Redesigning Mathematics Curriculum, thoughts?

[RaijuRainBird]

I've had a pretty poor experience going through the standardized education system in California, and now that I'm in college, I'm really fed up with how mathematics is taught (even at the college level). With this said, I thought it would be fun for me to redesign the entire math education curriculum from scratch exactly the way I would want it to be. I think this might be a fun general discussion about how math is taught in the US.

Here is what I came up with: (The wording is a little strange, but that's because I wrote this as an answer to someone on Quora who was interested in relearning mathematics from the ground up -- something I've always wanted to do).

"

1. Go online and read the short paper “The Three Crises in Mathematics: Logicism, Intuitionism and Formalism” by Ernst Snapper.
2. Learn formal logic with: Introduction to Formal Logic by Peter Smith
3. Go online and read the Scientific American article “Dispute over Infinity Divides Mathematicians” by Natalie Wolchover (also in Quanta Magazine)
4. Learn Set theory with Karel Hrbacek and Thomas Jech. Introduction to Set Theory. Don’t get too bogged down with this, just enjoy the read and move on when you feel ready. Go back to it when the situation arises that you need it to move forward with math.
5. Read a little bit about (don’t read it all the way through; just enjoy it until you get tired of it; go back to it as you work through math and see how it all fits together) Category Theory with Lawvere, Conceptual mathematics: a first introduction to categories, 2nd Edition, 2009
6. Go online to the CSUSM Spring 2009 Math 378 course website by Prof. Aitken and download all of the class lecture notes (Ch. 0 - 10). Save them before they’re taken down, and work through these excellent notes as if they were a textbook. Learn it all as if you were given the Sports Alimak in Back to the Future series in 1985; it’s literally that good. This is the most important step in this entire list; if you do nothing else, at least do this.
7. Get a copy of Russell’s Principles of Mathematics on amazon: https://www.amazon.com/dp/0393314049/?tag=pfamazon01-20 Like the books on Set and Category theory, read it, but not like your life depended on it. If you mastered Smith’s book on formal logic, you will master this book too, and it will help you clarify things that seem like magic in mathematics. But know, Russell’s work isn’t the end all be all.
8. Learn about non-classical logic. Question the law of excluded middle; think for yourself — does it make sense to you? Do you believe physical reality follows this rule? All of formal mathematics from this point on, including calculus, is built on the idea that the law of excluded middle is right. In fact, even the books by Smith and Prof. Aitken, as well as all of Set Theory assume this notion. Maybe just let this question simmer in the back of your mind and continue to read about more mathematics. Don’t forget that it’s still a valid philosophical question.
9. Read Principles of Mathematical Analysis by Walter Rudin; accompany this read with lecture notes and free online midterm exams from Stanford’s Math 19, 20 and 21 and Harvard’s Math 1a, 1b and 112. Just google the course websites and use what you can find. You should realize that Prof. Aitken’s lecture notes should make this transition seamless. After all, his notes could could well be called “Analysis of the Natural Numbers, Arithmetic and Algebra.” He even covers some real and complex number stuff, so when you see Rudin, you should be in a very, very solid position to blow this material out of the water. Do it. When you have, congratulations, you’ve probably surpassed the majority of college graduates understanding of mathematics. But don’t stop here, you need to understand more than just two dimensional mathematics after all.
10. Read A First Course in Topology by James Munkres. This should be tons of review by this point. You should recognize things from set theory, real analysis and logic popping up everywhere. This should be an easy A, and it comes in handy as you move up to more than two dimensions.
11. Read Abstract Algebra by Dummit and Foote. Steps 9–11 could probably be done in any order you like, or ever simultaneously. This book should build on set theory, Aitken’s lecture notes (it’s impossible to understate how good these are) and topology should seem relevant here as well. All the stuff taught in high school math is explained here with sets and axioms.
12. Read Linear Algebra, Vector Calculus and Differential Forms, 5th edition, by Hubbard and Hubbard. Much like Rudin should have flowed seamlessly from Aitken, Hubbard and Hubbard should role off the tongue like butter to you now. You should easily grasp this material, and you should learn it because it’s important in real life. Accompany your reading with lecture videos of Math 3500/3510 by Shifrin on youtube (excellent lectures of an honors class covering multivariable math). Supplement Hubbard and Hubbard with either: 1) Linear Algebra by Levendosky, 2) Vector Calculus by Marsden and Tromba or 3) Multivariable Mathematics: Linear Algebra, Multivariable Calculus and Manifolds by Shifrin. It’s hard to say which is better. Don’t waste money buying all 3. Personally, I’d probably buy Shifrin based on his lecture videos, and also because Marsden and Tromba is on Scribd online. You can pickup Levendosky cheaply on amazon ($30 or less). If that sounds like a super good deal, buy it. It’s excellent. One last thing to add, the case could be made that Rudin will cover enough of this material to not bother with these books — that’s fair, I’d grab at least one of these just to get exposure to it though; if for no other reason than to understand physics and economics applications.
13. Go ahead and solidify your linear algebra because it’s really important from now on. Have Linear Algebra done Right by Axler at hand and take Berkeley’s Math 110 midterm and final exams before opening Axler’s book (you can find them online easily enough). If they are easy for you, just scan the table of contents of Axler and read anything that sounds unfamiliar; skip the rest unless you want to read it. If you want to do the HW from 110 as well, then pick up a copy of Linear Algebra by Friedberg, Insel and Spence (optional). Don’t spend too much time on this. Just make sure you have Hubbard and Hubbard, and Math 3500/3510 down really well (youtube). Glaze through Axler to patch up anything not covered in Abstract Algebra and multivariable mathematics.
14. Now you have gotten to the point where you can go online and buy any math book that interests you, and you should be able to just learn it with ease. Explore whatever you want. Algebraic topology, differential geometry, differential topology, complex analysis, physics, cryptology, computer science, statistics, anything at all. My advice, try to learn about multilinear algebra and tensors in depth. I don’t know why, but multivariable math textbooks don’t teach it, in fact the only school that I know of that teaches it is Stanford in their Math 52h class. The sky is the limit man! Have fun!

"

I feel like somewhere between steps 5 and 6 there should be a lesson on linguistics and the human limitations of semantic understanding, and how this shapes our ability to understand anything that's linguistic, like mathematics. And to highlight the difference between linguistic things and the things they represent.

 

[Lucas SV]

Nice list! First I was a bit skeptical because the list begins with a lot of logic oriented material, and category theory, which might be off-putting for some people, particularly because the applications cannot be seen just yet. But you did tone down a bit on this, saying not to get too bogged down with some of the material.

Then it starts going to Rudin's book which is personally more familiar territory for me (I'm a theoretical physicist, not mathematician, but still really like mathematics). Well of course all those topics are very important. I am assuming this would make a pure mathematics course of course, because of the omission of applied modules. Although you could say that these are optional courses at later years. My point is for students who do care about the applications, they would of course want to learn about them in the degree, and may be impatient to wish to learn it sooner rather than later. For example dynamical systems, differential equations, statistics, variational calculus, optimization, ... not to mention subjects that relate to the sciences.

I myself will use some of the material you stated to solidify my mathematical education, so thanks for that. Interests are algebraic topology, differential geometry, functional analysis, and in particular their applications to certain function spaces which appear in field theory.

 

[RaijuRainBird]

Nice list! First I was a bit skeptical because the list begins with a lot of logic oriented material, and category theory, which might be off-putting for some people, particularly because the applications cannot be seen just yet. But you did tone down a bit on this, saying not to get too bogged down with some of the material.

Then it starts going to Rudin's book which is personally more familiar territory for me (I'm a theoretical physicist, not mathematician, but still really like mathematics). Well of course all those topics are very important. I am assuming this would make a pure mathematics course of course, because of the omission of applied modules. Although you could say that these are optional courses at later years. My point is for students who do care about the applications, they would of course want to learn about them in the degree, and may be impatient to wish to learn it sooner rather than later. For example dynamical systems, differential equations, statistics, variational calculus, optimization, ... not to mention subjects that relate to the sciences.

I myself will use some of the material you stated to solidify my mathematical education, so thanks for that. Interests are algebraic topology, differential geometry, functional analysis, and in particular their applications to certain function spaces which appear in field theory.

