A couple of "kids these days are bad at math" stories crossed my feed reader last week, first a New York Times blog post about remedial math, then a Cocktail Party Physics post on confusion about equals signs. The first was brought to my attention via a locked LiveJournal post taking the obligatory "Who cares if kids know how to factor polynomials, anyway?" tack, which was obvious bait for me, given that I have in the past held forth on the importance of algebra for science students (both of these are, at some level, about algebra).

Of course, these articles aren't about science students, so you might want to say that the same arguments don't carry over. I would tend to claim that algebra is still an important part of education. The example that comes to mind about this is always an incident from the house I lived in in grad school, involving circuit breakers.

When I was in grad school, I did my lab work at NIST, which is about half an hour from the UMD campus on a good day, and considerably more if traffic is bad. Accordingly, I moved closer to Gaithersburg once I was more or less done with my class work, and ended up renting a room in a crackerbox Cape Cod house on the edge of a bad neighborhood in Rockville, MD. The rent was nice and cheap, but since the house was so far from the local academic institutions, the other rooms were occupied not by other grad students, but by a rotating cast of people in low-wage jobs, or no job at all. This led to a number of colorful stories that I used to amuse folks on Usenet with back in the day, but the relevant one here is pretty simple.

The kitchen of this house had some slightly dodgy wiring, I believe in the hanging lamp over the table, which would occasionally trip the circuit breaker controlling the lights, the microwave, and the stove (which was gas, but had a clock and an electric starter). One morning, I woke up to find a note under my door from one of the other tenants, who we'll call W, saying that he had gotten up in the middle of the night to get a snack, and found the power out in the kitchen. So he lit the stove with a match, cooked what he wanted, then wrote me a note and went back to bed.

This was, as you might imagine, a little boggling. When I asked him about it, he explained that he didn't know where the fuse box was, but he knew I did know, and would fix it eventually. At least he hadn't woken me up at 3am, right?

Even once I had explained to him where the box was, though, we still had problems. The next couple of times, he went down there, and couldn't figure out which breaker was tripped, so he just turned each one off and back on in turn, resetting all the clocks in the house in the process. On another occasion, he tripped a breaker in his own room, and turned off the breaker for my room while he was trying to figure out which one was his. I ended up taping a sign with detailed instructions to the outside of the box, and putting big arrows pointing to the two breakers that most frequently went out.

What's this got to do with algebra? The fundamental problem here is a lack of systematic thinking. The first time I had had a problem with the power in the kitchen, it wasn't too hard to figure out by running through a sort of mental checklist. Since the power was on in the other rooms, I figured it had to be a breaker. I didn't know where the box was, but I knew it wasn't likely to be in one of the rented rooms, which meant it pretty much had to be in the basement. Since it wasn't in the laundry room, that meant it had to be in the storage room, and hey, there it was. I found the breaker that was off, turned it back on, and there you go.

W. was, apparently, incapable of figuring this out. Neither was the other housemate he enlisted to help him one day when I was at work. They needed me to tell them where the box was, and even then couldn't work out how to get the right circuit. Confronted with a problem and nobody to point them to the right answer, they just gave up.

That's one of the things you're supposed to get out of taking math classes like algebra in school. There's nothing all that alien and intrinsically difficult about algebra, it just requires you to systematically apply a few very simple rules. When you're asked to solve an equation like

x

^{2}- 2x - 8 = 0

you shouldn't need to have the quadratic formula programmed into your graphing calculator, or even committed to memory. You know you're looking for something like:

(x ± a)(x ± b) = 0

Since the sign of the final term in the original equation is negative, that means the two factors need to have opposite signs, so we have

(x + a)(x - b) = 0

We also know that a times b has to be eight, which means one is two and the other four. The negative sign on the middle term of the original equation means that the negative term must be the larger of the two, so we have:

(x+2)(x-4) = 0

This is satisfied if x=-2 or x=4, so those are your solutions.

There's nothing sneaky or arcane about this. You don't need to know how to do anything other than basic arithmetic to solve this problem, you just need to approach it systematically.

The breakdown that leads most students to give up on this type of problem-- usually saying "I don't remember the quadratic formula"-- is just a breakdown of systematic thinking. It's the same failure that leads to nonsense like "I didn't know where the fuse box was, so I left you a note." It's a failure to systematically attack a problem beyond the most superficial level. Hell, I'd be happier to see them start plugging integers in for x in hopes of finding a solution that way.

