Iowa Council of
Teachers of Mathematics
Mathematic
Modeling
Journal
Volume 36 Fall 2009
Teaching for
Understanding
Connected
and
Coherent
Content
Core
Problem
Based
Instructional
Tasks
Curriculum
Meaningful
Distributed
Practice
Another Bite at the Core
Rigor and
Relevance
Deep
Conceptual
and
Procedural
Knowledge
Essential Characteristics
Effective Use
of
Technology
ICTM Executive Board
2009-2010
President:
Judith Slezak, Swisher
President-Elect
Ruth Avazian, Creston
Treasurer:
Laura Brincks, Waukee
Executive Secretary:
Maureen Busta, New Hampton
Vice Presidents:
Elementary:
Jane Haugen, Dubuque
Middle School:
Diane Royer, Coon Rapids
Secondary:
Mike Dillon, Postville
Post Secondary:
Catherine Miller, Cedar Falls
Conference Chair
Dave Blum, West Des Moines
Pre-Conference Chair
Deb Tvrdik, Gowrie
Conference Program Co-Chairs
Brooke Fischels, Ottumwa
Cheryl Ross, Mason City
Governmental Liaison
Edward Rathmell, Cedar Falls
NCTM Representative:
Deb Tvrdik, Gowrie
Journal Committee:
Lori Mueller, West Point: Editor
David Burrow, Algona
Marlene Meyer, Cedar Rapids
Vicki Oleson, Cedar Falls
Ed Rathmell, Cedar Falls
Deborah Roberts, Corning
Diane Royer, Coon Rapids
Dept. of Education
Mathematics Consultant:
Judith Spitzli
Newsletter Editor/Webmaster
David Burrow, Algona
Conference Exhibits
Travis Nuss, Gowrie
Regional Directors are listed
on the back cover.
Iowa Council of Teachers of
Mathematics Journal
The ICTM Journal is published annually by the Iowa
Council of Teachers of Mathematics.
Non-profit organizations are granted permission to
reprint any article appearing in this journal provided
that two copies of the publication in which the
material is reprinted is sent to the Journal Editor.
Opinions expressed in the articles are those of the
authors and not necessarily those of the Iowa Council
of Teachers of Mathematics.
Check out the ICTM website at: www.iowamath.org
Iowa Council of Teachers of Mathematics
38th Annual Conference
Another Bite At The Core
Friday, February 19, 2010
7:30 – 3:30
Valley Southwoods Freshman High School
West Des Moines, Iowa
Register now at www.iowamath.org
Reserve your hotel room now.
ICTM Executive Board
2
About this issue…
It was a joy to put together the articles for this issue of the ICTM Journal because the articles are
so timely. Grab a cup of coffee and a cookie then take time to peruse the articles on teaching ideas that
can be used in the classroom as well as articles about the Core Curriculum. The 2010 Journal will have
a focus of teaching in the Core Curriculum. We are looking for teachers to share their experiences of
getting ready for and using the Core Curriculum in their mathematics classroom. We want to hear from
you! Your experiences can help other teachers throughout the state to be successful. Go to the ICTM
website at www.iowamath.org to find out the format for an article for the journal. Together we can all
be top notch educators.
I want to hear from you,
Lori Mueller
[email protected]
An Invitation to Share…
You are invited to share some of your teaching experiences through an article in the ICTM
Journal. The article can be as short as a page or as long as 12 pages; it could include student work or
photographs from your classroom. Your article might be a research paper that you wrote while working
on an advanced degree. If you are uncertain about a particular idea for publication, contact me to
discuss it.
Manuscripts should be computer generated in Microsoft Word and be single spaced, size 12
Times New Roman font. All figures such as diagrams, tables, or photographs should be ready for
reproduction. Manuscripts may be submitted in CD or attached to an email. Please also mail a hard
copy to me at the address below.
Lori Mueller
2001 130th Street
West Point, IA 52656
319-469-6414
[email protected]
Table of Contents
Articles
Articles
4
Meaningful Distributed Instruction4
Developing Number Sense
Edward Rathmell
92
The Iowa Core Curriculum and Me:
How My Teaching of Mathematics
Methods Will Change
Catherine Miller
22
Conceptual Previews in Preparation
For the Next Unit of Instruction
Michele Carnahan
and Dr. Bridgette Stevens
100
Problem Based Instructional Tasks
Larry Leutzinger
103
Top Ten Things I Wish I Had
Known When I Started Teaching
33
Japanese Lesson Study: A Brief
Explanation and Reflection
Brooke Fischels
38
Health Literacy in the Mathematics
Classroom: The Iowa Core
Curriculum as an Opportunity to
Deepen Students’ Understanding of
Mathematics
Dr. Elana Joram, Dr. Susan RobertsDobie, and Nadene Davidson
55
Updating Math Teaching and
Learning for Pre-Service Elementary
Education Teachers
Jeanette Pillsbury
72
Skittles Chocolate Mix Color
Distribution A Chi-Square
Experience
David Duncan and Bonnie Litwiller
76
Distributed Curriculum and
Mathematical Discourse: A Little Bit
Goes a Long Way
Angie Peltz
edited by Dr. Bridgette Stevens
3
Departments
37
Web Bytes
71
Conference Flyer
91
Grant Opportunities Information
105
Membership Form
106
Fun Fridays
Magic Squares
Lori Mueller
Meaningful Distributed Instruction—
Developing Number Sense1
Edward C. Rathmell
What Is Number Sense?
The primary goal for elementary and middle school mathematics is to help
students learn to use numbers meaningfully, reasonably and flexibly in everyday
life. This means that students must develop a deep understanding of
numbers and operations,
when operations can appropriately be used to solve problems, and
judging the reasonableness of their solutions to problems.
They also need to develop attitudes so they
believe they can make sense of mathematics, and
habitually try to make sense of mathematics.
In other words, students need to develop number sense.
Understandings Needed for Number Sense
To develop number sense related to a specific topic students need to have a
deep understanding about
1. when to use that topic in everyday situations,
2. how to represent that topic in multiple ways, and
3. how to think and reason with that topic in multiple ways.
Fraction Example
1. Know When Fractions Can Be Used To Solve Problems
Grade six students who deeply understand fractions will know that
fractions can be used to solve a variety of problems when a whole is split
into equal-sized parts. Problem contexts might be linear, involve area or
volume, or include wholes that are comprised of individual items.
Students need to learn that fractions can be applied in a variety of different
contexts with a variety of different problem structures.
2. Flexibly And Fluently Use Multiple Representations Of Fractions And
Their Relationships To Other Topics,
When students first begin learning fractions, they often directly model the
structure of the problem. This leads to different representations of
fractions. Students must make sense of these different representations.
Adapted from a manuscript that will be published by the National Council of Teachers of
Mathematics.
1
4
Halves, fourths, sixths, and eighths of a pie or pizza are familiar situations
to many students. Experiences like this lead many of them to associate
fractions with area models, and more specifically splitting circles into equal
parts.
Area models can also involve any other shapes that are split into equal
parts. Each of the following illustrates the fraction 3/4.
Linear measurement and the use of rulers lead to a number line
representation of fractions. In this case a unit length is split into equalsized parts.
|
0
|
¼
|
½
|
¾
|
1
Measuring volume leads to a three-dimensional representation of
fractions. Measuring cups used in cooking are typical examples.
All of the representations above are connected or continuous. Fractions
are also used in discrete situations. Groups of items, which are split into
equal-sized parts, lead to different representations.
5
In contrast, students who only represent fractions in symbolic form often
have difficulty using fractions in all of the different types of situations
shown above. This limits their ability to know when to use fractions in
everyday situations.
These representations also help students make connections between
fractions and other topics. For example, using a unit square split into 100
equal-sized parts by forming 10 rows and 10 columns provides a natural
way to connect fractions to decimals and percents. Forty-five of 100 equal
parts illustrates 45/100; forty-five hundredths illustrates 0.45; and forty-five
per one hundred illustrates 45%.
45/100
0.45
45%
6
Ratios, which are comparisons of a part to the whole, can be represented
by a model that also illustrates a related fraction. If the ratio of the number
of brown candies to the total number of candies is 2 to 5, that can be
represented by a model to illustrate 2/5. Two-fifths of the candies are
brown.
ratio
fraction
2 browns to
5 candies
2 rows of brown
out of 5 rows
2//5
3. Flexibly And Fluently Use Multiple Reasoning Strategies With
Fractions And Their Relationships With Other Topics
In order to make sense of fractions and use them effectively to solve
everyday problems, students need to think about the following just to
understand the meanings of the fractions.
What is the whole (unit)?
Are there equal-sized parts?
How many parts are you considering?
How many parts are there in the whole (unit)?
Students with number sense understand the ―size‖ of fractions. Just like
they understand that 58 is close to 60, they understand that 9/10 is close
to 1 and that 4/7 is a little more than 1/2. When solving problems, it is
often much easier to use a nice number that is close to a fraction than to
use the fraction itself. One-half is often much easier to use than 4/7.
Size of Fractions
fraction
2/13
benchmarks
0
1/35
0
4/7
1/2
11/25
1/2
8/9
1
14/12
1
thinking
Two parts is very small compared to 13 parts in the
unit. So 2/13 is not that much more than 0.
One part is very small compared to 35 parts in the unit.
So 1/35 is not that much more than 0.
Four is little more than half of 7, so 4/7 is a little more
than 1/2.
Eleven is a little less than half of 25. So 11/25 is a little
less than 1/2.
Eight parts is almost the same as 9 parts in the unit, so
8/9 is almost 1.
Fourteen parts is a little more than 12 parts in the unit.
So 14/12 is a little more than 1.
Reasoning that is used to compare fractions is much more complex than
reasoning to compare whole numbers. One thing that confuses many
7
students is that fractions with larger numbers aren’t necessarily greater
than fractions with smaller numbers. For example, 5/8 is not as great as
3/4. But, 7/9 is greater than 2/3. Sometimes the fraction with larger
numbers is greater and sometimes it is not. If you cannot decide which of
two fractions is greater by comparing the size of the numbers involved,
how can you decide?
Comparing Fractions
Which is greater?
1/2 or 1/2
strategy
compare the
size of the
units
thinking
If the unit for one fraction is greater, 1/2 of that
unit is greater.
3/8 or 5/8
common
denominators,
then compare
the size of the
numerators
If the fractions have the same size unit and they
have the same number of parts per unit
(common denominators), then you can
compare the number of parts you are
considering (numerators). Three parts is less
than 5 parts, so 3/8 is less than 5/8.
3/4 or 3/8
common
numerators,
then compare
the size of the
denominators
If fractions have the same size unit and they
have the same number of parts being
considered (common numerators), then you
can compare the number of parts per unit
(denominators). Fourths are bigger parts than
eighths, so 3/4 is greater than 3/8.
3/8 or 4/7
compare to
benchmark
fractions
6/7 or 8/9
compare the
―missing‖ parts
to make a unit
or benchmark
fraction
Three is less than half of 8, so 3/8 is less than
1/2. Four is greater than half of 7, so 4/7 is
greater than 1/2. So,
3/8 < 1/2 < 4/7
Six-sevenths is only 1/7 away from 1. Eightninths is only 1/9 away from 1. Since sevenths
are greater than ninths, 8/9 is closer to 1 than
6/7. That means 8/9 is greater.
Fractions are different from whole numbers in many ways. There is no
whole number between 23 and 24. But there are an infinite number of
fractions between any two given fractions, no matter how close together.
Fractions are dense.
8
fractions
between
2/3 and 3/4
2/3 and 3/4
Strategy
thinking
change both to a
common
denominator
(much larger than
the least common
denominator), then
choose a
numerator
between
average the two
fractions; that is,
add the fractions
and divide by two
2 20
x
3 20
40
60
3 15
x
4 15
45
60
41/60; 42/60; 43/60; and 44/60 are all fractions
between 2/3 and 3/4
8 9
12 12
17
12
(17/ 12) ÷ 2 = 17/24
So, 17/24 is a fraction between 2/3 and 3/4. In
fact, it is ―halfway‖ between 2/3 and 3/4.
Using the standard symbols there is only one way to write each whole
number. With fractions greater than one, they can be written as a mixed
number or as an improper fraction, with a numerator greater than the
denominator. Changing a mixed number to an improper fraction or back
relies on making sense of 3/3, 5/5, 16/16, etc. Each of those fractions is
just 1.
Changing a Mixed Number to an Improper Fraction
mixed number
1
3
4
thinking
4/4 is 1, so three times that or 12/4 is 3.
There is an extra 1/4, so it is 13/4.
improper fraction
13/4
Changing an Improper Fraction to a Mixed Number
improper fraction
11/3
thinking
3/3 is 1, so 9/3 is three times as much or
3. That leaves 2/3.
mixed number
3
2
3
With fractions there are an infinite number of ways to write the same
number. For example, 3/5 is the same number as 6/10 or 15/25 or 24/40
or (3 x n)/(5 x n) for any non-zero number.
9
Equivalent Fractions
equivalent fractions
fraction equivalent to
2/3
strategy
multiply both
numerator and
denominator by the
same non-zero
number
thinking
When you double the number of parts
in the unit and double the number of
parts you are considering, the amount
is the same. The same is true if you
multiply both numerator and
denominator by the same non-zero
number. For example,
x4
2
3
8
12
x4
Is 6/8 equivalent to
9/12?
cross multiply and
compare products
Since 6 x 12 = 8 x 9, the fractions are
equivalent.
6 12
x
8 12
72
96
9 8
x
12 8
72
96
The numerators are simply the cross
products. You don’t have to compare
the denominators because they are
the same.
To add and subtract fractions, you have to be combining or separating
parts that are the same size, that is, they have a common denominator.
How do you find a common denominator?
Finding Common Denominators
find common
denominator
2/3 and 3/4
3/5 and 2/3
strategy
list multiples of each
denominator
list multiples of the
greatest denominator,
check to find a multiple
of the other denominator
10
thinking
3, 6, 9, 12, 15, 18, ...
4, 8, 12, 16, ...
The first common multiple is 12, so
12 is a common denominator, in
fact, it is the least common
denominator.
5, 10, 15
Three does not divide 5 or 10.
Three does divide 15, so 15 is a
common denominator, in fact, it is
the least common denominator.
5/12 and 3/21
factor the denominators
into prime factors, then
combine factors to
insure that both
denominators divide the
product.
7/12 and 7/18
multiply the
denominators and divide
by the greatest common
factor
12 = 2 x 2 x 3
21 = 3 x 7
Factors of a common denominator
must include at least 2 twos and 1
three so that it is a multiple of 12; it
must include at least 1 three and 1
seven so that it is a multiple of 21.
So, 2 x 2 x 3 x 7 = 84
Factors of a common denominator
must include at least 2 twos and 1
three so that it is a multiple of 12; it
must include at least 1 two and 2
threes so that it is a multiple of 18.
12 x 18 = 2 x 2 x 3 x 2 x 3 x 3
There is one extra two and one
extra three in that product, 2 x 3 is
the greatest common factor of 12
and 18. So, (12 x 18) ÷ 6 = 36
Counting by fractions, just like counting by multiples of 4 can be helpful in
adding and subtracting fractions. Counting by fractions eliminates finding
a common denominator, converting both fractions to equivalent fractions
with that common denominator, then adding or subtracting. However,
counting by fractions is complicated because some multiples of fractions
are whole numbers
1 2
3 3
1
1 2
1
3 3
Counting by fractions is even more complicated because often a multiple
of a fraction is equivalent to a simplified fraction.
1 1 3
1 1 3
, , , 1, 1 , 1 , 1 , 2, ...
4 2 4
4 2 4
Counting by Fractions to Add or Subtract
computation
1
2
2
1
3
4
3
4
3
8
strategy
count on by one-fourths
count back by one-eighths
thinking
1
2 ,...
2
1
3 ,...
4
3
1
2 , ,3
4
4
1
7
3 , ,2
8
8
Often fractions can be related to decimals, money, or even percents.
Some students might find it easier to think of 1/4 as 0.25, a quarter, or
25%. Similarly, 2/5 can be 0.4 or 40%. Common fraction decimal
equivalents include the following.
11
Common Fraction Decimal Percent Equivalents
fraction
1/2
1/3; 2/3
1/4; 3/4
1/5; 2/5; 3/5; 4/5
1/10; 3/10; 7/10; 9/10
decimal
0.5
about 0.33; 0.67
0.25; 0.75
0.2; 0.4; 0.6; 0.8
0.1; 0.3; 0.7; 0.9
percent
50%
about 33%; 67%
25%; 75%
20%; 40%; 60%; 80%
10%; 30%; 70%; 90%
Other fractions can often be converted to decimals and/or percents by
using the equivalents listed above.
Other Fraction Decimal Percent Equivalents
fraction
1/6
3/8
7/20
decimal
1/3 is a little more than 0.33, so 1/6 is half of that or
about 0.167
1/4 or 2/8 is 0.25, so 1/8 is half of that or 0.125
3/8 is 0.25 + 0.125 or 0.375
1/10 is 0.1, so 1/20 is half of that or 0.05
7/20 is 7 x 0.05 or 0.35
percent
16.7%
37.5%
35%
Every fraction can be converted to a decimal, simply divide the numerator
by the denominator. But because decimals only have denominators that
are powers of ten, some fractions have terminating decimals and others
have repeating decimals. If a fraction, in simplest form, has a
denominator with prime factors that only include twos and/or fives, it will
have a terminating decimal.
Converting a Fraction to a Decimal
fraction
3/20
strategy
multiply
numerator
and
denominator
by the same
number to
get a power
of ten in the
denominator
41/250
multiply
numerator
and
denominator
by the same
number to get
a power of ten
in the
denominator
thinking
20 = 2 x 2 x 5 If the number of twos matches
the number of fives as factors, then they can be
paired up to make tens. If you multiply the
denominator by 5, then 20 x 5 = 2 x 2 x 5 x 5 =
(2 x 5) x (2 x 5). That is 10 x 10 or 100, a power
of ten. You can multiply both numerator and
denominator of 3/20 by 5 to get 15/100.
Because the fraction has a denominator that is a
power of ten it can be written as fifteen
hundredths.
250 = 2 x 5 x 5 x 5 If the number of twos
matches the number of fives, then they can be
paired up to make tens. If you multiply the
denominator by 2 x 2, then 250 x 2 x 2 = 2 x 2 x
2 x 5 x 5 x 5 = (2 x 5) x (2 x 5) x (2 x 5). That is
10 x 10 x 10 or 1000, a power of ten. You can
multiply both numerator and denominator of
41/250 by 4 to get 164/1000. The fraction can
be written as one hundred sixty-four
thousandths.
12
decimal
0.15
0.164
0.8333
6 5.0000
48
20
18
20
18
20
18
2
5/6
When you divide by six, there are only six
possible remainders for each division, 0- 5. This
division will never have a remainder of 0
because six has a factor of 3. That means it
cannot be multiplied by any whole number to get
a power of ten. Powers of ten only have factors
of twos and fives. So this division will repeat.
As soon as the same remainder appears a
second time, the quotient will begin repeating.
In this case when 20 is divided by 6, the
remainder is 2, which becomes 20 when the
next place value is added (―brought down‖) to
that remainder. The quotient will repeat from
that point on.
0.8333...
Students should learn to compute using standard fraction algorithms, but
they also should be able to use mental computation and estimation
strategies to solve problems, with exact answers and with estimations.
Variations of many of the strategies that students use with whole numbers
work with fraction computation.
Mental Computation and Estimation With Fractions
computation
3
3
2
1
4
4
5
1 3
1
4 8
1
1
3
4
3
2
1
5
2
3
4
7
4
1
3
8
7
3
5
4
1
5 16
2
Strategy
use a unit
use a unit
thinking
3
Start with 2 and add enough to make
4
1
1
1
3. That leaves 1 .
3 1
4
2
2
2
Subtract 1/4 to get 5. Then subtract the
1
8
remaining 1 .
front end, then
adjust
front end, then
adjust
use benchmark
fractions
use benchmark
fractions
5 1
1
8
3
7
8
2 + 3 is 5
1/4 + 1/3 is a little more than 1/2
so, it’s a little more than 5 and 1/2
5 – 2 is 3
2/3 – 1/4 is a little less than 1/2
so, it’s a little less than 3 and 1/2
1 and 7/8 is about 2
3 and 4/7 is about 3 and 1/2
2 plus 3 and 1/2 is 5 and 1/2
4 and 3/5 is about 4 and 1/2
1 and 5/16 is about 1 and 1/4
4 and 1/2 minus 1 and 1/4 is 3 and 1/4
Models to represent operations with fractions are often extensions of
models to represent operations with whole numbers. For whole numbers,
