“I don’t understand! What do they want me to do?”
Do you remember the voice of Charlie Brown’s teacher? We aren’t sure what she’s saying but the sounds of “wah, wah, wah” wash over us as she’s talking. For some students, the abstract notation of mathematics, whether 1/2 + 1/3 = 5/6 or f(x) = x2 + 3x + 1 or Area = length × width, is a visual version of this effect. These symbols and notation may feel like squiggles on the page, random visual images with little meaning. The power of using and connecting multiple representations is that we give meaning to these squiggles, these abstract mathematical symbols, when we show students how they are connected to other representations, representations which might make more sense to and have more meaning for our students.
Elementary students are expected to master 15 different problem situations related to addition and subtraction. Many elementary students find situations involving comparisons particularly difficult. Let’s explore how the use of multiple representations makes these problems more accessible for all learners. I encourage you to ask three questions when using multiple representations to help your students solve problems in this way.
Erin has 452 books in her library. That is 28 more books than Sara has.
How can we create a physical representation by acting the problem out with real objects or with manipulatives?
Not easily, as written, but we could if we make the numbers smaller. Try this: Erin has 10 books in her library. That is 3 more books than Sara has. How many books are in Sara’s library?
Now, two students representing Sara and Erin can take books off the class bookshelf and act out this problem to reason about a solution. They can also build a model to represent the simpler problem using counters or Cuisenaire Rods. All of these are physical representations.
How can we create a visual representation that will work for the numbers in the real problem?
By now students have reasoned the solution to our simpler problem; there are 7 books in Sara’s library. The Cuisenaire Rod model above is built to represent the smaller numbers literally (a black rod is 7 units long, the light green is 3 units long, and the orange rod is 10 units long) and it is labeled in more general terms.
To make sense of comparison problems involving addition and subtraction, students need to think about three elements in the problem: a smaller quantity, a larger quantity, and the difference between them. A word problem will give two of these pieces of information and ask students to find the third one.
In order to transfer from the smaller numbers to the real problem, we can relabel the Cuisenaire Rod model with the actual numbers for the problem or we can sketch a bar model diagram for the problem. I like using Cuisenaire Rods as a physical representation of a visual bar model because the students have a small leap to make in transferring to a visual sketch on the page.
How can we create a symbolic representation, an equation, which will help us solve the problem?
The two equal bars in the model, the top & bottom rows, represent the two sides of a number sentence. Students can write the number sentence as Smaller Quantity + Difference = Larger Quantity, filling in the numbers they know from the problem. Let’s use the letter S to represent the books in Sara’s library. This gives us S + 28 = 452 for this problem.
If a student is unsure about this, encourage them to use the numbers from our simpler case (S + 3 = 10 so there are 7 books in Sara’s library and 7 + 3 = 10). To solve the actual problem, we are using the same “slots” in the equation, representing the same elements (smaller quantity, larger quantity, and difference) and inserting the larger numbers from the given problem. This is one way the strategy of trying a simpler case can be helpful in problem solving.
What number, when added to 28, gives us 452? Or, said differently, take 28 away from 452 to find the number of books in Sara’s library. [There are 424 books in Sara’s library, by the way.]
This short walk down an instructional path from physical representations to visual representations to symbolic representations shows several ways students can reason through solving a problem like this. You may be saying, “This is nice, Sara, but it seems long and convoluted.” It does to us, because we already have more efficient strategies for finding solutions and we understand the structure of the problem. We know how to talk about it and how to find a solution. For a second or third grade student who is just learning about these problems, the use of multiple representations gives them an opportunity to grab onto the representation that makes the most sense to them. Then, they can work, with our help, to connect that representation to others. Instead of relying only on the verbal, as Charlie Brown’s teacher does, we provide multiple opportunities for sense-making and reasoning.
For our elementary students, we are making visible three important phases of learning (Hattie et al, 2016):
- Surface learning, including labeling the parts of the problem and reading the equations;
- Deep learning, seeing the underlying structure common to all comparison problems; and
- Transfer learning, identifying and using this problem structure in a variety of settings.
Mastery of comparison problems involving addition and subtraction takes time and experience with many examples. Students need practice to build fluency with the surface features of the problem, rich experience to see the deeper underlying structure, and then opportunities to transfer this learning to new situations and new numbers (problems involving fractions, for example).
I invite you to consider ways to increase your use of multiple representations of mathematics in your classroom. What manipulatives do you have available to use as physical representations? What visual representations can you help your students learn to use? How do you use language and context to connect these representations to the symbolic “book math” we want our students to understand? How does this work lead to deeper student understanding?
Hattie, J. M., Fisher, D., Frey, N., Gojak, L.M., Moore, S.D. & Mellman, W. (2016). Visible Learning in Mathematics. Thousand Oaks, CA: Corwin Press.