I would have the applied stuff come after this list in the later years as you said. This way everyone knows exactly what is going on, and can fully grasp the applications with total understanding. For example, probability theory and differential geometry can come next, followed by a high level statistics course. Diff. equations should be easy enough to grasp after doing all this. It would probably suffice to hand out supplementary lecture notes to cover the material in a short time (like a few weeks, or simply just cover the material throughout a course that makes use of differential equations, presenting the material as a special case of a more general mathematical framework students already know).

So, for example, variation calculus can be learned within the physics class that contains its use. And more generally, regular old calculus should also be learned in physics class rather than in math class. In math class one should learn mathematics! In physics class, one should learn physics! Etc. I think this sort of curriculum would be very complementary and would produce graduates who knew their stuff extremely well.

I would hope that emphasizing the journey of understanding pure logic before mathematics, and the interesting things this approach teaches about human cognition, linguistics and linguistic understanding, would lead to a more fulfilling learning experience for students anyway, circumventing the desire for immediate applications.

 

[Lucas SV]

I would have the applied stuff come after this list in the later years as you said. This way everyone knows exactly what is going on, and can fully grasp the applications with total understanding. For example, probability theory and differential geometry can come next, followed by a high level statistics course. Diff. equations should be easy enough to grasp after doing all this. It would probably suffice to hand out supplementary lecture notes to cover the material in a short time (like a few weeks, or simply just cover the material throughout a course that makes use of differential equations, presenting the material as a special case of a more general mathematical framework students already know).

In my opinion, the applications are best learned on the fly after learning the logical build up that justifies them. So, for example, variation calculus can be learned within the physics class that contains its use. And more generally, regular old calculus should also be learned in physics class rather than in math class. In math class one should learn mathematics! In physics class, one should learn physics! Etc. I think this sort of curriculum would be very complementary and would produce graduates who knew their stuff extremely well.

I would hope that emphasizing the journey of understanding pure logic before mathematics, and the interesting things this approach teaches about human cognition, linguistics and linguistic understanding, would lead to a more fulfilling learning experience for students anyway, circumventing the desire for immediate applications.

Yes I understand this opinion, and to some extent I shared and still share it. But I'm not entirely convinced. The proper understanding of variational calculus, for instance is highly technical. I wouldn't want to deprive someone of using variational calculus without a fuller understanding, although you can argue that this is something they can do in their own time. However I certainly wouldn't want to wait until I can derive equations of motion and field equations from action principles.
In any case at least in physics it is true that we need to be able to apply mathematics without knowing the full theory, and even researchers do that, although often they need to improve their mathematics in order to deal with technicalities. One argument in favour of 'shortcut' learning at least for physics is the lack of time.

Now in the case of mathematics, it is a different story. But then what is meant by a math course? Nowadays it is common to do joint courses, say mathematics with economics. Also universities will have titles for the course such as Mathematics, Appllied Mathematics, Pure Mathematics. I do think the flexibility is a good thin; one is able to choose what they want. So one thing is to argue that there should exist a good number of courses like you describe. It is a much more extreme statement to say that all courses should be like that. I don't think you would get much support for the extreme statement.

Now whether you solution is the best way to learn mathematics (even if one wishes to apply it in the future), i don't know. Personally only experience will tell, if I compare the two approaches in the future, when I'm more acquainted with both, I will be able to judge better.

It may be interesting to look up the calculus trap.

 

[RaijuRainBird]

Yes I understand this opinion, and to some extent I shared and still share it. But I'm not entirely convinced. The proper understanding of variational calculus, for instance is highly technical. I wouldn't want to deprive someone of using variational calculus without a fuller understanding, although you can argue that this is something they can do in their own time. However I certainly wouldn't want to wait until I can derive equations of motion and field equations from action principles.
In any case at least in physics it is true that we need to be able to apply mathematics without knowing the full theory, and even researchers do that, although often they need to improve their mathematics in order to deal with technicalities. One argument in favour of 'shortcut' learning at least for physics is the lack of time.

Now in the case of mathematics, it is a different story. But then what is meant by a math course? Nowadays it is common to do joint courses, say mathematics with economics. Also universities will have titles for the course such as Mathematics, Applied Mathematics, Pure Mathematics. I do think the flexibility is a good thin; one is able to choose what they want. So one thing is to argue that there should exist a good number of courses like you describe. It is a much more extreme statement to say that all courses should be like that. I don't think you would get much support for the extreme statement.

Now whether you solution is the best way to learn mathematics (even if one wishes to apply it in the future), i don't know. Personally only experience will tell, if I compare the two approaches in the future, when I'm more acquainted with both, I will be able to judge better.

It may be interesting to look up the calculus trap.

I've never heard of the calculus trap until now, but it sounds very relatable!

I also agree that sometimes there just isn't enough time to do both math and, for example, engineering, or a science like physics, and learn them both really well, which is really unfortunate because then I think you get people who go on to be physicists or engineers without really understanding the mathematics too well, and this might lead them astray as they get more and more advanced; or worse, in might lead their intuition to funky places that just wouldn't make sense if they learned the math right from the begining.

As far as curriculum goes, and implementation, I agree that this might be a little impractical for everyday use, but at most schools something like this isn't even an option (like the calculus trap explains), and that's just too bad because I think it would serve some people really well.

Now, on the other hand, even if it's extreme, why not introduce formal logic in elementary school? Set and category theory too. I mean, category theory could easily be taught in a music class when teaching music theory (which is an extremely cool way to learn music). And set theory could be taught as an extension of logic, then students could be weaned off of constant problem solving and learn a little about the peano axioms, and what mathematical proof is all about, as well as learn that it actually is the foundation of everything that anyone who uses any math does with or without their knowledge of that fact.

Philosophy electives in high school can take this foundation and run further, teaching epistemological notions of consciousness for example. Metaphysics is also a great place to go from a school system like this, students shouldn't have their grades punished for these classes or anything, nor should they be required, but to have them just to encourage free thinking, and to bring back class discussion and a focus on the socratic method rather than standardized testing would be nice in my opinion.

______

But anyway, if you're actually interested in trying to learn the stuff from this list and see how it goes I'd be curious to hear about the results!

 

[Lucas SV]

Actually I think introducing logic, some further set theory and category theory in schools is a great idea. Again one needs to tone it down a bit, so one needs new material invented, new books, ... For example the first time i looked at category theory it was offputting because many examples I did not know at the time. Now I know some more. So bearing in mind the lack of mathematical experience, it is certainly a challange to create such material, but not impossible.

One argument one may have against this is saying that some students might never use this in their life. Well you could question this assumption itself, but another counterargument is that there are many things students learn in high school that they do no directly use in their life (although it may have an indirect influence) anyway, in other subjects. Moreover learning some logic could be benefitial for critical thinking, which is prety much applied everywhere. The International Baccalaureate, which I did, is a program that has a compulsory part called Theory of Knowledge, which does include logic and philosophy of mathematics. But it would be nice to also see logic in a maths class. Oh yes and as you said, philosophy can take this even further.

I do think that in school the mathematical education needs to change a lot. Personally, here in the UK, I see so many unmotivated students of maths. I believe this is because of how it is taught. At least students should know that mathematics is not just about a tool. Some people may go through high school without awareness of the existence of pure mathematics. People are not too keen in becoming calculators, they want to use their creative power.Yes, I can tell you the results of my learning, although it may take some time. I'm quite busy at the moment.

 

[Mark44]

I've had a pretty poor experience going through the standardized education system in California, and now that I'm in college, I'm really fed up with how mathematics is taught (even at the college level).

How so? I went through high school and some college in California, albeit lots of years ago, but it served me well. What exactly is the problem you're trying to fix?

The first 6 or 7 items on your list in the OP have to do with logic and set theory. The so-called "New Math" of the 60s also had these areas as priorities, and were taught with mixed success, to be charitable, as many of the teachers in public schools did not have strong enough backgrounds in mathematics to understand this stuff, let alone teach it.

And more generally, regular old calculus should also be learned in physics class rather than in math class. In math class one should learn mathematics! In physics class, one should learn physics!

Are you asserting that calculus isn't mathematics? Many of the applications of calculus are drawn from physics, but the underlying concepts are mathematical in nature.

Philosophy electives in high school can take this foundation and run further, teaching epistemological notions of consciousness for example. Metaphysics is also a great place to go from a school system like this, students shouldn't have their grades punished for these classes or anything, nor should they be required, but to have them just to encourage free thinking, and to bring back class discussion and a focus on the socratic method rather than standardized testing would be nice in my opinion.

