The Math Myth (2016)
econlib.org
econlib.org
I've used a lot of math. Most of it built on high school algebra and geometry. Does that mean that's all I needed to learn?
No.
The key thing I do use is more ephemeral: Mathematical maturity. In my current job, I use math I never learned in graduate school. I'm able to learn it quickly because I learned a lot of math back then. My math classes were a way to develop mathematical maturity.
Which specific math I learned in graduate school was almost incidental. What I picked up was the ability to learn new math.
https://learningcurrents.weebly.com/visible-learning-surface...
It takes a minimum of about 3 semesters to be able to get linear algebra to the stage of transfer learning. There are slower ways to do it (which most people follow), but there aren't shortcuts -- there is no way high school Algebra 2 can get you there in the time allotted.
That kind of high school surface learning is not a step you can skip, though:
"Surface learning is critical because it provides a foundation on which to build as students are asked to think more deeply."
There are a lot of shortcuts which can cut back on the number of hours, mostly by spreading out the number of years. Early surface exposure helps a lot, as does revisiting topics in different contexts.
https://en.wikipedia.org/wiki/Spacing_effect https://en.wikipedia.org/wiki/Spiral_approach
I feel like the whole point of "math" is to abstract these natural patterns so we can like... write them down and investigate them further. We have a limited mental capacity so we abstract it into a syntax/system of meaning so that we can let the paper or the computer do some of the memory work for us.
If you're doing it in your head, sure that's a skill, but is it really math?
It's never felt like math to me. If it did, I probably wouldn't be able to do it. I'm not especially good at math and feel damn near dyslexic when I try to read math-notation-heavy writing. Programming, meanwhile has always come easily and naturally.
... except the languages that try to look like math. Looking at you, Haskell. I get the concepts just fine, but I can't stand the style.
As someone with a background in math, I think that makes total sense. I don't feel like there's any magic reason why practicing reasoning under (one of the many paradigms of) mathematics would automatically carry over to the type of reasoning you practice while programming any more than the previously mentioned exercises.
The only times math has helped me is when the programming I'm doing straight up references mathematical concepts I've practiced with (matrices, group theory, etc). Which is a pretty uncommon occurrence in my current position.
If you want your background in something else to help you with programming I think you'll have to put conscious effort into identifying and using the relevant abstractions and metaphors
Microprogramming was the kind of thing where you're trying to, for example, cut down an assembly loop to run in 6 cycles rather than 8, design a numerical algorithm, an operating system scheduler, or similar. This is very mathematical in style of thinking.
Macroprogramming is what happens when you write a database-backed web application, for example. It's much more linguistic. It's about understanding other people's code, writing your code to be understood, and gluing a lot of stuff together.
Both have their place. Macroprogramming is taking over now, but I enjoy microprogramming more.
I don't think that's a good way to define it. But if we do define it that way, then saying "x is math" becomes a very weak and much less interesting statement.
Is it not? Adding numbers in my head is certainly math. Algebra in my head is certainly math. Why would it stop being math just because I'm doing it more abstractly?
Advanced math typically starts with logic and structures, not numbers. Even when they study numbers, they tend to be focusing on their qualitative properties: compactness, convexity, primeness, etc.
Which isn't that bad a thing to do.
Many programmers could use more formal logical thinking (or, just, logical thinking to begin with). DailyWTF situations, which most of us have seen in our jobs, are often the case of non applying basic logical principles properly.
But a lot of real world "logic" (decision theory, etc) is statistical in nature, often with a big inductive component and often based on "axioms" that only approximate the real world.
In other words, the field of Analytic Philosophy, when not using the proper amount of math and statistics (especially Bayesian statistics) tend to either lead to doubting everything (if they know what things they do not know) or drawing bad conclusions (if treating their axiomatic assumptions as absolute truths)
They have similar problems with science, and maybe Physics in particular, because of the abstraction of modern physics. By interpreting statements made by some physicist too literally, they may (falsely) end up with conclusions that go way beyond the domain of validity of the original statement.
Imho, Philosophy is fine as a side-project for people in academia, but I think philosphers who do not study math, science, psychology, etc at a level comparable to their philosophy work are at risk of ending up in lala-land.
I would estimate that if someone studies only philosophy in college, they will reach a point after about 1 year where their knowledge of philosophy come to a point that requires more understanding of math, science etc than they had when they started. Kind of like a physicist that doesn't take college level math.
Inference rules, axioms, language?
Whether it is using symbols, pebbles or categories, it is neither better nor worse.
But they make people think that “maths” is just a terrible abstract construction.
Same for when a builder chooses to use a different type of fastener when constructing something. It could be borne of experience, but I would not necessarily call it material engineering at that point. Even if the exact same practice is how materials exploration happens, at a superficial level.
What's interesting is when the real, practical world runs into the theoretical, abstract math world.
https://petapixel.com/2019/07/05/goodbye-aberration-physicis...
The kind of person who is useful in engineering is that person who gets to the bottom of things. They not only find a way to understand the problem, they also find a robust solution. You come across this kind of thing on HN often, like that article about the guy who opens safes with a device hidden in a marker pen. For 99% of people, that level of effort is incomprehensibly deep. I mean you really have to know a heck of a lot about the subject, and you have to be willing to hurdle quite a lot of issues in the solution as well.
Math is the ultimate get-to-the-bottom-of-things subject. That kid who asks you to explain Euler's identity, who wants to know why only 23 people are needed for half of such groups to share a common birthday, and wonders whether there's something deeper than complex numbers when looking at polynomials. The kid who diligently expands his binomials just to see that the formula is actually right.
That kid is also the one who will figure out why your rocket is malfunctioning, how your car will drive itself, how to fix carbon dioxide, and every other problem that we call a technical problem.
So yes, you don't need advanced math to do a lot of these things, and maybe doing advanced math doesn't even help you do these things. But math is a sort of flag that suggests you can get to the bottom of whatever problem you're faced with.
I love math and am decently good at it.
But I don't agree with this statement. I don't think math is the ultimate get-to-the-bottom-of-things subject, I think it's an ultimate get-to-the-bottom-of-things subject. And I think that makes a big difference.
It's easy to set up a filter which only very intelligent/whatever people pass, and say "yep, this is the correct filter, everyone passing this filter is intelligent". But we might be missing a lot of people we'd get otherwise! What if someone is totally uninterested in math, but she happens to love biology, and spent her whole life "getting-to-the-bottom-of" how bodies work? Or substitute in chemistry, exercise science, psychology, or any other field. People can be totally uninterested in and bad at math, but still geniuses of their respective fields.
Note: I'm not sure you disagree with what I wrote, you yourself wrote " math is a sort of flag that suggests you can get to the bottom of whatever problem you're faced with."
The thing about math is you can plumb the depths without much resource. You don't need to check your ideas against the physical world, something that costs time and money that most kids will not have access to.
So it's more like if someone is good at math, that is a good sign that they are the kind of person were talking about, but if they're not that doesn't mean they aren't.
In some of those cases, you only require basic math. Some require intermediate math and some relatively advanced math.
But if you want to be a "genius" if either biology, chemistry or psychology, you most likely will need more math than most people learn in high school. Almost every kind of scientific experimentation requires a firm grasp of statistics (way too many papers are written and even published with sub-par stats in many fields). And in most STEM fields, before you can get "genius" level specialization, you will need a relatively broad mastery of the field, and both Biology and Chemisty, there are plenty of things to learn that require at least some entry level collage math.
There is a vast swath of research effort directed to modern geometrical foundations of physics that look pretty much like pure math.
I would say that String Theory has been more similar to Math than Physics for a while, in that it hasn't really led to many interesting experiments.
Why is 1 + 1 = 2? This isn't really a mathematics problem (I know about axiomatic systems, they were invented relatively late in the timeline of mathematical history, invented as a fiction to avoid getting to the bottom of the messier things, i.e. the social context of how mathematics was actually invented and is used). Sidenote: I have written an article (not in English) about how numbers were a linear to log(N) time complexity reduction trick for recording information, and that 1+1=2 presumes a reductionist and somewhat capitalist view of the world...
