This seems so strange to me, this sharp distinction between algebra and non-algebra. I might have gone to school in the Soviet system so everything was backwards and such but we were introduced to https://en.wikipedia.org/wiki/Elementary_algebra very early and just did simple expression transformations, variable substitutions. That was a lot early than 8th grade. I even taught my 2nd grader a few of those.
I have observed the curriculum so far in US, at least up until the 2nd grade and it's a hot mess. Even for double digit addition problems they jump right into "cool tricks" and "mathematical thinking" while there are just really a few steps to memorize and, guess what, kids are great at memorizing stuff. Later on it's time to show a few short-cuts and kids even discover those themselves.
Now going back to algebra, I am firm believer that a simplified version of this has to be introduced much earlier. It's a bit like a spiral, do basic arithmetic, geometry, number line manipulation, simple word problems, next year do the same but more complicated, add some algebra in there and so on.
But I guess that's just too boring and teachers, and probably higher level authorities, decide that they need to do "something" so they start making changes just to they can slap something on their resume. Developers do that too some extent, "here let me rewrite all this..." we can certainly understand that, but doesn't mean it's a good thing.
I read somewhere, don't remember where, that when a business isn't going well, people start grasping at "process improvements" like it will bring about a huge transform in the way business is going. In reality it doesn't, but it's the sort of busy-work that makes people feel like they are being useful.
I think it's very similar in education. There are vast swaths of the education system that are doing absolutely terrible. Much of the problems are outside of school administrator's control (broken families, gangs, poverty, etc), so they reach for "process improvements" and hope that makes a difference.
> [a study looked at] the relationship between eighth grade enrollment [in Algebra] and National Assessment of Educational Progress math exam scores at the state level. In short, they didn't find one... [In a different study], they found that early exposure was actually associated with a net decrease in average student math score exams within a given district.
Maybe we're looking at the wrong metrics. Obviously introducing the average kid, whose average abilities change much more slowly than "process improvements," would have a decrease in today's math exam.
Maybe there's something wrong with the exam! And not in the sense of what questions it asks, but more like, who the hell cares?
I know what the answer is, I'm not stupid. But eventually, we'll have to square away the multitudes sitting inside of parent's brains: that they want their kids to actually learn something, while simultaneously having the highest scores / grades / metrics possible. The two cannot be true and honest at the same time.
But the tension isn't learning versus evaluation. It's honest versus dishonest. Under the early algebra scheme, you get a more honest evaluation (students perform worse at harder, more realistic material) and more learning (they're not rehearsing stuff that's easy for them).
The real dynamic is that no one is going to enroll their kids in a school with low test scores.
Haha, my dad advised me to sell stock in companies that embark on a massive headquarters project. It's not just a curse - management spends their effort designing the HQ rather than working on the business.
Curriculum, most especially through 6th grade, is driven very much what whatever fad catches hold in the education establishment. It changes every 2 - 3 years. It's really just a symptom that there is very little accountability for the people making these decisions. Put out a curriculum with poor results? Who's going to notice? If anyone does, who will have the power to do something about it?
Do you think that is only a problem in government? Fads are a fundamental part of human nature.
In tech, we have many examples: Agile, Ruby on Rails, React, RISC, OOP, microservices, and so on and so on. All have some real benefits, but also have or have had huge hype cycles.
You can make the same argument for most other industries (if not all of them). And then there are things like diets, religions, etc.
School boards and other elected officials get voted out. Not every time, but businesses that latch onto fads don't fail every time either, as anyone can see.
And I think it's extremely harmful to delay these concepts for children like we do in the US.
I wasn't particularly smart growing up, but my parents always kept me up to date with the math and science curriculum they had at their respective ages in their own country. At least up until the end of middle school, the difference was astounding and had a dramatic effect on my academic (and thus future) success.
I wasn't particularly smart growing up, either, and I failed to place into the middle-school algebra classes like most of my peers. Somehow clicked for me in high school, though, and I raced through the curriculum at an accelerated pace, finishing with Calculus my senior year. A few years after that I graduated from MIT with an aerospace engineering degree.
