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My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much

My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult. The analytical type of thinking that proof-writing is certainly useful, but you can make much the same argument of many other curricula, and besides, it's not like most intro calc courses even do any proofs. The vast majority of them, I would assert, are simply pre-med weed-out courses. I still remember freshman year, showing up to the standard intro calc course, and dropping it as quickly as I could in favor of my uni's equivalent of Math 55 (i.e. the hardest u/g intro math course) because of how asinine I found the content...

antishatter 5y ago
Statistics converges to calculus at its core. Source: am statistician
jll29 5y ago
Calculus and linear algebra seem to _totally_ dominate the curriculum in most (all?) countrie. What about meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? In the post, the fieds under "electives" are by far the most interesting ones, IMHO. And I fully agree, in-depth knowledge of probability theory as well as descriptive statistics and of course the application to systematic and sound decision making is absolute key, and ought to be taught to anyone from medic to policy makers (scary: Gigerenzer showed that medics tend to be confused about the difference between P(A|B) and P(B|A) - the very people whose job it is to diagnose whether you have cancer or not!).
- eftychis 5y ago
  Anything besides a cursory layperson's outlook on a lot of these topics (besides basic logic and history/philosophy of math -- although not sure how you would teach the last two/what you have in mind as curriculum) requires calculus and/or linear algebra. There is a reason they say you can never learn too much linear algebra. And yes probability and statistics are fundamental. I was shocked a bit when I learned it was not taught in highschools world wide (i.e. not in the U.S.A.). But then again I had gotten numb with the current average level in the taught topics people arrive at undergrad at. Note there is a lot of interconnectivity. To understand a new concept you might need concepts in another. E.g. number theory and probability.
  - jltsiren 5y ago
    In my experience, mathematics can be roughly divided in two: topics that use real numbers all the time, and topics that rarely need them. In the former, almost everything builds on calculus and/or linear algebra. In the latter, they are just two topics among many others. The true foundational classes in the typical undergraduate mathematics curriculum are logic and abstract algebra. People rarely start with them, because the usual way of teaching mathematics is applications before foundations. You learn linear algebra before abstract algebra, proofs before formal logic, and axiomatic probability before measure theory. And there is definitely such thing as too much linear algebra. Once upon a time, I wanted a decent mathematical background for theoretical CS and continued (at least) until the first graduate-level class in most major topics. Graduate linear algebra was "foundations without applications" for me, as I've never worked on anything building on it.
  - t-3 5y ago
    A decade+ ago when I was in school, the policy was to push honors and AP classes for "college-bound" students. Stats (not offered as honors or AP in my school) was one of the fun courses that co-op kids got to take while we were stuck in pre-calc.
  - caddemon 5y ago
    I think a layperson's outlook on these topics in high school could be very interesting though. Use this outlook to help motivate learning calculus and linear algebra. With a mix of engineering survey and math survey there are plenty of kids out there that would find parts of the course that stimulate their natural interest. Then the "calculus has many practical applications" thing would feel way less contrived. I mean of course I know now that it is true, but in high school I didn't feel it because I didn't have enough broader context.
- dwohnitmok 5y ago
  > meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? Honestly all of those feel more niche than calculus. I agree with you and joatmon-snoo on the usefulness of statistics and would probably support bumping calculus in favor of statistics, but meta-mathematics, topology, logic (which bleeds into meta-mathematics), combinatorics (which is kind of covered by stats), number theory, discrete mathematics, and graph theory are all much less useful even in adjacent STEM fields (discrete mathematics and graph theory matter more in CS, but far less for day-to-day programming). History of mathematics is effectively an entirely separate discipline and philosophy of mathematics has meta-mathematics/mathematical logic as a prerequisite. Calculus unlocks much of physics and engineering (and lots of stats!). Large cardinal theory does not unlock any other field to the best of my understanding.
  - caddemon 5y ago
    I think the problem is that students don't know these fields exist unless they took an especially strong liking to the standard school math curriculum. Going into college I couldn't have told you what most of the fields mentioned were about even broadly. Truly I had no idea what math was, because we barely talked about proofs through high school. The only reason I swung back around to learning some of it is because my upper level college CS classes introduced me to ideas that then made me want to revisit my math curriculum. There ought to be some way to encourage late high school/early college students to "survey" the field without necessarily taking full courses in these topics. This could also give some earlier understanding in how the different fields relate - for example you could present a toy graph theory problem within linear algebra as a matrix problem, later presenting the same problem in graph theory section and walk through it using graph representation. I think high school courses struggle with memorability precisely because most units are taught basically in a vacuum. Regardless though, the point of the survey course wouldn't be to remember details so much as to find topics of interest for further study/be generally aware of their existence in case a relevant problem comes up in the future.
- edflsafoiewq 5y ago
  Calculus and linear algebra continue to dominate in applied mathematics. "How can I turn this into a problem in linear algebra?" is probably the most fruitful mathematical technique that has ever existed.
  - JadeNB 5y ago
    > "How can I turn this into a problem in linear algebra?" is probably the most fruitful mathematical technique that has ever existed. And, from that perspective (with which I agree), calculus itself is just another instance of trying to turn a non-linear problem into a problem in linear algebra!
    - Koshkin 5y ago
      True: the differential at a point is a linear map; the integral is a linear form (on the vector space of functions).
- ghaff 5y ago
  Interestingly the ascent of linear algebra is a relatively recent thing. I have an engineering degree but while I was certainly exposed to basic matrix stuff, never took a linear algebra class. When I was an undergrad, you took differential equations in addition to basic calculus for engineering. (You needed for system dynamics among other things but it was very cookbook.) Linear algebra became a lot more interesting once you had cheap computers and Matlab.
  - antishatter 5y ago
    Computers made it insanely useful where before it was an interesting view of systematic math.
  - bigger_cheese 5y ago
