For an example, the replication crisis in the social sciences. These studies, containing flawed statistics, are carried out by social scientists with PhDs who have been required to take courses in statistics designed for their discipline.
If they can’t get the statistics right, how can we expect it from teachers with far less education?
Because it’s their job—and if the statistics curriculum is mandatory, then the teachers might spend inservice days developing statistics curriculum, going to workshops to learn how to teach statistics, etc. Teachers will develop statistics curriculums and share them with each other. With NSF grants, you can fund teacher outreach programs, put statistics exhibits in science museums, etc.
A lot of teachers will struggle to teach various subjects. That’s why we have support networks in place, to help develop curriculums and provide training for teachers. I know that the support network has a lot of problems—but many teachers would struggle to teach, say, biology, history, or algebra, too, without support.
Will they teach discrete and continuous probability distributions? The binomial, Poisson, and normal distributions? Dependent and independent random variables? Bayes' Theorem? Measures of statistical significance and hypothesis testing? Chi-squared and student-t distributions? Confidence intervals and p-values? Maximum likelihood estimation?
Highly doubtful, since many of those topics are built on top of university-level calculus.
Most problems I see with moderns statistics aren’t of the form “ohhh, they fooled you by using a subtly wrong statistical metric to ascribe significance” but “the way the data was gathered/interpreted is fundamentally wrong and made to mislead”
Many midlevel statistical practitioners suffer from a holier than thou complex, where a “correct” approach to statistical analysis might buy a little more precision at the expense of a lot of comprehension.
Box plots or Bar charts with error bars, using randomized data collection. That’s like 90% of the interpretive value right there. Statistics is a UI for math and it could use improvement if we expect so much from it.
See Brett Victor’s “Kill Math” for more context on why we should expect more from our mathematical interfaces. http://worrydream.com/KillMath/
Form that Brett Victor.
Wow
Now I've read / skimmed all of it and it was interesting, I hope the new methods he wants to use for teaching maths will work fine. (I'm sceptical, but still seems like worth to give it a try.)
I think his project title does his project a disservice: "Kill maths"? That sounds silly to me. And how it starts -- I got annoyed and stopped reading (until I went back two days later).
Another more positive project title maybe could be "Maths for everyone" or "A new approach for teaching maths"?
The problem is that those things are not "immediately" needed, so students don't learn them or immediately forget them if they do. What students "immediately need" is to run some test in some application and check if some value passes some magical threshold.
These students then become researchers and these researchers become professors.
I'd go as far as to argue that has already happened. The reason math doesn't really teach problem solving and instead opts for working through things with formulas etc is precisely because that's a dumbed down version of math.
Not all topics are equally easy to teach and assess in a deep way, with limited time and resources.
Statistics is IMO a lot like security. Unless it's at a very high level, just follow a basic check list and don't do anything creative. Calculus is more like algorithms - you can get to deep and creative levels at an earlier stage.
The class had a calc 3 prereq I found the computation generally the easiest part. Truly grasping the topic takes patience and work but it's pretty rewarding. That said, it must be difficult to find genuinely insightful instructors who can make the material remotely interesting because good god it's necessary.
In any case, we did cover multivariable as a pretty straightforward extension of single-variable calculus, without making it a separate course. Do I likely have some huge blindspot as a result of not spending a full course on multivariable calc?
(All of my formal education was in the U.S., for what that's worth. Though, it was an accelerated magnet program teaching middle school students algebra and trigonometry and covering geometry, calculus, linear algebra, and differential equations in high school.)
The main thing you lose then is you don't know how to apply calculus on non linear coordinates like spherical coordinates and so on. It is useful for data analysis if your data is easier to work with after a non linear transformation, but if you don't work with that sort of thing then probably not very useful.
We did integrals in cylindrical and spherical coordinates in the integral and differential calculus course.
So, I guess the standard multivariable calculus got smeared across my first calculus course and my linear algebra course.
By the time the curriculum gets reformed, it'll be like 2025. By which time students can ask GPT5 for any statistics question.
