principles and Standards
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document and the American Statistical Association’s report Guidelines for Assessment and
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Instruction in Statistics Education (GAISE) (Franklin et al. 2007) describe statistical problem
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solving as an investigative process that involves the following four components:
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Formulate a question (or questions) that can be addressed with data
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Design and employ a plan for collecting data
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Analyze and summarize the data
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Interpret the results from the analysis, and answer the question on the basis of the data
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The common thread throughout the statistical problem-solving process is the focus on the
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variation in the data. The goal is not only to understand the variation but also to offer possible
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explanations for the variation. This process provides a framework for teaching and learning
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statistics in the schools, and meaningful tasks that employ this process should permeate the
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statistical education of our students.
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Key elements of reasoning and sense making with statistics and probability include the
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following:
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to analyze and describe patterns and relationships in the variation.
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Modeling randomness. Developing probability models to describe the long-run behavior in
outcomes from a random process.
•
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Data analysis. Using graphical and tabular representations and numerical summaries of data
Connecting statistics and probability. Recognizing probability as an essential tool of
statistics, and understanding its role in statistical reasoning.
•
Interpreting statistical studies. Drawing appropriate conclusions from data while accounting
for the variation.
We address these key elements in more detail in the following sections.
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Data analysis
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Data are observations or measurements collected on one or more variables to address a
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statistical question. In statistics, a variable is any characteristic that can be expected to take on
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different values between individual observations. Consequently, data vary and, once collected,
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need to be summarized in ways that foster meaningful insights for addressing the question under
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study. The analysis of data includes exploring various representations of the data distribution
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(the values of the variable that can occur and how often each value occurs from the data
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collected) so as to summarize and describe patterns and relationships in the variation.
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Technology provides the opportunity to effectively analyze data by examining a variety of
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representations of the data distribution and to identify important characteristics of the
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distribution. For numerical data these characteristics include center, spread, and shape.
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Example 19 shows how statistical procedures for analyzing data that are developed in
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earlier grades continue to be useful in high school. A detailed description of this activity is given
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Example 19: Meaningful Words
(Adapted from WGBH Educational Foundation [2001])
Task
Scientists are interested in human recall and memory. Is it easier to memorize words that have
“meaning?” To study this problem, two lists of 20 three-letter “words” were used. List A
contained meaningful words (e.g., CAT, DOG), whereas List B contained nonsense words (e.g.,
ATC, ODG). A ninth-grade class of thirty students was randomly divided into two groups of
fifteen students. One group was assigned to memorize words on List A, the other group was
assigned to List B. The number of words memorized by each student was tabulated, and the
resulting data are as follows:
Scores from List A: 12, 15, 12, 12, 10, 3, 7, 11, 9, 14, 9, 10, 9, 5, 13
Scores from List B: 4, 6, 6, 5, 7, 5, 4, 7, 9, 10, 4, 8, 7, 3, 2
1. Provide a display for summarizing and comparing these data sets.
2. On the basis of your display, what observations can be made regarding how students did on
List A compared with List B? Write a paragraph summarizing what the data and your
analysis reveal about the question “Is it easier to memorize words that have meaning?”
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p. 65
In the Classroom (a class of high school students is discussing this problem)
Several groups of students created comparative boxplots, as shown below.
Teacher: Looking at the boxplots, on which list did students generally do better? How do you
know? Give some specific reasons for your answer.
Sherry: List A. The median for List A is 10, and the median for List B is 6. The box for list A
goes from 9 to 12, while the box for list B only goes from 4 to 8.
Teacher: Is every summary measure for list A larger than the summary measures for list B?
Serena: Yes, the maximum for List A is larger than the maximum for List B.
Jerry: Yes, that’s true for the “minimums” as well!
Teacher: What about the quartiles?
Miguel: Yes, the first quartile is bigger for List A, it’s 9. The first quartile for List B is only 4.
Teacher: Does this mean that all students performed better on List A?
Miguel: No, there is an overlap in the boxplots.
Teacher: What does the overlap mean?
Serena: The overlap means that some students got the same scores in both lists.
Teacher: Can you tell how many students did better on List A?
Serena: Yes, about half the students in List A did better than all students in List B.
Teacher: What part of the graph tells you that?
Miguel: The median for List A is 10, and the median is the middle number; so half of the
students’ scores are from 10 to 15, but the highest score in List B is only 10.
Teacher: Now that we have looked at individual summary measures, what about the range of
scores for each list?
Jerry: The range for List A is 12; the range for List B is 8.
Teacher: What about the outlier? What would the range for each list be if we removed the
outlier?
Jerry: Now the ranges are closer. The range for List A is now 10 and the range for List B is 8.
Teacher: Since the range for List A is bigger, we say that there is a more variation in the scores
on List A than in the scores on List B.
Teacher: Now, compute the interquartile range for each list.
Jerry: The interquartile ranges of both lists are the same! They are both 3, so that means that the
variation is the same for both lists.
Teacher: The variation is the same for what part of the scores?
Miguel: The middle 50 percent of scores for both groups have similar amounts of variation.
Teacher: Each time we have looked at data distributions, I have asked you to notice three
characteristics—center, spread, and shape of the distribution. What can you say about the
shape of these two distributions?
Jerry: They are not very skewed, but they are not exactly symmetric either.
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p. 66
Teacher: Without computing each mean, what can you tell me about the mean score for each
list?
Serena: Well, for List A the mean would be lower than the median because of the outlier, but not
much lower. I think the mean for list B would be about the same as the median, around 6.
Teacher: What do these results suggest about memory and meaning?
Serena: Since students mostly did better on the list with meaningful words, it’s easier to
memorize words that mean something.
Teacher: Do you think it would be appropriate to say that all ninth graders would get exactly the
same results?
Jerry: Sure, it’s easier to memorize words that really mean something.
Serena: Well, we would not get exactly the same scores! But we might still see that words with
meaning will be easier to memorize.
