Journey to Interleaved Practice #1
“Learning to pair problem types and procedures is especially challenging in mathematics because different problem types are often superficially similar.” -Doug Rohrer
This semester I decided to create a study on interleaved practice with my second-semester calculus class. By no means is the study empirical in nature – I am not using any controls, and haven’t thought much about confounding variables. The study is more observational in nature, with the goal of collecting student solutions to analyze how students are answering specific questions.
The idea came about through an email discussion with Yana Weinstein of the Learning Scientists (and University of Massachusetts, Lowell), and Doug Rohrer of the University of South Florida. I had been interested in using some of the interleaved practice tools that Yana had helped develop in our Slack team, and she thought it would be nice to touch-base with Doug, as he thinks a lot about how interleaved practice affects students’ learning in mathematics.
There were two specific papers that I remembered reading, this being one of them. I thought the discussion on discrimination rather appealing, and something that I tended to see each semester. Roughly speaking, we teach mathematics in a particular way, scaffolding from one idea to the next, with practice questions always coming from specific chapters. As students practice, they always know the strategy that they need to use in order to solve the question (ie. they think “the questions are at the end of the lesson on the Pythagorean Theorem, so I probably have to use the Pythagorean Theorem to answer the question”). They don’t, however, get much practice mixing the different strategies that they learn. Unfortunately, this means when they come to a summative test, extra effort has to be initially put in to determine what strategy to use to solve a question.
My main goal is to do a bit of observational research around discrimination on summative tests. I have developed a schedule of interleaved homework and interleaved lessons for my integral calculus class, and we are currently off to the races. When I check back in next time, I will share some of the tools I am using to create the interleaved homework.
For the Love of Maths 
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There is quite a large knowledge base telling us that an interleaved approach to learning is more beneficial than a blocked approach. However, the irony is that most students like the blocked approach more because they *feel* as though they learn more – even though interleaved practice leads to better learning (in the sense that knowledge is more flexible and longer-term). Either way, how we space the practicing of concepts is definitely important to flexible and long-term learning…
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Hm. I thought I’d be able to delete this comment (since I posted it as a non-reply) – but apparently I cannot…
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I don’t think that changing the order of presentation of problems does much to help students learn—the important thing is to have multi-step problems that require using already learned knowledge (in addition to new material, not in place of it).
Some books are better than others about avoiding the “this is section x, so the method must be x” problem. The old Singapore Primary Math books were very good about making sure that older material continued to be used, for example, gradually ramping up multi-step problems through the first 6 years.
At the high-school/early-college level, the Art of Problem Solving texts often require using prior learning to solve problems. Of course, those books are aimed at the top 1–5% of high-school students, so the problems are too challenging for average students (they avoid all routine drill and ask only challenging questions, many taken from math competitions).
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There is quite a large knowledge base telling us that an interleaved approach to learning is more beneficial than a blocked approach. However, the irony is that most students like the blocked approach more because they *feel* as though they learn more – even though interleaved practice leads to better learning (in the sense that knowledge is more flexible and longer-term). Either way, how we space the practicing of concepts is definitely important to flexible and long-term learning…
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