How much math should most college students be able to learn?

August 25, 2016 2:02 AM

I have read quite a few articles that state that there's no such thing as being bad at math, that often it is just the case that you haven't had a specific subject explained to you in a way that made sense to you, or that you just have to work harder at it. That resonates with me, but clearly there's a limit to that and I wonder what that limit is.

I know that theoretical math at the university level is one of the hardest subjects and that even people who are the best in their class in high school often have trouble with that. So, I wonder if there is any consensus about what a level of math is that most college students could reach, given enough motivation and good explanations. I'm interested in both experiences from math teachers/people who took higher level math classes in college, and articles that are written about this.

I'm in Europe so general explanations would be preferred because I have no clue what Americans learn in which classes. I'm assuming college students, so people who have the IQ/skills to go to college, but not necessarily people who had advanced math classes in high school.

posted by blub to Education (15 answers total)

The upper tier of American universities typically requires calculus to graduate, regardless of major. This is generally a first year college course and all students would be expected to have enough math in high school to jump in to an introductory calculus class. So in the US, "calculus" would be your "smart but terrible at math" baseline a college student at a "good" college would be expected to attain without TOO much trouble.

I assume most people could go farther, but America's college administrators seem pretty united in the consensus that any reasonably smart person can manage to at least struggle through calculus without flunking out of their liberal arts program.
posted by Eyebrows McGee at 2:57 AM on August 25, 2016 [2 favorites]

I was "good at math" right up until integral calculus and that's where I hit a wall. Differential calculus wasn't hard to grasp at all, but my brain just couldn't handle doing it backwards.
posted by Jacqueline at 4:22 AM on August 25, 2016 [1 favorite]

As for what "calculus" is in a USA university sense, I suppose you could begin with the curriculum overview of the AP Calculus Test. A pretty good overview is at
Khan Academy for AP Calculus AB

In the US, AP Courses are meant to be college-level courses taught in high school, and students can take a test at the end of the year. A passing score in the test can allow students to receive credit for those subjects at some schools.

At my liberal arts school, the math requirement for graduation was "two semesters of mathematics. At least one semester must include a course containing some calculus." My school granted credit for both semesters worth of math to students who received passing grades in the AP Calculus test.
posted by kellygrape at 5:14 AM on August 25, 2016 [1 favorite]

The usual limiting factor is not ability, but time. My experience from teaching math at a regional public university in the US is that most students have some sort of gap in the prerequisites, where they've seen a concept but don't really understand it fully. Depending on the class, that might be the concept of a variable (lots of students can solve an equation for x, but don't know how to interpret formulas that use multiple letters), how fractions work (trigonometry and calculus are both far more mysterious if you aren't sure which moves for simplifying a rational expression are legal), or the meaning of a logarithm. University-level math classes have set curricula and move quickly, so there often isn't time to go back and address more fundamental misunderstandings.

(For reference, graduating from a regional public university like mine typically requires algebra through about the quadratic formula, followed by one "college-level" math class, which might entail more algebra, a non-rigorous calculus course, introductory statistics, basic combinatorics, or a "topics in modern mathematics" course touching on things like fractals and modular arithmetic.)
posted by yarntheory at 5:19 AM on August 25, 2016 [13 favorites]

Another data point that might be helpful and could give you some research starting points. Many US institutions have core curriculums of some type. The core curriculum lists the expectations of coursework that all students will need to complete before graduation.

At Penn State University (the State University in Pennsylvania, currently 70,000 undergraduates at 20 campuses throughout the state), the core curriculum is called General Education and the "math" requirement is referred to as Quantification. Here is a listing of all the courses that satisfy that requirement. There's a pretty broad range of courses not requiring some type of calculus, including all the MATH courses from MATH 017 - MATH 082, and some of the statistics and programming courses.
posted by kellygrape at 5:27 AM on August 25, 2016 [1 favorite]

Best answer: I think I'm struggling to answer this question because it assumes that there's a hierarchy of math that aligns with the traditional primary/secondary education curriculum (that is: arithmetic, pre-algebra, algebra/geometry, pre-calculus, calculus, etc). We've certainly created that hierarchy and a whole generation of people who can't see it any other way, but I would be hard pressed to come up with anything similar to describe the rest of the courses I took in college (I have a bachelors in math). There are courses that are considered "higher level" because of their location in the curriculum that I would consider "easier" than others, and vice versa.

The example that comes to mind is Algebra (or sometimes called at the college level "abstract algebra"). Technically this is a "harder" course as it's further along in the sequence of study for a math degree, but introductory abstract algebra is a topic that engages a totally different through process than calculus -- doesn't even require knowledge of calculus -- and could appeal to and delight a whole different kind of person (potentially someone who thought they were "bad at math" all along), to say nothing of whether it satisfies the requirement of simply being "learnable."

