I know I’m late to the game here, but I can’t resist taking a moment to dive in to the furor surrounding yesterday’s appalling NY Times op-ed, “Is Algebra Necessary?”. (Yesterday was my birthday, and I just couldn’t face reading something that I knew would make me so angry.)
In case you haven’t seen it yet, let’s start with a quick look at the argument:
A typical American school day finds some six million high school students and two million college freshmen struggling with algebra. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal? I’ve found myself moving toward the strong view that we shouldn’t.
My question extends beyond algebra and applies more broadly to the usual mathematics sequence, from geometry through calculus. State regents and legislators — and much of the public — take it as self-evident that every young person should be made to master polynomial functions and parametric equations.
There are many defenses of algebra and the virtue of learning it. Most of them sound reasonable on first hearing; many of them I once accepted. But the more I examine them, the clearer it seems that they are largely or wholly wrong — unsupported by research or evidence, or based on wishful logic. (I’m not talking about quantitative skills, critical for informed citizenship and personal finance, but a very different ballgame.)
Already, this is a total disgrace. The number of cheap fallacies in this little excerpt is just amazing. To point out a couple:
- Blame the victim. We do a lousy job teaching math. Therefore, math is bad, and we should stop teaching it.
- Obfuscation. The author really wants to make it look like math is really terribly difficult. So he chooses a couple of silly phrases that take simple mathematical concepts, and express them in ways that make them sound much more complicated and difficult. It’s not just algebra: It’s polynomial functions and parametric equations. What do those two terms really mean? Basically, “simple algebra”. “Parametric equations” means “equations with variables”. “Polynomial equations” means equations that include variables with exponents. Which are, of course, really immensely terrifying and complex things that absolutely never come up in the real world. (Except, of course, in compound interest, investment, taxes, mortgages….)
- Qualification. The last paragraph essentially says “There are no valid arguments to support the teaching of math, except for the valid ones, but I’m going to exclude those.”
and from there, it just keeps getting worse. The bulk of the argument can be reduced to the first point above: lots of students fail high-school level math, and therefore, we should give up and stop teaching it. It repeats the same thing over and over again: algebra is so terribly hard, students keep failing it, and it’s just not useful (except for all of the places where it is)
One way of addressing the stupidity of this is to just take what the moron says, and try applying it to any other subject:
A typical American school day finds some six million high school students and two million college freshmen struggling with grammatical writing. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal? I’ve found myself moving toward the strong view that we shouldn’t.
My question extends beyond just simple sentence construction and applies more broadly to the usual english sequence – from basic sentence structure and grammar through writing full-length papers and essays. State regents and legislators — and much of the public — take it as self-evident that every young person should be made to master rhetoric, thesis construction, and logical synthesis.
Would any newspaper in the US, much less one as obsessed with its own status as the New York Times, ever consider publishing an article like that claiming that we shouldn’t bother to teach students to write? It’s an utter disgrace, but in America, land of the mathematically illiterate, this is an acceptable, respectable argument when applied to mathematics.
Is algebra really so difficult? No.
But more substantial, is it really useful? Yes. Here’s a typical example, from real life. My wife and I bought our first house back in 1997. Two years later, the interest rate had gone down by a substantial amount, so we wanted to refinance. We had a choice between two refinance plans: one had an interest rate 1/4% lower, but required pre-paying of 2% of the principal in a special lump interest payment. Which mortgage should we have taken?
The answer is, it depends on how long we planned to own the house. The idea is that we needed to figure out when the amount of money saved by the lower interest rate would exceed the 2% pre-payment.
How do you figure that out?
Well, the amortization equation describing the mortgage is:
Where:
- m is the monthly payment on the mortgage.
- p is the amount of money being borrowed on the loan.
- i is the interest rate per payment period.
- n is the number of payments.
Using that equation, we can see the monthly payment. If we calculate that for both mortgages, we get two values, 
But whoa…. Quadratic equations!
According to our idiotic author, expecting the average american to be capable of figuring out the right choice in that situation is completely unreasonable. We’re screwing over students all over the country by expecting them to be capable of dealing with parametric polynomial equations like this.
Of course, the jackass goes on to talk about how we should offer courses in things like statistics instead of algebra. How on earth are you going to explain bell curves, standard deviations, margins of error, etc., without using any algebra? The guy is so totally clueless that he doesn’t even understand when he’s using it.
Total crap. I’m going to leave it with that, because writing about this is just making me so damned angry. Mathematical illiteracy is just as bad
I agree with Arthur Benjamin’s brief Ted Talk about math literacy — we should be making Stats the pinnacle of HS math, rather than Calculus. Calc is awesome, and powerful, and really opens up the world through a mathematical lens, but stats is far more universally useful and would benefit humanity more if it was mandatory.
That said — algebra still fits into that equation (pun not intended).
Plus, algebra also gives you a solid foundation in abstract thinking. I still remember struggling with the idea of “the portion of a unit circle that is not ‘x’ is ‘1-x'” in 7th grade Algebra, and how happy I was when I finally grasped that, and later concepts.
Statistics, in the sense that you are thinking of, is not a replacement for calculus. It’s a scientific discipline, unless you are referring to the mathematical discipline of statistics, which requires knowledge of probability theory, which requires knowledge of calculus.
Discrete probability theory does not require calculus. And it’s shockingly under-emphasized, even for college math majors.
It does, however, require algebra.
I have a PhD in math and I have taught a couple of cross-disciplinary course, so I feel like I need to contribute.
I think, discrete probability and statistic are very good classes to take for people who are interested in doing well in undergrad social sciences (econ, finance) or “soft” natural sciences (like bio). Good understanding of calculus might be required for graduate work in these fields. The reason why calculus is a pinnacle of HS math is that it combines and (in some ways) unifies algebra and geometry.
Of coure, I completely disagree with NYT article. I have experienced HS math in US, coming from one of the former soviet union republic. From my experience, the problem is with the teachers. I have seen a lot people who have never taken math in college teach math in high school. There are a lot of logic behind the techniques in hs algebra and geometry which teachers (and consequently the students) just miss completely. I suspect that’s why a lot of people find math so difficult.
Last note. Probability theory is usually taught as uppler level math undergrad (or low level grad) course. It requires a lot of “calculus”. But some calculus can certainly help with understanding of basic statistics.
Wouldn’t it be nice if journalists (if this guys fits into that category) were required to have an understanding of a subject before they were allowed to write about it?
I may have googled the wrong ‘Hacker,’ but I believe he’s a political science professor, not a journalist.
This, of course, makes his editorial all the more offensive, as he either doesn’t use statistics when his field demands it, or he is subtly arguing that the peasants don’t need to know certain things to do their jobs.
The more I reread the piece, and the responses to it, the more I think that Andrew Hacker doesn’t quite understand what “algebra” means.