Key Elements of Mathematics
Reasoning with Statistics and Probability—Data analysis
Reasoning Habits
Analyzing a Problem—determining whether a statistical solution is appropriate; looking for
patterns and relationships
Initiating a strategy—selecting concepts, representations, or procedures; organizing a solution
Seeking and using connections—connecting different contexts
Reflecting on one’s solution—interpreting solution; justifying or validating
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Modeling randomness
Probability modeling helps students make sense of many of the seemingly random events
26
they experience and hear about in their lives. Probability models reinforce the foundational
27
principle that although an individual outcome from a random process cannot be predicted with
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certainty, patterns emerge in the frequency of outcomes over a large number of repetitions of the
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process. Fostering students’ understanding of random processes and probability requires
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engaging them in the development of both mathematical models and simulation models that
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illustrate the long-run behavior in outcomes. Example 20 explores a variety of developmental
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strategies for determining probabilities.
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p. 67
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Example 20: Random Committees
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Task
A high school club has fifty members: ten girls and forty boys. A committee will be formed by
selecting two students from the club at random. How often will the committee have no girls?
One girl? Two girls? That is, what is the probability for getting zero girls on the committee? One
girl on the committee? Two girls on the committee?
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n!
x!(n ! x)!
counts the number of “subsets” of size x from n objects.
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In the Classroom (ninth grade)
The class is instructed to design a hands-on simulation for estimating the probability of each of
the different possible results for “the number of girls” on the committee. Prior to this activity,
students should have had several experiences performing simulations using various random
devices and random-number generators in graphing calculators. After the task is read, the teacher
can ask the students how they might simulate the situation of choosing two students for the
committee.
The teacher can lead a discussion to help the students design the simulation. Students may
propose to use a standard deck of cards (they would remove two face cards and represent the
girls with the ten remaining face cards and the boys with the forty non−face cards). For each
trial, they would shuffle the deck several times and deal two cards at a time. The number of face
cards dealt represents the number of girls on the committee. Students would then need to decide
how many times they should perform the simulation. From previous experiences, students know
that the relative frequencies tend to be close to the actual probabilities if they perform a large
number of trials. Together, the class performed two hundred simulations. The results are
summarized in the table below.
Number of Girls
0
1
2
Frequency
121
70
9
200
Experimental Probability
.605
.350
.045
1.00
In the Classroom (eleventh grade)
Students can revisit this problem in the eleventh grade after they have learned counting rules.
The class is instructed to use counting rules for determining the probability of each of the
different possible result for the “number of girls” on the committee and to interpret the
probabilities. Revisiting the same problem at a higher level gives students an opportunity to
make sense of their earlier simulation and strengthens the notion that a probability indicates how
often a result from a random process should occur if repeated a large number of times.
Students are reminded that
C(n, x) =
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For this problem, C(50, 2), or 1225, “equally likely” committees (subsets) of size two are
possible from the club of fifty students. The number of committees with —
exactly 0 girl (and exactly 2 boys) is C(10, 0)C(40, 2), or 780;
exactly 1 girl (and exactly 1 boy) is C(10, 1)C(40, 1), or 400;
exactly 2 girls (and exactly 0 boys) is C(10, 2)C(40, 0), or 45.
Thus, the mathematical probabilities are as seen in the following table.
Number of Girls
0
1
2
Mathematical Probability
.637
.327
.037
These probabilities indicate that if two members are repeatedly selected at random from the club,
then approximately 64 percent of the committees should have no girls, 33 percent should have
one girl, and 4 percent should have two girls.
In the Classroom (twelfth-grade statistics)
Students can revisit this problem in a statistics class after they have learned about tree diagrams
and rules of probability. The class is instructed to construct a tree diagram for this situation, to
use probability rules for determining the probability for each of the different possible results for
the “number of girls” on the committee, and to interpret the probabilities. Revisiting the same
problem at a higher level gives students an opportunity to make sense of their earlier simulation
and strengthens connections with their existing knowledge.
The tree diagram below shows the possible ordered results for gender when selecting two
students at random from the fifty students.
Gender of Second Student
Girl
9/49
Gender of First Student
Number
Outcome of Girls
Probability
(Girl, Girl)
2
(10/50)(9/49)
Boy
40/49 (Boy, Girl)
1
(10/50)(40/49)
Girl
10/49 (Girl, Boy)
1
(40/50)(10/49)
Boy
39/49 (Boy, Boy)
0
(40/50)(39/49)
Girl
10/50
Boy
40/50
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Connecting statistics and probability
p. 69
P(No Girls) = P(Boy on First and Boy on Second) = (40/50)(39/49) = .637
P(One Girl) = P(Girl on First and Boy on Second) + P(Boy on First and Girl on Second)
= (10/50)(40/49)+(40/50)(10/49) = .327
P(Two Girls) = P(Girl on First and Girl on Second) = (10/50)(9/49) = .037
When selecting random samples of two students from the club of fifty students, the number of
girls in the sample will vary from one sample to another. The probabilities above indicate that in
repeated sampling, approximately 64 percent of the samples will have no girls, approximately 33
percent of the samples will have one girl, and approximately 4 percent will have two girls. Note
that the estimated probabilities from simulating repeated selection of random samples of two
students in the first task are close to the mathematical probabilities.
Key Elements of Mathematics
Reasoning with Statistics and Probability—Modeling randomness
Numbers and Measurements—Counting
Reasoning Habits
Initiating a strategy—selecting concepts, representations, or procedures; organizing a solution;
making logical deductions
Monitoring one’s progress—changing or modifying strategies
Seeking and using connections—connecting different representations
Reflecting on one’s solution—interpreting a solution; justifying or validating
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By incorporating randomness into data collection, probability provides a way to describe
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the random variation in sample results from one sample to another. The sampling distribution of
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a sample statistic (e.g., the sample mean) summarizes the long-run behavior of the statistic from
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repeated random sampling. The sampling distribution provides a mechanism for describing the
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variation expected in sample results and to determine whether observed data are reasonable from
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chance variation or are due to some other factor. Sampling distributions are the link between
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probability and statistics. The previous example describes the sampling distribution for the
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number of girls in a random sample of two from a club of fifty students with ten girls and forty
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p. 70
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boys. The activity “Sampling Rectangles” in Navigating through Data Analysis in Grades 9−12
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(Burrill et al. 2003) illustrates important ideas related to the sampling distribution of the sample
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mean. Sampling distributions provide a way to quantify the uncertainty associated with the
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notions of statistical estimation and statistical significance, which are components of the
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following key element.
