As for computing science, those of you who claim to never need math in your job are probably benefiting from the fact that you use a bunch of high-level libraries, and your job is to just glue them together and slap a UI on top. That’s what a lot of programming has become these days. But where do you think those libraries come from? Someone has to write them, and those people need higher skills. That cool library call you can use to rotate text required a programmer who understood how to do matrix manipulation. That cool game engine with the awesome physics modeling required a programmer somewhere to know calculus, physics, and all kinds of higher level topics.
The point to high school is to allow you to make the choice to become a game programmer or a biologist or a doctor or an electrician after graduation, and not be hamstrung by a lack of basic knowledge. It’s also to ensure that you’re a functioning citizen with the intellectual tools needed to understand the issues of the day, read the newspaper and get the gist behind the articles, and solve household problems that you may run into.
School has been dumbed down enough. Let’s not look for more ways to degrade the education of young people.
If one cares about mathematics, almost zero effort.
If one does not, it requires more mental energy than splitting an atom.
One more observation that I saw on the web some years ago about getting students to “internalize” and “critically think” :
There was a fresh college professor that held high ideals about teaching mathematics the “right way.” He thought he could prepare the students with the right “motivations” with a little “history, context, real-world applications, etc” He spent extra effort to integrate the “beauty” of math into his teaching but when he did so, many of student evaluations were negative with comments like, “I wish he would skip all the math cheerleading and just tell me exactly what I need to pass the exam!”
He eventually caved in and just taught the class as brainless mechanical manipulation to cater to the plug-n-chug students (which are the majority). The student evaluations improved after that. I had to laugh when I read his defeat.
I don’t know the exact google keywords to bring up that story but it’s out there somewhere. If any of you mathematics professors had different experiences of student expectations, please chime in. Perhaps that jaded math professor was trying to teach truck drivers at a community college; I don’t remember the school he was at.
On another note, there’s some irony in the math enthusiasts bewilderment as to why the math phobics don’t “internalize” stuff beyond mnemonics. I have a friend that’s gifted at mathematics. He had memorized “The Gettysburg Address” and various other poems (Robert Frost, etc) (I knew those details because we went to the same school as kids.) I asked him if he knew what the Gettysburg Address was actually about – what did it actually say? He said he didn’t know. He didn’t know it at the time he memorized it and he doesn’t know now. He’d have to look it up on wikipedia like most of us. To him, memorizing the the Gettysburg Address was simply stringing together a bunch of phonemes for class recitation and get a passing grade.
People will pay attention to whatever they really want to pay attention to and use rote memorization as a coping mechanism for everything else. The math guys included.
How about understanding compound interest and knowing how it affects your loans, savings and investments? I know there are calculators online to do this for people, but don’t they want to know how it works?
Or doing excel spreadsheets to calculate budgets, savings and retirement? I use a formulas to calculate my monthly income when I retire based on my pension, 401k, standard savings, home value and so on. How does someone do that without knowing algebra?
I’m sure I could think of a few more examples of algebra I use often, but those two just scream out to me.
Whose rule? My proposal was that mathematics, sentence diagramming, art classes, chemistry, biology, and Shakespeare all be available courses of instruction in high school (and college and post-college…), but not, beyond some very basic point, compulsory for students.
As for everything else: your goal can be universal mastery of all of mathematics, and every other subject to boot. But to what extent does simply mandating the coursework actually accomplish this goal? It’s no use pointing at your hopes to justify means which fail to achieve them.
Yes, schools do a shitty job teaching math, and that is a problem which should be corrected. But part of why do they do a shitty job teaching math (and so many other subjects) is the shitty job they’ve been tasked with, of having to offer the service not only to those who want it, but also to those who would really rather do something else with their time, and who, even after all the compulsory math education they are forced to endure, will continue to do something else with their time.
If we poured all the world’s resources into math education, we could no doubt jam in proficiency with, say, basic calculus into every non-mentally-deficient child, against the resistance of their own lack of interest. We could do the same with violin training or French. All it takes is a large amount of time and effort. But would this be a reasonable allocation of resources?
It seems to me worrying so much about making sure everyone pursues mathematics just wastes the teacher’s energy, annoys the reluctant student, and drags down the curriculum for everyone else. A few skills are so ubiquitously vital that they are worth jamming in against childish resistance (for example, literacy and arithmetic fluency); most aren’t.
I would finally return to noting that “algebra” can mean a spectrum of different things, about which I would feel differently as to the reasonability of ensuring universal knowledge and comfort.
