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Math and Science Popular With Students Until They Realize They're Hard
First time accepted submitter HonorPoncaCityDotCom writes "Khadeeja Safdar reports in the WSJ that researchers who surveyed 655 incoming college students found that while math and science majors drew the most interest initially, not many students finished with degrees in those subjects. Students who dropped out didn't do so because they discovered an unexpected amount of the work and because they were dissatisfied with their grades. "Students knew science was hard to begin with, but for a lot of them it turned out to be much worse than what they expected," says Todd R. Stinebrickner, one of the paper's authors. "What they didn't expect is that even if they work hard, they still won't do well." The authors add that the substantial overoptimism about completing a degree in science can be attributed largely to students beginning school with misperceptions about their ability to perform well academically in science. ""If more science graduates are desired, the findings suggest the importance of policies at younger ages that lead students to enter college better prepared (PDF) to study science.""
Whats needed is good educators, like Richard Feynman was. What passes for "good educator" these days is pathetic.
"His name was James Damore."
hard is merely the fact that often, the theories and equations taught are quite abstract. It is very important to have a solid grasp of concepts, but in the end, the material could be improved with visual and/or tangible results which have some values and/or association to the abstract concepts.
I've had dozens of college profs and the ones which stood out were the ones who were good listeners as well and perceptive of what students struggle over. Generally I found when I thought a course was 'hard' I knew 80% or more of the material or concepts, but I was struggling over one or two things which blocked conceptual understanding of things further on.
Subbing, as a TA once in a programming class I was perplexed how people couldn't wrap their heads around the idea of a Variable (think of it as a name on a bucket, into which I add or remove apples, yet they were still stumped).
Things do tend to be more 'hard' when the student spends more time listening to their nay-saying peers than their instructors. When you actually believe Math, Chemistry or Physics is 'hard' your belief is your own largest obstacle to learning.
A feeling of having made the same mistake before: Deja Foobar
No. Math is hard because it's like running long distances. Few people actually like running, or any kind of exercise. Many people do it for utilitarian reasons while hating it. Some people like it inherently, though. I had a gym teacher once who was addicted to running to the point that it was bad for his health.
In almost any skill that has to be learned, there's often a fairly rapid and abrupt transition from "I can't do that" to "I CAN do that and since I now know how to it's actually easy".
I think a lot of people get discouraged when they're unable to get through that transition on their own the first time they try it, and "I can't do that right" can be appear to be an impossible mountain to climb, even if you're not far from the top.
I think we need to be challenging kids from an early age to learn things that are "hard" so that they become intimately familiar with this progression from impossible to trivial. Too often I see kids these days try something that looks interesting to them a couple times and then decide "nah, that's too hard" and quit.
It's not specifically teaching perseverance, but more about learning to recognize that progress is almost never linear toward a goal and many times you won't recognize you've reached your goal until you're actually there.
Additionally, we ought to be able to get better at helping people fight through these places they get stuck, rather than just leaving them with a failing grade in a math class and a feeling that that they're not up to the task. Early recognition of students who are having difficulty and focused tutoring and other help getting through the hard parts to the point that they achieve their needed breakthrough.
I don't think any undergraduate subject should be so inherently difficult that anyone who can get into the university in the first place shouldn't be able to do well in it.
G.
Feynman was fantastic at inspiring people and giving them an intuition for physics with simple drawings.
Do you think he understood partial differential equations, functions in a complex space, matrix math, group theory? Sure he did. If he wrote some of that on a blackboard in a 60 minute talk, would the audience struggle to keep up?
I am still not sure I understand using 4x4 matrices to do transforms in three space. I can write the code though (slowly).
My wife (English and Drama) said the biggest party people were the liberal arts students because they did not need as much time to study. And when they were studying they mostly were reading.
A good educator can make learning calculus better than a poor one, but there it is still hard (well for me anyway).
Make sure you include the requisite grain of salt. The blog is based on a study from over a decade ago - performed at a liberal arts college. Quickly perusing the school's website, I do not see a strong emphasis on STEM programs (I don't even see a B.S. offered, even the CS degree is a B.A.).
