The current state of math education (vanity)

Yahoo answers ^ | 10/28/2010 | self

[Bob]

On Yahoo answers today, I came across a disturbing question asked by, presumably, a high school math student. Here is the question:

Please, please explain these answers to me!! I am so frustrated!?

No matter how hard I try, I can not understand domain and range!!! The definition in my book says domain is all the x alues and range is all the y values. That's it. Soo, how the heck do you figuire this out?

1. f(x) = x+2/x
Domain: x does not equal zero
Range: f(x) = any real #

2. f(x) 2/x
Domain: x does not equal 0
Range: f(x) does not equal 0

< snip >

Here's my response:

The domain of a function is the set of values to which the x variable can be set.

Since problems 1 and 2 involve dividing by x, the domain does not include the value zero since dividing by zero is an undefined result.

< snip >

The range of a function is the set of values that the function can result in.

In problems 1 and 2, since x can never be zero, neither can the value of the function be zero. I think that both problems should list f(x) does not equal zero.

< snip >

[added] With all of those errors in the listed ranges and domains, it's no wonder that you're confused. Didn't your teacher know that the errors were there? If not, why not? It looks like your math teacher doesn't know much math.

Sorry for the snips. Since this was originally posted on a Yahoo site, I don't want to violate excerpting requirements.

[Bob]

There were quite a number of errors in the "correct" answers that were listed. Am I being unnecessarily harsh in expecting that math teachers should know the math that they're teaching? I'm no mathematician and I could certainly be wrong in my explanations. I'd appreciate any feedback pointing out any errors that I've made.

Source link for the full question and my full response:

Question on yahoo answers

[Resolute Conservative]

That is because most don’t know math. I argue all the time that math skills are not close to what they were 2 generations ago when we put a man n the moon and created computers. Now the teach how to build “T” charts in elementary instead of memorizing the times tables etc... There is no foundation. They teach for the standardized tests.

[cartan]

The definition in my book says domain is all the x alues and range is all the y values.

That is the most braindead definition I’ve ever heard. Did anybody really print a book containing such nonsense?

[netmilsmom]

You either “get” math or you don’t. I don’t.

Thankfully, we have worked with Saxon and now my daughters “get it”.

But then again, I taught them times tables and touch math for easy calculations.

[domenad]

Wow, that math teacher made some unforgivable mistakes. If the teacher is making the mistaks, the student can’t learn. Like in question 1, for example, the range cannot be the real set {R}, because 0 is excluded ( f(x) =/ 0 for any value of x).

4) is also wrong, because the absolute value of |x-2| will always be a positive number, so the range is from -infinity to 4.

[the invisib1e hand]

for the luvofPete, that's not a math question so much as it is a simple definition -- semantics.

For your reading pleasure...Safety in Numbers...

If you're not a one, you're a zero....

[Hodar]

It only gets worse. Houston Independent Schools have waivers for students to bring calculators to tests, because the students/parents are too lazy to learn/teach both the addition and multiplication tables.

So we have a generation of High School students who cannot perform simple, basic mathematics without a calculator. The fact that many cannot read, write or speak English is already an accepted fact.

[17th Miss Regt]

The range of a function is the set of values that the function can result in.

In problems 1 and 2, since x can never be zero, neither can the value of the function be zero.

The second statement here does not follow from the first. Certainly it is true that both problems should list "f(x) does not equal zero". But that is not because x cannot equal zero.

In problem 2 the sign of f(x) is always positive for x > 0 and negative for x < 0. Since zero is excluded from the domain, f(x) is not zero for any finite values of x. We note that the function gets arbitrarily close to zero as we approach infinity (in both directions) but we assume we are not in the extended real numbers as this is presumably a high school level problem. Problem 1 can be dealt with in a similar manner.

[hannibaal]

In the first question f(x) = x + 2/x
f(x) does include 0 in this case if x = i * sqrt2
Therefore f(x) does include all real numbers including 0
x cant be 0 because 2/x yields an undefined result

In the 2nd f(x) = 2/x
x cant be 0 as it would yield an undefined result aswell
f(x) cant be zero but it can get approximately close with x being infinite

I miss the days when I was doing math this easy :) Currently doing control systems specifically bode plots and z transforms.. ugh so boring.
And no you arent being harsh on teachers, alot of them are really unqualified in imparting knowledge on their students. They may know the material but they dont know how to teach, I have come across alot of teachers like that both at school and university. Someday when i leave the private sector I want to go into teaching.

[Bob]

The definition in my book says domain is all the x alues and range is all the y values.

That is the most braindead definition I’ve ever heard. Did anybody really print a book containing such nonsense?

I think that the kid was misstating what he read. The domain does consist of x values and the range does consist of y values, just not necessarily all of them.

[Catphish]

Actually, the range of y=f(x)=x+2/x is the set of all y whose absolute value is less than or equal to the square root of 8.

You just set 2+2/x=y, turn it into a quadratic equation in x and then compute discriminant, y^2-8.

[Catphish]

I meant “whose absolute value is greater than or equal to 8”

[libertarian27]

It’s all Greek to me

You lost me at f(x)

[Bob]

They may know the material but they dont know how to teach

Given the number of errors that the teacher apparently didn't catch, I'd have to disagree with the assertion that this one, at least, may know the material.

[Catphish]

er, square root of 8!

[ebersole]

“Now the teach how to build “T” charts in elementary instead of memorizing the times tables etc... There is no foundation.”
It’s nice to see some else write what I’ve been struggling with in my kids schools!!! They have no expectations that the kids know the most basic math facts of addition and substraction. I just had to come to the conclusion that they will not be receiving an adequate math education in their school and have them enrolled in a math program (and home study as well) outside of the school district.

[Bob]

Actually, the range of y=f(x)=x+2/x is the set of all y whose absolute value is less than or equal to the square root of 8.

You just set 2+2/x=y, turn it into a quadratic equation in x and then compute discriminant, y^2-8.

Sorry, you lost me at "turn it into a quadratic equation in x". How would you do that? (As I said earlier, I'm no mathematician.)

[cartan]

I was unhappy about the term “x values.” What are the “x values” of the function

f : ℜ → ℜ
t ↦ t2

It’s not like variables being called x or y was an essential part of the definition of a function.

[hannibaal]

youre right but actually y can be less than or equal to -sqrt 8, greater than or equal to sqrt 8, and 0 for x=isqrt2

[Catphish]

You just multiply through by x, so x + 2/x = y becomes x^2+2=yx. Then you get -x^2+yx-2=0. The discriminant of that is y^2-8. Since you are looking for real values that y can be y^2-8 has to be greater than or equal to 0, i.e., the absolute value of y has to be greater than or equal to the square root of 8.