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. Guiding Questions. What does it mean to understand fractions?
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Susan Empson
The University of Texas at Austin
Smart Start Conference  July 13, 2006
Discuss and record your answers on “Ally’s Mathematical Thinking” in handout packet
*Generative  leads to new concepts, strategies, procedures, and so on
“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.”
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?
Record your observations on “Video Notes” handout
“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.”
“Each child gets 1 third from the first candy bar. this one, what are they likely to think about fractions?
“Each child also gets 1 third from the second candy bar. That’s 2 thirds for each person.”
“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.”
2 ÷ 3 = 2/3
1/4 of a cup That’s 2 thirds for each person.”
Sample children’s strategies5 cups
4 cups
1 cup
2 cups
3 cups
“…so 5 cups altogether.”
1/4 of a cup That’s 2 thirds for each person.”
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.
1/4 + 1/4 + 1/4 + 1/4 = 1 That’s 2 thirds for each person.”
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)
4
3
1
2
5
6
“Ohkee can make 6 snow cones.”
“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.”
Q: What’s a number sentence for this problem? 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.”
A: 4 ÷ 2/3 = 6 (there are others)
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.”
Child 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.”’s Strategies

Understanding
Problems
Mathematics
What do teachers need to know to develop fractions?See “Fundamental Concepts of Fractions” in handout packet
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?
(from Saxe et al., 1999)
There are 3 candy bars for 4 children to share equally. How much candy bar can each child have?
notebook
 messy part
 neat part
12 children want to share 9 pineapple cakes so that everyone gets the same amount. How much cake can each child have?
a third + a third
See handout in packet
such as 1/2 or 1/4:
“it’s more than a fourth, but less than a half”
“it’s smaller than a quarter”
fractional quantity to unit instead of number of pieces.
“how many of this piece would fit into the whole candy bar?”
instead of “how many pieces is the candy bar cut into?”
Write an equal sharing problem that a child could solve entirely by repeated halving.
Write an equal sharing problem that could involve the fractions 2/5 and 4/10 in the possible solutions.
Write an equal sharing problem that a child could solve entirely by repeated halving.
Example: 8 children are sharing 6 quesadillas so that everyone gets the same amount. How much can one child have?
Write an equal sharing problem that could involve the fractions 2/5 and 4/10 in the possible solutions.
6 children are having breakfast at a pancake restaurant. The waitress brings them 20 banana pancakes to share. If everyone gets the same amount, and they eat all of the pancakes, how much pancake can each child have?
Tom has ___ dog biscuits. His dog, Harmony, eats ___ biscuits a day. How many days will it take for Harmony to eat all of the dog biscuits?
(7, 1/4) (12, 1 1/3)