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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. Guiding Questions. What does it mean to understand fractions?

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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

Guiding Questions Issues and Insights for Teaching Fractions

• 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?

Reflection Issues and Insights for Teaching Fractions

• List two things that struck you as important from Dr. Franke’s talk yesterday

• Choose one thing from the table to report – one sentence only

Some main points Issues and Insights for Teaching Fractions

• 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

I. What does it mean to understand fractions? Issues and Insights for Teaching Fractions

Video Clip: Fifth grader Ally Issues and Insights for Teaching Fractions

• 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

Video Clip: Fifth grader Ally Issues and Insights for Teaching Fractions

Circle the bigger fraction Issues and Insights for Teaching Fractions

Write as improper fraction or mixed number Issues and Insights for Teaching Fractions

1. What does Ally understand about fractions? Issues and Insights for Teaching Fractions

What she said: Issues and Insights for Teaching Fractions

• “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”

Her understanding: Issues and Insights for Teaching Fractions

• 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

• 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?

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

They think… this one, what are they likely to think about fractions?

• 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.”

Let’s watch two more students solving fractions problems this one, what are they likely to think about fractions?

• 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?

Video Clip: Fourth grader Ebony this one, what are they likely to think about fractions?

• What does she understand about fractions?

Record your observations on “Video Notes” handout

Video Clip: Fifth grader Crystal this one, what are they likely to think about fractions?

• What does she understand about fractions?

Ebony’s and Crystal’s understanding this one, what are they likely to think about fractions?

• 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

II. this one, what are they likely to think about fractions?What kinds of problems help children develop their understanding of fractions?

Let’s solve some problems this one, what are they likely to think about fractions?

• Purpose:

• To practice listening to and understanding each other’s thinking

Think-aloud problem-solving activity this one, what are they likely to think about fractions?

• Pair up

• One of you solves problem, thinking aloud as you go

• 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

Problem #1 this one, what are they likely to think about fractions?

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

Sample children’s strategies this one, what are they likely to think about fractions?

“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.”

Mental strategy 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.”

Mental strategy That’s 2 thirds for each person.”

2 ÷ 3 = 2/3

Problem #2 That’s 2 thirds for each person.”

• 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?

1/4 of a cup That’s 2 thirds for each person.”

Sample children’s strategies

5 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)

Problem #3 That’s 2 thirds for each person.”

• 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?

Sample children’s strategies That’s 2 thirds for each person.”

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)

Problem #4 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.”

• 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?

Sample child’s strategy 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.”

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.”

Problem #5 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.”

• 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?

What do teachers need to know to develop fractions? 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.”

• 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?

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?

Problem types for fractions 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.”

• 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)

Problem types for fractions, cont’d 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.”

• 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?

• Write down 1 thing that children can learn about fractions by solving problems like these

• Hint: Think about the strategies you used

• 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

• 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

• A fractional amount can be expressed in many ways

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?

• 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)

Developing children’s understanding problems?

• 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

• 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

There are 3 candy bars for 4 children to share equally. How much candy bar can each child have?

• 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

• Problem solving

notebook

- messy part

- neat part

Classroom video clip: Listening to the details of children’s thinking

12 children want to share 9 pineapple cakes so that everyone gets the same amount. How much cake can each child have?

• First, solve and discuss at your table.

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!

Helping children symbolize fractions children’s thinking, cont’d

• 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

Symbolizing fractions, cont’d children’s thinking, 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?”

Writing problems children’s thinking, cont’d

• 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.

Possible problems children’s thinking, cont’d

Write an equal sharing problem that a child could solve entirely by repeated halving.

• 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 could involve the fractions 2/5 and 4/10 in the possible solutions.

Continuing your learning… children’s thinking, cont’d

• Now let’s plan for using problems like these back at your school

Problems children’s thinking, cont’d

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)

To consider… children’s thinking, cont’d

• 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?

Website children’s thinking, cont’d

• http://www.edb.utexas.edu/empson

Mathematical proficiency children’s thinking, cont’d

• Thinking mathematically involves

• Conceptual understanding

• Procedural fluency

• Strategic competence