Another teacher asked me an interesting question today. "What do you think about this idea of replacing calculus with statistics. After all statistics is more useful."
That's probably true on the surface, but I can't say I'm a huge fan of the "replace calc with stats" craze... even as I lament the poor understanding many people have of stats. It's obvious we need more education on interpreting datasets and visualizations.
But, there is a subtext to this argument I dislike. 1/
The subtext is that statistics is somehow "easier" than calculus since it's more connected to the "real world" --
But, nothing about real world problems is easy. Real world problems are harder, require more experience, more analysis, are less susceptible to canned techniques and strategies.
And more importantly it's hard to really understand much about statistics without knowing some calculus. Not a popular opinion. I know. 2/
But there is an even more fundamental danger in trying to quickly pull away from calculus to some other math subject as the "capstone" of most programs. I don't think many of the people considering the change understand the magnitude of what they are trying to do.
Calculus has been refined and developed as subject for math education for decades and decades. From theory to problem sets huge sets of material have been refined by thousands of people-- stats isn't so well developed. 3/
If the argument is that stats ought to be that well developed I agree.
I also think that working with data sets, and reading data presentations should be a bigger part of social sciences education, and science education.
But it's not like there is a fully formed educational version of statistics that exists that could be dropped in to replace all the things that calculus is trying to do... at the moment.
@futurebird Well put. I also wonder if there's room for linear algebra. I don't think linear algebra is any "harder" than calculus, and it seems to gain more relevance all the time in our digital world. For me at least, linear algebra introduced an element of "beauty" to math which I hadn't appreciated before. It's abstract in a way that taught me, at least, why abstractions are important: because they help you generalize, for example from 3 dimensions to 100 dimensions.
@futurebird I'm puzzled by the claim that there isn't "a fully formed educational version of statistics." It's confusing to me because it seems like statistics is already offered at the high school level as an elective in some cases and regularly offered at the college level, with some form often being required for biological and social science students. So it seems like the educational methodology should be reasonably well developed. But, admittedly, you would know better than I.
I have advocated for adding stats to the base curriculum because I feel like it has become critical to exist in our society. Feels like even a weather forecast requires a little understanding of stats. Much less things like elections.
But having a well thought out curriculum is key. And I don't think stats should ever be a capstone for math students. Everyone needs it.
Do you think there is a way to fit "stats literacy" in without replacing another course?
When I graduated high school, with the top mark in pre-calc math, none of the following had been taught to me:
mean vs median vs mode
distributions (normal/skew)
basic probabilities or inclusion/exclusion
confidence intervals
Less than 50% of adult Americans have done Post-secondary. When you use the word "average", most adults don't know if you're talking about the median or the mean. In fact, many of them have never learned that both things exist. 1/
@futurebird to me, the bigger problem is that people graduate with a degree in statistics, but believe that statistical significance is sufficient to prove a result, and the concept called effect size (or clinical significance) is entirely left out.
@futurebird it feels like a false dichotomy. Why can't kids learn calculus and statistics. My high school had a tiny bit of probability mixed in with the "algebra 2" class. Why don't we formalize how we teach stats and do a better job of including it in the curriculum, instead of teaching less of something else.
@futurebird I really don't like the framing of "replacement" or treating stats as somehow "easier" at all.
Like I've seen people who have masters degrees in other disciplines trip over themselves with even basic probability and stats.
I'd love to see greater literacy here (I could see an entire class based around Utts's "What Educated Citizens Should Know" paper https://ics.uci.edu/~jutts/AmerStat2003.pdf )… but it isn't a capstone, a replacement for calc, etc… and as you point out we don't have the buildup.
I'm going with stats. After considering since seeing your post (yesterday?)
In my zillion years ago experience, I discovered that calculus was really fun (as was physics, dang it my dad was right and I ignored his advice quite a while for reasons you don't want to hear).
But I think some level of statistics should be in the scholastic pathway of citizens. (Also civics.)
High school should prepare all kids, not just those going to college.
