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I've heard a good argument that algebra specifically should be moved the other way: teach it in elementary school, when kids are first exposed to arithmetic.

I think the core problem here is trying to teach all kids math in the same way. Not all kids are equally capable of learning math.

Some smart kids can learn algebra in elementary school. There are plenty of students capable of going through the standard curriculum and finishing calculus by the time they are 13 or 14. For these students, the faster the better. Go into even a "good" second-grade math class and you will find some of these students, acing tests and completely bored with everything taught in class.

On the other hand, some students will never be able to have a good understanding of pre-calculus. It doesn't matter if they take it when they are 17 or in college, they will never really get it. Or at least, there is no known teaching method capable of doing it.

How can you possibly design an education program that takes this into account? You just can't teach all 15-year-olds the same math curriculum without wasting the time of many of the students, either it's too hard or too easy.

It's not really a problem that not everyone will get it the first time. There's always the second, or third attempt. The latter handled by differing levels thru high-school math.

It's notable that Cupertino has some remarkable public schools, with high schools that have about half of their math faculty as Calculus teachers, and half of the students incoming from junior high are on track to finish Calculus by their junior year. Some students continue their math education at local colleges.

Part of the success is most certainly the student culture, which pushes far harder than the faculty. It's one thing for a school to offer opportunity for advanced math (and in that sense Khan Academy already offers this); it's quite another for peer pressure to demand it.

> with high schools that have about half of their math faculty as Calculus teachers

What does this mean? Is there some special qualification or licence to be allowed to teach calculus?

Most of the student body is taking/has taken some level of Calculus.

Both my parents were public school math teachers. There are gradations of teaching qualifications* in some districts and, even where there aren’t, the math department head usually knows which teachers are incapable of teaching beyond Algebra II or incapable of teaching specifically Calculus. Presumably, if a teacher has ever been assigned to teach a Calculus class, they are a Calculus teacher.

* Concrete example: our district had a shortage of high school qualified teachers. Middle school teachers were invited to apply with the understanding that they’d teach only general math, pre-Algebra, Algebra I, and Geometry.

I don't really get how you can be a mathematics teacher and not know enough to be able to teach calculus? Like surely you've done calculus yourself?

One of the high school math teachers explained to my brother “the square root of 2 is irrational because it says so on page XYZ” (true story). I’d say it’s fair to conclude that there exist math teachers who’ve taken Calc who can’t teach it.

Just because you know it doesn't mean you have the pedagogical skills to help somebody else know it. By way of example, I am more than competent at introductory computer science yet I am wholly incompetent at teaching it--were a student to need help with something that basic in CS, I wouldn't know how to go about helping them understand, since I don't understand why they might not be understanding something.

This past trimester I taught a Computer Organization & Design class to middle school students for the first time, with curriculum that I developed.

Observations:

* Wow-- the brighter half of MS students can do nearly as well as bright undergraduates on this material if you go a little slower and make some allowances for attention span. The neuroplasticity, recent memory of learning arithmetic, and lack of fear of the material really helps.

* Holy crap-- I had no idea before trying to teach this how much number theory that I take for granted. Like, I forgot to cover how many possibilities there are for n-digit numbers in different bases and some early lessons on number systems took this for granted. Figuring out what you need to cover, and how, for something to make sense later is hard.

I think a few years ago I could have written a comment like this, but now I've been a teacher for a couple of years now.

It is hard to teach something. Being able to do it is isn't anywhere near the level of expertise needed to:

* Do it up on a board at a moderately quick but consistent cadence, with engagement and narrating the whole part through.

* Not be taken off track by inevitable distractions.

* Understand the stack of necessary knowledge in the ways needed to figure out what a student is missing.

* Completely reinvent your lesson plan when you find the students unexpectedly lost, providing whatever the missing bit of knowledge is to get your class back on track.

* Handle unexpected diversions and requests for deeper knowledge on the fly.

I didn't realise teachers went under some kind of assessment to teach particular parts of their subjects.

In the UK a mathematics teacher is just a mathematics teacher. They all have to teach calculus because that's part of the curriculum. If you couldn't teach calculus then you couldn't be a mathematics teacher.

(That's not a boast - we're poorer and less effective at maths than you are.)

