Can calculus be taught without differentiating or integrating by hand?

Depends if you want to teach a mathematician or an engineer or a physicist.

If you only need an engineer, numerical approximation is good enough. You teach about slopes, and area under the curve. Automatic differentiation and numerical quadrature.

Alternatively you can teach about function representation, by introducing Taylor and Fourier series, and then you have an easy closed formula to integrate and differentiate a function.

If you want to teach a mathematician, integration by part is a great first introduction to the freedom of choice that you encounter when playing the find the proof game. There may be multiple paths to the solution.

If you want to teach a physicist, you have to teach change of variables, and how to strike the infinitesimals to simplify them. Also teach them variation of constants and separation of variables.

If you want a statistician or financier, just teach Monte-Carlo integration and Ito's formula.

Education is pick and choose, depending on what you need it for.

I believe that the only way to find the answer is to get a group of volunteers and try to teach them calculus without teaching them to differentiate and integrate by hand. It is futile to ask people who was taught the other way, they absolutely predictable will say that it is important to learn calculus the way they were taught.

People do not know how education works. People just do to younger generations what was done to them. They can try small variations, and some variations stick. It is like a local search for the maximum. I believe that there is something ingrained deeply into a human mind, that makes human such native educators who can reproduce their skills in new generations. For example, parents tend to reproduce their own parents while parenting. It is a big problem for those, who is trying to become better parents, because they need to carefully monitor themselves to not slip into parenting approaches they are trying to eliminate.

So, no one knows how education works, now one knows what is important, why it is important, and is it possible to replace it with something else. But people tend to have pretty hard opinions on the lines of "the truest way to educate is how I was educated".

I tried that in Geometry for Programmers https://www.manning.com/books/geometry-for-programmers

I needed a short chapter on calculus so I go on and explain Bezier curves and splines, but I couldn't afford a full introduction. So, I only explained how differentiation works and gave a few examples. The rest was delegated to SymPy.

Funny thing. There are errors in differentiation formulas in the final edition of the book. Some final edits went wrong or something. Anyway, so far, we had exactly 1 (one) complaint about that.

Apparently, programmers don't care about formulas if the code is already there.

In SICP (Structure and Interpretation of Computer Programs) an early distinction is made between "declarative" and "imperative" ways of knowing. If you teach the declarative aspect in mathematics you are teaching what I would consider the "language" of mathematics. It is in fact a language and has value on its own. Now, later we can put on the programmer's hat and work with algorithmic aspects (the imperative part). This seems very valuable to me.

Yes. I regularly solve (mostly motion control) problems that require integration and differentiation and I absolutely cannot and have never been able to do those things by hand but I understand the concept and with a computer by my side that's plenty.

I'm sure there are things that I'm missing out on by having this gap in my knowledge but there's a lot of knowledge out there and I spend most days learning, I may get to it one day.

Yes, it can. The biggest failure in math education is that we spend an inordinate amount of time solving equations without understanding how to apply them to solve real-world problems. Thanks to the calculator we can now spend less time figuring out how to solve x+1=7 and put that time on how to understand when the equation is needed in the real world or how to create an equation that will help solve a real-world problem.

The same can be said about calculus. Given the abilities that computers give us, it's much better to teach applied calculus than to teach students how to solve equations by hand. Calculus is a tool, and it should be available to as many people as possible.

For most students, Calculus should be seen in the same light as drills. We understand when we need a drill so we get one and use it but we never have to go out and learn how to build one just because we have a need to use it. Calculus should be similar. We need to learn when we need it and then use a computer to get an answer that helps solve a problem that advances our needs.

I'm probably an atypical calculus learner. I'm middle aged and brushing up on my math(s)/physics so I can do audio DSP, understand the acoustics/physics of musical instruments, and explore patterns in nature - all for personal/artistic reasons rather than job-related (for now :-)).

I had an "a-ha" moment when I came across Ivan Savov's "No Bullshit Guide to Math and Physics" about 6 weeks ago in terms of finding a resource that meets my needs perfectly. I personally find it useful to work through a certain number of problems by hand so I understand the concepts, as well as how folks did the work before the advent of computers. I'm also the kind of person who will type out examples by hand when learning a new computer language so I can get a feel for it.

https://minireference.com/static/excerpts/noBSguide_v5_previ...