@futurebird this is the best argument in favor of keeping calc over stats that I've heard. I never took a stats class in HS, nor did I take an applied stats class in college (with T-tests and such). I took calc-based stats and mathematical modeling, which dealt with lots of stats fundamentals, but I imagine it's much more advanced than HS stats.
My HS calc curriculum was pretty bad (as were my HS trig and pre-calc curricula), so I never considered that stats would be worse.
@futurebird I think a lot could be gained already if there was more interest in evaluating the learning curves of different calc intros in practice to actually pick the better fitting ones for specific cohorts. Also, emphasizing connections to other fields more would provide motivation for everyone who’s not very interested in the pure puzzles calc itself provides. An integrated calc/physics or calc/stats intro also seems like worth trying, if one takes the time needed for both.
(Also I don't know if all teens are developmentally ready to learn calculus... especially if they aren't naturally "in to" math. By the time a person is in their early 20s they probably have the executive function skills to manage the subject ... but it's unreasonable to think every teen will be ready and silly. )
@futurebird@andyhilmer calculus was a class in my high school. i wasn't in the right stream for it, and when it finally was taught, i did not understand it very well. i kind of understand what it's for, but like trig, i found the mechanics confusing.
@futurebird i went to a very … not the best school in what was the 49th in the nation at the time for public schools (arizona), and when i got to college i went into computer engineering because i always had an inclination towards computers. this unfortunately did not extend to mathematics.
@futurebird@andyhilmer Also, it would be better if we started the algebra & geometry concepts in elementary school, when kids' brains are more active & less alienated. (When I taught 1st grade we had a framework that did just that but of course they dropped it before teachers got a chance to learn it) That would help them be ready for calc earlier. Instead we keep giving them harder & harder arithmetic & ill-conceived word problems.
@freemo@futurebird@andyhilmer I think that "banned" is the wrong term to uses in most school districts. They simply lack the capacity to teach advanced math, including Calculus, even if they thought it was important and wanted to do so. Starting with early grades, they think of math as a series of drills to be memorized. Who would teach the needed abstract reasoning concepts?
It is the lefty liberal districts, that do have some math capable teachers who do actually ban access to higher level math. They do so because they are worried about "gaps", the early sorting of students into high and low math tracks, a practice that disproportionately disadvantaged Black and Latino students and made it harder for them to access advanced courses down the line. If you prohibit Algebra until 9th grade, some students are either getting outside supplementation, or they are being slowed down. Opportunities to burst forward are also not accessible. https://www.edweek.org/teaching-learning/san-francisco-insisted-on-algebra-in-9th-grade-did-it-improve-equity/2023/03
Of course the issue isn't just school sorting, it is social and educational privilege that enables some families to provide their offspring with enriched environments, starting at the preschool level.
It is hard to create catch up opportunities in public schools, especially when so many elementary school teachers have week abstract learning/math skills themselves.
And we are a far cry from being a society where most adults understand at least the basic concepts of rate of change or approaching a limit. People don't understand that Calculus might actually have some relevance.
It's not really banned. It's just not a course they can take. The pre requisites aren't offered they don't have the right teachers or small enough classes to do it. If we wanted to every high school could offer it but they just don't because we don't want to invest in it. The end.
@futurebird@grumpasaurus@andyhilmer Can confirm. If your kid wants to get into a top school—even a top state school—calculus in HS is a must.
Lucky for our daughter I was a math major in college and could tutor her through calculus in her senior year. Our son is now a senior—same deal, although he’s more gifted in math than his sister.
@futurebird This is an excellent "professionals study logistics" point that I just would not have considered in the light of educational materials, thank you. It feels like a point that should be extended to any mature process/documentation pairing in some general way.
@futurebird On the other hand, is there a risk no one ever learns stats? I know I haven’t, not really. No stats in high school. Only a little exposure in undergrad where it was needed for quantum or thermodynamics. Nothing in my masters, despite it being a math degree. And very little in my phd, despite large parts of my field being built upon statistics (the parts I mostly avoided).
For what it’s worth, Calculus could not have been avoided in this way.