3 rows with 2 in each row represents 3 x 2. Below, the shaded rectangle
with a height 3/4 of the unit rectangle (outlined with a heavy border) and a
length of 2/3 of the unit rectangle is 3/4 x 2/3 of the unit rectangle. That is,
3 rows of 2 shaded parts out of 4 rows of 3 parts in the unit rectangle is 6
parts out of 12 parts or 6/12 of the unit rectangle.
13
Similarly, division of fractions can be represented on a number line just
like division of whole numbers. Six divided by 2 and be represented by
splitting the 6 into groups of 2.
|
0
|
1
|
2
|
3
|
4
|
5
|
6
|
7
|
8
Two divided by 2/3 can be modeled in two steps. First, how many thirds
are there in 2? Since 3/3 is a 1, there are three times as many thirds as
the unit. 3/3 is 1, so 6/3 is 2. Multiplying the denominator by the number
of units determines the number of thirds in the total number of units.
|
|
|
|
0
|
|
|
1
|
|
|
3
2
Since you need to know the number of 2/3 in 2, you can simply divide the
number of thirds by 2. Dividing the total number of thirds by the numerator
determines the number of 2/3 in the number of units. The number line
below illustrates there are 6 thirds in 2, but only half that many groups of
2/3.
|
0
|
|
|
|
|
1
|
2
|
|
|
3
Students, who understand what is described above, will be able to
Confidently and meaningfully use fractions,
Flexibly use appropriate and efficient representations and reasoning
strategies with fractions, and
Perform extremely well on any fraction items in any assessment, from
skills through problem solving.
14
What Is The Current Status?
In contrast, many sixth grade students
Have many misconceptions about fractions,
Do not understand the relative size of the number or which nice fractions
are close, and
Almost always have memorized symbolic procedures without
understanding,
Most curriculum materials in grade six include two, three, or four units of
instruction on fractions. If each unit takes about three weeks, that means
students must learn
all of the different situations when fractions can be used,
multiple ways to represent fractions, and
multiple reasoning strategies for solving problems with fractions.
The examples above include over 30 different reasoning strategies. And
students only have about nine weeks to make sense of them. This is an
impossible task for both the teacher and the students.
First, it needs to be recognized that our current programs are not successfully
helping students develop a deep understanding of fractions, and they certainly do
not help students develop number sense with fractions.
Second, no current curriculum materials include sufficient instructional time to
help students make sense of all of the ideas shown in the examples above.
Because there are so many different contexts for problems, so many different
ways to represent those problems, and so many different reasoning strategies
used to solve those problems, students need an extended amount of time to
understand.
Third, making sense of concepts and reasoning strategies simply takes too long.
With current curriculum materials students do not have the opportunity to learn
and to develop a deep understanding of fractions.
How Can We Improve Achievement?
Organizing Curriculum to Develop Number Sense
One alternative is to spend more time on each of the current units of instruction.
This approach fails in two ways. First, our current materials focus on only a few
of the ideas shown in the example above, so they still will not provide a complete
instructional program. Second, this means the focus on at least some topics
other than fractions will need to be diminished, if not ignored. Developing
understanding of one topic at the expense of another is a questionable practice.
15
Another alternative is to distribute the instructional experiences throughout the
school year, a few minutes each day. Since this approach cannot be used for
every topic, one or two key topics need to be selected for each grade level.
Daily five-minute lessons on that key topic will provide time for students to
develop number sense and still leave time for the current curriculum. The extra
twenty-five minutes of instruction each week will provide students repeated
opportunities to make sense of multiple representations, thinking strategies, and
contexts for using the key topic.
Research has clearly established that distributed practice is more effective than
massed practice. A few minutes of practice each day is more effective than
spending a similar amount of time practicing once a week.
The same principle applies to mathematical concepts and to the development of
reasoning strategies. Five minutes of daily conceptual instruction throughout the
school year is more effective than a three-week unit of instruction—even two,
three or four units of instruction. Meaningful distributed instruction provides
valuable time for students to make sense of mathematical concepts and thinking
strategies.
Meaningful Distributed Instruction
Guidelines for these brief five-minute lessons include
1. Use a problem-centered approach
2. Focus on conceptual ideas and routine problem solving
3. Expect students to make sense
4. Encourage students to develop fluency and flexibility
1. Use A Problem-Centered Approach
Students should have repeated opportunities to choose their own way to
represent problems and to choose their own idiosyncratic solution
strategies—ones that make sense to them. With repeated experiences
and with many opportunities to model and see illustrations and to hear
other explanations, students will gradually enlarge their repertoire of these
representations and thinking strategies. This permits them the luxury of
choosing an appropriate and efficient representation or strategy for each
situation. This allows them to develop flexibility in their use of the key
topic.
Students should also communicate about their solutions of these
problems. They should have repeated opportunities to explain their
representations and solution strategies. These explanations should
include both how the representations were used to solve the problem and
why they were selected. Students should also be encouraged to actively
listen and ask for clarification as other students explain their solutions.
16
2. Focus On Conceptual Ideas And Routine Problem Solving
These five-minute lessons are meant to develop understanding; they are
not intended to provide extra time for practicing skills. Manipulatives,
diagrams, drawings, and other concrete models should be used to
illustrate solutions. These visual presentations together with explanations
help students understand the concepts and the reasoning. This extra time
spent on helping students understand concepts is time well spent on
helping students make sense of symbolic procedures that are introduced
later.
Routine word problems enable students to apply a topic in a variety of
everyday contexts. These distributed experiences through the school year
enable students to decide when a topic can be used. The different
contexts in these routine problems also encourage a variety of
representations and thinking strategies. They provide the stage for
students to develop a deep understanding of a key topic.
3. Expect Students To Make Sense
Students should be expected to try to solve these daily problems. They
also should be expected to make sense of multiple representations and
multiple thinking strategies and ask for clarification when they do not
understand. Students should be actively involved.
4. Encourage Students To Develop Fluency And Flexibility
The distributed curriculum is designed to promote flexibility. As students
become familiar with and understand various representations and thinking
strategies, they are able to make strategic choices, which are efficient.
Ultimately, it is important for students to be reasonably fluent with skills. If
students need to hesitate and rethink a procedural step in their
performance of a skill, they are less likely to use it in everyday life. While
practice is important in developing fluency, it is much more effective after
students have a deep understanding of the underlying concepts. The
daily five-minute lessons are great preparation for skill fluency, but
practice needs to be delayed.
Five-Minute Lessons
These lessons are designed to provide the experiences needed to help students
develop deep understanding. These lessons need to become a routine in the
classroom—a routine for the teacher and a routine for the students. Everyone in
the classroom needs to know what to expect.
It is important to create an appropriate classroom climate. Students need to try
to make sense of these problems, try to represent them appropriately, and try to
use an efficient thinking strategy to solve the problem. Students also need to feel
17
comfortable trying new approaches, explaining their solutions, and asking for
clarifications.
It should be noted that routine word problems are suggested. While they are
only one kind of problem, they can be used to provide a variety of contexts, a
variety of problem structures, and a variety of situations where efficient thinking
strategies can be used. More complex problems are important too, but they
cannot be completed in the amount of time allotted for these lessons.
Typically the lessons are structured as below.
1. Present a routine problem to the students (about 20 seconds)
2. Give students time to think and solve the problem (about 30 seconds)
3. Ask two or three students to explain their solutions (about 2 minutes)
4. Highlight and illustrate one of the explanations (about 1 minute)
5. Ask students to try using that highlighted strategy to solve a new problem;
have one student explain using that thinking (about 1 minute)
Each of the five parts of these lessons are more fully described below.
1. Present A Routine Problem To The Students (about 20 seconds)
Select a routine problem with nice numbers. As students gain proficiency,
more complex numbers can be used
Usually this problem will be read to the students. It might be projected on
a screen so students can read the problem as well.
2. Give Students Time To Think And Solve The Problem (about 30
seconds)
All of the students need to be thinking about how to solve the problem. If
it helps to keep all students involved, paper and pencil may be used. This
also allows the teacher to check any drawings or diagrams the students
might have used and their solutions.
Students should not be using manipulatives at this time. It’s not that
manipulatives aren’t helpful; they should be used at other times. It simply
takes too much time for students to use manipulatives during these brief
lessons.
3. Ask Two Or Three Students To Explain Their Solutions (about 2
minutes)
Two or three students should explain their solutions. Students should
develop an expectation that their solution strategies need to be different
from any previous explanations. Students should expect to actively listen
and ask questions to clarify any misunderstandings. As they gain
experience, students will communicate better and feel more confident in
their explanations.
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4. Highlight And Illustrate One Of The Explanations (about 1 minute)
After the explanations, the teacher should select one of the efficient
strategies and highlight it. This choice should be made to promote a
strategy, which will help the class make progress in their collective
thinking strategies.
Once the choice has been made, that thinking needs to be explained once
or twice more—perhaps explained by a different student and perhaps
explained in conjunction with a teacher demonstrated use of models or a
diagram along with a student or a teacher explanation.
5. Ask Students To Try Using That Highlighted Strategy To Solve A New
Problem; Have One Student Explain Using That Thinking (About 1
Minute)
The ―same‖ problem, but probably with larger numbers, can be presented
again. This time students should be directed to try to use the highlighted
strategy. ―Everyone try to use Marissa’s thinking to solve this problem.‖
After about ten seconds to think, ask one student to explain using that
same thinking. Ask another student if the highlighted thinking was used.
Repeat that thinking one more time.
Sample Five-Minute Fraction Lesson
Teacher
―Listen carefully and solve this problem.‖
Jonas ate 5/6 of a small pizza. His dad ate 7/8 of a small pizza. Which
one ate more of their small pizza?
Wait about twenty to thirty seconds.
Teacher
―Cam, will you explain your solution?‖
Cam
―His dad ate more. 7/8 is bigger because if you change them to a common
denominator 7/8 is 42/48 and 5/6 is only 40/48.‖
Teacher
―Did anyone solve it a different way? (brief pause) Yes, Maddie.‖
Maddie
―Jonas had 1/6 of his pizza leftover. His dad had 1/8 of his pizza leftover. 1/8 is
less than 1/6, so his dad had less leftover. That means he ate more.‖
Teacher
―How do you know that 1/8 is less than 1/6?‖
Maddie
―If you cut 8 slices of pizza instead of 6, there are more pieces so each piece is
smaller.‖
Teacher
―Ok, today, let’s focus on Maddie’s way of thinking. Erin, can you explain how
Maddie solved the problem?‖
19
Erin
―She looked at how much was leftover. Jonas’s dad had a smaller piece leftover,
so that means he ate more.‖
Teacher
While Erin is explaining, the teacher is drawing a diagram on the board.
Teacher
―So, Maddie thought about the parts that were leftover. She knew that 1/8 is less
than 1/6, so Jonas’s dad had less pizza leftover. That means he ate more.
Sometimes you can compare fractions by checking to see how close they are to
a benchmark or nice number. Let’s all try that same thinking on this problem.
Kaitlyn ate 3/8 of a large pizza. Hayden ate 4/10 of a large pizza. Who
ate more pizza?
Wait about ten seconds.
Teacher
―Who can explain Maddie’s thinking with this problem? (brief pause) Marissa,
will you try this?‖
Marissa
―You can compare these to one-half. 4/8 is one-half. Kaitlyn ate 3/8, so she ate
1/8 less than one-half. 5/10 is one-half. Hayden ate 1/10 less than one-half.
Since 1/10 is less than 1/8, Haydn ate more.‖
Teacher
While repeating the explanation, draw a diagram showing this same thinking.
Look at the diagrams. 3/8 is 1/8 less than one-half. 4/10 is 1/10 less than onehalf. Which is less, 1/8 or 1/10?‖
4/8
3/8
5/10
4/10
Felipe
―1/10 is smaller because it has more pieces in the pizza.‖
Teacher
―You can compare fractions by comparing them to benchmarks. Sometimes it is
easy to look at the ―missing‖ part like we did today.‖
Summary
Key number and operation topics can be identified at each grade level. To
develop number sense for those key topics, students need to have a deep
understanding about
when to use that topic in everyday situations,
how to represent that topics in multiple ways, and
how to think and reason with that topic in multiple ways.
Given the structure of current curriculum materials, most students do not have an
opportunity to develop that deep understanding. There are just too many
20
applications, too many models to represent a topic, and too many ways to think
and reason with the topic to make sense of all that in only two, three or four units
of instruction.
Meaningful distributed instruction provides students a much better opportunity to
develop number sense. A few minutes of instruction each day keeps the ideas in
the foreground all year long. Instead two or three conceptual lessons on a topic
prior to practicing related skills, students use and see representations daily.
Instead of three or four lessons on a specific thinking strategy, which might be
spread over five or six months, students will use and listen to that strategy being
used repeatedly throughout the school year.
Routine problems in the five-minute lessons provide a variety of
contexts for using a topic,
problem structures to represent in different ways, and
situations to develop more efficient ways of thinking and reasoning.
As students begin to develop understandings of these ideas, they become more
flexible in their use. With a minimal amount of practice on ideas they understand,
they begin to develop better fluency.
Meaningful distributed instruction, as a supplement to any regular mathematics
curriculum, can help students make sense and help them learn to use their
number sense in everyday life.
Edward C. Rathmell is a Professor of Mathematics Education at the University of
Northern Iowa in Cedar Falls, Iowa.
A Few References:
Everyday Mathematics. (Summer 2000). Distributed practice: The research base. Retrieved from:
http://everydaymath.uchicago.edu/educators/references.shtml
Grouws, D. A. & Cebulla, K. J. (2000). Improving student achievement in mathematics. Geneva,
Switzerland: International Academy of Education.
Hiebert, J., (2003). What research says about the NCTM Standards. In J.Kilpatrick, W.G. Martin,
and D. Schifter (Eds.), A research companion to principles and standards for school
mathematics (pp. 5-23). Reston, VA: National Council of Teachers of Mathematics.
Mathematics Learning Study committee, National Research Council (2001). Conclusions and
recommendations. In J. Kilpatrick, J. Swafford, & B. Findell (Eds.). Adding it up: Helping
children learn mathematics (pp. 407-432). Washington, DC: The National Academies
Press.
Rea, C. P. and Modigliani, V. (1985). The effect of expanded versus massed practice on the
retention of multiplication facts and spelling lists. Human Learning, 4, 11-18.
Willingham, D. (Summer, 2002). Allocating student study time: ―Massed‖ versus ―distributed‖
practice. American Educator, 26(2). Retrieved from http://www.aft.org/pubsreports/american_educator/summer2002/askcognitivescientist.html
21
Conceptual Previews in Preparation for the Next Unit of Instruction
Michele Carnahan and Dr. Bridgette Stevens
Understanding meanings of operations and how they relate to one another is an
important mathematical goal for students in fourth grade (National Council of Teachers
of Mathematics, 2000). Using pictures, diagrams, or concrete materials to model
multiplication helps students learn about factors and how their products represent various
contexts. The foundation of understanding how operations of multiplication and division
relate to one another deepens the understanding of the composition of numbers.
Discussing different types of problems that can be solved using multiplication and
division is important, along with the ability to decompose numbers. When students can
work among these relationships with flexibility, conceptual understanding strengthens.
With this said, a group of fourth grade students were introduced to conceptual previews
as a new approach of making sense out of multiplication and division so to develop
strong images of what happens when numbers are multiplied and divided in small fiveminute mini-lessons before the specified unit on multiplication and division.
The Development of the Conceptual Previews
Akin with mini-lessons described by Fosnot and Dolk (2002) and distributed
practice by Rathmell (2005), conceptual previews are an instructional practice where a
teacher identifies a key concept or big idea from an upcoming unit and develops 10 to 15,
five-minute mini-lessons where students explore a concept informally. The mini-lessons
may consist of two or three questions related to modeling a concept, representing the
concept through the use of a diagram, connecting prior knowledge with the concept,
and/or preparing students with some of the basic ideas related to the key concept or big
22
idea before the upcoming unit begins. Through these mini-lessons, it is the intent for the
teacher to bring to the forefront student’s prior knowledge and assist in their first steps in
the development of new knowledge based upon this existing knowledge when the formal
unit begins.
The fourth grade unit, ―Packages and Groups‖ (Investigations in Number, Data,
and Space, 1998), an investigation in multiplication and division, was the unit of
instruction after the use of the conceptual previews used in my classroom for the purpose
of this paper. There are three main objectives during this unit. First, students solve and
create cluster problems by using the distributive property. Another objective is for
students to become familiar with landmark numbers to solve problems. An example
would be to use landmarks like 5 or 10 because they are easier to work with or another
strategy would be the use of partitioning large numbers to make them manageable for
multiplication. Last, students will learn how to solve double-digit multiplication
problems and how to solve division problems using multiplication.
For the ten days of previews I developed for the unit, students used the model of
an array to begin solving multiplication problems. The students were familiar with
making arrays of multiplication problems from a previous unit; however, they did not
