These days, a lot of students receive extra help for their maths. So why aren’t they doing better?

Back when I was a mathematics teacher at a selective London prep school, it was hard to be unaware of an unseen and ghostly army, toiling away behind the scenes. Their silent work could be detected in the margins of the exercise books I marked every week, though rarely did they have much influence in the classroom. On the whole, their impact was probably more positive than not, though in some cases it was closer to neutral, and in one or two actively harmful. I refer, if you hadn’t guessed, to private tutors.

It was difficult to understand, from my position in front of the blackboard, what exactly these shadowy figures were doing wrong. But surely it was something – if I could get a class of twenty-five students through Common Entrance and Scholarship 13+, why couldn’t they get their one-to-one charges to even the middle of the class? With the classroom teacher’s traditional sense of martyred superiority, I put the whole thing down to some miserable combination of apathy, incompetence, and ignorance of what the syllabus actually consisted of.

It took a few years of full-time tutoring to realise that this wasn’t quite right. Admittedly, some tutors really are incompetent, and there does exist a depressing minority who don’t care as much as they should.^{1)}Acceptable answers to “what’s your day-job?” are “teacher” and “private tutor”. Unacceptable answers: “student”, “aspiring actor”, literally anything else. But, at root, this isn’t the real problem. What is?

The answer lies in that terribly ambiguous request, to “support” the child with his or her maths. This is asked always with the very best of intentions, yet it can be poisonous to effective teaching.

Good tutoring builds a mathematician the way that an architect builds a skyscraper. There’s no point putting up the next floor until everything below is rock solid – you’d only be making a shaky structure even more rickety. Instead, the focus is on shoring up the foundation, brick by brick, until it makes sense to proceed upwards. Gradually, the pupil becomes more and more comfortable with the concepts that lie at the root of school mathematics.^{2)}Number bonds & tables, fractions, decimals, percentages, linear algebra. And as the student masters these, they suddenly find learning from their classroom teacher a completely changed proposition. Concepts suddenly make sense – they scarcely need further tutoring at all. They have become *good at maths*.^{3)}Something that pupils often say to tutors is “it makes so much more sense when you explain it”. Good tutors know what this means. It means that they haven’t yet done a good enough job.

So this is what good tutoring looks like: guiding the student towards mastery, one fundamental at a time. That certainly wasn’t what my students were receiving – indeed, they seemed to find each new concept harder than the last. What was happening instead? Something like this, I’d imagine:

*Our well-meaning tutor arrives at the client’s home, to discover that the pupil is learning, this week, about Circle Formulae. They’re a mystery to our student – he has no idea how they’re meant to work. The tutor guides him through the memorisation of these obscure and magical invocations, after which the student, much encouraged, finds he can get 60% of the questions correct. The methods taught may or may not match those encountered in class; when the topic is revised in a couple of months, the student may or may not be condemned to forever muddling up two equally arcane processes for solving the questions. As the lesson winds to a close, pupil and tutor do the week’s homework “together”, which means that the tutor “reminds” the student how to tackle the harder questions. The school’s mathematics teacher may be able to delude herself that this is the pupil’s unvarnished work; the student is happy, and continues to feel that if only Mrs Chalk could explain things like Mr Hourly, learning maths would be so much easier. *

*Money changes hands. Nobody has made any progress whatsoever.*

How did it go so wrong? His parents did the right thing in hiring a tutor, and the tutor was almost certainly genuine in wanting to help the student. But keeping a child afloat with their schoolwork should never be the aim of out-of-hours work. Ironically, unless the child is a very high achiever, it virtually condemns them to sink.

Transforming a pupil into a good mathematician is hard work for both teacher and student. It’s a long-term process, and it can be a hard sell for the parent. It’s probably also the most wonderful, life-changing gift they can ever give.

Notes

1. | ↑ | Acceptable answers to “what’s your day-job?” are “teacher” and “private tutor”. Unacceptable answers: “student”, “aspiring actor”, literally anything else. |

2. | ↑ | Number bonds & tables, fractions, decimals, percentages, linear algebra. |

3. | ↑ | Something that pupils often say to tutors is “it makes so much more sense when you explain it”. Good tutors know what this means. It means that they haven’t yet done a good enough job. |

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