In my district, we are fortunate to have core curricula that provide meaningful learning opportunities, that have helped us build coherence in terms of both content and practices across K-8. There are practice opportunities built into each curriculum, but it’s often not quite enough for some students. So how can we use information about students to give them more targeted practice opportunities?
Retrieval practice focuses on deliberately recalling previous learning, geared towards encoding it into long term memory. Students are given time to ‘forget’ the information before being asked to retrieve it, and this process of retrieving helps strengthen the brain’s memory of it.
Retrieval practice is suggested the day of new learning, the day after the learning, and in two week intervals after that (McConchie 2022; Roedinger et al. 2011). We are about two weeks out from our unit on linear equations and linear systems — deep into our study of functions — with some students still demonstrating a clear need to practice some skills and concepts.
Solving Equations in Middle School
One that seemed essential to address is how to interpret a coefficient. represents the product of and , not And a few students needed more experience working with and practicing this idea. They had some ideas about how to work with coefficients, but had not encoded a mathematically accurate understanding in their long term memory. So I decided we would play a few rounds of Number Boxes. [more about the game]
Setting Up The Game
I modeled a game board: ❒x = ❒ ❒, with two throwaway boxes in the upper right corner.
Then, I announced the target: the greatest possible value for $latex x&bg=ffffff$.
The students would take turns rolling a 10-sided die to generate random digits, and placing them in one of the five possible boxes. The placement of previously placed digits cannot be changed once a new number has been rolled/generated.
The first digit: 5.
Round 1: Can we compare without calculating?
We continued to generated random numbers until all five boxes were full. Here are the results from two students:
Which of these equations has a greater value for ?
“Ohhh, they are both equals a number, so that’s five times some number equals 53, and five times the other number equals 57. That number is bigger. It wins,” Tamara explained.
“Do we need to calculate the value of each to check?”
“No,” Inaya shook her head, casually glancing down at her new shoes. “It sounds right. So I won. That’s cool.” She looked quickly over her shoulder, and added, “you know, we could play again.”
Round 2: Where do you place the 9?
We decided to play again with the same board, and the same target: the greatest possible value of x.
The first digit we generated: 9.
Tamara put hers in the tens digit of the two digit number. Inaya put hers as the coefficient for : .
“Oh, okay, so I put my 9 in the two digit number because I want my x’s to be worth a lot,” Tamara explained.
“And I want a lot of those x’s. A great number of them!” Inaya said.
They agreed that it was time to generate another number: 2. Tamara quickly placed it as the coefficient for the variable.
2x = 9 ❒, with two more throwaway boxes available. Inaya paused, and eventually placed it in the throwaway boxes. “I’m doing the opposite of Tamara,” she said crisply. “I want big numbers.”
The next digit we generated was a 7.
Tamara chuckled to herself, clearly delighted but not wanting to give away her strategy just yet. Eventually, they ended up with the following:
Which has a greater value for ?
Tamara smiled, and explained how she knew that she had won without calculating. “I have two x’s, and they make a bigger number than your 9 x’s. So each x must be worth more!”
I asked them to prove this by solving for : compared with .
Inaya realized how she could turn this around, and asked to play another round. “But this time, let’s have the winner be the one with the smallest value for .”
It’s like she read my mind.
One Last Round: Adding another step to the equations
We played two more rounds with the goal of generating the smallest value for . Then I switched it up one more time: this time, there would be two steps to the equation: one multiplicative, one additive.
We set up the boards: ❒ + ❒ = ❒ ❒, with two throw away boxes. The goal: have a value of x as close as possible to 10.
The first number we rolled: 2.
“Wait, this one feels very different,” Tamara said.
“Yeah. I don’t know how to make it be close to 10,” Inaya added.
One of the reasons I love this game is because it provides meaningful practice and also encourages students to “look for and make use of structure.” (SMP7) As someone who has had a wide variety of experiences with linear equations, I can parse the equation, and see meaningful chunks. If we ignore the addition in there, ❒ = ❒ ❒ for a value of that is approximately 10 means that the coefficient should be approximately the value of the two digit number. So it would be helpful to have the same number for the coefficient on the left and the tens digit on the right.
Both students placed their 2 in the same spot, followed by a 9 in the same spot: the coefficient.
“So we have and we want a value of that is as close as possible to 10. What are you hoping to get for the two remaining digits?
“90… 2. 92!” Inaya called out excitedly.
Our next roll gave us a 3, which Tamara judiciously placed as the ones digit.
“I’m going to throw that one away,” Inaya said with a bit of pride. “I don’t want us to have the same answers.”
The game continued until they had generated the following equations:
“We’re both really close!”
“…but who is closer?” I always have to spoil the fun, lest they quit too early.
Both Tamara and Inaya worked to solve their equations.
“Ugh, those are both so close. Which one is closer?”
“Well, if were exactly 10, we could express it as .
“Mine is so close! is one away from .
“One what away?”
“Yes! And how far away is yours from , Inaya?”
Tamara let out a frustrated groan. “Did we actually tie?” She crinkled her nose and narrowed her eyes.
I smiled. “Yes.”
Two days later, in math class, I saw Inaya and Tamara working with some coefficients. They interpreted it as multiplication as if it were no big deal. We may end up playing again later, to make the long term memory even more durable, but I’d say this is a good sign.
When playing Number Boxes, I often invert the goal every single round. We don’t want to get too comfortable.
Interleaved practice offers problems that are arranged so that “consecutive problems cannot be solved by the same strategy.” (Rohrer et al. 2017). This offers students retrieval practice, and a chance to figure out which strategy to employ. Sometimes, problems can look deceptively similar:
…yet require very different thinking.
I did not arrange our rounds purely as interleaved practice. I gave the students the opportunity to do two rounds trying to maximize the value of before moving on. And we were only working with equations written in a similar format. Truly interleaved practice might have asked students to solve a linear equation, then find the volume of a cone, then determine a percentage change when given the start and end values. But this was baby steps towards flipping around some of our work involving practice.