    When did you do your Engineering degree? When I did my BE Degree (2004 to 2008) Linear Algebra was included. We were introduced to the basics of linear Algebra in year 12 of high school (simple matrix manipulation, intro to vectors, dot and cross products that sort of thing). I'm Australian if that makes a difference. The Math in my 4 year engineering degree was structured something like this: First Semester Year 1 two math courses: Calculus 1, Linear Algebra Second Semester Year 1 two math courses: Calculus 2, Sequences and Series (this one was probably least useful all I remember from this 15+ years later is Taylor Series and Binomial Theorem) First Semester year 2 two math courses: Differential Equations, Statistics for Engineers. From second semester year 2 onwards there were no more discrete math classes this was where the degree really specialized into various engineering streams, Mech Eng, Chem Eng, Civil etc. Had their own courses. I studied Materials Engineering some courses were shared with other eng students (For example I shared Thermodynamics with Mechanical Engineering students) but others such as Non-ferrous metallurgy were pretty deeply specialized. A lot of subject used built on earlier math (Fluid Dynamics was backed up by lots of differential equations for example, stuff like Gamma Function would come up in a lot of places. Solid Mechanics had a lot of integrals second moment of area etc.), Linear Algebra I can remember from Fracture mechanics and crack propagation (Stress and Strain tensors etc.)
    - ghaff 5y ago
- diffeomorphism 5y ago
  These come as a second step. In the first year (non US) you learn linear algebra, some real analysis and how to write proofs as well as important basics. In your second year you can choose all these electives, which then don't have to spend time introducing natural numbers, induction etc.
- caddemon 5y ago
  I fully agree with your general point, but in the US I've found calculus to be much more dominant than linear algebra. I went to an "elite" high school that still does not offer any course in linear algebra, despite having enough juniors in AP calc to fill a room. Senior year options for those students are AP stats or multivariable calc. At my college, everyone was (and still is) required to take through multivariable calculus, but there's no such requirement for linear algebra. I have some appreciation for calculus now but I really did not enjoy it much in high school or even in college. It turned me away from learning more math for some time which is unfortunate - linear algebra isn't my favorite either but I liked that much more off the bat, so I wish I had some exposure to it in high school. Then again, maybe the high school teaching style is what made me dislike calc to begin with.
dhosek 5y ago
Or the math covered in finite math classes which is a mix of combinatorics, stats, probability and linear algebra. I remember doing my time in the math tutorial room during my grad school days and helping the kids from the business calc classes who were learning math they weren't going to use from books written by people who didn't understand the domain that they were trying to teach math for. Seriously, what business use is there for f(x) = x^1.3 or the indefinite integral thereof?
- ghaff 5y ago
  When I tutored in business school--mostly related to math-related things (which itself says something about the level of math knowledge among those who weren't engineering undergrads)--pretty much the only calculus that came into play was finding maxima and minima of curves in economics which was both simple differentials and mostly pretty academic anyway. A little later I did some more complex optimization problems but that was done by software (LINDO at the time) anyway.
dwohnitmok 5y ago
FWIW (and I know you're not making this claim) I don't believe Math 55 is a good example of how mathematics should be taught at large, nor do I think calling it an intro class accurately conveys what it is. It's effectively the compression of an entire undergraduate degree into a single freshman course. I say "effectively" because the material covered depends heavily on who's teaching it, but certainly anyone coming out of Math 55 can pick up any undergraduate math content trivially. For example, undergraduates who have taken it are generally explicitly prohibited in course descriptions from taking further undergraduate mathematics classes (because it would be free credit for retreading the same ground) and can only take graduate courses from then on out. It's strongly self-selecting and as a result can afford to cover a truly insane amount of ground. The overwhelming majority of people who take it drop out (the usual dropout rate from people who take it in the first week is probably > 90%), but the people who stay almost all get As. And you will need to be almost entirely self-motivated because a lot (maybe most) of your waking hours will be thinking about math. There's a very small minority of students for whom this is an optimal way of learning. For most students this is the quickest way to make them run screaming away from mathematics even faster than they already do.
- C-x_C-f 5y ago
  > For example, undergraduates who have taken it are generally explicitly prohibited in course descriptions from taking further undergraduate mathematics classes (because it would be free credit for retreading the same ground) and can only take graduate courses from then on out. Not at all, you are only prohibited from taking "freshman courses"[1]—that's what 55 is supposed to cover (definitely not all of the undergrad math curriculum), though some professors go beyond that. Many students go on to take at least some undergrad courses, with those in the 140's range being mathematical logic gems with no real counterpart in the graduate department. [1]Students from Math 55 will have covered in 55 the material of Math 122 and Math 113. If you have taken 55, you should look first at Math 114, Math 123 and the Math 131-132 sequence. https://legacy-www.math.harvard.edu/pamphlets/courses.html https://legacy-www.math.harvard.edu/pamphlets/courses.html
  - dwohnitmok 5y ago
Bostonian 5y ago
I wonder how much statistics someone can understand without calculus. For example, how do you explain what a continuous probability density function (such as the Gaussian) is without calculus?
- pmyteh 5y ago
  I teach introductory quantitative research methods to communication undergrads, most of whom have no calculus. (Which is essentially optional in the UK as it's taught at A Level after most students have specialised away from maths). They don't seem to have a problem intuiting what a plotted PDF is showing. I think that's because in some sense it can be read analogously to a histogram. Of course, they don't have the tools to generate or manipulate one. But that's honestly not something that applied social science researchers have to do often when using traditional methods.
jldugger 5y ago
> My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult. Strong agree, but engineers probably need both. I'm currently watching a course on causal inference, and the tools are very much calculating gradients. And even if you just use someone else's MCMC, even in the models a differential equation or integral can randomly appear usefully. In retrospect I should have taken a stats class in high school when I had that 1 hour gap for 1 semester, just to build a better intuition around the basic concepts.
- qumpis 5y ago
  Which course, if I might ask? I only recall Brady Neal having his videos online
  - jldugger 5y ago
    this one: https://xcelab.net/rm/statistical-rethinking/ https://xcelab.net/rm/statistical-rethinking/
Melatonic 5y ago
Not sure I agree with that - you also have to take into account that calculus is a requirement for many other sciences. I suppose you could just of course make it a pre-req for things like that but I found that in highschool rarely did they go that deep. Physics becomes a hell of a lot easier with basic calculus for example. If anything should be dropped from a highschool level its all of the insane memorization you have to do for some of the lower level math classes - I found that totally useless. You then learn some basic calculus and realize "I just wasted so much of my life" and never need to memorize those things again.
- JadeNB 5y ago
