On “Lack of Confidence” and Times Tables

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Here’s a line I often hear from the parents of younger children, and from far too many teachers:

{Child} knows their tables really, but he/she just lacks confidence and must believe in themselves. 

Lack of confidence is something I’ve heard about a lot since I became a full time tutor, and I always almost respond in the same way:

Ask him when his birthday is.

I’ve taught kids who have to stop and think about that one, but they’re definitely a minority. For most children, when they’re asked something that they really truly know, there’s no confidence issue at all – if they furrow their brow at all, it’s to wonder why you’re even asking them something so obvious.

So why don’t they answer the same way when mummy asks them for three times seven? Why the pauses, why the silly mistakes? Because they don’t know it well enough. And, almost always, that’s a matter of practice.

Here’s how most people teach times tables, as far as I can tell:

Parent: What’s 4 x 7?
Child: Um. Er. 28?
Parent: Good! What’s 8×8?
Child. Um… . I think I’ve forgotten – can you help me on this one?
Parent: It’s 64.
Child: Oh, yes! I knew that one.

And on they go, both parent and child allowing themselves to believe that only some strange psychological blip prevented perfect recall of the eight times table. It’s very possible to do this for months without getting anywhere at all.

By contrast, the right way to teach tables is based on a simple principle:
The child should not be getting questions wrong.

Here’s the sort of conversation I’d have in a times tables session:

Tutor: What’s 4×4?
Child: Um –
Tutor: It’s 16. What’s 4×4?
Child: 16.
Tutor: Good. What’s 4×4?
Child 16.
Tutor: What’s 4×2? (deliberately an easy one)
Child: 8
Tutor: Good. What’s 4×4? 
Child: 16
Tutor: What’s 16?
Child: Four times four.
Tutor: Good. Okay, 4 x 5 is 20. What’s 4×5?
Child: 20
Tutor: Good. What’s 4×4?
Child: 16

At the end of 30 minutes of that, the pupil will know their four-times-table perfectly for the next week or so. You’ll need to revisit it with reasonable regularity over the next six months, but if you do, he or she will have made a friend for life. A quick tables session fits perfectly into a car journey, and once they’ve mastered all seven of the tricky tables (3s, 4s, 6s, 7s, 8s, 9s, and 12s) you can relax – there’s nothing more to do but maintain perfection.

Learning the multiplication tables, properly and thoroughly, pays off in a thousand ways – five years later, when their GCSE teachers inform them that 4y multiplied by 7y is 28y², they’ll have 100% of their brain free to concentrate on mastering the algebra, while their classmates are losing the teacher’s thread in the second-and-a-half it takes them to figure out where the 28 came from.

Some children are shy and introspective – others are born as bold as brass. But any child who feels confident about their birthday can feel confident about their multiplication tables.

The Next Question

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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. [1]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.

Notes

Notes
↑1 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.

Rote Learning and Mathematics

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Most people who care about what they do, in any profession, can suggest a couple of ways in which they feel that their field of expertise is being ill-served by its conventional wisdom. Sometimes, these will be cries to tear up the rulebook, new and progressive ideas about how everything could be moved forwards.  Just as common, however, are rather more reactionary ideas – a conception that their peers and colleagues have gone ‘too far’ in some particular direction. In my own case, this contradiction comes to the fore in the matter of mathematical rote learning.

When I taught at Bruern Abbey, a specialist school for pupils with learning difficulties, I often thought about a sign that hung on the wall of the staffroom. “If a pupil can’t learn the way you teach,” it read, “can you teach the way he learns?” This is splendid advice, and though it might be challenging to apply in a class of 40 pupils, it’s very important for smaller groups and virtually a moral requirement for one-to-one work. Those of us who were true believers in the Bruern Way talked a lot about ideas like kinaesthetic learning, and enjoyed thinking about how far ahead we were, or thought we were, of traditional, talk-and-chalk pedagogues.

