# Represent! Part 2

In my previous post, I shared the first few questions I asked at a recent #ElemMathChat I hosted. Today I’d like to continue talking about using and connecting mathematical representations with a focus on fractions.

Before reading on, pick one of the models yourself and analyze it.

• How does it represent 2/3?
• Where is the numerator represented in the model?
• Where is the denominator represented in the model?
• Can you answer these questions with all three models?

First it might help to differentiate the three models. The top left corner is an area model, the top right corner is a set model, and the bottom middle is a number line.

If you look at the area model, you’ll see that the whole rectangle – all of its area – has been partitioned into three equal parts, each with the same area. When we divide a shape or region into three parts with equal area, we actually have a name for each of those parts: thirds. Those thirds are countable. If I count all of the thirds in my area model, I count, “1 third, 2 thirds, 3 thirds.”

Two of them have been shaded orange. So if I count only the parts that are orange, “1 third, 2 thirds,” I can say that 2 thirds, or 2/3, of the whole rectangle is shaded orange.

If you look at the set model, you might think at first that this model is the same as the area model, but this representation actually has some different features from the area model. In the set model, the focus is on the number of objects in the set rather than a specific area. I used circles in the above image, which are 2D and might make you think of area, but I could have just as easily used two yellow pencils and one orange sharpener to represent the fraction 2/3.

I can divide the whole set into three equal groups. Each group contains the same number of objects. When we divide a set of objects into three groups with the same number of objects in each group, we actually have a name for each of those groups: thirds. Those thirds are countable. If I count all of the thirds in my set model, I count, “1 third, 2 thirds, 3 thirds.”

Two of the groups contain only pencils. So if I count only those groups, “1 third, 2 thirds,” I can say that pencils make up 2 thirds, or 2/3, of the objects in this set.

Finally, we have the number line model which several people in the chat said is the most difficult for them to make sense of. While we have a wide amount of flexibility in how we show 2/3 using an area model or set model, the number line is limited by the fact that 2/3 can only be located at one precise location on the number line. It is always located at the same point between 0 and 1.

In this case, our whole is not an area or a set of objects. Rather, the whole is the interval from 0 to 1. That interval can be partitioned into three intervals of equal length. When we divide a unit interval into three intervals of equal length, we actually have a name for each of those intervals: thirds. Those thirds are countable. If you start at 0, you can count the intervals, “1 third, 2 thirds, 3 thirds.”

However, what’s unique about the number line is that we label each of these intervals at the end right before the next interval begins. This is where you’ll see tick marks.

• So 1/3 is located at the tick mark at the end of the first interval after 0.
• 2/3 is located at the tick mark at the end of the second interval after 0, and
• 3/3 is located at the tick mark at the end of the third interval that completes the unit interval. We know we have completed the unit interval because this is the location of the number 1.

This quote sums up what I was aiming for with this discussion of representations of 2/3:

“Helping students understand the meaning of fractions in different contexts builds their understanding of the relevant features of different fraction representations and the relationships between them.” – Julie McNamara and Meghan Shaughnessy, Beyond Pizzas and Pies, p. 117

The bold words are very important to consider when working with students. What is obvious to adults, who presumably learned all of these math concepts years and years ago, is not necessarily obvious to children encountering them for the first time. What children attend to might be correct or it might be way off base. One common problem is that children tend to overgeneralize. A classic example is shared in Beyond Pizzas and Pies. Students were shown a model like this:

They overwhelmingly said 1/3 is shaded. The relevant features to the students were shaded parts (1) and total parts (3). They weren’t attending to the critical feature of equal parts (equal areas).

I’ll close this post with a Which One Doesn’t Belong? challenge that I shared during the #ElemMathChat. (Note: I revised the image of the set model from what was presented during the chat.) As you analyze the four images, think about the relevant features of the area model, set model, and number line; look for relationships between them; and then look for critical differences that prove why one of the models doesn’t belong with the other three.

