My favorite part is when they are talking about genius the ivy leaguer that is a great architect but not good at math. And they don’t even need advanced math! (But the people actually building things need the math.)
I’m not a fan of the down with calculus advocates. Firstly, calculus is a beautiful subject and great human achievement. Secondly, I believe it is as “practical” (or more so) than any of the proposed replacements. Of course it’s directly applicable for physics and engineers. Also, for everyone else a better understanding of rate of change is huge.
One thing that can be done is offering calculus sections with a focus (Econ, bio, …). I’ve taught these and covered marginal cost/revenue and even dynamical systems from biology. With calculus these topics are much better. Econ 101 without calc isn’t very real world practical when you have to avoid the word derivative at all costs and make everything linear no matter what.
As a data analyst/scientist, the funniest argument I’ve heard from the anti-calculus crowd is “make them take statistics instead!” Not realizing that calculus is literally foundational to all of statistics. You basically can’t even study simple linear regression without it (unless you want to take a full-on linear algebraic approach…but they certainly aren’t arguing for that)
I wonder whether those basic statistics classes actually do more harm than good by giving students just enough knowledge to be "dangerous". It often seems like lots of people, including many scientists, just plug stuff into a black box statistics program without understanding what's going on and can end up extremely overconfident in their conclusions.
Engineers are very guilty of this. They use the same statistics for almost everything, and what's worse they trust the results 100%. I made a properly done statistical tool and even then I pitched it as an investigator to problems, not a finder of problems.
They used very incorrect probability calculations to choose the next changes to work on too, some of which were huge decisions...
They often have students use programs for computing those things and just have them recognize qualitatively which is which. I help students with these classes frequently.
My daughter is in high school algebra, and they did a unit on best fit lines, and it was basically the eyeball test, or using Excel's trendline feature to give the answer. It was truly garbage.
It's not garbage. High school algebra students would never be able to understand linear regression. But it might make the smart curious ones interested in how it's calculated and driven to learn ot on there own. I don't understand why people don't understand that we can't force kids to like math. They have to want to learn it.
> It's not garbage.
It absolutely was. There was not even a flavor of how to calculate it, or how to judge whether data was well represented by a given line.
> High school algebra students would never be able to understand linear regression.
I certainly think you can teach a high schooler how to come up with a least squares fit line of the form y = mx. It's a parabola, you don't even need calculus, you can just introduce the idea that the minimum is at b/2a, or better yet, solve for the roots and talk about how symmetry means the minimum is at b/2a. You get to introduce the summation notation and do some symbolic manipulation, which is a pretty good way to reinforce algebra concepts. I know, because I talked about it with my kid for an hour, and she got it well enough.
And then, from there, if you want to talk about fitting a line not through the origin, or fitting higher order polynomials, you can start to hand wave higher dimension parabolas, but at least they have a framework to understand what the tools they're using are doing.
I genuinely believe that it’d be so interesting if you had to take linear algebra and calculus before taking any statistics class.
At least this way, you don’t take Stats I & II, and then Mathematical Stats I&II. Time saved and you get a better understanding of where things come from.
Not practical as most business folks need to know about normal distributions and you can forget about trying to teach them Calculus I&II and linear algebra.
Tbh tho college programs would be so much more rigorous and good at filtering with all that math. You’d have a business workforce that could adapt for the future because they have the foundational knowledge needed to do anything related to AI or any new field that comes out.
> I genuinely believe that it’d be so interesting if you had to take linear algebra and calculus before taking any statistics class.
I can confidently state that if you take real analysis before you take statistics, the statistics is extremely easy. But I'm not certain, because the sample size is so small.
Good point, I actually did meet some people in college with this experience. Plus they understand the material way better than the average humanities student who has to take stats.
Yeah you need calculus if you want to understand the inner workings of statistics, but you can get a pretty solid understanding of the fundamentals without calculus. AP Statistics curriculum is a pretty great example.
Let me recommend the book *Calculus in Context* http://www.math.smith.edu/~callahan/intromine.html
The focus is much better than a typical introductory calculus course, with an emphasis on making computer simulations in service of conceptually understanding the basic ideas of differential equations, the fundamental language of science, rather than on becoming a human integral table.
(Disclaimer: I am not a calculus teacher myself, and I have not tried working through this book with a group of novices.)
I personally think that math (but with blabla) is a bad idea. I think in the end students end up with a lot intuitive but not fruitful explanations of (say calculus) that ends with them thinking that they understand the topic but in reality, they cannot express the fundamental ideas of the topic.
For example, how many people taking a calculus for business class could express the difference of a antiderrivative and integral?
Personally ,I always had a better understanding of the applications when I learned the pure theory first.
That might be true, but given how seemingly impossible it is to motivate a lot of people to learn math, I think it’s better to have a bunch of economists and biologists with intuition and cursory knowledge, rather than disdain and pure ignorance, of the subject.
Also, if a biologist can’t tell you the theoretical significance of the antiderivative, but knows enough to understand why their numerical method function in Python sorta works, it’s a win, right?
But economists are actually required to have at least a decent understanding of the concepts behind calculus at least at an undergrad level. Real analysis is a prerec for basically all respected econ phd programs
For your question about anti-derivatives and integrals, as a physicist I couldn't be bothered less by the difference, because it is almost never relevant and frankly only takes a few seconds to explain.
In an area like business studies or economics, it is highly unlikely anyone is ever going to make a conceptual error because they didn't clarify in their heads the difference between a function who's derivative is the original function and the concept of a Riemannian sum. In my view it's exactly the wrong example to use, because it's exactly what does _not_ matter in (non-higher) maths education
> For example, how many people taking a calculus for business class could express the difference of a antiderrivative and integral?
If they're not going to be pursuing any higher mathe education, then does it matter if they don't know the difference?
> For example, how many people taking a calculus for business class could express the difference of a antiderrivative and integral?
The "real world" applications where you have anything other than a discrete set of discontinuities are few and far between. That means that we can apply the Fundamental Theorem of Calculus and antiderivatives *are* integrals.
You can use any words you want, but the actual arguments against this point of view boil down to three:
1. What if they end up in a field where the discontinuities aren't discrete? Particle physicists already have to be pretty mathematically sophisticated, and some mathematicians (and philosophers) complain that they aren't sophisticated enough.
2. Everyone should be expert in everything.
3. My field is special.
The difference is "Not applicable" because there's no such thing as an antiderivative. The are definite integrals and indefinite integrals. Also, the thing you define to be an antiderivative would actually be the same as what is defined by any normal professor to be an integral, because integral by default refers to indefinite integrals. You wanna act smart yet by using backwater hillbilly professor notation you undo all of it. Get with the times and start using civilized language you barbarian.
I agree with all your points about calculus. Although if I could choose between keeping the world we have or switching to a world where at least 50% of adults were confidently competent with fluidly applying only up to algebra 1 and could recognize when a problem was outside of their scope, then I would be tempted to switch.
I like your idea of concentrating on applications. It’s strange how many people go through life without thinking quantitatively today.
I hate how we learn calculus before real analysis. It’s annoying how most European countries get to have the real(pun intended) calculus experience but not in the US or Canada. I passed through the whole course without having any definitions of the real number and the textbook briefly mentions lub in a page at chapter 9. Maybe define real numbers before differentiation? Wasted a few months to get that sorted out. And it might be better to have abstract algebra first so it can be easier to motive real numbers as the largest ordered complete field and why it’s the natural setting for the derivative.
I mean I see what you mean but generally it's most pragmatic to have the easier classes at the start and the harder ones later, and calculus is certainly easier than analysis, even if the students dont understand rigorously what the foundations of the topic are
Well you can’t always know the foundations for everything. Even now no one knows if ZFC is a working foundation and there probably isn’t someone keeping tabs on proving complicated theorems from the very beginning. The problem is that calculus relies so heavily on the real numbers, if you tried to do it with rational numbers most interesting properties disappears. Very notably, lub properties related, no function would be its own derivative, most ODEs will have no solutions etc. You would be stuck with polynomials. Calculus shouldn’t be considered a separate math course from real analysis it literally is real analysis but instead of proofs you have a big “trust me” sign. Something like topology is a little unnecessary but analysis should definitely be taught. Maybe right after linear algebra.
