The case for ending calculus requirements for science majors(quillette.com)
quillette.com
The case for ending calculus requirements for science majors
https://quillette.com/2022/09/15/math-for-future-darwins/
141 comments
You can't hope to understand first year physics without calculus. To be a reasonable civil engineer, don't you need to understand structural engineering, including dynamics, which requires differential equations?
On the other hand, what do I know? The civil engineering plans for our place had glaring errors that the construction workers caught after doing half the earthmoving.
Even after it was fixed, the results lasted less than one rainstorm. When we went back and checked the contracts we signed, they made it clear they assume zero liability for incompetence. Hiring a lawyer is more expensive than renting an excavator, and fixing it ourselves.
Anyway, I'm not impressed with the corner of that field I interacted with, and would prefer they tighten the educational standards.
On the other hand, what do I know? The civil engineering plans for our place had glaring errors that the construction workers caught after doing half the earthmoving.
Even after it was fixed, the results lasted less than one rainstorm. When we went back and checked the contracts we signed, they made it clear they assume zero liability for incompetence. Hiring a lawyer is more expensive than renting an excavator, and fixing it ourselves.
Anyway, I'm not impressed with the corner of that field I interacted with, and would prefer they tighten the educational standards.
> You can't hope to understand first year physics without calculus. To be a reasonable civil engineer, don't you need to understand structural engineering, including dynamics, which requires differential equations?
Yes, but memorizing a bag of tricks for hand solving integrals and ODEs that you will never see in symbolic form without more than a passing exposure to the underlying concepts doesn't help anyone understand anything.
Learning mathematical thinking and getting into the groove of that style of problem solving is equally as important (although much less concrete in its applicability for an engineer). But conflating rote memorization of the 'trick' to integrate cos(x)sin(x) with either helps nobody.
A first year physics course is a better venue for teaching actual understanding of what derivative means than a purely symbolic presentation of the chain and product rules followed by presentation of a 'quotient rule' as a separate thing to memorize rather than a trivial combination of the other two. As is implementing some simulations of simple dynamical systems. As is intro to epidemiology or ecology. As is playing cookie clicker.
Get students to visualise things, and actually experience them rather than memorize symbols. Then formalize themand introduce the language so they can express what they've learnt
Yes, but memorizing a bag of tricks for hand solving integrals and ODEs that you will never see in symbolic form without more than a passing exposure to the underlying concepts doesn't help anyone understand anything.
Learning mathematical thinking and getting into the groove of that style of problem solving is equally as important (although much less concrete in its applicability for an engineer). But conflating rote memorization of the 'trick' to integrate cos(x)sin(x) with either helps nobody.
A first year physics course is a better venue for teaching actual understanding of what derivative means than a purely symbolic presentation of the chain and product rules followed by presentation of a 'quotient rule' as a separate thing to memorize rather than a trivial combination of the other two. As is implementing some simulations of simple dynamical systems. As is intro to epidemiology or ecology. As is playing cookie clicker.
Get students to visualise things, and actually experience them rather than memorize symbols. Then formalize themand introduce the language so they can express what they've learnt
The connection between calculus and the quality of subcontractor work is dubious.
There's a big difference between applied calculus and theoretical calculus. Colleges teach theoretical and then wonder why students struggle to apply it.
The same argument could be made that what you interacted with was someone designing because their head is lost in the abstract porcelain tower of calculus proofs instead of the practical properties of the materials they were dealing with
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What's needed for that is the U.S. Navy calculus book from WWII. The Navy needed people to understand calculus because navigation and fire control systems are devices that do integration, analytic geometry, and trigonometry. Since they still do that, with more software and fewer gears, there's a current version of the book.[1]
From the book: "The proofs of theorems shown in this section will be omitted in the interest of brevity."
You need calculus for anything that involves calculating the physics of the real world. But not as a proof system.
The same is true of geometry as taught in schools. There's an emphasis on proofs because those were a big deal when geometry was being invented. But if you need to solve geometry problems, you seldom if ever do proofs.
I've worked on proof of correctness systems, and there are times when you need heavy formalism. But it's a means to an end, not an end in itself. The end is programs that don't fail, not more proofs. I used to argue this with mathematicians when I was working on that sort of thing. There's still a tendency in software proof of correctness to fall in love with the formalism as an end in itself.
[1] https://archive.org/details/US_Navy_Training_Course_-_Pre-ca...
From the book: "The proofs of theorems shown in this section will be omitted in the interest of brevity."
You need calculus for anything that involves calculating the physics of the real world. But not as a proof system.
The same is true of geometry as taught in schools. There's an emphasis on proofs because those were a big deal when geometry was being invented. But if you need to solve geometry problems, you seldom if ever do proofs.
I've worked on proof of correctness systems, and there are times when you need heavy formalism. But it's a means to an end, not an end in itself. The end is programs that don't fail, not more proofs. I used to argue this with mathematicians when I was working on that sort of thing. There's still a tendency in software proof of correctness to fall in love with the formalism as an end in itself.
[1] https://archive.org/details/US_Navy_Training_Course_-_Pre-ca...
> In twenty years of engineering practice I have never once had to take a difficult limit, and can probably count on one hand the number of times I've had to take a limit of any kind.
In principle, I agree that calculus doesn't need to be mandatory. However this apparently respectable counterargument is misframing the situation. A bit like me saying that a given soldier never placed top 3 in a race, or indeed had to run anywhere in a combat situation, and therefore efforts at army fitness training were useless.
Everyone knows that nearly no engineers use calculus in their day job. Therefore, the purpose and measure of success isn't whether an engineer uses calculus in their day job. It is a great signal of technical problem solving ability, it is really powerful having a large pool of people who are trained in working with rates, areas and margins.
Plus, don't underestimate the impact of forcing people to acknowledge that there is exactly one right answer and that it can be proven. There are people out there who genuinely believe that everything is up for negotiation to a limit that just can't fly in a math course and something needs to screen them out before they get anywhere near anything involving high pressures. People who don't have engineers for parents need to be inducted into the culture from a couple of angles.
In principle, I agree that calculus doesn't need to be mandatory. However this apparently respectable counterargument is misframing the situation. A bit like me saying that a given soldier never placed top 3 in a race, or indeed had to run anywhere in a combat situation, and therefore efforts at army fitness training were useless.
Everyone knows that nearly no engineers use calculus in their day job. Therefore, the purpose and measure of success isn't whether an engineer uses calculus in their day job. It is a great signal of technical problem solving ability, it is really powerful having a large pool of people who are trained in working with rates, areas and margins.
Plus, don't underestimate the impact of forcing people to acknowledge that there is exactly one right answer and that it can be proven. There are people out there who genuinely believe that everything is up for negotiation to a limit that just can't fly in a math course and something needs to screen them out before they get anywhere near anything involving high pressures. People who don't have engineers for parents need to be inducted into the culture from a couple of angles.
"Everyone knows that nearly no engineers use calculus in their day job. Therefore, the purpose and measure of success isn't whether an engineer uses calculus in their day job. It is a great signal of technical problem solving ability, it is really powerful having a large pool of people who are trained in working with rates, areas and margins."
I'm not saying that calculus is useless in an engineer's education - far from it.
I'm arguing that knowing the algebraic tricks required to take the limit of (x^2)sin(2x)/sqrt(x) as x->0 is useless. It's just a display of the mastery of very intricate algebra, exactly the same as all the other displays of mastery of intricate algebra she has had to master and will continue to be required to master.
What she actually needs is a solid understanding of the physical meaning of derivatives and integrals, how we arrive at them via numerical approximations, the fact that the velocity of a rigid body can be described as the integral over the forces that body is experiencing, etc.
That connection is literally never made in calculus.
The connection is of course made in the parallel calculus-based physics class she is taking. Which has its own problems, but is far superior as an introduction to calculus than the actual calculus class is.
I'm not saying that calculus is useless in an engineer's education - far from it.
I'm arguing that knowing the algebraic tricks required to take the limit of (x^2)sin(2x)/sqrt(x) as x->0 is useless. It's just a display of the mastery of very intricate algebra, exactly the same as all the other displays of mastery of intricate algebra she has had to master and will continue to be required to master.
What she actually needs is a solid understanding of the physical meaning of derivatives and integrals, how we arrive at them via numerical approximations, the fact that the velocity of a rigid body can be described as the integral over the forces that body is experiencing, etc.
That connection is literally never made in calculus.
The connection is of course made in the parallel calculus-based physics class she is taking. Which has its own problems, but is far superior as an introduction to calculus than the actual calculus class is.
It sounds like your daughter might enjoy reading my MATH+PHYS book. It covers mechanics and calculus in an integrated manner. I don't do all the proofs, so might not be a good fit for a proofs-based, course, but lots of illustrations and intuition.
Preview here: https://minireference.com/static/excerpts/noBSmathphys_v5_pr...
Concept map here: https://minireference.com/static/conceptmaps/math_and_physic...
Also be sure to check out the SymPy, in case you don't know already: https://minireference.com/static/tutorials/sympy_tutorial.pd... very good for checking answers...
Preview here: https://minireference.com/static/excerpts/noBSmathphys_v5_pr...
Concept map here: https://minireference.com/static/conceptmaps/math_and_physic...
Also be sure to check out the SymPy, in case you don't know already: https://minireference.com/static/tutorials/sympy_tutorial.pd... very good for checking answers...
Looks great. I added it to my cart for a future purchase and recommended the linear algebra one to my friend who just started a new degree.
> The connection is of course made in the parallel calculus-based physics class she is taking...
The physicists as always (a) are pragmatically out-front but (b) don't worry about whether their tools are internally sound so long as they make useful predictions.