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Interpreting statistical studies
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Statistical reasoning involves making conclusions from, and decisions based on, data
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from observational studies, sample surveys, or experiments. Data from studies that properly
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utilize randomness can be used to estimate population parameters or determine when a result is
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statistically significant. The following example illustrates the role of probability in assessing
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statistical significance when random assignment to treatment groups is used in data collection.
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Example 21: Meaningful Words—Continued
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Task
In Example 19, the difference between the medians (Words – Nonwords) is 4, and this result
lends evidence that students tend to do better when memorizing words compared with nonwords.
Because random assignment was used to divide the class into two groups, probability can be
used to assess the strength of this evidence. Specifically, “Is the observed difference between the
medians due to the chance variation expected to occur from random assignment, or is this
difference due to the fact that List A has words and List B does not?”
Design and implement a simulation to address this question.
In the Classroom
After a class discussion, students agreed that if different lists (words and nonwords) have no
effect on memorization, then the observed difference between the medians is due only to the
variability expected from the random assignment to groups (i.e., chance variation.) If so, then a
student’s score does not depend on the list received, and consequently, her/his score would be
the same regardless of the list assigned. This situation can be simulated by writing each of the
thirty scores on an index card. Shuffle the cards and deal out fifteen cards. The fifteen cards dealt
represent scores for List A; the remaining fifteen cards represent the scores for List B. Determine
the median for each list and the difference between the two medians (A − B). How often will this
simulation produce a difference between medians of 4 or more? Repeating this simulation 100
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p. 71
times produced the empirical sampling distribution shown in the figure below for the difference
between the two medians.
On the basis of the simulation, when there was no difference between the effects of the two lists
on memorization, a difference of 4 or more between the medians of the word and nonword lists
occurred only two times out of 100 trials. This result suggests that a difference between medians
that is this large or larger is unlikely to occur and thus is not plausible when there is no
difference between the effects of the two lists on memorization. In this situation we say the
difference between the medians is statistically significant (cannot be explained by chance
variation), and the data provide fairly strong evidence that the use of “words” does help when
trying to memorize.
(A more detailed illustration of students’ reasoning in this problem is presented in the
forthcoming topic book specific to statistics and probability.)
Key Elements of Mathematics
Reasoning with Statistics and Probability—Modeling randomness; Connecting statistics and
probability; Interpreting statistical studies
Reasoning Habits
Analyzing a problem—determining whether a statistical solution is appropriate; making
conjectures; looking for patterns and relationships
Initiating a strategy—selecting concepts, representations, or procedures; making purposeful use
of procedures; making logical deductions
Seeking and using connections—connecting different mathematical domains
Reflecting on one’s solution—interpreting a solution; justifying or validating; understanding the
nature of a statistical conclusion
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Example 21 illustrates concepts underlying the notion of statistical significance. Sample
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surveys based on random sampling provide a way to introduce basic concepts of statistical
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inference (making conclusions about a population based on a sample). Example 22 is described
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in detail by Scheaffer and Tabor (2008) and illustrates one of the primary forms of statistical
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inference, estimating a population parameter on the basis of results from a random sample.
p. 72
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Example 22: Margin of Error
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Task
Students in a twelfth-grade statistics class wanted to know the percent of students at their school
who have a curfew. The students selected a random sample of 100 students from a list of
students in the school. In the sample, 53 of the 100 students reported having a curfew. The
students realized that the true proportion of all students in the school is not likely to be 0.53, and
that another sample of 100 students would likely produce a different result. This idea underlies
the notion of a sampling distribution—that variation will occur in the sample proportion from
one sample to another. Consider the following questions:
 How much variation can be expected in the sample proportion from one sample to another?
Design and implement a simulation to address this question.
 Use the results from the simulation to estimate the margin of error for estimating the true
percent of students who have a curfew.
 Use the estimated margin of error to report an interval of plausible values for the true percent
of all students with a curfew.
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In the Classroom
Assuming that the true population proportion is 0.53, students can simulate the repeated selection
of samples of size 100 from a “theoretical” population that has a 53 percent chance of generating
a favorable outcome (several methods are proposed in Scheaffer and Tabor [2008]). The figure
below shows a simulated sampling distribution of sample proportions from 200 such samples
with a normal distribution superimposed.
This empirical sampling distribution has a mean of 0.53 and a standard deviation of 0.05, and
appears to have an approximately normal distribution. For a normal distribution, about 95
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percent of measurements are known to lie within two standard deviations of the mean. This
measure of two standard deviations to either side of the mean is called the margin of error. In
this example, the margin of error is 2(0.05) = 0.10 and would not change much even if the
assumption of 0.53 for the true population proportion is a little off. Finally, the plausible values
for the population proportion, those values that could have produced the observed sample as a
likely outcome, lie in the interval 0.43 and 0.63 (0.53 ± 0.10). That is, we are 95 percent
confident that the true population percent of all students that have curfews is between 43 percent
and 63 percent.
Key Elements of Mathematics
Reasoning with Statistics and Probability—Modeling randomness; Connecting statistics and
probability; Interpreting statistical studies
Reasoning Habits
Analyzing a problem—determining whether a statistical solution is appropriate; making
preliminary deductions and conjectures; looking for hidden structure; looking for patterns
and relationships
Initiating a strategy—selecting concepts, representations, or procedures; purposeful use of
procedures
Seeking and using connections—connecting different mathematical domains
Reflecting on one’s solution—interpreting a solution; justifying or validating; understanding the
nature of a statistical conclusion
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In our increasingly information-intensive world, proficiency in statistics has been
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recognized as vital for both workplace and college readiness (American Diploma Project 2004;
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Ganter and Barker 2004; CUPM 2004). Statistical problem solving offers ways for developing
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the reasoning skills required for a quantitatively literate citizenry, for college preparation, and for
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Equity
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Mathematical reasoning and sense making must be evident in the
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mathematical experiences of all students.