  > If anything should be dropped from a highschool level its all of the insane memorization you have to do for some of the lower level math classes - I found that totally useless. You then learn some basic calculus and realize "I just wasted so much of my life" and never need to memorize those things again. What sort of memorizations do you have in mind that you no longer need to memorize once you know calculus?
  - wizzwizz4 5y ago
    s = ut + ½at²
  - p1necone 5y ago
    Knowing the basics of integration/derivation makes all sorts of very common concepts in physics much more intuitive - velocity, acceleration, area etc.
  - ghaff 5y ago
    Pretty much all the classic Newtonian formulae you memorize in high school physics classes can be derived (relatively) easily once you bring calculus in. A general problem with high school classes in particular--still somewhat true in certain university classes but much fewer--is that there are dependencies across classes. So you end up with a lot of "Memorize this thing because you don't have an advanced enough background to understand why $XYZ is the case."
staindk 5y ago
Super agree. I took calc 1 and stats 1 & 2. Much preferred the stats and it set me up for understanding all kinds of science lingo in articles and papers. I also indirectly use stats fairly often at work.
- ghaff 5y ago
  There are different types of stats. I took a stats course in my grad engineering program. Not being the best person at math I think I probably struggled with the math sufficiently to get distracted from the concepts. When I breezed through a stats course when getting an MBA I understood the concepts much better--though the professor was almost certainly better as well.
Koshkin 5y ago
> within the US All fifty states or just continental?
Siira 5y ago
Statistics needs calculus as a prerequisite. Heck, a non-introductory treatment of probabilities needs measure theory.
geebee 5y ago
I'm really wary of this. House wiring would be more useful than Shakespeare. That's not to knock house wiring, or statistics. I'd love to know more about both. But don't deny yourself an understanding of the meaning of limits. Almost all mathematics before calculus leaves you with a misimpression that neat formulas exist to solve problems. In reality, you've learned to draw straight lines with a ruler, and maybe a few curves with a compass. Before Calculus, you might actually believe that numbers that can be expressed as the ratio of two integers are typical, and that numbers like pi and the square root of two are "irrational" rarities (and until calculus, you probably don't know about Euler's constant unless it was introduced in precalc as another one of those odd and rare numbers). Look out at nature, where are the triangles, rectangles, and circles? Maybe a wasp nest? Nah, not really. Try to draw a cloud, a tree, a tiger, or a human face. How useful is that straight line or compass? How useful is a line at all, other than to hint at something you can't actually draw, maybe by implying it exists as an ever vanishing limit from above and below? Math required calculus the instant humans decided to describe the world as it is, rather than by the limits of what we impose on it. Also - in stats, how do you know what the area is under the probability density function?
- SamReidHughes 5y ago
  Well, AP statistics covered the area under a probability distribution function and people seemed to understand that -- you look up the answer in a table (or use the TI-83 function). Presumably they'd do the same for a cumulative distribution function.
  - bee_rider 5y ago
    Highschool is pretty early to teach people that math is too complicated for them to actually evaluate their expressions.
    - adgjlsfhk1 5y ago
      Teaching people that there are lots of problems where solving them numerically is the best way of solving them is often just the truth. Also, you then get to discuss some of the really cool numerical techniques which tend to get short-changed in traditional math classes.
jpz 5y ago
You can’t understand much about statistical inference without calculus. It’s all integrals over pdf’s. A lot of integration by parts and other things in the core curriculum.
pjbeam 5y ago
My prob and stats courses required the calc sequence (US based math undergraduate).
com2kid 5y ago
The ideas behind Calculus are much more useful IMHO. The idea of rate of change, or change over time, converging and diverging series, these all have practical application to the real world. And the basics of physics makes next to no sense without calculus, and even parts of intro chemistry make more sense after having taken calculus. I remember in one economics class, which didn't have calc as a pre-req, the prof said "alright, for those of you who have taken calculus, this is an integral, you can now leave the classroom and come back tomorrow. Everyone else has to stay." The ideas of calculus just seem so fundamental to me. It is sad that the American schooling system is so slow, and expectations so low, that it isn't taught to everyone. Meanwhile in other countries, everyone, artists to engineers, learns calculus.
bee_rider 5y ago
I think I'd rather we lose geometry than calculus. Geometry made more sense as an intro to the idea of 'proving stuff' when everybody's dad had a workshop, compass, etc.
- caddemon 5y ago
  Agreed, also the way my high school taught it made proofs seem like a boring subset of Geometry. The unit was poorly connected to other units and the problems I found by far the easiest in the course as we got very formulaic ones. It's entirely possible my teacher was just awful but I definitely walked away with some misconceptions about what proofs are. Perhaps it would have been more appropriate at an earlier stage of development.
- bick_nyers 5y ago
  I never thought about it like that, thanks for that insight
smugma 5y ago
A few years ago USC’s MBA program replaced statistics with data science, which is further down the pragmatic chain. Stats is not a generally useful skill (the concepts are, but can be taught in a data science course) but understanding how to work with data is. Discrete math seemed to be the most applicable math class to my CS curriculum. I never took graphics so didn’t use linear algebra, and definitely never touched anything related to DiffEq.
2143 5y ago
> at least within the US Because America has straight lines everywhere and no curves, unlike other places. Calculus is absolutely required for most engineering tasks. Maybe for CS students there could be more emphasis on statistics as well, without diluting the calculus. Infact, where I studied, calculus was a prerequisite for certain statistics and probability classes. For good reason.
thaumasiotes 5y ago
> My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult. I think math people may have a view of this question that is skewed in an interesting way. Statistics is very useful and its common techniques are not difficult to apply. But they seem to be very difficult to apply correctly. We have entire academic fields that are built mostly on the spurious application of statistical methods in contexts that make the whole project invalid. And this sort of "techniques without theory" approach is what's being advocated for upthread and represents a failure mode that math people are unlikely to consider -- because they know that part of being able to apply a technique is being able to tell whether or not applying it is valid. Math people are unlikely to fall into the trap this approach sets. But the same people who want this approach are also likely to end up being hurt by it.
- bick_nyers 5y ago
  My thoughts exactly. What you want to teach ultimately before anything else is mathematical maturity, which in my opinion, calculus is currently doing the best job at providing.
kjasdfj3 5y ago