Sounds about right. It is important for the aforementioned daughter to know this cultural difference between physicists and mathematicians if she is to be a good engineer. I believe this can only be learned firsthand if it is to be appreciated.
The physicists as always (a) are pragmatically out-front but (b) don't worry about whether their tools are internally sound so long as they make useful predictions.
Sounds about right. It is important for the aforementioned daughter to know this cultural difference between physicists and mathematicians if she is to be a good engineer. I believe this can only be learned firsthand if it is to be appreciated.
You don't need tricks to solve the limit of (x^2)sin(2x)/sqrt(x) as x->0, only to know equivalent infinitesimals, so that sin(2x) can be replaced by 2x in this limit and 2x^3/sqrt(x) can be simplified to
2x^(3-1/2) so the limit is zero. Moreover, you should know that x^2 and sin(2x) are really smaller compared to sqrt(x) so that the limit is zero. Knowing how to compare two infinitesimals can help you to replace a difficult process for an easier one, just like in differential equations sin(x) is replaced by x for small vibrations. That kind of knowledge allows you obtain an approximate solution in many real processes. So I don't consider that knowledge a bag of tricks. Furthermore, replacing sin(2x) for 2x is just the process of taking the linear approximation (the tangent line near x=0 for y=sin(2x)), so this is just an example of the power of calculus to replace a difficult question with an easier one. Also you learn to explore the world near a point (like a miscroscope) by the process of taking the linear approximation or taylor expansion of a function near a point, then you iterate this procedure to nearby points just like what happens when you solve a differential equation by Euler method. So I should say that there is a framework to solve problems via approximate solutions and computing limits is just a good first step to enter the realm of Calculus.
If your daughter is not going to learn that 1) x^2 is much smaller than sqrt(x) for x small. Or to know how 2) how to computer the tangent line of y=sin(2x), then I think she will not be able to appreciate how to approximate the solution for a problem and bound the error of that solution, and that are very useful skills. If the learns those two facts I think she will be able to solve that limit and appreciate how rules compose to apply your knowledge more broadly.
I wouldn't describe it as a useless trick to know that sin(2x) ≈ 2x for small x. I'd describe this as a basic part of understanding what it means to say sin'(0) = 1.
> Plus, don't underestimate the impact of forcing people to acknowledge that there is exactly one right answer and that it can be proven.
There's plenty of classes in a civil engineering degree to instill that particular lesson. Why does it have to be calculus? It's just legacy - it used to be that limits and integrals could only be computed by hand, so any aspiring engineer would have to practice that skill a lot. Now computers can do the tricky algebra on their own (and I'm not even mentioning numerical approaches, which are more broadly useful but can be very tricky in their own way), yet the calculus courses have barely changed.
There's plenty of classes in a civil engineering degree to instill that particular lesson. Why does it have to be calculus? It's just legacy - it used to be that limits and integrals could only be computed by hand, so any aspiring engineer would have to practice that skill a lot. Now computers can do the tricky algebra on their own (and I'm not even mentioning numerical approaches, which are more broadly useful but can be very tricky in their own way), yet the calculus courses have barely changed.
> Why does it have to be calculus?
It doesn't, but it has to be something. So the proposal isn't 'get rid of calculus' as much as 'replace it with X which will be better at achieving the outcomes we think are important'. Like the original article did with statistics, which is a decent proposal.
That being said if my civil engineer hadn't ever studied calculus I, personally, would prefer to find a different civil engineer. It makes sense to give the will-it-stay-up? job to someone unusually well educated and if they haven't studied those sort of mathematical basics it is pretty suspicious.
It doesn't, but it has to be something. So the proposal isn't 'get rid of calculus' as much as 'replace it with X which will be better at achieving the outcomes we think are important'. Like the original article did with statistics, which is a decent proposal.
That being said if my civil engineer hadn't ever studied calculus I, personally, would prefer to find a different civil engineer. It makes sense to give the will-it-stay-up? job to someone unusually well educated and if they haven't studied those sort of mathematical basics it is pretty suspicious.
The mathematical foundations of "will it stay up" questions are a lot more challenging than what you'll find in any calculus course. I'm all for teaching less calculus, and more theory of structural engineering. (Setting statistics aside, which does require calculus but not of the pointless-algebra-tricks kind.)
> statistics aside, which does require calculus but not of the pointless-algebra-tricks kind
https://en.wikipedia.org/wiki/Gaussian_integral
https://en.wikipedia.org/wiki/Gaussian_integral
I both agree and disagree.
A university degree is basically a sort of intellectual cave diving exercise. You need to be comfortable exploring in the dark, and sometimes you need to hold your breath under water until you emerge on the other side of the tunnel.
There may be no purpose at the end of a given cave. In fact most university courses are like this, you won't use the content later in life.
Others can be abbreviated, but on purpose are not, precisely because we want to practice hard things. For instance why do we need to do arithmetic beyond the point where we understand the uses and limitations of a calculator? Similarly for calculus, why do more trig substitutions beyond where you understand how a symbolic calculator works?
In the end the purpose isn't to have the knowledge you need, but to be able to find it. For instance, I didn't know anything about option math when I finished, so I read some books on my first job learned it that way.
Having said that, I think the attitude of "why on earth do I need this" is unhealthy. It makes it hard to concentrate when you know that you can forget about it the moment the exam ends. Perhaps it's better to develop an appreciation that calculus is at the bottom of a lot of things, and might be a starting point for unknown adventures later on.
A university degree is basically a sort of intellectual cave diving exercise. You need to be comfortable exploring in the dark, and sometimes you need to hold your breath under water until you emerge on the other side of the tunnel.
There may be no purpose at the end of a given cave. In fact most university courses are like this, you won't use the content later in life.
Others can be abbreviated, but on purpose are not, precisely because we want to practice hard things. For instance why do we need to do arithmetic beyond the point where we understand the uses and limitations of a calculator? Similarly for calculus, why do more trig substitutions beyond where you understand how a symbolic calculator works?
In the end the purpose isn't to have the knowledge you need, but to be able to find it. For instance, I didn't know anything about option math when I finished, so I read some books on my first job learned it that way.
Having said that, I think the attitude of "why on earth do I need this" is unhealthy. It makes it hard to concentrate when you know that you can forget about it the moment the exam ends. Perhaps it's better to develop an appreciation that calculus is at the bottom of a lot of things, and might be a starting point for unknown adventures later on.
I sure hope university isn't like cave diving—it is one of the riskiest things a human can do, and (unlike either caving or diving in isolation) if a person chooses to do it I will question their judgment.
>unlike either caving or diving in isolation
Caving, the activity where people crawl in spaces barely fitting and if take a wrong turn there's a high chance of getting stuck and dying even when it's found they're missing and try to pull them out? Wouldn't really say it's much better. Though cave diving is that plus carrying a big oxygen tank.
Caving, the activity where people crawl in spaces barely fitting and if take a wrong turn there's a high chance of getting stuck and dying even when it's found they're missing and try to pull them out? Wouldn't really say it's much better. Though cave diving is that plus carrying a big oxygen tank.
There's definitely some kinds of caving that are riskier than others.
Computer science education contains lots of quite tricky algorithms, datastructures, and runtime proofs. So far in my career I haven't had to use anything more complicated than a balanced tree and a hashmap and I certainly didn't have to open my copy of Concrete Mathematics and find bounds for any recurrence equations. On the other hand I do have to be familiar with various idiosyncrasies of operating systems, networks, and an awful amount of tooling, which I managed to almost completely avoid during my time in university.
However, I wouldn't want to change the university curriculum. I believe universities should not be seen as preparation for jobs, but as preparations for research. It is quite unfortunate that institutions that just teach the part of the field that is really needed in the job have such low reputations. Two years at a trade school would be more than sufficient to learn everything you _really_ need to know as a programmer in all but the most specialized jobs.
However, I wouldn't want to change the university curriculum. I believe universities should not be seen as preparation for jobs, but as preparations for research. It is quite unfortunate that institutions that just teach the part of the field that is really needed in the job have such low reputations. Two years at a trade school would be more than sufficient to learn everything you _really_ need to know as a programmer in all but the most specialized jobs.
I think a rigid foundation is a good thing for reasons that might not be obvious. I have never had to use much calculus for any job I had over the past 30 years and the rigorous Dijkstra formal methods that were punched into me at university are not often practical, however, they do help me to understand things on a deeper level and allow me to document things in a clearer fashion. I recently needed proofs to show why one of our solutions was more secure than any existing solution in payments and the auditors were delighted it was written on 1 A4 pdf in terse formulas instead of the gibberish they usually get (they said they often get 80 page presentations (yes, actual powerpoint) they have to distill into something that even makes logical sense let alone passes, but that is what they get paid for). It is good for the mind imho.
It’s a very complicated IQ test.
Most people don’t need weed out level college math. But on the whole, those that pass the bar are more intelligent than those that don’t.
Maybe more useful to just take IQ tests to prove credentials than jump through these kind of hoops?
Basically none of the “hard” stuff I learned in computer science is part of my career (3rd year physics, linear algebra, etc). But I got to jump those hoops!
Most people don’t need weed out level college math. But on the whole, those that pass the bar are more intelligent than those that don’t.
Maybe more useful to just take IQ tests to prove credentials than jump through these kind of hoops?
Basically none of the “hard” stuff I learned in computer science is part of my career (3rd year physics, linear algebra, etc). But I got to jump those hoops!
I went to a math oriented boarding school. When one started school, a test was administered to determine one’s initial math course. Students began their studies in a range of classes from Geometry all the way to MVC/Analysis. Having seen the standardized test scores and GPAs of many thousands of students at this school, and knowing many of these individuals personally, there is, at best, only a weak relationship between their terminal math course and their intelligence. We had kids who started in geometry and received perfect GPAs and SATs and were brilliant. We also had kids who started in Calc 2 or MVC and barely graduated.