Not that I entirely disagree with the premise, but I think a study at a school with a broader academic base would provide more worthwhile results.
+1 Disagree
I have a most wonderful proof of that assertion, but sadly the limited character set of the slashdot text editor will not allow me to present it!
Having talked to East Asian co-workers, we came to the conclusion that while rote memorization was by far in favor of the Asians, solving unseen problems went to the Americans. They were constantly astounded at how easily we could solve problems that we had never heard of before and credited the American education system. So, I would say not dumb, just a different focus.
Why would I care about doing the lightning-speed mental arithmetic? I have a calculator for that.
Peter predicted that you would "deliberately forget" creation 2000 years ago...
While my intuition tells me that high school grads are, on the whole, not as well prepared as they should be, there is certainly some improvement that could be done at the college level.
One problem I faced on the path to my EE degree was that in mathematics classes and some engineering classes (particularly electromagnetic fields, communication systems theory, and stochastic signal analysis -- which of course are some of the most math/calculus heavy of the EE curriculum), was that I lacked an intellectual model of what the mathematics was accomplishing. While concepts like derivatives and integrals made a degree of sense because they could be related to velocity, acceleration, position, area, and volume, when I got to the point I was dealing with eigen-this and eigen-that and hermetian-something-or-others I had lost any real-world connection, and my understanding suffered as a result.
The most frustrating and poignant instance of this was the first day of my linear algebra class, which I was taking only as a pre-req for CS class on GUIs, which only needed it to the extent that rotation, translation, and scaling using matrices was involved, and I already knew that much. Anyway, the mathematics professor walks in and announces "I do not care, even one little bit, what this material is used for in the real world. I am here to instruct you in mathematics alone." I looked around the room. In a class of about 25, I believe there were 20 science/engineering students, 4 math students, and one photography major (she was one of those brilliant types who took upper level classes in sciences, math, philosophy, or anything else just for fun). I was somewhat incredulous at the professor's utter disregard for his students' background, abilities, and interests. And just as I expected the course was utterly miserable and tedious, and then there were the bad days.
I contrast that with the math classes I took for Calculus II-IV, and Numerical Systems Analysis. The professors (thank heavens I avoided graduate students) who taught those classes were totally on top of the situation, and made it very clear what we were trying to accomplish with real world examples, or at least didn't veer too incredibly far from intuitive models. I think it helped that in Calc II-IV I had the same professor all through, and he was teaching a pilot course that integrated calculators into the material, so there was a lot of approachable material throughout. This was a stark contrast from the previously mentioned Linear Algebra as well as the Differential Equations I courses.
To this day I hate Linear Algebra and Differential Equations, and I'm 100% convinced it's due to the terrible instructors I dealt with. Which is a shame, because I loved mathematics in high school, and would go beyond my coursework to explore what I could on my own without much additional help from my (incredible) high school teacher, and I had a blast doing it. If I hadn't developed a strong interest in aeronautics and computers I most likely would have pursued a math degree.
The biggest problem I faced throughout my mathematics education, as well as many engineering classes, is that as the course would progress it was building taller and taller upon a shaky foundation. While my arithmetic was bedrock, my algebra was concrete, and my trigonometry was 2x4 construction, the rest was a lot less solid. Calculus felt a lot like building with Tinker-toys, and by the time I got to anything past that it was toothpicks stuck together with Sticky-Tack. As more and more material was piled on top, a lot of it kept slipping off because the stuff underneath it was crumbling. I would have benefited greatly from either better construction (i.e. better instruction), or a lot more hands-on experience with those shaky bits such that they were strongly reinforced.
Cyrano de Maniac
Care to offer some evidence for that assertion?
Of course not! He was talking about math, not science!
I guess technically he needs to provide a proof.
In high school a very smart student can get honours marks with minimal effort. In high school an average student can get honours marks by working very hard.
In engineering school a very smart student needs to also work very hard just to get by. If you are diligent about doing all the problem assignments, hand in all the labs, study efficiently (in a small group really worked for me), be very strategic about obtaining all possible marks, you can do reasonably well. In engineering school an average student can't get by on hard work, because the workload is too high, and will likely fail.