Anyway, to me, mathematics is a subject that invents abstract concepts and rewards those who are capable of understanding and believing in them. That kind of thinking is probably useful in a lot of areas that have business value. The make-believe part is, I would humbly suggest, quite an important feature too. It really fits nicely into our capitalist framework playing the money game where they'd want you to believe $1+$1=$2 is an axiom, and it's best not to reach out beyond the closed, axiomatic system and ask where those $ comes from or the messy social implications of the "math".
Hopefully, not all Chinese engineers think that...
> mathematics is a subject that invents abstract concepts and rewards
Not according to much of the history of math. Mathematics was borne of practical needs, and it has been extremely useful in science and engineering. (Sure, you can think of the number 2, say, as an abstract concept, but to most people it is a very concrete thing - as is the definite integral, etc.)
Yes, a philosophical journey can be a journey down a rabbit hole, so one should be careful...
:) The main point of divergence between "Western Capitalism" and "Socialism with Chinese characteristics" isn't whether it's ok to accumulate capital, but who controls it :) I think both agree that capital is a great idea for building strong nations.
> Not according to much of the history of math. Mathematics was borne of practical needs, and it has been extremely useful in science and engineering. (Sure, you can think of the number 2, say, as an abstract concept, but to most people it is a very concrete thing - as is the definite integral, etc.)
I agree, and... that's kind of my point. I intended to say modern mathematics is a subject that invents abstract concepts... Historically numbers were invented to solve a practical need, and then some hobbyists came and had too much fun with it, ending up with funny things like Peano and ZFC. I've seen too many people trained in mathematics that forgot the social-historical context came first, and jump straight to the newly invented axiomatic systems to explain things as basic as 1+1=2, which to me doesn't fit well with the "ultimate get-to-the-bottom-of-things" description.
I mean, to cite a specific example, while the exercise is probably entertaining to some, anyone seriously believing 2+2=4 requires thousands of lines of "proof" is probably not very good at getting to the bottom of things in a general sense. (ref: http://us.metamath.org/mpegif/mmset.html#trivia )
> Why is 1 + 1 = 2? This isn't really a mathematics problem
Umm, so first of all, depending on what you mean, "why is 1+1=2" is definitely a maths problem. It's exactly in the foundations of math, as you said. Or rather, it makes more sense to think of it as "what is our system for deriving true statements, that lets us do things like 1+1=2?".
> (I know about axiomatic systems, they were invented relatively late in the timeline of mathematical history, invented as a fiction to avoid getting to the bottom of the messier things, i.e. the social context of how mathematics was actually invented and is used)
That's not really why axiomatic systems were developed. It was part of trying to give better foundations to math, because mathematicians ran into a lot of actual problems, mostly beginning with calculus. They managed to reach paradoxes, weird conclusions, etc, and wanted to start being more rigorous in how they treated everything in math, which has led to lots of flourishing in maths, so it was probably a good idea.
> [...] 1+1=2 presumes a reductionist and somewhat capitalist view of the world...
This I don't understand at all. What does this have to do with capitalism? The rest of your comment is equally weird to me. Mathematicians like to invent concepts and play with them, sure, I'm fine with that description, but how does this have anything to do with the capitalist system?
> [From your other comment] I mean, to cite a specific example, while the exercise is probably entertaining to some, anyone seriously believing 2+2=4 requires thousands of lines of "proof" is probably not very good at getting to the bottom of things in a general sense. (ref: http://us.metamath.org/mpegif/mmset.html#trivia )
I think you're confusing two things here. For the most part, most mathematicians don't need to bother with "foundations" questions, or reach a point where they're proving 1+1=2. It can be helpful to "peak under the hood" of various definitions to understand how they are defined rigorously, and more importantly, it can be fun. But most mathematicians use numbers as a black box, never needing the detailed definition of how numbers work.
As for taking 2000 lines to prove this, that's probably only when you want to rigorously use a computer proof or something from first first principles. Given a set of axioms (e.g. Peano), I'm pretty sure it's easy to prove 1+1=2 in only a few steps. And again, this is mostly of interest to people who actually want to work on this kind of foundational thing, not the average mathematician I wouldn't think.
For instance, without stats, decisioning processes quickly become "gut feelings". But a little stat can be misleading, as seen in problems like p-hacking. To use statistical thinking generically, not by following some recipe, you need to understand calculus.
Algerbra (linear- and abstract) similarly help structuring logical problems. Linear algebra builds an intuition for organizing problems into dimensions, and and can be useful for semmingly unrelated efforts, such as organizing work (by finding activities that are orthogonal or linearly dependent), and more related effort, such as vectorizing code.
Abstract algebra is useful for thinking about symmetries and for transforming concepts into each other. It creates bridges between areas of mathmatic, like complex numbers and matrices. (Things like Euler's identity becomes easy to grasp if considering the unit circly on the omplex plane as a group, and this in turn remove the "magic" from complex numbers.)
In computer science, algebra can provide intuitions for seemingly unrelated tasks, such as refactoring. Basically, tranform the "algebraic structure" of your code to a simpler one.
Precisely how much of this kind of benefit people will get from knowing math probably depends on their talent for logical thinking. At the lower end, anything beyond basic arithmetic may not provide much value.
But the close someone come to the gifted side of the spectrum, the more secondary benefits may be gained from knowing more math. A gifted person with the same education as an average person may be twice as productive, buth add some math, and that becomes 4x.
Now, if the hypothesis above is true, math would turn a linear curve into a parabolic curve. Math/stats could both be used to test that, but even just coming up with the hypothesis requires some inclination to think mathmatically, beyond mere arithmetic.
And, obviously, if the different between average and good math education is the same as the difference between the area under a parabolic vs linear curve, someone who learned calculus would instantly know how massive the economic benefit might be.
But economically, I think it's much like any advanced level STEM course. In such courses you come across a huge range of things, but the main benefit is that you now know that X is a thing that people have investigated. If you know that X is a field of some depth, that completely changes your own course of action.
The thing is I'm not sure the actual meat is necessary. Yes it sounds criminal, but what you just indexed everything and came back to learn it properly when needed? When I look at HN type people that seems to be the actual way things happen, rather than learn X -> apply X. It's more like people kinda sorta remember there was something that sounded like it might be useful in this other area, then they circled back, read up on the details, and applied it.
I agree. Almost all of those would be some kind of applied mathematics, though.
I was not arguing above for the importance of studying pure math exclusively. For most people, math is a force multiplier when combined with some other knowledge/skill.
> It's more like people kinda sorta remember there was something that sounded like it might be useful in this other area, then they circled back, read up on the details, and applied it.
Absolutely. By having a good and broad generic understanding of a field, you can circle back to it. Understanding something in a more abstract way makes that easier, as it promotes a deeper understanding rather than just memorizing facts.
For instance, if you think of euler's identity in terms of a rotation group it becomes very easy to prove it (at least for someone with a mostly visual understanding of math an physics). But just as important, the rotation group gives a much better understanding of many cases where complex numbers are useful, ranging from physics to electronics to electrical engineering. Quantum wave functions, capacitors/inductors and electrical motors all have this kind of rotational symmetry in common.
So while some colleges may teach quantum mechanics, electronics or electrical engineering with a shallower math curriculum as the base, some abstract algebra can help gain deeper understanding of each, compared to memorizing the formulaes without fully understanding them.
Also, even if many with a STEM degree often end up in a different field than what they studied in collage, the ability to apply math and statistics can often be useful way outside of the field studied initially.
(At least, that's my experience as a former physicist. While I rarely need quantum mechanics, the ability to manipulate probability density function, find and describe symmetries, and have developed a quantitative intuition that spans a 50 orders of magnitude remains useful, both professionally and even as a citizen. )
I wish. I wonder what proportion of people reading your post don't see or feel the beauty in it. I think you need a special and probably quite rare 'sixth sense' to perceive it.
That kid is also the one who will enter academia and spend the rest of their life taking part in intradepartmental warfare as they desperately strive for tenure.
Equally possible, right?
Actually, I think it is a different personality type. The kid who does well at math because rules are provided clearly and success is guaranteed somehow is different from the kid who wants to get to the bottom of it from a curiosity perspective. Both are driven partially by ego but one is more fear driven and the other more joy driven.
Of course, years of academic training might drive the joy out of anyone, but many end up in academics because they just wanted to keep getting the positive validation through clearish rules. These people struggle in entrepreneurship but can do great with corporate advancement.
