Teaching mathematics.

ThePerfectHacker · #1 $Old$ January 18th, 2006, 03:40 PM

I am in 12th grade and I help students with Calculus class that we have. I did the same thing when I was in 10th and 11th grades. There were a number of things which I realized. Firstly, students have great difficulty with parantheses. They place it incorrectly or omit it completely. For example, find the derivative of $e^{x^2+x}$ students omit the parantheses and write $2x+1\times e^{x^2+x}$ by using the chain rule. Secondly, students always have problems with the negative and positive signs. This is why in the posts I always before opening parantheses write "Watch the negative signs". Finally, and this hurts me greatly. Students rely on their calculator (especially the graphing calculators). They think it has all the answers. The thing that hurts me is that they cannot do $\sqrt{0}$ they need a calculator for that. Other examples include, $(-1)^{-1}$ and $\frac{0}{1}$ . I think that these calculators should be used only for complicated and long computations. Because they corrupt the numerical thinking of students. I only use a calculator for taking complicated transcendental functions. If I ever become a math teacher I would forbid the class to use the calculator for the above reasons.

Any stories you have based on mathematical teaching?

Treadstone 71 · #2 $Old$ January 20th, 2006, 05:08 AM

In most high schools, students learn, or think they learn, that a decimal answer is the best form of answer. Probably because in other science classes, decimals are used primarily. So, if the answer to a math question is $\sqrt{2}$ , they instinctively try and give some decimal expansion of it, thus explaining the dependence on the calculator. However, this isn't necessarily a bad thing: high school math prepares students for applications, and the "real" life, where an analytic solution might not be needed or desirable; only those who are truly interested in math move on to learn real math.

CaptainBlack · #3 $Old$ January 20th, 2006, 05:39 AM

Quote:

Originally Posted by Treadstone 71

In most high schools, students learn, or think they learn, that a decimal answer is the best form of answer. Probably because in other science classes, decimals are used primarily. So, if the answer to a math question is $\sqrt{2}$ , they instinctively try and give some decimal expansion of it, thus explaining the dependence on the calculator. However, this isn't necessarily a bad thing: high school math prepares students for applications, and the "real" life, where an analytic solution might not be needed or desirable; only those who are truly interested in math move on to learn real math.

Those of us who regard Maths at humanities greatest cultural achievement
can only find it sad that the education system sets out to deny the majority
of humanity any sight of real Mathematics.

Its like teaching children to paint walls but refusing to let them see
real paintings.

RonL

Treadstone 71 · #4 $Old$ January 20th, 2006, 07:43 AM

Classical geometry that is being taught in high schools is actually first of "real" math classes for students that will be going into mathematics. Unfortunately it's also the last for most. (By "real", I mean there are theorems and students are required to prove things a la Euclid.)

ThePerfectHacker · #5 $Old$ January 20th, 2006, 09:54 AM

I agree with you CaptainBlack, students should learns pure mathematics. It is has beauty without explanation, it also makes people smarter-like chess. But the problem I think is that pure mathematics is far TOO difficult for high schools students to understand.

earboth · #6 $Old$ January 21st, 2006, 10:20 PM

Quote:

Originally Posted by ThePerfectHacker

... The thing that hurts me is that they cannot do $\sqrt{0}$ they need a calculator for that. Other examples include, $(-1)^{-1}$ and $\frac{0}{1}$ . ...

Any stories you have based on mathematical teaching?

Hello,
how about this:

You can simplify $\frac{16}{64}$ by dividing numerator and denominator by 6, so you get $\frac{1}{4}$

or:

To calculate the power of a number is a commutative(?) operation because:

$2^4=4^2$

Do you know the story about Sysiphus? Some (but only few) of your students will never learn math - and some of them even later.

Bye

Treadstone 71 · #7 $Old$ January 22nd, 2006, 11:54 AM

That reminds me of a joke:

$\frac{\sin x}{n} = 6$ .

