Math is, generally, poorly taught (Engineering student's perspective)

[Elbobo]

[I'm more concerned with how mathematics is taught in university in this topic]

I'm a junior Mechanical Engineer and have taken Differential & Integral Calculus, Multivariable Calculus, Differential Equations, and Linear Algebra. Just from these classes and talking with a variety of peers in STEM majors, it's pretty clear to me how poorly mathematics is introduced and driven into us.

At least for the non-mathematics major.

Just because we're engineers or scientists doesn't mean the mathematical theory shouldn't be expounded upon. I'm not necessarily talking about proofs (which engineers and scientists usually DO hate), but more about the consequences of some theorems, how a theorem came into conception and what it enabled, what it actually means versus being just some formula, etc.

My Linear Algebra instructor took the route of skipping formal proofs and instead tried to get us to understand the subject and theory itself rather than just solving rote, systematic problems (though we did do some of that, of course). By elaborating on concepts in class, he took our thinking to a deeper level and my interest piqued, and thus I strove to do better in the class and get a better understanding, which ultimately led to a very satisfying Math class that I haven't been able to have in forever. Just because I didn't major in Mathematics doesn't mean I don't appreciate math.
My only gripe is that he would seldom relate the many applicable concepts to engineering or the sciences, such as bases or eigenvectors.

Differential Equations was by far the worst. I learned the methods. I learned how to use integration factors, Fourier series, Laplace transforms to solve different problems. But did I learn why I did what I did? Did I ever learn what a Wronskian truly meant? Or that the the conception of the Fourier Series enabled the heat equation to be solved because it turned complicated oscillations into sines and cosines? Or what the hell "s" was in a Laplace transform? NO.
This shouldn't be optional for the curious student; it's pivotal in understanding why a certain method is used and why it's used the way it is. If you understand that, you have a much better chance of retaining that knowledge, a deeper grasping of the theory, and possibly an ability to apply it creatively to unknown territory.

I can now say I have a firm grasping of intro Linear Algebra, while I can't say the same for Differential Equations. A lot of my friends who took the same LA professor will say the same. DE was a worthless class because it consisted of crunching out God-given methods without a good theoretical understanding.

So in essence, this is an engineering student's plea to anyone who sets up Math curricula for engineers/scientists or teaches it:
TEACH US THE THEORY. Get us to appreciate it. Get us to understand it from an abstract, qualitative perspective rather than just part of the problem-solving drudgery. That doesn't necessarily mean make us do a formal mathematical proof, but rather make us see mathematics in a meaningful light.

[sorry, I may have rambled a bit. Just trying to get people to understand what is wrong with mathematics in university today.]

 

[Sankaku]

DE was a worthless class because it consisted of crunching out God-given methods without a good theoretical understanding.

Yup. How do you think these type of classes feel to a math major if they are forced to take them?

TEACH US THE THEORY. Get us to appreciate it. Get us to understand it from an abstract, qualitative perspective rather than just part of the problem-solving drudgery. That doesn't necessarily mean make us do a formal mathematical proof, but rather make us see mathematics in a meaningful light.

I think this is certainly possible, as your linear algebra teacher showed. However, there is a point at which you have to see proofs to get insight into what is going on in mathematics. It is kind of like reading popular science books on black holes, cosmology or quantum gravitation. The story is fun and motivates you to want more, but without the math and physics to back it up, everything is waving hands.

I agree that some professors in mathematics could do a better job of teaching proofs and intuition hand-in-hand, but the traditional method in math is to show the proof and make the student grapple for the intuition on their own. To compound this, the math curriculum for engineers is designed to pack as many disjointed tools into as short a time as possible so that you can get on with building things. No wonder engineers don't want to see any proofs.

Could it be improved? Most definitely. It comes down to good teaching skills, though, and those are in short supply in any field...

 

[intwo]

Mathematics courses that neglect theory can cover more topics and focus on computation and applications. Most science and engineering students will never need to know what a supremum is, but they'll need to know how to tackle a variety of problems.

