An update on the campaign to defend serious math education in California
scottaaronson.blog
scottaaronson.blog
While well-intentioned, we believe that many of the changes proposed by the CMF are deeply misguided and will disproportionately harm under-resourced students. Adopting them would result in a student population that is less prepared to succeed in STEM and other 4-year quantitative degrees in college. The CMF states that 'many students, parents, and teachers encourage acceleration beginning in grade eight (or sooner) because of mistaken beliefs that Calculus is an important high school goal.'
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.
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.
[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.
> 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.
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.
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.
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.
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."
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..
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.
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.)
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
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.
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?
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.
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.
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.)
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.
[1] - I take it as a fact that different people have different talent levels for different things, but not everyone agrees with that, and disagreement on this point is a big driver (but not the only driver) in the "everyone gets exactly the same" approach that is trending now.
Edit: in the new version it has been changed to "high-level mathematics achievement is not dependent on rare natural gifts, but rather can be cultivated"
I mean, I would hope this is true.
I'm not "naturally" gifted at mathematics, but like reading, writing, and other things, I can learn them in school and got quite good at them.
Public education is like mass transit. Not everyone gets their own Lamborghini. Most have to take the bus. Its goal should be providing the best general education it can for all people and making as many people as possible productive.
If you looked at society 500 years ago you could assume that only certain people were smart enough to read and write.
Holding kids back is really the opposite of cultivating mathematical achievement. To use the reading analogy, do you think a kid that can read at 4th grade level should be forced to only read 1st grade books anyway because that's what their age is? I'm not sure what that accomplishes.
It'd be one thing to make a resource allocation argument but that's not even what this is. This curriculum is clearly a philosophical statement and personally I don't get it.
When I was in school in the U.K. we did the first and second things, including eg multiplying matrices, eigenstuff, diagonalising them, inverting small matrices, some determinant/cross product stuff, and we maybe did the thing where you solve a first order linear ODE system by converting to matrix exponentiation, though I don’t quite remember. I think we just called it vectors and matrices.
There was some useful stuff there. The problem is that it was at a course so close to the leaves of the ‘x allowed to depend on material from y’ tree that we didn’t get to apply that much (related example: we had to waste a bunch of time on silly equations in physics because they couldn’t depend on us knowing about the y’ = kx ODE)
At university we did some courses in vectors and matrices / vector calculus that went down the practical route towards physics things and useful tools, and we had a course called ‘linear algebra’ that covered the abstract algebra side of things, where everything was lemmas/theorems/proofs beginning with e.g. suppose e1, e2, …, en is a basis for a vector space V over F, …. However it is certainly possible that the US terminology (linear algebra for everything) was more common outside of the courses I took.
It wasn't until I'd dropped out of college and taught myself math, because of the interviews in this industry, that I learned to enjoy math again. My point is that you're really fucking with the social firmware of kids when you do that. Also, reading between the lines of my life, not being in that upper level math class clearly had no impact on the latter parts of my life.
I'm trying to figure out how this relates. It sounds like you had a bad experience with tracking, and there is a fundamental issue with tracking where some amount of people will have bad experiences, but ultimately you were able to achieve your potential anyway, so why make the experience bad with tracking if people will eventually get to the right level over time - is that a fair interpretation of your comment?
In the year 2081, the 211th, 212th, and 213th amendments to the Constitution dictate that all Americans are fully equal and not allowed to be smarter, better-looking, or more physically able than anyone else
On the other side of the argument you have people from the Department of Education who specialize in Mathematics Education who seem happy to lower the bar as far as possible in the name of equality.
When I was in University the Department of Education was the most woke department on campus, except for perhaps the Department of Gender Studies. We are now seeing policies that favour wokeness ahead of the best interests of the students affected by the policies.
They should rename the Department of Education the Department of Equity. By pushing frameworks like this, they show that they are less interested in education and more interested in achieving equity. They'll even privilege equity at the expense of actual education.
Do we need to mail copies of Stand and Deliver to the entire California school board? Or am I the only one that recalls that movie... based on something that actually happened... in California.
When you're highly privileged it's extremely easy to imagine that most things are positive sum and not zero sum competition because when you're wrong and your cooperation was actually a disadvantage it's still no big deal for you.
