Some time ago, I asked here, non-rhetorically: What use is handwriting?, and I got a lot of very useful answers, such as that techies can communicate very well if they can hand-write, in ways that just wouldn’t work with any gadget more complicated than a pencil or felt-tip pen. By attaching labels to hastily sketched diagrams or graphs, for instance.
Now, for similarly pedagogical reasons I ask: What use is algebra? I refer to the most primitive sort of algebra, where you merely tiptoe into the swamp of abstraction and say things like: if a is 2 and b is 4, then what is a plus 2b? What is the specific value of writing out algebraic equations with small letters in them, and then either substituting particular values for those letters, or else deducing some of those values? Why go into letters, if all you then do is get out of them again, which seems to be the rule when you first start out at algebra.
I’m guessing – guessing because it is decades since I myself did any of this – that there is value to an equation, as a generalisation, quite lacking in the mere specifics of what happens in the particular case when a is 2 and b is 4. An equation specifies a general relationship, and one that is often worth understanding, and impossible to understand without this on-the-face-of-it peculiar and regressive diversion out of arithmetic and back into mere letters. But can the commentariat rephrase, correct, expand on that?
Ideally, they would do this in a way that might convince a twelve-year-old whose ambition is to get rich – perhaps by being a Something in The City (assuming there still is a City for him to be a Something in when he reaches his twenties) – and who now gets up before 6am every morning to do a paper round. By the time I get around to teaching him things like algebra, he is tired. What’s the point of this?, he asks. I would like to be able to give him some better answers than I have managed so far. I both like and admire this boy, and would really like him to do well.
We meet every Tuesday night, so my next chance to pass on such things will be tomorrow evening.
UPDATE Tuesday lunchtime: Many thanks for all the comments, most useful. Lots to pass on and to think about, and not just this evening.
Well, as I’m a software developer and pay the rent by selling code, I would hazard its useful to me every day.
You need to understand the basic principles of algebra to be able to get beyond the beginner level with Excel or similar spreadsheets.
FV = PV * (1 + ( i * N ) ]
I = PV * i * N
FV = PV * ( 1 + i )N
FV = PMT * [ ( ( 1 + i )N – 1 ) / i ]
PV * ( 1 + i )N = PMT * [ ( 1 + i )N – 1 ] / i
These algebraic equations represent respectively:
Simple Interest Future Value
Simple Interest
Compound Interest Future Value
Compound Interest
Annuity and
Simple Interest Amortized Loan Formula
These are the financial basics. To really get somewhere, you will need to excel at statistical modelling and calculus, all of which are impossible without a foundation of algebra.
But understand, you can’t even do a basic mortage calculation without algebra.
The brain is a muscle, it needs to be probably the smartest move I ever made and I’m prtrained. Latin works well, as do languages generally, but apart from that pure mathematics is about the best thing there is. Doing Further Mathematics A level was etty sure I’ve got little dumber every week from my first year of university onwards.
Case in point!
What I meant to write was this
The brain is a muscle, it needs to be trained. Latin works well, as do languages generally, but apart from that pure mathematics is about the best thing there is. Doing Further Mathematics A level was probably the smartest move I ever made and I’m pretty sure I’ve got little dumber every week from my first year of university onwards.
Algebra is a way to think. Symbolic logic. Many real world problems cannot be successfully analyzed with words or feelings. For example, if 3 men can do a job in 5 days, how many days will be needed if 4 men do the job?
Alegbra, like geometry, is actually very useful in the real world. Unlike, say, calculus.
if you’ve ever tried to read a book of philosophy, you will know that trying to follow an argument written in words is really hard going. All such arguments-in-words are pretty much virgin forest and they have to be hacked through with enormous effort. (This is why charlatans and postmodernists are able to survive with their “arguments” undemolished – few readers have the patience to hack their way through the tangle of deliberate obfuscation.)
