# NCTM 2016 : Justification Presentation

This year was the first time I’ve presented at the NCTM annual meeting. In one session, I talked about my work on helping students build justification with fractions. In the other, I presented with Megan Gundogdu about how we have been using lesson study to build capacity at a middle school in the Bronx.  I really enjoyed sharing the work and I loved meeting some of you in person and learning more about what you are all working on.

I included the slides below from the justification talk for those who might want to take a closer look.

The main takeaway was that if we want to help students build justification, we need to explicitly create learning opportunities for students to:

• Reason about mathematics (using manipulatives, diagrams, contexts)
• Make connections between one’s activity with these tools and the solution
• Practice developing justifications
• Evaluate evidence used for justifications
• Revise justifications

I’d love to hear your feedback!

# PD: A Math Task for Teachers

I’m planning a PD for a group of elementary math teachers that I’ve never met before. This makes things difficult because I don’t know anything about what they know or what their experiences are.

I was told that one of the sessions should be about engaging students in doing mathematics–which is what I think all PD sessions should be about:).  I’m trying to use a math task to engage teachers in doing math before we talk about how to engage students. My goal is to use the task to build some content knowledge around fractions and to have a shared learning experience that we can generalize from. Here’s the task I’m planning on using during the session:

I used the guidelines I posted last time to help me select and modify this task.  Here’s what I was thinking as I planned:

1. The task starts by asking for a prediction or estimate of the final answer:  I added this in because I wanted to get a quick read of the group and to make sure that everyone understood the task. I’m debating about using notice/wonder instead.

2. Knowing the formula is not enough to complete the task: It’s not enough to know the procedure to find equivalent fractions to complete all parts of the task.

3. I can provoke an interesting discussion based on a common misconception. The numbers in the task allow me to pose the following misconception: “The first three situations are equivalent since there is always 1 sandwich fewer than the number of people.”  I’m hoping this will lead to an interesting conversation about why this thinking doesn’t work and what you might do with students that think it does.

4. They have to prove WHY something is true. I like the idea of having them create a poster with convincing evidence that supports their solution. The discussion that comes out of comparing different posters will allow us to discuss what it means to convince or prove in math.  I’m going to encourage using diagrams and I also plan on handing out connecting cubes that they can use to work through the problem.

5. It can be used with students with minor tweaks. The task will need to be modified based on the level of students it’s used with, but I think it could be used across grades 3 to 5.

After we complete the task as a group, I’m planning on using these prompts as a reflection:

• Doing mathematics: Write a about the activity from the perspective of a learner. Think about the learning processes. What helped you as a learner? What helped you sort out the mathematics
• Teaching mathematics: Write about the activity from the perspective of a teacher. How is this activity different than other lessons on fractions? What do you like about it? What are some concerns?

The comments you all left last time were so helpful. You pushed me to think about now about how I will decide what warm up task I’ll use, what type of reflection prompts might be helpful and to keep in mind that some teachers that might be uncomfortable with the content.

I’m excited to hear your feedback on this task and any suggestions on how you might change things before I actually try it out.

# Using pictures as a tool to justify

One of the things that is particularly interesting to me right now is how we help young students develop the ability to justify why things work in math. Often, kids know the correct rule but have no idea why the rule works. When I ask them to explain why a rule works, they wind up just listing the steps of the procedure.

Part of developing a conceptual understanding in math is being able to anticipate what procedure to use and why that procedure works.  For example, when I am trying to convert a mixed number into an improper fraction, I don’t simply need to know that I multiply the denominator by the number of wholes and add the numerator.  I also need to know why that rule works and why it will give me the correct number of parts in the mixed number.

I’ve found that children (and even adults) can have a really hard time writing or even explaining out loud their justification for why something works.  They often say “I know why but I can’t explain it.”

Recently, I have noticed that drawing diagrams or pictures is one way to help them begin to justify.  It’s as if the diagram allows them a way to make explicit what they are doing when they perform a calculation.