> My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult. I'm questioning your math pedagogy. Calculus is the single most important math anyone, of any field, can learn as it's the first "practical math" you actually learn. Life behaves like calculus and in order to think about real life you need the concept of limits, derivatives, integrals, differentials, etc. It's patently absurd to say this should be replaced by statistics, which done to any rigor requires up to 2 years worth of calculus (through diff eq.) to even appreciate. I'm shocked that you're a math major and didn't take away the biggest thing from learning analysis - the ability to think clearly through a problem and prove it correctly. While you may not be asked to vomit cantor's diagonalization onto paper for an interview the ability to think about problems you learned from doing these proofs translates to so many different fields, jobs, and life skills that I take the complete opposite view. If you want to understand anything you need to learn how to proof. I don't care if you're a nurse or an accountant. A rigorous proof based math course will change your life. If by "learn statistics instead of calculus" you mean being able to mindlessly vomit today's new machine learning paradigm without understanding a thing then I think I can understand where you are coming from. Otherwise, I think this is some absurd parody of someone who studied math.
- mangodrunk 5y ago
  I think calculus and trigonometry shouldn't be taught to non-math majors. You have the perspective of someone who is passionate about calculus, which is great, but considering many people aren't as passionate, what you're suggesting (status quo) doesn't work. Or maybe you have ideas on teaching calculus better than how it's currently done? Calculus is obviously extremely important, but I don't think it's useful to drill a few things that most people will quickly forget after the test. I do think logic is important, and teaching that instead of focusing on math proofs might be a good alternative.
  - dchftcs 5y ago
    No, most of STEM should learn calculus at some point. In engineering and physics they are useful; in CS, chemistry and biology they are reasonable foundational knowledge and used in some specializations. Trigonometry is used even in some of the trades. For science-oriented high school curriculums, calculus is necessary, and should only be optional in a few cases where the student already really knows what they want to do. By the time a student wants to specialize at say 20, they should have already had some foundational ability for that specialty, otherwise they wouldn't be able to compete internationally. For people who don't end up using calculus, it's a waste, but there is a trade-off. As a result, much of the high school curriculum is to build a broad foundation to prepare for multiple possible specializations.
peterhalburt33 5y ago
I’ll admit I’m biased on this issue, but I’m very wary of this type of reasoning. There are plenty of courses that are more useful than calculus, but kids/young adults won’t be convinced to learn a subject purely out of its utility in adult life (same reason I’m against replacing math with a “life skills” class); in g TV fact Calculus was the course that changed my whole college and career trajectory. Up until I took Calculus, I recognized the importance of math, but hated every dull second of it - there was no motivation that appealed to me, and it seemed that math could only be used to solve the most contrived problems (word problems that resulted in linear systems, simple roots of trig or polynomial functions, etc). The only reason I took calculus is because it was the next math course after pre-calc and I still had a year left in high school. I didn’t even realize that there was a type of math that could be used to exactly calculate quantities related to continuously changing processes, but I was absolutely fascinated by how many real world problems calculus could solve - I realized that the problems given in algebra were contrived out of necessity to resolve in a neat manner. Now I had access to an entirely new vocabulary that allowed me to describe the world as it is, not as it needed to be to neatly fit in a 10th grade word problem. Until that point I was interested in psychology (and had a vague notion that I’d drop out of school to make a career in music somehow), but I immediately dropped everything to take as many math and physics courses as I could. 14 years later and I work in a very math heavy engineering oriented field. I know my story is probably atypical, and I have no clue what I’d be doing right now if I hadn’t taken calculus, but it’s one of those things that I look at in hindsight and think that I was that close to giving up on STEM, but for being forced to take Calculus. Instead, I earned my Ph.D. in applied math, and there’s not a day where I don’t use calculus of some kind.
toto444 5y ago
There is one problem that I have not seen mentioned. Statistics might be much more useful but except for the first chapter (discrete probability) they are much harder. As someone who wanted to work in data I have tried hard to understand probability distributions, I have watched 10s of videos by different professors, done exercises and sure I was able to pass but the truth is I never really understood it. The intuition behing statistics is non trivial. Sampling is very hard to understand. Even on HN during COVID there were mathematically literate people trying to make some back of the enveloppe probability calculation and all they were able to do was show their misunderstanding of how statistics work (to be fair I would not have been able to do the task myself but I could feel something in their reasoning was wrong).
q-big 5y ago
> My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult. > The analytical type of thinking that proof-writing is certainly useful, but you can make much the same argument of many other curricula, and besides, it's not like most intro calc courses even do any proofs. The vast majority of them, I would assert, are simply pre-med weed-out courses. A statistics course would be a much more brutal weed-out course.
theptip 5y ago
Strong agreement here. I'd go further and say that introductory stats should replace trigonometry too. An understanding of statistics is simply required to function optimally as a citizen in modern society, whereas trigonometry and calculus are niche. I'm all for giving tasters of as many different "beautiful ideas" as possible in school, but I think we should be elevating practical statistics into the top tier of subjects that we require kids to go through.
bick_nyers 5y ago
Statistics without calculus relies pretty heavily on memorization to get the right answers. I'm remembering my AP Statistics class (high school) where most of the time students got lower grades because they looked up numbers in a table wrong, or used the wrong table. Never mind when you start using regression models and need to start understanding axioms of the models you are using like homoscedasticity. There's some intuition there, but generally speaking, calculus is where most people "mature" into mathematics. I would argue that most people (NOT engineers/doctors/etc.) don't really get the point of math until around calculus where it starts to really open up for them. While statistics would certainly be useful, and is a must for modern day ever-connected society, it would just be another math subject that needs to be memorized like trigonometry. Some people mature in math a lot younger, but it entirely depends on the quality of their math teachers when growing up (and the curriculum/testing focus in which they are subjected to).
nsriv 5y ago
I'm a high school math teacher, and I think statistics education is desperately lacking in graduating seniors. I think the space for it should come from cutting pre-calculus as a yearlong course though, with some of the concepts backfilled to Algebra 2. There's nothing a basic Stats 1 course needs calc grounding for (perhaps Riemann sums under a normal curve but that's more for deriving than the concept itself). As it stands now, AP Stats is used for kids that don't want to progress to an AP Calc, but want a math or need to hit a school or county requirement.

