Someone has a plea to teach real-world math and statistics instead of “derivatives, quadratic equations, and the interior angles of rhombuses”

Robert Thornett writes:

What if, for example, instead of spending months learning about derivatives, quadratic equations, and the interior angles of rhombuses, students learned how to interpret financial and medical reports and climate, demographic, and electoral statistics? They would graduate far better equipped to understand math in the real world and to use math to make important life decisions later on.

I agree. I mean, I can’t be sure; he’s making a causal claim for which there is no direct evidence. But it makes sense to me.

Just one thing. The “interior angles of rhombuses” thing is indeed kinda silly, but I think it would be awesome to have a geometry class where students learn to solve problems like: Here’s the size of a room, here’s the location of the doorway opening and the width of the hallway, here are the dimensions of a couch, now how do you manipulate the couch to get it from the hall through the door into the room, or give a proof that it can’t be done. That would be cool, and I guess it would motivate some geometrical understanding.

In real life, though, yeah, learning standard high school and college math is all about turning yourself into an algorithm for solving exam problems. If the problem looks like A, do X. If it looks like B, to Y, etc.

Lots of basic statistics teaching looks like that too, I’m afraid. But statistics has the advantage of being one step closer to application, which should help a bit.

Also, yeah, I think we can all agree that “derivatives, quadratic equations, and the interior angles of rhombuses” are important too. The argument is not that these should not be taught, just that these should not be the first things that are taught. Learn “how to interpret financial and medical reports and climate, demographic, and electoral statistics” first, then if you need further math courses, go on to the derivatives and quadratic equations.

81 thoughts on “Someone has a plea to teach real-world math and statistics instead of “derivatives, quadratic equations, and the interior angles of rhombuses”

  1. Applying the same logic to the introductory statistics course: what should be taught first? Traditional approaches involve probability theory, sampling distributions, hypothesis tests, etc. More modern approaches involve basic programming. How about starting with interpreting basic graphical information, measurement, interpreting predictive models instead? Personally, I would like students to see a range of predictive models (including regression, classification trees, neural nets, etc.) before exposing them to the traditional topics found in introductory statistics books. I agree that statistics is one step closer to application than most math courses, but I’m not sure that step is capitalized on much.

    • I think what you see in a typical college statistics class taught to non-STEM majors is making sure that students can calculate percents and interpret bar graphs.

    • Trees and nets seem a bit more advanced/specialized to me, but I do think that shunting the entire glossary of statistical tests to a more advanced course, and instead focusing on the linear model and variants (and how they can be used to do similar/identical things!) would benefit students. “How can I test for the equivalence of proportions” is a much more google-able question than “how can I tell if A is related to B in this data, and how can I estimate if that’s true in other data, too”

    • I think we start with developing intuition, focused more on recognizing key concepts in real-world data rather than mathematically precise calculations. That’s what I try to do with my two books.

  2. In his wonderful book titled “The Atomic Nucleus”, the late Prof. R.D. Evans included a three-chapter section at the back titled “STATISTICAL FLUCTUATIONS IN NUCLEAR PROCESSES”. This became the basis for a full-semester graduate-level course, which I enjoyed immensely. It does have some mathematical material, for example to explain Gaussian and Poisson distributions, but a lot of the material and especially the problems tackle practical matters – well, practical matters for laboratory-oriented nuclear scientists, anyway.

    The book freely available at

    https://library.psfc.mit.edu/catalog/online_pubs/books/evans_atomic_nucleus.pdf

  3. I am not convinced by Thornett. He lists Darwin as a scientist who did not like mathematics but shows us that Darwin needed mathematics and outsourced the math to Galton. I can easily believe that there is much to learn from a rotting tree stump, but there are lots of other things to study as well. He says that a math curriculum should include geometry; wouldn’t that get you into a rhombus corner? What are “students drawn to science but not to math, labs, and Petri dishes” going to study. I know that he focuses on calculus, but isn’t his appeal simple anti-intellectualism? Drive those pointyheaded professors into the fields to do honest work!

  4. How to interpret financial and medical reports and climate, demographic, and electoral statistics are all challenging problems that don’t have obvious right answers in the important cases. What does a fraudulent 10-K look like? Should we stent this seemingly healthy guy who has high amounts of plaque in his arteries? Those are easier questions than the climate, demography, and electoral ones since people typically don’t get emotional about 10-Ks and cardiology. I worry that teaching those things well requires either very talented teachers or very motivated students. Otherwise you are just going to get students regurgitating whatever the equivalent of “the mitochondria is the powerhouse of the cell” is for climate, etc. Maybe that’s an improvement, but it seems marginal to me.

    Derivatives and geometry have the advantage that they have right answers. Are they relevant to most students? No, of course not, but aside from basic literacy and the ability to add and subtract very little of what students learn in school is useful unless they go into a handful of professions. I think the real problem is that modern schooling isn’t well matched to the needs of the students. For many students school is basically a place to park them until they are old enough to work. For a smaller portion of the population school is used to train them and select them into specialized fields that they have aptitudes or interests for.

    • And formulas were not created to deceive and mislead you, but public-facing statistical reports often were: “our company is totally sound” or “after cutting testing we have found very few new cases of this disease so we can remove public health measures” or “crime rates in this neighbourhood are rising (because we sent people there to find and record ‘crimes’) so we need more resources.” Dealing with people who are trying to deceive you is a hard problem because verifying is expensive and rarely brings rewards comparable to the rewards of a successful deceiver.

  5. Why should passing calculus be required to major in biology any more than passing biology should be required to major in math?

    Calculus is a tool used to describe the behavior of dynamic systems, and biology is the study of the most dynamic systems we know of. You can do it by learning to write simulations instead, but you need some method to work out the consequences of your theory.

    The lack of calculus/programming training is actually a huge obstacle for anyone who wants to go beyond checking if two groups are different. Actual science (like what brought us the technological advances we enjoy today) looks completely foriegn to many researchers in these fields.

