Perspectives of a Curriculum
Developer
Glenda Lappan
Michigan State University
Contrast these two views of learning
and curriculum.
 View One:
A series of skills and procedures learned
and practiced.
 View Two:
A set of experiences with important
mathematics that stresses meaning, use,
connections, AND proficiency with
related procedures and algorithms.
Two forms of problem solving:
Search your memory for available
algorithms or procedures.
 Make sense of the situation, represent it
in a useful way, estimate the
approximate size and nature of a
solution, put together ideas to form a
solution path—perhaps in creative or new
ways, carry out the plan, and examine
the result in light of the original
problem.
45 ÷ 7
6.4285714
A ticket to the Blue Rock Concert costs $7.
How many tickets can you buy if you have $45?
6 tickets
A mini van can carry 7 school children. How many
vans are needed for a field trip for 45 children?
7 vans
Sue got paid $45 for mowing lawns last week.
She worked 7 hours. How much per hour did she
get paid?
$6.43 per
hour
A swimming pool needs to be cleaned every
45 days. How many weeks is that?
6 weeks
and 3 days
What mathematical
occasions arise in classrooms
that teachers have to
navigate?
The need for teachers

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
to handle unexpected mathematical
situations.
to conjecture what a child has in mind
when he or she says something.
to examine the mathematical range of
possibilities.
to ask questions that help a child
reason about a situation without taking
away the mathematical challenge.
Young Children’s Reasoning
First grader who knew the meaning of
division and a few simple division and
multiplication facts.
What’s 42 ÷ 7? Well, 40 divided
by 10 is 4, and 3 times 4 is 12,
and 12 and 2 is 14, and 14
divided by 7 is 2, and 2 plus 4 is
6, …..so its 6.
Source: San Diego State University- Judy Sowder
Design a monograph to
show what you know about
3/4.
Source: San Diego State University- Judy Sowder
Sally:
Source: San Diego State University- Judy Sowder
Sam’s response
1.
2.
3.
4.
5.
3/4 is bigger than 5/8
3/4 is smaller than 1 whole
4/4 is bigger than 3/4
13/16 is bigger than 3/4
32/16 is 20/16 bigger than 3/4
Source: San Diego State University- Judy Sowder
Sandy: I
found them
all!
Source: San Diego State University- Judy Sowder
Connected Mathematics

The overarching goal of Connected
Mathematics is to help students and
teachers develop mathematical
knowledge, understanding, and skill
along with an awareness of and
appreciation for the rich connections
among mathematical strands and
between mathematics and other
disciplines.
Connected Mathematics


is organized around important
mathematical ideas and processes,
carefully selected and sequenced to
develop a coherent, connected
curriculum.
is problem-centered to promote deeper
engagement and learning for students.
Key features: CMP
connects mathematical ideas within
a unit, across units and across
grades.
 provides practice with concepts and
related skills.
 is for teachers as well as students.
 is research-based.

The single mathematical standard that
has been a guide for all the CMP
curriculum development is:

All students should be able to reason and
communicate proficiently in mathematics. They
should have knowledge of and skill in the use of the
vocabulary, forms of representation, materials,
tools, techniques, and intellectual methods of the
discipline of mathematics, including the ability to
define and solve problems with reason, insight,
inventiveness, and technical proficiency.
Curriculum Challenges

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Identifying the important concepts/big ideas
and their related concepts and procedures
Describing developmental trajectories
through which students have opportunities to
learn the requisite mathematics
Designing sequences of mathematical
problem tasks to develop these identified big
ideas
Organizing the problem tasks into a coherent,
connected curriculum
Focus of Development Tasks

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Engage student in mathematical
exploration
Support students’ learning to participate
in mathematical conversations
Promote analytic thinking--knowing both
how to, why to, and when to
Encourage reasoning and justification
Enactment Challenges

The negotiation between students and teacher
around mathematics tasks often results in a
genuinely challenging problem becoming an
exercise.

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Teachers do not like to see their students struggle.
Parents do not like to see their student struggle.
Yet, learning without struggle is unlikely.
Dilemmas for teachers

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Scaffolding learning without denying
children time to think and reason their
way through a mathematical task or
challenge.
Establishing the expectation that
students can and will persevere in
finding solutions to challenging
problems.
Motivating students to do so.
End Goals for grade 8 Math



What should a student know and be able to do
in each mathematics strand at the end of
grade eight?
What are the intermediate, related
mathematical ideas and techniques that should
be developed earlier in the middle grades to
support these ending goals?
How are these ideas supportive of or
supported by other strand development work?
Student Engagement

Ideas must be explored in sufficient depth to
allow students to make sense of them.

Mathematical tasks are the primary vehicle for
student engagement with the mathematical
concepts to be learned.
Posing mathematical tasks in context provides
support both for making sense of the ideas and
for cognitively processing them so that they
more easily can be remembered.

