Every Child Mathematically Proficient from the Learning First Alliance, 1998

1. Reconceptualizing Mathematics: Courses for Prospective and Practicing TeachersSusan D. NickersonMichael Maxon San Diego State University

2. Teachers matter.From a carefully chosen sample of 2525 adults, representing a cross-section of U. S. adults, an overwhelming majority agreed that improving the quality of teaching was the most important way to improve public education.

3. Every Child Mathematically Proficient from theLearning First Alliance, 1998 Before It’s Too Late: A Report to the Nation from theNational Commission on Mathematics and Science Teaching for the 21st Century, 2000 What Matters Most: Teaching for America’s Future and No Dream Denied from the National Commission on Teaching and American’s Future, 1996 & 2003 The Mathematical Education of Teachers from the Conference Board of the Mathematical Sciences, 2001 Mathematical Proficiency for All Students from the RAND Mathematics Study Panel, 2003 Educating Teachers of Science, Mathematics, and Technology from the National Research Council, 2001 Undergraduate Programs and Courses in the Mathematical Sciences from MAA

4. Consider the following: A teacher wants to pose a question that will show him whether his students understand how to put a series of decimals in order from smallest to largest. Which of the following sets of decimal numbers will help him assess whether his pupils understand how to order decimals? Choose each that you think will be useful for his purpose and explain why. .123 1.5 2 .56 .60 2.53 3.14 .45 .6 4.25 .565 2.5Or would each of these work equally well for this purpose?

5. There is a growing understanding that the mathematics of the elementary and middle school is not trivial, and that teachers need more preparation and different preparation than has been common.

6. Looking aheadRecommendationsOur focus with examples of content and instructional possibilitiesGuiding QuestionsExamples of our PD structures and delivery

7. In 1998, the National Research Council (NRC) appointed a committee of mathematicians, scientists, mathematics and science educators including K-12 teachers, and a business representative to investigate ways of improving the preparation of mathematics and science teachers.

8. The Committee’s 2001 report, Educating Teachers of Mathematics, Science, and Technology, includes recommendations about the characteristics teacher education programs in mathematics and science should exhibit.

9. Programs should have the following features: • Be collaborative endeavors developed and conducted by mathematicians, education faculty, and practicing K-12 teachers; • Help prospective teachers to know well, understand deeply, and use effectively the fundamental content and concepts of the disciplines that they teach; • Teach content through the perspectives and methods of inquiry and problem-solving

10. Programs should have the following features: • Be collaborative endeavors developed and conducted by mathematicians, education faculty, and practicing K-12 teachers;• Help prospective teachers to know well, understand deeply, and use effectively the fundamental content and concepts of the disciplines that they teach;• Teach content through the perspectives and methods of inquiry and problem-solving

11. Collaborative endeavors are a part of our:• Pre-service courses • Professional development: summer courses, usually at the university (3 days to 2 weeks) partnership with several districts, urban & rural classes through the university on-line hybrid courses

12. Integrating Learning Mathematics with Practice Director of Mathematics Site Mathematics Administrators Mathematics Instructors (SDSU) Math Teacher Ed Instructors (SDSU) Teachers inprogram All Teachers at site Math Resource Teacher

13. Programs should have the following features: • Be collaborative endeavors developed and conducted by mathematicians, education faculty, and practicing K-12 teachers;• Help prospective teachers to know well, understand deeply, and use effectively the fundamental content and concepts of the disciplines that they teach;• Teach content through the perspectives and methods of inquiry and problem-solving

14. What content preparation do teachers need to have to teach middle school mathematics well? The 2001 document from CBMS, The Mathematical Education of Teachers recommends that for teaching middle school mathematics: At least 21 semester hours of mathematics, some of which should include a study of elementary mathematics, some of which should focus on the middle grades, and some of which include the study of mathematics proper.

15. Content recommendations in MET are made in four areas: 1. Number and Operations2. Algebra and Functions3. Measurement and Geometry4. Probability and Statistics

16. Number and Operations• Understand and be able to explain the mathematics that underlies the procedures used for operating on whole numbers and rational numbers.

17. Example 1: Multiplication of Fractions“Juanita had mowed 4/5 of the lawn, and her brother Jaime had raked 2/3 of the mowed part. What part of the lawn had been mowed and raked?”

19. Example 2: Why do we invert and multiply when dividing fractions? Write a story problem that can be represented and solved by 2 1/2 ÷ 1/3 • First, divide by the unit fraction: 1 ÷ 1/3 • 1 ÷ 1/3 is 3, or 3/1 • So 2 ÷ 1/3 is twice 3, or 6, or • in general k ÷ 1/n = k n

23. 2. Algebra and Functions• Understand and be able to work with algebra as a symbolic language, as a problem solving tool, as generalized arithmetic, as generalized quantitative reasoning, as a study of functions, relations, and variation, and as a way of modeling physical situations.

