Are hard math problems of value? Why, why not?

Question

In regards to Math:

1.What makes a problem hard?

2.Can they be harmfull to you, why or why not?

3.If you were a math teacher , would you assign hard problems?

4.Speaking of political correctness and accomidating the disabled; should teachers hold different students to different levels of accountability in regards to critical thinking?

Answer 1

1... I think four things...
(a) As you progress, you get told that you can't things... only to find out later, that you can. "You can't subtract 2 from 1." Sure you can, but the teacher wasn't ready to teach you about negative numbers yet. "You can't divide 2 by 3." Of course you can, but the teacher wasn't ready to teach you about rational numbers, or "fractions," yet. "You can't take the square root of -1." Of course you can, but the teacher wasn't ready to teach you about complex variables yet. Some things you really can't do. For example, "You can't divide by zero." But the teacher didn't think you were ready to learn why... or maybe he or she didn't know. So, maybe "can't do that" is in the back of a student's mind.

(b) Because math problems have discrete answers. That is, an answer is either wrong or right. You might get credit for a correct approach, but the final answer is either correct or it is not. This is not what most people are used to. In most courses you can get a good grade for effort. Or you can get a good grade for being able to justify your point of view. So, if a student doesn't get the right answer quickly, they just quit, thinking it's too hard.

(c) Because mathematics is kind of an esoteric study. It deals with concepts that are sometimes far outside ones daily life. It's not something they can "fake." They have to learn it. That takes work. And work can be hard.

(d) Math skills are cumulative. In algebra you find you have to call on stuff you learned in arithmetic. In geometry you have to use rules you learned in algebra. In trig you have to rely on your geometry knowledge, And so it goes. It's not like history where you can spend a a week or so on the Greeks, and then can kind of forget that while studying the Renaissance. Students who tend to "cram and forget" their way through school probably don't find math easy.


2. Math is an incredible mental discipline. It teaches objective and logical thought. It teaches the application of rules of thought (theorems, postulates, axions, laws, etc.) to problems that are not within ones usual frame of reference.

It teaches one to form theories and to prove them, logically, and without emotion. It teaches one to recognize theories that are not so proved... or that may not be provable at all.

It stretches ones mind. Some people brag about being able to "think outside the box." People who work in higher mathematics don't brag about it. They do it all the time. I believe that either their minds have stretched the box to such an extent that thinking outside it is precluded, or perhaps have removed the box entirely.

And math teaches one to search for and select the knowledge necessary to solve a problem, and to apply that knowledge to the problem. That's called research. Often the knowledge is beyond their current capabilities... that's called learning. Both are minimal requirements for education.

The only problem is that tend to be emotional rather than rational. They look at a logical, objective answer as impractical,, or impossible. Historically, logical and rational thinkers have been persecuted, or ignored as crackpots.


(3) As a math teacher I have assigned difficult problems. If a teacher only assigns easy problems, how are students supposed to learn?

If a teacher assigns homework that is easy for everybody, then the class would be run for the benefit of the poorest student. Education is discriminatory. It discriminates against those who will not, or cannot learn. That's as it should be. If it were not so, there would be more morons in college than there already are.

Besides "hard" is a subjective thing. Stuff you found hard in 1st grade are (hopefully) trivial by the time you get into high school. Perhaps that brings us to part 4.


(4) As has been said, in math you don't get credit for being a nice person, or trying. This is the way it is in "real life." If one gets a job. it doesn't matter how often he is at his desk, how many hours a day he works, or if he's a nice guy. If he doesn't get the job done, the company finds somebody who can. It's unfair to the company to wish them to do otherwise.

So it is with math--and as it should be with education in general. If you can't do the work, you don't pass. Two examples from my personal experience...

I was one of about 45 people enrolled in a required math class. At the end of the semester there were 5 left and 3 of us passed. Should the teacher have required less so mediocre students could pass? The three that passed went on to get Masters degrees. The others didn't. Should the others have been allowed to continue to work toward and eventually get their degrees without understanding the subject?

I was tasked to teach what was nominally supposed to be a "pre-calculus" class. I walked in and wrote down the commutative and associative properties for addition and multiplication, the distributive property of multiplication over addition, the properties of one and zero, and the concepts of additive and multiplicative inverses. The class looked at me as I were speaking a foreign language. They didn't understand elementary algebra. They were not ready for pre-calculus. I told the head of the school that they either had to rename the course, or I'd have to flunk the whole class in pre-calculus. They renamed the class. Apparently their previous teacher either didn't know the subject, or passed them for showing up to class. Should I have done the same? Most courses have a "curriculum" that includes what the teacher is expected to teach. If a student fails to learn the material why should the teacher have to say that he did by giving him a passing grade? If you say it's the teacher's fault, I'd say, the student needs to ask questions in class, seek help outside of class, study with friends, or hire a tutor.

