Current state of math education

For the discussion of math. Duh.

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Dason
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Current state of math education

Postby Dason » Sat Feb 06, 2010 7:53 pm UTC

So this thread got me thinking about our current state of math education. I think we all know a lot of people who "hate" math and yet really have no idea what math is all about.

It makes me sad that most people judge math on the most elementary branch of math (arithmetic) and yet I think most of us would agree arithmetic is actually the least interesting branch. Now I don't blame them since the education system currently force feeds that particular branch exclusively almost all the way up until high school. I guess I wish we would have more classes that were more focused on the creative side of math. Maybe before we "require" everybody to take some algebra and geometry let them take a class where they explore those topics on their own and try to figure some things out before being exposed to the classical results? I don't know... I just feel that by sucking all the creativity out of the education system we give the wrong impression. Honestly I wouldn't mind if we did away with almost all of the math we have through high school. I don't think most students get anything out of it and all it does it cause resentment toward the subject.

Now I am torn about my previous statements because I do feel the classic topics (algebra, geometry, trigonometry, ...) should be introduced around this time (especially for students that enjoy math and consider going onward with their studies at a university) to give a fighting chance while at university. On the other hand if we introduced math in a more exploratory setting I think it would be easier to pick these topics up quickly.

I look at other subjects (like music and art) and marvel at how they're taught. What do we do? We let the students explore the subject. They get to play music or they get to make art. The students get to explore the beauty of the subject. I mean... I don't really hear anybody saying "Music is so friggin useless... why do we even have to learn about it?" or "I hate art so much... that class just pisses me off... when am I going to USE art?" What do we do in math? We say... "This is how you do the problem now do it 80 times and then we'll repeat with a similar but slightly different problem" so the students reply with "What's the point in math... when am I ever going to USE this?"

Now I know it's not easy to get changes implemented in the school systems so I guess this is really just my lament about the current state of mathematics in primary schools. I would really like to hear other people's thoughts on the subject. Do you guys have an good suggestions? Have you seen math taught in primary schools differently and thought it worked well?

Note: I know not everybody here is from the US but that's where I'm from so I guess I'm more or less commenting on the state of affairs here in the US (if it's taught differently elsewhere I'd love to hear about it). Also I realize this has probably been discussed somewhere on the forum at some point but I didn't really know what to search for so if there is already a thread for this I apologize.
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Re: Current state of math education

Postby psychosomaticism » Sat Feb 06, 2010 8:38 pm UTC

Needs to be started earlier. I think a lot of time in the grades 4-7 or so are not spent as productively as possible. You could introduce so much more if the subjects were taught by more specialized teachers in lower grades. It isn't a solution, but I think it'd be a step in the right direction.

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Re: Current state of math education

Postby ThomasS » Sat Feb 06, 2010 8:42 pm UTC

There is much that I could say on this subject, but I think that Lockhart has already said it better than I would.

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Re: Current state of math education

Postby Talith » Sun Feb 07, 2010 3:35 am UTC

I've read that article a couple of times now and I agree with it more every time.

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Re: Current state of math education

Postby Josephine » Sun Feb 07, 2010 9:13 am UTC

*sigh* that article is depressing.

So, we know the problem. Is there any way to fix it?
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Re: Current state of math education

Postby z4lis » Sun Feb 07, 2010 6:23 pm UTC

One depressing fact is that the system is so bad, it discourages those teachers described in the article from even thinking about entering it and trying to fix it, because we already know our hands will be tied. I'm an undergraduate right now who would probably really enjoying teaching middle or high schoolers about elementary algebra, geometry, discrete mathematics, and logic. That natural movement through subjects that a teacher takes with his class is something that would move me with meaning and joy, and I know that the children currently wrapped in the education system need it, badly. They need someone who adores "abstract" mathematics, relishes its elegance, too much to actively participate in its undoing in the K-12 classrooms.