I find learning calculus & physics together is a good mix for me especially in light of my musical/DSP interests. FWIW just in these 6 weeks - I've been able to read through things like Claude Shannon's classic Theory of Information or Manfred Schroeder's books on chaos/fractals and number theory and get the gist of the math in a way I couldn't before.

Calculus isn’t solely about learning how to derive and integrate equations using algebraic rules. There are underlying principles behind differentiation and integration that are missed if one only focuses on learning symbolic manipulations. Sure, many could get by with just a computer algebra system, but eventually someone will run across a problem that a CAS won’t handle. That person will need to have an understanding of the deeper mathematical principles to solve the problem.

I think a course like that would need to be called something like “Mathematics Appreciation,” in analogy to “Music Appreciation.” I would never hire someone whose only music education was the latter to actually play in a band or write a movie score, and I would never hire someone who didn’t know how to do differentiation or elementary integration by hand to do any work that required an understanding or use of calculus.

Now, there is such a thing as overkill in assigning calculus homework; integration problems can be made arbitrarily difficult, and past a certain point I don’t think there’s much educational payoff.

But learning advanced math is hard and requires hard work. Sorry about that. Math students often start off in elementary school finding everything easy and intuitive, but then at some point everyone hits the wall. Same thing can happen when learning a language or developing a physical skill. At that point there is no alternative to working hard in order to make progress past that wall.

I'm currently teaching myself Calculus with the Fifth Edition of Larson, Hostetler and Edwards' Calculus.

What do you mean "by hand"? If you mean, you don't have to do things like the product rule, chain rule, etc. then I think this is probably a terrible idea because the "by hand" bits are the foundational concepts behind Calculus.

I may not have understood the question though - can you clarify?

Numerical Methods already made it pointless for me to learn how to solve different kinds of integrals.

I don't need to master any of the dozens symbolic integration techniques.

I am not sacred of any equation looking too complex to solve.

I just write less than 30 lines of python and have the solution to the problem via numerical methods.

At the end of the day what I care about is that answers, not symbolic techniques.

I'm a math professor. In my opinion, no. This would be possible, but I don't believe it would be a good idea.

I firmly believe that, if you want to learn any math subject, whether elementary or research-level or anywhere in between, you have to do a bunch of computations. Trying to skip this would be like trying to master chess by reading books and learning theory, without ever playing against live opponents.

I think it's very plausible that we could do with much less symbolic computation by hand, and more use of computers and calculators. Doing integrals and derivatives by hand is often overemphasized, because that's the easiest thing to test for. But I believe that even in an ideal world, this should have a prominent place in the curriculum.

You can build a much more abstract model of mathematical analysis by starting with open and closed sets instead of ye olde derivations and integrals. This changes the balance somewhat; instead of defining continuity of a function through limits, you say that a continuous function is a function for which preimage of every open set is open.

AFAIK this isn't taught often, especially not to not-mathematicians by trade. The tradition is to start with infinetesimal calculus, Newton and Leibniz style, which also produces much more "tangible" effects (you can solve some equations and tasks like that), plus can be easily tested in a written exam.

Here in CZ, one guy (Petr Vopěnka) spent a lot of time on trying to build mathematical models in this abstract way; his work included set theory and analysis among others. He is highly regarded, but no one adapted his way of teaching AFAIK.

Pedagogy has at least two aspects. The rote component, which people usually deprecate, and the inductive/reasoned component, where you learn "how it is" not "what to do"

The thing is, the rote component is (in my personal experience) much more important than people give credit to. You need to learn some axiom applied rules to be able to use "mental muscle memory" to perform things, and then once you can operate as a bit of a black box, you can get the uplift.

Sure, in practice the "why" comes as the stories, the instances of calculating acelleration over time, or distance travelled under non-constant speed, or whatever the analogy-instance is you're using to say why being able to compute derivitives and integrals matter, but if you can't do recall on 2+2 = 5 then this is a bit of a mixed bag.