If his argument was about calculus, rather than algebra, it would make a little more sense. Only those who work in a mathematical field (science, engineering, economics or what have you) really need to know calculus.
For everyone else, we’d be far better off if we laid off the calculus and taught them more statistics. Not everyone needs to be calculus-literate, but everyone needs to be statistically literate to a certain level of competence these days.
The calculus was first published in the later years of the seventeenth century, about one hundred years after Shakespeare’s works were published. The significance of its discovery or development to modern learning and society is inestimable.
One might readily make the argument that without knowledge of the calculus, a person today is really not fully educated, because one would do so if a individual was ignorant of Shakespeare.
The calculus was first published in the later years of the seventeenth century, about one hundred years after Shakespeare’s works were published. The significance of its discovery or development to modern learning and society is inestimable.
One might readily make the argument that without knowledge of the calculus, a person today is really not fully educated, because one would do so if a individual were ignorant of Shakespeare.
For another perspective on why we suck at teaching math: http://www.ams.org/notices/200502/fea-kenschaft.pdf
Just ignore the bits where the author is busy straining his arm to pat himself on the back about how good he is at striving for racial equality.
TL;DR version: we fail so hard before kids get anywhere near algebra that by the time they get there there’s nothing that can be done.
Thank you for this. It was a good read. It’s a bit funny, I was just talking with someone about the need for improved elementary math education at a very fundamental level.
I will pass this article on to others. Thank you again –
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I agree with everything you’ve said, but I also agree with this guy: http://mikethemadbiologist.com/2012/07/30/its-not-the-algebra-its-the-arithmetic/. How can people expect to master algebra if they can’t do arithmetic?
To those responding that we shouldn’t have the math curriculum lead up to calculus, but to statistics instead: You do realize that statistics is based on probability theory and that probability theory requires a foundation in calculus right? Just putting that out there.
@Aaron H
I disagree with your argument for two reasons.
First of all the pinnacle of high school science is physics, the class is usually the senior year class and that class requires calculus to be properly understood. Any high school that doesn’t get to calculus at least by the time it is teaching physics is, I think, failing its students.
Second, the basis of statistics is calculus. Much of statistics is derived using calculus. The statistics of a continuous variable needs calculus, as integrals abound. Fitting functions using least squares methods needs derivatives.
I found math easier to learn when we learned it in the order that you build it up. With arithmetic you build algebra, with algebra you build calculus. To build statistics, you need calculus.
One thing that bothers me about this article is that it is another example of the open contempt that many people with humanities degrees seem to have developed towards math and science. If a STEM graduate were unable to read, their illiteracy would likely be a shameful secret, and were it to become widely known, it’s unlikely they would be considered an educated person regardless of their abilities within their actual field. Yet our society is perfectly comfortable with supposedly educated journalists flaunting a shocking degree of innumeracy. Some of them almost treat it like a badge of honor. The fact this attitude is so common in our academic communities should horrify us.
You know what’s even more frustrating? For STEM majors taking a college composition course and never being taught how to write a STEM research paper. Thankfully, I’m past the point of college composition, but I never liked having to write research papers where the topic had to be something completely unrelated to my interests.
I find it interesting how, in that context, there’s the double standard of difficulty in the fields. Many insist that STEM and humanities are equally challenging, yet oddly STEM majors have to achieve at least a moderate competency in the humanities (how many STEM majors have to pass a history or writing class for GE?). But the humanities majors don’t have similar STEM requirements levied on them, and as you point out they often see no problem with their lack of understanding.
Yes, they do. Most colleges do require non-science majors to take some science GEs, typically including both life and physical science. And the amount of writing that humanities majors have to do is substantial.
Yes, but many of them offer alternate science classes that STEM majors cannot take for credit, at least not towards their major. I’ve even taught a few of them. They’re frequently touted as overview courses, and to say they merely skim the surface would not be an understatement. Still, I think they’re a good idea because, depending on the instructor, they provide some scientific literacy in our scientifically illiterate society. My students were required to not only cover the mandated competencies that are not under my control, but they were required to examine various related issues that can affect their everyday lives, or that are prominent in the news on a semi-regular or regular basis. Not all faculty are like me, however, as I discovered when I got a look at another faculty members PowerPoints for the same class.
We had those at my university too, with colorful names like “Dinosaurs and disasters,” which fulfilled the science requirements for non-science majors. They seemed like fun courses (I didn’t get to take them as a science major), but ultimately rather empty.
How can any of you even suggest that we need not teach our youth calculus!!!……the biological sciences are finally poised to come up with a true quantitative understanding of physiologic and molecular function that will certainly not be made by Americans with this attitude!! (BTW we rank 23rd in math on the last survey of OECD countries academic performance)….let alone the physics chemistry and engineering sciences which already have a reasonable appreciation for the need for calculus and differential equations…..appalling….
Agree with most of the above. Topics in maths can’t be easily isolated, and the problem is in the foundations. Maths are not useful to some people because they don’t know much and can’t apply them, and they mistook the situation believing that it’s maths which are useless. The solution is of course education, not ignorance.
I love changing subjects as a way to show how the formal structure of the reasoning is wrong, without even needing to evaluate the truth of the content. I liked it.
Happy post-birthday, by the way 🙂
“Would any newspaper in the US, much less one as obsessed with its own status as the New York Times, ever consider publishing an article like that claiming that we shouldn’t bother to teach students to write? “
Nope. However, several such newspapers could, and probably have, published articles claiming we should rid the world of the 5 paragraph essay though. Or de-emphasize Shakespeare. Which is far more closely analogous to what Hunter is suggesting. Proposing to remove algebra as *the* essential required mathematical hoop for all students is not advocating abandoning all attempts at fostering quantitative literacy.
Look, to be honest, if I wanted to solve your mortgage problem I’d look up a calculator for it on the internet. And no offense, but if you are trying to make the case algebra ‘isn’t so terribly hard’, your description would not be how I would go about it (side note: WHY do mathematicians and scientists so commonly put the equation FIRST and THEN the explanation of variables?)
Nom de Plume- go read some Leon Lederman. The order in which we present material can be rationalized in various ways. I suspect different styles are easier for different teachers, and better for different students. Nothing I learned in calculus was worth a lick of anything except for that we reviewed in physics- for a student like me, having them integrated into one course would have been ideal, but co-requisites would work better than pre-requisites. I do acknowledge that might not be universal, but I think it’s underappreciated how many people it applies to.
Actually the point is valid. Algebra is a necessity if you want to comprehend geometry, trigonometry, calculus, statistics, biology, chemistry, physics, engineering, finance, accounting, computer science and even logic. It is much more fundamental than reading Shakespeare or writing a form of essay.
Also, the “just use a calculator” excuse is great. Put in a typo or experience a flaw in the program and your number could be off 10- or 100-fold or even more. I’ve seen students (college-aged) do it all the time to get ridiculous answers, but because their calculator or computer told them that it was the correct answer, they believed it. Or, as many people have experienced, at a store you give the clerk some money and he gives the wrong change because that’s what the register said to give.