My grandfather never advanced beyond trigonometry in high school, but went on to become a moderately famous theoretical physicist.
So if we're looking at anecdotes, it's hard to form a clear conclusion. Today I teach high school mathematics and I see accelerated kids who fall to pieces their junior year, and also kids like me who arrive in the 9th grade knowing nothing, but somehow end up in an accelerated track by the time they graduate.
Yeah it's interesting that grade 9 or 10 is kind of the point where students can make a complete turnabout in scholastic achievement and enter the accelerated track and get to a good university, etc.
In my case I started getting serious in grade 10, entered accelerated track for grade 11 & 12. Unfortunately I sorta peaked in grade 11 and started coasting in grade 12. I probably shouldn't have gotten serious until grade 11.
> And I think it's extremely harmful to delay these concepts for children like we do in the US.
I wonder how many people grow up hating math, like I did in elementary school, because they just don't like how rote arithmetic is and how trite simple word problems are because they steer away from algebra. It wasn't until I got to Algebra that I really, really started to love math.
I was an algebra teacher and tutor for years. It isn't the subject matter. It's the contextualization into the student's life. How is this relevant? I once taught a girl who was a soccer nut how to use algebra to budget for an imaginary soccer team. Soon it wasn't just basic algebra. We created a whole spreadsheet to calculate the entire thing. Change a number and it cascaded down to the final budget. She was jazzed.
It is vital not just to teach to the curriculum, but to teach to the student.
And that's where a lot of modern education falls apart.
It's not just algebra though. History. English (from literature to grammar). We're just failing to connect.
Why are you advocating memorization? That just gets people to treat mathematics as a set of magical spells you have to receive from the holy scriptures and priests, and that problems just can't be solved without having received the appropriate spell from the masters. (It also encourages strange beliefs like that mathematics is a subjective human construction, like art and music.)
Conveying that there's a deeper conceptual "mathematical thinking" seems much more important than memorizing the algorithm for "completing the square" or whatever.
Because it's something kids do really well and it's something to take advantage of when learning at that age. In other words, it's better to memorize the multiplication table then find insights into how it work later, and do proofs and make connections with calculating an area of a rectangle etc. A lot of the tricks and insights don't mean anything to the children if they didn't already do many examples the rote, repetitive way and in a lot of cases the "insights" are often a distraction as well. Those should come later. Even more interesting is when children see or discover these tricks or rules on their own, then they become really memorable to them.
I don't have many samples work with, but I have observed this over the years based on my own experience, my kids and my extended family members and I have noticed the same patterns.
Memorizing the multiplication table is a useless waste of time and I recognized as much when they tried to make me do it and I refused. Nobody needs to know 8x7 off the top of their head. What's really important is estimation techniques to tell if something passes a sanity check, like reducing things to powers of 10/etc and seeing if they're close to the result you got on a calculator, like how 123x456 should be close to 50000 (100x500)
> Memorizing the multiplication table is a useless waste of time and I recognized as much when they tried to make me do it and I refused. Nobody needs to know 8x7 off the top of their head. )
Me too! I thought I was so smart as a kid, realizing that stuff I didn't want to do was useless.
However, I was wrong. An over-dependence on calculators due to a lack of arithmetic fluency has sabotaged all my encounters with mathematics ever since. I had a hard time understanding and developing any kind of fluency because I was forced to push everything through a black box that also distracted me.
I now realize that memorized facts are the foundation for knowledge and advanced thinking. Dismissing memorization, especially of fundamental facts, is like choosing not to use RAM because you can always swap to disk.
This is exactly what I tell my kids right now as they learn the tables by rote: knowing the answers instantly by drawing from your subconscious makes it vastly easier to solve harder problems later on.
Making it competitive between the two of the kids helps too... after all, sports is “just” being better at getting the ball in the goal better than the other team, right?
Higher math isn't harder arithmetic, it's increasingly more abstract and sometimes doesn't even involve numbers. A student with their times tables down isn't going to do any better in a calculus course than one who relies on a calculator.
