By: Keith Lyle

Keith Lyle is an Associate Professor of Psychological and Brain Sciences at the University of Louisville. He received his B.S. in Psychology from Indiana University and his M.A. and Ph.D. in Psychology from Yale University. You can find him online at http://louisville.edu/psychology/lyle or on Twitter @KBLyle.

We all have a dream job. When I was growing up, mine was private investigator. I wanted to solve mysteries, like Sam Spade in The Maltese Falcon. I sort of got my wish. I ended up being a cognitive psychologist – a learning scientist, specifically. As a scientist, I still get to solve mysteries (or try to, at least). But not everyone is so fortunate. For example, many high school students dream of becoming engineers, but about a third of all students who enter an engineering program in college fail to complete it (1). Their dreams are broken.

Why this happens is a mystery fit for a learning scientist. Two of my colleagues in the Department of Engineering Fundamentals at the University of Louisville came to me a few years back and told me they thought inadequate mathematics knowledge was a major culprit. Many students simply can’t do calculus well enough to succeed in engineering courses, and my colleagues wondered whether learning science could help students master more of the mathematics they were taught. My graduate student and I decided to take the case.

Learning scientists know that one key to learning and remembering information is spaced retrieval practice (if you’re a regular reader of this blog, you might know this, too). If you’ve ever learned a locker combination, you’ve probably experienced the power of spaced retrieval practice. Locker combinations are usually arbitrary combinations of numbers – a type of information people might say they have trouble remembering. But what do you do with a locker combination? You bring it to mind from time to time in order to open your locker. You practice retrieving the combination in a manner that is spaced out over time and voila! you soon know it by heart.

Math classes also tend to feature a lot of retrieval practice involving numbers. Instructors present information and then students practice retrieving it on homework, quizzes, and exams. But retrieval practice on a given topic is not necessarily spaced. It may all occur in a fairly small window of time: Students might be presented with the steps in performing a mathematical operation, perform a bunch of problems to practice those steps, and take an exam covering the operation – all in a week or two of class. After the exam, students may not need to retrieve the information again until the dreaded cumulative final exam, which often reveals that students no longer remember how to perform the operation. If retrieval practice were more spaced out, students’ memories would be more robust and they would be more likely to remember the information over the long term, not only on the final exam, but also when they advance into subsequent classes that build on what they were previously taught.

With this in mind, my colleagues and I designed a simple research project to test whether we could help engineering students acquire the calculus knowledge they need to turn their dreams into reality (2). We decided to study students in a precalculus course for freshmen in engineering because if students can’t remember what they learn in precalculus, they’re probably doomed to fail in calculus. My colleagues in Engineering Fundamentals honed in on the most important mathematical operations students were taught in the precalculus course.

In previous iterations of the course, three questions about each operation appeared on a quiz administered soon after the operation was first introduced in class. In fall 2014, my colleagues and I decided to take those three questions and manipulate how they were spaced. For some randomly selected operations, the questions were massed, meaning they all appeared on the first quiz following the operation’s introduction (in other words, standard practice). For other randomly selected operations, the questions were spaced so that one question appeared on the initial quiz, one question appeared on the next quiz, and the third question appeared two more quizzes down the road. In the end, questions in this condition were spaced over about four weeks (see table below for an example). 

All the quizzes were administered online, which made it relatively easy to manipulate the massing or spacing of questions (or, more accurately, it was easy after one of my colleagues put in many hours of prep work). Online administration also allowed us to run some students—taught side by side with their peers—through the entire course without spacing any of the questions. Basically, these students comprised a control group—taught exactly like in past semesters—that we could compare to our experimental group, in which some questions were spaced. The control and experimental groups were essentially identical in terms of demographics, average high school GPA, and average Math ACT performance.

We examined performance on the final exam in the precalculus course, on which students had to perform all the important operations learned throughout the semester. Because we had manipulated spacing within participants (the experimental group practiced some material with spacing, and other material without) and also between participants (the experimental group got some spacing, and the control group got none), we were able to look at the data from two perspectives.

Looking at just the experimental group alone, we compared performance on questions targeting spaced and massed operations within participants. We expected students in the experimental group to do better on exam questions targeting spaced operations than massed ones, and that’s exactly what we found. On average, students did about 3% better on spaced operations than massed ones. This may not sound like an enormous effect, but it translates into about a third of a letter grade. That would be enough to change a student’s grade from, say, a C+ to a B-.

Looking across the two conditions, we were able to compare performance on the same operations when they were spaced in the experimental group versus massed in the control group. We expected better performance on operations when they were spaced in the experimental group versus when they were massed in the control group and, again, that’s what we found. In this analysis, the benefit associated with spacing was about 8%, suggesting that students taught in a traditional all-massed format might be underperforming by nearly one whole letter grade what they could achieve with the help of spacing.

An especially nice feature of our study is that we were able to follow students in the precalculus course who advanced into a calculus course the next semester. If spacing really helped students retain their hard-won precalculus knowledge, then it should have set them up to do better in calculus. Note that we didn’t manipulate anything in the calculus course. It was taught as it had been in past semesters with no meddling by us. We examined performance on the cumulative final exam as a function of whether students had been in the experimental group or the control group in precaclulus. The result was striking: Students who had been in the experimental group performed, on average, 10% better than students who had been in the control group. Moreover, 68% of students who had been in the experimental group earned at least a C in calculus, versus only 48% of students who had been in the control group (but note that this difference, although practically important, did not reach statistical significance, most likely because we were limited to analyzing only the 54 students from the precalculus class who advanced into the calculus class and actually finished the calculus class).

So, what’s the solution to the Mystery of the Failed Engineering Student? One answer might be that collegiate instruction typically doesn’t involve enough spaced retrieval practice. Our study showed that spacing can increase retention of a complex body of mathematical knowledge and thereby improve performance in genuine college classrooms. The notion of a simple, no-cost intervention that could help more students earn an engineering degree is, like Sam Spade says in The Maltese Falcon, the stuff that dreams are made of.

References:

(1) Hopkins, R. F., Lyle, K. B., Hieb, J. L., & Ralston, P. A. S. (2016). Spaced retrieval practice increases college students’ short- and long-term retention of mathematics knowledge. Educational Psychology Review, 28, 853-873.

(2) Pearson, W., & Miller, J. D. (2012). Pathways to an engineering career. Peabody Journal of Education: Issues of Leadership, Policy, and Organizations, 87, 46-61.