Bingo. It's well intentioned, but the intentions aren't to ensure that America can keep up with a rising China.
It's shocking to me that people in California aren't more worried about this. About 15 years ago, I was talking to an engineer at Juniper/Cisco. We were joking about how Huawei had copied one of their router designs down to the silk screened assembly instructions (in English!) on the PCBs. Fast forward to today, Huawei is making fully custom equipment down to state of the art switch and router chips, and Chinese companies are white boxing lower end products made by American brands.
There's a big bet out there that the U.S. can survive on software and social media alone. I would think the success of Tik Tok would have blown even that rationalization out of the water.
On the general point of U.S. math education: my cousin who lives in a nice California suburb was complaining that the math education her early high school student is receiving is several grade levels behind what she got--in Bangladesh. My mom, who also went to school in Bangladesh (in the 1960s!) was deeply unhappy about the math education in our affluent Virginia suburb, until I got into a top STEM magnet high school. My own kids go to an expensive private school, but are still getting math tutoring on the side. Math is just a shockingly low priority for Americans.
I'm not trying to suggest that the US is fine and we shouldn't fix anything, but if you look at the world by comparing test scores and grade levels in mathematics, you're going to come to some very warped perceptions about what is important. I'm speaking as someone passionate about STEM education, who got a B.S. in mathematics.
The whole situation is warped. The USA accounts for 4% of the world population, and 40% of the top 100 universities in the world. That's fucking weird. I don't have an explanation for it. I'm just saying that the different signals we use for evaluating how good our education system is functioning are giving us radically different pieces of feedback, and our understanding needs to be correspondingly sophisticated.
There are all these narratives about how China is going to eat our lunch (like Japan in the 1980s, or the USSR in the 1960s) and while I don't feel comfortable betting on long-term US hegemony, and while I do think we should put more work into our mathematics education, I do think that looking at the world through high-school mathematics test scores is going to give you anxiety more than it's going to give you an accurate picture of what are problems really are.
To take another statistic into account, there are actually many STEM graduates in the US. What do we do with this information? How do we change our policies? It's unclear.
We lost entire industries of good paying jobs in consumer electronics and appliances to Japan, Taiwan, and Korea. It wasn’t the total eclipse people predicted—in part because our population control evangelists helped derail Japan’s development—but our shift to software and IP didn’t replace the kinds of jobs we lost.
Finance, insurance, social, and professional services will keep our GDP numbers afloat long past the point where the real economy has been surpassed. A better comparison is Britain. Once the real economy moves, most of the ancillary services do too. Banking and insurance keeps Britain going, but it’s actually kinda poor compared to America. (The US is to the UK as the UK is to Hungary or Poland.)
No there aren't, at least if you only count homegrown STEM graduates. USA can be a tech powerhouse thanks to brain drain, not thanks to its STEM education. USA is one of the worst in OECD.
https://i.insider.com/5661f406dd0895ff628b46bb?width=600&for...
What would our STEM university departments look like if the next generation of Indian and Chinese kids with good K-12 math educations decided opportunities were good enough at home that they didn’t need to leave their families behind and immigrate to America?
that is just inertia, carried over from the time when the US was the only superpower. More and more Chinese universities enter this list every year.
For those of who were around in that era and watched as the US auto industry and consumer electronics industry were devastated as their Japanese counterparts ran rings around them, Japan did eat America's lunch and with considerable gusto. Sony products were everywhere, as were Toyotas and Hondas.
Japan doesn't support your case. (Neither does China really.)
Yes, because of tariffs. They wouldn’t do it if imported cars weren’t heavily penalized for being manufactured elsewhere.
No it isn’t. The best researchers want to go to the best universities and the universities are ranked on research performance. Its a feedback loop.
Should math be a higher priority in the US? Should working hours in the US be the same as in China? Should the academic pressure on kids be as high in the US as in China?
The US is much smaller population-wise, would we actually need to try five times harder than China?
Is it not enough to compare today's high school overachievers with those of 20 years ago, all still fighting for the same universities but with all similarly-inflated resumes? Do we actually need to push them even further?
Do we instead want to be more like the European countries that currently put themselves under less pressure than the US?
TJ? :P
1. Blank slate - All humans are of equal ability
2. Any observable differences between humans are merely the result of social factors
3. Any observable differences in outcomes between groups of humans are the result of oppression from the majority group
4. If you observe differences at your org/institution, it's your moral duty to create policies which disfavor groups of humans performing better and to favor groups of humans performing worse, as those performance differences are due to oppression.
If these beliefs undergird your worldview, and your social groups/information environment reinforce and reward these beliefs, it is of no surprise that we'll see a lot of people soberly propose the types of policies we see here. I can empathize that they really do think they are fighting the good fight, and are doing the right thing for society.
These proposals come from committees and groups of people, and it's just not realistic to write off the entire group of people behind these proposals as having some uniform set of beliefs like that, especially when they give other rationales for the proposals!
The current school system makes decisions in middle school (8th grade and earlier) which determine whether or not each particular student will be able to take calculus in high school. This is, simply put, insane.
Because it's obviously insane, when you introduce questions of race and class into the mix, then it's easy to apply pressure to the department of education to come up with a proposal that changes things. And then you end up with bad proposals... why? Because these proposals are produced by poorly-shepherded committees full of government employees under political pressure, and it's much easier to come up with a bad proposal that responds to political pressure than it is to come up with a good proposal.
There's just no need to try and explain that this proposal is bad because the people who made it have bad beliefs. I'd characterize this as fundamental attribution error here... "the committees made a bad proposal because of wrong beliefs" versus "the committees made bad proposals because it's easier to respond to political pressure than to write a good proposal".
They ignore the fact that the degree is supposed to be a proof of "This person has X skillsets at minimum". It no longer is. There is basically no job in the United States that cares if you have a high school diploma or GED other than government jobs. They are repeating this process with College right now. The College Diploma is now the new High School Diploma, and pretty soon jobs are gonna want Masters or Doctorates for entry level.
This is a bad belief, they don't (or won't) understand that they are ruining and devaluing a thing by their actions and beliefs.
Yes pretty much every college requires "college algebra". But some schools have "college algebra for stem majors" vs "college algebra for non stem majors". Guess which one is easier, has lots of bonus credit and extra curricular stuff to earn extra credit. (You attended the college showing of "vagely related math movie?! Here's 10 points on your final!") And also grades on a curve. Also you only need 69.5 (and sometimes just a D!) to graduate.