Somebody could be very intrinsically driven by the pursuit of truth, but still appreciate the importance of politics in increasing earnings and status and thus outwardly appear to your 2nd type.
Those same people might "struggle in entrepreneurship" with regards to inventing wholly new ideas, but be good at fundraising or copying. (Let's make it an NFT...) Just something to consider.
Not equally, since more PhDs end up in industry than academia.
Also, those in academia routinely consult and solve these problems for industry, they make startups, they solve problems that further advance industry. Many bounce between careers on both sides.
It's silly to think those in academia don't work on and solve real world problems and only spend time fighting politics. Those research grants are for producing research, and those corporate grants are for producing items useful to those companies.
Emmanuel Derman wrote about the mass of PhDs in the 1980s who went into finance. This will just keep happening.
Plus you don't need to get a PhD at all to be one of those guys who solves problems.
The article also seems to be focused on a certain kind of math. Boolean algebra is a type of math that is used regularly in CS and EE. It is so fundamental that if you don’t get it you probably can’t code anything non trivial.
Better to know the math and not use it for your particular role, than be unqualified because you can’t
You can find Nino Schurter’s gym workouts on YouTube (elite mountain biker). Same for some top race car drivers.
This to me is the real problem with over specialization. You have less than gifted parents living out their athletic dreams through their kids but if the parent wasn't that good at the sport there will be correlation with the kid not being that good.
Kids shouldn't specialize in sports too young because there is such a great chance they are not in the right sport at 11.
But to play HS sports for example at most schools, either will do the job. And for most people playing the sport they love in high school will be worth it. Even among pro players, their HS experience is often their best experience.
Everything you might be doing in the gym is in support of the task you're trying to accomplish, not the other way around, so I really doubt that the gym is 'the bare minimum' for most athletes.
AAU basketball literally has leader boards ranking the top 6th and 7th graders in the country. And goes all the way down to 3rd grade.
I did not say anything about juniors being prohibited from competing, healthy competition has immense benefits early on.
Key word being playing. It helped me learn teamwork and good sportsmanship, which is supposed to be the point.
When my son was playing youth sports, many of his peers started doing speed/coordination training with a coach in 6th/7th grade. This was usually 1-2x/week and applicable to most sports. Focus was more on first-step speed, running form, and full body movements (jumps, "suicide" sprints/shuttle runs, etc). I thought it was pretty crazy at the time, still do, but at least it was generic.
These days, neighbors have kids in with pitching coaches at age 8, playing travel league baseball 3+ nights/week, plus 2-3 games/weekend. 9 months of the year. That is REALLY crazy to me. How about some swim team, or soccer, or anything else?
Back when I was in high school (mid-90s), the best athletes were all multi-sport athletes. Football/basketball/track-field was a common combo. No idea if that's true today, but I suspect there's a lot more single-sport athletes at the prep level, which is kinda sad.
Today, I wish I'd done a lot more math and statistics before focusing on exclusively physics, even if it'd cost me another 1-3 semesters.
I was spending way too much time trying to re-invent the wheel (statistics) in the experimental part, and particle physics really assumed quite a bit more math than what was listed as prerequisites, simply because the university didn't want to delay progression. (To compensate, the bar at the exam was set pretty low.)
1) I bet a far fewer percentage of people use Shakespeare, Steinbeck, or any other literature in their jobs. Is this an argument for eliminating literature from the curriculum? Why or why not?
2) The percentage that do use higher level math are extremely useful to society, but the infrastructure that educates that smaller percentage needs to be large and widespread and accessible to all. A system that selectively educates a small amount of kids in valuable skills probably turns into a privilege filter and decreases the amount of talent educates massively.
The participation in calculus class seems to have been the best predictor for who would “get out the hood” now that it’s been 15 years since we graduated.
Shutting that pipeline down because the managerial class doesn’t see the point reeks of privileged nonsense — like removing AP classes because they offend the racial sensibilities of that same managerial class.
How about you stop destroying lives to express your privilege?
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I also just don’t buy the premise: a lot of people use concepts from advanced math routinely, even if they don’t set up equations.
For example?
- the actuaries and economists I worked with used calculus, statistics, and linear algebra (even if software did the calculations)
- the mechanical and electronics engineers I worked with used calculus, control theory, and signal theory (even if software did the calculations)
- SDEs use type theory and category theory (and the equivalence of them), as well as implementing the math above (even if libraries did the calculations) and things such as discrete math and graph theory to reason about data structures
For people I haven’t worked with, but know:
- biotech workers use calculus concepts and statistics
- Boeing engineers use calculus, statistics, and complicated simulations
- physicists have a whole bucket of mathematics they use
That non-exhaustive list of my personal experience only covers a few percent of the adult population.
How many people do you believe use any particular skill?
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Fun aside: according to the BLS, around 1% of working adults are software developers. (A little over, actually.)
https://www.bls.gov/news.release/pdf/work.pdf
https://www.bls.gov/ooh/computer-and-information-technology/...
For example, I use multivariate regressions in Excel quite frequently to help us understand some data I’ve gathered. Yes, it’s just a few keystrokes, but if I didn’t have a basic understanding of what it were doing, it would just be magic, and I’d never be able to tell if the output was bogus or not.
Knowing the basic ideas behind complex numbers helped when I was doing some computer graphics work on a game a few years ago.
Having a basic intuition about what a derivative is comes up all the damn time when doing just about anything involving stuff that changes over time. I don’t use a lot of calculus, and I especially don’t calculate things by hand much, but knowing the general concepts helps a lot.
Statistics I use massively and I think is super important. Every single day, I need to at least understand what a standard deviation is, how basic probability works (I like poker), bayes theorem, etc.
Boolean logic and discrete math comes up quite a bit when I write code.
Non-Euclidian geometry came up a lot when I was writing code related to mapping and calculating distances on a curved Earth.
All that being said, I still can’t remember how to do long division :)
People who don't learn statistics, are doomed to think in terms of anecdotes, or to trust some authority. Math is necessary to do a proper independent evaluation about a lot of questions in our world, be they political, economical or scientific. In the office, you cannot double check the work of the "experts", making it very hard to lead effectively, handing them all the power. In elections, those who don't understand statistics and science (that require math) are vulnerable to charlatans and demagogues. Economically, not knowing math means you have trouble evaluating risks and expected benefits of investments and exposures.
https://academia.stackexchange.com/questions/9602/rediscover...
You may not know it, but you're definitely using the calculus "intuitively" - that's why the paper doesn't land right in front of you, or into the next room over, even if you miss the basket.
> There are two counter-arguments. The first is that higher mathematics is central to a serious higher education. [snip] It is both unreasonable and unworkable to insist that all students get such training. Of course, such training should be available to all who desire it.
So they argue for not removing higher level maths from the curriculum. Just not making it compulsory. They might feel the same way about Shakespeare and Steinbeck, or learning a foreign language, or pretty much any other high school subject, really.
I'm sure everyone knows that Shakespeare is used in no more than a tiny sliver of jobs. To me the question is, does reading Shakespeare's plays and writing essays on them make for better humans? Or, as Caplan would probably say, is it just yet another "good worker" signal implying some combination of social class, intelligence, compliance, and conscientiousness/industriousness?
At least in Germany, school is for getting general education and university/tertiary education is for specializing in the area that you chose as your degree course. So, in my opinion it indeed makes sense to make this non-compulsory at the university level if you don't choose a humanities degree course where this is subject matter.
Such that, I don't think you would get disagreement from the people that you are writing towards.
How do you figure? I think you should prove that one before you try and pin a point to it.
Instead, higher mathematics allows you to add powerful new features to your mental models, and reason about these features accurately and effectively. A key example: the Fourier transform. Understanding a signal in terms of its frequency components is immensely valuable for understanding how periodic phenomena work, and since periodic systems are ubiquitous, this often leads to concrete insights into concrete problems. With this more sophisticated understanding of the problem, you can write down a solution that can be implemented with “8th grade algebra and a little programming”. Other similar examples: singular value decomposition, spectral decomposition, probabilistic reasoning (MLE, Bayesian inference).
It’s important to equip people with the tools required to build powerful mental models of the world around them, so they can solve hard problems. Higher mathematics offers many such tools.
