Ask HN: Can calculus be taught without differentiating or integrating by hand?
Maybe the focus could be on solving calculus problems with the help of a symbolic algebra system instead?
101 comments
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.
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.
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.
The whole education can be seen as a sieving process routing people towards their natural abilities as needed by the economy.
It starts very early with things like mental calculus, where you try to get kids to compute things like 99 times 101, 98 times 102, finding tricks and rules to make the computation "easier". We could instead teach them to follow mentally some multiplication algorithm, but instead we try to promote some exploration and looking for pattern.
In chess, similarly you can train for puzzle to develop some vision without having to calculate, like multiplication table.
The education system must also take care not to prevent it from adapting to further changes by being too rigid. If the cat can't get out of the box, the box won't ever grow bigger.
But you can't have too many cats getting out, or you'll soon have an empty box and hangry cats.
I don't know what the goals should be, far from me the idea of proposing to teach, I only show paths that can be taken.
It starts very early with things like mental calculus, where you try to get kids to compute things like 99 times 101, 98 times 102, finding tricks and rules to make the computation "easier". We could instead teach them to follow mentally some multiplication algorithm, but instead we try to promote some exploration and looking for pattern.
In chess, similarly you can train for puzzle to develop some vision without having to calculate, like multiplication table.
The education system must also take care not to prevent it from adapting to further changes by being too rigid. If the cat can't get out of the box, the box won't ever grow bigger.
But you can't have too many cats getting out, or you'll soon have an empty box and hangry cats.
I don't know what the goals should be, far from me the idea of proposing to teach, I only show paths that can be taken.
I'm not sure i understand your answer, to be honest, especially the cat fable.
> as needed by the economy
Even if that is the purpose of education, a mathematician, engineer or physicist who has a deep understanding is worth more to the economy, than one who can remember and perform algorithms and computations, isn't he?
> as needed by the economy
Even if that is the purpose of education, a mathematician, engineer or physicist who has a deep understanding is worth more to the economy, than one who can remember and perform algorithms and computations, isn't he?
From the individual intellectual point of view it's better to have a better education, but from the collective point of view having too many wild cards is not good for anyone. If there are too many physicists, they won't earn enough to justify the cost of the studies, and won't be happy in a menial job after spending years learning advanced mathematics.
If anybody can do everything interchangeably, it also tends to create a monoculture which is a more fragile system.
Often it's better not knowing how the sausage is made, and trusting that people who have specialized know what they are doing.
Wild cats ("overeducated population") are dangerous.
If anybody can do everything interchangeably, it also tends to create a monoculture which is a more fragile system.
Often it's better not knowing how the sausage is made, and trusting that people who have specialized know what they are doing.
Wild cats ("overeducated population") are dangerous.
Interesting point of view. I have to think about it.
> 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.
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.
Rather than talking about engineers, I am mostly speaking about the subject which is calculus, aka the study of continuous change, aka infinitesimal change, aka derivatives and integrals.
At first order the engineer need to be able to provide some sort of solution to any problem and get something done. The numerical toolbox is kind of an easy reliable Swiss knife that will get you through anything related to derivatives and integrals. It
Ten years down the line when your nuclear core is melting and you need to pull the right lever will you be confident enough to trust your rusty manual computations tricks and not miss some minus sign somewhere in the transcription of your derivative.
The derivative and integral game, is kind of an interesting mathematical artefact about the study of functions from before the time we had computers. As far as I know it's not yet a solved game. It's more about heuristics and picking rules than using a specific formula.
Rule based Integration https://rulebasedintegration.org/ is a kind of collection of all the tips and tricks to play this game, but even then it's not foolproof.
Closed formula are nice to have but hard to get by in real problems.
The thing which makes the difference between something that is nicely behaved or not, is whether or not the operations are closed ("closure" concept). When it's the case you can build some nice algebra with nice properties. The computations becomes deterministic and formulaic. When it's not you end-up playing a NP-complete game to find the solution.
For example in the case of the study of infinitesimal rotations, the natural extension of calculus for physics, we have the Lie Algebra. Even there your numerical toolbox will be more useful than your pages of Feynman Diagrams.
At first order the engineer need to be able to provide some sort of solution to any problem and get something done. The numerical toolbox is kind of an easy reliable Swiss knife that will get you through anything related to derivatives and integrals. It
Ten years down the line when your nuclear core is melting and you need to pull the right lever will you be confident enough to trust your rusty manual computations tricks and not miss some minus sign somewhere in the transcription of your derivative.
The derivative and integral game, is kind of an interesting mathematical artefact about the study of functions from before the time we had computers. As far as I know it's not yet a solved game. It's more about heuristics and picking rules than using a specific formula.
