Math education in the USA

[mississippichem]

So there's been talk around the internet for a while now about what is wrong with mathematics education in the USA. Lockhart's Lament, is a good read if you want a good synopsis of some of the ideas that have been floating around. I thought this would evoke some good forum discussion.

 

One problem I see is that much of the necessary mathematics for a good high school science education get introduced way too late, or not at all. Take for example the calculus/physics problem. Many high schools don't introduce calculus until near the senior year so students never get a chance to take a good physics course. Instead they get a primarily algebra based physics course that may teach some good things but doesn't do much to really ingrain the fundamental concepts from a mathematical standpoint. I remember taking physics and calculus my senior year of high school. There just wasn't much to the physics class really. I'm sure some schools do this differently, but from reading around this seems to be a fairly common situation around public schools in the US. This also sets up a situation where high school students don't get any mathematics above differential (and some integral) calculus.

 

So what can we do to fix such a conundrum? Or maybe, what is the problem? Are we introducing algebra too late? Or are we spending too much time on arithmetical computation in elementary schools? I think the problem is that we have basically no specialization of curriculum in our secondary schools, most of the other problems are corollaries to that.

 

I'm speaking quite generally here, I just wanted to hear some thoughts from the mathematically minded of SFN.

[DrRocket]

So there's been talk around the internet for a while now about what is wrong with mathematics education in the USA. Lockhart's Lament, is a good read if you want a good synopsis of some of the ideas that have been floating around. I thought this would evoke some good forum discussion.

 

One problem I see is that much of the necessary mathematics for a good high school science education get introduced way too late, or not at all. Take for example the calculus/physics problem. Many high schools don't introduce calculus until near the senior year so students never get a chance to take a good physics course. Instead they get a primarily algebra based physics course that may teach some good things but doesn't do much to really ingrain the fundamental concepts from a mathematical standpoint. I remember taking physics and calculus my senior year of high school. There just wasn't much to the physics class really. I'm sure some schools do this differently, but from reading around this seems to be a fairly common situation around public schools in the US. This also sets up a situation where high school students don't get any mathematics above differential (and some integral) calculus.

 

So what can we do to fix such a conundrum? Or maybe, what is the problem? Are we introducing algebra too late? Or are we spending too much time on arithmetical computation in elementary schools? I think the problem is that we have basically no specialization of curriculum in our secondary schools, most of the other problems are corollaries to that.

 

I'm speaking quite generally here, I just wanted to hear some thoughts from the mathematically minded of SFN.

 

I am generally not in favor of high school calculus. I once had this discussion with an extremely well known professor of engineering at MIT, and he and I agree. The difference between an average high school calculus class and a university calculus class lies in the understanding of the basic concepts, and not in facility in pushing symbols and doing calculations. To teach calculus properly requires an in-depth knowledge of real and complex analysis. Not many high school teachers have that understanding. The local university does not give credit for AP calculus below a "5" level on the AP test, and that strikes me as about right.

 

I would much rather see high school students who really understand basic algebra and trigonometry. The problem is that all too often they don't. In fact many can't add fractions or compute the area of a rectangle.

 

To address the problem what is needed are high school teachers with understanding of mathematics and science. In my state a senior emeritus professor of mathematics has taken on the task of developing a program to educate ex-professional people to suit them to teach high school science. Believe it or not one of the participants is a rather intelligent fellow of my acquaintance whose previous employment was in the National Basketball Association (above average height for a science teacher).

 

Bottom line: I would rather see high school students learn algebra and trigonometry well than calculus superficially. Algebra and trig are quite sufficient for high school chemistry and introductory physics. Quite advanced physics can even be presented with little else. See "Physics for Future Presidents".

[Bignose]

I am generally not in favor of high school calculus. ...

 

I disagree, far too often a calculus class is the only quality math class a lot of high schoolers get, because it may be the only one that challenges them. The challenge is more important than the specific of the mathematics that is learned, in my opinion. If a university does not think that a high school class should get credit for a college course, then I think that a good score on the AP test earns them the right to attempt to test out of the college class, and maybe that's it.

 

In addition, how many people taking AP calculus are actually going into a profession where knowledge of calculus is required? Especially knowledge at the level you cite: "in-depth knowledge of real and complex analysis"? Unless they are becoming mathematicians, I think that this is overkill. Engineers just need to know what the tools do and what their limitations are, not necessarily where they come from, for example. And, most undergrad level engineers don't use a whole lot of calculus in their day-to-day lives. It is anecdotal, and I haven't kept in contact with every member of my AP calculus class, but I am pretty sure I am the only one that even chose a semi-math-centric profession in engineering. I don't think that having a teacher who barely knew calculus (and I've kept in contact with him, I know he did indeed barely know calculus) impaired anyone's career. Again, I really think it is the fact that the math class was a challenge is much more important at the high school level than the subject itself.