None of them can see the clouds; The polished wings don't care.
Funny, marriage is like that, too.
You are welcome on my lawn.
Or because running long distances requires a constant amount of effort. You can't show up to a marathon 13 miles in, think it's over in less than 4, and expect to win anything or even get a sense of accomplishment.
Math and science build, it starts very early, and it keeps building up. By high school most people are already severely disadvantaged. By college, the game is over but for the most dedicated. I will give these people a little credit, I think they truly want in and see the value, but get lost in college material and pacing, and don't even understand how they went wrong, They end up with retarded cop-outs like "i'm dumb at math" or "science makes no sense", which sometimes become self-fulfilling prophecies. They have to be approached like a physical fitness program: you start out easy or you will hurt yourself, and you work up to the serious stuff. There's no cramming for it, you can't jump in and be awesome, it takes a long time.
Most of the other subjects covered in that article can be easily picked up to "beyond the average bear" levels by just reading some books for a few weeks. It's not a surprise then if you're looking for a piece of paper in 4 years and you do not already have some skill in STEM, you go for something easier.
> I am still not sure I understand using 4x4 matrices to do transforms in three space. I can write the code though (slowly).
That's just proof that you had a bad/crappy teacher. :-( Here is one explanation:
In 3D computer graphics we use a 4x4 matrix to conveniently and compactly represent _two_ things:
a) orientation, and
b) location (or position) within a single variable.
Where:
R = 3x3 orientation matrix, and
T = 3-dimensional position vector.
To understand how this comes about let us start with something a little more basic: 2D Affine Transformations. Namely: Rotations, Translations, Scaling.
Given a point P = we can write it in matrix form as either [ x y ], or
[ x ]
[ y ]
How would we write the equation for a point that is rotated around the origin (or z-axis.)? We will eventually want to write a matrix equation where the matrix represents a change in orientation. That is by definition:
x = R * cos(A), and
y = R * sin(A)
x' = R * cos(A+B), and
y' = R * cos(A+B)
Where:
R = radius of the angle,
A = initial angle,
B = the relative change in the angle,
A+B = the absolute final angle
We don't always know R, so let us rewriting these in terms without R:
x' = R * cos(A+B)
= R * {cos(A)*cos(B) - sin(A)*sin(B)}
= {R*cos(A)} * cos(B) - {R*sin(A)} * sin(B)
= x * cos(B) - y * sin(B)
Similarly we do the same for y.
Now, we would also like to write the equation for the Translation of a 2D point:
x' = x + dx
y' = y + dy
Likewise Scaling is pretty straightforward:
x' = x * sx
y' = y * sy
These 3 different operations require 3 different functions and order of operations! This sucks. It sure would be nice if we could unify these operations into one equation! We actually have two choices for how we could write/calculate this:
a) Pre-multiply the column vector (ignore the '.' it is whitespace due to /. being lame.)
b) Post-multiply the row vector
At the end of the day it doesn't matter which convention you pick just as long as you are consistent.
Since /. is lame and doesn't like an _informative_ MATH post I'm breaking it into two parts...
Engineers want to be physicists.
Physicists want to be mathematicians.
Mathematicians want to be God.
God is an engineer.
> I am still not sure I understand using 4x4 matrices to do transforms in three space. I can write the code though (slowly).
Part 2 since /. ecode formatting is still so gey I am including a bunch of whitespace filler text '.' to align things up in columns.
Now, expressing the Rotation equation in Matrix form. Remember we ended up with these two equations:
x' = x * cos(B) - y * sin(B)
y' = x * sin(B) + y*cos(B)
We can literally "transcode" them from algebraic form into matrix form without too much difficulty. We end up with this:
And expressing the Scaling in Matrix form:
Likewise expressing the Translation in Matrix form:
The problem is that a 2x2 matrix form won't do! We need to extend the problem from 2D to 3D !
The exact same _principle_ is used for 3D. We extend a 3x3 matrix (orientation) to a 4x4 matrix so that it expresses BOTH a orientation AND translation.
Hope this helped!