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.'
The updated CMF looks better, but I just don't see how an educator who knows math or how to teach math could come to such a conclusion (that Calculus should not be a goal). If it is well-intentioned, what was the intention... to dumb down math in high school? Perhaps we need to educate those who are coming up with the math frameworks in math and science, or to get people who care on the California Department of Education?
> If it is well-intentioned, what was the intention... to dumb down math in high school?
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.
In the 1960s, it was the USSR. In the 1980s it was Japan. Now it's China.
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.
If we look at countries beyond the US and China... what are they surviving on?
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?
I don't doubt that the people crafting these proposals care. I think they truly believe they are doing the right thing. I personally think it's just increasingly popular, mistaken moral beliefs that inform these types of proposals. Some of the underlying beliefs:
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.
I don't think that those beliefs are a workable explanation here.
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".
> 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.
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.
I never understood why people try to make “blank slate” into a binary thing.
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 hard for me to square your claim that there's a dominant belief that all humans are blank slates of equal ability with the sheer volume of messaging I see in both government-sponsored and private media about embracing differences, follow your own goals, find your talent, etc.
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).
You are leaving out who is involved and what commercial interest will be benefitted from these policies. It is likely those commercial interests are the ones sponsoring and pushing these by finding sympathetic folks.
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.
I don't understand how these people could consistently ignore facts. Case in point, I could earn way more than Scott Aaronson or had way more social privilege than him, but you'd think I'm crazy if I claim that I can be as good at maths or quantum computing as Aaronson.
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.
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.
The assumptions above should result in ensuring educational resources are available for people who are struggling. They should also show people in which careers calculus is useful, to motivate people. Making calculus mandatory for everyone might also help on the margin, though this bears empirical investigation.
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.
Do you disagree with the following re-formulations:
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."
1. This is kind of a founding principle of the country, so...
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).
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.
I don't think it has to be related to any point you put in here. I think when STEM people comment on math education, they easily forget K-12 math education is for all students, not future college STEM students.
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.
Is calculus an important highschool goal? I feel like I may have benefited more if that time was spent on statistical literacy than calculus. I encounter stats very often in my adult life, calculus style problems are rare and I don't remember the formulas offhand, so end up just looking up what I need.
It's true that to someone familiar with collegiate mathematics, it doesn't feel too important to make that the goal - sure, why not statistics (except perhaps that a thorough understanding of statistics requires some calculus), or why not discrete mathematics, or number theory, or linear algebra, or set theory...there are lots of topics! Mathematics really is a tree with many branches, and you're correct that the high school track towards calculus just develops one trunk with a couple stunted growths, which is definitely unfortunate.
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.
The point isn't that every single student should take calculus. The goal is to make it so different students can choose the path that's best for them. California's proposed changes make it much tougher for students to take calculus in 12th grade, let alone earlier.
Stats requires calculus; you can't even define a probability distribution without some pretty advanced notions of real analysis. Discrete math, linear algebra etc. are viable alternatives.
I'm certainly very happy that I learned calculus in high school. I then got a 5 on the AP calculus test and tested in to third quarter calculus in college. It was important as an engineer to understand those concepts. I don't use them too often and I forgot a lot of it, but I really enjoyed it and I think it would be unfortunate if other students did not have that opportunity. I agree stats would be great too!
I think if the policy was "calculus isn't an important goal, we should actually teach stats" the reaction would be different.
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.
The abstract concepts of calculus are useful and will shape the way you think about and go about solving problems, even if you don't explicitly employ an integral or derivative. Rates, sums, areas, volumes, etc.
Learn the nuts and bolts in highschool, use the intuition for the rest of your life.
I would've done better in college - and probably have a better understanding of calculus today - if I hadn't tested out of University-level Calc I due to AP credit. I didn't really know what I was doing in the high-school course.
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.
The quality of mathematical arguments presented in AP Calculus are significantly higher than other "standard" high school courses. So to the extent that it is a nationwide program that promotes some careful learning, it is a big plus.
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 had calculus before college, got a 5 on the AP calculus exam, and it did not seem to have prepared me in any way for college-level calculus. I always thought that had been a total waste of one of my senior year class periods.