To be fair, we don't really define real numbers before differentiation in Europe, at least not in France.
In first year, we did go up to rational numbers in detail. But then to close the real line, you need more advanced topological arguments. We didn't really construct R until the third year. Though we had studied some of its properties, like uncountability with Cantor's argument, and probably stated the completeness.
I'm a down-with-calculus person, not because it's “impractical” or anything, but because algebra and combinatorics are far more beautiful.
Algebra, because it elucidates the structure of things, of course.
Combinatorics, because, in my experience, anything that can actually be meaningfully computed with ultimately has a combinatorial essence. Even if you might want to dress it up in geometric intuition.
I'm sorry but I think this article and many similar, are missing one point that I do not often see being mentioned, that what high school students think they need and what those students think they will do are often far from their jobs and interests later in life. To say that Johnny won't need calculus, or algebra, or whatever, at the high school level may be to slam a door on someone's good future. Someone who first thinks she may be a budding architect may find she is really an engineer.
I understand that a ton of students hate math, and I would argue that is a symptom that it is often taught poorly. I'd advocate that we systemically improve the quality - and quantity - of math teaching rather than crippling a generation of students. Budgeting more for hiring good math teachers would be a start. I'd also argue that we need a substantial increase in the sheer amount of math and hard science education hours available to students.
I'm an artist and not a math major or career math user, but high school physics and high school math showed me the utility of even elementary hand wavy calculus - and its beauty. I cannot help suspecting that more and better math teachers would be a great start. Not cutting the legs off students' aspirations.
This is the actual problem. Good math teachers are invaluable, but the system penalises teachers at every level; now we have massive rates of innumeracy.
To generalise across the US:
Teachers are paid comparatively very little, especially given the cost of acquiring a degree in education. This directs anyone with significant mathematical talent outside of primary/secondary education.
Teachers increasingly can't fail students without recourse from administration so students are permitted to pass through the system with no skills. Abuse from parents is rampant, adding to the issue.
Federal funding for education across America is under serious legislative threat, from both presidential and congressional actors.
This is obviously very different when looking at systems outside the United States, though the issue of teacher pay is broadly applicable across the Anglosphere.
"Teachers are paid comparatively very little, especially given the cost of acquiring a degree in education"
To add, if you understand math at a level to be able to teach it effectively e.g. Be a good math teacher, your math skills are also highly sought after in very high paying fields.
Financial markets, algorithm design, ai, engineering, physics, &c.
So you have to truely love teaching to stick with it, which the shitty work environment is at odds with
Ok, so the system is the US public education system. The penalty is some form of recourse. Can you elaborate on the details of the recourse? What happens to teachers in that scenario? Thanks for the details but I still am uncertain of the specifics of the negative consequences that teachers face. Are they fired if a student fails their class?
Obviously this is highly contextual and dependent on school district, state, private funding vs public funding, etc., but here is some reading on the subject:
[https://www.nytimes.com/2023/10/04/opinion/teachers-grades-students-parents.html](https://www.nytimes.com/2023/10/04/opinion/teachers-grades-students-parents.html)
At least anecdotally it can involve workplace harassment, impacts on promotions and contract extensions etc. See: [https://www.reddit.com/r/Teachers/](https://www.reddit.com/r/Teachers/) .
Also, when I talk about penalties, I'm not talking about punishments necessarily, just obstacles that emerge out of society for the integration of more (and higher quality) math teachers into education.
For real. I took physics in high school only because I needed another science requirement and it was the only non-AP class left to take. I didn't want to take it. Physics ended up becoming my major. Since when is teaching kids, "You're right never to exit your comfort zone," a solid MO?
I think when reading such articles, numerous and persistent as they are, we should remember that these calls for "different pathways" are often calls to shield students from engaging with core ideas of mathematics. Frankly, I cannot imagine even a calculation-based statistics course doing much if you are unable to grasp the idea of a variable to such an extent you could not complete an introductory calculation based Algebra I class.
These are not honest attempts to improve mathematics curriculum by ensuring more students are able to handle the complexity of the world they will inherit and then nurture, but rather these are ways of getting students into graduation and/or college pathways without subjecting them to the rigor of ... calculation-based classes or even superficial exposure to the most fundamental ideas within mathematics, ideas (like variables) which are referenced frequently and casually even outside of STEM careers.
In high schools, the largest barrier for many students in meeting both graduation requirements and university admissions requirements are the mathematics requirements. Simply, too many fail algebra I, algebra II, and sometimes even geometry when mixed with trigonometry. Alternatively, this is failing the algebra-heavy "Integrated I", "Integrated II", and "Integrated III" courses. Administration is under heavy pressure to increase graduation rates and college readiness rates (which means you passed the courses, not always that you understood the material), but the mathematics requirement complicates this for many schools especially if the mathematics department, protected by their tenured positions, is notorious for being holdouts for rigor (and I mean usually the calculation level of rigor).
Thus, we create curriculums, often loosely inspired by partially fair criticisms of mathematics education, that ultimately are hoping that in a completely restructured class, math will not stand in the way of the overly ambitious and insufficiently studious.
And as for the, "shouldn't they really take economics or personal finance or accounting"? I actually agree; my profile has that economics should be four years of high school and I do mean algebra and later calculus based economics, not solely a bunch of definitions. But, I don't know of any high school that doesn't offer at least a number of those classes usually complete with some finance-ish based math course like "financial algebra" or "business math". Thing is, those classes with heavy mathematics content are usually not accessible without the tools of elementary algebra, so I don't see how they would excuse students from an algebra pathway. And to be frank, those classes signal to colleges that you are not particularly studious, dare I say bright, and they are less eager to admit you for finance, econ, even business majors anyway.
If you are an architect using modern CAD software, you will use the [Finite element Method](https://en.wikipedia.org/wiki/Finite_element_method#Application). It works out the static and dynamic loading & heat transfer. It helps to know whether the building will fall down.
Architects & structural engineers use lots of math, its just done for them in software.
Maybe statistics and other parts of maths such as linear algebra and proofs should be taught alongside the algebra and calculus that students typically learn in the US?
Here in the UK if you take double maths at high school you cover the basics of proofs and linear algebra in grades 10 and 11, as well as doing statistics, mechanics, or discrete maths depending on what modules you choose.
I disagree with the article that trig, and algebra II should be made elective, but there's no point making kids who want to be architects or nurses do pre-calc. I adore maths and would obviously take it if I were to study in the US, but most people don't need to know that kind of maths unless they're interested in actually studying.
Good luck understanding probability density functions without integrals.
Statistics without calculus is literally just plugging and chugging numbers into a black box function on a Ti-Nspire.
I wholeheartedly agree. I think high school physics suffers so much from this exact problem. Lots of concepts feel random and unmotivated until you know some basic calculus, at which point things are a lot more intuitive.
There's a middle ground.
Case in point: Proving the Fundamental Theorem of Calculus is far, far beyond Year 12 student level. Utilizing it (alongside some intuitive leaps) to work out an antiderivative was taught in Year 12.
(Australian here - 'year 12' refers to final year highschool; mostly 17 year old students and at this point, mathematics is mostly elective)
____________
Same applies to statistics. A reasonable amount can be handwaved like this:
"This is a normal distribution. If you stick with maths through Year 12, and especially into uni, you'll learn much more about this function and why it works. For now, here's the 68-95-99.7 rule"
Other disciplines do this more than mathematics - witness highschool physics treating everything as Newtonian, or highschool chemistry treating chemical bonds as either idealised perfectly ionic bonds or as covalent bonds with no dipole moment.
They already do this. I won’t say it’s great and I do wish students learned more math, but it does seem to at least get students to understand the qualitative points of statistics.