We are in the midst of a financial crisis that had its start in people getting loans that they could not pay back. There is a lot of talk about people getting the loans not understanding what they were getting into. The math behind mortgages is really easy to derive and understand with high school algebra. If you don’t understand algebra you are reduced to talking the word of people who don’t have your best interests at heart tell you how your finances work.
A lot of programming jobs (most?) are LOB (line-of-business) applications. Programming software to track sales orders, order fulfillment, etc. You don’t need algebra or calculus for it.
The other programmers in fields like video games (physics engines), financial trading (derivatives, differential equations), artificial intelligence (matrices) requires heavier math.
In other words, the usage of mathematics is domain dependent.
Nope. I’m Canadian from Ontario; which is why I was so disappointed that a native English speaking math teacher with a very British name couldn’t spell Asymptote. She was one letter off from the Italian, but she was educated at U of T, so I doubt she even knew she was accidently close.
Yup. As I said before, universal compulsory math education just wastes the teacher’s energy, annoys the reluctant student, and drags down the curriculum for everyone else.
[None of this is really specific to math, though. That’s just the particular topic on which discussion happened to be raised. Actually, I wish it weren’t raised in this particular way, because algebra (which, as I keep saying, covers a spectrum of topics) is just about at the border of where I would consider math to switch from “Everyone needs to know this; you’ll all thank me later” to “Forcing everyone to go through this is a waste of time; large quantities of you will get nothing out of it but resentment”.]
I’m only a lowly math grad student, but this all jibes perfectly with my experience teaching math. As you noted, it can be surprisingly smooth work with students who want to be there, and frustratingly Sisyphean with those who don’t. I don’t resent the latter; I sympathize with their wishes to be set free. Alas, the mandate to calibrate coursework so that even the uninterested will put up with temporarily muddling through them forces the curriculum to be watered way down for everyone.
You’re completely comfortable trusting your financial future’s calculations to a template you downloaded off the Internet? It doesn’t bother you in the slightest that you won’t understand the mechanisms of how money works?
Isn’t this kind of ignorance that contributed to the recession? People not understanding the basics of how loans and interest rates work?
Well, I don’t really know what your background is, still. If you feel comfortable with algebra, then I don’t think introductory calculus is unreasonable, if that is a subject you are interested in. If you don’t feel comfortable with algebra, you will definitely want to patch that up before diving into calculus.
(I also don’t really know what your mathematical motivations are; it could be that there is some other area of mathematics which could be more appealing or useful for you to study right now. I could give better advice with more information. But if your motivations are simply (and not unreasonably) “I don’t know much math, but I wish I knew more. I’m keen to learn anything.”, then calculus is a fine way to go)
I have a bachelor of music with a minor in German, and a master’s in musicology. I’m in law school now.
During my M.A., I had a teaching supervisor position that had me working with grad students in math, engineering, and physics. The Math grad was doing all this quantum computing stuff, and in one of our lunch chats, he casually told me that even should i have aced all the stuff in high school grade 12 on, “that’s really still just skimming 18th century math. There’s a whole two and a half centuries of math you don’t do until you get to university.” I found that incredible to sink in; I couldn’t fathom being three centuries behind on all of literature, music, history, geography, etc…but my math skills are stuck in antiquity!
So yeah, my interest in math is as a dilettante, so I can’t really give you any practical applications where I could see myself using it. Perhaps this was part of the problem of motivating me in high school…
I wasn’t talking about myself. I was speaking in general of how people cope with “gaps” in their knowledge. My 2nd job out of school was programming software to calculate loan portfolios. Spreadsheet templates weren’t available back then so I had to use my trusty slide rule.
What I’m saying is that people know and learn the things they desire to know and learn and mentally outsource everything else.
Millions of people board airplanes without understanding the Bernoulli principle.
Millions drive cars without knowledge and theory of combustion engines or how their steering column rotation translates to the differential axle.
Corporate business planners depend on inventory optimization software even though they are not familiar with Hill Climbing algorithm.
Any piece of knowledge can be justified as essential for everyone. But billions of us just cope with the knowledge gaps and move on with our lives.
There are PhD economists that don’t understand how money works! (Or to be more accurate, the various PhD economists at different schools of thought disagree with each other how money actually works. )
I see. Then what exactly is the value of a high school diploma, if not a guarantee that students have been exposed to a wide range of subjects and have demonstrated reasonable proficiency in them? How am I to know if the person asking me for a job has a diploma that included studies in the things I need him to know, or whether he skated through taking basket weaving and tie-dying?