The problem is, I honestly feel sick even thinking about it. I'd show up and be handed a textbook and a lesson plan. I'd have about three weeks of material to stretch over a whole semester, none of which is really mathematically interesting, relevant, or inspiring. Rather than spending my algebra course investigating real problems in the theory of numbers, curves, equations, and how elegantly they fit together (perhaps even some group theory - these kids are forced to memorize what the associative and commutative laws are, and then are never shown a structure in which they don't hold - what good is that?), I'd be forced by the school and the children's parents to "prepare" them adequately. Honestly, I think that if I scraped down into something deeper than what the curriculum is, I'd be automatically teaching the elementary stuff I'm supposed to be mindlessly drilling them on. And they wouldn't even know it.

So, I'm essentially afraid that the system will not only crush them, it'd crush me, as well. This fear directly leads to the fact that I will never, ever consider teaching at a K-12 school. If I do, I will do it the way it should be done, hope for fantastic results from the student's attitudes, probably receive complaints from parents and administration, and get fired after a semester. At least I'd be honest to myself and my students, that way.

If they don't think this sort of teaching style, a more natural, investigative method works wonders on people, let me give a bit of my story. I was in a calculus class in high school, and had at that point flat out refused to pay any attention or do any work. The material was too easy, repetitive, and dry for me, so while the other students were working on something, the teacher gave me a little paper about the Koch snowflake, a blank sheet, and a ruler. An infinitely long line bounding a finite area? That's interesting! I drew the thing up, and immediately wondered if I could express the area of the snowflake in terms of the length of the side of the original triangle. A pretty elementary calculation, no doubt, but not so much for my early high school self.

When I got home, I rushed to the computer and looked up the snowflake on Wikipedia to confirm my formula was right. It was, and I got that rush of pure satisfaction that only comes to me through overcoming that mathematical frustration. If I had to pick a single moment in time where I decided to become a mathematician, I'd have to choose that one. Other things, of course, helped tipped me towards math from physics, but it was that simple calculation that showed me that thinking about odd, strange things, picking apart their properties and how they work, could be so rewarding.

Perhaps I'm just an oddball case, destined in some sense for that choice of mathematician, and my little experience with the Koch snowflake had no bearing. I like to think otherwise. I'd also like to think that if one simple, trivial adventure into something outside of the curriculum could have such a deep impact on me, the impact of a whole curriculum taught in that investigative style would be phenomenally uplifting for the students. However, I have to shamefully admit that I don't believe I'll ever see it.
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Re: Current state of math education

Postby Dasboard » Sun Feb 07, 2010 6:48 pm UTC

I guess I wish we would have more classes that were more focused on the creative side of math


Wait, wait, math has a creative side?

I'm now in my last year of high school and haven't noticed anything creative about math yet. All we do is learn certain tricks to solve some problems and practice till we actually get how they work.

Examples, calculating chanches and things like "Here's a line, make a formula for it.". Pretty much really basic and repeative stuff.

Now I'm pretty horrible at maths, so it might not be as fun for me as for other people but with the knowledge I got I got to conclude that the way maths is being teached at my school is pretty much crap. But that goes for any subject actually.
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Re: Current state of math education

Postby Josephine » Sun Feb 07, 2010 7:35 pm UTC

dashboard, it sounds like you didn't read the article posted earlier. I highly encourage it. What you were being taught wasn't math at all.
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Re: Current state of math education

Postby Dasboard » Sun Feb 07, 2010 7:43 pm UTC

nbonaparte wrote:dashboard, it sounds like you didn't read the article posted earlier. I highly encourage it. What you were being taught wasn't math at all.


