Mathematics 24x7

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Why most students all over the World consider mathematics as most difficult subject and all ways be sceptical about this subject?

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I guess what is meant by the first question is to verify that N is prime one only needs to check as factors primes below sqrt(N). This is an essential point but whether it's a "key" or why I could not say.

Squaring yout second equation one gets 2xx + 2sqrt(xx - 2x + 1) = 2. For x in [.5, 1] this does not give extra values.
Rewrite as sqrt(xx - 2x + 1) = (1 - x), which is true due to (1 - x)(1 - x) = xx - 2x + 1. This is not difficult. Electrical engineering students may run into more difficult problems. Whether it is useful, or was given out of context - I do not know.

One thing I object to is the generalization: What is wrong with us math teachers ...? I do not believe you think that anything is wrong with you as a math teacher. There are good teachers and bad teachers. In fact, it is not obvious to me that either of the two teachers you referred to is bad. In the first case, the question may have come from a text book that the teacher disliked but was forced to teach from. In the second case, it is also unlcear that the teacher was bad. Perhaps your son's girlfriend should think of another profession. The problem is trivial - just 1.5 steps and does not require to determine any domain. How is it useful to put everyone into a single bag?
I learned something from your exchange. You are both concerned about other people. That is actually a good thing. The first asked: What is wrong with us math teachers that we ask these questions?
The second responded: One thing I object to is the generalization: What is wrong with us math teachers ...? I do not believe you think that anything is wrong with you as a math teacher. There are good teachers and bad teachers. I know that we are allowed to disagree while not making it or taking it personal. But because you both care about other people I see that as the benefit of the exchange. Thank each of you for having the exchange so that I could benefit.
Linda, what is wrong with AB's answer (other than vagueness)?

For question #1, "How is the root of a number the key to the Sieve of Eratosthenes and finding prime numbers?", if we want to find the prime number up to N, we need only eliminate multiples of numbers below sqrt(N) in the sieve.

I would defend this question (if not its wording) by noting that in the Sieve of Eratosthenes, it might not be obvious to students from the procedure that they wouldn't have to cross off all the multiples up to 89 know that 97 is prime. Depending on the context, it's a problem either requiring recall from what was said in class or requiring high-level thinking skills.

For question #2, "prove that: sqrt(x+sqrt(2x-1))+sqrt(x-sqrt(2x-1))=sqrt(2) on [0.5,1]":
For x \in [0.5,1], \sqrt{x} \in [0,1]. We might be concerned about the domain, but we can solve the inequality x-\sqrt{2x-1}\ge 0 to see that this is not a problem on [0.5,1]. Then, square both sides of the equation to get
x+\sqrt{2x-1}+2\sqrt{x^2-(2x-1)}+x-\sqrt{2x-1}=2
and then simplify and rearrange and divide by 2 to get
\sqrt{(x-1)^2}=1-x
which is of course
1-x=1-x
One can verify that reversing the steps is legal.

The interesting thing about this question is that while it may be obvious when plugging in x=0.5 or x=1 that it works, for x=3/4, it is a different matter. I would have had no idea that \sqrt{3/4+\sqrt{1/2}}+\sqrt{3/4-\sqrt{1/2}}=\sqrt{2}. This shows us that our representations of algebraic numbers do not always clearly reveal themselves, which is why Galois theory is so interesting.

I do agree with AB that generalizing the failure of a few problems to an indictment of mathematics teaching is probably not a good idea. I also agree with you that these problems could be dread-inducing for beginning students. I would argue that the reason has more to do with the fact that most math teachers give little time to the affective needs of their students and less to do with the difficulty of the problems. Consequently, they haven't learned how to approach a problem that initially frustrates them.
Mathematics can be difficult, of course, but there is no reason for students to be as intimidated by it (or hating it) as much as many do. Except, of course, for how the subject is taught. When taught badly or at least in ways that are teacher-centric rather than student-centric, it's easy for students to think they aren't good at math or that math is incredibly intimidating and inaccessible.

