Back-to-front maths

I have some reservations

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I was recently contacted by a subscriber to this Substack who was concerned that their school has committed to implementing ‘Back-to-front maths’ next year. This was a label I had not come across. However, when I began looking into the resources, I recognised it as an approach to teaching maths developed by researcher Tierney Kennedy that I have seen mentioned before.

My subscriber is, perhaps unsurprisingly, a practitioner of explicit teaching. Back-to-front maths, on the other hand, is something else. The Back-to-front maths website suggests that it is a ‘balanced’ approach that includes an ‘experimental problem’ and ‘explicit teaching of flexible strategies’. It links to a conference paper published by ‘Kennedy Press’ that discusses direct instruction and ‘the use of challenging tasks’ before alighting on a ‘conceptual change’ approach first outlined in another Kennedy Press paper available on the website.

The conference paper presents standardised results from a ‘PAT-M’ numeracy test. These results come from three primary schools in South Australia that adopted Kennedy’s approach. The results are then compared to a strange control group, the South Australia Department for Education and Child Development’s (DECD) Standards of Educational Achievement (SEA). These state that to meet whatever benchmark these standards are intended to represent, a child in Year 3 should obtain a PAT-M score of 101 or above and a child in Year 4 should obtain a score of 110 or above. Kennedy computes the difference between the two floor scores i.e., 110 - 101 = 9 and claims this is the ‘expected gain’ between Years 3 and 4. She then compares this with the gains made by students in the schools following her program (14.7 in the first year of the study and 12.9 in the second year). She concludes this is a statistically significant result in favour of her method.

This does not seem right to me. I cannot find out how South Australia calculates their expected standards, but assuming the difference between the floor standard at Year 3 and the floor standard at Year 4 is the ‘expected gain’ seems unfounded. What if most students sitting at 101 in Year 3 score above 110 in Year 4? What if most kids are performing well above or well below this? Without further evidence, I see no reason to think the comparison Kennedy makes is meaningful.

How do we know that my subscriber, implementing explicit teaching in their current classes, would not generate an average gain in excess of those students in Kennedy’s intervention? There is no reason to assume that this weird control group represents my subscriber’s classroom, or any representative group of children at all.

Setting aside the evidence as presented, does Kennedy’s program align with what we know about more and less effective classrooms? I don’t think so.

My subscriber has access to a training video featuring Kennedy and has sent me screenshots of the transcript. In this video, Kennedy talks about a ‘problem-based’ approach to teaching mathematics through a cycle of lessons. In the first lesson of this cycle, she advises setting a problem so difficult that about 80% of students answer it incorrectly. This is described as ‘starting well’. She then advises not to summarise at the end of the lesson. Instead, that should wait for lesson two. This summary is then something to ‘bounce off’ to do explicit teaching.

However, I cannot see where any explicit teaching is described. Instead, it seems to be about questioning students and asking them to demonstrate their own solution methods to each other, which seems a little worrying if 80% of these are wrong.

At other points, she advises telling students that, if they want, they can copy from each other.

What do we do with the students who don’t figure it out? Again, it doesn’t look to me like explicit teaching. Instead, Kennedy advises giving a prompt, or giving them a wrong answer to prove wrong.

Clearly, I have not seen this enacted in the classroom, but at face value, this seems like very poor advice. In the 1960s, researchers conducted process-product studies where they observed teacher behaviours and looked for gains in outcomes for students. They then identified the behaviours that were associated with greater gains. Rosenshine’s Principles of Instruction summarise these findings, along with supporting evidence from experimental studies. Rosenshine suggests the reverse of Kennedy’s advice—that teachers should aim for an 80% success rate.

Kennedy justifies her approach with an appeal to inducing ‘cognitive conflict’. This is a popular idea, particularly with advocates of ‘constructivist’ teaching—an approach that seems to have inspired much of Kennedy’s work. It suggests we ask students to draw on prior knowledge to formulate a view or solution to a problem, then present them with the evidence that proves they are wrong. As a result, the assumption is that they will change their views.

Anyone who has spent some time on Twitter/X may have reason to doubt this assumption. As we may expect, the evidence on the effectiveness of inducing cognitive conflict in the classroom is mixed at best, with a potential for this strategy to backfire—people don’t like the idea they are wrong and this can provoke a number of alternative responses to changing their views. In my opinion, the evidence is certainly not to a standard to justify abandoning explicit teaching, for which there is much more robust evidence.

I cannot see how Back-to-front maths balances explicit teaching with problem-based or constructivist teaching. In fact, from what I have read, it appears to be a quite extreme form of the latter. I am often told that it is incorrect to characterise problem-based learning as ‘minimally guided’, but what would you call this?

I have every reason to assume Kennedy is sincere and that she truly believes the wider adoption of her teaching methods will lead to improved outcomes for students. However, I don’t think it will and I think it would be back-to-front to insist my subscriber abandon explicit teaching in its favour.

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