And that's one of the benefits people ought to be getting from taking math classes. There's a lot of repetitive drill in math classes, specifically to establish the habit of thinking systematically about problems. If you can learn to work through simple algebra problems, you ought to be able to apply that same sort of reasoning to real-life problems, even ones like "Why doesn't the kitchen light come on?"

The problem we have is two-fold: 1) students compartmentalize to a remarkable degree, and often don't even connect math to science, let alone apply the systematic thinking of math and science to other areas of their life, and 2) as a society, we enable this sort of compartmentalization by shrugging off poor math performance as just one of those things. I bang on this a lot, but it's incredibly annoying that a student who doesn't like to read represents a major crisis, but a student who doesn't like to do math is steered to special classes for people who don't like to do math. If the Times were to write about a remedial class for students who can barely read, I doubt it would get quite as flip a treatment as the math class did.

That's a problem, because the skills you learn in math go well beyond learning to plug numbers into the quadratic formula. Math is about systematic thinking, and systematic thinking is what built human civilization. Without the ability to think systematically, we'd all be stuck huddling in caves, hoping the lions didn't eat too many of us tonight.

Or at the very least, sitting in a dark kitchen, waiting for somebody else to fix the lights.

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"students compartmentalize to a remarkable degree, and often don't even connect math to science"

Its not really the main point of your post, but I'll add that they compartmentalize to such a degree that if you take students that can do the problem you posted easily, but change the variable from x to (say) g...they freak out just as much. I had this come up several times in my science class. I would give an equation like that (but with non-x vars) and students would ask "what do I solve for?", "where is x" and all kinds of things like that.

I have a similar story that happened to me a couple of months ago where even a trained electrician was unable to find the solution, but I was able to by applying a bit of systematic thinking.

One day, the circuit breaker controlling all the power outlets in my apartment got tripped. Since I knew nothing about household electrics, I got an electrician on the phone who thought it was most likely to be a faulty appliance. He suggested unplugging all the appliances, flipping the breaker and then plugging the appliances back in a room at a time, so we could narrow down which room the appliance that breaking the circuit was in. We did this, but the trouble was that the circuit didn't break again once we plugged everything in, so this gave no information. Having not found anything, we just continued with the daily routine for another hour or so, after which the circuit breaker tripped again. When this happened, I called the same electrician again and he suggested that we do the same thing, but again there was no joy. After that the electrician was completely stumped and suggested that we should call them out the following day.

However, we didn't do that because I came up with a cunning plan. The next time the breaker tripped, which was about an hour or so later, instead of unplugging all the appliances at once, I unplugged them one at a time and tried to flip the breaker after each one. It has a failsafe so it won't go on if there is a bad circuit, which meant that as soon as it turned on we had found the faulty appliance. It turned out to be a lamp in the living room which my Dad was turning on every now and then to look at the TV guide, which explains why it was happening intermittently.

In hindsight, this was a completely obvious solution, but it took me a long time to even think of it because I trusted that the electrician should know what to do better than I would, especially since I am a theoretical physicist studying the most completely impractical area of physics ever devised by humankind. I think the problem was that the electrician had memorized a "rule book" for things to do in common situations and he wasn't used to reasoning from basic principles to derive or go beyond the rules he had memorized. I think this is strongly analogous to the way that many students just want to learn "the method" for solving different problem types in algebra rather than learning how to think about them systematically from first principles.

There's a lot of repetitive drill in math classes, specifically to establish the habit of thinking systematically about problems.This doesn't really make sense to me. How does repetitive drill get you to think systematically? Repetitive drill, in my experience, involves setting up a pattern for solving some specific kind of problem, and then having the student solve a bunch of instances of that kind of problem, using the established pattern.

For most students, this is essentially the extent of the skills they learn in their mathematics classes. A set of arcane and impenetrable rituals to be invoked for an exam and then quickly forgotten afterwards. The old adage that studying mathematics promotes critical or systemic thinking is demonstrably false as by now the vast majority of people in the Western world have had signifigant exposure to mathematics, yet I would not describe the general public as critically thinking(This observation can also be extended to most academic communities in relation to paper publishing via for profit Journals.).

The Western world (or at least the anglophone part) has not taken a long hard look at how we teach mathematics. I tend to think that despite all the grumbling, arithmetical instruction is not so bad. But everything beyond this is more or less atrocious.

If you've spent much time volunteering in elementary school classrooms, you quickly find out that elementary school teachers (in general) don't like math and aren't really all that good at it, especially at the kind of systematic thinking Chad is talking about. Nobody would be hired as a teacher who admitted not liking reading, but in our society, most people don't like math, so it's okay.