@futurebird
Here's a stats story for you. When much younger I ran a computer system for a university research group. I also participated as a mathematician.
A programmer supporting a postdoc came and asked me if I knew of a method to get a figure of merit for a correlation coefficient. I found it in my MBA quantitative methods book, and gave it to him.
He came back later and returned the book. The postdoc didn't want to use it as a reference because it was a business book.
@futurebird I took my first semesters of General Physics and Calculus at the same time. I think that the synergy made both subjects easier to understand.
@futurebird I'm sympathetic to the idea that statistics is a more practically useful thing to teach than calculus (for people not intending to go into STEM), but I would never claim that it's easier.
While I realize that calculus is needed to explain some concepts in probability and statistics, it seems like you could still teach a lot without needing it. For example, I have taught algebra-based physics, and while it limits what you can cover and prevents you from proving some things, it can still convey a lot of understanding of the topic. And it seems like calculus is more central to physics than to statistics. Moreover, couldn't one argue that concepts like measure theory or Stieltjes integration are needed to really understand probability? Yet I don't think most students taking statistics understand those.
@futurebird didn't know this is a craze but I often feel like the thing I miss in my work as a software dev is statistics. I don't need any real math skills but doing any type of analysis tend to be stats heavy and pretty naive without a robust stats knowledge
in what sense is calculus required for understanding statistics? Isn't the core concept probability?
probability is massively important in daily life and poorly understood (eg the lottery, "hundred year storms", risk pooling, etc.)
What's the analogue for calculus?
you think that simple, engaging concepts for teaching calculus are common? Hoo boy, not my take (and I was an A student in math and calculus.)
@taoish@futurebird Continuous probability distributions (like "bell curves") are governed by the same rules as other continuous functions: calculus.
Just to stick with that example, the equation for a "bell curve" is of the form e^(-x^2). What's "e"? It's the number which is the solution to the equation d(e^x)/dx = e^x for all x. As soon as you're dealing with "e" you need calculus!
And lots of other equations about continuous distributions are derived using calculus.
@futurebird
I think one motivating factor for disciplines outside of math is that we have almost entirely taken calc out of the curriculum but retained basic stats in labs. So it is hard to explain/justify why students need calc, when they don’t even need it for the classes we teach. But we can pre-req and justify stats because they will run some t-tests or regressions for their lab projects.
@futurebird I would be for swapping stats for calc if you also had to take classes on rhetoric and then swear a blood oath not to use anything you learned and to take up fishing or somesuch.
I've seen business majors do things with statistics... thinking they knew it well since they took the "Statistics for Business" class (No calc required!) that made me blush and also become paranoid about reading contracts.
They would use all kinds of distributions, and "methods" and it was just... vibes based.
It's not the students as much as the typical Business program heads, who want their students to have math, but also don't want them to have hard math class or give math enough space. So you end up with "X for Y majors" courses which are always questionable.
My favorite students when I was at a community college for a time were the nursing majors.
Yeah, I think my prof was more upset about the truncated curriculum than the actual students.
I do also remember taking "Math for Engineers" through first year, and noticing a big difference in second year when Eng Phys streamed out into the math proper courses with the math students. Very different relationship to the subject.
@futurebird@CedarTea@ciggysmokebringer I thought my physics degree could do with more statistics, and then thought it was really strange that almost all of the statistics teaching was embedded in major-track courses in biology, public health, public policy, and so on. There was only one pure statistics course that wasn’t for math majors! I know there are reasons why those departments do it that way, but I don’t like that distribution at all
I know... but nobody wants to listen to me explain it all.
You can prove things about random variables and data sets that will always be true for all time and forever.
It's just as solid as any other math. All the structure is there... and most stats classes just give recipes and rules of thumb that work "most of the time" and that's OK... but it also makes people think it's not real math.
@futurebird@barrygoldman1@ciggysmokebringer
I feel like it's a more fundamental thing than that. Like as soon as you say "chance" or "probability", human overgeneralization kicks in and people start thinking that everything connected to this thing must be completely up in the air. Into stats classes may very well be taught using heuristics, but also I feel like it's an uphill battle from the start.