I think that schools should offer two flavors of mathematics courses: computational and theoretical. Then students who only care about the tools can take the computational version and students who are interested in the development can take the theoretical version.

 

[homeomorphic]

I am teetering on the edge of giving up on being a math prof (currently, I'm a PhD student, struggling to possibly graduate this year), chiefly due to student complaints (which are a deal-breaker because the powers that be in math departments get so panicked about them). When I started on my path towards being a mathematician, I liked the idea of teaching because I just thought of it as, "explaining math". When I found out that it was not about explaining math, but more to do with putting on an act to prove to the lower level students think you are good enough for them, the whole career has lost its appeal for me. The question is whether I can learn how to put on this act for the lower level students, and whether the upper level teaching and research is worth putting up with it for.

Anyway, my greatest fear if I do decide to continue in academia is that I will present things conceptually and intuitively and the students will just PREFER to just plug and chug because that is what they are used to. Then, to avoid their dreaded complaints, I would have to do the ultimate sell-out, dumb things down, and make it all cookbook. Essentially, when I taught trig/precalc, that is exactly what happened to me. Maybe this fear is unjustified. But I have heard some math profs express this sentiment about teaching linear algebra--that the students just want cookbook. I hope it's not true.

So, instead, I'm thinking about just finding whatever job outside academia I can get that pays the bills and leaves me with plenty of free time, so I can create the best math website on the web, and then most of the unmotivated, unintuitive, conceptually shallow math woes will be solved for everyone once and for all. Just look it up on my website. Give me 5 years, and we'll see if I can't make it happen. Whether or not I go for academia, this is one of my life goals, and a much greater contribution to mathematics than what would come through researching new topics.

 

[twofish-quant]

Anyway, my greatest fear if I do decide to continue in academia is that I will present things conceptually and intuitively and the students will just PREFER to just plug and chug because that is what they are used to.

And that's what their employers want them to do. Ninety percent of people that take lower division math are doing it so that they can get a piece of paper that they can give to an employer and convert to cash, and they are not interested at all in the beauty of math or the deep concepts.

But I have heard some math profs express this sentiment about teaching linear algebra--that the students just want cookbook. I hope it's not true.

It's true, because the students want what employers want, and in most jobs employers would rather do cookbook math. Employer pay the money so that students pay tuition so that your salary gets paid.

So, instead, I'm thinking about just finding whatever job outside academia I can get that pays the bills and leaves me with plenty of free time, so I can create the best math website on the web, and then most of the unmotivated, unintuitive, conceptually shallow math woes will be solved for everyone once and for all.

There's also another way to do it. You can give no-credit, optional seminars on topics that you are interested in, and get involved with undergraduate math clubs. If you give a no-credit seminar, then 90+% of the students will skip it, but the people that show up will be interested in what you have to say.

Websites, and youtube are also good for this. Just don't expect any of this to be counted toward tenure review. On the other hand, I've found that sometimes doing stuff like this is necessary to just keep sane.

On the other hand, we really can't expect our students to do things that are not in their career interests out of pure academic curiosity if we ourselves aren't willing to do the same thing.

 

[Sankaku]

I agree with twofish that the strange beast that Calc I,II,III, ODEs and LA have become is directly driven by the get-it-done-quickly mentality. You have to be a pretty patient teacher to deal with 120 disinterested students. That doesn't stop a teacher from at least trying to convey some of the context of what is going on. For profs, upper-level courses are much more attractive to teach because most of the students actually want to be there.

...and leaves me with plenty of free time, so I can create the best math website on the web, and then most of the unmotivated, unintuitive, conceptually shallow math woes will be solved for everyone once and for all. Just look it up on my website. Give me 5 years, and we'll see if I can't make it happen. Whether or not I go for academia, this is one of my life goals, and a much greater contribution to mathematics than what would come through researching new topics.

