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Most students find math unfathomable, labyrinthine by the time of univariate integration (Reddit). Even overachievers – who ace undergraduate math without studying – will eventually be convoluted by math, like in post-doctoral abstract algebra or number theory. But exactly why’s (abstract) math way knottier and thornier than law or medicine? I’m comparing math with law and medicine as they’re highly sought non-math subjects.

Rule out these non-answers because they also appertain to law and medicine.

Pre-suppose no dyscalculia. Pre-suppose no ineffective teaching that cannot be the decisive reason, because math still bewilders and overwhelms students even when taught skillfully by teachers who have PhDs in math.

Information overload and quantity. Information changes more swiftly in law and medicine. Laws are daily created, amended, repealed by administrative authorities, civil servants, or politicians. Old medications stop working (e.g. drug resistance), new medications are patented, and new diseases crop up.

Incrementalism. Law and medicine “also build on each other in a way that most topics don’t, especially before college”. Law schools make contract law a prerequisite for commercial or maritime law.

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    $\begingroup$ What precisely makes you think that math is "way knottier and thornier than law or medicine"? I can't see evidence for this in the question. $\endgroup$ Dec 19, 2021 at 21:50
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    $\begingroup$ You are likely going to get a lot of answers that say "no, it isn't" or "what do you mean by it being harder?" The links in your first sentence don't clarify what you mean by math being harder than law; they are about why antiderivatives are less straightforward than derivatives. It would be great if you could clarify exactly why you think math is more difficult than law or medicine. $\endgroup$
    – Chris Cunningham
    Dec 19, 2021 at 21:51
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    $\begingroup$ Maybe I'll flesh this out more as an answer later, but my initial reaction (in the U.S.) is that unlike law and medicine, pretty much everyone has to study math at least through the first year of university (high school requirement, university graduation requirement and additional math for many majors), so law and medicine have a huge filter there. Then there's the much greater society/social prestige for law and medicine, and the much greater salary for law and medicine (especially when those who leave math for some allied field such as comp. Sci. or finance are excluded). (continued) $\endgroup$ Dec 20, 2021 at 14:39
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    $\begingroup$ There are some major social and cognitive differences between math and the professions of law and medicine. In the US, there are about 26,000 medical and osteopathic graduates each year, and about 40,000 law graduates each year. These are large numbers compared to math doctorates, even if you include computer science and statistics. Both math and law require cognitive changes to develop ways of thinking that are specialized, and organized memorization of a large body of facts. I think the memorization task is much smaller in math. $\endgroup$ Dec 20, 2021 at 15:13
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    $\begingroup$ There is no evidence given that the fundamental premise of this question is true, and I, personally, find the premise of the question highly suspect. Without the premise, though, the question is non-sensical: the question "why is X true" is meaningless if X is false. I'll give just one example: the legal case Oracle v. Google is about a very simple question: are these 11000 lines of code in Google's system copied from Oracle's system? The legal case has been going on for 11 years now, and went all the way up to the Supreme Court. The Supreme Court ruled 6–2 in favor of Google, but crucially … $\endgroup$ Dec 21, 2021 at 8:56

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The perceived difficulty of abstract math is due to two factors:

  1. You learn math at school, but it is actually very different from what you do at university. In school you are applying rules to get some numerical result. And the rules are easily mapped to concepts you encounter in real life. This changes in university. Suddenly you work almost exclusively with proofs, and real world applications are often only an after thought.

    This change of pace causes pupils that are good at math in school to struggle at university, because they got into something different then they originally thought.

    The common idea of what studying law or medicine is, is much closer to the real thing.

  2. Abstract math is abstract. If you pick up some law text you can understand the basic ideas directly. You will certainly miss some finer points, but most of the words used will have a meaning similar to the one they have in every day usage.

    Medicine is a little more difficult. There are many special terms that you have never heard before – like Latin names for every organ, tissue and part of your body. But most of these terms can be easily explained with a single paragraph of text in common in English and an image. So again you can read and understand the gist of it as long as you can access Wikipedia. Of course, this changes once you get into things like e.g. biochemistry.

    Compare that to a university level text book on math. After a few pages of introduction, you'll be confronted with pages of pages with very little of actual English. Most will be special symbols which have absolutely zero meaning in real life. In order to understand it, you cannot just look up certain terms, you have to look up every single symbol.

    Take for example Euler's identity: $e^{i\pi }+1=0$.

    Every symbol has a very specific meaning, and even the positioning of terms above and to the right, or right next to each other have meaning. And this is a short and simple example. And at first glance, this has nothing to do with your real life. There is no experience from your life to which you can tie this new concept. And tying new information to existing information is something very important for learning. So studying math easily ends up being like learning new concepts – while at the same time – being like learning a new language to express these concepts in a language, using a very different alphabet and grammar than the one you are used to.