@futurebird@barrygoldman1@ciggysmokebringer Frequentist statistics uses math, but it’s a different way of knowing. The difference is most visible in phrases like “confidence interval”. Bayesian stastics might be math; it’s certainly moving that direction, but I don’t know if it’s there yet.
My old, beaten up Applied Multivariate textbook is sobbing in the corner right now.
/What/ recipes?
Yes, it is often mistaught that way, but it doesn't have to be, and there are a ton of good books and classes in the field, it is just that doing it that way is often much more difficult than simply "use this formula in this situation" approaches, and requires more foundation.
@futurebird Hot Take on math subjects: stop breaking it into subjects that we do now - instead work all of math in a basically western historical sequence of Greek/Roman - Muslim Caliphate - Renaissance - Modern while blending Stats, Algebra, Geometry, Trigonometry, Calculus (possibly accounting) and showing the dependencies of each as math evolves.
i think that the focus on "real world", both in pre-college and college is something we'll all regret. whether it's a deliberate attempt by the far right to dumb down the general population or not, failing to educate everyone in broad areas that foster critical thinking is vital. sure, teaching stats would be a useful adjunct, but the real underlying focus should be in teaching kids to be critical thinkers, people who will embrace learning their whole lives; not just "be useful to the economy" or "get a job".
@futurebird so in a small midwestern town of around 2000 people in the late 80s, my high school did algebra I in 9th grade, geometry/trig in 10th, and algebra II in 11th grade. after that, math was optional for senior year -- which is when they taught calculus, if you opted to take it. i did not.
@futurebird hmm...when I was in highschool we DID do stats before calculus...although I think it may have been a split course with only one semester of stats...most students did not get to calculus unless they were in the higher track for math. Not entirely sure if we even had a calculus class that wasn't AP...I think everyone else may have stopped at "pre-calculus and discrete math"...that was about twenty years ago in Pennsylvania.
@futurebird Do people think statistics is just the pretty charts media pushes out? I found the underlaying mathematics of statistics just as rigorous as calculus. IIRC, there was calculus there also.
@futurebird
I watched Stand and Deliver too many times as a kid. It's hard for me to separate my relentless pursuit of calculus in high school and college from wanting to prove to the Mr. Escalante in my head that I had ganas.
I feel like "more useful" should be heavily qualified - more useful to whom, exactly? I would imagine engineers and architects need calculus far more than they do stats.
@futurebird
Trying to teach stats before calculus is probably a non-starter. Kids should be exposed to stats earlier, sure, but that's realistically more of a comp-sci topic for 90% of applications, and you can't teach stats-as-math without calculus.
In any case trig is the worse gate-keeping subject imo. How tf are we still using the classic names for "sin cos tan" etc instead of something usefully descriptive? Most kids abandon math long before they even attempt calc.
@futurebird@tlariv@australopithecus the most recent place I found a slide rule was -- a quilting shop! It was a circular slide rule, for computing sizes after scaling.
@futurebird
100%, but also really trig should be taught with vectors. ”r•sin(θ) gives the y component, r•cos(θ) gives the x component" is much easier to grasp and appreciate than "here are some weird transformations you can do to triangles for no reason."
Plus trig is really about circles anyway, not triangles at all, and it's way too easy/common for teachers to know just enough trig to make it harder.
@futurebird@australopithecus
I loved trig, especially all the relationships between circles and the curves of trigonometric functions. And there's so much of the world around us that relates to trig, ranging from simple things like how the pitch of a roof affects the difficulty of keeping your footing on the roof, to rainbows, sound, computer graphics, and cyclic things. I can understand skipping it if the resources are insufficient, but without it so many things become inaccessible.
@futurebird@NYSloth At the risk of being 'that guy' who Kool-Aid-Mans into a convo to answer a (likely?) rhetorical ?: the artificial scarcity in the US educ system, created by generations of underfunding, which is caused by a vocal minority which doesn't want any public resources or goods to accrue to the benefit of people who aren't white?
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