I heartily applaud this. I have watched the explosion of university mathematics lecture videos onto the web in the last 3 years. I am hopeful that essentially everything up to lower graduate level will be there in the next 2 years. The teaching quality still varies quite a bit and there is an issue of helping people find the right path through the material. However, I think the game of knowledge is changing quickly.

This, of course, is only useful for those who aren't interested in plug-n-chug. Also, students still need face-to-face courses with research-level professors, especially as they get further along their education. That said, I speculate that, for non-professional degrees, there will come a point where the debt load becomes so high for a degree, and free information of high quality becomes ubiquitous enough, that something is going to break. I hope that the traditional university model doesn't melt down too much, as there is a lot to lose.

 

[Elbobo]

The core problem is that educators believe, when choosing quantity over quality, all those techniques and methods STICK.

What good is learning 20 different ways to tackle an ODE if you easily forget them right after you take the final exam? That's the point I'm trying to make. A conceptual approach yields greater retention and a better ability to apply those techniques to various problems.

There were two LA professors teaching the semester I took it; my professor took the more conceptual approach while the other one chose to cover more topics with less focus on the theory. In the end, her students couldn't tackle the problems as well as we could.

Plug-and-chug-education only works so long as you have tests and quizzes and homeworks to keep it fresh in your mind. Once that incentive is gone, how are you going to plug-and-chug data that you have no idea what to do with? That's what a conceptual approach corrects, to a certain extent of course.

Too much theory, I believe, can be bad for such engineers and scientists, but if the educator can find a happy medium between elaboration and sparsity, students will benefit most. The issue now is that it's weighed much more heavily on the side of sparsity than theory.

 

[mathwonk]

If you think math is poorly taught, try some physics classes. When I was a student the math dept had the most brilliant and most skillful and dynamic lecturers teaching the math class I was in, but the physics dept had the oldest, poorest most boring lecturer in the world teaching the physics class. Objections ran so high that they had to actually replace him with someone good after the first term. By then I had quit caring or trying.

Of course I was in the super honors math class and the regular physics class. Against all logic they often put the best teachers in the honors class instead if the other way around as they should. But you might try taking an honors math class.

 

[D H]

TEACH US THE THEORY.

In general I agree with that perspective. Way back when I was in college fellow students would ask me for help on their math. I cringed a bit when those fellow students were engineers. For the most part they didn't take math courses from the math department. The engineering college taught math the way they thought it should be taught: Cook. Book. Yech.However, as a counterpoint, the absolute worst math course I have ever taken delved way too much into theory. The class was on optimal control theory. The instructor was a pure mathematician. He balked at the light treatment in the text on uniqueness and existence. Light treatment, my eye. The book spent 60 pages on uniqueness and existence. Most texts on optimal control theory spend at most 10-20 or so. Some, much, much less (paraphrasing): "Sometimes there is no solution, optimal or suboptimal. Sometimes there are multiple optimal solutions. Sometimes there is just one optimal solution. Live with it."

This was one of the most theoretical texts on optimal control theory I have ever come across, and yet it was too light for him. We spent a good 2/3 of the class on those 60 pages on uniqueness and existence. We never did learn control theory, let alone optimal control.

 

[mathwonk]

sometimes you need to remind the professor that while he is primarily interested in why the solution exists, that you are more interested in how to use it once that is established.

 

[dacruick]

But I have heard some math profs express this sentiment about teaching linear algebra--that the students just want cookbook. I hope it's not true.

I took a poorly taught linear algebra course in first year that fits the cookbook description. Nothing was in-depth, heck, nothing was even anything...
What I mean to say is that I not want this style of teaching. That course, or lack-there-of, gave me a terrible foundation for quantum mechanics which still haunts me to this day. I wish you had a really naive outlook and thought that you could change the world, because it would at least change a class.

 

[homeomorphic]

I took a poorly taught linear algebra course in first year that fits the cookbook description. Nothing was in-depth, heck, nothing was even anything...
What I mean to say is that I not want this style of teaching. That course, or lack-there-of, gave me a terrible foundation for quantum mechanics which still haunts me to this day. I wish you had a really naive outlook and thought that you could change the world, because it would at least change a class.