understand the connection between the arrays and the multiplication algorithm when it
came to double-digit multiplication. So to begin the previews for the first day, students
used graph paper to represent a multiplication problem like 6 x 8 and then decomposed
the numbers in order to think about the factors associated with each number. For the
previews on day two and three, students used graph paper again with more difficult
multiplication problems and decomposition once again. On the fourth day, students
23
continued to use graph paper for 10 x 16 purposefully so they could see the patterns of
ten. For days five and six, students solved double–digit multiplication problems and they
could use the array or a visual representation of the array, and were gradually weaned
from the graph paper during the five-minute conceptual previews.
Days seven through ten, students used word problems and were encouraged to
solve the problems with the visual representation of the distributive property. The
previews were designed to scaffold students’ knowledge of arrays, a series of addition,
and connect it to the visual model of the distributive property or decomposition of
numbers, as a method for solving difficult double-digit multiplication problems.
My Fourth Grade Class
My fourth grade class was located in a Midwestern Iowa town that has
experienced an influx of immigrants. I taught in a dual-language school with forty
students in two mathematics classes. Twenty-eight of my students were English
Language learners. Fifteen percent of my students received special education/resource
services. The biggest factor affecting my students was poverty. Eighty-one percent of my
students qualified for federal free or reduced lunch. I taught only in Spanish.
Pre-/Post-Assessment of the Conceptual Understanding
I developed a short assessment instrument to gauge the effectiveness of the minilessons. If I was going to continue to devote five minutes every day to conceptual
previews, I needed to determine if it is worth the time and investment. The following
four questions served as the pre-/post-assessment. Not only was I interested in the
correct solutions, I also paid particular attention to the type of strategies students used to
solve the problems. Additionally, I kept a journal with notes on the activities that took
24
What does 32 x 21 mean? Use pictures, words, or numbers.
What strategy or strategies can you use to solve 32 x 21?
Estimate the answer to 32 x 21 by rounding both numbers.
Write a list of easier problems that will help you solve 32 x 21.
place each day and journaled reflective comments. I then analyzed for common themes
to aid in interpreting student activity during the five-minute mini-lessons and my
thoughts as the activities transpired.
Analysis of Implementation and the Results
After the third day of the mini-lessons, I was ready to quit using the conceptual
previews. Fortunately for me and my students, we pressed forward. It was a small
commitment to make each day so it was worth the time to try something new that
research has shown to improve students’ conceptual understanding.
Before the fourth day, the students were decomposing numbers, but not making
the connection of how it made solving the problems easier. Some were not selecting
―nice‖ numbers based on place value or groups of two or five. Along with the previews, I
used the opportunity to highlight various strategies students used to solve each problem,
yet most students seemed disinterested in learning from others. Day four they continued
to use graph paper as a way to represent 10 x 16. I chose this model so they could see the
patterns of ten. After the fourth day I felt more confidence in the previews when many of
the students recognized the problem 10 x 16 as 10 x 8 + 10 x 8. We had made progress
with this example! Day five and six students solved double digit multiplication problems
25
and they could use the array or a visual representation of the array, and I gradually
weaned them away from the graph paper during the previews.
By day seven, I saw different chunking methods generated by the students and
they were listening to one another and creatively trying to think outside of the box. For
example, there was the problem 25 x 22. Some students used (10 x 22) + (10 x 22) + (5 x
22). One student recognized 5 x 22 was half of 10 x 22. Others used (25 x 20) + (25 x 2).
Some used 25 x 10 + 25 x 10 + 25 x 2. It was exciting to listen to students seek out
numerous number sentences.
On day seven students struggled with interpreting the word problem; however,
they did not struggle with finding a strategy and solution for the double-digit
multiplication problem. Keep in mind, the students have not learned the standard
algorithm for double-digit multiplication, but they have a strong visual representation for
understanding how to represent its meaning. Days eight through ten proved most
difficult when students were asked to work with larger non-five numbers that did not
easily separate into halves either.
Results of the Pre-Assessment
For the results of the first question (see table below; an asterisk denotes a correct
solution or solution method), many of the students recognized the problem as repeated
addition. Others understood the question but responded with 32 groups of 21, or the
number 32 twenty-one times and added the numbers. Some attempted a multiplication
problem, but did not arrive at the correct solution. I expected students to think of the
problem as repeated addition, a typical strategy for fourth-graders; yet one that will prove
to be inefficient in the later years.
26
# Students
*2
9
*12
1
4
7
2
3
Response
Multiplication with correct answer
Multiplication with incorrect answer
Repeated addition
Skip counting
Used an array
Wrote 32 groups of 21 with no attempt to solve
Addition
No answer
For the second question, most of the students attempted to separate the number
(30 x 10) + (30 x 10), but could not complete the steps. The most common error came
when students tried to take 30 x 20 and 2 x 1. Some tried repeated addition and arrived at
the correct answer.
# Students
*3
12
*9
*2
4
1
1
9
Response
Distributive property with correct
answer with incorrect
Distributive property
answer
Repeated
addition
Array
Some use of 32 and 21
Division
Skip counting
No answer
For the third problem that asks to estimate the answer, some students correctly
choose 30 x 20, but could not complete the multiplication and answered 30 x 20 = 60.
# Students
*7
6
*7
9
11
Response
Estimation correct
Estimation incorrect
Repeated addition
Repeated addition (60)
No answer
27
Last, to address the issue of cluster problems, the fourth question asked for a list
of easier problems. A cluster problem is a set of problems partitioned out of the original
equation that make the equation easier to solve. This is a move beyond the visual model
and includes various number sentences of the student’s choosing. Some students thought
of 30 times 20, 20 times 30, 1 times 2, and 2 times 1, but did not know what to do with
the equations. Other students separated both numbers; for example, 7 x 16 and 3 x 16 or
2 x 32 and 1 x 32. Most of them left the answer blank.
# Students
*3
15
1
6
15
Response
List of cluster problems
Cluster problems no
Wrong cluster problems
Addition
answer
No answer
The information from the pre-assessment offered insight into places where my
students might struggle in the regular unit of instruction, so the two weeks of five-minute
conceptual previews should help fill the gaps and hopefully start everyone on the same
page feeling success going into the new unit on multiplication and division.
Results of the Post-Assessment
With the first question, I was pleased to discover several students used the array
as a model for 32 x 21 (see table below for results of their strategy use; again, an asterisk
denoted a correct solution and/or strategy). Others tried the distributive property. I
believe my wording of the question, ―what does it mean‖ should read, ―use a model or
picture to show what 32 x 21 means‖ so students have a clear understanding that I am
looking for a diagram; although the second question leads to this idea, as well.
28
# Students
*10
*11
2
10
4
1
2
Strategy
2 strategies: The visual model and distributive
2 strategies:
distributive
property
1 strategy: distributive
1 strategy: distributive incorrect answer
1 strategy: visual model incorrect answer
Skip counting incorrect
Blank
As with the curriculum I implement and the beliefs I hold towards teaching
mathematics, I ask students to provide two strategies to show our thinking when solving
most problems. I believe it helps in developing flexibility in their thinking and will lead
to making stronger connections across the mathematics discipline and lead to improved
use of representations for conceptual development. In the second question on the post-
# Students
*12
*10
*9
5
3
1
Strategy
Wrote 32 groups of 21
Array
Distributive property correct answer
Distributive property with errors
Skip counting
―ways to separate numbers to make them easier to solve
assessment, many students solved the problem with two different strategies. Seven
students improved in this two-week period.
For the third question, the array model greatly improved students’ estimation
skills. More than 75% of the students correctly estimated the product of 32 x 21
compared to 35% on the pre-assessment. In the pre-assessment, most students thought 30
x 20 = 60, but the array model assisted them in thinking about the size of these quantities.
Last, in stark contrast to the pre-assessment, 77.5% of the students used the
correct number sentence and solved it correctly, as well. I was not expecting such growth
29
from five-minute mini-lessons in just two weeks! I did not catch the three students who
wrote cluster problems without answers, but if I had, I think they would have been able to
solve them, too.
# Students
*25
*3
8
1
2
1
Strategy
Cluster problems correct
Cluster problems w/o answers
Cluster problems wrong
Algorithm
wrong answer
answers
Another strategy correct answer
Blank
Conclusion and Implications for Future Work
I felt the conceptual previews and the discussions that took place helped students
make the connections between the array as a model for multiplication and the use of the
distributive property, or breaking apart the numbers, to make the problems easier to solve
an effective approach for accessing students prior knowledge before the unit of
instruction begins. My students hit the ground running and felt confident from the
beginning, which often is not the case for a classroom with almost three-fourths English
language learners. Compared to years past, the use of just five-minute mini-lessons
enabled students to grasp the idea of the array, use it, and progress to the abstract level
with the distributive property in just ten days. More students were able to construct
various cluster problems or different ways of separating the number sentence because of
the conceptual previews than my experience in years past. They were able to develop
alternatives and were not stuck with just one way of thinking. The visual model served
as a foundation during the previews and subsequent lessons. This allowed for the
exploration of other forms of decomposing the numbers; for example, halves, tens, and
30
fives. Some students who did not have a working strategy on the pre-assessment had two
strategies by the end.
I think the unit was clearer to me once I had my specific thinking strategy
planned. I continued to build and refer to the model with a sense of purpose. Having the
students post and discuss their strategies assisted the students in visualizing other ways to
represent numbers and they were able to learn from each other.
Often times when a new unit of instruction begins, it takes several days for
students to get into the swing of what they are learning. Mathematics textbooks either do
not help in developing prior knowledge when chapters skip around from algebra to data
or geometry, or they overload the text with too much review making it impossible to
know where to begin. Conceptual previews offer an approach for teachers to assess what
students know and understand about the upcoming unit and plan activities accordingly.
In reflecting upon my approach in using conceptual previews for multiplication and
division before the unit ―Packages and Groups,‖ I would give the pre-assessment first to
understand what my students understand and then tailor the ten-day previews to fit their
needs. The pre-assessment should be given first with the development of the conceptual
previews after results of the pre-assessment are understood.
It is difficult to know if the mathematics students were learning during the current
unit of instruction while the conceptual previews were taking place suffered in anyway.
End of unit assessments did not indicate any unusual affects as a result of redirecting five
minutes of math class to other concepts and skills. It is also important to note whether
the upcoming unit took less time to teach and whether scores on end of unit assessments
were vastly different than years prior. Unfortunately, data that could address this was not
31
collected. Therefore, one implication of this research would be for future researchers to
administer an end-of-the-unit assessment after the conceptual previews but before the
upcoming unit to assess pre-unit conceptual understanding. This information may shed
light on the opportunity to lessen the length of time needed to teach the unit as the result
of the conceptual previews, offering additional time in the year to incorporate topics often
eliminated.
Creating conceptual previews forced me to think critically about the mathematical
concepts and representations that are important in teaching for understanding. The
previews promoted class discussion and an opportunity to discuss efficient ways of
multiplying. With many strategies, the students were exposed to multiple levels of
thinking thus encouraging flexibility in their thinking. Multiplication and multiplication
of two-digit numbers are big ideas for fourth grade and require a great deal of time
developing the concepts and skills. This is one reason why I chose this unit assuming the
conceptual previews might shorten the time spent or strengthen the conceptual
understanding of multiplication.
Michele Carnahan teaches sixth grade in Bigfork, MN. Her interests include mathematics
education for struggling learners. [email protected]
Dr. Bridgette Stevens is a Professor of Mathematics at the University of Northern Iowa
in Cedar Falls, Iowa. [email protected]
References
Fosnot, CT & Dolk, M (2002). Young Mathematicians at Work: Constructing Fractions,
Decimals, and Percents. Portsmouth, NH: Heinemann.
National Council of Teachers of Mathematics (2000). Principles and Standards for School
Mathematics. Reston, VA: NCTM.
Rathmell, R (2005, Spring). Successfully Implementing a Problem Solving Approach To
Teaching Mathematics Iowa Council of Teachers of Mathematics 32. Des Moines, IA: ICTM
32
Japanese Lesson Study: A Brief Explanation and Reflection
Brooke Fischels
Professional development has revolutionized the way I teach in my classroom. It
has changed the expectations I have for my students, the depth of mathematical content I
teach, and increased the connections and relevance I provide for my students. Initially,
this transformation began five years ago with Iowa’s secondary mathematics initiative,
“Every Student Counts,” where I learned how to create or revise problem-based
instructional tasks that include an enticing launch, an exploratory section for rich
discovery, and a summary to help students synthesize the content we had learned. Every
lesson began to have more meaning and I felt that I had more tools at my disposal to help
get us from Point A to Point B in a 45-minute class period. Although this professional
development significantly improved my instruction, my involvement with a Japanese
Lesson Study project really took it to another level. I will outline the process, but it is
important to note that everyone can implement these strategies even if you are not part of
a formal lesson study project. It is the process of working together and looking at lessons
with a critical lens that help facilitate improved practice.
Background
Two years ago, I joined Important Mathematics and Powerful Pedagogy (IMAPP),
a Japanese Lesson Study project made possible by the Mathematics and Science
Partnership, the Iowa Department of Education, and the Iowa Board of Regents. This
particular project focuses on high school mathematics instruction and included
participating teachers from a variety of school districts in Southeast Iowa. Teachers
involved in the project must work in teams comprised of teachers who work in the same
33
building. Each summer the teams attend a 1-week institute on important high school
mathematics content. The sessions are rigorous and our instructors push us to dig deeper
to strengthen our individual mathematics knowledge. Although each summer is different,
the institute focuses on the four disciplines of mathematics found in the Iowa Core
Curriculum: Algebra, Geometry, Statistics and Probability, and Quantitative Literacy.
We are fortunate to have two outstanding instructors, Cos Fi, a mathematics education
professor at the University of Iowa and Eric Hart, a mathematics professor at Maharishi
University of Management as well as a contributor to the Iowa Core Curriculum.
A second 1-week institute in the summer focuses our attention to powerful
pedagogies. Many of these signature pedagogies are found throughout modern
mathematical literature, including NCTM’s Focus in High School Mathematics:
Reasoning and Sense Making and Schoen & Charles’ Teaching Through Problem
Solving. During this institute, teams also develop specific lessons that they will teach
during the school year that will infuse powerful pedagogies, and reviewed with the lesson
study approach.
What is lesson study?
In a nutshell, each team chooses a topic for a lesson. Before the lesson is created,
focus questions are formed to provide direction. Our lesson study team learned early on
that the focus questions are extremely important. They should not be too broad and
should hone in on specific objectives. I think of them as essential questions that the
students should be able to answer after the lesson is taught. The focus question isn’t the
solution to the problem posed in the lesson, but rather a question that would assess
whether or not the content had been learned. Good focus questions can help get a team
34
back on track if the lesson begins to wander away from the objectives or becomes too
long to teach in one or two class periods. If the lesson needs to be slimmed down, our
team was able to use the focus questions to detect which parts of the lesson were
unnecessary or could be taught at a different time.
After the focus questions are created, the team works together to develop the
lesson. A lot of rich discussion takes place to determine how the lesson should unfold
and how students would perceive the problem. It is important not to get too attached to a
specific part of the lesson. During the revision of the lesson, the team will look critically
at each part. If something is not helping the lesson progress, makes the lesson too long,
or does not address the focus question, it has to be removed. Since you’re working as a
team, go in with an open-mind and be willing to compromise parts of the lesson to
achieve the objectives.
Once the lesson is designed, it is taught by a teacher in the team to a set of
students. All other teachers in the team or lesson study cadre observe the lesson. The
cadre could include administrators, math consultants, or project leaders, as well as other
math teachers. Observers keep track of the timing of the lesson, students’ reasoning, and
the effectiveness of the lesson. None of the observers are allowed to talk or help students
during the lesson; they merely absorb the information for the team discussion after the
lesson is taught. Observers should not be overly critical of the lesson; instead they should
focus their attention to what the students are able to do well and what they struggled with
during the lesson. It is important for observers to write down student comments and
interactions. The lesson is recorded so that it can be reviewed.
35
After the lesson, the team debriefs together. Debriefing sessions can take 1 – 2
hours, and they are also recorded. Each session begins with each member giving their
initial impressions of the lesson. Once everyone has had the chance to speak, the
debriefing discussion will focus on each part of the lesson, what students said, what went