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    $\begingroup$ "Of course, this changes once you get into things like e.g. Biochemistry." Please expatiate and elaborate? Why does Biochemistry make medicine so much LESS accessible and intuitive? $\endgroup$
    – NNOX Apps
    Dec 20, 2021 at 18:13
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    $\begingroup$ @evolva I think chemistry shares many properties of math, by being very abstract and using a language that is very far removed from normal spoken languages. $\endgroup$ Dec 21, 2021 at 7:04
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    $\begingroup$ Euler's identity is at least still using numbers! Set theory is commonly said to be the foundation of mathematics, and is so abstracted that lots of discussions of it don't even use numbers. $\endgroup$ Dec 21, 2021 at 11:44
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    $\begingroup$ @curiousdannii Almost all graduate level maths uses letters/ symbols which do not even represent just numbers but much more abstract concepts. Even large chunks of number theory are not about numbers in a sense a lay person would think of as numbers. $\endgroup$
    – quarague
    Dec 21, 2021 at 16:03
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    $\begingroup$ @evolva See here. $\endgroup$
    – J.G.
    Dec 22, 2021 at 21:10
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Univariate calculus — e.g. integration (see also Reddit) — is when most students find math unfathomable and labyrinthine.

Well, not really. Actually most students never reach this level of math, and most students who have difficulty with math have difficulty with much more basic math than this.

Suppose, for example, that I tell you that a bedroom community called Havensleep has an area of 17,000 acres, and I also tell you that the average housing lot in Havensleep is 0.43 acres. Can you figure out how many lots there are in Havensleep?

In my experience, most college students are unable to solve this, even with the aid of a calculator. They're given the numbers $x$ and $y$, but they don't know what arithmetic operation to do with them. Should they do the product $xy$? The sum $x+y$? Should they find $x/y$ or $y/x$?

This is supposed to have been grade school math, but the way grade school math was taught, one was always told what operations to do.

Why is math way more difficult, puzzling, abstruse compared to other subjects like law and medicine?

I don't think this is true at all. I would have a terrible time with premed classes such as o-chem, for example, because I'm bad at memorization. Ditto for a lot of the graduate curriculum in law school. Law can be very difficult and abstract. Can you explain what a tort is? Can you explain why Roe v Wade is important constitutional law even though Casey has completely replaced its criteria for regulation of abortion? Do you know which constitutional amendments Roe invokes and how this makes it different from previous case law? Can you make a coherent argument as to whether it does this in a valid way or not?

In general, people just tend to be bad at things that they're never given practice doing. The Havensleep example is one such thing that students are not given practice with. But in most public K-12 schools in the US most students get very little exposure to things like recognizing parts of speech, using the metric system, recognizing the sounds of different musical instruments, or breaking up their writing into paragraphs.

Ineffective teaching cannot be a reason, because students are still bewildered, overwhelmed by math even when taught well by teachers who have PhDs in math.

Subject matter knowledge is necessary but not sufficient to make someone a good teacher.

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    $\begingroup$ "Subject matter knowledge is necessary but not sufficient to make someone a good teacher." Indeed. You can have a PhD in differential geometry and still be terrible at teaching, say, an introduction to derivatives to 16-year olds. And on the other hand, you can be a math-loving recent high school graduate and be awesome at it. It may be likely that an arbitrary doctor is better than an arbitrary high schooler at that particular task. But it is not a foregone conclusion. $\endgroup$
    – Arthur
    Dec 20, 2021 at 14:05
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    $\begingroup$ Your Havensleep example actually needs some out-of-the-box thinking, because we don't know what % of the community is occupied by lots and how much space is occupied by streets, parks, playgrounds, etc., so the answer to the question is: no, you can't. Unless "lot" means something else than I think it is, but that's not a maths question. Is that the answer you expect? $\endgroup$
    – gerrit
    Dec 20, 2021 at 14:25
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    $\begingroup$ @gerrit: Something I've had to teach my kids as I help them with math homework is... "The question has all the information needed to find the answer." If one is supposed to factor in space for streets, etc, then that information will be provided. If the problem is presented in a "Community Planning" class, then it's a different problem altogether. $\endgroup$
    – James
    Dec 20, 2021 at 14:47
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    $\begingroup$ @James Which is actually a little bit of a shame. To quote some TED talk from 10 years or so ago, "What problem have you ever had to solve in real life where you had all the information you needed presented to you, and no superfluous information?" Yes, if the goal is to teach concrete mathematical tools, then there is merit to curating the given information. But if we want students to be prepared for using their math the real world, then maybe one ought to train them a little in gathering and filtering the initial information as well. $\endgroup$
    – Arthur
    Dec 20, 2021 at 15:48
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    $\begingroup$ @Arthur: I can see some merit in your argument, but consider a math class homework problem... You have a 2m long wood board and wish to cut it into 20cm pieces, how many pieces will you have? Now to answer that accurately, the student must research saw blades and assume a kerf width. Then the answer will be something like "Assuming a 1mm kerf thickness, one may obtain 9 pieces of length 20cm." I suggest that only a woodworking class should expect/accept an answer like this. But I'm fine with the more complex answer if the math teacher includes the kerf thickness in the question. $\endgroup$
    – James
    Dec 20, 2021 at 19:26
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Many good answers already, but here's one more thought: the bar in math is set much higher.