For example, the other day I was working with a group of elementary school teachers on fractions.  They worked in groups to draw the pictures like the one below to show why the calculations they performed worked.

It was an interesting experience for them because they had to think deeply about what it really means to say, multiply fractions, as opposed to just remembering the formula.  It also gave them something to refer to when explaining to the rest of the class.

I’m currently digging through the research on this, but I’m curious to hear what your experience has been. What do you do to help student justify in math?

# Using Diagrams to Make Sense of Fraction Word Problems

I’ve been spending a lot of time in schools and classrooms these past two weeks using diagrams with students and teachers. As a result, I’ve been neglecting this blog but I’ve been learning so much. Hopefully, I’ll start making more time to write about what I’m doing.

Here’s the latest problem I’ve been tackling with students:

Each lap around the lake is 3 ¾ miles long. Nicora runs two-thirds of a lap and then stops.  How many miles did she run?

Some students have no idea how to get started. Others add the two fractions. Others convert the mixed number to an improper fraction and then multiply.  Of the ones who chose to multiply, many make mistakes–they convert the mixed number to an improper fraction incorrectly, they forget the rule for multiplying fractions or they forget how to convert the product to a mixed number.

Students with a strong understanding of fractions and good number sense could do the problem in their head. They thought about decomposing 3 ¾ into 3 parts and then taking two of them. The million dollar question is “why don’t the other students see the problem this way?”

In an effort to help them see the problem differently, this is how we worked on it together.

First, I asked them to tell me about the problem in their own words.  Was it about cupcakes?  Unicorns?  What is going on in the story?  I’ve found that many students need to be forced to think about what the problem is asking them to do because they are so used to circling numbers and randomly picking an operation to solve them.

I then asked them to draw a bar that shows how far a lap is.

Next, I asked them to show how far two-thirds is.

Finally, I asked them to figure out how many miles two-thirds of the lap was.  Some decomposed the three and then the three-fourths.  Others used guess and check.

After we did a series of similar problems, I asked students to solve the problem without actually drawing the diagram. When they had to picture the diagram in their head, they started to develop shortcuts. One student told me that she could just divide the whole number into three and divide the fraction into three and then add them together to get the amount for one-third.  To find two-thirds, she thought about doubling the amount for one-third.

I wanted to share this example with you all because I think it shows how representations can help make explicit what may be obvious to other students. I also think it shows a nice progression of how students can use a representation to build a strategy to solve the problem.  How is this different than telling them the procedure?  Because the students build a process to solve the problem that makes sense to them.  Furthermore, they can always go back to the diagram if they forget the procedure.  So often, I find that students don’t forgot things because they are lazy, but rather because the procedure never made sense to them to begin with. My goal is to help students build on what makes sense to them, not to show them an algorithm that makes sense to me, but may not make sense to them yet.

# Using tape diagrams to solve fraction problems

Read the following problem and think about how a student might solve it.

There are 250 students in the fifth grade.  Three-fifths of them speak two languages.  How many speak two languages?

As I’ve mentioned before, I’ve been introducing tape diagrams, or bar models, to students and teachers recently as a way to solve word problems. These models have been popular in Singapore and Japan and are slowly working their way into some US programs. Essentially, the models use bars of different lengths to represent the size and relationships between different quantities.

Students who worked with tape diagrams previously solved the problem in the following way:

First, they drew a bar to represent the 250 students.

Next, they partitioned it into 5 equal parts to represent the fifths.

Then, they tried to figure out what number would go in each box. What was interesting at this stage was that students used different strategies depending on their abilities. Students who were fluent with division immediately divided 250 by 5. Students who were less comfortable with division tried to find what number times 5 would equal 250. They used guess and check until they arrived at 50. Other students used repeated addition and tried to find what number could be added five times to equal 250.

They then added the number to the diagram:

Finally they either multiplied 50 times 3 or added 50+50+50 until they arrived at the answer.

What’s neat about using this model is that students can approach the problem at different levels. Lower level students can use guess and check or repeated addition until they build up their multiplication and division skills.