Statistics is so unbelievably broadly useful at low levels, compared to calculus. Understanding 1. Selection bias 2. Normal distribution + standard deviation 3. Central limit theorom Is a massive help in modern society. You don't even need any math equations, just understanding them on a rough conceptual level would help.
If we encourage them to let AI do their thinking for them, there's much less need to worry about wrongthink...
Will students get unlimited free access to GPT5?
> If they can’t get the statistics right, how can we expect it from teachers with far less education?
This isn’t really a statistics education issue, much like Enron wasn’t an accounting education issue.
Social scientists abuse statistics because the incentives are all aligned with abusing statistics. To get a good job you have to publish lots of papers. To publish you have to have statistically significant results. To get into top journals you need surprising results. You see other people in your field playing fast and loose with their data and getting rewarded for it. Why not exclude that one problematic subject from your data analysis? Without them the p-value (from a regression on a carefully chosen eight of your eleven measured variables) drops to .038, and you can publish...
It’s not that nobody thought to teach the scientists about corrections for multiple comparisons or the dangers of picking observations to exclude as outliers based on what gives you the result you want. They learned all those things. But it’s so much harder to get results when you’re not willing to play games with the data, and they need results. The people who do try to play by the rules wash out, either by choice after getting fed up with the fraud they see all around them or by failing to publish N papers in top journals.
They're both wrong but I'm not sure they're so comparable
Poor replicability of social sciences is multifactorial. To some extent, what they're studying is a moving target that is not all that amenable to scientific methods, which tend to assume static reductionist laws dictating system behavior. The dynamics of how fundamental forces dictate everything from gravity to covalent molecular bonding don't change from culture to culture as well as over time as trends. When you're studying human behaviors and preferences, I'm sure some of it is governed by more or less immutable eternal laws, but some of it is semi-random diffusion of learned trends and what is true today of one group of people may not be true of any other group or even the same group at some later point in time.
Some of it is statistical illiteracy and not understanding the limitations of the techniques you're applying.
Some of it is outright fraud.
There is also interplay between those two because outright fraud is facilitated by peer reviewers not having the statistical maturity to be able to detect it.
Just to be clear, nobody actually said the social sciences are roughly equivalent to Enron. I used Enron as a hyperbolic example to emphasize that there are reasons for poor behavior other than lack of education. I was not implying that academic fraud is at the same level as Enron fraud, and after rereading what I wrote I’m comfortable with how I worded it.
> But every energy company had the same incentives as Enron. Not every energy company published fraudulent financial statements.
Accountants and corporate executives have substantial disincentives against publishing fraudulent financial statements, like going to jail. Academics mostly do not have similar disincentives against abusing statistics. There have been a few high profile embarrassments, but for the most part even people who are widely known to have p-hacked their way to dozens of questionable publications are still sitting comfortably in their tenured professorships.
While there are definitely still papers being published as a result of p-hacking (intentionally or not), psychology has been undergoing a renaissance in this area for a few years now. See the "Reproducibility Project: Psychology": https://osf.io/ezcuj/wiki/home/
In short, there is a movement to validate past results, starting with those most influential in the field. A lot of progress has been made in that area.
Other fields are involved, as well: https://osf.io/collections/rpcb/discover
Yet, society wants to pretend like we can.
Larger groups and timescales are (in practice) outside the domain of science (in the most rigorous form) and must be studied with other methods and epistemological criteria. This also leads to major abuse of statistics as the inference has to be done with unrealistic assumptions and unwieldly models.
The "brand" of science is so strong that many fields, especially economics, want to appropriate it even though they don't and can't do science in the strict definition.
It's largely a cargo cult. For example p-values are reported as t(N) = x, p < threshold but very few understand why and keep on doing it and even demanding it from others.
(The why is because the p-values had to be read from tables before computers. It makes no sense nowadays.)
So i would argue that this is more damning - essentially a misunderstanding of probability and what p values tell us at a fundamental level.