Calculus is important, because in high school you only superficially know the breadth of the field you're interested in, where you may fit, and what you need to get there.
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.
Lets say you need to find the probability of something happening 10% of the time to 40% of the time, you need to perform definite integration of the curve ( lets say normal curve ) from 0.1 to 0.4 on the x axis multiplied by the normal curve function. This is one of the easiest examples I could remember from my undergrad. We could solve these problems with ease at undergraduate level because we grinded hard during our high school. And also these type of problems were just a subset of the huge variety of problems presented during our undergraduate. But lets say they started teaching calculus only during Undergrad it would have become a tremendous task just to first learn about calculus then start with applying it on other subjects. I am all in for teaching calculus during high school.
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.
Goodharts law is in effect. They have a target they are trying to hit and are aiming a different way towards it.
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.
The intention was to dumb down mathematics. If you lower the bar sufficiently you can get everyone over it. Then we'll all be equal, which is the goal of this curriculum.
It's almost a case of inmates running the asylum --but in this case, it's not even the inmates but their caregivers who in their maternalistic view seem to think they know what's best for the "inmates" and are guiding them to the path to "hell" --hell being a reduced education in an increasingly competitive labor market.
I think you’re looking at it the wrong way. The job that the administrator is hired for is “make Black and White test scores identical.” With only the leverage of the school, and no broader socioeconomic levers, the only way to make this happen is to reduce all assessment to 1+1=? (Multiple choice)
I’d rather linear algebra and discrete math be the goal. Calculus is greatly overrated. I mean, sure you should take it, but IMO linear algebra is considerably more useful in the real world and most people never take it. Knowing how to integrate and differentiate in continuous space isn’t nearly as useful as learning how to count in discrete space. Most people operate in a discrete world.
My guess is the intention is to be able to say that the framework benefits disadvantaged people. But like almost any policy like this all it does at best is pull down people at the top, at worst pulls down everyone making the situation worse for everyone.
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.
> 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.
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.
Being able to integrate and differentiate is less important than understanding that P(B|A) does not, in general, equal P(A|B), and you can learn that latter fact without learning how a probability distribution is defined. Wikipedia has a proof of Bayes' Theorem without calculus:
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.
I could see replacing Calculus with Statistics as a goal for High School. It seems more broadly applicable, and both are offered at the AP level. But that doesn't seem to be this.
The perception of math by the public has led to a cognitive predisposition that math (especially calculus) is beyond the ordinary person. I wish pop culture would transform this.
What does "calculus" here mean? I'm not American, no idea what's included in that word and what's outside, in this context. Does it mean limits, derivatives, integration (Newton), maybe even some high level talk about ODEs for the "very best" schools? Anything more, anything less?
Also, in case anyone is also wondering, 8th grade means 13-14 years old.
> limits, derivatives, integration (Newton), maybe even some high level talk about ODEs
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.
At my US school around the year 2000, precalc was one option for seniors, which primarily focused on limits and derivatives. The more advanced pace AP calc course also went into integrals. Beyond that I don't really recall, but by that point you also had gone through courses focusing on basic geometry, basic algebra (using variables, factoring), and a course dedicated to trigonometry (mostly memorizing the rules around figuring out angles).
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.
Calculus without analysis, so the mechanical rules and recipes of real analysis of well behaved scalar functions that an engineering course might use (and that were used by the developers of Newtonian mechanics in the pre-modern era), limits on intervals, Riemann integration, etc.
I don't think calculus should be a goal to be honest. Or at least not as taught. Calculus could be greatly condensed to a shorter theoretical view. The ideas of understanding differentiation and integration are great. Memorizing the rules of doing it is pretty painful and likely won't stick. But that's the bulk of classroom time, homework, and testing.
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.
SF CA resident, parent of two school age children chiming in. The direction with math seems pretty dismal, in that as of right now, everyone is singly tracked together for math through freshman year of high school. This results in children who have higher aptitudes[1] to not be well served by schools. The majority of people I know who have the means opt out of the public school system, which probably makes the problem worse generally but solves a pain point for them.
[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.
I don't know if it's still there in the revision, but in chapter one of the earlier draft of the California framework it said, in a prominent place "we reject ideas of natural gifts and talents"
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"
> "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.
