Tuesday, December 08, 2009

Mathematics what is it good for?

A discussion with Hugh about Darwinism reminded me of something that happened with the Bratii over the weekend.

Yes what's Evolution got to do with Maths, bear with me. Somehow in a discussion while opening presents simple mathematical questions were posed to Minor such as 14+20, which he did with ease. Someone then tried the harder 100-88.
"..." though Minor.
"22" said Major.
"12" said Minor at the same time.
Major laughed at Minor until pointed out that he was the correct one.
"Oh I got confused" he flustered.
"Okay" I asked. "How about the sum of all numbers from 1 to 100?"
Blank looks all around. So I turn to Bratus Pater with raised eyebrows.
"Yes I know there's a formula but I can't remember it" he replied.
"Okay how about 1 to 10?"
"Oh I know that it's 55."
Still silence from the Bratii.
"It's n/2(n+1)."
"What's 'n'?" Major asked.
"The last number you're summing. So for 1 to 100 it'll be 100/2 equals 50 multiplied by 100 +1 or 101. So 101 times 50 or 5050. Works for any number to sum from 1. If you want to do from say 5 to 100 then it's different it's (n²-m²+m+n)/2".
"What's 'm'"?
"The number you're starting from"
"But what's the point of that?" asked my father.
"Because it forms patterns with numbers that reveals other patterns. It's like every number is a unique product of primes"
"So what?"
"Well it's important for encryption" added Bratus Pater
"But for anyone else?"
"No not really"

At this point the discussion turned to the point in teaching these sorts of things at Middle/High School level as they're never going to be used in "real life". I agreed, but stated that it forms a basis for learning. I then put forward two points. The first was that in theory you should only teach this sort of thing to those who are going to use it later, expect how do you know who those are going to be?

My second argument is where the reminder to Evolution comes in and that's the Lies to Children. The simplification of things at certain ages. The classic being that an atom is just like a mini-solar system. This is fine when introducing such concepts, but with the ability to leave school at 16 and not having to take a Science subject into GCSE level means that large numbers leave education with that 'truth' embedded.

Normally not a problem in times gone by, but with the communications revolution that the internet has brought these same people are able to promote and argue their ideas based on, well not even, High School science. Sadly we all know that no-one really likes their ideas to be challenged so when someone who knows more about what's being discussed comes along things get hostile.

The only solution I see is that when ever one of these Lies to Children is used it's also emphasised that "This isn't really what it's like. It's just a simple model you can use at the moment". Might save us all some hardship down the line.


Dan H said...

There's another way to think about this problem. Consider the numbers in pairs, taking the largest and the smallest. 0+100 is 100, 1+99 is 100, 2+98 is 100, 3+97... and so on. Each pair contributes 100 (that's n), and there are 101/2 pairs (that's (n+1)/2 ). Multiply these two quantities together, and you get the formula.

It's said that Gauss came up with this method when he was a schoolboy. The schoolmaster set this supposedly time-consuming but easy problem, hoping to get an hour's relaxation while the kids struggled with it. Gauss (the story goes) thought about it for a minute and then wrote down the correct answer, much to the schoolmaster's disdain.

The story means that this problem is a great example to talk about teaching maths in schools. The number-theoretic stuff is not that useful in everyday life, but it's a good way of teaching how to think about maths, how to prove results, and how to problem-solve. Don't forget also that maths is the language of science. Already, the reduction in how mathsy maths teaching is makes it much harder to teach sciences at A-level and at university level, because without knowledge of algebra, and how to reason about mathematical statements, you just can't communicate scientific results. You've often posted about misunderstandings over statistics, and how the man in the street doesn't ask questions like "what's the sample size?" "what's the effect size?" "does this meet a p=0.05 significance test?" and that's exactly the kind of reasoning citizens in a democracy need in order to make informed decisions on animal testing, schooling, medical treatment, and climate change.

Anyway, taking a more anecdotal view: I did the maximal amount of maths at school. I learned all the maths they were willing to teach me. I kept learning more maths as a student, and now, in my professional life, I learn new maths all the time. There's only one subject that learning more of at school would have helped me in my everyday life. You guessed it: maths. Spending less time on history and geography wouldn't have affected my knowledge today - I've learned far more about those subjects outside the classroom - but learning those things took time away from learning more of the tools that I have to use every day.

FlipC said...

Well I didn't get chance to explain why this is true, if I had I would have used a visual model writing 1 - 10 vertically then folding the paper in half so that the 10 touched the 1 etc. Basically the same way I gave a proof of the Pythagorean Theorem.

One of the things I missed of the original discussion was a statement that the poor teaching of maths now is due to the poor teaching of maths in the 60/70s. I added to the pot the strangeness of trying to teach basic set theory as an introduction to the subject.