antishatter 5y ago

Statistics converges to calculus at its core. Source: am statistician

jll29 5y ago

Calculus and linear algebra seem to _totally_ dominate the curriculum in most (all?) countrie. What about meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? In the post, the fieds under "electives" are by far the most interesting ones, IMHO. And I fully agree, in-depth knowledge of probability theory as well as descriptive statistics and of course the application to systematic and sound decision making is absolute key, and ought to be taught to anyone from medic to policy makers (scary: Gigerenzer showed that medics tend to be confused about the difference between P(A|B) and P(B|A) - the very people whose job it is to diagnose whether you have cancer or not!).

eftychis 5y ago
Anything besides a cursory layperson's outlook on a lot of these topics (besides basic logic and history/philosophy of math -- although not sure how you would teach the last two/what you have in mind as curriculum) requires calculus and/or linear algebra. There is a reason they say you can never learn too much linear algebra. And yes probability and statistics are fundamental. I was shocked a bit when I learned it was not taught in highschools world wide (i.e. not in the U.S.A.). But then again I had gotten numb with the current average level in the taught topics people arrive at undergrad at. Note there is a lot of interconnectivity. To understand a new concept you might need concepts in another. E.g. number theory and probability.
- jltsiren 5y ago
  In my experience, mathematics can be roughly divided in two: topics that use real numbers all the time, and topics that rarely need them. In the former, almost everything builds on calculus and/or linear algebra. In the latter, they are just two topics among many others. The true foundational classes in the typical undergraduate mathematics curriculum are logic and abstract algebra. People rarely start with them, because the usual way of teaching mathematics is applications before foundations. You learn linear algebra before abstract algebra, proofs before formal logic, and axiomatic probability before measure theory. And there is definitely such thing as too much linear algebra. Once upon a time, I wanted a decent mathematical background for theoretical CS and continued (at least) until the first graduate-level class in most major topics. Graduate linear algebra was "foundations without applications" for me, as I've never worked on anything building on it.
- t-3 5y ago
  A decade+ ago when I was in school, the policy was to push honors and AP classes for "college-bound" students. Stats (not offered as honors or AP in my school) was one of the fun courses that co-op kids got to take while we were stuck in pre-calc.
- caddemon 5y ago
  I think a layperson's outlook on these topics in high school could be very interesting though. Use this outlook to help motivate learning calculus and linear algebra. With a mix of engineering survey and math survey there are plenty of kids out there that would find parts of the course that stimulate their natural interest. Then the "calculus has many practical applications" thing would feel way less contrived. I mean of course I know now that it is true, but in high school I didn't feel it because I didn't have enough broader context.
dwohnitmok 5y ago
> meta-mathematics? Topology? Logics? History of mathematics? Philosophy of mathematics? Combinatorics? Number theory? Discrete mathematics? Graph theory? Honestly all of those feel more niche than calculus. I agree with you and joatmon-snoo on the usefulness of statistics and would probably support bumping calculus in favor of statistics, but meta-mathematics, topology, logic (which bleeds into meta-mathematics), combinatorics (which is kind of covered by stats), number theory, discrete mathematics, and graph theory are all much less useful even in adjacent STEM fields (discrete mathematics and graph theory matter more in CS, but far less for day-to-day programming). History of mathematics is effectively an entirely separate discipline and philosophy of mathematics has meta-mathematics/mathematical logic as a prerequisite. Calculus unlocks much of physics and engineering (and lots of stats!). Large cardinal theory does not unlock any other field to the best of my understanding.
- caddemon 5y ago
  I think the problem is that students don't know these fields exist unless they took an especially strong liking to the standard school math curriculum. Going into college I couldn't have told you what most of the fields mentioned were about even broadly. Truly I had no idea what math was, because we barely talked about proofs through high school. The only reason I swung back around to learning some of it is because my upper level college CS classes introduced me to ideas that then made me want to revisit my math curriculum. There ought to be some way to encourage late high school/early college students to "survey" the field without necessarily taking full courses in these topics. This could also give some earlier understanding in how the different fields relate - for example you could present a toy graph theory problem within linear algebra as a matrix problem, later presenting the same problem in graph theory section and walk through it using graph representation. I think high school courses struggle with memorability precisely because most units are taught basically in a vacuum. Regardless though, the point of the survey course wouldn't be to remember details so much as to find topics of interest for further study/be generally aware of their existence in case a relevant problem comes up in the future.
edflsafoiewq 5y ago
Calculus and linear algebra continue to dominate in applied mathematics. "How can I turn this into a problem in linear algebra?" is probably the most fruitful mathematical technique that has ever existed.
- JadeNB 5y ago
  > "How can I turn this into a problem in linear algebra?" is probably the most fruitful mathematical technique that has ever existed. And, from that perspective (with which I agree), calculus itself is just another instance of trying to turn a non-linear problem into a problem in linear algebra!
  - Koshkin 5y ago
    True: the differential at a point is a linear map; the integral is a linear form (on the vector space of functions).
ghaff 5y ago
Interestingly the ascent of linear algebra is a relatively recent thing. I have an engineering degree but while I was certainly exposed to basic matrix stuff, never took a linear algebra class. When I was an undergrad, you took differential equations in addition to basic calculus for engineering. (You needed for system dynamics among other things but it was very cookbook.) Linear algebra became a lot more interesting once you had cheap computers and Matlab.
- antishatter 5y ago
  Computers made it insanely useful where before it was an interesting view of systematic math.
- bigger_cheese 5y ago
  When did you do your Engineering degree? When I did my BE Degree (2004 to 2008) Linear Algebra was included. We were introduced to the basics of linear Algebra in year 12 of high school (simple matrix manipulation, intro to vectors, dot and cross products that sort of thing). I'm Australian if that makes a difference. The Math in my 4 year engineering degree was structured something like this: First Semester Year 1 two math courses: Calculus 1, Linear Algebra Second Semester Year 1 two math courses: Calculus 2, Sequences and Series (this one was probably least useful all I remember from this 15+ years later is Taylor Series and Binomial Theorem) First Semester year 2 two math courses: Differential Equations, Statistics for Engineers. From second semester year 2 onwards there were no more discrete math classes this was where the degree really specialized into various engineering streams, Mech Eng, Chem Eng, Civil etc. Had their own courses. I studied Materials Engineering some courses were shared with other eng students (For example I shared Thermodynamics with Mechanical Engineering students) but others such as Non-ferrous metallurgy were pretty deeply specialized. A lot of subject used built on earlier math (Fluid Dynamics was backed up by lots of differential equations for example, stuff like Gamma Function would come up in a lot of places. Solid Mechanics had a lot of integrals second moment of area etc.), Linear Algebra I can remember from Fracture mechanics and crack propagation (Stress and Strain tensors etc.)
  - ghaff 5y ago
diffeomorphism 5y ago
These come as a second step. In the first year (non US) you learn linear algebra, some real analysis and how to write proofs as well as important basics. In your second year you can choose all these electives, which then don't have to spend time introducing natural numbers, induction etc.
caddemon 5y ago
I fully agree with your general point, but in the US I've found calculus to be much more dominant than linear algebra. I went to an "elite" high school that still does not offer any course in linear algebra, despite having enough juniors in AP calc to fill a room. Senior year options for those students are AP stats or multivariable calc. At my college, everyone was (and still is) required to take through multivariable calculus, but there's no such requirement for linear algebra. I have some appreciation for calculus now but I really did not enjoy it much in high school or even in college. It turned me away from learning more math for some time which is unfortunate - linear algebra isn't my favorite either but I liked that much more off the bat, so I wish I had some exposure to it in high school. Then again, maybe the high school teaching style is what made me dislike calc to begin with.