Furthermore, isn’t part of the ‘compulsory’ aspect of education a recognition that minors may not be the best judge of what they should be studying, and that the decision can’t be left to the parents, because there are too many uncaring/uninterested/absent parents?
You are supposed to graduate from high school with a base level of education in certain common topics in part so that you are prepared to go where life takes you without too many doors slammed in your face from the outset. A student who skates through high school taking easy subjects and avoiding the tough ones would quickly find out that there are many career paths and opportunities completely closed off to him. Woe to the person who gets the basket-weaving diploma and then discovers at age 19 that he has a burning desire to be an engineer, or a doctor, or a biologist, or a technician, or any number of career options that require more knowledge than how to make sure you don’t get ink on yourself when making that gnarly graphics pattern on your T-shirt.
Furthermore, what do you think will happen to the unemployment rate of people with only a high school diploma, when the diploma ceases to signal anything to employers other than that someone baby-sat you for 12 years while you played with your friends and tinkered around with what was ‘fun’ to you?
And here I thought we had these people called ‘teachers’ whose JOB it is to do that? If they can’t do it, fire them. Hire someone who will. Maybe it’s time to re-think the way we credential and employ teachers, and how we give them lifetime tenure and make it impossible to fire the incompetent ones.
That argument would carry more weight if it wasn’t for the fact that there are many schools that DO manage to teach students math (and how to write and read and do the other things that are required). It would also carry more weight if there weren’t at least 25 other countries doing a better job of teaching math in high school. Canada has much higher math scores than does the U.S., despite our population cohort being fairly similar and our kids being subjected to the same kinds of materialist and cultural pressures.
Wow, talk about moving the goalposts and inserting straw men. First, we’re not talking about making a herculean effort here - we’re talking about teaching the same math concepts that other generations of kids have managed to learn in the past, and which children all around the world manage to learn today. And we’re not talking about kids who are significantly mentally deficient. Special needs children get a different curriculum. We’re talking about the average high school kid, and what the expectations should be for a minimum level of mathematical understanding and facility from those children.
Here in Canada, we do have multiple math tracks. If you’re headed for the trades or other non-college career, you can get a diploma by taking the applied math track which focuses on arithmetic and life skills, or you can take the college-prep math track, and then you get everything up to and including basic calculus and linear algebra. You can also take the highest-level math track in IB or AP, and focus more on number theory, formal proofs, and the like.
ALL of those tracks require a significant amount of Algebra. There are no options for graduating high school without algebra. The simplest math curriculum you can take in Canada is the Apprenticeship and trade math program, and that’s not even offered anywhere. Here’s the curriculum: Canada Apprenticeship and Trade math program. You don’t have to solve quadratic equations, but there’s still heavy emphasis on statistics, geometry, slope and rate of change, equations of various line types, etc. All of which requires algebraic manipulation, algebraic reduction, and other intermediate algebra topics.
And if you study computing science in university, are you planning to only go into ‘line of business’ programming? Do you really think we should be graduating people from CS who are wholly incompetent in many fields of computing science?
This problem is near and dear to me, because I take part in the hiring of new people for our various programming teams. The quality of knowledge I’m seeing from them is abysmal. These days, they get exposed to many different languages, applications, and operating systems so they can check off all those ‘experience in..’ boxes on applications, but their basic knowledge is pathetic.
The other day I was reviewing some code by a new hire, and it had a lot of very convoluted logic in it, like this: if (Not A AND Not B AND Not C OR A and Not B OR C and Not (A or Not B). It was incomprehensible and buggy logic. I asked him if he’d tried to reduce the logic, and he said he didn’t know what I meant. I said, “You know, logic reduction. DeMorgan’s theorem, etc.”. He looked at me like I was speaking Swahili.
Or, a programmer who gets a job coding in a language that doesn’t have one of those awesome libraries available. Or someone hired to maintain legacy code which contains some of that math. Or a QA testing that kind of code and needs to know how to set up a test for marginal cases. Or someone who gets promoted to team lead and has to hire someone to write graphics routines and needs to know what to look for. Or someone hired by Apple to work on IOS 7, and is asked to improve the performance of the scaling algorithms. Or…
Yep. And in the programming field, it’s really hard to know what ‘domain’ you’ll be working in next year, let alone through the course of your career. Hence the need for a broad education in CS concepts.
