When doing long divsion with DS11, so far the only math practice he abhors, we had an important realization. He can divide, with accuracy, faster in his head than he can writing out all the steps. So...how to convince him to do division the way the schools want him to do it?

Get him to do things like 7834095.56 / 143.8 ? Or can he really do that in his head? I'm serious; make the sums hard enough that he sees the benefit of writing it out. It's great that he has strong mental arithmetic too. Sometimes he'll have to grin and bear the requirement to write out working for something he can do confidently in his head, but if for now he needs practice, it's probably better to practise on things he can't do in his head, because that's less frustrating! (At the same time, encourage estimation and other sanity checks.)

Can you be more specific about the kind of numbers involved?

How many digits is the divisor (the b in a/b)?

Yup.

DD balked this way at writing out dimensional analysis when she was about 8-- but her dad and I both know that there is wayyyyyy more at stake there than being able to convert units properly-- basically about 90% of quantitative determination is being able to recognize when you have the right quantities in the right locations in calculations, and following the units and making sure that they "come out" right is the KEY to doing them.

So we wanted her to very definitely view equivalent values as fractional conversion factors which equal one, and to WRITE.THEM.OUT.

She was very resistant because she could do the (simple) assigned conversions in her head just fine. Except, of course, when she got something turned upside down, which was about 20% of the tine.

Eventually, I came up with a story about Henrietta* the chicken, who was having a dinner party, but wanted things "just so" for her many guests... and needed the proportions of guests to cockroach appetizers, musical numbers, etc. to be perfect.

It was quite a complex problem-- far more complex than I've ever given to a freshman college student struggling with dimensional analysis, but it worked with DD.

Yes to making the problems HARD ENOUGH TO REQUIRE what you're seeking from them.

* Henrietta was a sort of barnyard Martha Stewart on a budget, as it happened... so she needed to make sure that she had planned out her shopping with her purchasing power in mind. (My own DD responds best to problems which are quirky or surreal. IMMV.)

In DD's case, she needs to have problems which are difficult enough that she makes a LOT of mistakes trying to do it without writing things out.

Ah, yes. I get it now. And admittedly, these were at most three to four digits divided by two to three digits. So tomorrow's "lesson" will be on showing ds the value of steps with more difficult problems.

Thank you, all.

KADmom: sometimes math proofs are really fun to read. Mabye it can help inspire your DS?

HowlerKarma: It sounds like your DD is the same. Perhaps she also needs more challenging problems that can encourage her to write steps out.

Mabye your children can eventually progress to unsolved math equations, or work on Fermat's theorem?

I'm curious of some specific examples of divsion problems that "he can divide, with accuracy, faster in his head than he can writing out all the steps". I'm wondering what exactly is it that makes them "easy" enough to do mentally (where "easy" is understood as a relative term).

In your example 7834095.56/143.8 it doesn't divide evenly (as a terminating decimal). When do you stop calculating, and in what form is the anser to be given?

When I use a calculator I get 54479.10682 (a 10 digit approximation, not exact). But when you use long division, do you know a way of getting that the first digit of the quotient is 5 straight away without some trial and error (and subsequently the other digits the same way). Suppose, when you ask how many 10000's of 143.8 go into 7834095.56, if you carelessly guess 4 of them instead of 5, and subtract those off, you'll realise you haven't subtracted off enough 10000's of 143.8 and you have to subtract off one more before proceeding to the next digit of the quotient. The only way of avoiding this that I can see is to know the multiples of 143.8 up to 9*143.8, but I'd be curious to here other ideas. Can it be made purely mechanical. Of course it would be easier if we all used base 2.

DS 5.5 is reluctant to writing out anything as well. Same thing, 2-4 digit division. Sometimes I will MAKE him do it to show me how he came to this answer and to make sure he knows how to "show his work". If I am not mistaken schools are big into this..... I like the idea of giving them really difficult problems so they see the benefit of writing it out. Great idea!

For instance in this kind of problem, 769 divided by 47, I think he uses estimation to come to the answer. He ends up with the correct answer with a remainder more quickly than he could write out the problem and i take nearly the same amount of time to check his answer using traditional long division--but i compute very quickly. I've never computed in my head though.