There's also other cheats/hacks. Like lots of state schools will let you transfer from a community college with credit for your "core courses", and some of those have questionable standards. There's also the fact that college algebra usually has some kind of test out or online option. There was a whole sub industry of "pay you to take the online test for me" at colleges for stuff like college algebra. Some of those courses did have some kind "you have to take 1 test in person so we know its you" rule. But they didnt check super heavily that you were actually that person other than a cursory examination of your drivers license name matched who was supposed to take the test. (and oh boy let me tell you about how covid and masks interacted with all of that)
Dont get me started on the "Statistics for Sociology" that was different than actual "Statistics" (but fulfilled the Stats requirement for the degree)
This is also ignoring that taking College Algebra to begin with IN COLLEGE. Was a major sign you were not a Stem Major. Stem Majors took that in high school and were taking at minimum precalculus. (and even that was viewed as the slow lane, you should be talking Calc 1 as a red blooded STEM freshman)
An acquaintance asked me if she should take "Calculus for Artists" after I suggested she take a calculus course. I laughed, and said that such a course should be named "Pretend to Learn Calculus". She should take a real calculus course, which she did, and did well in it.
If you're in college, stick to the real math classes, not the "math for losers who are forced to take a math class". You'll be with other students who want to learn math, and you'll have a prof that wants to teach math (the loser math course has a prof who doesn't want to be there, either). It'll be a much more pleasant experience.
Hey, if I was running a college, I'd have the two track math system, too. That way the students who want to learn won't be bothered by the ones who don't.
Was that based on actual knowledge of the course content?
It's like "Flight School for Artists". There's no such thing.
dae STEM has got to be one of the most exhausting memes on the internet. Hell, I’m 3/4 of the letters and even I find this aspect of our culture insufferable.
E.g., there can be a course on compilers that teaches automata theory. There can also be a course that just jumps in and codes a compiler. Both courses teach different, but valuable skills. Now, having a course “Compiler for Artists” where you’re taught such useful gems as “a compiler translates between human-level languages and machine code,” not so much.
Both have their place.
So, yeah, in the real world, it doesn't work out so good.
I eventually learned to appreciate that Caltech never taught how to use formulas, but instead taught where the formulas came from. I recall a class on jet engines (really, an awesome class!), where the prof spent the entire lecture deriving the formula for a jet engine's performance. It was breathtaking. I knew where every term came from. I finally understood how the damned thing worked. All those other handwavy explanations mystified me.
If I was just handed the formula, it wouldn't have meant much of anything to me.
- get you used to the general feel of the topic
- make you practice proving things
The specific theorems students prove are not really the point.
Right, that's why they should skip the calculus and start students directly on Real analysis. Then they might actually have some use for what they prove in the class.
Caltech didn't. In fact, they expect you to know calculus before entering. I didn't, and that nearly capsized my college career before it left the dock.
Sad to hear that. It suggests their admissions process has succumbed. When I attended, it was very rare for the admissions people to admit someone who couldn't do the work. I only knew one who couldn't.
Colleges should only admit those who pledge themselves to the Guild of Software Engineering at 13, so we can keep the noble profession of building CRUD apps in JavaScript free of riffraff.
The calculus requirement was not listed as an admissions requirement. The other students were astonished that I didn't simply assume it was required.
Fortunately, physics prof Ricardo Gomez invited me to his office for some impromptu calculus tutoring, which saved my bacon. I owe him.
This does not in any way imply that those should be the only kind of places to exist.
This does not in any way imply that there's anything wrong with people who started to recently to have relevant background.
We also do split students as early as after the 4th year (end of elementary school) in three different school systems with about 50% of students attending the highest university track tier. The other two tiers might not have calculus or only in a limited form, but also don't qualify for university without further education. Thus setting students on different paths based on ability/performance early on seems very natural to me. Students always can up/downgrade schools based on performance at any year later on.
For cheating we proctor all of the tests, which are worth by far the highest portion of the grade. I’m sure people have cheated on the proctored tests, but there’s not rampant purchasing of passing grades. This is specifically to rebut your implication that students are cheating or being carried by the grading structure, not necessarily to discount your position in the overall discussion.
It was well known there was a side hustle for some of that. And certain proctors would at times be told to drop it if they pressed. It was notoriously an issue with the non english speaking population. They would show up with an id that was clearly not them, proctors would be told to just let them take the test, same student would still be at classes months later. Though to be fair foreign students and schools chasing that money is probably its own kettle of fish.
That's probably true. But there's just not much that you can do to teach years of junior high and high school math in a single college class, to students who never got the chance to seriously engage with that material because their school replaced actual math courses with this new "data science" thing (or for countless other reasons which would fully apply even today). "Weedout course" might not be the intention, but it's definitely the real-world outcome.
I don't have a specific comment on the policy under discussion, I'm not a californian and I'm not familiar with the specifics - but telling people "You'll be fine without a degree" isn't going to go over well and, honestly, is asking the recipient to accept a large risk regarding their future while you, an employed person, have already passed that hurdle. American colleges definitely have issues with over enrollment but even if wanting that state is a "bad belief", it's certainly an accurate belief.
I don't disagree at all that it's mostly BS, but, much like the stock market, it is functioning primarily on the social valuation - and currently degrees are seen as close to a necessity to enter the job market.
1. The only thing I've seen on this front is some advertising for trades work which can pay equal or better to non-SF software engineers.
The boss of my first internship at Cisco majored in Geology.
[0]: https://www.sfgate.com/bayarea/article/Stanford-professor-Ka...
A lot of students in the US learn physics without calculus, which is basically useless. It's just a set of magic formulae to plug and chug. We should not go further in that direction but instead go in the other direction.
I have an issue with the parent's gossipy put-down of Professor Boaler.
I think it's important to emphasize that genuine intellectuals who have put serious thought into this proposal support it. I would prefer more serious engagement with it, even if on HN the majority disagree.
It’s an account of how a very diverse group of scholars advising the United Nations ignored research contradicted their ideological views.
One of the scholars even admitted in private some of their recommendations were “wrong.”
The group was chosen for its expertise and diversity.
I’ve served on school boards. They’re heavily susceptible to group think.
Once the culture wars get involved, peoples minds turn off.
There is however a very broad movement of people who believe that unequal outcomes are a manifestation of racism and so they act accordingly in their roles in government etc..
Many of these people are in the civil service and so this will influence their view.
It would be 'insane' to ignore this movement, it's one of the most powerful social forces in the US right now.
But even conceding that racism is the dominant factor in performance differences, the framework proposed here is plain ridiculous. In essence, what they’re advocating is that because disadvantaged groups underperform in advanced math, advanced math should be scrapped altogether in public schools. The effect of this will be to handicap disadvantaged groups even further, as they’re less likely to attend private school or have access to private tutoring, and thus will be locked out of advanced math entirely while advantages groups continue to receive advanced math instruction in private schools.
Since I have no such 'scientific racism' views and claims of 'proto-fascism' are ridiculous at face value ...