This viewpoint of mine is hardly new or unique. We can choose to make math inviting and accessible or forbidding and arcane. I'm not suggesting that category theory or partial differential equations on manifolds can be or should be everyone's bag of tea. I am suggesting that it's obscene that so many students are pushed away from learning about some of the very beautiful, useful, and accessible ideas in a wide range of mathematical areas.
Subhashree, I wonder if we can learn something about your question from other phobias. Here's an abstract about Spider Phobia, that concludes "The latter finding may indicate, that the acquisition of spider fear is facilitated by specific parental disgust reactions when confronted with spiders." Here's an online science paper about Fears and Phobias of Snakes. That paper mentions that (i) parental influences lead to phobias in their children. (ii) People of all ages can develop fears because of traumatic experiences. These references are not directly relevant to mathematics education, but sometimes lateral thinking helps us to look at our problems in new ways. Do you use the word "phobia" to dramatise your question, or do you believe that many students have a real fear of mathematics?
Is there any question that many students and adults have a true phobia surrounding mathematics? I know adults who go "blind" faced with a page of math problems (not necessarily anything past high school algebra). Who find that they simply can't think straight when asked to do any math. Whose hearts race, blood pressure rises, and who generally suffer from a wide range of other physical and mental symptoms associated with real fear.

I am not any promotional agent for any website/toolbar/technology integration applications/any news channel, neither an theatre arttist so that I will dramatise my question! I belong to a country where basic infrasrtucture in elementary Education still not provided; there I teach mathematics , secondary & higher secondary levels since last 12/13 years. I face this problem every single day. Students, even from some of the most prestigeous, high prifile schools, specifically ''GIRLS'' are truely phobic about mathematics. When I propose a surprise test/ even give an assignment of 10 miscellaoneous questions, their heart beats rise, blood presssures go up, and they start conviencing me not to have that. I often observe they don't get hiper for any other subject's surprise test! parents frequently request our math teachers' group to councell students over this topic. Maximum complains come from both parents and teachers side about this subject only. I may not be perfact in using this word '' phobia'' but I think I am 100% sure about relevance of associating it's meaning with maths. I need help from teacher's who practically deal with teaching mathematics through text book curriculum , so that I could help myself , then my students ultimately. Here we are discussing to cla rify/share views, not to sensitise it. There are many site for dramatising, by the way I don't need them.
Colin McAllister said:
Subhashree, I wonder if we can learn something about your question from other phobias. Here's an abstract about Spider Phobia, that concludes "The latter finding may indicate, that the acquisition of spider fear is facilitated by specific parental disgust reactions when confronted with spiders." Here's an online science paper about Fears and Phobias of Snakes. That paper mentions that (i) parental influences lead to phobias in their children. (ii) People of all ages can develop fears because of traumatic experiences. These references are not directly relevant to mathematics education, but sometimes lateral thinking helps us to look at our problems in new ways. Do you use the word "phobia" to dramatise your question, or do you believe that many students have a real fear of mathematics?
I was looking into the same question and found this thesis to be helpful (despite the poor print quality):
http://etd.ohiolink.edu/send-pdf.cgi/Mavis%20Joni%20E.pdf?def128162...
Regarding the two problems posted by L F-S:

Out of context, the Sieve of Eratosthenes question seems bizarre. But, I am assuming you were not in the classroom when the lesson was being taught. As a math teacher, I have had to field many complaints from parents who CLAIM, according to their child, that I never explained the (fill in the blank). When in REALITY, I have explained the (fill in the blank) in several ways and the student wasn't paying attention or didn't understand but didn't ask for clarification.

The second problem with square roots requiring proof should not be too difficult for a COLLEGE level Engineering student. In the region of California, where I teach, students can prove that kind of problem in an 11th grade pre-calculus class.

As to Math Phobia... This is indeed a problem. It is often where I START a lesson, especially at the beginning of the school year when students have packed their math knowledge away for the summer and return to school in freak-out mode. I agree with Collin McAllister. Often the original math fear is generated by parents vicariously reliving their unpleasant school experiences through their child. How many times have I heard, "He gets it from me. I never was good at math either."? And, unfortunately, these same parents often blame the teachers instead of examining themselves.
Subhashree, I accept your response to my accusation that you may have used the word "phobia" to be dramatic. Indeed, Michael has identified that the physiological symptoms of fear of math are indistinguishable to those of arachnophobics. One reason I would make such an accusation is that I enjoy mathematics, so it is hard for me to imagine that other people fear it. When I teach a subject, such as Computing or Physics that has math content, I should be more sensitive, and realise that some of my students are not comfortable with the subject, or might even be totally lost. Math Chique described "parents vicariously reliving their unpleasant school experiences through their child". That is exactly the mode in which fear of spiders is transmitted. These additional viewpoints validate your personal experience of math phobia in the classroom.
If I am afraid to comment on this thread because several of my friends seem to fight, does it count as math anxiety?

;-)
I am sorry - I guess when I was trying to fix my "double" registration, I removed myself and then all of my comments got removed. It was not intentional :)

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