When you're doing law or medicine or languages or whatever, there's a fairly broad spectrum of "OK". You don't need to know stuff perfectly, you can also arrive at acceptable solutions which are not ideal. You can also rely on understanding some things only partially or intuitively.

In contrast, maths is very strict. It has to be, because it's completely abstract and talks about absolute truths and falsehoods. You cannot "sorta kinda" prove a theorem, it has to be an ideal, watertight proof, or else it doesn't count at all. There's still plenty of room for intuition, but intuition alone is not enough - you need to follow it up with precise, concrete logic.

And human mind is not very well suited for that. It prefers to categorize the world in vague, fuzzy patterns, and this razor-sharp ultra-precise thinking does not come naturally to it. You need more training to get your brain to accept and memorize and utilize these kinds of patterns.

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    $\begingroup$ @evolva - No, no, not unfit. This isn't about how well you've mastered something, but rather what kind of thinking you need to master it. Unfortunately I have little experience with medicine or law, but from what I understand medicine is mostly about memorizing things. There's not much that you can logically deduce in medicine, 90% is empirical knowledge or results of long and complicated scientific studies. And of course the practical side which relies on muscle memory. That's why medical students study so long - they need lots of repetition. $\endgroup$
    – Vilx-
    Dec 20, 2021 at 18:19
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    $\begingroup$ I am afraid I am unconvinced. "the bar in math is set much higher." "When you're doing law or medicine or languages or whatever, there's a fairly broad spectrum of "OK". You don't need to know stuff perfectly, you can also arrive at acceptable solutions which are not ideal." "In contrast, maths is very strict." Pls expatiate and elaborate? Competent lawyers and medicines CANNOT simply be OK. An unfit lawyer can lose your case, or get you innocently convicted or held liable. An unfit physician can harm or kill you. $\endgroup$
    – NNOX Apps
    Dec 20, 2021 at 18:20
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    $\begingroup$ @evolva - Similarly for law. You have to memorize plenty of details and principles, but after that it's down to "who's the most eloquent talker" and "what bits of the things I've memorized can I use to get as close as possible to my desired result?" But there's very little hard logic where you have to come up with 30 precise consecutive steps that will give you an EXACT result which you need. $\endgroup$
    – Vilx-
    Dec 20, 2021 at 18:26
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    $\begingroup$ @evolva - Yes, maths students need lots of repetition too. The difference is that medical students memorize the final answers - in case of X do Y. Sometimes a patient has multiple issues and then you have to do a bit of thinking or basic maths to choose the best option, but no more than that. In contrast, maths students memorize tools that they need to combine in myriad of ways to get the actual answers. You very rarely get to memorize the solution to a specific equation; rather you memorize formulas that are broadly useful, but which you still need to combine for any particular equation. $\endgroup$
    – Vilx-
    Dec 20, 2021 at 18:35
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    $\begingroup$ Another aspect might be that law and medicine deal with real things in the real world. They're familiar and easy to conceptualize and visualize. Maths on the other hand are completely abstract. There's no quadratic equation lying on the side of the road. Your mind needs to come up with completely new ways of visualizing and understanding the concepts in math, often from scratch, without anything familiar in the real world to build upon. $\endgroup$
    – Vilx-
    Dec 20, 2021 at 18:41
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Ineffective teaching is absolutely part of it. Math is about understanding and problem-solving. Problem-solving is hard. And students who aren't already into math want it easier. So the teacher "helps" by showing them steps (me included!), and then they aren't really learning as much.

K-6 teachers teach elementary school because they love kids (most, anyway), but that doesn't mean they like math, or understand it at all. Their mathphobia gets passed along to the students.

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    $\begingroup$ I am afraid this answer dodges, and doesn't answer, my question. "Ineffective teaching is absolutely part of it." Yes in reality. But I'm pre-supposing, and narrowing my question to, mathematical hardship not caused by ineffective teaching. $\endgroup$
    – NNOX Apps
    Dec 20, 2021 at 18:03
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    $\begingroup$ I certainly was not attempting to dodge. Your suppositions are contrary to fact, so it's hard to imagine what the answer would be. You might want to ask folks who've worked in educational systems that deal successfully with math. $\endgroup$
    – Sue VanHattum
    Dec 20, 2021 at 20:45
  • $\begingroup$ Sorry!!!! did I outrage you with "dodge"? "Your suppositions are contrary to fact" You're quite correct. But I want to focus on the academic or intellectual differences in complexity between math vs. law, medicine. I hope you understand now? $\endgroup$
    – NNOX Apps
    Dec 21, 2021 at 0:11
  • $\begingroup$ No outrage here. Just noticing that we are not communicating well. $\endgroup$
    – Sue VanHattum
    Dec 21, 2021 at 2:44
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"Even overachievers — who ace high school calculus without studying — will eventually be puzzled or convoluted by math, like in undergraduate or graduate Analysis or Abstract Algebra. But why exactly is (abstract) math way knottier and thornier than law or medicine?"