Of course they think their work has no real world application. It'll take two or three generations worth of dumbing it down before us mortals can turn it into a nice computer vision algorithm or something like that.
Communication with the general population is key.
You don't "use" it because there isn't demand for its use. The advances we dream about in the sciences are retarded by a widespread lack of understanding regarding how those advances are made, how long they might take, what they cost, and what costs would be offset by their achievement.
If anything needs to change it's the teaching of mathematics primarily as a language to describe everyday life more accurately. The question every grade school child asks, "why do I need this", requires an irrefutable answer that doesn't need several years worth of prerequisite knowledge. Then we need to get on with improving our civilization.
This. This right here.
You can tell kids that math is super-important but they're just going to look at all the barely-numerate adults in their lives who seem to be doing just fine without it, and at all the nothing they encounter in their own lives that seems to call for much more than basic arithmetic, and decide you're a liar or some kind of weirdo, like the dude who's super into model trains and keeps telling them how fun it is (yeah, sure buddy, I bet it is)
Reading doesn't need to be sold. It's useful everywhere, constantly, and you don't need to look for uses. It's even very useful for stuff that kids really want to do, like playing video games. You don't do 13 years of it and then find that almost the only part you ever use is the alphabet, and wonder why all those reading nuts wasted so much of your time with it and why some other lunatics are telling you that now you still have to do even more to get a college degree.
For most kids, math needs to be sold to them. And that's not happening.
Before even talking about math, teachers just need to explain where the problem came from, who was thinking about it and why (if it's known specifically, like cardano's 3rd degree polynomials, even better).
If there's one thing that gets me (and by my own observation, the high school students I tutored) motivated, it's this by far.
That could perhaps be changed though.
Reading gives us compelling stories right away.
Not a flippant suggestion—I actually think that might improve the value most students get out of their later math classes. Might even make "I hate math" a less common sentiment.
What does this refer to? And how would it affect physical sciences and biology?
A broad base of at least some of stats, linear algebra, and others seems like it would be more useful to most even in STEM to me. I've met folks who were great at those who struggled with geometry and/or calculus, and vice versa. Yet plenty of need for people good at all sorts of those.
Stats is a non-starter without calculus. You'd just be doing the "ram a bunch of rules for derivation and integration into your head." method since first principles require calculus. Linear algebra is less reliant on calculus, but the fun stuff still requires calculus. I learned basic linear algebra in high school for what it's worth, just enough to solve systems of linear equations.
Teach it the way you suggest and it would just be another useless math class that people would tune out of instead of grounding it in their everyday life experiences.
("Deriving from first principles" isn't the interesting stuff about calculus, either. And we're not requiring quantum physics to derive gravity from first principles in high school...)
I use linear algebra every day in my work as a graphics programmer. A huge portion of what I do depends on it. While I frequently need calculus it's still a minority of the time.
Derivates obviously show up everywhere: velocity, acceleration, gradients, ...
But it seems to me only the narrow portion relating to physically-based rendering or simulation requires an understanding of integrals. At the moment I can mainly think of how it's used to define continuous functions before introducing the discretization used to actually compute them.
Agreed that “memorize rules” is not a great learning experience for students… but it is easy to present.
Point1. Education system which would strive for bear minimum outcome would be something like “not everybody needs addition, cashiers can take care of that”. (You will cut calculus, somebody else will cut multiplication table)
Point2. which iq range education system is optimised for? (Fair discission; though hard politically). Gifted people are random in population. If you teach and test only the bare minimum content, which 99.9% of population passes with flying colors, then gifted are undertrained and also harder to spot, thus under-utilised.
Point3. Memorization is useful. Even if one understands the concept deeply, the “cheat sheet” inside the head (i.e. memorised info) makes things much much faster. Memorization is abit like muscle-memory and or caching.
The problems with rote learning are obvious, and I don’t mean to dismiss them. But I do think the pendulum has swung so far that today, there is excessive and harmful stigma attached to it. Rote learning is a bit of a leap of faith, and it requires trusting that it will be useful in the end (in other words it comes with an upfront cost)… and this is incredibly powerful for taking some academic shortcuts in a limited amount of time.
Now, as an adult, I've noticed how much faster I can get past a lot of tricky issues in my work because the patterns I see a lot are familiar and I can glance over them because I know them by heart and don't get stuck on every little detail.
Disagree. It definitely changed the way I think, and I'm no mathematician.
Also calling something 'nonsense' this way is an argument smell; no justification + ad-hom = "in my opinion".
Problem is, you have to be at a very high level of maths[1] and be in the right place to use that before it becomes useful, but then it becomes vital at an industry or country level.
Personal opinion: change the way maths is taught. It is so appallingly taught so often, starting with abstractions and no examples, not acknowledging that so many (like me) are competent but aren't brilliant minded, and we are not abstract thinkers, so root your abstractions in the concrete to get them over. Smart people just don't understand that we don't understand.
[1] I'm going to regret this, but here's me making an idiot of myself over a maths paper yesterday, just getting the wrong end of the stick massively https://news.ycombinator.com/item?id=31643296
[2] Added full quote after, sorry
A good foundation in understanding scientific claims and evaluating proofs seems essential in such times. Maths is an essential part of that.
Rather, as the US Constitution has bluntly put it, if we want to keep democracy, "the right of the people to keep and bear Arms, shall not be infringed." The above examples seem to support this.
In fact, I believe that the invention of firearms democratized military might by making the professional soldier (knight, man-at-arms or Swiss style mercernary) obsolete. In a time where military might was determined by how many men could be drafted into service, a government could only retain power through the support of those drafted men.
A rule by "all drafted men", in other words, is the foundation of democracy. Like most societies, a democracy is rule by the predominant warrior caste. In the greek democracies, all male citizens were serving as hoplites. That was teh basis for their "democracy". The Roman republic had legions of citizens (until Marius), enabling another form of citizen rule. Then came the Marian reforms, with professional soldiers controlled by the elite. Almost directly after, Sulla, Caesar and Octavian introduced the Roman Empire.
500 years later, the Romans in Western Europe were overrun by Germanic chiefs. Their elite soldiers became the armored knights of the Medieval Europe, and they introduced Feudalism, where power was held by the nobility (and sometimes shared with a king, who was often quite weak).
Then came the age of the pike armed Swiss mercenaries and similar non-noble military formations more similar to the legions of Imperial Rome. This gave many European monarchs full controls.
The founding fathers saw this, and rightly predicted that armed citizens was an ingredient in ensuring governance by the people. In Europe it would take another 150 years or so for this to play out.
Many on the left will not realize this. They think that democracy is self-sustaining regardless of potential for the elites to be corrupted and sieze the power.
The founding fathers had studied their history. They knew about Marius and Sulla and how the Roman elites quickly had taken control once the citizens were no longer the backbone of the army.
The founding fathers were not "common people", but rather highly educated members of the elites. Their existence had been made possible by a monarchical system, and they bore many similarities with the elites of the Roman empire (the senators).
The guns may give the power to rule, but not the wisdom, and people like Thomas Jefferson understood this. And this is where the understanding of the populist right falls short.
Thomas Jefferson in particular was a proponent of quality education for the citizenry, he thought that "no other sure foundation can be devised for the preservation of freedom and happiness".
In other words, not only guns but also books were needed to keep the people free (and happy).
https://en.wikipedia.org/wiki/Thomas_Jefferson_and_education
When citizens completely lose the ability to understand the world in which they live. He promoted the teaching of "Science and Virtue", but was concerned about intellectuals from "tyrannies" in Europe.
Maybe he anticipated something like the Frankfurt School?
The only way to prevent demagogues (like Ceasar) who promise to people that which is impossible to deliver, is to: 1) Make sure the people know enough Science to be able to know the possible from the impossible. 2) Make sure people know enough history to recognize what values are true Virtues that lead to healthy societies, and which are hollow lies, more suitable for use by would-be-dictators to gain control.
The Civil War was the ultimate aim of the Second Amendment. State militias banded together to fight the United States. It was never about ensuring democracy.
The interpretation that the Second Amendment is about enabling individuals to protect their personal interests with guns is very new. And this idea that the Second Amendment is supposed to be about guns maintaining democracy is a perversion of even that. Coercion via the threat of violence from random, unelected strangers doesn't feel very 'democratic' to me.
