Perspectives of a Curriculum Developer Glenda Lappan Michigan State University.

Perspectives of a Curriculum Developer

Glenda Lappan

Michigan State University

View One:A series of skills and procedures learned and practiced.

Contrast these two views of learning and curriculum.

View Two:A set of experiences with important mathematics that stresses meaning, use, connections, AND proficiency with related procedures and algorithms.

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.

Two forms of problem solving:

Search your memory for available algorithms or procedures.

45 ÷ 7

A ticket to the Blue Rock Concert costs $7.How many tickets can you buy if you have $45?

A mini van can carry 7 school children. How many vans are needed for a field trip for 45 children?

Sue got paid $45 for mowing lawns last week. She worked 7 hours. How much per hour did she get paid?

A swimming pool needs to be cleaned every 45 days. How many weeks is that?

6.4285714

6 tickets

7 vans

$6.43 per hour

6 weeks and 3 days

What mathematical

occasions arise in classrooms that teachers have to

navigate?

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.

The need for teachers

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 14divided 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

1. 3/4 is bigger than 5/82. 3/4 is smaller than 1 whole3. 4/4 is bigger than 3/44. 13/16 is bigger than 3/45. 32/16 is 20/16 bigger than 3/4

Source: San Diego State University- Judy Sowder

Sam’s response

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 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

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. 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 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 mathematicsStudents 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 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

Operating on numbers: Putting together Taking apart Duplicating Sharing Measuring etc.

Properties and relationships + and – ; x and ÷

Computational Algorithms

Number sense and estimation skills

Situation that give rise to using, operating with, and interpreting numbers

An Example of Student Work

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”?

The Orange Juice Problem

What strategies are students using? Which are efficient and generalizable?

• 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?

Mixing Juice

Mix W Mix X 5 cups 8 cups 3 cups 6 cups concentration cold water concentration cold water

Mix Y Mix Z 6 cups 9 cups 3 cups 5 cups concentration cold water concentration cold water

Compare these four mixes for apple juice.

a. Which mix would make the most “appley juice?

b. Which mix would make the least “appley” juice?

Mix W Mix X 5 cups 8 cups 3 cups 6 cups concentration cold water concentration cold water

Mix Y Mix Z 6 cups 9 cups 3 cups 5 cups concentration cold water concentration 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 percente. Suppose you make only 1 cup of Mix W.

How much water and how much concentrate do you need?

• 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.

Decide whether each is accurate. Give reasons.

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.

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?

QuickTime™ and aPhoto - JPEG decompressor

are needed to see this picture.

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 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

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 = and 2.3 =

so,

0.52 2.3 =

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?

€

52

100

€

23

10 or

23

10

€

52

100×

23

10

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 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: 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.