Furthermore, I think it helps students make sense of the problem and the mathematical operations they are using to solve it. After enough practice drawing the models, students can be asked to visualize what they would draw in their head.  This will help them to focus on what operations they are performing. It can also help them generalize about what they are doing to solve these types of problems.

Eventually, students will be able to solve the problems without drawing the models. However, starting with the models allows them to develop meaning for why they divide the set by the denominator and multiply by the numerator.

I’ll talk more about what the research says about these models next time, but I’d love to hear what your experience has been.

# Making Fractions Real: An RME Task

Recently, I’ve been investigating Realistic Math Education (RME).  I like the idea of building on what is real to the student.

Because I spend a lot of time thinking about fractions, I wanted to know how RME approaches them.  Luckily, Streefland wrote about his three-year teaching experiment in the Netherlands.

In the experiment, students were introduced to the “Fractured Family.”  The family encounters many experiences that require fractional thinking and proportional reasoning.  For example, they need to divide an omelet at lunch or share apples after school or bake cookies from a recipe.

It’s not necessary that students have experienced these situations themselves, but rather that they can imagine the adults and children in the family doing them.

Here’s one of the initial tasks students encounter.

When Anja and Monica Fractured come home from school they may have an apple each.  But what do you do about such a difference in size?

• It’s a great context for introducing fair sharing.  Students know that it’s not fair if one child gets the big apple and one gets the small apple.
• It’s a great context for talking about the unit.  Is one-half of the small apple the same size as one-half of the big apple?  Why are they both called one-half?
• Using an imaginary family allows students to connect sharing with specific people they can imagine as opposed to the more abstract idea of sharing with unnamed people.
• Students are encouraged to draw how they would share the apples.  At first students create very detailed drawings–drawing leaves and stems for the apples. But as they do more tasks, they move away from detailed drawings and use circles or rectangles to represent items.   The drawings become representations or models of the situation.  Eventually they become a mental model.

I’m not a big fan of calling the family fractured, but you can adapt this task and name the family anything you want.

I’m still making my way through the book, but I’ll be sure to share any other tasks that I find interesting.  I’d also love to know if anyone has any experience using RME tasks in their classes.

Want to read along with me?

Streefland, L. (Ed.). (1991). Fractions in realistic mathematics education: A paradigm of developmental research (Vol. 8). Springer.

# Let students break things

One of the most important things you can do when you introduce fractions is to let students break things.   Give them lots and lots of experiences where they have to break up things evenly and share things fairly.  For example:

• Have them fairly share 3 cookies with 2 friends.  Have them fairly share 5 granola bars with 4 friends.  Have them fairly share 1 granola bar with 3 friends.
• Have them fairly share large pizzas and small pizzas.
• Have them fairly break up groups of items.
• Have them evenly break up spaces between numbers on the number lines.

Have them break whatever else you can think of.  Change the numbers of items and the number of people sharing them.  Change the things to be shared.    Just let them keep breaking things.

While they are breaking things, build on their concept of fairness to develop a notion of what equal shares mean.  Ask them questions like:

• Does equal mean the same number of pieces?  Is it fair if we both get three pieces but mine are bigger than yours?  Why or why not?
• Do equal shares have to be the same shape? If we both have the same size brownie and I cut my brownie into 2 pieces by making a diagonal line and you cut yours into 2 pieces making a horizontal line, did we both make fair shares?  Why or why not?

And then let them break some more things.

So what do we do after they break all these things?

Let them put things back together.  Give them a broken piece and tell them it is the piece one person got after a candy bar was shared between 5 people.  Ask them to figure out how big the candy bar was.  Do this with the pizzas and the granola bars and the groups of items.

After enough breaking things and putting them back together, talk to them about how we name the size of the pieces we made.  If we break a cookie or a granola bar or a group of objects into five pieces, we made fifths.   Ask them questions like:  If I break something into fifths and you break something into fifths, can my fifths be bigger than yours? Show me. Why are they both called fifths?