To an extent the trouble is the emphasis on the 'practical' rather than showing why the "mathsy maths" is, in many ways, just the same. So what if I don't know Pythag' when am I ever going to need it? Well maybe not, but it shows you how things are structured; as we both agree it teaches you how to think, how to structure.

And yes statistics is a particularly practical subject for the creation of an informed society something that wasn't really touched upon at my school until A-level. So anyone leaving education prior to that pfft what's a sample size?

To an extent that's why I'm trying to fill in some of the future blanks of the Bratii education, to try to get them excited about algebra, statistics, etc. to see how they work, how to pull them apart and put them back together in different ways. Even in 'stupid' ways such as pointing out how the AI works in games path-finding etc.

But it does depend on what path you follow career-wise, basic knowledge in a wide set of subjects allows for a more well-rounded individual. While knowledge of crop-rotation and enclosure laws aren't that helpful currently it does allow me to pick parallels with current events.

Likewise understanding the nature of flood plains etc. is certainly a hot topic at the moment.

Sure you can study these things now if you'd chosen not to in the past, but you still need the framework to draw upon to slot it all into place.

I also see a danger in early specialisation in that it's in the nature of humans to judge that which they enjoy to be the most important. Think of the Hard Scientists sneering at the Soft Scientists and imagine what they'd be like if they'd only really been taught hard science from an early age?

I can't look back at any lessons and think "Wow that was a complete waste of time" even PE was useful in a psychological field experiment type of way :-)

Dan H said...

Certainly, I completely agree. I would hate to see more specialisation at an earlier age - I remember I would have done two more A-levels if my school had let me (and if I'd had time), and I had to fight to do one more than anyone else did anyway. I'm a generalist at heart, and I guess that's part of why I enjoy maths so much - it underpins everything else I do, whether that's drawing with linear perspective, proving properties of type systems with Tarski's fixed-point theorem, thinking about the impact of word choices using Zapf's Law, or optimizing the use of fabric in clothes patterns with bucket-filling algorithms.

It's also because I'm a generalist that I can sympathise with your feeling that no lessons are a waste of time, and I can agree that learning anything is worth it; but for me, there were lots of lessons that taught me things I already knew, lessons that taught me things I already knew were false, and teachers who tried to discourage me from seeing the parallels between situations and experimenting with new ideas. There were years of "art" lessons that taught me to recognise the salient features of a few artists and the use of the same techniques in architecture and modern design, but didn't teach me to draw or paint. There were years of "English" lessons that made reading a displeasure for me by making me read at the pace of other people, and read things I was too young to appreciate. There were a few terms of history and geography where I learned more in the field trip to the local museum at the end of term than I did from a whole term of lessons.

No learning is a waste of time. But lessons are often a waste of time. I've made one suggestion for how that time could have been used more effectively; I'm sure you can think of lots more.

FlipC said...

But maths is boooring; all dry numbers and complicated formulas. It's not like chemistry where you can blow things up, physics where you mess with springs, or geography where you get to go on field trips; it's yawn city.

At least that's how it was taught to me and to an extent that's why I cn see people complaining that it's not practical, that's because it's not taught that way. Ah I've said this before so never mind me it just irritates.

Anyway as a fast reader I can sympathise with being forced to stay at the same pace as the others, I can only be thankful that due to numbers I was placed in the second stream for English and thus missed out on Shakespeare and Dickens etc.

Neither wrote for children so why are they still touted as classic curriculum material? Oh sure they tell us about the times they were written during, but that's history or social studies not English Lit.

Heretical though it may seem I don't consider Dickens to be a good writer, a good social commentator perhaps; but the limitations of writing periodically along with the heavy-handed social message puts me off.

It's similar to what you mention about art; it's not teaching you 'art' it's teaching you art appreciation, but like the yawnmaths is doing so in a rigid, forced institutionalised manner which unless it hits a spark is likely to put off any student from going near these things ever again.

Even the way lessons are compartmentalised doesn't help, as I say with Dickens and history; for art why was the Renaissance called that? Why was it so important, what was going on around it at the time?

Maybe it's because I'm a generalist too, but I don't think you can just pluck on aspect from something and really gain anything from it that way?

Orphi said...

Ah yes, Mathematics. Without it (or Science for that matter), modern civilisation would be virtually impossible. And yet, the general populus seem to regard both of these subjects as a completely pointless waste of time.

I guess the problem is that you personally don't need to know how to calculate the bending moment of a 27-tonne weight at the end of an 11-meter lever just to drive over London Bridge. Somebody, somewhere else, has already done that calculation for you. This is probably why normal people don't seem to understand how utterly vital Mathematics is.