Rule based Integration https://rulebasedintegration.org/ is a kind of collection of all the tips and tricks to play this game, but even then it's not foolproof.
Closed formula are nice to have but hard to get by in real problems.
The thing which makes the difference between something that is nicely behaved or not, is whether or not the operations are closed ("closure" concept). When it's the case you can build some nice algebra with nice properties. The computations becomes deterministic and formulaic. When it's not you end-up playing a NP-complete game to find the solution.
For example in the case of the study of infinitesimal rotations, the natural extension of calculus for physics, we have the Lie Algebra. Even there your numerical toolbox will be more useful than your pages of Feynman Diagrams.
The challenge is you often have all 3, and more, all om the same class.
Then it's more about gardening, than mathematics. Weed-out bad elements and behaviors, provide rich fertile grounds and nutrients, a direction to grow towards and let them grow by themselves. Promoting curiosity, building inner representation of abstract concepts and learning by doing with examples is usually a good recipe.
But if you are more into agriculture, a combine harvester also works great.
But if you are more into agriculture, a combine harvester also works great.
> If you only need an engineer, numerical approximation is good enough.
Fluid dynamics enters the chat.
And whaddya mean "only" need an engineer?
Fluid dynamics enters the chat.
And whaddya mean "only" need an engineer?
Even in fluid dynamics the game is only as hard as you make it for yourself :
https://en.wikipedia.org/wiki/Lattice_Boltzmann_methods
You have well defined operations and conservation laws well behaved by construction.
Don't let Navier & Stokes enter the chat, and you will be fine.
The whole problem of integration is that mathematicians historically introduced new functions, and try to relate them together, in interesting way.
Polynomials are good and well behaved. Algebras too.
https://en.wikipedia.org/wiki/Lattice_Boltzmann_methods
You have well defined operations and conservation laws well behaved by construction.
Don't let Navier & Stokes enter the chat, and you will be fine.
The whole problem of integration is that mathematicians historically introduced new functions, and try to relate them together, in interesting way.
Polynomials are good and well behaved. Algebras too.
[deleted]
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".
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'.
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'.
Pedagogy cannot know how knowledge is generated in the human mind. Pedagogy can advise on how to construct the process of education, like classrooms or grades, but it cannot know what knowledge of calculus is.
Moreover, no one knows what is knowledge, how it works, and how it can be acquired. If it was known, then AI developers would already have build a general artificial intelligence. There are some known rituals, if you do enough of them with uneducated youth then some of them become knowledgeable. But no one knows how it works and why it doesn't work every time.
Moreover, no one knows what is knowledge, how it works, and how it can be acquired. If it was known, then AI developers would already have build a general artificial intelligence. There are some known rituals, if you do enough of them with uneducated youth then some of them become knowledgeable. But no one knows how it works and why it doesn't work every time.
I feel like you are trying to border on philosophical 'knowing' rather than the very established body of evidence around cognition and understanding, and the ability to share and pass information to another.
> no one knows what is knowledge, how it works, and how it can be acquired
Chimps literally teach each other to use rocks as tools. Clearly knowledge or 'how to accomplish a task' is something that can be communicated to another. When it is communicated the nueral activity and memory in the other persons brain are clearly doing 'something' to store and be able to recall concepts and steps needed.
> But no one knows how it works and why it doesn't work every time.
Those are massively different though, clearly transmitting the same knowledge via different approaches is possible, i.e. OP's question about teaching calculus in non-traditional ways.
> no one knows what is knowledge, how it works, and how it can be acquired
Chimps literally teach each other to use rocks as tools. Clearly knowledge or 'how to accomplish a task' is something that can be communicated to another. When it is communicated the nueral activity and memory in the other persons brain are clearly doing 'something' to store and be able to recall concepts and steps needed.
> But no one knows how it works and why it doesn't work every time.
Those are massively different though, clearly transmitting the same knowledge via different approaches is possible, i.e. OP's question about teaching calculus in non-traditional ways.
> Chimps literally teach each other to use rocks as tools. Clearly knowledge or 'how to accomplish a task' is something that can be communicated to another.
Yes, and no.
Lets take calculus as an example and the skill to take derivatives. If a teacher show you how to take derivatives, you'll kinda get the idea but if you try it, you would face a lot of problems and you probably would need teacher's help.
To become proficient with derivatives, you need to teach yourself by taking a lot of derivatives. But... wait... you are taking derivatives, you're not doing something special that can be called "learning".
Chimps also communicate knowledge by showing the right "ritual" how to accomplish things, students reproduce ritual and somehow learn how to do it properly.
The whole education works mostly in this vein, the method is to make people to do something, and while they are doing it they'll learn how to do it. Magic.