Much of this seems very impractical to me, especially such esoteric subjects and epistemological notions of consciousness and metaphysics. A major problem at too many U.S. high schools is that students are graduating and are unable to do simple algebra or even grade-school arithmetic. I don't see teaching them metaphysics as being a viable alternative.

I would hope that emphasizing the journey of understanding pure logic before mathematics, and the interesting things this approach teaches about human cognition, linguistics and linguistic understanding, would lead to a more fulfilling learning experience for students anyway, circumventing the desire for immediate applications.

I agree that some knowledge of logic is necessary for students to be successful with proofs in geometry, and much later, linear algebra and beyond, but I don't see how human cognition and linguistics tie into or are important in a mathematics curriculum.

 

[Lucas SV]

A major problem at too many U.S. high schools is that students are graduating and are unable to do simple algebra or even grade-school arithmetic.

How do you think this issue can be resolved?

 

[RaijuRainBird]

How so? I went through high school and some college in California, albeit lots of years ago, but it served me well. What exactly is the problem you're trying to fix?

The first 6 or 7 items on your list in the OP have to do with logic and set theory. The so-called "New Math" of the 60s also had these areas as priorities, and were taught with mixed success, to be charitable, as many of the teachers in public schools did not have strong enough backgrounds in mathematics to understand this stuff, let alone teach it.

Are you asserting that calculus isn't mathematics? Many of the applications of calculus are drawn from physics, but the underlying concepts are mathematical in nature.
Much of this seems very impractical to me, especially such esoteric subjects and epistemological notions of consciousness and metaphysics. A major problem at too many U.S. high schools is that students are graduating and are unable to do simple algebra or even grade-school arithmetic. I don't see teaching them metaphysics as being a viable alternative.
I agree that some knowledge of logic is necessary for students to be successful with proofs in geometry, and much later, linear algebra and beyond, but I don't see how human cognition and linguistics tie into or are important in a mathematics curriculum.

First of all, the logic in geometry is, in fact, logic. It's no different from algebra.

Now, the idea is that the people who struggle with basic algebra and arithmetic are struggling for a reason, and I don't think it's systematically teachers' fault, or students' lack of effort, rather I believe that the curriculum is to blame. Maybe most of the students who struggle with basic algebra struggle because they can't see the logical framework underlying the algebra. Is it such a radical idea to teach abstract algebra to the students who don't seem to be accepting the notion that they just have to believe their teacher without question? Why not teach them group theory after logic? Show them that functions are the mapping between two sets, the domain and codomain. Ordered pairs are the cross product of two sets; ordered pairs are usually things that are just casually tossed around in high school without ever explaining what they are. Right? "They're just the X and Y coordinates" ... "Look, let's draw a cartesian plane, see, the ordered pair is a point on the plane -- rise over run." But what they don't tell you is that the real numbers are a field, and that the cartesian plane is the set R^2. Thus the cartesian plane is the set of ordered pairs formed by R cross multiplied with R. This is why a set of basis vectors spans the space, since the set of all linearly independent basis vectors can take on any ordered pair (in dim = 2). This is why dimension is defined as the cardinality of the set of basis vectors that span the space -- Ie, dim 2 = ordered pairs, dim 3 = ordered triples, dim 4 = ordered quadruples. This is why the number of components in a vector corresponds with dimension, an element of the basis of R^3 is a vector of the form [x, y, z] -- this is no coincidence. This is why dimension is often written as R^1, R^2, R^3 and R^4, since R^4 is R x R x R x R, therefore R^n is R x R ... n times. And if you think of a line as a set of points and the set of real numbers as the "real number line (which is the set of points in R)," then R^n represents a set of lines -- a set of real number lines, or the cardinality of the set of basis vectors of a space -- since the span of a vector goes off in either direction like a line. In other words, R^n represents the coordinate axis, and it's why R^2 is drawn with two real number lines. This is why the cartesian plane is justified and drawn like a cross. And in fact, the cartesian plane in two dimensions is {(x, y) | x,y e R}, and R^2 = {a[x] + a[y] | a e R}.

Why do we delude our students? Why can't we just tell them the truth so it's not surprising where the rules and properties in problem solving come from?

I believe that regular curriculum can be taught as special cases of the truth of mathematics (Gasp, what a novel idea!), and the philosophy classes are there to look at justifications for the axioms that are used without justification in mathematics -- of course they are assumptions, but to students who don't accept algebra, I can assume they'll also ask, say, "how can you be sure of the axiom of infinity?" Or, "How does it work in real life if there is no explanation for the axiom of infinity?" These questions are clearly metaphysical, so why not educate people on metaphysics? It seems like a plenty worthy subject to understand.

Finally, I do believe that Calculus is more like physics than mathematics. Calculus is to math is like what oranges are to orange juice in a cup. There's nothing interesting about calculus, you just take the concept of dividing by a number close to zero -- big deal. You find areas by stacking a bunch of rectangles next to each other and add them all up -- big deal. These are things that are needed primarily for physics; I mean, Newton believed there was no difference between math and physics when he created calculus. If you read his "Princpia Mathematica" you'll see that it was all based on euclidian geometry, and the idea that geometry was reality, and that reality was geometry. Therefore, by studying geometry he was studying physics, and by inventing calculus the intention was to explain the reality of physical objects. He was making the philosophical assumption that geometry is metaphysically true. Now, it seems like there are people who are in extreme disagreement over whether or not Mr. Newton was right about his metaphysical assumptions. In fact, nobody even knows today. As an aside, isn't this kind of embarrassing for academics? People don't even know what mathematics really is; ie, created vs discovered argument that has no answer because people actually do not know the answer. The approach I suggest is the logicist's approach, because to me it seems that the laws of logic are more fundamental and obvious than anything else. Unfortunately for me, 1) the law of excluded middle is questionable (though all mathematics assumes it's metaphysically true), and 2) most professional mathematicians either believe, like Newton, that physical reality is simply math, or they somehow believe that math is just a set of rules, and apparently the content of these rules don't matter. Either way, right now in today's world mathematics seems to be working, and right now everything is based on a set of rules, which is obviously ZFC. The point is to teach students about ZFC and why everything is currently the way it is in the field of mathematics, because there are no other reasons. I don't understand why students are left in the dark here.

The problem of educators not being educated enough to teach what needs to be taught is an alarming problem, but it all has to start somewhere. It doesn't make sense to constantly delude students by lying to them year after year as the story changes in mathematics with things like "oh, last year we just told you X because it's easier to understand. We actually lied to you, it's really Y." After a while of this it's no surprise that kids grow up hating math, and not learning it.

I think my solution is better.

 

[Mark44]

A major problem at too many U.S. high schools is that students are graduating and are unable to do simple algebra or even grade-school arithmetic.

How do you think this issue can be resolved?

I should say that my meaning was some students graduate without being able to do simple arithmetic. There are also significant numbers of them who are functionally illiterate. I base this on my years in a community college, and the large number of remedial classes in algebra and below and English.
I'm not sure that it can be solved, short of significant changes in (U.S.) society. My wife is a school psychologist, and she reports that there are many parents who don't place any value on education. Our schools here (U.S.) don't provide a track for jobs in the trades, and the students who don't do well don't have the intention or aptitude to succeed in college.

 

[Mark44]

First of all, the logic in geometry is, in fact, logic. It's no different from algebra.

I disagree. In algebra there are a number of properties and laws, such as the addition property of equations, and the laws of exponents, to name a couple. To my mind, these are quite different from and less sophisticated than such concepts of proofs as contrapositives or contradictions.

Now, the idea is that the people who struggle with basic algebra and arithmetic are struggling for a reason, and I don't think it's systematically teachers' fault, or students' lack of effort, rather I believe that the curriculum is to blame.

Based on what evidence?

Maybe most of the students who struggle with basic algebra struggle because they can't see the logical framework underlying the algebra. Is it such a radical idea to teach abstract algebra to the students who don't seem to be accepting the notion that they just have to believe their teacher without question?

Yes, that is a radical idea. Good teachers will give an explanation for why some property or law works the way it does.

Why not teach them group theory after logic? Show them that functions are the mapping between two sets, the domain and codomain. Ordered pairs are the cross product of two sets; ordered pairs are usually things that are just casually tossed around in high school without ever explaining what they are. Right?
"They're just the X and Y coordinates" ... "Look, let's draw a cartesian plane, see, the ordered pair is a point on the plane -- rise over run." But what they don't tell you is that the real numbers are a field, and that the cartesian plane is the set R^2.