I think maths is like cookery. You need basic knife skills and a discipline about order and time and sequence. For veggie stew it's less important but for baking its everything or the cake doesn't rise. Well, maths is the same but with more egg on your face and less sugar coating.

TL;DR you need to do things by hand. Thats what learning is, sometimes. I don't personally think teaching without doing some hand examples works as well. BTW I consider myself functionally illiterate in maths, but reasonably competent in arithmetic. Calculus is the dividing line. The massive mountain range which I climb on, but never cross.

My experience is the opposite can be true: I learned how to differentiate and integrate in high school but did not get a deep intuitive understanding until 10 years after when I got back at Math for video games and machine learning. If I were to teach someone I would first work through practical problems with acceleration, speed and position, going from one to the other with computer calculations (I remember the concept really clicking with the use of accelerometers inputs), and only going into the symbolic notation and hand calculation after already being able to use them. But that’s just my way of learning.

At schools with a broad range of majors, you often find a Calculus for X majors class. Mostly for majors where they need to have some exposure to Calculus, and may need to find slopes and areas under curves and such, but are not expected to do any further math courses, and so don't need to really hold all those symbolic concepts in their head for too long. And anyway, since you can't have Calculus and Arithmetic in your head at once, and they'll be better served with Arithmetic, best not to have them integrate by hand too many times.

I've taught undergrad and grad math. I think in principle you can, but only if your audience will either not need it in the future, or only need it superficially. If a student continues their path and become a practitioner they will need to know how the lower level stuff works, in order to be effective at their art. Just like a regular hacker, a "math hacker" won't be able to do magic without an intimate knowledge of the inner workings of their tools.

At my university this was called 'business calculus'. It's enough to get you through most of economics and finance, but not enough to understand basic physics. You can memorize crap about pendulums, but you won't understand Newton's theory of gravity in the end, and you'll be even more lost on the fancier stuff like Maxwell's equations. Sure, you don't need that level of understanding for most computer stuff, but I frequently find that it helps.

I would love experimentation and innovation in this direction.

I see students solving complex differentiation and integration problems without understanding what's going on, or being able to apply calculus to the real world problems.

I also wish for Lagrangian Mechanics to be the default instead of Newtonian in middle and high school. In my understanding, the core thing in the way is that Lagrangian uses calculus much more prominantly than Newtonian. Being able to teach calculus in a simpler way could be useful for that too.

Why would you though?

Differentiating by hand is not difficult at all, and graphing a curve and its derivative is a really good way to teach what a derivative actually is.

Integration is harder, obviously, and it is a pain point for sure. Even the idea that an integral is the area under the curve is clearly not trivial to students.

But I don’t see a clear benefit trying to avoid integration. Integration is more difficult, so it should be pointed out as such, but then it’s still really useful to know how to do it…

UK A-levels certainly used to teach calculus quite successfully without ever deriving a single differential by hand. Students specializing in maths will have done differentiation using product and composition rule, integration, by parts and by substitution, along with trig identities. Basic classes of ODEs are also solved.

This is pretty good for the 80% who don't go on to study maths, and still a useful intro to the ones who redo it properly at Uni.

Years ago, my college multi variable calculus and linear algebra courses were both taught primarily using course materials that were interactive Mathematica Notebooks.

We had access to all of the symbolic algebra tools and were even expected to use them regularly for both courses. It was great!

I'm not sure how well this would extend to introductory courses though, especially if the standardized tests still expect integration by hand.

I feel like the point of learning integration isn't to know how to integrate, but rather how to use math to solve open ended problems within a firm set of rules. Plus you have to be able to check whether a numeric integration method is actually correct and idk how you do that without some knowledge of hand integration

At the college I went to there was a class that involved solving differential equations in Simulink. The exams involved mostly creating diagrams on how the variables were related. The only prerequisite for that class was Calculus 2 so only a conceptual knowledge of Calculus was needed

I think it could be, but I found realizing the correlation between differentiation and integration to be just beautiful. I think offloading those calculations to a computer system, at least while initially learning, would be a disservice.

dual numbers[0] in lisp/lambda calculus context aka let a=lisp car and b-epsilon be lisp cdr. Then type of number/calculation can also be symbolic. Although, per Lewis Caroll's concept of imaginary numbers with no concept of 'time', this doesn't quite work per functions/lists needing 'time to compute' a result. Guess why Lewis Carrol's 'turing' rabbit was always late.