Finally I think this conversation is incomplete without some mention of structural issues in math education. I don't think I'm comfortable using the "able to learn" vs. "not able to learn" dichotomy in this context because it presupposes that every student has an innate limit, that you seem to suggest is related to their "skills and IQ," and that motivation and teaching skill can augment or diminish that. There are lots of data to suggest that whether a student learns a subject or not also has a lot to do with things like structural sexism, racism, and ableism as well as early childhood education or lack thereof. We don't have any people who were raised in a vacuum without these influences so I don't think it's possible to say how much math they can learn based on "skills and IQ" alone.
posted by telegraph at 5:56 AM on August 25, 2016 [19 favorites]

For a counterpoint to all the calculus, my liberal arts college used to teach logic and proofs as an introductory college math class. We started with the axioms of set theory and built up the integers, the rational numbers, and then the real numbers. Building the fundamentals of mathematics with proofs. I suppose in some way it was an undergraduate version of Principia Mathematica.

It was a great course. It felt much more like "real math" and appropriate to the liberal arts education. Calculus is much more useful, since it's a tool used in every physical and engineering discipline. (For that matter, so is statistics). But the logic class was much more about what math itself is actually like. It was reasonably accessible to all the students, particularly since it didn't assume any particular foundation of knowledge.
posted by Nelson at 7:05 AM on August 25, 2016 [1 favorite]

I know that theoretical math at the university level is one of the hardest subjects and that even people who are the best in their class in high school often have trouble with that.

This is maybe re-iterating what telegraph said, but at least in US+Canada, most of what I'm guessing you might mean by theoretical math isn't taught before university. There's barely anything even preparatory to the bulk of university level math. So I think a lot of high school students think they know what math is (calculus) and show up and encounter entirely different kind of things, both within math departments and elsewhere -- proof-based classes (even calculus-derived areas are really different when you have to prove things), abstract algebra, lots of discrete math, graph theory, probability, logic, algorithms, etc.

I teach an intro class in discrete math for my department (topics: set theory, discrete structures, graphs, automata theory, all applied to cognitive science), and I start from scratch. I'm also not sure the question of what they are "able to learn" makes a lot of sense, and they are certainly able to learn this stuff; the limits to what they could learn are more about structural factors, as well as interest/motivation (which aren't independent from structural factors). But most of them have never encountered anything remotely like this material as a high school student. When I was in high school in the 90s we did learn some elementary proof techniques in a geometry class, but from talking to my students, they don't seem to even get that any more.
posted by advil at 7:20 AM on August 25, 2016 [4 favorites]

Oddly enough, some forms of dyscalculia make the mathematics in late elementary school and high school harder to learn than the later mathematics taught in University. If the dyscalculia is based on short term memory issues or transposition errors eg. seeing the numeral 93 and write it down as 39, a higher level math class will not deduct points. Students may be graded entirely on getting their process right, and not on inability to remember times tables. The students are expected to write down every step, whereas in earlier grades they are expected to do some calculations in their heads.

It is also worth noting that some studies have found that people develop the ability to process higher abstract reasoning at different ages and in many women it's impossible to follow at about age nineteen, but pretty straightforward by age twenty-four. So if someone is struggling with absorbing or following a math course it can be that their brain has just not matured enough. Interestingly, another study suggested that girls lagging in math will catch up if they are given an iron-rich diet.
posted by Jane the Brown at 8:17 AM on August 25, 2016 [4 favorites]

Best answer: Yes I definitely agree with telegraph above. When you say "level of math" - to me there's not a straight line from counting/arithmetic to calculus and beyond. I have a bachelors in math, and when I think back to all the math classes I took at my university, some were just very different. Yes, there was calculus, multi-variable calculus, differential equations....these were all kind of building off each other. But then I took things like abstract algebra (crazy different class, not the usual algebra) and complex analysis (an entire class about imaginary numbers!). Math branches out in multiple ways, and some people were awesome in some classes, but struggled with others and vice versa.

Also, for me, once I got past multi-variable calculus, nearly everything at a "higher level" was easier. Like, way easier. It was like getting over a hump. Most of the students I was with at the time felt the same. Until we got to a course called Real Analysis which was all about proofs and that class was one of the hardest. But also that professor was horrible so that probably didn't help.

