High School Math Education?

Question

What exactly are they teaching kids in high school these days?

I was tutoring a freshman college student recently.
Basic problem: integrate e^(-0.2x) dx
There really is nothing ever so hard about this.

Her first instinct was to pull out her TI89 calculator and do the integration on that. The calculator gave her:
-5*0.81873^x

Clearly since 0.81873 is the evaluation of e^(-0.2), this answer is not wrong, but it's so unintuitive.

Why are students trained to let their calculator do all the work for them? Doesn't anyone do math by hand anymore?

Opinions please.

Answer 1

I don't have a calculator which would be capable of doing integration. I use a scientific or basic calculator when the nature of questions is such that they call for a numerical solution rather than one as a formula, or based on surds.

Even that is a big advantage over anything available when I was at school or university, in the days of tables, slide rules and (for statistics) the University's hand-cranked adding machines.

Maths questions at A level were designed so that the numerical aspects of the solution were minor compared to the mathematical principles involved. However, things seem to have changed radically. I notice other differences on here, such as emphasis on naming laws such as distributive, commutative and associative used in everyday calculations, and an almost total absence of any questions in the field of what used to be called applied mathematics and theoretical mechanics, which was divided into statics and dynamics. That formed 50% of the A-level syllabus.

Answer 2

Well, she did get it right and if she became an engineer rather than a mathematician that's all she needed to know: the answer.
This question treads on a very thin line however

A calculator is a tool. Once a math or science student passes calculus and understands what an integral represents, shouldn't a calculator be allowed thenceforward?
A pen or pencil is a tool as well, and performing integrals without them would be overly harsh, wouldn't it?

The point seems to be whether your student had in fact understood what an integral actually represents. And yet that too may be irrelevant: a student of calculators (for example) may need to know nothing of integrals and yet be able to answer every such question perfectly with it. If we put a person in a black box and submit a question into the box and we get a correct answer, which person is in the box? A mathematician with a pencil, a student with a caculator, a computer, etc? Does our society value or care?

I recomment reading:
"The Feeling of Power" by Isaac Asimov, a very short story
about how an advanced society rediscovers the joys of multipying numbers BY HAND, a forgotten art!

http://www.themathlab.com/writings/short%20stories/feeling.htm

Note that this was written 50 years ago!

It makes one believe that this trend will be the inevitable outcome of at least a portion of our population (if it hasn't already).

Answer 3

The skills required to be a caveman living in 100,000 BC Europe were extremely complex and time-consuming and required a great deal of skill and ingenuity. Yet nobody has these skills any more.

Fifty years ago, it was a valuable skill to be able to add and subtract long lists of numbers manually and exactly. You would even be paid a good wage to do so. Not any more.

Using a calculator correctly is an important skill now, and can be just as complex a task as long division. It's more important today to be able to *estimate* the result of a calculation than to do it by hand exactly right, so that you can check the calculator. (But alas they cannot do that either). The correct use of a calculator or Mathematica would be just peachy by me.

But the problem is that they can't even use a calculator correctly, just as you say!! Arghe!

There are many ways to gain insight into mathematics, and learning algorithms for addition multiplication and integration is only one way. There's no reason to suspect that approaching mathematics differently would necessarily lead to less understanding or appreciation of the subject.

In other words, there is a range of skills in arithmatic, including estimation and translating statements from English (or whatever vernacular) into formalism (more important in 2010) and manual algorithms (more important in 1950) as well as number theory, set theory, etc.

I think the main problem is that more students need college degrees now just to get jobs, and people that don't have the skills to be in college are nevertheless enrolling.

Answer 4

In the high school I work at, we do not allow kids to use calculators till MIDDLE of grade 10 and it has to be a simple calcculator.

They cannot have integration facility. In grade 12, only in term 2 we allow them to use a graphing calculator and the exams are 2 parts, one with calculator and one without. We have to let them use the calculator as we are trainig them to sit for the AP.

Yes Dr D, I have to agree with you. The kids need to do things without dependence on calculator for every little thing.