24. 2. Algebra and Functions• Understand and be able to work with algebra as a symbolic language, as a problem solving tool, as generalized arithmetic, as generalized quantitative reasoning, as a study of functions, relations, and variation, and as a way of modeling physical situations.

26. Coping strategies:1. Just add2. Guess at the operation to be used

27. Limited Strategies3. Look at the number sizes and use those to tell you which operation to use.4. Try all operations and choose the most reasonable answer.5. Look for “key words.”6. Decide whether the answer should be larger or smaller than the given numbers, then decide on the operation.

28. Desired strategy:7. Choose the operations with the meaning that fits the story. Perhaps draw a picture to help understand the problem.

29. Consider this problem: Dieter A: I lost 1/8 of my weight. I lost 19 pounds.Dieter B: I lost 1/6 of my weight, and now you weigh 2 pounds more than I do.How much weight did Dieter B lose?

30. First, make a list of relevant quantities • Dieter A’s weight before the diet • Dieter A’s weight after the diet • Fraction of weight lost by A • Amount of weight lost by A • Difference in weight of A and B before diet • Difference in weight after diet • Etc.

32. Now we consider the values of the quantities • Dieter A’s weight before the diet: ? • Dieter A’s weight after the diet: ? • Fraction of weight lost by A: 1/8 • Amount of weight lost by A: 19 pounds • Difference in weight of A and B before diets: ? • Difference in weights after diets: 2 pounds (A is 2 pounds less than B • Etc.

33. Dieter A’s original weight (before diet) Before diet Shows weight loss of 1/8 of original diet. Weight lost was 19 lb. After diet

34. A’s weight before diet: 19 x 8 = 152A’s weight after the diet 152–19=133B’s weight after the diet: 133 + 2 = 135135 is 5/6 of B’s weight before the diet so B weighed 162 poundsB lost 162 – 135 = 27 pounds

35. 2. Algebra and Functions• Recognize change patterns associated with linear, quadratic, and exponential functions.Example: Growing Dots

36. 3. Measurement and Geometry• Identify common two-and three dimensional shapes and list their basic characteristics and propertiesExample here GSP quadrilateralsvenn

37. 4. Data Analysis, Statistics, and Probability• Draw conclusions with measurements of uncertainty by applying basic concepts of probabilityEx: three card poker

38. Programs should have the following features: • Be collaborative endeavors developed and conducted by mathematicians, education faculty, and practicing K-12 teachers;• Help prospective teachers to know well, understand deeply, and use effectively the fundamental content and concepts of the disciplines that they teach;• Teach content through the perspectives and methods of inquiry and problem-solving

39. Cynthia has 1 cup of sugar and each recipe requires: a) 1/2, b) 1/3, c)2/3, d) 3/4, e) 4/5 of a cup of sugar. How many recipes can she make?Model and draw each case. What number sentence describes each?Later generalize for more or less than one cup of sugar.

40. In a report by the MAA Committee on the Undergraduate Preparation in Mathematics (2004):“..does not mean the same thing as preparation for the further study of college mathematics. For example, while prospective teachers need knowledge of algebra..the traditional college algebra course with primary emphasis in developing algebra skills does not meet the needs of elementary [and middle school] teachers.”

41. Looking aheadRecommendationsOur focus with examples of content and instructional possibilitiesGuiding QuestionsExamples of our PD structures and delivery

42. Decisions about what to include rely on answers to guiding questions • Pre-service courses• Professional development: summer courses, usually at the university (3 days to 2 weeks) partnership with districts classes through the university on-line hybrid courses

43. Guiding Questions: What is the content at their grade level? With what have they had only superficial exposure? What would be difficult to learn from the curriculum? How does this content need to be extended? How does the research literature characterize the difficulties & student misconceptions? Does this group of teachers have particular needs?

44. What is the content at their grade level? With what have they had only superficial exposure? What would be difficult to learn from the curriculum? Ratio and proportional reasoning Minimal and sufficient definitions of quadrilaterals

45. Procedures can sometimes be learned from the curriculum. When we talk about content we are talking about conceptual understanding and procedural fluency.

46. 2) How does this content need to be extended?

47. 3) How does the research literature characterize the difficulties & K-12 student misconceptions? Specialized knowledge for teaching

48. Difficulties in algebra include • Misunderstanding the equals sign • Comprehending use of literal symbols as generalized numbers or variables • Expressing relationships in a variety of ways such as tables, graphs, and equations • Understanding the role of the unit • Ratio and Proportional reasoning

49. Misunderstanding the equals sign Students tend to misunderstand the equal sign as a signal for “doing something” rather than a relational symbol of equivalence.

50. Comprehending the many uses of literal symbols • A=L x W • 40=5x • 2a + 2b = 2(a + b) • Y = 3x + 5