I read the article and decided to post how it was here in a country which wasn't the US. ( Maybe I should've stated that. )

And I know very few of maths itself and the subject is called "maths" here so I pretty much assumed that what they teached us in those lessons was somewhat related to math. I think I'll do some research.
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Re: Current state of math education

Postby Josephine » Sun Feb 07, 2010 7:47 pm UTC

yeah, the math education issue seems pretty global to me.
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Re: Current state of math education

Postby Dasboard » Sun Feb 07, 2010 8:10 pm UTC

Yeah, seems like it.
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Re: Current state of math education

Postby eSOANEM » Sun Feb 07, 2010 9:30 pm UTC

Dashboard, being in a similar situation to you (I'm 16 and at secondary school) but having done quite a lot of research (not as in original research, probably more like self-teaching) in my own time I can probably offer a greater insight.

What you're taught at school is not maths any more than literacy is literary analysis. What you are taught are the tools with which to do the real maths, this is called maths because maths is its main use but it would be better described as numeracy (it's all basically teaching you how to use numbers, and symbols in the same way as numeracy is teaching you to use letters).

The real maths lies in taking what is already known and producing something unexpected or unknown from that. This is the creative part and the most important part of maths.

Someone can tell you that every even number greater than two can be written as the sum of two primes because they've seen this pattern. Seeing this pattern is creative and is also maths, what separates the mathematician from the normal person is that the mathematician tries to prove that this will always be the case. Coming up with a correct proof from the reams of tools available to a mathematician is incredibly creative and requires a stroke of genius (this statement for example has remained unproven for over 250 years).

If you wanted to find the gradient of a the tangent at a point on a curve and you had no knowledge of differentiation it would take huge creativity to work out and was only generalised in the mid-late 1600s.
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Re: Current state of math education

Postby BlackSails » Sun Feb 07, 2010 10:58 pm UTC

The problem I think starts in trig classes. First, you dont need an entire year to learn trig. You need a month, tops, embedded into a geometry class. Second, its often the student's first introduction to proofs, and they are done in the silliest, most asinine way possible.

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Re: Current state of math education

Postby polymer » Mon Feb 08, 2010 12:59 am UTC

The current state of the math education is an interesting subject. Lockhart's lament is one of the better essays I've read on the subject. There's a particular theme in it that I particularly appreciate. The thing about mathematics is that as a subject there generally is a precise answer for any question you throw, however the concepts being discussed are abstract. Consequently, almost by definition your going to have numerous physical and mathematical interpretations for the same answer. These numerous interpretations almost always imply other consequences as well, which may be more or less emphasized depending on your interpretation. Because you can say the same thing in so many different ways, you have this situation where people can agree, or disagree, over what's the best way to see a particular theorem or mathematical statement. I would appreciate it if the education system made it more evident that the students can disagree with a mathematics teacher's, or the textbook's explanation. It'd be even better if the kids were able to talk to their teachers about math. Solving lists of problems is fine and all, but that will never show if the kids thinking.

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Re: Current state of math education

Postby RogerMurdock » Mon Feb 08, 2010 7:34 am UTC

BlackSails wrote:The problem I think starts in trig classes. First, you dont need an entire year to learn trig. You need a month, tops, embedded into a geometry class. Second, its often the student's first introduction to proofs, and they are done in the silliest, most asinine way possible.


Hell yeah.

I took a Precal/Trig class and for the trig part I passed the tests, learned everything they wanted me to, but I DID NOT UNDERSTAND TRIGONOMETRY. Like, I didn't understand the the "sin(x)" of an angle gives you the y component....I was just memorizing facts/formulas. Not until physics rolled around where I was required to think of things in a physical/geometrical sense (who'd have thunk it?) did I understand the relationships between all the functions and how useful trig is.

