HS Algebra and String Theory

[Steven G]

You are 100% correct when you say that it is all algebra. I think that 99.999% of people who fail calculus do so because of their weak algebra skills. I completely shock people when I tell them that calculus is actually easy--at least the calculus part is! It is the algebra that will kill you. In your example above for the calculus part you just needed to right lim many times (although, understandably you didn't) and in the end plug in 0 for h. Now most any calculus student could do that. However doing all algebra is another thing.

I have a question for every one. If some fails calculus do you think that they should retake calculus the next semester or retake algebra/pre calculus?

 

[JeffM]

Jomo

I have a radically different view. I DO agree that most of the hard mechanical work in calculus is just algebra. I think I finally became totally comfortable with algebra after studying calculus. (In all likelihood, I'll become totally comfortable in integral calculus after I finish studying differential equations.)

But I think the hard conceptual work in calculus is the sort of introduction to analysis that is forced down the student's throat before learning any calculus itself. It is unmotivated, abstruse, and woefully fragmentary. Some students are baffled at this beginning and never catch up.

It's as though we taught algebra by teaching the Peano Postulates, graduating to real analysis, and finally introducing linear equations and the Cartesian plane.

 

[topsquark]

Jomo

I have a radically different view. I DO agree that most of the hard mechanical work in calculus is just algebra. I think I finally became totally comfortable with algebra after studying calculus. (In all likelihood, I'll become totally comfortable in integral calculus after I finish studying differential equations.)

But I think the hard conceptual work in calculus is the sort of introduction to analysis that is forced down the student's throat before learning any calculus itself. It is unmotivated, abstruse, and woefully fragmentary. Some students are baffled at this beginning and never catch up.

It's as though we taught algebra by teaching the Peano Postulates, graduating to real analysis, and finally introducing linear equations and the Cartesian plane.

Sounds to me like you are describing how a string theorist would explain things...

-Dan

 

[Romsek]

I sort of agree with JeffM but also have a radical view he probably won't like.

It took a while but calculators are now accepted as ok to do arithmetic on as long as the students understand and can perform the operations long hand, at the end of a plank if necessary.

I maintain the same thing should be accepted for algebra and basic calculus, provided the subject being taught is not the mechanics of either of these.
But in other courses that use these tools I say take it for granted the student understands algebra, let them use software to grind through it, and get on with the conceptual matter at hand.

This probably is most applicable with physics, but it is with math as well. When teaching for example integration, why force the students to slog through 5 pages of algebra that they showed they were able to do at the end of that plank. This isn't the 1300s, and time is valuable.

Once they hit real life they will be expected to use these tools to their full extent before having to sit down and grind stuff out themselves. And even if they do manage it their computations will be suspect until verified. We don't really have to worry about software screwing up algebra calculations any more.

Anyway let the flames fly, I think this is an important thing to get squared away w/regard to the future of teaching STEM.

 

[JeffM]

Romsek

Actually I somewhat agree. I think it is silly in an algebra class to ask someone to solve to 3 decimals without a calculator.

[MATH]x = \dfrac{243}{137}.[/MATH]
You can leave it in exact form or use a calculator.

Similarly, I see little pedagogical value in making people do a lot of algebra by hand in calculus. The fact that my skiils in algebra were honed by taking calculus was not meant as an encomium for how I was taught algebra.

I beleve we should spend far more time on word problems, which teach how to translate problems into math, than mechanics, especially with today's technological tools.

With that said, I cannot believe that it makes sense to say that a student should need technology to find 124/4 or to solve 3x + 17 = 5x - 3 or to differentiate cos(x). The reason for that is that having to put everything into a black box will cost time rather than save it.

In the meantime, students are still asked to do all these mechanics. We might as well help them out.

 

[Steven G]

I have a question to ask. Why do you think that the calculus sequence is a prerequisite for more advanced math courses? I'd love to read your responses.

I will start with mine. Calculus is a prerequisite for advanced courses because most math departments members feel that one needs the mathematical maturity from Calculus before taking these advanced courses. Now I am not saying that students can't get their maturity in other ways besides taking calculus, but you need to take that into account before you let them use software. If we let students do arithmetic (perhaps after learning it formally) with a calculator, not require students to use algebra after passing an algebra class and have them push buttons on some software program to solve integrals then where are they getting their maturity from to start doing proof based courses? Why even teach them how to do proofs when we can just give them the theorems and just let them do mechanical problems using these theorems?

Now here is what really scares me. Helpers here see the importance on seeing why something is true compared to just memorizing it. I feel (since I am apparently old fashioned to the core) that if we allow students to use electronics then the beginning lessons of mathematics is lost. I remember clearly showing to myself why the integral from a to b is equal to the integral from a to c plus the integral from c to b for any value c. I never would have done that if I was using a calculator to do my integrals. Doing these little things made me realize that I loved mathematics and that I should get my degree(s) in it.

Now I am not saying that using electronics can't be done correctly but it will be difficult. It needs to be done in a way that conventual math majors of the past would still be excited about math and pursue advanced degrees if they had been exposed to math using the new way (software math). Why would anyone want to be a math major if they do not see the beauty in or if they do know what it really is.? I remember a student asking my classmate why he was studying Topology, my classmate started to tell him why and I interjected and stated that we are studying Topology for the beauty in it. My classmate felt ashamed that he did not say that as he felt about math as I did. I believe that (sometimes-think fractals) computers take the real beauty out of math and if student does not get exposed to that beauty early on I fear that math will die. This will really hurt our society as almost all progress these days have their roots in math.