My response to the blog wars about multiplying negative numbers. Mostly inspired by Eric’s comment on Mike Croucher’s Walking Randomly.
Big image, links to a PDF (of vector goodness).
I wanted to put the Inkscsape SVG source inside the PNG image. But it turns out wordpress.com “optimises” the image and means my klever hack doesn’t work. Bad wordpress.com.
2009-10-05 at 15:25:20
I prefer the “money” explanation. Having a negative amount of money means owing the money (anyone with a bank account will understand this) so to say I have -£1000 in my account makes perfect sense.
If I am paying a mortgage of £600 a month, I can work out what my balance is (absent other transactions) by multiplying by number of months from now. 2 months ahead gives me 2 * -600 pounds, i.e. -1200 pounds. 2 months in the past gives me (-2) * (-600) pounds, i.e £1200 more.
At the time I gave this any real thought (when I was doing a PGCE) I did indeed have some serious financial problems so this was foremost in my mind.
There is some suggestion that Indians (who used numbers for accounting) found -ve numbers easier to conceptualise (and therefore use) than Greek based mathematicians who were used to numbers as measures of geometric extent.
Your geometrical explanation (which is great by the way) does lapse into alegbra. I don’t think it needs to but you can see how the -ve is so much more obvious with money (which can be naturally negative) than distance (when you have to understand co-ords to do the job).
Double entry bookkeeping is of course something you use if you are a bit nervous about minuses 8-).
2009-10-05 at 15:57:45
Wikipedia on the history of negative numbers suggests that confusion about negative numbers was widespread among Europeans (even European mathematicians) as late as the 18th century: for example, in 1758 the British mathematician Francis Maseres claimed that negative numbers “darken the very whole doctrines of the equations and make dark of the things which are in their nature excessively obvious and simple”.
In some sense the “minus times minus” question is an improvement on the 18th century situation, since it basically assumes the existence of negative numbers, with at least some understanding of them, even if complete understanding of how to operate on them is still missing.
Maybe at some point we will reach this level of public understanding of imaginary and complex numbers? I am cautiously optimistic.
2009-10-05 at 19:01:04
The trouble with complex numbers is motivation. Why bother? At school level you can try to do this via the completion of a field idea – complex numbers are sufficient to supply all the roots you need for any polynomial.
Of course those roots don’t look like they appear on any graph but…
Sadly, a rather slender reason for the complex plane.
The real motivation for complex numbers is that complex *analysis* is so wonderful. Sadly you can’t see that until you’ve learned some analysis – I am told that this is now solely post 16. Maybe in a generation only university students will learn it. A shame.
2009-10-05 at 16:01:18
Leo Rogers identifies three sources for the confusion:
[ed: I added LI markers to make the three points typographically clearer; sorry about the bold]
2009-10-07 at 19:45:48
Of course I had <ul>…<li>… in the original. As usual, WordPress destroyed my markup.
2013-05-21 at 21:49:01
hi
“algebraiclly we know (a-b)(c-d) = ac + a(-d) + (-b)c + (-b)(-d) ”
how did u know this ,the proof of it might contain (-) * (-) u should prove it without using (-) * (-)
2009-10-05 at 17:43:48
Using algebra to explain this is a non-starter, at least the way that maths is taught here. Anyone who hasn’t deeply internalized this result (minus times minus is plus) either is too young to have encountered algebra or has been so turned off maths that the algebra will bring them out in hives.
Nice pictures though.
2009-10-05 at 18:25:19
I totally agree, FWIW.
2009-10-05 at 18:51:29
Part of the difficulty here is that your areas aren’t directed. If they were (say) elements of an alternating algebra, then you might make some progress. You could teach directed areas to children pre-algebra, but no-one does.
2009-10-05 at 17:48:21
I think Gareth has the right idea here: negative numbers are different from subtracting. Failing to mentally distinguish them is the same kind of mistake as converting “10C warmer” to “50F warmer” (which category error is seen disturbingly often).
2009-10-05 at 18:53:26
I like it. Good for people with a strong geometric intuition.
2009-10-05 at 18:58:44
One of the things I found very positive (in terms of improvements in pedagogy) during my PGCE (c.1999 I think) was the strong emphasis on building on the intuitions that one’s students happened to have and the methods that worked for them.
Of course this means you have to be a much better mathematician and understand many ways of looking at the same problem, but all the better (and all the more interesting) for the teacher.
It will just happen that there is probably a way to help the student “get” what you are trying to say, even if others fail.