The communication doesn't transfer all the knowledge, and it is pointless to try, because people are different and edge cases are different for different people. For example, I'm always confusing left and right, it was a pain to deal with the driving instructor who would say "turn left at the next crossing" and then I will dutifully turn right. I can deal with it most of the times, I just need to pay more attention to "left" and "right" things. But it is my specific edge case, if I teach others I do not try to communicate the necessity of paying attention for words "left" and "right", and when someone teaches me they do not communicate this to me. There are a lot of such individual edge cases and the more difficult the knowledge the more individual it becomes.
Of course there are textbooks with selected problems in the selected order that works better than other problems or in other order. We can even explain why these problems and why this order, but I do not believe these explanations, they are explanations in hindsight, they look suspiciously as rationalizations, and they either lack the predictive power, or there is no research trying to falsify their predictions.
It maybe a philosophical take on the problem, but if it is so than pedagogy and psychology is even worse, because it should be their task to explain what happens, how it happens, why learning is indistinguishable from doing. Why these problems in this order? Shouldn't we personalize problem sets and their order? How to personalize? Pedagogy says you just try standard approach and if it doesn't work, then try something else, and keep trying different things until something works.
Pedagogy mostly like this, shamanism and black magic. There was alchemy that nowadays considered unscientific, I hope that at some point in the future modern pedagogy will be considered unscientific because it was transform into a science, like alchemy was transformed into chemistry. The science that can take any knowledge, dissect it, and produce a plan of educational activities, that will teach this knowledge to others. And not just this, but I hope the future pedagogy will be able to prove that this plan is the best possible plan, all others will be worse, will need more time, more effort or whatever. I hope that the future pedagogy will forget an idea of a "great teachers" like Feynman, because it will be able to educate anyone to be as a great teacher as Feynman was. It will be a new era, when colleges will easily educate people into Einsteins, Newtons, Archimedeses (how to properly make Archimedes plural in English?), Shakespears, you just name it.
I'll add a couple of anecdotes, to show how little we know on how our mind works.
Sometimes I cannot explain how I managed to understand some topic, and why I struggled with it initially. The last example of this is borrow-checker in Rust. It is simple and obvious now, but at first I fought battles with it and I've lost most of the battles. What have changed in my mind between then and now? I don't really know, I have one guess, but a very vague one: code patterns which I learned the hard way. It is a vague answer because I cannot enumerate the new patterns I learned and the old patterns from C that I was forced to reject. It was just like something clicked in my mind, and now my mind can write Rust code easily. Magic! (Moreover this guess is not my invention, I've read something somewhere in internet, got this idea and it kinda fits in the sense it doesn't contradict to any facts I know. I don't know what happens in my mind when I'm writing Rust and what was happening when I couldn't write Rust.)
There was another example from my undergraduate studies. I struggled with group theory for several months and I couldn't understand what exactly I didn't understand. It seemed like I understood everything and somehow at the same time I understood nothing. At some point something clicked in my mind and I really understood everything. But with this example I know exactly what was wrong: my mental geometric model of groups, factor groups and these things was constructed wrongly. As I fixed it, it started to work wonderfully. And no pedagogy could help me with this, it was a geometric model for internal use only. I'm not sure it is possible to communicate it, and I bet that no teacher would ask me to communicate it. Though if I just talked with someone knowledgeable about group theory in "informal" mode, without strict definitions and proofs, it could probably help me to find my mistake earlier (in "formal" mode I'd catch my mistakes by formal methods and patch them, no one would notice them except me). How to debug something like this in a mind of a living person?
Yes, and no.
Lets take calculus as an example and the skill to take derivatives. If a teacher show you how to take derivatives, you'll kinda get the idea but if you try it, you would face a lot of problems and you probably would need teacher's help.
To become proficient with derivatives, you need to teach yourself by taking a lot of derivatives. But... wait... you are taking derivatives, you're not doing something special that can be called "learning".
Chimps also communicate knowledge by showing the right "ritual" how to accomplish things, students reproduce ritual and somehow learn how to do it properly.
The whole education works mostly in this vein, the method is to make people to do something, and while they are doing it they'll learn how to do it. Magic.
The communication doesn't transfer all the knowledge, and it is pointless to try, because people are different and edge cases are different for different people. For example, I'm always confusing left and right, it was a pain to deal with the driving instructor who would say "turn left at the next crossing" and then I will dutifully turn right. I can deal with it most of the times, I just need to pay more attention to "left" and "right" things. But it is my specific edge case, if I teach others I do not try to communicate the necessity of paying attention for words "left" and "right", and when someone teaches me they do not communicate this to me. There are a lot of such individual edge cases and the more difficult the knowledge the more individual it becomes.
