Math instructors do it all wrong. I’ve taken 10 college level math classes, and it really despises me the way the material is introduced. Math is a great tool, but it’s boring. Only when it’s applied does it become fun. So, instead of teaching me what a Taylor Series is, or how a Heaviside Function works, why don’t you spend the first 5-10 minutes of the lectures telling me who these guys were, and what they were trying to accomplish?
For example, my most influential instructor (granted, it was an engineering class) told a story about Rayleigh observing the overhead shafts in mills. Since the river was always flowing, the shaft was always spinning. To transfer power from the shaft to your device, you pulled a lever which engaged a leather belt. Rayleigh noticed that the two dimensional projection of the wave changed shape. He used this informaton to develop a mathematical model for different types of waves, and even got a type of surface wave named after him.
Sure, that instructor spent about 1/3 of the class telling cute little stories. But I guarantee that the other 2/3 of the time, I was more attentive than in any other class I’ve ever had. I actually learned why the mathmatical relationships were developed, instead of simply how they worked.
Aaaah, I wondered that for years, specifically after 9th grade, where (quote the best student I’ve ever met) “all the math we’ve learned, we’ve learned it in Chemistry and Physics.”
I think part of it is that there’s so many math instructors who got saddled with math by default. The few times I’ve been able to speak with a mathematician in depth (in as much depth as you can get when one of the two people doesn’t speak the language, that is… I don’t) have been FUN!
I’m the same way - I like the background of maths, chemistry, whatever. It helps me know what to look for in my solutions.
However, in a college classroom there’s limited time and a lot of people there for the grade. (This may be different in a math program.) They don’t care about the why, they care about the calculations, the tests, etc.
Besides that many math professors don’t have the capability to lecture as well as your one professor on many things that aren’t math related - I think that many of them have either internalized the background and don’t think about it or aren’t good at bringing it out in lecture.
The problem isn’t limited to college. Actually, I think a lot more people would “get into science” if we were better at making it interesting when it’s first encountered.
But, college comparison:
Lilbro and me both had a 9-month, 3h/week Statistics class in college. For me, it was one of the classes I loved. For him, it almost cost him his degree.
He finally was able to pass his class - by studying with my classnotes!
We covered the exact same functions. But our class was called Applied Statistics; even though the applications were in Chemistry and he was Business, being able to see “what the heck is an ANOVA for” and “why do we have both G and t” made the different functions a lot easier to remember.
I suspect that this is an issue that if the person was interested in the application of mathematics, they wouldn’t have become teacher. They’re more in to math for math’s sake.
Yeah, the best math teachers I had did that. I flunked out of algebra II twice before I had a teacher who thought to explain how these concepts were used and how they came about. I’m not sure if the story about Pythagorus and the pyramid is true, but it helped me learn it. I’ve never once not been able to find the side of a right triangle since.
In Dutch highschool, for math, chemistry or physics, I had straigth A’s with one teacher and E’s the next year with another teacher. And vice versa.
Mu guess as to the “why” is that Beta-teachers are so hard to find, schools have to hire appallingly bad teachers as well.
I haven’t noticed this difference in quality with teachers in, say, history or French. Schools just seem to have a larger pool of alpha-science teachers to choose from.
The problem is that students expect to be entertained. Mathematics and Science involve definitions, and technical proficiency requires disciplined learning. While the human back-story of the scientists and mathematics may well be an entertaining way to avoid actually doing any of the science or mathematics, the point of the class is to develop proficiency and understanding of the technical material: the actual mathematics and science.
It is the student’s job, especially at the collegiate level, to motivate themselves. It is the instructor’s job to have technical proficiency in the subject matter, and to create a learning environment that allows students to learn the material. At the adult levels, the students drive the learning, the instructors provide the learning environment and knowledge.
Analytically challenging courses may well be boring, but in many cases, the course is of necessity about learning lots of deep stuff in a short time-frame, where those skills are needed for a related course or sequence of courses.