Developing the mathematics
 Students need examples and
encouragement to learn to ask themselves
questions that guide their thinking in new
mathematical situations.
 The mathematics must be accessible,
engaging, and yet demanding in ways that
promote students’ view of their own learning.
 The mathematical story line in a sequence of
developmental problems must be
transparent to both teachers and students.
Problem Criteria
･ A good problem has important, useful
mathematics embedded in it.
･ Investigation of the problem should contribute
to conceptual development.
･ Work on the problem should promote skillful
use of mathematics and opportunities to
practice important skills.
･ The problem should create opportunities to
assess what students are learning and where
they are experiencing difficulty.
Problems chosen

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Engage students and encourage classroom
discourse.
Allow various solution strategies or lead to
alternative decisions that can be taken and
defended.
Solving requires higher-level thinking and
problem solving.
Content of the problem connects to other
important mathematics.
Our Curriculum development goal

To create curriculum materials that
support teachers in bringing
mathematics and students
together so that students “learn.”
Curriculum analysis

What are some big ideas in
developing concepts and
procedures related to number?
Meanings and use
of whole numbers
Meanings and use
of rational numbers
Situation that give
rise to using, operating
with, and interpreting numbers
Operating on numbers:
Putting together
Taking apart
Duplicating
Sharing
Measuring
etc.
Number sense and
estimation skills
Properties and relationships
+ and – ;
x and ÷
Computational
Algorithms
An Example of Student
Work
The Orange Juice Problem
The Orange Juice Problem
Mix A: 2 cups concentrate, 3 cups cold water
Mix B: 1 cup concentrate, 4 cups cold water
Mix C: 4 cups concentrate, 8 cups cold water
Mix D: 3 cups concentrate, 5 cups cold water
Which recipe will make juice that is the
most “orangey”?

 Which recipe will make juice that is the
least “orangey”?
What strategies
are students using?
Which are
efficient and
generalizable?
Mixing Juice
• What can each part of the problem
contribute to students learning?
• What conceptual difficulties might
students encounter?
• What are the important connections to
other ideas and concepts, e.g.
knowledge packets?
Compare these four mixes for apple juice.
Mix W
5 cups
8 cups
concentration cold water
Mix Y
6 cups
concentration
9 cups
cold water
Mix X
3 cups
concentration
6 cups
cold water
Mix Z
3 cups
concentration
5 cups
cold water
a. Which mix would make the most “appley
juice?
b. Which mix would make the least “appley”
juice?
Mix W
5 cups
concentration
8 cups
cold water
Mix Y
6 cups
concentration
9 cups
cold water
Mix X
3 cups
concentration
6 cups
cold water
Mix Z
3 cups
concentration
5 cups
cold water
c. Suppose you make a single batch of each mix.
What fraction of each batch is concentrate?
d. Rewrite your answers to part (c) as percent
e. Suppose you make only 1 cup of Mix W.
How much water and how much concentrate do
you need?
Decide whether each is accurate. Give reasons.
•
Mix Y has the most water, so it will taste the least
“appley.”
•
Mix Z is the most “appley” because the difference
between the concentrate and water is 2 cups.
•
Mix X and Mix Y taste the same because you just add
3cups of concentrate and 3 cups of water to Mix X to
turn Mix X into Mix Y.
•
Mix Y is the most “appley” because it has only 1 1/2 cups
of water for each cup of concentrate. The others have
more water per cup.
A large table seats 10 people. A small table seats 8
people. Four pizzas are served on each big table
and 3 pizzas on each small table.
QuickTime™ and a
Photo - JPEG decompressor
are needed to see this picture.
The pizzas are shared equally by everyone at the
table. Does a person sitting at a small table get
the same amount as a person sitting at a large
table. Explain your reasoning.
Which table relates to 3/8? What do the 3 and
the 8 mean?
Selena uses the following reasoning:
10 - 4 = 6 and 8 - 3 = 5 so the large table is better.
Do you agree or disagree with Selena’s reasoning?
Suppose you put nine pizzas on the large table.
 What answer does Selena’s method give?
 Does this make sense?
 What can you say now about Selena’s method?
Our Theories

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Engagement matters
Coherence of mathematical trajectories
in materials matter
Mathematical discourse matters and
must be learned
Teacher’s questioning matters
And the MATHEMATICS CHOOSEN
matters
What is the basis for these
theories?
Examples of useful research from
cognitive science
Jim Greeno

For many mathematics is a collection of
propositions and procedures —some of
which they know and others they do not.
When these people encounter a
problem, the question is whether they
can tell what procedure to use and
remember how to do it.
Jim Greeno

When mathematics is treated as a set of
things to remember, its main affordance for
activity involves showing who has acquired
which pieces of knowledge.
Jim Greeno

When mathematics is treated as a domain
of interrelated concepts, its affordances are
much broader. They include sense making
and reasoning within the domain of
mathematics and in other domains, with
mathematics as a useful resource.
Bob Siegler, Carnegie Mellon

Research results suggest that
conceptually oriented teaching before a
focus on procedures is more effective
than a focus on procedures first.
Bob Siegler, Carnegie Mellon