You have used the term "overachievers" to describe people who ace high school calculus without studying, but within them, there's some who even ace undergraduate and graduate-level analysis and abstract algebra courses. They will still run into challenges eventually, if for example they go on to become a researcher who wishes to publish new findings or solutions to unsolved problems in those areas. But that is also true for medicine and law students at that level (i.e. people who are actually practicing the subject professionally for a living, will find that there's a lot of challenging unanswered questions, like why is this patient not responding to this medication like the textbooks would suggest, and what legal reasoning can I use to acquit the defendant I'm representing?).

For the most elite of the "overachievers" (to use your term!), it will often not be hard to get through medical school, law school, and graduate-level analysis and abstract algebra courses, and the biggest challenges will come when trying to advance the cutting edge of the subject (proving unsolved conjectures in analysis, discovering a cure for cancer, reforming the legal system).

Perhaps it's the "mid-range overachievers" where the question becomes a lot more interesting. these are people who aced high school but struggle in analysis and abstract algebra courses. I could probably write forever on the subject (and since this is my first time posting on this site, I'll leave space for others to weigh in too!) but I do agree with you that the "mid-range overachievers" in mathematics will struggle more than the "mid-range overachievers" in medicine or law. Out of the many things I can say, perhaps one stands out: I've always been able to pick up a law or medicine textbook and understand what I'm reading, and learn whatever new terms I come across by looking them up further (as long as it's written in standard English), but a random math textbook will rarely read like a novel, and will have tonnes of symbols and notation that vary from author to author, and most of all, I find myself needing to read extremely slowly and carefully with many hours of thinking in between to understand the proofs, whereas in medicine and law I am typically just reading a regurgitation of facts without having to try to reproduce the proofs of those facts on my own. In analysis and abstract algebra courses, even at the lowest of undergraduate levels, we want students to actually do the proofs on their own, not just to know what the theorems/facts are.

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  • $\begingroup$ thank you! "there's some who even ace undergraduate and graduate-level analysis and abstract algebra courses. They will still run into challenges eventually" Yes I know. Do you think I should remove the example of graduate-level analysis and algebra? $\endgroup$
    – NNOX Apps
    Dec 20, 2021 at 2:46
  • $\begingroup$ your last para. grasped my intent and intention completely. I'm happy to see that "I've always been able to pick up a law or medicine textbook and understand what I'm reading, and learn whatever new terms I come across by looking them up further". $\endgroup$
    – NNOX Apps
    Dec 20, 2021 at 2:46
  • $\begingroup$ @evolva Thanks for the kind words, and thanks for accepting my answer! It's my first time posting on this site :) I don't think you need to remove anything from the question. It's all good! I'm glad you were able to connect well with my last paragraph :) $\endgroup$ Dec 20, 2021 at 3:04
  • $\begingroup$ You are very welcome. Do you know any studies or articles or even just anecdotes to substantiate that math is more enigmatic than law or medicine? $\endgroup$
    – NNOX Apps
    Dec 20, 2021 at 3:08
  • $\begingroup$ @evolva I could check, but perhaps in the future people will refer to this thread when they ask themselves that same question :) $\endgroup$ Dec 20, 2021 at 3:27
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The standards in math are way higher because of supply and demand.

Lawyers and doctors directly help people. We need lots of them. Mathematicians do some difficult to understand work that might possibly be useful to someone somewhere sometime. We need some mediocre lawyers and doctors just so that all the necessary legal and medical work in our society can get done. We don't need mediocre mathematicians (though some people who might have been mediocre mathematicians end up being engineers (or lawyers!)).

Is mathematics easier than creative writing or philosophy, which are fields with a similar level of demand? I seriously doubt it.

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  • $\begingroup$ That is a good point! You basically say "they struggle because indeed it is much harder", and explain why. $\endgroup$ Dec 20, 2021 at 12:07
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    $\begingroup$ We don't need mediocre mathematicians - We do need many K-12 math teachers though. While some of them could be called good mathematicians, many are "mediocre". $\endgroup$
    – Kimball
    Dec 20, 2021 at 16:20
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    $\begingroup$ @Kimball - and, in reality, we don't expect all our high school math teachers to actually understand real analysis or abstract algebra. In the states that require high school math teachers to have a math degree, universities respond either by creating an alternative math major largely devoid of proofs and abstraction or by making it possible to pass analysis and algebra without coming up with even the simplest proof. $\endgroup$ Dec 20, 2021 at 17:08
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    $\begingroup$ @evolva - I was once a juror for a trial. After half a day I understood the case better than one of the lawyers. No mathematician is that mediocre; they wouldn't get into a PhD program in the first place, and if they did, they wouldn't get a job. $\endgroup$ Dec 20, 2021 at 20:15
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    $\begingroup$ We don't need mediocre mathematicians – As a mediocre mathematician, I disagree. Even within pure mathematics, there is clearly a spectrum of the quality of mathematical work (as perceived by pure mathematicians) going by how relevant, innovative, or challenging it is. I see no reason to assume that this spectrum has a natural bottom, but instead it gradually shifts to people who go for the things “better” mathematicians don’t care about or connect mathematics to application. […] $\endgroup$
    – Wrzlprmft
    Dec 21, 2021 at 7:47
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Using only my personal experience: I think it is indeed that mathematics as a science is entirely abstract, as Jens said in another answer.