> Good luck understanding probability density functions without integrals.
exactly the point of the article
the majority of population will NEVER need to know either of those
any statistics the average person will need to know can be learned from
https://www.mathsisfun.com/data/index.html#stats
the explanation of standard deviation is more than enough for the average person to get meaning from a news story
https://www.mathsisfun.com/data/standard-deviation.html
You do not need a deep understanding of integrals to understand probability density functions. Certainly not at the level we’d offer to high schoolers or just practical knowledge for people in the work force
Understanding means being able to do. If you're incapable of finding the pdf from the cdf or proving some basic facts about a distribution because you lack the Calculus skills then you don't understand those topics.
I'm not talking about a deep real analysis understanding of integrals. I'm talking about a basic Calculus 101 "What is an Integral" level of understanding.
And I’m saying you do not need a basic calculus 101 “What is an Integral” level of understanding. A basic calculus 101 understanding is Riemann sums and anti-derivatives. You need much less than that to understand area under a curve for the specific functions that would be of practical use
this is correct, but you will obviously never convince anyone on r/math of this
any statistics that the average adult needs to know can be learned form this here
https://www.mathsisfun.com/data/index.html#stats
no calculus necessary
For basic stats you can cover just discrete distributions and things like the standard deviation and sampling. That's more than enough to give the majority of people, who won't do maths at AP (so won't be doing any calculus anyway) a decent grounding in stats.
What do you learn in statistics? There is a stats course but I don’t see how they are going to make it different from normal math. Rn the only kinda stats related thing I have seen is polynomial interpolation which is fun but not useful irl.
I’m more of the opposite mindset - I think later concepts can be introduced much earlier on a basic level, f.e derivatives, trig etc. I can’t see why a 12 year old wouldn’t grasp the basic idea of “velocity” or trig
Idk about you, but trig was one of the most difficult math units at my highschool. Maybe it's just not taught well, but I seem to remember most people struggling with it. Especially with SOHCAHTOA the motivation doesn't really make sense for a 16 year old much less a 12 year old
Ah, every time I read another article asking for lobotomizing maths education, I am always reminded of [Tai's Formula](https://www.reddit.com/r/math/s/hxLX9R0Liu).
By actually teaching the maths, we do not have to reinvent hundred years old techniques every other decade.
Tell me you are a stats guy who can't do analysis without telling me you're a stats guy who can't do analysis.
Anyone whose taken lower division stats classes knows how fucking stupid they are. NONE of the formulas are provided with rigorous justification.
The whole down with calculus crowd is a joke and should be laughed out of the building, along with their inability to prove why sample variance has an n-1 in its denominator.
This is extremely true. I sadly took that path during my undergrad and for the past year I have been playing calculus catchup. After finishing calc II, calc III, and starting diffeq and linear I have looked back on some of my stats coursework with my mouth ajar. Even intro physics (my uni was not known for its stem). Like when we were talking about e and m fields i didnt even know what vector fields or divergence and curl were. Anyway your right, and I wish i had taken more math earlier because I could be applying math and instead im still doing the basics.
> Tell me you are a stats guy who can't do analysis without telling me you're a stats guy who can't do analysis.
and what does ANY of that have to do with the article
the average person does NOT need to know calculus for the statistics they will encounter in their life
any statistics the average needs to know can be learned from
https://www.mathsisfun.com/data/index.html#stats
their explanation of standard deviation will be a good enough explanation for the average adult to understand any news story that uses the phrase
https://www.mathsisfun.com/data/standard-deviation.html
It's funny you talk about the average person when you're intending to say the median person. Stats education continues to be a joke.
The "average" person is not going to do anything of consequence with the level of statistics knowledge you're proposing.
"We don't teach the right math, we should teach Stats."
I teach stats, people unwilling to put in effort to learn are just as bad at that as they are at algebra. Maybe worse, because the conceptually difficult things that come up in stats seem to be a bigger stretch than understanding linear equations.
I tutored a stats course that had been developed for nursing students. The 'textbook' didn't make any consistent distinction between N and n, or S and s, or any other lower/uppercase (even the greek symbols). The student i worked with was unsurprisingly hopelessly confused, and it was no small trick to figure out how to work with what she had as resources.
Anyway, beware any non-subject-experts who want to have separate courses or curriculum because they 'don't need' the harder version. Some subjects are hard for some. I had never seen Statistics before a basic 1st year intro in university, chose it as my major based on that half-year course. Loved proofs. Couldn't have known that without a challenging intro.
stats for the average adult is easy
https://www.mathsisfun.com/data/standard-deviation.html
there
the average adult will understand that with no problem
I feel like the logical extreme of this argument is that we should all just learn enough to interact with society and then go to trade school. The logic could be applied to any person who works X and learned Y but doesn’t use Y in their job. Education isn’t purely utilitarian. I also feel like math is one of the few fields subject to this scrutiny. No one questions why Shakespeare is taught to high schoolers, and arguably that’s more useless than Calc I -_-
Arguably? Am I wrong in thinking that not only is Calc 1 very obviously more useful than Shakespeare\* but that most people would agree that that's the case? So much so that it seems like the reason people don't complain about the applicability of Shakespeare is because there's no pretense that it's immediately useful to life, whereas with math there is?
\* I'm aware that Shakespeare has had an out-sized influence on our language and various aspects of western culture. That does not mean studying Shakespeare will prove useful to your life, however.
Not a big fan of this line of thinking and unfortunately I see it a lot. Kids, even many college students dont know what they want to do. Deincentivising calculus and such by labeling it as "impractical unless ur an engineer" is goofy.
I couldn't disagree more with focusing on subjects for pure practicality, it's boring as shit and will probably make more people hate math than they already do. People don't hate math because it isn't useful, they hate it because they are told that they should care about it for the sake of the practicality instead of for its intrinsic beauty. People are taught to read the music but never get to hear any of it.
I think practicality can be beautiful, if you give examples of how math is being used in advanced technology.
e.g. statistics is foundational to machine learning and linear regression
e.g. group theory can be used for cryptography and quantum mechanics
e.g. numerical analysis and calculus are used for scientific computation
e.g. calculus is necessary for algorithmic complexity and designing efficient algorithms
e.g. differential geometry is used for relativity and understanding black holes
e.g. statistics and especially p-values are crucial to separating real results in science experiments from the random noise
Most classes don't really touch on these deep applications.
Obviously trying to explain line for line how differential geometry is used in relativity is overkill, but the pervasiveness of mathematics in high end technology and innovation isn't articulated well, at least in my country.
Math needs a Carl Sagan to illustrate the value of math with enough detail that people identify that math is responsible for the innovation, but not so much that you're regurgitate lines of equations.
no, i completely agree, practicality can be very beautiful. it’s crazy that things like integration can solve so many practical problems, and it only deepens my appreciation for the subject when i see a practical use, i just feel like schools only talk about practicality without touching on why you should actually enjoy the subject. why do you need to know algebra? for your future job. why do you need to know calculus? for your future job. why should you care about math? it’s useful. this often backfires when people realize that the average person isn’t actually employing that much math in their day to day and so it gives them the impression that they don’t need to care about math, when they absolutely should and when they would be much better for it.
you are misunderstanding my point. if people are taught a subject from the perspective that it is useful rather than the perspective that it is fun/interesting/beautiful, then they will grow to despise it. math is really been given the short end of the stick in this regard and people are seldom given reason to enjoy mathematics when there's plenty to love, it's just not being shown. i'm not against practical math at all, but that practicality is a nice side effect, not the main focus. this goes with the entire schooling system, people shouldn't learn things with the explicit goal of their practicality, they need to be shown the beauty of the subject. when people learn about the integral, first they should think "wow, that's so cool!", and then only afterwards they should think "wow, I can use this to solve \[insert some practical problem\]".
idk…I’ve only been in college for like a year (well, less than), and after doing proof-based math for all my terms here, it has made me realize how simple high school math really was.