Of course there are textbooks with selected problems in the selected order that works better than other problems or in other order. We can even explain why these problems and why this order, but I do not believe these explanations, they are explanations in hindsight, they look suspiciously as rationalizations, and they either lack the predictive power, or there is no research trying to falsify their predictions.
It maybe a philosophical take on the problem, but if it is so than pedagogy and psychology is even worse, because it should be their task to explain what happens, how it happens, why learning is indistinguishable from doing. Why these problems in this order? Shouldn't we personalize problem sets and their order? How to personalize? Pedagogy says you just try standard approach and if it doesn't work, then try something else, and keep trying different things until something works.
Pedagogy mostly like this, shamanism and black magic. There was alchemy that nowadays considered unscientific, I hope that at some point in the future modern pedagogy will be considered unscientific because it was transform into a science, like alchemy was transformed into chemistry. The science that can take any knowledge, dissect it, and produce a plan of educational activities, that will teach this knowledge to others. And not just this, but I hope the future pedagogy will be able to prove that this plan is the best possible plan, all others will be worse, will need more time, more effort or whatever. I hope that the future pedagogy will forget an idea of a "great teachers" like Feynman, because it will be able to educate anyone to be as a great teacher as Feynman was. It will be a new era, when colleges will easily educate people into Einsteins, Newtons, Archimedeses (how to properly make Archimedes plural in English?), Shakespears, you just name it.
I'll add a couple of anecdotes, to show how little we know on how our mind works.
Sometimes I cannot explain how I managed to understand some topic, and why I struggled with it initially. The last example of this is borrow-checker in Rust. It is simple and obvious now, but at first I fought battles with it and I've lost most of the battles. What have changed in my mind between then and now? I don't really know, I have one guess, but a very vague one: code patterns which I learned the hard way. It is a vague answer because I cannot enumerate the new patterns I learned and the old patterns from C that I was forced to reject. It was just like something clicked in my mind, and now my mind can write Rust code easily. Magic! (Moreover this guess is not my invention, I've read something somewhere in internet, got this idea and it kinda fits in the sense it doesn't contradict to any facts I know. I don't know what happens in my mind when I'm writing Rust and what was happening when I couldn't write Rust.)
There was another example from my undergraduate studies. I struggled with group theory for several months and I couldn't understand what exactly I didn't understand. It seemed like I understood everything and somehow at the same time I understood nothing. At some point something clicked in my mind and I really understood everything. But with this example I know exactly what was wrong: my mental geometric model of groups, factor groups and these things was constructed wrongly. As I fixed it, it started to work wonderfully. And no pedagogy could help me with this, it was a geometric model for internal use only. I'm not sure it is possible to communicate it, and I bet that no teacher would ask me to communicate it. Though if I just talked with someone knowledgeable about group theory in "informal" mode, without strict definitions and proofs, it could probably help me to find my mistake earlier (in "formal" mode I'd catch my mistakes by formal methods and patch them, no one would notice them except me). How to debug something like this in a mind of a living person?
> To become proficient with derivatives, you need to teach yourself by taking a lot of derivatives. But... wait... you are taking derivatives, you're not doing something special that can be called "learning".
Actually, that is the process of learning. Doing it yourself and checking your results is part of a feedback loop essential in learning anything. That's why the answers in the back of most math textbooks are so important, if you can't tell you've got the problem wrong, then you have "taught" yourself something wrong.
Actually, that is the process of learning. Doing it yourself and checking your results is part of a feedback loop essential in learning anything. That's why the answers in the back of most math textbooks are so important, if you can't tell you've got the problem wrong, then you have "taught" yourself something wrong.
A charitable interpretation of what he said is that we need to do real world experiment instead of of relying on the opinion of supposed experts. Real data based on well made experiments trumps the opinion of any particular person.
I mean, should we be charitable when confronted with pure nonsense?
Research on education, cognition, memory, and learning is definitely much more
> Real data based on well made experiments
and not just
> relying on the opinion of supposed experts
I don't think it's acceptable to throw your hands up and say 'no one knows' when clearly people have spent entire careers thinking about this stuff and testing it in experiments, and plane-scale educational techniques. We're far closer to 'knowing' than at any point in history, and true knowing is a philosophical topic anyway without a satisfying 'answer'.
Research on education, cognition, memory, and learning is definitely much more
> Real data based on well made experiments
and not just
> relying on the opinion of supposed experts
I don't think it's acceptable to throw your hands up and say 'no one knows' when clearly people have spent entire careers thinking about this stuff and testing it in experiments, and plane-scale educational techniques. We're far closer to 'knowing' than at any point in history, and true knowing is a philosophical topic anyway without a satisfying 'answer'.