The Bruern Abbey maths department put a great deal of emphasis on understanding the underlying logic of things like multiplication tables. This is essential for dyslexic learners, for whom the ability to calculate six times seven is important and achievable, but the prospect of memorising the result is likely unrealistic. On the whole, these methods worked tremendously well, and seeing the lights coming on for students who had almost given up on learning mathematics remains among my proudest memories. [1]I’d highly recommend Dorian Yeo’s Dyslexia and Maths to anyone keen on helping dyslexic students do well at the subject.

However, not all of our students were dyslexic. Some had processing difficulties, or specific issues with short-term, rather than long-term, memory. For such students, an ability to memorise facts by rote was among their primary academic strengths. How did we teach such students? Well, we focused on understanding the underlying logic of multiplication tables…

In other words, by trying to apply an approach that was so helpful for the dyslexic members of the class, we were failing other students. It was oddly easy to do so; images of draconian, cane-wielding pedants, chanting multiplication tables at cowering students, made it easy to miss the fact that this approach could actually help some of our students.

For it’s uncomfortable but undeniable that there are areas of mathematics in which rote learning can really, really help. Of these, one of the most under-appreciated is the degree to which a background of facts, learnt by rote, can actually help students to assimilate new ideas and concepts. I’ve touched before on the idea that one of the major difficulties in understanding a teacher’s explanation is all of the background processing that a student has to perform at the same time. Ask a class of Year 7s to add 5x to 6x, for example, and students who arrive easily at the ’11’ will have an enormous advantage in deciding what to do with the ‘x’.

What should be learned by rote? For some students, such as those with dyslexia, almost nothing. For students with an opposite set of difficulties, as much as possible. For most students, however, a set of key facts represents a happy medium: multiplication tables to 10 or 12, number bonds to at least ten, key decimal / fraction equivalences such as three quarters being equal to 0.75 and one-fifth converting to 0.2.

And if this post reads as a cautious defence of rote learning, at least in certain circumstances and for certain students, it remains no excuse for bad teaching. To teach rote facts requires as much creativity and sensitivity to the student as any other method. To pore over a list might be right for one or two students, but walking around the room will help many more, while others learn faster from games or even songs.

There are general principles to effective rote learning, but these are largely the same as elsewhere. Try not to muddle the mixture, or delude yourself about how well students really know the material. Make sure the student enjoys it – a sense of growing mastery should be compensation for their hard work.

Above all, teach rote learning in the way that your student learns. A kernel of unforgettable mathematical knowledge, once memorised, will help students in every aspect of the subject for many years to come. If it’s in your power to give such a kernel to a young learner just starting their mathematical journey, it would be a crime to keep such a gift to yourself.

Notes

Notes
↑1 I’d highly recommend Dorian Yeo’s Dyslexia and Maths to anyone keen on helping dyslexic students do well at the subject.

Three Good Questions about Fractions, Decimals and Percentages

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I’ve talked before about the importance of the next question, the principle that every question one asks of a pupil should be intelligently tailored to that pupil’s individual needs at that exact moment. Not all questions, however, are created equal. Here are three that I’ve found particularly good over my time helping my students to understand fractions, decimals and percentages, along with the principles that I believe that they demonstrate.

1)    3/4 + 2/5

Principle: Don’t muddle the mixture at the beginning.

When I teach addition of fractions, and introduce the idea that sometimes the answer may be more than ‘one whole’, this is almost always the first question that by pupils solve by themselves. It’s actually too good to waste on a worked example, because it’s almost unique in how friendly it is to the inexperienced learner. I’d invite readers to try this question themselves, to see just how smooth the process is.

All four numbers are different so there’s no ambiguity about which is which. Every multiplication required (4×5, 3×5, 2×4) is one that even pupils with weak tables will know by rote, allowing them to give 100% of their focus to the topic. Best of all, however, once they arrive at the ‘topheavy’ answer of 23/20, converting this to the correct answer of “One and three-twentieths” is not only easy but intuitive. Subtraction can be tricky, but since any pupil can solve ‘23 minus 20’ in a heartbeat, the learner will be able to devote 100% attention to the new material (the whole number part), rather than attempting to apprehend it with half their brainpower spent harnessing a jumble of numbers. That, in turn, means that the new principles will stick with them a good deal better.