Every college class is an IQ test, that's pretty much inherent in any sort of post-K12 education. Calculus is just really badly taught and has a misplaced focus on material (like computing closed forms of tricky integrals) that's less and less relevant in the computer age.
Just because you have a fast computer doesn't mean you don't also need good well tuned software. For computers of the squishy human variety we install this software via learning and practicing. Doing college level maths refines thinking patterns and provides new categories of thinking. Even if you never take a limit or solve a system of equations directly, you know the concepts of them and they can deepen your understanding of many problems.
College math professors may be the original "woke" crowd, trying to slip formal methods into their math classes in a desperate attempt to indoctrinate yet one more mathematician. My college experience:
Freshman Fall semester: abstract algebra was the first math course (yes, the full blown groups, rings, lattices, fields, etc.). There was no opt-out (this was a small college) and all science students took the same course. We were mystified - why study this? The three declared math majors were delighted.
Freshman Spring semester: we were finally offered calculus. In the classroom physics, chemistry and biology professors were stunned to find students had no knowledge of calculus! When my physics prof first drew an integral on the blackboard, the class unanimously protested: "We haven't been taught that yet!" [To his credit and our undying relief, he taught us calculus for the next 3 classes and did a beautiful job of it. Turns out he had once been a math professor!]
Faculty meetings ensued with much shouting. The verdict: an unspoken assumption among faculty was that the first freshman math class would be calculus. But the university had, in the summer hiatus, hired two new mathematics professors, both proponents of a particularly rigorous school of thought. One of them, the new mathematics department chairman, had decided without consultation to change the math curriculum. Ergo abstract algebra as an intro course.
The mathematics department chairman was fired and a new chairman installed ASAP. Normality was restored.
Freshman Fall semester: abstract algebra was the first math course (yes, the full blown groups, rings, lattices, fields, etc.). There was no opt-out (this was a small college) and all science students took the same course. We were mystified - why study this? The three declared math majors were delighted.
Freshman Spring semester: we were finally offered calculus. In the classroom physics, chemistry and biology professors were stunned to find students had no knowledge of calculus! When my physics prof first drew an integral on the blackboard, the class unanimously protested: "We haven't been taught that yet!" [To his credit and our undying relief, he taught us calculus for the next 3 classes and did a beautiful job of it. Turns out he had once been a math professor!]
Faculty meetings ensued with much shouting. The verdict: an unspoken assumption among faculty was that the first freshman math class would be calculus. But the university had, in the summer hiatus, hired two new mathematics professors, both proponents of a particularly rigorous school of thought. One of them, the new mathematics department chairman, had decided without consultation to change the math curriculum. Ergo abstract algebra as an intro course.
The mathematics department chairman was fired and a new chairman installed ASAP. Normality was restored.
My college used calc 1/2 as “weed out” classes. The best part was the professors were married, the same calc book was used for calc 1/2/3, and they revved the book every semester, basically rearranging the problem sets so you had to buy a ~200 book for 3 semesters, or make friends with someone who would let you photocopy the problem sets.
My entire faith in academia was broken as a result.
My entire faith in academia was broken as a result.
Do you mean that you had to buy the same book 3 times?
(or otherwise copy)
That's astonishingly bad.
(or otherwise copy)
That's astonishingly bad.
Married?
I think they meant it literally, to further reinforce the fact that they were colluding to extract profit out of their students via yearly textbook revisions.
You would probably be surprised at how little money a textbook author makes per copy. Think $2.00 on a $200 book.
It can add up to a significant sum if you're lucky enough to have your textbook adopted by many large universities nationwide, but I'm dubious that anyone is getting rich from a textbook used at only one university, even a large one.
It can add up to a significant sum if you're lucky enough to have your textbook adopted by many large universities nationwide, but I'm dubious that anyone is getting rich from a textbook used at only one university, even a large one.
The confusion arises from “married” not necessarily meaning “married to each other”, in this context, especially since the number of professors is unknown. I did not even think about it as a possibility until I read your comment.
Aye I did mean literally.
I may be biased, but I cannot see how one can fully comprehend concepts in Dynamics, Solid Mechanics, and other core courses in an undergrad civil/mechanical engineering curriculum without ~six credits in calculus. Yes, you would never face those tricky limits they expect you to solve in high school, but I tend to see those as games that shape the students' frame of mind. Once you pass core engineering courses, you hardly ever have to calculate the forces, torques, and moments by analytical integration either, but having gone through the concepts is what prepares you for their application in practice.
The article advocates for teaching statistics in place of calculus. Surely any stat course has to go further than teaching the sample mean and standard deviation. I cannot see how the students would calculate CDF out of PDF without background in calculus.
The article advocates for teaching statistics in place of calculus. Surely any stat course has to go further than teaching the sample mean and standard deviation. I cannot see how the students would calculate CDF out of PDF without background in calculus.
Adding my own personal story as an anecdote -
I never completed my bachelor's (double major in physics and CS). I did okay on my physics classes, great on my CS classes (often in the top 2%), yet failed badly on a couple of advanced calculus classes.
It got to the point where I was done with most of the requirement and elective classes on the CS and physics side, was already taking graduate courses in CS and lecturing in a course I helped create in CS - yet failed out of my bachelor's.
In the end, the industry offers for me to go all into tech got too good, and I managed to advance to middle management with no need for that degree, so I never went back to finish it. I still think academics are important and I am thankful for the stuff I learned there, but I think the fact calculus and "hard maths" are such a strong focal point early on serves no purpose other than serving as an arbitrary filter for most people.
In the end, the industry offers for me to go all into tech got too good, and I managed to advance to middle management with no need for that degree, so I never went back to finish it. I still think academics are important and I am thankful for the stuff I learned there, but I think the fact calculus and "hard maths" are such a strong focal point early on serves no purpose other than serving as an arbitrary filter for most people.
I was a civil engineer for a few years and left to do something that interested me more. I was glad I had the broad university treatment rather than a narrow practical one, it meant I had the knowledge to go on and do what I wanted when I figured out what that was.
You may find the following books useful for your Daughter;
1) Who Is Fourier?: A Mathematical Adventure - A very nice intuitive introduction to Calculus and more.
2) Higher Math for Beginners (Mostly Physicists and Engineers) by Zeldovich et.al. - https://archive.org/details/HigherMathForBeginners
3) Elements Of Applied Mathematics by Zeldovich et.al. - https://archive.org/details/ZeldovichMyskisElementsOfApplied...
1) Who Is Fourier?: A Mathematical Adventure - A very nice intuitive introduction to Calculus and more.
2) Higher Math for Beginners (Mostly Physicists and Engineers) by Zeldovich et.al. - https://archive.org/details/HigherMathForBeginners
3) Elements Of Applied Mathematics by Zeldovich et.al. - https://archive.org/details/ZeldovichMyskisElementsOfApplied...
So you need better teachers.
I can show you statistics no human could understand taught to… Business Admin majors.
I can show you statistics no human could understand taught to… Business Admin majors.
Perhaps you don't need to solve any limit because computers can do that for you, but when you try to solve a problem in your mind if you know limits you don't need a computer all the time, you just need to know that some approximations give you an acceptable solution to certain problems and in many cases the approximation is based on a limit. For example in many problems one can foresee the capacity of a process and what kind of error are acceptable is that setting, so you are able to obtain local solutions or assymptotic expansions that save you job.
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succeeding in college calculus is highly dependent on a good precalculus foundation. (it's the actually hard part). consider picking up a good precalculus textbook. (if the author or publisher of the calculus text has one, definitely get that one.)
avoid test prep or quick refresher books. you want the longform text that has examples of all the tricks needed.
avoid test prep or quick refresher books. you want the longform text that has examples of all the tricks needed.
> Every proof has a trick. If you can see the trick, you can generally figure out how to do the proof.
Not a good way of teaching a mathematical mindset... How are you supposed to know if your "trick" actually goes through? You'd probably need a full course in real analysis to actually learn the fundamentals.
Not a good way of teaching a mathematical mindset... How are you supposed to know if your "trick" actually goes through? You'd probably need a full course in real analysis to actually learn the fundamentals.
> I hold a PhD in aerospace engineering with a minor in applied mathematics.
...
> And as part of teaching it, they are attempting to instill a "mathematical mindset". ...
> I told her that what she actually needed to understand about calculus was how to use it to solve engineering problems, not how to manipulate polynomial fractions, chain trig identities, and do substitutions of variables.
I am really surprised that a PhD in aerospace engineering does not see value in "substitution of variables" and also mocking "mathematical mindset". I feel your emotional despair but your criticism seems somewhat misplaced.
In USA there is widely held resentment and frustration against STEM in general and mathematics in particular. Math is not an easy subject, it requires concentration and logical thinking. Teaching math is hard. Just few points I learned teaching college level math and also teaching math to my kids:
1. Math thinking should be cultivated in children from early age (starting at 6 or 7 years old) and continue uninterrupted throughout high-school. There is nothing wrong with gently teaching kids basic algebra and geometry concepts.
2. Stop using useless examples of difficult and boring math (I am looking at you trigonometry). Math exercises should be practical and fun, memorizing math relations is not math. AP Calculus A,B,C are not for everyone and the text-books are awful.
3. Proofs are central to math and should not be avoided. Proofs teach kids logical thinking and this ability is useful in life even if they never touch math again in their life. And, depending on the task, proofs can be very simple.
Unfortunately, the way math in taught in America on all levels from school to college is a travesty of education. I have been teaching math to my kids myself (including college level calculus and linear algebra) to spare them the aggravation of main-stream math education.
DISCLAIMER: Ph.D in Physics and Applied Mathematics. Was teaching math at two American colleges.