And how would knowledge of the properties of a field help them solve an equation like 3x + 7 = 16, something that many of them struggle with?

Why do we delude our students? Why can't we just tell them the truth so it's not surprising where the rules and properties in problem solving come from?

To paraphrase Jack Nicholson in "A Few Good Men": "They can't handle the truth!"

I believe that regular curriculum can be taught as special cases of the truth of mathematics (Gasp, what a novel idea!), and the philosophy classes are there to look at justifications for the axioms that are used without justification in mathematics -- of course they are assumptions, but to students who don't accept algebra, I can assume they'll also ask, say, "how can you be sure of the axiom of infinity?" Or, "How does it work in real life if there is no explanation for the axiom of infinity?" These questions are clearly metaphysical, so why not educate people on metaphysics? It seems like a plenty worthy subject to understand.

What exactlyl is the "axiom of infinity?" I have a bachelor's and a master's, both in mathematics, and this is not a term I've ever heard. I disagree strongly that metaphysics is something that needs to be taught or that philosophy be taught as a prerequisite to mathematics. I should warn you that discussions of philosophy aren't permitted at this site.

Finally, I do believe that Calculus is more like physics than mathematics. Calculus is to math is like what oranges are to orange juice in a cup.

The rate of change of a function and the area under a curve are purely mathematical. As I said before, the physics comes when you attach these concepts to some application.

There's nothing interesting about calculus, you just take the concept of dividing by a number close to zero -- big deal. You find areas by stacking a bunch of rectangles next to each other and add them all up -- big deal.

The concept of the limit, which is used in the definitions of both the derivative and the definite integral, wasn't crystallized until long after Newton (and Leibniz) did their work. These are purely mathematical concepts.

These are things that are needed primarily for physics; I mean, Newton believed there was no difference between math and physics when he created calculus. If you read his "Princpia Mathematica" you'll see that it was all based on euclidian geometry, and the idea that geometry was reality, and that reality was geometry. Therefore, by studying geometry he was studying physics, and by inventing calculus the intention was to explain the reality of physical objects.

It's well known that physicists and other scientists use mathematics as a tool. What you're saying sounds to me like you're equating the hammer a carpenter uses with the stairs that he builds.

He was making the philosophical assumption that geometry is metaphysically true.

I would say this as, "He was making the philosophical assumption that geometry is metaphysically true grounded in reality."

Now, it seems like there are people who are in extreme disagreement over whether or not Mr. Newton was right about his metaphysical assumptions.

What we do know is that a lot of Newton's writings were on mysticism, which are irrlevant to mathematicians generally.

In fact, nobody even knows today. As an aside, isn't this kind of embarrassing for academics? People don't even know what mathematics really i; ie, created vs discovered argument that has no answer because people actually do not know the answer. The approach I suggest is the logistics approach, because to me it seems that the laws of logic are more fundamental and obvious than anything else. Unfortunately for me, 1) the law of excluded middle is questionable (though all mathematics assumes it's metaphysically true)

We don't add the word "metaphysically."

, and 2) most professional mathematics either believe, like Newton, that physical reality is simply math, or they somehow believe that math is just a set of rules, and apparently the content of these rules don't matter. Either way, right now in today's world mathematics seems to be working, and right now everything is based on a set of rules, which is obviously ZFC. The point is to teach students about ZFC and why everything is currently the way it is in the field of mathematics, because there are no other reasons. I don't understand students are left in the dark here.

The problem of educators not being educated enough to teach what needs to be taught is an alarming problem, it all has to start somewhere though. It doesn't make sense deluding students, constantly, in effect, lying to them year after year as the story changes in mathematics with things like "oh, last year we just told you X because it's easier to understand. We actually lied to you, it's really Y." After a while of this it's no surprise that kids grow up hating math, and not learning it.

I think my solution is better.

And in my opinion, much of what you propose is completely unworkable.

 

[RaijuRainBird]

I disagree. In algebra there are a number of properties and laws, such as the addition property of equations, and the laws of exponents, to name a couple. To my mind, these are quite different from and less sophisticated than such concepts of proofs as contrapositives or contradictions.
Based on what evidence?
Yes, that is a radical idea. Good teachers will give an explanation for why some property or law works the way it does.
And how would knowledge of the properties of a field help them solve an equation like 3x + 7 = 16, something that many of them struggle with?
To paraphrase Jack Nicholson in "A Few Good Men": "They can't handle the truth!"
What exactlyl is the "axiom of infinity?" I have a bachelor's and a master's, both in mathematics, and this is not a term I've ever heard. I disagree strongly that metaphysics is something that needs to be taught or that philosophy be taught as a prerequisite to mathematics. I should warn you that discussions of philosophy aren't permitted at this site.
The rate of change of a function and the area under a curve are purely mathematical. As I said before, the physics comes when you attach these concepts to some application.
The concept of the limit, which is used in the definitions of both the derivative and the definite integral, wasn't crystallized until long after Newton (and Leibniz) did their work. These are purely mathematical concepts.
It's well known that physicists and other scientists use mathematics as a tool. What you're saying sounds to me like you're equating the hammer a carpenter uses with the stairs that he builds.
I would say this as, "He was making the philosophical assumption that geometry is metaphysically true grounded in reality."
What we do know is that a lot of Newton's writings were on mysticism, which are irrlevant to mathematicians generally.
We don't add the word "metaphysically."

And in my opinion, much of what you propose is completely unworkable.

Well, this is the axiom of infinity: https://en.wikipedia.org/wiki/Axiom_of_infinity It's what justifies mathematical induction by asserting, axiomatically, that at least one infinite set exists. In other words, it's impossible to know. We can only count finitely, and so, it requires faith. Constructivist mathematicians reject this axiom and carry on without it, this means any proofs that rely on mathematicall induction become invalid. The difference between constructivist math and regular math is a metaphysical distinction, in particular, the axiom of infinity. I say students deserve to know that this controversy exists, and they deserve to know why, think for themselves, and come to their own conclusions. Hence, metaphysics is something good to teach in high school.

As for how abstract algebra helps a student find "3x + 7 = 16," for a start the definition of a field justifies addition and multiplication, for which subtraction and division can then be derived. Here is the wiki on fields: https://en.wikipedia.org/wiki/Field_(mathematics)

 

[Mark44]

Well, this is the axiom of infinity: https://en.wikipedia.org/wiki/Axiom_of_infinity It's what justifies mathematical induction by asserting, axiomatically, that at least one infinite set exists.

Any comments on the other questions I asked?

As for how abstract algebra helps a student find "3x + 7 = 16," for a start the definition of a field justifies addition and multiplication, for which subtraction and division can then be derived. Here is the wiki on fields: https://en.wikipedia.org/wiki/Field_(mathematics)

Students are typically exposed to problems like this one in the 8th or 9th grades (it was 9th grade algebra for me, back in the late 50s). And your intention is to teach them about the field axioms before they solve problems like this? Good luck with that...

 

[RaijuRainBird]

Any comments on the other questions I asked?
Students are typically exposed to problems like this one in the 8th or 9th grades (it was 9th grade algebra for me, back in the late 50s). And your intention is to teach them about the field axioms before they solve problems like this? Good luck with that...

This is exactly the intention. I don't see why it has to be any different from what we do today. We would just introduce sets early, and teach basic logic like "if a then b" stuff early as well. Then teach them about the rules of sets, then introduce group theory. That way when one approaches regular algebra there is no need for confusion. Set theory is a branch of pure logic, it shouldn't be hard to grasp because there's really not much going on in set theory. It's very general.

As for the calculus stuff, well, analysis is mathematical, but computational calculus classes that don't teach anything but following the rules to get a numerical answer is not really mathematics to me. In fact, I'd argue that it actually isn't math, it's whatever the math is being used for in the particular computational problem being solved. If it's a physics question, then it's physics. If there is an economics question on a HW, then the student is doing economics, not math. If it's a statistical question, then are doing statistics, not math. Anyway, I'd say just teach real analysis instead of "calculus for engineers, mathematics and scientists" as it's normally done, because "calculus for engineers" implies that people don't want to understand math; and it never even tries to teach for understanding.

Moreover, I feel like today people's understanding of math is artificial. There's just something wrong with that.