Being able to check yourself or understand what is going on gives you a great advantage when it comes to life.
First, to be clear- I would argue against algebra classes as THE crucial make-or-break point of high school education, not against all teaching of algebra.
Algebra is very handy for intro chemistry (at least the way it’s typically taught), and bits of it are useful for trig. Bits of trig are useful for navigation of ships (that’s why it was invented after all), and very small bits of trig are useful for calc. Bits of algebra are also critical for calc, which is useful for physics. But until I got to the physics, I didn’t care about the rest of it well enough to get particularly good at solving problems. Teaching mathematics in isolation from everything else is dull for students like me (imagine if every writing assignment were to write only about English usage). One option would be to integrate it more into other subjects (writing across the curriculum, just for math).
That said, although algebra is broadly useful, it’s not quite so broadly as you state. To quibble- geometry is often taught before algebra, and the Chinese were doing geometry 300 years before the Islamic world was working on algebra. What you’re saying is like claiming one “needs” Hemingway to understand Shakespeare. It’s historically backwards. I’m not shocked comprehension of history is apparently unaided by algebra, but I am a little surprised your relatively low comprehension of possible causality.
Also, I have a PhD in molecular medicine. I’m pretty sure I comprehend biology adequately to opine on it (also, much of relevance in chemistry). Being able to *do* algebra has been of almost no value to me.
On the other hand, being able to *understand* algebra has sometimes been valuable for me, I just didn’t need a class for that. I left school when I was 11. I enrolled at a community college when I was 14 precisely because I didn’t think I could teach myself algebra. Turns out? I almost tested into college-level precalc. What’s covered between fifth grade and college is pretty easy for a bright 14 year old to get intuitively- no instruction needed. You can use good math basics and logic to do many of the same things (in a much more cumbersome way) that the algorithmic procedures taught in most algebra classes accomplish. That doesn’t mean it’s not worth it to learn how to do algebra problems, but there’s a good chance it won’t ‘stick’ with students unless they actually want to solve those problems on a regular basis. Which is frankly unlikely unless they are engaged in science or engineering (or perhaps accounting/computer science- those aren’t my fields, so I don’t know how essential they are. I do question how many high schoolers are using algebra for them, though).
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I find myself making a similar, although more specific, argument as the author quite regularly. The way that algebra is currently taught is worse than useless. Algebra is currently taught so that students can manipulate patterns of interest at a later date. However, being taught a way of manipulating patterns is useless without access or interest in those patterns. This is disastrous to student interest, effectively destroying any natural interest in mathematics without any benefit to mathematical ability. Therefore, algebra as it is currently taught should be removed from the curriculum. It should either be taught for the sake of its own beauty or it should be delayed until it is necessary for a purpose.
The latter is how things already are, which is obvious to anyone who has taught lower level physics courses. Most students don’t have enough algebraic ability to be useful despite years of algebra based math courses. We should stop wasting all that time which could be put to better (hopefully mathematical) purposes.
Think that should be an n, not a 2, as the exponent in the numerator.
Making algebra in high schools optional (I assume that is what he wants, abandoning it completely is even worse) is a horrible idea.
Students are lazy and algebra requires some discipline and work. Furthermore, students (and parents to a lesser extent) are completely incapable of judging what skills they may need later on in life. Basic statistics for socials sciences still requires an ‘x’ occasionally, to the great surprise of students. If all these students had dropped algebra, reasoning they would not need it, the end result would be colleges being forced to teach (even more) high school material.
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From Hacker’s conclusion: “Yes, young people should learn to read and write and do long division, whether they want to or not. But there is no reason to force them to grasp vectorial angles and discontinuous functions.”
My response: http://www.quickmeme.com/meme/3qaokc/
I think one could have a reasonable discussion about altering the math curriculum to teach perceived crucial skills first, etc. But the thing that really bothers me about Hacker’s argument is that it basically boils down “Algebra is too hard for the masses”. Not only is this profoundly elitist (which rubs me the wrong way), but it also sends the message that hard things are not worth doing. Many children (and, sadly, adults) already have this tendency. I see it in my college students everyday. Hell, I see it in my graduate students.
I expend a great deal of effort teaching my daughter exactly the opposite – that many hard things are worth doing. And I would never ask her to give up the pride she feels when she puts effort into something hard and comes out being able to do it, and even excel at it. That is an experience everyone should have and, frankly, probably be forced to have several times in their schooling. They should have this with athletics, music, reading, writing, science, AND math because they are all different ways of interacting with the world. The details – Hamlet vs. Moby Disk, statistics vs. calculus – are probably less important.
From the article: “But it’s not easy to see why potential poets and philosophers face a lofty mathematics bar.”
Being a math major in undergrad, there were some axiomatic set theory classes in our philosophy department. They taught (and quite rigorously so) ZFC. Of course, learning this requires a high degree of symbolic logic and proof writing.
I would imagine that (all?) philosophers in a 4 year institution have to learn symbolic logic to some extent. I would also image this would be very difficult without a full understanding of symbolic manipulation that comes with algebra.
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I’ve never understood the fuss made about math. It is easy, if you have decent teachers and if you put in hard work. The same is true for any other subject – science, language, geography, you name it. If you don’t put in effort, you are going to suck anyway.
I suppose the author of the NYT column would like to dispense with education altogether. After all, nobody is going to use everything they learnt at school. That’s why it is called basic education because you don’t know beforehand what you are going to be doing later and what you are going to need for it. To stretch a point, it’s like saying that we don’t need to teach our kids how to walk, run or ride a bike because they are going to drive a car when they grow up anyway!
There is a considerable math complex in the US and also in Europe where people think you need some sort of God-given gift to be able to do basic math. Get real people! Basic math, high school math are easy and anybody can do them and master them with a little bit of work. Just take a look at other countries in Asia where nobody believes that math can’t be learnt or girls can’t do math. No wonder they all can do math as a result.
We are not expecting all our high school students to win the Fields medal, are we?
As a mathematician and someone who has taught these subjects I fully agree with the NYT point of view.
First, it is simply sloppy thinking (if not prejudice) to refute an argument by indignation (how can you even think of not teaching math) or simply listing a couple insulting suggestions as to why your opponents might unreasonably take their position.
Teaching is a zero sum activity. Time spent on one subject is time not spent on another. More time learning algebra is less time in programming class or shop or learning writing. The burden isn’t on the doubters to show that algebra is worthless, it is on the advocates of teaching algebra to show that it gives a greater benefit than other studies it competes with. Frankly, it drives me nuts that mathematicians who in most respects are so scrupulous about unjustified inferences think this issue is settled by a few vague remarks about the benefits of math and the unthinkability of not teaching a particular subject. If algebra is truly a worthwhile subject to teach to everyone (surely some people need it) then ultimately it’s a claim that needs to be backed up by a strong theoretical argument or better yet, empirical data.