  6. The problem with this ongoing discussion is that it makes a lot of assumptions.

    The problem is not that it completely oversimplifies the math curriculum. Almost all of the algebra 1, geometry, algebra 2, precalculus, calculus sequence of the middle class, college bound high school student is better taught using a data-rich, modeling approach. And since most of the selective colleges do not allow high school calculus (AB) to replace Calculus 1, we should just admit that for most students. Changing the end of the sequence to a data-rich, applied statistics would make a lot of sense, but changing the whole sequence would be even better.
    However there are many issues with this. First for many students, more than we college-educated people who were seen as college-bound high school students, math ends with one year of algebra.

    Another issue is that the typical high school math teacher has never had any experience taking a class like that. They’ve had the one course in statistics required for math majors who have already completed at least a year of calculus. And that course often does not include any work with data because they are doing proofs and learning probability. My college provides a huge percentage of the teachers in the Bronx, and that is the situation here. And when they teach statistics they often teach it as “a math problem” meaning applying formulas and doing things like calculating the median of 10 decontextualized numbers.

    But there is a more important thing going on at the college level. There is a very strong push to the “stat way” as an alternative to the “math way” focus on just getting students to check the box of the general education math requirement by taking a statistics course that leads no where. Not to more statistics, not to algebra or precalculus, no where. That class has, in essence, had most of the algebra removed. You cannot learn about regression if you don’t understand that lines can be represented by equations. Not learning about logs similarly really creates barriers for many different majors.

    Here’s a kind of generic outline for Algebra 2 topics from Khan Academy.
    https://www.khanacademy.org/math/algebra2
    Huge numbers of students don’t get these. So while I think that a data focused approach to teaching these can be excellent (and interpreting a graph with a logged axis can make a lot more sense to students than hearing about decontextualized logs), I think we really need to be careful about making sure that they get through a lot of that content. Otherwise their “stat way” class will be a deadend.

    • And let’s be realistic, who wants a good curriculum and will pay to have it created? Certainly none of the typical students, they just want a grade in the required courses and a sheepskin. And the professors are busy applying for research grants and don’t have time to teach well. Besides, a huge fraction of college undergrad classes are taught by adjunct faculty, and a crazy proportion of them are homeless
      https://www.theguardian.com/us-news/2017/sep/28/adjunct-professors-homeless-sex-work-academia-poverty

      For the most part kids are actually learning stuff from YouTube lectures anyway.

      Academia is broken beyond repair, it’s only people embedded in it who don’t see that.

    • elin– At my school they are called “math pathways” (a term that I have seen elsewhere), and the point seems to be to get students into a course which a) they can pass easily, and b) be the last math class they will ever take. It really seems to just be a way check a box for getting accredited: “our college has a math requirement”. Since algebra was seen as a “barrier to student success”, a “stat way” was created to avoid algebra. But it is, as you say, a “deadend”. Although natural science departments were allowed to keep algebra 2 and, for some, calculus, social science departments were bludgeoned into giving up algebra 2 as a prereq for their stats and methods classes.

      • Correct 100%.
        That’s why social science departments have to teach their own statistics courses.

        And what they have done as a result is forced the removal of anything with math content from much of the curriculum.

    • I agree that the comments on this post have lots of assumptions (often that how things were done in one US state when the author was in university is how they are done everywhere for all time) but algebra and logarithms are secondary-school math not university math in the system I went through. What would they be doing on a university curriculum in the place you have in mind except as a remedial course?

      • Sean:

        Logarithms are difficult for many people. Just about all the students in my classes have learned logarithms back in high school in their pre-calculus class or the equivalent, but most of them don’t have a good gut sense of logs. For example it’s not obvious to them that log(1.1) is approximately 0.1. They will struggle even more with problems such as elasticities, for example log y = a + b log x corresponds to y proportional to x^b. Students can learn this stuff, but it takes some time.

        And it’s not just students. Professional social scientists often fit regression models on the linear scale in settings where the log scale is obviously better, to anyone who understands logs. For example, you’ll see country-level regressions with predictors such as population, GDP, etc.

        So, yeah, these students do need remedial education in logarithms if they’re going to use them for anything.

      • Remedial education aka developmental math would be _before_ algebra 1. College Algebra is a course that is often offered once students have done remediation. Students who have already passed Algebra 1 are placed in precalculus or statistics (what I refer to as statistics without algebra). The physics students get trig somewhere.

        When I learned logarithms we used them to do multiplication and division and spent a lot of time practicing interpolation. These days it would be insane to do that. Just like students don’t apply the same skills to trig tables or the normal distribution.

        One thing you do also need to remember is that there are as many students in community colleges as four-year colleges or universities.

        • Elin:

          At the universities I’ve attended or taught at (Maryland, MIT, Harvard, California, Chicago, Columbia), intro statistics has had three tracks:

          1. Two-semester sequence of mathematical probability and statistics. Traditionally this was straight math, although I guess they mix in some computing too. Derivations, proofs, uses calculus, the whole thing.

          2. One semester “statistics with calculus.” Does not actually use calculus, or even algebra. There are some mathematical derivations in class and in the textbook, but students aren’t required to reproduce these. This is basically the statistics-without-calculus class but with higher expectations for the students—they’re supposed to be able to solve slightly harder problems.

          3. One semester “statistics without calculus.” Nominally has precalculus as a prerequisite but doesn’t really use it. Again, no algebra, no solving for x or anything like that. The precalculus requirement is just there I guess to screen out students whose math background is so weak that they’d have trouble dealing with expressions such as 1/sqrt(n).

        • Andrew:
          Yes this is basically true everywhere except in non elite institutions there would be a 4th level (or the third level and your third level would not exist) which does not require precalculus. This might sometimes be what is called a statistical literacy class. The whole “mathway” versus “statway” approach is premised on the idea that students who have not had any algebra can pass an easy statistics class.

          At the places you have taught there are essentially no students admitted who have not taken precalculus in high school. Someone without precalc would not do well enough on the SATs or the Chemistry and Physics classes that they are also taking to get into Columbia, MIT or Maryland. That’s the middle class, college bound track. But for low income kids, that is not an assumption. In fact, there is soon to be AP precalculus.

        • Elin:

          Yes, at Columbia they all took precalculus in high school, indeed I guess that most of them took either statistics or calculus in high school too! But the statistics courses they take here (with the exception of the two-semester sequence of mathematical probability and statistics) don’t really use any pre-calculus. They don’t even use algebra.