What do you do if you're a mathematics teacher but can't do the calculus part? Handover to someone else for just that part of the year? Must be a bit professionally embarrassing?

> I didn't realise teachers went under some kind of assessment to teach particular parts of their subjects.

This is hugely variable within the US.

> In the UK a mathematics teacher is just a mathematics teacher. They all have to teach calculus because that's part of the curriculum. If you couldn't teach calculus then you couldn't be a mathematics teacher.

I think there's a lot of math teachers who would be a bit iffy teaching Calculus that have department chairs that schedule them to teach earlier math instead.

> (That's not a boast - we're poorer and less effective at maths than you are.)

I don't feel like our mathematics system is very good at all.

Sounds like the difference is in the UK a teacher teachers a cohort of students through all subjects, not a subject for all cohorts of students. We have one maths teacher who teaches you everything.

Maybe they're terrible at half of it? I don't know.

> Sounds like the difference is in the UK a teacher teachers a cohort of students through all subjects, not a subject for all cohorts of students. We have one maths teacher who teaches you everything.

Really? That's very interesting.

I had a different math teacher every year / level. I'm at a smaller independent school, where you may have the same math teacher a couple times but people definitely specialize-- we have the teacher who teaches mostly 6th grade math and pre-algebra, and a teacher who mostly teaches algebra I and geometry, and so on.

On the one hand, you lose continuity. But you also get a fresh start each year, with a new teacher who probably teaches the things you had trouble with in a different way...

This was not my experience in the UK - at a grammar school with a specialist mathematics designation (in the late 90s). There was a wide ability gap between the mathematics teachers, and very few could practically teach calculus.

Further, at that point in time, calculus was only part of the A’ level curriculum, and most people would never encounter it unless electing to do that course.

Most high school math teachers don’t know calculus, it’s as simple as that. Even at good public schools, there is often just one in the math department that can teach it, but maybe also some science teachers could, too. In my school, some of these were great, highly engaging teachers for more basic math classes when teaching average and below average students.

A typical American high school will not have enough students enrolled in calculus to require more than one calculus teacher.

Is it that their parents are indifferent to their kids' education? Or they don't realise that calculus is important in many fields? Or they just want their kids to become a lawyer?

I am not aware of any surveys, but I would guess that many parents think it is the school's job to motivate students to study further math, while many schools think their only job is to teach enough students to pass the basic standardized tests, and neither side has properly communicated their beliefs.

Other parents likely believe that further math isn't necessary because they have done well without it and think their children will as well. They do not understand that a large portion of the world that was impoverished when they were growing up now have functioning education systems and are now able to compete for the same cushy jobs.

Seems like it's already part of the California standard for 1st grade: https://www.ixl.com/standards/california/math/grade-1

E.g. "Find the missing number: _ + 1 = 5". Same thing with subtraction.

Based upon my experience with my 6 year old (in a public California elementary school), they actually seem to teach math in a huge variety of ways in 1st grade. Word problems, number problems, relationships between plus and minus, and boatloads of ways to solve any given problem.

I think they introduced it in Kindergarten too. But it was way beyond her at the time.

> Not a huge leap from there to the concept of equations, inverse operations, and solving for unknowns.

Unfortunately, for a child it typically is a huge leap.

There are certain abstractions that young children simply cannot do. The ability to do certain abstractions is distributed along a bell curve (edit: of age), and maybe your child is on the very far left of that curve, but most kids won’t be able to handle the abstractions you speak of until at least a little later.

To be fair, that “later” is definitely not 8th or 9th grade (much earlier), but it’s also not immediately after learning basic arithmetic.

The only thing distributed along the bell curve is adults who are willing to help the kids. Yes for some kids it will come pretty easy, even at very young ages, but even very young children can learn these concepts if you would just spend some time teaching it to them.

When you have a system set up to force one adult to teach 30+ children you will never be able to spend the time required to ensure everyone gets the concept, but that neglect comes at the cost of compressing the learning in later years which still only benefits those who have high aptitude. Which is exactly why they are pushing it back further, not because it will help kids learn more, but because more children will have the ability to learn it without much interaction from an instructor.