This already has been done. It was denonstraitable. Project seed and a brilliant educator, Dr Steven Giavante.
The person who explained how it worked,without ever seeing it, was Dr Fernando Flores.
The person who explained how it worked,without ever seeing it, was Dr Fernando Flores.
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.
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.
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.
What exactly do you do? Discrete approximations like trapezoidal rule or finite differences? Or do you reach for a symbolic math tool like Maxima?
Discrete approximations, I'm mostly working on embedded platforms. Though admittedly a lot of the time you don't really need to even do the math as much as you need to intuit that something correlates to the area under a piece of curve or have that you want a particular vibe to the derivative. (energy in a system/ controlling acceleration)
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.
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.
The vast majority of so called real world problems aren’t things people who use math in their jobs actually do.
I guess you are referring to the "Steve has 17.5 credits, how many pizzas can he buy?" type problems?
For me, when learning calculus it was that it seemed pointless. They were teaching me to do this mechanical task, but why? Why not another task like increasing every other number in the equation by 6? It wasn't until I learnt about velocity and acceleration that it all started to make sense. The task of differentiating/integrating seemed far less important than the understanding that functions have derivatives and anti-derivatives and what that means.
For me, when learning calculus it was that it seemed pointless. They were teaching me to do this mechanical task, but why? Why not another task like increasing every other number in the equation by 6? It wasn't until I learnt about velocity and acceleration that it all started to make sense. The task of differentiating/integrating seemed far less important than the understanding that functions have derivatives and anti-derivatives and what that means.
I'm sorry, but if you don't know how to solve x+1=7, you don't actually understand maths.
Maybe that's fine? I mean I'm sure a lot of people who fix stuff up in their homes know very little of the physics behind their hammers, but they can still use them just fine.
Maybe a good tool does not need to be understood on a deeper level to be used.
If we gave students more exposure to maths as a tool to be used, rather than arcane formulae and symbolic manipulations, they could build an intuition and appreciation for it that allow them to use it even if they cannot perform the mechanical transformations of equations on their own.
I'm not saying x+1=7 is a good example of that -- my son, who is not even literate, could answer that in the blink of an eye. But recently I needed to get the implied marginal probability out of a partition of conditional probabilities, i.e. solve for p in something like e = ap + b(1-p). Could I have done it manually? Sure. Did I plug it into a symbolic solver? Absolutely.
For me the insight lay in (a) knowing that was the shape of the equation I knew, and (b) that I needed to rearrange it to get p as a dependent variable. Actually going through the motions is not that important, in my mind.
Maybe a good tool does not need to be understood on a deeper level to be used.
If we gave students more exposure to maths as a tool to be used, rather than arcane formulae and symbolic manipulations, they could build an intuition and appreciation for it that allow them to use it even if they cannot perform the mechanical transformations of equations on their own.
I'm not saying x+1=7 is a good example of that -- my son, who is not even literate, could answer that in the blink of an eye. But recently I needed to get the implied marginal probability out of a partition of conditional probabilities, i.e. solve for p in something like e = ap + b(1-p). Could I have done it manually? Sure. Did I plug it into a symbolic solver? Absolutely.
For me the insight lay in (a) knowing that was the shape of the equation I knew, and (b) that I needed to rearrange it to get p as a dependent variable. Actually going through the motions is not that important, in my mind.
> I'm not saying x+1=7 is a good example of that.
Exactly, it's not. I'm not arguing with what you're saying, I'm arguing with what GP wrote.
Exactly, it's not. I'm not arguing with what you're saying, I'm arguing with what GP wrote.
I made it an easy equation to make a point. Pick your favorite multivariable set of equations. Depending on the situation a computer can solve it in a fraction of the time and with better accuracy than a human with pencil and paper. And it can create thousands of variations in a matter of minutes.
> I made it an easy equation to make a point
It was a bad example then. Obviously there are equations that are difficult to solve by hand and you don't need to remember all tricks in order to do successful modeling, but if you're not able to solve x+1=7, it means you don't even understand what an equation is, so you won't be able to use it for modeling anything.
It was a bad example then. Obviously there are equations that are difficult to solve by hand and you don't need to remember all tricks in order to do successful modeling, but if you're not able to solve x+1=7, it means you don't even understand what an equation is, so you won't be able to use it for modeling anything.
I agree, but how far do you take this? Do you say if you don't know the Peano axioms you don't understand maths?
No.
Why?
Because knowing how you derive arithmetic from axioms isn't necessary for mathematical modeling. Mostly only logicians, theoretical computer scientists etc . need to know about PA. Euler and Gauss didn't know about PA (it hadn't been invented yet), clearly they were still great mathematicians.