well, and what were misconceptions the students had. Notes are taken as ideas formulate
on how to improve the lesson to overcome misconceptions and increase the effectiveness
of the lesson.
After the debriefing session, teachers in the team decide what to remove from the
lesson, what to change, and what to add to it. The new lesson might be similar or entirely
different than the original lesson. The revised lesson is taught by a teacher on the team
and observed by the group so that the process can repeat. Observe, debrief, revise. The
lesson study is not complete until an entire cycle is finished, which means each lesson has
at least two revisions and is taught twice. Our team found the revised lessons went much
more smoothly, and students were able to perform at even higher levels.
Frequency
Obviously, the process is intense and necessitates a lot of group trust.
Administrative support is also needed so that teams can observe each other’s classrooms.
So, how often could a collegial team be able to complete this process? The IMAPP
project asks teachers to complete the lesson study cycle twice during the school year.
The process is intense, but I can personally attest that even when I was working on my
own individual lessons they improved as well. My students were more prepared and I
had honed my critical skills enough to improve my own lessons without the full support
36
of the team. The lessons designed and revised by our team are some of the richest
lessons I have ever had the privilege to teach, and I look forward to another successful
year with my colleagues.
I strongly encourage you to seek out lesson study opportunities or create your
own teams at your own schools. There is a growing body of literature on lesson study in
mathematics education journals as well as online, so guidance is readily available.
Brook Fischels teaches mathematics at Ottumwa High School in Ottumwa, Iowa.
Iowa Council of Teachers of Mathematics offers information about the organization as
well as links to resources for teachers. Information about grant opportunities can also be
found here. www.iowamath.org
National Council of Teachers of Mathematics offers enormous resources for teachers to
use. From classroom lessons to Washington legislation, it is all at www.nctm.org
Sodoku-sodoku.com This site has Sudoku puzzles online (with printable versions) plus
tutorials on how to solve the puzzles. www.sodoku-sodoku.com
Gamequarium This site has many math games, puzzles, and riddles. www.gamequarium.org
Learn With Math Games We all learn better when we enjoy what we are learning. Math
games help put the fun into math while building important math skills at the same time.
www.learn-with-math-games.com
37
Health Literacy in the Mathematics Classroom: The Iowa Core Curriculum As An
Opportunity to Deepen Students’ Understanding of Mathematics
Elana Joram, Ph. D, Susan Roberts-Dobie, Ph. D and Nadene Davidson, Ed. D
By 2012, all high schools in Iowa will be required to incorporate the new Iowa Core
Curriculum, followed by elementary and middle schools in 2014 (Iowa Department of
Education, 2009). The Iowa Core Curriculum addresses the question: "How is Iowa's
educational system preparing our youth for successful lives in the 21st-century global
environment?‖ (Davidson, 2009). It consists of core content standards, and identifies
essential concepts and skills for content areas. The Iowa Core Curriculum also includes
the ―21st Century Skills‖ of ―health, financial, technology, and civic literacy, and
employability skills. These skills are to be infused into existing subject matter rather than
taught as separate stand-alone subjects.
Clearly, incorporating these newly identified essential concepts and skills is a daunting
task for today’s teachers in Iowa, in part, because they are not yet accompanied by
suggestions for specific ways they can be implemented. In this paper, we discuss
potential linkages between mathematics and one of the 21st Century Skills, health
literacy, and provide suggestions for how Iowa’s mathematics teachers can incorporate
this aspect of the Iowa Core Curriculum into their lessons. Our discussion and
suggestions are intended to serve as an example; similar points could be made about the
relationship of mathematics to financial literacy, another 21st century skill. In this way,
we hope to move the discussion about the Iowa Core Curriculum forward, from
identifying essential concepts and skill sets as ―big ideas,‖ to thinking about specific
38
issues of classroom implementation. We intend to show how the Iowa Core Curriculum
can be viewed as an opportunity to take mathematics into an out-of-school context that
will ultimately deepen students’ understanding of mathematical concepts.
The 21st Century Skills
Health literacy is one of the 21st Century Skills that form part of the Iowa Core
Curriculum. The Iowa Department of Education has identified expectations for the 21st
Century Skills, specific to each skill and grade band. For example, the following
expectations are listed for the grade 3-5 grade band for Health Literacy (Iowa Department
of Education, 2009):
obtain, interpret, understand and use basic health concepts to enhance personal,
family, and community health;
utilize interactive literacy and social skills to establish personal family, and
community health goals;
demonstrate critical literacy/thinking skills related to personal, family, and
community wellness;
recognize that media and other influences affect personal, family and community
health; and
demonstrate behaviors that foster healthy, active lifestyles for individuals and
the benefit of society.
To meet these expectations, teachers would teach health literacy in the context, for
example, of existing health, mathematics, science, and language arts classes. In this
paper, we provide ideas for how health literacy can be integrated into the K-12
mathematics classroom.
39
Definitions of Health Literacy and Numeracy
The currently accepted definition of health literacy from Healthy People 2010 is "the
degree to which individuals have the capacity to obtain, process, and understand basic
health information and services needed to make appropriate health decisions" (US
Department of Health and Human Services, 2000). A branch of health literacy that deals
with numerical information in health and medical contexts has been referred to as health
numeracy, and this is where we see many fruitful possibilities for the integration of health
literacy with the K-12 mathematics classroom.
Peters et al. (2006) define health numeracy as ―the ability to process basic probability and
numerical concepts‖ (p. 407). Golbeck and colleagues (Golbeck, Ahlers-Schmidt,
Paschal, & Dismuke, 2005) advance a more specific definition of health numeracy
adapted from the definition of health literacy given above: ―Health numeracy is the
degree to which individuals have the capacity to access, process, interpret, communicate,
and act on numerical, quantitative, graphical, biostatistical, and probabilistic health
information needed to make effective health decisions‖ (p. 375). This can range from
relatively simple tasks such as comprehending quantitative concepts imbedded in texts,
such as nutritional information, to more challenging situations such as making medical
decisions based on risk and probability information. It is clear, from these definitions,
that knowledge of mathematical concepts is essential for accomplishing just the first Iowa
Core Curriculum expectation listed above for health literacy which is to ―obtain,
interpret, understand and use basic health concepts to enhance personal, family, and
community health.‖
40
How is Health Literacy/Numeracy Related to Mathematics?
We suggest that the mathematical concepts that are essential for health literacy/numeracy
are shared with the typical K-12 mathematics curriculum (National Council of Teachers
of Mathematics, 2000); in other words, there are no new mathematical concepts that need
to be taught in order to be health literate or numerate. What is unique to health literacy
and numeracy, however, is that individuals must be able to effectively apply these
mathematical concepts to health and medical contexts. For over 100 years, educational
psychologists have consistently found that transfer, the application of knowledge gained
in one context to a different context, is often very difficult, and is certainly not automatic
nor guaranteed (National Research Council, 2000). We can assume, therefore, that
simply learning mathematical concepts in the context of mathematics classes may not be
sufficient to ensure that students are able to successfully transfer these concepts to
relevant health and medical contexts.
Recent research has revealed that many adults have difficulty with basic health
literacy/numeracy tasks such as accurately estimating portion sizes (e.g., Huizinga et al.,
2009), correctly reading and interpreting prescription labels (Davis et al., 2006;
Schillinger, 2006; Wolf et al., 2007), and using probabilistic information to make medical
decisions (Peters et al., 2006). Not surprisingly, researchers have found that individuals
with high levels of health literacy/numeracy perform much better on these tasks than
those with low health literacy/numeracy levels (Nelson, Reyna, Fagerlin, Lipkus, &
Peters, 2008). Therefore, it is crucial to teach students to interpret and use health
information so that they can maintain a healthy lifestyle, and later on be able to prevent
and manage disease. Health and medical contexts that are relevant to K-12 students can
41
also provide interesting and relevant everyday contexts in which students can flex their
mathematical muscles. For example, understanding serving size information on
nutritional labels may involve measurement, computation, and problem solving.
In summary, students who become fluent with the application of mathematical ideas to
health and medical contexts will be better prepared to navigate their own health,
including and the self-care of diseases they may encounter now and as adults, and the
health of their families and communities. In addition, through explorations of
mathematical concepts in health contexts, students may deepen their mathematical
knowledge. Below, we offer some suggestions for ways that teachers can incorporate
health literacy into the mathematics classroom.
Mathematics Instruction and Health Literacy/Numeracy: The Case of Nutrition
In this section, we provide an example of how health literacy/numeracy can be integrated
into mathematics curricula, with respect to major standards and expectations in
mathematics for different grade bands (National Council of Teachers of Mathematics,
2000). We focus on nutrition, which is a very rich context for examining a broad range
of mathematical concepts, from the early elementary grades through high school. A
greater number of suggestions are presented for elementary-level students than middleor secondary-level students because more curricular material has been developed for
these students. We have selected several mathematics standards to focus on, where the
fit between nutrition education and mathematics education seems particularly fruitful.
Standards and expectations (National Council of Teachers of Mathematics, 2000) are
indicated below by bold text.
42
In addition to our own research and experiences in mathematics classrooms (e.g., Joram,
2003), we draw on lessons that have been developed for several programs in which
nutrition education has been integrated with mathematics and science education:
FoodMASTER (Duffrin, Phillips, & Hovland, 2009), the Science of Food and Fitness
(Moreno, Clayton, Cutler, Young, & Tharp, 2006), and The Science of Energy Balance
(National Institutes of Health, 2009a). Although the Science of Food and Fitness and The
Science of Energy Balance were designed to teach science, some of the lessons cover
topics that overlap with the K-12 mathematics curricula, such as measurement, and are
therefore appropriate for the mathematics classroom. Alternatively, elementary-level
teachers could teach an integrated mathematics/science unit that incorporates health
literacy/numeracy, and middle- and secondary-level mathematics teachers may be
interested in partnering with science teachers for the same purpose.
Materials discussed here are available, free of charge, either on the program websites or
by placing a request on the website for teachers’ manuals and student exercises (Duffrin
et al., 2009; Moreno et al., 2006; National Institutes of Health, 2009a, 2009b). In
addition to the programs above, we discuss lessons that have been posted on the National
Council of Teachers of Mathematics Illuminations website (National Council of Teachers
of Mathematics, 2009).
Measurement Standard – Example for Elementary Level Students
From prekindergarten through grade 12, the National Council of Teachers of
Mathematics measurement standard specifies that students should be able to
―understand measurable attributes of objects and the units, systems, and processes
of measurement,‖ and they are also expected to be able to ―apply appropriate
43
techniques, tools, and formulas to determine measurements‖ (National Council of
Teachers of Mathematics, 2000). Nutrition offers an ideal context in which to work on
these expectations because food is often in the form of continuous or mass quantities that
must be measured rather than counted, and because measurement serves an important
real-life purpose in nutritional contexts, such as cooking or controlling portions sizes.
Although most available lessons we reviewed are designed for grades 3 and up, we
suggest that Pre-K-2 students can explore measurement in nutritional contexts by getting
a sense of the relative magnitude of different foods and liquids with respect to their
weight, liquid capacity, and volume. One activity that we have used successfully with
students is to have them place foods of different weights, arranged from lightest to
heaviest, on a long table. This will help them work on the standard ―compare and order
objects according to attributes” (National Council of Teachers of Mathematics, 2000).
Students at this age may have difficulty separating weight from volume, and may assume
that a food object with a larger volume will also weigh more. Having students pick up
objects of increasing weights, that vary in volume, should help them distinguish the two
measurement attributes of weight and volume and lay a foundation for later representing
these attributes numerically.
Both lower and upper elementary students can also pick up each food item and then
estimate and check its weight. In this way, they will gain experience connecting the
perceived weight to its numerical representation, and gain practice in using a scale (using
tools to measure). Our experience shows that students usually enjoy this kind of ―guess
and check‖ activity, especially if presented as a game, where students in groups see who
can come closest to the actual measurement with their estimate. These measurement
44
activities can be completed using both U.S. Customary and metric units, thus giving
students practice in grounding measurement units for both systems in real world
referents, as suggested by the National Council for Teachers of Mathematics (National
Council of Teachers of Mathematics, 2000). This activity should help students meet the
measurement expectation: ―become familiar with standard units in the customary and
metric systems‖ (National Council for Teachers of Mathematics, 2000).
Using benchmarks for measurement estimation, which is part of the measurement
standard, can easily be integrated with nutrition – in fact, health educators often use
benchmarks to represent appropriate portion sizes, such as a deck of cards to represent an
appropriate portion size of meat (Iowa Department of Public Health, n.d.). In Activity 4
of The Science of Food and Fitness (Moreno et al., 2006), students are introduced to
―Quick Hand Measures‖ for common foods, such as the tip of one’s thumb to represent a
teaspoon of butter. Some of these benchmarks are presented along with their standard
measurement while others are not. We suggest that it will be important to include
standard measurements with all benchmarks that are used, so that students learn how to
represent the measurement as well as the appropriate food portion. For example, instead
of simply equating an appropriate serving of meat with a deck of cards, teachers can
identify the portion size as 3 ounces, and in this way, the benchmark (i.e., the deck of
cards) will represent the measurement of 3 ounces, which in turn is the appropriate
portion size. A chart found in The Science of Energy Balance, Lesson 1 ―Burning It Up‖
(National Institutes of Health, 2009a) is helpful in this regard: it lists benchmarks for
portion sizes, accompanied by their measurements.
45
Using benchmarks for measurement estimation promises to enhance students’ meaningful
representation of standard units of measurement as well as help them gain knowledge of
appropriate food portions. This is an excellent way to build connections between
measurement units and their referent quantities. In addition, learning to estimate
appropriate portion sizes addresses the Iowa Core Curriculum expectation that students
should ―obtain, interpret, understand and use basic health concepts to enhance personal,
family, and community health‖ (Iowa Department of Education, 2009). Although we
have presented these activities in the context of the elementary classroom, they would be
appropriate for older students as well.
FoodMASTER includes a set of lessons entitled ―Measuring Up,‖ that consists of baking
activities through which students explore measuring dry and wet ingredients, using both
customary U.S. and metric units for making chocolate chip oatmeal cookies. Although
the second lesson culminates in actually baking the cookies, students can measure and
mix ingredients for a no-bake recipe, if teachers do not have access to a stove. In
addition to learning about standard measurement units, students can learn about
appropriate units in this context – for example, teachers can ask students what unit
should be used to measure flour for a given recipe: teaspoons or cups.
Numbers and Operations Standard: Example for Middle Level Students
Percents, fractions, and decimals can be introduced in the context of nutrition, for
example, by examining the percentage of the US Recommended Daily Value (DV) that a
serving of a given food provides. Students can bring in packaged food containing food
labels for these activities. Interesting problems can be posed to students, for example,
asking them what it means to read on a nutrition label that the DV of Vitamin C provided
46
by an orange is 110% (“develop meaning for percents greater than 100 and less than
1”).
The National Institutes of Health website has an activity called Portion Distortion, in
which pictures of portion sizes 20 years ago and now are shown, and the viewer has to
identify what the difference in calories is between the two (Department of Health and
Human Services: National Institutes of Health, 2009). For example, the website informs
us that a bagel 20 years ago that measured three inches in diameter had 140 calories, and
then asks us to choose either 350, 250, or 150 calories for today’s much larger bagel
which is shown. After selecting the correct answer (the website immediately provides
feedback about which response is correct), students could also compute the percent
increase in size of the bagel from 20 years ago, as well as the mean percent increase in
food portion sizes for all the foods on the quiz.
Planning a meal for a larger or smaller number of people than for a typical recipe would
be an excellent way to introduce middle school students to the measurement standard
―solve problems involving scale factors using ratios and proportions.‖ Students could
be asked to half, double, or increase a recipe by 2.5 times to prepare needed quantities of
food. This is an authentic activity in our experience, because one often has to adjust the
quantities of recipes for different numbers of people.
Problem Solving: Example for Secondary Level Students
The National Council of Teachers of Mathematics (2000) has identified the following
standards for problem solving for secondary students: “solve problems that arise in
mathematics and in other contexts; apply and adapt a variety of appropriate
strategies to solve problems.” Food labels provide an excellent authentic context for
47
problem solving and computation for secondary-level students. For example, a food label
may state that a single serving of chips has 120 calories and that there are 2.5 servings per