# Represent! Part 1

This week at #ElemMathChat I had the pleasure to lead the chat. I used the opportunity to talk about using and connecting mathematical representations, a topic that has been on my mind a lot this school year.

I kicked off the chat with this quote:

“Because of the abstract nature of mathematics, people have access to mathematical ideas only through the representations of those ideas.” –National Research Council, 2001, p. 94

and this question:

What does it mean that people only have access to mathematical ideas through representations?

I wanted this to be our guiding question throughout the rest of the chat.

I immediately followed up with this question:

As expected, the folks in the chat remarked that the symbolic form of this number does not convey anything about the number seven. Even if someone told you this is the number seven, what that means to you will vary depending on what you already understand about that number. Just being able to see this symbol and say the word, “Seven,” does not necessarily mean a person understands anything about the number seven or the quantity it represents.

But what if I show you this?

So what do these representations convey to you about the meaning of the number 7? Before reading on, take a moment to analyze the different representations. Do they all represent the same thing about the number seven? Do some representations give you different understandings than others? How many different things can you learn about the number seven from these representations?

Here are some of the things these representations convey to me:

• 7 can be made with combinations of smaller numbers: 1 and 6, 2 and 5, 3 and 4.
• At first I usually see a specific combination within a representation, like 4 and 3 in the domino or 5 and 2 in the math rack.
• After spending time looking at them, I start to notice multiple combinations within some representations. The teddy bears show me 4 and 3 if I look at the rows. However, I also see 6 and 1 if I look at the group of 6 with 1 teddy bear hanging off the end.
• I also see that 7 can be made with combinations of more than two numbers: 3, 3, and 1 for example as shown in the matches and the teddy bears.
• The number track shows me where 7 is in relation to other numbers. I can see that 6 is just before 7 and 8 is just after 7.
• I also see how 7 is related to 10. The math rack, number path, and fingers all show me that 7 is 3 less than 10.

This is hardly an exhaustive list of all the ways the meaning of 7 is conveyed, but hopefully it serves to demonstrate the point that the more representations of 7 I have access to, the more robust my understanding of the number 7 may become. The same applies for any number.

I followed up with this quote:

“There is no inherent meaning in symbols. Symbols always stand for something else. The meaning a symbol has for a child depends on what the child knows and understands about the concepts the symbol represents.” — Kathy Richardson, How Children Learn Number Concepts, p. 20

and this question:

Sometimes it’s hard to put ourselves in the shoes of our students, but doing so can help us better understand our students’ struggles and frustrations. We have been seeing numeric symbols for years and years. We see 7 and immediately have access to meaning. When in our adult lives might we encounter symbols we don’t understand?

For me it’s any time I encounter writing that doesn’t use the Roman alphabet. Even if I can’t speak Spanish or German, I can at least read the words I see (despite any horrible pronunciation problems):

• Buenos días.
• Por favor hable más despacio.
• Entschuldigen Sie bitte.
• Lange nicht gesehen!

And if there are any cognates involved, I just might be able to make some sense of what I’m reading.

But when I encounter writing in Hebrew or Chinese?

• בוקר טוב
• נעים מאוד
• 你好嗎?
• 我很高興跟你見面

These symbols have absolutely no meaning to me. They are inaccessible. Visiting Israel several times for work, it was always disconcerting to be bombarded by street signs, advertisements, and menus and have no way to even map any sounds to the text I was seeing.

Now am I saying that teachers are not currently providing students access to multiple representations of numbers like 7? No.

But that doesn’t mean it isn’t worth reflecting on our practices to ensure we are providing students access to these concepts via multiple and varied representations and that we aren’t rushing to the use of a symbol because that’s our “goal.” There is nothing inherently more mathematical about a symbol like 7 than a collection of dots on a domino or seven fingers on my hands. What numeric symbols do allow for is efficiency of representing quantity, especially once the place value system comes into play. But that efficiency is lost on students, especially those who struggle, if they do not have a solid foundation in the concepts the symbols represent.