          That’s one reason I’m sympathetic to the don’t-teach-algebra-by-default position, as exemplified by the article referred to in the above post. On one hand, yeah, algebra is super important; on the other hand, even students who’ve taken algebra and pre-calculus aren’t ready to use it and aren’t expected to use it, even in classes with calculus or precalculus as a prerequisite. So I could easily see the courses being taught in reverse order: first statistics or some other applied class, then follow up with algebra. I don’t really know, and I’m not going to make any general recommendations here, I just want to register that I see some reason for the argument.

        • Andrew:
          I’d be totally interested in the idea of statistics first. I also totally agree about the reality the the statistics classes that require calculus at most do one thing with calculus, and it is never in the homework. It’s just gate keeping.

          Then actually the poor students who take the two semester hard sequence don’t do any work with data because they are busy doing proofs.

        • Yeah, I should have clarified what I meant by “it”.

          Specifically my idea was more about what elin said… Reconfigure the whole curriculum to teach math as application oriented together with various sciences and with a basis of building models from experimental data. And IMHO explicitly with Bayes for the stats.

          So, you’d teach quadratic equations by having people design some explanation about the amount of paint needed to cover some building as a function of the dimensions of the walls, and calculating how long a two story warehouse would have to be before it required the same paint as a single story one or etc etc.

          Make everything an application, and use graphs of real world data everywhere and never ever under any circumstances calculate a p value.

          Of course the teacher would also show examples on the board and you’d discuss how to manipulate symbols and such but the science would be forward throughout… It’d take a teacher who has both math and science skills.

        • Give kids an optical theodolite and a tape measure and have them survey the school, each team would collect their own data, design their own survey, use their own control points…. Calculate stuff using trig, include uncertainty associated with tape measure inaccuracy and angular measurement error and such. You could spend a month on that one problem going more and more in depth

        • Take data from the Mythbusters on walking vs running in the rain, and model the results as a function of body size measurements and speed of running. Use only data on kids under 5ft tall. Do graphs and try to create a dimensionless measure of rain exposure. Then after you have that, see how well the model extrapolates to adults 5ft + tall…

          Build bridges using popsicle sticks and then break them, design models that predict the maximum load, and the deflection of the bridge… Etc

          I think all of these things are doable in high school with appropriately well designed curriculum and support.

        • Ah Daniel, I see. Very PhD in engineering of you.

          Setting aside the claim that this doesn’t work as well as it seems like it should, there is another option. Just make math open to all, required of none, and for the few.

        • I think the world suffers when people are innumerate. And people are exceedingly innumerate. Heck even apparently elsewhere in this thread someone called a “brilliant economist” couldn’t do very basic trig calculations for woodworking.

          What should be the high school graduates level of math competency? I’d argue you should be able to look at a scale drawing of a building and calculate the diagonal of a room, or the required length of baseboards around the room excluding doors… If 3in of rain falls in the parking lot how many liters of water need to be absorbed by the dirt swales? If a 10% reduction in price is offered how much less will the pavement cost to install? Etc etc

        • How do you know:

          (1) How valuable “greater numeracy” would be?
          (2) Whether it’s the most valuable use of a student’s time?
          (3) Whether numeracy can be improved much?
          (4) The best way to improve it?

          I claim the answer to all these is in serious doubt. Two things that can be said for certain are:

          (a) An enormous number of math education reforms have been tried, from huge well funded changes, to grant funded pilot programs, math ed phd thesis, and math ed experiments. It’s clear from these that people’s deeply held opinions on the subject are shockingly, and consistently, poor guides for what works.
          (b) While answering (1-3) in general may not be easy, people can at least answer them for themselves. They have the most skin in that particular game after all. And they consistently decide that striving for greater numeracy isn’t the way to go.

        • My biggest concern about numeracy is how easy it is to manipulate people who don’t understand quantitative things. Eventually the private equity firms and the regulatory capture lobbyists and the bankers run everything…

          I’m not opposed to your suggestion that we not make math “required” or whatever, I just think if we make math much more about solving real world quantitative problems people will choose to engage more of it and be less gullible.

        • But that assumes “make math much more about solving real world quantitative problems” implies “people will choose to engage more of it and be less gullible.”

          I deny that implication,

        • My only real data is that on reddit multiple times a week ill see posts saying “I was terrible at math in high school but now I really want to use math for (insert real world purpose) how can I learn the math I need?” On r/learnmath and several other subreddits its very common. and hundreds of people come out of the woodwork to say how they also did a similar thing and what techniques they use.

    • Absolutely. Though Lockhart is saying the opposite—that math is intrinsically valuable and should be taught like other intrinsically valuable subjects (like art).

      I think the difficulty with math pedagogy is that students are motivated by different things. Some students are most motivated by practical applications (though I think fewer than the number of students who complain about math impracticality). Others like math because it’s fun: something like a puzzle or a game or, yes, art.

      And then there’s a middle path—some students are most motivated by scientific or engineering applications of math. I think I fall in this category. Probably a lot of readers fall into this category, too. I think these kind of students are best served when math and science are integrated and taught together.

  7. If I say nothing else on this topic, which I’ve ruminated on for decades, I should tell this story: It was in the early 1980s, and I had signed up for a weekend workshop presided over by none other than Paulo Freire. (It was at Cornell.) I joined the “technical education” group, and we spent many hours fruitlessly trying to apply Freirean dialogical methods to teaching math, econ, stats, etc. Finally, near the end of the workshop, the Great One paid a visit to our group. We laid out our problem to him, he thought for a minute and said, “There is a moment of induction in all education, but it is a Hegelian moment. It can last for years.” A sigh of relief could be heard around the room.

    Also, in the early 90s I went through a training in teaching reform calculus. I actually wasn’t much impressed with the tools they were pushing, but I won’t forget a factoid (which may or may not be true) about what was wrong with the then- (and I think still-) current situation. A standardized math test was given to students in an initial year and again in a later year. On average, the more math a student had taken in between, the more their score went down. The test was devised to measure “understanding” in some fashion. Assuming no measurement issues (yeah, big assumption), the explanation given was that every math course further reinforced a destructive habit of eyeballing a problem, seizing on a formula that seemed like it might apply, and then shoehorning the problem into the formula, giving no thought to whether the result made any sense. And I’ve seen this countless times with my own students; it’s what they think “doing math” means. The idea of reasoning through a problem to see what it might take to solve it is alien to them.