dhosek 5y ago

Or the math covered in finite math classes which is a mix of combinatorics, stats, probability and linear algebra. I remember doing my time in the math tutorial room during my grad school days and helping the kids from the business calc classes who were learning math they weren't going to use from books written by people who didn't understand the domain that they were trying to teach math for. Seriously, what business use is there for f(x) = x^1.3 or the indefinite integral thereof?

ghaff 5y ago
When I tutored in business school--mostly related to math-related things (which itself says something about the level of math knowledge among those who weren't engineering undergrads)--pretty much the only calculus that came into play was finding maxima and minima of curves in economics which was both simple differentials and mostly pretty academic anyway. A little later I did some more complex optimization problems but that was done by software (LINDO at the time) anyway.

dwohnitmok 5y ago

FWIW (and I know you're not making this claim) I don't believe Math 55 is a good example of how mathematics should be taught at large, nor do I think calling it an intro class accurately conveys what it is. It's effectively the compression of an entire undergraduate degree into a single freshman course. I say "effectively" because the material covered depends heavily on who's teaching it, but certainly anyone coming out of Math 55 can pick up any undergraduate math content trivially. For example, undergraduates who have taken it are generally explicitly prohibited in course descriptions from taking further undergraduate mathematics classes (because it would be free credit for retreading the same ground) and can only take graduate courses from then on out. It's strongly self-selecting and as a result can afford to cover a truly insane amount of ground. The overwhelming majority of people who take it drop out (the usual dropout rate from people who take it in the first week is probably > 90%), but the people who stay almost all get As. And you will need to be almost entirely self-motivated because a lot (maybe most) of your waking hours will be thinking about math. There's a very small minority of students for whom this is an optimal way of learning. For most students this is the quickest way to make them run screaming away from mathematics even faster than they already do.

C-x_C-f 5y ago
> For example, undergraduates who have taken it are generally explicitly prohibited in course descriptions from taking further undergraduate mathematics classes (because it would be free credit for retreading the same ground) and can only take graduate courses from then on out. Not at all, you are only prohibited from taking "freshman courses"[1]—that's what 55 is supposed to cover (definitely not all of the undergrad math curriculum), though some professors go beyond that. Many students go on to take at least some undergrad courses, with those in the 140's range being mathematical logic gems with no real counterpart in the graduate department. [1]Students from Math 55 will have covered in 55 the material of Math 122 and Math 113. If you have taken 55, you should look first at Math 114, Math 123 and the Math 131-132 sequence. https://legacy-www.math.harvard.edu/pamphlets/courses.html https://legacy-www.math.harvard.edu/pamphlets/courses.html
- dwohnitmok 5y ago

Bostonian 5y ago

I wonder how much statistics someone can understand without calculus. For example, how do you explain what a continuous probability density function (such as the Gaussian) is without calculus?

pmyteh 5y ago
I teach introductory quantitative research methods to communication undergrads, most of whom have no calculus. (Which is essentially optional in the UK as it's taught at A Level after most students have specialised away from maths). They don't seem to have a problem intuiting what a plotted PDF is showing. I think that's because in some sense it can be read analogously to a histogram. Of course, they don't have the tools to generate or manipulate one. But that's honestly not something that applied social science researchers have to do often when using traditional methods.

jldugger 5y ago

> My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult. Strong agree, but engineers probably need both. I'm currently watching a course on causal inference, and the tools are very much calculating gradients. And even if you just use someone else's MCMC, even in the models a differential equation or integral can randomly appear usefully. In retrospect I should have taken a stats class in high school when I had that 1 hour gap for 1 semester, just to build a better intuition around the basic concepts.

qumpis 5y ago
Which course, if I might ask? I only recall Brady Neal having his videos online
- jldugger 5y ago
  this one: https://xcelab.net/rm/statistical-rethinking/ https://xcelab.net/rm/statistical-rethinking/

Melatonic 5y ago

Not sure I agree with that - you also have to take into account that calculus is a requirement for many other sciences. I suppose you could just of course make it a pre-req for things like that but I found that in highschool rarely did they go that deep. Physics becomes a hell of a lot easier with basic calculus for example. If anything should be dropped from a highschool level its all of the insane memorization you have to do for some of the lower level math classes - I found that totally useless. You then learn some basic calculus and realize "I just wasted so much of my life" and never need to memorize those things again.