But I do have a naive outlook and think I can change the world. As I indicated, the solution, at least for me, seems to be not to work within the confines of the current system at all, but to just do it in my spare time, so I don't have to answer to anyone. With my website idea, I won't have to cover specified material or be held accountable by the students for anything, and I will reach a much wider audience.

I know many students don't like the cookbook approach. But there are a lot of students who just want the piece of paper and don't really care about learning.

 

[mathwonk]

look on my website and you will find several free linear algebra books. i spent years writing them. but will you read any of them? sighhh...

 

[homeomorphic]

look on my website and you will find several free linear algebra books. i spent years writing them. but will you read any of them? sighhh...

To me, it's worth writing, even if no one reads it, but me. Partly, I write for myself. But, I think once I get going, I think I'll have some fans. In theory, your audience is the whole world, so it shouldn't be hard to getter a bigger audience than you would teaching one class. I think there are some internet tricks you can do to get more hits. I plan to cover a lot more than linear algebra. I'm not even sure if I will write about linear algebra at all.

 

[mathwonk]

from "change the world" to "have some fans" is a little adjustment.

 

[DrummingAtom]

To me, it's worth writing, even if no one reads it, but me. Partly, I write for myself. But, I think once I get going, I think I'll have some fans. In theory, your audience is the whole world, so it shouldn't be hard to getter a bigger audience than you would teaching one class. I think there are some internet tricks you can do to get more hits. I plan to cover a lot more than linear algebra. I'm not even sure if I will write about linear algebra at all.

When you do write whatever you plan, please let us know. I have a funny feeling that your approach to math will be a pleasure for me to read.

To the OP, I'm an EE major and I had the exact same experience just last semester. I took a combined Linear Algebra and Differential Equations class and it made me hate math for a short time. The beginning of the semester seemed really cool then the plug and chug began and never stopped. My linear algebra experience consisted of 2 weeks of lectures and two problem sets that occurred early in the semester. We learned how to calculate RREF, vector spaces, determinants, eigenstuff, basis, span, Wronskians, and Jacobians in that 2 weeks.

About 2 months in, I grew extremely frustrated with the way the class was taught and spent a lot of time on my own trying to learn what it all meant. It's sad to say but the things I learned on my own felt like I discovered their true meaning, but really I just made some sense of the concepts my professors brushed under the rug. I'm not one to make excuses but because my frustration grew so great and I had to venture out a lot just to learn the concepts, my grade slightly suffered. I'm very confident though that I now have a far better understanding than the vast majority of what my classmates learned from that class. The self learning was a very worthwhile experience but I still feel cheated on what I could have learned if they approached the subject differently.

 

[Nano-Passion]

I agree, I want to to know the theory! And I want to know the big picture of things and the context of what I'm learning in the grander scheme.

look on my website and you will find several free linear algebra books. i spent years writing them. but will you read any of them? sighhh...

Actually, please do share. Its a shame, you sound disappointed that more people don't partake or appreciate on something you worked on so hard.

 

[homeomorphic]

from "change the world" to "have some fans" is a little adjustment.

Well, truth be told, I just don't know what will happen, one way or another. Maybe, I will have a few fans, maybe I'll change the world. I can't say how successful it will be ahead of time. There are many variables involved, including how much time I will have available to devote to it.

If I can affect just one person's life significantly with my work, I think that will justify it.

 

[mathwonk]

so,,, get on with it.

 

[Sankaku]

Actually, please do share.

mathwonk has linked to his books on the forum here many times. Unfortunately, the link is currently down:

http://www.math.uga.edu/~roy/

 

[Chunkysalsa]

I agree that lower division math courses should focus a bit more on theoretical material and less on computation, to a degree. This will help serve both the engineers, physicists and mathematicians. However in an engineering class, which is where I learn a lot of the math I use, there simply isn't enough time to go over the proofs and ideas behind a mathematical construct. I too learn math better when I understand the meaning behind it. For example I've learned Laplace Transforms in a math environment and Fourier Transforms in an EE environment. I am much more comfortable and knowledgeable about laplace, simple because I've seen the various proofs and theory behind everything I use. I like my Signals book because when explaining various mathematical Ideas they'll usually provide a short informal proof and historical footnotes, enough for me to get a bigger grasp on the material without getting bogged down with the formality of math.