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p. 74
In this document, we build on the concept of equity as defined in Principles and
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Standards for School Mathematics (NCTM 2000a), in which equity is described in terms of
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having high expectations of, and providing support for, all students. This population includes
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students of all races and cultures; students with low socioeconomic backgrounds; students whose
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first language is not English; both girls and boys; both gifted and low-achieving students; and
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students with learning disabilities, emotional problems, and behavioral problems. All too
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frequently, one or another of these groups is deemed too difficult to teach. Often, inequity is
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enacted by well-meaning teachers and administrators through practices that create a bias they do
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not intend. High schools can monitor equity by paying focused attention to phenomena that pose
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potential barriers to engaging every student in the activities of reasoning and sense making.
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These factors include the following:
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•
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and sense making.
•
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Courses that students take have an impact on the opportunities that they have for reasoning
Students’ demographics too often predict the opportunities students have for reasoning and
sense making
•
Expectations, beliefs, and biases have an impact on the mathematical learning opportunities
provided for students.
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p. 75
Courses
Over the past two decades, enrollment in advanced and college-preparatory mathematics
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courses at the high school level has increased noticeably (Planty, Provasnik, and Daniel 2007).
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At the same time, high schools have continued to employ some form of tracking or ability
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grouping (National Center for Educational Statistics 1994). With the tracking of students, high
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schools run the risk of promoting inequitable opportunities for mathematical learning for those
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students placed in remedial classes or in classes on “low” or “regular” tracks. High schools that
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use tracking or ability grouping can be proactive by diligently ensuring that at every level, all
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mathematics courses offer students opportunities for reasoning and sense making. In addition,
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with proper school support and access to such opportunities, students do not have to be relegated
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to one track for their entire high school career. Successful high schools make sure that moves to
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different tracks are both offered and actively supported (Education Trust 2005).
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Schools with a curricular structure in which multiple levels of the same class are offered
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(e.g., Informal Geometry vs. Honors Geometry) have a responsibility to engage all students in
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every class in reasoning and sense making, thereby ensuring that these students are learning the
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same major concepts regardless of the course title.
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Students’ unpreparedness for algebra in the ninth grade is a current problem for high
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schools and often leads to schools’ needing to provide some sort of remediation or additional
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academic support. Schools deal with this problem in a variety of ways, such as stretching algebra
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over two years or placing students in a ninth-grade prealgebra class. These strategies can be
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counterproductive, especially if the result is that students are confined to the low track with
22
limited attention to reasoning for the duration of high school and without the opportunity to catch
23
up to their peers. High schools can support students who are underprepared for first-year algebra
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p. 76
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in ninth grade without limiting their future progress. For example, some high schools take early
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action with struggling students by offering courses in the summer before high school or as part of
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after-school programs (Education Trust 2005). Other schools provide freshman students with
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general academic support in the form of teacher and peer tutoring, Saturday programs, and
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double course periods for some courses (Huebner, Corbett, and Phillippo 2006). In all instances,
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these high schools provide support early on for those students entering high school below grade
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level. Algebra presents great opportunities for sense making and reasoning, and lays a foundation
8
for further study in mathematics. Thus, ensuring that underprepared ninth graders are supported
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in ways that allow them to become proficient with algebraic concepts is of crucial importance.
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To ensure that this document’s vision becomes a reality, teachers must hold high
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expectations for all students and find strategies to ensure that all students, at any level and in any
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course, can reason mathematically by engaging in challenging mathematical tasks. One such
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strategy is using problems that have multiple entry points so that students at different levels of
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mathematical experience and with different interests can all engage meaningfully in reasoning
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about the problem. Using problems with multiple entry points allows students to use their
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individual mathematical strengths in approaching the problem and gives students opportunities to
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make sense of the reasoning of other students.
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Several examples in chapter 3 of this document describe tasks that have multiple entry
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points. For example, Example 6 presents a problem that can be solved with or without the use of
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equations. If students are able to use equations flexibly, as Alice can in the example, they can
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represent the described situation symbolically, solve the equation, and find the solution.
22
However, other students who are either less confident, have had limited experience with
23
equations, or are unable to use them flexibly might approach the problem by first drawing a
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picture to represent the scenario, then move toward writing an equation. Others might use a more
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concrete reasoning approach, such as the one used by Barbie, to find the answer. Exploring
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multiple approaches to solving a problem can forge important connections between different
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mathematical domains and strengthen the reasoning abilities of all learners. For this reason,
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students should be given opportunities to discuss their different approaches with one another and
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work toward understanding how each approach relates to the others.
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Student Demographics and Course Placement
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Discrepancies in achievement and resources between student groups have been noted and
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discussed in policy documents for years. Current data from NAEP suggests that schools continue
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to face these issues. High schools are in a particularly challenging position because they have to
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ultimately deal with the consequences of the discrepancies that students may have faced at the
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elementary and middle school levels. For example, as reported on the mathematics portion of the
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2003 NAEP (Lubienski and Crockett 2007), 52 percent of Hispanic and 61 percent of African
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American eighth-grade students scored below the basic level compared with 20 percent of white
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eighth-grade students. Only 7 percent of African American students and 12 percent of Hispanic
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students scored at or above the proficient level in comparison with 37 percent of white students.
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The same study found a statistically significant difference between the percent of mathematics
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teachers who majored in mathematics in college who were teaching eighth-grade white students
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as compared with the percent teaching eighth-grade African American and Hispanic students.
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White (91 percent) students were more likely to have fully credentialed teachers than African
21
American (80 percent) and Hispanic (83 percent) students. Students of teachers who majored in
22
mathematics in college scored seven to fifteen points higher on NAEP than students with
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teachers that were non−mathematics majors (Lubienski and Crockett 2007). Every year high
2
schools open their doors to minority students whose opportunities to learn might have been
3
limited in prior years and who as a result are far below grade level. High schools must ensure
4
that such students are provided with the support they need to succeed mathematically, rather than
5
continue to perpetuate these same inequities.
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Presently, more students of color, in particular African American, Hispanic, and
7
American Indian students, are taking more advanced mathematics than in previous decades.