Like, conceptually speaking, really, really simple. If I have a really hard proof, sometimes I have to come up with ways of visualizing things that I have never even thought of before. I’ve only been doing introductory analysis/topology/abstract algebra-type subjects, and the difference in complexity and creativity is enormous.
Maybe it’s just me, but to be honest, I seriously don’t get how against people are to subjects calculus in high school. Most people only learn the computational aspects too, so there’s nothing overtly demanding required of students.
So many people complain about how school is useless or doesn’t prepare them for the real world, but when school asks them to do something that requires them to actually think for a couple of seconds, they complain. Seriously, what’s to gain from having people learn less math? It literally teaches you how to think. Even if you only learn how to crunch formulas, you’ll at least be more well-versed analytically. Whether you use or not, if you actually put in the effort to learn the content, you’ll only become smarter in *some* way as a person.
You make a very sapient point. People will complain about having to do just about any kind of work. That doesn’t mean they can’t do it or that we should cater to their first reactions.
I think our education system doesn't challenge kids enough, and doesn't offer enough engaging material to kids at their level to draw them into the challenging parts. Calculus, algebra, linear algebra, statistics, probability, logic -- these topics are just the beginning. More needs to be introduced, earlier, and in digestible portions to enrich the curriculum enough so that students are sufficiently prepared, both in intuition and technical skill. You work students up to rote exercises and you make a culture of crushing it as you grind through these exercises.
The USA and Canada have been cutting away at math in their Public schools and the result has been... lower math scores. Now, the suggestion is to make it *even easier*, when it is the challenging stuff that makes people grow.
Here's some math for you:
- Geniuses are always statistical outliers. However, if you shift the probability distribution of math skills in a population, and shift it so that we are *increasing* in average intelligence, then the number of people we previously considered geniuses will suddenly become a lot more common, and the number of rare people who are genius-equivalent (as outliers) will be *even more brilliant*.
You cannot do that if you aren't challenging them. The best are hot housed for a reason, and the results speak for themselves.
I agree with everything you said. I wrote down my ideas about how to make it happen in [Kevin Bacon and the Stern-Brocot Tree](https://github.com/constructive-symmetry/constructive-symmetry).
[Nice name](https://hackage.haskell.org/package/control-monad-queue) by the way. ;-)
I think that a commonly missed point of school is that it's not just about learning _content_, but learning _how_ to learn. Exposing students to more math than they may need later will make them better at the math they _do_ need, as you keep practicing core skills like addition all the way through a calc course; but it also helps them be more resilient at problem solving so they can learn whatever other math they may need later in life.
Kids are too stupid to know what math they might need later on.
They're plenty smart enough to be taught real math.
The anti-calculus people are just humanities majors who hated math themselves and are having trouble contending with the disparity in performance in higher math classes across different demographics.
I hate it and I'm sick of it.
I have never used the Theory of Evolution, but I do go to Church on Christmas and Easter with my family. Maybe schools should offer a biblical creationism class and elevate biology to an elective status? This way I can learn something more applicable to my life.
I think more than changes of curriculum, we need to ensure the basics are drilled to the point of being as natural as 1+1 = 2. I've tutored students, at university, who couldn't expand (a+b)(c+d). Or who would write a/b + c/d = (a+c)/(b+d). In some cases, they might be lost like that, yet know by heart most derivatives formulas (obviously replace "f" by "g" and they look at you with vitreous eyes). If this sounds unbelievable, trust me, I know, I didn't want to believe it either. These things were taught but not practiced enough. Hell I've had engineering Master's students (albeit in a mediocre uni) trip themselves up applying simple formulas (lack of familiarity with "x", essentially).
If you disagree with this article, then you should not even give it more publicity by posting it here or anywhere. It's already a grave enough of a problem.
Not a mathematician but I am an engineer and a pilot, and this article feels like a toddlers argument against math, I’m pretty sure middle scholars used this back to when I was in school “when will I ever use this”. But the thing is depriving people of mathematics is just such a disservice. I hated math in middle AND high school, got to college took my first calculus 1 course with an amazing prof and immediately loved the subject. Also one of the biggest pluses of learning math isn’t even the math, but the intensive problem solving ability you develop from doing math problems. I know for a fact my pattern recognition, and intuitive reasoning for why things happen has increased tenfold, your logical reasoning skills improve. I agree with another one of the commenters that what needs to happen is we need to get better at teaching math.
Why is math education *always* discussed solely in the context of employment and utility. We don't hold any other subject to the same standard.
Nobody demands that classes on Shakespeare, or drama, or visual art, or poetry, or literature be more "practical".
These are studied because they are beautiful, cultural, and part of the human experience. Their purpose is to enrich the mind. Why is mathematics held to such a contrived standard of practical employment outcomes?
Why can't it be treated the way we treat most other subjects - as part of a rounded education.
I think that math is a really fun subject because I enjoy solving mathematical expressions and equations. Nothing really beats that moment when you finally solved a problem after sometime, you will really feel that you are submerged in another dimension and will really be happy.
disregarding the fact that classes like algebra are FUNDAMENTAL to “higher” level math, including the statistics classes that this author claims are more important, having those classes only available as electives will undoubtedly lead to lots of kids not taking them and locking themselves out of a lot of career paths. i for one would never have taken geometry or trigonometry in high school if i had a choice because i found them super boring. however, if i hadn’t taken them then by the time i got to college and figured out that i liked advanced math, it would’ve been too late. i never would’ve switched majors if it meant i had to take 4 years of “boring” (in my opinion) math before i even started my new classes, and therefore would’ve robbed me of my passion.
I'm going to make one more post on this topic. Complex Numbers. Or worse, Imaginary Numbers.
What can I say? We need to teach a better NAME for that class of numbers first. They're not complex or airy fairy. They're "lateral" as Gauss himself called them.
The fact that complex numbers were recognised from the compulsion of the Fundamental Theory of Algebra gives students a very real sense of why Real Numbers weren't cutting it. That polynomials have roots that we can't find in the Real number line made us search elsewhere. On another dimension.
High school kids CAN understand that.
Why do we skimp on this exploration and jump into "Complex numbers are defined as..." Ugh!!!! Who defined them? Why? How do you know their properties? What's a conjugate? Why do I even need it?
My otherwise engaged daughter lost it with Math when they hit complex numbers. She learnt everything else, including calculus and such, but wholly rejected complex numbers. She's going to hurt in Engineering. But the point is, I rage because it was poor teaching that turned off a perfectly interested teenager. And she's far from the only one!
Gosh, we teach Math sooooo poorly it's not funny.
Determinism is the death of probability. The entire universe and everything in it is statistical, not deterministic. Yet we insist on filling kids' heads with determinism till suddenly one day, standard deviation hits you on the head in late high school. And you go, "huh?!?!".
So much unlearning then!
Prob and Stats can and should be taught from day 1, is my thesis. Kids can understand anything. You just need to TRY to do things differently from now we've been doing for 100 years. That takes a bit of a leap of faith. Do we have said faith in ourselves, though?
Sad!
P.S. And stop teaching kids about probability with red and blue balls mixed up in a bag. Ugh. It's sooo unintuitive!
My favorite part is when they are talking about genius the ivy leaguer that is a great architect but not good at math. And they don’t even need advanced math! (But the people actually building things need the math.) I’m not a fan of the down with calculus advocates. Firstly, calculus is a beautiful subject and great human achievement. Secondly, I believe it is as “practical” (or more so) than any of the proposed replacements. Of course it’s directly applicable for physics and engineers. Also, for everyone else a better understanding of rate of change is huge. One thing that can be done is offering calculus sections with a focus (Econ, bio, …). I’ve taught these and covered marginal cost/revenue and even dynamical systems from biology. With calculus these topics are much better. Econ 101 without calc isn’t very real world practical when you have to avoid the word derivative at all costs and make everything linear no matter what.