... that a common person would take some rather mundane comment somewhere to mean those things, implies that they've been radicalized in some way in the manner that I've described.
"Racists is everywhere, under my bed, in the jingle for that product, in our schools! Math is racist! We Need Action Now!"
'Anti Racism' is a reasonable concept at face value, but the issue has obviously created groups of wayward ideologues in large swaths in the US, who are more likely than not to be involved in the civil service, particularly in education.
Since 'improving education' is a perennial issue of contention anyhow - if we throw 'math is racist' into the fire, along with legions (at least a critical mass) of fervent supporters - and then finally allow the hollow politicians and media to misrepresent and aggrandize all of it in bad faith (votes, money, attention, power etc.) ... then we have yet another toxic cocktail of public malaise and dysfunction distracting us from dealing with the core issues.
> Many of these people are in the civil service and so this will influence their view.
It influences the political pressures placed on people in the civil service more than anything else. People in the civil service are in the civil service for a long time, typically. Often they are in the civil service for decades. On the other hand, elected officials and political appointees rotate in and out much more quickly.
I know people in the civil service, they have a much more long-term and level-headed view of things, and my impression is that they "weather the storm" of changing political pressures. At least, the ones who survive in the job do. Political appointees and elected officials are much more mercurial. Appointees know that their position is (somewhat) an extension of their own political power to begin with.
Christians are fond if it: "You have an innate, direct relationship with God, in his eyes you are equal to the The King" etc..
But we've only had 'governance' in the broad sense for 100 years.
And we've never really tried to apply such principles into education until the 1960s.
Now we have actually made incredible progress on social issues, we have our 'wars' in Social Media with Holy Anti Racism Fanatics trying to do their best - because 'racism is bad' - which of course it is - and 'systematic racism exists' - which of course it does - but the 'kernel of truth' of these issues drives people into ideological fervour as though it's some giant overwhelming issue, when really it's not. Racism is still pernicious, but it's not fundamental.
And FYI don't think it's all rubbish.
For example - 'Math' is heavily based on prerequisites. If you 'fall back' in Grade 4, you may never be able to 'catch up'. While that's true in general, it's not as acute as in math.
Poor kids might be far more likely to 'fall of the bandwagon' and a lot of poverty might be due to systematic racism, and so the 'Hard Requirements' for certain things may not be ideal.
You could have 'summer school' or 'after school' or 'accelerated catch up' programs.
Those would be 'reasonable' solutions in my view - and FYI these are mostly issues of poverty, not race, they are couched as racial issues because that's what fires people up.
Edit: and yes, I agree 'most people in the civil service' are level-headed. Most people actually are. But some groups have outsized voices, amplified by 'allies' elsewhere.
The 'Anti Racism Agenda' is a 'fundamental pillar', like a religion, of 25% of the US population, and they are pretty active about it. And the actions of the most extreme 5% end up really upsetting other people. Much like a very powerful fool using made up constitutional manoeuvres to try to take over the government would upset a lot of people as well.
Good intentions, surely, but that doesn't make them right.
How did you reach this conclusion in the face of 300+ years of verifiable, well-documented federal, state and local racist legislation and policies? If it’s pernicious now then it’s always been pernicious since generations have persevered irregardless of these policies… but that doesn’t make sense to me.
Systematic racism only exists today to positively discriminate against people belonging to so-called disadvantaged races, at the expense of those belonging to so-called privileged races. That's why White and Asian kids are discriminated against in college admissions.
Shitty taxation policies fuck all new home owners and have little to do with race.
I assume your example is talking about Prop 58. [1] Is Prop 58 racist? Of course not, the CA Constitution prohibits racist and sexist laws and the courts are very good at enforcing that prohibition.
Prop 58 means a lower property tax bill can be inherited by the children with the property. By your reasoning, isn’t allowing property inheritance at all racist?
To me this is a meaningless injection of “equity” into a discussion on equality. Racism is not a word that means “inequity” - as in - everyone not consuming, producing, contributing, and owning the exact same quanta. That of course is a bizarre fiction.
We are talking about a tax policy which generationally preserves a tax break. It had a meaningful basis in law and policy which does not discriminate based on race, but it certainly provides a significant tax benefit to generational land owners.
Now again, that’s a drop in the bucket compared to the Federal estate tax lifetime exclusion amount, which provides orders of magnitudes greater tax benefits to preserving generational wealth, so Prop 58 is just a bit player in this scheme.
If the argument is that a tax policy which is advantageous to anyone with generational wealth is “systemically racist” then I think the argument has gone off the rails.
There are important justifiable public policy goals to ensuring a certainly level of generational wealth. There is no and never will be an important justifiable public policy goal of being “racist”.
If someone wants to argue for wealth confiscation, redistribution, and equity (e.g. some form of communism) then by all means give it a shot. But I don’t abide calling people racists if they disagree with that as a policy goal. If the argument is “merely” that there should be no tax benefit which preserves generational wealth, you actually immediately end up in exactly the same place.
The answer is of course more likely along the lines of “we do both”. We have policies that provide for protecting and maintaining a certainly level of generational wealth. And we also have policies which provide a massive welfare system for people without any generational wealth.
As to estate tax exemptions, they aren't nearly as egregious. There can certainly be an argument for reparations of ill-gotten gains to level the playing field, but it's at least possible for me to accumulate wealth myself and pass it on to my children. In contrast, it's simply impossible for me to acquire and pass on the same property tax rate that my white neighbors have.
I think saying you can’t go back in time and capitalize on an investment 60 years ago that also comes with an accrued property tax advantage is true, but has nothing to do with race. Anyone can start accruing their own tax advantaged status under the exact same Proposition, and no one can ever try to take it away from them on the basis of a protected trait, right?
I guess what I’m asking is, is there anything specific about this tax policy that accrues benefits to a historical capital investment which makes it “racist” over any other tax policy that accrues benefits based on historical capital investment?
There are so many race neutral policies that provide accrued benefits like this. Even social security benefits accrue based on historical wages. But also things like Roth IRAs which provide compounded tax free growth. You can’t contribute the same dollars today to a Roth IRA and get the same tax advantages as someone who contributed the same amount but over the last 60 years, the benefit can only accrue over time, just like the CA property tax benefit can only accrue over time.
Not everyone pays the same tax rate on their income or their property, nor are they provided the same level of government benefits. In many cases that tax rate or benefit level will be based on historical personal or family wealth. Often times it works out to be a progressive tax, sometimes it’s regressive depending on so many policy goals that are often in conflict with each other.
Simply put, any of your neighbors who have owned their (family) property for several decades have accumulated a several thousand dollar a year tax benefit for being a long-time owner. You too can accrue a several thousand dollar a year tax benefit the longer that you hold your property. You don’t get to buy in and get that benefit on Day 1, and neither did they.