People think that physics, economics and statistics is math, but it isn't. But all they've been doing in all their supposed "math" classes all this time actually hasn't been math, it's been a smorgasbord of assorted questions in other fields.

Okay, but there's nothing wrong with physics and the likes. In fact, I love physics, but I don't understand why math has to be the slave to the sciences and engineers at the expense of understanding. If we taught math in math class and economics in economics class (in terms of the general math theory, again, what a novel concept) I believe our education would be greatly improved. Eventually, since we'll start teaching math in math class, the economists will be able to teach economics in terms of the general framework. This way the students can focus on the core concepts, of, say, economics and spend little time applying math to it. Likewise, in math class students don't get bogged down with economics questions.

The curriculum should not be biased in this way as it currently is. Math should stand on its own and its classes should teach mathematics.

___

Here is an example of a HW assignment that can be done in the 8th grade: "Suppose that a and b are two elements of a field F.
Using only the axioms for a field, prove the following:
– ∀a ∈ F, 0a = 0.
– If ab = 0, then either a or b must be 0.
– The additive inverse of a is unique."

This is totally feasible.

 

[Lucas SV]

My wife is a school psychologist, and she reports that there are many parents who don't place any value on education.

That is a good point. Probably the greater part of the problem is a social issue. I do think teachers should try to motivate the students as much as they can, be there is only so much they can do. Part of the responsability will lie with the parents.

 

[RaijuRainBird]

That is a good point. Probably the greater part of the problem is a social issue. I do think teachers should try to motivate the students as much as they can, be there is only so much they can do. Part of the responsability will lie with the parents.

For the record, this is a separate, but very important issue. I totally agree with this.

 

[Lucas SV]

This is exactly the intention. I don't see why it has to be any different from what we do today. We would just introduce sets early, and teach basic logic like "if a then b" stuff early as well. Then teach them about the rules of sets, then introduce group theory. That way when one approaches regular algebra there is no need for confusion. Set theory is a branch of pure logic, it shouldn't be hard to grasp because there's really not much going on in set theory. It's very general.

Ok I agree that basic logic is important and I personally never found it too difficult. It is in fact introduced often in say, books on analysis, because of the need for proofs.

Now, the idea is that the people who struggle with basic algebra and arithmetic are struggling for a reason, and I don't think it's systematically teachers' fault, or students' lack of effort, rather I believe that the curriculum is to blame. Maybe most of the students who struggle with basic algebra struggle because they can't see the logical framework underlying the algebra.

Well that is a bit a generalization. Often it is the teachers fault, and/or the student lack of effort. You can certainly ask what are the causes of these issues, and I think they are sociological, cultural and psychological in nature, for the most part.

It depends on what you mean by logical framework. If you mean what I think you mean, i.e. learning the required subjects for the construction of the real numbers as in done in analysis, even before one can solve equations, I think is not effective. I have two main reasons for this.

Firstly there is a logical framework in doing algebraic manipulations. There are some rules, and by following these rules one learns how to manipulate abstract symbols. A computer indeed does the same thing, and just because I said we don't like being computers, doesn't mean computation isn't a very significant part of mathematics. And when we are doing manipulations, we are doing proofs too, so I think this is what needs to be emphasized.

The field axioms for real numbers is certainly known to students, even though they don't know it is called 'field axioms' or that mathematicians may think of a multitutde of examples of fields. They are the basic rules which are taught early on about algebraic manipulations. I do think the teacher should point out the name 'field', and maybe even give a definition. This is tricky however, because of the important fact that not many examples of fields are known to the student at this point, so it can be confusing.

Secondly, logicians also do manipulations. They also manipulate symbols in logic. There are also rules of logic they are following. They also compute, even if what they compute may be more abstract. Even logic has computation.

It is easier to learn how to use a set of rules to do manipulations with objects that are more concrete. Hence why solving equations is taught first. A number 5.4 is certainly more concrete than a mathematical proposition.

So I'm in favour of teaching basic logic simultaneously with solving equations, so that the connection between proofs and the algebraic manipulations can be seen. However I am not in favour of teaching logic and set theory strictly before teaching how to solve quadratic equations, because of the lack of concrete examples.

Actually this is a point I have been mentioning in these posts: the examples. We learn much more, and in a better way, by knowing examples and doing exercises. Well of course you should know this (actually do you? How long have you been in college for?). This is not logic's fault. It just so happens that are our brains have evolved in that way. Also, not only we remember concepts that we learned by example and exercise, we also remember what has been learned visually. There are many concepts I can remember in topology and analysis precisely because I have a visual picture of them. I usually remember a definition after I remember the picture or concept. That is why I remember a lot, not because I have memorized. Actually things i do not remember tend to be correlated with things I have not done many exercises in.

So if you are going to design a class where it is not even posible to come up with examples because the students simply do not know them, you will be in trouble.

 

[Lucas SV]

Finally, I do believe that Calculus is more like physics than mathematics. Calculus is to math is like what oranges are to orange juice in a cup. There's nothing interesting about calculus, you just take the concept of dividing by a number close to zero -- big deal. You find areas by stacking a bunch of rectangles next to each other and add them all up -- big deal.

Can you really say you can do computations in calculus if you have not practised? Sure, we learn how to explain those computations more precisely after courses in analysis, but I feel like ommiting calculus is wasteful, since even historically analysis was motivated by calculus in the first place. Also, if you know some analysis\calculus you should know computations in calculus are not as easy and trivial as you described. They can be extremely difficult. The rigorous logic behind the subject serves to make the problems of calculation more visible and clear, but they do not necessarely make it simpler.

That actually gives me an opportunity to say that historically both the development of physics and mathematics are highly interconnected. While mathematics provided great tools to solve physical problems, physics provided great motivation to develop further mathematics. Yes, Newton did invent calculus as a tool, because he wanted to calculate the gravitational pull of the moon by the Earth and the mathematics of his day was not good enough to solve this problem. I think you should delve more deeply into the history of mathematics and you will see what I mean.

Recently string theory has motivated and even produced a ton of new mathematics. Many calculations mathematicians were struggling with were solved by string theorists even before they could catch up - and yes the string theorists did invent new mathematics. In fact, one of the fathers of string theory, Edward Witten, was the first (and so far the only) physicist to be awarded the fields medal.

When I go into research I want to communicate with mathematicians, even though I am a theoretical physicist. I have strong faith in this interplay which historically has been so successfull. It is inefficient for us to isolate ourselves from each other, when so very often our problems are very similar in nature, even though the language describing the problems and the motivation behind doing the problems might be different.

 

[RaijuRainBird]

Can you really say you can do computations in calculus if you have not practised? Sure, we learn how to explain those computations more precisely after courses in analysis, but I feel like ommiting calculus is wasteful, since even historically analysis was motivated by calculus in the first place. Also, if you know some analysis\calculus you should know computations in calculus are not as easy and trivial as you described. They can be extremely difficult. The rigorous logic behind the subject serves to make the problems of calculation more visible and clear, but they do not necessarely make it simpler.

That actually gives me an opportunity to say that historically both the development of physics and mathematics are highly interconnected. While mathematics provided great tools to solve physical problems, physics provided great motivation to develop further mathematics. Yes, Newton did invent calculus as a tool, because he wanted to calculate the gravitational pull of the moon by the Earth and the mathematics of his day was not good enough to solve this problem. I think you should delve more deeply into the history of mathematics and you will see what I mean.

Recently string theory has motivated and even produced a ton of new mathematics. Many calculations mathematicians were struggling with were solved by string theorists even before they could catch up - and yes the string theorists did invent new mathematics. In fact, one of the fathers of string theory, Edward Witten, was the first (and so far the only) physicist to be awarded the fields medal.

When I go into research I want to communicate with mathematicians, even though I am a theoretical physicist. I have strong faith in this interplay which historically has been so successful. It is inefficient for us to isolate ourselves from each other, when so very often our problems are very similar in nature, even though the language describing the problems and the motivation behind doing the problems might be different.

I do like how you bring up the interconnectedness of math and physics. I guess I have to concede to this point. I also know that the calculations can be very difficult.

As a theoretical physicist, do you ever think it's possible to know the explanation before the result? I think the root of what I'm saying involves what might be a fact that understanding probably is always historically forced to come after doing. Since it's impossible to explain before experiencing. Maybe math is taught in this same spirit, that first you must accidentally, or mysteriously, figure out something and then you must go back to figure out why -- where as I'd rather try to determine what's logically possible before trying anything. And then, once all possibilities are known through the laws of logic itself, then anything that happens is explainable. But you know, this is probably never how it can be. So I'd understand it if those who designed our curriculum wanted to train people for real work in the field, that they'd feel that students have to get used to trying first and explaining later. So this might be the actual reason for the current way math is taught.