As a student who participated in Math Olympiads throughout middle and high school, having to be on the same track as everyone was downright painful. This type of thing really shouldn't exist.
That would drive my wife up the wall. Our school district in the Pittsburgh suburbs has 5 math tracks from grades 4-12. They just added linear algebra because so many kids were maxing out the available math curriculum.
I think I can respond to your footnote. I went to a Catholic school that split us up into separate tracks for maths specifically in grades 4-8. I was in the upper level math class for a year before they moved me. I had a teacher who celebrated and encouraged bullies, slapped children with a ruler, and threw a chalkboard eraser at me from the front of the classroom because I appeared to be falling asleep. When my grades fell the knee jerk reaction was that I was wrongly assigned to this class and it was expected to have below some magical threshold of attrition. The ramifications for me were that my old friend group would no longer interact with me the way they used to, I was immediately bored in our lower maths class, and I was now a "dumb" kid.
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.
First, I'm sorry you had a bad experience with that teacher and that it caused you problems for many years down the line.
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
It’s honestly a fucking nightmare to live here, especially with kids. I’m a crazy “leftist” where I grew up but SF/CA is bonkers (e.g. re: [1] — are you fucking kidding me? Have you not met a single real fucking living human being? My family has a “curse” we ascribe to all the clumsy folks of our blood and I assure you there are more than can be reasonably blamed on chance).
(personally I don't understand what your saying / what the point is, but I'm also tired. I'm not GP btw. -- in what ways is it a nightmare for example? What's a real human and a not real human? "the clumsy folks of our blood" -- I don't understand, how are they clumsy for example)
Something is rotten in the state of academia when looking at the evidence there are mathematicians, scientists and social scientists standing bravely in favour of a high bar and a quality education program in mathematics.
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.
> 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.
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.
The worst thing about the CMF effort is that it would only deepen the disparities between rich and poor. Public education is often the only shot poor kids have to gain knowledge and skills that might propel them into STEM fields.
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.
Well, why do you think the elites are unanimously supporting the recent equity initiatives? Because in reality they penalize the potential contenders from the rank-and-file class, while the top ladder plays by their own playback.
I don't think that's the motivation. Elites don't mind if some bright poor kids get a good education-- all the better: more advanced education means that there is a larger hiring pool to fill complex jobs at the elite's companies.
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.
Sorry for being out of the loop and asking a (possibly) dumb question: What is the stated reason that the CMF suggested these changes?
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.
I believe the gist of the argument is that when you split students into 'normal' and 'advanced' classes at a young age, the students who are not put into the advanced classes will believe they are just naturally not good at math and will give up on trying to get better because they will think they just "don't have a math brain". Here is a short blurb about the idea:
> 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.”
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.
I was in the "middle" tier math program in high school. But around sophomore year I wanted to get more into science/engineering but you can't switch tiers or catch up to those ahead of you, no matter if you're making extra effort and doing well. It was frustrating.
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.
I know it's considered good form on HN to beat around the bush a bit when it comes to sensitive culture war topics, but I think it's worth pointing for the non-Americans here that the only reason this discussion is even happening is because of the demographics which are observed after the "split" occurs.
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?
Yep, tracking students into systems like high/low early on makes it very hard to ever escape that track, as they're a sort of self-perpetuating system. That has downstream effects for one's entire life. It's a crude method of personalizing education within the context of factory education.
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.
> “When kids struggle, they immediately say ‘I don’t have a math brain,’” Boaler said. “That changes how the brain operates.”
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.
Sounds like more woke nonsense. Sounds nice and easy to a layman from a super high level but not practical or put through any kind of rigorous rational thought.
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.
> we don't want to send the message that some kids are just bad at math
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.
Why wouldn't you just let everyone take the same class and the same exams, but let the kids who have interest do extra work? Want to do calculus a year early? Here's the book, here's the exercises, why would I stop you?
From my understanding, they revisit this framework every 8 years. California is doing poorly in 8th grade math scores, so I think they want to make changes to improve that.
It seems like such a bonehead solution to the problem. Of course if you’re doing poorly in math scores, you can make math easier in the hope to increase scores.
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.”