> And as part of teaching it, they are attempting to instill a "mathematical mindset". ...
> I told her that what she actually needed to understand about calculus was how to use it to solve engineering problems, not how to manipulate polynomial fractions, chain trig identities, and do substitutions of variables.
I am really surprised that a PhD in aerospace engineering does not see value in "substitution of variables" and also mocking "mathematical mindset". I feel your emotional despair but your criticism seems somewhat misplaced.
In USA there is widely held resentment and frustration against STEM in general and mathematics in particular. Math is not an easy subject, it requires concentration and logical thinking. Teaching math is hard. Just few points I learned teaching college level math and also teaching math to my kids:
1. Math thinking should be cultivated in children from early age (starting at 6 or 7 years old) and continue uninterrupted throughout high-school. There is nothing wrong with gently teaching kids basic algebra and geometry concepts.
2. Stop using useless examples of difficult and boring math (I am looking at you trigonometry). Math exercises should be practical and fun, memorizing math relations is not math. AP Calculus A,B,C are not for everyone and the text-books are awful.
3. Proofs are central to math and should not be avoided. Proofs teach kids logical thinking and this ability is useful in life even if they never touch math again in their life. And, depending on the task, proofs can be very simple.
Unfortunately, the way math in taught in America on all levels from school to college is a travesty of education. I have been teaching math to my kids myself (including college level calculus and linear algebra) to spare them the aggravation of main-stream math education.
DISCLAIMER: Ph.D in Physics and Applied Mathematics. Was teaching math at two American colleges.
Another comment on this thread talked about the difference between a mathematical mindset and an engineering mindset, and I think that's exactly right.
As a researcher and controls engineer, I certainly appreciate the necessity of rigorously proving the behavior of new algorithms. On occasion I develop those proofs myself. I'm not very good at it, and I appreciate those who are.
But.
Compared to the amount of time I spent in grad school honing my ability to generate proofs (courses in both real and numerical analysis, on top of the very applied math-flavored controls courses), the amount of time I spend either reading or producing stability proofs is minuscule. On the other hand, I spend an enormous amount of time solving constrained optimization problems. Both require calculus. But guess which topic I never got formally taught in grad school?
The point being that while I appreciate proofs and mathematical rigor and clever algebraic manipulations, including changes of variables, in practice not only do I virtually never use those things, but the mindset I need when approaching an engineering problem is really quite distinct from the mindset I had to adopt when working my way through real analysis.
I'll go one more step and say that in a lot of cases the two mindsets can actually be mutually exclusive. My advisor was brought up in a nonlinear controls lab where rigor was everything. Today, controls is being consumed by deep reinforcement learning, which is driven primarily by experiment. We get an idea for a new approach that might work, we try it out and see whether it works. We then go back and try to understand mathematically why it works. My advisor is unable to work that way; he must have things proven first, and only then will he even look at experimental results.
He's been completely left behind by the field. He's unable to brainstorm about new things to try, because he can't accept that experimentalists can move faster than theoreticians. We've had vigorous arguments about this. I think that if enough experimentalists can show repeatable results, that's a good indication that mathematically there's something there worth investigating. He just thinks the experimentalists are all building a house on sand.
As a researcher and controls engineer, I certainly appreciate the necessity of rigorously proving the behavior of new algorithms. On occasion I develop those proofs myself. I'm not very good at it, and I appreciate those who are.
But.
Compared to the amount of time I spent in grad school honing my ability to generate proofs (courses in both real and numerical analysis, on top of the very applied math-flavored controls courses), the amount of time I spend either reading or producing stability proofs is minuscule. On the other hand, I spend an enormous amount of time solving constrained optimization problems. Both require calculus. But guess which topic I never got formally taught in grad school?
The point being that while I appreciate proofs and mathematical rigor and clever algebraic manipulations, including changes of variables, in practice not only do I virtually never use those things, but the mindset I need when approaching an engineering problem is really quite distinct from the mindset I had to adopt when working my way through real analysis.
I'll go one more step and say that in a lot of cases the two mindsets can actually be mutually exclusive. My advisor was brought up in a nonlinear controls lab where rigor was everything. Today, controls is being consumed by deep reinforcement learning, which is driven primarily by experiment. We get an idea for a new approach that might work, we try it out and see whether it works. We then go back and try to understand mathematically why it works. My advisor is unable to work that way; he must have things proven first, and only then will he even look at experimental results.
He's been completely left behind by the field. He's unable to brainstorm about new things to try, because he can't accept that experimentalists can move faster than theoreticians. We've had vigorous arguments about this. I think that if enough experimentalists can show repeatable results, that's a good indication that mathematically there's something there worth investigating. He just thinks the experimentalists are all building a house on sand.
The top priority in teaching students is to produce future professors for the department, or similar departments at other schools. But that is a vanishingly small percentage of the students being taught.
So when the math dept teaches engineering students, there is a mismatch. None of these students have any interest in being a math professor. They are not interested in a mathematical mindset because it will not help them be an engineer.
They need an engineering mindset, which is different than a math mindset.
A friend of mine went to an engineering school in Sweden, and the math was taught by engineering professors rather than math professors. He ended up with a better understanding of math than I (or any of my classmates) did getting math from math professors. I went to a top tier school in the US.
So when the math dept teaches engineering students, there is a mismatch. None of these students have any interest in being a math professor. They are not interested in a mathematical mindset because it will not help them be an engineer.
They need an engineering mindset, which is different than a math mindset.
A friend of mine went to an engineering school in Sweden, and the math was taught by engineering professors rather than math professors. He ended up with a better understanding of math than I (or any of my classmates) did getting math from math professors. I went to a top tier school in the US.
Something used in only one proof is a trick. If it is used in two proofs it becomes a lemma.
Calculus is used as a filter. It is not taught because you will need it in the future, it is used to detect second rate minds. It is debatable whether this is an appropriate use .
Calculus is used as a filter. It is not taught because you will need it in the future, it is used to detect second rate minds. It is debatable whether this is an appropriate use .
This is exactly why I gave up on higher level calculus. It was full of highly illogical "tricks" in a way diametrically opposite to go how physics is rarely illogical. That plus the fact that even with fairly low power computers available then, it was apparent numerical methods would very soon become good enough.
Fully agree mathematics for non math majors the should not be taught by mathematics departments in college. Today in a very similar way I find data science is not effective when taught by college professors and their TAs and instead we get our own courses created by people with practical experience.
Fully agree mathematics for non math majors the should not be taught by mathematics departments in college. Today in a very similar way I find data science is not effective when taught by college professors and their TAs and instead we get our own courses created by people with practical experience.
> It is being taught by mathematicians. And as part of teaching it, they are attempting to instill a "mathematical mindset".
I think this is a problem not just with calculus, but really with most math instruction. Math is taught by people who are already in love with the special insider world of math. To students who don’t share that love, it’s often unappealing at best - and exclusionary at worst. It weeds out people who don’t need to be weeded.
When math is taught in a way that develops practical intuition, it is so much more useful and rewarding.
I think this is a problem not just with calculus, but really with most math instruction. Math is taught by people who are already in love with the special insider world of math. To students who don’t share that love, it’s often unappealing at best - and exclusionary at worst. It weeds out people who don’t need to be weeded.
When math is taught in a way that develops practical intuition, it is so much more useful and rewarding.
Maybe the OP's daughter would be better off as a business major.
Yeah tbh undergraduate calculus courses are really not that difficult.
Thank you for this. My failure at calculus in my undergrad really set the tone for my mech eng. degree. I felt exactly what you mentioned: that you needed to memorize the "trick" and then you could solve the equation.
I am still embarrassed at my poor performance in undergrad.
I am still embarrassed at my poor performance in undergrad.
> But calculus is not being taught to her by engineers.
30+ years ago, when I started undergrad, calc for engineers, pre-meds, and scientists were 3 different classes. Seems like your daughter should choose a more enlightened school.
(Math majors all avoid calc thanks to AP.)
30+ years ago, when I started undergrad, calc for engineers, pre-meds, and scientists were 3 different classes. Seems like your daughter should choose a more enlightened school.
(Math majors all avoid calc thanks to AP.)
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This is spot on, but should be extended down the stack. For those with less direct need of math should get "math literacy" with principles and real world application of same.
I studied computer science. It came with some math - four semesters of calc, diff eq, linear, discrete, etc. FOUR semesters of calc!
It’s only use has been to tutor my daughters.
It’s only use has been to tutor my daughters.
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Are episilon delta proofs even necessary if limits can be reconstitutes as sums of infinite series?
I can help her out. ; )
I vaguely remember the treatment of statistics in my high school/college science classes where calculus wasn't used. It was confusing and tedious. They have you memorize quantiles of the normal distribution so you can answer questions referring to the central limit theorem without ever understanding it, because it is not really possible to understand it without understanding integration. Combined with the fact of the general incompetence of people designing high school curricula, I'm fairly convinced that statistics sans calculus as a mandatory requirement would be a waste of time, joke of a class that would the average kid hate math even more than they do today.
Yes, you need to know about integration to understand the central limit theorem, or to do any statistics with continuous values. But most of the time spent in calculus class is not learning what limits, derivatives, and integrals are. It's mostly learning how to take a function defined as some algebraic expression and finding the limit, derivative, or integral as an algebraic expression.
How many jobs in science involve pushing around algebraic expressions?
How many jobs in science involve pushing around algebraic expressions?
> How many jobs in science involve pushing around algebraic expressions?
Most engineering stuff. SW is an exception because SW testing is more about: if the input value can be any int value but customer requires that the function , when given 5 as input returns 3 as output then the function is tested only with 3 as input.
