California's math overhaul focuses on equity
latimes.com
latimes.com
They claim that this is elevating those lower on the rung, but rather, this is doing everyone a disservice. Either those of lower ability get thrown into the deep end without a float, or those of higher aptitude are forced into wasting their life doing things they already know.
If you’re a talented kid in public schools with busy working class parents, I guess you’re SOL?
How do those morons get into decision makers roles? They can't see 1h into the future.
And now race theory creeps into the math curriculum. So relieved my own children didn’t get experimented on by “educators.”
For those going the vocational training path you'd want to have a program where students in their 1st and 2nd years are spent sampling a variety of trades and vocations so they can determine where they want to focus for their 3rd and 4th years. Let's face it - even computer programming can be taught in this manner and people would be able to graduate from High School and get a job programming.
It is agreeable that data science is interesting, but one must ask what is data science without Linear Algebra or Calculus / Analysis.
In general, many believe that statistics education suffers because of the focus on calculus. See for example this StackOverflow post from 7 years ago, https://matheducators.stackexchange.com/questions/2099/empha... .
It's not hard to find other links, like https://www.edweek.org/teaching-learning/calculus-is-the-pea... , with more information on the problems with emphasizing calculus:
> For a picture of how severe that inequity can get, one only has to look at calculus. ... Until about 1980, calculus was seen as a higher education course, primarily for those interested in mathematics, physics, or other hard sciences, and only about 30,000 high school students took the course. That began to change when school reformers glommed onto calculus as an early example of a rigorous, college-preparatory course, ... despite the rapid growth of calculus as a gold standard, university calculus experts argue it is a much weaker sign that a student is actually prepared for postsecondary math in the science fields than it appears
Indeed, the L.A. Times piece describe it as "in high school, when they can choose advanced subjects, including calculus, statistics and other forms of data science" (emphasis mine).
Note too how it classified statistics as part of data science. IMO, data science without statistics is even worse than data science without calculus.
"Analysis" has different meanings in mathematics than in general use.
When you wrote "or Calculus / Analysis", I assumed you meant "the branch of mathematics dealing with limits and related theories" ( https://en.wikipedia.org/wiki/Mathematical_analysis ). The real analysis course I took as an upper-level undergrad focused on the fundamentals of calculus. I did a lot of ε/δ proofs that semester.
Statistics does not require that sort of analysis, so it cannot the "analysis" you refer to.
"Analysis", less mathematically speaking, means "the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it" (quoting https://en.wikipedia.org/wiki/Analysis ). That goes back to at least Euclid ("Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth") (Ibid.)
Historically, mathematics education taught Euclid's Elements - geometry - as a way to learn analysis. Lincoln, for example, "kept a copy of Euclid in his saddlebag, and studied it late at night by lamplight; he related that he said to himself, "You never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father's house, and stayed there till I could give any proposition in the six books of Euclid at sight"." (quoting https://en.wikipedia.org/wiki/Euclid%27s_Elements ).
Clearly, "analysis" doesn't require calculus, and surely can be taught independent of calculus. Indeed, Fermat and Pascal laid the foundation for statistics before Newton was a teenager.
"Linear algebra" is also a very broad topic. When I learned to solve a problems like {given x + 2y = 5; 3x - y = 12; solve for x and y} in 9th grade, I was doing linear algebra, though it wasn't specifically told it was linear algebra.
When I learned about eigenvalues and eigenvectors in college course actually named "linear algebra", it of course covered things in much more depth.
But why does an introduction to statistics course requires knowing about eigenvectors, null spaces, and orthonormal basis?
I mean, I vaguely remember my college statistic course, and it didn't require most of the mathematics I learned from linear algebra, real- and complex- analysis, or group theory?
What I remember most strongly about that course was Bayes' theorem. That's such an important concept yet it wasn't taught in my math courses until taking a required class mostly for stats and math majors.
