I’ve heard a distinction made between problems and exercises. If you can come up with the answer just by following rules or working through a specified method, it’s an exercise. A problem is something that you have to figure out how to solve, maybe by some flash of insight or creativity, or by trying different things until you come up with something that works.
Depending on the math class, students may not be expected to solve any problems in this sense of the word. And if so, they’re missing out on where many math people find the real fun of the subject.
You have just described me. Surprisingly I have a reputation for being really good at math. Wrong, I suck at actually doing it. What I AM really good at is setting_up the equations to describe real problems…then I totally suck at simplifying/solving those equations, making many “careless” errors. I’m also pretty good at coding math based algorithms, so often I can let computers do the part I suck at.
Conversely, I find many people who are really good at doing math get stumped when it comes to even fairly simple “word problems”: 6x50? no sweat. “Santa is 6 feet tall. If each foot weighs 50 lbs. how much does Santa weigh?” Deer in headlights. (this is an actual word problem I remember from 4th grade) So even though these people are pretty good at math, it is a useless skill for them. Whats to like?
Many people don’t see mathematics as having any use in their day-to-day life. And for most people that’s true.
Most people never have a need to do any mathematical operation more advanced than addition and subtraction. Maybe multiplication and division once in a while.
Sure, there are people who use higher mathematics on a regular basis: engineers, scientists, building contractors, insurance actuaries, bookies. But your average person can go for years without ever having to use algebra, calculus, geometry, probability, trigonometry, etc.
And neither is there one of music. Yet still, our culture is suffused in music – which is the reason why bands with little explicit musical knowledge can nevertheless start a whole scene: they’ve picked up their musical knowledge by cultural osmosis – it’s not that they had none, it’s that despite their lack of formal education, music is just so omnipresent that one can’t avoid picking up enough musical knowledge to know what ‘sounds good’ (granted, to a certain extent, that’s probably just a part of the brain’s wiring).
Of course, maths doesn’t necessarily lend itself to such suffusion – you can’t really make maths records, or maths concerts (who said the thing about maths rather than math upthread? Curse you. ;)). But consider games, for instance – a lot of them are essentially maths already, think checkers or similar. That connection, however, is never really made.
That’s not to say that arithmetic isn’t important, but teaching it by way of introducing god-given rules and rote memorization just might not be the best way to go about it. Lockhart gives a few great examples in his essay.
And yet, people, avowed mathophobes even, play Sudoku, just for the heck of it, or similar games. It’s just that they don’t know they’re doing maths, because to them, maths means horribly contrived rules, equations, symbols, boring, repetitive memory tasks, and who knows what else. But actually, maths is just doing puzzles.
I have a question for those advocating that math be taught in a more applied style instead of simply lists of rules and how to use the rules. As an example, here is an abstract rule from algebra:
The difference of two squares, a[sup]2[/sup] - b[sup]2[/sup], can be factored as (a + b)(a - b)
Now, one can learn basic factoring, and derive the above property from other rules. But if we’re teaching algebra in an applied way, based on the real world, how would we teach the above rule? It doesn’t precisely fit into any physical scenario that I can envision, like figuring a tip or finding the area of a floor. But we have to learn this rule to do algebra.
Another example from algebra might be the quadratic formula. A bad teacher will just write it down and say “memorize it!” A good teacher will show its derivation from the general form of a quadratic equation by completing the square. I certainly agree that the latter is better than the former, because it allows you to understand whence the formula arises.
And while one can certainly come up with applied problems that result in having to solve a quadratic equation, at the end of the day, you still have to commit that damn formula to memory in order to solve the problem. So how do we teach these things in an applied way, with context?
I suspect the situation you just mentioned is less of a math issue than a reading comprehension issue. I never saw the issue with word problems, since I’m extraordinarily good with language, and being able to break it down into a formula was easy.
I also have 2.5 years of high school math. The reason I don’t have 3 years is that I flunked my last semester. I was forced to take every year of it. I didn’t hate math I just didn’t really like it.
I went into college to pursue what I did like (Chemistry) and was forced to take up through the Calculus series. I didn’t mind it as much as high school math. I found Calc more interesting…but I didn’t love it. I did like it though.
Then, for some reason, we had to take Linear and modern Algebra. Linear was fascinating. However, Modern Algebra…oh wow…oh wow…oh wow. It was so deliciously ABSTRACT. They had intuitive proofs. My fellow chemistry students curled up and died in that class. I loved loved loved loved it.
I loved it so much I went to the math department and asked if there were more classes like it. They looked at me with amazement and said that was what math WAS. All the stuff before was just…background. One prof let me sit in on his Real Analysis class and I was hooked. Shit, I never even finished my Chemistry major.
The real world application isn’t about necessarily have a real world application for every rule directly, but understanding why you may want to do that relative to other things you already may want to do. For instance, all you really need to know is why you might want to know how to factor polynomials. Again, building on previous concepts, knowing why you might be working with quadratic equations, then knowing you can use factoring to find roots for that.
There are a lot of processes in math that don’t directly equate to any physical process, but are part of relating one abstract concept to another. As long as the basis for that concept are understood, then building on it shouldn’t be so much of a step. I think the problem comes when building on concepts with the assumption that that is a solid base, when it isn’t. That’s when it confuses people.
This is exactly how it should be taught. When I learned it, my teacher didn’t just derive it, but she had the class actively engaged in trying different ways to derive it, we’d try something, it wouldn’t work, and then we’d try something else. Rather than just saying “here’s the derivation”, we actually worked for it, and we were able to understand where it came from and why. I can’t remember when the last time was I used it, but to this day I can still recall it correctly because of that lesson.
