Math 221–F07
Reading Questions
Van de Walle, Chapter 1: Teaching Mathematics
in the Context of the Reform Movement
The NCTM is currently revising the Standards.
There is a web site devoted to providing information
and getting feedback about the current draft. The
document I handed out in class is from this web site.
The eventual resulting document is already being referred to as Standards 2000. (But this may change,
notice that NCTM refers to it as Priciples & Standards.)
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(g) According to the Technology Principle,
technology (1) influences what mathematics is taught, and (2) enhances students’
learning.
Give an example of each of these 2 aspects
of technology.
6. What are the five Content Standards (five
strands)?
7. Which content strand receives the most emphasis in lower elementary grades (K–5)?
1. Answer the questions in the first column on
page 3.
8. What are the five Process Standards? Why are
they called process standards?
2. What does NCTM stand for? What kind of
organization is NCTM?
9. What was the role of the Process Standards in
the original 1989 Standards document?
3. What is Priniciples & Standards for School
Mathematics? What three documents does it
largely replace?
10. What is the relationship between the Process
Standards and the Content Standards?
4. Principles & Standards lists five goals for students.
(a) What is the main point of the Problem
Solving Standard?
(b) What question might be the most common
question asked by a teacher following the
Reasoning and Proof Standard?
(c) Why is communication important for
learning mathematics?
(d) What are the two aspects (thrusts) of the
Connections Standard?
(e) What are some ways that mathematical
ideas can be represented?
(a) Based on your experience in school mathematics classes, rate yourself in terms of
each of these goals.
(b) Which goals do you think will be hardest
to have your students achieve once you are
a teacher?
5. Principles and Standards lists six principles
fundamental to high-quality matheatics programs.
(a) What is meant by the Equity Principle?
(b) What are the three components of the
Curriculum Principle?
(c) What two (or three?) things does the
Teaching Principle require of teachers of
mathematics?
(d) What three things does Van de Walle say
teachers must do to provide high-quality
mathemaitcs education?
(e) Upon what two fundamental ideas is the
Learning Principle based?
(f) Exlain the Assessment Principle in your
own words.
11. Let’s look at the Process Standards.
12. According to Principles and Standards, what
are the four purposes of assessment?
13. Besides NCTM and its Standards and Principles document, what three other things does
Van de Walle say have had a major impact on
the way school matheatics is taught? What affect have they had?
14. What is NAEP? What is the good news from
NAEP? What is the bad news from NAEP?
15. What is TIMSS? What are its most widely reported results? What are some of the differences between 8th grade math classrooms in the
US and Japan, according to the TIMSS video
study?
Math 221–F07
Reading Questions
16. Summarize the discussion of state standards.
17. If you opened a mathematics textbook, what
would you look for to determine if it is a
traditional textbook or a reform textbook?
Which kind(s) of textbooks did you use in your
schools?
18. Look at the list of five shifts in classroom en-
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vironment (p. 9). How do the mathematics
classes you had in school compare? Are they
more like the “towards” list or the “away from”
list? Which shifts do you think you would have
appreciated most as a student? Which do you
think would be hardest to implement as an instructor?
Math 221–F07
Reading Questions
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Van de Walle, Chapter 2: Exploring What it Means
to Do Mathematics
and effort.” What are the tools, materials and
effort we use to build understanding?
1. What is your reaction to the second paragraph
(beginning “The description of...”)?
5. How does Van de Walle define ”understanding”? How does understanding differ from
knowledge?
2. What does Van de Walle say is the most basic
idea in mathematics? How is this idea important for a classroom teacher?
3. In the second edition of this book, the section
on traditional views of school mathematics was
entitled “Math Does Not Come From the ‘Math
God’”. What do you think was meant by that?
Why do you think there is no longer a section
with that title?
4. Make a list of a couple of (mathematical) things
that you know (or knew) how to do, but don’t
(or didn’t) “understand”. Do you wish you understood them? If you have forgotten how to
do them, do you think you would have remembered better if you had understood better?
5. In school, were you ever asked to solve a mathematical problem of a type you had never seen
before? How did you feel (or would you have
felt)?
6. Do you agree that doing “paper and pencil computation” is not “doing mathematics”? Why
or why not? What is “mathematics”? How can
you tell when someone is “doing mathematics”?
7. Van de Walle says he will not provide solutions
to the problems in his book. How does that
make you feel?
Van de Walle, Chapter 3: Developing Understanding in Mathematics
1. Who is Piaget?
2. What does Van de Walle say understanding is a
measure of? (Put another way, what increases
as someone understands something better?)
3. What is the ”basic tenet of constructivism”?
4. Van de Walle says that constructing ideas is
analogous to building something in the physical world, in that it ”requires tools, materials,
6. Briefly describe ”relational understanding” and
”instrumental understanding.” These 2 ideas
are sometimes described as ”rules without reasons” and ”knowing what to do AND why”;
which is which, do you think?
7. List the benefits Van de Walle claims accompany relational understanding. Choose two
that you think are especially important and explain why those two benefits should result from
an instructional approach that emphasizes relational understanding.
8. Briefly describe procedural knowledge of mathematics and conceptual knowledge of mathematics. Which is more important? Why?
Which does should come first? Why?
9. What type of knowledge is particularly susceptible to instrumental learning?
10. How are conceptual and procedural knowledge
related to relational understanding?
11. On page 32, Van de Walle states that ”It is incorrect to say that a model ’illustrates’ a concept.” But look at the caption of Figure 3.8 on
p. 33; weird, huh?
12. What 3 types of representations for concepts
are presented in Figure 3.9? (There are actually 5 representations depicted, but we’ll reduce
that to 3.)
13. What are 3 ways that models can be used in
the classroom?
14. How can models be used incorrectly? What is
the result?