Both law and medicine are sciences that have a strong connection to real world events or items that everybody can relate to. In effect they deal with failures of humans and failures of their everyday interactions. That is unknown to exactly nobody; solving the problems of either field is of interest to everybody. That's why everybody talks at parties about their medical problems or their divorce proceedings, but very few people outside of department Christmas parties have small talk about Abelian groups.

The case is very different with the part of math that is directly related to real-world issues. You can have party talk about folding paper to get to the moon, or about an exponentially growing number of infections, or the chances to draw a flush on the river. Few people have trouble understanding math up to, say, 10th grade where you count things, divide amounts, have parabolic trajectories of balls etc. That's the kind of math that is at the level of law and medicine. You have a real-world problem and a systematic approach to solve it. Medicine and law then continue to heap on more of the same; what is difficult for students is the sheer amount of stuff to cram into their heads, not the increasing abstraction.

That's different with math. You don't have another 42 similar laws after the commutative and distributive one. You also don't need to learn multiplication tables for the first 52 natural numbers — but that's what law students must do in the United States, and medical students with the bones and blood vessels in the body. That is why those subjects are hard. That is not why math is hard. In math you see that there is a general pattern in the laws governing the concrete operations (addition) and sets (apples) you started with. Math then goes on defining and making statements about these abstract patterns. It's as if M.D.s, after looking at three human specimens, would start making generalized assumptions about the self organization of matter in a space-time matrix governed by certain natural constants where the matrix loci are only allowed certain parametrized interactions with their neighbors. That's not what doctors do: They try to save their patients' lives. That is as if math professors would be concerned with counting change in the supermarket or engineer bridges. That is not what mathematicians do.

The secret to understanding abstract math is, I think, to become acquainted with the abstract building blocks so that they take on the role of things the student can relate to, as if they were part of the real world. Series and sequences, limits, operations and relations and so on become almost tangible objects about which calculus and abstract algebra make statements. I think it's a bit like a muscle memory of the brain: You make certain concepts your own and use them with ease so that they can form the basis for the next abstraction. This "mental muscle memory" is built with practice. To some, of course, this practice doesn't feel like a chore, very much like a gifted musician builds muscle memory on the side by doing what they like best, while others loathe their piano practice, or their math homework.

But however you get there: You need to be familiar with the concept of a function and of derivation to understand the concept of a differential equation, and so on.

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    $\begingroup$ "law and medicine are sciences" Law is a science??? $\endgroup$
    – NNOX Apps
    Dec 20, 2021 at 18:26
  • $\begingroup$ @evolva Well, it's complicated: In America it's more of a (liberal) "art" -- but so are the natural sciences! In German we call jurisprudence plainly "law sciences". It's taught at universities, there are academic titles ... Admittedly, most work in the field is to "law science" what engineering (or plumbing!) is to physics, but still, it's as defensible as saying "political science". $\endgroup$ Dec 20, 2021 at 19:03
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    $\begingroup$ a good friend of mine becomes rather indignant when someone calls medicine a "science". He insists that medicine is an "art". His standard for science is too high for medicine. $\endgroup$ Dec 21, 2021 at 16:48
  • $\begingroup$ I've long considered mathematics an art, not a science. $\endgroup$
    – ryang
    Dec 21, 2021 at 16:51
  • $\begingroup$ @JamesS.Cook The solution to that is certainly to up the standards of medicine! Other fields are much more preposterous in assuming that label, like psychology and economics. Even when they go through the motions associated with science it's more like a cargo cult (reproducibility crisis). $\endgroup$ Apr 25 at 7:28
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I think a large part of this is that the media bombard us from an early age with the message that maths is boring and difficult. Neither of these things are true, the point is that the media industry is dominated by people with arts and humanities degrees, so they are just telling us that they found maths difficult and boring, because their interests lay elsewhere.

Another part of the problem is that maths, unlike e.g. art, has a stage where we have to do a lot of groundwork with no immediately useful benefit to us. However we can't get to the more interesting and useful aspects of maths until we have got past that stage. This is a reason why more children play football than cricket. All you need to be able to play football is some basic level of fitness and co-ordination. You can play without technique or strategy. That doesn't work in cricket, if you can't play a solid defensive shot, you won't bat long against a bowler that knows what they are doing. This means that it can take you a lot longer to get to the point in cricket where you really understand what is going on and enjoying it fully.

For students to learn any subject effectively you need them to be interested in learning, so if the majority of them have been told for years that maths is difficult and boring, and they can't see the point in e.g. algebra, because it isn't useful in their daily lives, as a teacher you have a much more difficult job on your hands than e.g. an art teacher.