But an argument that can be put into algebraic form relies on consistent, easily manipulable, logical arguments, pre-approved and pre-checked by much greater minds than our own, that represent cleared paths through the logic thickets.
In arithmetic, the numbers are known. An arithmetic equation applies the simple dyadic operators + – x ./. to known numbers to arrive at a result.
In algebra, the numbers are not known beforehand. What is known is a relationship in which letters are used to symbolise unknowns (or variables, in applied maths). Solving an algebraic equation involves finding values for the letters which make the equation true. Think of the letters as a sort of meta-number.
To start off by saying a is 2 and b is 4 makes this rather pointless. Better to start by saying (a+b)=6; what is the value of a? Answer: lots of possibilities but don’t know, because we need two equations to identify two variables. Then introduce your simultaneous equation. And so on…
Applied maths is essential to most sciences and algebra is used to represent the variables. Most of the Latin and Greek alphabets have been assigned to various items: t for time, m for mass, f for force, a for acceleration etc. “f=ma” allows you to immediately apply Newton’s first law. “Force = mass x acceleration” instead leads to the question “so what?”
The progression from scalar quantities to vectors is also a good example. Think of flying an aeroplane in a wind – what direction should you go to reach your destination?
Maths can be fun if you can see the point to it. In the 60s in my prep school algebra consisted of 3 years factorising polynomials – an (almost) complete waste of time. The fun came later when I had an x kg tiger leaping at me at a speed s and a rifle with a muzzle velocity v : what weight of bullet would stop it with one shot?
Hope this helps!
Next time the little tyke whines “Why are we doing this”, clip him round the ear and say “Because I said so!”
Worked for me.
Seriously, when I was learning maths, algebra was fascinating to me, whereas geometry was boring. All the way up to advanced calculus, I was fascinated by how the relationships took shape.
If the youngster doesn’t find it fascinating there are at least a couple of possible reasons.
a) You’re doing it wrong.
b) He’s not mathmatically inclined,
c) The way he’s learning maths at school is so divorced from the way you learnt that he has no common point of reference to find a way in.
The point of Brian’s query is not to find examples of the value others have found in having a grasp of some of Algebra – the point is how to convey to a tired adolescent the “why” of a particular study.
Perhaps the so-called Socratic Method offers a useful, if not always effective approach:
1. Have you found it useful to know how to count?
Why is that useful, and how to you expect it to
continue to be useful? Are there some things you
use counting for that you could not do if you did not
know how to count?
2. Do use ever use Addition, subtraction,
multiplication – even division? If things are to be
divided up how do you know what your share
should be?
3. So, if you now see uses for those parts of Arithmetic,
which you learned – even if not to perfection –
earlier, and basically by memorization, can you see
how learning something more – using numbers and
symbols for numbers can turn out to be very, very
useful; and mean a lot to what you can do as you
get on in living?
4. Now, do the symbols – the Xs, Ys, As and Bs put you
off, or look confusing or mysterious? Well, do the
symbols or signs you see every day, like red for
stop and green for go make things run better?
When you count aren’t the numbers really just
symbols for the actual things you are counting
whether it’s money or time or anything else?
5. Has using some arithmetic, the adding, subtracting,
multiplying, dividing been useful? How useful can
it be to learn other ways you can use what you
know about numbers, counting and arithmetic that
can make it possible for you to do more and
different things, and do them faster, to have better
work, more free time and more opportunities?
Where would you expect to be if you had not
learned to recognize the numbers, the letters, to
count to do arithmetic?
Can you think, now, of how Algebra can be useful?
For anything other than the simplest thoughts, we need models to help our thinking along – the models being simpler than the real thing. Within the models, we usually want to explore relationships to get an idea of the relative importance or influence of various factors. If we can capture this in a mathematical model, we make a dramatic step forward – and algebra, and calculus, become crucial to our explorations. demonstrations, and experimental planning.