But yes, the interpretation of p-values and confidence levels are wildly misunderstood. p > alpha is often taken as "evidence of absence" of an effect, which is just wrong. Or when for some quantity p1 < alpha and other p2 > alpha, it's often intepreted that the quantities differ.
It's a mess.
Analysing data is similar. I'm all for it, it's clearly useful on the days when my main working tools are slack and powerpoint, but it's not clear to me how teach that without math. One of my math textbooks even had the word "analysis" in its title. ("Calculus and analytic geometry" perhaps?)
GCSE starts after the third year of secondary school in the UK so most people will have completed their GCSE in the year they turn 16. Then 16-18 year olds wanting to study STEM subjects go on to do maths A-level, which includes more topics that in the US would be considered precalc as well as some calculus, more probability and statistics and some mechanics. Here’s the A-level syllabus from the same board. As you can see, for stats it includes distributions and hypothesis testing, so much more depth https://www.aqa.org.uk/subjects/mathematics/as-and-a-level/m...
The ideas of standard deviation and confidence intervals can be taught visually, for instance. You needn't be able to calculate them to understand what they mean when they're presented.
The extent of my statistics education in high school was "mean, median, mode". That was it, and I exhausted every math course that was offered at my school by the time I entered 11th grade.
My (mostly) calculus classes were even called "analyse" (analysis).
If your class of 12 year old physics students has just timed a block of wood sliding down a sloped plank at different angles, and plotted an X/Y chart of their results, you can just have them put a best fit line through the points by eye.
No need for matrices or differentiation or X-transpose-X-inverse-X-transpose-y - just bang a line through the data by eye.
I'm more concerned with the populace at-large being able to understand and apply fundamental statistical concepts. For example, another commenter mentioned Bayes Theorem, and how it's a very powerful idea and not that difficult to grok. Related, predictive value positive and negative, and how they are calculated from (but also very different from) sensitivity and specificity, are extremely valuable concepts to understand.
To what you point out, I think the concept of "p-hacking" is super important to understand, but I'd be less concerned about a student needing to hand-calculate the steps to run a t-test (that's what a college-level class is for). That said, I decided to look up T-test on Wikipedia while writing this comment, and I found this interesting tidbit. Great example of the applicability of statistics, and something that I think would peak the interest of many high-schoolers:
> Gosset had been hired owing to Claude Guinness's policy of recruiting the best graduates from Oxford and Cambridge to apply biochemistry and statistics to Guinness's industrial processes.[13] Gosset devised the t-test as an economical way to monitor the quality of stout.
I’m unconvinced that the incentives pull them towards “better use proper statistical methods” or penalize them when they don’t.
This is, IMO, not insufficient education; if it’s not, the premise that math teachers given incentives to get it right couldn’t accomplish it because they have less education than a PhD sociologist is flawed.
Metascience is a growing field at many top universities because of this issue and the belief that modern science may be very flawed right now.
So we ram language constructs into people’s brains, which we started doing centuries ago to preserve knowledge.
Yet it comes with none of the warnings by the long dead mathematicians who initially built out statistical tooling[1]. Statistical tools are intentionally crude to help communicate the complexity of stats and yet we build society on crude leaky abstraction. Not out the obvious day to day right in front of our faces.
Modern science isn’t flawed. Society is. Boomers and GenX are proper gold star for nothings who lucked into living in the only country that could manufacture anything after WW2. They dismantled the New Deal that propped them up and made us their serfs.
They didn’t fight the war or build the economy. They have no muscle memory for doing anything “real”. They went to college (much less rigorous in the 50-70s), memorized the cliff notes and recited the catechisms. Neither generation struggled materially as any generation before them (yes yes distributions, ranges, gradients of truth; relative to material conditions before 1950s).
New generation comes along with no awareness of that time and doesn’t feel much obligation to status quo. GenX and Boomers are big mad about it, never mind they walked away from religious life. This time the future will stick to our script.
Society wants less to do with their hyper industrialized life due to war time manufacturing habits they had rammed down their throats by a much more imperialist society of decades gone.
