Teaching Mathematics through Problem Solving Emma Ames Jim Fey Mary Jo Messenger Hal Schoen 1

2 Problem Solving Problem solving... can serve as a vehicle for learning new mathematical ideas and skills.... A problem- centered approach to teaching mathematics uses interesting and well-selected problems to launch mathematical lessons and engage students. In this way, new ideas, techniques, and mathematical relationships emerge and become the focus of discussion. Good problems can inspire the exploration of important mathematical ideas, nurture persistence, and reinforce the need to understand and use various strategies, mathematical properties, and relationships. (Principles and Standards for School Mathematics, National Council of Teachers of Mathematics 2000, p. 182)

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4 Selecting Classroom Tasks - Basic Questions R. Marcus & J. T. Fey  Will working on the tasks foster students’ understanding of important mathematical ideas and techniques?

5 Selecting Classroom Tasks - Basic Questions R. Marcus & J. T. Fey  Will the selected tasks be engaging and problematic, yet accessible, for many students in the target classes?

6 Selecting Classroom Tasks - Basic Questions R. Marcus & J. T. Fey  Will work on the tasks help students develop their mathematical thinking—their ability and disposition to explore, to conjecture, to prove, to represent, and to communicate their understanding?

7 Selecting Classroom Tasks - Basic Questions R. Marcus & J. T. Fey  Will the collection of tasks in a curriculum build coherent understanding and connections among important mathematical topics?

8 Interesting Variations on a Basic Problem Goldenberg & Walter Find the mean of 7, 4, 7, 6, 3, 8, and 7. 1.What if only five of the seven data are given?Can we determine the missing data if we know the mean of the original seven? 2.What if we compute the mean of each possible combination of only five of the given seven numbers? (How many such combinations are possible?) What could we learn from, say, a histogram of those means?

9 Interesting Variations on a Basic Problem Goldenberg & Walter Find the mean of 7, 4, 7, 6, 3, 8, and 7. 3.What if the original seven numbers are sampled from a population consisting of eight numbers? What might we reasonably infer about the eighth number? Do ideas from problem 2 help answer that question? 4.What if we know the mean but none of the data? What, if anything, could we say about the data? What possible sets of data would fit?

10 Some Questions That Promote Understanding - D. A. Grouws  How did you decide on a solution method to try?  How did you solve the problem?  Did anyone solve it in a different way?  How would you compare these solution methods?

11 Some Questions That Promote Understanding - D. A. Grouws  Which of the solution methods do you like best? Why?  Can you tell me how you solved the problem without saying the answer?  Does this remind you of any other problems you have solved?

12 Teaching Mathematics through Problem Solving: Research Perspectives M. K. Stein, J. Boaler, & E. A. Silver  The research on TMPTS and on curricula designed to support it suggests both the feasibility and efficacy of this approach.  When TMPTS is implemented effectively, students (compared to those taught traditionally) are likely to better understand mathematical concepts, to be willing to tackle challenging problems, and to see themselves as capable of learning mathematics.

13 Teaching Mathematics through Problem Solving: Research Perspectives M. K. Stein, J. Boaler, & E. A. Silver  TMPTS is challenging and to do it well teachers need support, including good curriculum materials and strong professional development.  TMPTS can work with a wide range of students, but the level of student support required may differ depending on the students’ mathematical background and interest.

14 Teaching Mathematics through Problem Solving: Research Perspectives M. K. Stein, J. Boaler, & E. A. Silver  Which of the solution methods do you like best? Why?  Can you tell me how you solved the problem without saying the answer?  Does this remind you of any other problems you have solved?

15 Some Questions That Promote Understanding - D. A. Grouws How can we change the problem to get another interesting problem?  What mistakes do you think some students might make in solving this problem?

16 In addition to learning mathematics, students learn to be good problem solvers. What Happens in the Classroom When Mathematics is Taught Through Problem Solving?

17 What Happens in the Classroom When Mathematics is Taught Through Problem Solving?  Thinking and problem solving are the fundamental part of our lessons.