Yes algebra has masses of uses, but you need to understand its application and that is the part that seems to be missing from the teaching of algebra. It’s no good trying to convince children that this is logic and it works. I was useless at even basic algebra because of poor teaching. Only when I bought my first computer (ZX81) did I find an application. I wanted to program my new toy so I have to learn but having a goal and practical application unlocked the whole thing.
The big problem with algebra teaching is that the teachers normally work to the fastest pace in the class. They also have a habit of showing you how to do a simple equation and then giving you one to do which differs (eg contains a negative or a third term) and so your first solo attempt is a disaster. The complexity is ramped up too quickly.
I think applied maths for the majority of the population is really useful. Let the brainboxes do their pure maths, but if you’re not mathematically minded, you need some connections to be made for you.
To teach abstraction and pattern recognition.
Also, with all things you learn in school I guess it’s a bit of the ‘just to show you can’ idea.
Two fundamental uses of algebra-which are distinct:
1)to encapsulate rules (as formulae). Examples include
area of a circle- pi r^2
volume of a sphere – 4/3 pi r^3
volume of a box – dwh
sin^2 x + cos^2 x=1
Stirling’s formula- n!=n^ne^(-n) sqrt (2 pi n)(1+O(1/n)).
In order to encapsulate the rule we need a generic name for the quantities which appear in it, rather than particular values;
2) to deal with (initially) unknown quantities. Examples include simultaneous equations (a+b=5, 2a-b=4), quadratic and higher polynomial equations, Pythagorean triples (a^2 +b^2=h^2).
Both these techniques are extremely powerful ways of handling complex information. For eample the Babylonians (I think) left reams of calculations to solve the simple equation
x+x/2=24,
which they expressed in words as (roughly) the question
“what quantity when added to a half of itself results in 24?” and then solved essentially by trial and error.
For that matter, compare Pythagoras’s Theorem in words and symbols:
“the (area of the) square on the hypotenuse is equal to the sum of the (areas of the) squares on the other two sides (of the right angled triangle)
vs
a^2+b^2=h^2.
When we deal with the `unknowns’ type of algebra, the symbols are empowerig because we can midlessly apply the rules of arithmetic to them. This is a good example of the general power of mathematical abstraction-as we abstract, we pack more and more information into smaller and smaller space, but in such a way that we can still apply standard rules to manipulate it.
I always wanted to know what the mysterious notation meant – the attraction of occult knowledge.
The utility of basic algebra is twofold, and pretty straightforward:
1) You can handle a quantity you don’t know by letting the label stand for it until you figure out what it is.
2) It is a form of shorthand that lets you can write down rules neatly in ways that make them easy to remember.
“symbols are empowerig because we can midlessly apply…”.
None of the above protects us from carelessness, however. I recommend avoiding doing things midlessly and looking out for mathematicians claiming to be empowerig.
The question falls into two categories, scientists/engineers who need all forms of abstract math of which algebra is the root, and the layman.
If the question why (for the layman) is because we have so many tools today that will do the thinking for us there are two quick answers-
-teaching people the basics of “abstract” math is necessary from a pedigogic starting point. Many people can’t add a column of numbers without practice so abstract math needs to have some exercises too, at least to determine if that person wants to pursue certain careers that make the tools that the rest of us will use.
– as a CPA there are times when I need to attack a problem with a couple of variables, and if it’s rather simple it’s easier just to dash it out on a piece of paper. Of course if it’s more complicated, or I want to play around with the variables, or I want to save it, I’ll use Excel. But there are plenty of times where I’ve used pencil and paper rather than some amortizer software or Excel function.
On the subject of Pythagoras, I was looking round a secondary school a couple of years ago with my daughter and found a maths GCSE paper which asked what the pupil could say was special about a Pythagorean triple in which the lowest number was odd.
The maths master couldn’t tell me the answer. It took me a while to realise (what I had never been taught) that all odd numbers give rise to a Pythagorean triple in which the larger two numbers are one apart: (3, 4 and 5), (5, 12 and 13), (7, 24 and 25) etc. Curious.