Instead, I think it's just cheap feel good measures, virtue signaling, and not wanting to take the costs/risks of bucking a fad. If you don't know whats good or bad, well cheer for the popular change and pat yourself on the back. If you do know that the initiative is bad, speaking against it may get you called a racist-- better to say nothing and relocate to where your kids will get a good education.
Someone sufficiently wealthy never has to worry about a known bad policy resulting in a poor education for their kids-- they can always apply money to the problem, one way or another.
Is it about budgets? Is it because some people might think these classes "aren't that important?". The open letter seems to suggest that it's about closing gaps between privileged and less privileged - is that it? Honest question - I'm not trying to stir the pot.
> The framework would not forbid districts from accelerating students in middle school. It does, however, recommend that middle-school students all take the same sequence of “integrated” math classes that blend concepts from arithmetic, algebra and other subjects with the goal of cultivating a foundation and comfort level with numbers.
> On top of that, the framework recommends that schools postpone offering students Algebra 1 until 9th grade or later, when it says more students are likely to be able to master the material.
> “When kids struggle, they immediately say ‘I don’t have a math brain,’” Boaler said. “That changes how the brain operates.”
https://calmatters.org/education/k-12-education/2021/11/cali...
I am sympathetic to the idea that we don't want to send the message that some kids are just bad at math, but it does seem to be a bit of throwing the baby out with the bathwater by holding back the other kids who are doing well. Even if you keep the advanced kids in the same class, the kids are are struggling are going to be well aware that some of the kids are getting it really quickly.
In my case I got a letter about summer school at a local university. So I pre-calced over summer school to get moved into calculus in high school. It honestly changed my path. I get having tiers, but once placed into one its hard to move. If I wasn't self motivated, and had the opportunity to try I would be in a different place.
Black and Latino students are overrepresented in underperforming math classes, while White and Asian students are overrepresented in the high-performing math class. That's literally the only reason there's any controversy, and if said disparity didn't exist, or if the races were reversed, then we wouldn't be having this discussion whatsoever.
If you approached athletics with the same strategy, you'd end up with a similarly wonky outcome. Consider that Asian students have always been highly underrepresented in high school football.
> “When kids struggle, they immediately say ‘I don’t have a football body,’” Boaler said. “That changes how the body operates.”
Is the equitable conclusion to change the rules of football so Asian students perform better? Surely not, right?
Downsides are that kids develop at different times, have different educational needs, have home life issues that can temporarily derail progress, etc and if those happen around the time kids are getting tracked, they may not reach their full potential.
A good education system would offer students a way to rise up whenever they're ready to rise up, let them learn at their pace, focus on mastery, build upon knowledge gained rather than schedule followed, etc. There's a lot of edtech out there that incorporate these concepts but school models struggle to integrate it into the (literally) old school way they operate. It's quite difficult to reorient school around these new concepts at scale, it has to be done school-by-school, leader-by-leader, school board by school board.
Agree its complex, as it may be the 'best of the worst' option for certain contexts. Anything involving balancing equity/access/etc is like that.
This really jumped out at me.
I didn't read any context, but students CAN and SHOULD learn to struggle. Productively. Without thinking they are failing.
Imagine you thought everything should come easily? That's not my experience in the world.
The fact that students (are reported to) shut down when faced with difficulty is a failing of the educational system and something that should be worked against.
I didn’t think our understanding of the brain was that advanced yet. AFAIK we run some experiments and observe results, but we can’t explain why those results were observed.
Which is useful and awesome from a learning perspective, but extremely worrying we use it to craft public policy.
But some kids are just bad at math. Some kids are bad at sports, music, dance, etc. Some kids are good at some things and kids are good at different things.
It’s sad that the state is proposing these changes. I remember in school there were kids who argued “algebra is stupid, who needs it, why waste time” and there were one or two sympathetic teachers who would respond “well, I rarely have to use algebra to balance my checkbook” or something silly. It seems like those kids have grown up, gained power, and are literally pushing the argument that this math isn’t important.
> "Since achieving a solid foundation in mathematics is more important for long-term success than rushing through courses with a superficial understanding, it would be desirable to consider how students who do not accelerate in eighth grade can reach higher level courses, potentially including Calculus, by twelfth grade. One possibility could involve reducing the repetition of content in high school, so that students do not need four courses before Calculus. Algebra 2 repeats a significant amount of the content of Algebra 1 and Pre-calculus repeats content from Algebra 2. While recognizing that some repetition of content has value, further analysis should be conducted to evaluate how high school course pathways may be redesigned to create a more streamlined three-year pathway to pre-calculus / calculus or statistics or data science, allowing students to take three years of middle school foundations and still reach advanced mathematics courses."
