My Struggle with Math: Understanding Real-World Applications

[micromass]

Good points, micromass. There are, at least, two separate skills required for a teacher, "class management" being the one that must precede the other.

During my years in grade and high school, the teachers had no problem with this. It was against a background of good class management that they, never-the-less, could not connect most math to my, or anyone else's, reality. One high school algebra teacher in particular answered my query about this with the vague information that engineers use algebra a lot and then he changed the subject, which gave me the impression he had no idea what specific sorts of problems an engineer might actually be confronted with.

I feel he failed me when I was most receptive. If he could have gotten me interested in some real world engineering problem at that point, it could have lit a fire under me to be hungry for more math.

Interesting. I agree that the system failed you when you were most receptive. But let's analyze this situation a little further, because it is very interesting.

1) Let's first assume that he did actually know what problems an engineer might be confronted with. I personally don't know specific problems, but I can certainly justify to you that high school algebra is useful. Nevertheless, I would have given the same reply as your teacher? Why, because you were most likely not mathematically mature enough to grasp the application. Furthermore, if you're a teacher, then you don't teach to one person, but rather to an entire class. And the class has weak and strong students. So while giving a discourse on applications is fun for the strong (maybe!), it will be very useless for the weak. Furthermore, you have a series of topics to cover in a limited time, so it is unwise to give a very long application that might prevent other topics from being seen.
I do sympathize with you and with the students asking for applications, but things are just not that easy sadly.

2) What if he didn't know any applications. That still doesn't mean he's a bad teacher. Class management and connection with the student is more important than knowing advanced mathematics (although the teacher should know the basics of his field). It is still unwise to say that "he personally failed you". Rather, it would be better to think of solutions to this problem. You can't blame the teacher for not knowing this, and you can't blame the student for asking this. On the other hand, I feel some cultural shift must happen so that both parties are more satisfied. For example: textbooks and courses that are better and cover some applications (I mean really, most textbooks these days are horrible), or additional schooling for teachers, or additional classes for the interested students, etc.
I know of some program in scandinavia where university students are able to eliminate their student debt by teaching for 2 years. This is an excellent idea, but only privded that they are properly guided and get the audience they deserve.

 

[micromass]

You make an excellent point.

I used to have to tell people in my English class to shut up so I could hear the teacher. One kid even started eating a thanksgiving turkey during class. I would actively encourage the teacher to give disrespectful students detentions, but he told me that that didn't work, in his experience. I don't blame him one bit for the failing grades that many of those students received.

I am lead to believe that the problem is more of a cultural one, and needs to be addressed at the familial level. Children should be taught at a young age to respect knowledge, and by extension, those with more knowledge than them. Namely, teachers. American society, from what I have experienced, places little value on knowledge compared to collectivist cultures. And it certainly shows.

It's certainly a cultural problem, but not only an american. I guess it happens everywhere where education is mandatory. You won't see this kind of behavior in third world classes where the children don't have much other opportunities. They really value their education! On the other hand, in the first world, children have everything and don't value education as a means to get further in life, or at least: don't value knowledge for what it is.
I have always felt that we put too much emphasis on getting students to go to school until they are 18. This doesn't work for some students. And for some other students, what they learn in the later years is pretty useless and something they will never use later in life. I can't blame them for disliking this. The problem is that they will put their negative attitude on the other students. So I feel this is something that will need to be fixed.
Of course, there are also the parents who think that their child is a failure if he doesn't finish his high school education. Or the employers who won't hire people with a high school education. So there needs to be an attitude change from those people too that just because you didn't see much use for school later in life does not mean that you're a failure or a bad employee.

 

[micromass]

As a(n) (unemployed) teacher and student, I don't think much useful can be reached in this (nor, really most, if not all others) by making this into an "us against them" discussion. Not a brilliant insight, but sometimes discussions (de)evolve in this direction.

We are not having an "us versus them" discussion. I feel that most people on this thread just bring up valid concerns, such as: math is taught too abstractly, or: teaching applications is too difficult. Those are both sides of the coin. If you want to sole this problem, you're going to have to look at all sides.

I mostly wish to give people a full perspective, and not only the perspective of the student. Furthermore, I wish this thread would eventually go towards giving out certain solutions to the problem, and not just the naive "the teacher should be better" or "the students are entitled brats".