So I completely agree with you, which is why I've also decided to double major in Mathematics along with EE.

 

[Nano-Passion]

mathwonk has linked to his books on the forum here many times. Unfortunately, the link is currently down:

http://www.math.uga.edu/~roy/

Thanks!

 

[Woopy]

out of curiosity mathwonk, why do you assign required homework? One thing I had this year in calc 2 was problems for homework, but were not required. It put a lot less stress on me to work on learning how to solve the problems, rather than having to worry about whether I did my HW or not. No HW is the best!

 

[mathwonk]

well remember my job is not to lessen your stress but to teach you the material.

i have tried every possible combination: homework graded for credit, homework graded but not required, homework graded and only counted if it helps your grade, homework counted only for quantity of effort (measured by number of problems attempted and pages of work done), no homework problems but class presentations required but I let you practice them on me first and i write your lecture notes up for you in advance, no homework and no tests, courses where i give everyone an A no matter what they do,...

there is no system that works for everyone. but basically most people perform better when expectations are higher.

 

[Angry Citizen]

The problem with assigned homework typically revolves around manpower, at least at large universities. You just can't grade anything but a 'representative sample' of homework problems assigned. Thus, the expectations can't be considered 'higher'. It's just more hand-holding in my opinion. Some people benefit from that hand-holding. Others do not. I like quizzes, myself. They're more directly relevant to timed exams, whereas I think actual assigned homework would benefit a class with take-home exams. Just my thoughts.

I also agree that engineering mathematics is taught way too superficially, but I think there's just a need to keep things as condensed as possible. At my school, I (as an aerospace engineering major) am allowed to take two electives, and both of those electives must be technical electives, and one of those technical electives must be directly related to aerospace engineering. That doesn't give the department much flexibility in requiring a greater number of math classes, or a greater number of topics covered. My school already doesn't require a linear algebra course for engineers; it's apparently crammed into several other courses as necessary, which likely means I'll miss out on the oft-discussed 'beauty' of the subject (with which I am completely unfamiliar).

So in a sense, the handwaving done in engineering mathematics is likely the result of practicality. As an engineer, you have to realize that quality must sometimes be sacrificed in order to ensure other conditions are given proper emphasis - in this case, time required for an engineering degree; stress on the student; relevancy of the information; and so on. I personally plan to study mathematics as much as possible, because mathematics is the foundation of physics and engineering. One of my electives will be a partial differential equations course. I'd also like to tackle real and complex analysis, linear algebra, and possibly find out what in the blazes abstract algebra is, after I've graduated. But again, I have to work within the constraints of my degree, unfortunately. I think most engineering educators feel the same pressures.

 

[Kholdstare]

Math is, generally, poorly taught (Engineering student's perspective)

Alexander Calandra and barometer

 

[resolvent1]

I agree that lower division math courses should focus a bit more on theoretical material and less on computation, to a degree. This will help serve both the engineers, physicists and mathematicians. However in an engineering class, which is where I learn a lot of the math I use, there simply isn't enough time to go over the proofs and ideas behind a mathematical construct. I too learn math better when I understand the meaning behind it. For example I've learned Laplace Transforms in a math environment and Fourier Transforms in an EE environment. I am much more comfortable and knowledgeable about laplace, simple because I've seen the various proofs and theory behind everything I use. I like my Signals book because when explaining various mathematical Ideas they'll usually provide a short informal proof and historical footnotes, enough for me to get a bigger grasp on the material without getting bogged down with the formality of math.

So I completely agree with you, which is why I've also decided to double major in Mathematics along with EE.

Out of curiosity, what was the signals book you used? Thanks.