bag. Students can figure out how many calories they would consume if they ate the
entire bag of chips, or what quantity of chips they should eat if they only want to
consume 100 calories. The article entitled ―The Newest Vital Sign‖ includes a food label
with accompanying questions – although designed as an assessment of health literacy, the
questions pose interesting problems that students could solve, requiring them to calculate
calories, number of grams of fat, etc. based on a food label describing the nutritional
components of ice cream (Weiss et al., 2005). In addition to providing students with
problem solving experiences, becoming aware of food labels and how to perform such
computations mentally can help them estimate and control the amount of food they eat.
Websites such as the US Department of Agriculture’s Nutrient Data Laboratory (U.S.
Department of Agriculture, 2009b) provide a wealth of information from which problem
solving activities can be designed. For example, students could keep a food diary for one
week, and then compute the mean number of calories, fat, sodium, etc. they consume
each day, comparing these to the recommended amounts (Iowa Department of Public
Health, n.d.; U.S. Department of Health and Human Services: U.S. Department of
Agriculture, 2005). Activity Seven, Nutritional Challenges, of the Science of Food and
Fitness (Moreno et al., 2006) includes a list of dietary requirements for individuals who
have different dietary needs and restrictions, for example, a pregnant woman, someone
who is lactose intolerant, or a person who has Type II Diabetes, and students are asked to
construct a one-day menu for these individuals. The Food Guide Pyramid website (U.S.
Department of Agriculture, 2009a) has nutritional information available for different
48
ethnicities, so students can also work on developing healthy menus for people who eat
foods common to a certain ethnicity, thereby integrating an awareness of ethnic diversity
with mathematics and nutrition education.
Similarly, the Science of Energy Balance (National Institutes of Health, 2009a, 2009b)
includes lessons entitled Burning it Up and A Delicate Balance, in which students are
asked to keep food and activity diaries and enter their data onto a website. They can then
examine the data for patterns and make predictions based on these patterns. In the webbased student supplements for Burning it Up students are introduced to profiles of
teenagers and their ―energy in/energy out‖ patterns, and are asked to make predictions
about their weight gain and loss over time.
Activities such as those described above will give secondary-level students opportunities
to ―formulate and refine problems because problems that occur in real settings do not
often arrive neatly packaged‖ (National Council of Teachers of Mathematics, 2000, p.
334). As recommended by the National Council of Teachers of Teachers of Mathematics
(2000), teachers can present students with the goal to be achieved, for example, to come
up with a menu that is appropriate for a person with specific dietary restrictions but that
meets the Food Guide Pyramid requirements, allowing students to specify the
information and the source of that information, that is relevant for solving the problem.
Additional Resources
The National Council of Teachers of Mathematics’ website, Illuminations (National
Council of Teachers of Mathematics, 2009) has a number of lessons available on
nutrition and mathematics, for grades K through 8. Clicking on the ―Lessons‖ tab, and
searching for the word ―food‖ reveals 19 lessons that relate mathematics and nutrition.
49
However, for the purposes of promoting health literacy in the mathematics classroom, it
is important to make sure that lesson objectives include both health and mathematics.
For example, a lesson that uses M & Ms to teach counting to young students may offer
benefits in terms of mathematics concepts learned, but not in terms of nutritional
information.
An example of a lesson that promotes both mathematics and nutritional concepts is
―What is the Best Chip?: Conducting a Sales and Marketing Investigation,‖ geared for
Grades 3-5. Students choose quantitative dimensions such as amount of fat, sodium, and
calories, and then compare different brands of chips on these dimensions. Students
discuss relative values on these dimensions in terms of their nutritional properties, and
compare the nutritional value of the chips to other foods, with reference to information on
websites that describe nutritional information such as the food guide pyramid (U.S.
Department of Agriculture, 2009b, 2009c). Explorations like this one promise to enhance
students’ understanding of both mathematics and nutrition.
Conclusions
Although we anticipate that many teachers may first feel overwhelmed when hearing that
they now must incorporate health literacy into their curriculum, we have tried to show
how teachers can embrace health an interesting and relevant context in which students
can examine a broad range of mathematical concepts. We have discussed one rich
context in which students can explore the intersection of health and mathematics, and we
have provided examples of activities for different grade bands that address several key
mathematics standards. Many more mathematics standards could be addressed for the
purpose of integrating the 21st Century Skill of health literacy into the mathematics
50
classroom, such as data analysis and probability, connections, etc. There are also many
contexts in addition to nutrition in which mathematics and health intersect include, for
example, examining probability and statistics through education on risk behaviors such as
smoking, which would be appropriate for middle and high school students.
In addition to those activities described above, there are many others that teachers can
make use of in their classrooms that have already been developed, although there are
relatively more activities available at the elementary level than for middle or secondary
level students. As we suggested above, teachers can create activities themselves, making
use of information such as that provided in the Nutrient Data Laboratory (U.S.
Department of Agriculture, 2009b), or that students find themselves (on food labels, the
internet, etc.) and bring to their classroom.
Incorporating health literacy into the mathematics classroom allows teachers to go
beyond simply teaching mathematics concepts plus health concepts. Because health
represents an applied, personally relevant context, integrating mathematics instruction
with health literacy has the potential to increase students’ number sense, or in the
example of measurement applied to nutrition given above, their measurement sense
(Joram, 2003). For example, teachers can discuss the different needs in precision when
measuring certain ingredients in recipes (e.g., ―a pinch of salt‖) in contrast to measuring
medicine, which requires a high level of precision.
Higher levels of health literacy and numeracy can also have direct and immediate
benefits for students’ current level of health and well being, in addition to enhancing their
mathematical understanding. For example, a recent assessment conducted in the spring of
2007 revealed that a mean of 37.2% of Iowa’s 3rd through 5th graders were overweight or
51
obese, a statistic that has doubled for girls and tripled for boys just since Fall 2005 (Iowa
Department of Health, 2007). Teaching children to more accurately estimate appropriate
food portion sizes, as described above, may lead to a reduction of childhood overweight
and obesity, leading to a healthier life today and in the future.
Seeing a personally relevant, real world application of mathematics to health contexts
may increase students’ motivation to learn mathematics. In addition, working on
mathematical problems in real world contexts can make the mathematics more
meaningful for students. This should have the effect of increasing their understanding of
mathematical concepts, making it more likely that they will transfer these concepts to
other situations. The ideal outcome of incorporating health literacy into the K-12
mathematics curriculum is that students’ knowledge of mathematics will be deepened and
in addition, they will be much better prepared to navigate everyday health and medical
situations in the 21st century.
Dr. Elana Joram is a Professor in the Department of Educational Psychology & Foundations at the
University of Northern Iowa in Cedar Falls, Iowa. [email protected]
Dr. Susan Roberts-Dobie is Assistant Professor for the School of Health, Physical Education and
Leisure Services at the University of Northern Iowa in Cedar Falls, Iowa [email protected]
Dr. Nadene Davidson is Interim Department Head, Department of Teaching & Assistant Professor
Office of Student Field Experiences at the University of Northern Iowa in Cedar Falls, Iowa.
[email protected]
52
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Peters, E., Vastfjall, D., Slovic, P., Mertz, C. K., Mazzocco, K., & Dickert, S. (2006). Numeracy and decision
making. Psychological Science, 17, 407-413.
Schillinger, D. (2006). Misunderstanding prescription labels: The genie is out of the bottle. Annals of Internal
Medicine, 145, 926-928.
U.S. Department of Agriculture. (2009a). MyPyramid.gov: Steps to a healthier you. Retrieved June 3, 2009,
from http://www.mypyramid.gov/
U.S. Department of Agriculture. (2009b). Nutrient data laboratory. Retrieved May 4, 2009, from
http://www.nal.usda.gov/fnic/foodcomp/search/
U.S. Department of Agriculture. (2009c). Nutrition.gov: Smart nutrition starts here. Retrieved May 5, 2009, from
http://www.nutrition.gov/
U.S. Department of Health and Human Services: U.S. Department of Agriculture. (2005). Dietary guidelines for
Americans 2005. Retrieved June 2, 2009, from
http://www.cnpp.usda.gov/Publications/DietaryGuidelines/2005/2005DGPolicyDocument.pdf
US Department of Health and Human Services. (2000). Healthy people 2010: Understanding and improving
health. Washington, D.C.: US Government Printing Office.
Weiss, B. D., Mays, M. Z., Martz, W., Castro, K. M., DeWalt, D. A., Pignone, M. P., et al. (2005). Quick
assessment of literacy in primary care: the newest vital sign [Electronic Version]. Annals of Family
Medicine, 3, 514-522. Retrieved Nov-Dec from
http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&dopt=Citation&list_uids=1
6338915
Wolf, M. S., Davis, T. C., Shrank, W., Rapp, D. N., Bass, P. F., Connor, U. M., et al. (2007). To err is human:
patient misinterpretations of prescription drug label instructions. Patient Education and Counseling, 67,
293-300.
Join Iowa Council of Teachers of Mathematics today!
It has never been more important to be a member of the Iowa Council of Teachers of Mathematics
(ICTM). ICTM offers you the chance to be a member of an organization that is working to help improve
mathematics education in Iowa.
Being a member of ICTM offers you the ability to network with other mathematics educators from
across the state and to help contribute to mathematics education. ICTM also holds an annual conference,
centrally located in the state, to give members a chance to communicate with recognized leaders in
mathematics education and to network with peers.
Register at www.iowamath.org
54
UPDATING MATH TEACHING & LEARNING
FOR PRE-SERVICE ELEMENTARY EDUCATION TEACHERS
Jeannette N. Pillsbury
The focus of this article is the reform of an ―elementary math methods‖ course as
influenced by two national studies published in 2008: (1) Final Report of the National
Mathematics Advisory Council and (2) No Common Denominator: The Preparation of
Elementary Teachers in Mathematics by America's Education Schools from the National
Council on Teacher Quality. It begins with a brief overview of the history of the reform
in K-12 math education and follows with highlights from each of the two reports. I then
present a description of my math methods course in the context of the reform, including
the two recent studies.
INTRODUCTION: CONTEXT FOR REFORM
Since the USSR launched Sputnik in 1957, math education in the United States
has been analyzed and tinkered with in the hopes of developing cohorts of
mathematicians and scientists to ensure the leadership of the US in the world of
discoveries and advancements. During the past 50 years, the focus has been on the
curriculum itself with limited attention to the preparation of K-12 math teachers. Until
recently, that attention has centered on secondary math education. In March 2008, the
National Mathematics Advisory Panel published its Final Report that includes
recommendations for elementary teacher education, and in June 2008, the National
Council on Teacher Quality completed its study, No Common Denominator: The
Preparation of Elementary Teachers in Mathematics by America’s Education Schools.
55
As a K-8 administrator, for several years I supported math teachers in their
classrooms. In 1988, as a director of instruction, I had opportunity to enlist the high
school math department chair to teach a series of in-service classes for the K-8 math
teachers in which she related the learning in Algebra I to the math in the K-8 curriculum.
Both she and the K-8 teachers were reminded of the substantial foundation for Algebra I
that is laid in elementary math. Unfortunately, I cannot report significant changes in the
pedagogy of the K-8 teachers.
In 1995, as a teacher educator in a liberal arts college, I taught a ―math methods‖
course to pre-service elementary teachers. I used the original NCTM standards,
Curriculum and Evaluation Standards for School Mathematics, and other NCTM
standards-based publications as the foundation for the course. These future teachers let
go of some of their math phobia and discovered that math could be ―fun.‖
A few years later, when I returned to the principalship, I used NCTM materials
with my faculty, also. From 2001-2004 I taught the K-5 teachers in my school an
advanced ―math methods‖ course using the revised NCTM math standards as described
in Principles and Standards for School Mathematics. Our work together helped to
change the conversation amongst several of the teachers; excitement developed between
teachers as they shared examples of their students’ learning and mathematical thinking.
Gradually teachers became less defensive when they did not understand some of the math
concepts. They began to invite each other ―to play with the math‖ to gain conceptual
understanding.
In 2004, I returned to pre-service teacher education. I had learned a lot about
supporting elementary teachers as they learn the curriculum and pedagogy of standards-
56
based mathematics. I built on that learning in the hopes of developing a ―math methods‖
course that would make it possible for my future elementary teachers to break the cycle
of ―math phobic teachers begetting math phobic students‖ and of ―insecure math teachers
being unable to recognize the excitement and challenge of learning mathematics.‖
In 2008, the National Mathematics Advisory Panel and the National Council on
Teacher Quality (NCTQ) published their reports. Their findings and their
recommendations provided me both affirmation of much of what I had been doing and
research-based suggestions for changes I could make.
2008 NATIONAL MATH EDUCATION STUDIES
The National Mathematics Advisory Panel
President Bush established the Panel via Executive Order 13398. The Secretary of
Education, Margaret Spellings, was assigned oversight of the committee and appointed
its 19 members. The members represented 26 institutions, including several public and
private universities, a public middle school, think tanks, and one liberal arts college. The
president of the National Council of Teachers of Mathematics was a member, as were at
least three university professors well known to math teacher educators.
The Panel worked for 20 months, receiving public testimony and investigating
questions like the seven noted in the introduction to the Final Report.
What is the essential content of school algebra and what do children need to know
before starting to study it?
What is known from research about how children learn mathematics?
What is known about the effectiveness of instructional practices and materials?
How can we best recruit, prepare, and retain effective teachers of mathematics?
How can we make assessments of mathematical knowledge more accurate and
more useful?
57
What do practicing teachers of algebra say about the preparation of students
whom they receive into their classrooms and about other relevant matters?
What are the appropriate standards of evidence for the Panel to use in drawing
conclusions from the research base?
The six elements below represent the essence of the recommendations in the Final
Report.
1. Streamline curriculum in PreK-8
2. Children should have a strong start:
conceptual understanding
procedural fluency & automatic (quick & effortless recall of facts)
recognition that effort, not just inherent talent, counts in math achievement
The Panel cautions that to the degree that calculators impede the development
of automaticity, fluency in computation will be adversely affected…
3. Good teaching includes both ―student-centered‖ and ―teacher-directed‖
instruction
4. Teachers must use formative assessment regularly
5. Explicit instruction: teachers provide clear models for solving a problem type
using an array of examples, that students receive extensive practice in use of
newly learned strategies and skills, that students are provided with opportunities
to think aloud (i.e., talk through the decisions they make and the steps they take),
and that students are provided extensive feedback.
6. Algebra, algebraic thinking as emphasized in the section of the report entitled
“Critical Foundations of Algebra”
For the complete Final Report from the National Mathematics Advisory Panel, please see
http://www.ed.gov/about/bdscomm/list/mathpanel/index.html
National Council on Teacher Quality
The National Council on Teacher Quality (NCTQ) advocates for reforms in a broad
range of teacher policies at [all] levels in order to increase the number of effective
teachers. It is committed to increasing public awareness of the impact on teacher quality
58
by the federal government, states, teacher preparation programs, school districts, and
teachers' unions.
In March 2007, NCTQ began a yearlong analysis and evaluation of the mathematics
preparation of elementary teachers. Eight (8) educators made up the Mathematics
Advisory Group that led the study. This group included math professors, an elementary
school math coach, two (2) central office K-12 administrators, and one (1) president of an
educational support group. Below is the opening paragraph to the Executive Summary of
the final report: No Common Denominator: The Preparation of Elementary Teachers in
Mathematics by America's Education Schools, June 2008.
In this second study of education schools, the National Council on
Teacher Quality (NCTQ) examines the mathematics preparation of
America’s elementary teachers. The impetus for this study is the mediocre
performance of American students in mathematics compared to their
counterparts around the world. Through improving American students’
relative performance depends on a variety of factors, a particularly
critical consideration must be the foundations laid in elementary school
because mathematics relies so heavily on cumulative knowledge. The link
from there to the capability of elementary teachers to provide effective
instruction in mathematics is immediate. Unfortunately, by a variety of
measures, many American elementary teachers are weak in mathematics
and are too often described, both by themselves and those who prepare
them, as “math phobic.”
Below is a summary of the recommendations from the NCTQ report.
Aspiring elementary teachers must begin to acquire a deep conceptual knowledge of
the mathematics that they will one day need to teach, moving well beyond mere
procedural understanding.
1. Elementary teacher candidates should demonstrate a deeper understanding of
mathematics content than is expected of children.
2. Elementary content courses should be taught in close coordination with an
elementary mathematics methods course that emphasizes numbers and operations.
3. Education schools should require coursework that builds towards a deep
conceptual knowledge of the mathematics that elementary teachers will one day
59
need to convey to children, moving well beyond mere procedural
understanding…
4. …Algebra must be given higher priority in elementary content instruction.
A deeper understanding of elementary mathematics, with more attention given to
the foundations of algebra, must be the new “common denominator.”