Analytic thinking refers to a set of processes
for identifying the causes of events. He
distinguishes identifying causes of events
from features that usually accompany
events—the steps in an algorithm versus
features that are essential for the procedure
to apply.
Bob Siegler, Carnegie Mellon

Analytic thinking is both a cause and a
consequence of a second useful quality:
purposeful engagement. When Children
have a specific reason for wanting to
learn about a topic, they are more likely
to analyze the material so that they truly
understand it. In that sense analytic
thinking is a consequence of purposeful
engagement.
Sidney Pogrow: HOTS Revisited
Phi Delta Kappan, September 2005

Curriculum is designed to lead students
into key cognitive processes that
underlie all learning: 1) metacognition,
i.e. the ability to think about, develop,
and articulate problem-solving
strategies; 2) inference from context; 3)
decontextualization, i.e. generalizing
ideas and information from one context
to another; and 4) information
syntheses.
Sidney Pogrow: HOTS Revisited
Phi Delta Kappan, September 2005

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Lots of teacher talk and little student talk
does not work for disadvantaged
students.
This pattern needs to be changed. But
talk alone is not enough. Not all
conversation can stimulate powerful
student learning.
Curriculum Challenges

To take from what seems
convincing in theories to create a
stance on curriculum for thinking
and learning mathematics
Influence of Theory and Research:
Social Constructivism —

Our agreement with this theory is reflected in
the authors’ decision to write materials that
would support student-centered investigation
of mathematical problems and in our design of
problem content and formats that encourage
student-student and student-teacher dialogue
about the work.
Conceptual and Procedural
Knowledge —

We have interpreted research on the interplay of
conceptual and procedural knowledge to say
that sound conceptual understanding is an
important foundation for procedural skill, not an
incidental and delayed consequence of repeated
rote procedural practice.
Input from a variety of sources
personal learning and teaching
experiences, knowledge of theory and
research, and imagination of the authors;
 advice from mathematicians, teacher
educators, curriculum developers, and
mathematics education researchers;
 advice from experienced teachers and
teachers and students who used pilot and
field-test versions of the materials.

Compute 0.52  2.3
Will it be less than or greater
than 2?
Will it be less than or greater
than 1?
Compute 0.52  2.3
Would changing the form of the numbers help?
0.52 =
52
100
and 2.3
=2
3
10
or
23
10
so,

0.52  2.3 =
52
100

23
10
What is the product equal to in fraction notation?

What is the product equal to, when written as a decimal?
How does knowing the product as a fraction help you to
know how many places will be in the decimal form of the
product?
How is this related to the short cut of counting the
number of decimal places in the factors?
No clean slate exists in school
Children come to us having ideas,
not as blank slates nor with fully
formed viable ideas.
Curriculum Developers

Face the challenge of creating
experiences that push children to
examine their beliefs in the form of
misconceptions or ill-formed
conceptions
An example


Children learn about multiplication and
division first with whole numbers. They learn
to believe that multiplication makes larger and
division makes smaller.
How do we focus on the incorrectness of this
belief in a way that allows students to
confront their misconceptions and make
progress toward building new more accurate
conceptions?
Operations

Can you find two numbers whose product is
larger than either factor?

Can you find two numbers whose product is
smaller than either factor?

Can you find two numbers whose product is
between the two factors?
Operations

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Can you find two numbers whose quotient is
larger than either the dividend or the divisor?
Can you find two numbers whose quotient is
smaller than either the dividend or the
divisor?
Can you find two numbers whose quotient is
between the dividend and the divisor?
Misconceptions

Opportunities to promote theories on the
grow

Perhaps we need to think in terms of
incomplete conceptions rather than
misconceptions.
Contexts

We seek to create context in which rich
mathematical discussion can take place.
Contextualized problems are the vehicle.
But for us, contexts are of at least three
kinds:

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Real world problem contexts
Fantasy problem contexts
Mathematical problem contexts
Contexts: real world example

Deciding the best phone company
to choose given two company plans
with different conditions.
Fantasy context

You are standing at the origin in an
infinite forest with a tree the width of a
line at each lattice point. Can you walk
out of the forest in a straight line and
never hit a tree?
Mathematical context


What is true about the number between
any two twin primes greater than 3?
Why did I restrain the problem to twin
prime greater than 3?
Mathematical Connections

Curriculum writers have to purposefully
organize curriculum so that problem
tasks focus students attention on what
we want them to see and what
connection we want them to make
among what they have already
experienced and the mathematics in the
new task situation. Connections are not
made without effort.
Where are we heading?

One serious constraint in the reform
vision is the changes in pedagogy
required of teachers and the
acceptance of a different view of
mathematical activity and competence.
The BIG curriculum question-
What is accessible to students with
what kind of support from the
curriculum, parents and teachers?
The BIG research question-
What is learned by which students from
what curriculum in what kind of
classroom environment with what level
of support for the teacher and the
parents?
Penultimate comment:

We can create very beautiful and
compelling descriptions of what we are
attempting, what our conceptual
frameworks are, etc., and then usually
have to spend decades growing into or
changing our descriptions.