I don't believe you're correct. As my 2nd & 4th graders' teachers tell them, they need to learn a set of level-appropriate "math facts" ... but they're not forced to start by memorizing those facts (my son, for example, has a printed multiplication table up to 15x15 to reference while practicing decomposition of 3 & 4 digit multiplication problems). I truly appreciate the Common Core principle of focusing on teaching methods over facts, and -- at least in my children -- it is apparent that it sets them up for problem solving success IRL. YMMV. It's not a perfect curriculum and is still tailored to the lowest common denominator student, but I do think there's value there. And again, in a classroom with a strong teacher who can take time for differentiated instruction, there are ample opportunities for advanced students to go beyond or to practice their learned skills by helping their classmates.
That said, I think your statement is absolutely incorrect in the context of higher math (and experimental science). The more facts you know -- whether literal facts, axioms, proofs, applied example or theories -- the more facile problem solving will be. This holds true in all disciplines (and not just math & hard sciences, but also engineering, social sciences & business).
A student with their times tables down isn't going to do any better in a calculus course than one who relies on a calculator.
A linear algebra course, on the other hand, is going to make mincemeat out of any student who can't do basic arithmetic in their head at high speed. Gaussian elimination, matrix multiplication, determinants, diagonalization; all of these tasks are extremely arithmetic-intensive. If someone needs a calculator to multiply 8x6 then they are not going to be able to solve a linear system in 4 unknowns on an exam that disallows calculators (which is all math courses at my university, besides stats and act sci).
Such a math course is being taught wrong if passing a test relies on churning through a million arithmetic steps instead of demonstrating understanding. I did fine in my course. Had to use the technique of mentally placing objects around a familiar path to memorize a silly list of matrix properties though.
The computational part was only 60% of the exams. The rest was proofs. For most of us, we needed every mark we could get in the computations because the proofs were really hard.
> Higher math isn't harder arithmetic, it's increasingly more abstract and sometimes doesn't even involve numbers. A student with their times tables down isn't going to do any better in a calculus course than one who relies on a calculator.
IMHO, that's wrong in many cases. For instance: there's arithmetic in algebraic manipulations, and a lot of that is in the times-table memorization range. I definitely had to rely on my calculator in calculus class (and specifically swapped professors to one that would allow the use of one for that reason).
There's also the fact that neglecting memorization as a "waste of time" at the start of your math career sets up a bad precedent and bad habits that will continue throughout it, barring substantial corrective effort.
I, personally, only stopped having problems due to my deficit when I got to discrete math, which was the one math class I took that didn't involve numbers, like you said.
That's narrowing the argument I made, I said "harder problems", of which higher math is just one aspect.
Even sticking to just higher math; you're going straight from arithmetic to... calculus? What about algebra, trig, geometry; all much easier when your arithmetic chops are blazing?
Except memorizing multiplied numbers is not foundational knowledge for any higher math, knowing what the concept means is. Learning this involves working through some examples but it doesn't mean you need to know an entire table. Not knowing 7x8 never held me back in higher math/math-relevant classes like Calculus I-III, Discrete Math, Linear Algebra, Physics I-II, Chemistry, Macroeconomics, Computer Science (anything from basic programming to language processing or algorithm proof), Music Theory. Had a fun and mostly easy time in all those. Once in a blue moon on a test I'd have to do a multiplication and count it out (8x5, add 5 8 times) because it was a product in a derivative or something and that class had an (unrealistic for the real world) no-calculator policy. In fact I'm pretty sure mathematicians are notorious for poor arithmetic.
Additionally to your point, I think forcing arithmetic memorization on students is harmful because it makes math seem boring and hard.
> Except memorizing multiplied numbers is not foundational knowledge for any higher math
But my point wasn't that it's foundation in general, category theory, set theory, etc are foundational, but that it's important to start from it. Just because something is foundational doesn't mean it's a good starting point.
> it doesn't mean you need to know an entire table.
Kids at an young age are usually very good at memorizing things and once they have memorized then they can build on that and learn new tricks, relationships, abstract and foundational principles.