At face value, that suggests that the root problem is that students reaching middle school Algebra 1 aren't ready and need more remedial math instruction. As an Electrical Engineering professor, I can definitely attest to the fact that students reaching higher level classes with a precarious foundation are rarely as successful as those whose foundation is more solid. I suspect Scott would also agree that barely passing calculus in high school is not an adequate preparation for a career in data science. As a parent of a kindergartner and a second grader, I can also see that there is opportunity to push more math further down, but even at that age there are kids who have a huge variability in how they view their math.
With regard to resources, I thought this statement in the CMF was particularly insightful:
> “While early tracking of students into low-level courses has been problematic, there is evidence that thoughtful grouping of students to ensure they receive high-quality instruction geared to their needs at a moment in time can be helpful. This includes students who need to fill in gaps in their prior learning and high-achieving students who are ready to be more intensely challenged. It is also true that teaching heterogeneous classes requires greater skill for differentiating supports than teaching in classes where the range of performance may be narrower, and should be accompanied by high-quality professional development to enable success.”
I agree with that!
Want to know the one resource that schools can never give students? Parental involvement. It also happens to be the #1 variable in success for students.
If parents aren't checking their children's grades on a daily basis, asking what they're studying, staying in frequent communication with teachers and the school, there is nothing that is going to replace that.
My oldest son was on a robotics team during high school in which we were a minority. All the other parents, their kids were studying calculus by the end of high school. I asked every one of them how their kid was able to do that and each one of them shrugged and said, "nothing".
To them, "nothing" was their child spending 2 hours per day at Kumon after school and a few hours on the weekend in addition to their constant checking of their work and insistence on academic excellence.
I can understand how an electrical engineering professor like yourself is rightly concerned about your incoming students having less calculus proficiency. On the other hand, there many (possibly far more numerous than EE) education and career paths that would benefit from better general numeracy but not necessary calculus.
The terrible thing here is that these goals should be set against one another due to resource limitations. The blame for that lies with the broader societal inequities and their reflection on the educational funding system.
It's saying that it is impossible to get the bulk of students (with the teachers we have) to complete the standard mathematics curriculum by the end of high school. This has always been true, in all countries, hence "streaming". But now they're saying let's do away with the advanced stream, therefore students can't complete the last part of the US mathematics curriculum (which is called "Calculus"). Rather than justify that move in terms of cost or fairness, we're going to say "because Calculus isn't important now".
This is obviously completely bogus. If their assertion that Leibnitz-style calculus isn't important now, they could replace it with Linear Algebra, Number Theory, or some other "important now" subject.
Add to that, the fact that in the US the names of high school mathematics classes are by convention. "Geometry" isn't all geometry, for example. And "Calculus" isn't all calculus. The classes are really : Math 1, Math 2, Math 3, Math 4, AP Math.
I get the sense that such rote methods are no longer encouraged and a lot of the "new math" in Common Core is aimed at approximation and reckoning so that students won't rely on memorization.
1. Make it possible for HS students who are interested in Calculus to take the course under the instruction from a college profess on the high school campus. That way it would be set up for students to get college credit for the class and they would not need to travel to a college campus or deal with the AP exam system.
2. Make it possible for HS students who are not interested (Yet) or at all to graduate with out taking the class. Lots of student are ready for Calculus until college anyway. No need to force them on a single path IMO.
https://www.khanacademy.org/math/ap-calculus-ab
When we focus narrowly on what brick and mortar Public HS should and should not be teaching in regards to math curium we sideline all the pathways for learning available outside of this bureaucratic model.
Folks in most places in the United States a least can check out a Chromebook from a local library and use their free internet to access this information.
Students could also take additional ge's at college outside of school hours, and I entered University with about 80 credits
> 5,972 Math Professors in USA [2]
1. There are simply not enough college math professors for this to work.
[1] https://www.zippia.com/high-school-mathematics-teacher-jobs/...
