Perspectives of a Curriculum Developer

1. Perspectives of a Curriculum Developer Glenda Lappan Michigan State University

2. 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.

3. 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.

4. 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?

5. What mathematical occasions arise in classrooms that teachers have to navigate?

6. 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.

7. Young Children’s Reasoning

8. 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

9. Design a monograph to show what you know about 3/4. Source: San Diego State University- Judy Sowder

10. Sally: Source: San Diego State University- Judy Sowder

11. 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

12. Sandy: I found them all! Source: San Diego State University- Judy Sowder

13. 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.

14. 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.

15. 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.

16. 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.

17. 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

18. 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

19. 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.

20. 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.

21. 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?

22. 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.

23. 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.

24. 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.

25. 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.

27. Our Curriculum development goal • To create curriculum materials that support teachers in bringing mathematics and students together so that students “learn.”

28. Curriculum analysis • What are some big ideas in developing concepts and procedures related to number?

29. 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

30. An Example of Student Work The Orange Juice Problem

31. 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”?

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

40. 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?

41. 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?

42. 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?

43. 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.

44. 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?

45. 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?

46. 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

47. What is the basis for these theories? Examples of useful research from cognitive science

48. 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.

49. 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.

50. 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.