Well that's stupid. If anything they should introduce it sooner. There's tons of research that show the sooner it's introduced the better they are able to digest that (as in it becomes more second nature and are later able to reason with it and build off that knowledge better).
It's hard to introduce Algebra much earlier than 8th grade - the students just don't have the mathematical maturity for it. What you can do though is challenge and expose them to slightly more complex applications of ordinary math (often stated via increasingly elaborate "word problems") that make the introduction of algebra a lot more natural when the students are finally up for it. You tend to see this approach in the various "Singapore Math", "Russian Math", what have you.
> the students just don't have the mathematical maturity for it.
Doesn't this get at the root of the problem though? Some students have the mathematical maturity for it. Others don't. The arguments seem to be between "we should present these concepts early for the benefit of the students who are ready for it" vs "we should delay these concepts until all students are ready."
But different students have different levels of mathematical maturity. the problem seems to stem from working in a paradigm where everyone at a particular age has to learn the same thing. It seems we should be moving in the direction of more personalization rather than less.
> The arguments seem to be between "we should present these concepts early for the benefit of the students who are ready for it" vs "we should delay these concepts until all students are ready."
Yes, I'm saying that the argument should be quite a bit broader than that. There's much that could be improved in how we expose students to more advanced math in early grades, and there's also much to learn from these well-established teaching approaches. If you do it badly, it's less likely to work.
> Some students have the mathematical maturity for it.
I think part of the problem here is definitional.
* My oldest son's a few years ahead in mathematical understanding. He understood algebraic concepts very early, as presented in Singapore math and then through enrichment and bantering about various problems with his mathematically-inclined parents. Kids can absolutely get algebra and learn the rules early.
* But even if your 4th grader understands all the rules of symbolic manipulation, and the general concepts behind them... that's only part of what is taught in an algebra class aimed to 13 year olds. There's an emphasis on systematic process, checking for mistakes, carefully matching terms that is likely to be unnecessarily frustrating to younger kids.
* Many programs for gifted youth go in exactly the wrong direction: emphasizing more rigor for the gifted youth, harder problem sets, etc. He took Algebra I with CTY and the number of opportunities for sign mistakes or mismatching coefficients per problem were dizzying.
I believe we should be:
* Throwing ideas at primary kids, with small numbers of degrees of freedom to make the problems manageable for a population that developmentally has less discipline. The complexity of the ideas involved can scale based on what the kid knows so far.
* Throwing deep process and carefulness at older kids. The complexity of the ideas involved can scale based on what the kid knows so far, but the process and accuracy expectations can scale mostly with age.
Seems entirely arbitrary (if not false?). I was taught the basics of algebra and even trig (at least, how pi and radians work) in the 4th and 5th grade
You shouldn't optimize for the average student; you should stratify instruction so students of different levels are pushed to their limits and beyond.
Sure, some students will not be able to handle algebra as early as others. That doesn't mean you hold back the students who can. That's disastrous public policy, at the very least.
> You shouldn't optimize for the average student; you should stratify instruction so students of different levels are pushed to their limits and beyond.
While I personally agree with your point, I think your statement makes the case for the folks who developed this policy.
The “problem” that these policymakers see is that students, when stratified, are not stratified across certain groups in a proportion that is similar to the population.
For folks who focus on equality of outcome, this is a problem.
For folks who focus on equality of opportunity, it is not a problem.
I agree, but stratifying too early can be harmful.
I believe what's likely to be best for everyone is:
* Keep everyone on the same track through early elementary, but we need to work hard on making the classrooms encourage everyone to be curious and stretch themselves. Games and puzzles are the answer here.
* In upper elementary, start to offer differentiated instruction within a classroom.
* In middle school and beyond, have true stratified tracks (which this article recommends not doing).
So you propose we handicap the geniuses to make the rest feel better? Who do you propose will invent great things to keep our world thriving if you’ve made all the geniuses average?
If you're going to teach math to kids in a normal school and not an academy for geniuses, before stratification based on level makes sense (and there's evidence doing this too early is harmful), you need to figure out how to do it in a way that works for the majority of students.
And what is your policy for gifted students ? The below is un-acceptable.
"A key sticking point in the approval process has been the framework’s recommendation that teachers refrain from labeling students as “naturally talented” in math."