 

On the topic of the first post:

 

to me, the main issue with the mathematics curriculum is that the practicing of problem solving is not emphasized nearly enough. Mathematics gives you can environment where practicing problem solving is very nice. You are given a set of tools that do a very specific job. If the problems are checked and well-written, you get problems that have only right and wrong answers. The idea is to practice problem solving in a very structured and logical environment, because eventually solving their problems aren't going to be so well-behaved.

 

In school, English composition class is at its heart problem solving: the problem being that you need to write an essay that makes the reader feel what you want them to feel. But, this you are given imperfect tools: words mean different things to different people. And, while there are rules for spelling and grammar, those don't help in choosing the right words or setting the right mood to convey the author's feelings exactly.

 

And then in real life: problems are going to be things like you have a flat tire, and your daughter has to be at soccer practice at 5:00 and you have to clock into your work at 5:30. Your tools in this case may be a fellow parent to call to give your daughter a ride and knowing the bus schedule to take the bus to work. Or, there may not even be a solution: you may not get to work that day. But, the point is that you practice solving problems with the nice tools of mathematics, and the nice environment, so that you are practiced when the real problems arise. It is just like when you starting physics class, you do the equations of ballistics, but assume no air resistance -- because the air resistance problem is harder. You build up to tougher and tougher problems.

 

Per the linked-to article: math has become a set of things to memorize. I don't think that you'll get rid of having to introduce a bunch of varied things, but if they are coached are new tools which help solve new problems (think if someone handed you a power drill instead of a hand crank auger!) I think that it would help.

[DrRocket]

 

 

In addition, how many people taking AP calculus are actually going into a profession where knowledge of calculus is required? Especially knowledge at the level you cite: "in-depth knowledge of real and complex analysis"? Unless they are becoming mathematicians, I think that this is overkill. Engineers just need to know what the tools do and what their limitations are, not necessarily where they come from, for example. And, most undergrad level engineers don't use a whole lot of calculus in their day-to-day lives. It is anecdotal, and I haven't kept in contact with every member of my AP calculus class, but I am pretty sure I am the only one that even chose a semi-math-centric profession in engineering. I don't think that having a teacher who barely knew calculus (and I've kept in contact with him, I know he did indeed barely know calculus) impaired anyone's career. Again, I really think it is the fact that the math class was a challenge is much more important at the high school level than the subject itself.

 

 

 

This is in large part a matter of experience and opinion.

 

So, one one side we have an MIT professor of engineering and a PhD mathematician with an MS in engineering and about 25 years of industrial experience.

 

On the otherbside I hear the standard response of undergraduate engineering students who just want to "plug and chug" and who in industry have to be watched closely, arguibg in favor of learning a subject only superficially.

 

I think I'll stick with the side I am on.

[Bignose]

Classic appeal to authority! -- 1) you don't know my credentials (because I don't share them) and 2) that does nothing to refute my points. Do you have any non-fallacious arguments?

[Marat]

I went to a private high school in the U.S. and took AP calculus in my final year, and I also took the basic science courses offered. Friends of mine at other private and public schools in the area seemed also to have about the same math and lab science knowledge that I had. I then went on to study at a German university, where I was shocked to find that many of the new German science students (about 30%) had not yet had calculus, and so had to have certain concepts explained graphically rather than mathematically. I was, however, very unpleasantly surprised at how easy the other students found all the labs (which for me were far too advanced) since, as they explained to me, "they had already done these labs in high school."

 

So my personal experience suggests that U.S. secondary education is adequate in math but defective in laboratory science.

[DrRocket]

Classic appeal to authority! -- 1) you don't know my credentials (because I don't share them) and 2) that does nothing to refute my points. Do you have any non-fallacious arguments?

 

Plenty.

 

As I said this question is really a matter of opinion and experience. It is not a question of established fact. The opinion of people who have both taught and practiced mathematics and engineering at a high level is certainly germane. Everyone has a right to an opinion, but in a question like this all opinions are not of equal value.

 

Apparently you don't understand logic, the nature of fallacy, any better than you understand this problem. I did not mention your credentials, and I frankly don't care about them. I have heard your arguments many times, from whining sophomores and from mediocre practicing engineers. That does not imply that you are one of them, but it does serve to identify a common source of that argument and a common end product of that approach.

