I have a student (age 14) who is valiantly striving to improve his algebra. He knows the "rules", and he more-or-less manages to apply them correctly, and hence is generally able to tackle the "simplify this" which his curriculum routinely throws at him.

But then he encounters something like:

$$-14 x + 21 = 7 x^2$$

and he completely fails to notice that he can divide every term by $7$.

When I pointed out to him that cancelling big numbers out of his expressions (and in this context, "big" seems to be anything greater than or equal to $2$) his reaction was: "How can I tell what number to cancel by?"

And at that point it occurred to me that was unable to recognise that, in the expression above, he could not tell that $7$ was a divisor of each of $14$ and $21$.

Back in my day, we learned our "times tables", and hence recognising by sight every $2$-digit number's prime factors is instant, so I never even thought that would be a problem.

Now I realise that because pupils are no longer expected to learn arithmetic, they are also unable to "recognise" numbers in this way.

Apart from giving this dude lots of practice doing basic arithmetic (which doesn't work because he doesn't do it), what strategies can I suggest for him so he can gain facility at "cancelling down" composite numbers like this?

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    $\begingroup$ I read "facility" as "faculty", I didn't know what to make of it. I think you already answered your question through, he needs to do a bunch of arithmetic. He doesn't do it ? Well, you can lead a horse to water, but you can't make it think. $\endgroup$ Oct 13 '21 at 13:36
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    $\begingroup$ FWIW, I have a speed-quiz site for things like times tables that are under-emphasized nowadays. I suggest 5 minutes of practice a day. One thing I find is that if I ask "Do you know your times tables?", students say yes, but really mean they can do sequential additions. The presence of the limited timer is eye-opening for some, as they were never informed it was expected to be that automatic: www.automatic-algebra.org $\endgroup$ Oct 14 '21 at 1:48
  • $\begingroup$ Just because you know the multiplication table doesn’t mean you instantly know the factorization of every 2-digit number. For example, does the factorization 91 = 7 x 13 leap to mind based merely on knowledge of the multiplication table? Even Grothendieck (in)famously tripped up on the factorization of a 2-digit number! $\endgroup$ Dec 17 '21 at 20:07
  • $\begingroup$ Every student of mathematics should write out, from scratch (not just copying), the prime factorizations of the numbers from 2 to 100 at least 500 times (with no more than once a day counting toward this goal). $\endgroup$ Dec 17 '21 at 20:10

If he won't practice on his own, then I suggest the following. At the beginning of each tutoring session, give him a blank 10 by 10 times table and have him fill it in. He can then refer to it while he is working. If he needs to know the factors of 63, then he can find them.

I have done this with students who had difficulty learning the times tables. Every time they fill out the table they get faster. If he doesn't know something, he will have to figure it out and you can give him different tips that may help him in the future. For example if he is doing the 8 times table, he can add 8 to each previous number, or double the 4 times table. When doing 9 times a number, he can subtract that number from 10 times that number or notice the pattern in the table.

If he balks at doing this, explain that his choice is to learn the facts, or to do this each time he sits down to work with you. Do NOT accept one that was filled in before, because you have to see him fill it in each time.

Hope this helps.

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    $\begingroup$ This is a very nice answer. $\endgroup$ Oct 14 '21 at 9:28
  • $\begingroup$ @Joel Reyes Noche: Efektive. “Kiu cedas al sia infano pereigas rin per propra mano.” $\endgroup$ Dec 17 '21 at 20:27

I am guessing that you were hired by his parents to tutor him. If that's the case, he isn't necessarily on board. That's a tough situation to be in.

You made a few errors in your post

  • "Back in my day, we learned our "times tables", and hence recognising by sight every 2-digit number's prime factors is instant" That's not true. There are many people who have "learned their times tables" but still have problems with factoring. The times table is one direction, and factoring is another.
  • "Now I realise that because pupils are no longer expected to learn arithmetic..." I'm curious why you think this is so. I do not see any evidence of this, and am in touch with lots of teachers (and a few students).

What I hear is the basic problem is that you know he needs to practice some basics, and he's not into it, and won't. I recommend fun. There are plenty of educational video games for multiplication and some for factoring. There's a game you could play with him (not online), called either Divisor Miser or Tax Collector. (I can't believe I've never blogged about that. Here's a reasonable write-up of it.) There's a lovely board game called Prime Climb. I'm sure there's lots more.

I will add more to this answer if I think of more.


Why is it a priority to give him the ability to do these simplifications? I can do them and naturally go that direction. But, like you, am to the right side of the bell curve on math topics.

Seems to me that the priority should be to get him to do the algorithms first (add like quantities to each side, isolate Xs on left and numbers on the right, etc.). Once he has total facility with that, sure then (THEN!) work on the recognition of numbers and the like. But first things first.

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    $\begingroup$ Not cancelling common factors makes the standard algorithms have much more difficult arithmetic... which is his problem... $\endgroup$ Oct 14 '21 at 3:07
  • $\begingroup$ I donno. To me, it feels like trying to juggle 3 balls before learning with 2. Like learning the basic practices of arithmetic first (several digit multiplications, long division) first. THEN learning little tricks like how to quickly calculate 38*42. I think, for a weaker student, you're making it harder, not easier to get into these tricks and recognitions before just learning the brute force algorithms. Which, for all your disdain, work fine in either situation. $\endgroup$ Oct 18 '21 at 21:53
  • $\begingroup$ I can do the "trick" to calculate 38×42, but it is a contrived example for textbooks. In real life it would be 37×41 or something, and I would just square 40 for a quick estimate, and use calculator for exact number. School texbooks push calculator usage starting from middle school anyway. $\endgroup$
    – Rusty Core
    Oct 20 '21 at 1:38

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