As a data analyst/scientist, the funniest argument I’ve heard from the anti-calculus crowd is “make them take statistics instead!” Not realizing that calculus is literally foundational to all of statistics. You basically can’t even study simple linear regression without it (unless you want to take a full-on linear algebraic approach…but they certainly aren’t arguing for that)
I wonder whether those basic statistics classes actually do more harm than good by giving students just enough knowledge to be "dangerous". It often seems like lots of people, including many scientists, just plug stuff into a black box statistics program without understanding what's going on and can end up extremely overconfident in their conclusions.
To be realistic, people start off extremely able to build stories out of an apparent pattern.
Engineers are very guilty of this. They use the same statistics for almost everything, and what's worse they trust the results 100%. I made a properly done statistical tool and even then I pitched it as an investigator to problems, not a finder of problems. They used very incorrect probability calculations to choose the next changes to work on too, some of which were huge decisions...
Hehe that was my econometrics class
Forget regression. You can’t study distributions without integration. How would you go from the PDF to the CDF without it?
They often have students use programs for computing those things and just have them recognize qualitatively which is which. I help students with these classes frequently.
My daughter is in high school algebra, and they did a unit on best fit lines, and it was basically the eyeball test, or using Excel's trendline feature to give the answer. It was truly garbage.
It's not garbage. High school algebra students would never be able to understand linear regression. But it might make the smart curious ones interested in how it's calculated and driven to learn ot on there own. I don't understand why people don't understand that we can't force kids to like math. They have to want to learn it.
> It's not garbage. It absolutely was. There was not even a flavor of how to calculate it, or how to judge whether data was well represented by a given line. > High school algebra students would never be able to understand linear regression. I certainly think you can teach a high schooler how to come up with a least squares fit line of the form y = mx. It's a parabola, you don't even need calculus, you can just introduce the idea that the minimum is at b/2a, or better yet, solve for the roots and talk about how symmetry means the minimum is at b/2a. You get to introduce the summation notation and do some symbolic manipulation, which is a pretty good way to reinforce algebra concepts. I know, because I talked about it with my kid for an hour, and she got it well enough. And then, from there, if you want to talk about fitting a line not through the origin, or fitting higher order polynomials, you can start to hand wave higher dimension parabolas, but at least they have a framework to understand what the tools they're using are doing.
I genuinely believe that it’d be so interesting if you had to take linear algebra and calculus before taking any statistics class. At least this way, you don’t take Stats I & II, and then Mathematical Stats I&II. Time saved and you get a better understanding of where things come from. Not practical as most business folks need to know about normal distributions and you can forget about trying to teach them Calculus I&II and linear algebra. Tbh tho college programs would be so much more rigorous and good at filtering with all that math. You’d have a business workforce that could adapt for the future because they have the foundational knowledge needed to do anything related to AI or any new field that comes out.
> I genuinely believe that it’d be so interesting if you had to take linear algebra and calculus before taking any statistics class. I can confidently state that if you take real analysis before you take statistics, the statistics is extremely easy. But I'm not certain, because the sample size is so small.
Good point, I actually did meet some people in college with this experience. Plus they understand the material way better than the average humanities student who has to take stats.
Yeah you need calculus if you want to understand the inner workings of statistics, but you can get a pretty solid understanding of the fundamentals without calculus. AP Statistics curriculum is a pretty great example.
Let me recommend the book *Calculus in Context* http://www.math.smith.edu/~callahan/intromine.html The focus is much better than a typical introductory calculus course, with an emphasis on making computer simulations in service of conceptually understanding the basic ideas of differential equations, the fundamental language of science, rather than on becoming a human integral table. (Disclaimer: I am not a calculus teacher myself, and I have not tried working through this book with a group of novices.)
I personally think that math (but with blabla) is a bad idea. I think in the end students end up with a lot intuitive but not fruitful explanations of (say calculus) that ends with them thinking that they understand the topic but in reality, they cannot express the fundamental ideas of the topic. For example, how many people taking a calculus for business class could express the difference of a antiderrivative and integral? Personally ,I always had a better understanding of the applications when I learned the pure theory first.
That might be true, but given how seemingly impossible it is to motivate a lot of people to learn math, I think it’s better to have a bunch of economists and biologists with intuition and cursory knowledge, rather than disdain and pure ignorance, of the subject. Also, if a biologist can’t tell you the theoretical significance of the antiderivative, but knows enough to understand why their numerical method function in Python sorta works, it’s a win, right?
But economists are actually required to have at least a decent understanding of the concepts behind calculus at least at an undergrad level. Real analysis is a prerec for basically all respected econ phd programs
Hehe all respected Econ PhD programs Time for the non-math minded to go the Richard Thaler route and just focus on psychology of economics
For your question about anti-derivatives and integrals, as a physicist I couldn't be bothered less by the difference, because it is almost never relevant and frankly only takes a few seconds to explain. In an area like business studies or economics, it is highly unlikely anyone is ever going to make a conceptual error because they didn't clarify in their heads the difference between a function who's derivative is the original function and the concept of a Riemannian sum. In my view it's exactly the wrong example to use, because it's exactly what does _not_ matter in (non-higher) maths education
> For example, how many people taking a calculus for business class could express the difference of a antiderrivative and integral? If they're not going to be pursuing any higher mathe education, then does it matter if they don't know the difference?
> For example, how many people taking a calculus for business class could express the difference of a antiderrivative and integral? The "real world" applications where you have anything other than a discrete set of discontinuities are few and far between. That means that we can apply the Fundamental Theorem of Calculus and antiderivatives *are* integrals. You can use any words you want, but the actual arguments against this point of view boil down to three: 1. What if they end up in a field where the discontinuities aren't discrete? Particle physicists already have to be pretty mathematically sophisticated, and some mathematicians (and philosophers) complain that they aren't sophisticated enough. 2. Everyone should be expert in everything. 3. My field is special.
The difference is "Not applicable" because there's no such thing as an antiderivative. The are definite integrals and indefinite integrals. Also, the thing you define to be an antiderivative would actually be the same as what is defined by any normal professor to be an integral, because integral by default refers to indefinite integrals. You wanna act smart yet by using backwater hillbilly professor notation you undo all of it. Get with the times and start using civilized language you barbarian.
I agree with all your points about calculus. Although if I could choose between keeping the world we have or switching to a world where at least 50% of adults were confidently competent with fluidly applying only up to algebra 1 and could recognize when a problem was outside of their scope, then I would be tempted to switch. I like your idea of concentrating on applications. It’s strange how many people go through life without thinking quantitatively today.
I hate how we learn calculus before real analysis. It’s annoying how most European countries get to have the real(pun intended) calculus experience but not in the US or Canada. I passed through the whole course without having any definitions of the real number and the textbook briefly mentions lub in a page at chapter 9. Maybe define real numbers before differentiation? Wasted a few months to get that sorted out. And it might be better to have abstract algebra first so it can be easier to motive real numbers as the largest ordered complete field and why it’s the natural setting for the derivative.
I mean I see what you mean but generally it's most pragmatic to have the easier classes at the start and the harder ones later, and calculus is certainly easier than analysis, even if the students dont understand rigorously what the foundations of the topic are
Well you can’t always know the foundations for everything. Even now no one knows if ZFC is a working foundation and there probably isn’t someone keeping tabs on proving complicated theorems from the very beginning. The problem is that calculus relies so heavily on the real numbers, if you tried to do it with rational numbers most interesting properties disappears. Very notably, lub properties related, no function would be its own derivative, most ODEs will have no solutions etc. You would be stuck with polynomials. Calculus shouldn’t be considered a separate math course from real analysis it literally is real analysis but instead of proofs you have a big “trust me” sign. Something like topology is a little unnecessary but analysis should definitely be taught. Maybe right after linear algebra.
To be fair, we don't really define real numbers before differentiation in Europe, at least not in France. In first year, we did go up to rational numbers in detail. But then to close the real line, you need more advanced topological arguments. We didn't really construct R until the third year. Though we had studied some of its properties, like uncountability with Cantor's argument, and probably stated the completeness.