Edited June 13, 2011 by DrRocket

[Bignose]

Well, then here's a PhD engineer (with a thesis topic and many papers written using some very advanced applied mathematics) who also has extensive industry experience disagreeing with you. I hope that I am not too egotistical to say that I really don't think that I am a "mediocre" engineer.

 

I have taught many fluid mechanics classes, and I have worked with people who do fluids computations many times a day. I do not think that any undergraduate student nor most practicing engineers need to know a wit about existence and uniqueness of solutions to the Navier-Stokes equations. Now, I have read papers on that topic, as much as for the mental exercise, but even in one of my research areas of writing multi-phase computational fluid dynamics simulations, the analysis done about uniqueness and existence of solutions aren't really relevant. I think that a great deal of a good fluids class for a group of to-be-industry-ready has very little to do with even solving the N-S equations. An industry-ready engineer needs to know more about how to size a pump and calculate losses through a valve than the N-S equations in my mind.

 

There are many well-verified correlations and rules-of-thumb that give pretty darn good answers to problems like these. How does any kind of "in-depth knowledge of real and complex analysis" help in this case? I have an algebraic estimate of flow through a squared-out orifice, as a function of fluid density, pressure, and orifice size published as a guideline by ASME. I can use this algebra to solve for what size orifice I need without any need for knowledge about real and complex analysis.

 

Which isn't to say that the N-S equations are unimportant, and with the advent of CFD becoming more and more of a useful tool, general knowledge of the behavior of the N-S eqns are becoming more important. But, I think that there are even more important topics such as sizing a pump or flow rate through a packed bed that need to be introduced to a group of students who are to be industry-ready.

 

There have been trillions of dollars of infrastructure built in this country -- and I can personally speak of refineries and chemical plants -- that have been built on algebraic relationships. Things like the McCabe-Theile algebraic method of calculating a good deal of information about distillation have been be the basis of design for almost countless distillation towers built out there.

 

Are there tough problems out there? Of course. Are there problems where exploring the basis of the derived equations are needed? You bet! But, as someone in this world, my opinion is that this is what graduate school and students and researchers are for. I don't see it as issues that even very experienced and good engineers with only an undergraduate degree need to be aware of.

 

Or, to put it succinctly, as someone who had taken some and self-studied a fair amount more of real and complex analysis -- it is interesting stuff to me -- but I do not think that in-depth knowledge of it would help more than a tiny percentage of the day-to-day work that most engineers in this country do.

 

And lastly, I don't think that it is just a question of opinion. I think it can be a question of fact, though I suspect that no one has bothered to study it. I think it can be a question of fact to find out how many concepts from real and complex analysis would actually apply to the day-to-day calculations an average engineer does, someone just needs to study it.

 

Obviously, your mind is made up, but if you have some examples of day-to-day calculations that would be benefited significantly from in-depth knowledge of real and complex analysis, I'd like to know what they are. I just don't know of any that an average undergrad level engineer working today would benefit greatly from.

[baric]

I do not think that any undergraduate student nor most practicing engineers need to know a wit about existence and uniqueness of solutions to the Navier-Stokes equations.

 

You never know you need the knowledge unless you have it already.

 

Obviously, your mind is made up, but if you have some examples of day-to-day calculations that would be benefited significantly from in-depth knowledge of real and complex analysis, I'd like to know what they are. I just don't know of any that an average undergrad level engineer working today would benefit greatly from.

 

I am a software developer and an in-depth knowledge of calculus certainly helps with my job even though it's not part of the description.

 

Advanced analysis abilities are like a rising tide that lifts all ships. You don't always realize that you need them until you are in a situation wherein what looks like an intractable problem to your coworkers actually has a straightforward solution. That makes you the indispensable guy.

[DrRocket]

Well, then here's a PhD engineer (with a thesis topic and many papers written using some very advanced applied mathematics) who also has extensive industry experience disagreeing with you. I hope that I am not too egotistical to say that I really don't think that I am a "mediocre" engineer.

 

I have taught many fluid mechanics classes, and I have worked with people who do fluids computations many times a day. I do not think that any undergraduate student nor most practicing engineers need to know a wit about existence and uniqueness of solutions to the Navier-Stokes equations. Now, I have read papers on that topic, as much as for the mental exercise, but even in one of my research areas of writing multi-phase computational fluid dynamics simulations, the analysis done about uniqueness and existence of solutions aren't really relevant. I think that a great deal of a good fluids class for a group of to-be-industry-ready has very little to do with even solving the N-S equations. An industry-ready engineer needs to know more about how to size a pump and calculate losses through a valve than the N-S equations in my mind.