I'm not American, and I'm not writing specifically for the US. But after reading up on most periods of Western history, I did notice that whatever group of people served as the core of the army, tended to be the same group that controlled the country. And that this pattern seems to be older than history itself.
> And this idea that the Second Amendment is supposed to be about guns maintaining democracy is a perversion of even that.
The idea itself is almost certainly not new. Around the time of the American Revolution, Europe had been through a period of around 1800 years of mostly some form of tyranny.
For most of that period, the "right to bear arms" (which tended to mean military grade weapons and armor) was restricted to the nobility or other privileged group, who would often take advantage of the monopoly for coercive purposes. (Kind of like the Sheriff of Nottingham in the Robin Hood legend.)
People living 200-250 years ago were certainly aware of that, as it was recent history for them. The concept that was new to them, was that of "democracy". Instead, they tended to care about "liberty" vs "tyranny".
For instance, St. George Tucker in 1803:
"A well regulated militia being necessary to the security of a free state, the right of the people to keep, and bear arms, shall not be infringed. Amendments to C. U. S. Art. 4. This may be considered as the true palladium of liberty ... The right of self defence is the first law of nature: In most governments it has been the study of rulers to confine this right within the narrowest limits possible. Wherever standing armies are kept up, and the right of the people to keep and bear arms is, under any colour or pretext whatsoever, prohibited, liberty, if not already annihilated, is on the brink of destruction."
People at that era, just as now, had many different opinions, and I don't know the discussions well enough to know the precise intent of the Second Amendment.
But whether it was the intent or a side effect, I believe one effect of an armoed populace (privately armed or drafted), is that it makes it harder for a would-be tyrant to take away all liberties.
One example that may be more relevant than the USA, right now, is Russia. Russia still has a system partially based on the draft. But Putin is not willing to pull in all reserves the way Ukraine has done. A reason for that may be that it could increase the risk of the army turning against him.
But I do hold the concern you hint at, and with an increasing tendency towards fewer, more expensive and soon maybe automated weapons platforms, the risk may increase of some kind of move towards some alternative system. (Monarchy, Aristocracy, Oligarchy, Theocracy, Autocracy, etc)
Which is why I (who used to be too liberterian to like the idea of a forced draft) now support a system based on the draft, simply because of the stability it provides.
But if and when the most effective type of military once again becomes wieldable by a small minority, it will become harder and harder to prevent a shift.
I don't claim simply learning maths is sufficient - I mean math in the form of what is a proof, statistical reasoning, things like that.
In general I don't think we ever even looked at a paper or "study" and tried to find tell-tale signs of bad science. That should change.
How could managers evaluate my performance, make decisions about where to apply my skills, or understand my value if they knew nothing more than high school math? How would I remain sane with no one around to rubber duck proofs or concepts to, to commiserate or bounce ideas with? Sufficiently advanced technology is equivalent to magic; deskilling of the workforce makes the black boxes even blacker.
Every colleague may not know the precise details of my work, but that this remains a strategic decision and not a limitation due to their lack of skill is essential.
When you look at the most successful startups of the last 50 years or so, it seems to me that most of them were founded by people with a strong STEM background (and math talent), even if several dropped out before finishing college.
On average, I think these founders had at least a quite strong pre-graduate level understanding of math, if not more. I believe that level is roughly the minimum required to lead a team of highly educated tech workers.
Who cares if they never “use” calculus in their dreary jobs. The don’t “use” Shakespeare either.
And who are these “engineers” who don’t use math, but just plug numbers into Excel? I am afraid of China. I’ve met many of their engineers and scientists, and they sure as hell know math.
Chinese engineering education is far inferior to Western according to every person I’ve ever met here who hired fresh graduates. I live in China.
The author makes a fairly compelling case against the way education is directed at kids today. And a lot of it is hinged on just how poorly adults show that they have learned in school. This includes in the area of critical thinking, where college students don't perform that much better than you would expect from non college level students.
But I guess his point is rather that most graduates of engineering schools don't even work as engineers at all, which might be true, I don't know (I even have difficulty believing that given the number of people I know who weren't even trained as engineers and are now working as software engineers).
Similarly, is a quarterback running ballistics equations to determine if they can complete a pass? All while running an optimization on the timing of various other players on the field to know if they will have time to make the pass?
Programming is a more interesting issue. If this blog had argued that computer science should be marginalized within software engineering then he might have found something to say (but which has already been said many times by others).
It's not that those jobs are without challenges but they are in other area's, such as requirements gathering and project management.
Are scientists all using 8th Grade math? Maybe that subset of math became 8th grade math, because it was the subset that was most used.
Maybe the big top level planning already happened via lots of micro-transactions and came to the right answer.
edit: commented before I got to the end, I think this is what they're arguing, though they're not being smug about it.
I think I'd agree with them more if they weren't dissing Excel.
Don't use it myself, but it (and it's predecessors/analogues) are powerful tools. Widely used and abused, for good and bad.
Complete lack of knowledge left people vulnerable to superstitions.
Using everyday as a tool is not the benchmark for whether it is having impact.
Higher mathematics should be offered and taken by those who need it, or want it; but never required of all students.
I studied economics and computer science at a top-20 university. I did a grand total of one course in the math department (1st semester calculus). Everything else was specific to my areas of study. I don’t recall anybody being forced to take high level mathematics outside of engineering programs. Even within the hard sciences, the requirement was usually something like “take 2 semesters of math, plus statistics.” Or something like that.
Would the author also argue that since I don’t use history at my job I should never have taken government, architectural history, or a survey of western religions? Those all contribute to my understanding of the world around me. As does math.
If his point is we need more generalists with a broad understanding of the world, including social sciences and history, that could be a valid point, but it’s not the one he made.
Sure -- most engineers shouldn't use advanced math, because that's easy to screw up. But either they are aware that their linear models came from some more complex equations that have been simplified, or they are living in a magic demon-haunted world where equations are just runes handed down by mystic sages. Hopefully it is the former!
Teaching engineers the hard math serves dual purposes -- it helps them understand where their models come from, and somebody is going to have to invent the models. They are one population that has already shown at least some mathematical aptitude.
Yes, but in "software engineering" this has been a deliberate goal for decades. Every new language and platform is intended to make things easier and safer.
One can debate what educational backgrounds are actually legitimate requirements for doing web frontends and such, but that's a debate for the recruiters to have. It shouldn't impact the curriculums.
When I didn't understand calculus I thought that was pretty unessential for being a good programmer. Strange thing though, after learning calculus I find myself using to help understand or model problems a couple of times a week at least.
When I didn't understand linear algebra I didn't see how useful it was for anything outside of the obvious niches like graphics programming. But once I learned it I was able to reduce a remarkable number of complex problems to literally 1-3 lines of code.
Even when I knew statistics, I didn't see much use in all that fancy advanced theoretical stuff. Now I find one of the most useful, practical tools when modeling problems is the Laplace approximation, and knowing the Cramer-Rao bound let's me realize the limits of the results I get from this.
Perhaps most famously, the great mathematician G.H. Hardy, in 1940, discussed how permanently impractical and "useless" number theory was (to be clear, he cited this as a virtue of number theory) in his Mathematicians Apology. Just a few years later Alan Turing (and several others) realized that number theory was the key to securing (and undermining) cryptographic systems.
The lesson for me has been that programmers (and I'm sure many other professions) have a habit of finding good uses for the math they've learned.
And it does feel like there's a lot of labor and energy going into teaching subjects that for the most part, people never use again.
Some people argue that it teaches you the meta-skill of learning how to learn, but if that was true, you could literally just teach the meta-skill itself, instead of teaching it indirectly.
And in some ways I sort of get that at the end of the day, what you're really trying to do is predict the future; what skills will be relevant to which people and so you go with a shotgun approach and go deeper and broader than most people will need, just in case.
Maybe efficiency is overrated. Maybe it's better to waste time learning about things you may not necessarily use. But I do wonder what the opposite approach would look like.
If that was his central point, I would agree.
>Some people argue that it teaches you the meta-skill of learning how to learn, but if that was true, you could literally just teach the meta-skill itself, instead of teaching it indirectly.