 

[micromass]

Not that I entirely agree with this person, but it is a good watch and he has some good points:

 

[AlephNumbers]

I wish this thread would eventually go towards giving out certain solutions to the problem, and not just the naive "the teacher should be better" or "the students are entitled brats".

Okay, here is an idea. I'm building on the ideas I have read in this thread, those of the guy in that video, and what I know of some foreign (to me) educational systems. This solution does, however, have a few major flaws.

All students take roughly the same classes up to seventh grade. All students take the typical math courses up through pre-algebra. Students are evaluated throughout their academic career, and their aptitudes are thoroughly tested. Depending upon these two factors, and partially the student's own wishes for their future, their secondary school experience is heavily modified. Those that show little interest in academics are selected to go to a trade school. They only have to take whatever classes are relevant to their trade, but they can take a few more if they want.

Math class is abolished after seventh grade, except for those who wish to pursue mathematics, applied mathematics, and physical sciences. Students of physical science take specialized applied mathematics courses alongside their science classes. There could definitely be some overlap between the mathematically intensive sciences (such as physics) and mathematics.

The flaws are probably pretty obvious. How can one expect a child to know what they want to pursue as a career? How do you know that they won't change their mind at some point? Is it wrong to make children who perform exceedingly poorly on a test go to a trade school, even if they really want to be a physicist?

 

[micromass]

Okay, here is an idea. I'm building on the ideas I have read in this thread, those of the guy in that video, and what I know of some foreign (to me) educational systems. This solution does, however, have a few major flaws.

All students take roughly the same classes up to seventh grade. All students take the typical math courses up through pre-algebra. Students are evaluated throughout their academic career, and their aptitudes are thoroughly tested. Depending upon these two factors, and partially the student's own wishes for their future, their secondary school experience is heavily modified. Those that show little interest in academics are selected to go to a trade school. They only have to take whatever classes are relevant to their trade, but they can take a few more if they want.

Math class is abolished after seventh grade, except for those who wish to pursue mathematics, applied mathematics, and physical sciences. Students of physical science take specialized applied mathematics courses alongside their science classes. There could definitely be some overlap between the mathematically intensive sciences (such as physics) and mathematics.

The flaws are probably pretty obvious. How can one expect a child to know what they want to pursue as a career? How do you know that they won't change their mind at some point? Is it wrong to make children who perform exceedingly poorly on a test go to a trade school, even if they really want to be a physicist?

Yes, it is flawed, but it's not a bad suggestion. In fact, this is almost the system that is in place in my country (belgium), but not so drastic as you propose it.

Some other thoughts on math education:
1) I don't really think that schools nowadays really teach math in a beneficial way. You are being taught to solve equations by going through a number of steps. But this isn't really math. Math is a creative endavour, but this is not really stressed nowadays. Proofs and reasoning are essential to math, but this is rarely taught in school. Instead, they just memorize what they need to do. In this case, I feel that math education is essentially useless. If you want to make math education mandatory, then at least do it in a way that actively engages the creative thinking process of the student. Again, I don't agree with everything here, but he has good points; https://www.maa.org/external_archive/devlin/LockhartsLament.pdf

2) Maybe there should be different math classes covering the same topics. For example, you could have different classes teaching algebra, but with different emphasis. For example, you could have fairly theoretical algebra class for the bright students, and a more applied-oriented approach for the students who just wish to use math as a tool.

In either case, I feel math education needs a very broad reform. We need to think about why we really need to teach math in schools, and then adjust education to suit that need. (this doesn't only apply to math, but history, literature and others as well).

 

[wabbit]

I guess I was lucky to have taken math before the "new math" came out. The elementary school math many younger people took is not the math I took. Math in my day made sense.

Same here, and it was not a matter of practical applications but of actually teaching math. Is still remember fondly my Euclidian geometry classes from junior high school, that was probably when I learned what a proof is and that seemed magical. What I sometimes see today in high school (no generality, speaking only of what I saw) is watered down "math" which I find as uninteresting and useless as OP did, and I wouldn't miss that.

 

[AlephNumbers]

Proofs and reasoning are essential to math, but this is rarely taught in school.

A little off-topic: I have yet to take a college level calculus class. When I do, I am going to insist that my professor prove every single rule. So help me god, I will walk up there and prove the quotient rule myself if they say no.

 

[micromass]

A little off-topic: I have yet to take a college level calculus class. When I do, I am going to insist that my professor prove every single rule. So help me god, I will walk up there and prove the quotient rule myself if they say no.