And yet Bayes' theorem doesn't require analysis or linear algebra, and neither does it entail eye-balling lines SPSS memorization.
At https://nzcurriculum.tki.org.nz/The-New-Zealand-Curriculum/M... you see some of the goals for teaching statistics to secondary school students in New Zealand:
> Carry out investigations of phenomena, using the statistical enquiry cycle:
> determining estimates and confidence intervals for means, proportions, and differences, recognising the relevance of the central limit theorem using methods such as resampling or randomisation to assess the strength of evidence.
> critiquing causal-relationship claims / interpreting margins of error.
> calculating probabilities of independent, combined, and conditional events
> applying distributions such as the Poisson, binomial, and normal.
Clearly it is possible to have meaningful statistics education without calculus and linear algebra as prerequisites, so I don't understand your objections at all.
I meant to summarize Calculus and Analysis as basically the same subject matter. Mathematical Statistics definitely requires Calculus / Analysis as pre-reqs (many textbooks make zero attempt for review) and Linear Algebra is a well-defined course at the undergraduate level, even if every other university has its own 20% difference on what it chooses to emphasize.
Linear Algebra at the undergraduate level can be considered a replacement for Euclidian geometry; it is the formalization of that space. As for the actual course of Geometry in the US, it is an isolated anomaly in the HS to undergrad curriculum in that it is never really touched again. Its proof-heavy style and construction from geometric primitives will not be continued upon.
If you mean Statistics as in the introductory undergraduate stats course, that would murder most children's interest because it involves memorizing distributions as collections of facts, and it most definitely doesn't involve understanding cause vs effect. And Stats AP is generally not trusted by universities.
I talked about linear fitting as an example for data science, and I pointed out that without understanding Linear Algebra, linear fitting means to use your eyes and draw a line, and the "linear" part will literally mean a line. And what is the difference between a correlation and a linear fitting?
"If you mean Statistics as in the introductory undergraduate stats course ..." While I mentioned my being in an undergraduate stats course, the LA Times piece and most of the references I gave refer to secondary education, not tertiary. So, no I don't mean that.
The AP Statistics test in the US doesn't require calculus. "Recommended Prerequisites: A second-year course in algebra" at https://apstudents.collegeboard.org/courses/ap-statistics , suggesting it's "College Course Equivalent [is] A one-semester, introductory, non-calculus-based college course in statistics".
To double check, https://tb2cdn.schoolwebmasters.com/accnt_42975/site_42976/M... shows Statistical Reasoning with Algebra 2 as a prerequisite, and AP Stats require only pre-calculus. https://www.pike.k12.in.us/userfiles/15934/my%20files/2018-2... requires Algebra 2.
Which means statistics is already being taught in high schools, without calculus or linear algebra. Do you really think those courses are 'murdering' the interest of most students any more than any of the other classes they take?
The argument, quite clearly made by the math education advocates in California, is that the traditional math education progression is just that - tradition. The pieces can be put together in different ways and still give a good education, only with a balance which is more appropriate for more people.
I find knowing about Bayes' Theorem to be much more useful than integration by parts, but the former was never taught in my high school. And I haven't used the latter in over 20 years.
No, a high school stats course isn't going to be rigorous Mathematical Statistics. But then again, a high school calculus course also isn't rigorous Analysis.
In high school I used cosine and log functions on my calculator, without knowing how they were computed. I certainly couldn't have done it by hand outside of a few well-known special cases. From what I infer looking at things like https://www.northernhighlands.org/cms/lib/NJ01000179/Centric... , the goal of stats education is to understand how to use and apply a tool like linear fitting, and what concepts like "correlation" and "residual" mean, not the full underlying mathematical treatment.
Entirely appropriate for high school, yes? Otherwise, when and how should students learn that sales prediction, height prediction, etc. are not esoteric concepts but things they can do themselves?