JadeNB 5y ago
> If anything should be dropped from a highschool level its all of the insane memorization you have to do for some of the lower level math classes - I found that totally useless. You then learn some basic calculus and realize "I just wasted so much of my life" and never need to memorize those things again. What sort of memorizations do you have in mind that you no longer need to memorize once you know calculus?
- wizzwizz4 5y ago
  s = ut + ½at²
- p1necone 5y ago
  Knowing the basics of integration/derivation makes all sorts of very common concepts in physics much more intuitive - velocity, acceleration, area etc.
- ghaff 5y ago
  Pretty much all the classic Newtonian formulae you memorize in high school physics classes can be derived (relatively) easily once you bring calculus in. A general problem with high school classes in particular--still somewhat true in certain university classes but much fewer--is that there are dependencies across classes. So you end up with a lot of "Memorize this thing because you don't have an advanced enough background to understand why $XYZ is the case."

staindk 5y ago

Super agree. I took calc 1 and stats 1 & 2. Much preferred the stats and it set me up for understanding all kinds of science lingo in articles and papers. I also indirectly use stats fairly often at work.

ghaff 5y ago
There are different types of stats. I took a stats course in my grad engineering program. Not being the best person at math I think I probably struggled with the math sufficiently to get distracted from the concepts. When I breezed through a stats course when getting an MBA I understood the concepts much better--though the professor was almost certainly better as well.

Koshkin 5y ago

> within the US All fifty states or just continental?

Siira 5y ago

Statistics needs calculus as a prerequisite. Heck, a non-introductory treatment of probabilities needs measure theory.

geebee 5y ago

I'm really wary of this. House wiring would be more useful than Shakespeare. That's not to knock house wiring, or statistics. I'd love to know more about both. But don't deny yourself an understanding of the meaning of limits. Almost all mathematics before calculus leaves you with a misimpression that neat formulas exist to solve problems. In reality, you've learned to draw straight lines with a ruler, and maybe a few curves with a compass. Before Calculus, you might actually believe that numbers that can be expressed as the ratio of two integers are typical, and that numbers like pi and the square root of two are "irrational" rarities (and until calculus, you probably don't know about Euler's constant unless it was introduced in precalc as another one of those odd and rare numbers). Look out at nature, where are the triangles, rectangles, and circles? Maybe a wasp nest? Nah, not really. Try to draw a cloud, a tree, a tiger, or a human face. How useful is that straight line or compass? How useful is a line at all, other than to hint at something you can't actually draw, maybe by implying it exists as an ever vanishing limit from above and below? Math required calculus the instant humans decided to describe the world as it is, rather than by the limits of what we impose on it. Also - in stats, how do you know what the area is under the probability density function?

SamReidHughes 5y ago
Well, AP statistics covered the area under a probability distribution function and people seemed to understand that -- you look up the answer in a table (or use the TI-83 function). Presumably they'd do the same for a cumulative distribution function.
- bee_rider 5y ago
  Highschool is pretty early to teach people that math is too complicated for them to actually evaluate their expressions.
  - adgjlsfhk1 5y ago
    Teaching people that there are lots of problems where solving them numerically is the best way of solving them is often just the truth. Also, you then get to discuss some of the really cool numerical techniques which tend to get short-changed in traditional math classes.

jpz 5y ago

You can’t understand much about statistical inference without calculus. It’s all integrals over pdf’s. A lot of integration by parts and other things in the core curriculum.

pjbeam 5y ago

My prob and stats courses required the calc sequence (US based math undergraduate).

com2kid 5y ago

The ideas behind Calculus are much more useful IMHO. The idea of rate of change, or change over time, converging and diverging series, these all have practical application to the real world. And the basics of physics makes next to no sense without calculus, and even parts of intro chemistry make more sense after having taken calculus. I remember in one economics class, which didn't have calc as a pre-req, the prof said "alright, for those of you who have taken calculus, this is an integral, you can now leave the classroom and come back tomorrow. Everyone else has to stay." The ideas of calculus just seem so fundamental to me. It is sad that the American schooling system is so slow, and expectations so low, that it isn't taught to everyone. Meanwhile in other countries, everyone, artists to engineers, learns calculus.

bee_rider 5y ago

I think I'd rather we lose geometry than calculus. Geometry made more sense as an intro to the idea of 'proving stuff' when everybody's dad had a workshop, compass, etc.

caddemon 5y ago
Agreed, also the way my high school taught it made proofs seem like a boring subset of Geometry. The unit was poorly connected to other units and the problems I found by far the easiest in the course as we got very formulaic ones. It's entirely possible my teacher was just awful but I definitely walked away with some misconceptions about what proofs are. Perhaps it would have been more appropriate at an earlier stage of development.
bick_nyers 5y ago
I never thought about it like that, thanks for that insight

smugma 5y ago

A few years ago USC’s MBA program replaced statistics with data science, which is further down the pragmatic chain. Stats is not a generally useful skill (the concepts are, but can be taught in a data science course) but understanding how to work with data is. Discrete math seemed to be the most applicable math class to my CS curriculum. I never took graphics so didn’t use linear algebra, and definitely never touched anything related to DiffEq.

2143 5y ago

> at least within the US Because America has straight lines everywhere and no curves, unlike other places. Calculus is absolutely required for most engineering tasks. Maybe for CS students there could be more emphasis on statistics as well, without diluting the calculus. Infact, where I studied, calculus was a prerequisite for certain statistics and probability classes. For good reason.

thaumasiotes 5y ago

> My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult. I think math people may have a view of this question that is skewed in an interesting way. Statistics is very useful and its common techniques are not difficult to apply. But they seem to be very difficult to apply correctly. We have entire academic fields that are built mostly on the spurious application of statistical methods in contexts that make the whole project invalid. And this sort of "techniques without theory" approach is what's being advocated for upthread and represents a failure mode that math people are unlikely to consider -- because they know that part of being able to apply a technique is being able to tell whether or not applying it is valid. Math people are unlikely to fall into the trap this approach sets. But the same people who want this approach are also likely to end up being hurt by it.

bick_nyers 5y ago
My thoughts exactly. What you want to teach ultimately before anything else is mathematical maturity, which in my opinion, calculus is currently doing the best job at providing.

