为什么数学老师这么不擅长解释(一)Why are math teachers so bad at explaining things?
2022-09-17 汤沐之邑 3908 0 1 收藏 纠错&举报
Why are math teachers so bad at explaining things?
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Some of these other answers are laughably egotistical and negative. None of them consider the uniqueness of mathematics as a discipline. The primary answer to this question is just that – that mathematics is genuinely a hard subject to teach.
Think about any realization you’ve had while studying math. Some problems seem daunting, but once you’ve had a look at the answer often times it seems like it should’ve been obvious. Usually because there exists a nicely elegant and perhaps rather creative solution being presented.
Think about every time you’ve seen that “beautiful solution” and convinced yourself the problem is easier than it actually is. Well duh, everything seems easier in hindsight. A similar thing happens when teaching math. The more mathematics you understand, the harder it is to gauge how difficult the material is. 3Blue1Brown makes a good point of this in his one video
by the end he’s talking about the author of math textbooks who must come up with practice problems. He noted that:
“Across a wide variety of contributors there is one constant: Nobody is able to tell how difficult their exercises are. Knowing when math is hard is waaayyyy harder than the math itself.”
And I think this is something to keep in mind both as an instructor and as a student.
Quick answer: culture and administrative practices.
America has both awe and disdain for mathematics. On one hand, people are impressed by long, complicated-looking equations on blackboards; on the other, they say “What the hell am I gonna need this for in real life?” This influences the outlook of many math teachers: one subset will look to impress; another will look for the easiest way out—both at the expense of efficacy.
District guidelines strangle K-12 teachers in public schools; so, even the ones who are good are compromised by bureaucracy. Whatever passes the most students is the enforced teaching methodology. In most cases, that’s memorization.
In college, professors are just as susceptible to the memorization disease as their students. To make it worse, they’re often responsible for reaffirming that ideology and oiling the gears for another cycle when their students become instructors.
Memorization leads to poor explanation.
Math professors work at research universities for the prestige, for churning out papers and securing funding. It’s a show to impress, not to teach. As such, they’ll cut every pedagogical corner they can, conveniently blaming their students when it all goes down.
“The course is hard”—All the more reason to prepare to address the points of difficulty as smoothly as possible.
“Students don’t read the textbook”—A lot of good reading the textbook will do them if the professor is incapable of answering questions coherently.
Efficient, clear communication skills are apparently neglected in math departments. For several months, I worked for a proofreading company that services even top-ranking colleges in the United States, and most of the mathematics submissions I encountered were atrocious. I actually lost that job because I could not bring the last paper up to standard without rewriting it, and my suggestions for improvement “weren’t indulgent enough” for the professor.
When I was a math instructor in college, I was sorely disappointed at how lazy and apathetic many of the other instructors and professors were. Yes, there are students who don’t want to lift a finger, but too often this image is wrongly assumed, especially by professors. They’re minimizing the flaws of their in-group and maximizing the flaws of an out-group.
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Without agreeing with your premise, I will throw out one common issue. Most pre-college math is taught by people who are not particularly interested in or good at math. There are many exceptions, of course, but math is a requirement in every year of pre-college schooling, and there aren’t enough math-lovers who want the jobs to teach them all.
Imagine a music teacher with no appreciation and talent for music. He could tell students how to hold their instruments and how to produce different notes. He could lecture on musical styles, history and theory. All these things can be learned and taught by rote. But he couldn’t explain the nature or value of music, nor would he be likely to inspire students nor extract enjoyable performances from them.
While there are good and bad teachers in every subject, it’s easier to find English teachers who love literature, history teachers who are fascinated by the past and gym teachers addicted to sports—than math teachers who appreciate math.
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I read a lot, and something stuck with me. . . If you can explain it to an 8 year old, you understand it PERFECTLY. None of these teachers can explain it to an 8 year old , they can barely explain it to a college student and they all have access to answer keys.
Why hasn’t anyone come to the conclusion that these teachers just don’t understand calculus very well, and come up with a bevvy of excuses for the professors who have both tenure and play judge jury and executioner with our futures?
Math and physics teachers always get a lot of criticism and hate from students. Here is what’s at issue. Most students never learn how to study with the aim of understanding, at best they just memorize a bunch of garbage for the upcoming test. Unless you genuinely study and keep up in a math or physics class, reading the relevant chapter BEFORE the lecture, you will get very little out of the lecture… it will go right over your head. This is not a defect of the lecture, in general, it is a defect found in much of the intended audience.
Some think just because they find math fascinating and/or fun everyone else does. Or worse, they don’t find it fascinating or fun. For many people it is scary, hard, or boring. It has to be made interesting and fun at least through introductory calculus. (Advanced math in college is different though it still can be made more interesting than most professors made it). A physics teacher I had in HS would, for example, draw a cliff and stick figures with names of students. Then he’d ask, if Alice pushed Bob with a force of X off a 50′ high cliff, where will Bob go splat?