> Not knowing 7x8 never held me back in higher math/math-relevant classes like Calculus I-III,
Neither did me either. Embarrassed to say, I had forgotten a lot of it. But I would have been struggling early on in the 4-9th grades if I didn't know how to add or multiply numbers then. I would have been wasting time on tests doing it the slow way, instead of thinking of more interesting problems.
So to summarized, yes, it is not foundational but it all depends on age, it's better to take advantage of what the brain already knows how to do well at that particular state in time.
> In fact I'm pretty sure mathematicians are notorious for poor arithmetic.
Could be but if they are poor they would be left behind in early grades and might never becomes the mathematicians they are. There is also a bit survivorship bias in the sense that if there mathematicians who are poor ar arithmetic, they'd stand out and become memorable, but those that are good are not noticeable because they are expected to be good with numbers.
Do the memorisation by spaced repetition and make it competitive, and it takes almost no time to learn them. Of course if the kids begrudge it, they’ll hate it and fail to work at it.
The opposite extreme is when a smart child cannot solve a math problem, because despite having interest and some cool insights, and making a few hopeful steps, they predictably fail at one of the "2 + 3 = ?" steps.
Also, smart kids who hate memorizing are often really bad at languages. Which are mostly memorization; there is no way to derive the language from the first principles. But I guess if one's first language is English, this is not a big problem.
I get it: memorizing stuff doesn't signal how smart you are. But at some moment, the lack of factual knowledge is going to overcome the advantage of being smart. You will start making logically correct conclusions from factually wrong premises, and you will be proud that you didn't fill your brain with garbage.
I agree. That’s why there I some truth to the stereotype of Chinese kids being better in Math.
Mandarin has 5 tones and having a keen ear to identify each tone and then recognizing it and memorizing it and matching sound to meaning is an enormous advantage.
Also...teaching music..to recognize diff sounds/tones..musical notes(Bach would do) also gives mandarain/language/math students an edge.
I had a hard time getting in the habit of learning French but I adapted to it when I had to. However, as I've mentioned in another comment, shoddy arithmetic in particular was never an issue for me in real math contexts.
Being able to do mental arithmetic for small numbers is actually a very useful life skill. Sure, everyone has a calculator in their pocket these days (despite what my maths teacher told me), but that overhead of having to pull your phone out does slow you down.
But I don't believe that rote learning your times tables up to 12 is the way to do it. Mental arithmetic should be incorporated into the curriculum in a more integrated way. The problem is that school systems love testing and grading people. And it's quick and easy to grade people on their times tables by giving them a speed test to see how many equations they can complete in 5 minutes. You can say that little Jimbo is doing his times tables at a third grade level because he knows his 2, 3, 5 and 10 times tables. It's a terrible method of testing student progress though, I managed to get all the way to high school and was in the top stream class and never learned my times tables.
The actual concept of multiplication tables is an important concept for children to learn though, as it does help solidify fundamental mathematical concepts.
I do mental math all the time. Nobody needs to memorize tons of math but it's laughable that you would take that to such an extreme that you can't do basic multiplication without a calculator.
Can you give the example of the last mental math calculation you did? I almost never do it and I still almost never did it even when I was in college taking higher math. Tip calculation is all I can think of.
I may chime in with another story. I‘m teaching operating systems at university and use these skills (for small numbers, and written multiplication/division for slightly larger ones) all the time. My general consensus after overseeing a few hundred students is that those that are unable to quickly do simple math in their head also struggle with the rest of the curriculum. Especially if they need a calculator for computing 57:2 everybody gets distracted and we have like 2 minutes idle time while everybody starts entering the equation. I’d rather spend my time teaching actual os stuff, it’s a shame enough that I have to teach how to decimal <-> binary <-> hexadecimal base conversion.
I work in big data/cloud stuff so I do estimates all the time to figure out what size resources I need/how long something will take. Less than 5 minutes ago I did mental math to see how long a job would take (I knew how long it would take for one read, and I had 2 days of reads over 3 data sources each with 8 data partitions => 48 reads each taking about 90 seconds => about 1.25 hours).