    As for the remedy, I still believe in Deweyan constructivism, at least in a general way, but I think it’s a framework, not a formula. Of course, it has implications for the practicality of the subject matter as well as the role of communication between those learning it. I agree with Freire that there are times to pull back from this framework and backfill a chunk of technical stuff so students can get back to topics that concern them. One piece I’d add is spending more time with gaps and errors. Mistakes in practical problems are consequential, and it’s useful seeing how mistakes play themselves out. This can motivate students to try to understand the thinking that can avoid these mistakes — also, the methods we routinely rely on today were often developed precisely because older methods failed. There’s a failure-solution dialectic, as in Lakatos.

    I admit that I fall way short of these ideals in my own teaching. I sure wish there were textbooks that modeled this approach though.

  8. I suspect that probability and statistics are either something humans are inherently bad at, or something neurotypicals are inherently bad at. Most of the statisticians I know agree that most academics who use statistics in their work are terrible at it, even though they are selected for being good at academic learning and have lots of specialized training. If my suspicion is correct, expecting most humans or most neurotypicals to become good at probability and statistics if you just train them harder is a fool’s errand, like expecting them to not engage in motivated reasoning or nepotism if you tell them not to. At the very least, it might require a revolution in teaching methods, like how in the late 20th century coaches learned how to teach any healthy adult to run a marathon.

    Geometry and algorithms are very practical in some areas eg. surveying, woodworking, estimating volumes of storage spaces, business.

    • People have trouble with statistics class not because theyare “neurotypical” but because they are taught a byzantine set of ad hoc rituals that make no sense. I always liked this Deming quote:

      Example 3. The Panel on Statistics distributed at the meeting of the American Statistical Association in Montreal in August 1972, in the pamphlet INTRODUCTORY STATISTICS WITHOUT CALCULUS, the following statement (page 20).

      A basic difficulty for most students is the proper formulation of the alternatives H0 and H1 for any given problem and thc consequent determination of the proper critical region (upper tail, lower tail, two-sided). (Here H0 is the hypothesis that u1= u0; H1 the hypothesis that u1/=u0. )

      Comment. Small wonder that students have trouble. They may be trying to think.

      https://deming.org/wp-content/uploads/2020/06/On-Probability-As-a-Basis-For-Action-1975.pdf

      • Anon:

        Wow—“the proper formulation of the alternatives H0 and H1 for any given problem and the consequent determination of the proper critical region . . .”! How horrible.

        I guess we’ve advanced a bit from 1972. Yes, I think this crap is in almost every introductory statistics textbooks, but at least there are no ASA panels recommending it, right?

        • My idea/hope is that the internet is the printing press of the current age, so that type of stuff will eventually go the way of catholic theology to be a niche topic. But we are not there yet, it may take another generation.

          The main obstacle is people refusing to accept so many highly regarded and educated people could be teaching and basing their careers on something so silly. The argument from authority and consensus heuristics are powerful.

      • For evidence that human beings are not good at thinking statistically, see also the big chunk of the psychology of cognitive biases and heuristics which apply to probabilistic thinking. The pop version of that research does not address neurotype and when researchers repeat the experiments with a sample of autists or ADHDers I suspect they will find that some of their biases are specifically neurotypical traits (one obvious example: depressive realism could also be framed as a flaw in thinking which affects the non-depressed). But any program of mass education has to start from the strengths and limits of neurotypical minds.

        • I’m pretty sure the average person can understand basic concepts like coinflips. Here I ended up writing essentially a draft for a intro to a stats book. I really think if anyone can understand this theyre good (of course, it won’t help anyone understand the nonsensical stuff taught as stats[1]), the rest is implementation details.

          Anyone can understand coinflips. Ie, while the probability of tails is 50%, any given set of flips need not be exactly 50%. But if you see far from 50% there might be something else going on (eg, both sides are heads… but then if you see one at least one tails it can’t be that explanation either..).

          That is the basic idea. Then some people bothered to think more deeply about the problem and worked out that if you flip a coin, eg 3 times you can get various outcomes.

          Three heads/tails and zero of the other:
          HHH
          TTT

          Two heads and one tails:
          HHT
          HTH
          THH

          Two tails and one heads:
          TTH
          THT
          HTT

          So there are eight total possible outcomes, and three correspong to two tails. Thus, if there is 50% chance of tails/heads, the probability of seeing two tails after three flips is 3/8 = 0.375.

          Further, people found a pattern here that saves us the trouble of writing out all the possibilites for four flips, five flips, etc to as high as you want to go. The equation is:

          1) prob(n_tails) = [n_flips!/(n_tails! * n_heads!) ] / 2^n_flips

          Eg,

          2) prob(n_tails = 2) = [3*2*1/((2*1)*(1)) ] / 2^3 = 3/8

          […Then you can explain this is a special case for binary outcome and how to derive it if they are interested. But the point is that probabilities are just comparing the number of combinations of one type to the total possible…]

          Above we wrote prob(n_tails), but actually this is shorthand because we left out that this should only be true *if the chance of heads/tails is always 50%*. To be clearer we should write prob(n_tails|A), where “|” denotes “given” and “A” represents the assumptions used to derive our model/hypothesis (equation 1).

          These types of probabilities, ie prob(observation|model), are referred to as “likelihoods”. They tell us the probability of seeing the data we got given our model (and hence assumptions) is true.

          However, there is another probability we are often more interested in: prob(A|n_tails). This is the probability our assumptions are correct given we observed two tails, and is referred to as the “posterior” probability. To calculate this value we need to use the likelihood calculated above, divided by the sum of the likelihoods of all the models we can come up with (for now assume all models are equally likely). Now using A to refer to assumptions/model A and B for model B, etc we get:

          3) prob(A|n_tails) = prob(n_tails | A) / [ prob(n_tails | A) + prob(n_tails | B) + prob(n_tails | C) ]

          The denominator should ideally include every possible explanation, but in practice we have only thought of a few. Here we can consider two other obvious possibilities: the coin is B) double headed or C) double tails. Then we need to determine the likelihoods for these models. In this case it is easy, since we observed a mix of heads/tails. So we know: prob(n_tails| B) = prob(n_tails| B) = 0. Therefore our posterior for model A is equal to one. No other possibility was considered that could explain our observations.