To anyone interested in maths I recommend the Penguin Dictionary of Curious and Interesting Numbers byDavid Wells. Many hours of fun.
RW: 33 56 65
Michael: 33 544 545. There are no doubt lots of relationships, the one I spotted works because the series of odd numbers is the series of differences between successive squares.
To predict the larger two members of the triple, we need – algebra!
In algebraic notation, let an odd number be m.
Since m is odd, let m^2=2n +1
Then (m, n, n+1) are a Pythagorian triple.
Any other predictive rules – I’m interested?
I’m a software developer and as such I obviously like to read things that explain new ways to do things in software, yet because I don’t understand the maths the really advanced articles are using, I’m cut off from a whole world of useful stuff that I’d quite like to know about. The nearby articles that I can follow can be fascinating, useful and profitable in different amounts so I believe I’m regularly denied inspiration, functionality and money as a result of my ignorance.
I speculate that the amount of cool stuff described using basic maths is bigger than the amount of cool stuff described using complex maths. While its harder to keep track of the reasons you did understand something than the reasons you didn’t, I reckon if I didn’t know algebra I’d be on the dole queue by now and really quite bored. Since I have a very low tolerance for boredom I’d hate to think where I’d have ended up without the maths I do have.
Simon
PS if you can tolerate occasional cheesy American moralizing then you could always give him a Numb3rs DVD. Guns? Check. Hot chicks? Check. Maths? Check. Sorted 😉
If you don’t have a strong, almost-instinctive ability for ‘basic’ algebra – if, for example, you can’t write out the solution of the quadratic equation without even thinking about it – you will not be able to master any of the math required for any of the major engineering disciplines – physics, thermo, anything to do with electricity, statics, dynamics, calculus, and the list goes on.
As said, you won’t be able to drive database/analysis programs – even the relatively-straightforward Excel and its brethren – unless you understand the notations and methods of algebra.
Basic algebra is (in many cases) merely a formalized method of precisely notating and working problems that we actually all do all the time. Like trig, the fancy Greek name should not be allowed to be off-putting.
llater,
llamas
RW, what is the case is that if the triple is (a, b, c) then one of (a, b) is odd and c is odd (we consider only primitive triples i.e. those whose GCD is 1 and are thus coprime). In the case where the two larger differ by one, the smallest is odd, but the converse is not necessarily true, as Michael Jenning shows. One can construct a triple from any odd number with the two larger differing by one, as I will show:
Every odd perfect square gives rise to a triple with the two larger numbers differing by one. If we write a² + b² = (b + 1)² then we see that a² = 2b + 1. Thus pick an odd square, e.g. 83484769 = 9137², subtract 1 and halve it to get 41742384, and we immediately know that 9137² + 41742384² = 41742385².
In the case of 33, 33² + 544² = 545².
Alternatively, we can note that for a given a, the other numbers are 1/2 √((a² – 1)²) and 1/2 √((a² + 1)²). These are only integral for odd a.
There are many properties that hold true for Pythagorean triples e.g.:
The area of a Pythagorean right triangle is an integer (because one of a, b is even)
Exactly one of a, b is divisible by three
Exactly one of a, b is divisible by four
Exactly one of a, b, c is divisible by five
At most one of a, b is a square
For each integer k, there exists at least k Pythagorean triangles with different hypotenuses and the same area.
All prime factors of the hypotenuse are of the form 4n + 1
The difference between the hypotenuse and the even leg is a perfect square, as we saw above, and the difference between the hypotenuse and the odd leg is twice a perfect square. This implies:
There are no Pythagorean triples in which the hypotenuse and one of the legs are legs of another Pythagorean triple
There are no triples where the hypotenuse and one of the legs differs by a prime number greater than 2
This Pythagorean triples tangent [:-)] we’re onto is really intesting. I’ve noticed that although (as has been pointed out already) RW’s statement about odd numbers always giving rise to a Pythagorean triple where the larger two numbers are precisely one integer apart is untrue, it is true for all prime numbers (those less than 100, anyway). In fact, once you get to 67 there are three successive pairs of one-integer-separated numbers which form a Pythagorean triple with every base prime, and at 91 that increases to 5 successive pairs. This is fascinating to me; does anyone know of a proof for it?