Anyway, I'm rambling. My opinion on this subject is that most people I've encountered can learn this stuff, given the right circumstances but it's hard to separate that out from the structural issues telegraph discusses regarding math education. But yeah, people I've met who tell me they are bad at math usually get the concepts when I explain things simply to them from another angle. In many cases they just had a shitty teacher, or not enough time to practice and understand concepts.
posted by FireFountain at 8:56 AM on August 25, 2016 [4 favorites]

In university, I took four classes of calculus during my Bachelors of Engineering, obviously along with a ton of other math-based science classes. My marks got progressively worse over those calculus classes, because I had a hard time with the purely theoretical problems. I went to talk to my professor about it, and he pointed out that I was failing the "easy" questions - the short, purely theoretical ones, but was getting perfect marks on the longer application questions, the ones where math was applied to solve a hypothetical problem. I studied so hard, but still just didn't manage to do well on those pure-theory questions in the end - got 50% in the last class, a definitive mercy-pass.

As my engineering classes progressed into the later years, my marks improved greatly, as did my enjoyment of the classes - they became applied science classes, where all the calculations were centered around real-world applications and I could envision what was happening and how things work. My ability to understand how to perform a calculation is heavily weighted upon being able to relate it to a tangible problem.

Then there's my Brother-In-Law who ended up taking Electrical Engineering (very intensive on the math theory) because he found theoretical math a breeze, far easier than memorizing a bunch of stuff for Dentistry.
posted by lizbunny at 9:36 AM on August 25, 2016 [1 favorite]

I did well enough in math in high school to finish calculus my junior year, and was able to test out of the university math requirements for general education. FWIW, I have dyscalculia, and make horrible transposition errors. It makes arithmetic a living hell for me and is a constant thorn in my side in daily life — but calculus was infinitely easier, as Jane the Brown mentioned above.
posted by culfinglin at 10:49 AM on August 25, 2016

Best answer: but clearly there's a limit to that and I wonder what that limit is.

I don't believe that this is true.

First of all, the embedded assumption here is that learning mathematics is a linear progression of increasing difficulty: you learn simple math, and then build on that through more and more intellectually challenging math. That's not true, or even close to true. Some of the math children learn is taught that way, but relatively little. How much probability did you use in introductory calculus?

There are tools that are common across different areas of math, and it's frequently very interesting for research mathematicians to apply tools and techniques developed in one area of math to other areas, but the world of math in general is a series of disconnected theoretical islands, for the most part.

Currently, this doesn't really become clear until most students get to university-level math, because schools don't offer much choice in progressions, but there's no reason younger students couldn't branch off into number theory instead of doing calc, for instance (I personally believe a lot more time should be spend on discrete math as well as probability and statistics for young kids instead of a lot of the stuff they do now that leads to calculus). At university, though, mathematics education doesn't really proceed in 'levels'. But someone who wants to take an applied-oriented class in partial differential equations doesn't need to study number theory or algebraic topology. At the university level, there's only really a few successive courses in any one direction you can go before you end up in 'reading the research papers' level stuff. Most of the material doesn't build on the other material.

I guess my point is that there can't be a 'limit' if you aren't going in a direction, and you're not. It only looks like you are because there is no choice in the direction mathematical education takes from age 5 to 18 or so, and that's arbitrary. The typical US math education (I'm not familiar with Europe) from age 5-18 basically goes arithmetic with real numbers, algebra, a weird diversion into geometry that's probably meant to introduce the idea of mathematical proof but rarely succeeds, 'trigonometry' which is a pretend branch of math only found in high schools that's just specific types of geometry problems, and then differential and integral calculus of one variable. Some of the very basic arithmetic and numerical concepts are actual building blocks here that the calc education depends on, but a lot of it is not. There's no reason why the whole year spent silently reciting 'soh cah toa' to oneself while hunched over a hundred pictures of triangles couldn't be spent on, say, statistics, combinatorics, number theory, whatever, and then calculus could pick up after that. You don't need any of that triangle stuff, or the concept of proof and construction learned in the year of 'geometry', to do calculus the way calculus is taught (that is, as a tool used to solve engineering and physics problems), beyond motivating the idea of the slope of the tangent line or the area of a rectangle under a curve. If you switched the order in which the age 12-18 courses were taught, and just adjusted the difficulty level in terms of problem-volume, it would have no pedagogical difference in my opinion. There's absolutely nothing on the USA's AP Calculus AB exam (in the US, this is the advanced-placement 'calculus of one variable' test that ~16-18 year olds take at the end of the year) that depends on a student having spent a year doing trigonometry (however, this stuff is handy for the problems they put on high school physics tests, which is probably the reason they do it in this order, but that doesn't affect the actual question you are answering about the limits of math-learning).