Most engineering stuff. SW is an exception because SW testing is more about: if the input value can be any int value but customer requires that the function , when given 5 as input returns 3 as output then the function is tested only with 3 as input.
I have one point to add to the topic.
The author claimed that calculus is required to biology major in many colleges. I would argue the opposite, I thought at two different universities and talked to many people. They always find it strange that it is not required for most biology fields. There is even an American invention called algebra based physics. This for people in biology who never saw calculus and your try to teach them physics and mechanics without calculus (Good luck with that, that is at best lost opportunity to understand physics) I thought in algebra bases physics classes many times and it was always horrible experience for both sides. Also calculus is not mandatory in US pre-college.
In general, you don't understand physics without calculus, period.
The author claimed that calculus is required to biology major in many colleges. I would argue the opposite, I thought at two different universities and talked to many people. They always find it strange that it is not required for most biology fields. There is even an American invention called algebra based physics. This for people in biology who never saw calculus and your try to teach them physics and mechanics without calculus (Good luck with that, that is at best lost opportunity to understand physics) I thought in algebra bases physics classes many times and it was always horrible experience for both sides. Also calculus is not mandatory in US pre-college.
In general, you don't understand physics without calculus, period.
Definitely. Biology is no calc, biochem takes a good degree of calc but these fields have quite a different depth of underatanding
Am I the only one who loved that this article talks about an author named Andrew Hacker.. writing an article about myths of science.. on HN..?
But regardless, while I disagree with the article’s argument it’s valid to debate this topic. My undergrad background is in engineering so I’m very biased here (and work as an engineer and have never used calculus for work). I do think a math major should take biology as much as a biology major should take math (calculus)! Folks often talk about the goal of a liberal arts education being learning a “breadth of topics” or the “classical teachings” and that’s fair and important. But I also think the same for individuals studying a STEM field. Maybe bio majors don’t need the level of calculus rigor required for a physics major but I personally think it’s such a fundamental topic they should learn it.
But regardless, while I disagree with the article’s argument it’s valid to debate this topic. My undergrad background is in engineering so I’m very biased here (and work as an engineer and have never used calculus for work). I do think a math major should take biology as much as a biology major should take math (calculus)! Folks often talk about the goal of a liberal arts education being learning a “breadth of topics” or the “classical teachings” and that’s fair and important. But I also think the same for individuals studying a STEM field. Maybe bio majors don’t need the level of calculus rigor required for a physics major but I personally think it’s such a fundamental topic they should learn it.
As far as I know, in the US, a bachelor of science major does require a variety of science courses. I had to take at least 2 biology courses with accompanying labs, 2 chemistry with accompanying labs, multiple physics, thermodynamics, plus a certain number of humanities (equaling at least 1 per semester I believe, so 8 courses).
Math was probably the most numerous universal course requirement, between multiple calculus and probability, statistics, discrete, and analysis courses. But that kind of makes sense given the foundational nature of it being the fundamental tool one needs to verify information in all the other sciences.
https://xkcd.com/435/
Math was probably the most numerous universal course requirement, between multiple calculus and probability, statistics, discrete, and analysis courses. But that kind of makes sense given the foundational nature of it being the fundamental tool one needs to verify information in all the other sciences.
https://xkcd.com/435/
I majored in CS and minored in Math (graduated in 2013) as someone who both struggled with math and was not the biggest fan of it. My biggest problem was simply caring about the subject matter. None of the instructors I've ever had for math courses put any effort in to showing what the particular subject was useful for. It was all just mechanical "Here's a type of problem", "Here's 100 examples of solving it" (of course I'm exaggerating this part but the point remains). Want my interest? Show me use cases. Don't just shove a ton of context-free examples in my face.
That bugged me about math as well, no context. You mean to tell me the guy had an apple drop on his head and went to solving the three-body problem, unsuccessfully, and so thats why we are deriving integrals with no end in sight to the complexity?
And then no matter what was said for 3 hours in class, the homework immediately throws curveballs at you in each problem, instead of reinforcing what was learned, hoping to teach you exceptions that were not covered
There is not a large population this style of learning will hit with
And then no matter what was said for 3 hours in class, the homework immediately throws curveballs at you in each problem, instead of reinforcing what was learned, hoping to teach you exceptions that were not covered
There is not a large population this style of learning will hit with
It's worth remembering that math departments have a large service component: They teach courses for majors in other departments, because that is what those other departments want - and in fact, teaching other majors produces most of the credit hours taught in math departments. So they tell us what math courses their majors will take, and sometimes even what topics their majors need to see. We just do what we're told! :-)
Sometimes the math is needed for courses in the other major; for example, if an earth science major is taking a fluid dynamics course in their major, they'll need 3 terms of calc, and maybe differential equations and linear algebra. This is not to say that all majors in a department will use all the math they're required to take. I once asked a colleague in chemistry how often they actually needed to do derivatives or integrals; she replied that many chemists (e.g. experimental chemists) would rarely need calculus, but people in theoretical chemistry would. The simplest thing is often to require all majors in a given field to take the same math courses, even if only a few will use all of it.
The CS department at the school I retired from seems to have shifted its math requirements from 2 terms of calc to one term, plus stuff like statistics and linear algebra. (I don't recall all the details.)
Another reason for math requirements being what they are is that certain math courses are required for other departments to have their programs accredited. A while ago we were revising the syllabus for the calc course which was taken by business majors. I asked the business department chair if they had any comments or suggestions; he thought everything was okay, but wanted to verify that certain topics were in there (I think maxima and minima was one). He said their program accreditors wanted to see it.
I think other departments gradually adjust math requirements for their majors; it's in their interest to ensure their majors are prepared for jobs or grad school So if you graduated a while ago and you feel that the math you took wasn't that useful (and you have suggestions for alternatives), take a moment and mail your old department and let them know - they'll appreciate it, and you'll be helping future students.
Sometimes the math is needed for courses in the other major; for example, if an earth science major is taking a fluid dynamics course in their major, they'll need 3 terms of calc, and maybe differential equations and linear algebra. This is not to say that all majors in a department will use all the math they're required to take. I once asked a colleague in chemistry how often they actually needed to do derivatives or integrals; she replied that many chemists (e.g. experimental chemists) would rarely need calculus, but people in theoretical chemistry would. The simplest thing is often to require all majors in a given field to take the same math courses, even if only a few will use all of it.
The CS department at the school I retired from seems to have shifted its math requirements from 2 terms of calc to one term, plus stuff like statistics and linear algebra. (I don't recall all the details.)
Another reason for math requirements being what they are is that certain math courses are required for other departments to have their programs accredited. A while ago we were revising the syllabus for the calc course which was taken by business majors. I asked the business department chair if they had any comments or suggestions; he thought everything was okay, but wanted to verify that certain topics were in there (I think maxima and minima was one). He said their program accreditors wanted to see it.
I think other departments gradually adjust math requirements for their majors; it's in their interest to ensure their majors are prepared for jobs or grad school So if you graduated a while ago and you feel that the math you took wasn't that useful (and you have suggestions for alternatives), take a moment and mail your old department and let them know - they'll appreciate it, and you'll be helping future students.
I remember hearing in college (a couple decades ago) that Chemical Engineering students needed to know a lot about solving differential equations.
In the math department, I was told by professors that a lot of other departments were requiring the "pre-calculus" course without requiring calculus as a means of weeding out poorer performing students.
In the math department, I was told by professors that a lot of other departments were requiring the "pre-calculus" course without requiring calculus as a means of weeding out poorer performing students.
> I remember hearing in college (a couple decades ago) that Chemical Engineering students needed to know a lot about solving differential equations.
That stands to reason. Describing the dynamic behavior of a chemical system -- e.g. the amounts of different reactants and products over time, the temperature and pressure of the system, etc -- is exactly the sort of task that differential equations are perfect for.
That stands to reason. Describing the dynamic behavior of a chemical system -- e.g. the amounts of different reactants and products over time, the temperature and pressure of the system, etc -- is exactly the sort of task that differential equations are perfect for.
I only really understood the intuition behind differential equations once I did the control theory course. I didn't need any prior math course.
How about the case for making science majors more rigorous instead of less? Calculus touches so much of science and engineering (not to mention everyday life) that suggesting removing it from the curriculum seems not just ignorant, but downright malicious. Talk about setting a generation up to fail.
Perhaps a better discussion is how to teach calculus more effectively?
Perhaps a better discussion is how to teach calculus more effectively?
I agree the most with this take. We (the United States) can teach math much better at all levels than we are currently doing. I would rather try that than give up.
Interesting point. I am not academically inclined. I got a BS in Physics at UCSB 50 years ago. I was in a special research project (10% of my incoming Freshman class) where we had no course requirements to graduate except for our major. I ended up taking a calculus class every term for four years. I was amazed the Math department offered so many different calculates classes. So, I took all my units in physics and math/calculus classes except for a few required chemistry and biology classes, and for fun I took a philosophy class and three anthropology classes.
I feel really sorry for the young woman mentioned in the article having to take what was for her an unnecessary class. As a non-academic, my casual four years to get a BS was probably really good for me.
I feel really sorry for the young woman mentioned in the article having to take what was for her an unnecessary class. As a non-academic, my casual four years to get a BS was probably really good for me.
> Similarly, Thomas Edison said that as a boy he had a “distaste for mathematics.” But this did not stop him from becoming one of the most famous scientific inventors of all time. “I can always hire a mathematician,” said Edison, “but they can’t hire me.”
The perfect solution, assuming you happen to be wealthy enough to hire mathematicians without worry.
The perfect solution, assuming you happen to be wealthy enough to hire mathematicians without worry.