Oops, I was wrong about the successive pairs of numbers (I didn’t go out enough decimal places). However, I still think I’m right that this works with all prime numbers. Anybody have a proof?
If someone can’t do simple algebra it means they aren’t too bright. That’s one use for it.
School exams once used this reasoning to sort the wheat from the chaff to help univerisities and employers choose between suitable applicants.
Yer actual mathematician will consider a diversion out of symbols and back into mere arithmetic to be the regression. Arithmetic allows you to solve examples; algebra and the advanced mathematics it represents allows you to generate proofs and as such is far more interesting and productive. Also, without algebraic concepts it would be impossible to handle complex numbers, which would not only mean you would be prevented from contemplating the wonder that is:
but you’d also be stymied in doing any analysis of, for example, alternating current ciruits.
Laird, the point is that every odd number a serves as the smallest of a triple (a, (a² ± 1)/2) i.e one in which the two larger are consecutive. It’s just that for some a‘s there are more than one triple in which it is present (i.e. there are more than one pair of perfect squares whose difference is a squared, e.g. 85² + 132² = 157², 85² + 3612² = 3613²).
Algebra, like most maths, helps adolescents to develop their reasoning abilities. And to keep them preoccupied with thoughts other than humping and/or wanking themselves to death.
Algebra, like most maths, helps adolescents to develop their reasoning abilities. And to keep them preoccupied with thoughts other than humping and/or wanking themselves to death.
Usually, this sort of question has to be tailored to the questioner. What do they see value in? And is the problem really that they don’t know why it’s useful, or could it be that they are just seeking an excuse not to do it? If it’s the latter, then you need to address that first before explaining the finer points of algebra.
The other thing to think about is whether the way it is done when you first start algebra is the right way to do it. If they’re bored and don’t see the point, then something’s wrong. Again, it’s often the case that you have to tailor the approach to the way the student thinks – something teachers with a classroom of 30 rarely have the luxury for. The ideal is to ask questions that make them invent the algebra on their own. They do it because it helps them.
Using letters is a way of describing complicated calculation methods (algorithms) compactly. If there’s a long calculation you know how to do and want to tell to somebody else, using algebra is easier and less ambiguous than trying to describe it in words. It can also help if you have to do it repeatedly – this can be a good technique, to ask for the same formula over and over but with different numbers substituted into it. Eventually, they’ll invent the formula themselves, as a way of saving effort.
They make it easier to manipulate algorithms – to combine them, to find easier ways to do it, to understand how and why they work. To spot patterns in them that make calculations and methods easier to remember and understand.
They make it possible to handle more complicated calculations than you could otherwise manage, with a lot less effort.
They enable you to effectively write down lots of results at once, and to describe the relationship between sets of numbers all at once.
They are one way to do geometry: y = ax+b is a line, for example. The intersections of lines and circles and other shapes can be translated into algebra, and quickly solved. Often, they allow you to do geometry problems quickly where the old methods the Greeks used would take ages.
They enable you to prove that a useful relationship always holds. For example, you can do a shortcut for multiplication by eleven where you simply add each digit to its neighbour. 4321×11 is (0+4) (4+3) (3+2) (2+1) (1+0) = 47531. It’s easy to check out a few examples, but how do you know it will always work? The answer is to use algebra. Because algebra deals with whole piles of numbers at once, it lets you find these handy shortcuts and be absolutely certain they’ll always work.