We keep letting people who cheered on conquest of other countries to stay ahead of them keep running things. Everyone is too apathetic to tell grandpa it’s time for hospice. We let elders implicitly engage in ageism against youth.
Everyone is acting shocked the old warlords are looking for idiot soldiers to serve their fiefdoms? All they knew for their formative years was war time …hustle. Embedded deep.
Stop with the meta bullshit. Day to day life is just this. The abstract mental models are not helping. They’re distractions apathetic people escape into to avoid reality. Boomers down to apathetic centrists who love their material privilege despite the environmental toll… this culture is a joke
[1] Willful Ignorance: The Mismeasure of Uncertainty https://a.co/d/agWe8LT
Yes, it is. A lot of the issues found in modern science with replicable studies comes down to the publish-or-peril approach. If you have academics on temporary postdocs having to publish X papers to get an extension, find a new postdoc, or maybe get a professorship then you're going to have issues. Add to this the lack of incentives to replicate papers under this stress and people build up on top of them rather than validating them first. Especially when the studies are expensive/time-consuming with MRI machines etc.
> Stop with the meta bullshit
The impact bad research has both financially and in society is huge. For example the issue recently with Alzheimer's where loads of work was built up on a 2006 seminal study that wasn't replicable (because of academic fraud). Finding incentives to catch bad science early is important.
I have no idea what the rest of your message is about.
The rest of my post was to suggest where the flawed incentives come from; educationally outdated meat suits and apathetic voting public who think they’re off the hook to society. Elder politicians enable such perverse incentives because they care about fiat currency flow, not science. Ignore it and dig into vacuous meta theory because the public is kowtowed by threats by the seniles
It’s amazing to me how many think choices today are guaranteed to matter tomorrow. You have no idea if what you say is possible given the state changes that occur constantly reshaping global society
In the end you’re peddling high minded BS
We can’t get the world to agree on climate change. Surely we’ll keep all hackneyed science from propagating, certainly we’ll keep the costs from ballooning to serve any perverse incentives that pop up as the public lacks any real command of the political system and it’s pork spending
Sure, sure. Musk will have a full colony on Mars first
But when I got the about two-tailed tests and p-values, it was all just so opaque. I didn't feel it was that useful, plus the whole argument about rejecting or not rejecting the null hypothesis felt too philosophical and contrived. And why 95% confidence?
I hated statistics throughout undergrad because of those philosophical contortions, which seemed very arbitrary to me.
It wasn't until grad school when I discovered the statistical learning side of things (PCA/PLS) that statistics became exciting to me -- because suddenly statistics became useful and able to predict things.
I'm still convinced the 80% of people don't need to know p-values and null hypotheses. Some might say, but what! These are used all over the social sciences! Umm no.
Let the people who need to learn that learn, and actually teach statistical learning (linear regression, logistic regression, Bayesian statistics) to the rest of us in school.
Everyone should know Bayes' theorem. It's not that hard to grasp, but it's a wildly powerful idea. Even if you aren't plugging-and-chugging, just use it to dissect logical fallacies.
That is basic probability, not statistics. Basic probability is taught before high school in many parts of the world and is very easy to teach, there is no reason to put a lot of focus on this in high school.
You have the kids write down probability graphs from events, that takes a few lessons. Then you do some basics about confidence intervals and sample sizes. Basically every kid learns this at some point, just that they forget and now they think it is missing. Kids forget about what they learned about statistics since it doesn't seem relevant to them.
If anyone who learned that sees Bayes theorem later it is intuitively obvious, so there is really no need to bring it up, and there is no reason to take a course to understand Bayes theorem later as an adult. Those things are easy thanks to what you learned in middle school.
If you are teacher, you learn pretty quickly that every time you invoke magic or "trust me it works this way" you lose students. Students are clever and have self-respect and they don't like being lied to, or when information is hidden from them. You have to present a coherent picture, and I think teaching statistics without calculus is incredibly difficult to keep coherent.
[1] I am not sure if it's possible to argue without limits that the normal distribution extends to infinity, yet has a finite area under the curve.
