Perspectives of a Curriculum Developer

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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.

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### Perspectives of a Curriculum Developer

Glenda Lappan

Michigan State University

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.

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

6 weeks

and 3 days

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

What mathematical

occasions arise in classrooms that teachers have to navigate?

The need for teachers

• 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.

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. 3/4 is bigger than 5/8

2. 3/4 is smaller than 1 whole

3. 4/4 is bigger than 3/4

4. 13/16 is bigger than 3/4

5. 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
• 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 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
• 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

Number sense and

estimation skills

Operating on numbers:

Putting together

Taking apart

Duplicating

Sharing

Measuring

etc.

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 themost    “orangey”?
• Which recipe will make juice that is the least     “orangey”?

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 WMix X

5 cups 8 cups 3 cups 6 cups concentration cold water concentration cold water

Mix YMix Z

6 cups 9 cups 3 cups 5 cups concentration cold water concentration cold water

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

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

Mix WMix X

5 cups 8 cups 3 cups 6 cups concentration cold water concentration cold water

Mix YMix 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 percent
• 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.

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
• 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.
• 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.
• 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.

Will it be less than or greater than 2?

Will it be less than or greater than 1?

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?

### 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.