 

There are many well-verified correlations and rules-of-thumb that give pretty darn good answers to problems like these. How does any kind of "in-depth knowledge of real and complex analysis" help in this case? I have an algebraic estimate of flow through a squared-out orifice, as a function of fluid density, pressure, and orifice size published as a guideline by ASME. I can use this algebra to solve for what size orifice I need without any need for knowledge about real and complex analysis.

 

Which isn't to say that the N-S equations are unimportant, and with the advent of CFD becoming more and more of a useful tool, general knowledge of the behavior of the N-S eqns are becoming more important. But, I think that there are even more important topics such as sizing a pump or flow rate through a packed bed that need to be introduced to a group of students who are to be industry-ready.

 

There have been trillions of dollars of infrastructure built in this country -- and I can personally speak of refineries and chemical plants -- that have been built on algebraic relationships. Things like the McCabe-Theile algebraic method of calculating a good deal of information about distillation have been be the basis of design for almost countless distillation towers built out there.

 

Are there tough problems out there? Of course. Are there problems where exploring the basis of the derived equations are needed? You bet! But, as someone in this world, my opinion is that this is what graduate school and students and researchers are for. I don't see it as issues that even very experienced and good engineers with only an undergraduate degree need to be aware of.

 

Or, to put it succinctly, as someone who had taken some and self-studied a fair amount more of real and complex analysis -- it is interesting stuff to me -- but I do not think that in-depth knowledge of it would help more than a tiny percentage of the day-to-day work that most engineers in this country do.

 

And lastly, I don't think that it is just a question of opinion. I think it can be a question of fact, though I suspect that no one has bothered to study it. I think it can be a question of fact to find out how many concepts from real and complex analysis would actually apply to the day-to-day calculations an average engineer does, someone just needs to study it.

 

Obviously, your mind is made up, but if you have some examples of day-to-day calculations that would be benefited significantly from in-depth knowledge of real and complex analysis, I'd like to know what they are. I just don't know of any that an average undergrad level engineer working today would benefit greatly from.

 

1. I said that a knowledge of real and complex analysis is needed to be adequately prepared to TEACH calculus, not to use it in the monkey-see monkey-do manner that you describe. It is needed for more sophisticated application of physical principles, most assuredly including fluid dynamics.

 

2. I have see quite a few applications in which the Navier-Stokes equation and the nature of solutions are very important. I have seen millions of dollars spent on such problems with billions at risk.

 

3. That MIT professor to whom I referred is a world-recognized expert in aerodynamics, and the Navier-Stokes equation is rather important to his work. His opinion is at odds with yours. I agree with him.

 

4. I have encountered quite a few PhD engineers who are woefully deficient in their understanding of mathematics. One more is not a big surprise.

 

5. You are quite right that the "average engineer" uses very little calculus. The "average engineer" also has a tendency to find himself in over his head with non-routine problems. I have been involved in several major failure and accident investigations as a result. These issues put lives at stake.

 

6. I am well-acquainted with refineries and chemical plants. Your argument does not impress me.

 

7. "Sizing a pump or flow rate through a packed bed" are useful topics, but of little use in the case of supersonic flow involving compressible gas. There are many facets to fluid dynamics and a head-in-the ground perspective that limits one's horizons is counter-productive to a broad understanding of the discipline.

 

8. I would certainly not hire anyone with the sort of blinders that you describe in what was my fluid dynamics group. There multi-phase analysis of reacting flow is not just an academic exercise, it is an everyday analysis task.

 

9. If you would like to see applications of real and complex analysis to air foil design you need look no farther than the work involving conformal maps or the work of Abraham Robinson.

 

10. Deep understanding of physics by virtue of understanding the associated mathematics may not reflect the work of an "average undergrad engineer", but "average undergrad engineers" do not handle much beyond routine problems. To do exceptional work, you need exceptional understanding.

 

11. The problem with your statement that you "don't see it as issues that even very experienced and good engineers with only an undergraduate degree need to be aware of" (referring to the basis of derived equations) is that unless one is at least "aware of" such issues one does not recognize when a situation arises in which they are crucial and the result can be catastrophic. This attitude is precisely why an understanding of the fundamentals is important. At the very least the average engineer needs to recognize when help is needed.

