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:

Have you ever encountered symbols in your adult life that had no inherent meaning for you?

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.