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.
So you propose to teach them different methods of solving.
I think, while your mapping from solution methods to professions makes sense, the act of solving should be taught as an afterthought.
The goal should be to build a robust mental model. Personally i find playing around with computer visualisations helps a lot with that.
The same is even more important for differential equations, imo. I wish i would have spent less time studying algorithmic DE solving techniques in my school days. The beauty and power of expression of these has nothing to do with the way you solve them, if you ask me.
> If you only need an engineer, numerical approximation is good enough.
No. Completely wrong. Engineers and scientists are given the very same rigorous fundamentals of mathematics. We (EE/CS) also had to take the very same weeder courses as mathematicians including abstract mathematics and concrete mathematics, in addition to modern physics for physicists and engineers, followed by the entire computer science and electrical engineering series. We did learn the Monte Carlo method for problems that may not be directly computable in reasonable time, but these are approximations that can be achieved with a qualified amount of error. In fact, I helped port a FORTRAN-based nuclear reactor simulator for UNIX to Windows that was, at its core, a quadruple integral Monte Carlo approximation considering configurations of voxels of matter and voids over time with their various properties.
We started with the fundamental in high school on paper with area under the curve and learning mathematical transformations of integrals, and derivatives as rate of change. The Erable CAS found in the HP 48 helped provide transformations and reductions sometimes, but not every time for basic calculus, vector calculus, and ordinary diff eq. (We didn't have MAPLE, Mathematica, or GIAC which are far more powerful.)
Finally, in college, we ended up with discrete FFTs and Laplace transforms in EE and the Schrödinger eave eq for hydrogen in physics. This isn't as far as physicists or mathematicians go in grad school such as groups, fields, and such.
So, I think you need to reevaluate your assumptions about curricula that seems wholly inadequate.
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".
> 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.
Those are some pretty baseless statements, considering pedagogy is an entire field https://en.wikipedia.org/wiki/Pedagogy and it's an active area of research. I think it's reductive to end with 'no one knows'.
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.
Students don’t like “real world” problems. They say they want more of that but when you actually do those types of problems they complain or do poorly on them. Word problems are even more confusing to students than non-word problems.
The vast majority of so called real world problems aren’t things people who use math in their jobs actually do.
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.
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.
This analogy doesn't make any sense. Music is not like math and the music industry is not like industries that are related to calculus.
Music theory is NOT required to play in a band or write a movie score. Check Gustavo Santaolalla's for an example of a great composer with ZERO musical bg.
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?
If you use numerical methods routinely, you need to know about error propagation too, otherwise you may get results that are completely wrong. That's theory too, although of a different kind than the typical symbolic calculus.
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.
You can develop muscle memory without rote memorization by doing exercises. There, you end up memorizing facts by solving problems and therefore building intuition. I agree that you need some level of memorization when you first "perform things," but if you're performing, it's not rote memorization.
Rote is memorizing integrals with flash cards. Building intuition is doing exercises. Might seem pedantic, but the difference is huge pedagogically.
I switched schools in tenth grade (1994). Till then, I was mostly interested in the science subjects. We had Physics, Chemistry and Biology as separate subjects. I had a pretty good chemistry set (compared to the crap you get these days) at home, I had a small microscope, I could create slides, do basic chemical analysis etc. That was sort of my domain. Math always seemed too abstract for me and while I liked some of the topics, I was never too good at it and treated it as "below me". Science subjects, I could reason from first principles and didn't have to study many things by rote. I rationalised this saying that it was because I was too intelligent to waste my time memorizing stuff and considered the ability to reason way more superior.
After switching schools, we had a mathematics teacher (who since passed away). His style of teaching was to make us practice with harder and harder problems. He had a tower of books with example problems in all the topics that were relevant to us and he made us do all of them. No calculators. It was tedious work and doing large integrals and complicated equations can be a chore. However, because of the sheer effort, I broke through some kind of wall and saw it was that I was really doing. I suppose it's analogus to an inductive rather than deductive way of learning. I was able to derive several formulae using basic Calculus which I had previously learned by rote in my physics classes. All the hidden patterns started to make sense and I learned to appreciate the subject very much and started seeing places where I could use it.
I've forgotten some of the topics which I don't use on a daily basis but the basic idea of doing grunt work to master a subject stuck with me. I also emphasise it when I teach my kids math. Lots and lots of problems. Till it becomes second nature. It's how I learnt how to code. It's how I learnt calligraphy. It came full circle when I started studying martial arts. The repetition of forms and drills to "learn" a technique is quite different from "learning the theory". It helps to have a teacher (coach) who pushes you but the basic idea that "doing is the best way to learn" is foundational.