The statistics which is currently taught in HS is no more respected than an AP CS course. Colleges do not trust them and are reluctant to offer credit, and this is a very big deal. The undergraduate intro course to statistics, such as the ones given to social science majors, is a big grab bag of memorized facts. It is soul killing by cultural reputation.
And whether we're talking about social sciences like econ or math, cs or physics, for anyone seeking continued education, students are well served by getting a boost in Calculus, especially when there's a stratum of HS infrastructure which can deliver results that colleges are willing to trust. This isn't true for stats.
I would argue that the point of math and foundations isn't rigor. Rigor is a tool for professionals to make sure long bouts of work don't go to waste. But we're talking about pedagogy.
IMO the point of mathematical foundations is: (1) it will provide the pedagogical bridge to how mainstream resources talk about a subject, (2) it will connect more smoothly to college programs anywhere (3) you want to develop the mental machinery which generates new facts and thoughts about math, rather than memorize what feels like large and incoherent bags of facts.
Perhaps Linear Algebra should be what is taught. Linear Algebra is one of the most petite subjects at the undergraduate level which packs a disproportionate amount of firepower. And it would be an excellent foundation for studying data.
That was taught in my undergrad stats class in 1991. I still remember one of the examples was about using the true/false positive/negative rates in AIDS testing to estimate that if someone tests positive for AIDS, do they likely have AIDS?
> The statistics which is currently taught in HS is no more respected than an AP CS course.
I fail to understand the relevance of this point. You like HS calc but not HS stats, and colleges like neither?
> The undergraduate intro course to statistics
Except we're talking about HS math pedagogy, not college. HS English classes are also taught rather different than college English classes. Plus, HS classes are mostly taught by teachers trained in pedagogy, while most college ones are not.
I still can't believe how my grad school dropped wet-behind-the-ears grad students like me into TA'ing courses, and making up our own quizzes for students, without any real guidance on how to teach or make tests.
You make broad assertions about soul killing. I point to existing HS classes in statistics. Are you really saying that those courses are soul killing? On what basis do you make your strong assertion?
> especially when there's a stratum of HS infrastructure which can deliver results that colleges are willing to trust
You just said that the calculus courses are no more trusted than the statistics courses. You can't have both ways.
> This isn't true for stats.
Where is your evidence?
Your pedagogical views:
(1) is a matter of cultural change. If people believe statistics education is important - which it is - then there's a new bridge. Just like how Euclid's Elements was once that cultural bridge, replaced by a focus on calculus. Now the emphasis is that focusing on a single bridge is wrong as it dissuades students interested in other bridges.
(2) assumes nearly everyone will go to college. For those who don't go to college, will statistics training be a better education than calculus? Indeed, part of the complaint is that this pedagogical emphasis on calculus overly biases education towards a specific type of college-track student - those with money, or those not afraid of taking on possibly life-crippling debt
Based on my experience some 30 years after college, I think there should have been more statistics training in my education and less calculus.
(3) I pointed you to various resources which, to my mind, demonstrate that the goal is indeed the development of mental machinery. You have only continued to repeat your assertions, not provide evidence.
I took a college course in Linear Algebra. I have not needed to compute an eigenvalue in nearly 30 years, and rarely do I need matrix multiplication for my work. I need to know statistics much more often than that.
How would learning the basics of linear algebra for most people (medical doctor, social worker, plumber, lawyer, fire fighter, or news reporter, mechanic, office manager, etc.) be better than learning the basics of statistics? Surely these are all people who need the mental machinery to understand if evidence is credible and to make predictive models.
Either way, what a shame that this is even being considered.
We’re fortunate enough that we don’t have to contemplate sending our kids to public school but if we weren’t this might be a push factor in those kind of decisions.
of course everyone is going to argue about equity or lowering standards or whatever... but personally i think bringing more applied math to k-12 will be a boon, especially in the age of computers.
maybe by demonstrating how fun it can be to use the tools, more kids will be willing to suffer through the abject monotonity of rote memorization for doing hand calculations.