Sadly, I can recall quite a few occasions during my K-12 education when my experiences with math were negatively influenced by the educator. One particular memory involves a piece of software that delivered progressively more difficult problems as the student gave correct answers. At one point I exceeded the type of math I had been taught (fractions) and the program moved on to percentages.

The question was: "Billy is taking a multiple choice exam with 4 possible answers to 1 question. If he guesses the answer what is the likelihood that he will get the question correct?"

Clearly the answer the software was looking for was 25%, but my 5th grade brain raced with a million questions: "what was Billy's field of study?" "what was the topic of the test?" "was he able to absolutely eliminate any of the other answers?". How could I possibly account for all the variables!? I didn't even know this Billy kid!

I began to panic. When I asked the teacher for an explanation (since I was stumped at that point) her response was vague and essentially amounted to "because that's the way it is". Which is a response I continued to get from math educators through middle school and high school.

Luckily, I can attribute a great deal of my ability to think logically to some terrific science teachers. Once I hit college I discovered math professors that actually GOT math. :)

Where is there a bad neighborhood in Rockville? Or have things gotten better since you were there?

How does repetitive drill get you to think systematically?That may have been a bad choice of words-- I don't mean that you do examples of exactly the same type over and over. There are a huge number of variations on algebra problems, though-- the test linked from the Times piece contains a bunch of them. You can't use exactly the same procedure for all of them, but you use the same core principles over and over.

Math homework generally consists of lots of problems covering variations of the type of problem being studied, with the idea being to give students enough practice that the process becomes familiar if not comfortable.

Where is there a bad neighborhood in Rockville? Or have things gotten better since you were there?Here. One side of the street was little Cape Cod houses, the other was public housing and run-down apartments. The people across the street got busted for dealing pot at least once, and we had a guy beating up his girlfriend on our porch one night (she ran to our house to try to get away-- we called the cops, and he ran off).

Two blocks away in just about any direction were nice houses, but that one street was pretty bad.

Brain, I know the type of students of which you speak. I tried to use money as a way to demonstrate unit conversions, thinking I'd start with something they knew. (1 dollar = 4 quarters = ten dimes...) One student said "but that's not what I do". To which I relied, repeating myself for the third time or so, 'you don't do each step individually anymore but this is what you do in your head, just spelled out mathematically.' Another said "money isn't math."

Unfortunately you can do well in most math classes (k-12 anyway) without learning how to think.

For many years I taught calc and physics as a one-person team. We started with just algebra in physics, and as we learned more calculus we would incorporate those ideas into our physics chapters.

We had to learn over and over that algebra is spelling, grammar and sentence structure for describing how the world works (Physics!). The quote was "If you can say it (with algebra), you can do it (in physics and calc).

Having the same kids for two hour-long periods every day was great from an educational perspective. I could schedule whatever I wanted into those periods. Having so many motivated students was a treat. I hope I did them justice. One of my gang ran by during the Falmouth Road Race and saw me in the crowd, so he yelled "Hey, Coach Mac, I'm a ____ing engineer!"

You put up with a lot of crappy paychecks to get a payoff like that!

On the one hand,unless you are using a very specific definition of "systematically" that I am not aware of, I'm not sure you're literally correct.

The guy who left you a note went through the following steps: "hmm, there's a problem here. It's not that the power is out in the whole house, so it must be a circuit. How has that gotten solved before? Oh, right. That one guy knows how to fix the circuit board. I don't even know where it is. But it's 3am. I don't want to wake him up. Well, I only need to use the stove. That's gas! I can start it manually, and ask him to solve the circuit problem later." There's a lot of steps there, and the reasoning reflects a perfectly reasonable recognition of the relation of parts-to-a-whole. The fact that *you* were a part (the circuit-fixing part) in the system is perhaps annoying for you, but it's a perfectly valid analysis. So it's systematic, sort of. It's also easy to see how it's a solution that fits a local low-effort well, if you will. Sometimes good enough is good enough.

