How 3 Techniques From Cognitive Psychology Reinvigorated My Math Classroom

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How 3 Techniques From Cognitive Psychology Reinvigorated My Math Classroom

By Torre' Mills     Apr 28, 2020

How 3 Techniques From Cognitive Psychology Reinvigorated My Math Classroom

This article is part of the report: Education in the Face of Unprecedented Challenges.

If a college baseball coach wants to improve the hitting average of his team, what should he do in batting practice? Is it more effective to have the players hit 15 fastballs, then 15 curve balls, then 15 change ups? Or is it better to have them hit a random assortment of 45 pitches consisting of 15 fastballs, 15 curve balls and 15 change ups?

Researchers at California Polytechnic State University conducted a study in 1994, which showed conclusively that the group that was given a random assortment of pitches did much better in the post test than the group that received blocked pitches—several of one kind followed by several of another—and the control group.

Here’s the rub: the group that received the random assortment of pitches did not feel like they had improved. Even when they were presented with the data to show the improvement, they still wanted to do more of the blocked practice.

Cognitive psychologists call this anomaly desirable difficulty. Simply put: some practices that feel challenging for learners lead to effective and efficient learning.

So, where does this come into play for me as a math teacher? For as long as I can remember, I have loved everything about math—especially teaching young people and seeing the lightbulb come on in their eyes. But, sadly, after 20 years teaching math to middle and high schoolers, I have experienced the pain of watching students struggle. Some of my most brilliant students have grappled to comprehend concepts, to remember processes and to perform well on tests. I’ve seen hardworking, eager, bright-eyed students lose hope and perform below standard and barely average in my classroom.

As an educator, my two biggest challenges have been pacing and prepping for state standardized tests. Every spring, I start gauging the amount of time I have remaining until the tests arrive. I know I need to build in at least three weeks of review time, because as resilient as students are, they have forgotten the material that was taught in the beginning of the year. No matter how much time we spend on the material, which is what causes me to fall behind, my students don’t seem to retain as much of the information as I’d hoped.

The repeated cycle of students forgetting what I worked so hard to teach has been defeating. In 2019, after moving into a position teaching Algebra I and II at a high school in a new district, I began to search for a solution. I knew there had to be a better way. After exploring new curriculum and sequencing changes, I wasn’t finding what I needed, so I pivoted to dive into research related to cognitive psychology—a field that is concerned with mental processes (such as perception, thinking, learning and memory). Unexpectedly, when I started exploring how the brain learns, I found my answer. I now incorporate frequent retrieval practices in my curriculum, that incorporate interleaving the math content and spacing out when topics show up on the retrievals.

The problem? Though it’s effective, the process of frequent retrievals and discriminating between similar looking problems is tough to do consistently because it’s hard. It’s a desirable difficulty.

During the first semester of the school year—when my classes were meeting face-to-face—I began putting these practices in action. Like many educators, I’m still contemplating how to best adapt my teaching to online learning. The natural question might be: Will these techniques work in an online environment? My answer is yes, I believe so. By assigning units in Khan Academy that we had already covered prior to the closing of our school, I’m incorporating many of the research-based practices discussed here. The fact is, these techniques are sound for instruction in general, though we might have to think creatively about how to implement them virtually for maximum effect.

Experimenting With Three Techniques

When I first started exploring how the brain learns best, I attended a presentation about the science of successful learning during an AP Statistics training in Atlanta. Shortly after, I read “Make It Stick,” the book the presentation was informed by and started reviewing academic research related to the book.

I learned that there are certain processes through which we learn information and there are three techniques that improve retention more than anything else: retrieval practice, spacing and interleaving. The best part was that these techniques are inexpensive, and with small tweaks, they can be applied to any curriculum. So, as an experiment, I decided that I would implement the techniques of retrieval practice, spacing and interleaving into my current curriculum.

Retrieval Practice

According to Dr. Pooja Agarwal, a cognitive scientist and expert on how students learn, retrieval practice is “a learning strategy where we focus on getting information out of students’ heads, and not just cramming information in.” It sounded like it would help with recall, but how could I do it and what should it look like? One example that resonated for me is what Agarwal calls a “Brain Dump,” meaning at the midpoint of a unit, as an opening or bellringer, I could ask my students to write all they know about the unit.