[2] https://www.zippia.com/math-professor-jobs/demographics/
Which country do you mean? As far as I'm aware, it's the opposite: countries like China and South Korea have math as a required subject on their college entrance exams for all students. The non-STEM students may have fewer requirements, but I'm fairly sure they've still learned more than the average American high school student.
https://en.wikipedia.org/wiki/Gaokao#Reform_of_the_National_...
https://en.wikipedia.org/wiki/College_Scholastic_Ability_Tes...
https://medium.com/@yujia_jo/2016-jiangsu-gaokao-national-hi...
I do know there are countries which allow specialization at the high school level (India, Italy, Romania I believe).
I found this interesting list, which is more complete than I can be on this. I don't think it proves that countries with great mathematical cultures always allow specialization in high school, but it does have some examples of this.
https://en.wikipedia.org/wiki/List_of_secondary_education_sy...
I think you are right that we need more flexibility on the high school level though, at least for juniors/seniors that have safely passed the earlier requirements. I like that we have a chance to dabble in various subjects in college here before committing to something, but it would be a lot more efficient if some of that could be done in late high school.
I thought about this for a bit. Suppose we kept the number of teachers (or courses) fixed, but eliminated the gen-ed math requirement. You would end up with a better teacher : student ratio, and I believe this would benefit the serious math learners. And so by not doing this, I believe we may be implicitly impeding them.
My guess is there are lots of ways wasted resources (and I do believe any class which could be perceived as "rocks for jocks"-type is wasted resources, and may actually be a damaging use of resources - I say this as a TA for the pre-major courses*, where the students are constantly and overwhelmingly negative about the whole thing) are implicitly impeding the serious students.
(And, as a positive note, I think the students not characterized as "serious students" for the above discussion could have a chance at meaningful and valuable contribution elsewhere, if we weren't putting so much effort into wasting their time.)
Edit: Changed "lower level courses" to "pre-major courses"
I came to that realization a decade after college when I was digesting 3Blue1Brown's series on Calculus for fun and had it finally click. Before then I was basically a Chinese Room that was able to solve calculus problems via pattern matching (i.e. "oh, this problem fits the shape of these rules, etc.") without really understanding how calculus works.
I think this is the key. No, calculus isn't incredibly applicable in day to day life. But understanding how it works builds critical thinking skills that absolutely are. Math should be about critical thinking, not memorizing. Common core seems to be moving us in the right direction.
There were three factions in WW2 and the Cold War. It was about preventing the assault of extremists on both sides.
One side (Equitable Math) is engaged in a discussion of white supremacy in Californian math pedagogy. The other side (Moses Charikar, Scott Aaronson, et al) is arguing against a weakening of math standards.
It's a lot easier to just move the goal post than to actually achieve the goal.
The first is The Dawn of Everything, by David Graeber. I’m left with the notion that what we now call science and mathematics emerged from tinkering, persistent experimentation done mostly by women, and that what we have now disciplined into mathematics emerges from the systematic study and production of pattern (basket making, the organization of communal structures), and Graeber seems to argue against the hierarchical gender divides when viewed across the broad stretch of human history.
Rachel Thomas https://www.fast.ai/2022/03/15/math-person/ also makes a case that math is something that all people do.
I think that the larger point both are making is that disciplines don’t have to be the way they are constructed now.
My only “political”’statement would be the hope that states (particularly the u.s.) would invest as deeply in mathematics education at the primary and secondary level, for all of its communities, at the level of investment in big science and big military projects.
How do I know that? I was one of such students. I made it because my teachers pushed me. They used carefully designed homework to show me that I didn't really have the deep understanding of math as I thought. They used homework to show me that my understanding of Newton physics was inadequate. They used midterms to show me that I didn't really intuitively grasp algorithm analysis (I didn't know how to rigorously and succinctly argue that two pivots and one pivot are really the same in quicksort). I grew to enjoy STEM and CS precisely because I was lucky to have tough yet encouraging schools. Dumbing down courses would have taken away the push I needed, and I would likely have a lot more failures in my life.
How we got to this point is perhaps a lot more complicated and politically fraught, but it has to be dealt with. Administrators and state education leadership are often simply responding to the incentives and avoiding dire outcomes suggested by the data. They have to craft palatable excuses for it, and it's ultimately a waste of time to engage those excuses on face.
I think you are touching on theory I have that Americans are becoming increasingly resentful that technical skills are becoming more necessary for middle class living. This is a huge driver of the pervading sense of precariousness. For some reason there’s a huge math phobia in this country
I remember the huge blowups over “Common Core” years ago, which included new ways of teaching math concepts. I got to see some of it in action during COVID as I sat in on elementary Zoom school with my kid. I have to say I was impressed; they used techniques I did not recognize, but they seemed to work well.