They enable you to see relationships between what appear at first glance to be entirely different problems. Adding apples together is one thing, adding distances is another, adding cake mix ingredients by weight something else. But a lot of the rules are the same, and tricks useful in one can be used in another. This is one of the most powerful features of abstraction: the ability to use what you know in one subject area that you know well, and apply it in another that you are still learning. It’s like the difference between a complete socket set with dozens of spanners of every conceivable size, and carrying one adjustable spanner about with you. It saves a huge amount of effort.
Algebra is the first step on the road to things that are more complicated, more interesting, and if you have the temperament for it, a lot more fun. Footballers spend a lot of time practising basic moves, like dribbling round traffic cones. It’s boring doing that for hours on end, but later on, you get the enjoyment of running rings round all the players that didn’t practice when you do it for real on the football pitch. Playing football well is lots more fun – but to be able to do it you have to practice the basics. Maths is like that only more so. Not everyone can be a world-class footballer, and not everyone can be a mathematician, but you can get good enough to make either of them enjoyable.
TomJ wrote
E to the pi times i
OK, I guess it’s obvious, if you don’t know the answer.
As an engineer, it’s all algebra all the time. But not the simple stuff. My reason is, it improves thinking.
Did you know what CO2 is really useful for? Photosynthesis and cellar respiration. Without it carbon lifeforms would be history. You do know what carbon lifeforms rare, don’t you?
What do you get when you add one-half critical mass of plutonium to another one-half critical mass? How many liters of liquid do you have if you pour one liter of alcohol into one liter of water? Then there’s the Stooges math something to the effect that if I give you an apple, and your father gives you an apple, how many apples do you have? OK, that’s not even arithmetic, but in surprising ways it’s still mathematical.
If you like Pythagorean triples, all of them can be generated by the formula C^2 = 2*A*B, where in the resulting triple X = (A+C), Y = (B+C), and Z = (A+B+C). Relatively-prime triples require that A and B have no factors in common; either A or B is a square.
Math is a language, and helps simplify the process of thinking about certain things. I suspect that the typical teacher fails primarily in not treating word problems as verbal constraints on the manipulations of objects, and getting students to practice potential manipulations against those constraints. (It really irks me, however, that conventional constraints are not even mentioned as being present, where unconventional ones are interesting. It’s like that Sci-Fi quiz where the question was how you could tell the height of a building if you had a barometer.)
If the math is understood, its relationships to the objects and their context makes it possible to infer certain other things without doing any hard work. It makes it possible to borrow solutions from similar problems. Practice in recognizing the pattern of relationships makes it easier to recognize them in new settings, and as components of more complex relationships. Math in general, including algebra, if done right, makes it easier to think.
“If someone can’t do simple algebra it means they aren’t too bright. That’s one use for it.”
That’s one of the most pompous things I’ve heard in a long time. Incredibly blinkered, too!
Joel:
You can’t do any serious engineering without Calculus.
Try to do something really simply, like calculate the draining time for a large round storage tank without using it.
You can’t.
There have been several good replies so far. I think the best was from RRS at February 9, 2009 02:35 PM – a Socratic dialogue is probably the best way to go about it.
I have a PhD in Maths, and wrote a thesis on Group Theory (that’s one of the nicest flavours of algebra), so like a true mathematician I’ll start by generalising your question. I think the question has two parts: what use is abstraction? (this is essentially what maths is) and why is further abstraction (of arithmetic) useful to this particular 12 year old? RRS’s Socratic dialogue seeks to answer both these questions. I feel that a few of the replies have taken too much of a “grandfatherly” attitude: “well son, when I was your age I didn’t see the point of all that algebra but I’ve found it really useful since I grew up”. I think this is not the approach to take.
So what is the use of abstraction? It increases your power; you can think about and solve current problems faster and with less (sometimes much less) effort, thus avoiding boredom, you can also think about and solve (some times much) harder problems than you could tackle before. That’s the easy question.