The need for limits and such arises when you try to differentiate the exp function. But we don't need differentiation for basic statistics. Just integration.
(There is a geometric definition of exp that I know of, but it's that it turns a vector into an integral curve, so not so useful without calculus or limits)
You also need limits to be able to talk about the central limit theorem/when a normal even ought to be used. Otherwise you get confused people thinking everything is normally distributed by default.
> The usual definitions of e involve a limit.... I don't know of a definition of exp that doesn't involve a limit ...
Yes, I agree with you that if you want to define e as interesting itself, you need to use limits. Similarly, the way exp(x) was introduced to me in high school was as a function whose derivative was equal to itself (i.e. as an interesting function) - which also requires limits but I think my teacher/curriculum just handwaved away that part.
But in our hypothetical curriculum, I am indeed proposing that we just say that e = 2.718..., since we are not interested in e, but in its usage for defining continuous probability distributions. Then to compute something like e^2 you just plug it into the calculator (like you do sin/cos) and it will give the answer. But again, we will have to put in effort to argue that something like e^(5/4) or e^pi is a computable real number.
> You also need limits to be able to talk about the central limit theorem
Indeed, but I think rigorous usage of the central limit theorem is quite beyond high school mathematics.
Even Bayes' theorem is, IMO, most obvious in the continuous setting where you can interpret it in terms of relative areas, which gives a nice, easy picture. Making big tables and trees obscures the basic geometry.
And I observe that even from friends that are reasonably well educated but have done little to no statistics.
This lack of basic stats skills does lead to bad public policies.
And in the case of BLM. I am not deeply involved with following what the development of the movement but I think at it core (at least when it was formed) is because statistically speaking, you are more likely to be killed by police id you are black. If you look at US prison distribution per capita and race, you can draw a similar conclusion. Stereotypes are a cause not an effect. You get some of that in part because some people are racist and have racial stereotypes.
And it is great that you are saying everyone should be treated as a unit itself. But we all know what this is not the case when you have a significant portion of the people think of people from their lens of stereotypes., Black peoole, muslims, Asians, Jews..etc and this have real consequences on the life of this people.
Good luck telling an TSA officer that you should treat me as individual and don't pick me "randomly" because I have have the wrong color to not be suspicious.
But it’s even more mundane things. People are focusing on extreme percentiles of distributions, like % of engineers at google, or board members of major companies, ie looking at 0.001%-ish percentiles, where even minor differences in distributions may have a dramatic effect. And this leads to people reacting to this with “so you are saying that women engineers at google are less capable” (read many times on HN during the google memo controversy) which is absolutely not what those differences in distribution mean. They just mean you may see more of one group and less of another, but if the recruitment process is fair, all the people who passed the threshold are equally capable. And then you have to compare that to the distribution of people who actually apply, which has its own biases.
I think you meant more likely not "less". And even in thay case, this [1] is statistics up to before covid that shows that it is actually 3 times ratio between black and white risk of being killed by a police. This does not constitute "slightly".
> But there are vastly more interactions of black people with the police, and those are aligned with crimes committed, including crimes which stats are unlikely to be affected by policing practice (eg murders).
That would be true of any race or group of people. They have much more interactions with police other than those which will result in death.
> People are focusing on extreme percentiles of distributions, like % of engineers at google, or board members of major companies, ie looking at 0.001%-ish percentiles, where even minor differences in distributions may have a dramatic effect
Yes, and that's actually how personal stereotypes works. Your experience with people is very limited to form an opinion about race, nationality...etc. It is not even extreme percentile like top or bottom but it is a small percentage that people use to build their opinions.
> but if the recruitment process is fair, all the people who passed the threshold are equally capable
If the process involve human judgment or evaluation, then biases start to be a huge factor that prevents establishing a clear threshold. Unless you have something like a standard exam that will be taken without human interactions. You will always be working under the assumption that the process is not 100% fair. That is actually hard problem to solve especially at big companies. I don't know a solution that can give the best outcome and I don't envy people who have to do it.
