# 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.

# Using manipulatives to build on what students can do

In my last post, I talked about building on what students can do instead of focusing on what they can’t do.  One way many of us try to do that is by using a variety of manipulatives in our classes.

I love using concrete objects with students of all ages.  But I can remember several times when other teachers or administrators made fun of me for using “toys” with my 8th graders to teach algebra.

Manipulatives can be a great way to provide access into lessons for all students and when used well they can help build deep understanding.

But as two researchers, Stein and Bovalino (2001) pointed out: “manipulatives do not magically carry mathematical understanding.”

They identified two problems that can occur when using manipulatives.

1.  They can sometimes be used in a way that only requires students to mindlessly follow what they see the teacher do.   As a result, students mindlessly move around blocks just as they would mindlessly use a formula without meaning.

2.  They can lead to unsystematic and non-productive exploration.  If the mathematical goal is not clear and the activities are not carefully designed, the manipulative does not serve as a tool for developing the concept the teacher intends.

The researchers contend (and I would agree) that manipulatives need to be used in a way that links an activity a student can do to a new concept by allowing them to take in new information and give it meaning.

I’ll give an example.  I recently worked with young students on how to find the area of rectangles.  I gave out square tiles and asked them to find the area of a rectangular region.  All of the students could complete the task by simply covering the region with the square tiles.  What did they learn by doing this?  Not much.  At this point–they haven’t learned anything new.

It wasn’t until I asked them to how to find the number of squares without counting all the tiles that they began to develop new understandings.  Some counted the number of rows and the number of tiles in one row and multiplied.  Others counted the number of columns and the number of tiles in one column and multiplied.  Others used repeated addition. As we worked on different sized regions and shared different strategies, they began to develop a rule for finding the area.  Eventually, they were able to find the area when the dimensions were given without using any tiles.

Manipulatives allowed them to access the activity and have a concrete way to justify their answer but using the manipulatives alone did not lead to the new understanding developed.

If the student could do the activity before the lesson, he or she did not learn anything.   Learning occurs when students use what they can do to develop a new understanding.

Or as the researcher Doug Clements (1999) says:  “Students may require concrete materials to build meaning initially, but they must reflect on their actions with manipulatives to do so.”  Encouraging that reflection is a difficult task and one that I think often goes missing from lessons.

How do you use manipulatives?  How do you encourage your students to reflect on their work with them?  I’d love to know.

What to know more? Read what the research has to say:

Clements, D. H. (1999).  Concrete manipulatives, concrete ideas. Contemporary Issues in Early Childhood, 1, 45–60.

Stein, M. K., & Bovalino, J. W. (2001). Manipulatives: One piece of the puzzle. Mathematics Teaching in the Middle School, 6, 356–359.