This can be trivially demonstrated: ten times more men are arrested and incarcerated than women, yet no one would claim this suggests the US criminal justice system is systemically sexist toward men.
watch “the news” or go on twitter
"The news" is dominated by organizations which are trying to maximize their social media engagement metrics so they can get more money from advertisers, so they're subject to the same forces that drive Twitter.
Neither of these sources reliably give you a picture of national affairs. Right now the best I can do is get a picture of local affairs by talking with people that I happen to encounter because I live near them.
Political pressure is a manifestation of the population's problems / beliefs / perceptions. It's wrong to bow completely to political pressure, but it's also wrong to ignore it.
If a political decision-making process concludes "men account for more road traffic deaths than women, but we've decided that gender can't be taken into account when pricing car insurance" that's fine; the facts have remained the facts, and a political decision has been made by a political process.
But if the same process concludes "men and women cause the same number of road traffic deaths" the process has gone off the rails despite the fact the outcome is the same because in one situation the facts have been acknowledged and the decision to act contrary to them made knowingly; while in the second situation that isn't the case.
It's not clear that this is the case. The "current approach" is to offer Algebra I in middle school, which ought to leave plenty of time for students who want to shape up in math and be prepared for HS calculus to do so. Push advanced math later in the curriculum, and you just expose the students to even higher-stakes dilemmas. Lowering standards is no solution, since you'll just end up with lower education quality for all students, that will make it even harder for them to catch up to reasonable levels. This is the broad background of OP's letter.
You don't have to write off entire groups of people - just the few at the top. They got there by sucking up to those who were already at the top. Everyone else just has to keep quiet if they want to keep their job. Worse, they're expected to express visible support for their "superiors" and their ideas if they want to keep their jobs.
What is being recommend is that no one take calculus in high school.
Both of these ideas are bad, but only one of them informs the California Math Framework.
It’s very commonly taught in 10th grade in France, Germany, Singapore and Taiwan (where I used to teach). It’s not universal by any means but as far as I can tell, the idea that calculus should be delayed until university is a nearly uniquely American idea.
https://www.mathvalues.org/masterblog/calculus-around-the-wo...
I can’t speak with certainty about other places but this directly contradicts my experience living most of the past 20 years in Taiwan (and speaking with friends who taught mathematics in the public school system).
Many, many people not on a STEM track learn calculus in high school.
Imho, 50% or more of students don’t even need to take algebra.
For that matter, that same number pretty much don’t even need to go to high school (although some sort of voluntary coop with practical skills training might be better than current high school or nothing).
HS is glorified baby sitting for most students. It sometimes is useful for people who want to go to “competitive” schools or enter stem fields in slightly less competitive schools.
As for your solution, learning math starting at algebra or pre-calc in a CC is fine if that is what the person needs for their chosen career path. I think the general idea is that the lack of calculus in high schools prunes the career path tree fairly aggressively in some relatively well-paid fields (esp. STEM).
I would never describe someone's beliefs as "circumstantial", and I would also never think of being on a committee as something "intrinsic".
I don’t know anyone who seems to disadvantage high-achieving individuals. Only people who wish to raise the performance of as many as possible to the same level. And yes, I see oppression everywhere, including education.
It would be a net loss to society to deliberately reduce the performance of anyone.
Can people not have varying degrees of physical and neural plasticity? Perhaps some people are more like blank slates and can adapt more readily than others? Maybe plasticity changes with age?
It's just a 'general set of principles', intuitions, pop culture ideals that lead a large group of people to assumptive believe that 'unequal outcomes are driven mostly by systematic racism' and that's that.
Ergo 'the world is deeply racist' and 'there is racism around every corner' including in your math textbooks and teaching.
Ad nauseum.
Some of it is actually correct.
Some of it is reasonable from an intellectual perspective, but it's hard to take anything from it.
Much of it just ends up being toxic.
I see a lot more stuff that would lead a kid to believe "it's ok that I'm not good in math" rather than "I could be good in math if I wanted to be."
Frankly, I think this is actually worse educationally than what you suggest.
We need to find more ways to reward effort instead of pre-existing ability (regardless of how that pre-existing ability is gained... the kid whose parents got him ahead of the curve through high school math and then bombs out after taking university-level Calculus is similarly harmed by the current system as the one who's shunted away from ever being challenged).
The important thing to note here is- if you reduce the bar in high schools, a lot more students will end up in college - more money will spent, more loans will be written out etc.
Or political interest?
I wrote sth about that here:
https://news.ycombinator.com/item?id=31176276
(Namely, foreign state psyops? -- I do not think so, just a thought)
In reality it's not quantum computing that we're talking about, it's high school calculus and algebra. You don't have to be a hardcore blank slate proponent to believe that most people can learn it. And that is what these people don't believe.
It's important to consider the goals of this committee. They propose this reform because they oppose the blank slate theory. The current structure really isn't appropriate to most people. Because it relies on a wrong form of the blank slate theory. They offer the wrong solution in my opinion, because they end up going too far the other way.
You have been to a remarkably bad college of education, if that was your experience. Which is saying something, because most of them aren't great, but for very different reasons.
Not that taking calculus takes a genius, as evidenced by more sensible countries.
Not only does the earlier version of the framework explicitly reject this view, it cited specific empirical studies that the broad approach targeted (which I gather had not changed in the revisions which is why the complaints remain despite some revision to details) was better for people across the ability spectrum.
Similar points apply to each of your bad-faith assumptions about the underlying beliefs.
Can you share that explicit rejection of the idea that there are not innate differences in ability, in the CMF? I have not seen it myself, thank you ahead of time.
To share what I've read, and colors my views a bit, is the following, from 'Chapter 1: Mathematics for All' of the Second Field Review [1]:
> An aim of this framework is to respond to the structural barriers put in the place of mathematics success: equity influences all aspects of this document. Some overarching principles that guide work towards equity in mathematics include the following:
> All students deserve powerful mathematics; high-level mathematics achievement is not dependent on rare natural gifts, but rather can be cultivated (Leslie et al., 2015; Boaler, 2019 a, b; Ellenberg, 2014). ...
> All students, regardless of background, language of origin, differences, or foundational knowledge are capable and deserving of depth of understanding and engagement in rich mathematics tasks.
> Hard work and persistence is more important for success in mathematics than natural ability. Actually, I would give this advice to anyone working in any field, but it’s especially important in mathematics and physics where the traditional view was that natural ability was the primary factor in success. —Maria Klawe, Mathematician, Harvey Mudd President (in Williams, 2018)
> Fixed notions about student ability have led to considerable inequities in mathematics education.