8
However, persistent gaps in course taking and achievement remain between these students and
9
their White and Asian/Pacific Islander peers. In 2004 approximately 6 percent each of African
10
American, Hispanic, and American Indian high school students graduated having completed
11
calculus, as compared with 33 percent of Asian/Pacific Islanders and 16 percent of white
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students (Planty, Provasnik, and Daniel 2007). However, Hoffer, Rasinski, and Moore (1995), as
13
cited in Tate and Rousseau (2002), found that when African American students and white
14
students from different socioeconomic backgrounds completed the same mathematics courses,
15
they had comparable achievement gains. This powerful finding makes clear the need for high
16
schools to be diligent in supporting all students, especially those students who are often
17
underrepresented in high-level mathematics classes, to take more and more advanced
18
mathematics. Once these students are in high-level classes, they should be supported by teachers
19
in ways that will promote success.
20
For example, teachers can support the development of mathematical reasoning in
21
bilingual (ELL) students by encouraging them to discuss their mathematical thinking verbally on
22
a daily basis and by providing them with a variety of opportunities to communicate
23
mathematically. Mathematical activity in which students are asked to reason is a vehicle for
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1
enhancing the linguistic skills of students who lack English vocabulary, thus avoiding their
2
placement in remedial mathematics classes and preventing the delay of reasoning activities in the
3
mathematics classroom until they have mastered basic linguistic skills (Moschkovich 2007).
4
Instruction that centers on promoting group interaction, integrates language, uses multiple
5
representations, and emphasizes context is important for the success of bilingual (ELL) students
6
(Bay-Williams and Herrera 2007).
7
High Expectations
8
9
In Principles and Standards for School Mathematics (NCTM 2000a), the Equity Principle
describes the importance of teachers’ holding high expectations for all students. Teachers can
10
motivate their students to perform at high levels by having and communicating high
11
expectations, just as they can lower students’ confidence in their mathematical abilities by
12
exhibiting low expectations. Teachers’ self-awareness of personal bias about who can and cannot
13
do mathematics is an essential part of holding high expectations for all students. All too often,
14
teachers’ and administrators’ views of students’ ability, motivation, behavior, and future
15
aspirations are influenced by their beliefs associated with such student identifiers as race, gender,
16
socioeconomic status, native language, and home life. In turn, those beliefs can have serious
17
consequences for the opportunities that a school provides to its students. By working to become
18
aware of subconscious attitudes, teachers and administrators can strive to overcome those
19
notions and act purposefully against social inequalities.
20
Although holding high expectations for all students is crucial, expectations alone are not
21
enough. By taking specific action in the classroom, teachers can truly support students in
22
meeting those high expectations. One such action is to encourage all students to share their
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1
thinking and listen to the thinking of others. Establishing a community of learning in which a
2
multitude of approaches and solutions are encouraged and respected is a way to effect positive
3
change (NCTM 2008). Using tasks with multiple entry points is one way to encourage this
4
student interaction. Subtle teacher behaviors also demonstrate the belief that all students can be
5
successful. For example, Cousins-Cooper (2000) suggested that teachers can encourage African
6
American students when they get stuck on a problem by asking probing questions instead of
7
showing the students how to do the problem; such behavior is important for teachers of all
8
students. This behavior communicates to the students and all who observe the interaction that the
9
teacher believes that her or his students can be successful in making sense of the problem and
10
working toward a solution. Teachers must be constantly aware of the messages they
11
communicate to students both verbally and nonverbally.
12
Creating an inclusive environment in which all students—regardless of race, gender,
13
socioeconomic status, or perceived motivation and learning ability—are engaged in reasoning
14
and sense-making activities communicates to students a belief that everyone is expected to be
15
successful in mathematics. Researchers have noted a tremendous difference in success rates
16
among high-minority, low-socioeconomic student populations when a culture of high
17
expectations and a shared goal of success for all students is adopted not just by individual
18
teachers but by the entire school, including parents and the community (Gutierrez 2000;
19
Education Trust 2005).
20
Conclusion
21
22
In general, teachers and schools across North America are working tirelessly toward
educating all students sitting in their classrooms. However, in the shuffle of high school life,
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1
some students are inadvertently overlooked or underserved. Moreover, often the same students
2
are among those who are overlooked—those of color, those of low or exceptionally high
3
achievement, and those of low socioeconomic status. Teachers, teacher leaders, and
4
administrators, along with parents and the community, must constantly reflect on how school
5
practices may work to support or fail to support giving all students the opportunities with
6
reasoning and sense making that they need to be successful as they exit high school. Some
7
questions to reflect on in trying to reach this goal include these: Are the students enrolled in
8
advanced mathematics courses at a school representative of the demographics of the school? Do
9
students move between tracks flexibly and with the support of teachers and administrators? In
10
any given mathematics course, are students engaged in reasoning and sense making on a daily
11
basis?
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1
5
2
Coherence
3
Curriculum, instruction, and assessment form a coherent whole in order to
4
support reasoning and sense making.
5
Increasingly, calls are made for consistency in the high school mathematics curriculum
6
across the country (National Science Board 2007). Several sets of recommendations have been
7
proposed reflecting a range of goals and priorities (American Diploma Project 2004; College
8
Board 2006; ACT 2007; Achieve 2007a, 2007b). This document does not present a list of
9
content topics delineated by grade level. Rather, its intent is to argue that content development
10
and student learning at the high school level must be grounded in reasoning and sense making, as
11
well as to promote the notion that reasoning and sense making provide criteria for structuring
12
curriculum and for evaluating various curriculum proposals for sequencing content topics in high
13
school mathematics.
14
To achieve the vision set forth in this document, the many components of the educational
15
system must work together for the good of the students they serve. Curriculum, instruction, and
16
assessment can be designed to support student reasoning and sense making. A coherent and
17
cohesive mathematics program requires strong alignment of these three elements.
18
Curriculum and Instruction
19
20
Classrooms that support the goals of this document prepare students to view mathematics
as a connected whole across various mathematical domains that consistently involves reasoning
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1
and sense making. Integrated curricula are particularly well suited to meet these goals because
2
they are designed to emphasize connections across mathematical domains. Regardless of the way
3
a curriculum is organized, those involved in the mathematics program must do their best to
4
ensure that all students are being given a strong preparation in mathematics that emphasizes the
5
curricular connections needed to support effective mathematical reasoning and sense making.