Incredibly uncommon for 30 kids a class in elementary school. Also many students can barely handle arithmetic without massive handholding. Algebra in elementary school is out of the question

You are correct that 30 is on the high side, but I have seen classrooms in Philadelphia with 30 students to a single teacher ( 3rd grade, during the 2005 school year so it may have changed since then). However even at lower ratio's the point remains, you need to hand hold the kids, and we don't give our teachers the time or flexibility to do that.

> but even very young children can learn these concepts if you would just spend some time teaching it to them.

Many decades of cognitive science disagree with you.

See Piaget for one of the earlier researchers on this topic. Even though his original ideas have been refuted or refined, the basic premises of his ideas are still prominent among the large group of modern researchers that some people dub neo-Piagetians.

Many decades of the lived experiences of children in other countries disagrees with that research. Children in eastern European countries learn advanced concepts from as early as second grade. In asian countries they are introducing Algebra in 6th grade or earlier. This is the problem California finds its self in. They have wonderful "research" showing that kids pick up the concepts better if you wait till they are older, but they ignore the lived reality of the rest of the world, that with the correct teaching you could actually have them doing this work at a much younger age.

Even here in America, tutoring schools such as Russian Math or Kumon are successfully teaching advanced topics to children at much younger ages than the public schools. I know the temptation is to just say its its high aptitude kids that are being sent to these after school programs, but it's not. It is just rich kids whose parents want to give their children a leg up. Almost all children could learn the topics, but we won't put in the effort

> Many decades of the lived experiences of children in other countries disagrees with that research.

Based on your reply, I’m fairly certain that you are not familiar with the research and your not familiar with what is actually happening in Eastern Europe, Kumon, etc. in relation to that research. As a professional in this area of research, I feel fairly qualified to say that I am.

I am happy to agree to disagree about what the problem is here, but I strongly encourage you to look deeper into what is actually happening in the research and the learning environments you are praising/promoting before you propagate a loose collection of anecdotes as data that points to a clear solution.

I am glad that we can disagree, and I hope you would spend some time considering that while the research has progressed we have experienced a rapid decline in the subjective and objective quality of education.

Just because I completely reject the ideas you may be involved with researching doesn't mean I am ignorant of them, I just haven't been convinced that they are based in reality.

> we have experienced a rapid decline in the subjective and objective quality of education.

Interestingly, on this we agree. That’s why throwing more bodies at the problem (your initial suggestion) is not a solution that I think would work. When a typical educator is asked why a certain teaching or learning method works, their answers tend to be both incorrect/incomplete and remarkably shallow (e.g., it worked for them when they were in school).

The research available points at multiple tried and true ways of learning, but getting these ways of learning implemented in households and in the public school system in the US is a Sisyphean task.

Why would we "see" Piaget if his work, as you correctly note, his theories have been refuted? Educationalists, who have never personally and successfully taught anythingmarried simply wrong theories of learning, are a good 50% of the problem.

You left out the “refined” part of my comment.

Most of the broader ideas that he espoused and is famous for were refined in a way that made them accurately generalizable. Typical research timeline where an idea is introduced and is refined over time.

Some of the specific ideas he stated (and typically backed with data) was not generalizable to broader populations and was therefore refuted as an absolute. The data and conclusions were not necessarily wrong for the sample of the population studied, but they were not found to be as broadly true as initially believed.

This additional research led to the refinement of his theories, to great effect, imho.

Counting and arithmetic are themselves abstractions. The concrete reality is "this is a raspberry. this is another raspberry" etc, and that's about where my kid was at 2 (his loveys are named "Owl" and "Another owl"). Then we learn that there can be 5 raspberries just like there might be 5 ducks or 5 bricks, and that taking away 1 raspberry is like taking away 1 duck or 1 brick.

I could believe that the ability to abstract develops at different times in different kids. But I'd caution about taking Western research and generalizing it across all cultures, or generally assuming that kids are less adaptable than they are. There are existence proofs for kids learning algebra at much younger ages than most American schools do (3rd and 4th graders do in eastern Europe, as do Americans who participate in RSM).

I hate to be that guy, but... it's not irony.

In this case, it's design. Increasing social injustice is the whole point, because that ensures increased political power for those who campaign on social justice.