On the flip side, you only need to be a little bit good at maths to gave a genuinely useful skill (as it is relatively uncommon), where as you have to be extremely good at English to have a similarly useful level of skill.

BTW, I am not an overachiever. I got a D at A-level (not good), but these days I am happy writing/reading about reproducing kernel Hilbert spaces. People learn things at different rates at different times in their lives, and a lot of it has to do with how much they want to learn at the time.

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    $\begingroup$ I think this is one of the best answers! I signed up on Maths Educators just to upvote $\endgroup$ Dec 20, 2021 at 13:17
  • $\begingroup$ don't your first two paras. contradict each other? first you deny that math is boring and difficult. but then you explain why maths, like cricket, has a way thornier and knottier barrier requirements and entry. "the media bombard us from an early age with the message that maths is boring and difficult. Neither of these things are true". "maths, unlike e.g. art, has a stage where we have to do a lot of groundwork with no immediately useful benefit to us". $\endgroup$
    – NNOX Apps
    Dec 20, 2021 at 18:19
  • $\begingroup$ @evolva no, I don't think they do. Learning the basics of maths is not particularly difficult, but part of the problem is that to learn you need to bring some enthusiasm with you. We can't expect our teachers to provide motivation for everything we do and learning to do groundwork is one of the more useful things we can do at school (but this is rarely taught). The barrier requirements are not thornier or knotty, it is just that they exist in a way that they don't for e.g. art. Learning maths is no more difficult or boring than learning grammar or metre in English. $\endgroup$ Dec 21, 2021 at 8:03
  • $\begingroup$ "The barrier requirements are not thornier or knotty, it is just that they exist in a way that they don't for e.g. art. " Can you please expatiate this? I will use music as an example that also has barriers? Just consider PhD's in music composition. $\endgroup$
    – NNOX Apps
    Dec 23, 2021 at 4:59
  • $\begingroup$ @evolva I am an amateur musician and I know from experience that there isn't much of a barrier to enjoying playing an instrument, just to being able to play an instrument well (which I haven't managed yet). Everything is difficult and has barriers at PhD level, if it wasn't it wouldn't be worth a PhD. Whether someone finds a topic particularly difficult is more a statement about the person than the subject, mostly interest but also talent - it is hard to learn anything you are not really interested in. $\endgroup$ Dec 23, 2021 at 6:59
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Sorry for one more answer...

Math needs a special way of thinking, different from our everyday practice (and, to some degree from law and medicine).

In everyday life, we often use similarities and analogies. If something is similar enough to an established truth, we accept it. To my understanding, the whole system of precedents in law is based on that kind of thinking. And in medicine, a close match in symptoms often is enough to correctly diagnose an illness.

Our brain is hard-wired by evolution to excel at "pattern matching", at classifying things by similarities.

In math, there is no such thing as "similar to truth". Something is either correct or incorrect, with nothing in-between.

And here comes my criticism on (some approaches to) math education. Students can achieve high math grades by doing things the "pattern matching" way, by just finding the "most similar formula" and blindly applying it. Give them a quadratic equation using c as the variable instead of x, and they're lost. But we don't do that, we always use x, and so we support the non-math pattern matching approach instead of the exact thinking process needed in math.

And then, later on, when exact thinking "the math way" is needed, it has not been trained enough.

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    $\begingroup$ "a special way of thinking", "similarities and analogies"? Like counting apples, or measuring something with a tape measure or clock? There're not much better "similarities and analogies" than between numbers and real life! I don't think this argument hold water. $\endgroup$ Dec 20, 2021 at 13:20
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    $\begingroup$ I do think this is a really good answer. Pattern matching in fields like law and medicine are fuzzy, with gradients of truth. I think that's what this answer is about. $\endgroup$ Dec 20, 2021 at 15:52
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    $\begingroup$ Doesn't math require pattern recognition too? See also encyclopediaofmath.org/wiki/Pattern_recognition. $\endgroup$
    – NNOX Apps
    Dec 20, 2021 at 18:59
  • $\begingroup$ @LeonardoHerrera Thank you for suggesting the word "fuzzy". That is what I meant. $\endgroup$ Dec 21, 2021 at 10:01
  • $\begingroup$ @evolva - pretty much anything where our brain is present requires pattern matching, but that's beside the point. It's just an idea that for some disciplines you don't need an exacting brain, so doing some very inexact pattern matching and appropriate reasoning you can thrive and excel. I, for one, cannot remember what I did last week and have a really bad memory for exact stuff, but I can find my way around just by associating concepts and playing a mean google game. $\endgroup$ Dec 21, 2021 at 16:47
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TL;DR: The problem with math is that it is layers of abstraction. You need some familiarity (not just bare understanding) with one layer before you can progress. Most students don’t realise this and thus become frustrated when they fail to understand something even though it is clearly defined.

Many things have already been said, but I need to challenge your second assumption:

Rule out these non-answers because they also appertain to law and medicine.

