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Like many teachers, I see the students reach for the calculators to solve problems - now graphics calculators are indeed very useful for modelling in topics such as optimisation.
However, I notice students also using the calculators to calculate basic times tables parts of their working out.

When not using the calculator, many students struggled with these basic skills. Particularly with the 7 and 13 (and higher) times tables.

So, my question is how to reinforce mental times table skills within a calculus (optimisation) context?

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  • $\begingroup$ Times tables require memorization, there are "shortcuts" but they are not as useful as pure memorization. Frankly, I don't think you can spend time to try and teach this as a calculus teacher because this is something they should have learned in elementary school--you can take time to quiz them but that's about the best you can do. $\endgroup$ – Jared Oct 21 '14 at 9:33
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    $\begingroup$ Frankly, I wonder what use this has. Granted, I would expect a calculus student to be able to do 7*12 in their head (84??--I had to check it to make sure) but having said that, what does it have to do with calculus? Certainly they won't be expected to do such calculations on the non-calculator part of the AP test. $\endgroup$ – Jared Oct 21 '14 at 9:38
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    $\begingroup$ @Jared The question asked "How to...?" you comment "Don't!" Please, have a look at our meta discussion on best practices related to this situation meta.matheducators.stackexchange.com/questions/368/… $\endgroup$ – quid Oct 21 '14 at 12:40
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    $\begingroup$ @quid: it is perfectly valid to answer with "don't". If the question was "how to make students memorize all square roots up to 100 to 5 decimals", would we be allowed to answer with "don't"? $\endgroup$ – Andrej Bauer Oct 22 '14 at 14:27
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    $\begingroup$ @Jared I did not want to imply that you did anything that was out of line. But, I think that the tone of your comment was somewhat harsh (in particular towards a new user). In any case, OP left because of it, which is rather unfortunate. Please, do not read this as me blaming you individually for the loss of a user, but rather as a general reminder that we should try to be welcoming. $\endgroup$ – quid Oct 23 '14 at 13:16
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Memorization of multiplication tables is a skill which used to be valuable, but is much less so nowadays. I do not actually see a valid reason to force students to learn multiplication tables, other than it is a minor convenience to be able to multiply numbers in one's head. Bear in mind there are always a couple of iPhone's sitting around that can do much more than multiply numbers.

The students behave rationally, given the environment they live in.

Why don't you force them to learn the periodic table of elements? Do you think they should know that? Should they know how many years it takes for 100 dollars to double at 1% interest rate? Should they know the speed of light? Memorization is as obsolete as being able to skin a rabbit. If and when the civilization goes to ruins, people will quickly master the old skill, no worries about that.

Lest I be accused of violating a policy: you cannot force students to learn multiplication tables directly. You may be able to motivate them though, if you can somehow demostrate that it is advantageous to them to know this stuff. For instance, you could have a set of problems which do not ask for exact answers, but for approximate answers calculated mentally (no paper) in a very short time. You can try arguing that's an advantage in real life.

Addendum: I asked my 9-year old daughter why she thought it was useful to learn the multiplication table (which she did last year). She said right away "because then you can answer when the teachers asks you". (I find this answer very sad by the way). I pressed on, I wanted to know where outside school multiplication was useful. She said something like "then you can tell how many coins you need if you need three by six coins". I think she was just trying to get rid of me. Can your daugher do better? Can you? I can't.

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  • $\begingroup$ I agree with most of this answer, but I think that the extrinsic motivation of "I need to get good at mental arithmetic so I can do well on this pointless quiz" is not as good as trying to find ways to foster students intrinsic motivation. $\endgroup$ – Steven Gubkin Oct 22 '14 at 16:43
  • $\begingroup$ Yeah, I am not happy with that part either, I wanted to suggest something positive. $\endgroup$ – Andrej Bauer Oct 22 '14 at 21:19
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    $\begingroup$ "so when a clerk in a store rings up a 25% discount wrong, I know he's made a mistake, and I won't get ripped off." $\endgroup$ – JoeTaxpayer Oct 22 '14 at 21:48
  • $\begingroup$ One reason that memorizing the times table can be helpful is that it can free up cognitive resources to tackle more challenging questions. If your daughter's teacher only asks questions of the form, What is $3 \times 6$?, then I agree that the value is lower in the digital age. However, for students who have understood the times table, it may be nice to use it as a starting scenario for pattern exploration. See matheducators.stackexchange.com/a/1750/262 for suggestions on elementary school problems, and math.stackexchange.com/q/776447/37122 for a question that is too hard for me! $\endgroup$ – Benjamin Dickman Oct 23 '14 at 1:36
  • $\begingroup$ @BenjaminDickman: I agree with you about "freeing up cognitive resources", that's probably the best answer I've heard so far. The proposed questions in your first link are going to be boring for kids. Who wants to "estimate the number of different numbers in the 10 by 10 multiplication table"? And what's the kid supposed to think when faced with "explain prime numbers using multiplication table"? $\endgroup$ – Andrej Bauer Oct 23 '14 at 12:08
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One thing that has been very helpful for me is reinforcing how valuable quick mental arithmetic can be. I recently taught about some great mental math shortcuts for squaring any reasonably sized number ending in 5, and using the difference of squares to quickly multiply any two numbers that are centered around an "easy" square (and now 5s are in that list as well as 10s). Upperclassmen in particular have tended to be fascinated by these "unlocks" - and I use them to illustrate how important the memorization of times tables is (up through 12s, at least, which was always good enough for me). At the Calculus level specifically, it might be worthwhile to assign some basic derivatives with higher exponents, and make them do it in class without the use of a calculator. Even though we normally only run into variables with relatively low exponents (especially at the HS level), this can be a good way to sneak in some times tables drilling without deviating from your curriculum.

Since my students respect my expertise on the subject, I try to communicate to them the mental techniques I use when doing a problem in my head. Every teacher has their own stable of tricks; have you shared yours with your students? They seem to take notice of that, and when I tell them they shouldn't be using a calculator for a simple problem, they usually agree with me. Once they feel that way, the next step is teaching them how we made it easy for ourselves - so, you'll have to remember the methods you used that worked in your own mind.

Another technique that has worked for me, if you can spend some time on it in class, is to hammer the memorization in as many different ways as you can: Flashcards, timed drilled written problems, any sort of game involving rapid mental calculation, etc. I know it's not technically your job in a calculus course to review elementary ideas, but a single day spent on this can be fun for students if done right, and will end up benefiting them much more than another day of over-relying on the calculator. I use the analogy with my students that the calculator is like a crutch, and you won't build your mind up to be strong enough if you lean too heavily on it.

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  • $\begingroup$ +1 - I wish the square-diff trick were taught formally at an appropriate time. Whenever I ask if a student 'checked her work' after an exam and hear that there was no time, I can't help but think how much faster the mental is, and the time saved can result in a full grade higher. $\endgroup$ – JoeTaxpayer Oct 22 '14 at 15:57

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