I tried this out during a lesson on complex numbers and their operations, asking my students to write everything they remember/know about complex numbers in three minutes. This activity served as a review of what the student knows about complex numbers without limiting what they should recall and showed them what they remember and what they are unsure about.

I used this as a formative assessment for my classes. I was able to lead a discussion in each class on “things we know about complex numbers” and have each student give one new fact from their list until the collective knowledge of the class had been exhausted and was written out on my Smartboard. Next, I asked the students, “what did we not mention? And what did we miss?” From there I was able to ask each of my three Algebra 2 classes, “what do we need to learn next?” This was helpful because it allowed me to get a snapshot of what each class had retained, and what was not sticking in the middle of the unit.


Next up is interleaving, which in math occurs when students are given a variety of problems at one time; essentially mixed exercises. This approach isn’t new. In fact, the Saxon math curriculum my teachers used when I was in middle school over 25 years ago leveraged interleaving. Cognitive psychologists believe that interleaving is helpful because it allows students to determine the appropriate strategy to implement for similar looking problems.

These activities take no more than 15 minutes and are graded so students are incentivized to try in earnest. The greatest value is that I am able to track which skills students are—or are not—retaining. The skills that are not retained are reviewed in mini-lessons and reassessed, as are those that students are remembering how to do.

How is this different from what I have done in the past? I used to teach a unit on say, adding complex numbers and then assign homework on just that skill. Next, I would teach subtracting, multiplying and dividing complex numbers, each in isolation. And then I would give the students a quiz or unit test and have all four types of problems mixed together on the test. In the past, I have been puzzled by why my students struggled through these tests, but now I see that I hadn’t given students an opportunity to differentiate between types of similar problems.

I’ve learned my lesson and now, leveraging the interleaving method more consistently, I make sure that classwork and homework assignments provide opportunities to practice mixed skills. I see now that it is important for students to have practice opportunities to differentiate between similar skills when the stakes are low so that they can be more successful on high stakes assessments like quizzes and unit tests.


The third technique is spacing. The spacing effect is when material is distributed over a period of time to strengthen learning. Earlier this year, for example, during a unit on complex numbers, I introduced addition of complex numbers on the first day. Then as I proceeded through the curriculum to subtracting, multiplying and dividing complex numbers, I would be mindful to put in a bellringer on addition. The main idea of this process is to allow students to revisit topics over a period of time, which in turn strengthens learning. The process that I use for my retrieval practices and my cumulative exams allows me to implement the spacing effect without any additional planning or effort. This provides an added boost to my student’s retention.

Do These Techniques Really Work?

These techniques sound good, but do they really help students learn? In my class, I’ve seen results. At the beginning of the year, one of my students, S.J., told me that she wasn't really good at math and then shortly after, missed over a week of school for medical reasons. When she returned,she appeared lost and frustrated. I encouraged her to keep working hard and that the processes in place would help her to understand more over time. On one especially tough day, she wrote me a note explaining that she thought I was a good teacher but that she felt like she would fail the class.

Fast forward a few months, she scores at 87 percent on a comprehensive midterm that covered so much information that practically all of my students took two days to complete it. She has always been a hard worker who will persist and keep trying, so what changed? The regular retrieval practice activities featuring interleaving and spacing allowed her to learn the content from the first semester over the entire year and not just during one unit.

So if this method is so good why doesn’t everyone start using it instantly? This process of frequent retrievals that force students to discriminate among several strategies to solve problems is tough to do consistently. Remember the baseball study? It’s a desirable difficulty.

Much like how the baseball players had trouble perceiving their improvement, my students have trouble sticking with retrieval practice because it's difficult and frustrating, and at times it doesn’t always feel like it is working. My toughest challenge is to consistently implement the techniques in the face of their frustration and to convince them, show them and sell them on the fact that they are indeed improving.

My desire to find an answer to my 20-year struggle to help kids retain information has brought me to a new hurdle: helping my students (and parents) trust the process enough to persist through difficulty so they can see the benefits on the other side. Coaches do this all the time with student athletes, so I am spending time with some of the coaches at my school to learn more about getting kids to persist through challenges—for me, that’s the next piece of the puzzle.

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