Second, it was widely believed until not that long ago that topics like calculus were so abstruse that only a handful of incredibly devoted students of mathematics would become familiar with them. Simplifying greatly, calculus as we know it was invented in the 1600s. 150 years or so later, we were at the point where students at Harvard might spend three months getting a very brief intro to the most basic concepts in calculus. It was only in the 1960s that people even started to consider teaching it to high school kids in the US (fun fact: this was because of Cold War fears that the USA wouldn't have enough engineers to ensure military dominance of space over the Soviet Union)! Now, it's completely routine, and its commonplace for a Western student to have progressed through a good chunk of multivariable calculus by age 16 or 17.

We get better at teaching things. Formalizing math into teachable, digestible forms is a kind of technology and it's one that is being continuously improved.

Third, the way most students are encouraged to continue to study math at university does create a bizarre feeling of "hitting a limit". At least in the US, most students believe the natural progression of study is to go into more years of calculus study, where calculus continues to be taught primarily as a tool for solving problems in the physical sciences and engineering. This, to me, really makes absolutely no sense. It bares very little resemblance to what makes up the whole of the rest of math: almost no other area of math taught at the university level will consist of students 'solving problems' the way they do in the increasingly advanced calculus courses, which is really a kind of elaborate plug-and-chug requiring no real mathematical thinking. There's nothing more conceptually difficult about most honors track second-year-university multivariable calculus course material. If anything, its likely that the only new concepts that have been introduced since the Big Ideas of Calculus (roughly four years of school before this point) are things like Taylor series expansion. There's just more shit to grind through. Instead of "calculate the area under this curve", it becomes "calculate the volume between these two surfaces, in the region that is bounded by this ring of touching-different-sized spheres". The second problem doesn't take any more intelligence to understand, it just takes more diligence to solve correctly, and it's boring as hell. Even for the kinds of students who were fascinated when the teacher lead them through the invention of calculus, there's nothing interesting here, it's just grunt work.

If most students, on getting to university, completely switched things up and signed up for a basic course in any other area of mathematics, there would be a big adjustment in realizing that they weren't going to be required to hand in problem sets where they calculate a specific number very often, but they might find that the work was actually quite a bit easier (for some definition of easier) than the work in the Advanced Calculus for Physics Majors II or whatever, or at least more interesting.

Now, when we consider the outer edges of the known world, things are different. That's humanity as a whole's current limit, and I don't believe diligent study means that any person can eventually solve any open problem in mathematics. This is where the pros have hit their limit, at least for now. I think one of the things that separates professional mathematicians from the rest of us is their comfort with failure. They fail at almost everything they try to do, almost all the time. But they keep trying. And they are working on new stuff: discovering things that no one has discovered before. Anything below that I believe is learnable by anyone of normal intelligence with sufficient time and patience (and maybe with sufficient years gone by to figure out the correct way to package it and teach it to make it tractable for the non-geniuses among us to learn in a human lifespan).
posted by jeb at 8:46 PM on August 25, 2016 [2 favorites]

Prior posters have addressed the objection that there is no single hierarchy (total ordering?) of math-topic difficulty. Nevertheless, the order in my (white, middle-class, suburban*) school district was Pre-Algebra, Algebra I, Geometry, Algebra II, Trigonometry / Pre-calculus, Calculus, and Statistics.

[*] In the USA, you select your "public" schooling by the location / price of your home.

If you started Pre-Algebra in 7th grade, you would hit Calculus in 12th grade, and this was considered "solidly college-track". Some people took Statistics in lieu of calculus. "Pre-calculus" was mostly discrete math. If you did not start Pre-Algebra in 7th grade, I believe you were in a course that prepped you for the state exams.

I'm assuming college students, so people who have the IQ/skills to go to college, but not necessarily people who had advanced math classes in high school.

This being America, with our vast inequality: academically-challenged children of the wealthy go to college because their parents want them to. It doesn't really matter what they learn in college because parents' social capital is adequate. These kids will probably take something like College Algebra (y = mx + b) for a business degree and be done with it, or a Statistics, for a psychology / sociology degree.

Finally, if you want an institutional view of this question, lookup some practice SAT math exams. That indicates what you're expected to be able to solve by the time you apply to college (11th or 12th grade).
posted by batter_my_heart at 1:01 AM on August 26, 2016 [1 favorite]

Response by poster: Thanks for all your answers. I did not know that in the US all students get math in college. Where I live, only people who study things that require math get it in college. I had definitely not realised that there's no hierarchy in math complexity. I had never looked at it that way, just took it as a given that there was. I am happy to have been corrected about this assumption.
posted by blub at 4:15 AM on August 27, 2016