On replacing calculus with statistics

Russ Roberts had this to say about the proposal to replacing the calculus requirement with statistics for students.

Statistics is in many ways much more useful for most students than calculus. The problem is, to teach it well is extraordinarily difficult. It’s very easy to teach a horrible statistics class where you spit back the definitions of mean and median. But you become dangerous because you think you know something about data when in fact it’s kind of subtle.

A little knowledge is a dangerous thing, more so for statistics than calculus.

This reminds me of a quote by Stephen Senn:

Statistics: A subject which most statisticians find difficult but in which nearly all physicians are expert.

Related: Elementary statistics book recommendation

 

 

14 thoughts on “On replacing calculus with statistics

  1. I remember reading about a study in which students were pre-tested before taking a stats course, and the post-tested afterwards. Maybe you have a reference? Reliably, what happened was that students were sure they learned something, but the post-test showed that they scored far worse after taking the course than before taking the course when they thought they knew nothing. Very dismaying.

  2. It sounds plausible. There are certain wrong ideas about statistics that require training to instill.

    Maybe there should be an educational term analogous to the medical term “iatrogenic harm.”

  3. I’ve never taken a statistics class, but I’ve read a lot of books on the subject. And while I learned how to do various tests — t, X^2, etc. — I don’t think I really understood what I was doing until I was able to simulate random data (beginning with one of the early spreadsheet programs). So, I think it would be possible to get students to understand statistics intuitively if the class involved a good deal of experimentation.

  4. I agree that a large simulation component would help.

    Another problem I have with “statistics instead of calculus” is that it’s hard for me to imagine understanding statistics at any depth if you don’t know calculus. And not just being able to do do calculus homework problems but having a good conceptual understanding of calculus.

  5. I think its a great idea. And I don’t think you need calculus to understand what would be useful in a high school stats class. I think what would be really useful is for kids to have a better intuitive understanding of probability and expectation. We also don’t start teaching variability early enough. My daughter’s middle school science fair was all about averaging trials with no mention at all of quantifying variability. Lets just teach a class in understanding variability.

  6. Why don’t we make both Stats and Calculus available? I would agree simulation would make Stats more intuitive. ProbabilityManagement.org is trying to do this through the use of Excel. Dr. Sam Savage’s team has put together something for the middle and high school kids.

  7. Peter Thiel gave a presentation at SXSW a few years ago where he describes people’s opinion of the future as either optimistic/pessimistic and either determined/indeterminate.

    One of his examples of a shift from a determined to indeterminate future was rise in importance of statistics vs. calculus. So, the importance of probability vs. arriving at a knowable answer.

  8. In high school, I found statistics (well, probability theory actually) boring and opaque. Examples in textbooks often involved coin flipping, heights and playing cards. I found them exceedingly artificial and divorced from real life.

    I came back to statistics later in life (after taking lots of graduate-level math courses) and it finally dawned on me why statistics was useful.

    As an engineer, I’m used to seeing models of the form y = f(x) (where x, y are multivariate and f is some arbitrarily complicated, often nonlinear/non-smooth, mapping).

    That’s fine and well for deterministic values of x, except x is often fuzzy in real life. Wouldn’t it then be useful to know how y fuzzes (behaves) given that x is fuzzy? To me, that is one of the central questions that statistics is able to answer, because based on the variability in the inputs, I can robustify the system against variability in the output. But many instructors gloss over this point.

    In many stats courses, so much time is spent characterizing the random variable x and not f(x). I think if that connection was made early on and repeatedly, all the surrounding theory (e.g. Jensen’s inequality, higher moments, etc.) would suddenly acquire a level of practicality not often seen in stats instruction.

    I believe NN Taleb makes a similar point elsewhere too: focus on the f(x) (exposure/effect), not the x.

    p.s. that said, for most non-trivial models, f(x) is not something that is easily worked out by hand. Simulations can definitely help in this regard.

  9. It’s interesting that you say you need to understand Calculus to understand statistics. A common position of the “Down with Algebra!” crowd is that we should teach more statistics and less “Algebra” in high school. They apparently believe you don’t need to know much algebra to understand statistics, which seems absurd to me.

    Lots of students take both courses at our high school, and we are fortunate to have good instructors for both. But at least for technically-minded students, I would never suggest stats as a course to take instead of Calculus.

  10. I would rather see high schools teach probability than statistics.

    Probability is less subtle than statistics, so it’s easier to teach and easier to understand. And it’s a prerequisite for statistics.

  11. I’d like to see probability and statistics taught in place of trigonometry and the very basics of trigonometry taught as part of first year physics. I’d also like every kid take one programming class as a freshman or sophomore and then when functions and matrices are discussed in Algebra 2 kids would be writing programs to make theses ideas more concrete. When prob and stats comes up the next year kids could be writing up simple simulations. There’s no way I’d replace calculus though, it’s too important and too much fun and there are much better choices of things to replace.