15. What does Van de Walle say is “the single most
important principle for reform in mathematics”? (p. 36) for effective learning”? What does
Van de Walle mean by this? What is the opposite of this? Do you think he has given a good
name to this? Can you come up with a better
name for what he is trying to express?
Math 221–F07
Reading Questions
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16. Briefly explain in your own words the 4 fundamental principles of constructivism discussed
on page 36.
6. What are some suggestions given to engage all
class members in in class discussions after students have spent time working on a problem?
17. Why might a Christian teacher be especially attracted to the instructional approach presented
in this chapter?
7. In teaching through problem solving, teachers
have to resist the urge to “help” their students
too much, to solve the problems for the class
instead of letting students struggle to find their
own solutions. But a teacher doesn’t want the
students to just “spin their wheels” and waste
class time because students are missing some
essential information.
18. On pp. 36–37, seven “strategies for effective
teaching” are discussed. The first one, “create a mathematical environment”, is obviously
related to all 5 shifts in classroom environment
we encountered in Van de Walle Chapter 1. For
each of the other strategies discussed here, tell
how it links to one (or more) of the 5 shifts in
classroom environment from Chapter 1.
(As further practice in constructing your own
understanding regarding these ideas, you could
also try to link each of these 7 strategies to one
or more of NCTM’s 5 goals for mathematics
students that we also saw in Chapter 1.)
Van de Walle, Chapter 4; MSB, Chapter 1:
Problem Solving
1. What (according to Van de Walle) is the difference between a problem and an exercise?
2. In this chapter, Van de Walle discusses teaching
through problem solving (the first 17 pages of
the chapter) and teaching about problem solving (the last 3 pages of the chapter). What’s
the difference between these two perspectives
on mathematical problem solving?
3. What is the main point of Figure 4.2 on p. 44
in VdW?
4. An extensive discussion of a “three-part lesson
format” to be used in teaching through problem solving is included in this chapter. What
are the key features of the teacher’s role in each
of the “before”, “during”, and “after” phases
of such a lesson? Do you find any of Van de
Walle’s suggestions strange or objectionable?
5. Why does Van de Walle suggest that teachers
use praise cautiously in class discussions after
students have spent time working on a problem?
(a) According to Hiebert et al, what are 3
types of information that it is OK for
teachers to give their students in such situations?
(b) What suggestions does Van de Walle give
to reduce how often this situation (of students not able to make any progress on a
problem) happens?
8. What is Polyá’s four-step framework for
problem-solving? What (at least 3 things) are
meant by “looking back”?
9. Several mathematics education researchers
have claimed that listing Polyá’s four stages or
explaining them to students does very little to
help them become better problem solvers. Why
do you think this is the case? If this claim is
true, why are they important?
10. Briefly describe the 3 areas mentioned in the
NCTM Principles and Standards with regard
to teaching about problem solving (first column
of p. 57 in VdW).
11. The first chapter of Musser, Burger, & Peterson presents 6 general strategies or heuristics
that are often helpful in trying to solve problems. What are they? Van de Walle (on p. 58)
also presents 6 ”plan-and-carry-out strategies”;
are there any differences between the MBP and
VdW lists of strategies?
12. What is “meta-cognition”? What is its role in
problem-solving? How can a teacher promote
meta-cognition on the part of the students?
Math 221–F07
Reading Questions
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13. Describe one or two ways to help students develop positive attitudes toward problem solving.
2. What is the difference between structured and
unstructured sets of attribute materials? What
are some advantages of structured sets?
14. Is it important that a mathematics teacher continue to solve problems herself? Where can a
teacher find problems to work on? Where can
a teacher find problems for students?
3. If a set of structured attribute materials has 4
attributes each with 3 values, how many items
will be in the set?
M. Frank: Problem Solving and Mathematical Beliefs
1. What five beliefs of students about mathematics does Martha Frank say interfere with their
ability to become good problems-solvers? What
is wrong with each?
4. What is the fundamental difference between the
activities in the section entitled “Learning Classification Schemes” and those in the section entitled “Solving Logic Puzzles”?
5. Do activity 18.11 (page 366).
Van de Walle, Chapter 19, pages 384–394:
2. Do you agree with the author that most stu- Patterns
dents hold these beliefs? Why?
1. What advantage do physical materials have
(compared with worksheets) when doing pat3. Give an example of how beliefs about matheterning activities?
matics can interfere with problem-solving.
2. What is a ‘core’ of a pattern? What kinds of
4. What does the author suggest can be done to
patterns have a core? How does Van de Walle
improve the situation?
classify cores?
G.H. Wheatly: Problem Solving Makes Math Scores
Soar
1. What were the goals of of the Marion, Indiana,
elementary school mathematics program?
2. Make a list of changes made in the way mathematics was taught.
3. What evidence is given that the program was
successful? Are any other explanations possible?
4. The claim is made that the students who scored
in the 82nd percentile in 1983 had scored in the
16th percentile in 1979. Find the data in the
table that justify this claim.
5. Do you think this program promoted conceptual understanding? relational understanding?
Explain.
Van de Walle, Chapter 18, pages 362–367:
Classification
1. What are “attributes” and what are “values”?
3. Give an example of a repeating pattern and
identify its ”core”. Give another example of
this same pattern but using different materials
or a different medium.
4. What is the technical term for what Van de
Walle calls a ‘growing pattern’ ?
5. Do all growing patterns grow? If not, give an
example that doesn’t grow. If so, explain why.
6. How do repeating patterns and growing patterns differ?
7. Give an example of a growing pattern. What
is another term that can be used for growing
patterns?
8. Make a numeric chart for your example from
the previous question. How many items will be
in the 30th ”frame” for your example? Did you
use a ”one frame to the next” relationship, or
a relationship between frame number and number of items?