If you studied a videotape of one of my classes, you’d notice that for much of the time, I don’t seem to be teaching at all. To all outward appearance, all I’m doing is watching the pupil attempt the current question, leaning slightly forwards, saying nothing whatsoever. And yet, I’m going to argue, it’s what happens in those 5-to-10-second chunks of silence that separates good maths teachers from bad ones.

While I’m sitting there watching, there are two things on my mind. The first is this: *should I say something here*? This is a rather subtle and complicated question, and I’ll discuss it in more depth in a future article. But I’m also furiously pondering something else, something I consider every bit as important, if not even more so. What? I’m trying to decide *the next question*.

One of the most powerful advantages of the one-to-one tutor (or parent) over the classroom teacher is the power to set the exact question that the students needs, right now. For this reason, you should be extremely sceptical of any maths tutor who shows up bearing a worksheet – they’re voluntarily relinquishing what’s probably the single most powerful weapon in their arsenal.*

*: Exam papers are a little better, but only if the tutor’s been hired to prepare for a specific, definite exam in the near future. Otherwise, same thing.

To understand what I mean here, let’s start with an example of what not to do. Here’s a possible exchange between an adult and a child, after the two have been studying the decimal system:

Adult: What’s three-tenths as a decimal?

Child: Um. Hm. Nought-point-three?

Adult. Yes, good. What’s two thirds as a decimal?

This is a disaster. Not only is the second question much harder than the first (any child who might get it right would trivially get the first one right as well), but it actually requires somewhat different skills (rote memory of key fractions, as opposed to deducing them from understanding of the decimal system). The child has communicated (though hesitation, then a correct answer delivered in an uncertain manner) that they are right on the cusp of understanding, but not quite there yet – a few more similar questions, and it’ll become firm knowledge. Instead, the adult in this example has chosen to follow up with entirely the wrong question – and will likely end up spending the next ten minutes explaining something for which the child isn’t ready.

On the other hand, here are some questions with which I might follow up the first question (‘What’s three-tenths as a decimal?’), depending on the context:

a) ‘What’s nine-tenths as a decimal?’

b) ‘What’s three-hundredths as a decimal?’

c) ‘What’s nine-hundredths as a decimal?’

You can probably guess that I’d use question ‘A’ with a child who’d struggled with the original question (or even had to think about it at all). But what’s the difference between ‘B’ and ‘C’? The answer is that ‘B’ contains a hint – the child has just been asked about 0.3, with reference to a different fraction, so he or she knows that this can’t be the answer. On the other hand, ‘C’ offers a plausible, but wrong, answer, in the form of ‘0.9’, making it harder. Of these three options, I’m aiming to pick whichever choice the student will get right around 80% of the time – and while they’re working out three-tenths, I’m desperately trying to deduce which one that is.

This is the kind of precise control that I cherish in a one-to-one setting. The best learning happens when you can consistently set questions that stretch the child, make them think, but don’t require a break in the flow of examples that’s progressively granting them mastery of the topic. That’s impossible with a worksheet, or with any pre-prepared list of questions, because a piece of paper can’t know how easy or difficult a student found the previous question. Working one-to-one with a student, you know just this, and it’s a crime not to use that information.

This isn’t exactly a revolutionary idea – in fact, it’s the Socratic method, plain and unvarnished. And it doesn’t require a teaching qualification, or even all that much experience. Next time you want your child to really understand something, start with something they know. Then ask something that’s just a tiny bit harder, the smallest increment you can manage. Trickier numbers, an extra decimal place, a subtraction instead of an addition. But only one of those things at a time – so that the pupil is odds-on to get it right, every single time. Keep doing it. You’ll be surprised how far they get.

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