Now, having said that I will present some personal experience that makes me doubt the benefit of teaching algebra to everyone. I agree this does not amount to a demonstration that algebra should be removed from the standard curriculum but it is surely enough to cast the issue into doubt and show that data (or theory supported by data) is needed to resolve the question.
First, almost all algebra is learned entierly by rote. The abilities of reasoning and inference that mathematicians so prize tend to be banished during algebra lessons not reinforced. Worse, students learn to hate math as a result and view it as boring rote work (yes, I have really known students entering elite colleges who thought mathematicians sat around adding up big numbers).
Sure, you might say that this is because the math is taught poorly but that, frankly, is bullshit. No matter how you teach algebra (at least) half the students will feel they suck at the subject…no amount of encouragement can overcome our natural human inclination for comparison and you can’t stop students from realizing when they have a harder time doing algebra than their friends. None of us like to work creatively or try things that might fail (esp publicly) when we don’t think we are good at the subject. So try as you might you simply can’t both instill knowledge of algebra in most students and teach from an abstract point of view that requires student reasoning and inference not rote memorization. No matter how gifted the teacher you either give up on a significant fraction of the class mastering the subject or you break down and give them rote rules.
Besides, it’s silly to assume that great teachers are an infinite, cost-free resource. There just aren’t enough great math teachers to go around at any affordable salary so even if (which I doubt) you personally can somehow motivate every student to learn algebra by logic and inference most students will inevitably learn it by rote.
Learning algebra by rote would be fine and good if most students needed the skill but whenever I talk to professionals (lawyers, doctors, etc.. ) who have been out of school for more than a few years they can’t remember the first thing about algebra (with a few exceptions). Likely the situation is even worse in fields where less advanced math is required (such as service jobs). So at the end of the day we have students who spent a bunch of time memorizing some rules for manipulating lists of symbols which they then forget. We might as well have them memorize the rules for phrenology for all the good that is doing them.
This isn’t to say that I think quantitative learning should be totally abandoned. Personally, I think algebra should be replaced with computer programming. For starters, it is a practical skill for far more people while still emphasizing quantitative thought (and even rules for arithmetic…perhaps even the solving of algebraic equations can be smuggled in). Secondly, feedback is immediate and comes from an impersonal machine which encourages more trial and error and minimizes the disincentives for trying to think out the problem for yourself (the computer won’t laugh and you can erase it before anyone sees).
I don’t expect you to be convinced but I do hope I’ve raised enough doubts that you agree that real data is needed to settle this question.
Computer programming seems to me an excellent example that defeats your whole argument, Peter. How on Earth can you teach even elementary programming techniques without any grasp of algebra? Without ever coming across the concepts of functions, variables and constants, order of operations? Try to teach this stuff on the fly in the computer lab if you want, but you *will* have to cover it, and it seems to me you’d be going about it the hard way.
In fact I’d go even further and say that without a few years using the basic principles of logic that algebra exposes you to, you’d have a nigh on impossible task getting students to understand the same concepts formalized in code.
As a programmer, I can say for sure that programming requires a lot of algebra. If you can’t deal with a variable in algebra, you can’t deal with a variable in code. Now if you want to argue that the classes should be combined, I might agree. Programming as a motivation for algebra might work. (Everyone wants to make games, but no one has any idea how hard that really is, or how much math is involved. 3D games especially require a ton of linear algebra, which of course requires regular algebra.)
Really? So you are saying you don’t know anyone who learned to program before they learned algebra?
I did. I know plenty of others who did as well. Sure, advanced programming ends up using algebra but that is to the advantage. It is learned as needed in a context where students see it’s use and importance.
Well they learned coding, not programming. Memorizing a computer language is not programming.
Teaching is not a zero sum game. Time spent on one field helps across all fields to some degree.
That doesn’t undermine my point. It’s still the case that given a fixed block of hours in which to teach every minute spent on one class is a minute that can’t be spent on another.
The fact that increasing the total amount of time spent learning in class/homework by increasing the amount of time spent in say math may benefit learning in history is irrelevant. Presumably, we want to discover the optimal amount of time per day for students to spend studying and then divide that time up appropriately between disciplines. Once the total amount of study time has been fixed that division is zero sum.
Let me add that merely thinking that all educated persons should know algebra is an incredibly bad argument.
Go through history and look at what we expected all knowledgeable persons to know at various times. Widespread latin teaching (esp at elite schools) survived far to long into the 20th century and there has been far too much time spent making students memorize (mostly arbitrary) biological classifications because it is what educated people should know. The same can be said for the time spent reading Heidegger, Freud and other great thinkers who are read not because they give the most up to date information on the subject but simply because they are famous for thinking it up first (in math we don’t go back to Newton’s original proofs we read the modern improved versions and it is no different in other fields). Hell, it wasn’t that long ago that everyone at our greatest institutions of learning was taught christian theology.
One final point:
You suggest that algebra is quite useful and really not that hard. That’s great from a subjective point of view but I suggest that as a matter of on the ground fact the overwhelming majority of Americans (even college educated Americans) couldn’t perform such a computation on their own. Something you can’t do doesn’t provide any benefit no matter how useful or easy.
Given that some of the brightest minds in both math and education have been laboring since the introduction of the subject to improve general math education it seems highly doubtful that massive improvements can really be expected given reasonable resource constraints. So unless you have data which contradict my (admittedly anecdotal) claim that most americans quickly forget how to do useful algebra or a master plan to finally do what all those past math reformers couldn’t why should we believe that the vast majority of students who take algebra won’t ever perform those useful tasks you mention (even if they remember it is with distaste and they avoid doing it with the same enthusiasm they would avoid doing taxes if they could).
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<3 Peter.
To add to his many good points, I would just bring up again the girl mentioned in the original NYT article. She's 17, working at subway for minimum wage, and she's about to flunk out of high school. People are saying "how will she ever become a scientist or an engineer without learning algebra?". Well guess what, SHE WON'T. Her chances of getting those jobs are about the same as my own chances of being drafted by an NBA team : zero. The example this article uses ("she can use algebra to refinance her mortgage!") is similarly laughable.
There is no job she could feasibly get that would require algebra. Even with a high school degree she's going to struggle to make ends meet, but if you insist on flunking her just for not knowing algebra (kind of hard to study when you're also spending all your time working fast food btw) then she will likely never get a job much better than minimum wage.
The thing is that you can apply the same argument to virtually everything on the curriculum. You can do a minimum wage Subway-type job without anything taught after grade 5 or so; the remaining grades are therefore useless to her.
If that’s actually Hacker’s argument, then he’s done a bad job of expressing it by focusing on just one of the subjects, rather than including things like English Literature (Shakespeare et al), he’s suggested it’s about a single subject that’s too hard for the majority of students.
