Meant to get back to this. I like this way of thinking, though one has to be careful.
A couple of extras: when you divide a tiny number by a tiny number, the result might not be tiny. That’s why we can’t just translate the dy/dx to 0/0 (which breaks things anyway). Also, a big+tiny number minus the same big number leaves us the tiny number. Although tiny, if it’s the only thing that’s left, it’s still important and we can’t throw it away. But we can still get rid of tiny*tiny numbers because they’re extra-tiny compared to ordinary tiny numbers.
One nice thing here is that students can experiment around with the tiny numbers on their calculator. Plug in e[sup]0.0001[/sup] and get 1.000100…[a tiny bit extra]. Then see how the error goes down even further as you add more zeroes.
Of course, one wants to be careful that the students don’t get the idea that this is the only thing going on–however many zeroes you add, infinitesimals are smaller yet–but it does give a nice bit of concrete intuition.
Better yet would be a computer program with a kind of “infinite zoom” system, like the 3Blue1Brown videos do. With that, it should really sink in that you can always zoom in enough that the slope looks constant (except for non-differentiable functions).
As for formally developing infinitesimals in order to teach calculus, as Hari Seldon explained, sure, why not? But keep in mind that there is going to be some conservation of difficulty in defining, e.g., hyperreal numbers, and the aforementioned physicists may not care deeply about the formal logic of it, and it does not impact their symbolic calculations either way.
Formally developing them? No need for that; calculus is already on solid footing. The students don’t need that any more than they need to construct the real numbers or develop set theory from axioms. That can be done later; early on, they can be introduced just like the complex numbers, where we add a new number to the system and then work through the standard arithmetic operations.
I’m talking about the very earliest parts of calculus: where the usual starting point is lim[h->0] (f(x+h)-f(x))/h. And from there, showing where the power rule, the chain rule, the product rule, the transcendental derivatives, and the others come from. Starting with implicit differentiation feels both simpler and more elegant to me.
When I was a TA for a PDP-11 assembly language class over 40 years ago we started with an assignment, done in Pascal, which we graded more on beauty than on correctness. Beauty in this sense was following the rules of the then new structured programming movement. We figured that if the students didn’t learn it early, there assembly language code would be even more of a mess than it turned out to be.
Definitely true when I took it in high school a long time ago and when my daughter took it not such a long time ago. I can’t imagine how anyone could teach integration first.
Interesting. Upon reflection, I suppose there was probably a lower limit on beauty in my classes–I have to think that if my assignments in Pascal or C++ were entirely written with conditional gotos, I’d have lost a few points. So most likely, structured programming was considered a minimum (though perhaps not: this is entirely speculative). But higher level languages deserve a higher standard of beauty than just that.
It’s been a while since I was learning calculus, but if you start with implicit differentiation, don’t you make it much harder to see the geometric interpretation (finding the slope of the line)? Which will make it much harder for students to understand the basic idea of what differentiation is? I think once you’re comfortable with the idea and concept, working with infinitesmals is easier, but I think the limit helps in getting comfortable with the concept.
No headache but I’m in a place possibly worse; I had Calc I and II (more than 40 years ago now) and I get just enough of the discussion that I had to make some notes and do some calculations/equations myself and follow it along. That, I think, is why there will always be some form of SDMB; where else can threads like this so easily mix with the range of other topics we discuss? This is really terrific!
This is true to history. The derivative was discovered because math peeps of the time were struggling with understanding integration. For teachers who aren’t afraid of front-loading their pedagogical effort, it makes some legitimate sense to introduce the ideas in the historical order, which makes the derivative a fuckin revelation, rather than just “Here’s the easy part, which we’re going to do before we do the hard part.”
Integration first makes the underlying reasoning behind the fundamental theorems really stick in the mind. I know for a damn fact that for me on first exposure, calculus was not the beautiful interrelationship between two seemingly different concepts. Quite to the contrary. Calculus was only a list of formulas to be memorized. Why these formulas? I could not have honestly told you. It didn’t matter enough for my grade. I think going against the historical grain feeds exactly that attitude. Derivatives-first is a great way of telling students that the integral simply does not matter as an independent idea, and all they have to do is reverse the derivative to get the right formula to get a decent grade in the course.