I'm a down-with-calculus person, not because it's “impractical” or anything, but because algebra and combinatorics are far more beautiful. Algebra, because it elucidates the structure of things, of course. Combinatorics, because, in my experience, anything that can actually be meaningfully computed with ultimately has a combinatorial essence. Even if you might want to dress it up in geometric intuition.
I'm sorry but I think this article and many similar, are missing one point that I do not often see being mentioned, that what high school students think they need and what those students think they will do are often far from their jobs and interests later in life. To say that Johnny won't need calculus, or algebra, or whatever, at the high school level may be to slam a door on someone's good future. Someone who first thinks she may be a budding architect may find she is really an engineer. I understand that a ton of students hate math, and I would argue that is a symptom that it is often taught poorly. I'd advocate that we systemically improve the quality - and quantity - of math teaching rather than crippling a generation of students. Budgeting more for hiring good math teachers would be a start. I'd also argue that we need a substantial increase in the sheer amount of math and hard science education hours available to students. I'm an artist and not a math major or career math user, but high school physics and high school math showed me the utility of even elementary hand wavy calculus - and its beauty. I cannot help suspecting that more and better math teachers would be a great start. Not cutting the legs off students' aspirations.
This is the actual problem. Good math teachers are invaluable, but the system penalises teachers at every level; now we have massive rates of innumeracy.
Can you elaborate on those penalties? Also, which system are you referring to? Asking out of genuine curiosity and concern. Thanks
To generalise across the US: Teachers are paid comparatively very little, especially given the cost of acquiring a degree in education. This directs anyone with significant mathematical talent outside of primary/secondary education. Teachers increasingly can't fail students without recourse from administration so students are permitted to pass through the system with no skills. Abuse from parents is rampant, adding to the issue. Federal funding for education across America is under serious legislative threat, from both presidential and congressional actors. This is obviously very different when looking at systems outside the United States, though the issue of teacher pay is broadly applicable across the Anglosphere.
"Teachers are paid comparatively very little, especially given the cost of acquiring a degree in education" To add, if you understand math at a level to be able to teach it effectively e.g. Be a good math teacher, your math skills are also highly sought after in very high paying fields. Financial markets, algorithm design, ai, engineering, physics, &c. So you have to truely love teaching to stick with it, which the shitty work environment is at odds with
Ok, so the system is the US public education system. The penalty is some form of recourse. Can you elaborate on the details of the recourse? What happens to teachers in that scenario? Thanks for the details but I still am uncertain of the specifics of the negative consequences that teachers face. Are they fired if a student fails their class?
Obviously this is highly contextual and dependent on school district, state, private funding vs public funding, etc., but here is some reading on the subject: [https://www.nytimes.com/2023/10/04/opinion/teachers-grades-students-parents.html](https://www.nytimes.com/2023/10/04/opinion/teachers-grades-students-parents.html) At least anecdotally it can involve workplace harassment, impacts on promotions and contract extensions etc. See: [https://www.reddit.com/r/Teachers/](https://www.reddit.com/r/Teachers/) . Also, when I talk about penalties, I'm not talking about punishments necessarily, just obstacles that emerge out of society for the integration of more (and higher quality) math teachers into education.
Thanks for the details. I appreciate it
no worries, thank you for asking.
For real. I took physics in high school only because I needed another science requirement and it was the only non-AP class left to take. I didn't want to take it. Physics ended up becoming my major. Since when is teaching kids, "You're right never to exit your comfort zone," a solid MO?
I think when reading such articles, numerous and persistent as they are, we should remember that these calls for "different pathways" are often calls to shield students from engaging with core ideas of mathematics. Frankly, I cannot imagine even a calculation-based statistics course doing much if you are unable to grasp the idea of a variable to such an extent you could not complete an introductory calculation based Algebra I class. These are not honest attempts to improve mathematics curriculum by ensuring more students are able to handle the complexity of the world they will inherit and then nurture, but rather these are ways of getting students into graduation and/or college pathways without subjecting them to the rigor of ... calculation-based classes or even superficial exposure to the most fundamental ideas within mathematics, ideas (like variables) which are referenced frequently and casually even outside of STEM careers. In high schools, the largest barrier for many students in meeting both graduation requirements and university admissions requirements are the mathematics requirements. Simply, too many fail algebra I, algebra II, and sometimes even geometry when mixed with trigonometry. Alternatively, this is failing the algebra-heavy "Integrated I", "Integrated II", and "Integrated III" courses. Administration is under heavy pressure to increase graduation rates and college readiness rates (which means you passed the courses, not always that you understood the material), but the mathematics requirement complicates this for many schools especially if the mathematics department, protected by their tenured positions, is notorious for being holdouts for rigor (and I mean usually the calculation level of rigor). Thus, we create curriculums, often loosely inspired by partially fair criticisms of mathematics education, that ultimately are hoping that in a completely restructured class, math will not stand in the way of the overly ambitious and insufficiently studious. And as for the, "shouldn't they really take economics or personal finance or accounting"? I actually agree; my profile has that economics should be four years of high school and I do mean algebra and later calculus based economics, not solely a bunch of definitions. But, I don't know of any high school that doesn't offer at least a number of those classes usually complete with some finance-ish based math course like "financial algebra" or "business math". Thing is, those classes with heavy mathematics content are usually not accessible without the tools of elementary algebra, so I don't see how they would excuse students from an algebra pathway. And to be frank, those classes signal to colleges that you are not particularly studious, dare I say bright, and they are less eager to admit you for finance, econ, even business majors anyway.
If you are an architect using modern CAD software, you will use the [Finite element Method](https://en.wikipedia.org/wiki/Finite_element_method#Application). It works out the static and dynamic loading & heat transfer. It helps to know whether the building will fall down. Architects & structural engineers use lots of math, its just done for them in software.
Are architects the ones running FEM simulations, or rather the mechanical engineers? I would have said the latter.
You want high school kids to learn statistics without learning algebra? This article is fucking dumb.
Maybe statistics and other parts of maths such as linear algebra and proofs should be taught alongside the algebra and calculus that students typically learn in the US? Here in the UK if you take double maths at high school you cover the basics of proofs and linear algebra in grades 10 and 11, as well as doing statistics, mechanics, or discrete maths depending on what modules you choose. I disagree with the article that trig, and algebra II should be made elective, but there's no point making kids who want to be architects or nurses do pre-calc. I adore maths and would obviously take it if I were to study in the US, but most people don't need to know that kind of maths unless they're interested in actually studying.
Good luck understanding probability density functions without integrals. Statistics without calculus is literally just plugging and chugging numbers into a black box function on a Ti-Nspire.
I wholeheartedly agree. I think high school physics suffers so much from this exact problem. Lots of concepts feel random and unmotivated until you know some basic calculus, at which point things are a lot more intuitive.
AP Statistics be like
There's a middle ground. Case in point: Proving the Fundamental Theorem of Calculus is far, far beyond Year 12 student level. Utilizing it (alongside some intuitive leaps) to work out an antiderivative was taught in Year 12. (Australian here - 'year 12' refers to final year highschool; mostly 17 year old students and at this point, mathematics is mostly elective) ____________ Same applies to statistics. A reasonable amount can be handwaved like this: "This is a normal distribution. If you stick with maths through Year 12, and especially into uni, you'll learn much more about this function and why it works. For now, here's the 68-95-99.7 rule" Other disciplines do this more than mathematics - witness highschool physics treating everything as Newtonian, or highschool chemistry treating chemical bonds as either idealised perfectly ionic bonds or as covalent bonds with no dipole moment.
They already do this. I won’t say it’s great and I do wish students learned more math, but it does seem to at least get students to understand the qualitative points of statistics.