As to the math education, I speak from the point of view of someone as a senior in High School in Calc BC currently, and I would have to agree with a lot of the points brought up here. I never remember really "hating" math because it was always something that came natural to me, but I think why people do hate it is really kind of a self-fulfilling prophecy of sorts. They can't do something....they assume that they are "bad" at it....they grow tired to attempts for teachers to teach them material they know they are "bad" at and never will get, and soon grow to hate the subject that people are trying to "make" them learn. This happens in middle school, when some of the earliest algebra/geometry classes are taken. I don't know any kids in elementary school who "hate" math. It's just multiplying numbers at that point, and I have fond memories of the class hooting and hollering along with "math games" in 2nd grade or so. The way geometry is taught is particularly terrible. Many of you may not remember it or have the same kind of class I had, but I have never talked to a single person in any math class I've EVER had that said they enjoyed geometry. Even the word "Proof" sends shivers down their spines to this day. Heartbreaking, really.

The main problem I've had with most classes is the heavy focus on procedure versus concept. The concept should be the first and last thing you talk about every day you learn something new, and the numbers/variables are only thrown in when everyone is comfortable with WHY what you're about to do works. Most lessons in my first calc class started with my teacher drawing a grid, a squiggly line to represent a function, and then circling a point and asking us a very poignant question about that day. "Is the function continuous at this point? Why?" or perhaps "If I wanted to find the area from here to here, how would I do it?" And class discussion would usually erupt from there with everyone throwing out opinions and possible answers and whatnot. No numbers, no symbols, just talking. And students in high school like to talk, even if they're wrong at first. Only once we got a consensus down about the answer to that initial question would we start working out problems. Best teacher I ever had right there.

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Re: Current state of math education

Postby Dasboard » Mon Feb 08, 2010 6:27 pm UTC

eSOANEM wrote:Dashboard, being in a similar situation to you (I'm 16 and at secondary school) but having done quite a lot of research (not as in original research, probably more like self-teaching) in my own time I can probably offer a greater insight.

What you're taught at school is not maths any more than literacy is literary analysis. What you are taught are the tools with which to do the real maths, this is called maths because maths is its main use but it would be better described as numeracy (it's all basically teaching you how to use numbers, and symbols in the same way as numeracy is teaching you to use letters).

The real maths lies in taking what is already known and producing something unexpected or unknown from that. This is the creative part and the most important part of maths.

Someone can tell you that every even number greater than two can be written as the sum of two primes because they've seen this pattern. Seeing this pattern is creative and is also maths, what separates the mathematician from the normal person is that the mathematician tries to prove that this will always be the case. Coming up with a correct proof from the reams of tools available to a mathematician is incredibly creative and requires a stroke of genius (this statement for example has remained unproven for over 250 years).

If you wanted to find the gradient of a the tangent at a point on a curve and you had no knowledge of differentiation it would take huge creativity to work out and was only generalised in the mid-late 1600s.


Ah, this makes a lot more sense to me. But what I'm wondering now, why didn't I ever get told what you just told me? I think math students in High School will be a lot easier to motivate, just by showing what you're actually learning. Before I started reading these comics and forums I pretty much had no idea about actual maths.

In classes I never ever got a definition of what we were learning, or what Maths actually is. All we got were some vague lines like "You are confronted with Maths every day." which didn't make sense to us, as we never really got Maths. Next up, we learned the tools ( what we're learning in High School, which I now know thanks to your explaination. ) which we of course never used. Thus leading us to the conclusion that Maths was pretty useless.

Looking at it from this way, my conclusion is that this is one of the biggest issues in the Math's education given in my school.

I would appreciate it if the education system made it more evident that the students can disagree with a mathematics teacher's, or the textbook's explanation. It'd be even better if the kids were able to talk to their teachers about math. Solving lists of problems is fine and all, but that will never show if the kids thinking.


This would really help, of course this really depends on the teacher. Personally I got a teacher who's exactly what he shouldn't be. All he does is say the assingment and write ou the answers. If you ask a question he either looks at you with a weird You-Should-Understand-This-Look / Ignores you / Rabbles something which makes it even more complicating.

The way my teacher is "teaching" Maths is like Polymer said, not teaching us to actually think, which is another big issue.