In a limited time-frame, even students with the appropriate prerequisites and core skills still require lots of detail in a technically deep course.
It may well be better to use a softer approach in non-major, service courses, but if the course is part of building an analytic skill-base, the focus needs to be on the actual stuff.
Having said that, enlightened instructors might be well-advised to provide supplemental “soft” materials as appropriate for off-class reading.
Having said that, lecture is a horrible way to teach mathematics. A better way is to preassign reading, so that the tedious definitions and foundations are preloaded by the students, and to then build on that in-class with group-mediated case-work. Active learning is a better proposition in analytic courses, especially when the goal is working proficiency for the next course or courses.
My first year of college I tested into a calculus course. I never had calc in high school. It was an engineering calculus course and it just taught the math of it and didn’t apply it to anything. I did so horrible in that class that I ended up auditing it before I flunked out of it.
The next semester (with the same professor) I took a business calculus course. It made so much more sense this way that I aced the class and got the highest grade in the class.
That’s exactly why, IMO, I did much better in Physic than Math in high school. We’re talking about 2 grade letters better (final grade for Physics B+, Math D+).
Math class is all about drop the equation on you Okay, you take this number this number and this number and use them to come up with that. It doesn’t matter how we came up with them, just that you know how to use them.
Physics was here’s what we’re doing, how it applies, this number actually means something and isn’t pulled out of thin air and once you do all this it tells you this. Then I knew how to use them (and was able to use them in math class with less headaches).
I didn’t need a huge amount of background info (though it was interesting) I need to know why we are doing things this way (at least a general idea of it), how we got the number and what it means. Otherwise it was just meaningless babble, and why should I learn it then?
I think it’s an aspect of the dichotomy between ‘pure’ science, or math, and ‘applied’ science or math. With so-called ‘pure’ studies, what matters is the knowledge and study, itself. With ‘applied’ studies what matters is what things can be used for. And, except for rare cases, the two views don’t often meet. It’s a way of looking at the world, and often ways that are antagonistic towards each other. I’ve seen chemical engineers and chemists get into huge, flaming arguments about the morality or propriety of using processes that are well-defined but poorly understood.
Both views are important, I believe, but there are a lot of people in the ‘pure’ fields who hate the idea that one might simply want to use their tools, rather than simply understanding and appreciating them. And so try to avoid any mention of practical applications.
This isn’t to say that cerberus, doesn’t have a point - there is only so much time in any given course. But I do believe that there are competing institutional attitudes at work, as well - where those people who are most enamored of ‘pure’ studies actively avoid mentinoing anything that might be ‘applied.’
Odd. I always liked how my teachers taught math, with the exception of a geometry teacher. She would occassionally try to bring mathematician biographies into the math, and I usually found that tedious. The others tended to have a very methodical step-by-step approach to showing how to solve something. Some were better than others at it, but the best would essentially get out of the way of the math and just let the math stand on its own.
But then, I don’t think math is boring. I actually like math! (Although, for the record, I’m not that great at it.)
Another odd thought - I suspect that most of us who’ve mentioned a preference for ‘applied’ math were the sorts who loved word problems.
Which puts us firmly in the minority. I seem to recall that word problems are a major hassle for most students - because they don’t think about what they can do with the math, just wanting to know how to do it.
Well, I’m an engineer and Lilbro is in business administration, so we both definitely like knowing “what is it for, what problem does it solve” and “why are there several that look similar” (the answer is often “because they have different uses, they are solutions to different problems”).
But also, the first time I remember getting frustrated at math was the demo on how to solve a second degree equation. My memory insists on labeling it “the first demo we had to learn” but no: it was the first one I had to learn by rote, with no explanation of “why” you had to follow that particular sequence of steps and no understanding of the sequence. Previous demos had been about geometry, which comes naturally to me, so I hadn’t had any problem seeing their logic. For that one, I didn’t see the logic and the teacher’s response to my "why"s was “because, you just study it!”