It depresses the shit out of me every time I hear another mathematician say that it’s OK to not expect everybody to have basic mathematical compentency because he believes (also without any support by good objective data) that some large fraction of the population congenitally and irreversably sucks at math. You should be ashamed, and I feel sorry for your students. It stuns me that you don’t recognize that your attitude contributes to the pathetic situation that makes idiots like Hacker seem credible.
I actually enjoy teaching high-school level algebra even though I’m “overqualified” for it because I strongly believe it’s exactly the students who have been failed so badly by crappy and incompetent mathematics teaching in the past who deserve good teaching now. What’s bullshit is the statement that algebra needs to be taught by rote or needs to be learned by rote unless you have some special talent for it, which, by your estimation, describes at least 50% of the population.
I’m not going to touch your “is it useful” argument because that’s been done elsewhere. But I will say that Latin was for centuries the universal language of the educated, and that’s why it was taught. It has since been supplanted by English, which is taught for the same reason. But algebra has been rendered obsolete by what? Programming? “Citizen statistics”?
Can a tone deaf person learn to sing opera?
Isn’t it at least possible that some, and perhaps many, people lack the ability to think algebraically?
I completely agree with Elizabeth. It still surprises me how can a teacher believe that some people can not learn. Really.
In any case, I really can’t see any real argument in Peter’s post. He says people don’t like algebra, but probably I can safely say that most people don’t like to study hard (you know, it’s hard), so no argument against math here, maybe against education.
I don’t believe that elementary algebra is learned by rote (and, as most people here, I once were a student and spoke to my fellow students about these things), in which most of the post is based. Again, that’s more an argument against education in general (history is learned by rote, more than maths, for example), not specially about math.
About substituting elementary algebra with programming, what kind of serious programming can you do without elementary algebra? (We’re not speaking about advanced algebra here). I believe that Peter’s perspective is just too narrow: he researches computability, so it’s cooler than anything else. I still prefer my children to know algebra than Prolog or Scala. Better both.
Peter’s most potentially interesting idea (as far as I can see, of course) is the zero sum game argument, and maybe programming is more useful. But probably most people here can probably argue by themselves why elementary algebra is useful for programming (we’re not speaking about advanced algebra here). What kind of serious programming can you do without algebra?
Peter’s explanation about programming to be better because it’s impersonal, it’s evaluated by a computer which doesn’t embarrass people, etc. is fallacious, because he is not speaking about a topic. I can also teach algebra with an trial-and-error play-based approach. Instead of giving a boy BASIC (does BASIC still exist today?), I can give him DERIVE (does DERIVE still exists today), and he can learn algebra without anyone embarrassing him, playing, trying things. In fact, I can teach any subject this way. You may be a not very resourceful teacher, but you should not mistake the method with the subject. I can also teach programming by giving my audience a very boring theoretical lecture about preconditions, postconditions and invariants. Believe me, they will also hate me as they hate you for algebra. So no argument here.
So, as I said, I can’t see any real argument there.
With respect to Argh, I still believe that a mathematical education can help the girl to find a better job. But even if you lost hope in her, a lack of education will not only tie her to those low-paid jobs, but also her children. If he hasn’t any opportunity, maybe her children have.
But my main argument here is, those people who don’t wanting math to be universally taught, would you be happy if your own children don’t learn math? Can I safely guess that Peter wants his children to learn math? I’m pretty sure you’re never speaking of your own children when you say math should not be taught to everybody. Understand that I’m not appealing to feelings here. I’m speaking about equality of opportunities.
I just have to ask you one thing.
All of these discussions in support of teaching algebra to everyone are made by people who do understand algebra.
What is if were actually true that a substantial portion of the population were actually incapable of learning algebra?
I teach high school chemistry. I used to teach a class for those kids who have very low math skills, and I can teach the basic concepts of chemistry with almost no math. But when I shifted to teaching honors chemistry, I ran into those kids who are fairly bright, but had very poor math skills. They literally cannot solve PV=nRT for T. And this is not just a few kids. In some honors classes it was more than a third of the class.
What if that is an inherent (and rather common) disability? Would that change your position? Should otherwise bright kids who just can’t get algebraic thinking be excluded from success in fields that don’t really require understanding math?
What makes you think that there can be people who can go to school and can’t learn math? O_o
It sounds to me as if you ask me to consider that maybe there are people who can go to school but can’t learn their native languages, and maybe we should consider taking this fact into account when designing our children education. This is, completely absurd.
English (or any natural language) requires more a lot more neurons than solving some stupid elementary problem in algebra. Really. Basic math is a looooooot easier than irregular verbs, grammar, syntax, phrasal verbs, conjugations and all that. And any four year old boy can speak acceptably well.
There is an alternative hypothesis to thinking that maybe some people have their brain incapable of solving for X. That those people are not motivated, have a bad foundation, have bigger problems in life, don’t see the usefulness, are told that they’re incapable of learning, believe it’s of no use, are more interested in other things, have to work and have no time for learning, and a long list of possibilities. I don’t understand why people always bring the “maybe some people are incapable” (implying that there are some kind of supermen) when there are so obvious explanations.
I would argue that English grammar is a lot easier than algebra. But you see, I have always had a knack for verbal thinking, and algebra did not come so easily to me. In fact, contrary to most other students, I found geometry much easier than algebra. I think people who are themselves good at mathematical thinking need to realize that it may not be that way for everyone.
If we can get more students to master basic mathematical skills with better instruction, I’m all for it. One step might be to require at least a year of college calculus for all teachers, even pre-K. Another might be to take all of the calculators out of the classroom until students can do basic arithmetic operations, at which point the calculator is the convenience it ought to be and not the crutch it has become.
But there are certainly people who are incapable of learning to sing on pitch, for instance. Other’s like me, could have learned to sing much better if we had been better motivated, or trained, or whatever. I really believe that’s the case with mathematical thinking. Of course we can do better with most of them, but I really believe that some will just not get it.
But speaking as a high school teacher, I can’t address the inadequacies of students’ preschool environment, their inadequate elementary education, or the factors in their lives that make learning academic tasks so hard. I wish they read something worthwhile instead of the young adult crap that passes for literature. I wish their middle school teachers would quit telling them that the blood in their veins is blue, but I have to deal with what they send me.
ISTM that the best course of instruction gives all students the best outcomes. The very brightest can excel in whatever they are good at, and and more limited students can do whatever they can be encouraged to do. What we have now is an environment where some of the students with great potential do not learn, because instruction has been geared to each and every student learning before moving on. The kids I was talking about (who can’t solve for a variable in a simple equation) had all passed “algebra II.” It is a prerequisite for honors chemistry. Clearly the instruction they got was not what I would call “algebra II” level.
I strongly suspect that the insistence that every student pass is the reason the standards are so low. But if the standards are high enough to encourage the best results for the brightest students, then some will not meet them. Do we relegate them to the garbage heap?