 

[Lucas SV]

I do like how you bring up the interconnectedness of math and physics. I guess I have to concede to this point. I also know that the calculations can be very difficult.

As a theoretical physicist, do you ever think it's possible to know the explanation before the result? I think the root of what I'm saying involves what might be a fact that understanding probably is always historically forced to come after doing. Since it's impossible to explain before experiencing. Maybe math is taught in this same spirit, that first you must accidentally, or mysteriously, figure out something and then you must go back to figure out why -- where as I'd rather try to determine what's logically possible before trying anything. And then, once all possibilities are known through the laws of logic itself, then anything that happens is explainable. But you know, this is probably never how it can be. So I'd understand it if those who designed our curriculum wanted to train people for real work in the field, that they'd feel that students have to get used to trying first and explaining later. So this might be the actual reason for the current way math is taught.

I think it is possible to have some insights about the explanation, but one must work hard to have the full explanation, if the problem is difficult. I also thinking there are layers to understanding. When Newton discovered calculus, he did not suddenly know everything there is to know about calculus. It was a basic level of understanding which was improved and deepened as time went by. So now we understand much better, thanks to the work of many individuals. And by the way, it is conceivable, that maybe we will never have the 'full' knowledge of a certain subject.

Yes you are probably right that the way mathematics is taught mimics the historical discoveries, to some extent. That is why we learn things that have been discovered in the 17th century first, before learning what was discovered in the 20th century, and this trend is not only true in mathematics.

The idea of determining what is logically possible before trying anything is excellent, and this is why it is so widely used nowadays. Even physicists use it. Particle physics is based on some constraints (like Lorentz invariance) which determine what is possible and what is not possible. We try to logically derive what is impossible given the assumptions of our theory, but sometimes it is very hard to be rigorous. This is certainly one good reason to communicate with mathematicians.

So even the problem of determining what is logically possible is far from trivial, and the technicalities highly depend on what you are studying, but this approach is very fruitful in research. That doesn't mean that research focused on calculation is also not fruitful - it is.

And yes, real work requires good problem solving skills which can only be acquired by practise and being able to solve complex problems with the tools you have at your disposal - and not falling for the calculus trap. Of course one day, someone comes along, and decides to invent a new tool to solve a problem, which is much more effective than the tool used by peers, and everyone is amazed!

 

[Mark44]

This is exactly the intention. I don't see why it has to be any different from what we do today. We would just introduce sets early, and teach basic logic like "if a then b" stuff early as well. Then teach them about the rules of sets, then introduce group theory. That way when one approaches regular algebra there is no need for confusion. Set theory is a branch of pure logic, it shouldn't be hard to grasp because there's really not much going on in set theory. It's very general.

As I mentioned already, this was tried back in the 70s or so, and it wasn't very successful, to my recollection. All of this was new to the football coach who also taught some math classes, so things ended up in a hash.

As for the calculus stuff, well, analysis is mathematical, but computational calculus classes that don't teach anything but following the rules to get a numerical answer is not really mathematics to me.

That's an opinion that probably isn't shared by many people. Finding a limit using the $\delta - \epsilon$ definition of the limit is by just about anyone's definition, mathematics, as is taking the derivative of a function by using the limit definition of the derivative. In addition, finding an antiderivative of a function is not just "plug and chug" as you seem to imply.

In fact, I'd argue that it actually isn't math, it's whatever the math is being used for in the particular computational problem being solved.

I disagree, for the same reason I gave before -- a carpenter uses a hammer to build some stairs. The stairs and the hammer are different things.
Being able to apply mathematics to other fields is a good thing, and helps to motivate students in ways that straight math problems don't. How many "find the slope of the tangent line" problems or "find the area under the curve" problems can you do before total boredom sets in?

If it's a physics question, then it's physics. If there is an economics question on a HW, then the student is doing economics, not math.

No. The student is using mathematics to answer a question about physics or economics or whatever.

If it's a statistical question, then are doing statistics, not math. Anyway, I'd say just teach real analysis instead of "calculus for engineers, mathematics and scientists" as it's normally done, because "calculus for engineers" implies that people don't want to understand math; and it never even tries to teach for understanding.

Horsefeathers.
I taught calculus for many years in a college to students who were primarily going into engineering. If they wanted a decent grade out of the class, they had to understand the mathematics.

Moreover, I feel like today people's understanding of math is artificial. There's just something wrong with that.

People think that physics, economics and statistics is math, but it isn't. But all they've been doing in all their supposed "math" classes all this time actually hasn't been math, it's been a smorgasbord of assorted questions in other fields.

Which is not a bad thing, as I already mentioned.

Okay, but there's nothing wrong with physics and the likes. In fact, I love physics, but I don't understand why math has to be the slave to the sciences and engineers at the expense of understanding. If we taught math in math class and economics in economics class (in terms of the general math theory, again, what a novel concept) I believe our education would be greatly improved. Eventually, since we'll start teaching math in math class, the economists will be able to teach economics in terms of the general framework. This way the students can focus on the core concepts, of, say, economics and spend little time applying math to it. Likewise, in math class students don't get bogged down with economics questions.

What's the harm? A question that comes up in economics is the point where the supply curve intersects the demand curve. The concept is pure economics, but finding the intersection point requires the ability to solve a pair of simultaneous equations, which is a concept in mathematics.

The curriculum should not be biased in this way as it currently is. Math should stand on its own and its classes should teach mathematics.

___

Here is an example of a HW assignment that can be done in the 8th grade: "Suppose that a and b are two elements of a field F.
Using only the axioms for a field, prove the following:
– ∀a ∈ F, 0a = 0.
– If ab = 0, then either a or b must be 0.
– The additive inverse of a is unique."

The last one would be tricky for an 8th grader, as it would pretty much require that they know how to do a proof by contradiction. Pretty sophisticated stuff when some 8th graders can't figure out 7 x 8 without a calculator or how to divide 140 by 10 in their heads.

This is totally feasible.

 

[RaijuRainBird]

Being able to apply mathematics to other fields is a good thing, and helps to motivate students in ways that straight math problems don't. How many "find the slope of the tangent line" problems or "find the area under the curve" problems can you do before total boredom sets in?

You're right, instead of asking to find slopes of the tangent line 50,000 times why don't we use that time to learn about the set theoretic definitions of functions, the domain and codomain, inverses, images and preimages? How about the intermediate value theorem? Why not learn about fields and ordered pairs. It might be nice to learn what this thing called the cartesian plane they keep using is, and how it's related to dimension and functions. This would be a great time to learn about the set theoretic definition of a line and a plane, and vectors.

A blurb from an earlier post I edited in after the fact:
"Ordered pairs are the cross product of two sets; ordered pairs are usually things that are just casually tossed around in high school without ever explaining what they are. Right? "They're just the X and Y coordinates" ... "Look, let's draw a cartesian plane, see, the ordered pair is a point on the plane -- rise over run." But what they don't tell you is that the real numbers are a field, and that the cartesian plane is the set R^2. Thus the cartesian plane is the set of ordered pairs formed by R cross multiplied with R. This is why a set of basis vectors spans the space, since the set of all linearly independent basis vectors can take on any ordered pair (in dim = 2). This is why dimension is defined as the cardinality of the set of basis vectors that span the space -- Ie, dim 2 = ordered pairs, dim 3 = ordered triples, dim 4 = ordered quadruples. This is why the number of components in a vector corresponds with dimension, an element of the basis of R^3 is a vector of the form [x, y, z] -- this is no coincidence. This is why dimension is often written as R^1, R^2, R^3 and R^4, since R^4 is R x R x R x R, therefore R^n is R x R ... n times. And if you think of a line as a set of points and the set of real numbers as the "real number line (which is the set of points in R)," then R^n represents a set of lines -- a set of real number lines, or the cardinality of the set of basis vectors of a space -- since the span of a vector goes off in either direction like a line. In other words, R^n represents the coordinate axis, and it's why R^2 is drawn with two real number lines. This is why the cartesian plane is justified and drawn like a cross. And in fact, the cartesian plane in two dimensions is {(x, y) | x,y e R}, and R^2 = {a[x] + a[y] | a e R}. "

What's the harm? A question that comes up in economics is the point where the supply curve intersects the demand curve. The concept is pure economics, but finding the intersection point requires the ability to solve a pair of simultaneous equations, which is a concept in mathematics.