If I know “why”, I can rebuild the “what”, I don’t need to remember it.
Continuing the example with Lilbro and statistics, for us it’s easier to remember that the equations that include “n-1” are those where “n is finite” and the ones that include “n” are those where “n is infinity or treated as if”, than to “just remember it.” It makes sense, because “n-1” would still be “n”, when “n” is infinity.
I don’t think the problem is with applied versus pure math. Pure math can be interesting, IMO, if they teach you how and why it works the way it does. For instance, in elementary arithmetic, they teach you that if the digits in a decimal number all add up to nine, then the number is divisible by nine, but they don’t prove out to you why it works, and, later, in algebra, they don’t bring it up again and show how it can be generalized to polynomials in any x. Consequently, if you don’t figure these things out entirely on your own, math becomes just a collection of seemingly unrelated rules and tricks. By the time you get to college, the only calc you can hope to pass is that offered in the non-rigorous track, where you don’t have to understand or derive the formulas, but only apply them.
For me, math was much easier and more beautiful without applications. I didn’t need it to do anything, it just needed to be. (This would occasionally be a problem in science classes where I could easily see that the answer was “7” but had no idea what “7” meant. Other people could figure out that we were trying to find a certain property or value - but couldn’t figure out exactly what that value was. )
Language questions, I figure I could ask in GQ instead of hijacking but then again, you guys look like a good pool of possible responders:
what do you call that step where you extract the equations from a word problem? I don’t have my Big Dictionary here and none of my online ones gives a translation that feels right… they insist on saying that “plantear un problema” is “to raise an issue,” which is a different animal.
would this be an example of a “word problem”, or not?
Given two variables, x and y, the sum of x and y is 10 and their product is 24. Find the values of x and y
To me it’s a “problema matemático”, a math problem. I’m used to distinguishing three levels, I guess, where that one would be the intermediate. One is where they just give you the equations directly. That’s the monkey-do level; so long as you can figure out which tool to use (and many people are able to do this, without being able to explain why) you can solve them.
Then there’s where you have to extract the equations from the problem, like this one I gave in 2). This requires analytical thought.
And there’s where the equation, which may have been given to you or extracted from the problem, is not in terms of X and Y but of whatever-other-symbols. Recognizing [Na+] as your “x” (or being able to say, “aha, this business scenario is like those concentration problems my sister had!”) requires abstract thought.
The steps to solve an applied problem would be:
get equations
identify variables
solve
The steps to solve a math problem would be:
get equations
(variables are already identified, but they’re abstract)
solve
Either of these two steps can be terribly difficult or it can be helpful, depending on who you’re talking about. Lilbro and myself, we find the applied teaching easier because the knowledge behind “how the tools came to be” (which does not require a course on “history of mathematicians”, just a “this is for when you can get enough data to assume infinity” and “this is for when you have very limited data”) helps us choose the right tool. Once we know the applied version, we can work the abstract one. If all we’re given is “here, a box of tools” and “here, a bunch of nuts and bolts and screws”… we have to go to trial and error, pretty much.
I believe that both sorts of problem you’d suggest, Nava meet the definition of a word problems as they’re considered here in the US. It’s simply a general term for describing any mathematical conundrum with words, instead of giving the student the equations having the student solve it.
To my mind, the best word problems were those that presented some kind of real-world situation, and left the student to solve something about it, with maybe a hint.
An example of a nice multidisciplinary word problem might be:
This resembles a word problem I recall from chemistry, but involving a number of algebraic calculations. With small changes in the information included in the problem statement, it could become a valid exercise for a regular math class, as well.
This isn’t to say that the intermediate problem you describe isn’t in use in American texts, just that I don’t recall them giving classmates fits like this example would have.
Actually, the word problem is “You’ve just bought a house with a rectangular yard. When doing home improvements, you covered the yard with 24 sq.ft. of sod and you used 20 ft. of fencing to keep the neighborhood kids away. What is the length and width of your yard?”