I understand the point made by some posters here for the calculator, but frankly, this is such an easy problem that IT SHOULD BE QUICKER to do it without. I would understand if it was a complex integral where the student knows how to get the answer but to save time, they use a calculator.

Answer 5

I've been wondering that myself, in light of some of the answers I see people giving here in the math section. Though even when I was a teaching assistant in grad school ('97-98) I was still seeing the signs.

The most common problem I see, which your example here shows, is the act of always putting in a rounded, numerical solution when an analytical solution is what's really needed. For example, the solution to 3x =2 is x = 2/3. It's not "0.667", "0.6667" or "0.7"! There's a time and a place for decimals and rounded-off answers, and "solve 3x=2 for x" isn't one of them.

Calculators are great for doing numerical computations that would otherwise be needlessly time-consuming, whether it's heavy arithmetic or looking for an exact answer to something like log(2.2). But they shouldn't be used as a substitute for fundamental principles. Trying to solve x^2 + 2x + 1 = 0 by graphing the equation and eyeballing the intersection points is just a bad way of going about things. Yet I've actually seen a student solve x^2 - 4 = 0 "by graphing" and submit "1.99" as the answer.

Calculators also have their computational limitations. For one thing, they're only accurate up to so many decimal places. I see people who whine about doing arithmetic with fractions, but fractions are more accurate. When I was teaching linear algebra, students had to do some fraction arithmetic to solve matrixes. Some of them just skipped this and did everything in decimals, with their calculators. But after enough calculations the round-up error significantly built up, and their answers ended up being wrong as a result.

Answer 6

I agree with the person above (Zaphod). If they understand what the problem represents (intuitively), who cares. If I told you to multiply 3542345 * 897626640 - would you bust out a paper and pencil or a calculator? Exactly.

I am an engineer, and in college (graduated '06) we HAD to know how to do it long hand before using a calculator to understand how we can use integration in our field. When we were given a STATICS or DYNAMICS test (in regards to the person above me), we only had 1 hour --- that hour should NOT be filled crunching integrals. Calculators eliminate that time.

I do feel that students DO rely on calculators too much and let the "understanding of the problem" get away from them, but that solely goes back to the teacher...

Answer 7

That is the biggest flaw in the way they teach Math in high schools. Using a calculator really makes a person handicapped. I never used a calculator in my previous school. But after i came to US, i saw that the people here made it mandatory to have a calculator. And the irritating part is that you can do everything with a programmable calculators like the Ti's. I used to do most the calculations mentally. Now i have started to rely on a calculator to do some of the things. It can easily make u fell stupid.

Answer 8

I don't have a Tl89 calculator, so I do much of my work by hand (not very good at integrating though. When I get one I might start using it =D).

My school doesn't teach us to use the calculators, although it's more like we aren't allowed to. I don't know what high schools actually do, but it's a REQUIREMENT to have a calculator for math, or whatever (based on my bro, not so sure).

I use my calculators (a less mathematic one that let's you add and calculate trigonometric functions) intuitively now just like those kids for math, "outside" of school.

Yahoo! Answers have too much competition. They taught me to use my calculator everyday =D.

And wow, I couldn't actually have integrated that intuitively. I must be pretty stupid, but it doesn't concern me so much.

My opinion is that students are not all trained. They just do it. We're too modernized to use our hands and our laziness makes us feel more like using calculators.

I'm sure EVERYONE has probably used their calculators for math.

People also just might like using electronics to do math. When I don't feel like using paper for math, I use my computer to keep a list of the things I've been doing.

________________________________________

You actually tutor people?

Answer 9

When I was in high school my teachers didn't allow us to use calculators until there was such a hugh outcry from the students and parents of students that " in the real world people has access to calculators" that the teacher just gave in. Suddenly students forgot how to do long division, factors, roots.

I'm currently studying for the GMAT and on the GMAT you are not allowed calculators. I'm relearning a lost skill. =(

Answer 10

They'll probably be MORE likely to take you. A math major is much more rigorous than a math education major. Either way, the answer is yes.