Authentic Interleaving In Mathematics Lessons

One of the most frequently mis-interpreted ideas from research on memory is that of interleaving.  I know that I have previously fallen into the trap of confusing Interleaving and Spacing . This is easily done as they are related ideas!


What it is not:
  • Spacing practice over time, that is called Spaced Practice
  • A curriculum design tool for chopping up lots of topics, that is also an application of Spaced Practice
  • An excuse to shoe-horn fractions, decimals, angles, negatives or whatever else into every single exercise.  That is also an application of Spaced Practice. 

What it is:
  • A consideration at the time of initial study, where related ideas are taught in close proximity
  • This is a desirable difficulty as described by Bjork(1) 

Spacing is revision throughout the course, whereas interleaving is switching between ideas while you study.

Interleaving might work because it helps students learn to distinguish between concepts and learn when to apply which strategy (2).

Interleaving completely unrelated ideas results in no gain in learning (3)

Much research is needed to interpret how we can specifically use interleaving in the mathematics classroom. However, Rohrer and Taylor have done some interesting work in this regard. (4)

Applying it 

The task below is something I was working on with a pupil last night. This is a pupil studying SQA Higher Maths, here in Scotland. This pupil could already do basic differentiation and integration. The new learning was on basic chain rule for differentiation, integration of linear expressions to a power, and integration and differentiation of trig expressions.

We worked through a few example problem pairs of differentiation and integration, which I followed with the set of questions below. I would normally have taught all of these related ideas separately, however the premise of interleaving is the create “interference” of some sort. By making the learning harder in the initial stages there is long term gain. This is uncomfortable for teacher and pupil alike.

For the pupil, there is the struggle involved in switching back and forth between the two different, but related ideas. Similarly, for the teacher the lesson may not have “seemed” to have gone as well. However, this sort of thinking is a result of confusing instantaneous performance with learning. They are not the same thing!

Rather than blocking these 3 or 4 skills over separate lessons and then brining together at the end, the idea is to work on all of them together from the start of the learning episode (sequence of lessons).

Some thoughts on this task

1. The pupil I was working with managed to demonstrate some technical proficiency with this task, making errors along the way which I was able to discuss with in, in the moment. In a classroom setting this sequence of questions might best be executed using mini whiteboards.

2. The task is clearly about technical details rather than conceptual understanding. However this is important. Experience has shown me that many pupils struggle with these more involved areas of calculus.

3. The task is rooted in the idea of doing and undoing . Once we had completed the grid I asked the pupil to go back and work in the opposite direction. Integration and differentiating his answers to get back to my initial questions. He jotted down “power to front -> one off power -> multiply by derivative of bracket”. It was interesting to see him re-trace his steps backwards along this journey when looking at integration. This doing and undoing allowed him to be work independently, with self directed feedback, as he worked through the task for the second time, in reverse.

4. Of course, with integration, it is not the derivative of inside the brackets. It’s the coefficient of x in the linear expression. Two of the examples were deliberately put in, such that they could only be differentiated but not integrated at this point in time. I wanted him to see there was a need for more sophisticated techniques such as integration by substitution. I didn’t want him to over-generalise the idea.

5. The task is one of a sequence of tasks which would be used. I would drill down deeper into the technical and conceptual aspects of these ideas using focused tasks. Perhaps using a couple of the tasks from:

Other ideas

There are many other areas of the curriculum where related ideas could potentially be interleaved: area and perimeter, volumes of shapes, sine and cosine rules etc.  

One thing I frequently think about, in terms of research, is that we have to test it in our won experience.  My experiments with interleaving will continue.  A tension I am conscious of, as are many teachers I’m sure, is the balance between ensuring pupils have high level of success and making the learning hard enough such that the long term performance is secure. 

I have previously written about dabbling with research in relation to teaching calculus here.  It might be of interest! This was a couple of years ago and my thinking has evolved.

In terms of research, I find much of it gives a language to describe things that many excellent, experienced teachers already DO.  As such, I’ll finish with this lovely sequence from the master himself, Don Steward:


(1)Bjork, R.A. (1994). “Institutional Impediments to Effective Training”. Learning, remembering, believing: Enhancing human performance.

(2) Rohrer, D. (2012). Interleaving helps students distinguish among similar concepts. Educational Psychology Review, 24, 355-367.

(3) Hausman, H., & Kornell, N. (2014). Mixing topics while studying does not enhance learning. Journal of Applied Research in Memory and Cognition, 3, 153-160.


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