I think it is. And I sucked at math in school. I think it is a meta-skill every bit as much as reading is and nearly as important. But so are some of the other soft skills that we are not teaching kids.
Math is workout for the brains and also teaches how to analyse and solve problems. Maybe that's just a useless skill, what do I know, I was good at math.
Math is one of the basis of science and engineering, so it is important for someone to teach you that. I think we should make a bit of space for programming now, not because we will all be coders, but because it is the basis of how computers work, and computers are everywhere, and like math, programming is a good way to teach rigorous and precise thinking. Computers are stupid, they do what they are told, not what you think you told them, that make them good training material.
The usefulness of math, engineering, or computer science education in industry is typically in knowing how your tools work under the hood and being able to apply that knowledge to infrequently optimize them or extend them. It also helps to be able to transfer between similar tools/jobs/occupations with less lead time.
From a society wide view, I'd argue we as a society benefit from pushing everyone toward STEM type education because even if 75% of people who go that route never utilize their training the 25% who do have an outsized impact on our collective well being. Overcommiting to STEM is necessary because we don't know who will be become our technical innovators a priori, so we try to make sure everyone with the potential to be a technical innovator is equipped with the tools to do so.
The problem is that "it helps you think better" applies to lots of things, like being fluent in a second language, understanding ancient history in depth, studying great literature, etc. We don't require them, beyond an introductory class or two.
We also have the ratio between calculus and statistics entirely wrong, with much more of the latter required in most engineering degree programs, whereas the latter is much more often going to come up in work (and the rest of life).
FAANG engineer here, having worked in multiple companies you would recognize as a principle (or higher) engineer. I've also taken Discrete math when I was a CS undergrad.
I've never used any of the advanced math learned in school, and I've had the pleasure of working on some of the largest and most complex systems ever made. Lots of basic Excel. The "math" I have needed for work (such as TLA+ modeling, percentile distributions, etc) was always learned "on demand".
As a trivial example, to say that you have never used Discrete Math in your work means that you have never for example write if(a && b) or have never done if(!a && !b) and judged it to be cleaner than if(!(a || b)). This also means that you never used a finite state machine, strings, combinatorics, trees, graphs or modulus.
I would believe that you have not explicitly sat down and worked out Discrete Mathematics proofs. But the ideas of Discrete Math are pervasive and unlikely that you have never used them.
There are several algorithms (in for example graphs and crypto) that you can not understand or implement without discrete maths. But even then, only the basic level is required.
Maybe only guy ever needs to know these things in a FAANG? (Meaning, could implementing these algorithms keep more than guy busy?)
finite state machine => usually taught in a theory of computation class
strings => wtf?
combinatorics => would be taught in a discrete maths class but I can totally believe that a software engineer has never used combinatorics aside of solving toy problems
trees, graphs => more likely dealt with in a data structures and algorithms course. That said there is a discrete maths slant on how these topics can be approached, but they involve proofs that nobody really cares about when writing software
modulus => those are usually dealt with under number theory classes, besides, unless you're implementing RSA or some cryptographic function, usually the grade 8 level understanding of modulus (i.e. remainder of a division) suffices
Math is a good abstraction, such that it can be brought to describe the work many other fields do. It doesn't stand that what the others are doing is automatically math, though.
To me, that was very disheartening. Philosophy is the foundation of all academia, and will work as a basis for all things both in your academic life, and life in general.
But that was very much a product of the "If it's not math/engineering/hard science, then it must be a wishy-washy nonsense without real-life applications"
Which is also what a lot of technocrats seem to believe. Whether or not something is "useful", depends on how you look at it, and apply it.
I'm sure a lot of STEM students would greatly benefit having some liberal arts classes (or similar). If nothing for making you a more well-rounded person.
Dickens used Professor Gradgrind to mock this attitude back in the 1850s in Hard Times.
It’s just a grownup’s version of “will this be on the test?”.
Some things never change.
The problem I have is that he is looking at averages. In my experience, there are five tiers: counting, arithmetic, algebra, calculus, and higher math. The USA is simultaneously sending more people to the top two and bottom two, with the bottom two going from a small few to a real population segment. I don't even know if the worst public schools teach anything anymore or if they are just pseudo-prisons to capture taxes.
I won't even go into the woeful state of logical and statistical knowledge. I don't know why those aren't core requirements for graduating high school.
I've heard of Science professors who use English scores to sift through PhD applicants. They reason this to be an effective way to find the strongest amongst otherwise equal candidates. (There is also a benefit in terms of writing papers, but writing strengths cannot make up for scientific weakness. Thus, writing is not the central point, as mathematical ability is not the central point in this econlib.org essay.)
I've not seen "gold standard" (large, randomized, double-blinded, ...) studies of the effectiveness of this selection strategy. And I don't know how such studies could even be done: in research departments, it costs a lot to take on PhD students, and a lot rests on their success.
> I must study Politicks and War that my sons may have liberty to study Mathematicks and Philosophy. My sons ought to study Mathematicks and Philosophy, Geography, natural History, Naval Architecture, navigation, Commerce and Agriculture, in order to give their Children a right to study Painting, Poetry, Musick, Architecture, Statuary, Tapestry and Porcelaine.
Somewhere along the line it seems we got stuck at what Adams wanted his sons to study, and never really moved on to what he hoped his grandchildren could study.
A big issue with the motivation to learn math is that it doesn't tell a compelling story for students for many years. But perhaps that could change.
Side note, the same could be said for Chemistry and Biology -- while interesting in the abstract maybe, the actual utility is minimal. I have never had to balance a stoichiometric equation nor do I expect to to see one any time soon.
I would agree, as a guess, that learning "higher" math helps you think more clearly or in a more focused way for a longer period of time.
Except to understand the context of the work those people do, which touches most of everyday life.
* If daily user growth is increasing linearly, total user count is exponential over time.
* If I have a radially symmetric shape it's center of gravity is going to be centered on the axis of symmetry - if the density is uniform. But if the density isn't uniform, where will the center of balance shift?
This is single dimension and multi-dimensional calculus, respectively. I bet most engineers use at least the former, at an intuitive level, on a regular basis.
Exponential growth would be when daily growth is proportional to the current number of users, which is different.
cracking me up here
Ironic that a hypothesis that most people won't need mathematics should require solid statistics to resolve.
The funny thing is that I started realising I'd need a lot of math even before working professionally. As a kid I wanted to understand game engines and ray tracers, so trigonometry and linear algebra were what was lacking.
At some point I worked on a search engine of the bigger kind. Back in the day this meant I had to touch statistics and machine learning, distributed systems, computer science theory.
I enjoy understanding my programming languages on the deepest level possible. And there's a lot of math-based and comp-sci stuff in compilers as well!
In these days, original whitepapers are available for anybody to read. Not wikipedia, not story retold - the real stuff. Not being fluent in math means not being able to consume the most interesting part of the treasure.
Surely, math is not directly applicable to web-dev or whatever the latest flavour of UI building is now. And that's a lion's share of work available today.
The secret is that math is not like some kind of specialised tech. It's more like a language. Being able to read it gives access to more knowledge the same way reading is a way to get access to written knowledge.
So? How does this follow:
"Acceptance of the conjecture should have revolutionary educational implications . In particular, it undermines the legitimacy of requiring higher mathematics of all students. Such mathematics is actually needed by only a minute fraction of the workforce."
The conjecture (if it holds) just states that only few use higher mathematics.
Not that things wouldn't improve if more used them, and also if more "commoners" had a better understanding of basic mathematics (because even "basic algebra" and "excel" are way above tons of people)...
Starting with:
"The math myth is the myth that the future of the American economy is dependent upon the masses having higher mathematics skills".
And concluding that it is false, because "even the vast majority of scientists just use little and lower level math" is a non sequitur.
I think this take would have benefit from some mathematical experience. Mathematicians are mainly adept at discarding their beliefs. The rule of consistency, ie that my beliefs should not allow me to reach contradictions, is slippery to apply, and generally requires a lot of practice. Practice that is by far most easily obtained through a formal math curriculum, where it is least difficult to overcome the first hurdle of getting someone to agree that this rule exists in the first place, and the second hurdle of convincing someone that the contradiction does in fact follow from the assumptions. There is a lot of finesse involved, and that finesse comes with experience.