I don't think the previous standards of rigor were ever so high for most students.
When I was growing up in the 80s and 90s in the suburban working class Midwest, the vast majority of the students in my high school didn't advance beyond algebra. This was in a well resourced school district.
But that was a time when it was felt that most people didn't even need to know anything past basic math to be employable. Times have changed of course, but the vast majority of educational paths, even STEM paths, don't require calculus.
However a great many do require basic data analysis abilities, so it's reasonable to emphasize those.
This shouldn't be done at the cost of offering calculus as an option for students who are prepared for and motivated to do it, though. Of course in the end this is about cost. Assuming the same resources, to teach a broader set of students data science will require reducing the availability teaching resources for something else.
I also recall that in the 1980s in some places there were programs that taught calculus at public community colleges for students who were on an accelerated academic path in high school. That is another option to consider.
How many folks are really willing to do that? The problem is that actually discussing the perceived root causes of this issue will get you shunned out of polite society.
Why is this a rational fear? It seems shocking to me that students can't perform at the same prior standards given their parents are more educated than the prior generation's parents.
You can pass and pass and pass people, but eventually an employer is going to need someone to do the job, and they will make sure they get that someone. So there will be a hard standard sooner or later. The only choice is whether we give all students a chance to meet that standard.
It's a warning to any parents of younger children: Unless something has changed radically in the past 8 years, your child will be put into a math track in 6th grade: Separated into standard, accelerated and advanced classes. Which track you're in is determined by grades, standardized tests and teacher input after 5th grade.
This track determines which classes you can take in 7th and 8th. If you were in the advanced class, you will have finished Algebra 1 by the end of 8th grade. This allows the student to begin 9th grade taking Algebra 2, and then extending from there so that by their senior year they can take AP Calculus.
If you want little Suzy to be in more advanced classes, you better be prepared to be the most vocal Tiger Mom Karen you can imagine, because you'll have plenty of competition. As a result, almost no child moves between tracks. And in fact, in my opinion, the difference between normal and accelerated is so little, I'm pretty sure it's there just to give those children somewhere to go.
In other words, if your child doesn't demonstrate math skills as an 11yo, they will unlikely be able to take AP calculus 7 years later without doing something extra like taking summer classes, redoing an entire school year (an option a fellow parent I know took), or extraordinary effort like that.
Even if the MVLA education system isn't exactly the same now, or you live in a district that does something totally different, or even if you're in another state, I suspect this sort of thing is happening everywhere.
I personally was happy my son was in accelerated classes, right up until 9th grade when I realized how this circumscribed his future options for classes. In the end he would never have wanted to take AP Calculus, so it was fine. But I personally felt like I had fucked up as a parent because I simply wasn't paying attention. Planning out your kid's future math classes in detail in the 5th grade never crossed my mind, or if it had, I would have dismissed it as ludicrous overparenting! Had I known, I might have sent him to a math camp or something if I had realized how important the difference between a B+ and an A in math was at that moment. And he my have really gotten into mathematics as a subject. I really don't know.
So anyways, that's my experience. California is such a massive change from where I grew up in rural NH, I honestly can't imagine where to begin to fix a system with so many millions of children from such a varied socioeconomic and cultural backgrounds. I barely got my one kid through the system unscathed, and I live in one of the wealthiest districts in the country.
I absolutely loved learning Calculus in high school in Math and 1st and 2nd year of University. I consistently got 97+% on my grades.
And I've never had to use it in my Computer Science degree or my 20 year Software Engineer career since.
Am in a bubble because I don't spend much time in the Machine Learning domain?
If you take the position that calculus is the concept of limit and all its consequences, then things like exp() and log() are calculus and it's hard to get anything done in CS without those. In this view, saying that quicksort is O(n log n) is a statement of calculus.
If you say that calculus is derivatives and integrals, then I'd say that calculus is not that important in a digital world, and that discrete math is much more useful. However, discrete math is harder than calculus, but you can use calculus as an approximation to the discrete answer (i.e., compute the integral if you don't know how to compute a sum, or use a derivative to approximate a difference). Ironically, this is the opposite of the old attitude that the continuous answer was the true one and the discrete answer was a poor man's approximation to the true one.