I guess I'm surprised because this is my verbal kid who finds math concepts exciting but not the computation.

Great idea. It would show him what math could be.

I contemplated getting up a calculator and trying it, but didn't, because I reckoned that if it didn't give a terminating decimal it would be even more interesting to get the child in question to ask such questions!

Well, it *is* purely mechanical: you just gave the procedure yourself. Binary admittedly easier, if longer, though!

This does relate to various interesting issues (that might make good discussion points with interested children) though. For example, we can get an approximation of the answer by using approximations of the arguments, and we can bound the error, but nevertheless, knowing when we have been precise enough to be sure about the first digit can be tricky. If you don't have access to full information about the arguments, it can be more than tricky, and one can go off into study of the on-line computable functions (which do not include division, IIRR)...

Balanced ternary notation is a cute and accessible thing to talk about in this context...

(Disclaimer: none of this is remotely my field, but I think I kept this vague enough not to be wrong...)

Not only do they expect the kids to show work, but I've found some unnecessary steps have been added to make the solution "clearer" but those seem to me to impede clarity. Of course that could be an issue of my age and I'm more of a verbal person...

Okay. So it's not that there are tricks or shortcuts to this (like there would be with, say, 2017/125). It's just that it's not hard enough, relative to his mental skills, to force him to pencil and paper. Harder problems will convince him that it's sometimes necessary.

Also there are a couple of later reasons where the concepts are useful, e.g. rewriting x^3/(x+2) as x^2-2x+4-(8/(x+2)), which is analogous to rewriting 769/47 as 16 17/47.

Also, if this is really busywork, which "show your work" can sometimes become, maybe see if the teacher will accept a couple of "homemade" harder problems in place of writing all the details for the easier ones.

Right. It's just that I've never seen it explicitly suggested. My son's courses presented it as partly trial and error. I don't remember what I did as a kid, but it could be that the problems weren't hard enough that I couldn't get each digit of the quotient right first time.


Ha! I have been led to exactly these mental meanderings inspired by my son's 4th/5th grade maths (except I don't know the definition of on-line computable functions, but I can guess based on the context).

I've been frustrated by the presentation of long division. Since there is a mechanical method, they should present it, even if laborious. You can still use faster methods (and/or mental arithmetic) where appropriate. Most questions in mathematics can't be solved by mechanical methods, but when they can, it's worth knowing about.

Exactly.

Quit teaching parlor tricks in elementary arithmetic, and teach the procedural ways that work FIRST.

DD always finds the laborious, old-school explanation of things like this to suit her learning needs better.

It's been horrifying to me that ten minutes worth of live instruction in a procedural method (including an quickie explanation of where it comes from and why it works that way) do more than WEEKS of problems based on individual special case examples.

This nonsense of 'construct your own basic understanding' through 'seeing' a variety of approaches is just... weird.

I shudder to think how little the average learner gets out of it. It's no wonder DD sees algebra students who can't work with fractions.

As to the long division, I would encourage him to continue to practice the mental skills. Then use the long division only as explanation or proof. I'm still a little peeved almost forty years later that school broke my ability to do complex division quickly in my head when they forced me to do long division. I think at some point later I would've understood it well enough to bullet proof it against their methods, but without continuing to use it I lost it. It didn't occur to me at the time that I would need to do something to preserve the skill.

Thanks for your perspective and reminder.

Love the Henrietta problem.

Long division is pretty much obsolete except you do need to know how to do it well for algebra when you divide polynomials. Tell him that. There is something about long division that is different from other math - it's very procedural and it's about lining things up and writing things out a certain way. My non-mathy DD actually pick it up more easily than my somewhat mathy DS although she was 8 and he was 7 due to his acceleration. By picking it up, I don't mean getting the right answer, but being able to write everything correctly step by step all lined up.

I am not certain what level of calculation your ds is able to do mentally - I have encountered some savants who could do calculations like a computer. However, if you are only talking about double digit divisors, then simply add a decimal and a place value or two so that he will need to write it out.