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.
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.
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.
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.
> music is not like math
That's going to age beautifully
That's going to age beautifully
Yeah, my bad, I was mostly referring to "music industry". Music is 100% like math. Music is math, and math is music.
Edit: inb4, I did read Hofstadter's Gödel, Escher, Bach.
Edit: inb4, I did read Hofstadter's Gödel, Escher, Bach.
No, music is not like math in that sense. Music is like math in the sense that carpentry is like music or math, or sports are like music or math. You can't be a spectator, you have to grind thousands upon thousands of exercises to be good in it.
Hehe, honestly I just assume everyone here has a copy of GEB (or has infused it into their being)
But yeah, I read TC's post as being less "you can't play music without formal theory" and more "taking a music appreciation class won't (solely) imbue you with the ability to play violin"
...nor would a course which provides a cursory pass over advanced concepts give you applicable (let alone employable) skills without at least a little of the manual labour needed to really feel the intuitions (?)
There probably _are_ some undiscovered clever ways to teach foundations such that you form useful or higher level synaptic links faster - but does seem you still need to then add some brute force to gain adequate fluency..
PS. I don't think that excludes the possibility of Ramanujans or Dizzy Gillespies existing - whether your hard work is unstructured or structured, top-down or bottom up, the key point is - beyond a superficial level there are eventually "no shortcuts"
But yeah, I read TC's post as being less "you can't play music without formal theory" and more "taking a music appreciation class won't (solely) imbue you with the ability to play violin"
...nor would a course which provides a cursory pass over advanced concepts give you applicable (let alone employable) skills without at least a little of the manual labour needed to really feel the intuitions (?)
There probably _are_ some undiscovered clever ways to teach foundations such that you form useful or higher level synaptic links faster - but does seem you still need to then add some brute force to gain adequate fluency..
PS. I don't think that excludes the possibility of Ramanujans or Dizzy Gillespies existing - whether your hard work is unstructured or structured, top-down or bottom up, the key point is - beyond a superficial level there are eventually "no shortcuts"
> I don't think that excludes the possibility of Ramanujans or Dizzy Gillespies existing
I wholeheartedly agree with this.
I wholeheartedly agree with this.
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?
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 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.
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.
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.
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.
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.
Rote is memorizing integrals with flash cards. Building intuition is doing exercises. Might seem pedantic, but the difference is huge pedagogically.
I have a related story that affirms your point.
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.
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.
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.
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.
> only fractions with denominators with only 2 and 5 as factors can be expressed exactly in binary without using custom datatypes and arithmetic logic.
You mean decimal, not binary. For binary, it’s only 2.
You mean decimal, not binary. For binary, it’s only 2.
This is getting offtopic, but back when I was in grad school I TAed some physics labs and the number one thing that annoyed me to no end was students didn’t understand or care about significant digits at all. They had no idea that 2.1cm they measured was in fact (2.1 \pm 0.05)cm, they just put 2.1 in the calculator and wrote down 9.7831m/s^2 the calculator spit out. I would point it out, and next week, same thing. Thankfully I never TAed an electronics lab to see what they read from digital meters. Btw that was in Princeton, the students weren’t dumb.
Weird, I was taught about the importance of significant digits ("sigdigs") in my first year of high school science class.
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.
Sure, for somebody who is already on board with inductive reasoning and abstraction the best path in might be code. Personally I suspect "hand" has merits but I won't say there aren't other approaches.
I remain unconvinced theory-first, and "theory, the machine does it" would work out ok.
Reminded of my pascal lecturer at uni in 79 "this compiled cleanly but I haven't run it" for every coding example. He really did behave like syntactic and basic semantic checks were all you needed to prove correctness. I am sure he also knew runtime is everything.
I remain unconvinced theory-first, and "theory, the machine does it" would work out ok.
Reminded of my pascal lecturer at uni in 79 "this compiled cleanly but I haven't run it" for every coding example. He really did behave like syntactic and basic semantic checks were all you needed to prove correctness. I am sure he also knew runtime is everything.
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.
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…
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.
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.
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 can't imagine doing proofs like Cauchy's integral theorem without touching a paper.
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
[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?
>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?
Dunno if I agree with this, just sounding it out seeing how it feels.
>>Can literacy be taught with hitting students with a stick when they err?
>And then the answer is yes, as this is how it is done. (19th century).
>Now, why would be want to do otherwise? Why would we ever want people to learn less?
Maybe you don't learn less by not being hit? Even if my grandparents all had vastly better handwriting than myself. Perhaps I learned some other things not being met with such hostility in my education? I don't know for sure.
At some level of complexity we learn "too hard for me to integrate by hand" and also stop learning how to perform more and more complex integrals. We stop. Where is the most productive place, to maximise useful learning, to stop?