 

You have done nothing but convince me that more attention to fundamentals, not less, is what is needed. Your attitude is a symptom of the problem, not the solution.

 

Your last statement that "Obviously, your mind is made up," is an unwarranted statement, commonly adopted by someone who wishes to deflect attention that their own mind is closed.

[Bignose]

4. I have encountered quite a few PhD engineers who are woefully deficient in their understanding of mathematics. One more is not a big surprise.

 

I could do without the personal attack here. You have no knowledge of my mathematics ability, and I have no further wish to have any kind of discourse with you.

[Cap'n Refsmmat]

I'd argue that above deep mathematical understanding, scientists in difficult fields need the ability to engage in debate with others of differing opinions without resorting to insult or personal attack, particularly when your opponent is attempting to be civil.

 

DrRocket: Escalation and insults will persuade nobody in a debate. I'd like for this to stop now, so the thread can get back on topic, please.

[A Tripolation]

Or are we spending too much time on arithmetical computation in elementary schools?

 

This is what I agree with. And not just elementary schools. In high school as well. I took Calculus as a junior, and excelled. But it was only because it was what I see now as "algorithm execution". I didn't understand WHY my math was working, as I do now. I was just good at taking a series of steps, and manipulating them to find an answer. There was always an answer.

 

I feel that this has hindered my ability to understand more advanced maths, as I have trouble with some very basic abstract concepts. Intermediate vector analysis only makes sense if I think about it VERY carefully. I don't believe this is normal, and I attribute it to being taught to value my skill at carrying out a series of repetitive, meaningless steps.

[DrRocket]

This is what I agree with. And not just elementary schools. In high school as well. I took Calculus as a junior, and excelled. But it was only because it was what I see now as "algorithm execution". I didn't understand WHY my math was working, as I do now. I was just good at taking a series of steps, and manipulating them to find an answer. There was always an answer.

 

I feel that this has hindered my ability to understand more advanced maths, as I have trouble with some very basic abstract concepts. Intermediate vector analysis only makes sense if I think about it VERY carefully. I don't believe this is normal, and I attribute it to being taught to value my skill at carrying out a series of repetitive, meaningless steps.

 

That is exactly the problem with many high school calculus courses.

[Cap'n Refsmmat]

It also exactly describes the linear algebra course I took two semesters ago, which was explicitly designed for engineers and scientists. Only the math majors took the "matrix theory" course that taught the details.

[mississippichem]

Alright, so we've got some good replies here. Even though we had some disagreement; it seems that everyone agrees with the notion that high-school calculus courses lack rigor. I agree for the most part. When I took high school calculus, the focus was also very "symbol-pushy", I did well but probably couldn't have told you what a limit was at that time.

 

I've also noticed that in many of my undergraduate math courses, the higher calculus courses and differential equations, many of my rather intelligent class mates don't really seem to understand what is going on. Every time solving a problem requires additional logic or reasoning beyond an algorithm, some get drastically confused and feel cheated. I'm somewhat mathematically minded, so I do pretty well as far as my curriculum goes, but I feel that I meet too many students that have this problem. For example, I know a lot of students that can compute curl, but none of them seem to have any intuition as to what the curl is beyond [math] \nabla \times \vec{F} [/math].

 

So how do we fix this problem? We have a bunch of math zombies walking around who can compute whatever operation you need but have no clue what the operations are or mean. When you dump these folks into a higher level chemistry or physics, you get a disaster.

 

Do we teach deeper maths to younger children without symbolic manipulation, like giving intuition about limits, vectors, and things like that? Or is the problem in the college curriculum? Instead of teaching exhaustive use of algorithms do we try to instill mathematical intuition or reasoning? These questions are quite open ended still so don't be afraid to throw something out there.

[baric]

So how do we fix this problem?

 

You don't fix it. Some students actually enjoy math (I was one) and will dabble with numerical methods in their spare time. A desire to understand deeply cannot be taught.

 

What we want to do is readily identify those students who have that desire and ensure that it is properly fed.

[A Tripolation]

Do we teach deeper maths to younger children without symbolic manipulation, like giving intuition about limits, vectors, and things like that?

 

Again, this is what I agree with. Children have such a large capacity to learn new ideas. If we introduced the more abstract side of math when they were younger, instead of mindless computation, then I think these students would excel in the higher level courses.

[DrRocket]

Alright, so we've got some good replies here. Even though we had some disagreement; it seems that everyone agrees with the notion that high-school calculus courses lack rigor. I agree for the most part. When I took high school calculus, the focus was also very "symbol-pushy", I did well but probably couldn't have told you what a limit was at that time.