[1] https://www.forbes.com/sites/joewalsh/2021/09/30/study-us-po...
Criminal statistics are a record of who is arrested and prosecuted, not a record of who commits the most crime. If a black criminal is more likely to be arrested than a white criminal, then the statistics will reflect that.
What confuses me is that this is now painted as racist and somehow Republican. I clearly remember that back in the 90s it was the Democrats, at the urging of black leaders, who supported harsher sentences and funding for more police (most prominently the Clinton crime bill [1] and three strikes laws in Washington and California [2]).
And we can see that black people still want the police to focus on their communities, as they elect leaders like Eric Adams (over the objections of white progressives), and a Gallup poll in 2020 found that 81% of black people want police to spend same amount of or more time in their area.[3]
That naturally leads to more interactions with police and more incidents when terrible things happen. Policing is always a trade off between the harms that police cause (through mistakes, misconduct, and misunderstandings) and the harms they prevent.
---
1: "the largest crime bill in the history of the United States... provided for 100,000 new police officers, $9.7 billion in funding for prisons". "Then-Senator Joe Biden of Delaware drafted the Senate version of the legislation".
https://en.wikipedia.org/wiki/Violent_Crime_Control_and_Law_...
2: "The first true "three-strikes" law was passed in 1993, when Washington voters approved Initiative 593. California passed its own in 1994, when their voters passed Proposition 184[16] by an overwhelming majority, with 72% in favor and 28% against."
https://en.wikipedia.org/wiki/Three-strikes_law
3: https://news.gallup.com/poll/316571/black-americans-police-r...
Politicians now know that "more police" did not solve crime issues. Why would a politician in 2023 suggest we repeat the mistakes of the past?
But this is a good example of the problem with statistics and the public. Fryer’s analysis was highly touted, but not a single person I talked to could explain it to me or say why his results differed from other analysis.
What I am saying is that you find a similar black over representation in crimes that are less likely to be overpoliced (homicides which stats should be fairly reliable, ie every incident accounted irrespective of the race of the author, and which if anything would be less policed in poor neighbourhoods where gang violence is more common and less effort is made to identify the author).
In fairness to Fryer (and yourself), we may not have the tools to determine this definitively either way today. And really my more important point is that as a society we should approach with caution statistical claims where we don't have a good understanding of the methods and the pros/cons of the methods. After becoming familiar with Fryer's methods in this study I'm probably leaning toward his conclusions being wrong -- but I wouldn't wager large sums of money on my leaning.
They're both. Some are a cause, others like for example "black people can't swim" are an effect of racist laws. Changing how you treat a group creates its own set of stereotypes. Then you can also get a multilayered stereotype pile from things like cake walk.
It was mind-numbing how easy it was by comparison. We got to the end of the semester and I realized we'd only just reached some of the principles from the first week of calculus. I don't remember why he was so proud to point out that we could now calculate derivatives (it was a long time ago), but I was thoroughly unimpressed.
By the second semester, I had my college acceptance letter, so I didn't bother with the other half.
At my relatively average Bay Area public high school, we offered AP Stats, AP Calc AB, AP Calc BC, and AP Physics E&M. This was the kind of HS where 10% of the student base finished BC by 10th grade and began taking Multi and Linear Algebra from our local community college or Cal.
Our AP Stats teacher integrated a lot of Calc and E&M concepts into the curriculum and stressed Probability theory quite heavily, the same way our Calc and Physics teachers dug deeper into Numerics/Interpolation than the AP Board demanded.
That said, the AP Stats test was an absolute joke.
If you don't understand the concept of "integrating a function" they how can you possibly make sense of virtually any part of statistics? 90% of practical statistics can be boiled down to understanding the basic algebra of normally distributed random variables and then doing basic calculus on the result.
Statistics without calculus is the worse kind of statistics where students are taught to blindly throw tests at a problem without having a clue as to why they are doing this. Statistics without understanding is worse that no statistics at all.
The useful parts of calculus are typically the easiest for students, and could be taught in a semester. Instead, your typical calculus track is 2 whole years.