Require both.
because sure, p hacking, but i’ll tell you this for free: you will not get far in any discipline before you are whacked in the face with a differential equation based model of a system, and if you cannot grok that you are totally hosed.
because sure, p hacking, but i’ll tell you this for free: you will not get far in any discipline before you are whacked in the face with a differential equation based model of a system, and if you cannot grok that you are totally hosed.
Yeah, I agree. I lamented my lack of early education in statistics later in life. Now, at the time I was studying calculus-based courses, I didn't like them either. But it didn't take long into various quantitative fields where it clearly made a difference.
The lack of statistical understanding by the general population should not come at the expense of calculus-based mathematics. Furthermore, simply increasing the knowledge of statistics in peoples' minds won't solve all the problems that people see with innumerate individuals; a significant problem is that statistical reasoning often runs counter to human intuition and emotion. No amount of knowledge defeats those in all cases.
The lack of statistical understanding by the general population should not come at the expense of calculus-based mathematics. Furthermore, simply increasing the knowledge of statistics in peoples' minds won't solve all the problems that people see with innumerate individuals; a significant problem is that statistical reasoning often runs counter to human intuition and emotion. No amount of knowledge defeats those in all cases.
Hm. I think it’s a great thing to debate, but I think the issue, as identified by the author, is how calculus is taught to various audiences. I think it is a bit short-sighted to suggest people with ambitions in the life-sciences do not need to know how calculus can be used to describe the world.
I think my experience suggests that even outdoor types who are drawn to field work should learn calculus in some functional way if they want to do science. I did an undergrad in environmental sciences and biology, but understanding the role of modeling in theory and practice is key to those fields and makes heavy use of calculus. I went on to get a PhD in ecology, but I used calculus explicitly and implicitly routinely. There is great satisfaction in writing out ideas and hypotheses about the world and then translating them into math and then code so that you can play with those ideas. Sometimes that just requires algebra, but it usually pulls in calc.
The other risk is that by not learning broadly about the fields that are foundational to conducting science you simply limit your options. Maybe that field tech job loses its luster after a while and you start to want to understand how plate tectonics affected how the dirt got this way. Need calc. Personally, I switched to industry and do a combination of data science and engineering and calculus continues to be useful to me in practice and as a way of reasoning about the world.
On the flip side, you can always learn it later when it does become explicitly needed. But again I think the problem is teaching geologists and biologists to do epsilon-deltas but not showing them how to specify and simulate a model based on rates of change of growth or propagation of waves or whatever. A lot of coursework is not thoughtfully conceived or delivered and the concept of weeder courses is super lame. But thats not the fault of calculus.
+1 for requiring statistics too but virtually all of the above applies there too.
I think my experience suggests that even outdoor types who are drawn to field work should learn calculus in some functional way if they want to do science. I did an undergrad in environmental sciences and biology, but understanding the role of modeling in theory and practice is key to those fields and makes heavy use of calculus. I went on to get a PhD in ecology, but I used calculus explicitly and implicitly routinely. There is great satisfaction in writing out ideas and hypotheses about the world and then translating them into math and then code so that you can play with those ideas. Sometimes that just requires algebra, but it usually pulls in calc.
The other risk is that by not learning broadly about the fields that are foundational to conducting science you simply limit your options. Maybe that field tech job loses its luster after a while and you start to want to understand how plate tectonics affected how the dirt got this way. Need calc. Personally, I switched to industry and do a combination of data science and engineering and calculus continues to be useful to me in practice and as a way of reasoning about the world.
On the flip side, you can always learn it later when it does become explicitly needed. But again I think the problem is teaching geologists and biologists to do epsilon-deltas but not showing them how to specify and simulate a model based on rates of change of growth or propagation of waves or whatever. A lot of coursework is not thoughtfully conceived or delivered and the concept of weeder courses is super lame. But thats not the fault of calculus.
+1 for requiring statistics too but virtually all of the above applies there too.
Maybe for biology, especially for the bio majors who are pre-med, but probably not for chemistry, and certainly not for physics, whose laws are expressed in terms of calculus. Force equals mass times acceleration, and acceleration is the second derivative of position wrt time.
Doesn't Biology require calculus to express, for example, rates of reproduction? R_0 comes to mind[0].
[0] - https://en.wikipedia.org/wiki/Basic_reproduction_number
[0] - https://en.wikipedia.org/wiki/Basic_reproduction_number
Physical chemistry, physical organic chemistry, physics... These things need far more than basic calc. I also think that basic algebra is something that people can use in daily life. Not quadratic equation in the like, but determining things like unit prices, handling interest, etc. Simple things that take a little more than a calculator. And I recall needing variables when doing statistics - isn't that algebra?
And it's real hard to program a whole lot of stuff on a computer w/o algebra.
And it's real hard to program a whole lot of stuff on a computer w/o algebra.
Part of understanding science is understanding where the ideas come from. It's not enough to know that the derivative of x^2 is 2x, but you have to know how and why this is the case. You have to understand how to model physical systems using math. And it turns out scientists occasionally need to build novel mathematical models. It is much better in these cases to have been exposed to more math, not less.
Where I went to school, "real" stats and experimental design topics were (mostly) offered as graduate courses. Psychology majors were given enough stats to read a psych paper. Physicists and Chemists were given enough stats to understand stat mech / pchem. Biologists and Geologists were given a much less rigorous intro to stats.
I believe the OP is advocating for more stats, not less calculus. Or at least a more consistent approach to teaching stats & experimental design at the undergrad level.
Where I went to school, "real" stats and experimental design topics were (mostly) offered as graduate courses. Psychology majors were given enough stats to read a psych paper. Physicists and Chemists were given enough stats to understand stat mech / pchem. Biologists and Geologists were given a much less rigorous intro to stats.
I believe the OP is advocating for more stats, not less calculus. Or at least a more consistent approach to teaching stats & experimental design at the undergrad level.
Basic differential and integral calculus should be taught in high school.
> The real question is: Why should passing calculus be required to major in biology any more than passing biology should be required to major in math?
Interesting question, why should I be required to pass biology?
There are a lot of good points addressed thoroughly in other threads here, but (while calculus may fill a bit of a different role) education to a little bit of depth in a variety of areas does seem to be part of the role of a college (at least at present). I think requiring statistics to be taken more broadly would be great for our society, but calc (especially when taught well) definitely touches on areas of thinking that other subjects don't cover.
(Obviously slightly biased perspective from a math major.)
Interesting question, why should I be required to pass biology?
There are a lot of good points addressed thoroughly in other threads here, but (while calculus may fill a bit of a different role) education to a little bit of depth in a variety of areas does seem to be part of the role of a college (at least at present). I think requiring statistics to be taken more broadly would be great for our society, but calc (especially when taught well) definitely touches on areas of thinking that other subjects don't cover.
(Obviously slightly biased perspective from a math major.)
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Interesting article. I think it would be a nice experiment to run an introduction to proof course, including something like Martin Gardner's puzzles, to show how proof works, and the power of thinking carefully and building up arguments (and using some elementary arithmetic to help). I think that'd go a long way towards increasing mathematical thinking with minimal suffering, like memorization of integrals. It should provide a more gentle introduction to the heart of maths.
20 something years ago, GT had constructing proofs as an early CS course. It doesn't seem to be required anymore. Here's a 2005 course page for it: https://faculty.cc.gatech.edu/~vigoda/1050-Fall05/index.html
I don't recall it being terribly popular, but it was good as I recall (and useful when I later added a math degree, I think I had a leg up on most of my math major peers because of it).
I don't recall it being terribly popular, but it was good as I recall (and useful when I later added a math degree, I think I had a leg up on most of my math major peers because of it).
At my university in the late 90s, I was required to take calculus for my CS curriculum, but no statistics courses, which I always found funny because our department was "Computer Science and Statistics". With "Data Science" (i.e. statistics) and "Machine Learning" (i.e. statistics) becoming so commonplace, I sure hope that has changed.
> The common requirement to pass calculus in order to major in a science is a killer of students’ dreams. And it unnecessarily limits the pool of future scientists.
So: you need better syllabi and teachers.
Do they teach measure theory and the Radon-Nykodim theorem in those Statistics courses? And the central limit with proofs? No.
So: it is not the Calculus. It is the teaching of it.
So: you need better syllabi and teachers.
Do they teach measure theory and the Radon-Nykodim theorem in those Statistics courses? And the central limit with proofs? No.
So: it is not the Calculus. It is the teaching of it.
I tried to get a degree in electrical engineering, but calculus halted that. I’m just not good at calculus. I would’ve really liked to of been an engineer. The professors were not good, they were all mathematicians, and it was all very rote and bland.
I like calculus and I've always liked it more than other branches of mathematics. That's not to say that I consider myself a true expert in the subject as I'm not (I still find some aspects of the subject difficult).
I like calculus because it allows us to put a measure on how things in our world change, it greatly simplifies our calculations of the rate at which these changes occur than we would otherwise be without it.
Things in this universe are rarely fixed, unchanging and immutable and the fact that they're not is why the universe works as it does, and it's why we exist and are around to actually study it. Calculus is a principal tool in helping us make observations about how the universe works.
From my experience, probably the single biggest problem with calculus is its frightening reputation. There's likely nothing better to give someone a lifelong phobia of calculus than for a teacher to say to him or her as a kid 'you must be very good at arithmetic if you are ever to be good at that advanced subject calculus'.
When I was a kid calculus had a frightening and awesome reputation for being difficult so by the time we'd reached highschool and were confronted with learning it many kids already had preconceived notions that they weren't going to do well in the subject.
This notion about calculus being difficult isn't new and it goes back a long way. Only a few days ago I read a HN story about physicist Richard Feynman learning calculus as a teenager and that he learned it from an early type of 'teach yourself' book titled Calculus Made Easy (1914) by Silvanus Thompson. I'd not seen this book previously so out of curiosity about how the great man came to learn the subject I downloaded a copy and it was an eye-opener.