Maybe not being hit with a stick cost me beautiful copperplate handwriting. I feel it was more productive for me to not be hit. I can see someone taking a different view on that with regards themselves. "Never did me any harm" is still a pervasive way of thinking about what others call child abuse. Not many wish they were hit more.
What do you get out of integrating by hand? At each step ratcheting up complexity from y=x, through each increasingly complex function type to where you stopped? Are there diminishing returns? Or do you get past a threshold opening up whole new areas of understanding otherwise not experienced? Something else?
I note anyone can integrate y=x and get 1/2x^2 + c so I don't think avoiding this integration by hand is the point.
>>Can literacy be taught with hitting students with a stick when they err?
>And then the answer is yes, as this is how it is done. (19th century).
>Now, why would be want to do otherwise? Why would we ever want people to learn less?
Maybe you don't learn less by not being hit? Even if my grandparents all had vastly better handwriting than myself. Perhaps I learned some other things not being met with such hostility in my education? I don't know for sure.
At some level of complexity we learn "too hard for me to integrate by hand" and also stop learning how to perform more and more complex integrals. We stop. Where is the most productive place, to maximise useful learning, to stop?
Maybe not being hit with a stick cost me beautiful copperplate handwriting. I feel it was more productive for me to not be hit. I can see someone taking a different view on that with regards themselves. "Never did me any harm" is still a pervasive way of thinking about what others call child abuse. Not many wish they were hit more.
What do you get out of integrating by hand? At each step ratcheting up complexity from y=x, through each increasingly complex function type to where you stopped? Are there diminishing returns? Or do you get past a threshold opening up whole new areas of understanding otherwise not experienced? Something else?
I note anyone can integrate y=x and get 1/2x^2 + c so I don't think avoiding this integration by hand is the point.
> At some level of complexity we learn "too hard for me to integrate by hand" and also stop learning how to perform more and more complex integrals. We stop. Where is the most productive place, to maximise useful learning, to stop?
I preface this by saying I find math very difficult. However, it is the struggle to understand in mathematics that leads me to understand the concepts. It often seems too hard, but I found that if I stuck at it I actually started to understand what was going on.
Not popular advise, I realise, but those who can stick with it seem to find maths almost fun, despite the pain.
I preface this by saying I find math very difficult. However, it is the struggle to understand in mathematics that leads me to understand the concepts. It often seems too hard, but I found that if I stuck at it I actually started to understand what was going on.
Not popular advise, I realise, but those who can stick with it seem to find maths almost fun, despite the pain.
The concept you're looking for is desirable difficulty, not too hard but not too easy either.
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.
I'm an Australian and we don't seem to have the same subdivision of Calculus into Calc 1, 2 & 3... what does each involve?
curious, what was covered in Calculus 4? Quadruple integrals? More vector valued stuff shading into proper linear algebra? Diff Eq?
Calculus 3 in the U.S. is multi-variable calculus. Sometimes and introduction to linear algebra is given in this course. Calculus 4 is differential equations. It’s almost never officially called Calculus 4. Sometimes an introduction to linear algebra is given in this course.
Okay that makes sense, that's how the classes went for me in the US, with Calc 3 being multi variable (and as you say, a brush with linear alg via vector-valued functions), and 2 sideways steps from there, one being a course on PDEs and the other on linear algebra, both called by those names rather than listed as a step in the calculus sequence itself.
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?
When I was doing an associate's in engineering, our calculus and differential equations courses were like this. We'd learn some math, do some problems by hand, then we'd have a lab component where we were introduced to either methods in a computer algebra system or some numerical methods. The problems we solved there were word problems that had the higher level physics already set up for us, so that we ended up just having to solve the calculus or differential equation portion of the problem.
The calculus books we used were not set up like this and the books that focused on learning the CAS or numerical methods weren't structured any better. I think this only worked because it was a small program aimed at technical education with a faculty that cared about developing a unified curriculum.
When I transferred to a different university to finish a degree as a stats major, all of our courses and most of the textbooks were structured in a way to use R. We did some problems on simple linear regression by hand, but very quickly it becomes impractical do to it any other way. This seemed very natural to me, but apparently it was not the typical experience of studying statistics.
Perhaps there are some calculus books out there that do a good job of both teaching calculus concepts and using CAS / numerical methods, but my narrow minded view is that calculus is a tool for physics, engineering, or other applications, and you'll be bogged down in teaching the relevant domain knowledge to get interesting examples. If you're looking for your own examples, perhaps this could be done purely through the differential calculus topics of related rates and optimization or the integral calculus topics of simple ordinary differential equations.