18 What Happens in the Classroom When Mathematics is Taught Through Problem Solving? Technical reading, writing, and communicating are emphasized. Just look at this work young man. Just look at at this work young man. You’ve got some explaining to do. Einstein as a boy

19 Team Work

20 What Happens in the Classroom When Mathematics is Taught Through Problem Solving? Real-world problems are used frequently and answers are given in terms of what makes sense for any given situation. What is a Problem?

21 Problems must have meaning for students.

22 Teaching Equation Solving and Inequalities Through Problem Solving Cable TV (CPMP Year 1) X = 75 – 2.5X

23 Teaching Equation Solving and Inequalities Through Problem Solving Cable TV (CPMP Year 1) 30 = X

24 Teaching Equation Solving and Inequalities Through Problem Solving Cable TV (CPMP Year 1) 75 – 2.5X > 40

25 Teaching Equation Solving and Inequalities Through Problem Solving Cable TV One way to solve the equations or inequality is to make tables and graphs of (time, share) data for the two models and look for key points in each.

Tables and Graphs 30 = X X = 75 – 2.5X 26 XY1Y 

27 Teaching Equation Solving and Inequalities Through Problem Solving Lines (CPMP Year 1) The next diagram shows linear models from four rubber band experiments, all plotted on the same grid. What does the pattern of those graphs suggest about the similarities and differences in the experiments?

28 Teaching Equation Solving and Inequalities Through Problem Solving Lines (CPMP Year 1)

29 Teaching Equation Solving and Inequalities Through Problem Solving Lines (CPMP Year 1) (a). Sharing the work among your group members, make four tables of (weight, length) pairs, one table for each linear model, for weights from 0 to 10 ounces. (b).According to the tables, how long were the different rubber bands without any weight attached? How is that information shown on the graphs? (c).Looking at data in the tables, estimate the rates of change in length for the four rubber bands as weight is added. How are those patterns shown on the graphs?

30 Teaching Equation Solving and Inequalities Through Problem Solving Lines (CPMP Year 1)

31 Teaching Equation Solving and Inequalities Through Problem Solving Lines (CPMP Year 1)

32 The Bears Problem

33 The Bears Problem  Various Levels  Middle School  Algebra  Precalculus

34 The Bears Problem

35 The Bears Problem

36 The Bears Problem

37 The Bears Problem

38 The Bears Problem

39 The Bears Problem

40 The Bears Problem

41 The Bears Problem

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43 The Bears Problem

44 Learning Through Problem Solving Students Actively Participate, Reason, and Explain to Others

45 Teaching Through Problem Solving Establish the norms that students’ responses should include a rationale, students should strive to make sense of their own methods and those of their classmates, and students should ask questions and raise challenges when they do not understand.

46 Time to Reflect

47 Frustration is Part of a Real Problem

48 The Satisfaction of Solving the Problem “ Well, then when you get your grades up to a B average, THEN you can choose your own wallpaper.

49 Teaching Through Problem Solving Always be aware of who is doing the thinking, the teacher or the student.

50 Byproducts  Self esteem  Motivation  Better Understanding

51 Materials to Support Teaching Mathematics Through Problem Solving 51 Projects at All Levels: The K – 12 Mathematics Curriculum Center ( Elementary Projects: The ARC Center ( Everyday Mathematics ( Investigations in Number Data, and Space TERC ( Math Trailblazers (

52 Materials to Support Teaching Mathematics Through Problem Solving 52 Middle School Projects: The ShowMe Center ( Connected Mathematics Project ( Mathematics in Context ( MathScape Curriculum Center ( MATHThematics Project ( Pathways/MMAP Curriculum (

53 Materials to Support Teaching Mathematics Through Problem Solving 53 High School Projects: COMPASS ( Core-Plus Mathematics Project ( Interactive Mathematics Project ( MATH Connections ( Applications /Reform in Secondary Education ( SIMMS Integrated Mathematics (

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59 Your Challenge! “Education is not the filling of a pail, but the lighting of a fire.” William Butler Yeats