Do you really believe than learning a language is easier than learning to solve for T in PV=nRT? O_o
Hell, that’s an odd assertion. I didn’t expected that. Do you claim to be able to learn a new language faster than elementary algebra? How many languages do you speak?
Anyway, it doesn’t matter. I still have to know a normal person who can’t learn their native languages. Even those who “better at math”. It would take you a lot of evidence for you to convince me that some people can’t learn to solve for T in PV=nRT. A lot. I’m willing to make a “my fair lady” kind of bet.
It also surprises me a lot the conclusions you extract from the situations you present me. Really.
You, as a teacher, have to deal with students that, after passing Algebra II, can’t solver a variable. Of course, you can’t teach them chemistry. And you believe the problem is that someone let them pass Algebra II.
Hell, that surprises me. I believe anyone would say that the problem is that nobody in that Algebra II class taught them to solve for an X in a simple algebraic equation.
You excuse the teacher by saying that maybe those people couldn’t learn how to solve an X at all. Solving an X from a simple algebraic equation is something completely mechanic. So you suggest than it’s impossible for some people to learn to solve T from PV=nRT! And you suggest the solution is to leave these children behind.
You may not like it, but someone have to tell you the truth: you’ve got a terrible Algebra II teacher in your institution XD. He probably deserves imprisonment XD Come on, PV=RT. That’s quite a bad teacher, despite the students.
From the infinite ways to teach Algebra II, from where do you infer that it’s the children who are wrong?
From my perspective, the solution for the low standards is not to do nothing and just leave children who don’t learn behind. The solution is to actually do something to fix what prevents them from learning. I believe it’s society responsibility to provide every child with an environment right for education. It can be difficult, but that’s why we build schools and pay taxes, isn’t it?
Motivate children. Children work best with immediate rewards. Most children have no clue about why they’re taught math, so they’ve got no interest in them. No interest, no learning. I couldn’t learn Klingon, really, because it doesn’t interest me. Make quality lectures adapted to their level and interests. Make people write interesting books. Explain them the usefulness of education. Show them cool science, cool history. You know, teachers shouldn’t be just a passive filter. They’re expected to actually do something. Like teaching to solve PV=nRT.
But it’s not only teachers, they can’t do much. Build more libraries. Promote reading. Provide school material to those lacking it. Intervene in problematic families. Give them a good environment. And don’t be defeatists.
If you just filter students in a good situation for learn, you’ll perpetuate ignorance. Well educated people will have better resources and better motivation to provide their children with the right conditions. If you give every children equal opportunities, the average education will rise, despite slowly.
So, in the end, it’s just a question of small government versus big government. But at least don’t blame math, like the original article. We should want children to learn math!
“I would argue that English grammar is a lot easier than algebra.”
paging Dunning and Kruger…
I know how we can settle the question about difficulty.
I could write a (admittedly very long) post explaining the algebra required to solve equations like PV=nRT (that a variable is a number you don’t know, that an equation means that the left part is the same thing than the right part, and that if you do the same thing on the left and the right part, you’ll end with the same things again; but in a longer explanation). That is, what your fellow teacher couldn’t teach your students.
And you could write a post (maybe a bit longer) explaining what every child can learn: English XD
Then we can compare the relative difficulty by the number of rules and concepts one should memorize. But you should do something else, because every child can learn English 🙁
Oh, wait! There’s a better way. I could program a computer to solve simple algebraic equations (I feel able to) and you could program it to speak English! There are several possibilities.
(I know a child is not a computer, but come on! You even can check if your solution to an equation is right).
If learning English is not easier than learning algebra, then how come almost everybody learns their native language without any explicit instruction at all and nobody learns algebra without instruction?
Our brains are set up to communicate and relate within a social group. Here’s an interesting example of how we handle logically identical problems very differently if they are in a context our brains are set up for, btw.
All of us have brains set up to learn languages. My claim is that not all of us have brains set up to learn algebra.
Yes, the algebra instruction is lousy, although I would blame the goal of making every student pass a standardized test more than lousy teachers, not that I don’t know of a few of them as well.
Can’t nest replies any deeper…
Everybody learns their native language without (much — to say “any” is willful blindness) explicit instruction because they’re immersed in it from day 1. If people had nearly the amount of environmental exposure to algebra from birth, they’d pick that up too.
Arrrghh, again I didn’t expect this. You insist on that idea. I believe you’re a bit lost here, John. Truly, because the answer is completely obvious.
We learn English and not algebra because we’re exposed to at least 12 hours every day of people speaking it every day. Even then, a child needs several years to get it right.
I can guarantee you that if you expose a child since its infancy to algebra for 12 hours a day, he will learn it. Most children get it (algebra) in a year with just a few hours a week. That’s a lot less effort than algebra.
And I’m not even talking about motivation here. You’re saying that learning English is easier than algebra even knowing that a child needs a lot more hours with English than with elementary algebra.
I can’t believe this is not absolutely obvious.
The proof? Try to learn a new language without that kind of exposition. I can tell you: I studied Japanese for four years and I can only speak a little of it. My English is still horrendous, because I don’t practice it much.
Again I ask you: how many languages do you speak?
Again John Armstrong was faster and clearer :S
” Here’s an interesting example of how we handle logically identical problems very differently if they are in a context our brains are set up for, btw.”
By the way, you’re making up the conclusion! The link you cite does not say or suggest that.
Obviously, people think faster about common day situations. The link you cite says that we should use that to teach math. Just that.
Reread it, please.
About the link:
When you tell the children the rule about alcohol and the 21 year limit, you’re telling them something they already have assimilated. They know and understand the rule, having interiorized it before you test them. They don’t even need to hear you, you could just tell them the situation and you they would tell you who’s wrong.
With cards, you’re telling them a new rule they don’t know. They have to think about it and understand it, and can doubt about language ambiguities.
The result is just the expected: even if formally the situations are the same, people reason faster about common day situations they already know.
And that’s the writer conclusion: that we should take advantage of this to teach math.
You’re just making up an unrelated conclusion. You could do the experiment with the bouncer situation but changing the rule to something people don’t already know. The results will change.
You’re arguments (this one, the impossibility for some children to learn how to solve for T in PV=nRT, that English is easier than algebra) are against evidence.
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It doesn’t matter what anyone wants to teach in schools, if the teachers are incapable of teaching it and the system is not designed for it. Math taught in most schools in the US is infernally boring and designed, when any purpose exists, with the assumption that kids are going to be scientists or engineers or at least have to pass a multivariable calculus course for the distribution requirements on their way to a history degree. There’s just no need for that, and most kids don’t retain what they supposedly learned anyway. So while algebra is clearly important, the way it’s taught is pretty indefensible. I like Paul Lockhart’s article: http://www.maa.org/devlin/LockhartsLament.pdf
Algebra teaches abstract thinking, meaning that you can learn some truths about x without actually knowing the specific value of x; and if you can deduce enough truths about x you can figure out what x must be. This is an extremely important ability. It is hard to imagine someone being very successful in any profession without the ability to think abstractly in this way.