We had a completely incomprehensible “explanation” for – x – = + when I was taught that involved the number line (a dreary place to start I feel) and though we learned it I don’t think it helped at all.
The *easiest* way to understand it – but only if you are teaching your children in a really off the wall way – is that rotating through an angle of \pi twice is obviously going to leave you facing the way you started. I know of no-one that starts using the complex plane before multiplication is taught, but you could.
2009-10-05 at 19:39:44
Of course, there are many audiences for these explanations. You might be talking to a class of kids, or one interested friend over a glass of wine.
So I agree that ideally you’ve got many ways to understand the same thing.
For example: Some of the other blog explanations for this topic use reflection about zero on the number line. This is a way to describe your rotation in the complex plane, but without discussing complex numbers.
2009-10-05 at 19:43:18
All reflections are pale shadows of higher dimensional rotations. It seems a shame to hide the real truth from one’s interlocutor. Sadly, needs sometimes must.
2009-10-05 at 19:12:40
To use the complex plane, one has to abstract number away from reality. Three isn’t three apples or £3 or three fingers. Once you have abstracted numbers into these entities which have their own properties, which are fun to play with in their own right – independent of the apples, oranges, money, sweets, or football scores – then the step to complex (and to other kinds of numbers) is easy.
But: this abstraction isn’t really taught in primary school, although it certainly could be and a lot of children do learn to abstract number at a very early age (I could stretch to a claim that some of these children are actually not learning not to). So a lot of people never really learn to abstract numbers.
2009-10-05 at 19:16:01
“not learning not to”: I taught my son to do long multiplication and division in about an hour, when he was about 5 or 6. A couple of years later, over a ridiculously long period of time, he was then taught some bizarro-world crippletastic method of multiplying numbers of 2 or 3 digits together (only ever 2 or 3 digits). In the process, of course, he unlearned long multiplication and division, and essentially learned that multiplying and dividing are hard things best done with a calculator.
2009-10-05 at 19:33:26
One superb thing about my early maths education is that my teachers actually gave a commentary on competing algorithms.
So she preferred using “equal addition” as a subtraction algorithm, but explained to us how to use decomposition and also why she thought that equal addition was superior algorithmically (empirical evidence suggests decomposition is easier to teach but more error prone).
Having that sort of education is very useful (if you are bright enough to follow it) because it means that you aren’t confused by the way that other people do things.
It sounds like your son’s maths tuition was terrible. You should never overwrite methods they already know and that was very strongly “policy” in ’99. Sadly I was aware that lots of my fellow students (at a top maths PGCE course) did not care much for “policy” with disastrous consequences.
The same bunch had one student who couldn’t accept that 1=0.9 recurring. No really.
2009-10-06 at 09:51:51
Re subtraction algorithms: What is “equal addition” and “decomposition” ?
2009-10-07 at 09:07:02
“bizarro-world crippletastic” was maybe too harsh. It was really just long multiplication with every little intermediate sum written out in its own separate box. When multiplying 435 by 27, I learned (and taught my son) to do this:
435x
27
—
8700
3045
—-
11745
wherein the only marks one makes on the paper are those, with possibly a tiny carry digit in 3 places (depending on one’s fluency).
The “boxes method” would involve drawing a grid of (I guess) six boxes, doing each digit multiplication in a separate box, then adding down the columns, then adding across the rows. Maybe. Something like that. So you’d write this:
400 30 5
20 8000 600 100
7 2800 210 35
10800 810 135
10800
810
135
11745
Something like that. I can see that this method has pedagogical value: it shows the exact mechanics – what is going on under the hood in a multiplication. But actually using this technique for doing a set of sums involves lots more writing, and lots of opportunity to make elementary errors (in particular, losing track of columns, so the 600 would be 60 or 6000). So one often gets the wrong answer, therefore multiplication is difficult and mysterious.
2009-10-07 at 09:12:14
But yes, maths teaching at primary schools is crap.
2009-10-07 at 09:49:31
And in fact one multiplies 435 by 27 by saying “400 25s is 10k, 35 25s is 750 and 125, 2 435s is 870, that makes 11k7 and some change”. I’m not sure this is possible to teach to any child who doesn’t do a lot of mental arithmetic. Which they are not required to do. So I suspect that this is a dying art, possibly inaccessible to anyone under 30.
2009-10-07 at 09:56:56
I saw some pretty good mental arithmetic stuff when I was doing teaching practice. There was a move towards making sure children did get exposed to mental arithmetic and the SAT tests had a mental element to them (no working on the paper for instance).
A plus of this is that you can teach estimation. Efforts to do so without a mental arithmetic test fail because the kids use a calculator and then round off.