The hard question is what’s the relevance of algebra to this particular 12 year old? Why further abstract the arithmetic he already knows? Why invest time and energy learning complicated things when he could be doing something else? RRS’s dialogue gets us part way there – by drawing out past examples where increased abstraction increased power, the problem is what if he answers the last questions with: “I can see how I’ve benefited in the past, but I think I know enough and would like to learn some other things.” That’s not a bad answer, and you need an alternative reply to: “well son, I know more than you, and I’m telling you that you do need to know.”
To pull him away from this response you need examples of problems that are relevant, or will seem interesting to him, which either he cannot solve with his current knowledge, or can only solve in a very laborious way. Only you know him, so only you can answer these questions. But here are a couple of suggestions:
* Can he add up all the numbers from 1 to 100 in his head? Does he think its possible to do this? How about 1 to 1000, or 1 to 1000000? A little algebra makes these sums easy. See http://en.wikipedia.org/wiki/Gauss#Early_years_.281777.E2.80.931798.29 for the story.
* How high can he count on his fingers? Ten? Think again. You can count to over 1000 on your fingers. Using binary (finger extended for 1, finger curled in towards the palm for 0) you can count up to 2^5-1 = 31 on one hand, and up to 2^10-1 = 1023 on both hands. This is how computers count. He should be able to see by inspection that he can count up to 31 on one hand, but checking that you can count to 1023 on two hands requires abstract thinking or a really boring exercise where you might miscount and get the wrong answer.
I’m having difficulty coming up with problems that would be within the grasp of a 12 year old, but still be interesting – can any one else think of others. A practical problem would be nice.
Also, as well as immediate interesting problems point out the high peaks in the distance, and emphasise that, just like mountaineers, footballers, or any other elite group you need to practice the basics to be able to scale the peaks and feel the exhilaration that comes with doing that. Some of my favourite peaks are:
* Godel’s theorem. You can write down statements about arithmetic and nobody can tell you whether they are true or not because its impossible in principle to tell. Closely related is the fact that there are logical limits to what a computer can do, no matter how powerful. If we are computers then these limits apply to us. (Good lesson this, because its about the limits of human powers of reason.)
* Imaginary numbers. Anyone who doesn’t think these are cool is just weird.
* Calculus and Newtonian mechanics. Who would have thought that a falling apple, the motion of the stars across the sky, and the tides were all describable by one theory?
* Arrow’s Impossibility Theorem. There are severe limits to how well a democratic vote can reflect the wishes of the individuals doing the voting.
* Secret codes. The story of Bletchley Park, and a discussion of Number theory – as if being beautiful were not enough, spies use this to make virtually unbreakable codes.
* Cantor’s theory of transfinite cardinals. Is there a biggest number? No there are infinitely many numbers. But is there only one infinity? Cantor showed that there were infinitely many infinities.
* Probability and Stochastic Calculus. A triumph of 20th century mathematics, and the basis for much finance and risk management. Perhaps a very simple example of risk-neutral valuation might be within the grasp of a 12 year old; the difficulty would be making it relevant.
* Fractals. Lots of pretty pictures, and also the fact that time series for the price of traded assets are fractal in nature.
* The lives of mathematicians. Galois, Ramanujan, Cantor, Gauss, Erdos, G. H. Hardy’s “A Mathematician’s Apology”.
Lots of non-algebra there, but a knowledge of basic algebra and the relevant technical skills is necessary to gain access to any of these peaks.
As this is for a 12 year old: without algebra, how would you calculate the best weapons load to use in any given situation when playing your favourite first person shooter?
cuic cuic wins the thread thread twice twice.
But, of course, there’s a catch…
‘If someone can’t do simple algebra it means they aren’t too bright. That’s one use for it.”
That’s one of the most pompous things I’ve heard in a long time. Incredibly blinkered, too’
It is true though. An inability to do even such easy algebra doesn’t say much for a person’s abstact thinking. Schools should find out each pupils capabilities and educate accordingly. Algebra is just one of the many ways they can do this, so making sure that the universities and employers are supplied with recruits suitably graded by academic ability as well as other personal qualities.
