I am really excited about our second hop in the Fly on the Math Teacher's Wall series. (You can still visit our first hop about place value.) We have even more math bloggers joining us this time to talk about fractions. I want to start with an example of student misconceptions.

"A teacher drew a picture of a brownie that had been cut with a slice removed and asked a first grader and a third grader to decided how much had been eaten.

The first grader studied the picture for a moment and then said the missing piece was 'half of a half.'

The third grader said it was an impossible amount, because the pieces were not all the same size and it therefore could not be '1 out of 3.'" (Epson, Susan B., and Levi, Linda. Extending Children's Mathematics: Fractions and Decimals. 2011: Heinemann, Portsmouth, NH, xvii)

How does this happen? The authors' research suggests that young "children have some conceptually sound understanding of fractions, even before instruction", but "children can learn to ignore this understanding in favor of models introduced in school that portray fractions in narrow ways." (xvii-xviii) This actually blew my mind. Students ignore the conceptual understanding they bring with them from real world experiences in favor of what we teach in the classroom, even when their prior understanding is correct. How? Why? Our teaching can actually hurt a student's understanding? That's a lot to consider.

I'm going to pull a lot of information from this book to help all of us reconsider how we teach fractions. I can't even begin to touch on all the great ideas in this book. (If you are interested in reading the book, click on the cover to go to Amazon.) This book starts with an amazing forward written by Thomas Carpenter whose research on cognitively guided math instruction has been so influential. He says, "Much of the research on fractions has documented students' misconceptions and computational errors. As a consequence, it has generally assumed that children's natural insight into whole-number arithmetic does not carry over to fractions and decimals and that there are inherent difficulties in learning even basic fraction and decimal concepts and skills." (xi) He continues to say, "By holding back on the introduction of fraction symbols, instruction offers a chance for students to ground their experience with fraction symbols in conceptual knowledge and avoid the errors resulting from attending to superficial features of fraction notation." (xii)

I love that last sentence. Here's my paraphrase of it: by holding back on our introduction of fraction symbols we can give our students a chance to ground their experience with fraction symbols in conceptual knowledge. When I teach addition and subtraction, I never go straight to the equation. Yet when I teach fractions I start by showing them the symbols. In fact just last year I blogged about my introduction of fractions using cookies. Feeling a little embarrassed now. It was a great lesson and it will still be a great lesson. Someday. Yes, someday when I'm teaching comparison of fractions not when I'm introducing fractions for the very first time. Putting those Chips Ahoy cookies back on the shelf for another day because "teachers often do not devote enough time to helping children build meaning for fractions before moving on to equivalencies, comparisons and operations involving fractions." (p. 5)

So my first question is how do I introduce fractions without the symbols? Start with word problems. A foundation built upon story problems that require equal sharing are perfect for helping students draw "on their intuitive understanding of sharing, measuring, grouping and distribution." (p. xix) In fact, solving and discussing equal share problems as early as kindergarten and first grade "prepares students to learn from a variety of other problems involving fractions."(xxi) So, new plan this year..lots and lots of problem solving involving fractions. Here is an example from the book:

So why is this problem a good way to introduce and develop fractions?

Here are a few more examples that use both proper fractions (less than one) and improper fractions (mixed numbers):

The first grader studied the picture for a moment and then said the missing piece was 'half of a half.'

The third grader said it was an impossible amount, because the pieces were not all the same size and it therefore could not be '1 out of 3.'" (Epson, Susan B., and Levi, Linda. Extending Children's Mathematics: Fractions and Decimals. 2011: Heinemann, Portsmouth, NH, xvii)

How does this happen? The authors' research suggests that young "children have some conceptually sound understanding of fractions, even before instruction", but "children can learn to ignore this understanding in favor of models introduced in school that portray fractions in narrow ways." (xvii-xviii) This actually blew my mind. Students ignore the conceptual understanding they bring with them from real world experiences in favor of what we teach in the classroom, even when their prior understanding is correct. How? Why? Our teaching can actually hurt a student's understanding? That's a lot to consider.

I'm going to pull a lot of information from this book to help all of us reconsider how we teach fractions. I can't even begin to touch on all the great ideas in this book. (If you are interested in reading the book, click on the cover to go to Amazon.) This book starts with an amazing forward written by Thomas Carpenter whose research on cognitively guided math instruction has been so influential. He says, "Much of the research on fractions has documented students' misconceptions and computational errors. As a consequence, it has generally assumed that children's natural insight into whole-number arithmetic does not carry over to fractions and decimals and that there are inherent difficulties in learning even basic fraction and decimal concepts and skills." (xi) He continues to say, "By holding back on the introduction of fraction symbols, instruction offers a chance for students to ground their experience with fraction symbols in conceptual knowledge and avoid the errors resulting from attending to superficial features of fraction notation." (xii)

I love that last sentence. Here's my paraphrase of it: by holding back on our introduction of fraction symbols we can give our students a chance to ground their experience with fraction symbols in conceptual knowledge. When I teach addition and subtraction, I never go straight to the equation. Yet when I teach fractions I start by showing them the symbols. In fact just last year I blogged about my introduction of fractions using cookies. Feeling a little embarrassed now. It was a great lesson and it will still be a great lesson. Someday. Yes, someday when I'm teaching comparison of fractions not when I'm introducing fractions for the very first time. Putting those Chips Ahoy cookies back on the shelf for another day because "teachers often do not devote enough time to helping children build meaning for fractions before moving on to equivalencies, comparisons and operations involving fractions." (p. 5)

So my first question is how do I introduce fractions without the symbols? Start with word problems. A foundation built upon story problems that require equal sharing are perfect for helping students draw "on their intuitive understanding of sharing, measuring, grouping and distribution." (p. xix) In fact, solving and discussing equal share problems as early as kindergarten and first grade "prepares students to learn from a variety of other problems involving fractions."(xxi) So, new plan this year..lots and lots of problem solving involving fractions. Here is an example from the book:

### "Four children want to share 10 brownies so that everyone gets exactly the same amount. How much brownie can each child have?" (p. 6)

So why is this problem a good way to introduce and develop fractions?