Algebra is a very concise and tangible way to teach abstraction, because it has a precise notation and very clear definitions. In that sense, it is probably one of the least difficult ways to teach abstract thought.
In any field, if you want to contribute to that field in some way, you must be able to hypothesize about a solution to a problem in that field, figure out some truths about the solution without knowing the exact answer, and use those truths to validate or invalidate your hypothesis.
This is true in science, math, history, language, biology, or any other field that I can think of. For example in history, you might want to analyze the reasons for a particular war, and to do that you will need to know facts, and you will also need to make hypotheses, and know how to support those hypotheses or invalidate them, using facts, axioms, rules, generalizations and deductions.
In literature, you might want to suggest a new meaningful interpretation of an author’s oeuvre. To do that, you have to hypothesize about some new interpretation, narrow in on a definition with evidence, and think abstractly about whether your interpretation fits within the rules of what is known about the author’s life and work.
Algebra teaches these abstract thinking skills. If you can’t do algebra, you probably won’t be able to think abstractly in any number of ways, and you won’t be able to contribute to any field at a very advanced level.
That isn’t to say that anyone can succeed at anything. I am not ready to say that everyone can succeed at algebra. It may not be politically correct to say this, but not everyone is equally smart. Some people are simply better at abstract thinking than others.
But I’m pretty sure that if you can’t succeed at basic algebra, you will also not be making big contributions as a historian, or linguist, or lawyer, or doctor, or military officer, or sociologist, or economist, either. Abstract thinking is very important to advancing in any field, and algebra is a very good and simple way to teach abstract thinking.
If you don’t teach algebra because some kids cannot succeed at it, you are doing a great disservice to the kids who can succeed at it, because it will help them in any field they go into.
To the people saying we shouldn’t teach all kids algebra:
How do you choose which kids to teach algebra to, and which kids not?
All students should be taught the basics of math. Some will be able to go on, and some will not. Its their ability that would sort them, not some external decision.
What I’m mostly advocating is not that we don’t teach algebra, but that there might be some kids who simply will not get it, and that, in and of itself, doesn’t mean they are relegated to the dustbin of life.
I have met quite a few rather bright students who are hopeless in math. Maybe that’s bad elementary instruction, but what if it is simply human variation in an ability we were never evolved for? There are plenty of other things they can learn and learn very well.
It sounds as if you’re saying that we should use arithmetic as an indicator as to how successful someone will be at algebra, is that correct? From what people have been saying, it sounds as if the abstractness (symbolic nature) of algebra is what makes it difficult for students. If this is the case (and I’m not asserting that it is), is arithmetic really a good indicator?
I know a surprising number of people (some very smart and pursuing PhDs and JDs) with dyscalculia who are truly awful at arithmetic. Their brains are wired to have difficulty with arithmetic in the same way that a dyslexic is wired to have trouble with reading: they frequently transpose numbers, have trouble reading clocks, can’t estimate sums, have trouble reading maps.
Curiously, the ones who got past algebra found geometry much less difficult, and I suspect that some would do fine in the basic proofs course I took at the start of my BA in math. Just because they have trouble with some kinds of math doesn’t mean they shouldn’t learn mathematical reasoning.
To me, what you’re saying is equivalent to “People with dyslexia can’t even learn to read, so they’ll never be able to write an essay. There’s no point in trying to teach them to read; they’re just naturally not good at it.” Can you understand why that’s fucked up?
One last comment, then I’m gone.
This whole discussion, as far as I can tell, is being carried out among people who are in fact quite good at math. It is rather like a group of eagles wondering why penguins don’t just try harder to fly, or for that matter, penguins talking about why eagles don’t try harder to swim.
My basic question is, if there really is a learning disability WRT learning math, no matter how common or uncommon it is, is it reasonable to keep people expressing that disability from achieving in other fields?
Absent any hard data on the actual ability of children across the board to learn higher math, it remains a lot of uninformed opinion.
Let’s consider a slight variation on that.
“My basic question is, if there really is a learning disability WRT learning to write, no matter how common or uncommon it is, is it reasonable to keep people expressing that disability from achieving in other fields?”
The point is, you’re trying to claim that somehow, math is fundamentally different from any other field that we teach in school that some kids have trouble with. Some kids have trouble learning grammar. Some kids have trouble learning to write. Some kids have trouble learning to read. Some kids have trouble learning math.
Should we abolish *all* requirements in school, because some children will have trouble with them? If not, why is requiring minimal mathematical literacy any more onerous or awful than any of the other requirements?
Exactly; to put my own spin on it: just because dyslexia exists means we should abolish reading and literature classes?
The most depressing thing about all of this is that Mr. Krehbiel’s actually in a better position than any of us to put his backwards, regressive views into practice.
So I can’t defend that there are no such a thing like eagles and penguins because, being myself an eagle, I can’t have an opinion about penguins. Can’t you see the fallacy in your arguments?
I basically agree with you; there are eagles and there are penguins. Some people are not very good at math, some are not very good at writing, some are not very good at art, etc. People have different abilities, and you can’t mandate ability. That’s one reason courses have grades.
But that isn’t really the gist of the issue. As I see it, the issue is whether algebra is a meaningful and essential element of a complete high school education. Sometimes it is helpful to think of the inverse situation. Rather than asking “should a bad student be required to take algebra,” ask “should an excellent student be able to graduate without taking algebra”. They are flip sides of the same coin.
If an excellent student, who has the potential to learn from a good algebra course, doesn’t take it, he would be missing out on an important part of his education, and I think it might leave him at a disadvantage. We don’t generally design curricula solely for the students with learning disabilities. We try to make sure they help the excellent students and the average students succeed as well. I think it is acceptable if an average student has to struggle a little and put some effort into learning a subject that would help him succeed in later life. The math kids should be required to struggle with Shakespeare, too.
By the way, I don’t think it is really fair to call algebra “higher math”. That seems a bit like calling essay writing “advanced literature”, or calling an understanding of atomic valence “advanced chemistry”. It is just foundation work for that subject matter.
I’m a parent with two kids. One is an Eagle, and the other is a Penguin. Now, i’m an Eagle, even though i can swim. Now, the Eagle gets flying the first time, even if listening to Industrial Metal on headphones cranked to the max in the next room, with lessons taught out of order. So i end up spending way more time helping the Penquin fly, who is now flying quite well compared to most of the Eagles. It turns out, the Penquin is helping the Eagle swim. All this to say, not all students are equivelent.
An interested parent with no teaching experience can totally beat a great teacher. For one thing, a dedicated tutor can easily outperform a classroom experience.