Different people learn through different senses and methods. A friend’s kid was having trouble learning the multiplication tables. He told me they had tried drills, flash cards, etc. Nothing worked. I told a great math teacher I knew about it. From my descxtion she immediately knew what to do. She said, have the kid write out his own flash cards. A few days after I told the friend he said it was a miracle. His child had learned them almost immediately. It was the combination of physical (writing the flash cards) and visual (seeing the cards he had written out himself) that made the connection. You can’t teach every person the same way. You need to present the material in several different ways. Then, if several don’t get it yet, work with them individually to find out what works for them. (See “Engage the students” below).
Make it relevant. Give real world examples of applications for the subject matter. Algebra, trig, geometry, calculus all have lots of fascinating applications in the real world. Yet they are all often taught simply as dry equations rather than applied to our lives and to nature. There are thousands of amazing applications all around us. Show them how math is everywhere and they won’t ask “Why do I need to know this?”.
A lot of lower level math teachers really don’t know their math. Some don’t even seem to enjoy it. If you can’t see the beauty in math, you shouldn’t be teaching it. Its one thing if you are teaching arithmetic: +. -, *, /. But once you hit algebra, trig and geometry you should be totally proficient. Sure, there will be those students who will stump you with a question. But you should be able to come in the next day with an answer. Yet there are a lot of teachers who really don’t know the subject matter very well. You should know these subjects inside out, be able to derive any formula and explain why it is so. I also believe you should know, and know well, at least one level above whatever it is you are teaching. And again, you must know the real world applications and be able to explain them clearly.
The worst teachers I had were the ones who just regurgitated the same problems as those in the text books. That does nothing to help students see alternative ways to think about and solve a problem, see different forms of problems, and it certainly does nothing to bring math to life. Most text books are dry and formulaic. The teachers who taught that way were the same.
The shame of it is I had very few math or science teachers who were great. In fact, most were not even good.
One last item that really peeved me in college. A teacher or professor has to be able to speak the language clearly and intelligibly. I had foreign-born professors in college who were nearly unintelligible. If you have to spend all of your energy just trying to understand the words, you miss the meaning. I would immediately transfer to another class.
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This isn’t true for all maths teachers, of course. But let me give a few possible reasons:
Some teachers may think that if something is obvious to them, then it’s obvious to everyone. From my experience teaching mathematics, this is something I used to believe is true that I later learned is very far from the truth. Perhaps your teacher thinks that sin2x and 2sinx are obviously different, so it’s useless explaining the obvious. But you cannot see why you can’t write sin2x instead of sinx+sinx .
Some teachers believe that everyone learns through step-by-step guides. People who are good at mathematics tend to be sequential, structured thinkers. But not everyone has a brain that functions sequentially and orderly. Not everybody requires a set of instructions to build a piece of furniture, for example. If you are in the ‘not requiring instructions to do things’ category, you may find your teacher’s step-by-step instructions to solve a particular equation confusing or even maddening.
Some aspects of mathematics are difficult, or even impossible, to teach. I wonder how Art teachers teach art. How do artists know where to place the brush, which colours to use, whether to use brush or pencil? I don’t, and I’ll never know. The same thing is true for mathematics. If you’re stuck solving a trigonometric equation, then perhaps you have chosen the wrong brush. That’s something your teacher cannot always teach you.
Some teachers can be just plain bad. Perhaps teaching mathematics was never what they intended to do, or they are unmotivated for some reason or another. In other words, they hate their job. No wonder they are terrible at explaining things — they’d rather be in a different job.
In short, the problem may be the teacher, it may be you, it may be the nature of the subject, or it may be a combination of all three.
Probably because they wrongly assume that:
(a) your brain works the same way theirs does,
(b) you have the same prerequisite knowledge in your head that they have, and
(c) you care about math as much as they do.
For example, this math joke…
…is funny only if:
(a) you’ve got the kind of brain that likes puzzles and likes to check the math whenever you see numbers, and
(b) you know that a nickel is worth five cents and that 43 is not evenly divisible by five; and
(c) you cared enough to recall the necessary information and do the required processing to get a mere chuckle.
Note also that most teachers these days don’t know the difference between teaching and testing. This is how we explain the difference to our Plain English teachers:
When you tell a student exactly what to do, and he does it, you're teaching — transferring tried-and-true neural patterns from your working brain into your student's fledgling brain. So you tell the student over and over exactly what you want him to know and do; and before long you find him saying, without prompting, "Alright already! I've got it." This is teaching.
Asking a student to solve problems that he hasn't seen before is not teaching; it's testing. Asking him to hypothesize about future results is not teaching, it's testing. In fact, the whole so-called "Socratic Method" is not teaching; it's testing. And when you use such techniques, you're essentially asking the student to "reinvent the wheel" — to intuitively come up with something that you already know. Why not simply let him in on the secret?
So when you're teaching Plain English programming, don't ask your students to solve problems, or guess what will happen if this or that code is run — tell them. Show them. Have them type it in (so it will enter their brains through their hands as well as their eyes and ears), and have them run the programs they've entered (for the fun of it); but don't ask them to "come up with" what's already in your head: hand it to them on a silver platter. Teaching is telling.
If a student wants to experiment on his own, that's fine. But don't hand out an assignment unless the answer is provided and the student is told to study and copy the answer (not try to reinvent it on his own). A student who is typing in a correct solution is being taught; Teaching is telling.
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