I'd recommend that we try in elementary to make the current math curriculum reach a broader set of students through games, puzzles, and in-classroom competition that doesn't absolutely favor the strongest students. Maybe Tom is farthest in math and wins a lot, and maybe Amy has a natural talent for computation that makes her strong, but there's also some randomness and the ability for gambits in the game to let others have a chance of winning. The result is that everyone tries hard. Sorting students by level prematurely has been shown to be bad, so I think having levels before late upper elementary or early middle school is bad.
I think the link's recommendation of not sorting students based on level in middle school is bad-- my 7th grader is doing precalc now. But he was on the normal math path through elementary with some enrichment and diversions.
Gifted kids are hard to deal with in math in elementary, because they may have a good intuitive understanding of math, but they generally are not so developmentally ahead in focus, accuracy, etc. So while you may have some 9 year olds that can understand work intended for 14 year old students, they generally cannot do difficult problems with any degree of accuracy. They make too many mistakes, swapping coefficients and signs.
> The below is un-acceptable.
> "A key sticking point in the approval process has been the framework’s recommendation that teachers refrain from labeling students as “naturally talented” in math."
Gotta disagree with you on this one point. Labelling someone as having a natural talent helps no one. Label them as ahead, or having worked hard.
"Natural talent" may be the truth, but as a label is toxic for everyone. It's toxic for everyone else, because it's not something they can hope to have: why try? And it's toxic for the labelled-- anything that is hard can be threatening that this label of natural talent could be stripped away if they try and don't do well-- so why try.
Studies show that praising kids for "talent" or "intelligence" is actually demotivating.
Let's not rush to apply labels, but instead try to create environments where everyone can be motivated to try hard, excel, and grow. In middle school and up this can be through tracks. In late elementary this can be through differentiated instruction. And throughout elementary, we need to just focus on keeping it engaging and interesting and speaking to curiosity of everyone in the room, instead of drilling the poor kid who's struggling on arithmetic facts incessantly.
Another issue is that the labels are not overwhelmingly predictive. The students who are considered weakest in elementary school can improve a lot. And many of those who continue struggling may do so because they've internalized a message of being weak at math-- or internalized short term coping strategies imposed by teachers (e.g. given up on understanding and instead are trying to learn the correct sequence of juggling symbols by rote to pass this next class).
I think a whole lot of kids can get algebraic ideas early.
They may not be able to have the attention span and accuracy to factor some 6th degree polynomial in 2 variables with mixed signs.
But the idea of an equation; of invertible operations; of doing things to both sides of an equation... If you word the questions right and make it interesting, most 7-8 year olds can do this stuff no problem.
I think a whole lot of the math whizs who post on HN grew up around people who loved math and figured out how to share interesting tidbits with them very young.
E.g. why are my kids all terrifically accelerated at mathematics? Is it because of some magical genetic thing (maybe a small part of it is)? Or is it because we, as parents, value and enjoy it?
I want to figure out how to bring more of that magic to ordinary classrooms.
Why/how exactly do you think people become a math whiz? For many of us it was because we were introduced to subjects at an early age and had supportive teachers/parents that encouraged our academic success even when (especially because!) it meant surpassing our peers.
Anecdote: I’ve tutored elementary school kids (in Massachusetts) who are doing basic algebra with shapes and emoji instead of letters.
> 5 + $basketEmoji = 7
> What number is hiding in the basket?
They have no idea they’re doing algebra or what algebra even is but they’re doing it. And they understand the concept on a fundamental level.
I don’t remember anyone explaining a variable to me so explicitly. I just remember showing up one day and having to deal with random letters mixed into my math homework.
I, along with 30 other kids in my grade, took our first algebra class in 7th grade (early 90s). After a few weeks of struggling, I was entirely fine with it. I really think you're underestimating what kids can do if taught properly. If students don't have the "mathematical maturity" for algebra until 8th grade, or later, then that just means instruction in prior years was lacking.
Not everyone took algebra in 7th grade; those who did not, took it in 8th grade.
We were introduced to algebra in 6th grade. Talking about Indian subcontinent in the 90s. I (and a lot other kids) did fine with it. Well there were many that didn't and they switched to humanities or business studies later. But even if lives are not good at math, they should be taught math. Doing otherwise is how they grow up to become conspiracy theorists.
My elementary school was introducing algebra in 4th and 5th grade. Questions like "7 * ? = 21" aren't explicitly teaching algebra - student's aren't being taught the multiplicative property of equality, they're just remembering their times tables - but it's laying the foundation.
I introduced my kid to Algebra in gr.2 during that first stretch of covid last year. He's not particularly adept at Math(doesn't struggle with it - it was gr.2 though so simple + and -), but it really didn't take much.