Researchers have talked a lot about the importance of these activities in developing a strong concept of fractions.   They use the terms partitioning and iterating for the two activities.   Partitioning is the act of breaking a unit into pieces.  In the case of fractions we talk about equi-partitioning or breaking a unit into equal pieces.  The unit can be the pizza or the candy bar or the space between two numbers on a number line or a group of objects.    Iterating is what we do when we put the pieces back together (or later on when we put pieces together to form other fractions).

I’ll talk more about why these activities are important (and what to do next) another time, but if you want to know now, check out the research below.

Lamon, S. J. (1996). The development of unitizing: Its role in children’s partitioning strategies. Journal for Research in Mathematics Education, 27(2), 170-193.

Mack, N. K. (1990). Learning fractions with understanding: Building on informal knowledge. Journal for research in mathematics education, 21(1) 16-32.

Olive, J. (2002). Bridging the Gap: Using Interactive Computer Tools To Build Fraction Schemes. Teaching Children Mathematics, 8(6), 356-61.

Siebert, D., & Gaskin, N. (2006). Creating, Naming, and Justifying Fractions. Teaching Children Mathematics, 12(8), 394-400.

# Assessing what students really understand about fractions

Do the following tasks look familiar?

What fraction of the circle is shaded?

Shade in 2/3 of the rectangle below.

Are most of your students successful at these types of tasks?  I would be willing to bet that they are after trying a couple of them.

However, I’d also guess that once you move on to more complicated concepts like comparing fractions or adding fractions, suddenly students are confused.

Why?

I think there is a misconception that if students can name a fraction in a picture and shade in a fractional amount in picture, that they understand what a fraction is.  But being able to do these tasks does not necessarily mean they have a strong conceptual understanding of what a fraction is.  And if they don’t have a strong foundation, topics like comparing or adding fractions is incredibly difficult.

Why can students do the tasks above yet not really understand fractions?

1.   The tasks do not require students to pay attention to the equal size of the parts because they are already equally partitioned for the student.  Students didn’t create the pieces themselves.

When asked to shade 2/3 of a bar that is already partitioned into 3, students can do this just by coloring in 2 boxes.  They do not need to pay attention to the fact that the boxes are equal.

In fact, if given uneven size boxes, students will often shade in 2 and say that it is 2/3 because 2 are shaded and there are 3 boxes in total.

2.  Young children will often focus on the shape of the pieces being the same and not on the size of the pieces being the same.    Those students then have problems identifying the fraction when the pieces are equal sized but different shapes.

Van de Walle offers a great assessment problem to see what your students really understand about fractions.

Which of the shapes below are correctly partitioned into fourths? Why?  Which are not correctly partitioned into fourths?  Why?

Items b and c will help assess if students are focused on the number of pieces and not the size, while e and g will help assess if students are focused on the shape and not the size.

How can you avoid these misconceptions?  What tasks might help develop a strong understanding of what a fraction is?

I’ll share my ideas about that in my next post, but I’d love to hear what you think.

Want to know more?  Read Chapter 15 of  Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2007). Elementary and middle school mathematics: Teaching developmentally.

# 5 reasons you should NOT talk about fractions as “out of”

One of the most common ways to introduce fractions to young students is to talk about ¾ as “3 out of 4.”

And it’s part of the reason why students have so much trouble with fractions later on.

Researchers Siebert and Gaskin (2006) wrote about why introducing fractions as “3 out of 4” instead of “three-fourths” creates a problematic image for students.

When the fraction three-fourths is talked about as “3 out of 4”, children picture 4 items and then taking 3 of them.  The numerator and denominator are thought of as whole numbers.

The four is not a fourth.  It does not refer to a certain size piece that was made by cutting up a unit into 4 equal pieces.  The three does not mean 3 pieces that are one-fourth in size.  The three and the four are whole numbers that happen to be written in fraction notation.

Why is thinking about the numerator and denominator as whole numbers a problem?