As for why infinitesimals get downplayed in the standard pedagogy? It’s pretty obvious, really.
For math profs who are tasked with teaching introductory calculus, “calculus” is simply not an independent tool, in and of itself, for use in engineering and the sciences. That is not how they see it. For people who live their lives in higher maths, calculus is interpreted as merely the students’ first taste of real analysis, dotted with real-world examples for pedagogical convenience. Real analysis is done with real numbers. Real numbers handle infinities with limits, not infinitesimals. Therefore, calculus as the first exposure to real analysis should be done with limits. Non-standard analysis comes later, for those who are interested. But it is exactly what they call it: non-standard. It’s a fascinating addendum to real analysis, but not the fundamental thing.
People whose first big-time, thrown-into-the-deep-end-of-the-swimming-pool exposure to theorem proving was the famous intro real analysis class are not-not-NOT going to teach calculus with infinitesimals. That’s not how they think of their own intellectual progression, and so it’s simply not how they’re going to teach.
All you seem to be asking is why Calculus is not taught the way 3Blue1Brown does it, where he tries to make calculus more intuitive, and less about memorizing formulas and just “doing the math.” I’d say it’s because the whole “common core” idea hasn’t made it up to the higher mathematics.
I would guess this is because those types of math are not seen as something everyone needs to be able to do. Common Core ideas seem to be about making math accessible to wider groups of people. If Calculus is not considered important for that goal, they don’t touch it.
Though it may also just be that things are slower to change the higher up you go in the chain. Maybe you need the students who learned using Common Core and teachers who taught using it before they can make calculus more accessible.
All I know is that watching his videos, I actually understood what I was doing when I differentiated or integrated for the first time, and saw why so many things actually work. It felt akin to when I figured out why the formula works for the area of a circle.
The relationship between a circle’s circumference, radius and area — an elementary result in integration — was allegedly first discovered by … Archimedes of Syracuse.
The very first thing you study to learn calculus is how to realize and work with the real numbers; see Nava’s post for example. If you want hyper-reals or dual numbers, then you need to introduce those instead— it’s not that bad.
But that was not my point. My hypothetical working physicist knows that taking the derivative of a function means that you find the best linear approximation to it. So he or she writes f(x + h) = f(x) + h f’(x) + o(h). Whether o(h) is small, or formally infinitesimal, she does not care.
Note, ideas from non-standard analysis and synthetic geometry may be quite helpful and useful. But the gritty details of the formal basis of the theory do not normally rear their head so that the person differentiating a function needs to worry about whether limits were used to prove a theorem versus some other technique. Of course, I may be wrong, and would be happy to hear more about it from scientists and engineers and mathematicians.
To play devil’s advocate, why should this be taught at all? In my various careers in retail, banking and insurance operations, including data analytics, I have never needed this level of math. I would daresay that 90+% of the population falls into the same category. Pre-college/university education can’t address everything, wouldn’t that time be better spent teaching things that the majority will use directly or indirectly in their life/career?
Were you forced to take mathematics courses against your will? People who are not required to study, and have no interest in, mathematics are presumably not enrolling in these classes.
Still, we are discussing basic calculus, which is not too esoteric and arguably quite useful for banking, data analytics, economics and many of the subjects you name.
Not at all–the interpretation I proposed is, if anything, more geometric. It’s just that I don’t have a nice video series or whiteboard to express what’s going on. Also, I’m not that practiced in teaching.
To clarify a bit, the steps in the process I proposed are:
Identify two points on the curve. I specified coordinates of (x, y) and (x+dx, y+dy), but this was just convenience: I could have used something else. But this works well because we know we’re eventually going to treat dx and dy as being small.
Substitute our two points into the implicit curve to get two equations.
Algebraically simplify these two equations. Most likely, we want to rearrange until we have a dy/dx on one side, but that’s not the only option.
Note that the first step is all geometry: draw our curve on the whiteboard, label the points, draw a line through the points showing that it’s a good approximation for the tangent when dx and dy are small, label the differences in x and y, show that the slope is going to be some form of (y2-y1)/(x2-x1), etc.
There’s no way to get rid of the algebra–that’s necessary with the standard approach, too. But the implicit approach shows that there’s nothing really special about the (f(x+h)-f(x))/h formula. That’s really just a special case of solving for the slope of a line going through two points.