Its Chapter I titled To Deliver You from the Preliminary Terrors, only confirmed the fact that not only my generation of kids had been forewarned of calculus' 'terrors' but also so had previous generations of kids - even those long before the author's time. Thompson, an electrical engineer, professor of physics and educator, was well aware of its 'terror' factor ipso facto the chapter's title. In only one half pages of the most delightfully written prose for a mathematics text he attempts to alleviate the reader's fears with simple straightforward explanations. Therein he explains the dreaded 'd' as simply meaning 'a little bit of' so dx means a little but of x. (In my opinion all educators of the subject would do well to read this chapter.)
Despite Thompson's best efforts to dispel fears of calculus they still remain with many of us today. Perhaps the reason why the students referred to in the article failed calculus is that they've carried this preconceived notion of its difficulty with them from their earliest days and that the level of (or the subject material) wasn't appropriate to their courses.
I'm not in favor of removing calculus from courses because I believe it is necessary to understand what it teaches us about the world, as it explains the underpinnings of almost everything we do, especially so the physical world, and we need to have some knowledge of how that happens.
If I could I would go even further than Thompson and teach calculus to primary school kids. Get in early enough and kids wouldn't have time to develop a fear of the subject.
For skeptics who say we can't teach anything meaningful about calculus at that age then they ought to think again.
For instance, getting kids to learn Simpson's Rule by showing them how to work out the area under a curve by having them cut out rectangles of cardboard and best-fitting them to the area is well within their capabilities. Couple this with a stress on how important this is in real world siutations and we'd be well on the way.
Then there's the old fable about the frog on a stone in the middle of a pool who wants to jump to the outside bank. At first he jumps half way and then a quarter and then an eighth and so on as he tires. Every kid knows this story and that the poor hapless frog never makes it.
Now teach kids the frog really doesn't perish after all - because our hero Calculus says his feet are too big. Happy ending.
(Apologies to any mathematicians whose sensibilities I've offended.)
___
Edit: I should have stressed that in professions that have a less rigorous dependence on mathematical calculations, their calculus courses should place more emphasis on the meaning of what it teaches us rather than the manipulation of symbols and equations per se.
(Humans have always had difficulties in recognizing small and large rates of change - exponential growth etc. - especially in their early stages (when often still manageable).
We should always emphasize to students why calculus is an essential tool for processing the mathematics of rates of change. In many cases understanding the underlying reasons is more important than doing calculations (if one actually understands the problem then one can always call on someone with better mathematical knowledge - after all, at times even the best did this - Einstein for instance).
I like calculus because it allows us to put a measure on how things in our world change, it greatly simplifies our calculations of the rate at which these changes occur than we would otherwise be without it.
Things in this universe are rarely fixed, unchanging and immutable and the fact that they're not is why the universe works as it does, and it's why we exist and are around to actually study it. Calculus is a principal tool in helping us make observations about how the universe works.
From my experience, probably the single biggest problem with calculus is its frightening reputation. There's likely nothing better to give someone a lifelong phobia of calculus than for a teacher to say to him or her as a kid 'you must be very good at arithmetic if you are ever to be good at that advanced subject calculus'.
When I was a kid calculus had a frightening and awesome reputation for being difficult so by the time we'd reached highschool and were confronted with learning it many kids already had preconceived notions that they weren't going to do well in the subject.
This notion about calculus being difficult isn't new and it goes back a long way. Only a few days ago I read a HN story about physicist Richard Feynman learning calculus as a teenager and that he learned it from an early type of 'teach yourself' book titled Calculus Made Easy (1914) by Silvanus Thompson. I'd not seen this book previously so out of curiosity about how the great man came to learn the subject I downloaded a copy and it was an eye-opener.
Its Chapter I titled To Deliver You from the Preliminary Terrors, only confirmed the fact that not only my generation of kids had been forewarned of calculus' 'terrors' but also so had previous generations of kids - even those long before the author's time. Thompson, an electrical engineer, professor of physics and educator, was well aware of its 'terror' factor ipso facto the chapter's title. In only one half pages of the most delightfully written prose for a mathematics text he attempts to alleviate the reader's fears with simple straightforward explanations. Therein he explains the dreaded 'd' as simply meaning 'a little bit of' so dx means a little but of x. (In my opinion all educators of the subject would do well to read this chapter.)
Despite Thompson's best efforts to dispel fears of calculus they still remain with many of us today. Perhaps the reason why the students referred to in the article failed calculus is that they've carried this preconceived notion of its difficulty with them from their earliest days and that the level of (or the subject material) wasn't appropriate to their courses.
I'm not in favor of removing calculus from courses because I believe it is necessary to understand what it teaches us about the world, as it explains the underpinnings of almost everything we do, especially so the physical world, and we need to have some knowledge of how that happens.
If I could I would go even further than Thompson and teach calculus to primary school kids. Get in early enough and kids wouldn't have time to develop a fear of the subject.
For skeptics who say we can't teach anything meaningful about calculus at that age then they ought to think again.
For instance, getting kids to learn Simpson's Rule by showing them how to work out the area under a curve by having them cut out rectangles of cardboard and best-fitting them to the area is well within their capabilities. Couple this with a stress on how important this is in real world siutations and we'd be well on the way.
Then there's the old fable about the frog on a stone in the middle of a pool who wants to jump to the outside bank. At first he jumps half way and then a quarter and then an eighth and so on as he tires. Every kid knows this story and that the poor hapless frog never makes it.
Now teach kids the frog really doesn't perish after all - because our hero Calculus says his feet are too big. Happy ending.
(Apologies to any mathematicians whose sensibilities I've offended.)
___
Edit: I should have stressed that in professions that have a less rigorous dependence on mathematical calculations, their calculus courses should place more emphasis on the meaning of what it teaches us rather than the manipulation of symbols and equations per se.
(Humans have always had difficulties in recognizing small and large rates of change - exponential growth etc. - especially in their early stages (when often still manageable).
We should always emphasize to students why calculus is an essential tool for processing the mathematics of rates of change. In many cases understanding the underlying reasons is more important than doing calculations (if one actually understands the problem then one can always call on someone with better mathematical knowledge - after all, at times even the best did this - Einstein for instance).
that's funny, I ended up buying the book based on exactly the same HN comment and received it today. The idea that calculus could be learned based on the dual numbers is really intriguing to me, especially because I've been studying them in conjunction with their cousins the double (or split-complex) numbers, and trying to understand the whole Cayley-Klein geometry framework, for some years now.
It's sort of a never-ending spiral of research, where I never seem to be able to get to the core of the subject (geometry). Today I am looking at Stillwell's "The Four Pillars of Geometry", in yet another attempt to understand the cross-ratio, and in particular, Von Staudt's reconstruction of arithmetic based on projective geometry. Of course one could go deeper, to Klein and Russell's criticisms of Von Staudt and so on...
Each time I think to close my hand on the golden bird, she flits from my grasp.
It's sort of a never-ending spiral of research, where I never seem to be able to get to the core of the subject (geometry). Today I am looking at Stillwell's "The Four Pillars of Geometry", in yet another attempt to understand the cross-ratio, and in particular, Von Staudt's reconstruction of arithmetic based on projective geometry. Of course one could go deeper, to Klein and Russell's criticisms of Von Staudt and so on...
Each time I think to close my hand on the golden bird, she flits from my grasp.
Yeah right. I'm not a mathematician by profession so much of my time has been spent just plugging values into equations. The trouble with that is twofold, first one can forget those less frequently used esoteric tricks one was taught, and second, one can overlook whole swathes of the subject - those parts outside one's normal orbit.
I never seem to have enough time to sit down and do nothing but mathematics for its own sake.
I never seem to have enough time to sit down and do nothing but mathematics for its own sake.
>not only my generation of kids had been forewarned of calculus' 'terrors'
I think the real terror as a youngster is that you could fail a timed exam and have to repeat the whole course.
I think the real terror as a youngster is that you could fail a timed exam and have to repeat the whole course.
That, I agree, is very true.
I recall the horrible stress and anxiety I experienced early on in high school when I ran out of time in a mathematics exam and thus couldn't complete all of it. Panic started to set in because I faced the dilemma of having to decide which of the questions I could best answer (and that was far from being clear). The alternative option was to sketch out parts of more of the questions and leave each partially unanswered (on grounds of obtaining some marks fór at least having some knowledge of the material).
I cannot remember exactly what I did but I think it was a mixture of those two approaches. What I vividly do recall however was that I wrote a somewhat sarcastic comment on top of my examination paper with words to the effect that the examination was too long to do within the allotted time.
Under the circumstance, that wasn't the wisest of moves, it didn't win me any Brownie points with the teacher. Also, I was ridiculed and humiliated by my classmates as they laughed and crackled at me when the teacher stopped to read my comment out during his handing out of the marked papers.
I can't remember what marks I received except that it was borderline fail (and I can't even remember what side of the line the marks were on). I do recall the only consolation I had was that some kids received lower marks in the examination than I did.
There are two reasons for why that exam remains particularly memorable, the first is that it was first [but not the last] time I'd done badly in what I considered an important and key subject (unlike languages which back then I considered of little importance—much to my later chagrin and regret); and second, I knew that I could have done much better if I had had more time to compete it. Still that's hardly a reasonable excuse for being poorly prepared (which was a fact).
Here, I'd also make a distinction between examinations where one simply doesn't know answers to questions and those where one cannot provide satisfactory or adequate answers in the time available (for whatever the reasons), as marks won't necessarily provide an accurate assessment of a student's actual knowledge of a subject.