6
Fostering depth of students’ mathematical knowledge requires classrooms in which
7
students are actively involved in solving problems that allow them to make connections among
8
content areas and to develop mathematical reasoning habits. Effective instruction requires having
9
and communicating high expectations for all students. In such learning environments, students
10
continually explore interesting mathematics problems, either individually or in groups, and
11
communicate their conjectures and conclusions. Pedagogical techniques and carefully designed
12
instructional materials further ensure that the focus of the classroom is on student thinking and
13
reasoning, and provide opportunities for all students to participate.
14
Vertical Alignment of Curriculum
15
All too often students experience discontinuities as they move from one level of
16
mathematics to another (Smith et al. 2000), from middle school to high school and from high
17
school to postsecondary education. The alignment of mathematical expectations related to the
18
goals of this document is important as students move from one school level to another. To
19
achieve a coherent whole curriculum, an open dialogue is necessary among prekindergarten
20
through grade 8 teachers, secondary mathematics teachers, mathematics teacher educators, and
21
mathematics and statistics faculty in higher education, as well as from client disciplines, to
22
ensure continuing support and development of students’ mathematical abilities. Schools, parents,
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1
policymakers, and others need to see evidence that reasoning and sense-making abilities are
2
shared goals among elementary school mathematics, high school mathematics, and the
3
undergraduate curriculum. The time is right to build strong partnerships among the various
4
constituencies, and to recognize the potential beneficial consequences resulting from a
5
mathematics curriculum that focuses on reasoning and sense making as articulated parts of the
6
continuum from prekindergarten through grade 16.
7
Alignment with prekindergarten through grade 8 mathematics
8
9
This document takes the recommendations of Curriculum Focal Points for
Prekindergarten through Grade 8 (Curriculum Focal Points) (NCTM 2006) as its starting point
10
for ensuring that proper alignment occurs between middle grades and high school. For each
11
grade level, prekindergarten through grade 8, Curriculum Focal Points outlines “focal point”
12
content areas and describes objectives in additional areas of “related connections.” Each chapter
13
of Curriculum Focal Points begins with two reminders:
14
•
The “curriculum focal points and related connections … are the recommended content
15
emphases for mathematics….” That is, the objectives outlined in the additional “related
16
connections” are also required material. Otherwise, students will not receive the breadth of
17
mathematical content required for high school, particularly in the areas of data analysis,
18
probability, and statistics.
19
•
“It is essential that these focal points be addressed in contexts that promote problem solving,
20
reasoning, communication, making connections, and designing and analyzing
21
representations.” That is, teaching skills without providing a foundation in, and without
22
giving attention to, important mathematical processes, including reasoning and sense making,
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1
will not adequately prepare students for the kinds of thinking expected in high school
2
mathematics.
p. 85
3
This current document assumes that all students entering high school will have a sound base of
4
mathematical experiences as described in the broadest interpretation of Curriculum Focal Points.
5
Taking algebra in middle grades has become increasingly popular (Lee, Grigg, and Dion
6
2007). This document contends that whenever new content is encountered—in elementary,
7
middle, or high school—the foundation for conceptual understanding is laid, in part, through
8
opportunities for students to reason with and about the content topics and to connect them with
9
prior knowledge. Therefore, courses that contain significant algebraic content must build on and
10
foster student reasoning and sense making—including making connections across various
11
mathematical domains. For students without the content background described in Curriculum
12
Focal Points and with insufficient experience in reasoning and sense making, taking algebra
13
prematurely will result in missed opportunities for broadening and strengthening their
14
understanding of fundamental mathematical concepts. Taking algebra too early may be
15
counterproductive if the foundation necessary for developing reasoning and sense making with
16
symbols in high school mathematics is weak. Additionally, middle-grades students who take
17
algebra without a focus on reasoning and sense making are more likely to develop negative
18
attitudes toward learning mathematics, thus hampering their study of more advanced topics
19
during high school.
20
Alignment with postsecondary education
21
The recommendations in this document are consistent with recommendations by the
22
Mathematics Association of America (Ganter and Barker 2004; CUPM 2004), which proposes
23
that the entire college-level mathematics curriculum, for all students, even those who take just one
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1
course, should develop “analytical, critical reasoning, problem-solving, and communication
2
skills…” (CUPM 2004, pp. 5, 13). The commonalities between recommendations for the
3
undergraduate curriculum and the current document provide opportunities to strengthen
4
articulation between high school and postsecondary education. Furthermore, the framers of this
5
document believe that focusing on reasoning and sense making can move us forward on important
6
issues, such as those involving mathematics placement examinations used by postsecondary
7
institutions and articulation on the role of technology in high school and college mathematics.
8
Assessment
9
It is important to assess what we value. Assessments mandated by states, districts, and
10
schools that support the goals of this document will evaluate students’ capabilities in
11
mathematical reasoning and sense making and contribute to students’ progress in
12
mathematics. This endeavor is essential for at least two reasons. First, we will not be able to
13
gauge whether we are meeting our goals if those goals are not assessed. Second, high-stakes
14
testing that concentrates primarily on procedural skills without assessing reasoning and sense
15
making sends a message that is contrary to the vision set forth in this document and can
16
adversely influence instruction and student learning. Assessment that focuses primarily on
17
students’ ability to do algebraic manipulations, apply geometric formulas, and perform basic
18
statistical computations will lead students to believe that reasoning and sense making are not
19
important. Therefore, this document is calling for further progress in the design of many
20
high-stakes test items.
21
22
Assessing student progress in mathematics serves two distinct roles. One essential role
of assessment is to provide summative measures of students’ understanding of and ability to use
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1
mathematics. However, formative assessments are also an essential component of the learning
2
process for the student. According to Black and Wiliam (1998), formative assessment involves
3
providing students with learning activities and, on the basis of feedback from these activities,
4
adjusting teaching to meet the students’ needs. Various forms of formative assessment can
5
provide information about student reasoning and sense making. These include teacher
6
observations, classroom discussions, student journals, student presentations, homework, or in-
7
class tasks that ask students to explain their thinking. To ensure that their assessments reflect
8
curricular priorities, teachers must use both summative and formative instruments for assessing
9
student understanding of mathematics. Example 3 illustrates the use of an in-class assignment
10
for formative assessment. To better understand the students’ progress in understanding error and
11
relative error, the teacher might design a rubric, as in figure 5.1.