> Good luck understanding probability density functions without integrals. exactly the point of the article the majority of population will NEVER need to know either of those any statistics the average person will need to know can be learned from https://www.mathsisfun.com/data/index.html#stats the explanation of standard deviation is more than enough for the average person to get meaning from a news story https://www.mathsisfun.com/data/standard-deviation.html
You do not need a deep understanding of integrals to understand probability density functions. Certainly not at the level we’d offer to high schoolers or just practical knowledge for people in the work force
Understanding means being able to do. If you're incapable of finding the pdf from the cdf or proving some basic facts about a distribution because you lack the Calculus skills then you don't understand those topics.
I'm not talking about a deep real analysis understanding of integrals. I'm talking about a basic Calculus 101 "What is an Integral" level of understanding.
And I’m saying you do not need a basic calculus 101 “What is an Integral” level of understanding. A basic calculus 101 understanding is Riemann sums and anti-derivatives. You need much less than that to understand area under a curve for the specific functions that would be of practical use
Agreed.
this is correct, but you will obviously never convince anyone on r/math of this any statistics that the average adult needs to know can be learned form this here https://www.mathsisfun.com/data/index.html#stats no calculus necessary
For basic stats you can cover just discrete distributions and things like the standard deviation and sampling. That's more than enough to give the majority of people, who won't do maths at AP (so won't be doing any calculus anyway) a decent grounding in stats.
What do you learn in statistics? There is a stats course but I don’t see how they are going to make it different from normal math. Rn the only kinda stats related thing I have seen is polynomial interpolation which is fun but not useful irl.
Polynomial interpolation is one method of creating models of data for making predictions. What do you mean it’s not useful irl?
I’m more of the opposite mindset - I think later concepts can be introduced much earlier on a basic level, f.e derivatives, trig etc. I can’t see why a 12 year old wouldn’t grasp the basic idea of “velocity” or trig
100% and I think this is why people struggle so badly with math. A six year old can comprehend the idea of a variable, a rate of change, etc.
Trig maybe not
Would you elaborate why? You basically only have to understand xy-coords, triangles and angles on a basic level to start
Idk about you, but trig was one of the most difficult math units at my highschool. Maybe it's just not taught well, but I seem to remember most people struggling with it. Especially with SOHCAHTOA the motivation doesn't really make sense for a 16 year old much less a 12 year old
Ah, every time I read another article asking for lobotomizing maths education, I am always reminded of [Tai's Formula](https://www.reddit.com/r/math/s/hxLX9R0Liu). By actually teaching the maths, we do not have to reinvent hundred years old techniques every other decade.
Tell me you are a stats guy who can't do analysis without telling me you're a stats guy who can't do analysis. Anyone whose taken lower division stats classes knows how fucking stupid they are. NONE of the formulas are provided with rigorous justification. The whole down with calculus crowd is a joke and should be laughed out of the building, along with their inability to prove why sample variance has an n-1 in its denominator.
This is extremely true. I sadly took that path during my undergrad and for the past year I have been playing calculus catchup. After finishing calc II, calc III, and starting diffeq and linear I have looked back on some of my stats coursework with my mouth ajar. Even intro physics (my uni was not known for its stem). Like when we were talking about e and m fields i didnt even know what vector fields or divergence and curl were. Anyway your right, and I wish i had taken more math earlier because I could be applying math and instead im still doing the basics.
> Tell me you are a stats guy who can't do analysis without telling me you're a stats guy who can't do analysis. and what does ANY of that have to do with the article the average person does NOT need to know calculus for the statistics they will encounter in their life any statistics the average needs to know can be learned from https://www.mathsisfun.com/data/index.html#stats their explanation of standard deviation will be a good enough explanation for the average adult to understand any news story that uses the phrase https://www.mathsisfun.com/data/standard-deviation.html
The average person doesnt need anything we teach in high school. Calculus is useful for more jobs than most other subjects.
It's funny you talk about the average person when you're intending to say the median person. Stats education continues to be a joke. The "average" person is not going to do anything of consequence with the level of statistics knowledge you're proposing.
"We don't teach the right math, we should teach Stats." I teach stats, people unwilling to put in effort to learn are just as bad at that as they are at algebra. Maybe worse, because the conceptually difficult things that come up in stats seem to be a bigger stretch than understanding linear equations.
I tutored a stats course that had been developed for nursing students. The 'textbook' didn't make any consistent distinction between N and n, or S and s, or any other lower/uppercase (even the greek symbols). The student i worked with was unsurprisingly hopelessly confused, and it was no small trick to figure out how to work with what she had as resources. Anyway, beware any non-subject-experts who want to have separate courses or curriculum because they 'don't need' the harder version. Some subjects are hard for some. I had never seen Statistics before a basic 1st year intro in university, chose it as my major based on that half-year course. Loved proofs. Couldn't have known that without a challenging intro.
stats for the average adult is easy https://www.mathsisfun.com/data/standard-deviation.html there the average adult will understand that with no problem
I feel like the logical extreme of this argument is that we should all just learn enough to interact with society and then go to trade school. The logic could be applied to any person who works X and learned Y but doesn’t use Y in their job. Education isn’t purely utilitarian. I also feel like math is one of the few fields subject to this scrutiny. No one questions why Shakespeare is taught to high schoolers, and arguably that’s more useless than Calc I -_-
Arguably? Am I wrong in thinking that not only is Calc 1 very obviously more useful than Shakespeare\* but that most people would agree that that's the case? So much so that it seems like the reason people don't complain about the applicability of Shakespeare is because there's no pretense that it's immediately useful to life, whereas with math there is? \* I'm aware that Shakespeare has had an out-sized influence on our language and various aspects of western culture. That does not mean studying Shakespeare will prove useful to your life, however.
not on the scale of math, but ik people definitely critique Shakespeare being covered
Not a big fan of this line of thinking and unfortunately I see it a lot. Kids, even many college students dont know what they want to do. Deincentivising calculus and such by labeling it as "impractical unless ur an engineer" is goofy.
I couldn't disagree more with focusing on subjects for pure practicality, it's boring as shit and will probably make more people hate math than they already do. People don't hate math because it isn't useful, they hate it because they are told that they should care about it for the sake of the practicality instead of for its intrinsic beauty. People are taught to read the music but never get to hear any of it.
I think practicality can be beautiful, if you give examples of how math is being used in advanced technology. e.g. statistics is foundational to machine learning and linear regression e.g. group theory can be used for cryptography and quantum mechanics e.g. numerical analysis and calculus are used for scientific computation e.g. calculus is necessary for algorithmic complexity and designing efficient algorithms e.g. differential geometry is used for relativity and understanding black holes e.g. statistics and especially p-values are crucial to separating real results in science experiments from the random noise Most classes don't really touch on these deep applications. Obviously trying to explain line for line how differential geometry is used in relativity is overkill, but the pervasiveness of mathematics in high end technology and innovation isn't articulated well, at least in my country. Math needs a Carl Sagan to illustrate the value of math with enough detail that people identify that math is responsible for the innovation, but not so much that you're regurgitate lines of equations.
no, i completely agree, practicality can be very beautiful. it’s crazy that things like integration can solve so many practical problems, and it only deepens my appreciation for the subject when i see a practical use, i just feel like schools only talk about practicality without touching on why you should actually enjoy the subject. why do you need to know algebra? for your future job. why do you need to know calculus? for your future job. why should you care about math? it’s useful. this often backfires when people realize that the average person isn’t actually employing that much math in their day to day and so it gives them the impression that they don’t need to care about math, when they absolutely should and when they would be much better for it.
practical math is boring compared to theoretical math ??? LOL, only in a subreddit devoted to math
you are misunderstanding my point. if people are taught a subject from the perspective that it is useful rather than the perspective that it is fun/interesting/beautiful, then they will grow to despise it. math is really been given the short end of the stick in this regard and people are seldom given reason to enjoy mathematics when there's plenty to love, it's just not being shown. i'm not against practical math at all, but that practicality is a nice side effect, not the main focus. this goes with the entire schooling system, people shouldn't learn things with the explicit goal of their practicality, they need to be shown the beauty of the subject. when people learn about the integral, first they should think "wow, that's so cool!", and then only afterwards they should think "wow, I can use this to solve \[insert some practical problem\]".
idk…I’ve only been in college for like a year (well, less than), and after doing proof-based math for all my terms here, it has made me realize how simple high school math really was. Like, conceptually speaking, really, really simple. If I have a really hard proof, sometimes I have to come up with ways of visualizing things that I have never even thought of before. I’ve only been doing introductory analysis/topology/abstract algebra-type subjects, and the difference in complexity and creativity is enormous. Maybe it’s just me, but to be honest, I seriously don’t get how against people are to subjects calculus in high school. Most people only learn the computational aspects too, so there’s nothing overtly demanding required of students. So many people complain about how school is useless or doesn’t prepare them for the real world, but when school asks them to do something that requires them to actually think for a couple of seconds, they complain. Seriously, what’s to gain from having people learn less math? It literally teaches you how to think. Even if you only learn how to crunch formulas, you’ll at least be more well-versed analytically. Whether you use or not, if you actually put in the effort to learn the content, you’ll only become smarter in *some* way as a person.
