Relational Thinking

Part of the discussion that took place while I was at the CGI training last week was around the idea of relational thinking. One way in which relational thinking is used in mathematics is as a powerful tool to help students understand meaning of the equal sign. Rather than treating the equal sign as a signal to compute, students use relational thinking to view the equal sign as something that represents a balanced relationship between the two expressions on either side of the symbol. For example, suppose the following open number was given to a child:

9 + 7 = BOX + 8

According to math education research, a child who has a fragile understanding of the equal sign would likely say that 16 goes in the box. Or, even for a child who DOES know that the equal sign means "the same as," he might need to first compute the left side to determine that the sum is 16, and then find out what the box must represent so that the right side also has the same sum. The way in which a child might figure out the answer is by subtracting 16 - 8 to get 8, and then conclude that 8 must go in the box.

While we might be satisfied that a child is able to correctly determine the number that goes in the box by the strategy described above, we want to encourage him to use relational thinking: Is there a way to determine the number that goes in the box without first computing the sum of the left side? How can we determine the correct answer just by looking at the relationship between the expressions? It turns out that children who are often exposed to experiences with open number sentences will look at 9 + 7 = box + 8 and know that 8 must go in the box. (Note how there is a difference of 1 between the 7 on the left side and the 8 on the right side. In order for the number sentence to be true, the number in the box must be one less than 9.)

Once again, I was curious to see how my 9-year-old would respond to an open number sentence, so I asked him: "What number goes in the box so that this is a true statement?"

34 + 27 = 35 + BOX

(In hindsight, I think I should have written the equation as 34 + 27 = BOX + 35 to see if my son would say that 61 goes in the box.) I provided him no pen nor pencil, hoping that he would use the idea of relational thinking to do the problem. It didn't surprise me that he tried to find the sum of 34 and 27 mentally. I could tell that he was struggling to do the mental calculations, so I provided him with a hint:

"Is there a way that you can use the relationship between the numbers to help you find the answer?" There was a long pause. I decided to take a step back and give him the following:

34 + 27 = 34 + BOX

"27 goes in the box," he said. "Good," I replied. "Now go back to the first problem I gave you. What number goes in the box?"
"I know 35 is one more than 34, so the number in the box has to be less than 27," he said.
"How much less?" I asked. "I don't know...5 less?" he replied with hesitation. Hmmm...

I wrote the open number sentence differently this time:

34 + 27 = (34 + 1) + BOX

"Does that help?" I asked. "Yes, the box should be 26," he said. From this I learned that he needed to literally "see" the same quantity represented exactly the same way on both sides of the equation.

Here is the next open number sentence I gave him:

18 + 23 = 16 + BOX

"I know that 18 is two more than 16, so the number in the box has to be bigger," he said but still could not tell me the number that goes in the box. So I re-wrote the equation as:

(16 + 2) + 23 = 16 + BOX

Almost immediately, he wrote 25 in the box. I think he's almost there, so I'm going to challenge him one more time.

25 + 42 = 27 + BOX

This time I was pleasantly surprised that he adopted my strategy of decomposing the numbers to make them look the same on both sides.

25 + 42 = (25 + 2) + BOX

"It's 40," he said. No more hesitation, yeah! Let's try another one:

70 + 15 = 60 + BOX

This time he didn't need to rewrite the equation to make numbers match identically on both sides. "70 is ten more than 60, so the box is 15 plus 10, so it's 25." I could not believe how quickly he caught on! (And no, he's not gifted.)

I encourage you to try open number sentences with your young mathematicians!