When my children used to ask me what use a good physics or maths GCSE was going to be in later life I told them it showed an employer you are probably brighter than the average person and your CV might stay out of the bin that little bit longer.
Of course, with A grades for everyone this message is not as true as it was.
When I said that calculus was not useful in the real world, I was thinking not about engineers, who need a lot of math, but ordinary people, eg. farmers, doctors, teachers, carpenters, store owners, etc.
If somebody knew math up to simple algebra, solid geometry, trig, and very importantly, statistics, they would have enough math for almost any undertaking short of certain scientific disciplines and engineering. That is, a good high school math program is all you would need.
I used to love math, but gave it up to study medicine (Vietnam era.) Math was, and likely still is, beautiful. But, it was a good decision!
Algebra: good for telling whether your cellphone plan or box of cereal is a good buy, or a gyp.
llamas,
Basic algebra … Like trig,the fancy Greek name should not be allowed to be off-putting.
Wouldn’t that be a fancy latinized arabic name?
But I said above, indoctrination into a hidden mystery is surely part of the fun.
The value of calculus is that it allows you to look at the world through the eyes of change. If the old maxim is true, that the only constant in the world is change, then the power of understanding the manner in which change effects everything around you is very great indeed.
Why does falling off a bicycle going twice as fast hurt four times as much? Why does giving a bully your lunch money not solve your problems in the long term? What are negative feedback control loops, and why are they everywhere you look? These and millions of like issues are what the study of rates of change, i.e. calculus, is all about.
Oh, and you can’t very well understand calculus unless you can think algebraically.
Some twelve year old boys might be particularly interested in uses of algebra related to ballistics, artillery etc.
More useful in the day to day world would be calculations to do with buying and selling – and simple and compound interest, as DocBrown commented. For instance you might try to use algebra to work out a formula to help you decide whether to sell something on eBay by “Buy It Now” or Auction format. (Reminds me, I must re-do this now that they have changed the fee structure.)
The nice thing about algebra is that you only have to do the work of thinking once. Once you have the formula, all you have to do is plug any new sets of numbers in.
You answered your own question by making the point about generalisation and symbol-manipulation.
But to answer to your real question – why should this boy learn algebra – I would say he shouldn’t.
If he’s as enterprising as you imply, just tell him to go on being enterprising and forget algebra – before long he will be employing people who can do the algebra for him; they’ll be the wage slaves who spent four years getting into debt at Uni, and he’ll be the boss.
Unless credentialism stops him.
If you can’t count to well over 100 on one hand, you aren’t trying very hard.
I found that twelve year olds were entertained by simple visualization exercises and constraint expansion, despite, or perhaps in vivid contrast to, the utterly brain-dead school they were in. Pointing out the assumptions in questions, showing how two intersecting lines could be in (at least) two different planes — there’s quite a wealth of material at hand.
Which is greater: 5+2 or 2+5?
Which is greater: 203+1 or 1+203?
Which is greater: 534,365 + 783,892 or 783,892 + 534,365?
What is 534,365 + 783,892? Why could you answer the previous question quickly but not this one? That’s the difference between algebra and arithmetic.
May not be a good example for everyone, but it worked with my 6 year old.
this is so pointless.
The answer to your question is algebra has no use to the ordinary blue collar worker. Such as, the every day worker, probably has never taken algebra 101, but as you look on, to the higher end of the latter, such as scientist’s or even architect’s, who use algebra as an essential tool to create and solve formula’s and devolop plans and blueprint’s, respectively speaking. Colleges introduce it because who know’s, if the student excells in it, it then opens up their door to a broader opportunity on the job market. Now that student could end up in one of these above mentioned fields of study, and/or the list of fields goes on and on and so does your pay scale. Some people grasp it, and other’s just don’t and probably never will. I being one of them.