Very noble and ambitious! But also not a good idea. Let me explain why.

Proofs in calculus can be really difficult and subtle. If I teach you calculus and proof everything, you will get very confused and you will automatically start disliking it. After all, you wouldn't know why I make things so complicated! And believe me, rigorous proofs in calculus can be extremely complicated. You need experience to understand the big picture first, and the problems that I want to solve by rigor. If you just start off calculus, you will not have this experience.

Of course proofs should be present in calculus. But what is the use of proofs? In my opinion, proofs should be an argument that makes the result acceptable. In a calculus class, we should rely heavily on the intuition of the student, and we shouldn't attempt to proof things that already are intuitively obvious! (until we reach the point where we realize that obvious things are not that obvious :D )

Intuition is way more important than rigor. That's not to say that rigor doesn't have a place in math, but it has no place when first learning about a subject (of course some rigor should be present, but not a lot). If you want a course that proofs every single thing in calculus, then you will have to do analysis. But by then, you already know the big picture and the problems of calculus.

If you're interested in the deeper reasons for why something is true, then I suggest you self-study it. Classmates who are having trouble with calculus will not like you making the class harder by asking for rigorous proofs (after all, most see calculus as a tool and don't care whether they know why it works, that is not a silly attitude, they have a point). You can always contact me if you want to have a conversation on how to understand calculus in a deeper way.

 

[wabbit]

I think proofs are essential in math. But there are different levels of proofs, even if you read a research article in math not every proof will be spelled out in full rigorous detail. A proof translated into intuitive reasonning accessible to the student is better than a rigorous one he cannot understand - it is still valuable as it conveys the idea that there is a proof, and some understanding of the reasonning implied.
I agree that a course where the teacher would do every proof with full rigor would be impossible - it would take ages to do the simplest things, real concepts might be missed in the formalism, and most students (and teachers!) would shudder at the thought. No one (well, very few at least) uses Bourbaki as a textbook for undergraduates.

 

[AlephNumbers]

Classmates who are having trouble with calculus will not like you making the class harder by asking for rigorous proofs

I agree that a course where the teacher would do every proof with full rigor would be impossible - it would take ages to do the simplest things, real concepts might be missed in the formalism, and most students (and teachers!) would shudder at the thought.

You have made your points. I will be content to just sit in my chair and scribble in my notebook.

 

[micromass]

In my opinion, the saddest part about calculus education is that they don't teach it the way calculus was originally invented. They make it a lot more difficult and abstract than it really needs to be, for reasons that I don't find convincing (such as: we've always done it that way).

Calculus should be taught with a deep emphasis on infinitesimals. This has as benefit that it's way more intuitive. Face it, most calculus classes don't bother defining what a limit is. They just give some rules with working on it, and they explain it intuitively. This is how I would do it too. But some books do attempt to make things rigorously and give definitions of continuity like
$$\forall\varepsilon>0: \exists \delta>0:\forall x: |x-a|<\delta~\Rightarrow~|f(x)-f(a)|<\varepsilon$$
This is not obvious at all. It took me years to figure out exactly what it meant and why we used this. Although if you understand it, it becomes as obvious as saying that "if $x$ and $a$ are close, then so are $f(x)$ and $f(a)$". Well guess what? This is an idea you can perfectly explain THAT intuitively with infinitesimals.

I'm not saying that we should abolish epsilon-delta stuff at all. But there are certain things one must grasp in order to be ready for this. These things include historical and philosophical considerations. It took hundreds of years before this definition was formulated, and you can't understand it without first understanding the problem. It would be much better to develop calculus with infinitesimals first, and then gradually discuss the problems with it and the epsilon-delta solutions.

Furthermore, infinitesimals are way more useful in physics than epsilon-delta's. So also in that sense, we are not helping the students by not introducing infinitesimals.

 

[wabbit]

You have made your points. I will be content to just sit in my chair and scribble in my notebook.

That was not my point at all, but you are free to scribble to your heart's content :)

 

[wabbit]

In my opinion, the saddest part about calculus education is that they don't teach it the way calculus was originally invented. They make it a lot more difficult and abstract than it really needs to be, for reasons that I don't find convincing (such as: we've always done it that way).

Calculus should be taught with a deep emphasis on infinitesimals. This has as benefit that it's way more intuitive. Face it, most calculus classes don't bother defining what a limit is.