Tip is another thing. And I like to "min/max" grocery shopping so I typically do mental math when shopping to maximize things like grams of protein/$.
I do rough division all the time in my head, for instance figuring out how much each can of beer costs if a slab of 24 is $60. Or on the flip side, how much it's going to cost me to get a round of 4 beers at $8 each.
I can be claustrophobic..I don’t like being in a closed car..so when I am at traffic stops..I convert alphabets to numbers and numbers to alphabets of the license plates..and sometimes I add the numbers together to check if it’s a prime number. Calms me the fuck down.
The example of approximating 123 x 456 with 100 x 500 tends to undermine your point a little. It's too simplistic and the answer isn't even within 10%.
120 x 500 gives a much better approximation, and just needs you to know 12 x 5 = 60. Better still would be 125 x 460, which is 1/8 x 46 x 100000. That's within 3% of the true answer, and all it takes is quickly dividing 46 by 8.
It's very helpful to be on good terms with numbers less than about 100, which is why the multiplication table is taught. It's useful for reverse calculations (like 46 / 8) as well as forward ones. Decimal equivalents (1/8 = 0.125) are worth learning as well.
The key to being able to quickly approximate things is to populate your mind with memorized values. They work as landmarks (so the 7x8 example, in conjunction with the decimal equivalent of 1/8, means that you know almost immediately that, e.g. 55 x 125 is close to 7000).
You may be a bit embarrassed when you get older and a medical practitioner gives you a cognition test. Start with 100 and repeatedly subtract seven, announcing the results.
Actually, that's on a standard cognition test. Other questions include identify pictures of animals, "what date is today", "Remember these five words (and get asked them at the end of the test), an "a is to b as c is to" question, etc. The assumption is that people over 50 in the USA can do simple arithmetic, because they all went to elementary school. Sadly, that assumption may have to change.
Although I tend to mostly agree with your main argument, the example you presented ( memorizing multiplication tables ) is a bad one.
My mind was blown when I found out how they teach multiplication to children in Japan. Link: ( Vedic method ) https://www.youtube.com/watch?v=z4X98Mnj0tc
The superiority of the teaching method shows to be significant when compared to other teaching methods elsewhere and even remains noticeable when children encounter higher levels of math.
This is similar to the "Lattice Method" that they taught (still teach?) here in the US. I can't stand it. The students simply learn an algorithm, without developing a sense of place value and an intuition for "how numbers work." Sure, they can quickly multiply large numbers and it's a cool "party trick." But they aren't learning math. Might as well use an iPhone.
Fair point that memorizing an algorithm doesn't necessarily advance the student's intuition for how numbers work, but keep in mind that you're replying on a subthread about how learning a general algorithm is an improvement over memorizing multiplication tables.
Ideally, we teach children the intuition for mathematics. Just memorizing arithmetic algorithms isn't ideal, but it's surely better than memorizing finite arithmetic tables.
I don't think it is better to learn some shortcut algo at a young age. Memorizing your times tables up to say 10 * 10 or 12 * 12 is fundamental to understanding multiplication. Remember, the brain is not a turing machine, it's an associative memory machine. Facts are the foundation of knowledge.
First memorize the times tables. Then learn the long method. Then learn the shortcuts. IMO, of course.
I agree with you. To do multiplication by any non-parlor-trick method you have to be able to multiply single-digit numbers. If you don't memorize the table then every time you do larger numbers the process becomes that much more of a chore.
Now, that is not to say that you should forego understanding multiplication. But forcing student to memorize the basics (like forcing them to memorize verb forms) (1) makes their life easier later on and (2) gives them the chance to spot patterns themselves.
> The superiority of the teaching method shows to be significant when compared to other teaching methods elsewhere and even remains noticeable when children encounter higher levels of math.
I was mainly talking about addition and multiplication up to 10x10. Looking through all the tricks and hacks they teach the kids in an attempt to simplify ideas seems very wasteful as the brain at that age is so good at memorizing things. ut the schools don't seem to be taking advantage of that.
