Nicky Morgan’s comments today have started a debate over whether pupils really do need to have to learn their times tables by the end of primary. I think they should and I’m not going to rehearse the arguments here.

What I do want to do is to ask **what other maths facts** it’s useful for pupils to know by heart? The new national curriculum specifically says that pupils should memorise the number bonds up to 20 and the times tables up to 12, but are there other facts it is worthwhile memorising?

I’m going to start by saying fraction / decimal equivalences. I’m not talking about ones like 0.5, 0.25, etc, which are obviously helpful but which most pupils will just know (I hope!). I also think that memorising the decimal equivalences of less common fractions is useful: in particular, fractions with denominators of 6, 7, 8, 12 and 15. Very often newspaper articles and statistics you come across in everyday life are reported as fractions in these terms – for example, one in seven adults has a subscription to Netflix, or one in every 12 pounds is spent at Tesco (I made those up by the way). Being able to instantly flick back and forward from that to the percentage is really useful. The reverse is also useful. A lot of data are reported as precise percentages, and being able to easily mentally flick from this to a fraction often helps with understanding. If someone tells you that Andy Carroll wins 84% of aerial duels, it can help to think instead that that means he loses about one in every six of them. (Also a made-up stat).

It’s also a classic example of why it isn’t enough to know how to work it out. You might know how to convert a fraction to a decimal, but by the time you’d worked it out, you’d have forgotten what the context of the statistic was. The person who did know that 1/12 is 8.3% can move on to considering whether Tesco’s dominant market share is a cause for concern, estimating the share of other big chains and wondering what that might look like as an absolute sum of money. As ever, knowing stuff off by heart enables critical thinking rather than stifling it.

Some other suggestions: the 75 times tables. The person who suggested this one did so as a technique for winning the numbers game on Countdown. I wouldn’t recommend that we reorganise education around winning TV quiz shows (god forbid) but since I took this advice and learnt my 75 times tables, I have found them useful in more ways than expected. I suspect this is the case with a lot of these things – it’s only once you learn them that you fully appreciate how useful they are. A bit of the Dunning-Kruger effect, perhaps.

Any maths teachers out there, please leave your suggestions in the comments. Square numbers? Other times tables?

Robert CraigenFor older students: Standard angles and their trig functions.

I tend to think of formulas as “math facts”. Thus, the quadratic formula. Invert and multiply, and so on. When doing algebra it makes sense to just “know” some of these things and do them without thinking. This impulse to force students to do such stuff from scratch using “understanding” is really counterproductive and jams working memory in important tasks as you point out with converting to percentages and back.

Alfred North Whitehead wrote,

“It is a profoundly erroneous truism … that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them.”

The Wing to HeavenPost authorLove that Whitehead quotation! Also John Roberts: ‘The rapidity with which humanity has achieved so much since prehistoric times can be accounted for quite simply: there are many more of us upon whose talents humanity can draw and, more important still, human achievements are essentially cumulative. They rest upon a heritage itself accumulating at, as it were, compound interest. Primitive societies had far less inherited advantage in the bank. This makes the magnitude of their greatest steps forward all the more amazing.’

Robert CraigenGood JR quote. Even briefer Newton’s quip about standing on the shoulders of giants. I believe that (much of) our education world today has lost its compass by severing its prescriptions from generational wisdom — that learned by former generations. That applies both to subject matter and to the wisdom of the classroom. I’m glad to see this movement of (largely) younger teachers in the UK, of which you are a prominent figure, to restore those ties long-standing wisdom and which understands that taking seriously contemporary research doesn’t necessarily mean throwing everything that worked in the past into the dustbin.

Kris BoultonDunning-Kruger… The Baader-Meinhof Phenomenon is the one that’s come to my mind when thinking about how new knowledge proves frequently useful in ways you wouldn’t have predicted.

Michael TiddSome basic metric-imperial approximations remain startlingly useful in the UK.

discreetteacherSquare numbers certainly, reciprocals of numbers as multiplying to one is necessary, prime numbers. I would also add the sum of angles in a triangle.

The Wing to HeavenPost authorSome great suggestions from @mrreddymaths on twitter.

1. Subitising

2. Subtracting a single digit from number < 20

3. Factors of specific numbers, e.g. 36, 64 and 100.

4. Diff between factor & multiple

5. Meaning of prime

6. Metric unit conversions

7. Everyday metric > imperial conversions

8. Minutes in an hour, hours in a day, etc.

9. Names of polygons and solids.

The Wing to HeavenPost authorAlso from @TheOtherDrX

10^3, 4, 5 and 6 crucial for sciences, eg molarity, but rare known as fact

The Wing to HeavenPost authorReblogged this on The Echo Chamber.

Mark BennetThere is an issue here about what the facts are for. One use is for efficient computation, for which the more facts the better. Another is to exemplify patterns – then the pattern trumps the facts. For many, efficient computation will help, with diminishing returns as the computations become more complex. For pure mathematicians, the patterns, once noted, trump the arithmetic. Pure mathematicians who notice patterns quickly are less good at arithmetic than one would naturally think. That is why this is contested territory.

fortuitusthoughtsConcentrating on “atomic” facts is very dangerous – it takes the focus away from the why it is useful to learn the facts, i.e. their application. This is why times table is such a bad example of “what should be thought to the children”. Of course times tables are useful. Teach them by hard without anything to add colour to the task and you get a great recipe to put kids off math for life.

kal hodgsonHmmmm… the logical conclusion to these comments is to memorise everything. Why know how to calculate the integral of cos squared x? Just learn the answer. And tan cubed x. And cosh x. etc etc. Just memorise every mathematical fact in the world! Or am I being facetious?

Kris BoultonDepends on whether you intend to be! Some might interpret that as the conclusion… but no, it’s really far more nuanced than that. Memorising certain things is close to essential. Memorising others might not be essential, but might improve fluency and be useful – doing so certainly does not necessarily preclude learning the meaning behind the memorised fact, nor limit understanding (it can often facilitate it.)

Other things it might be a real stretch to bother memorising (e.g. cubic formula?) but then, occasionally some mathematician might find that there is benefit to doing so.

Far more nuance than the simplistic dull sense of ‘rote memorised facts’ that many people often have in their minds when they re-encounter the very notion of memorisation as a positive concept!