  12. Stats first, hands down.

    The “scientific process” made little practical sense to me in college (in terms of what I was learning and doing) until I took a physics lab that introduced me to the imprecision of the real world; how to detect, measure and deal with its effects and sources.

    This wasn’t just about taking data and applying canned analyses, but more importantly about using the results to improve the data acquisition process and to work with and extend the capabilities of the test apparatus.

    Our introduction to stats wasn’t to get the answer: It was to see if the answer was first relevant, and then useful. If an analysis indicated possible changes to the experiment, and the changes failed to improve the results, we typically blamed our experiment design, technique and analysis, then tried to develop new results and insights before iterating again.

    We started with a simple goal: Measure the local acceleration due to gravity with as little error as possible. We used approaches roughly in the order they appeared in history, starting with inclined planes and using our pulse (or other human-based perceptions of time passing) as a timer. By the end of the class we were using more sophisticated lab equipment: Rubidium timers, photo sensors, and precision triggers.

    This course was carefully interwoven with freshman and sophomore math and physics courses, but also had a class element of its own, the cornerstone of which was a basic engineering statistics text (whose name I forget: It was a half-inch thick paperback with the cover showing a French train that had plowed through the wall of an elevated station).

    While the course was enlightening (and a ton of fun), by the end I realized that 90% of the technique and knowledge that was taught was easily within the scope of high school physics and math curricula.

    What I was left with was a transformed vision of my relationship to the real world as both an observer, experimentalist and engineer. Even the most elementary statistics are of immense value when assessing the correctness of a control system, diagnosing its faults, predicting its failure modes, and ensuring its reliability.

    Quantum physics tells us there is nothing but statistics. Even the particles themselves are statistical fluctuations in their associated fields. While this becomes locally Newtonian at larger scales, the interactions quickly become too complex to handle, with thermodynamics being the first predominantly statistical area of science a student will typically encounter (starting with the definition of temperature itself).

    But neither quantum stats nor detailed thermodynamic stats are approachable in the high school context. But experimental statistics are immensely approachable, especially when combined with even a hand-waving introduction to Design of Experiments (DoE).

    Stats (with a pinch of DoE) helps turn students into critical observers and creative experimentalists. It directly involves the observer in the process, immersing the student in the true core of the scientific approach of learning about the universe we inhabit.

    The most valuable insight, IMHO, is that stats aid critical self-evaluation. Early in the lab class I realized that others in my lab group routinely obtained better data than I did usingthe same equipment (they were better experimentalists), but my results often yielded more useful insights (I was a better analyst). This spurred me to improve my experimental technique by observing others, and to share my thought processes during my analyses. Within weeks, our 4-member lab group was routinely obtaining the best results in the 200 person class (by the end of the course, an order of magnitude better). Yet not one in our group was exceptional in any way (especially from a GPA perspective).

    Done right, the stats don’t lie. Done wrong, there are no worse lies than bad stats. It’s not about “finding the numbers”: It’s about getting the right numbers the right way.

    In my experience, some of the worst published stats I’ve seen were in medical studies. Some physicians use cookie-cutter analyses without first validating their applicability. The most remarkable thing is that even a one semester high school stats course can impart enough skill to detect (or at least raise questions about) the most glaring of such errors.

    Being able to conduct such an analysis is tremendously empowering. A little bit of usefully applied stats knowledge can go a long way.

    Physics is merely one context within which stats may be learned and effectively applied. Once the fundamentals are in place, the applications explode wherever real-world data is to be found.

    Whenever I see an interesting or unusual claim in the popular press concerning conclusions made from data, I enjoy putting on my “stats goggles” and taking a closer look.

    I especially take deep pleasure in confounding the strongly-held “evidence-based” political convictions of my liberal and conservative friends alike, using only simple reasoning.

    Bottom line, when the stats are inconclusive, it prompts us to utter a difficult statement: “I don’t know.” As any guru will tell you, that statement is the beginning of all learning, knowledge and wisdom.

  13. I’m not familiar with any proposals to replace a calculus requirement with a statistics requirement, but I note immediately that this proposal seems to be about requirements, not about math.

    I _am_ familiar with Art Benjamin’s TED talk from a few years back, wherein he suggests that a curriculum that is designed to prepare students to eventually master calculus is much less useful FOR MOST STUDENTS than a curriculum designed to prepare them to eventually master statistics. A big part of that is that most of these students will never reach the end of that road — they will not master either calculus or statistics. His proposal has nothing to do with what future mathematicians or engineers or physicians or economists should be taught, and everything to do with what future journalists and elementary school teachers and plumbers and car salesmen and entrepreneurs (etc. etc.) should be taught.

    Also, as John suggests above, the road to statistics starts with learning about probability. If that’s all you get out of it, you have learned things that will be much more useful in your future life than if you start on the road to calculus and only get as far as high school algebra and trig.

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