I'm one of those people who find maths really easy. I've also had extensive experience working in mathematics education, from primary to tertiary levels. I've even written policy for our state education department.
You are spot on the money. I've seen "progressive" educators who believe that technology, including CAS is the answer to teaching calculus to high school and tertiary students. I think the tech is mostly useful _pedagogically_ after the student has a very firm grip on the content. Using it as a shortcut to "real world problems" leaves the student with no real understanding of the dynamics/functions involved, leading in turn to believing the machines are magic and always right, regardless of whether you've hit all the right buttons in the right order.
The most extreme example of this I've seen was in a classroom I was auditing. The students were learning about harmonic motion and trig functions by using a mass on a spring and an ultrasonic transducer. The students were told to "keep all the digits, because the machine gave them". Students were doing sums with 14 significant digits. When I pointed out that most of them were meaningless, being measures in the Angstrom range, the "teacher" said "the machine gives those figures".
Even when I pointed out that measuring 10^-14 m required a very high frequency, which meant high energy, all supposedly from AA batteries that last for months, he agreed that it did seem strange, but after all, the "machine gave the figures".
The man teaching (he was head of department AND author of the state mathematics syllabus) was unaware that only fractions with denominators with only 2 and 5 as factors can be expressed exactly in binary without using custom datatypes and arithmetic logic.
Now as I said, for me maths is easy. It's not for most people. Most people won't grasp it all as "but that's obvious", so for these people having the skills to check the result the machine gives is vital if they are not to cause all sorts of fun as bridges collapse, cars explode and mortgages go into default. Those skills are developed by doing it by hand, and learning what numbers change in what ways.
TL;DR Modern trend to use CAS in mathematics education is well intentioned but misled, in this retired maths educators very informed opinion.
A certain kind of student who has a lot of programming experience could write an automatic differentiator in the process of learning how to differentiate.
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.
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…
[+] [-] GistNoesis|1 year ago|reply
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.
[+] [-] lagrange77|1 year ago|reply
I think, while your mapping from solution methods to professions makes sense, the act of solving should be taught as an afterthought.
The goal should be to build a robust mental model. Personally i find playing around with computer visualisations helps a lot with that.
The same is even more important for differential equations, imo. I wish i would have spent less time studying algorithmic DE solving techniques in my school days. The beauty and power of expression of these has nothing to do with the way you solve them, if you ask me.
[+] [-] burnt-resistor|1 year ago|reply
No. Completely wrong. Engineers and scientists are given the very same rigorous fundamentals of mathematics. We (EE/CS) also had to take the very same weeder courses as mathematicians including abstract mathematics and concrete mathematics, in addition to modern physics for physicists and engineers, followed by the entire computer science and electrical engineering series. We did learn the Monte Carlo method for problems that may not be directly computable in reasonable time, but these are approximations that can be achieved with a qualified amount of error. In fact, I helped port a FORTRAN-based nuclear reactor simulator for UNIX to Windows that was, at its core, a quadruple integral Monte Carlo approximation considering configurations of voxels of matter and voids over time with their various properties.
We started with the fundamental in high school on paper with area under the curve and learning mathematical transformations of integrals, and derivatives as rate of change. The Erable CAS found in the HP 48 helped provide transformations and reductions sometimes, but not every time for basic calculus, vector calculus, and ordinary diff eq. (We didn't have MAPLE, Mathematica, or GIAC which are far more powerful.)
Finally, in college, we ended up with discrete FFTs and Laplace transforms in EE and the Schrödinger eave eq for hydrogen in physics. This isn't as far as physicists or mathematicians go in grad school such as groups, fields, and such.
So, I think you need to reevaluate your assumptions about curricula that seems wholly inadequate.
[+] [-] iancmceachern|1 year ago|reply
[+] [-] RhysU|1 year ago|reply
Fluid dynamics enters the chat.
And whaddya mean "only" need an engineer?
[+] [-] ordu|1 year ago|reply
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".
[+] [-] kingkongjaffa|1 year ago|reply
Those are some pretty baseless statements, considering pedagogy is an entire field https://en.wikipedia.org/wiki/Pedagogy and it's an active area of research. I think it's reductive to end with 'no one knows'.
[+] [-] ForOldHack|1 year ago|reply
The person who explained how it worked,without ever seeing it, was Dr Fernando Flores.
[+] [-] okaleniuk|1 year ago|reply
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.
[+] [-] 2ro|1 year ago|reply
[+] [-] thot_experiment|1 year ago|reply
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.
[+] [-] angra_mainyu|1 year ago|reply
[+] [-] WheelsAtLarge|1 year ago|reply
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.
[+] [-] skhunted|1 year ago|reply
The vast majority of so called real world problems aren’t things people who use math in their jobs actually do.