Note that my pointing to this doesn't mean I inherently disagree that hard work/education can't help improve outcomes. I show the above citations to show that the CMF is not explicitly denying the Blank slate theory, which is what you are suggesting. If anything, they go out of the way to view ideas of innate ability negatively. I'm happy to also look at the references that you alluded to but did not cite.
[1] https://www.cde.ca.gov/ci/ma/cf/ (it's in the bottom section, the format is .docx, so don't want to directly link to it as that format can sometimes be cause for concern on random links shared on the internet)
Absolutely no one I have met believes 1 and 2.
As for 3, "average observable differences" in mental ability can largely be explained by socio economic factors. Case in point, East Asian IQs stagnated behind US IQ averages 50 years ago, but are significantly ahead now. I dont think they underwent a general transformation in 50 years time.
4. Is again a strawman. Affirmative action is used for a limited number of historically discriminated populations in very limited contexts.
If you are interested in an actual debate don't invent a caricature of the opposing position.
Admittedly I am a little confused by your comment. You made the above statement, and then follow up with this statement:
> As for 3, "average observable differences" in mental ability can largely be explained by socio economic factors
If we are to assume you are being earnest here in your first statement in that you believe there ARE innate ability differences between humans and social factors don’t explain everything, can you explain how much innate ability differences are a factor in intelligence differences (using it since you brought it up)? And how do you determine it? What methodology are you using for that, and if you aren’t basing it on anything, what methodology should we use?
If you are going to dodge the question and suggest you were talking about averages between groups (for which the question does still apply) and that between groups are a special case, then please share how much of a factor innate ability differences are in intelligence within groups? And how you determined that?
1. If we are to assume you are being earnest
2. If you are going to dodge the question
etc.
I am a statistician and all of your questions have banal answers.
For individual differences you can plot the bell curve and the standard deviation gives you the individual variances.
I was clearly talking about population averages in 3.
> can you explain how much innate ability differences are a factor in intelligence differences (using it since you brought it up)? And how do you determine it?
AFAIK remember, heritable IQ/ height is estimated to be a bigger factor at an individual level. However, there is no existing causal methodology to determine it. Instead we have correlational regression models that make a lot of strong assumptions that are practical. There are various papers but it is clear that IQ is highly heritable like human height.
Also, I am going to replace "innate" ability with "inherited" ability because there is no methodology that can tell you how much of Roger Federer's tennis ability is "innate" and how much is "coached".
> talking about averages between groups
Your wording is ambiguous. But assuming you are talking about intergroup differences. When you sum up IQ or height within a group, the observation from regression models is that individual hereditary differences cancel out, and intergroup height and iq differences are easily explained by socio economic factors in a correlational regression model and heritable height/iq is an insignificant factor. Once again- this is not a causal model. Only correlational.
And this is just common sense. We know from life experiences that different people have differing IQs and heights from an early age. We are also aware of how important it is to have a school system - instead of just letting the kids figure things out on their own. The environment matters because we do have to work to nurture the inherent talents of a kid, otherwise that potential will be lost.
> please share how much of a factor innate ability differences are in intelligence within groups?
I expect the per sample standard deviation to be the same across different ethnic groups. For inherited "ability" i expect the total variance explained by "inheritance" factor to be the same across ethnic groups. I doubt if such an analysis has been conducted, because I don't see the value in this kind of subgroup analysis. Do you have any references to such studies? Have you seen data that contradicts this?
This is laziness of the moral clowns who know the easiest way to remove a difference in success is to eliminate the successful. As they don’t care about anything but the theatrical performance, they naturally take this route of least resistance.
Some observable differences are due to social factors.
Some observable differences by certain groups are the result of past actions by other groups.
You should favor policies to correct for the result of past harms.
One cannot reasonably claim that no groups in the US are still disadvantaged today due to actions taken on a centuries-long timescale. It seems willfully unfair to stick your fingers in your ears and just say "I'm not actively discriminatory, so there's no need to try to mitigate things, everything is peachy."
> Some observable differences are due to social factors.
> Some observable differences by certain groups are the result of past actions by other groups.
Agreed with this as well.
> You should favor policies to correct for the result of past harms.
Mostly disagree. I'm wholly convinced there's never going to be even remote consensus about ancestral oppression and victimhood. I think that if you want to correct harms, you should focus on the current social issues. Not only do I think that has much less shitty side effects (such as exacerbating group identity), but I also think it will be more effective at reaching those that have been mistreated in the past - both directly and their ancestors. The US in particular has, let's say, a decent amount of low-hanging fruit in terms of social issues to address (homelessness, lack of preventative health care, incarceration for victimless crimes etc etc).
> there's no need to try to mitigate things, everything is peachy
This is not the only option. You can recognize that things are fucked but have a vastly different opinion on how to improve.
I've run into it so often by a vocal minority who slander anyone who objects. Fortunately, the popularity is waning.
I wrote sth along those lines here:
https://news.ycombinator.com/item?id=31176276
(Foreign state psyops -- I do not think so, just a thought)
The alternative would be awful, no?
I really don't think it is. And if it was, it'd be only partly, since apparently lots of this is coming from people inside the US.
But what if there're foreign state controlled FB groups that try to manipulate people in the US, spreading these anti maths ideas? I'd be surprised but not super surprised.
Still, if you were Putin or Xi, why would you not want to damage STEM in the US? And even if you didn't start these ideas in the first place, why not try to amplify them, once they already exist
2. This follows from #1, if we rule out nature then it must be nurture. Also, you must not be a parent if you'll accept "some kids are just dumb" as an excuse
3,4. Replace "oppression" with "competition". I think it might sound better to you. But the conclusion is the same.
You want to prevent winners from accumulating an advantage, eliminating all others (which sounds vaguely genocidal in this context), then you have to handicap winners and support the others. And the fact that the wealth distribution in America is so uneven certainly suggests that the initial premise is true (i.e, winning allows you to accumulate and compound advantages with repeated victories).
Is it? Saying "all mean are created equal" has a lot of interpretations, of which many valid ones do not include "all men are born of equal ability in every regard." Given that some people are born to grow up to be 5'4 and weigh 100lbs and others are born to grow up to 6'7 and 320lbs, it should be clear that not everyone is "equal" at least in terms of their physical abilities. I'm pretty sure the Founders were aware of this, making it highly unlikely that their version of "created equal" meant "exactly equal in all terms of ability."
And I would agree. But your earlier comment seemed to imply a much more absolute stance. That's what I would disagree with.
It is ironic as you are sayings forcing kids down one path is bad, yet your solutions is to force kids down one path. The current system does not make you choose your path based on a some test at 10.
Just that they are 'equal' i.e. before God, or before the Law.
That one man is not from some superior lineage, that makes him a superior being.
