Redesigning Mathematics Curriculum, thoughts?

[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.

 

[RaijuRainBird]

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.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.

By truth I just meant logical justification. Not in the way it's implied here.

In any case, I do think common core math in the US is taking us in the right direction.

 

[micromass]

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.

Yes, the OP is wrong. Constructivists do not reject the axiom of infinity. Some do though, but most definitely not all.

Anyway, the plan in the OP is totally insane. Seriously, I recommend you to get more in touch with the the typical student. Why don't you try to make some money by tutoring high school or calculus students, or why don't you get a job as a TA of elementary math classes. I highly recommend you do this because then you will see that your very interesting math plans will fall on deaf ears with the students. It just can't be done practically.

Now, I live in a country where the curriculum is known to be more abstract than in the US. Not too abstract, but we see basic sets in elementary school and we cover some group theory in high school. In particular, we are shown that the real numbers form a field. And you know, I have always been mathematically inclined in my life, and of all the kids in my class I was probably the most interested and perhaps even the best at mathematics. And you know what? I found fields to be something quite abstract and something that made math more difficult than it actually is. It took me years before I could appreciate what a field is!

You seem to like a foundations first approach. But it is a very bad idea to teach mathematics this way. You can't possibly study category without studying abstract mathematics for years. You need to know group theory, ring theory, field theory, topology, etc. in order to really grasp category theory. Teaching it in the very beginning is being an abusive teacher. I am a category theorist by profession, and I wouldn't have understood one single bit of it without studying abstract mathematics for years already. The same holds true for logic. The same holds true for set theory. The same holds true for foundational issues like constructivism.

And about $\mathbb{R}^n$, you don't fully realize the genius of the notation I see. In set theory given sets $A$ and $B$, we have that $A^B$ is the set of functions from $B$ to $A$. In set theory also, the natural number $n$ is defined as a set containing $n$ numbers. More precisely, you can define recursively $0=\emptyset$ and $n+1 = \{0,1,...,n\}$. Then $\mathbb{R}^n$ is the set of functions from $\{0,...,n-1\}$ to $\mathbb{R}$. Given such a function $f:n\rightarrow \mathbb{R}$, we can write it symbolically as $(f(0), f(1), ..., f(n-1))$. So this is the full explanation of the $\mathbb{R}^n$ notation: it should not be interpreted as you say as $\mathbb{R}\times ...\mathbb{R}$ $n$ times. Not that I would tell high schoolers all of this, or even undergraduates in mathematics...

 

[micromass]

What you're proposing leads to situations like this (which actually happened): a pupil was asked how much $5+3$ was. The student said, I don't know but I know that $5+3=3+5$ by the commutative property.

I hope you realize how absolutely awful this is. I much rather prefer the answer to be " $8$ because you add $5$ to itself $3$ times, but I have no idea what the commutative property is".

 

[micromass]

And really, calculus not being a part of math but rather being a part of physics? What the hell is that?

 

[houlahound]

. I much rather prefer the answer to be 8 because you add 5 to itself 3 times,

Whooh dude.

 

[micromass]

Whooh dude.

Oh boy, doing basic arithmetic when sleep deprived is not a good idea... My apologies, but I hope you get the idea!

 

[RaijuRainBird]

And really, calculus not being a part of math but rather being a part of physics? What the hell is that?

Well, more specifically I should have said "Calculus for engineers" and "early transcendentals calculus" with an emphasis on problem solving.

Regardless, I argue calculus/analysis should be taught using logic notation and formal proofs. For example, when covering continuity it should be done like this: A function f: X --> Y is continuous if: (∀x)(∀ε>0)(∃δ>0)(∀y) ( (|x-y| < δ) ==> (|f(x) - f(y)| < ε) ). If students knew symbolic logic, they'd be able to understand this statement and manipulate it with negations, and be able to know if it's a valid statement syntactically.

This makes the distinction between continuity and uniform continuity easy to see. Just move around the quantifiers, (∀ε>0)(∃δ>0)(∀x)(∀y)((|x-y| < δ) ==> (|f(x) - f(y)| < ε)).

At the most basic level, this is the sort of thing I'm advocating. This along with real set theory, and the real underpinnings of algebra.