As the next question should demonstrate, I don’t believe that every question should be easy. But setting a hard question should be a deliberate choice with a clear aim in mind, and sometimes an easy one can lay the groundwork for a pupil to advance, step by step and with full understanding, onto much trickier problems in future.

2) ‘Give 7/250 as a decimal’

Principle: Know when your pupil really understands the material.

In contrast to the last example, this is what I like to think of as a graduation problem. It’s a question that includes so much of the Fraction-Decimal-conversion circuit that any pupil who can confidently answer it is one who is truly ready to move on.

Let’s review everything that happens in this question:

i)              The pupil has to know that to convert a fraction to a decimal, it must first be transformed into a fraction with 10, 100, 1000 etc as the denominator.

ii)             The pupil has to realize that ‘250 x 4 = 1000’. If they do, barring exceptional mathematical ability, it’s because you’ve given them sufficient practice at recognising that ‘25 x 4 = 100’.

iii)           Once they realise they need to be multiplying by four, the number 28 should be buzzing around the pupil’s head. This presents the final challenge – equipped with this number, which the pupil has worked so hard to acquire, the natural instinct of many students is to write ‘0.28’ – the wrong answer. Only a student to whom the principles of the number system have truly become intuitive will be able to arrive at the correct answer of ‘0.028’.

Why seven then?

If the question were one out of 250, the number buzzing around the pupil’s head in step ‘c’ would be 4, not 28, and confusing ‘0.4’ with ‘0.004’ is a far less likely error for an uncertain pupil to make than muddling ‘0.28’ and ‘0.028’.

If the question were two, four, five or six out of 250, the fraction could be cancelled down, which would contradict the frequent and pained insistence of all maths teachers everywhere that pupils should cancel down any fraction not already in its lowest terms.

Finally, if the question were three out of 250, the question would be similar, but the times-table multiplication would be easier (3×4 rather than 7×4). In this case, in stark contrast to the first question I discussed, this is not at all what I want – the pupil who has truly mastered the topic will be able to achieve a right answer even after expending 5% of their brainpower on a slightly harder calculation.

For a pupil to correctly answer this question doesn’t prove on its own that he or she understands fractions and decimals. But it’s a very encouraging sign.

3) Give 40% of 40

Principle: “Working memory matters” or “Understand what’s hard”

Many schools, such as Colet Court, now include within the their interview process a computerised test, designed to test which pupils are able to quickly and accurately perform key mathematical operations without writing anything down. Even those schools which stick strictly to pen and paper include in the interview process a ‘technical interview’ which is essentially a verbal test of the pupil’s mathematical savvy. For this reason, it’s vital for parents and tutors to understand which topics are considered ‘fair game’ for this kind of mental solving. Percentages of a number are very much in this category.

Once the principles are taught, it’s natural to leap all over the place: ‘20% of 80’, ‘90% of 30’, etc. And, certainly, examiners will do just this. However, I’ve found it extremely helpful, especially with younger pupils, to introduce the topic by repeating the number in the question twice – “30% of 30”, “40% of 40”, etc.

This is the principle of ‘understand what’s hard’. For a pupil encountering this topic, the steps that must be carried out to solve this kind of question require their full attention. As a result, when asked to find ‘80% of 60’, for example, they frequently forget about the 80 while calculating 10% of 60. That’s frustrating for both learner and teacher. These repeated-number questions, however, allow the pupil to put in the repetitions that they need for the ‘10% of’ process to become comfortable and quick. Once it is, they will find questions like ‘80% of 60 considerably easier, as the ‘6’ springs from the ‘60’ while the ‘80%’ is still fresh in their memories.