Also, I'd like to add that the way you guys talk about Maths is getting me pretty excited about it and pretty sad about the fact that I'm only learning tools to do maths.
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Re: Current state of math education

Postby Yakk » Mon Feb 08, 2010 7:57 pm UTC

One problem I suspect with a math education is that people who are good at math end up getting better paying, higher status, less annoying jobs than educating a bunch of elementary/high school kids.

I'm serious.

The few who are qualified, naturally seek the least annoying students to teach -- the upper-year students who elected to take mathematics courses, and who are at least somewhat interested (and possibly excited) to learn about something they are good at.

Even worse, apparently elementary aimed teachers are among the most math-phobic in the population. Which means that as they go from simple memorisation to actual math, they (as a group) cannot but help express their discomfort about the subject, and reduce it to more rote memorisation.

Math also happens to be objectively useful. Admittedly, proper English communication is useful -- but what communication styles are useful is a function of who you are communicating with. Regardless of who you are interacting with, getting the math right means that the cannon-ball lands where you want it to, and getting it wrong means it doesn't. The techniques of BS that work in a myriad of other disciplines tend to fall apart when used in a math context... (note that different techniques work in a math context!)

So when you don't understand something in math, it shows up. And this works on the teacher's side as well as the student's side. You cannot gloss over gaps in your mathematical knowledge very well as a teacher.

...

The way math is taught at the high school level, at least around here, is closer to engineering-style maths than to math-style maths. Engineers learn math in order to apply it to problems: mathematicians learn math in order to come up with new math to do. Engineers are well served by learning huge bibles of formula and problem solving techniques, if only to learn that they exist so they can look them up later. Mathematicians, even the applied ones, are trained to generate new mathematics. Engineers creatively use mathematics; mathematicians creatively create mathematics.

(Note that this, like any absolute statement, is rather false.)

So you are taught how to do a proof, not so that you can generate other proofs, but so that you can see a proof. You are taught trig, not so you can develop new mathematical theories similar to trig, but so that you can apply trig identities to toy problems.

To put this another way: you can go through an entire course of post-secondary mathematics education without touching a calculator or having to memorise a single formula. And those two things (the calculator, and formula memorisation) are core features of secondary and pre-secondary mathematics education.
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Re: Current state of math education

Postby agelessdrifter » Tue Feb 09, 2010 12:36 am UTC

I tutor Introductory Algebra through Calc II (which I'm currently taking) at the local community college, and it is really disheartening to see the number of people who come in with problems involving graphing (which as you know is most of the curriculum in Intro through pre-calc) and ask questions like "If I plug in a different value for x does that change the line?" indicating that they have no clue what the graph of a function is/represents. I mean people get all the way up into pre-calc (and sometimes even calculus) and do not understand the core concepts of the material of the two or three classes they managed to get all the way through with passing grades to get into the course they're in.

Generally I'll begin by trying to explain the underlying concept (not just that specific one, but whatever is relevant to whatever question is presented to me) to the person -- "you know this because of this/this can be demonstrated by that/when we say this, what it represents is that, therefore this/etc", and it's very fulfilling work when one of the students is responsive to this approach, but more often than not, I get a lot of head-nodding and "uh huhs" followed by "ok so what's the answer" or "ok so what do I do". These latter students come in and ask questions about every two or three exercises because they can't (won't) extrapolate the information beyond "when (some very specific x) do y". And I realize it is largely not their fault, but by the time people hit college age (and up -- many students are in their 30s and higher) it's likely so ingrained in them to behave this way toward math, I don't know how to get anything through to them. I mean this isn't even a question of "they're learning what they're expected to but not what's important," these people are so debilitated by what they've come to expect from maths that they can't/won't even learn what they are expected to learn.