          Imagine in a different set of coinflips we observe three tails, then the likelihood for model A (eq 1) would be p(n_tails=3|A) =1/8. For models B (double headed) and C (double tailed) we get p(n_tails=3|B) = 0 and p(n_tails=3|C) =1.

          The posterior probability would be:

          4) p(A|n_tails = 3) = (1/8) / (1/8 + 0 + 1) = 0.11

          Now the posterior for model A is only 11%, because model C is much better at explaining the observations. If we calculate the posterior for model C we get:

          5) p(C|n_tails=3) = 1 / (1/8 + 0 + 1) = 0.89

          So, if all we know is that three tails were observed and the only three possibilities are fair coin vs double headed/tailed, then the probability it is fair is 11% vs 89% that the coin is double tailed. But usually we know far more than this. Where did the coin come from? How does it look? And so on. Thus we should also weight each likelihood according to our background knowledge with the values prob(A), prob(B), and prob(C), referred to as the “prior probabilities”. Thus the posterior for model A is actually:

          6) prob(A | n_tails) = [prob(A)*prob(n_tails | A)] / [ prob(A)*prob(n_tails | A) + prob(B)*prob(n_tails | B) + prob(C)*prob(n_tails | C) ]

          For equation 3-5 above we assumed that prob(A) = prob(B) = prob(C), so these values all cancelled out. This special case is referred to as a “uniform prior”.

          From prior experience we know that most coins we get as change at the store will be fair, however this is much different than a coin bought at a magic shop. So knowledge of where this coin came from can drastically change the values assigned to our prior probabilities. Further, different people may have very different background knowledge and thus can validly assign very different priors. Thus the prior probability is said to be “subjective” and can take any value between zero and one.

          However, in practice it is sufficient to use rough values. Eg, if the coin came from the register at the grocery store prob(A) ~ 0.99 and prob(B) and prob(C) are near zero. Thus even if three tails are observed the posterior for model A would still be near 100%. But if it came from a magic shop perhaps a better prior would be the uniform one (all are ~33%).

          Note, however, that equation 3 was already subjective *before including the priors*. We only considered three models (A, B, and C). Other people may come up with further possible explanations which can drastically change the posterior. Perhaps the coin-flipper knows a trick to make it land on tails twice as often as heads. That could be a model D, for which we would need to derive a likelihood and assign a prior.

          [1] NB: Note this does not include any examples like comparing two groups to see if there is a difference. That is not a valid usecase for probability/statistics except in the rare case where zero difference is predicted by a theory of interest. It makes no sense and students are right to reject it.

        • Worse than that, the better people are trained at the classic view of statistics you find in an average master’s level stats book, the worse they will be at understanding the world and doing good science.

          Just read r/AskStatistics for a couple weeks.

        • Here’s some examples from just TODAY:

          https://www.reddit.com/r/AskStatistics/comments/126wmp3/how_to_calculate_optimal_matching_distances/

          “Right now I’m performing longitudinal research on households in problematic debts in 2017. My research question is to what extent household type (e.g.: one-person hh, married with children, etc.) influences the pattern of problematic debts for households over a time period of five years. I have yearly data from 2017 to 2021 (so 5 time points) and want to perform a sequence analysis, followed by a cluster analysis using Optimal Matching distances…”

          or

          https://www.reddit.com/r/AskStatistics/comments/126sw6s/appropriate_statistical_test/

          “I ran an experiment with 10 participants. Each participant had to complete trials. My analyses include:

          1 continuous dependent variable V.

          5 categorical independent variables : A (2 categories), B (8 categories), C (2 categories), D (2 categories), P (the participant, 10 categories). The combination of these 5 variables allow to uniquely describe each trial (e.g., trial X had: A category 1, B category 6, C category 1, D category 2, P participant 10).

          To note, each participant had the same number of trials with each category of the variables B, C and D, BUT the number of trials with each of the two categories of the variable A differed between participants.

          I’m interested in answering the question: does A affect V? But I want to control for the potential effects B, C, D and P might have on V

          Which statistical test would be most appropriate to answer this question?…”

          https://www.reddit.com/r/AskStatistics/comments/126j6fd/how_do_i_test_if_distributions_differ_from_each/

          “Hi! Maybe this is a stupid question, but I just can’t find the answer. I counted insects of different sizes in ~80 traps. The traps are split in three categories (A,F,H) and I have 5 size categories.

          Say my hypothesis is: The distribution of Insects (according to their size) in the A traps differs significantly from the distribution of insects in the H and F traps. How would I test that? From everything I’ve read I feel like I should do an Chi Square test but I honestly don’t understand how. (I’m working in R, if it matters)…”

          here’s one you should like Phil:

          https://www.reddit.com/r/AskStatistics/comments/126ges5/q_decorrelating_timeseries_data_used_for_ttesting/

          “I’m researching the correlation between different power purchasing strategies.
          In the area I’m researching, the power is traded through an exchange that sets the prices (Nordpool), and end-customers can then choose to either buy power at spot-price or to make deals for e.g. 1-, 2- or 3-year fixed price contracts (i.e. futures).

          My analysis is dependent on sorting data needed for whether it pays off to have a spot- or fixed price agreement, and this has previously been done using paired t-tests. Earlier research has noted that possible problems may be autocorrelation of their data (spot-prices), and that it may have generated invalid results because of that.

          How would I go about decorrelating this data and making the same comparison to fixed price afterwards?

          I thought about maybe fitting an ARMA (or SARIMA) model and using the corresponding values from that model in the same t-test?

          I’m working in STATA, any help or ideas is greatly appreciated!”

          ———–

          Each of these researchers is trained obviously in at least probably a year of “standard statistics” at the graduate level which is WHY they can’t do research to save their life… They are trying to shoehorn their real practical questions into bullshit questions that can be formulated in terms of frequentist tests of hypotheses, and especially ones their software can carry out at the push of a button or a couple lines of commands…

          It’s not “how do power purchasing strategies affect the outcome of total cost?” or “in what ways do the distribution of insects caught in different types of traps differ? it’s… “whats the correlation between power purchasing strategies and “whether it pays off”” so let’s discretize everything because it fits into the testing paradigm…or “is the distribution of insects in trap A different from B”

          or whatever. in each case here, and I promise you these are just a few of the cases from TODAY and these are totally typical of what you’ll see all day long, the researcher may not even be able to formulate an actual research question having broken their brain studying Frequentist stats to the point where it doesn’t even occur to them how to ask a straightforward question about a physical or economic process anymore.