Education is to educate. Those with the ability to learn the subject matter should be allowed to progress to higher levels.. Those who don't have the ability to learn the subject matter should not. Any other point of view destroys the concept of education.

Education is not a passive endeavor. The student has to learn on his or her own. If the student fails, the student fails. The parents don't fail. The teacher doesn't fail. The teacher is merely tasked to present the material. If a student is having problems, it is up to the student to seek special help. The teacher has a certain amount of stuff to cover. Is it reasonable to expect the teacher to gloss over, or skip, the hard stuff just because some students can't handle it?

Answer 2

1. The number of variables, the path a student must goes through to solve the problem, and the order of the equation.

2. No they cant, because I do like math, and I like mathematical and programming challange.

3. I will assign a hard problem only if I think that can make the student more creative. It is useless for example to give a student a linear equation problem with 10 variables, this kind of problem is better given to a computer. A problem given to a human being must be able to make the human being more creative.

4. I think there is a core mistake about the current system of education. Many teacher teach their student to get A in things that is other people domain, instead of teach their student to find where they can find A in themselves. A human being is a creative being not a computational system, or a data storage. We have plenty of computation power and hard disk space for that.

Answer 3

1.What makes a problem hard?

The difficulty in solving them. That can come from the complexity of the problem, the lack of skills by the problem solver, or a combination of both.

2.Can they be harmfull to you, why or why not?

Anything in excess can be harmful. If you were to lock yourself in isolation to solve hard problems, for example, your social skills would likely suffer.

3.If you were a math teacher , would you assign hard problems?

Challenging...they would require the student to understand why problems are solved the way they are, not just how to solve them. [By the way, I taught linear algebra at the MBA level.]

4.Speaking of political correctness and accomidating the disabled; should teachers hold different students to different levels of accountability in regards to critical thinking?

Nope, not at all. What I would do, however, is insist the students were well qualified to take the courses they entered. Once they qualified, the degree of difficulty and understanding would be the same for each student.

If there are students who are less qualified than others for hard problems, as you put it, they should go into courses where only easy problems are taught. Holding back qualitifed students while trying to teach unqualified students is unfair to the qualified students. Attempting to teach hard math problems to unqualified students is unfair to them. That's why there should be seqregated courses for qualified and unqualified students.

Answer 4

1) Computing is a problem solving culture! Technically 'easy or hard' problem-solving is having lesser utility. computing is usually done by linking with all (easy and complex) types of numbers, which is a reason to evolve hard problems!

2) Problems are hard by "a manner in which it is presented for solving"! Said manner is language sense inherent in problem and manner/order of presenting 'what is available and what is needed'! A good teacher can present a hard problem in a very simple manner!

3) No. Instead, I will give students 'too many problems to solve' in a manner students would like to do it happily!

4) Teachers can't afford to delegate critical thinking to any student (irrespective of great ability in a student the teacher has already sensed)! A Mathematics teacher may promote free thinking among able students and treating them respectfully is OK. Equally, weaker students should get the help 'they expect from a teacher'! Accountability is 'a matter of result' and teacher may control it in a suitable manner! A Mathematics teacher need 'freedom to act'!


Regards

Answer 5

1) I think what makes a problem hard is that you'll get a certain process ingrained into to your head and it gets difficult to steer away from that.

2) If the entire assignment was hard and you had to ask for help and still weren't getting any progress then it can destroy self confidence.

3) Yes, its how you get people to think

4) I don't think so they just approach a problem in a different way. There's more than 1 way to solve a problem

Answer 6

1-the complexity of the problem, (the ammount of calculations that must be done, addition of variables, etc)

2-if you try too hard, you might bust a blood vessel in your brain..maybe

3-yes, they are needed to encourage growth and intelligence.

4-yes, not every student is at the same level, just because of their age or grade. if one student is doing poorly, they should be put with another class at that speed, breaking down the problems and explaining the reasoning behind them a little more than the faster paced students class.

and, yes, and no. not everyone needs to know extreme rocket science math. but some engineers, scientists, physicists, and so on do. and since learning is easiest at a younger age, it should be incorporated into schools early on. (but perhaps not graded so harshly, since some brains just do not think in the same fashion as rocket scientists do.

Answer 7

This seems like an end-of-year homework assignment.

Anyway, here's my general answer:

Yes, hard math problems are important. They'll get harder as you go through your educational experience. Hard math problems are the norm in college. If you haven't done them before, then you won't be prepared for them when you get there.

And hard math problems come up a lot in real life in a lot of jobs, so again - if you haven't done them before, you won't be prepared for them.

Just because something is hard doesn't mean it's not worth doing. Challenging yourself and your skills is ALWAYS worth doing.