In the real world, people assume things without proof, and this makes the ability to heuristically generate counterexamples on the fly much more important.
All of my argument of course depends upon a world in which mathematics is actually the thing being taught, rather than whatever grotesque projection of it children are shown in school, but let's not pretend that that projection bears any resemblance to mathematics.
As a counterpoint, I use math on a daily basis as a principal engineer (architecting SpaceX's Starlink). In particular: lots of geometry/trig, optimization theory, linear algebra, Fourier analysis, quaternions, probability/statistics, control theory, dynamic programming, numerical methods.
This is typically in the service of modeling physical systems, modulating or demodulation RF waveforms, and controlling or compensating for electrical circuits, orbital dynamics, etc..
The value of math is that it is an especially portable skill-- you can work in many fields and it still applies. I've worked so far in bio-sciences, chip design, and satellites / space. In each case the value of math was very high.
US citizens are indeed underrepresented in mathematical degrees. In graduate school at MIT, the vast majority of my peers were not US citizens (EECS department). It's great the best and brightest from around the world study here, the US government should try hard to incentivize them to stick around (green cards, etc).
I needed to buy a bunch of precisely cut curved wood for a 2nd level (laser cut wood! So Awesome!), and had to actually re-learn the basics of trig in order to provide the details to have everything made.
In addition, that section of math was the most beautiful thing I've learned in my life.
Then later on I started learning Assembly and the lightbulb finally went on over my head. I finally understood it, and it totally made sense that what I'd learned in that class was super valuable.
Math is context.
It's easy to tell kids that mathematics is the secret language that explains the universe and how it works. Demonstrating that in a compelling way would be great.
Most of the pain behind decision making is discovering what the facts are. People hide the facts, twist the facts, or falsify facts. Without facts, advanced math is of little use. This is why we need experts who have seen a problem in their domain a thousand times. They can quickly spot "outliers" or fabrications.
The maths was quite a bit easier than what I was used to from pre-graduate physics, so I found it mostly a fun diversion. (I also learned some concepts that have been useful later, since I only did mathematical stats, not applied stats myself).
But even with this course, I would be worried about the ability of many of these doctors, even after a PhD to do a proper critical reading of a research paper. Indirectly this methodology would encourage p-hacking, and from memory, there was not much training in using judgment to see through such activity. And I don't remember if there were any domain-specific training in analysis beyond "statistically significant, P(H0)<0.05", such as calculating confidence intervals from effect sizes, in terms of medical prognosis, reductions in expected lifetime, etc.
And THESE would be expected to be the experts.
And for doctors WITHOUT this training, I think they have severe trouble digesting an original research paper themselves, which would leave them even more at the mercy of the advertising branches of Big Pharma.
Also, it's funny that you say so, because when I started my first calculus course, I got to know a really smart Med student who happened to think that just studying Medicine was too easy, so he enrolled in Math in the side, and over the first few semesters he was getting the best grades of all 400 or so Math-students, while still being a full-time Med student.
For people like that, adding some calculus is not a hindrance. Some can learn single and multi-variable calculus in a couple of weeks each. (And quite a few Med students are really smart.)
However, from the point of view of a software developer, there are a few branches of math that tend to be very useful even if you're stuck doing some boring front-end work. I especially mean discrete math: it really pays off to study the ideas and solve some problems. In your mind, you will immediately try to translate them into code. There are some intangible benefits of this that will materialize sooner or later.
Math is a way of thinking.
It's invaluable for solving all kinds of problems.
This is very much in line with Caplan's prior views. It holds true to a degree, but probably isn't universal.
Solving one scary calculus problem thrown at me as an undergrad - an integral - resulted in a solution which broke it into 16 separate integrals. Days of algebra were involved.
Do not concur.
The former is concrete; the latter abstract. The ability to connect dots of information in unexpected ways is indirectly connected with physical prowess, at best.
That seems incredible to me. I guess that means I live in a bubble.
Surely they would have learned concepts of compound interest by then?
The welfare cliff and similar issues wouldn't happen if people could communicate using continuous functions.
I use 1st year university-level math very frequently (stats, algebra, etc), but very very rarely anything more advanced.
PhD in CS, not from a school on his list.
The Math Myth - https://news.ycombinator.com/item?id=12418788 - Sept 2016 (313 comments)
And oh, the irony of dismissing the case for mathematics as a thinking aid as “self-serving nonsense” for which the author has “a very low opinion”!
Perhaps if the author had a better attitude towards education and was capable of appreciating the value of mathematics (or more generally, science) in everyday life (to say nothing about the importance of this in democracies), then he would not hold this infantile opinion.
Sorry, not sorry.
Edit: the point is not whether you “use it” when you’re out and about doing your groceries. If you’re judging the value of maths and science by this measure, you’re missing the point, and you’re at such a low level of insight that your perspective is useful to nobody. Rule of thumb: don’t listen to anti-intellectualists.
But I don't believe there was any name-calling, as I attempted to clarify in another comment: the "anti-intellectualist" label was intended to refer to the idea expressed in the article, not to the person who wrote it. (And even in the case where the label were attached to a person, it's not really an insult: I mean, it's definitely not a compliment, but to be clear, it says absolutely nothing of the person's intelligence, and rather speaks to their attitude towards "the pursuit of knowledge" in the abstract.)
I recognize that my intended meaning was not as clear as it should have been in the root comment, but I hope I have clarified this anywhere else. This was careless phrasing, but there was no ill intent.
"When disagreeing, please reply to the argument instead of calling names. 'That is idiotic; 1 + 1 is 2, not 3' can be shortened to '1 + 1 is 2, not 3." - https://news.ycombinator.com/newsguidelines.html
Thanks for the nice reply in any case.
I mean I personally think school is overrated, especially with high quality YouTube videos available for everything. But I don't think that is a common notion?
Edit: OK, you know what, why not - yes, abolish schools! They are just prisons for kids.
Seems pretty clear?
Maybe a more useful discussion would be how to measure success and how to determine what is necessary? This "8th grade maths is enough" is a bit too handwavy.
I can actually imagine that there would not have to be as much focus on doing calculations on paper or in one's head, as everybody has a calculator ready at all times these days. As one example.
> Anti-intellectualism is hostility to and mistrust of intellect, intellectuals, and intellectualism, commonly expressed as deprecation of education and philosophy and the dismissal of art, literature, and science as impractical, politically motivated, and even contemptible human pursuits.
https://en.wikipedia.org/wiki/Anti-intellectualism#Education...
We (North America) filter children early in the education system on the basis of their ability to do accurate arithmetic. Most jurisdictions don't even really teach math proper until high school. By the time kids get there, they've lost their self-confidence because the sheer tedium and boredom of the curriculum in the early grades.
Personally: Because of my primary and high school math grades I was barred from entering a computer science degree. But I have a 20+ year software engineering career only because I just really loved computers and pushed my way into the industry, and ended up at Google among other places, despite the lack of degree. But it was hard, really hard, to get into these jobs. And there's always been people in those jobs who gave me the sense they didn't think I belonged there with them.
The way we're teaching math to kids is on the whole really awful. I've seen both my kids struggle with it, despite being gifted intellectually and scoring incredibly high on the WISC-V.
The whole process seems geared towards grinding kids down in repetition and struggle.
Meanwhile we have machines to automate arithmetic and most math generally. In general, my sense of the way math is taught these days: it's an insult to the human spirit.
And people who excel in it and struggled through it and "won" have an intrinsic bias in seeing it as the "right" filter and "right" gating for what they're doing today, even when it clearly is not always directly related.
[1]: In my not-professional experience I see that people get lost in math down the road and will struggle in later classes because they have elementary troubles understanding even the basics of what an equation is. For example, I was baffled when one of my classmates argued that the x and y in an equation can't denote the same value.. But I also experienced the same thing when I tried to teach/help math to my sister. Math is simply not a subject like every other where not remembering the previous year's material is not a problem -- it necessarily builds on itself.
I certainly have NEVER used biology at work, and I didn’t do well in it in high school or college.
Math has a much stronger claim to be generically useful. Related fields, such as physics may in fact be seen something worth studing more because it teaches you how to apply math, than for the likelihood that you may end up in a job that requires the physics itself.
