We like to get into the flow when practicing something, and we like to have our students concentrate on one particular type of problem at a time until they have mastered it, before moving on to the next. But is that really the best way of learning? Spoiler alert: It is not!

In a 2014 study, Rohrer, Dedrick and Burgess show the benefits of interleaved mathematics practice for problems that are not superficially similar. If problems are superficially similar, it makes intuitive sense that one needs to – at least at some point – practice several types together, because clearly distinguishing different kinds of problems and choosing the appropriate approach to solving it is not easy since the problems themselves look so similar. But for problems that look already very different one might think that blocking similar problems and practicing on them until they are mastered, and then moving on to the next type of problem might be a good choice, since one can really concentrate on each type individually and make sure one masters it.

However, this is not what the data shows. Mean test scores in their study (on an unannounced test two weeks after a nine-week practice period) were twice as high for students who had practiced interleaved problems than for those who had been objected to blocked study. Why is that the case?

There are many possible reasons.

One not even connected to interleaving or blocking is that the spacing effect comes into play: just by learning about a topic spaced in chunks over a longer period of time, the learning gain will be higher.

But interleaving itself will help students learn to distinguish between different kinds of problems. If all problems students encounter in any given lesson or homework assignment are of the same kind, they cannot learn to distinguish this kind of problem from other kinds. Being able to distinguish different kinds of problems, however, is obviously necessary to pick the appropriate strategy to solving a problem, which in itself is obviously necessary to actually solving the problem.

So why can’t student learn this in blocked practice? For one, they don’t even need to look for distinguishing features of a given problem if they know that they will find its solution by applying the exact same strategy they used on the problems before, which will also work for the problems after. So they might get a lot of practice executing a strategy, but likely will not learn under which circumstances using this strategy is appropriate. And the strategy might even just be held in short-term memory for the duration of practice and never make it into long term memory since it isn’t used again and again. So shuffling of types of problems is really important to let students both distinguish different types of problems, and associate the correct strategy to solving each type.

If you are still not convinced, there is another study by Rohrer and Taylor (2007) that shows part of what you might be expecting: That practice performance of “blockers” (i.e. students who practice in blocks rather than mixed) is substantially higher than that of “mixers”. Yet, in a later test on all topics, mixers very clearly outperformed blockers here, too.

So what does that mean for our teaching? Shuffle practice problems and help students learn how to discriminate between different kinds of problems and associate the right approach to solving each kind!

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Rohrer, D., & Taylor, K. (2007). The shuffling of mathematics problems improves learning Instructional Science, 35 (6), 481-498 DOI: 10.1007/s11251-007-9015-8

Rohrer D., Dedrick R.F., & Burgess K. (2014). The benefit of interleaved mathematics practice is not limited to superficially similar kinds of problems. Psychonomic bulletin & review, 21 (5), 1323-30 PMID: 24578089

Alexander TscheulinTo put it in a more abstract way: isn’t it always good to narrow down from a broader perspective, understanding the problem and task and concentrating after that on the solving skills? Are strategies like that taught?

MirjamI’m not aware of very many courses that are taught that way even though there are some. And yes, I believe that’s the way to go. There are one or two that we are currently restructuring to include that exactly.

Alex, we miss you at ZLL!

Alexander TscheulinThank you!