On the other hand, this is my circuit board story:

"EEP! The power went out when I turned this on. No, only some power, that's still on over there. Hmm. Must have blown a fuse [NB: I grew up in a house with a fuse box, not a circuit switch board]. Hmm. I don't have extra fuses. Aghh! But there's no reasonable way to plug my computer in without this outlet, and I'm just going to put a major load on some other circuit if I try to reroute the plugs. Well, I guess I'll have to go to the store to get a new fuse. But I don't even know what kind of fuse I need! I guess I have to go down to the creepy dark cellar to find out. I can take the old one with me to the store and ask if I'm not sure. *finds flashlight, goes down to creepy dark celler* OH! That's a circuit board. I've never seen one before. I wonder how it works. There are a few labels telling me what parts of the house go with which switches. But I don't know which house the labels go to [NB- my apartment is one of three in an old house]- and most switches aren't labeled at all. And it's not like they are all pointing one direction. Darn. Hmm. I guess I could switch one that's labeled and see if it goes off in my apartment or if somebody starts yelling next door. *wince* Oh well. *flips switch, runs back up the creepy cellar stairs and into the house* Oh good! the labeled switches are for my house. *runs back down, flips switch back* Let me try this switch, which isn't labeled for the place I lost power in, but has a label that has things that are *nearby* the place I lost power. *flips switch, runs back up the creepy cellar stairs and into the house* VICTORY IS MINE! INTERNET FOREVER!"

So, *I* think I think systematically (and more functionally so than your housemate).

That said, I fail at systematic algebra. To me, factoring is about as "arcane and impenetrable" as the quadratic formula. Your list of rules for the signs seem particularly weird to me. At that point, it's just easier to plug in part of the equation.

Becca

I think I love you, at the least I love your take on it. Too often peopel do overlook how solving a problem may include utilyzing others as a part of a solution. As you very rightly pointed out it appears his housemate decided to use a solution more in line perhaps with an economist. He picked a solution that minimized the energy he would have to invest in order to get the outcome he desired.

Powers out in kitchen, so cook without it, later I might want power so let me leave a note for someone else who can fix that issue, and very likely will want the power to work.

No muss, no fuss and no stumbling around a storage room at 3am looking for a circuit breaker box.

@joemac53 -

I've often thought that getting this kind of idea through to the non-mathematically inclined who enjoy literature and languages might be a "point of entry" to their frame of mind, something to help them to begin thinking and reasoning mathematically.

Another idea I'd like to see circulating is that Mathematics is the language Nature speaks. A lot of people who don't care for math do care about nature.

There's huge potential, too, for using art to prod students to understand mathematics better. Art is all about proportion, and proportions are ratios. The same student, who can't reason that adding a 3 to the sequence {2, 4, 6, 8} makes the average of the numbers in that sequence go down, can yet intuitively grasp that making a background image much smaller than the foreground makes it seem proportionately farther away. Both are math. One observation is made voluntarily and with little effort. The other seems, to the student who's poor in math, like a far-fetched and difficult chore. I'm convinced that if kids could see the similarites, math would make so much more sense in so many ways.

Granted, they still have to drill and work problems - too often linking math to other subjects is done in a flaky way that seems designed to let students out of actually doing real math. The problem is using outside subjects to "enrich" students' understanding of math. What's so lacking is the sense in which math gets at the heart of things that move us and matter to us. Maybe math should be woven into the teaching of every other subject the way reading is - not some singular endeavor off by itself that you either can or can't do, but part of everything, part of life.

Chad, reducing the problem to a previously solved one is pretty much as mathematical as it gets, no?

Reminds me of an anecdote from one of Richard Feynman's book. "He fixes radios by thinking!"

Moreover - what's up with the grammar on that practice test?

Yeesh.

If you already know that factorisation will help you, then it's "obvious" that you should attempt to factor a quadratic equation. If you don't, then it's not. Otherwise, you wouldn't need to teach students about polynomials at all. (This was forcefully brought home to me at Cambridge when I started to learn about contour integrals - the sheer number of rules which "might help", from which I had to pick the one which would, floored me completely.)

Similarly, if you already know (or assume, perhaps?) that this is the kind of house in which the fuse box is not in one of the tenant's rooms, then it's "obvious" that an exhaustive search of the communal spaces will reveal it. If you don't, then it isn't. (Think about conversions of old houses for example.)

I'm afraid your article comes across to me as suggesting that algebra teaches you that trial and error is a useful technique, and I hope that's just because your command of English is worse than your command of maths. Quite apart from anything else, that's a technique which at school I associated more with chemistry ("identify this compound") than with maths.

To a certified Math ignoramus like me, a quadratic equation is an elegant way to describe ( is that the right word?) a parabola - and "solving" a quadratic helps you to find some points on that curve ... perhaps to draw it?

Well, I can do that, but nothing more. Knowing calculus woud presumably help me understand how some basic algebraic techniques might lead to useful understandings and application. At least that is what I BELIEVE some people have told me.