I don't know if that would work well pedagogically (probably depends just on how it is done) but that is an interesting option. Epsilon-delta is truly a tough nut to crack, and without an intuitive understanding of "what it really means" it can easily become dead letter - empty statements that one can follow formally through the steps of a proof only. And that is not learning limits.
So if the infinitesimal approach can do that it certainly has its place - some intuitive content must be brought in or math is just dead formalism.

 

[WWGD]

I agree that is more an issue of setting the right context in which things make sense than in finding applications o answering "what is this good for" and not just rigor and a big machinery for its own sake.

 

[zoobyshoe]

Interesting. I agree that the system failed you when you were most receptive. But let's analyze this situation a little further, because it is very interesting.

1) Let's first assume that he did actually know what problems an engineer might be confronted with. I personally don't know specific problems, but I can certainly justify to you that high school algebra is useful. Nevertheless, I would have given the same reply as your teacher? Why, because you were most likely not mathematically mature enough to grasp the application. Furthermore, if you're a teacher, then you don't teach to one person, but rather to an entire class. And the class has weak and strong students. So while giving a discourse on applications is fun for the strong (maybe!), it will be very useless for the weak. Furthermore, you have a series of topics to cover in a limited time, so it is unwise to give a very long application that might prevent other topics from being seen.
I do sympathize with you and with the students asking for applications, but things are just not that easy sadly.

2) What if he didn't know any applications. That still doesn't mean he's a bad teacher. Class management and connection with the student is more important than knowing advanced mathematics (although the teacher should know the basics of his field). It is still unwise to say that "he personally failed you". Rather, it would be better to think of solutions to this problem. You can't blame the teacher for not knowing this, and you can't blame the student for asking this. On the other hand, I feel some cultural shift must happen so that both parties are more satisfied. For example: textbooks and courses that are better and cover some applications (I mean really, most textbooks these days are horrible), or additional schooling for teachers, or additional classes for the interested students, etc.
I know of some program in scandinavia where university students are able to eliminate their student debt by teaching for 2 years. This is an excellent idea, but only privded that they are properly guided and get the audience they deserve.

I didn't give the full context, which would modify your understanding. This was at a college prep school. That's one of the reasons class management wasn't such an issue: most of the students were above average and 100% were intending to go on to college. To that end, I was struggling very hard this particular year to pull my math grade up so it would look better on college applications. This school was a boarding school; students and teachers all lived on campus, and this particular math teacher made himself available for off hours homework help. I availed myself of that frequently, and it was during one of these one-on-one homework help sessions that I asked about practical applications. The point being, no other student's time was being infringed on, it wasn't cutting into class time. He was already giving me three or more hours extra per week, so his patience was just about limitless. Time was definitely not the issue.

Rather, it would be better to think of solutions to this problem.

The solution is spread throughout my 'complaint,' and it obviously is: to prevent math from becoming abstracted from practical applications. Math is a requirement because it is a powerful and universally used tool in a huge variety of careers and occupations. It's not there in the curriculum for it's aesthetic properties (as mentioned by WWGD). Keeping it grounded in practical reality in the student's mind is such an obvious necessity it is migraine-inducing to contemplate that all teachers, and all members of all educational systems, don't automatically understand this and take pains to do it. As I mentioned in an earlier post, in the years after school, whenever I've encountered a real life use for math, it is suddenly clearer than it ever was in high school, and I feel absolutely enthusiastic about it's power as a tool. That taught me I wasn't born missing the math lobe. I feel I was failed: it was laid on me as an abstract, disembodied burden when it could have been a source of enthusiasm.

As for expert teachers; I don't think it's at all desirable to have math experts teaching (except at very high levels). I wouldn't want Gauss teaching me because Gauss had some kind of preternatural grasp of math that I'm sure he didn't understand and couldn't possible communicate. The average high school math teacher should be someone who once matched the average student he's going to be teaching, not the genius who could have picked it up himself in his spare time from a book. You can't teach if you're suffering from "the curse of knowledge." A teacher has to be able to psychologically put him/her self in the mind of the ignorant to understand what they need to be taught next, or to understand what miasma of confusion and cognitive dissonance is preventing them from understanding what's being taught. It's not expertise in the subject matter that's missing in teachers.