There’s a very nice response to Hacker’s article by Dan Willingham, who’s one of the foremost experts on cognition and education. His short answer is that yes, algebra is necesary, and he has some pretty good arguments as to why. Take a moment to read it.
Again, you are simply pointing at the goal and ignoring the question of whether the defended means achieve it… Fire and hire all the teachers you want; where in the world is a near-universal rate of genuine mathematics mastery beyond arithmetic achieved among the general populace?
I gave a simple example of analysis of rational functions earlier. I don’t think there was any time in the past, and I don’t think there is any place in the world now, in which there would be near-universal ability among the general populace to give the same analysis. (Convince me I’m wrong, and that there is a country where I could pick ten random adults off the street and confidently expect most of them to fluently tackle such a problem). I don’t think this is because the problem is hard, and I don’t think is because many people are incapable of grasping it; I think this is because many people, not unreasonably, have no interest in it, and thus it doesn’t stick.
I expect a great many people are capable of a great many things which they have no interest in and no need to be pushed to do. I could eventually teach anyone to solve cubics, given sufficient time and their compelled attendance and practice, and anyone could teach me to play the piano, again given sufficient time and compulsion. But we’ve wisely decided these goals do not deserve the level of compulsion required.
You accept this at some level, of course. You’ve studied different things in your life than I did, and continue to do so as well. You’ve just settled on a different cut-off for the point in life at which we should accept people’s pursuits of different interests (perhaps college or post-college…).
I do appreciate the breadth I’ve received in my education. But little of the education I appreciate is in the form of classes I was forced to take which I didn’t have interest in and could not have learnt about later as my interests evolved. The classes I was forced to take which I had no interest in, I did not get much from which lasted beyond the final exams. This is a near-universal experience which we oughtn’t ignore the reality of, and there does not seem to me evidence beyond wishful thinking that this would disappear if only teachers were fired more stringently.
Like I’ve been saying all along, “algebra” can mean various things, about which I feel differently. If the notion of “algebra” you have in your head does not even include solving quadratic equations, it’s quite possibly not the sense of “algebra” education that I have any qualms with. I agree students should be made familiar with the basic concepts of arithmetic and reasoning about unknown quantities.
(The one thing on your list that gets my hair up is “equations of various line types”. I don’t know what “various line types” means, but so far as studying equations for lines goes, in the form in which I am inclined to interpret this (bleary-eyed students made to write down “y = mx + b” and “ax + by = 0” and manipulate these co-ordinate equations under translations and reflections), I could tell a story under which this would be of use throughout the average person’s everyday life, but I will instead simply acknowledge the reality that it isn’t.)
On another note, I would like to say I do not intend to hitch my wagon to Hacker and his particular arguments. For example, much as I don’t care for the standard mathematics curriculum, I’m never swayed by the idea that it needs to be replaced by, in his words, “citizen statistics”. This idea always seems to be motivated by the wishful thinking “Oh, if only everyone understood statistics, political discourse would be so much sharper… [i.e., everyone would agree with me]”. I am also superficially amused that the man thinks there is something called “Femat’s dilemma”.
I have always believed that math could be taught more effectively if used in practical excersizes. Building catapults is a good one, calculating g forces and energy storage is a good start. Once the kids get their feet wet and are not so intimidated by the math the learning process can really begin.
I would be happy if the average person’s math skills were good enough to count a till and know that if they are 6 cents off it probably makes more sense to recount the pennies and nickles rather than the dimes and quarters. Understanding the nutritional information on food labels seems to be beyond most people’s scope as does knowing the difference between weight and volume. I bet that 75% of the population would get it wrong if asked what the price of a $1 product would be if you raised it 50% one day and then cut it 50% the next.
I’m old enough to have used a slide rule, and the most important thing you learn doing that is you had better estimate the range of your answer first. Anyone who blindly uses a template and inputs a number incorrectly is in big trouble if he can’t figure out what the answer is more or less.
We were playing around with a retirement savings calculator the other day. Because I understand this stuff I was able to more or less derive the way they were calculating needed retirement income, which was bullshit. (Not a mistake - it encourages you to run to the financial services company making it available.) That gives you a lot more power than looking at a few different calculators, getting different results, and not understanding why.
The big problem with templates is that while they are fine for the exact problem the template was designed to solve, they are inflexible and you are stuck if you have a slightly different problem.
I see this all the time. Spreadsheets are fine, but someone wanting to do some calculations on a million lines of data (we have those) is stuck if all they need is a spreadsheet. A five line Perl script can do what is impossible with the wrong tools.
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