My own experience is that math/stats is by far the most useful (non language) topic I studied in school/university, even more so that Computer Science (and I work in IT).
Reading? Now I have access to all these stories! And it's plainly useful in every single other class, pretty much constantly (even math class—go figure, this math book has more words than numbers in it!). I can read subtitles (on anime, say—it's gone pretty mainstream now, for kids). I can read video games without voice acting, and follow them (Breath of the Wild requires lots of reading, for instance). It is constantly, obviously useful, and enables access to lots of things that most kids want to do.
History? Social studies? Largely an application of reading—and hey, look, more stories! Great! Oh and now I understand that movie I watched better, or that game I'm playing.
Science? Well look at that, a ton more reading, and when it's taught halfway right it's like working through a mystery story, even. Fun!
Math? It really seems like you're just doing it for its own sake, and... that's it. The actual facts or what's intended don't matter—that is what it feels like. It's occasionally useful in other classes, but not constantly like reading is, and often that math is pretty simple compared to the stuff you're learning in math class. Kids aren't dumb and recognize quickly that word problems are contrived, pandering, and ridiculous, and meanwhile have a hard time finding applications for any of this even if they go looking for it. Their parents? They probably see them rely on reading (even just signs or directions) 1000x more often than they see them apply math beyond what's taught by 6th grade. Reading? Useful daily. Math? Only occasionally, and it's mostly the simple stuff.
To a student, it feels like math is about as useful as learning some constructed language that everyone's telling them is super important but in their actual experience sure doesn't seem to be—I mean, no adults they know even read, write, or speak it, and they seem to be getting by just fine. It feels like you're being gaslit or having some kind of practical joke played on you. Of course it's hard to care about it, unless you're the sort who just loves mathematical puzzles, like the way some people are really into crossword puzzles, but most people aren't.
I don't have any kind of solution to this, but IMO it's the fundamental problem with math education.
Reading? Yes, the intricacies of clause structure? No, no one uses that besides a few people.
Science? Almost no one I know uses chemistry or biology. I don’t even remember how to balance a chemical equation.
The same arguments against mathematics education can be used for almost anything.
If “Reading” is the example for English education, then “addition, subtraction, multiplication, and division” are the analogs in math, and they’re used by pretty much everyone in day to day life and in other classes. Hell, they’re even used when reading.
Do you ever play board games and try to come up with the best strategy? That's math.
And that's not even getting into things that involve numbers…
You must not know many people. The problem with trying to categorize K-12 mathematics (or other common, but non-uniform experiences) as good or bad or somewhere in between is precisely that the experience is non-uniform. Even being in the same classes as my twin sister (though sometimes not), we formed some wildly different opinions on different teachers and classes and even classmates. The experience was too different even sitting 5' from each other, let alone when speaking about this with people who attended schools in different parts of the country, or even different parts of the same state or city.
Not discussed is that statistics and logic are a much better use of instructional time than calculus for most students.
Caplan has a doctorate, teaches at a university, has written several books and home educated two of his sons to a level where they published academically before they entered university. He’s also more well read in philosophy and literature than average though that is a low bar to clear.
If you don’t think a professor of Mathematics, David Edwards, understands the value of Mathematics why do you think you do?
The author, as decorated as he may be, seemingly fails to appreciate the latter: in his own words, he has a “very low opinion of this self-serving nonsense”. The exercise of fleshing out the argument against the higher-order value of mathematics, you will note, is left as an exercise to the reader.
This is by definition an anti-intellectualist position: defending such a position, which goes contrary to what should be the default position in general, should require a very high burden of proof, not an offhand dismissal.
Directly from the article.
- He doesn't have a high opinion of it
- He makes a vague allusion to some unspecified conflict of interest (?!)
- He argues that first-order skill development is most effective (which may be true, but misses the point).
- He says that higher order value of mathematics is "unsettled".
As he is defending a fundamentally anti-intellectual position, I'm going to need a little more than that.
I’m sure the author has a high opinion of mathematics as an intellectual pursuit. He is a Math professor. That’s separate from his argument, that it has very limited practical use, even to most engineers and others you would assume would be highly selected for finding it useful.
If Mathematics was enormously useful for teaching argument and precise thought in a reliable way Economics would have eaten all the other social sciences already. Economists know far more Math than the others. Math is uncommonly useful but if it was that good at teaching people how to think, if the transfer of learning argument was true, it would not need to be argued. It b would be bloody obvious.
There's this thing called the burden of proof, in philosophy. When you take a difficult position, you must displace this burden. The author has not done so, and it is not my responsibility to show how he is wrong: he has not shown how he is correct.
Very eloquent phrase, I’m going to steal this!
And we do use that in everyday life. Learning higher maths doesn't help us solve problems we encounter all the time or everyday, at least not directly. But it helps us frame them and our thinking conceptually in a quantitative way, and maybe even sometimes in a qualitative one, which is incredibly helpful generally.
Lovers of physical exercise, the production of art or combat can sing the praises of their joys and the benefits according thereto. We don’t rearrange the world to force their interests on the world. If we’re going to do it for our interests we should have actual reasoning behind it, not hand waving.
This is not compatible with an inquisitive, academic or intellectual mindset, no matter that he is a professor or that he holds a PhD. He's free to advance arguments in support of this position, but he should be met with extreme skepticism, as he is defending a difficult position.
The world has millions of things to learn about. Why is Math specially subsidized? That’s good argument. That’s not anti-intellectual.
> He's free to advance arguments in support of this position, but he should be met with extreme skepticism, as he is defending a difficult position.
Again, you’re just assuming[edited] your conclusion. You and I and he are people who like ideas and arguing over ideas. He’s arguing that what he teaches isn’t that useful and asking why it gets so many resources. There’s nothing there about the beauty or joy of Math, which I presume he could wax rhapsodic about as a professor. You’re attacking a position that doesn’t appear in article. It’s about the practical value of advanced Math.
I'm not sure it is a good advice in general. All the quack "doctors" that go against the mainstream medicine are actually degree-earnt doctors, yet are just scams.
GP did not say "against their profession's interests", just their own.
If their kids were kept in normal schools with other kids, they may have published academically a few years later, but a lot of their peers would have had the privilege of studying with them and improved their academic lives because of that.
If you need other students to successfully study, that's either yours or the educational system's failure; but other children shouldn't be dragged down for it.
There is a large number of studies on the positive effects of integration in education, from more extreme cases such as not having special schools for disabled children to the more mundane socioeconomic mixing. The consensus seems to be that the positive academic effect a smart kid gets from being put in some advanced school is trumped by the negative effect on the average kids. European countries with a strong egalitarian bend in their schooling system seem to do much better than the American ones and not just on average.
Europe is not some fantasy-land where all the dreams of the American left come true. Most European countries track students into separate secondary schools based on their demonstrated academic ability.
https://en.wikipedia.org/wiki/List_of_secondary_education_sy...
I'm not sure; France and Russia and their selective systems, respectively classes préparatoires and elite schools, do not seem to do worse than other European countries.
You're barking up the wrong tree. At worst it's elitist, not anti-intelectual:
"The math myth is the myth that the future of the American economy is dependent upon the masses having higher mathematics skills."
Just doubting the need for higher math skills to be massified, not against their being necessary, perhaps to an elite.
Higher mathematics should be offered and taken by those who need it, or want it; but never required of all students.
Also the author is a mathematics professor. He isn’t anti-intellectual.
EDIT Wow. You posted ten angry comments in about 40 minutes. This seems to have hit a nerve.
It doesn't matter who he is: the idea he's advancing is fundamentally an anti-intellectual one. It would be an anti-intellectual idea even if it came out of Einstein's mouth.
Edit in response to edit: sorry if my comments give the appearance of anger, that is sincerely not my intended tone. I am admittedly very passionate about the issue of anti-intellectualism in society.
Hate to say it, but he looks like a Malcolm Gladwell wannabe with some attention-grabbing book titles. Perfect for a TED talk.
EDIT: I would add that Caplan is a working researcher who has made significant and original contributions to economics, where Gladwell is a journalist who interprets the work of scholars like Caplan. If anything, Gladwell is a Caplan wannabe (although I wouldn't characterize either of them in that fashion, I think they have different goals and methods)