          SMH

        • Daniel
          Those excerpts are ridiculous. We will have to be thankful for ChatGPT (which does better than that, flawed though it may be).

    • Anoneuoid: its possible that probability and statistics teaching is worse than calculus teaching or linear algebra teaching but where is your evidence? And why is this difference so consistent across at least the rich English-speaking world? If it were just chance, you would think that there would be universities which turned out people in the statistical science who had a consistently good understanding of probability (and someone would have tried to mathematize probability before the 16th century).

        • But there is not a crisis of people with engineering degrees who can’t differentiate x^2+10 like there is a crisis of people with social science degrees or PhDs who don’t understand statistics beyond p = 0.05 bad

          So either probability and statistics training is systemically worse than calculus training, or its inherently harder for humans or neurotypicals to learn.

        • For calculus you can mindlessly apply rules others have already figured out to deduce the right answer. Statistics is taught as if you can also do that, but you can’t. It requires abducing various explanations (that are then used to deduce the likelihoods) as well. This is more of a creative act that requires some degree of subject-area expertise.

          It isn’t hard to get from coin flips (near everyone has some degree of familiarity with that type of data) to Bayes rule. But you then need to think about what you are plugging into that rule. And most of the examples are of no value since no one believes the premises to begin with.

          I still do remember sitting in grad school biomed stats and wondering why I’m testing the “null” hypothesis rather than my hypothesis, but was very busy with other things so just did the pointless calculations. In fact, the class was treated as a blow-off even though everyone goes on to use that to determine what is “real” or not.

        • Sean
          We are not talking about people with PhDs here. We are talking about k-12 and the first year of college.
          I don’t think it is necessary to badmouth social sciences, that is probably where the best introductory statistics teaching for the general population of college students is found.

        • Elin said: “that is probably where the best introductory statistics teaching for the general population of college students is found.”

          You can’t possibly be serious.

          Elin: the last thing we should do is teach these highly unreliable mathematical and scientific methods to the students who have the least experience, knowledge and ability.

  9. I had a boss who was a brilliant economist and an amateur woodworker. He was telling me once about a woodworking problem he had and I said: “But that’s an easy trig problem.” He confessed that he’d never taken trignonometry and resolved to learn it (he was then in his early 50’s) in order to solve woodworking problems.

  10. The comments here are excellent — much better than Thornett’s essay, which I vaguely remember reading when it came out, and disliking. As others have noted: (i) things like calculus are, in fact, important for biology and this is *more* the case now than it was 20 years ago, since we have a better understanding of the relevance of dynamical systems — gene regulation, ecological interactions, etc. I’ve met a lot of very good biology students who wish their training had more emphasis on math, not less. (ii) we’ve already been trying the “teach people statistics” route for a while in higher ed (esp. biology, psycology, etc.) and now increasingly in high schools. It is not a success — taught poorly (as it often is) it relies on formulas and poor reasoning, and is (arguably) worse than not knowing statistics at all. At least with calculus and geometry one gets a sense of what mathematical reasoning is, even if one doesn’t use those particular tools later.

    Thornett’s tale of a geology student who likes working with rocks and keeps failing calculus is particularly annoying, since it makes the opposite point as what Thornett thinks: there are a lot of people who think geology is searching for rocks, and astronomy is gazing through telescopes. It is a unfortunate, and rather sad, shock to many to find that learning about these topics beyond an introductory level does, in fact, require a lot of math / physics / etc., especially if you want to make it your career. There should perhaps be terminal “science appreciation” degrees for these people (though of course, there are more resources than ever for hobbyist-level learning outside of formal education). One should not, however, pretend that the more difficult topics are unnecessary simply because one likes rocks.

    • “It is a unfortunate, and rather sad, shock to many to find that learning about these topics beyond an introductory level does, in fact, require a lot of math / physics / etc., especially if you want to make it your career. ”

      That’s because there’s almost no math in any intro level in science courses except physics and chem and because there is an extended suite of “science for non majors” courses that allow students to fulfill science requirements without math.
      “Science for non majors” courses should be eliminated. They never will be, because:

      1) the object of the university is no longer education, it’s just passing people through and collecting their fees
      2) this means that everyone has to be able to pass everything
      3) most students don’t want an actual education, they just want a degree
      4) faculty want classes that are easy and fun to teach where they can give everyone A’s and feel good (and also so they can get good teaching evaluations to keep the anti-education administration off their backs)

      The American education system has peaked and is now in decline. It’s not irreversible at this point, but it probably won’t be reversed. Selling faux education makes a lot of money for a lot of people who would have fewer and certainly less remunerative opportunities in the private sector.

      • I agree with the general pessimism and that school is a scam and a tax on society and whatnot, but this

        “Science for non majors” courses should be eliminated.

        sounds completely insane to me. A society where every electrical engineer has to take the physicists’ version of quantum mechanics and every doctor has to take the chemists’ version of organic and physical chemistry and everyone who wants to learn calculus has to take real analysis is a society with vanishingly few engineers or doctors.

        • “Science for non majors” courses are to provide science distribution requirements for people who aren’t in science majors. For STEM majors, they don’t count toward science distribution credits. So for example if you’re a chem major and you have to take, say, three 200+ level *science* courses outside chemistry, these must be from the STEM curriculum, and Geol 201, “The age of the Dinosaurs” wouldn’t be allowed to that requirement. Either that or it would be numbered outside that range, like “Geol 191”.

          I have no problem with doctors taking “the chemists version” of organic chemistry – not to mention analytic chem, biochem and genetics. Aside from the fact that they make decisions regarding organic / biochem ever day, If they’re going to make life and death decisions they should be able to handle some real chemistry.

        • I don’t know anything about chemistry, though the chemistry folks at Berkeley were very clear that the future doctors would get no real value added from taking the course for chemistry majors. However, I did take physics. The physics major version of introduction to electromagnetism spent a great deal of effort on how magnetism from moving charges can be viewed as an epiphenomenon from relativistic electrostatics, where the lorentz transformation from a moving reference frame would produce different electric field densities. This, of course, meant teaching us to use a minkowski spacetime on a hyperbolic 4-manifold instead of euclidean space. This didn’t come for free–we also spent less time solving realistic circuits and radiation models.