They think like adults in the sense of "oh let's introduce this abstract idea and prove it and derive the specific examples from it". Yes, a college kid or high school would do well there, but in earlier grades I haven't seen it work. It's a bit like learning a language. An adult might do well to start learning about grammar rules, tenses, cases etc, a kid might do better hearing a lot of example and deriving the rules on their own. Later they might find out that there are 8 tenses, 4 cases, predicates, adjectives etc.
Memorization is very important. You have to walk before you can run.
To perform well at algebra, you can't be stopping to punch numbers into a calculator. You can't be constantly making mistakes. The slowness and errors will make you stumble and lose your train of thought, preventing you from reaching any deeper conceptual "mathematical thinking".
Those old-fashioned methods for doing math are reliable and general. If you follow the rules, you get the answer.
That's what my typing teacher taught me about typing, that by not memorizing all the key+finger combinations I would never type (Owning a computer, I was self-taught and used all the wrong fingers). Twenty years and a law degree later, after authoring literally thousands of pages, I still cannot say which finger is supposed to hit the 'r' key. Somehow I muddled through. The point is that rather than memorize combinations of seemingly random numbers, the same knowledge will come naturally as students learn other aspects of math.
Seems to me that you just spent twenty years to slowly develop muscle memory, when you could have achieved the same results in a few weeks. (I did the same thing.)
> the same knowledge will come naturally as students learn other aspects of math
The key difference here is that being slow at typing for twenty years didn't discourage you from typing. Being slow at math can discourage kids from doing math, and the "twenty years of regular practice later" moment will never arrive.
Yes - When someone asks me what a particular key combination is in emacs or Eclipse, I put my fingers out like they're on a keyboard, think about doing the thing, and, look at what my fingers are doing.
To bring it back to the article: It's similar with a lot of things - you sort of need to memorize aspects of it to do it fluently. As part of calculus, you sort of need to know how to do polynomial factorization. Intuition is great and helpful, but you also need to be able to proceed methodically in order to produce or understand proofs. Scales on piano are boring, but you don't just tell kids to feel like they're Bach and they start playing Goldberg Variations. Why on earth we think the intuition is not only necessary but sufficient is unfathomable to me.
I recall research showing mathematics being one of the very few subjects where one can get better at through brute memorization. It gets you quite far in that field, as it reduces recall time for proofs etc, and then it becomes like muscle memory.
Practice/muscle memory in other fields, like the physical sciences, can not come from a page out of the book, since it requires interacting with nature in a multi-sensory way.
My impression is that mathematical thinking isn't really the goal of mathematics education until post-calculus University level. Before that it is mostly learning rules for doing symbol manipulation.
Although, this isn't entirely true, I remember doing proofs in Geometry class in Jr High School. So I was at least introduced to it, though we never really used it again.
Sorry that is pure BS. Memorization is a great technique to learn math of course it is not the way to do math. When you learn tables through rot learning you automatically get how they are being computed. At young age it is much easier to rot learn many things and understand the abstract concepts much later.
People who despise rot learning are either misunderstanding it or are simply lazy.
See Piaget's concrete operational stage [1] of human development: "Abstract, hypothetical thinking is not yet developed in the child, and children can only solve problems that apply to concrete events or objects."
In other words, you can only push the mathematical thinking as far as the developmental stage of the child allows you to. Memorization helps though.
My son is 6. He doesn't know the multiplication algorithm, but he does know that 2 x 2 = 4 or that 3 x 4 = 12. That's sufficient for him to do discoveries of his own. 3 x 4 = 3 x (2 x 2) ! Picture that!
I'm not seeing any advocacy of memorization? The only thing you might have to memorize, out of what parent listed, is the algorithms for doing written out arithmetic; and that too is in fact an exercise in mathematical thinking. After all, there's a reason we teach these algorithms, as opposed to just telling kids that they can punch numbers into a calculator.
Quote: "Even for double digit addition problems they jump right into 'cool tricks' and 'mathematical thinking' while there are just really a few steps to memorize and, guess what, kids are great at memorizing stuff."