I would be the founding fathers would have no problem if one man decided to 'discriminate' among others for some arbitrary reason - even if they were landholding men of high status etc..
As of now we effectively underwrite anyone who wants to go, often at the expense of the student racking up debt for a useless degree and later the taxpayer who will inevitably have to subsidize them.
curious to learn how you arrived at this conclusion?
Those compound advantages you're talking about are good things that we want people to have because they help them do more good for society. You seem to want to handicap them in the name of fairness. Where does that thinking end when you realize that the founding assumption (1. above) is false? Disfiguring beautiful people to prevent them accumulating the compounding advantages that come with beauty? Brain damaging intelligent people?
“ 1. This is kind of a founding principle of the country, so...”
yes those slave owning people really thought their slaves were the same as them, totally dude
Entitlement principles are why the West was the 1st place in the world to abolish slavery and then went about and abolished it throughout the world.
This is generally true.
People can find extreme examples that “disprove” this but they’re generally wrong. Most things people do aren’t that hard and people have the ability to learn to do them — they either choose a different path, have fewer choices, or just don’t care.
And yes, before you ask, this includes computer programming.
In _every_ activity I've ever participated in where I can observe many people's performance and progression - including powerlifting, bodybuilding, various ball sports, mathematics, chess, theoretical CS, software engineering, etc - it is transparently obvious that people's natural abilities vary dramatically.
Although it's not the most common scenario (training and experience do matter), I have seen many situations where someone with, say, 6 months' haphazard and lazy experience will absolutely crush the performance of someone with 3 years of serious and dedicated training.
Talent is real.
Talent is real, but generally speaking, ascribing failure or lack of progress to lack of talent is a mistake.
Not all failures are due to lack of talent, for sure. A blank statement like " ascribing failure or lack of progress to lack of talent is a mistake " in the context of our discussion is nonetheless a mistake as well.
Or, do you really think every 2-year old kid can teach their neighbor kids maths and then explained what groups are when he was 7-year old like Terence Tao did? I tried my own kids. Needless to say, I failed, miserably.
Without pointing to a vague notion of "we're different people", I think there's a few key things that were different, despite everything that was the same:
* She's the second child, I'm the first: there were some things in my education that my mother literally said "we are fixing that for her". (And I should note that I don't resent this: my mother was clearly doing the best she could with the information she had — and because she loved us. But she had more information during Round 2.)
* Education is a finite resource: in my home state, whether I got into a decent school (i.e., a magnet school) was dependent on the literal roll of a dice. (Literally literally. I.e., list of names goes in, gets shuffled, top n go to good school & gets educated, bottom m talent gets wasted.) In the worst case I was 5th? 6th? from the bottom of a several hundred person long wait list. She got in. She got a year in the magnet system that I didn't (I got entry a little over a year later). That missing year was an enormous detriment to my education and growth; it was such a clear detriment my parents were contemplating whether they could afford a Catholic private school (we're not Catholic) or simple home-schooling. Had they had the gift of clairvoyance, I think they would have the moment I was denied.
* Almost certainly the gift of a computer got me interested in CS. She didn't get one, and her interests are different. (She's still STEM, likely due to our parents.)
Education is following as similar course. Things like MITs open courseware, edX, etc. are making it increasingly easy to get the educational content from top teachers regardless of how limited these teachers are.
(Having access to an education is not the same as actually getting a degree, and getting a degree isn't always the same as getting an education.)
But there has probably never been a time in history where more people had free or inexpensive access to the top educational content in the world.
I see where you are coming from on access to labs, chemicals, and equipment. But someone who has fully availed themselves of everything they can learn from free/inexpensive online classes, books, forums, emailing people, etc. is headed on a path where they have a high probability of getting access to those types of things once that is the only thing blocking their continued education.
Not suggesting that, and it's wrong anyways: the point here is that we, as a society, want people to have a basic floor of an education by the time they reach adulthood. Not necessarily MIT, but not nothing either, and the track I was on for a bit in my childhood was closer to "nothing" than it was to "decent", and it was certainly way off course if I were to shoot for MIT.
For the students that never left that track… I cannot imagine that school did much, if anything, to produce a citizen with a high-school education.
That's true, but it's the small minority of tasks that require real intelligence that often matter the most. A regular person can probably be trained to 95% the skill of an anesthesiologist just by following instructions, but then they would kill the patient during edge cases.
The same thing applies to programming too. I had an internship at a regular company, and now work at a FAANG. The developers at the regular company are likely better at programming than me, especially when it came to regular tasks, but some of their technical decisions just didn't seem to make any sense.
You’re making the mistake of assuming anesthesiologists are something other than regular people with training.
Have you considered that maybe you're just unusually talented? I know most things people do are very hard for me, and the few things I do are very easy. Learning history is an uphill slog even though I love it, but math doesn't warrant effort. For a while I just figured people that failed math were lazy and people that passed history were geniuses, but it turns out people have different amounts of natural talent. It's not the start, it's the slope
Huh. To me it seems more narcissistic for someone to say that they have no special talent and that their success is entirely due to their superior work ethic and years of study and sacrifice.
I’m just like everybody else — can get pretty decent at a lot of things.
That's a very different claim than "innate talent doesn't exist".
Yes! This is why I never said innate talent doesn’t exist.
Lots of controversies in math education between STEM professors (especially mathematicians) and K-12 math educators/researchers are rooted in this. In the community of math and science education, we educators/researchers always focus on average students who will grow into future citizens, not STEM workers. This is really a different mindset to STEM professors.
“We only cover bad art — we don’t focus on students who go on to be professional artists.”
“We only focus on bad English — we don’t focus on students who go on to be professional writers.”
“We only focus on bad history - we don’t focus on students who go on to study history or social science.”
Each of those has an AP and IB track, competitions to find elites, etc — just as mathematics should for high performers.
If as an educator, you only teach to the lowest common denominator, then you’re failing the children you’re supposed to educate.
To me, your post reads as if you’re bragging about failing at your job.
Middle and high schools start auditioning and training students into “advanced” art, preparing them to go onto competitions and onto serious careers in programs differentiated from the casual art classes.
Learning calculus won’t be enough as a professional in STEM either — but AP Calculus is the equivalent of audition-only advanced art classes. (Which exist all over.)
Both had audition programs in art for “advanced” classes (both music and drawing), where students were matched with more serious training and where their bands went to competitions and drawings were entered in regional shows.
I think you’re confusing “good, on track for professional” with “absolute top tier” — many students from regular schools go on to, eg, be animators at a studio or choir directors.
The same split exists in math:
- advanced classes you have to place into exist at most schools, eg AP calculus
- but to be the “absolute top tier”, you’re talking about STEM schools and private mentoring programs
You need the first for engineers, scientists, etc — even if they’re not going to be Terry Tao.