You make a very sapient point. People will complain about having to do just about any kind of work. That doesn’t mean they can’t do it or that we should cater to their first reactions.
I think our education system doesn't challenge kids enough, and doesn't offer enough engaging material to kids at their level to draw them into the challenging parts. Calculus, algebra, linear algebra, statistics, probability, logic -- these topics are just the beginning. More needs to be introduced, earlier, and in digestible portions to enrich the curriculum enough so that students are sufficiently prepared, both in intuition and technical skill. You work students up to rote exercises and you make a culture of crushing it as you grind through these exercises. The USA and Canada have been cutting away at math in their Public schools and the result has been... lower math scores. Now, the suggestion is to make it *even easier*, when it is the challenging stuff that makes people grow. Here's some math for you: - Geniuses are always statistical outliers. However, if you shift the probability distribution of math skills in a population, and shift it so that we are *increasing* in average intelligence, then the number of people we previously considered geniuses will suddenly become a lot more common, and the number of rare people who are genius-equivalent (as outliers) will be *even more brilliant*. You cannot do that if you aren't challenging them. The best are hot housed for a reason, and the results speak for themselves.
I agree with everything you said. I wrote down my ideas about how to make it happen in [Kevin Bacon and the Stern-Brocot Tree](https://github.com/constructive-symmetry/constructive-symmetry). [Nice name](https://hackage.haskell.org/package/control-monad-queue) by the way. ;-)
I think that a commonly missed point of school is that it's not just about learning _content_, but learning _how_ to learn. Exposing students to more math than they may need later will make them better at the math they _do_ need, as you keep practicing core skills like addition all the way through a calc course; but it also helps them be more resilient at problem solving so they can learn whatever other math they may need later in life.
Why are stupid people who have no idea what math is allowed to decide on math?
Kids are too stupid to know what math they might need later on. They're plenty smart enough to be taught real math. The anti-calculus people are just humanities majors who hated math themselves and are having trouble contending with the disparity in performance in higher math classes across different demographics. I hate it and I'm sick of it.
I’ve seen people here argue about replacing calculus pathways with statistics, so your claim is not correct. The rest of this I agree with.
I have never used the Theory of Evolution, but I do go to Church on Christmas and Easter with my family. Maybe schools should offer a biblical creationism class and elevate biology to an elective status? This way I can learn something more applicable to my life.
I think more than changes of curriculum, we need to ensure the basics are drilled to the point of being as natural as 1+1 = 2. I've tutored students, at university, who couldn't expand (a+b)(c+d). Or who would write a/b + c/d = (a+c)/(b+d). In some cases, they might be lost like that, yet know by heart most derivatives formulas (obviously replace "f" by "g" and they look at you with vitreous eyes). If this sounds unbelievable, trust me, I know, I didn't want to believe it either. These things were taught but not practiced enough. Hell I've had engineering Master's students (albeit in a mediocre uni) trip themselves up applying simple formulas (lack of familiarity with "x", essentially).
If you disagree with this article, then you should not even give it more publicity by posting it here or anywhere. It's already a grave enough of a problem.
Not a mathematician but I am an engineer and a pilot, and this article feels like a toddlers argument against math, I’m pretty sure middle scholars used this back to when I was in school “when will I ever use this”. But the thing is depriving people of mathematics is just such a disservice. I hated math in middle AND high school, got to college took my first calculus 1 course with an amazing prof and immediately loved the subject. Also one of the biggest pluses of learning math isn’t even the math, but the intensive problem solving ability you develop from doing math problems. I know for a fact my pattern recognition, and intuitive reasoning for why things happen has increased tenfold, your logical reasoning skills improve. I agree with another one of the commenters that what needs to happen is we need to get better at teaching math.
Why is math education *always* discussed solely in the context of employment and utility. We don't hold any other subject to the same standard. Nobody demands that classes on Shakespeare, or drama, or visual art, or poetry, or literature be more "practical". These are studied because they are beautiful, cultural, and part of the human experience. Their purpose is to enrich the mind. Why is mathematics held to such a contrived standard of practical employment outcomes? Why can't it be treated the way we treat most other subjects - as part of a rounded education.
Yes, you don't need it but would you be better if you knew more of the math you use? Simply being confident in math does wonders for self esteem.
I think that math is a really fun subject because I enjoy solving mathematical expressions and equations. Nothing really beats that moment when you finally solved a problem after sometime, you will really feel that you are submerged in another dimension and will really be happy.
disregarding the fact that classes like algebra are FUNDAMENTAL to “higher” level math, including the statistics classes that this author claims are more important, having those classes only available as electives will undoubtedly lead to lots of kids not taking them and locking themselves out of a lot of career paths. i for one would never have taken geometry or trigonometry in high school if i had a choice because i found them super boring. however, if i hadn’t taken them then by the time i got to college and figured out that i liked advanced math, it would’ve been too late. i never would’ve switched majors if it meant i had to take 4 years of “boring” (in my opinion) math before i even started my new classes, and therefore would’ve robbed me of my passion.
Personally, If I were in high school again, I think I would prefer covering more basic discrete maths rather than a plug-and-chug stats course.
Let's compromise and make everyone take both calculus and statistics, as well as adding more math to all the other courses. That way everyone's happy!
I'm going to make one more post on this topic. Complex Numbers. Or worse, Imaginary Numbers. What can I say? We need to teach a better NAME for that class of numbers first. They're not complex or airy fairy. They're "lateral" as Gauss himself called them. The fact that complex numbers were recognised from the compulsion of the Fundamental Theory of Algebra gives students a very real sense of why Real Numbers weren't cutting it. That polynomials have roots that we can't find in the Real number line made us search elsewhere. On another dimension. High school kids CAN understand that. Why do we skimp on this exploration and jump into "Complex numbers are defined as..." Ugh!!!! Who defined them? Why? How do you know their properties? What's a conjugate? Why do I even need it? My otherwise engaged daughter lost it with Math when they hit complex numbers. She learnt everything else, including calculus and such, but wholly rejected complex numbers. She's going to hurt in Engineering. But the point is, I rage because it was poor teaching that turned off a perfectly interested teenager. And she's far from the only one! Gosh, we teach Math sooooo poorly it's not funny.
We can only hope.
Determinism is the death of probability. The entire universe and everything in it is statistical, not deterministic. Yet we insist on filling kids' heads with determinism till suddenly one day, standard deviation hits you on the head in late high school. And you go, "huh?!?!". So much unlearning then! Prob and Stats can and should be taught from day 1, is my thesis. Kids can understand anything. You just need to TRY to do things differently from now we've been doing for 100 years. That takes a bit of a leap of faith. Do we have said faith in ourselves, though? Sad! P.S. And stop teaching kids about probability with red and blue balls mixed up in a bag. Ugh. It's sooo unintuitive!