Teaching Mathematics through Problem Solving in the Primary and Intermediate Grades

1. Teaching Mathematics through Problem Solving in the Primary and Intermediate Grades Diana V. Lambdin Indiana University USA

2. Planning and teaching problem-based lessons • Identify mathematical goals • Select an appropriate problem • Plan and carry out the three parts of the problem-based lesson: • Launch • Explore • Summarize

3. Effective problems: Involve important mathematics Consider what the children may know Can be approached in more than one way.

4. Let’s look at some examples of problems

5. A sample task Think about the number 5 broken into 2 different amounts. Draw a picture to show ways that 5 things can be in two parts. Make up a story to go with your picture.

6. A sample task Provide a collection of coins (real, pictures, or a list) -- for example, 8 nickels (5¢), 9 dimes (10¢), 2 pennies (1¢), and 7 quarters (25¢). Ask students to find the total in two different ways. (Can lead to discussion about various ways to make 10s or 100s.)

7. A sample task Find ways to measure our Halloween pumpkin. Write a letter to our pen pals that will help them know how big our pumpkin is.

8. A sample task A fifth grader was trying to put numbers in order from smallest to largest. This is what he did: 3.4 3.38 3.45 3.4026 What would you tell him?

9. Now let’s consider a problem AND the lesson launch: Teacher goals for a 3rd grade class: Engage children in exploring multiplication/division (x/÷) concepts Assess levels of student understanding of x/÷ -- for planning future lessons

11. The Doorbell Rang The Doorbell Rang, by Pat Hutchins, is a story about children sharing cookies among friends. Copyright laws prevent us from including the complete story in this powerpoint posted on the www.

12. Suggestions for using children’s literature as a lesson launch above all, enjoy the story first read the book aloud let children respond personally encourage problem responses in a variety of modes (pictures, words, symbols, number sentences)

13. Benefits of using children’s literature to teach mathematics • Motivation - stories have universal appeal • Familiarity - most children have experience with reading or hearing stories • Structure - stories can provide a defined context where problems can be explored • Language - stories bridge the gap between informal, oral language and formal symbolic mathematical language

14. More benefits of using children’s literature in math • Integration - stories can bring many aspects of the school curriculum together (reading, math, science, social studies, art) • Shared experience - stories provide children a common starting point for discussing math ideas

15. Suggestions for using stories as problem launches • Try to pose problems with more than one answer, or more than one way to get to the answer (or both) • Encourage multiple approaches and use of multiple representations • Let children know they will be asked to share their thinking and to react to each other’s ideas

16. Selecting a problem: What kinds of multiplication/division problems could you pose from this launch?

17. The “sharing cookies” problem Suppose you had 18 cookies. How many different ways could 18 cookies be shared fairly with breaking any? Using pictures, words, or numbers and math sentences, show all the different ways the cookies could be shared. Show how many people would get cookies, and how many cookies they would get.

18. Teaching actions “during” the problem solving Circulate and listen to students Ask questions Encourage student ideas Resist explaining or telling Provide hints and extensions as needed

25. Teaching actions “after” the problem solving • Promote a mathematical community of learners • Encourage student-student questions & dialogue • Expect explanations with all answers • Listen actively. Do not evaluate. • Summarize main ideas.

26. Another problem-based lesson: Pattern Block Trains Teacher goals: perimeter, pattern recognition, use of variables, generalization Pick a pattern block shape. Build trains of that shape only. Find the perimeter of each train. Predict the perimeter of any train.

27. Lesson launch: Let’s try it for triangles!

28. Hexagon Trains • Compute the perimeter for each of the first 4 trains. • Determine the perimeter for the 10th train without constructing it. • Write instructions for finding the perimeter of any train without constructing it. • Find as many different ways as you can to compute (and justify) the perimeter of any train.

29. The “Summarize” phase of the lesson Ask students to share their formulas -- in words, or in symbols. Ask for explanations, justifications. If students have different formulas, compare them. Do they give the same result?

30. Does the formula make sense with the blocks? • Sometimes students generate formulas by looking at tables of values. • Ask them to explain where the numbers in their equations come from. • Example: P = 4N + 2 Why 4? Why 2?

31. Comparing different formulas • P = 4N + 2 • P = 2N + 2N + 2 • P = (2N + 1) + (2N + 1) • P = (6 - 1) + 4(N - 2) + (6 - 1) • P = 5N - 1(N - 2)

32. Discussing Student Ideas Amanda, a sixth grade girl said: To find the perimeter of any hexagon train, take the number of blocks in the train, add the next number, then double the result. Does this always work? Does it make sense?

33. Next part of the workshopReturn to our original room for wrap-up discussion