          Now, of course plenty of the electrical engineers could have done well in that class, but I do have to question what they would get out of it. And if all them had to pass the class to become electrical engineers, I’d wager we’d have fewer electrical engineers for no good reason. And the prospective doctors were also required to take introduction to electromagnetism–making all of them take that class would be a disaster.

          All this also applies for the math classes–I’m sure some of y’all disagree, but I think making every engineer prove Green’s theorem in n-dimensions when learning multivariable calculus would be a pointless exercise. And in computer science, lots of people could benefit from some basic scripting and plotting without knowing how to implement their own interpreter.

          I do think some “courses for non-majors” do more harm than good, especially statistics, and I don’t think “Age of the Dinosaurs” should be worth much credit. But lots of them are necessary because different people need to focus on different things. And why abolish “Age of the Dinosaurs” altogether? People should be able to learn trivia if they want.

        • There is nothing wrong with subject X for non-majors. We can’t all be majors in all subjects, so there had better be good courses for non-majors. The problem is the way these courses are designed and taught. Broad overviews of a vast subject come at the expense of learning what it is to think about things in the way that subject X professionals do. I personally thing the latter is more important than the former. The broad overview can be gotten by reading (e.g. read the Economist for an economics overview – but the principles course should leave students with some understanding of how economists view the world).

          The delivery of courses for non-majors is usually left to the least qualified faculty (with some notable exceptions). Majors are where the interests lie, for mostly bureaucratic and financial reasons. Just witness the fights over discontinuing certain majors taking place at many universities. I think the subject matter is more important than whether or not someone majors in it. In fact, I think this may be more true of statistics than any other subject. Statistics for non-majors is perhaps the most important course a person should take.

        • Somebody:

          I’m sure you can find thousands of examples where some specific thing is taught to some group of people in the course of their degree that they thought was useless. But education isn’t just the specific information. Education is being exposed to a wide variety of problems, learning what information is necessary to solve the problem, acquiring the information, then solving the problem. Information is only half of education. The other half is problem solving, and the more and wider variety of problems you have experience with, the better you will be at solving problems in the real world.

          I struggled through all of my math classes – Bs when I worked hard, Cs when I slacked. But for me the best thing about every math class I ever took is that when I took the next level up I finally started to get the previous level. By the time I scrapped through diff eq, I had a much better grasp on calculus. I came to understand calculus much better when I used it in specific applications in grad courses. So taking the next level course isn’t just about learning next-level material. It’s about being proficient enough with the material from the previous level to apply it to the new level. Struggled with Calc II? Take Calc III.

          Repetition, and in particular repetition in many different applications, serves an important function.

          BTW, I checked: where I did my undergrad, the easiest med school pathway – BS biology – currently requires organic chem and two additional upper level chem courses. A doctor may or may not directly use any of that in their entire career. It doesn’t matter. It’s still fundamental background information for how chemistry works. It’s not just a matter of the specific knowledge about chemicals. It’s a matter of obtaining a broader conceptual knowledge about the system one works in.

        • @chipmunk

          I’m fairly certain at this point that we’re talking about different things. You probably did not take the calculus class for math majors. Where I went to undergrad, introduction to multivariable calculus for “scientists and engineers” would have you apply stokes and green’s theorems in 3 dimensions. Introduction to multivariable calculus for math majors would have you prove green’s theorem in n-dimensions. Introduction to differential equations for scientists and engineers would teach you variation of parameters, integrating factors, you’d probably solve the heat equation at some point. Introduction to differential equations for math majors would have you prove the Picard–Lindelöf theorem for existence and uniqueness of solutions to differential equations.

          Medical students everywhere in the United States are required to take introduction to chemistry and organic chemistry. They are not required to take the versions of introduction to chemistry and organic chemistry that are designed for chemistry majors, which at Berkeley was a massive production worth more than the standard number of units and involving extremely lengthy lab work that would dominate one’s schedule. This was okay, because the chemistry students’ first year curriculum was designed to feature these classes as their primary focus. However, if the med students were required to do this, it would be quite literally impossible to schedule those courses simultaneously with the biology lab courses.

          The fact is that different versions of the courses serve different needs. They’re not even necessarily more or less rigorous or more or less content–the non-physics major version of electromagnetism covered some circuits and radiation stuff that we didn’t because we were bothering with relativity, because the engineers need that more. Trying to abolish the engineers’ version so they can work with special relativity would have been absurd!

        • Dale:

          I don’t see the point in “non-major” courses. If the “major” course is too far up the pre-req chain, whatever “non-major” information is necessary can be acquired in the context of the major or focused area of study. There are plenty of “regular” courses that are within the capability of the typical student. I took international relations, existential philosophy, P-chem and botany during my undergrad. None of them were for non-majors. The only place the “non-major” take seems to appear is in science, where making it a “non-major” course eliminates half the content. If you’re a reporter and you want to understand stats, take the real thing, not a half-baked substitute. If you take the real thing, you’ll understand the real problems.

        • chipmunk
          You are confusing two things: the idea of a course for non-majors and the currently designed non-major courses. With respect to the latter, I mostly agree with you. However, I don’t think making everyone use the courses designed for majors is much better – given the way those courses are actually designed. Statistics majors need to worry a lot more about assumptions behind techniques than non-majors do (there are limits, of course). For example, I don’t worry much about whether a business or psychology students doesn’t know much about homoscedasticity, but I think statistics majors should understand that. That’s my personal opinion, and you might not agree with it. My general point is that much of what is taught for majors, while important to majors in those subjects, are not particularly useful for non-majors. This is not an excuse for watering down the non-major courses, as is commonly done. I would maintain that the rigor in both types of courses should be similar – it is the focus that differs. Statistics majors should learn about application, but whether that is necessary in the first course is more open to debate (it really is a matter of the best sequencing of courses in the major). On the other hand, non-statistics majors – who may very well only take one statistics course – should (I’d say “must”) be concerned with application.

        • Somebody said:

          “You probably did not take the calculus class for math majors. ”

          But I did.