 

[WWGD]

But, Zooby, what is it that you found attractive: the potential applications or the fact that Math is a tool in so many areas (which is kind of amazing)? I found attractive the fact that so many different pieces of Math fit together coherently. That is a rare find.

 

[wabbit]

The solution is spread throughout my 'complaint,' and it obviously is: to prevent math from becoming abstracted from practical applications. Math is a requirement because it is a powerful and universally used tool in a huge variety of careers and occupations. It's not there in the curriculum for it's aesthetic properties (as mentioned by WWGD). Keeping it grounded in practical reality in the student's mind (...)

Applications are important both in math in general and in its pedagogy, and you make a good case for that. This depends on the student tough, it is essential for some, not so much for others.
But there is also another reason for teaching math, and for doing it right : it expands the mind. Math, like philosophy, history and other disciplines is highly formative and provides us tools to expand our thinking, as well as sharpen our deductive ang general logic abilities and capacity for abstract thought. As such it is highly valuable - if done right - in a curriculum, separately from its applications.

As for expert teachers; I don't think it's at all desirable to have math experts teaching (except at very high levels).

Agreed. That would be unnecessary and I wouldn't necessarily expect Perelman for instance to do well as a high school math teacher. Doing math research and teaching math, especially at introductory level, are different jobs requiring different qualifications (some excel at both, but one could almost say this is akin to the fact that a good ice skater can also be a great cook. : ) )

 

[zoobyshoe]

But, Zooby, what is it that you found attractive: the potential applications or the fact that Math is a tool in so many areas (which is kind of amazing)? I found attractive the fact that so many different pieces of Math fit together coherently. That is a rare find.

It is attractive to me when I, myself, have a specific use for it. Barring that, there has to be the clear prospect I could possibly have a use for it in the future. Despite the fact arithmetic was taught in a dry, unenthusiastic manner to me, it's future potential use became apparent at each step, and it was reasonable to pay attention. Likewise, the uses of geometry seemed always apparent to me. Somewhere into algebra, however, things departed from any practical application, and it became increasingly abstracted from there on. I went to a liberal arts college that was so heavily weighted in favor of the liberal arts that there was no math or science requirements for a degree, and high school algebra was the last math I ever had to study. It bothers me now that it was taught such that it was the last math I wanted to study. After those last two years of algebraic miasma, I was elated to be rid of it.

 

[zoobyshoe]

Applications are important both in math in general and in its pedagogy, and you make a good case for that. This depends on the student tough, it is essential for some, not so much for others.
But there is also another reason for teaching math, and for doing it right : it expands the mind. Math, like philosophy, history and other disciplines is highly formative and provides us tools to expand our thinking, as well as sharpen our deductive ang general logic abilities and capacity for abstract thought. As such it is highly valuable - if done right - in a curriculum, separately from its applications.

I agree, and it is in this spirit that I've gone back and taken a look at it all these years later, and found it to be, as you say, mind expanding. This is the spirit in which Archimedes and the other ancients pursued it: contemplatively, with breathing space, for it's benefits to the mind.

I think the modern student who somehow manages to consciously grasp that aspect of math does it on his own. I would say, too, that students exhibiting this bent of mind, or who see math as "aesthetic" might be in some danger of becoming mathematicians, and should be closely monitored. ;)

 

[wabbit]

There is one silver lining here, a benefit from a poor early experience with math - something I've experienced not in maths but for other things I hated at school : it leaves untouched the joy of discovery for when you later become interested.

 

[Tyrion101]

I would like to report that when I pass my current mathematics class it will be the end of math for me in college (at least as far as taking math specific courses!) Yay me. I had one plan and stuck with it., and so far it has worked.

 

[symbolipoint]

I would like to report that when I pass my current mathematics class it will be the end of math for me in college (at least as far as taking math specific courses!) Yay me. I had one plan and stuck with it., and so far it has worked.

Beware of the future need for more Mathematical skill and knowledge. Your current, formal terminal required course may not be enough for you to be competitive enough for what you want to accomplish; and you do not yet know what you wish to accomplish ten years from now.

 

[Tyrion101]

Well I simply meant the requirements for my degree, I don't need any extra math at this point, and I never thought I'd make it.

 

[symbolipoint]

Well I simply meant the requirements for my degree, I don't need any extra math at this point, and I never thought I'd make it.

As I said, BEWARE!

 

[Tyrion101]

I will keep that in mind, though I think I will concentrate on actual course load.