I have to disagree here. A lot of students are forced to memorize formulae, but I think that memorizing them is pointless if you don’t know what that formula means. I think to a lot of people, they just see plugging in certain numbers and magically getting out an answer, but they don’t really understand the relationship between the values.
To this end, I think memorization is WAY over emphasized, especially in today’s world. Going back to the quadratic equation, even if I couldn’t remember it, I can google it in 5s, so is it even really worth memorizing? The important thing is that I actually understand what it actually computes and how the various parameters relate to eachother and to the answer. Hell, anyone can memorize a formula, it’s knowing what to do with it that really matters and not understanding that part is what makes it seem hard.
An excellent question, and one that I’d like to see some people try to answer. The thing is, many of the things one learns in math aren’t directly applicatable to the Real World[sup]TM[/sup]. It’s more like, you have to know how to do this, so that you can do this, so that you can do this, so that you can do this, which has real-life applications.
But, for an “application” of that factoring formula, try this:
Pick any two numbers that are 2 apart from one another, like 1 and 3, or 5 and 7, or 8 and 10, or 352 and 354. Multiply them together. (1 x 3 = 3. 5 x 7 = 35. 8 x 10 = 80. 352 x 354 = 124,608.) Do enough examples, and you may notice that the product is always one less than a perfect square. In fact, it’s the square of the number that’s between the other two.
1 x 3 = 2[sup]2[/sup] - 1.
5 x 7 = 6[sup]2[/sup] - 1.
8 x 10 = 9[sup]2[/sup] - 1.
352 x 354 = 353[sup]2[/sup] - 1.
Why does this work?
Because of the factoring formula. (a-1) x (a+1) = a[sup]2[/sup] - 1, for any number a.
And you have a similar relationship when the two numbers that you multiply differ by some other amount, rather than by 2, from (a-b) x (a+b) = a[sup]2[/sup] - b[sup]2[/sup]
Granted, this is more of a “stupid number trick” than a real-world application. But it is an example of the kind of thing that at least some of the people who like math find interesting.
Of course there’s a concept of “sounds nice” in maths and languages. What do you think poetry is? And how do you account for the difference between elegant, easily understandable mathematical proofs that brighten your day, and ugly, technical proofs that only make you feel dumber? (And quite possibly actually make you dumber. That would explain what happened to me ever since I started studying math.)
Anyway, it’s been already said in this thread, but it bears repeating: I think most people only have the shallowest understanding of what mathematics actually are. Lockhart, in his essay, is claiming that math is basically an artistic endeavour in which anyone can participate (though of course, as in anything, to become good you need years of practice) and should be approached from this viewpoint, and I see what he’s talking about but I don’t agree 100% with him either. Many in this thread have said that math should be taught from a more applied standpoint, to make it more “relevant” to students, which I guess would appall Lockhart. But there is something to be said about showing, from a historical perspective, how mathematical concepts were developed in order to help solve applied problems. Of course, the stilted “word problems” of math classes aren’t going to do that.
But where Lockhart is definitely right, is that the current math education curriculum doesn’t really prepare people to actually work in mathematical research. In fact, it’s only after starting grad school that many mathematicians figure out what abilities are really required of them, and sometimes realise that is isn’t for them after all, even though they were “good at math” until then. (I’m currently doing a Ph. D. in mathematics, and I honestly couldn’t tell you if I’m good at it or not. All I know is that it’s hard.)
What I wonder is, is there any field apart from math where the difference between how people are introduced to it in school, and how professionals actually use it, is so large? This could help us figure out if math education should be reevaluated or if we’re only looking at a necessary effect of introducing a vast field of knowledge.
I think that’s a really interesting question. I mean, from what my dad (mathematician) tells me, once you get only a little further along in maths than I did, there really aren’t numbers to deal with any more (which, I’ll admit, I don’t grok, having not gotten to that point!) Yet any third grader will tell you that math is “what you do with numbers”. So that seems like a screamingly huge difference, right there.
I hate math because I never understood it. It was nothing but exercises out of a book, arcane vocabulary, and the knowledge that I would never need to know any of this shit once I got out of high school.* Even when I thought I understood the concept, when I would check my work the answer would be wrong and I would have no idea why. It wasn’t bad teachers for me, in fact I had the same algebra teacher through high school because he was passionate about math and I liked him, but the textbooks sucked and with 35 students per class he simply didn’t have the time to explain every minutia of abstract concepts to me. I don’t think in the abstract, I need concrete, visual examples to understand things.
Or the branch that made no sense to you. (i.e. no “real world” uses) For me, that branch was trigonometry. I inherently could see uses for algebra and basic geometry, but trig made no sense to me. (stats, which I took after, again made sense)
People do not have an interest in math because it is taught in isolation to anything relevant.
Differential calculus and advanced geometry are ingrained in our daily lives but are primarily taught in the “memorize this formula” format that makes it uninteresting. Only after you have spent many excruciatingly boring years learning basic formulas do you ever get to apply it to something useful, and that may be only one or two classes for those that are not science majors.
**
Life Is One Big Word Problem**, but we teach math in a vacuum.
The last thing mathematics needs is a revival of New Math. Math education should start with the basics. Then, after you’ve taught the facts, you can consider following up with the theory behind it.
Sudoku isn’t really math. It’s a game with numbers but there’s no mathematical operations involved. It could be played just as readily with letters or abstract symbols.