[…]

Incrementalism. Law and medicine "also build on each other in a way that most topics don't, especially before college". Law schools make contract law a prerequisite for commercial or maritime law.

Incrementalism is no binary thing: All fields have some of it, but there is a wide spectrum from deep fields (with lot of increment) to broad fields (with a lot of content). (Pure) mathematics is clearly on the deep side of this spectrum: For example, you can be a number theorist by only being good at this one thing and knowing about a few related mathematical subfields. The number theory that doesn’t require depth has already been discovered millennia ago. By contrast, a general medical practitioner needs to know about a lot of common ailments and a bit about biology, chemistry, physics, pharmacy, psychology, bureaucracy, and mathematics.

This is exacerbated by the abstractness of mathematics. In fact, I would argue that the core of mathematics is not proofs, but finding the right abstraction to be able to formulate useful theorems and facilitate proving them. And this abstraction comes in layers on layers. The problem with this layered abstraction is that you have to be somewhat fluent in one layer to grasp the next one. Otherwise you quickly have too many things you need to mentally juggle at the same time.

Consider your example of integration: To understand what integration is, you need to have a clear understanding of functions (amongst other things). To understand functions, you need to understand variables. To understand variables, you need to understand numbers and elementary arithmetics. And the last things are already something on which most children spend the entirety of primary school. (Yes, we also teach them stuff like multiplying with pen and paper, but that’s only a means to the end of understanding numbers and multiplication, since pocket calculators can do this better than any human.) And university-level mathematics puts layers of layers on top of that.

By contrast, consider MRNA vaccines (i.e., some top-notch medicine). To understand how these work, you need to first understand what cells are and how DNA works. On this basis, you can continue to understand what MRNA does, how viruses work, and how the immune system works. There are some layers here, but only some and they are not that abstract. For example, you can see cells under a microscope and you can explain DNA by analogy to blueprints. I can understand these things without any university-level education on these matters, and I can even explain them to a child in primary school. (Also see this XKCD.) Mind that I am far from being able to develop an MRNA vaccine myself, because I considerably lack breadth in medicine, pharmacy, molecular genetics, etc.

Now back to your question: Since mathematics requires you to wrap your mind around layers of abstraction, what determines your learning speed is how quickly you can understand abstract concepts and automatise them (so you can apply them without occupying too much of your brain). If you go too fast, you will inevitably hit a barrier where you run into something that is clearly spelt out to you, but you cannot understand because you just lack experience with the previous layer of abstraction. This is naturally frustrating, in particular if you do not understand why you are struggling (which is quite common because most professors do not realise or tell you these things; at least nobody told me). Something similar also happens in other deep fields, such as physics, philosophy, or linguistics.

By contrast, the limiting factor of learning speed in a broad field is the amount of stuff you can learn. You will hit barriers, but they are of a different and more obvious nature like bad explanations, being applications of another field, or you being tired from cramming too much knowledge into your brain since you last slept.

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  • $\begingroup$ Many good answers (incl. Vilx's) on this page, but I find the final three paragraphs here particularly resonant. $\endgroup$
    – ryang
    Dec 21, 2021 at 17:12
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    $\begingroup$ How the cake did I miss that XKCD? That is a fantastic supporting comic. $\endgroup$
    – William
    Dec 21, 2021 at 23:07
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Aspects of this answer may be specific to the US. I treat this as an empirical question about perceptions (i.e., what people think is more puzzling) than a question about what subject actually "is." I think three factors are particularly relevant.

The first:

A lot of people see math relatively early in life, while most people do not see law or medicine, or at least not until relatively later in life. This causes difficulties in math to be more readily apparent than other difficulties.

The question refers to "univariate integration" and this is a good illustration of the point. Integration is taught in most US high schools and almost every US college or university, where it is taken by many students who are not planning careers that directly involve math. There is no law or medicine counterpart to this! (Many students entering law or medical school in the US have no pre-existing legal or medical education, and "popular" treatments of law and medicine often smooth over details in the interest of storytelling in a way that a calculus instructor is under no obligation to do.)

People from all sorts of backgrounds might have a common experience of once having taken something like calculus. Calculus thus becomes a frame of reference for "something abstruse or difficult." There is no similar legal or medical touchstone. And there's no reason why the touchstone "has" to be math. Maybe 100 or 200 years ago, a better reference point for puzzling and abstruse might have been learning Latin or ancient Greek. That may have been more common reference point.

(It might be observed that calculus is not reflective of most professional uses of mathematics. This is true, but there is a closer parallel than there is between a history class and legal education/practice, or between a biology class and medical education/practice.)

The second:

Legal and medical education are accompanied by significant "buy-in" from students in the form of voluntary commitment of time and money. This creates selection effects (reducing the number of people who would find something abstruse), psychological effects (having committed to something, one may be less inclined to perceive difficulties as insurmountable), and social effects (having publicly committed to something, one may be less likely to complain about difficulties).

These effects are not present when people are forced to take a subject for a purpose other than pursuing a career that uses it (which is, on a person-hours basis, the vast majority of all mathematics education).