kjasdfj3 5y ago

> My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult. I'm questioning your math pedagogy. Calculus is the single most important math anyone, of any field, can learn as it's the first "practical math" you actually learn. Life behaves like calculus and in order to think about real life you need the concept of limits, derivatives, integrals, differentials, etc. It's patently absurd to say this should be replaced by statistics, which done to any rigor requires up to 2 years worth of calculus (through diff eq.) to even appreciate. I'm shocked that you're a math major and didn't take away the biggest thing from learning analysis - the ability to think clearly through a problem and prove it correctly. While you may not be asked to vomit cantor's diagonalization onto paper for an interview the ability to think about problems you learned from doing these proofs translates to so many different fields, jobs, and life skills that I take the complete opposite view. If you want to understand anything you need to learn how to proof. I don't care if you're a nurse or an accountant. A rigorous proof based math course will change your life. If by "learn statistics instead of calculus" you mean being able to mindlessly vomit today's new machine learning paradigm without understanding a thing then I think I can understand where you are coming from. Otherwise, I think this is some absurd parody of someone who studied math.

mangodrunk 5y ago
I think calculus and trigonometry shouldn't be taught to non-math majors. You have the perspective of someone who is passionate about calculus, which is great, but considering many people aren't as passionate, what you're suggesting (status quo) doesn't work. Or maybe you have ideas on teaching calculus better than how it's currently done? Calculus is obviously extremely important, but I don't think it's useful to drill a few things that most people will quickly forget after the test. I do think logic is important, and teaching that instead of focusing on math proofs might be a good alternative.
- dchftcs 5y ago
  No, most of STEM should learn calculus at some point. In engineering and physics they are useful; in CS, chemistry and biology they are reasonable foundational knowledge and used in some specializations. Trigonometry is used even in some of the trades. For science-oriented high school curriculums, calculus is necessary, and should only be optional in a few cases where the student already really knows what they want to do. By the time a student wants to specialize at say 20, they should have already had some foundational ability for that specialty, otherwise they wouldn't be able to compete internationally. For people who don't end up using calculus, it's a waste, but there is a trade-off. As a result, much of the high school curriculum is to build a broad foundation to prepare for multiple possible specializations.

peterhalburt33 5y ago

I’ll admit I’m biased on this issue, but I’m very wary of this type of reasoning. There are plenty of courses that are more useful than calculus, but kids/young adults won’t be convinced to learn a subject purely out of its utility in adult life (same reason I’m against replacing math with a “life skills” class); in g TV fact Calculus was the course that changed my whole college and career trajectory. Up until I took Calculus, I recognized the importance of math, but hated every dull second of it - there was no motivation that appealed to me, and it seemed that math could only be used to solve the most contrived problems (word problems that resulted in linear systems, simple roots of trig or polynomial functions, etc). The only reason I took calculus is because it was the next math course after pre-calc and I still had a year left in high school. I didn’t even realize that there was a type of math that could be used to exactly calculate quantities related to continuously changing processes, but I was absolutely fascinated by how many real world problems calculus could solve - I realized that the problems given in algebra were contrived out of necessity to resolve in a neat manner. Now I had access to an entirely new vocabulary that allowed me to describe the world as it is, not as it needed to be to neatly fit in a 10th grade word problem. Until that point I was interested in psychology (and had a vague notion that I’d drop out of school to make a career in music somehow), but I immediately dropped everything to take as many math and physics courses as I could. 14 years later and I work in a very math heavy engineering oriented field. I know my story is probably atypical, and I have no clue what I’d be doing right now if I hadn’t taken calculus, but it’s one of those things that I look at in hindsight and think that I was that close to giving up on STEM, but for being forced to take Calculus. Instead, I earned my Ph.D. in applied math, and there’s not a day where I don’t use calculus of some kind.

toto444 5y ago

There is one problem that I have not seen mentioned. Statistics might be much more useful but except for the first chapter (discrete probability) they are much harder. As someone who wanted to work in data I have tried hard to understand probability distributions, I have watched 10s of videos by different professors, done exercises and sure I was able to pass but the truth is I never really understood it. The intuition behing statistics is non trivial. Sampling is very hard to understand. Even on HN during COVID there were mathematically literate people trying to make some back of the enveloppe probability calculation and all they were able to do was show their misunderstanding of how statistics work (to be fair I would not have been able to do the task myself but I could feel something in their reasoning was wrong).

q-big 5y ago

> My strong opinion as someone who majored in math is that, at least within the US, the standard calculus requirement should be replaced with statistics. So much more useful and so much more important as an adult. > The analytical type of thinking that proof-writing is certainly useful, but you can make much the same argument of many other curricula, and besides, it's not like most intro calc courses even do any proofs. The vast majority of them, I would assert, are simply pre-med weed-out courses. A statistics course would be a much more brutal weed-out course.

theptip 5y ago

Strong agreement here. I'd go further and say that introductory stats should replace trigonometry too. An understanding of statistics is simply required to function optimally as a citizen in modern society, whereas trigonometry and calculus are niche. I'm all for giving tasters of as many different "beautiful ideas" as possible in school, but I think we should be elevating practical statistics into the top tier of subjects that we require kids to go through.

bick_nyers 5y ago

Statistics without calculus relies pretty heavily on memorization to get the right answers. I'm remembering my AP Statistics class (high school) where most of the time students got lower grades because they looked up numbers in a table wrong, or used the wrong table. Never mind when you start using regression models and need to start understanding axioms of the models you are using like homoscedasticity. There's some intuition there, but generally speaking, calculus is where most people "mature" into mathematics. I would argue that most people (NOT engineers/doctors/etc.) don't really get the point of math until around calculus where it starts to really open up for them. While statistics would certainly be useful, and is a must for modern day ever-connected society, it would just be another math subject that needs to be memorized like trigonometry. Some people mature in math a lot younger, but it entirely depends on the quality of their math teachers when growing up (and the curriculum/testing focus in which they are subjected to).

nsriv 5y ago

I'm a high school math teacher, and I think statistics education is desperately lacking in graduating seniors. I think the space for it should come from cutting pre-calculus as a yearlong course though, with some of the concepts backfilled to Algebra 2. There's nothing a basic Stats 1 course needs calc grounding for (perhaps Riemann sums under a normal curve but that's more for deriving than the concept itself). As it stands now, AP Stats is used for kids that don't want to progress to an AP Calc, but want a math or need to hit a school or county requirement.