A bit late to the game, i’ve noticed that not all students have an interested parent who is also an Eagle. An interested parent is still in trouble if they can’t grok the material. Not ever having had an original thought, is there an organization that matches willing Eagles with birds of need?
To those people out there who think math is hard and we shouldn’t teach kids math, I ask: how’s your credit card bill or your credit rating? Have you ever tried to take out a bank loan? Do you just pay minimum payments on them because it’s easier that way?
So far no one’s even mentioned the value in developing number sense (the ability to grasp what numbers mean without needing to work out all the details). And if you say you have good number sense, but can’t handle algebra, I’d say you’re lying.
Sephia.Mage:
Some people are well aware of the mathematical implications of paying minimum on loans. The problem is that when you don’t have much money, it’s safer to pay as little as possible to ensure that you have enough leftover money at any given time to absorb any risks that set you back (car breaks down, health problem, etc).
Anyhow, to address the topic at hand, I think it’s clear that math is important to a great number of fields. Just because kids have trouble with it doesn’t mean we should do away with it. We should instead focus on making it easier and more accessible to students.
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Just to be annoying I’ll point out that your mortgage example is not quite right. You forgot to take into account the fact that you will also build equity faster with the lower interest rate, so in fact you pay off the amount D faster because of that. If you take that into account you actually get a non-algebraic equation for k, which requires some more advanced technique probably using calculus (i.e. Newton’s method) to solve. So there you go, calculus is useful 😉
Just to be even more annoying I’d also point out that if you’re really serious you should include in the model the fact that if you don’t make the 2% payment you could put it in an investment, thus making the “don’t reduce interest rate” option more attractive.
I totally disagree with Hacker’s opinion. However, I thought he was playing a game about writing the most arguable words other than most correct words. He manipulated his words to maximize such an objective function that how his article will be discussed and read other than how correct his article is. That ‘s his “calculus”. In some sense, he won when he saw lots of people critize his article so that his name was to be remembered.
🙂
Zhiyong
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Hi:
I’m afraid there is some error in the mortgage payment formula you quote. At any rate, when I make a spreadsheet and up payments it generates for a $10,000 loan at 5% /year over 24 monthly payments, I get a monthly payment of $400.96. However, 24 payments of $400.96 gives me a total of $9600, not $10,000. Microsoft Excel says the payment should be $438.71, which is quite a bit more.
Could someone else please check these figures?
Thanks!
Daddyo
I’m afraid that in your attempt to be snarky, you’ve managed to instead just be wrong. Your make gross oversimplifications of the mathematical concepts.
First of all, parametric equations are not simply “equations with variables.” They generally refer to equations in which multiple variables vary according to other variables – for instance, in which both x and y are defined in terms of a variable (or “parameter”) t, rather than y being defined explicitly in terms of x. This is a concept usually not taught until precalculus, and what we teach involving them hardly falls under the umbrella of “simple algebra”.
Second of all, polynomial equations are not simply “equations involving variables with exponents.” A polynomial function (in one variable at least) must be of the form a + bx + cx² + dx³ + … — the powers of x must be positive integers. Because of that, your amortization equation, though it does involve exponents, is not a polynomial equation — it has powers of i in the denominator of a fraction. (Nor is your equation usually what we call a parametric equation.)
Perhaps you should do your own homework next time before you rail on somebody’s article and call him a “moron.”
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Hacker is right. Stop wasting educational resources trying to force feed abstract mathematical concepts with no context on high schoolers who don’t care and don’t get it. The most efficent use of resources would be to let those who fail algebra fail it, and allow them to opt out of any additional high school mathematics courses and pursue other subjects they have an interest or aptitude to complete successfully rather then feeding them into “remedial” math classes and forcing them to do it all over again.
The Sage of Wake Forest
I like math & physics, but I don’t see why people who don’t like it should have to learn it. A history of science & philosophy would probably be of far more importance & value. It has the advantage of avoiding equations. It could cover a history of symbolic notation across different cultures. It would also cover a lot more ground. It would show that the current physicalism is something that has arisen in different cultures over a wide period of time (the atomists of greece, the Nyaya school of logic in India, and I’m sure of many others).
All humans are good at conceptual reasoning, it comes with the territory, it is not exclusively tied to mathematics or even the sciences. One of the purposes of the exam system is to find hard-working conscientious people and bright, creative types who can make a novel contribution to an art or science. Its not hard to distinguish between the two. Nor to discover where a student appitude actually lies, or doesn’t.
So often, people make the mistake ‘you can do maths, therefore you must be clever’, and ‘only clever people can do maths’, to ‘if you can’t do maths, you can’t be clever’ (in certain circles). No it doesn’t, it shows only that you can do maths, and that all skilled activity requires a little intelligence.
If so many people who aren’t constitutionally able to stomach mathematics didn’t have to be force-fed it, it may have the beneficial effect of them not viewing mathematicians/scientists as masochistic nerds, but as scholars; and it might heal the divide between the arts/science a bit.
None of what I’ve said here diminishes the importance of advanced scientific/mathematical culture to a societies intellectual, cultural & economic growth. I’m sure legislators are aware of this (if not, they should); Just how many algebraic topologists does a country need – a million, surely not?
This comment is great. Right on the money.
Stop force feeding high school mathematics on those who don’t care or have no aptitude for it. There are currently several studies showing that math is “an aptitude” from an early age –some people have it and progress well, some don’t have it and do not progress well.
Why should we keep beating up the kids “who don’t have it” with remediation? Answer: We should not. Save the resources.
Of course the argument could be made that “they are not working hard enough.” Fine–but at the end of the day the result is the same–they fail or do poorly.
Could you cite one such study? Did Aristotle say it? It’s very difficult to argue when all you have is “I’m correct because authority says it”. Well, let’s take a look. What authority?
Because I can say the opposite. Authority say the opposite. Laszlo Polgar proved the opposite. Who’s right?
Turn what you’re saying around for almost any other subject we teach in school:
Many subjects come much more easily to some students than to others. But to be an educated member of society, there are some minimum skills that we consider essential.
For some reason, many people believe that math isn’t one of those. And yet, we expect people to be able to set budgets. To choose mortgages. To plan for retirement. To invest. None of those things can be done without math.
No one is suggesting that every student be required to study differential equations. But minimum basic mathematical literacy? It’s crazy that we’re even arguing about it.
These arguments are not new: Plato in his Republic when discussing the education of the ideal citizen, said ‘that arithmetic has a very great and elevating effect, compelling the mind to reason about abstract number…that those who have a natural talent for calculation are generally quick at every other kind of knowledge’ but he also goes
on to warn ‘They have in view practice only, and are always talking in a narrow and ridiculous manner, they confuse the ways of geometry with those of daily life’. All sounds familiar.
He also says ‘we are concerned with that part of geometry which relates to war…but for that purpose very little of either geometry or arithmetic will be sufficient’. Although he doesn’t say it, what is true for military purposes is also true for civilian purposes.
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