Sometimes, well-meaning parents might ask their son or daughter about the five and seven times tables, and on hearing right answers to the former and wrong ones to the latter, wrongly conclude that the remedy is more mixed tables. In reality, as I hope this question demonstrates, much of the secret of good maths teaching is to understand where in the question lies the difficulty for the student, and to set a difficulty in which that difficulty is all of the problem or none at all.

What Maths Tutors Do Wrong

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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

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.

The Cult of ‘Hard’

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Picture the scene: a parent and child, at the end of an hour’s tuition. Looking for confirmation as to how her son or daughter is progressing, she asks how the lesson went, and is told in a small, emphatic voice that “it was really hard!” Very possibly she’s pleased, and feels that this kind of challenge, these high expectations, are exactly what her child needs.

I’m going to say something mildly controversial here: if she does, she’s almost certainly wrong.

It’s easy to see how the misapprehension takes hold. Hard work, in the sense of repeated application of tiresome tasks, is vital to exam success. High expectations really do work wonders. And there’s the oft-peddled idea that modern pedagogy has become too “soft”, lacking in the strict classroom disciple that some consider essential to helping kids learn. So it makes plenty of sense that a parent might view a “hard” lesson as a good one.

It isn’t. In fact, if a child said that at the end of one of my lessons I’d be mortified, and I’d likely be spending the rest of that evening working out what I’d done wrong, and how to teach that topic better the next time. Put simply, if something was hard for a student to learn, it means that their prior knowledge didn’t support it. And if their prior knowledge didn’t support it, it was the wrong thing to be teaching them.

Essentially, good teaching is the art of making things feel easy. Kids aren’t consulting some sort of gigantic national database when they decide whether something is “easy” or “hard”. [1]If kids report that what they’re learning is “too easy”, if usually means that it fits so securely with what they know already that it isn’t actually expanding it. That’s still better than … Continue reading Rather, they’re reporting on how neatly the new information that they’ve been expected to learn fits into the framework of knowledge that they’ve already constructed. If it doesn’t fit, it won’t stick. They’ll remember it for a week, perhaps even pass the test, but then it’ll be gone. And the list of the things that they haven’t understood will be one item longer, for the rest of their educational career.

Pupils taught material in this manner are put into a horrible position, in which they must memorise incomprehensible techniques as if they were alchemical formulae, complex and arcane invocations that turn lead into gold for reasons entirely obscure. Once this method of learning has taken root, the spiral of incomprehension sets in. More and more future knowledge has to be built on these shaky foundations, and the day comes quickly when the whole subject has ceased to “make sense”. At this point, there’s little that even the best teaching can do but to start again from the point where the rot took hold, which can often be several years in the past.

The alternative to this, and the only cure for it, is to build knowledge step by gentle step, checking constantly that the foundations always remain firm. After all, one can’t forget what one has truly understood, and the best learning is an matter of understanding rather than one of memorisation. [2]This is true in all subjects, even those where memorisation is essential. When I teach Latin, I’m always struck by how often my pupils are being asked to learn complex forms like the passive and … Continue reading Teach at the right pace, introduce concepts at the exact point at which students become ready for them, and you’ll find that you’re able to communicate even “hard” ideas securely and quickly. Challenge your students, but make very sure that they’re equipped with the tools they need to rise to those challenges. It’s certainly tricky to get it just right. But when you do, the student won’t find your lessons hard at all. They may not be easy, but they’ll be something even better – they’ll be fun.

Notes

Notes
↑1 If kids report that what they’re learning is “too easy”, if usually means that it fits so securely with what they know already that it isn’t actually expanding it. That’s still better than too hard, because there’s a general tendency for children to over-estimate how well they know something, but it’s still bad.
↑2 This is true in all subjects, even those where memorisation is essential. When I teach Latin, I’m always struck by how often my pupils are being asked to learn complex forms like the passive and pluperfect without really having grasped what these even are. Equally often, they’re being moved to these tenses well before they have a secure grasp of simpler forms such as the perfect and future tenses.