Understanding Fourth Graders’ Mathematical Thinking: Issues and Insights for Teaching Fractions

1. Understanding Fourth Graders’ Mathematical Thinking: Issues and Insights for Teaching Fractions Susan Empson The University of Texas at Austin Smart Start Conference - July 13, 2006

2. Guiding Questions • What does it mean to understand fractions? • What kinds of problems help children develop their understanding of fractions? • How do you use the details of children’s thinking in your teaching?

3. Reflection • List two things that struck you as important from Dr. Franke’s talk yesterday • Share with your table • Choose one thing from the table to report – one sentence only

4. Some main points • Children use what they already understand to solve new problems. This leads to new understanding. • Understanding is generative • Teaching involves • Listening to children’s problem solving, • Figuring out what they understand, and • Building on that understanding

5. I. What does it mean to understand fractions?

6. Video Clip: Fifth grader Ally • What does Ally understand about fractions? What does Ally not understand? • Ally is an average fifth grader. What do you think accounts for how she thinks about fractions? • What does a teacher need to know to help Ally develop a deeper understanding of fractions? Discuss and record your answers on “Ally’s Mathematical Thinking” in handout packet

7. Video Clip: Fifth grader Ally

8. Circle the bigger fraction

9. Write as improper fraction or mixed number

10. 1. What does Ally understand about fractions?

11. What she said: • “1 is bigger than 4/3 because it’s a whole number” • “1/7 is bigger than 2/7, because usually (with fractions) you go down to the smallest number to get to the biggest number” • “1/2 is bigger than 3/10, because you just change the bottom number 1 more digit and it would be 1” • “1/2 is bigger than 4/6, same reason”

12. Her understanding: • Acts uncertain • Uses nonsensical rules • Believes all fractions are smaller than 1 • Relies on surface features of the symbol rather than understanding meaning of fractions to create equivalent mixed numbers and improper fractions • Not generative* *Generative -- leads to new concepts, strategies, procedures, and so on

13. 2. What do you think accounts for how Ally thinks about fractions? • A problem with Ally? • But so many students seem to have problems with fractions! She’s average. • A problem with the curriculum? • What is a typical approach to teaching fractions? • Does it support development of generative understanding?

14. A common curriculum approach to fractions:

15. If children learn fractions by doing lots of exercises like this one, what are they likely to think about fractions? • How much is shaded?

16. They think… • Fractions are pieces • Fractions are always smaller than a whole • Fractions values are determined by counting parts “A fourth is a little pie shape.” “4/3? That’s impossible!” “It’s 1/3 because 1 part out of 3 parts is shaded.”

17. Let’s watch two more students solving fractions problems • Here is the problem they solve: • Neither student has had direct instruction on adding fractions with unlike denominators Two sisters, Iris and Kathryne, are eating cookies. Iris has 3/4 of a cookie. Kathryne has 1/2 of the same sized cookie. If they put their pieces together to give to their mom, will it make more or less than 1 whole cookie? How much will it be?

18. Video Clip: Fourth grader Ebony • What does she understand about fractions? Record your observations on “Video Notes” handout

19. Video Clip: Fifth grader Crystal • What does she understand about fractions? • How does Crystal’s way of thinking about this problem compare to Ebony’s?

20. Ebony’s and Crystal’s understanding • Relationship between halves and fourths • 2 fourths can be put together to make 1 half • Halves can be cut into fourths • To add fractions, need to combine like units (fourths, halves) • Fractions can add to more than 1 whole • Understanding of concepts is somewhat separate from understanding of symbols (Crystal) • Used what they understood about fractions to generate new strategies for adding fractions

21. II. What kinds of problems help children develop their understanding of fractions?

22. Let’s solve some problems • Purpose: • To practice listening to and understanding each other’s thinking

23. Think-aloud problem-solving activity • Pair up • One of you solves problem, thinking aloud as you go • Read problem carefully • Then just start talking about the problem • “Hmm. I’ve never solved a problem like this one before. I think I’ll try… Nope, that didn’t work…” • Job is to keep going till it’s solved or you’re stuck • OK if unsure, make mistakes. • Other person listens • Say strategy back to first person, using your own words • Job is to understand what first person is thinking • Don’t help! (Listen, and do your best to understand.) • OK to ask clarifying questions as other person works • If time, switch roles and solve a second way

24. Problem #1 • 3 children want to share 2 candy bars equally. How much can each child have?

25. Sample children’s strategies “I cut the candy bars in half, to see if it would work and it did. Everybody gets a half. Then I cut the last half in three parts. Everyone gets another piece.”

26. “Each child gets 1 third from the first candy bar.

27. “Each child also gets 1 third from the second candy bar. That’s 2 thirds for each person.”

28. Mental strategy “I know that everyone can share each candy bar and get 1/3 of a candy bar. There’s 2 candy bars, so that 1/3, 2 times. It’s 2/3.”

29. Mental strategy 2 ÷ 3 = 2/3

30. Problem #2 • Eric and his mom are making cupcakes. Each cupcake gets 1/4 of a cup of frosting. They are making 20 cupcakes. How much frosting do they need?