12
13
14
15
16
17
18
19
4—Correctly uses the concept of relative error to establish an upper bound for the error.
3—Correctly uses the formula for relative error, but demonstrates some conceptual gaps—either
in establishing the upper bound or in applying his or her finding to an example— or makes a
serious computational error.
2—Demonstrates serious conceptual gaps—confuses absolute and relative error or misapplies
the formula.
1—Shows limited understanding of relative error.
20
Fig. 5.1. A rubric for assessing responses to Example 3
21
22
Conclusion
Alignment of curriculum, instruction, and assessment is a starting point on the road to
23
implementing the goals set forth in this document within the large and complex educational
24
system of the United States. These three components cannot be viewed in isolation. Rather,
25
decision makers or policymakers dealing with aspects of any one of these areas must ask
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1
themselves, “Is a coherent mathematics program being developed that takes into account all three
2
programmatic elements?”
3
Efforts to focus the mathematics curriculum on reasoning and sense making will require
4
responsible efforts of, and considerable work from, all stakeholders. Support will be required
5
through governing policy statements that clearly align with the goals set forth here. The
6
realization of these recommendations will take considerable time and resources to provide the
7
significant professional development required for school faculty and administrators. Many of
8
these efforts are discussed further in the next chapter.
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1
6
2
Stakeholder Involvement
3
Everyone involved must work together to ensure that reasoning and sense
4
making are central foci of high school mathematics programs.
5
This document sets forth an ambitious vision for the improvement of high school
6
mathematics by refocusing it on reasoning and sense making. This refocusing is not a minor
7
tweaking of the system but a substantial rethinking of the high school mathematics curriculum,
8
which requires the engagement of all involved in high school mathematics. Significant effort will
9
be needed to realign the curriculum to focus on reasoning and sense making, to provide teachers
10
with the professional development needed to develop new understanding of the curriculum, and
11
to ensure that all students are provided with the resources needed to prepare them for our rapidly
12
changing world.
13
The following sections outline questions that decision makers can ask themselves or
14
others as they work with different stakeholder groups to increase their engagement in improving
15
high school mathematics. These questions are intended to form the basis for extended reflection
16
on what is currently happening and what needs to happen, and to spur all involved to begin
17
moving toward the goals of this document.
18
Students
19
20
•
Do students see mathematics as a useful, interesting, and creative area of study focused
on reasoning and sense making that will open doors for them in the future?
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1
•
2
3
p. 90
Are students developing mathematical reasoning habits within the context of a rich
mathematics curriculum that prepares them for a range of future opportunities?
•
4
Do students actively participate in making sense of the mathematics they are learning
every day?
5
•
Do students regularly and actively communicate their reasoning in a variety of ways?
6
•
Do students look for relationships and connections, both within mathematics and to other
7
8
areas of inquiry—including other subjects and cocurricular activities?
•
9
10
Do students develop the reasoning underlying mathematical procedures and use those
procedures in a mindful way that conveys their understanding?
•
11
Do students exhibit a productive disposition toward mathematics and its importance in
with making decisions in other parts of their lives?
12
Parents
13
•
14
15
school?
•
16
17
•
22
Do parents understand why reasoning and sense making are important and that success in
mathematics requires more than the ability to carry out procedures?
•
20
21
Do parents understand the crucial importance of student effort in their children’s
mathematical development, and that no such thing as a “mathematics gene” exists?
18
19
Are parents engaged in the mathematics education of their children enrolled in high
Do parents have access to appropriate tools for supporting their children’s development
of mathematical reasoning and sense making?
•
When discussing mathematics, do parents ask their children to explain what they are
doing and why it makes sense?
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1
•
p. 91
Do parents value high school mathematics and recognize the “gatekeeper” nature of
2
success in high school mathematics with regards to college admission, success in college
3
generally, and in mathematics and science-related careers in particular?
4
5
Teachers
•
6
7
Are reasoning and sense making the foci of high school mathematics teachers’ instruction
every day and for every topic within every course taught?
•
Do teachers realize and demonstrate the importance of mathematics reasoning habits and
8
content knowledge as life skills that will ensure their students’ success in the future, not
9
just as the prerequisite for the next mathematics course their students may take?
10
•
Do teachers show awareness of the wide range of careers that involve mathematics—for
11
example, finance, real estate, marketing, advertising, forensics, and even sports
12
journalism?
13
•
Do teachers emphasize the practical worth of mathematics for addressing real problems
14
and seek contexts in which mathematics can be seen as a useful and important tool for
15
making decisions?
16
•
17
Do teachers see mathematics as a coherent subject in which the reasons that results are
true are as important as the results themselves?
18
•
Do teachers help students make sense of the mathematics for themselves?
19
•
Do teachers help students develop a productive disposition toward mathematics?
20
•
Do teachers strive for a balance among the areas of the high school mathematics
21
curriculum outlined in chapter 3 and help students find the connections among those
22
areas?
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1
•
2
3
p. 92
Do teachers use a range of assessments to monitor and promote reasoning and sense
making, both in identifying student progress and in making instructional decisions?
•
Do teachers recognize the importance of immediately beginning to focus their content
4
and instruction on reasoning and sense making while working with administrators and
5
policymakers toward broadly restructuring the high school mathematics program?
6
•
7
8
9
courses they teach?
Administrators
•
10
11
•
Do school districts provide teachers long-term, embedded professional development and
support that will help them reflect on their practice and work to improve it?
•
14
15
Do school districts and schools ensure that students receive a high-quality high school
mathematics curriculum that promotes reasoning and sense making?
12
13
Can teachers articulate how reasoning and sense making are integral components of the
Do professional development activities help teachers experience mathematics as
reasoning and sense making for themselves?
•
Do schools and school districts work with universities, including mathematicians and
16
mathematics teacher educators, to develop effective programs and courses that will
17
address teachers’ needs?