The argument presented in the article someone linked above was very interesting. I'd be very curious to see the outcome of the approach he proffers, but I also think it would take many years of refining to nail down. One point he never addressed that gave me pause is that, unlike an art class (where students can hardly make photocopies of each others' paintings to hand in), students in a "free-form" math class like the one he postulates can cheat. Not just off of each other, but off of the internet as well. Or their parents might just give them answers. It seems almost like any work (problems presented to be solved like the circle,right angle example) would have to be done in class, and in one class period.

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Re: Current state of math education

Postby BlackSails » Tue Feb 09, 2010 1:19 am UTC

Cheating is going to be pretty obvious I think, just based on how things are phrased.

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Re: Current state of math education

Postby njperrone » Tue Feb 09, 2010 3:03 am UTC

Not sure how much of a deviation from the conversation this is, but it is relevent. I think the reason the math education system, in the USA at least, is the way it is because most of the people who go onto college usually go into some specialization that does not involve a lot of upper division math so instead of taking 3 years of geometry and proofs and a year of algebra in high school they do generally 2 years of algebra, one of calc, and another of geometry and trigonometry because these are the most relevant tools to what is going to be used and perfected in college. But as for those who are math advocates the state of education is terrible. But for any change to happen there has to be a trade off in the sculpting of the tools, no matter how small.

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Re: Current state of math education

Postby LLCoolDave » Tue Feb 09, 2010 8:05 am UTC

njperrone wrote:Not sure how much of a deviation from the conversation this is, but it is relevent. I think the reason the math education system, in the USA at least, is the way it is because most of the people who go onto college usually go into some specialization that does not involve a lot of upper division math so instead of taking 3 years of geometry and proofs and a year of algebra in high school they do generally 2 years of algebra, one of calc, and another of geometry and trigonometry because these are the most relevant tools to what is going to be used and perfected in college. But as for those who are math advocates the state of education is terrible. But for any change to happen there has to be a trade off in the sculpting of the tools, no matter how small.


You could gain a couple of month on trig alone. There is no need to stretch it out that long, and some other aspects suffer from the same issue. Some parts of that tools education could easily be tought in signficantly less time while still having the same amount of valuable content. Besides, I think the ability to think abstractly, extrapolate your knowledge and think in a logically consistent way are much more valuable things to teach than stretching out trig to a whole course of repetition. Should somebody need more knowledge on that for his further education or job, the former skills will certainly help him aquire that additional knowledge fairly easily.

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Re: Current state of math education

Postby agelessdrifter » Tue Feb 09, 2010 7:22 pm UTC

They offer pre-calc/trig combo classes now at my school. I'm glad I took trig over a summer term -- six weeks instead of 18. It never really occurred to me how insipid that would have been stretched over three times the time period I took it in.

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Re: Current state of math education

Postby mdyrud » Tue Feb 09, 2010 10:54 pm UTC

Just restating what has been said before, trig should not be its own class. I took it independent study, and I procrastinated until the very end. I had two weeks to do a full semesters worth of work, minus one test and two or three homework assignments. It was stressful, but I got it done, as well as all the other homework I had. At our school, they have a new curriculum that starts precalc during the trig class. It is just taking effect, but there is already a difference in how far along the people are going into calculus.
Another thing I agree with is the statement that what you see in school is not math, it is teaching you the basic tools needed to do math. Math classes are almost entirely useless as anything other than preparation for more math until you hit calculus (aside from the higher order thinking skills they build). It is so unfortunate that people are coming out of high school after finishing precalc, thinking that math is no fun at all. So many people avoid math classes in college, simply because of past experiences with it, and they are quitting just short of hitting the fun math.

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Re: Current state of math education

Postby agelessdrifter » Wed Feb 10, 2010 5:19 am UTC

Since it's been more than three years since I took chemistry in highschool, I'm required to take an "intro to chem" class before I can get into gen chem. Sitting in Intro Chem today I realized that this same catastrophe is probably occurring pretty ubiquitously with science courses as well -- it certainly applies to the one I'm in. In fact I'd say this is probably a fairly wide spread problem in most subjects.


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