Excellent video series raises good question:

[BiGyElLoWhAt]

www.youtube.com/watch?v=oW4jM0smS_E
That's the video I'm referencing in particular, but 1 and 3 are necessary prereqs if you're new to the matter (as I am).

He goes through and derives the product rule and power rule for polynomials using algebra.

My question is this: why don't we teach calculus in this way first? It gives a more intuitive understanding in my opinion, and it uses readily accessible math that anyone in a first year calc course should be versed in.

Thoughts?

(I guess this is kind of a calculus question, but it's a differential geometry course... do what you will mods =D)

 

[Simon Bridge]

Reasons, and the education style, will vary between countries
- in NZ differentiation is taught first at secondary school level, while the algebra skill is pretty basic.
Many students won't have enough algebra to follow a purely algebraic path through calculus.

The common approach is abstract: differentiation by rule and integration as an anti-differentiation.
More recently there is a more geometric approach dealing with slopes and areas which seems to work quite well if integration is covered first... so history is also important.

afaik, though, the various rules for differentiation have always been taught by algebra, usually in the last year at secondary school or the first year at university.

However, usually, when students first encounter calculus, their algebra is not good enough to go right to proofs and derivations in that way.
It s very common you need to be able to use the tools before you have the skill level to understand how they work.

Most teachers teach the way they do largely because someone tells them to or because they have always done it that way.
The received wisdom in pedagogy will hopefully be informed by some sort of research for what seems to be effective (in terms of whatever the education goals are) for the most students in one go. A good teacher will try to teach at least two different ways, within the time/curriculum constraints, and then help the stragglers. The rest is politics.

Bear in mind: it is a mistake to generalize from personal experience - what is most intuitive for you will not work for everyone.

 

[BiGyElLoWhAt]

Reasons, and the education style, will vary between countries
- in NZ differentiation is taught first at secondary school level, while the algebra skill is pretty basic.
Many students won't have enough algebra to follow a purely algebraic path through calculus.

I take it you're from New Zealand?

The common approach is abstract: differentiation by rule and integration as an anti-differentiation.
More recently there is a more geometric approach dealing with slopes and areas which seems to work quite well if integration is covered first... so history is also important.

afaik, though, the various rules for differentiation have always been taught by algebra, usually in the last year at secondary school or the first year at university.

However, usually, when students first encounter calculus, their algebra is not good enough to go right to proofs and derivations in that way.
It s very common you need to be able to use the tools before you have the skill level to understand how they work.

This is the approach that I had to take, however I was a special case in the department, as I was 2 or 3 math courses behind for the first 2 years in the physics department, at least until I started doubling up on math courses and taking summer courses as well. Most students in my department, however, are through their calculus courses before they start to really need them.

Most teachers teach the way they do largely because someone tells them to or because they have always done it that way.
The received wisdom in pedagogy will hopefully be informed by some sort of research for what seems to be effective (in terms of whatever the education goals are) for the most students in one go. A good teacher will try to teach at least two different ways, within the time/curriculum constraints, and then help the stragglers. The rest is politics.

I know, and it's really unfortunate, some of the best teachers I've had were tenured and didn't give a crap, and just taught how they wanted to, and I think that you learn a lot more when the teacher is teaching from their mind, rather than from a piece of paper.

Bear in mind: it is a mistake to generalize from personal experience - what is most intuitive for you will not work for everyone.

Very true, and I'm not saying that it would be a simple process. My calc 1 class went something like this:
Review algebra and trig for a few weeks
start on redundant limits, just to get the idea (lim as x->1 of x, and stuff like that)
start to take limits in a meaningful way (lim as x->1 of (x+1)/(x-1))
introduce the limit definition of the derivative
introduce the rules (without proof, more or less by demonstration)
last week or 2 was anti differentiation by "guessing".