__

As for constructivism, I thought the argument was that since an infinite set can not be constructed it can not be shown to exist. I was under the impression they used a concept of potential infinity. I believe this is related to the rejection of the law of excluded middle.

 

[StatGuy2000]

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.

I think an important question is why so many parents in the US (and to an extent also in Canada) don't place any value on education. An education should be thought of as a stepping stone to advance further in one's society, both through employment and beyond, and all parents should place value in this. As someone who is of partial Asian heritage (and immersed in a culture that traditionally stressed the great importance of education), I find this attitude most baffling.

 

[axmls]

Regardless, I argue calculus/analysis should be taught using logic notation and formal proofs. For example, when covering continuity it should be done like this: A function f: X --> Y is continuous if: (∀x)(∀ε>0)(∃δ>0)(∀y) ( (|x-y| < δ) ==> (|f(x) - f(y)| < ε) ). If students knew symbolic logic, they'd be able to understand this statement and manipulate it with negations, and be able to know if it's a valid statement syntactically.

Watch how fast the engineering departments pull their students out of there and start teaching calculus themselves.

I think an important question is why so many parents in the US (and to an extent also in Canada) don't place any value on education.

Many people are successful with an education that stopped when they graduated high school. That doesn't mean they're right, but many pass on this attitude because it worked for them. Education is important for everyone--but that doesn't mean university education is important for everyone. There are many people who would be far better off learning a trade rather than going to college, but that's a topic for another discussion.

 

[micromass]

Well, more specifically I should have said "Calculus for engineers" and "early transcendentals calculus" with an emphasis on problem solving.

Regardless, I argue calculus/analysis should be taught using logic notation and formal proofs. For example, when covering continuity it should be done like this: A function f: X --> Y is continuous if: (∀x)(∀ε>0)(∃δ>0)(∀y) ( (|x-y| < δ) ==> (|f(x) - f(y)| < ε) ). If students knew symbolic logic, they'd be able to understand this statement and manipulate it with negations, and be able to know if it's a valid statement syntactically.

Symbolic logic is useless to get the intuition. If you read professional math papers, then symbolic logic like the definition you gave above is actually actively discouraged. Do you know why that is?

This makes the distinction between continuity and uniform continuity easy to see. Just move around the quantifiers, (∀ε>0)(∃δ>0)(∀x)(∀y)((|x-y| < δ) ==> (|f(x) - f(y)| < ε)).

Ah, and that's the point. If you see uniform continuity merely as "continuity with a few quantifiers moved around", then I'm afraid you don't grasp uniform continuity! Sure, students should definitely see that uniform continuity and continuity are related by just switching quantifiers, but then you're not telling them what uniform continuity really is and why we would be interested in it.

As for constructivism, I thought the argument was that since an infinite set can not be constructed it can not be shown to exist. I was under the impression they used a concept of potential infinity. I believe this is related to the rejection of the law of excluded middle.

That impression would be wrong then. See Bishop and Bridges "constructive analysis".

 

[RaijuRainBird]

Ah, and that's the point. If you see uniform continuity merely as "continuity with a few quantifiers moved around", then I'm afraid you don't grasp uniform continuity! Sure, students should definitely see that uniform continuity and continuity are related by just switching quantifiers, but then you're not telling them what uniform continuity really is and why we would be interested in it.

Well, if we understand symbolic logic correctly, then we actually do understand what the statement means. People can always symbol push with any notation. Besides, you have to communicate somehow. I don't see how writing it out in plain english is any better. This cuts out ambiguity and communicates exactly what is supposed to be communicated, plus it works when people don't speak the same language.

 

[micromass]

Well, if we understand symbolic logic correctly, then we actually do understand what the statement means. People can always symbol push with any notation.

Sadly that is not true. Just understanding the symbolic logic does not mean that you truly understand a concept. Symbolic logic is rather useless when it comes to professional math either. Nobody writes their results using symbolic logic (unless it's mathematical logic of course). And mathematics has been around centuries before symbolic logic became a thing.

 

[RaijuRainBird]

Sadly that is not true. Just understanding the symbolic logic does not mean that you truly understand a concept. Symbolic logic is rather useless when it comes to professional math either. Nobody writes their results using symbolic logic (unless it's mathematical logic of course). And mathematics has been around centuries before symbolic logic became a thing.