Germany separates people into vocational school and something closer to what we'd consider high school in the US by 9-10th grade. If you embrace the idea that some people are simply less suited for intensive math - whether it be because of work ethic, inherited IQ, lack of interest, etc. - and give them a path towards jobs that better fit their skillset, I think you'd see a lot less people drowning in college debt because they got a degree in sociology when they got weeded out of Calculus 101.
Depriving college-bound students of calculus, though, is a bad move. A lot of philosophy also involves calculus-related arguments (ie if we cut up space in really small pieces, what do we get?), so it has applications outside the STEM fields.
Unfortunately, I think it comes down to resource constraints: When I attended a relatively wealthy and large suburban school district that offered more courses than most other school system in the state, there were only 18 other students who took AP Calculus BC our senior years (and one anomaly who took it his junior year). There were a couple classes for Calc AB, mostly seniors and a few juniors. That special 18-student course was already pushing the limit on the minimum class size, a couple years prior they hadn't had enough students and didn't offer it at all.
If you'd split the curriculum into discrete math and statistics as well, there wouldn't be enough resources to support those branches. To take a chainsaw to the analogy, you wouldn't have the straight but sturdy tree trunk we have now, you'd have a stump or maybe a shrub.
Only in the same sense in which operating a car requires advanced knowledge of mechanical engineering, electrical engineering, aerodynamics, fluid dynamics, thermodynamics, etc. IOW, in no sense related to the everyday, ordinary practice of operating a car.
Sure, statistics requires calculus... and real analysis, and probability, and measure theory, and FSM knows what else - IF you're doing statistics research or trying to break new ground, or do really advanced things. But all of this is light years away from the level of statistics knowledge needed by Joe Q. Public to better understand (and not be misled by) the "statistics" frequently thrown out in news articles, government reports, etc.
Please, for the love of FSM, can we stop this HN "thing" of assuming that every mention of any mathematical topic implies that the goal of the user/learner is to do original research in the field?
Statistics is more complicated philosophically and computationally, as you do in fact need calculus to do statistics, since a fundamental tool of statistics is evaluating the area under a variety of distribution curves.
> Statistics is more complicated philosophically and computationally, as you do in fact need calculus to do statistics
This is just fundamentally false, as there are many statistical concepts and introductory statistics courses (given at some of the most renowned universities in the world) that can be taught without an underpinning of calculus. It's like arguing that you can't teach a programming class until someone understands digital logic design.
And again, you do need calculus to do statistics. Maybe you're saying: "If someone does the calculus part for you, you can use this one equation to solve this one question."
That's akin to calling a spelling test literature. You are confusing practice with knowledge. Anyone can bang on a drum. That does not make them a drummer.
Discrete distributions can be defined, even infinite discrete distributions (as sequences and series are taught in pre-calc). Continuous distributions can't be formally defined, but a lot of intuition can be given with hand-waving e.g. the area under this curve. Probably most of the class is spent with counting and probability-type problems, but plenty of actual statistics can be done without calculus - we can learn about distributions, what a statistic is, expectation, sampling, counting, probability... the list goes on.
As a professional data scientist I've never needed to use calculus unless you consider graphical reasoning on distribution diagrams to be calculus.
Most people who deal with data/statistics in their working lives do not need to know real analysis, do not utilize calculus to draw (correct) conclusions, and do not use linear algebra either. They learned all these things once and forgot them a long time ago, because they did not need them. They utilize statistics an order of a magnitude more often than they do calculus.
They are not statisticians, but people who need to deal with data as part of their job.
Sure, they may not be able to tell you what is a measurable function, but they can explain the way you should feel about a p-value.
It goes without saying this proposed class wouldn’t be based around memorizing unmotivated formulas.
To go back to the car analogy, I know why I need an engine, you might even say I know how to use the engine, but if the engine dies or I want to use the engine for some other purpose, I'm not equipped to do anything.
I don't have "literacy" with engines, I have rote memorization of a series of steps. I don't have enough information to know why the steps are what they are, nor could I know under what conditions the steps should change or what they should change to.
Will it result in students not knowing as many formulas? Of course, but who cares?
Then what? Are they going to incorrectly use anther one, a wrong one, because they understand it better?
(Searching not where one lost the key, but where the light is stronger)
Hmm, I wonder if you studied stats? It sounds as if you studied lots of math.
I did long ago, mostly forgotten, and, hmm, I don't think there's a single real life useful probability function one can understand, without knowing the underlying calculus? Except for understanding in the sense of trusting the teacher.
Which works pretty ok in medicine
You only need measure theory when working with something that is not easily replaceable by R^n, Z^n, or finite sets to meaningfully define integration, otherwise (in)finite sums and Riemann integration get you very far.
I am a bit rusty on my advanced probability theory, but IIRC the only thing that required* measures was defining conditional probabilities and expected values on zero-probability events.
Of course redoing that class without Lebesgue integration sounds excruciatingly painful.
* Not just to make proofs nicer and theorems more powerful
But what I really mean is that, as a person, I just really enjoyed calculus. I found I was very good at it, and that experience helped me understand why some people choose to focus their career on pure mathematics. I am happy I took calculus not as a means of training for the workforce, but because I found it enriching on its own. And I never would have taken all that time if it wasn't offered to me as a class in public school that counted as credits towards graduation.
Where I grew up btw we studied only calculus, integrals and stuff, no stats, in high school. (Stats in uni)
I give the edge to calculus because it allows students to go right into physics and be able to graduate college with an engineering degree in four years (saving them time and money), but any challenging quantitative material would be good for their development.
The big picture goal is to show them there is this big world of problems that can be approached with specialized knowledge and get them familiar with what it takes to gain that knowledge.
I think this is key. What exactly they study may not be quite as important as whether or not they are actually getting an opportunity to do some form of challenging mathematics.
Learn the nuts and bolts in highschool, use the intuition for the rest of your life.
Highschool calculus should be taught with computer algebra software. That's what you'll use in real life as soon as you find an even slightly difficult calculus problem. There's not enough time to teach both the symbol manipulation rules and the intuition.
But I was a slacker and that experience doesn't necessarily transfer.
And yes, I'd generally favor stats over calculus as an additional HS class; however, I am hesitant about discouraging the opportunity to take either.
I think statistical literacy is also important, but more than anything I think students benefit from learning how to think about hard(er) problems. If they learn that in statistics, great.
Generally I would say easy courses is the real problem, not content.
However, many many students enter college engineering programs with 1-2 semesters of calculus, so not having it could be a competitive disadvantage - presumably to your understanding of those first year classes.
I felt I was prepared for 2nd semester calculus when I took that as my first math course in college.