There is no harm in economics class -- everyone should already know how to solve systems of linear equations from math class.

As I mentioned already, this was tried back in the 70s or so, and it wasn't very successful, to my recollection. All of this was new to the football coach who also taught some math classes, so things ended up in a hash.

But, if the football coach had been educated properly himself, then this wouldn't have been a problem right?

The last one would be tricky for an 8th grader, as it would pretty much require that they know how to do a proof by contradiction. Pretty sophisticated stuff when some 8th graders can't figure out 7 x 8 without a calculator or how to divide 140 by 10 in their heads.

Well, perhaps they should know about number systems and group theory. Then they would understand where addition and multiplication come from. Perhaps they should also study some philosophy, such as, oh I don't know, metaphysics and epistemology so that they can think about what it means for a quantity to exist, and some symbolic logic so they can take a look at the peano axioms.

 

[Lucas SV]

R^n represents a set of lines

Off-topic, but this is the great concept of a quotient space.

Well, perhaps they should know about number systems and group theory.

Well, doing a proof in group theory requires more skill than doing 7x8 without a calculator, probably. Ultimately, the answer comes from experiment. So if you think someone can do proofs in group theory before doing 7x8 mentally, feel free to test it (within the bounds of law of course), or search for related experiments.

I am assuming by 'knowing about group theory' you really mean being able to do proofs.

Then they would understand where addition and multiplication come from.

Actually as obvious as it may seem to you, even this is questionable. The question of where addition of integers comes from looks suspiciusly similar to the question of where the axioms of group theory come from. Well, that is not too surprising. So by introducing the axioms of group theory have you really explained something deep about where does addition come from? You are not really giving new information. Maybe you can only claim this from peano's axioms.

Don't forget that the reason the axioms of group theory exist in the first place is to categorize. Mathematicians were both clever and lazy because instead of repeating the same kind of proof over and over for different mathematical objects, they categorized the objects as groups and said the theorems are valid for all such objects. However, for an 8th grader, there are not many objects they know of to categorize, so it is a bit pointless to introduce the general concept. But the proofs about addition of integers which are 'group theoretic' in nature will be essentially the same as in group theory.

 

[Drakkith]

So, for example, variation calculus can be learned within the physics class that contains its use. And more generally, regular old calculus should also be learned in physics class rather than in math class. In math class one should learn mathematics! In physics class, one should learn physics! Etc. I think this sort of curriculum would be very complementary and would produce graduates who knew their stuff extremely well.

I strongly disagree. Once you reach the point in physics where you actually need to use calculus to understand the material and solve problems, you already need to know how to use calculus! And not just basic, single-variable calculus, but usually multi-variable calculus with triple integrals, differential equations, and the like. You absolutely cannot try to learn both at the same time, not when you need several hundred hours of time just to learn the math portion before you can even start to understand the physics portion.

It doesn't make sense to constantly delude students by lying to them year after year as the story changes in mathematics with things like "oh, last year we just told you X because it's easier to understand. We actually lied to you, it's really Y." After a while of this it's no surprise that kids grow up hating math, and not learning it.

I doubt this has much to do with why kids hate math. It's far more likely that they hate math because it is difficult, requiring a lot of time and effort to understand, and, unique to math and subjects closely related to math, if you fall behind it is extra-punishing because you can't even understand the later material, leading to a larger and larger gap in your knowledge as time passes. Not to mention the fact that this occurs in many areas, not just math. It's commonly understood that you have to start simple and then move on to the complex.

Well, perhaps they should know about number systems and group theory. Then they would understand where addition and multiplication come from.

Well, unless they didn't learn number systems and group theory well enough to understand where addition and multiplication come from. Honestly I think you're expecting far too much from the average student. Most of the material I've seen you link or talk about appears fairly advanced and very abstract. I have a very hard time believing you can teach it to anyone below high school.

 

[Mark44]

Being able to apply mathematics to other fields is a good thing, and helps to motivate students in ways that straight math problems don't. How many "find the slope of the tangent line" problems or "find the area under the curve" problems can you do before total boredom sets in?

You're right, instead of asking to find slopes of the tangent line 50,000 times why don't we use that time to learn about the set theoretic definitions of functions, the domain and codomain, inverses, images and preimages?

They should already have been presented with most or all of these ideas and the class leading up to calculus. They certainly were in the college I taught at for 18 years.

How about the intermediate value theorem?

This is typically taught in the first quarter or semester of calculus. Aren't you aware of this?

Why not learn about fields and ordered pairs.

Ordered pairs -- precalculus. Fields -- not typically taught at the calculus level.

It might be nice to learn what this thing called the cartesian plane they keep using is

Precalculus...

, and how it's related to dimension and functions.

What's the harm? A question that comes up in economics is the point where the supply curve intersects the demand curve. The concept is pure economics, but finding the intersection point requires the ability to solve a pair of simultaneous equations, which is a concept in mathematics.

There is no harm in economics class -- everyone should already know how to solve systems of linear equations from math class. .

"Everyone should already know..." Unfortunately, that's not a fact you can count on.

As I mentioned already, this was tried back in the 70s or so, and it wasn't very successful, to my recollection. All of this was new to the football coach who also taught some math classes, so things ended up in a hash.

But, if the football coach had been educated properly himself, then this wouldn't have been a problem right?.

And if pigs had wings, they could fly, right?

The last one would be tricky for an 8th grader, as it would pretty much require that they know how to do a proof by contradiction. Pretty sophisticated stuff when some 8th graders can't figure out 7 x 8 without a calculator or how to divide 140 by 10 in their heads.

Well, perhaps they should know about number systems and group theory.

In eighth grade? Many of them still have trouble adding fractions. I say this based on the number of remedial math classes that were (and still are) offered at the college where I taught. Some quarters there were 25 sections of non-college-credit mathematics, ranging from plain old arithmetic at about the 6th grade level, to 9th grade algebra.

Then they would understand where addition and multiplication come from.

Perhaps they should also study some philosophy, such as, oh I don't know, metaphysics and epistemology so that they can think about what it means for a quantity to exist, and some symbolic logic so they can take a look at the peano axioms..

I strongly disagree that students need philosophy or epistemology to learn mathematics, and even more so that what's lacking is metaphysics.
I'd be willing to bet that you have never taught a class and had to deal with the reality of students.

One thing you seem to be overlooking is the concept of mathematical maturity. Most kids in middle school don't have the mathematical maturity to be able to understand the abstract concepts that you seem to be pushing. The saying is, you have to crawl before you can walk, and walk before you can run.

 

[Logical Dog]

I did A level maths and then I read the first chapters of "what is mathematics" by courant and also principles of mathematics, even with a good background in maths (we did calculus and calculus based mechanics) the books were abstract and hard to follow at first.

Unfortunately if this material was presented to me when I was in 8th grade I would have really struggled. In fact even some of these books assumes that you have basic algebra and calculus knowledge (and geometry). So some basic mathematical maturity is required first. Its a good list of books and papers IMO though, but it would be too hard (especially linguistically) on kids (and the material IS VAST!)...really vast.

 

[SSequence]

Well, this is the axiom of infinity: https://en.wikipedia.org/wiki/Axiom_of_infinity It's what justifies mathematical induction by asserting, axiomatically, that at least one infinite set exists. In other words, it's impossible to know. We can only count finitely, and so, it requires faith. Constructivist mathematicians reject this axiom and carry on without it, this means any proofs that rely on mathematicall induction become invalid. The difference between constructivist math and regular math is a metaphysical distinction, in particular, the axiom of infinity. I say students deserve to know that this controversy exists, and they deserve to know why, think for themselves, and come to their own conclusions. Hence, metaphysics is something good to teach in high school.

Are you sure about this? I don't know much about this stuff formally. But what I do know is that LEM (law of excluded-middle) is closely related to both incompleteness and how the humans mind mentally carries out processes (in a general sense).

Qualitatively (not going to every detail) a lot of what Brouwer said about mathematics seems to be spot on (in my opinion of course). I am not sure though whether Brouwer (or Bishop) ever wrote (I haven't read any of their technical work) that they doubt mathematical induction (for natural numbers). I would certainly be very surprised to see that.