How does the logic cut out the meaning? In the first case continuity is specified as for each x there is a specific delta, uniform continuity says there is one delta which works for every x. This is exactly what the logic is telling us. I don't see how this obstructs any understanding.

 

[StatGuy2000]

Many people are successful with an education that stopped when they graduated high school. That doesn't mean they're right, but many pass on this attitude because it worked for them. Education is important for everyone--but that doesn't mean university education is important for everyone. There are many people who would be far better off learning a trade rather than going to college, but that's a topic for another discussion.

If it is only with graduating from high school, that is one thing. What I have witnessed are families where even elementary or secondary school education is not valued, and where there is no shame attached to dropping out of high school. Granted, those experiences were what I witnessed back in the 1990s when I finished high school, and I have read various reports (which I'll post later) that the high-school graduation rates have been rising. So perhaps at least in Canada, the situation may be improving as far as importance of education is concerned(?)

As far as learning a trade rather than going to college or university -- it's worth keeping in mind that most skilled trades also require at least a certain element of post-secondary education or training. In Canada, for example, electricians are required to go through a formal apprenticeship program and are required to take select courses in community college.

 

[micromass]

How does the logic cut out the meaning? In the first case continuity is specified as for each x there is a specific delta, uniform continuity says there is one delta which works for every x.

I don't see how this tells you anything about the intuition of the concepts, why the concepts were invented, why the concepts are important, how to utilize the concepts in practice. The logic you mention gives rather superficial understanding at best.

 

[micromass]

Here is a nice question: "Is every bounded continuous function uniformly continuous?" If all you know of uniform continuity is the symbolic logic, you won't be able to solve this. If you have the intuition, then it becomes rather easy.

 

[Sherwood Botsford]

I come at this from a MUCH lower level. I taught high school math, mostly to kids who would not be using math in their daily lives after graduation.

I. Aspects of math that people NEED to know:
a. Basic numeracy: Estimating. While I don't think that being able to multiple 27 * 42 in your head is essential, I think that all people should be able to see that it's somewhere around 30 * 40 = 1200. So if you get .6something on your calculator, you hit the wrong key. Part of this is making reasonable assumptions. Problems like this can be fun: How many piano tuners are in Chicago? Goal is to get within 1 order of magnitude of the right answer.
b. How to lie (and detect liars) with statistics. At least average, and standard deviation. Always ask "Percent of what?"

Do most people need algebra? Geometry? I'm not persuaded.

II. Formal logic and logical fallacies.
Not sure that I would go to symbolic logic in high school, but I'd go at least for recognizing common fallicies, and the the normal way to use logic in discourse.
At present logic is not taught in most curriculum.

III. Creation and interpretation of graphs.

 

[houlahound]

Let us not forget the brain development of learners, most are limited to concrete and in later life develop a capacity for abstract thinking.

Psychology of learning gets lost in debates about which math topic and what sequence.

Psychology of learning and what age stage to introduce abstraction should inform syllabus IMO.

Math academics are some of the last people I would consult re school math curriculum, no offence.

 

[Mark44]

Regardless, I argue calculus/analysis should be taught using logic notation and formal proofs.

Watch how fast the engineering departments pull their students out of there and start teaching calculus themselves.

This is a very good point, one that the OP is probably not aware of. I estimate that I taught somewhere between 50 and 100 classes in calculus in my nearly 20 years of college teaching. The vast majority of the students in those classes were planning to go into engineering of some kind. I would also estimate that no more than 5% of my students would go on to pursue a degree in mathematics, and even that percentage could be an overestimate.

I agree that if calculus classes were taught using logic notation and formal proofs (or worse, metaphysics, as the OP has pushed for a couple of times), engineering departments would soon start their own calculus courses.

With regard to analysis courses, all the ones I took as an undergrad were completely proof based. AFAIK, this is how things are done in most university math departments, so I don't see the point of the OP's recommendation with regard to analysis courses.

 

[S.G. Janssens]

Whooh dude.

$5 + 3 = 5 + 5 + 5 \pmod{7}$.