The third:

Formalized mathematics seems at least potentially amenable to being "figured out," and less of a matter of opinion, than many legal and medical issues. People therefore have higher expectations about how much sense math ought to make, and these higher expectations are violated when one struggles with math in a way that expectations might not be violated when struggling with difficult issues in other areas.

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  • $\begingroup$ Wow, if you hadn't posted this answer, I would have. Thanks! I think you are spot on! $\endgroup$
    – amWhy
    Dec 20, 2021 at 23:09
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    $\begingroup$ "Maybe 100 or 200 years ago, a better reference point for puzzling and abstruse might have been learning Latin or ancient Greek." Isn't learning Latin or ancient Greek still puzzling and abstruse??? Just think of how many kids gripe about learning ANY second language!!! $\endgroup$
    – NNOX Apps
    Dec 23, 2021 at 4:53
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Because math is about truth rather than consensus.

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    $\begingroup$ True, but maybe "logical force" rather than (universal) truth per se. $\endgroup$
    – ryang
    Dec 21, 2021 at 17:13
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    $\begingroup$ "Math class is tough." --Barbie youtu.be/NO0cvqT1tAE $\endgroup$
    – Bill Cook
    Dec 22, 2021 at 0:10
  • $\begingroup$ Pls elaborate? Should this be a comment? $\endgroup$
    – NNOX Apps
    Dec 23, 2021 at 4:52
  • $\begingroup$ @evolva what is considered correct in both law and medicine is a matter of consensus. Both law and medicine involve a certain amount of ambiguity which makes logical deduction insufficient to determine what a deliberative body will qualify as correct. In contrast, when we find a new mathematical result, the question is just whether or not it is correct. The question is not whether or not the AMS approves, or whether or not the chief ruling council in the government approves the result. This is part of why math was incredibly robust in Soviet Russia. It has some natural immunity to politics. $\endgroup$ Dec 23, 2021 at 5:45
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    $\begingroup$ @evolva hey, I recommend the autobiography "Love and Math" by Edward Frenkel, his description of how math was a respite from Stalinism is my primary source for the comment. $\endgroup$ Dec 25, 2021 at 0:20
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My point of view on this is that it's all about whether progression in the subject gets more abstract or more concrete.

Law and medicine are both concentrations of much larger and more broad fields. Medicine is a particular corner of biology and law is a particular corner of the intersection between logic and political science. The further one goes in these fields, the more concrete and specific things get.

Math, unlike professional subjects, is the inverse. Math starts teaching you things that seem concrete, and then begins to generalize those relationships and algorithms.

Think about it this way: Arithmetic is algorithms for numbers, algebra is a generalization of arithmetic for that applies to large subsets of all numbers. Linear algebra is a generalization of algebra to multiple equations at once. Calculus is bridging the gap between discrete and continuous functions using algebra. And it just gets more general from there.

At some point, everyone runs out of the intellectual skill needed to think in ever more abstract terms (Some sooner than others).

TLDR: Professional subjects are concrete applications of general concepts. Math is generalizing from concrete observable use cases. Thinking in ever more abstract terms is a different problem with fewer systems to guide students, versus thinking in ever more concrete ones.

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I correspond somewhat to the profile invoked by the OP, having experienced difficulty with a part of the maths (and physics) in my bachelor's in Electronic Engineering.

I think the problem lies not in the abstraction of mathematics, but in its expression.

I reason and memorize verbally. At school, maths concepts are verbalized, which permits understanding by a wider audience. High school calculus is relatively easy to verbalize.

Higher maths (and physics) are much more symbolic. I find it difficult to self-educate because I can't verbalize the equations that I read. I find Physics Stack Exchange interesting, but my eyes glaze over when I arrive at the equations. On the other hand, I find accounting easy because numbers can be verbalized.

So I would say that the problem the OP describes arises when the verbalizers can no longer verbalize the equations they are required to understand and memorize. They need to develop a symbolic understanding in high school, or understand (as I did) that higher maths is not for them.

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    $\begingroup$ Regarding your last paragraph, I think the problem you're having is seeing the equations and such as something to memorize rather than as results having many interconnections with other things and often being logical necessities, rather than seemingly random results having little to no logical connection with each other or to much else. Compare this followed by this with trying to deal with some accounting task involving city, state, national regulations and tax laws. $\endgroup$ Dec 22, 2021 at 12:02
  • $\begingroup$ @DaveLRenfro, I class your first example as "high school maths", no problem. I like your explanation in the second example - you have succeeded in verbalizing it ! $\endgroup$
    – grahamj42
    Dec 22, 2021 at 12:23
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    $\begingroup$ Not quite the same thing we're talking about, but you might find my answer to Example of a very simple math statement in old literature which is (verbatim) a pain to understand amusing. $\endgroup$ Dec 22, 2021 at 15:32
  • $\begingroup$ @DaveLRenfro The math is considerably easier. $\endgroup$
    – John Glen
    Dec 23, 2021 at 1:25

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