For the complete report from the National Council on Teacher Quality, No Common
Denominator: The Preparation of Elementary Teachers in Mathematics by America’s
Education Schools, please see http://www.nctq.org/p/
A RESPONSE: ED 325--“ELEMENTARY MATHEMATICS METHODS”
Recent changes in our college’s Education Department provided opportunities to
achieve some of the recommendations of the two 2008 reports.
(1) The math and science methods courses were separated into two separate courses.
(2) The methods courses became a sequence: math and language arts methods
courses are taught first, in the fall semester, and the science and social studies
methods courses are taught in the spring semester.
[For most elementary
teachers, the ―professional semester‖ (student teaching) follows in the fall.]
(3) Instead of including the ―methods practicum‖ as part of the semester in which the
methods courses were taught, the students now spend the January-term full-time
in a K-6 classroom. Included in the expectation for this placement is the
development and teaching of a unit. The final product is a ―teacher work
sample.‖ Several of these students teach a math unit; all the students teach some
math lessons.
(4) The ―math for elementary teachers‖ course became a co-requisite for Ed 325, the
math methods course.
60
Students no longer take ―math for elementary teachers‖ at the start of their
college career, at least a year prior to taking any course in the Education
Department.
Although some students successfully complete ―pre-calculus‖ in high school,
they often choose ―math for elementary teachers‖ to meet the college’s liberal
arts math requirement because they think it will be an easier than a math
course with a higher number.
Now ―math for elementary teachers‖ is required for ALL students preparing to
teach in K-6 because it is understood that this course focuses on the conceptual
understanding math teachers need to have. ―Education schools should require
coursework that builds towards a deep conceptual knowledge of the mathematics
that elementary teachers will one day need to convey to children, moving well
beyond mere procedural understanding.‖ (NCTQ)
The textbook for ―math for elementary teachers‖ is Sybilla Beckmann’s text
recommended in the NCQT study.
During the summer of 2007, the ―math for elementary teachers‖ teacher and I
coordinated the two courses. This fall, 2009, will be the first time all our
students will be taking the two courses as co-requisites. There will be a
growing interdependence of the two courses.
At this point, the ―math methods‖ course cannot meet all the expectations of both reports.
The course focuses on the curriculum content and the pedagogy. Many of the
recommendations from the two studies relate to the mathematics understanding of
current and future teachers, not to the pedagogy, but with the interdependence of
61
the “math methods course and “math for elementary teachers, it is hoped that we
will move closer to meeting those expectations.
Our ―math methods‖ course is only a two-credit course.
Many of our future elementary teachers are not unlike other pre-service teachers; they
are unsure of their math knowledge and understanding. They are math phobic. They
were not taught in standards-based math programs; their K-8 math experience, more
often than not, emphasized procedure (algorithms) and not conceptual understanding.
Too many of these students demonstrate weak learning in the college’s ―math for
elementary teachers‖ course. This year, unlike previous years, I have noticed some
―math methods‖ students are dependent upon the use of a calculator; they do not
remember how to do a division algorithm.
Relating the study findings and recommendation and ED 325, ―elementary math
methods‖
The 2008 studies emphasize what elementary math teachers need to know and what is
important to include in the K-8 curriculum. A methods class includes acquainting the
students with the curriculum the future teachers will teach, the ―what,‖ and then teaches
them ―how‖ to teach it.
―Math methods‖ needs to include as much math as possible.
Too often elementary teachers do not understand the math they teach. They have
some procedural knowledge, but they lack the depth of the conceptual
understanding. (NCTQ) Due the limitations of a four-year undergraduate teacher
education program, almost all of our elementary education students will have just
62
one math course during their college career: a survey course, ―math for
elementary teachers.‖
Our future teachers will need to teach in ways they were not taught.
We expect our future teachers to teach in standards-based math programs. This
makes it necessary for me to teach them the math using a pedagogy that they did
not experience as students. It is a pedagogy that requires much more involvement
of the learner. Students’ mathematical thinking is very important. Math
discourse is at the center of this pedagogy. For college students who are, at the
very least, unsure of their math knowledge and understanding, this pushes them to
take risks they have not taken before. For students who are ―math phobic,‖ at
times this pedagogy feels unmerciful.
It is in this context that ―elementary math methods‖ is taught.
This past semester the two major improvements I made are the following:
1) Presentation of K-8 math curriculum that allowed the students
to identify the relationship and developmental aspects of the learning
described by the NCTM content standards and the curriculum focal points
(inter- and intra-relationships)
to appreciate the relationship of the math they will teach to the math their
students will need to know when their students leave elementary school
NCTM’s Curriculum Focal Points, the National Mathematics Advisory Panel’s
―Critical Foundations of Algebra,‖ and NTQ’s ―Exit With Expertise: Do Ed Schools
Prepare Elementary Teachers to Pass This Test?‖ are resources that significantly
guided the curriculum instruction I included.
63
2) Modeling the pedagogy of standards-based math learning
To be honest, I simply “bit the bullet” and taught math the way I know it needs to
be taught, facilitating student discourse, even though this was a change in the
culture of the students’ prior math learning. The challenge for me was very real
because I had to create a climate in the classroom that allowed students to take
risks. I had to help students move through “giving a wrong answer” in order for
them to realize that “wrong answers” can be supported to get the learner to the
understanding. I had to model good math teaching not just for “modeling sake,”
but in order for my ED 325 students to become comfortable with the math. I
honestly believe that as a result, I improved my own pedagogy.
Following is a list of the learning activities in which these students engaged.
Curriculum
Pre-Test based on the National Mathematics Advisory Panel’s ―Critical
Foundations of Algebra‖
NCTM content area guided reading questions for each content area K-12: number
& operation, algebra, geometry, measurement, data analysis & probability
Curriculum Focal Points/Connections learning sequences
―Doing the Math‖ for each content area—NCTQ test [See the NCTQ site.]
Data analysis activity [an application activity in which students use their own
GPA and PPST data]
―Content Notebook‖
64
Pedagogy
Small group presentation/mini-teaching for each process standard (by grade
cluster): problem solving, reasoning & proof, communication, connections,
representation
Co-teaching standards-based lesson
Follow-up lesson plan
Unit assessment analysis & evaluation [for the unit in which their co-taught lesson
is included]
Curriculum & Pedagogy
Feedback from students for each co-taught lesson
Books for use in a math classroom [connections to content standards]
Curriculum Focal point/standards-based activities
At the conclusion of this past fall semester, I asked the students for feedback.
Did the basic semester format support your learning: introductory
sessions that were teacher-directed; student-centered co-teaching with a
bit of teacher-directed instruction; concluding with primarily teacherdirected instruction? Why or why not?
20—Yes
2—No
What did you find most valuable in the list of learning activities for ED
325? Why?
65
PEDAGOGY
Mini-lesson co-teaching—11
CONTENT
―Books for use in a math
class‖—1
―Standards-based math
activities--1
Curriculum Focal Points--1
Observing peers teach—3
Reflection on one’s own
teaching--1
Writing math lesson plans—1
Listening to students’
mathematical thinking to support
math learning--1
―Doing the math‖—1
Content notebook—5
Other
Group work—2
Autonomy (due dates)—1
Too much emphasis was put on the NCTM standards—2
Liked emphasis on NCTM standards—1
In both sections (38 students in all), it is clear that the students most appreciated
the opportunity to practice teaching. The students still seem to shy away from the content
they will teach. Students affirmed their exposure to the math curriculum content by
citing specific activities they found helpful and by emphasizing they ―learned much in
this course,‖ but they demonstrated little evidence of being able to use that learning in
their own habits as math learners and math users. I remind myself that their learning
curve is great and they are just at the beginning. They still struggle with ―you can’t teach
what you don’t know.‖
I asked the students to respond in writing to three of the seven self-evaluation questions I
posed:
1) According to what you know and understand about what makes a good math
teacher, what are your strengths?
66
2) According to what you know and understand about what makes a good math
teacher, what do you believe will be your greatest challenge as a K-6 math
teacher?
3) What did you learn about being a math teacher that you had not thought about
prior to this semester?
Here are excerpts from some student responses:
As a math teacher, I think my greatest strengths will be that I need to
explain information in multiple ways. I am not very good at
math…Another strength I will have is that I know that not everyone is
good at math, so I need to be patient and offer extra help…As a math
teacher, my greatest challenge will be enjoying the subject and getting my
student excited about math. My challenge will be to not let students know
how much I do not like math and try to make math fun for them...In Math
Methods, I learned that math can be fun and as teachers, we need to make
it enjoyable and understandable for our students…In elementary school,
we are forming the foundation of math our students will use for the rest of
their lives.
…I can relate to those students who may become easily frustrated while
working with the math, and I feel that I will be more able to connect with
them…I learned that I can do it! I thought that I would be horrible at it;
however, after preparing and teaching a lesson I was able to see that it
came very naturally to me and that I should be a little more confident with
it…It’s important that I show my students that they can do it! And that
math can be fun once you figure it out…Also, it is important that I help
my students by providing them with the foundation necessary to succeed
as a math student in later grades.
I believe that my strengths are that I am good with using pictures and
different representations, looking over and talking to students about their
work to figure out where they went wrong or misunderstood, and I think
that I am motivated to help each learner understand math and like math. I
never liked math, and I don’t want any of the children that come through
my door to dislike math and have a bad view on it…I feel that my greatest
challenge as a math teacher will be feeling confident that I can do the math
work and do it correctly, efficiently, and fluently…I had never ever
thought about having to know and understand math beyond the grade I
wanted to teach…
67
These three students represent a significant portion of the class who see themselves as
insecure with the math and who, at a knowledge level, know that they must gain
confidence with the math and demonstrate that confidence and yet, they did not follow
through with the ―doing the math‖ problems. They did not attend extra help sessions
offered for the ―dong the math‖ work. They omitted problems and often did not ―show
their work‖ on the items for which demonstration of understanding was explicit.
It is clear to me that I have just begun on the improvements I desire.
1) Presentation of K-8 math curriculum that allowed the students
to identify the relationship and developmental aspects of the learning
described by the NCTM content standards and the curriculum focal points
(inter- and intra-relationships)
My students demonstrated this knowledge. They can talk about the
significance of the content standards and curriculum focal points relative
to their planning for teaching.
to appreciate the relationship of the math they will teach to the math their
students will need to know when their students leave elementary school
My students can state the importance of this relationship, but they still are
afraid of the math their future students will need to know as they get
started in the grades 7-9 curriculum. Too many of my students still avoid
the math. Too many still forget ―to do the math‖ as they plan their lessons
and assessments.
68
3) Modeling the pedagogy of standards-based math learning
I improved my own pedagogy with respect to the pedagogy I need to
model for my students. Some of the students who are most insecure with
the math found it difficult to understand why I called on students who did
not volunteer responses. Some of my math weak and insecure students
saw the involvement of these students as embarrassing to the students,
although the students gained much satisfaction when, as a part of the
discourse, they achieved understanding that they articulated!
Creating a nation of math achievers: a first step
The course revisions, primarily based on the concerns and recommendations of
the two 2008 math studies, have increased the self-efficacy of future elementary math
teachers. Upon their January-term practicum placement, I heard many comments about
the standards-based math textbook in their school. I heard ―learning from one’s
mistakes,‖ from not having ―done the math.‖ The former ―math methods‖ students talked
about the limitations of ―worksheets‖ and of the importance of students talking about the
math.
There is more work to be done regarding the math comfort level of future
elementary math teachers and their application of their lip service to ―doing the math‖ in
order to teach it, but the students’ feedback, their demonstration of just how great the
learning curve is from ―talking about it‖ to ―doing it,‖ and from their comments
following their January experience in the schools, I have some clues about what is
working. Next fall, the ―elementary mathematics‖ teacher and I will give our students
more math to do. We will continue our efforts to affect an entire generation of
69
elementary math teachers in the hopes of alleviating ―math phobia‖ and replacing it with
the excitement of doing mathematics! All this is an important to reaching the goal of
creating a nation of math achievers.
Dr. Jeannette N. Pillsbury is an assistant professor at Luther College in Decorah, IA. She has
been a K-8 principal, a director of instructor, and a coordinator of special education. She was
very involved with the math education of both K-8 teachers and principals in Boston, MA.
[email protected]
REFERENCES
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics.
Reston, VA: The National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school
mathematics. Reston, VA: The National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics (2006). Curriculum focal points for prekindergarten through
grade 8 mathematics: A Quest for coherence. Reston, VA: National Council of Teachers of
Mathematics, Inc.
National Council on Teacher Quality (2009). ―Exit with expertise: Do ed schools prepare elementary
teachers to pass this test?‖ Retrieved March 6, 2009, from
http://www.nctq.org/p/publications/docs/nctq _ttmath_testandanswerkey
National Council on Teacher Quality (2009). NCTQ Homepage. Retrieved March 4, 2009, from
http://www.nctq.org/p/
National Council on Teacher Quality (2008). No Common Denominator: the preparation of elementary
teachers in mathematics by America’s education schools; Executive Summary. Washington, DC:
National Council on Teacher Quality.
National Mathematics Advisory Panel (2008). Critical Foundations of Algebra. Foundations for success:
the final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department
of Education.
National Mathematics Advisory Panel (2008). Foundations for success: the final report of the National
Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.
70
presents
Another Bite at the Core
Featuring keynote speaker Dr. Deborah Ball
A high-profile figure in education, Dean of the School of Education at the
University of Michigan, a member of the National Mathematics Advisory
Panel, and one of the nation’s top experts on math education.
Friday, February 19, 2010
8:00 – 3:30
Valley Southwoods Freshman High School
West Des Moines, Iowa
Sessions will be available for all
levels of mathematics education. Look for
registration materials and program details
at www.iowamath.org after November 1st, 2009
Contacts:
ConferenceInformation:
Speaker Information:
Dave Blum
Cheryl Ross
[email protected]
[email protected]
Brooke Fischels [email protected]
Exhibitor Information: Travis Nuss
[email protected]
ICTM Office:
Maureen Busta
[email protected]
First time ICTM State Conference attendees should check the ICTM website
to look for free registration and other incentive information.
71
SKITTLES CHOCOLATE MIX COLOR DISTRIBUTION:
A CHI-SQUARE EXPERIENCE
David R. Duncan and Bonnie H. Litwiller
In teaching statistical processes, it is important that there be application to realworld settings and activities. When this is done, students are more likely to see the
meaning of the steps being developed.
One such activity involves using the Chi-Square statistical test and its applications
to counting Skittles Chocolate Mix candies. Many students are aware that these candies
come in five different flavors: Brownie Batter (BB), Vanilla (V), Chocolate Caramel
(CC), S’mores (S), and Chocolate Pudding (CP).
Let us first test the hypothesis that all colors are equally represented in this
product. We will test this distribution hypothesis, called the Null Hypothesis, with four
randomly selected 14-ounce bags of Skittles Chocolate Mix candies.
The following table reports the contents of these bags.
FLAVOR
BB
V
CC
S
CP
TOTALS
Bag 1
89
72
72
64
79
376
NUMBERS
Bag 2
Bag 3
65
83
117
75
72
73
46
74
73
73
373
378
72
Bag 4
65
114
57
41
92
369
Assuming generally equal numbers for the population of candies, color distributions
should be as follows:
FLAVOR
BB
V
CC
S
CP
TOTALS
NUMBERS
Bag 2
74.6
74.6
74.6
74.6
74.6
373
Bag 1
75.2
75.2
75.2
75.2
75.2
376
Bag 3
75.6
75.6
75.6
75.6
75.6
378
Bag 4
73.8
73.8
73.8
73.8
73.8
369
To test the Null Hypothesis, we shall use the Chi-Square statistic. Let us
construct Table 1 with column entries as follows for Bag 1:
O = The observed frequencies, the numbers of each color of Chocolate Mix
candies actually present in our bag.
E = The expected frequencies (if the Null Hypothesis were true).
(O-E)2/E = A measure of the discrepancy between O and E.
Table 1:
FLAVOR
BB
V
CC
S
CP
TOTALS
O
89
72
72
64
79
376
E
75.2
75.2
75.2
75.2
75.2
376
(O-E)2/E
2.53
0.14
0.14
1.67
0.19
4.67
In the last column (a measure of discrepancy), a small number indicates that O
and E are relatively close together, as is the case for V. A larger number indicates that O
and E are relatively far apart, as is the case for BB.
73
The sum of this discrepancy column, 4.67, is called the Computed Chi-Square
Statistic (CCSS). A determination must be made as to whether the CCSS is large
enough to cause us to reject the Null Hypothesis. To make this decision, a ―referee‖ is
needed. This referee is found in the Table Chi-Square Statistic (TCSS).
To read a Chi-Square table, the degrees of freedom must first be determined; that
is, one less than the number of categories (colors). In our case, the degrees of freedom is
5-1 = 4. This means that if the total number of candies were known, and the number in
each of four categories were known, the number in the fifth category could be calculated.
The significance level is the probability of rejecting a Null Hypothesis which is in
fact true. This could occur because the sample is not representative of the population.
From a Chi-Square table, we find:
SIGNIFICANCE LEVEL
10%
5%
1%
TCSS
7.78
9.49
13.28
The decision mechanism for the Null Hypothesis is:

If CCSS > TCSS, then CCSS is large in the “judgment of the referee.” If this is true,
reject the Null Hypothesis.

If CCSS < TCSS, then CCSS is small in the “judgment of the referee.” If this is true,
accept the Null Hypothesis.
Our CCSS of 4.67 is smaller than the TCSS’s for any of the SL’s. In other words, there
is insufficient evidence to reject the Null Hypothesis for all three significance levels.
According to our evidence, the assumption of equal numbers is accepted for Bag 1.
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Repeating this Chi-Square analysis for the same Null Hypothesis for Bags 2, 3,
and 4 yields CCS’s of, respectively, 36.42, 0.93, and 45.84. For Bag 3, we again accept
The Null Hypothesis of equal numbers in the product as a whole. However, for Bags 2
and 4 the CCSS’s are larger than any of the TCSS’s, leading us to reject the Null
Hypothesis and to conclude instead that the varieties are unequally represented in the
product as a whole.
What would happen if the contents of the four bags were combined? The
following chart results:
FLAVOR
BB
V
CC
S
CP
TOTALS
O
302
378
274
225
317
1496
E
299.2
299.2
299.2
299.2
299.2
1496
0.03
20.75
2.12
18.40
1.06
42.36
Since this CCSS of 42.36 is much larger than any of the TCSS’s, we confidently
reject the hypothesis of equal population numbers. Bags 1 and 3 might have suggested
to the contrary, but the pooled results decisively reject the Null Hypothesis.
The reader and his/her students are encouraged to investigate other
distributions using this Chi-Square process.
David R. Duncan is a Professor of Mathematics at the University of Northern Iowa
in Cedar Falls, Iowa. [email protected]
Bonnie H. Litwiller is a Professor of Mathematics at the University of Northern
Iowa in Cedar Falls, Iowa. [email protected]
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Distributed Curriculum2 and Mathematical Discourse:
A Little Bit Goes A Long Way
Angie Peltz, edited by Dr. Bridgette Stevens
Classrooms today, in general, are quite different when compared to classrooms of
the past. Historically, math classes were run exclusively by the teacher leading a 35minute lesson on a given topic that was followed by students working on a set of math
problems to check their ―understanding‖ of the topic. It was a textbook-driven
classroom. Today’s math classrooms are not solely focused on the completion of
problem sets. There is also considerable focus on lessons driven by student
understanding. Additionally, today’s classrooms are also different with respect to
classroom chatter. Traditionally, the teacher’s voice was the channel through which all
knowledge and instruction was communicated. Students typically only spoke when
either asking or answering questions. However, today, math classrooms are filled with
robust discussion where the students and teacher collectively wrestle with the topic at
hand. As such, students play an active role in learning and understanding mathematics.
This study documents the impact of a distributed curriculum focused on addition
and subtraction of fractions while analyzing its effect on student discourse in a fifth
grade, self-contained classroom. In collaboration with three peers in the Master of Arts
in Mathematics for the Middle Grades (4-8) Program at the University of Northern Iowa,
I developed a nine-week distributed curriculum focused on four models for representing
fractions to foster student understanding of addition and subtraction of fractions. Through
brief, five-minute daily activities, students were introduced to new ways of modeling
fractions and were supported in developing discourse for sharing their thought processes.
After much debate, Angie, Bridgette, and Ed Rathmell find ‘distributed practice’ and
‘distributed curriculum’ synonymous.
2
76
This paper provides an analysis of the specific activities and strategies that contributed to
the development of students’ fraction sense as well as how student discourse was
cultivated during this distributed curriculum activity.
A Review of the Literature
When students encounter a new topic or subject, their initial reaction is to draw
upon what they already know to begin making sense of the new phenomenon. In many
school subjects, students who lack an understanding of the fundamentals inevitably
encounter problems when more advanced topics are explored. Thus, it is important to
provide students with foundational knowledge. How to teach the concepts students need
to know is of critical importance. Gabriele and Rathmell (2006) contend not all students
benefit from a 45-minute, teacher-directed lesson on a given topic and recommend
teachers construct a curriculum that provides students with time to digest and understand
the concept at hand; a curriculum where students are introduced to or build upon existing
knowledge of a concept for a few minutes each day. They discuss three different
methods to promote students’ conceptual understanding, one of which is a distributed
curriculum.
Distributed Curriculum
Also detailed by Fosnot and Dolk (2002), distributed curriculum focuses on a
structured series of mini-lessons (e.g., approximately five-minutes) on a given topic that
occur daily over a period of weeks. While one could argue these discussions are delving
deeper into a topic, Rathmell (2008) contends a distributed curriculum is meant to take
five minutes. Of those five minutes, one minute is spent on presenting the problem and
the students working it out. Two additional minutes are spent asking two or three
77
students to explain it, followed by one minute on highlighting an explanation of choice,
and the final minute to solve a similar problem. Taking a big idea (or unit of study) and
distributing its concepts over several weeks in five-minute mini lessons can eliminate the
need to teach the unit, freeing up time for an already congested curriculum. Moreover,
these lessons typically use concrete materials, such as diagrams and manipulatives, to
provide an alternative to traditional instruction which equips students with new, perhaps
more efficient, thinking strategy.
Discourse
Researchers have found rich conversations about mathematics enhance student
learning. Manouchehri and Enderson (1999) identified several key elements that must be
present in a classroom in order for it to be rich in discourse. Two of the key elements
they document pertain to the role of the students and the role of the teacher. They
emphasize that a safe and comfortable environment must be cultivated so students are
willing to take risks as learners. Hiebert et al. (1997) stresses the importance of the role
of the teacher when it comes to encouraging students to share and discuss methods of
solutions because, ―when students’ intuitive strategies are made public, they can be
analyzed more deeply and everyone can learn from them‖ (p. 45). Unlike the
conventional teacher and student roles of the past, effective student discourse blurs the
lines between student and teacher.
Stein (2007) explores cognitive discourse within the mathematics classroom and
purports there is high-pressure and low-pressure cognitive discourse. In high-pressure
cognitive discourse, the teacher asks questions that help students make the connections
between mathematical concepts. In low-pressure cognitive discourse, the students (as
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opposed to the teacher) serve as the principal instructors pushing one another to the next
level of understanding. The author provides a discourse protocol (see Table 1) to help
teachers evaluate the levels of discourse occurring within their classrooms. By using a
discourse protocol, teachers are able to enhance the learning environment by monitoring
and coding daily lessons to see to what extent students are leading the discussion. Such
protocols challenge teachers to not only self-monitor and evaluate their teaching styles,
but also examine the learning styles of their students.
Table 1-Levels of Discourse in a Mathematics Classroom
Levels
Characteristics of Discourse
0
The teacher asks questions and affirms the accuracy of answers or introduces
and explains mathematical ideas. Students listen and give short answers to the
teacher’s questions.
1
The teacher asks students direct questions about their thinking while other
students listen. The teacher explains student strategies, filling in any gaps
before continuing to present mathematical ideas. The teacher may ask one
students to help another by showing how to do a problem.
2
The teacher asks open-ended questions to elicit student thinking and asks
students to comment on one another’s work. Students answer the questions
posed to them and voluntarily provide additional information about their
thinking.
3
The teacher facilitates the discussion by encouraging students to ask questions of
one another to clarify ideas. Ideas from the community build on one another as
students thoroughly explain their thinking and listen to the explanations of
others.
Researchers in the arena of student discourse have also focused on the dialogue
that occurs once a solution has been identified by students (Fang, Miller, Perry,
Schleppenbach, & Sims, 2007). The researchers contend, ―In extended discourse, a
student’s answer to a question serves as the beginning to a larger discussion about the
mathematical algorithms, rules, and reasoning needed to find that answer‖ (p. 381). The
authors looked specifically at the questioning used by teachers in order to move students
to a heightened level of understanding. By continuing the conversation even after the
correct answer has been stated, and by opening the discussion to look at different
79
methods of student thinking, students draw connections between their own logic and that
of their peers. Extended discourse allows students to learn how different students solved
a problem using different strategies. The researchers provide a method for coding
extended discourse to help teachers assess the type of discourse taking place within their
classrooms (see table 2).
Table 2- Content Codes for Function of Teacher and Student Statements in Extended
Discourse
Teacher Discourse Codes
Student Discourse
Codes
1. Request for computation
1. Computation
2. Request of procedure or method
2. Procedure or method
3. Request for reasoning
3. Reasoning
4. Request for rule or term recall
4. Rule or term recall
5. Check for student understanding
5. Indication of understanding
and/or agreement
and/or agreement
6. Request for short answer
6. Short answer
7. Teacher explanation
8. Restating student answer
9. Praise
In reviewing several sources on the topic of student discourse, there is a clear
theme that when students articulate their ideas and listen to the logic of their peers, a
powerful educational experience occurs through the active participation of students. As
the principal investigator, I was very eager to apply what I have learned from research in
my fifth grade classroom. Before delving into the results, I discuss a portion of my final
research paper for completion of an M.A. degree at UNI.
Composition of the Distributed Curriculum
As stated earlier, the distributed curriculum used in this case study was developed
as part of a group project for an assignment for a class in a graduate program at UNI.
Three fellow graduate students and I created a nine-week distributed curriculum unit on
fractions. By developing and implementing this unit, we were able to familiarize
80
ourselves not only with new ways to model addition and subtraction of fractions, but also
the skills needed to build that understanding in our students. What follows is the
sequence and purpose of the mini-lessons we created to foster students’ understanding of
fractions.
Area model. We began by focusing on the use of an area model. By doing so, we
would determine students’ background knowledge of fractions and thus be able to use
this knowledge to guide future planning. We spent two weeks asking such questions as,
―What does 2/3 look like?‖ to the use of diagrams to determine what fraction of the
diagram is shaded based on its attributes.
Clock model. Weeks three, four, and five were spent using the clock model to add
common fractions. The clock is a wonderful representation often overlooked. Using the
clock to envision halves, thirds, fourths, sixths, and twelfths, we started with helping
students recognize various fraction pieces. In week three, we began presenting fractions
as an addition problem using only unit fractions. Week four began with more complex
problems by taking out one of the unit fractions and replacing it with a larger fraction
(e.g., 7/12 + 1/6). In week five we returned to common fractions and began subtraction.
Set model. The next level of understanding addressed key concepts with a set
model. Students were given a set with which to work (e.g., pennies, apples) and asked to
show the fraction equivalent. During week six, students focused on identifying unit
fractions of a given set. In week seven, students were challenged with problems beyond
unit fractions (e.g., 7/8 instead of 1/8) as well as compute using mixed numbers. This idea
took us from always knowing the whole and introduced the idea of what to do when we
do not know the whole.
81
Measurement models. The final two weeks of the unit concluded with problems
focused on linear models and double number lines. We began with basic problems where
students envisioned themselves cutting a piece of string into four equal lengths. Then
students were asked to show ¾ + ½ which provided insight into their level of
understanding of fractions verses simply manipulating a ruler.
Implementation
Implemented of the distributed curriculum began three weeks into the school year.
This gave me enough time to establish classroom expectations, a sense of routine, and
rapport with my students. Each day, I wrote the mini math problem on the board for
students to see and think about. We did not have math until third hour, which allowed
students to think about it for a couple of class periods before discussing it in class. Once
class began, students had three minutes to work using their response log as a reference.
This form of documentation allowed me to monitor their understanding as needed, as
well as serve as a reference point for them from day-to-day and even week-to-week. I
videotaped each mini math segment to serve as a reference for myself as I progressed
through implementing the distributed curriculum. After students had time to write down
their thoughts and ideas, we spent three (or more) additional minutes sharing thoughts
about the problem as a whole group. Students were very respectful of their peers’
thoughts and feelings when challenging them to think about the problem in a different
way. They encouraged their peers to provide a new way of thinking about the problem.
As the results of this study will show, I was impressed by the way students interacted and
how discourse fostered an understanding of the problem.
82
Data Collection on Student Discourse and the Results
The first method, the daily response log where students showed their work and
explained their thinking, helped me monitor student thinking. Given each week, students
recorded their daily thoughts about each problem. Students often referred back to the
previous mini math lessons in constructing their reasoning about a problem during our
whole group discussion time. I analyzed the daily response logs for work shown,
correctness of answers, and type of strategies used to solve the mini math problems.
The second method, the video recording, captured the discourse taking place. I
coded the responses that took place using Stein’s (2007) Levels of Discourse, and
analyzed this data by rating the level of discourse and comparing how discourse
developed over time. I also coded the discourse with Fang, Miller, Perry, Schleppenback,
& Sims’ (2007) Content Codes for Function of Teacher and Student Statements in
Extended Discourse. I analyzed the data by counting the instances when each type of
response took place, looking for level three as the optimal level for classroom discourse
(see Table 1). For extended discourse (see Table 2), I strived to regularly achieve codes
two, three, five, and six. These four codes focus on the student providing more than just
the answer; they ask for student procedures, reasoning, understanding, and a short answer
response to the given reasoning.
The third method, a reflection journal, is where I kept notes about the events that
took place during the mini lessons, my reactions to students’ comments, reflective
thoughts while in the act of teaching and reflective thoughts on the act of teaching. These
notes were then analyzed for common themes that emerged as a result of how the mini
lessons unfolded each week.
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One may ask what is the benefit of taking five minutes out of a math class (which
is already too short) to teach a concept that is not taught until later in the year. After a
careful review of the collection of data, the benefits of this distributed curriculum
outweigh the concerns for shortening the time for mathematics instruction. The results of
the analysis show using a distributed curriculum had a positive impact on overall
classroom discourse.
The Results of Coding Discourse
In review of my notes from the large group reflection time and the coding of
student discourse, I found that when students have the opportunity to not only share their
thoughts, but also serve as resources for one another and enjoy and learn from the
thoughts of others, it becomes a truly empowering scenario for everyone involved. The
thoughts and connections my students made throughout the course of this distributed
curriculum unit were unlike anything I would have heard (or they would have
experienced) by solely using the district-adopted textbook. During the use of the text,
there is not the opportunity for such a rich discussion to take place. Some students try to
work ahead; others become distracted. The day-to-day routine sets in and students
become disinterested. The mini lessons have breathed new life into math class again.
Students are truly engaged and enjoying the opportunity to talk about math.
When coding the classroom discourse for the first week of mini lessons, using the
aforementioned coding schemes (see Table 1), the level of discourse was in the range of 0
to 1. I was not asking a lot of the students, but rather checking their background
knowledge of fractions and how they could verbalize that knowledge. The coding of
weeks three through five looked a little better on paper. The level of discourse these
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weeks rose from 0 to 1 and even to a 2 on mini math 2.3. The students were very
interested in a student’s explanation of an answer. It is the result of students taking
control of their learning. Their level of comfort was becoming apparent as they began to
challenge the thinking of others and asking for another explanation to a problem. When
students used the clock model to solve the addition problems by converting fractions to
different denominators (converted the fractions to fourths while others chose twelfths), I
found the discussion was richer than anticipated. Thus, when coding the extended
discourse, I found that all types of discourse were present. The class was not satisfied
with just one answer; we wanted to hear how others came up with their answers.
I discovered that when a problem came in at a lower level of discourse, as was the
case with mini math the first week, the extended discourse that took place was minimal,
which I attribute to the fact that the first week was more of an introductory or review
week of the curriculum used solely to discover the background knowledge of my
students. The discourse was more abundant (and more time intensive) as the tasks and
models used become more difficult. Students had more options in their responses and
how they viewed the problems so they wanted to share and ask questions when they did
not understand.
I found that throughout the distributed curriculum, we did not consistently reach a
level three, which I attribute mostly to the time restrictions. There were occasions where
our conversations went beyond five minutes, either due to a misconception or different
insights that were available for us to discuss or for the simple enjoyment of students
willing to share their thinking. A level three may be more easily achieved in
collaborative, small group scenarios.
85
As previously noted, my target codes for the level of classroom discourse was a
three. I found it very hard to reach a level three due to the time factor. I was elated to see
in the end that I reached a level of two or higher 67.6% of the time. My goal for extended
discourse was to have a code of two through six with the omission of code four. As seen
in Table 3, I found much success when it came to these four codes, especially in weeks
three through five when we were discussing the clock model.
Table 3-Extended Discourse Results
Extended Teacher Number of Lessons
Percent of Lessons
Discourse
(out of 34)
16
1
47.1%
31
2
91.2%
33
3
97.1%
22
4
64.7%
33
5
97.1%
34
6
100%
23
7
67.6%
30
8
88.2%
20
9
58.8%
In analyzing my role in the level of discourse, I found that codes two, three, five,
and six were present in more than 90% of the lessons. The student discourse in these four
codes were present 100% of the lessons with exception to code five, which dealt with the
indication of understanding which was present 73% of the time.
Changes in Implementation
In reviewing the data sources, I have found three areas in need of change: the
schedule for providing comments in the student response logs, a time restriction for the
mini lessons, and the omission of the textbook chapter on adding and subtracting
fractions. A brief discussion of each follows.
86
In reviewing student response logs, I would change this method of data collection
and analysis from collecting the logs weekly to daily. I have implemented another
distributed curriculum unit on area later in the year and found I was more successful in
collecting the response logs and marking notes nightly, allowing me to reiterate key
points missed by many the following day. This change would allow me to keep a closer
eye on students that were struggling on a daily basis and allow me to touch base with
them throughout the time that was spent on the mini math lesson each day. The response
logs, along with video recording discourse, served as the barometer for informing my
instruction and assessment practices.
The use of a timer will be implemented next year in order to limit the minilessons. While on average the mini lessons took 5-10 minutes, a review of the video
recordings shows there were days where we were so engrossed in the discussion, only a
small amount of time remained for the actual lesson of the day. I am encouraged that my
students want to talk about fractions for 25 minutes; however, as we all know, time is
limited. Focusing on just a couple of explanations will be the key rather than letting all
students share similar thought processes.
Because I am encouraged by the results of using a distributed curriculum
approach and highlighting various models to develop understanding, next year I plan to
administer the chapter text from the textbook prior to its start to see if there is a need to
teach it at all. It is my impression using distributed curriculum I can free-up precious
class time and teach an additional concept that I am rarely able to reach by the end of the
year.
87
Implications for Future Research
As I reflect on this experience, there are a number of aspects for which I am
pleased and a few things I would do differently, which I have already addressed. Despite
any shortcomings, the positive outcomes realized through this distributed curriculum
demonstrated the effectiveness and affirmed for me the value of this instructional
approach. The following three implications warrant further investigation to broaden and
deepen what we, as math teachers, know and understand about effective math curriculum.
Implication One: Models Enhance Student Understanding. There is clear evidence
students’ understanding of fractions is enhanced as a result of the use of models. My
students gained knowledge regarding the definition of fractions (equal parts of a whole),
the relative size of fractions, and the addition and subtraction of fractions using different
models. Specifically, by using different models, the students developed a better
understanding of relating diagrams with fractions written in symbolic form. While some
students may have developed a dependence on the clock model, it most certainly helped
to foster their fraction sense. Therefore, teachers should utilize models as a way to
improve student understanding. Further research should address the additional use of the
measurement model and possibly manipulatives, like fraction strips, to further student
understanding.
Implication Two: Student Discourse Enhances Student Understanding. It is clear
the clock model produced the highest levels of discourse. The discourse that took place
during these three weeks had an obvious affect on students’ use of this model throughout
their work as shown by 84% of the students increasing their scores on the post-test (not
presented in this portion of the paper). The use of models, coupled with robust discourse,
88
clearly contributed to my students’ fraction sense and overall understanding. Students
exhibited this preference not only during the distributed curriculum, but also referred
back to it on several occasions throughout the year. Additional research should be
considered to look at the possibility for ensuring the use of the clock model in all
elementary mathematics textbooks.
Implementation Three: Math Concepts Can Be Effectively Taught Through a
Distributed Curriculum. Through the use of mini lessons, I was able to enhance my
students’ knowledge of fractions. These mini lessons provided me an outlet to introduce
models and new strategies to help build knowledge of fractions, strategies they may not
have otherwise encountered. Through the use of this instructional approach, I am hopeful
in years to come to replace the unit in my textbook with this series of mini lessons and
thus be able to continue to cover a variety of math concepts more in depth. Additional
research should continue as teachers are interested in implementing a distributed
curriculum to alleviate an already over-crowded curriculum.
Conclusion
The distributed curriculum itself was very successful. The time and thought I put
into the construction of the unit proved successful as the students moved throughout the
unit. The knowledge and connections the students made from one day to the next with
regards to their understanding of adding and subtracting fractions was profound. By
providing these five-minute mini lessons, my students were free to take risks and explore
a concept they found intimidating when they began fifth grade. When the unit was over
the students were wondering what we were going to start exploring next. Since it was
seamlessly structured into our day, the students knew what was expected – reflection over
89
the problem and then sharing of ideas and procedures with their peers. It provided a great
transition into our regular math class and the environment I had hoped to build for our
class this year.
In addition to the growth and development experienced by my students, the
distributed curriculum helped me become a better teacher. This unit encouraged me to
look more in depth at my own teaching style and what I expected out of my students. By
coding each mini lesson to analyze the level of discourse, I was able to identify where my
strengths were in relation to encouraging students to share their ideas with one another. I
found that while I may not always provide the perfect environment for discourse to
naturally develop, I am consistently providing my students with opportunities to share
their thoughts, questions, and strategies. Together, this experience provided both my
students and me with a multitude of opportunities to grow and develop. From their
fraction sense to their written and oral communication skills to my teaching, everyone
involved benefited from this experience. I plan to use a distributed curriculum approach
for subsequent concepts in the coming years.
Angie Peltz teaches at Decorah Middle School in Decorah, Iowa [email protected]
Dr. Bridgette Stevens is a Professor of Mathematics at the University of Northern Iowa in Cedar Falls,
Iowa. [email protected]
90
References
Fang, G., Miller, K.F., Perry, M., Schleppenbach, M., & Sims, L. (2007). The answer is only the
beginning: Extending discourse in Chinese and U.S. mathematics classrooms. Journal of
Educational Psychology, 22, 380-396.
Fosnot, C.T. & Dolk, M. (2002). Young Mathematicians at Work Constructing Fractions,
Decimals, and Percents. Portsmouth, NH: Heinemann.
Gabriele, A. J., & Rathmell, E.C. (2006). Developing computational fluency, k-8 number and
operations. Unpublished manuscript submitted for publication.
Hiebert, James, Carpenter, T.P., Fennema, E., Fuson, K, Wearne, D., Murray, H., Olivier, A., &
Human, P. (1997) Making Sense: Teaching and Learning Mathematics with
Understanding. Portsmouth, NH: Heinemann.
Manouchehri A., & Enderson M.E. (1999). Promoting mathematical discourse: Learning from
classroom examples. Mathematics Teaching in the Middle School, 4, 216-222.
Rathmell, E.C. (2008). Meaningful distributed instruction – Developing number sense.
Unpublished manuscript submitted for publication.
Stein, C.C. (2007). Let’s talk: Promoting mathematical discourse in the classroom. Mathematics
Teacher, 101, 285-289.
Want to go back to school?
Got a new idea you would like to try in your classroom?
Interested in professional development?
Thought about attending an NCTM Conference?
ICTM is here for YOU!
There are opportunities to help you fund all of these: Conference Travel, Advanced
Tuition and Curriculum Grants are available to members of ICTM. By supporting our
members, ICTM is contributing to the mathematics education of Iowa students; help
us invest in the future by applying for one of these grants. Watch for next year’s grant
opportunities available on-line at www.iowamath.org
91
The Iowa Core Curriculum and Me:
How my Teaching of Mathematics Methods will Change
Catherine M. Miller3
It is an exciting time to be a mathematics educator in Iowa! We are joining the
other 49 states by having a set of state standards. In fact, Iowa is exceeding federal
expectations by having a curriculum to inform the work of teachers and school
administrators. Because of this, we enter an era of change in Iowa and, as we know,
change is never easy. To succeed in implementing the Iowa Core Curriculum (ICC) in
mathematics all teachers need to learn about it and have help in implementing its core
ideas and content. This includes teachers who will begin their careers in this era of
change; these are the folks I work with as a methods instructor at the University of
Northern Iowa. I need to prepare my teachers for the future, which now includes the ICC.
First, I need to be informed. To serve the future teachers of Iowa, I need to
understand the overall philosophy of the ICC and, in particular, be familiar with the
mathematics portion. Since I teach secondary methods classes, I’ll pay most attention to
that portion – but I will not neglect the elementary parts all together. Iowa students
experience education as a sequence of classes divided into elementary, middle
school/junior high and high school; this is done to organize education for teachers,
administrators and buildings. Teachers often find themselves isolated in one of these
areas. I think it is imperative that future teachers understand what comes before and after
the classes they teach so that what they teach and how they teach makes sense in the
continuum of education students experience. So, I must understand the elementary
3
The author would like to thank Lynn Selking for feedback on an early draft of this manuscript.
92
portions of the ICC to model this and prepare the future teachers in my class for this
continuum.
Visit http://www.corecurriculum.iowa.gov/ to find information about the soon to
be adopted (at high schools) ICC. Here, you can learn about the academic expectations of
the ICC, performance standards and essential skills (all searchable by content area). With
this information, I have updated the curriculum in my methods course; the changes are
highlighted below. Please do not think of this as a standard curriculum for methods
courses; it is my attempt to prepare mathematics teachers for their future work.
The Mathematics ICC Vocabulary
ESC, PBITs, MDPs, ―essential skills‖ and ―assessment for learning‖ are all
central to the mathematics portion of the ICC (Iowa Department of Education, 2009).
These acronyms and phrases will be ringing in the air at schools beginning this fall, if
they have not already begun to be used. I want my students to understand what each
means and its role in the ICC before they student teach. If the preservice teachers in my
classes are familiar with these terms, they can be part of the excitement associated with
change and keep up with the teachers they work with during their first years of teaching,
including student teaching, who will already have learned about the ICC and probably
have begun to implement some of it.
Every student counts
Every Student Counts (ESC) has been a statewide initiative created to improve
Iowa student achievement in mathematics. Until recently, there have been three levels of
training, elementary (addressing both grade level bands defined by NCTM), middle and
high school. These sessions have addressed teaching specific mathematical content and
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pedagogy. AEA personnel have been part of the training and are now, in many places,
rolling out their own ESC professional development opportunities for classroom teachers
and administrators. I was fortunate to have been involved with the middle school
planning and presenting team providing me some advance insights into the ICC. The
unifying principle of ESC is to teach mathematics for understanding with meaning. You
will find this resonating throughout the mathematics portion of the ICC. Because of this
alignment, teachers may talk about what they learn or relearn at ESC workshops as they
plan to implement the ICC. If there is an ESC session near you (check your local AEA
website) you might ask to attend as a guest. You can gain some insights into what will be
expected of teachers when the ICC is in place that can inform your work.
Problem based instructional tasks
An integral part of ESC and the ICC is Problem Based Instructional Tasks
(PBITs). It is important to note that these are not lessons, but specially designed tasks that
may take several class sessions to complete. These are designed to promote student
understanding of mathematics. Characteristics of PBITs include:

Help students develop a deep understanding of important mathematics

Emphasize connections, especially to the real world

Are accessible yet challenging to all

Can be solved in several ways

Encourage student engagement and communication

Encourage the use of connected multiple representations

Encourage appropriate use of intellectual, physical, and technological tools
Iowa Department of Education, 2009
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The use of PBITs is research based, calling on what mathematics educators have
learned about problem solving when teaching for understanding (e.g. Stein, Boaler, &
Silver, 2003, Kilpatrick, Swafford, & Findell, 2001, Grouws & Cebulla, 2000) and the
role of discourse in learning meaningful mathematics (e.g. Hiebert & Wearne, 1993).
NCTM has long supported the use of problem solving in mathematics class to promote
understanding (NCTM, 2000). The use of PBITs is not new, but newly highlighted as an
integral part of the ICC.
Meaningful distributed practice
Meaningful Distributed Practice (MDP) is also a core component of the
mathematics portion of the ICC. ―Practice is essential to learn mathematics. However, to
be effective in raising student achievement, practice must be meaningful, purposeful, and
distributed‖ (Iowa Department of Education, 2009). The purpose of MDPs is to provide
students an opportunity for quick, meaningful practice with big mathematical ideas.
Usually completed within five minutes at the beginning of class, students work
independently without manipulatives or technology to do a series of short tasks and then
report their findings. Often this time is used to preview or review important mathematical
ideas, or practice skills that must be maintained for future student learning. The
components of MDPs are defined as follows:

Meaningful: Builds on and extends understanding

Purposeful: Links to curriculum goals and targets an identified need based on
multiple data sources

Distributed: Consists of short periods of systematic practice distributed over a
long period of time
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Iowa Department of Education, 2009
Like PBITs, the rationale behind using MDPs is research based. See Hiebert
(2003), Willingham (2002), Kilpatrick, Swafford, and Findell (2001), Kilpatrick &
Swafford (2001), or Grouws & Cebulla (2000) for research related to practicing and
maintaining mathematical skills. These studies were among those used to inform the
ICC’s use of MDPs.
ICC essential skills
When I think of mathematical skills students need to master I think of
computational fluency, the ability to apply algorithms competently, etcetera. This is not
what is referred to as essential skills in the ICC. Instead, the writers of the mathematics
ICC documents considered NCTM’s Process Standards (2000) when defining skills
essential to understanding mathematics. These skills are:

Problem Solving

Communication

Reasoning and Proof

Ability to Recognize, Make and Apply Connections

Ability to Construct and Apply Multiple, Connected Representations
Iowa Department of Education, 2009
These skills are called for in response to the demands business and citizenship
now require of our students. ―Students need powerful skills to be successful in the
globally competitive workforce of the 21st century. Business and industry demand
workers who can solve problems, work in teams, and are able to apply learning to new
and changing situations, especially as workers change jobs and careers many times in
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their lifetimes‖ (Iowa Department of Education, 2009). This change in skill sets called for
in the ICC is also grounded in research (see e.g. NCEE 2006, NCTM, 2000, SCANS
1991).
Assessment for learning
Assessment has always been integral to teaching, but when teaching for
understanding is the goal it is even more important – especially formative assessment.
The authors of the ICC (Iowa Department of Education, 2009) documents and NCTM (in
Bush and Leinwand, 2000) use the phrase ―assessment for learning‖ in place of formative
assessment. By using this phrase in place of ―formative assessment‖ the need for
continuous assessment when teaching mathematics for understanding is emphasized.
Assessment for learning, which depends on teachers gathering evidence from multiple
sources and then acting on that evidence, can be one of the most powerful forces for
learning mathematics (Bush and Leinwand, 2000). Being able to formulate questions,
design and implement meaningful tasks, and critically listen to student discourse are all
parts of assessment for learning. More about how this is part of the ICC can be found in
the PBIT section of the ICC website.
Like the other aspects of the ICC, practicing assessment for learning is supported
by research. ―Listening to students, asking them good questions, and giving them the
opportunity to show what they know in a variety of ways are all affirmed by research to
be important ways of increasing student learning‖ (Wilson and Kenney, 2003). While
teachers have always known formative assessment is important, the ICC will increase the
need for these assessment practices.
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Implications for my Methods Curriculum
The methods class I teach is one of two required of all secondary mathematics
teaching majors at UNI. We already emphasize teaching for understanding, big ideas,
discourse and problem solving; I must do a better job when teaching assessment practices
to my students, something I do not think I have done well. I will focus more on
assessment for learning. Weaving formative assessment into a lesson is hard and I need to
prepare my students to do this effectively. Otherwise, how can they teach for
understanding?
Using ICC terms will also be part of my class. We will discuss PBITs and MDPs
and plan some as practice. Additionally, I will visit with AEA mathematics consultants or
AEA websites and find examples to share with my students. I may use the PBIT template
(not available online, but you can get it from your AEA or by emailing me) for some
lesson planning. My goal is that with the ICC vocabulary, emphasis on teaching for
understanding and preparation to assess for learning, my students will be prepared to
successfully launch their careers in this exciting and challenging time of change.
Catherine M. Miller is Associate Professor of Mathematics Education with the
University of Northern Iowa in Cedar Falls, Iowa.
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[email protected]
References
Bush, W. S & S. Leinward. (2000) Mathematics Assessment: A Practical Handbook for Grades 6
- 8. Reston, VA: NCTM
Grouws, Douglas A. & Cebulla, Kristin J. (2000). Improving Student Achievement in
Mathematics. Geneva, Switzerland: International Academy of Education.
Hiebert, James & Wearne, Diana (1993). Instructional tasks, classroom discourse, and students’
learning in second-grade arithmetic. American Educational Research Journal, 30, 393-425.
Hiebert, James (2003). What research says about the NCTM Standards? In J. Kilpatrick, W. G.
Martin, and D. Schifter (Eds.), A Research Companion to Principles and Standards for School
Mathematics, pp. 5-23. Reston, VA: National Council of Teachers of Mathematics.
Iowa Department of Education (Retrieved July 2009). Iowa Core Curriculum.
http://www.corecurriculum.iowa.gov/Home.aspx.
Kilpatrick, J. & J. Swafford. (Eds.); Mathematics Learning Study Committee, National Research
Council (2001). Helping Children Learn Mathematics. Washington, D.C.: The National Academies Press.
Kilpatrick, J.; J. Swafford & B. Findell (2001). Conclusions and recommendations. In Adding It
Up: Helping Children Learn Mathematics, pp. 407-432. Washington, D.C.: The National Academies Press.
National Center on Education and the Economy (2006). Tough Choices or Tough Times. Retrieved
July 2009. http://www.skillscommission.org/executive.htm.
National Council of Teachers of Mathematics (2000). Principles and Standards for School
Mathematics. Reston, VA: NCTM.
Stein, M. K.; J. Boaler; E. A. Silver. (2003). Teaching mathematics through problem solving:
Research perspectives. In H. L. Schoen (Ed.), Teaching Mathematics Through Problem Solving, Grades 612, pp. 245-256. Reston, VA: National Council of Teachers of Mathematics.
Willingham, D. (2002). Allocating student study time: ―Massed‖ versus ―distributed‖ practice.
American Educator, Summer.
Wilson, L. D. & P. A. Kenney. (2003) Classroom and Large Scale Assessment. In, J. Kilpatrick,
W. G. Martin & D. Schifter (Eds.), Research Companion to the Principles and Standards for School
Mathematics. Reston, VA: NCTM.
U.S. Department of Labor, The Secretary's Commission on Achieving Necessary Skills. (1991).
What Work Requires of Schools. Washington, DC: U.S. Government Printing Office.
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Problem-based Instructional Tasks
by Larry Leutzinger
Presenting problem-based instructional tasks and asking students to explain their thinking
results in higher overall achievement.
Tasks that ask students to perform a memorized procedure in a routine manner
lead to one type of opportunity for student thinking; tasks that require students to
think conceptually and that stimulate students to make connections lead to a
different set of opportunities for student thinking. The day-in and day-out
cumulative effect on classroom-based tasks leads to the development of students’
implicit ideas about whether mathematics is something about which they can
personally make sense and about how long and how hard they should have to
work to do so.
Evidence gathered across scores of middle school classrooms has shown that
students who performed the best on project-based measures of reasoning and
problem solving were in classrooms in which tasks were more likely to be set up
and implemented at high levels of cognitive demand. For these students, having
the opportunity to work on challenging tasks in a supportive classroom
environment translated into substantial learning gains on an instrument specially
designed to measure exactly the kind of student learning outcomes advocated by
NCTM’s (National Council of Teachers of Mathematics) professional teaching
standards. (Stein and Smith, 1998)
References:
Hiebert, J. and Wearne, D. (1993). Interactional tasks, classroom discourse, and
students’ learning in second-grade arithmetic. American Educational
Research Journal, 30(2), 393-425.
Mathematics Learning Study Committee, National Research Council (2001).
Conclusions and recommendations. In J. Kilpatrick, J. Swafford, & B.
Findell (Eds.). Adding it up: Helping children learn mathematics (pp. 407432). Washington, DC: The National Academies Press.
Stein, M.K., Boaler, J., Silver, E.A. (2003). Teaching mathematics through
problem solving: Research perspectives. In H.L. Schoen (Ed.), Teaching
mathematics through problem solving: Grades 6-12 (pp. 245-256).
Reston, VA: National Council of Teachers of Mathematics.
Stein, M. K., & Smith, M. (1998). Mathematical tasks as a framework for
reflection: From research to practice. Mathematics Teaching in the Middle
School, 3(4) 268-275.
100
Characteristics of Problem-based Instructional Tasks
 Help students develop a deep understanding of important mathematics
 Are accessible yet challenging to all students
 Encourage student engagement and communication
 Can be solved in several ways
 Encourage the use of connected multiple representations
 Encourage appropriate use of intellectual, physical and technological tools
Lesson Format for Problem-based Instructional Tasks
Launch: (Teacher)
 Discuss appropriate vocabulary
 Have students explain their understanding of the problem
 Provide a ―mini‖ problem*
Explore: (Students)
 Work individually, in pairs, and/or in groups
 Agree on the solution
 Ask clarifying questions (students and teacher)
Share: (Students)
 Provide a variety of answers or solution strategies
 Explain their thinking and justify their responses
 Ask questions or add comments
Summarize: (Teacher)
 Emphasize the key points involved in the problem solution
 Ask students questions related to the problem
 Clarify effective thinking strategies
Extend: (Teacher)
 Provide practice activities that emphasize use of effective strategies
 Look at the problem in a slightly different way
 Use different numbers or vary the instructions
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Creating a Mini Problem to Preview a Difficult Task
1. Read over the problem (task).
2. Create another problem (task) that contains the essential ingredients of the
original problem but is easier to solve.
3. Present the original problem (task) to the students and highlight how the sample
problem relates to this problem.
4. Highlight how the sample problem (task) is different from the original.
5. Have the students work in pairs or groups to solve the original problem (complete
the task).
Making a Routine Lesson a Problem-based Lesson

Turn the page around. Start with the last problems on the page.

Solve the introduction problem without showing or telling the student what to do.

Use the manipulatives recommended on the page, but have the students represent
the situations on their own.

Make a game of the problem on the page. Require decision-making and
reasoning.

Do aspects of the page mentally.

Differentiate the task for individual students.
The students should be the ones representing and solving problems, not watching
or reading how someone else solved a problem. The overarching goal is to have
the students develop thinking strategies while “doing mathematics.”
Larry Leutzinger is Professor of Mathematics Education at the University of
Northern Iowa in Cedar Falls, Iowa [email protected]
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Top Ten Things I Wish I Had Known When I Started Teaching
The following tips are from the series Empowering the Beginning Teacher in Mathematics, by
Cynthia Thomas. Reprinted with permission from www.nctm.org, copyright 2009 by the
National Council of Teachers of Mathematics. All rights reserved.
10. Not every student will be interested every minute. No matter how much
experience you have or how great you are at teaching, you will encounter times in
the classroom when no student is interested! The solution is to change your tone
of voice, move around the room, or switch from lecturing to some other activity.
Maybe you can even use a manipulative to increase the students’ understanding
and, possible, their level of interest.
9. If a lesson is going badly, stop. Even if you have planned a lesson and have a
clear goal in mind, if your approach is not working-for whatever reason-stop!
Regroup and start over with a different approach, or abandon your planned lesson
entirely and go on to something else. At the end of the day, be honest with
yourself as you examine what went wrong and make plans for the next day.
8. Teaching will get easier. Maybe not tomorrow or even next week, but at some
point in the year, your job will get easier! Try to remember your first day in the
classroom. Were you nervous? Of course; all of us were. See how much better
you are as a teacher already? By next year, you will be able to look back on today
and be amazed at how much you have learned and how much easier so many
aspects of teaching are!
7. You do not have to volunteer for everything. Do not feel that you always
have to say yes each time you are asked to participate. Know your limits. Practice
saying, ―Thank you for thinking of me, but I do not have the time to do a good job
with another task right now.‖ Of course, you must accept your responsibility as a
professional and do your fair share, but remember to be realistic about your limits.
6. Not every student or parent will love you. And you will not love every one
of them, either! Those feelings are perfectly acceptable. We teachers are not hired
to love students and their parents; our job is to teach students and , at times, their
parents as well. Students do not need a friend who is your age; they need a
facilitator, a guide, a role model for learning.
5. You cannot be creative in every lesson. In your career, you will be creative,
but for those subjects that do not inspire you, you can turn to other resources for
help. Textbooks, teaching guides, and professional organizations, such as NCTM,
are designed to support you in generating well-developed lessons for use in the
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classroom. When you do feel creative and come up with an effective and
enjoyable lesson, be sure to share your ideas with other teachers, both veterans
and newcomers to the profession.
4. No one can manage portfolios, projects, journals, creative writing, and
student self-assessment all at the same time and stay sane! The task of
assessing all these assignments is totally unreasonable to expect of yourself as a
beginning teacher. If you want to incorporate these types of exercises into your
teaching, pick one for this year and make it a priority in your classroom. Then,
next year or even the year after that, when you are comfortable with the one extra
assignment you pick, you can incorporate another innovation into your teaching.
3. Some days you will cry, but the good news is, some days you will laugh!
Learn to laugh with your students and at yourself!
2. You will make mistakes. You cannot undo your mistakes, but berating
yourself for them is counterproductive. If the mistake requires an apology, make it
and move on. No one is keeping score.
1. This is the best job on earth! Stand up straight! Hold your head high! Look
people in the eye and proudly announce, ―I am a teacher!‖
Mathematics Conferences Coming to a Location Near You
Another Bite at the Core
Iowa Council of Teachers of Mathematics Conference
West Des Moines, Iowa February 19, 2010
Connections: Linking Concepts and Context
NCTM Annual Meeting and Exposition
San Diego, California: April 21-24th 2010
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Join the Iowa Council of Teachers of
Mathematics
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105
Fun Fridays
Try Googling Magic Squares and a wealth of information will be available to you.
Benjamin Franklin had a keen interest in Magic Squares which are believed to have been
invented before 2200 B.C. in China. Try doing activities with your class that demonstrate
how to set up Magic Squares.
1. A 4 X 4 square is fun to work with. Draw a 4 X 4 square and go through the
boxes one row at a time, left to right, top to bottom, counting from 1 to 16, but
writing down the number of the box only when it falls on the diagonal.
1
4
Magic Sum is 34
13
6
7
10
11
16
2. Start at square one again and count backwards from 16 to one. Whenever there
is an open square write down the number that you are on when counting
backwards.
1
15
14
4
12
6
7
9
8
10
11
5
13
3
2
16
Note that the four numbers in
each quadrant will also add up to
the magic sum.
You can start with any number and count sixteen consecutive numbers to make
a Magic Square that will truly be magic. This is a fun activity for elementary
students and gets them interested in numbers.
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Regional Directors for The Iowa Council of Teachers of Mathematics
AEA 1:
Sue Runyon
601 Big Rock Rd.
Fayette, IA 52142
AEA 9:
Bryan Braack
313 W. LeClaire Rd.
Eldridge, IA 52748
AEA 14:
Deborah Roberts
2115 – 160th St.
Corning, IA 50841
AEA 267:
Vicki Oleson
AEA 10:
Marlene Meyer
1243 Apache Trail NW
Cedar Rapids, IA 52405
Great Prairie AEA:
Lynn Selking
502 West Jefferson.
Corydon, IA 50060
Northwest AEA:
Mike Baker
31423 479th Ave.
Akron, IA 51001
AEA 11:
Marcia Carlson
Student Director:
Hannah Peacock
Upper Iowa University
Fayette, IA 52142
AEA 8:
Linda Seeger
AEA 13:
Ann Doran
3028 Country Club Pkwy
Harlan, IA 51537
5744 Timber Ridge Rd
Cedar Falls, IA 50613
36 Ann Street
Milford, IA 51351
8355 Franklin Ave.
Clive, IA 50325
Iowa Council of Teachers of Mathematics
2382 Iowa Highway 24
New Hampton, Iowa 50659
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