Here are five misconceptions that it can cause:

1. Students think the pieces can be any size.  I worked with a class that had been introduced to fractions using “out of” language.  When I showed them the following image, many stated that the shaded area was one-fifth because it represented “one out of five pieces” in the rectangle.

2. Improper fractions make no sense.  When presented with 7/4, students are confused.  How can you take 7 out of 4 items?

3. Students do not see a fraction as an amount.  They have no idea where a fraction goes on a number line.  For example, they might place three-fourths between 3 and 4 on the number line.

4.  They apply whole number reasoning to fraction operations.  Take a look at this classic mistake @mpershan posted on www.mathmistakes.org.

Students add the numerators and the denominators because they are thinking of fractions as whole numbers.  It makes sense to them that having 3 out of 5 items plus 2 out of 7 items equals 5 out of 12 items.

5. They have difficulty comparing fractions. One-third is smaller than two-ninths because you only have one object instead of two objects.

It seems like a minor thing…saying “three-fourths” instead of “three out of four.”  But, when combined with the models students are using to think about fractions, it can make a big difference in avoiding misconceptions later on.

I’d love to hear what you think.

Want to know more?

Siebert, D., & Gaskin, N. (2006). Creating, Naming, and Justifying Fractions. Teaching Children Mathematics, 12(8), 394-400.

# How do you choose what model to use when teaching fractions?

The three representations above can all be used to model two-fifths.   Does it matter which one you choose when introducing fractions?

In my last post, I talked about why concrete objects or models can be helpful in building on what students can do.  What I didn’t talk about was how to choose a model when trying to foster a particular mathematical concept.

Fractions are a a good example since most teachers and researchers (Cramer & Henry, 2002; Siebert & Gaskin, 2006) agree that models are important in helping children develop fraction concepts.

But how to you know which one to choose?  Do you use pizzas, fraction strips, cuisenaire rods, number lines, chips?

The first thing to note is that different models offer different opportunities to learn. Let’s look at the three main models used for teaching fractions and what they offer:

1. An area model, like fraction bars, can help students visualize parts of a whole.
2. A model with discrete objects, such as counters or chips, can help students begin to think about fractions of a set.
3. A linear model, like a number line, can help students see that there is always another fraction between any two fractions.

As Susuan Lamon points out in her book Teaching Fractions and Ratios for Understanding:

“No one model is a panacea; every model has some useful features, it wears out at some point and it is up to the teacher to use it wisely.  If you have to spend valuable class time teaching students how to work with the model, it is wasted time that is not being used to teach fractions (p. 149)”

Contrary to what many outside the classroom believe, this confirms one of my basic beliefs about teaching:  It’s complicated.  There is no one right answer for how to do things.

Yes– it matters which model you choose.  Each model will foster certain understandings about fractions and mask others.  That’s why it’s important to be aware of the advantages and disadvantages of the model you choose.   That’s also why it’s important to expose students to more than one model.

But it’s not a simple matter of saying this is the right model or this is the wrong model or make sure students can use all of them.

Choosing which model or manipulative to use depends on your instructional goal.  If the goal is to work on partitioning a unit, an area model might work well.  If the goal is to work on viewing a fraction as a quantity, a linear model might work well.  If you have identified that students are struggling with fraction of a set, you may want to introduce a discrete model.  Being clear on your goal and the advantages and limitations of each model will allow you to choose the right model for you and your students.

There may not be one right answer when it comes to choosing a model for teaching fractions, but the choice does matter.

What about you?  How do you choose models for teaching fractions?

Want to know more?  Check out the resources below.

Cramer, K., & Henry, A. (2002). Using Manipulative Models to Build Number Sense for Addition of Fractions. In B. Litwiller & G. Bright (Eds.), National Council of Teachers of Mathematics 2002 Yearbook: Making Sense of Fractions, Ratios, and Proportions (pp. 41-48). Reston, VA: NCTM.

Lamon, S. J. (2005). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. Psychology Press

Siebert, D., & Gaskin, N. (2006). Creating, Naming, and Justifying Fractions. Teaching Children Mathematics, 12(8), 394-400.