A similar problem applies to Science. The engine oil in your car? Most people probably think you just dig it up out of the ground and put in into the engine. (Ha!) But, pray tell, how do they make it not burn at high temperatures? How do they make it not set like treacle on a freezing winter's day when there's 4 feet of snow on the ground? Eh? That would be, um, Science.

The other problem is modern culture. Now I'm no historian, but it seems that in times gone by, the great mathematicins, scientists and engineers of the day were regarded as legends. People like Isambard Kingdom Brunel had giant bronze statues dedicated to their worship.

Today, people regard scientists and mathematicians as sad, pathetic losers with nothing better to do with their pitiful, empty lives but spend them obsessing over cryptic equations that have no relevance to the real world. Too many Mad Scientists, Evil Geniuses, and Misunderstood Prodigies in films and books, methinks.

What can we do about all this? I have no idea. But it strikes me that trying to teach kids in school about something which modern culture says is “pointless” is a losing battle. (The UK education system is a whole other pile of brokenness, of course…)

People still build bridges, skyscrapers and computers of course. Sometimes I wonder where the **** they get the staff…

FlipC said...

I think it's a case of wonder-exhaustion. No-one thought it was possible to connect those two points by rail, build a tower that high. No-one thought you could describe gravity or equate energy and matter.

Those who could do so were feted, now it's a case of "Meh so what?" The large spectacle has gone and it's been replaced for the most part by incremental improvements that are taken for granted or invisible to those around them.

Of course to an extent the reason for that is because people don't understand what's being done and, as with the bending moment or oil, they don't need to.

Orphi said...

Damnit, we need to make science and mathematics exciting again!

Maybe we should come up with an idea for a crazy TV show, and go pitch it to the networks? Could make a little money on the way. ;-)

FlipC said...

Sadly that's what the networks have tried to do and their attempts have involved removing most of the science and mathematics from the programmes.

As I see it the problem lies in interactivity. Mathematics has always been a classic stand at the front of the classroom and lecture topic. Start trying to do applied maths and you start edging into Physics; which emphasises my distaste of the separation of subjects.

Television lacks that capability so you're back to wearing tank-tops, bringing in the chalkboard, painting everything brown and calling yourself the OU ;-)

Games are the way to go and there are plenty about, but again if you're not careful it ends up just being the one kid glued to the computer screen with no parents around. Just gotta get that interaction going.

Orphi said...

I think TV is the wrong medium for trying to seriously teach people science or mathematics, in any kind of thorough way at least. But it seems like a great medium for provoking interest and inspiring curiosity and excitement.

Remember Johnny Ball Reveals All? Local Heores? Rough Science? Horizon before it became policital?

Heh, I can still hear the physacist's voice in my head… “OK, this week we'd like you to make a thermometer.” “Right, we'll need a coil of wire…”

I think the key point is that you don't have to take all the science out of science to get people to watch it. You just need to make it more exciting than a guy standing there mumbling.

Dan H said...

That said, I think the Royal Institute's Christmas Lectures usually do a pretty good job. They're not always exciting, but they always show the real-world applications of things.

FlipC said...

I remember "Think of a Number" and indeed that's the sort of show that's needed. If you want to scare yourself take a look at the latest episode of "Numberjacks" from CBeebies. It's 15 minutes long and there's nothing about numbers until 7 minutes in.

The challenge - make 8 three different ways. At least they don't go down the 1+7 2+6 5+3 route because addition counts as one way and can't be used again.

So then we get subtraction 9-1 sorry "taking away". Finally multiplication 2*4., sorry "4 lots of 2 make 8"

All this took 4 minutes.

Then you get a recap of the story of which the number part took around 30 seconds.

And as a last challenge different ways of making the number 4. "You can make it any way you like - adding, take away, lots of, any way." another 20 seconds.

So a 15 minute programme of which less than 4 minutes had anything to do with numbers. Excellent educational value.

What's worse is that the blurb states "Animated numbers solve real world mathematical problems, with help from real children." yeah.

FlipC said...

Sorry that should be less than 5 minutes, for some reason I originally had the longest section pegged at 3 minutes and forgot to change the sum. I'm still being generous here on the times.

Oh and yes the RI Lectures are still good, but they're once a year and held at at time when they're competing with all the Christmas entertainment and presents.

That said I'll mention it to Bratus Pater and try to get the Bratii geed up for it.

Orphi said...

Even the RI Lectures seem to be slowly dumbed down.

On the other hand, some subjects are just really awkward to demonstrate in an interesting way to a room full of small children. It varies by year; some years they do really interesting stuff, and other years it's just dull. (Or the demonstrations don't really demonstrate anything. Like the bubble machine that was supposed to demonstrate that some people are genetically predisposed to build muscle more than others…)