[+] [-] Tainnor|1 year ago|reply
[+] [-] Optimal_Persona|1 year ago|reply
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.
[+] [-] linguae|1 year ago|reply
[+] [-] 7402|1 year ago|reply
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.
[+] [-] santiagobasulto|1 year ago|reply
Music theory is NOT required to play in a band or write a movie score. Check Gustavo Santaolalla's for an example of a great composer with ZERO musical bg.
[+] [-] chris_wot|1 year ago|reply
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?
[+] [-] max_|1 year ago|reply
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.
[+] [-] Tainnor|1 year ago|reply
[+] [-] impendia|1 year ago|reply
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.
[+] [-] inglor_cz|1 year ago|reply
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.
[+] [-] ggm|1 year ago|reply
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.
[+] [-] bokoharambe|1 year ago|reply
Rote is memorizing integrals with flash cards. Building intuition is doing exercises. Might seem pedantic, but the difference is huge pedagogically.
[+] [-] noufalibrahim|1 year ago|reply
I switched schools in tenth grade (1994). Till then, I was mostly interested in the science subjects. We had Physics, Chemistry and Biology as separate subjects. I had a pretty good chemistry set (compared to the crap you get these days) at home, I had a small microscope, I could create slides, do basic chemical analysis etc. That was sort of my domain. Math always seemed too abstract for me and while I liked some of the topics, I was never too good at it and treated it as "below me". Science subjects, I could reason from first principles and didn't have to study many things by rote. I rationalised this saying that it was because I was too intelligent to waste my time memorizing stuff and considered the ability to reason way more superior.
After switching schools, we had a mathematics teacher (who since passed away). His style of teaching was to make us practice with harder and harder problems. He had a tower of books with example problems in all the topics that were relevant to us and he made us do all of them. No calculators. It was tedious work and doing large integrals and complicated equations can be a chore. However, because of the sheer effort, I broke through some kind of wall and saw it was that I was really doing. I suppose it's analogus to an inductive rather than deductive way of learning. I was able to derive several formulae using basic Calculus which I had previously learned by rote in my physics classes. All the hidden patterns started to make sense and I learned to appreciate the subject very much and started seeing places where I could use it.
I've forgotten some of the topics which I don't use on a daily basis but the basic idea of doing grunt work to master a subject stuck with me. I also emphasise it when I teach my kids math. Lots and lots of problems. Till it becomes second nature. It's how I learnt how to code. It's how I learnt calligraphy. It came full circle when I started studying martial arts. The repetition of forms and drills to "learn" a technique is quite different from "learning the theory". It helps to have a teacher (coach) who pushes you but the basic idea that "doing is the best way to learn" is foundational.
Repetitio est mater studiorum as they say.
[+] [-] Perenti|1 year ago|reply
You are spot on the money. I've seen "progressive" educators who believe that technology, including CAS is the answer to teaching calculus to high school and tertiary students. I think the tech is mostly useful _pedagogically_ after the student has a very firm grip on the content. Using it as a shortcut to "real world problems" leaves the student with no real understanding of the dynamics/functions involved, leading in turn to believing the machines are magic and always right, regardless of whether you've hit all the right buttons in the right order.
The most extreme example of this I've seen was in a classroom I was auditing. The students were learning about harmonic motion and trig functions by using a mass on a spring and an ultrasonic transducer. The students were told to "keep all the digits, because the machine gave them". Students were doing sums with 14 significant digits. When I pointed out that most of them were meaningless, being measures in the Angstrom range, the "teacher" said "the machine gives those figures".
Even when I pointed out that measuring 10^-14 m required a very high frequency, which meant high energy, all supposedly from AA batteries that last for months, he agreed that it did seem strange, but after all, the "machine gave the figures".
The man teaching (he was head of department AND author of the state mathematics syllabus) was unaware that only fractions with denominators with only 2 and 5 as factors can be expressed exactly in binary without using custom datatypes and arithmetic logic.
Now as I said, for me maths is easy. It's not for most people. Most people won't grasp it all as "but that's obvious", so for these people having the skills to check the result the machine gives is vital if they are not to cause all sorts of fun as bridges collapse, cars explode and mortgages go into default. Those skills are developed by doing it by hand, and learning what numbers change in what ways.
TL;DR Modern trend to use CAS in mathematics education is well intentioned but misled, in this retired maths educators very informed opinion.
[+] [-] PaulHoule|1 year ago|reply
[+] [-] tarsinge|1 year ago|reply
[+] [-] toast0|1 year ago|reply
[+] [-] ziofill|1 year ago|reply
[+] [-] fat_cantor|1 year ago|reply
[+] [-] alok-g|1 year ago|reply
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.
[+] [-] d--b|1 year ago|reply
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…