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Sunday / July 22

Six Decrees of Mathematical Representation

This is a shameless play on “six degrees of separation.” I assure you Kevin Bacon will make no appearance, nor can I find any relationship between the two of us. Instead, I share some observations about representations in mathematics classes. Some are my own missteps. Others I’ve witnessed in my countless classroom visits. I can say that representations in mathematics education have come a long way in a short amount of time. But there is more to be done. These are my unofficial decrees of representation.

1. Know and Avoid Representational Bias

Pretend for a moment that you are a third or fourth grade teacher teaching multiplication. You’re working to represent 4 x 6. You’re considering the tools and/or representations in figure 1. Which one would you select? Which is the best representation of 4 x 6? Which makes most sense to you?

Figure 2

Clearly, there is no right answer as to which is the best representation of 4 x 6. However, your selection can be telling. In fact, it might tell about your representational bias. Essentially, this is the representation that makes the most sense to you. It’s what you prefer. There’s nothing wrong with having a preferred representation. But it is important to keep in mind that other students may perceive the mathematics concept in different ways. Featuring one representation predominantly might create an unnecessary challenge or disconnect for our students. A rather simple way to counteract representational bias is to be aware of it and intentionally design lessons or activities that make use of other representations.

2. Connect Representations to Contexts and Situations

Disconnections between representations and contexts may be even more damaging than our representational bias. Consider which of those representations would be “best” for the problem “Erin has 4 bags of 6 oranges. How many oranges does she have?” Any of the representations in figure 1 could be used to find the product of 4 x 6. However, the bags of oranges present an equal groups problem situation. Because of this, a comparison model (E) or an area model (B) don’t accurately represent the problem.

3. Connect Different Representations

I tend to think of connecting representations in a few different ways. One way to connect representations is to show their specific relationship to components of a problem. Another way is to connect physical models to drawings highlighting how the physical models can be represented with symbolic drawings (think sticks and dots for base ten blocks).

But there are two other critical ways to connect representations. The first is to make connections among clearly different representations so that students can move between representations when numbers or situations are problematic. For example, 24 bags of 37 oranges is an equal groups problem situation but using an area model may be more useful for finding the product using partial products. Understanding how the connections allows me to move between the different representations for multiplication.

This leads to the most critical connection. Simply, the connection between the physical model or drawing and the expression or equation must be clearly established and reinforced with every problem. We want our students to evolve to use expressions and equations for efficiency. This may not happen if they don’t understand where the symbols and operations come from or connect to.

4. Know When to STOP Representing

Number four probably reads with some level of blasphemy. But just because we can represent doesn’t mean we should represent. I am always impressed when a group of students can represent a mathematical idea in five different ways. And I’m equally struck that none of the students in the group ask “Why are we doing this?” Consider a problem like 29 + 1 in a second grade class. It’s wonderful that they can use base ten blocks, a number line, a hundred chart, or multiple ten frames. But really, I would prefer that they exclaim that 29 + 1 is just one more and STOP with all of the representing. This too is connected to number three. Once they can represent a concept or computation with pictures, diagrams, or physical models we want to help them move to more efficient approaches including equations and eventually mental mathematics.

5. Eliminate Representational Rules

There is a perception or in some cases a tradition that mathematics is nothing but a bunch of rules. And to be clear, there are rules in mathematics. But the rules that seem to be reinforced most frequently aren’t really rules. Karp and colleagues capture this well in their article “13 Rules that Expire.” (NCTM, 2014)

But there are other rules lurking below the mathematics surface. These are rules for working with representations and tools. And these rules can complicate or delay our students’ developing ideas about various concepts. One example might be that ten frames have to be filled from left to right and top to bottom. Consider the ten frames in figure 2. All of them show six. Students should be able to represent six in different ways to develop ideas about composition and decomposition. But a student’s deep understanding of six might be inhibited if they only encounter it as the upper left frame. In other cases, rote procedures for using tools and representations are taught. This new layer of unwarranted procedure further muddies the mathematical waters for our students.

6. Be Careful Calling Them Strategies

Drawing a picture is a problem-solving strategy. But representations aren’t really strategies for computation. Think about the students above who were representing 29 + 1. Counting on is a strategy. Counting back is a strategy. Adjusting and compensating are strategies. Representations show a strategy. Adding 29 + 13 by finding partial sums (20 + 10 + 9 + 3 or 29 + 10 + 3) is a strategy. Adjusting 29 + 13 to 30 + 12 is a strategy. Using base ten blocks to show 29 + 13 is a representation of a strategy be it counting on or finding partial sums. I see this all too often with Number Talks. Students will share that they did 25 x 33 with an array of base ten blocks. Then, they talk their teacher through how they “thought about it” ultimately describing a visual of a computation strategy.

As with anything, great mathematics teaching is a journey not a destination. These are some situations in which representations go bad. I’ve been guilty of a few in my career. I’ve learned. I know that all of these are still alive and well. And I’m guessing there are more decrees to be made. I would love to hear about hear about them.


Karp, K., Bush, S. & Dougherty, B.  “13 Rules That Expire: Overgeneralizing commonly accepted strategies, using imprecise vocabulary, and relying on tips and tricks that do not promote conceptual mathematical understanding can lead to misunderstanding later in students’ math careers.” Teaching Children Mathematics 21, no. 1 (2014): 18 – 25

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John SanGiovanni is a mathematics supervisor in Howard County, Maryland. There he leads mathematics curriculum development, digital learning, assessment, and professional development for 41 elementary schools and more than 1,500 teachers. John is an adjunct professor and coordinator of the Elementary Mathematics Instructional Leader graduate program at McDaniel College. He is an author and national mathematics curriculum and professional learning consultant. John is a frequent speaker at national conferences and institutes. He is active in state and national professional organizations and currently serves on the Board of Directors for the National Council of Teachers of Mathematics. John is the author of two Corwin books: Mine the Gap for Mathematical Understanding (3-5) and Mine the Gap for Mathematical Understanding (K-2)

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