By Zachary Champagne and Michael Flynn, Posted March 28, 2016 –

In our previous post, we discussed the importance of teachers “doing the math” before working with students. In this post, we exemplify what we mean. First we use one problem to show how a group of teachers could engage with the mathematics to consider how to use the properties of operations to justify student developed strategies for solving computation problems. (We encourage you to “do the math” before continuing to read here).

Solve this problem in at least three different ways: 56 + 45

Below is a small sample of strategies teachers might generate for solving these two problems.

56 + 45

56 + 40 = 96

96 + 5 = 101

56 + 5 = 61

61 + 40 = 101

56 + 4 = 60 + 45 – 4 = 41

56 + 45 = 60 + 41

60 + 41 = 10

We could spend the rest of this post exploring each of those strategies, but we would like to focus instead on one strategy to further explore the associative property.

56 + 40 = 96

96 + 5 = 101

Consider how the associative property plays a role in this strategy: It tells us that (a + b) + c = a + (b + c). Where do you see this property being used in this strategy? (Again, we would encourage readers to pause and consider this before reading ahead).

Notice that 45 has been decomposed into 40 + 5. Consider the steps that were actually used with this strategy.

• Step one is to substitute 40 + 5 for 45 in the problem. This can be expressed as 56 + 45 = 56 + (40 + 5).
• Step two is to use the associative property to first solve 56 + 40 which can be expressed as (56 + 40) + 5.
• Step three is to then add 96 and 5 to find the total.

By engaging with this mathematics, we can learn how the associative property is intuitively applied in this example. We can also see where this may be a roadblock for students. If students do not intuitively understand how this property works, using this strategy may be currently out of reach.

Considering different representations of the mathematics also reveals the nature of this property and gives us a chance to consider multiple ways in which students might come to understand what is happening. For example, by using cubes to represent both quantities, we can visualize the decomposition of 45 into 40 and 5. We can then see the associative property in action as we combine the 40 with the 56 and then add on the 5 remaining. This representation makes clear that although the groupings of the original configuration of cubes changed, the total number of cubes remained constant.

A second example of how doing the math can help us better understand our students and our teaching can be found when multiplying fractions. (We would encourage readers to try this one out first as well.) Consider this problem:

Create a visual fraction model that would prove that 3/4 × 2/5 = 6/20.

When considering this model, think about which models work for multiplying whole numbers. Consider using 4 × 5 as an example.

Let’s consider an array model for 4 × 5. This may look like the one to the left, with four rows and five columns. The answer is represented by the total number of whole squares in the array (20).

Now let’s consider what 3/4 × 2/5 would look like. First, consider that each of the factors in this expression is less than one. So, they will be representing a part of one whole. In this case, the rows would be partitioned into four equal parts, and the columns would be partitioned into five equal parts. Then three of the four rows are shaded, and two of the five columns are shaded. This overlapping area represents the area of a 2/5 × 3/4 rectangle. Each of the pieces in this case represents 1/20 of the whole rectangle, and six of those pieces are shaded. Therefore the product of 3/4 and 2/5 is 6/20.

  Of course, other models show multiplication of fractions, but again, by engaging with this content, we are able to gain new insight into the mathematical ideas behind multiplying fractions.

We encourage all our readers to “do the math” as they think about what their students are learning and how these two examples show this can be done. Check back again next week as we provide a suggested framework for how to “do the math” with colleagues in both face-to-face settings and online. We’ll also offer some suggested problems (by grade band) to try out and discuss.

Your Turn 

Now it’s your turn. What is your view of doing the math as a means of professional development? If a particular math problem has recently challenged you, tell us about it. In our next post, we will provide two example problems that groups of teachers may wish to explore.

We want to hear from you! Post your comments below or share your thoughts on Twitter @TCM_at_NCTM using #TCMtalk.

Zachary Champagne is an Assistant in Research at the Florida Center for Research in Science, Technology, Engineering, and Mathematics (FCR-STEM) at Florida State University. He previously taught fourth and fifth grade mathematics in Jacksonville, Florida, for thirteen years. He is currently interested in learning how young students think about mathematics and how to help them understanding that mathematics makes sense. He tweets at @zakchamp. Michael Flynn is the Director of Mathematics Leadership Programs at Mount Holyoke College in South Hadley, Massachusetts. He previously taught second grade in Southampton, Massachusetts, for fourteen years. He is currently interested in how primary and elementary students develop algebraic reasoning and how teachers can support that work. He tweets at @mikeflynn55.