I got to the point of writing my own toy neural network from scratch, seeing backpropegation, figuring that I'd have to use the chain rule myself on my forward pass, understanding what "automatic differentiation" was and why it's important, and decided "screw that I'm not putting myself through this hell again" and decided to look into https://en.wikipedia.org/wiki/Derivative-free_optimization
You can literally just find your own hetrodox subset of scholars in your field who are like "calculus? pfft!"
On the other hand, I feel psychology and stats are the biggest missing pieces in K12 education. We need to build greater awareness of human needs and fallibilities, and awareness of how to make decisions in uncertain environments by understanding probabilities. Both are about developing a nuanced perspective on life and making better, more sober decisions, and they build a great deal of empathy to boot.
Finally, psych & stats are inherently relatable - everyone deals with people and has to make decisions. So much of the K12 experience isn't relatable, which is why students often hate school.
I agree with this problem statement.
It's a MS Word document. How am I supposed to find lines 1226-1239? They might want to actually quote it.
https://www.groovypost.com/howto/quickly-go-to-a-certain-pag...
> We write to emphasize that for students to be prepared for STEM and other quantitative majors in 4-year colleges, [...], learning the Algebra II curriculum [...] in high school is essential.
Problems with math are one of the most common reasons why students encounter difficulties in STEM education and careers. The most common problem is difficulty with high-school level algebra.
I agree, fundamentally, with the relevant premise of the CA effort here (and agree with Aaronson's criticism of its implementation). That premise is that you shouldn't have to be on an accelerated track in middle school in order to take calculus in high school. And yet... the fact is, we get a lot of adults in college or graduate school pursuing STEM degrees, who have shaky foundations in high-school algebra.
Just looking at the "typical" math track in US high schools it does seem a bit arbitrary. Algebra I, Geometry, Algebra II, Pre-Calculus, Calculus--this is the most common math track I see, with accelerated students starting Geometry in 9th grade, Calculus in 12th grade.
The thing is... individual performance is highly variable in math classes, and to make sure that everyone gets good foundations in mathematics, we see high-school mathematics curricula that repeat the core algebra concepts in different classes. This repetition and focus on fundamentals is why the division between classes seems so arbitrary--what is presented as a sequence of classes is really more of a unified curriculum spread across multiple years. When you combine these two factors (variable performance, repetition in the curriculum), you end up with a population of high-school students who develop good foundations in algebra early on and are bored by the repetition, and a population of students who really benefit from the time spent mastering algebra, and it's hard to serve both.
I think we can figure out a way to let high-school students take AP calculus in 12th grade without expecting them to take Algebra I in 8th grade, and we don't need to push everyone into calculus faster in order to do it. And yet, my experience with high-school education in the US has left me very cynical about it. Letting students progress through the high-school math curriculum at the right rate requires a kind of "personal touch" that seems to only happen to individual students when their parents are involved, but not pushy. It's rare. The school system would rather do the easy thing (everybody moves in lockstep to the next class in the sequence), and parents are largely either uninvolved or overinvolved.
(This is more or less what the article says, I'm agreeing with the article.)
I feel as though there is too much focus on giving everyone more or less the same type of mathematical education in high school. This is probably due to limited resources (ie teacher availability and class sizes), but ideally there would be room for a more varied approach wherein students don’t need to have every year build on the next if the aren’t STEM tracked. Too many students fall behind and never are able to recover. Math class just becomes dead time, and those that do make it to college end up retaking the same subjects over again.
This sounds contradictory to me.
Moreover, how does IQ differ in this respect from "capacity and willingness to do hard work"?
In a excellent school district in a suburb of Chicago a district goal was adopted to reach equitable outcomes in higher Math. In a nutshell since black students scored statistically lower on AP Calculus this was seen as a failure in the school district. Despite increasing the number of black students able to pass AP Calculus the school district looked to cancel offering the course since Asian and affluent white students still scored statistically higher.
The idea that all races, genders, or whatever categorization you can come up with must have the same equitable outcome is a flawed goal. Education used to be about taking a student where they are and showing improved learning outcomes.
Equality used to be about striving for equal opportunity. The shift to conflating equality with equalized outcomes simply doesn't work.
Note that to the extent there was a shift, it took place in the 70s.
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(spywaresolution4) Gmail
watsPP + 1 412 227 3381
Dicord (spywarsolution#3558)