1. Differential and Integral Calculus
2. Differential Equations
3. Complex VariablesThis is true, and not just for high school but for college. Neither are out to teach specific vocational skills - rather they provide the underlying tools that students need to learn their vocation later on.
Sure we could stream kids from grade 1 for their "allocated profession", teaching them only with regard to their eventual job, but there's a reason we don't go that pretty dystopian route.
With that in mind we can argue that one curriculum is better than another for a specific path or other. But really that's a fruitless, and pointless argument. Future STEM students need Calculus, future programmers need Logic, future artists need Color Theory.
If I had to argue for a curriculum change I'd lobby for things like budgeting and basic accounting - but that's just me and everyone will have a different opinion.
You can read the standards here, everyone learns this before high school:
https://www.thecorestandards.org/Math/Content/7/SP/
And the same topics are taught everywhere in the world, it isn't like we spend 8 years just teaching arithmetics.
I'm more talking about the high school version of statistics, and the link you gave gives good info: https://www.thecorestandards.org/Math/Content/HSS/introducti.... That curriculum is more what I'm referring to, but as far as I know is not often taught to HS students in the US.
"For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be."
This is proper statistical understanding, since it includes how sampling is made and confidence intervals. They just don't use the names of theorems, they just teach the understanding for estimating values etc. I don't see the value in drilling math equations for calculating those values and forcing the kids to remember lots of names.
For people who know they don't want to go on the STEM track by then already, the change is a net positive. However, it might also mean STEM loses a few more people from underrepresented groups as a result, who would have done well in STEM but are now locked out of that track earlier.
Complex numbers also only work in 2D.
this is no longer the debate. the debate is whether kids should have to take math classes at all in order to graduate. And based on what i'm seeing, the answer the schools came up with is - No.
A solid foundation in linear algebra with proofs (instead of the horror that is high school geometry) will prepare both routes substantially better.
That said, I think proofs are an important thing we should teach in high school - its more a flavor of "real math", and logical thinking involved is a better transferrable skill than the symbol manipulation and bag-of-tricks-you-never-really-use-in-the-real-world you get from high school calculus courses. I'd just focus it on something easier to grasp, like basic number system construction.
"the middlemost estimate expresses the vox populi, every other estimate being condemned as too low or too high by a majority of the voters"
at a livestock weight guessing contest and conceiving of a measure to quantify normal variation: the standard deviation without the use of calculus.
To be sure this evolved to expressions that included an integral sign or two .. but the foundations were founded with no more than a Sigma and some division.
There's a stronger case for those than care to make it that statistics is more dependant on linear algebra, if one takes the view that statistics is about finding fewer lower dimensional fair representations of many higher dimensional values.
It's possible that you could do something like algebra (middle school)->geometry (maybe introduce the notion of a group here and focus more on symmetry and not so much on figuring out the missing angle/length in a complicated diagram)->linear algebra->probability/statistics. Concurrent with linear algebra, have kids learn calculus in physics class. After physics 1, they can do chemistry and/or e&m. Do vector calculus in e&m. Basically trim all the useless stuff out of high school and add the first couple semesters of college instead. Offer analysis as an elective after linear algebra/physics 1, and put a proper account of the n-D derivative and things like the Newton–Raphson method there.
Obviously that's a STEM bound curriculum, but at least in my school growing up, you only needed algebra 1 and geometry to graduate, which the honors kids did in middle school. So I assume any curriculum more advanced than that is for STEM bound kids.
As it is now, you have ~3 groups of students in most K-12 math classes. A group that is bored because they have mastered the concepts being presented, a group that is benefitting from the concepts being presented and a group that doesn't have the framework to benefit from the concepts being presented.
There are of course lots of teachers that will be doing what they can to address the gaps, but it needs to be systematic.
If we taught statistics the way we teach mathematics, the entire curriculum would consist of deploying Excel functions.
Though, if typical calculus courses don't spend the time to ensure a good understanding of the chain rule, that would explain why so many students seem to think calculus is a gigantic sea of random rules.