One problem is that until you have actually learned higher math (calculus or advanced statistics) the above has to be taken on faith. Another problem might be the very idea of "systematic" thinking. Designing bridges, guiding missiles, determining the actuarial bases for insurance policies, and predicting atronomical phenomena (or financial panics) are relatively rarified endeavors which currently often involve a lot of interaction with computers.

Unqualified as I am to think about these things it seems like Math is like a an infinite set of tools each with more or less specialized purposes. People in the UK who talk about "maths" might be more precise than Americans who refer to the megalithic (but mostly mysteriously unknown) singular "Math".

Another "certified Math ignoramus" here--

Seems that, if students have a brilliant math teacher, they're going to "get" math and enjoy it. The problem: brilliant math teachers are incredibly rare.

So: Would it be possible to write a math textbook that would help average teachers convey to average students the excitement of math? that would present repetitive problem sets in a way wouldn't bore and discourage average students?

I'm not sure if math teachers really need to be brilliant. Ok, they obviously should have an interest in math and in teaching and should have a certain basic understanding of math themselves. But beyond that - they should be honest about what they are doing and why learning certain stuff is necessary etc. I think that would already be a helpful step.

A lot of math is exciting, but not all! Some math you have to learn is really boring no matter what anybody says. Unfortunately this is true for a lot of basic algebra. But you must learn it anyway because the technical abilities are an intermediate step and necessary later. As cooks must know recepies.

But a teacher should say so and even try to explain where and why something might be useful.

Take the binomial theorem as a simple example. It's just a technical thing, easy to memorize from the left to the right ( a plus b squared equals ...). But more often it's needed the other way round. It's simply pattern recognition in expressions. And an 'application' might be to use it as a tool to obtain a solution for quadratic equation.

But on some reasons teachers rarely talk about this. Neither that it's something technical nor the way how it is used and interrelated to other things. They seem always take it for granted or as obvious or .... and their main achivement are totally confused and/or annoyed kids.

And some stuff - well it's good if you have seen it and if you know that there are ways to tackle certain problems. But no need to memorize the details. And again - a teacher should say so but they rarely do. Really a pity.

Finally mathematicians like big and important words from stupid old languages. Makes things sound important and difficult, even if the fact behind the word is really trivial. It is just a language sometimes even only just noise. Helpful for communication - common vocabularies for all - but thats it. Nothing more. Teachers should tell this.

To a certified Math ignoramus like me, a quadratic equation is an elegant way to describe ( is that the right word?) a parabola - and "solving" a quadratic helps you to find some points on that curve ... perhaps to draw it?

Well, I can do that, but nothing more. Knowing calculus woud presumably help me understand how some basic algebraic techniques might lead to useful understandings and application. At least that is what I BELIEVE some people have told me.

One problem is that until you have actually learned higher math (calculus or advanced statistics) the above has to be taken on faith. Another problem might be the very idea of "systematic" thinking. Designing bridges, guiding missiles, determining the actuarial bases for insurance policies, and predicting atronomical phenomena (or financial panics) are relatively rarified endeavors which currently often involve a lot of interaction with computers.

Unqualified as I am to think about these things it seems like Math is like a an infinite set of tools each with more or less specialized purposes. People in the UK who talk about "maths" might be more precise than Americans who refer to the megalithic (but mostly mysteriously unknown) singular "Math".

Chad, have a look at this 3rd grade "spelling contract," which includes an equation showing a total lack of understanding of the equals sign:

http://schools.sisd.net/oke/index.php?option=content&task=view&id=264&I…

I think our problems in this country start with teacher ignorance. Teachers don't understand the math to start with, or treat it as a black box, and then pass this on to the students. If the teacher has never taken a systematic approach to problem solving (as opposed to teaching an algorithm) then it takes a real leap on the part of the student to make the connection.

As a software engineer, I agree totally with you on the usefulness of systematic thinking - it's frankly what software is all about. As a preservice math teacher though, I think that there are way more interesting problems to learn this on than factoring (for example). I'd much rather have kids spend time learning problem solving a hand-on situation (debugging a program, getting a piece of equipment working, etc.) that is personally fascinating to them. Don't get me wrong. I think (I know) math is extremely important and useful, but since it is so useful shouldn't we spend time helping kids learn techniques to solve real problems, and instead of using math as an abstract brain exerciser?

When you can drop x^2 - 2x - 8=0 into http://www.wolframalpha.com - it's harder to defend a reason to go through systematic thinking for a subject so rote that it can be solved in 2 microseconds by a machine.