18
•
19
20
21
Do school districts hold themselves, schools, departments, and teachers accountable for
promoting reasoning and sense making within the high school mathematics curriculum?
•
Do administrators’ classroom observations of mathematics teachers focus on reasoning
and sense making and provide useful formative feedback?
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1
•
2
3
Are support systems in place to provide high school students with the needed assistance
to attain high expectations in mathematics?
•
Does the school district’s counseling infrastructure promote perseverance in taking
4
higher-level mathematics, supporting the goal of increasing students’ mathematical
5
reasoning and sense making?
6
7
Policymakers
•
8
9
Are reasoning and sense making embedded in state and district frameworks and standards
that guide curriculum decisions and assessment programs?
•
Do state and local curriculum guides reflect the difficult decisions about the curricular
10
priorities needed to promote reasoning and sense making and ensure a balance of
11
content?
12
p. 93
•
Do state and local assessments emphasize reasoning, either using authentic tasks in which
13
students show their ability to reason or using high-quality, machine-graded examinations
14
that can assess student reasoning (cf. Askey [2007])?
15
•
Do states and districts consider the impact of decisions related to teacher placement and
16
allocation of resources on the students’ opportunity to learn, particularly in schools
17
serving poor neighborhoods or enrolling large numbers of minority students?
18
•
19
20
Do states hold school districts accountable for promoting reasoning and sense making
across the high school mathematics curriculum?
•
Are adequate resources allocated for professional development to help high school
21
mathematics teachers develop the understanding needed to carry out and assess a
22
mathematics curriculum based on reasoning and sense making?
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1
2
Higher Education
•
3
4
Do colleges follow the recommendations of the CUPM Curriculum Guide (CUPM 2004),
which suggests a “move towards reasoning” as the basis for undergraduate mathematics?
•
5
6
p. 94
Do college entrance or placement examinations test students’ ability to reason
mathematically, as well as their ability to carry out mathematical procedures?
•
Do mathematics departments take seriously the recommendation of the Mathematical
7
Education of Teachers report (CBMS 2001) for the preparation of high school
8
mathematics teachers? In particular, do they provide “an opportunity for prospective
9
teachers to look deeply at fundamental ideas, to connect topics that often seem unrelated,
10
and to further develop the habits of mind that define mathematical approaches to
11
problems” (p. 46)?
12
•
Do colleges provide prospective and practicing high school teachers with a mathematical
13
preparation focusing on reasoning and sense making, including more than writing formal
14
proofs?
15
•
16
17
discrete mathematics, modeling, and strategic use of a full range of technology?
•
18
19
Do colleges give prospective high school teachers meaningful experiences in statistics,
Do college courses for prospective high school teachers help them reexamine the high
school curriculum from the perspective of reasoning and sense making?
•
Do institutions of higher education work with schools and teachers to develop courses
20
and programs that will support the continued learning of high school teachers as they
21
begin to refocus their instruction?
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1
2
Curriculum Designers
•
Do curriculum materials for high school mathematics include a central focus on
3
reasoning and sense making that goes beyond the inclusion of isolated supplementary
4
lessons or problems?
5
p. 95
•
Does the curriculum, whether integrated or following the course sequence customary in
6
the U. S., develop connections among content areas so that students see mathematics as a
7
coherent whole?
8
•
9
10
example, is more than an isolated unit?
•
11
12
13
Collaboration among Stakeholders
•
•
20
Is collaboration among stakeholders a central feature of efforts to improve the high
school mathematics program?
•
18
19
Are lines of communication among stakeholders being established to promote
mathematical reasoning and sense making as central goals of high school mathematics?
16
17
Does the curriculum emphasize coherence from one course to the next, demonstrating
growth in both mathematical content and reasoning?
14
15
Is a balance maintained in the areas of mathematics addressed, so that statistics, for
Is the importance of developing “learning communities” that collaboratively seek
continuing improvement across the educational system recognized?
•
Is each group of stakeholders identifying both long- and short-term action plans in
consultation with other stakeholders and beginning to implement those plans?
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1
2
p. 96
Conclusion
This document has demonstrated that all high school mathematics programs need to be
3
focused on providing all students with the mathematical reasoning and sense-making skills
4
necessary for success in their lives within the context of rich content knowledge. The need to
5
refocus the high school mathematics curriculum has been evident for many years (National
6
Commission on Excellence in Education 1983; Mathematical Sciences Education Board 1989;
7
National Commission on Mathematics and Science Teaching for the 21st Century 2000) and has
8
been reflected in the policies and priorities of the National Council of Teachers of Mathematics
9
for nearly thirty years, from An Agenda for Action (1980), to Curriculum and Evaluation
10
Standards (1989), through Principles and Standards for School Mathematics (2000a). Although
11
many high school mathematics programs have made significant progress toward achieving such
12
a refocusing, significant work remains to be done to make the deep-rooted, nationwide changes
13
that are vital in meeting the mathematical needs of all students.
14
Through this document, NCTM is providing a framework for thinking about the changes
15
that must be made and for beginning to consider how those changes might be accomplished.
16
However, many issues extending beyond what could be addressed in this document remain to be
17
answered. Over the coming years, NCTM will continue to provide resources and initiatives that
18
build on this framework. One of NCTM’s initial efforts is a set of topic books that set forth
19
additional guidance around particular content areas. Subsequent volumes will address additional
20
issues, such as ensuring equitable experiences for all students in reasoning and sense making,
21
using assessment to improve students’ growth in reasoning and sense making, and making
22
strategic use of technology across the curriculum. NCTM will also seek to partner with other
23
organizations concerned with high school mathematics.
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p. 97
Although NCTM can take a leadership role, all stakeholders must join forces and work
2
together in meaningful ways to ensure that the continuing story of missed opportunities to
3
significantly improve high school mathematics across this nation will not be told five years from
4
now, let alone in three decades. We simply cannot afford to wait any longer to address the large-
5
scale changes that are needed. The success of our students and of our nation depends on it.
Please access the on-line review form at www.nctm.org/highschooldraft.aspx by September 19, 2008.
Focus in High School Mathematics – Public Draft, August 28, 2008
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