 

[lpetrich]

Here is what I'd do if I could.

I think that the "New Math" got it backward. At least in the earlier years, I'd introduce math concepts in a quasi-empirical fashion. I'd also end the segregation of subjects, end spending 1 year on algebra, 1 year on geometry, 1 year on trigonometry, and the like. Each year, I'd teach a mixture of math topics, starting at simple ones and moving to more complicated ones. I'd start with simple versions of algebra and geometry and statistics, and move to more fancy ones in later years. I'd go slow on Euclidean-geometry constructions, but I'd get into coordinate systems and analytic geometry rather heavily, complete with making lots of graphs. As to reasoning, I'd teach the difference between deduction and induction, and how induction when treated as deduction is a fallacy: affirming the consequent.

I like the idea of education being tracked, so one can teach more advanced math in a "math and science" track. Stuff like infinite sets, formal logic, abstract algebra, and the like. One can introduce group theory with symmetries of physical objects like flowers, to have something easy to picture.

 

[Beanyboy]

Tell you what Mr. Rainbird, I'll have a go at what you're proposing and I'll check back with you to let you know how I'm getting on. I'm a teacher, background in the Arts, speaker of 3 languages other than English and currently trying my hand at AP Chemistry and Physics, primarily with Khan Academy. I'm fascinated by the linguistic elements of learning, semantics etc and recently thought I'd buy a tee-shirt which said "I speak Physics". Currently, I'd say I'm functioning at 8th/9th grade level Math. So, where do you propose I jump in? I'm impressed by your passion and erudition, let's see where we can take this.

 

[AaronK]

I at least agree with OPs comment on hating not knowing the "why" of the various algebra I did throughout my education--literally all of my math teachers would give me a set of rules to follow to get a correct answer (and indeed, sometimes I would intuit how I got to that correct answer through dutifully following the steps to get there) but they would never go into how those steps were pioneered--what bits of logic justified using those steps. Not educated enough to say whether the OP is right or wrong, but I do tend to respect micromass's opinion as far as intuition and clear & concise learning go. I can definitely see it being a problem going the way of OP for many students. I think what we're all not mentioning enough is diversity of preference for learning--I'm sure there are some who would prefer OPs approach for their personal learning, but certainly just as much (probably much more) who would hate having to delve into the abstraction and the rigorous logic before actually learning the specific math.

 

[micromass]

I literally all of my math teachers would give me a set of rules to follow to get a correct answer (and indeed, sometimes I would intuit how I got to that correct answer through dutifully following the steps to get there) but they would never go into how those steps were pioneered--what bits of logic justified using those steps.

That is a valid criticism of the curriculum. I think there are a lot of things wrong with the math curriculum in high school. Teachers giving you a set of ruls that you would have to follow without thinking, that's not math. It's anti-math. So that will definitely need to change. But I'm not sure that abstract and symbolic logic is the answer here...

You might enjoy this, which I more or less agree with: https://www.maa.org/external_archive/devlin/LockhartsLament.pdf
A point of view of education that I 100% agree with is Arnold's: http://pauli.uni-muenster.de/~munsteg/arnold.html

 

[Beanyboy]

Here is what I'd do if I could.

I think that the "New Math" got it backward. At least in the earlier years, I'd introduce math concepts in a quasi-empirical fashion. I'd also end the segregation of subjects, end spending 1 year on algebra, 1 year on geometry, 1 year on trigonometry, and the like. Each year, I'd teach a mixture of math topics, starting at simple ones and moving to more complicated ones. I'd start with simple versions of algebra and geometry and statistics, and move to more fancy ones in later years. I'd go slow on Euclidean-geometry constructions, but I'd get into coordinate systems and analytic geometry rather heavily, complete with making lots of graphs. As to reasoning, I'd teach the difference between deduction and induction, and how induction when treated as deduction is a fallacy: affirming the consequent.

I like the idea of education being tracked, so one can teach more advanced math in a "math and science" track. Stuff like infinite sets, formal logic, abstract algebra, and the like. One can introduce group theory with symmetries of physical objects like flowers, to have something easy to picture.

Speaking of "tracking", do you think an organization like Khan Academy could provide us with useful data on what people learn and how?