Now if some one doubts that numbers that are larger than scale of universe (in some sense) don't exist or make sense (or something like that), then it would be expected for them to doubt mathematical induction (for natural numbers). Anyway, what I wanted to point out (correct me if I am wrong) that this certainly isn't necessarily a consensus opinion about this among those working on some non-classical logic.

 

[axmls]

To teach math like this, you would have to have teachers who understand this mathematics. Most do not. This leads to misguided attempts to insert this stuff into the curriculum even if it isn't useful, because the teachers don't actually know why it's useful!

I will relate the following passage from Surely You're Joking, Mr. Feynman in which Feynman was asked to evaluate "new math" textbooks.

I understood what they were trying to do. Many people thought we were behind the Russians after Sputnik, and some mathematicians were asked to give advice on how to teach math by using some of the rather interesting modern concepts of mathematics. The purpose was to enhance mathematics for the children who found it dull.
I'll give you an example: They would talk about different bases of numbers--five, six, and so on--to show the possibilities. That would be interesting for a kid who could understand base ten--something to entertain his mind. But what they had turned it into, in these books, was that every child had to learn another base! And then the usual horror would come: "Translate these numbers,which are written in base seven, to base five." Translating from one base to another is an utterly useless thing. If you can do it, maybe it's entertaining; if you can't do it, forget it. There's no point to it.

Because this is what happens when teachers don't know the importance of the things they're teaching--and changing the curriculum won't help that.

Besides, why does the math curriculum need changing? Not everyone is going to be a mathematician or a physicist. Let's be honest with ourselves: most people will never solve an algebraic equation again in their life, and no amount of rigor when we teach how to solve them will help them remember if they haven't done one in 25 years. Are we not still producing fantastic graduates? The math curriculum isn't perfect, but it's not like none of our students get math. Ultimately, those who are most interested will learn it anyway, and those who aren't won't, plus or minus a few special cases.

 

[Lucas SV]

most people will never solve an algebraic equation again in their life

This depends highly on the school, well at least here in the UK, I don't know about the US. Most people from my high school went to college, and in college there is a rather large number of subjects that use maths at some point.

 

[Stephen Tashi]

This is exactly the intention. I don't see why it has to be any different from what we do today. We would just introduce sets early, and teach basic logic like "if a then b" stuff early as well. Then teach them about the rules of sets, then introduce group theory. That way when one approaches regular algebra there is no need for confusion. Set theory is a branch of pure logic, it shouldn't be hard to grasp because there's really not much going on in set theory. It's very general.

I sympathize with the plan of introducing mathematical logic early in the curriculum, but my prediction is that the mathematical progress of general population of students won't be dramatically improved by doing that. The general population of students isn't going to progress into advanced mathematics. I agree that introducing logic will help people who eventually go into law, philosophy or other fields where knowing the distinction between something that is "logical" and somethat is "plausible" or "true" is useful.

An assumption you are making is that if the students are taught logic in year X then teachers in year X+1 will be empowered to present mathematics "logically" in the sense of using precise definitions and demonstrations. That's a nice plan for training pure mathematicians and satisfying teachers who want to train them, but it isn't practical for the general student population and, probably, for the general teacher population.

Expounding mathematics using mathematical logic involves a cultural adjustment. We reject methods that use skills which are useful and essential and everyday life - e.g. intuition, tacit assumptions, broad analogies. An important fact about logic is that it can't provide answers to most everyday situations since such situations involve incomplete information and imprecise goals.

As for the calculus stuff, well, analysis is mathematical, but computational calculus classes that don't teach anything but following the rules to get a numerical answer is not really mathematics to me.

To you, the definition of "mathematics" seems to require that it is presented axiomatically. I agree with that definition of pure mathematics. However, if we try to forecast the content of the famous phrase "what students will need later in life", the forecast doesn't point exclusively to pure mathematics. To be fair, it doesn't point to practical mathematics either. (To me, the hazy forecast is that most students - including those that become engineers or accountants or bankers - will be using computer software to do their mathematical calculations. The utility of doing the calculations without computers will primarily be a skill needed to past certification exams.)

If it's a statistical question, then are doing statistics, not math.

Ha! Well, I sympathize with that bold statement because applying statistics is highly subjective. But applying any sort of math to real life is subjective.

Here is an example of a HW assignment that can be done in the 8th grade: "Suppose that a and b are two elements of a field F.
Using only the axioms for a field, prove the following:
– ∀a ∈ F, 0a = 0.
– If ab = 0, then either a or b must be 0.
– The additive inverse of a is unique."

It's theoretically feasible to teach that material to some 8th graders. It involves exposing them the "culture shock" that happens when a people are introduced to pure mathematics. They must put aside assumptions they have made in empirically using numbers and justify their conclusions axiomatically. They must understand the distinction between "existence" and "uniqueness".

Students (of all ages) can compartmentalize knowledge. In algebra class, there are certain ways that the text and teacher expect things to be done, in English class, the abbreviated style of exposition used in answering algebra homework is not suitable for writing essays, in chemistry class what is called a "group" is called a "group action" in math class. It might be useful expose students to the culture of pure mathematics at an early age and because students can compartmentalize knowledge, it wouldn't harm most of them and it wouldn't help some of them.

 

[Lucas SV]

To me, the hazy forecast is that most students - including those that become engineers or accountants or bankers - will be using computer software to do their mathematical calculations. The utility of doing the calculations without computers will primarily be a skill needed to past certification exams.

Yes this is the reality, unfortunately. I know some engineers, whose problems they are trying to solve, would be made easier if the knew some more maths and physics. I do not think for someone that uses a computer to achieve technical tasks, being ignorant of maths is a good thing. For example, understanding some maths may be crucial to understand algorithm optimization. Of course different parts of maths will be useful to different people, no doubt about that.

However to be fair, the people I know whose skills would be enhanced by mathematics - are all researchers anyway.

 

[RaijuRainBird]

I spent my k-12 schooling experience hating math because I was never able to get answers to my "why" questions. I never bought into the bs answers teachers always handed out; clearly they didn't know, or they thought the material would be too advanced to show me.

Now I'm realizing that I'd have to take graduate classes just to get the answers my 5th grade self has been searching for my entire life (22 years old). There is something seriously wrong if I can't even get basic answers about mathematics without first starting a PhD in the subject. Not everyone can even get admitted into graduate school, and not everyone has the time or money to commit to graduate school. Why, then, are we so insistent in withholding the truth from the general population?

I can not understand why or how it is a good idea to mislead kids into thinking they understand something when they don't know the first thing about said something. It all just seems horrible to me.

___

I'm taking graduate classes for my last two years as an undergraduate, and I'm finally starting to see all the stuff I've wanted to see since grade school. I knew I wasn't being given the entire picture then, but I couldn't figure out what was missing because no one told me about books I should read or the names of the subfields to look at. I really believe that my current age has nothing to do with my ability to understand. If I had seen the material I can finally see now 5 years ago I'd have been much happier.

I also don't think I'm alone in this camp. Many people who struggle in regular high school do so because they are trying to figure out the underlying logic on their own, at the pace the teacher presents material for us to simply take for granted without reasoning. What these people are wanting is graduate level mathematics. We should at least give people the right to choose an abstracted "path" (or whatever you like to call it) for students who aren't happy with not caring about understanding.

 

[axmls]

Why, then, are we so insistent in withholding the truth from the general population?

You said so yourself. It's because the teachers do not know the material like that. Changing the curriculum wouldn't change that. You'd have a bunch of teachers who don't know the material that well struggling to apply it correctly, like in Feynman's case.

Nothing stops bright students from learning the whys. Especially with the Internet existing nowadays.

 

[houlahound]

I am wondering if it is ethical to even have a syllabus mandated on an entire generation. I think we can do something a little more creative and modern than a factory model of education these days.

 

[Lucas SV]

clearly they didn't know, or they thought the material would be too advanced to show me.

I'm sorry that you were not fortunate enough for someone to push you foward from an earlier age. Please do not take out your disappointment on others. What may be easy for you may be hard for others (also you may not realize this, but what may be hard for you may be easy for others). I think you will only understand what it means to struggle in mathematics when you get closer to research/work.

Why, then, are we so insistent in withholding the truth from the general population?

I'm thankful not everyone in the history of mathematics was so fixed on a single 'truth'. Otherwise non-euclidean geometry would not have existed and we would have to say bye to the beautiful theory of general relativity.