If you can prove the rules by using algebra, however, or even just a solid demonstration, like in the video, drug out over the course of 2 maybe 3 classes depending on the students, I think that they could grasp this, and it would serve them well. I'm not saying we should nix limits, I'm not sure if all of calculus can be done without limits, but it at least deserves an honorable mention.

I guess I was just taken aback when I saw this. I literally had no idea that this was even possible, and without this video series, I very well may have gone ignorant for many years, if not the rest of my life. It's a shame. I believe this paper came out in 1787 is what the guy said? I might be off a few years without watching the video again. I just think it's neat that you can do calculus without actually doing calculus.
Kudos, guy in video.

 

[Simon Bridge]

Most students in my department, however, are through their calculus courses before they start to really need them.

This is usual - you usually cannot teach a course, like physics, that requires calculus until after a calculus course has been completed. In NZ we have a situation where, for instance, a change in the curriculum placed the needed maths for some parts of the physics course until a year after physics needed it. Teachers were instructed not to teach the required maths as part of physics (which stil gets done a lot anyway). The upshot was that this part of the physics course, which was still required to be taught, had to be rethought.

If you can prove the rules by using algebra, however, or even just a solid demonstration, like in the video, drug out over the course of 2 maybe 3 classes depending on the students, I think that they could grasp this, and it would serve them well. I'm not saying we should nix limits, I'm not sure if all of calculus can be done without limits, but it at least deserves an honorable mention.

The concept of a limit is central to calculus - you cannot have calculus without it. The concept of the limit is also more general than just calculus - students will need it in other areas. You should realize that calculus is not taught so students can learn calculus - but so they can obtain tools to solve problems.

The approach used in the video does not look much different from what I do.

It is a truism in education that any technique will be good for some students, most will muddle through, and some will find it counter-productive.
You are still generalizing from personal experience. Can you find any studies in the education literature which show that this approach is more effective in general than any other?

I know, and it's really unfortunate, some of the best teachers I've had were tenured and didn't give a crap, and just taught how they wanted to, and I think that you learn a lot more when the teacher is teaching from their mind, rather than from a piece of paper.

The really good teachers use a "student-directed" approach; this is where they start you out on a subject and guide your exploration - encouraging you as you go. It's very tricky - but it's how I stumbled on to a path to integration from an investigation of trigonometry (I was looking for a non-SOH-CAH-TOA approach.)

Usually the tenured teachers are either the best or the worst - very few in between.
It causes debates about accountability in the profession.

 

[BiGyElLoWhAt]

You are still generalizing from personal experience. Can you find any studies in the education literature which show that this approach is more effective in general than any other?

I am in no way saying that this method is superior, but it represents a school of thought that is clearly a minority, yet was supported (also derived) by lagrange and euler, 2 of the most brilliant minds in math. All I'm saying, is how come after 7 semesters of college (university, I'm not sure I really know the difference) math classes had I not heard of it until looking at graduate level maths in my own time?

There's something wrong with this picture, I guess is the point to be taken from this statement.
I was just seeing if there was something particularly wrong with this approach, as that would be an understandable reason for me to have not heard of it.

 

[Simon Bridge]

Oh that's easy... its not surprising you have not heard of this approach before, there's lots of approaches you won't have heard of. You didnt know before because you didnt look before and nobody had any reason to volunteer the information.

Whats "particularly wrong" with the approach is that its not all that useful as part of a crowded college education program. I think the main issue is that it "begs the question", and, at the time when you review calculus (freshman math) few students have the confidence to cope with the manipulations. Bottom line... there are more effective methods.

One of the goals of these forums is to encourage a more scientific way of communicating. In that spirit, if you want to say something like "there's something wrong" with something, you have to be able to support this with more than a personal impression/experience or gut reaction or it's meaningless. Why should anyone listen to you and ignore half a century of research?

 

[BiGyElLoWhAt]

We should start a thread on here called neat stuff that smart people did that youve probably never heard of. Problem solved.