 

I've also noticed that in many of my undergraduate math courses, the higher calculus courses and differential equations, many of my rather intelligent class mates don't really seem to understand what is going on. Every time solving a problem requires additional logic or reasoning beyond an algorithm, some get drastically confused and feel cheated. I'm somewhat mathematically minded, so I do pretty well as far as my curriculum goes, but I feel that I meet too many students that have this problem. For example, I know a lot of students that can compute curl, but none of them seem to have any intuition as to what the curl is beyond [math] \nabla \times \vec{F} [/math].

 

So how do we fix this problem? We have a bunch of math zombies walking around who can compute whatever operation you need but have no clue what the operations are or mean. When you dump these folks into a higher level chemistry or physics, you get a disaster.

 

Do we teach deeper maths to younger children without symbolic manipulation, like giving intuition about limits, vectors, and things like that? Or is the problem in the college curriculum? Instead of teaching exhaustive use of algorithms do we try to instill mathematical intuition or reasoning? These questions are quite open ended still so don't be afraid to throw something out there.

 

The problem that you have identified, the confusion between symbol pushing and understanding is ubiquitous, and is reflective of the distinction between superficial knowledge and the deeper knowledge of true understanding.. I doubt that anything can be done about it at the primary/secondary levels in the foreseeable future because it first needs to be corrected at the university level.

 

So long as authority figures in science and engineering departments encourage superficial application of mathematics -- see the post of BigNose for a clear example -- then students will continue to actively avoid learning real mathematics. The problem is highlighted when those whose understanding of mathematics is clearly deficient are seen as experts. I have seen this carried to the extreme as a proposal for the College of Engineering to teach its own mathematics classes -- blind leading the blind. So long as the attitude exemplified in the post of BigNose is reflected in the attitude of students (engineering students have stated it as "Skip the theory, we just want to plug and chug") the problem that you describe will persist. It is manifested in industry in the form of people who can operate computer codes with great facility as a "black box" but who don't understand the output and who accept nonsensical "answers" from the code. I have seen this even from PhDs and university faculty.

 

In fact, one reason that I left engineering after an MS to pursue a PhD in mathematics was lack of understanding of necessary mathematics by several members of the engineering faculty. To use mathematics effectively in research, or even in non-routine applications, it is necessary to understand it. For instance, one cannot apply the methods of conformal mapping to airfoils if one believes that complex analysis is irrelevant to fluid dynamics -- see the post of BigNose.

 

One sees this problem mollified in the unfortunately rare instances in which educators in using departments have knowledge and appreciation for the body of knowledge that is mathematics. Engineers and physicists like Rudolph Kalman, David Luenberger, Ed Witten, Gene Covert, Charles Desoer, Walter Thirring, Charles Misner, John Archibald Wheeler, Roger Brockett, and Gary Brown have helped to counter the attitude of the adherents of the philosophy represented by the post of BigNose and encourage curiosity and a quest for deep understanding.

 

Ultimately the issue lies in the curiosity of students and their desire to understand. This first and foremost requires that they understand what it means to understand. Faculty and senior practitioners can help or hinder. Some help, others hinder.

[khaled]

I am from middle east, studying mathematics was not good part of my high-school experience, although I worked out a mathematical model called the love formula, and published it as a small software in VB 6 .. anyway, I'd like to ask for an advice on a good place to get free online courses on mathematics, and algorithms ...

[John Cuthber]

I don't know about maths education, but the sub-title of this thread should be "affects" rather than "effects" and it shouldn't have a capital letter.

There's not much point knowing all this clever stuff if you can't write it down.

On the whole, I agree that you can't teach science in any depth without an understanding of calculus and the curricula should reflect that.

[mississippichem]

I don't know about maths education, but the sub-title of this thread should be "affects" rather than "effects" and it shouldn't have a capital letter.

There's not much point knowing all this clever stuff if you can't write it down.

On the whole, I agree that you can't teach science in any depth without an understanding of calculus and the curricula should reflect that.

Affect...effect. Never could get it right. I frankly don't care enough to check in this informal setting.

[John Cuthber]

Affect...effect. Never could get it right. I frankly don't care enough to check in this informal setting.

Oddly, it seems many politicians say the same about maths.

[DrRocket]

Oddly, it seems many politicians say the same about maths.

 

 

They admit to not getting mathematics (some with pride). More troublesome, thought not readily admitted, are their problems with logic and ethics.