- It uses equal sharing. This is something that children have a strong intuitive understanding of. Everyone has had to share the last few candies, the last piece of birthday cake or maybe even been involved in a fight over who gets the last homemade Oreo.
- The solution is actually a mixed number. While we don't traditionally start by teaching mixed number fractions, it is real life. Students learn from the start that "fractions are a type of number that 'fills in' the whole number, and they avoid the misconceptions that fractions are not numbers at all or appear only between 0 and 1." (p. 6)
- The number of people sharing (4) enables halving or repeated halving into fourths. Children can do this without formal instruction on partitioning because it comes naturally. It's real life.
- The problem itself only contains whole numbers. Students don't need to be taught to read or write fractions in order to solve the problem. Even young mathematicians will find a way to share the brownies based on their understanding of part-whole relationships.

You can also try 3 children sharing 7 brownies, 8 children sharing 30 brownies, or 3 children sharing 2 brownies. Use any "countable quantities that can be cut, split, or divided, such as candy bars, pancakes, bottles of water, sticks of clay, jars of paint, bags of sand, and so on." (p. 9)

Here are a few more examples that use both proper fractions (less than one) and improper fractions (mixed numbers):

**"4 children want to share 3 blueberry pancakes so that everyone gets the same amount. How much pancake can each child have?"****"4 children want to share 10 candy bars so that everyone gets the same amount. How much candy bar can each child have?" (p. 10)**

In order to devote enough time for problem solving using equal shares, I need more problems. I am planning to introduce fractions next month and will be writing some problems for it using the recommendations from this book. As you come up with problems for your class, here are some guidelines from the book to follow (p. 32-35):

### K-1 Students

- Use problems with multiplication, measurement division and partitive division of whole numbers to help students get used to grouping and partitioning. (These problem types are talked about extensively in Thomas Carpenter & al.
__Children's Mathematics.)__ - Use equal sharing problems with remainders that can be shared. Using 2 or 4 people to share should work for most children and use 1 or 2 remainders to be shared.
- Ask children about how they decided to split the remainders.
- Introduce fractions names for shares such as halves and fourths, but the actual written number notation is not necessary.

### Second Grade

- Use problems with multiplication, measurement division and partitive division of whole numbers to help students get used to grouping and partitioning.
- Use equal sharing problems with remainders that can be shared. The number of sharers should be 2, 4 and 3 for most children.
- Ask children about how they decide to share the leftovers.
- Introduce and reinforce fraction names such as half, one fourth, three-fourths, two-thirds.
- Encourage students to represent shares with pictures, whole numbers, and fraction words.
- Introduce the symbol for 1/2 once children learn to distinguish between halves, fourths and thirds in words.

### Third Grade

- Use equal sharing problems with answers that are mixed numbers and fractions less than 1. The number of sharers should be 2, 4, 8, 3, 6, and 10 for most students.
- Introduce standard notation for unit fractions after ample work with pictures and fraction words (1/2, 1/4, 1/3)
- Plan problems where students will get equivalent answers and discuss whether these answers represent the same amounts. (4 students sharing 2 cookies could result in answers of 1/2 and 2/4)
- Introduce equations to represent answers. (A child will get 1/4 of three different cookies and that is 1/4 + 1/4 + 1/4 = 3/4)

### Fourth Grade

- Use equal share problems with answers that are mixed numbers and fractions less than 1. Focus on 2, 4, 8, 3, 6, and 10 sharers, but include other numbers of sharers too.
- If your students have little experience with solving equal sharing problems, you may need to spend a couple of weeks letting students represent shares with pictures, whole numbers and fraction words before introducing and using standard fraction notation.
- Plan problems where students will get equivalent answers and discuss them.
- Represent your student's solutions with equations and an emphasis on addition.

### Fifth and Sixth Grade

- Use equal share problems with answers that are mixed numbers and fractions less than 1. Focus on 4, 8, 3, 6, 10 and 12 sharers but include other numbers of sharers as well, such as 15, 20, 100.
- Represent solutions with equations, with an emphasis on linking addition and multiplication and on equations that reflect a multiplicative understanding of fraction. (8 children sharing 5 hamburgers could link these equations: 1/8 + 1/8 + 1/8 + 1/8 + 1/8 = 5/8, 5 x 1/8 = 5/8 and 5 divided by 8 = 5/8)
- Represent word problem situations by using equations. (5 divided by 8 = ____ or 8 x ____=5)

Wow, this was a lot of information and I haven't even completely finished the book. I know that I will be changing my introduction to fractions this year and in the years to come. For more on squashing misconceptions about fractions for both students and teachers, hop on over to Lessons with Coffee and thanks for joining us for this installment of the Fly on the Math Teacher's Wall.