          From the current course catalog, abbreviated:

          “Sample Curriculum for the Bachelor of Science in Mathematics….4 MATH 1510 & 1510L (calculus I )”
          “Sample Curriculum for the Bachelor of Science in Physics with Astrophysics Option…MATH 1510 (calculus)”
          “Sample Curriculum for the Bachelor of Science in Chemistry….4 MATH 1510(calculus)”
          “Sample Curriculum for the Bachelor of Science in Technical Communication…4 MATH 1510 (calculus)”

          There is no separate course for majors, it’s all “calculus I” – math, astrophysics, chem – even technical communications!! This is a tech school. There are no “non-science” majors. The parents of many of my fellow students were researchers at nearby national labs.

          I believe you when you say that Berkley has a separate course for majors but I doubt that’s generally the case. Unfortunately I don’t have time to check – I’ve spent far too much time on this today already. Gotta get some stuff done.

        • I’m surprised to hear that. But the issue is this statement

          “Science for non majors” courses should be eliminated.

          Which denounces the very concept of having a separate course for majors and non-majors. The fact is that sometimes having no separate course for majors and non-majors is completely unworkable. In the extreme case, it can be a literal logistic impossibility, since being a biology major taking the introductory chemistry course would require you to be in two places at once.

        • They had honors/majors courses for calculus at my school but they didn’t teach them because there weren’t enough math majors. The math majors split off from the engineers and scientists at approximately 2nd semester of 2nd year where they taught a very different version of linear algebra and a somewhat different version of differential equations. After that, math majors continued with stuff most engineers and scientists never got to see, like combinatorics and probability, topology, set theory and formal logic, numerical methods in scientific computing, analysis (calculus with proofs), history of mathematical concepts, optimization theory, or whatever.

          Berkeley is maybe in a position where they get enough math majors to have a full separated sequence.

          I think the two of you are actually not really arguing just have different frames of reference. The course “chipmunk” took would have been considered “math for non-majors in the sciences” at Berkeley, and was considered “math for majors and scientists” at chipmunk’s school. This comes down to something about the demand and resources at those schools.

          Personally, I think even within math, there are kind of two tracks. There’s math for people who like math but don’t have any interest in applications… these people in high school are often doing stuff like math field day or reading about puzzles and recreational mathematics and then there’s math for people who want to do applied math, and maybe those people are programming computers to simulate physics problems or stuff like that. I was kind of on-the-fence on this and had a bit of a revelation about it a couple years after graduating from my math major. I like applied math more than I like fully abstract math. I’ve mostly stayed away from stuff like Category Theory for example, though it’s fine for other people with different interests I guess.

          The point is, even within the math major there’s “math for math majors who like puzzles” and “math for math majors who like applications” so it’s not even possible to say what the course sequence is like unless you know something about the composition of the school’s students.

        • Oh and I forgot to mention there’s often “math for people who want to be high school math teachers” and they take even a DIFFERENT set of courses… sometimes with more classical content, like plane/euclidean geometry and courses in pedagogy.

    • Raghu:

      The horrible shock that the real science world requires difficult math is also due to the Cartoon Factor: too many episodes of Magic School Bus and Scooby Doo, where complicated problems are solved in 30 min episodes by people with no science background or knowledge – and sometimes even thanked by scientists for figuring it out!!! :)))) It’s a lonely 8-year-old’s fantasy, which, interestingly enough, is reminiscent of a crusader we all know.

  11. I think this question is geared towards high school education, but I have had similar experiences in college.

    Physics majors end up learning math along the way, either because (i) students might not yet have taken the required Math course (diff eq is often taken concurrently with intermediate quantum mechanics); (ii) students might be rusty (“anyone know an easy solution to the wave equation?”); (iii) the math is not on the pre-req sequence (general relativity, contour integration for your first particle physics class). I’m sure this approach bothers mathematicians to no end since a lot of important rigor is ignored, but at least the applications are immediately obvious.

    I think there’s a similar trend in writing courses. It used to be that you studied Shakespeare or whatever in the English department and wrote essays and that was how you learned “how to write”. Now you can satisfy writing requirements via courses in your subject, so in physics you could be writing lab reports instead of another essay on Iago.

  12. Thornett quotes E.O. Wilson to back up his claims, but Wilson is writing about different interests and aptitudes, and how all can contribute. He is not arguing against having scientists-in-training learn about differential equations. Wilson’s most scientifically influential book, The Theory of Island Biogeography, written with Robert MacArthur, is chock full of differential equations. MacArthur was the engine behind most of the mathematics, but Wilson (and their many readers) had to be able to follow the argument. MacArthur, like Wilson, was also a keen field naturalist, and his detailed studies of the feeding habits of a group of closely related birds in New England earned the birds the sobriquet of “MacArthur’s warblers”.

  13. The couch problem (well, a slightly different one: what are the maximum dimensions of a couch that can turn a given corner with certain hall widths, and so on) is actually an unsolved problem in mathematics. Might be too challenging for high school students, but I guess you never know.

    • Adrian:

      Agreed, it would not make sense to pose an unsolved problem for a high school class, except to note that it is unsolved, which is interesting in itself, given that usual practice in school is to present only solved problems. There are easier versions of the general “couch problem” that are interesting and have applied relevance to real couches.

  14. > how do you manipulate the couch to get it from the hall through the door into the room, or give a proof that it can’t be done

    And while you’re at it, give a mathematical description of how movers were once able to get a couch into my apartment when I moved in, but not able to get it out when I moved out.

  15. I have objections to two of the statements made in that opinion piece.

    a) The claim that certain classes (be it calculus or organic chemistry) are dream killers is offensive to students who excel in those classes as well as teachers of those materials. This attitude seems to only infect critiques of academia – what about sports and entertainment careers? Not throwing a fastball hard enough has killed many MLB dreams, and so we should not teach how to throw fastballs?

    b) Their argument perpetuates a myth that “softer” science like economics and social sciences are easier than “harder” science requiring calculus, etc. Actually, reading financial reports is not easy, especially if one wants to pick out the creative tricks used to fool investors. Climate science is very complex and hard to understand, requiring, for one thing, deep understanding of how data are collected and measured. These materials are more relevant to daily living but they are not “easy”.

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