Again, that's unlikely to be a problem if one is sufficiently prepared. Nevertheless, there are some students who are particularly adept at doing well in exams and I'm quite envious of them or their ability. Especially, so those who've less interest in a subject than me and or who have demonstrated by other means that they've less knowledge of it than I have.
So be it, the world's not fair, and neither are examinations the ideal way to determine one's knowledge—but likely they're still the best method we have for said purpose.
I recall the horrible stress and anxiety I experienced early on in high school when I ran out of time in a mathematics exam and thus couldn't complete all of it. Panic started to set in because I faced the dilemma of having to decide which of the questions I could best answer (and that was far from being clear). The alternative option was to sketch out parts of more of the questions and leave each partially unanswered (on grounds of obtaining some marks fór at least having some knowledge of the material).
I cannot remember exactly what I did but I think it was a mixture of those two approaches. What I vividly do recall however was that I wrote a somewhat sarcastic comment on top of my examination paper with words to the effect that the examination was too long to do within the allotted time.
Under the circumstance, that wasn't the wisest of moves, it didn't win me any Brownie points with the teacher. Also, I was ridiculed and humiliated by my classmates as they laughed and crackled at me when the teacher stopped to read my comment out during his handing out of the marked papers.
I can't remember what marks I received except that it was borderline fail (and I can't even remember what side of the line the marks were on). I do recall the only consolation I had was that some kids received lower marks in the examination than I did.
There are two reasons for why that exam remains particularly memorable, the first is that it was first [but not the last] time I'd done badly in what I considered an important and key subject (unlike languages which back then I considered of little importance—much to my later chagrin and regret); and second, I knew that I could have done much better if I had had more time to compete it. Still that's hardly a reasonable excuse for being poorly prepared (which was a fact).
Here, I'd also make a distinction between examinations where one simply doesn't know answers to questions and those where one cannot provide satisfactory or adequate answers in the time available (for whatever the reasons), as marks won't necessarily provide an accurate assessment of a student's actual knowledge of a subject.
Again, that's unlikely to be a problem if one is sufficiently prepared. Nevertheless, there are some students who are particularly adept at doing well in exams and I'm quite envious of them or their ability. Especially, so those who've less interest in a subject than me and or who have demonstrated by other means that they've less knowledge of it than I have.
So be it, the world's not fair, and neither are examinations the ideal way to determine one's knowledge—but likely they're still the best method we have for said purpose.
We don't need calculous, software does most calculations. One has to wonder though, based on what principles does the software do those calculations. Black magic I presume?
They want to lower standards so they can increase diversity numbers without addressing the systemic racism in our educational system that leads to black Americans not having access to even moderately ok schools.
Math is too important to be taught by mathematicians.
What would Claude Shannon do?
Please, no.
Wait until someone points out to the author that statistics heavily relies on calculus...
This is a weird argument to make. Does someone need to understand the workings of an internal combustion engine to drive a car?
You analogy doesn't hold. You don't need to understand most things (if any) to drive a car. But applying doing. statistical calculation, interpretation and everything else needs to he done with understanding. To understand why you take this distribution function over the other, you will need to understand them which means you need to understand motr fundamental concepts. surprisingly, they depend on your calculus understanding.
If we take this to the extreme, should they have to study real analysis so they understand where calculus comes from as well, or would you prefer we start at set theory? Maybe we should delve into philosophy to understand the basis of mathematics so we can truly understand our statistics.
The point is that you have to pick a cut-off somewhere as your foundation to build upon. I doubt much understanding of calculus is required for the vast majority of statistical inference, but I'm certainly no expert.
The point is that you have to pick a cut-off somewhere as your foundation to build upon. I doubt much understanding of calculus is required for the vast majority of statistical inference, but I'm certainly no expert.
You don't need to understand where is calculus come from to tqke real analysis. Hell most academics who use calculus everyday did not study real analysis and they are fine. This extreme angle is not relevant. The difference is that you can't fot example understand physics without calculus but you can understand and study calculus without real analysis.
> Hell most academics who use calculus everyday did not study real analysis and they are fine.
That's an American perspective, I believe.
In many European countries, many STEM students routinely learn real analysis. In fact, the term "calculus" doesn't even exist in e.g. German, it's all called "Analysis". Sure, a course tailored for physicists, chemists or computer scientists may (or may not, depending on the institutions) have a different focus, may emphasise proofs less etc., but the underlying concepts (what are the real numbers, completeness) are generally taught.
That's an American perspective, I believe.
In many European countries, many STEM students routinely learn real analysis. In fact, the term "calculus" doesn't even exist in e.g. German, it's all called "Analysis". Sure, a course tailored for physicists, chemists or computer scientists may (or may not, depending on the institutions) have a different focus, may emphasise proofs less etc., but the underlying concepts (what are the real numbers, completeness) are generally taught.
I'my always confused by Americans and their calculus because I really don't understand the essence of what difference there is between real analysis and calculus.
The internet answer of "calculus is real analysis for engineers" doesn't help because you can have real analysis tailored for computer scientists and it is still real analysis.
The internet answer of "calculus is real analysis for engineers" doesn't help because you can have real analysis tailored for computer scientists and it is still real analysis.
As far as I understand, calculus = analysis without proofs - or at least without rigorous proofs. You might do some hand-waving and elementary transformations, but you're probably not gonna invoke the Least Upper Bound axiom of the real numbers.
I've a bachelor's in CS, and I've learned real analysis. As someone pointed out, maybe being in an European university led to it.
And it's probably, the most useful course I've taken. If you're going to perform a bunch of floating point ops, I hope you've taken real analysis. They're enough of a Russian roulette to begin with, let's make sure you aren't making my day worse hunting down for some weird ass rounding error.
And it's probably, the most useful course I've taken. If you're going to perform a bunch of floating point ops, I hope you've taken real analysis. They're enough of a Russian roulette to begin with, let's make sure you aren't making my day worse hunting down for some weird ass rounding error.
If you're going to perform a bunch of floating point ops and are not using interval arithmetic to precisely bound the result (and perhaps constructive real arithmetic, if you want guarantees of tight-enough bounds), you'll need a numerical analysis class as well. Real analysis will teach you about the theory of how real numbers are defined, but computation and the need for approximate results throughout bring challenges of their own.
Very much true. Have taken it as well.
My train of thought was more along continuity and such, since I'm mostly working on CG right now. But I guess Numerical Analysis has even greater value in the context I provided above.
Definitely yes, actually I'm studying computer engineering in Italy (still bachelor degree) and it's mandatory to have the exams of Mathematical Analysis I and II and also the exam of Numerical Analysis, where they teach you floating point numbers, Lagrange interpolation, Gauss method, non-linear equations and quadrature formulas.
Knowing the expected inputs, outputs, and assumptions a test makes is sufficient to use a function someone else wrote in python or R
[deleted]
A PDF only makes sense when integrated.... :).
Yet another article promoting a knowledge gap between the rich and the poor. rolls eyes
"This student had lost four credits’ worth of tuition money, in return for nothing but a lower GPA"
Haha, that's what it means when you fail these days.
"lost"
Wow
Haha, that's what it means when you fail these days.
"lost"
Wow
US universities have turned reinvented themselves into pay for play. Students have to shoulder more of the tuition and if you're paying for it you are the customer. And the customer always wants the best ROI.
Yes, I agree it is tragic that we have built a system where it makes sense for students to think in these terms
apparently, because his GPA, he can't find any job and the only thing can d is waiting for starve and die.
What companies even care about gpa
Absolutely NOT.
Or you need to stop calling biology, geology and other majors forms of science or STEM if you eliminate calculus.
Or you need to stop calling biology, geology and other majors forms of science or STEM if you eliminate calculus.
Remind me why there are no downvotes on articles again?
There's a "flag" option if you think it doesn't belong here. There's also a "hide" option if you just don't want to see it. These aren't hard to find, they're both available under the submission title.
The lack of an /s tag strikes again...
The comments in this thread are a nice case of survivorship bias.
This is a big no no.
There is a case for different education perspectives for different majors (trying to teach in a more familiar way for people with different backgrounds).
But dropping it entirely? Why are you in that major to begin with? It's like going into compsci and not expecting math.
There is a case for different education perspectives for different majors (trying to teach in a more familiar way for people with different backgrounds).
But dropping it entirely? Why are you in that major to begin with? It's like going into compsci and not expecting math.
Calculus should be replaced with a coding class. Any stem major you do you can speed up your analysis a lot more with a little bit of code. Even literature reviews:
https://www.ncbi.nlm.nih.gov/books/NBK25500/
https://www.ncbi.nlm.nih.gov/books/NBK25500/
I am currently trying to tutor my daughter through college-level calculus. She wants to be a civil engineer. Civil engineers, obviously, need to know calculus.
But calculus is not being taught to her by engineers. It is being taught by mathematicians. And as part of teaching it, they are attempting to instill a "mathematical mindset". Which means, in practice, that it is an entry to proofs.
Every proof has a trick. If you can see the trick, you can generally figure out how to do the proof. If you can't see the trick, you can't. The same thing is true with all of the very tricky limits they are making her learn to solve.
I hold a PhD in aerospace engineering with a minor in applied mathematics. I specialized in control theory. I use calculus every day of my life. In twenty years of engineering practice I have never once had to take a difficult limit, and can probably count on one hand the number of times I've had to take a limit of any kind.
I had a heart-to-heart with her about this today. I told her that what she actually needed to understand about calculus was how to use it to solve engineering problems, not how to manipulate polynomial fractions, chain trig identities, and do substitutions of variables. And she was not going to learn that in this class, but that she nevertheless needed to learn the tricky algebra and epsilon-delta proofs simply because this was a hoop she needed to jump through to get to the actual engineering classes.
There was a not inconsiderable amount of cursing, but I understood because I hit the same wall, just thirty years prior. Nothing has changed.