31. 1/4 of a cup Sample children’s strategies 5 cups 4 cups 1 cup 2 cups 3 cups “…so 5 cups altogether.”

32. 1/4 of a cup So, 5, 6, 7, 8 -- that’s 2 cups. 9, 10, 11, 12 -- that’s 3 cups. 13, 14, 15, 16 -- that’s 4 cups. 17, 18, 19, 20 -- that’s 5 cups. 4 of these is 1 cup… …so 5 cups altogether.

33. 1/4 + 1/4 + 1/4 + 1/4 = 1 1/4 + 1/4 + 1/4 + 1/4 = 1 1/4 + 1/4 + 1/4 + 1/4 = 1 1/4 + 1/4 + 1/4 + 1/4 = 1 1/4 + 1/4 + 1/4 + 1/4 = 1 5 cups Q: What’s a number sentence for this problem? A: 20 x 1/4 = 5 (there are others)

34. Problem #3 • Ohkee has a snowcone machine. It takes 2/3 of a cup of ice to make a snowcone. How many snowcones can Ohkee make with 4 cups of ice?

35. Sample children’s strategies 4 3 1 2 5 6 “Ohkee can make 6 snow cones.”

36. “2/3 plus 2/3 is 1 and 1/3. If I add 1 and 1/3 three times, I get 4. I remember this from another problem. So there are six 2/3s in 4. The answer is she can make 6 snow cones.”

37. Q: What’s a number sentence for this problem? A: 4 ÷ 2/3 = 6 (there are others)

38. Problem #4 • 4 children are sharing 10 pancakes, so that each child gets the same amount. How much pancake can each child have, if they eat all the pancakes?

39. Sample child’s strategy 1 1 1 1 1 1 1 1 1 1 “Each child gets 1 fourth from each pancake. There are 10 pancakes. So each child gets 10 fourths altogether.”

40. Problem #5 • 12 children are sharing 9 pineapple cakes, so that each child gets the same amount. How much cake can each child have, if they eat all the cakes?

41. What do teachers need to know to develop fractions? • What types of problems are these? • What kinds of strategies do children use to solve these problems? • What is the mathematics that can be learned by solving and discussing these problems? • What are the fundamental concepts of fractions? • How do you help children coordinate concepts and fraction symbols?

42. Child’s Strategies | Understanding Problems Mathematics What do teachers need to know to develop fractions?

43. Problem types for fractions • Equal Sharing (with remainder, answer > 1) • 2 children want to share 5 cookies equally. How much can each child have? • 4 children want to share 10 candy bars so that each one gets the same amount. How much can each child have? • Equal Sharing (answer < 1) • There is 1 brownie for 4 children to share equally. How much brownie can each child have? • 3 children want to share 2 candy bars equally. How much can each child have? • (Division is total divided by number of groups)

44. Problem types for fractions, cont’d • Addition (combining like units) • Janie has 3/4 of a gallon of blue paint left over from painting her room. John has 2/4 of a gallon of the same blue paint left over from painting a table. How much blue paint do they have? • Equal Groups • Eric and his mom are making cupcakes. Each cupcake gets 1/4 of a cup of frosting. They are making 20 cupcakes. How much frosting do they need? • (Backwards sharing context) 6 friends shared some cookies. Each person got 2 2/3 cookies. How many cookies did they have altogether? • Division (total divided by the size of a group) • Okhee has a snow cone machine. It takes 2/3 of a cup of ice to make a snowcone. How many snowcones can Ohkee make with four cups of ice?

45. What’s the mathematics in children’s solutions to these problems? • Write down 1 thing that children can learn about fractions by solving problems like these • Hint: Think about the strategies you used • Link to Arkansas framework?

46. What’s the mathematics in children’s solutions to these problems? • Meaning of fractions -- what does 1/3 mean? • 1 thing shared equally by 3 people, each person gets 1/3 • 1 candy bar for every 3 people • 1 ÷ 3 = 1/3 • 1 part, with 3 equal parts to make a whole • These meanings generalize to improper fractions too • 4/3 is … • Fractional units can be combined • 1 third from one candy bar plus 1 third from another candy bar is 2 thirds • 1/3 + 1/3 = 2/3

47. What’s the mathematics in children’s solutions, cont’d • Fractional units can be combined no matter how many there are • 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 + 1/4 = 10/4 • Fractional numbers “fill in” the whole-number line • 2 1/4 cookies is more than 2 cookies but less than 3 cookies • A fractional amount can be expressed in many ways See “Fundamental Concepts of Fractions” in handout packet

48. Video clips: Equal sharing strategies There are 6 cakes at Anthony’s party. 8 children have to share the cakes equally. How much cake can each child have? If each child at the party brings a friend, how much cake can each child have?

49. III. How do you use this information in instruction?

50. Three approaches to teaching fractions • Introduce procedures and explain concepts • Emphasis on student discovery, with no conceptual analysis of discoveries • Discuss and extend concepts and procedures that come up in children’s problem solving (from Saxe et al., 1999)