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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

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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?

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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

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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

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I. What does it mean to understand fractions?

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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

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Video Clip: Fifth grader Ally

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Circle the bigger fraction 1/61/31/72/7 14/33/101/2 3/61/2

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Write as improper fraction or mixed number 5 2/3 13/6

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1. What does Ally understand about fractions?

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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”

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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

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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?

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A common curriculum approach to fractions:

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

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

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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?

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

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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?

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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

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

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

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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

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

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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.”

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

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

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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.”

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Mental strategy 2 ÷ 3 = 2/3

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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?

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Sample children’s strategies 1/4 of a cup 1 cup2 cups3 cups 4 cups 5 cups “…so 5 cups altogether.”

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

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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)

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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?

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Sample children’s strategies “Ohkee can make 6 snow cones.”

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“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.”

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

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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?

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

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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?

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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?

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

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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)

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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?

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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?

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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

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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

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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?

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III. How do you use this information in instruction?

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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)

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Developing children’s understanding Use problem contexts to elicit children’s thinking about fractions Equal Sharing good place to start Then problems that involve combining like fractional units Equal Groups Division (Total divided by size of group) Ask children to solve problems in ways that make sense them. This helps children develop fundamental concepts with understanding

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Developing children’s understanding, cont’d Ask children how they solved problems. Probe their understanding. Introduce symbols, number sentences, and mathematical language to go with strategies Don’t rush using and manipulating symbols

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For example, instead of starting here…

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…start with an Equal Sharing problem Have materials for children to create fractions (nothing fancy) By drawing By folding or cutting Set expectation that children solve in way that makes sense to them (i.e., that builds on their understanding) Share and discuss strategies Use your own judgment about what to do and when to do it, by listening to children There are 3 candy bars for 4 children to share equally. How much candy bar can each child have?

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Solving problems and recording thinking Problem solving notebook - messy part - neat part

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Classroom video clip: Listening to the details of children’s thinking First, solve and discuss at your table. 12 children want to share 9 pineapple cakes so that everyone gets the same amount. How much cake can each child have?

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Classroom video clip: Listening to the details of children’s thinking, cont’d Then, watch teacher interact with two 5th graders who have solved this problem What do these boys understand about fractions? What does the teacher do to find out what the boys understand? What would you do next with these boys ? There is no one right answer!

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Helping children symbolize fractions Let children use physical materials to create fractional amounts (draw, fold, cut, shade): Use fraction words: 2 thirds of a candy bar a third + a third See handout in packet

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Symbolizing fractions, cont’d Relate unknown fractions to well known fractions, such as 1/2 or 1/4: “it’s more than a fourth, but less than a half” “it’s smaller than a quarter” Use language that emphasizes relationship of 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?”

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Writing problems To elicit children’s understanding of fractions To “steer” development of fractions 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.

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Possible problems Problems where # of sharers is 2, 4, 8, … (power of 2) Example: 8 children are sharing 6 quesadillas so that everyone gets the same amount. How much can one child have? 10 children are sharing 4 packages of modeling clay equally. How much clay can each child have? [20 children, 8 packages…] 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.

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Continuing your learning… Now let’s plan for using problems like these back at your school

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Problems 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)

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To consider… How do you think your students will solve these problems? How will you pose these problems to your students? Pull out a few students? Give to whole class? Etc. What could you learn from each student as you listen to their strategies?

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Website

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Mathematical proficiency Thinking mathematically involves Conceptual understanding Procedural fluency Strategic competence Adaptive reasoning Productive disposition

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