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.

‘How Does The Author?’

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If I had 30 minutes, a gun to my head, and a mandate to improve a ten-year-old’s comprehension mark by as much as possible, this is the question with which I’d start.

It’s hard to overstate just how frequently this question comes up on 11+ papers, and how heavily it’s weighted in their mark schemes. Take a look at this Sevenoaks paper, for instance – there are only 5 questions, and two of them begin ‘How does the author‘. Or consider this LIGSC paper, where the question (#7) is valued at a fearsome ten marks – 20% of the whole paper. When candidates practise other styles of questions, there’s a risk that these questions may not come up at all, but this one is as close to a certainty as they come.

It’s not just its frequency or mark allocation, however, that makes the ‘How does the author‘ question such an easy way to pile on extra marks. Rather it’s a simple and rather unfair fact – that most untutored students have no absolutely idea how to answer it. Teaching a candidate how to polish a 3-out-of-4 answer into a perfect one can be months of work, but here, even quite good students are usually starting out from a base of fewer than half marks. Considering how frequently the question comes up and how heavily it’s weighted, it’s not at all unreasonable for a proper answer to ‘How does the author‘  to be the difference between success and failure.

So, without further ado, here’s how to answer the ‘How does the author‘ question.

“How does the author do what?”, one might start by asking. In fact, however, it matters very little. The structure of a good answer is essentially always the same, whether we’re diagnosing how the passage was imbued with excitement or tedium, terror or joy.

In the first paragraph, go through the words that the author uses when they explicitly mention the state of affairs that you’ve been asked about. For example, suppose the question was “How does the author communicate Pamela’s excitement in this extract?”.

In this passage, Pamela is “thrilled” and “quivering with anticipation” at the thought of Christmas. She “cannot wait” until the day arrives.

This is all that’s required for half marks. A less polished answer (one without the “in-flow” quotations, for example) would probably score just under 50 percent;  two marks out of five or six, say, or three out of eight. This is the level of answer typically produced by an untutored student with some feeling for English.

A second paragraph is required for the other half of the marks. It looks something like this:

Furthermore, the author uses two techniques to help the reader feel Pamela’s excitement. Firstly, employs short sentences, such as “Christmas!” and “It was all too much”. These sentences rush past, and cause the reader to feel just as excited as Pamela. The author also uses personification, stating that “the days crawled past”. This image helps the reader to feel, like Pamela, that time is going unusually slowly as Christmas approaches.

The job of this second paragraph is to go beyond listing vocabulary, and instead list techniques that the writer uses to get their point across. A technique is really anything even mildly out-of-the-ordinary – it’s almost impossible to write a paragraph without using one or two. In this case, the techniques are Sentence Length and Personification.

For each technique, I’ve ensured maximum marks by also including a short follow-up, indicating what the writer achieved by using it. This is only required by the most challenging schools, but it’s a good habit for students to get into if they are aiming for the top.

The first technique I mentioned was sentence length. This isn’t a coincidence, even though I’m ‘quoting’ from a made-up passage. There’s always something to say about sentence length. Are some sentences short? Are some long? Unless the answer is “no, every single one is identically medium in length” (which would itself be rather remarkable), there should be something to comment on here. Mentioning sentence length in a how-does-the-author question is the closest thing to a free mark on the whole paper – it’s something that every student should be aiming to do. Given that one seldom needs to comment on more than two techniques for a high mark, and how few candidates actually know this trick, this information grants any student who’s been let in on the secret an enormous head-start on their competition.

For the second point, I used personification, but there’s countless other tricks to look for. In rough descending order of importance, students might partner the obligatory Sentence Length point with:

Sentence Structure (“Run-on”? Verbless? Lots of adjectives or adverbs?)
Repeated Words
Repeated Sentence Structures
Personification
Unusual Punctuation
Alliteration / Sibilance
Contrast
Similes & Metaphors

There are likely others, and I’d be lying if I said that students always find it easy to select one of the ‘right’ answers. That means that what I’ve written here isn’t a completely clear path to scoring ten-out-of-ten; some practice is still required before success is guaranteed. Sometimes candidates, no matter how well prepared, will slip up, grasp the wrong end of the stick entirely and only score 75%.

I really do believe, though, that this second paragraph will magically transform an answer that asks politely for half marks into one that requests the maximum. It’s two or three immediate extra marks on a question that comes up on every paper, every year. If you’ve got half an hour to improve your child’s mark in English, it’s hard to imagine doing better than that.

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.