Update - having re-read the first post about this, it seems that the issue of resources is exactly the problem. I think that opponents of the CMF would prefer to see More Resources put into careful, thorough elementary/middle school math so that middle schoolers would thrive in 8th grade Algebra. And in their view, the CMF simply lowers the bar, masking the need for more resources.
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.
The lack of resources for public education, coupled with the every increasing level of competency that students are required to internalize over time drives many of the fights in education.
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.
Ultimately the solution is better STEM foundation at the very very early ages. Universal pre school would probably be very useful. Empirically [edit: n = 1 or 2], it seems that drilling basic arithmetic and then multiplication tables early in pre K and earlier elementary will give students a better intuitive math foundation to do algebra very well. That would enable everyone to go into more advanced classes at the same time (earlier) rather than these policies which want everyone to go in at the same time (later).
As a math PhD with dyscalcula, I'm very skeptical. I was nearly held back as a child because of poor arithmetic performance, and really only started to be above average when we started on algebra. Poor arithmetic isn't that uncommon among the mathematicians I know.
On a similar note, I have a friend who majored in math at Harvard. He once told me that he came into Harvard being arrogant because in high school he was always at the top of his class in math. He enrolls in his first college level math course thinking he's got this, but he soon realizes that "higher math", which is largely proof-based, is a completely different subject than what he learned in high school. A month in he bombs the first exam. He went to the professor, who is originally from Italy, and explained his situation and how he was a star in high school. He responds in a thick Italian accent "that was not math, that was computation. In this course I teach math".
The math you typically learn in high school is very important, but I wish that we did a better job of explaining to high school students that what they are learning is completely different from what "real mathematicians" study (although I do think that computation is quite important in engineering, for example).
Did that dyscalculia prevent you from learning and attaining familiarity with the standard algorithms? That's the sensible goal of "drill and kill" in early grades, not doing routine arithmetic with high amounts of significant digits.
I wish we did probability with equal gusto as kids. I very occasionally multiply 7 and 6 in my head, but have to reason with probabilities and statistics all day long.
Agree. I think getting really good at addition subtraction and multiplication and then division/fractions is the way to go though. Have really strong math fundamentals and then learn algebra and then everything else is far easier to pick up.
Arithmetics is absolutely the most important topic to learn, since it is the basis for all other quantitative reasoning.
For example, it is really important to understand that 1 / 3 chance is the same thing as 3 / 9 chance. It is obvious to you now since you have done so much arithmetic's, but to someone who never properly learned it they wouldn't be able to properly compare those two and could think that one is a very different number than the other. Without basic understanding about quantities all other quantitative skills become worthless.
>"drilling basic arithmetic and then multiplication tables"
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.
> Empirically, it seems that drilling basic arithmetic and then multiplication tables early in pre K and earlier elementary will give students a better intuitive math foundation to do algebra very well.
This aspect of the Common Core was about recognizing deficits in conceptual understanding resulting from rote methods of drilling arithmetic.
The empirical evidence is the opposite of OP's assertion, but the end point of giving students a better intuitive foundation for higher level math is indeed the goal!
Signed,
An elementary school math teacher who has studied the 60 years of math reform in America, internationally, and worked very hard to ensure all students have a foundation to succeed in higher level mathematics
If you want to teach people methods to solve equations, limits, integrals etc. speed with basic algebraic operations is necessary.
Facility with those methods is then necessary to be able to adequately follow important proofs and gain understanding of more advanced concepts.
I don't know how you would teach people important results in their fields (physics, computer science etc., I'm not talking about actual mathematicians) without those skills.
More and more earlier and earlier doesn't comport well with child development and can backfire by making students feel incompetent. We need better teachers, probably by paying more so more talented people join the profession.
Other countries do a lot of "pre-algebra" in the later grades of primary education, when the kids are quite ready for it; "drill and kill" rote methods are generally focused on in very early grades, since they help build familiarity with the sort of rigorous, algorithmic thinking that's required for good math proficiency. This is what Russian Math, French Math, Singapore Math, etc. are built on, and the approach has stood the test of time indeed. The fuzzy "Common Core" approach pushes abstract content way too early, and ends up confusing kids as a result.