The calculus books we used were not set up like this and the books that focused on learning the CAS or numerical methods weren't structured any better. I think this only worked because it was a small program aimed at technical education with a faculty that cared about developing a unified curriculum.
When I transferred to a different university to finish a degree as a stats major, all of our courses and most of the textbooks were structured in a way to use R. We did some problems on simple linear regression by hand, but very quickly it becomes impractical do to it any other way. This seemed very natural to me, but apparently it was not the typical experience of studying statistics.
Perhaps there are some calculus books out there that do a good job of both teaching calculus concepts and using CAS / numerical methods, but my narrow minded view is that calculus is a tool for physics, engineering, or other applications, and you'll be bogged down in teaching the relevant domain knowledge to get interesting examples. If you're looking for your own examples, perhaps this could be done purely through the differential calculus topics of related rates and optimization or the integral calculus topics of simple ordinary differential equations.
> 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
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)
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
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
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/
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.
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.
I was exactly the same. I was good at integrating but had no idea why we were doing it. Then we did a physics-based module (called mechanics) and it was like you: a hundred light bulbs went on. Nowadays I "see" numbers in a totally different way. If I see a time axis I can't help but see the rates of change etc. I wonder if it could be taught in a more top down way: start thinking about rates of change in some discontinuous functions, like finance. How growth of assets over time relates to income. Then move to continuous with distance, speed, acceleration. Then do the thing where you differentiate symbolically.
You could teach calculus via a video game programming class, physics as well, just through the things you need to build a video game. Calculus and even physics never clicked for me until I actually used them productively.
This is also a good example of why we teach things that might seem "useless". You just don't know what someone will find useful in their life. The teachers probably didn't even think of video games having physics and needing to understand it for that. In my uni class I think I was the only one to use forces and masses (and therefore acceleration) instead of just speed. I initially used speed but then my school learnings came flooding back and I did a full physics model (it's really easy if all your objects are rigid bodies on perfect planes). I'm actually pretty sure I've used everything I've ever learnt in school at one time or another. But the the meme amongst normies is you don't use any of it...
I'd rather start the other way around: teach things steeped in the context that they are useful in. Everything is useful, so teach them usefully. Not all of us are great at storing away knowledge and accessing it much later when we find it actually useful.
But then you won't be able to teach as much. I think this whole thread boils down to pedagogy is difficult.
Interesting, I have had the complete opposite experience. I could never relate to the physics interpretation tag onto differential calculus. Only when I was introduce to the proper foundation of measure theory and topology that I really "got" multi variate calculus
Physics
Or association of lables/groups of actions with binary unit encodings[3]. aka programming languages (integration) and binary encodings (differentiation of groups of bits associate with programming language feature(s) into how relates/associates with underlying hardware aka compilation or on line by line basis via scripting).
programming language 'type' punning gives programmer way to use/access an associated encoding (typically binary) and programming language abstraction[2][3]. aka ieee floating point specification[1][2] -- programming language level -- just the typical human readable number notation vs. binary encoding is just linear group of bits, where portion is number, portion is exponent.
-----
"Modeling Practices in Calculus" approach ideas:
"string art generator" : https://github.com/halfmonty/StringArtGenerator
"Turing up the code" : https://news.ycombinator.com/item?id=30914603
assuming writing & finger presses don't count as 'by hand':
[1] : http://en.wikipedia.org/wiki/IEEE_754
[3] : https://en.wikipedia.org/wiki/Type_punning
programming language 'type' punning gives programmer way to use/access an associated encoding (typically binary) and programming language abstraction[2][3]. aka ieee floating point specification[1][2] -- programming language level -- just the typical human readable number notation vs. binary encoding is just linear group of bits, where portion is number, portion is exponent.
-----
"Modeling Practices in Calculus" approach ideas:
"string art generator" : https://github.com/halfmonty/StringArtGenerator
"Turing up the code" : https://news.ycombinator.com/item?id=30914603
assuming writing & finger presses don't count as 'by hand':
running code on paper using just the phone camera : https://www.youtube.com/watch?v=rb8kE-e2FXs
"stand-up comedy routine about spreadsheets " : https://www.youtube.com/watch?v=UBX2QQHlQ_I[1] : http://en.wikipedia.org/wiki/IEEE_754
vs. church encoding : https://en.wikipedia.org/wiki/Church_encoding
[3] : https://en.wikipedia.org/wiki/Type_punning
: punning with character : gawk chapter "11.3.11 And Now for Something Completely Different" program
: nil-punning : https://ericnormand.me/article/nil-punning
: https://en.wikipedia.org/wiki/Union_typePhysical experiments.
Yeah, would have been more useful to spend relatively more time on numerical approximations.