I unwittingly had a "low math" high school education, with pre-calculus only introduced spring of my last year. Freshman Engineering was a shock, with Calc, Physics, and Electrical Engineering using concepts I'd never heard of.
There's nothing worse than thinking you're accepted, prepared, and ready, and finding you're totally wrong.
No one does this in the real world. Not even professional statisticians who know calculus. In the old days they used table lookups. Today they use software.
You need to understand the concept of areas under curves. Calculus is just a means to compute the area.
There's a lot of competing / strange interests in school systems that can have well intended but BIZARRE outcomes.
My wife works in early childhood education. At one point it was recognized that the early childhood department should be more involved in helping students with learning disabilities as soon as possible. There was lots of outreach to parents to get them into free classes and education, and most importantly screening so they could get free services if they qualified / needed them.
However, it was noticed at some point by some very vocal parents that some students with specific backgrounds were refereed to these services more than others. These services were provided in and out of school, the kids weren't moved to another school or anything like that, but despite all their efforts... The result was deemed to be some sort of bias, or outright racism.
Therefore it was made very clear that they could not disproportionately "single out" students of some backgrounds for these services, that are free, to help them learn.
1. Math is a differentiating subject for getting into those competitive colleges, departments, and professions. In the meantime, the progressives simply refuse to believe that some people are just better at studying math. The logical choice, then, is to dumb down math to "level the play ground". It's the same unspoken reason why so many people pushed the magnet schools to use lottery to pick students (I actually think lottery with threshold can be a good solution, but that's another subject).
2. Progressive math educators have been advocating self discovery and that everyone can learn math in their own pace for years. What educators need to do, per the progressive argument, is to protect the fragile passion and creativity of the kids. Jo Boaler even argued that kids should discover all maths by their own. Naturally, we have to dumb down math courses, otherwise we would inevitably hurt the confidence and passion of some kids. As progressives always said: no kid should be left behind and some people got better at math only because they were socially privileged. I disagree with the progressive view of math education based on my personal experience, as so many classmates of mine simply were not interested in STEM, and maths in particular. I'm not sure why we don't accept that most people will hit a wall sooner or later when learning maths. To some it is arithmetic, to some it is calculus, to some it is abstract algebra, etc and etc. To me I definitely lost my drive when taking courses like model logic, and I certainly do not have interest or talent to get good at things like functional analysis or topology or algebraic geometry, but I make peace with it. I really don't understand why the progressives are hell bent on insisting that everyone can learn maths equally.
They are optimizing towards "High School Graduates" and "College Graduates". And if they need to destroy the value of being any kind of graduate to get there. So be it.
You can have separation of education by ability, and progress, or you can have equality, and everyone being pulled down to the same low level. And suffering for everyone. You can’t have both. Take it from someone who has direct experience with communism, which is the same mentality that drives this.
This seems a little off. What we're talking about (and what it seems like you're defending) is directing more resources towards the most gifted. It's fine to believe that, but it's an argument to give the most to those who have the most. Nobody is pulling anyone down, and communists are as happy to grant power and resources to those with aptitude and connections as capitalists are.
edit: with the constant attacks on teachers, it might be more realistic to stop aiming for calculus in high school. Any kid who manages it within a gutted public system would have gotten there anyway, no matter what situation they found themselves in. They can download calculus books and calculus lectures now; with the internet a feral education is within everyone's reach.
This might mean that there are additional costs for 'gifted' students. But that's no different than there being additional costs for mentally less advanced students: no matter what the students are, providing an education fit to their needs has costs... and a failure to provide it creates inequality when the most privileged families move their children (or relocate entirely) to places that will provide a more suitable education.
https://en.wikipedia.org/wiki/Bayes%27_theorem
Yes, it is ultimately founded on calculus. But we don't teach numbers to kindergarteners starting from Peano's axioms, and we don't need to gate the essentials of conditional probability behind calculus. In fact, doing so is positively harmful to society at large.
Also, in case anyone is also wondering, 8th grade means 13-14 years old.
* Calculus I: limits, derivatives, integrals
* Calculus II: More integration techniques (substitution, by parts, table), infinite series and convergence, basic numerical methods
* Calculus III: multi-variable calculus (partial derivatives, multiple integrals), vector calculus (gradient, divergence, curl, surface and line integrals)
ODEs were a class you could take after Calc II.
Is this uncommon in California schools?
(I think the idea that it's in any means required to be able to do STEM in college is ludicrous, but having the option is great.)
I took Advanced Calc and AP Stats senior year, which in retrospect was a mistake :P
My son is taking high school BC Calculus (one step above "AP" calculus) this year. It includes limits, derivatives, integration (including integration by parts and partial fraction decomposition), ordinary differential equations, infinite series and taylor/mcluarin series.
BC Calculus is the advanced version of AP calculus, not "one step above". Ideally people are moved into AB or BC Calculus based on skill level, but only take one AP course.
There were some other courses that had math involvement, but were more business oriented (finance / accounting type stuff) and I don't recall if they counted towards core math credit requirements.
Of course, that's not counting "mathamatical maturity" which is super important or if you're doing some specific thing that needs calculus (hello machine learning.)
- calculus if you want to do engineering
- discrete mathematics if you want to do CS
You can teach these in university, it's not a problem. Calculus doesn't need to be taught in high school to everyone but it should be available and it should be the goal state in terms of curriculum pace for everyone so that you should have no problem taking it by the time you are 17 or 18 (which is what we're talking about).
Anything else propagates back to a regressive dumbing down in an earlier year, from an already dumbed down curriculum by international standards.
> - discrete mathematics if you want to do CS
I'd guess the vast majority of software development jobs are like "gluing one API layer to another" and "writing simple-to-complex CRUD apps". Neither calculus or discrete mathematics really helps if your goal is to simply make a computer read data from database X and display it in webform Y.
I found all of the math required by my undergrad degree to be totally useless in real life programming. Whether you need any math at all will highly depend on the application domain you get in to. The most complex math I needed as a code monkey was vector arithmetic (3D graphics) and trigonometry (ocean and aero mapping navigation).
Just like it is safe to assume that most programmers aren't good at calculus. Or discrete math. Or proofs.
Most programmers will never design new solutions and are objectively terrible at their job.
I don’t think you know anything about professional programming if you think that makes them “objectively terrible at their job”. Sometimes businesses just need a website.
As an aside, those secretarial coding jobs will all go away within 50 years. They are only needed in the transition period where machines still depend on humans to talk to each other.
Many high schools offer mathematics through Calculus. You can typically get to that course at the regular high school pace if you were able to take Algebra 1 in 8th grade. If you were ahead of that pace, then you are typically left with few options outside of taking college courses.
Another way is to lower the standard to make the outcome easier to attain. It’s gross and racist.