[0] : https://www.youtube.com/watch?v=ceaNqdHdqtg

The question could be changed to a positive.

>Can Calculus be taught with differentiating or integrating by hand?

And then the answer is yes, as this is how it is done.

Now, why would be want to do otherwise? Why would we ever want people to learn less?

I always thought my calculus 1 & 2 courses focused too much on writing solutions without really understanding concepts. I was pretty lost in calculus 3 and 4.

The intuition and general concept can be taught without differentiating or integrating by hand but what calculus is can only be understood by doing it.

What exactly does this look like? Do you have example problems you can point to?

> Maybe the focus could be on solving calculus problems with the help of a symbolic algebra system instead?

Umm..

Like this -> https://www.mytutor.co.uk/answers/7336/A-Level/Further-Mathe...

Beyond the simple case shown in [1], by hand making sure the logical type/denomiator is consistent across an entire exercise/problem working beyond basics is extremely time consuming/difficult.

abstract case: Think y-combinator stuff and how quickly the information grows for one term to get rid of all free variables. aka simplified continuitiy/consistency of logic type use. is 1 imaginary, real, integer, vector, trig value, etc

[1] : https://www.youtube.com/watch?v=2OIiLu5xn-E

1) At hight school level, could always do a concise refresher linking algebra/geometry; algebra/gemetry/imaginary numbers as dual number concept staring with making 2 slide rules out of a 2 similar circular item via using compass (10 equal slices) . (aka identity unit & importance of continuity of units & number base / denominator in fractional notation / continutity)

   Exchange one circle with someone else in class -- can you sill generate same slide rule results 
   as other students?[0][1][3]

   note: square viewed on edge is a line!
   note: circle start/end implied vs. linear scale has explicet stop/end point. 
         Unit circle from trig is "1".
         
   Note: IF get rid of the 'circle arc' at bottom of triangle[2] & put into another "scale", then 1 "unit" is combined 'triangle' & circle arc fragment.  Comput the 1 circle arc unit 'size' using trig less one unit triangle area. 
        Or do appropriate 'fold' to push the circle arc into different dimentional plane. 
        aka you've just defined what lies  above pascal's triangle : https://www.youtube.com/watch?v=q2daqMR3l24

   1d pacman : https://news.ycombinator.com/item?id=38845510
   "15" is wrong : https://www.youtube.com/watch?v=FG0vtPa0UrM

2) Dual numbers and trig quadrants!

   a) What's the value of i? square root of i equal? : https://www.youtube.com/watch?v=2OIiLu5xn-E

      How does the resulting unit look in each of the unit circle quadrants? What would happen if added another line? aka instead of abcd, abcde?

'

   b) Automatic differentiation : https://blog.demofox.org/2014/12/30/dual-numbers-automatic-differentiation/

3) the easy stuff -> calculus delta & epsilon, integrands as triagle corners/log end points/circle end/start & evaluation as where dual numbers meet.

4) Time to inflate the log scale / line above into square (no euler yet!)

------

[3] category theory abstract nonsense : https://www.youtube.com/watch?v=igf04k13jZk / https://www.youtube.com/watch?v=DrldYpmwN5s

[2] what is the opposite of a set : https://www.youtube.com/watch?v=SrltwGJAiCM

[1] what happens if you add fractions incorrectly : https://www.youtube.com/watch?v=4d6YrTKmjfE

[0] make your own slide rule : https://www.mathed.page/calculator/super-sci/slide.pdf

Imho, the better question is should calculus be taught without physics?

None of it clicked for me, I stumbled to make any meaning out of any of the hand wavy explanations my poor teachers gave me… until I took my first calculus based physics class and a hundred light bulbs went on as to why any of this stuff needed to exist, but more importantly why.

Physics

Physical experiments.

Yeah, would have been more useful to spend relatively more time on numerical approximations.