Consider (a discussion I had with Gareth Rees recently) what volume of gas at 25 degrees C is occupied by 1 mole? The answer is (in Litres):
(8.314472*298.15)/101.325
In your head you will think (“that’s about 8 times 300 divided by a 100 or 24″). Kids can do that because there’s no way they can try to do anything else. Since the actual answer is just 24.4654313 we weren’t doing too badly.
So don’t despair of mental arithmetic too much.
2009-10-07 at 09:59:00
The paper (column) methods all require reasonable mental arithmetic, and this is one of the reasons I deliberately use paper methods: so I can practice my mental arithmetic.
2009-10-07 at 11:01:02
The main difficulty with mental arithmetic is remembering all the intermediate results (and in re-organizing the computation so that the intermediate results are small enough to fit in one’s feeble working memory).
On paper there’s no such difficulty.
2009-10-07 at 11:25:22
@Francis: interesting. I can usually remember that the molar gas volume is 24 (whereas I never knew the gas constant was 8 until now), but I have massive “K confusion” when it comes to physics. I can never remember whether burning 12 Kg of Carbon will give 24 litres or 24 kilolitres of CO2. It takes me a moment’s thought to realise it must be 24 kilolitres.
My intuition is very bad at visualising some large quantities and can easily be out by a factor of 1000.
2009-10-07 at 11:44:17
@drj: that is why I sneakily used J/L for atmospheric pressure rather than Pascals: three fewer 0′s.
I have the same problem too.
2009-10-05 at 19:52:48
On the last issue (0.9… ?=? 1), there’s a really interesting paper by Tall and Schwarzenberger, “Conflicts in the Learning of Real Numbers and Limits“.
Quote: “Teachers do not help the situation if they show clearly that they feel uneasy with the limit process and so pass on their fears to their pupils.”
2009-10-06 at 09:58:41
http://online.edfac.unimelb.edu.au/485129/wnproj/subtract/algorith.htm is a link to an explanation.
Basically when in a column you subtract larger from smaller you add a 10 to it, where does that 10 come from? In Equal Addition you add it to the next digit to subtract, in decomposition you “borrow” it from the next digit to the left. Decomposition means that you sometimes have to make repeated “borrows” for a single subtraction, in equal addition the same thing would cascade through each subtraction.
2009-10-06 at 10:38:32
Thanks. It seems I was taught (and use) “equal addition”. I always thought it was called “the standard column subtraction method”, but I guess “standards” can vary.
2009-10-06 at 10:45:44
Decomposition went through a phase of being very popular, such that for a while a generation of parents could not understand the algorithms their children were learning (if you see what I mean).
Of course since we read left to right the way you really subtract is to start at the left digit and work right, messier but hey.
2009-10-07 at 09:24:54
One of the views some of us came to on the PGCE (looking at the research evidence) is that some of the cognitive problems at age 11 (year 7) were entirely generated by the methods of teaching in primary school.
The classic (though there are many) is of course the equation 3 divided by 5 = 1 2/5 (from one bit of research a surprising number of children gave this answer).
This works thus: (i) FACT: you cannot divide a smaller number by a larger number (“it doesn’t go”); (ii) so we need to see how many times 3 goes into 5 => answer 1 remainder 2; (iii) what do we do with the 2? Well we were dividing by 5 weren’t we so it must be 2 fifths.
Hence: 1 2/5
The “doesn’t go” is a major block for some children, but it is just plain false. Of course you can divide 3 by 5 and most really quite young children can already do that (try it with a cake, chocolate bars, or something similar). They already have the right intuitions so telling them that “it doesn’t go” tramples on any real intuition they have about division and makes them think that fractions are somehow weirder things. You then have to teach them all this later on.
It means that “divide a half by a quarter” seems much harder than it needs to be. Again a surprising number of teens can do this if they don’t think about it but if they write it out it all goes pear shaped and they end up with stupid answers.
Recently my wife was revising some mathematics for an exam and I realised that she hadn’t understood (vulgar) fractions at school either. Asking for a “half of 2 thirds” at first left her blank though of course a “half of two apples” and a “half of two cats” made perfect sense. Easy when you understand what fractions are all about – impossible if you have been mystified by them.
At my PGCE interview I was asked to explain how to divide one fraction by another. Sadly this is done badly.
Why? One fundamental problem is that most primary teachers don’t understand what they are teaching, hence sowing further confusion. I know people who obtained their maths GCSE on the fifth attempt in order to qualify as a primary school teacher.