Desirable Difficulties - Interleaving

"In class they all got it! But in the test they didn't even spot it!"

The tortured cry of many a mathematics teacher, particularly of reasonably able Y9 students after they've been taught Pythagoras or right-angled trigonometry. In lesson they did a hundred questions, understood everything perfectly, could work backwards and forwards, they could even work out the length of the ladder needed to save that pesky cat in the tree...

Yet, weeks later, they're trying to use Pythagoras when trigonometry was needed, or vice-versa. Or, they're asked to find a volume of a right-triangular prism and suddenly a² + b² = c² is making a bizarre appearance - I mean, they see a triangle, they do Pythagoras, right?

But, delving even a tiny bit deeper, this isn't so surprising. In the lesson titled Pythagoras' Theorem, after your beautifully-crafted linked problem pair examples, of course they could do it. The lesson told them it was what they needed to do. Ditto SOHCAHTOA. Then, in your test, it contained a triangle. No title, no date, no learning objectives. Did you prepare them for this?

This is the fundamental underpinning of one of Bjork & Bjork's principles of Desirable Difficulties. Making the initial learning phase harder, perhaps by weaving in other topics (either closely related such as the link between Pythagoras and trigonometry, or even topics with only a surface level similarity such as Pythagoras and other topics which happen to involve a right-angled triangle), can lead to exceptionally better understanding, progress and retention in the long-term. Their short-term results may not improve, in fact they might even drop off slightly... but, the short-term is NEVER what we should be concerned with as teachers. If students gain a deeper appreciation for the connectedness of mathematics, become more able to recall a range of techniques and more effectively select the correct tool from the box for the job... then short-term pain is definitely worth it!

Further reading (as a starting point): https://bjorklab.psych.ucla.edu/wp-content/uploads/sites/13/2016/04/EBjork_RBjork_2011.pdf