ThePerfectHacker · #8 $Old$ January 22nd, 2006, 03:18 PM

Quote:

Originally Posted by earboth

Hello,
how about this:

You can simplify $\frac{16}{64}$ by dividing numerator and denominator by 6, so you get $\frac{1}{4}$

or:

To calculate the power of a number is a commutative(?) operation because:

$2^4=4^2$

Do you know the story about Sysiphus? Some (but only few) of your students will never learn math - and some of them even later.

Bye

You cannot be serious !!!

Treadstone I like your joke

I once did something so stupid, I was counting something and it was wrong each time. I was thinking what can possibly be wrong. Then I realized and made myself laugh. I started counting from zero.

ThePerfectHacker · #9 $Old$ January 22nd, 2006, 07:40 PM

30 minutes ago I just came back from teaching. My student was making a set of crazy mistakes which catches people.

This one is a classic, "Canceling from a sum" example,
$\frac{x}{1+x}$ cancels the x's and you are left with $\frac{1}{1+1}=\frac{1}{2}$

Another classic, "1=0", example $\frac{x+x^2}{x}$ canceling by x yields $2x$ . What happened with the 1? Oh, you told me I can cancel the x's so it is gone!

I think the problem is students have no idea what they are doing they just memorize a set of rules and use them improperly. This kind of stuff makes me laugh.

Here is another example, I purposely write this on the board because it catches almost everyone. $f(x)=e$ find its derivative. Students think that since $(e^x)'=e^x$ they make a guess that it is also something like that namely, $e$ . I tell them, NO! $e$ is a number. They tell me "OH! I see now". Immediately after that I write $\sqrt{e}$ , so what do they do? They write it as $e^{1/2}$ , then they use the power-rule.... You get the idea

.

If I ever become a math teacher for Calculus I will one day make a test filled with constants like in the previous paragraph. That would mean the answer to every problem in the test would be zero. And I bet that almost everyone will fail that test

dud · #10 $Old$ January 23rd, 2006, 12:33 PM

Talking about mathematics in teaching, perhaps it might be interesting to note how the situation is back where I come from, Norway.
Could it also be similar issues in other countries?

I'm sure we all are familiar with the mathematics/science "reforms" that the western countries conducted as a reaction to USSSR launching the Sputnik satelite successfully.
But to summarize very quickly; the west were stunned when they realized how vastly superiour Soviet was in science and mathematics. So most countries started focusing on educating good scientists and mathematics to quickly get levelled with the Soviets.

This also happened in Norway, but sometime during the last 30 years, we seemed to have slipped off target again.

Partly this is because that C and D average students can easily get into the teaching courses at universities here. This is again due to a serious shortage of tutors. Quite simply because they are paid way too little by the state schools, so noone really wants to tutor kids.

As a consequence of poor tutors, all through primary and high school kids learns science and math by people who struggled to get a D grade in mathematics!

The situation now is that most people in Norway, are absolutely dreadful when it comes to mathematics, in particular algebra.
But even the simplest equations like $\frac{1}{2}x = x - 2$ can cause a high school student go into a week-long depression.

Sad as it may seem, there are no real political will to change this in Norway, it's quite simply ignored as a case of classical SEP(Someone-Elses-Problem).

albi · #11 $Old$ February 6th, 2006, 02:11 PM

And another example, from my mathematics lesson:

We have a simple equation:

$\sin x = \frac{1}{\sqrt{2}}$

What is the solution of that, I asked:

$x = \frac{\pi}{4}$
she said.

Some examples later we had another, "more difficult" equation:
$\sin t = \frac{1}{2}$

She said, that is easy, it will be
$t = \frac{1}{2 \sin}$

Changing letter from x to t had made her very conffused.

Or another example (from science lesson):
$v_p = \frac{d \omega}{ d \beta} = \frac{\omega}{\beta}$

ThePerfectHacker · #12 $Old$ February 6th, 2006, 03:19 PM

Ha ha, I seen the mistake,
$\frac{dy}{dx}=\frac{y}{x}$ many times.