My point is that you are the exception in that you visualize math. Most people I know just follow one algorithm or another. I agree Long multiplication ( is that it’s actual name?) does not lend itself well to mental calculation. How does multiplication help with division? To my mind they are just reciprocals of each other so you can’t really separate them. Anyway, this is what I meant.
If you are dividing 1080 by 15 and have no inclination or knowledge of tricks you approach the problem like this.
Can anything times 15 go into 10? No.
Can anything times 15 go into 108? Yes
So now you guess how many times 15 goes into 108, write that number (7) above the 8 and multiply it by 15 giving you 105.
Subtract 105 from 108, bring down the 0 and repeat till you get to zero or whatever precision you are calculating to.
If you have a hard time multiplying you have an even harder time dividing.
I was interested as to how I would mentally solve the problem. The mental process is equivalent to considering it as a fraction then splitting the numerator into sums and products of easier multiples of 15, until all that’s left is something less than 15 (at which point you can carry on with shifted decimal places if needed).
Oh course, i don’t actually realise I’m doing that, it’s just the process is very close. The point is the approach is closely related to the example of multiplication in which you split the multiplicand into a set of sums and products, as described in the video.
I agree multiplication is a prerequisite to division, but I’m still not convinced of the algorithmic value of long multiplication in that (beyond it being another route).
I was taught long division, the tables up to 20, and old English money tables (quick, what is 7 times 4/3d?) by an old-school teacher over 50 years ago before I was eight. That’s what the old-schooling parents seem to want. Let me tell any of them here, it sucked. I could not get the idea behind numbers. The rote method was all that was taught, which was fine if you were going to be a double-entry clerk in a Victorian workshop. Some in my class were happy with this but I struggled (there’s always one awkward one in each class).
I was taken out of that school by my parents, and put into another, which tried to teach me why numbers were like that. I remember the stupendous delight of discovering decimals (you have the hundreds, the tens, the units; you put a dot and keep going). I remember learning counting to different bases, and realising base ten was nothing fundamental.
Come higher school, I had another ‘old school teacher’ who insisted we learned to derive all geometry be rote from Euclid’s postulates (this used to be a problem for a whole lesson: assume you can draw only one line parallel to a given line through a given point). The same guy taught me calculus, which I could not trust as he taught it (but I managed to figure out an explanation I could trust, and survived). Years later I found that the mathematician Kroneker had a very similar objection. Later still, I understood why the teacher always gave us Euclid’s fifth postulate, but we had to learn the others.
I suspect I may have always been the awkward one in maths class. If you have 30 others to teach, the sooner the weird kid flunks, the sooner the others can get on. But you may be failing the only one that might have become a mathematician. The others can use calculators these days.
Rote learning is not necessarily divorced from maths. I will take things from one side of an equation to another to simplify it, because I once learned to translate mathematical principles into typographical rules. We don’t go back to first principles all the time. But I always want to know I can.
As a 15 year old in the UK I learnt boolean algebra and binary arithmetic.
This got me a holiday job with a computer department after I passed my O level. My father casually remarked at lunch to the head of department that they were teaching children these strange things in maths nowadays - and the immediate response was “does he want a job?”
We have a 14 year-old and a 16 year old in high school. I think the way they are being taught math is generally excellent, for all the reasons the guy in the video explains. The methods are focussed on understanding, and not just on “the right answer”.
Our kids did/do learn that. They continue to work on fast approximations as part of their curriculum. It’s part of helping them visualise/understand what’s really going on, as well as a real world skill and a quick form of (gross) error checking.
~Fifty comments in and no one has linked to Lockhart’s Lament yet?
it is my favorite discussion of these topics, e.g. the difference between MATHEMATICS and COMPUTATION. Kids get force-fed computation algorithms before they’ve played with mathematics enough to naturally hunger for those algorithms. And many (most?) parents and teachers don’t even realize that there’s no math in math class, they think the students “should” be learning computation algorithms.
Imagine having to learn music notation, the circle of fifths, transposing keys, fugues, sonata form, etc, before being allowed to listen to actual music or to create some via undirected play? I think anyone who knows music (and I’m not even a musician) knows that music theory is a helpful-but-not-essential tool to create and understand music, rather than a foundation that is strictly required before someone could even grasp what music is. Math class is like music theory without any music.
I mean, I was suggesting that it be taught instead of other versions of multiplication. But your kids are very fortunate if they are taught to use math the way engineers and scientists use it.
I’m an engineer, and we use approximations as you described. Nevertheless, the “official” numbers are calculated precisely, and expressed to the appropriate number of significant figures.
Right! I still encounter situations in mathematics which are so new to me I can’t relate them to other things I know - they initially seem completely impossible to grasp then after thinking hard about them for a while they become internalised. I’m fairly sure it’s only the familiarity that allows me to “understand” it; the knowledge is really just acceptance of a new principle.
The Common Core (in the US) in math does seek to develop conceptual understanding and application of math concepts, so someone is listening. It does get wrapped up in politics and in the fear some parents have because they didn’t learn it that way.
I’m a programmer. These days, I’m also a substitute teacher. Last week, in a 5th grade class, the unit was on multiplying 2- or 3-digit numbers and the “area model” came up. Exactly what’s in this video.
I presented it as “there are many ways to do things. Here are two ways to do this problem”. Some of the kids prefer the “36 x 12” method. They thought the “area model” was complicated and overly complex. Other kids liked the area model.
It doesn’t bother me that the school or the teacher or the book teach different methods. It does bother me when they teach one method as the only One True Way.
I use the method that all the US public schools taught in the 1960s… but I really like the method that arranges
digits in a square and then reads out diagonally:
That is, you put the product of the each digit pair in the cells of the square array, with the digits offset as shown. Then you do conventional addition, proceeding downward and leftward through the diagonals. Still only one set of carries to manage, but more compact than the method you showed. I learnt this trick in the 1960’s from an elderly Estonian who had been taught to do it that way in “the old country.”
My children know math because I taught them. I know math because my mother taught me. My mother learned math in a 1-room schoolhouse in the 1930s in rural Virginia, not from her parents.
When my daughter was in 4th grade, every night she’d come home with homework she could not do because she’d not been taught, she’d instead sat (quietly and respectfully, btw, she was lauded as a “model student”) through an incomprehensible diatribe. So I’d teach her, using apples, pizza, sacks of pennies - whatever was needed. For me, my mom had to use flash cards.
Thanks. For some weird reason my brain knotted and thought i’d need a 3rd dimension if I wanted to do three digits in the area method. Probably because it reminded me of the method to minimize logical statements for Boolean operations.
Yes, though I was using the long form. In practice, we’d often use only one line per digit of the mulitiplicator, for example. Until we we old enough to calculate 123 x 344 as 1¼ x 36000 = 36000 + 9000 = 45000.