I think we could pay more, but only if we could actually get better teachers. So many teachers are bad and not worth even what they get paid (personal experience from a nominally quite good school district). My concern with paying more before figuring out how to weed out bad ones is that it would just be a waste.
Anecdote: my then 5 year old and I would "practice counting by different numbers" on the walk to school. By the end of kindergarten, she could count by everything up to 12s. In 1st grade, we started reversing it and asking how many 4s in 48 and the like, and by the start of second grade, we were firmly in adding and subtracting fractions with different denominators (though, on paper at this point, no longer mental math).
She had (has?) a solid grasp on numeracy. I recall asking her why, around 7th grade, "0.999..." is equal to 1. I was prepared to show some fancy algebra and she one upped me when she said "well, 1/9 is 0.111... so 9/9 is one and 0.999...".
She never liked math though. She spurned calculus.
In context I meant get really good at addition/subtraction starting pre K and then multiplication once +/- is mastered.
Though empirically, I don't know about age 4 but kindergarten is definitely not too young for learning up to 12*12. And once you figure out multiplication and eventually mental division, it's not too big of a leap to have one variable algebra with "move a plus to the other side to become minus" etc. The formalism can come later but it's fantastic to have some exposure to moving numbers and symbols around from an early age.
There is little evidence universal preschool would reduce academic variance later on. On cognitive measures (though not necessarily social/emotional), randomized trials of such programs tend to show fade-out (no difference between control and treatment groups) within several years.
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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.
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".
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?
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.
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.
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.
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."
I've run into it so often by a vocal minority who slander anyone who objects. Fortunately, the popularity is waning.
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The alternative would be awful, no?
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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).
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.
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.
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.
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.
Learn the nuts and bolts in highschool, use the intuition for the rest of your life.
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.
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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 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.
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.
See http://www.tnellen.com/cybereng/harrison.html
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.
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.
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.
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.
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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.
https://en.wikipedia.org/wiki/Jaime_Escalante
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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.
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> "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.
On a similar note, I have a friend who majored in math at Harvard. He once told me that he came into Harvard being arrogant because in high school he was always at the top of his class in math. He enrolls in his first college level math course thinking he's got this, but he soon realizes that "higher math", which is largely proof-based, is a completely different subject than what he learned in high school. A month in he bombs the first exam. He went to the professor, who is originally from Italy, and explained his situation and how he was a star in high school. He responds in a thick Italian accent "that was not math, that was computation. In this course I teach math".
The math you typically learn in high school is very important, but I wish that we did a better job of explaining to high school students that what they are learning is completely different from what "real mathematicians" study (although I do think that computation is quite important in engineering, for example).
For example, it is really important to understand that 1 / 3 chance is the same thing as 3 / 9 chance. It is obvious to you now since you have done so much arithmetic's, but to someone who never properly learned it they wouldn't be able to properly compare those two and could think that one is a very different number than the other. Without basic understanding about quantities all other quantitative skills become worthless.
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.
This aspect of the Common Core was about recognizing deficits in conceptual understanding resulting from rote methods of drilling arithmetic.
The empirical evidence is the opposite of OP's assertion, but the end point of giving students a better intuitive foundation for higher level math is indeed the goal!
Signed,
An elementary school math teacher who has studied the 60 years of math reform in America, internationally, and worked very hard to ensure all students have a foundation to succeed in higher level mathematics
Facility with those methods is then necessary to be able to adequately follow important proofs and gain understanding of more advanced concepts.
I don't know how you would teach people important results in their fields (physics, computer science etc., I'm not talking about actual mathematicians) without those skills.
She had (has?) a solid grasp on numeracy. I recall asking her why, around 7th grade, "0.999..." is equal to 1. I was prepared to show some fancy algebra and she one upped me when she said "well, 1/9 is 0.111... so 9/9 is one and 0.999...".
She never liked math though. She spurned calculus.
Though empirically, I don't know about age 4 but kindergarten is definitely not too young for learning up to 12*12. And once you figure out multiplication and eventually mental division, it's not too big of a leap to have one variable algebra with "move a plus to the other side to become minus" etc. The formalism can come later but it's fantastic to have some exposure to moving numbers and symbols around from an early age.
And you'll start seeing beauty in patters and sequences of numbers. The sooner the better.