The case for mixing modes of teaching: a mathematical model.

Here’s the truth.  We struggle to define the strategies that we deploy in our classrooms and, when we undertake research to identify the most effective strategies, there’s an awful lot of noise, error and variation between the conditions we’re studying in one place compared to any other.  However, various attempts have been made and we have a growing evidence base on which to base judgements; we have the possibility to develop ‘evidence-informed wisdom’ as classroom practitioners.

But let’s not ever forget the inexact nature of this research.  One thing I often consider is, given how hard it is to define and test just one strategy, how supremely difficult it is to test strategies in combination.  And yet, I do encounter some rather strangely absolute positions from some teacher-commentators, filled with talk of effect sizes, and opportunity costs and the like.

But let me throw an idea into the mix:  We don’t really know how some strategies might work in combination but let’s imagine we can model it. (If the maths leaves you cold, just scroll to the end for the conclusions!)

Imagine there are two strategies under consideration: Strategy X and Strategy Y.   We spend time x on X and time y on Y.  Our total time T is finite – so x + y = T.   Imagine that Strategy X has been shown in single factor studies to be more effective than strategy Y.  The question is: how much time to spend on each?  This depends on our model for how strategies work in combination.  Let me suggest three possible alternatives.

Scenario 1: Linear arithmetic.

The total Effect E depends on a combination of X and Y but, since X is more effective than Y, we spend more time  on X in a ratio a where a > 1.  We could write:

E = ax + y.

Here, as long as x and y add up to T, we could spend a mix of time on both, albeit slightly more on X than on Y.  But why would we?  If we want the maximum E, the best way to do this would be if y = 0, and we go all the way to x = T.

This is the scenario the opportunity cost people might imagine.  If would be true that time spent on Y is wasting an opportunity to spend time on X.

Scenario 2:  Non-linear arithmetic. 

Here, X is not better than Y by a factor of a.  It is a non-linear effect.  Here we say a is a power, not a multiplier.  The conditions a >1 and x + y = T still apply but the Effect might be written:

E = xa + y

In this scenario, we have the same situation as in Scenario 2 – it could be a bigger or smaller effect depending on the values but it is still true that, if y is zero, we get the biggest effect.  The difference here is that the closer y gets to zero, the impact on E is more and more significant.


We don’t know that this is how strategies interact.  I would suggest that in many situations we are not simply switching between independent activities, X and Y.  It is much more likely that X has an impact on Y and Y has an impact on X – even if X has a much bigger single-factor impact.  This is likely given the way learning can be shown to be consolidated through varied practice.   In this sense the variables operate in a different mathematical model:

Scenarios 3: Geometric multiplier. 

If we base this on Scenario 2,  we could write the effect E as follows:

E = xay

Here, immediately you can see that if either x or y is zero, the effect is zero.  We need some of each! The question becomes – what is the optimum time allocation.  A bit of simple algebra shows that, given that y = T -x, E = xa(T-x) and the maximum value of this will occur when x = [a/(a+1)]. T

If a = 1, X and Y are equal, we get x = 0.5 T: half the time each.

If  a = 2, then E = x2y and  the optimum is to spend 2/3 time on X; 1/3 time on Y.

If a = 4, then E = x4y and the optimum is to spend 4/5 time on X; 1/5 time on Y.

Even if a = 100, we would still need to spend 1/101th of our time on Y – just a little bit but not zero!


We don’t really know how strategies combine because our studies to-date don’t go anywhere near this level of detail.  The risk would be that Scenario 3 is the model that most accurately captures how learning strategies interact.  If we’re stuck in an ‘opportunity cost’ mindset, then we risk having very low impact if we abandon reasonable strategies simply because another one seems to be better on its own.   It’s a safer bet to hedge our bets and to combine strategies – to use a mix; a diet; a blend  – in proportions suggested by our empirical evidence from experience in our contexts:

Screen Shot 2019-02-24 at 19.38.50


“The use of precise mathematical formulae is for illustrative purposes only; the author takes no responsibility for any misunderstandings regarding the gap between this fiction and the reality they loosely represent due to excessive literal readings of the mathematics by purists and/or the twitter graph police.”


  1. Hi Tom, that’s interesting and does make a lot of sense. I’ll admit to having got a little lost with the masts but ignoring that fact I’ll add in another factor.
    I was reflecting last night on how education works within a social and philosophical context. How children (or adults) learn is not just a matter of x and y, there is the social context in which learning takes place. The societies norms and values, families and communities.
    Repeatedly politicians and academics have tried to take a system of learning from one country ad put this into another country. This is most prevalent in developing countries who look to the West using foreign aid to build their infrastructure and systems.
    However account needs always to be taken of the wider culture and values that exists around the learning, this will have a significant impact on the effectiveness of either a or b.

    Liked by 1 person

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