And rdtsc is entirely right about that. The Common Core math curriculum has been taken by many self-described "math teachers" and "educators" as an opportunity to just not teach the effective algorithms for doing arithmetic, and expect students to just "discover the results by themselves" via some mixture of trial-and-error, random guessing, and the rare "trick" or perhaps tiny flash of genuine "mathematical thinking". That was an unmitigated disaster, of course. It's probably part of the reason why Algebra is now being seen as way too difficult for Junior High students, and something that only HS students could approach effectively.
From your link, knowledge of the standard algorithm for addition/subtraction is only requested by Grade 4, and for multiplication by Grade 5. So the Common Core standard is basically relying on a hidden assumption that students can effectively learn all the other stuff that's expected of them up to Grade 4 and 5, without truly being fluent in addition, subtraction or multiplication! (I'm aware that the standards call for "fluency" slightly earlier, but that's mere wishful thinking until you teach the standard algorithms, or some cosmetic variant thereof such as lattice multiplication.) That's what so bad about the way CC is being applied in elementary education.
This debate comes up again and again, and any time one view is purported strongly over the other we just get education that is half-assed. This includes people with your point of view that deliberate attempts at committing propositions and procedures to memory should be avoided.
The issue is one of cognitive load. You can't get to mathematical thinking and appeal to both a higher-level "intuition" or "key idea" if you aren't fluent in the details. Else it becomes hand-waving that doesn't cash out to proof-ready reasoning. Likewise, you can't internalize the details if you don't have a principle which compresses them and lets you unpack them appropriately for the circumstance of the problem. This is fair, but first there has to be something to compress.
The question is, "How much detail can be advanced at this point in the student's growth?" If the key issue is to dispel the notion that mathematics is arcane, you should narrow down the scale of the problem so that you can quickly see how the details contribute to generalizations, and how a procedure can be taken from it. This achieves what you are looking for with less effort and less risk.
With that point of view illustrated with a simpler case, it can be developed or built upon with bigger cases. Bigger cases inevitably means more details to internalize. This protracts the phase where memorization would be appropriate. But the student, having seen how mathematics is developed in a simpler case, would understand that these details have a context that will be more adequately realized at a second or third pass of the material.
Eschewing the internalization of detail as you are purporting would encourage a sense that math is subjective, because the high-level impressions of mathematics would not have the memories of detail to back them up, allowing for more to be said, with less control.
Same. My grandmother taught me how to solve equations, and we'd do it for "fun" on the bus in Moscow; I was in, at most, fourth grade. And you know what, it was fun, and it wasn't hard.
Granted, she was also a programmer, and my grandfather was a mathematician, and both my parents were engineers - so maybe my baseline for fun and education was set differently.
I doubt you're a bad grandson, but I'll share this story with you.
I was raised by my grandparents, and I had enormous respect for my grandfather, who was the most badass human being I've ever known, though you would never know it to see him. He was very quiet, he spoke little, always watching and listening.
He was in his 50s when I was born and lived with them permanently when I was 5 years old. He and my grandmother were my parents.
They were always patient and kind, and always stoic about everything. One day while camping my grandpa was sitting next to a dart board, and one of the darts I threw went crazy wild; it hit and stuck deeply into his left calf. I just stood there with my mouth open while he puffed his pipe, staring at me. After a couple of moments, he calmly pulled it out, wiped the blood off of it, and carefully tossed it back to me, saying, "Careful where you throw those, boy."
I always had a lot of respect for them, though I didn't show it. In fact, by my actions, I took them for granted.
They both died of old age when I was in my 30s, even as I was just starting to truly appreciate them.
Now, many years later, I would give almost anything to be able to spend another day with them. There's so much I'd ask, so much to learn.
In summary: spend as much time as you can talking to your grandparents while you still have them, because once you're old enough to really appreciate their value, they might be gone.
Dragonbox produces an Algebra game for small children-- symbolic manipulation isn't that hard, you just have to be taught the rules: https://dragonbox.com/products/algebra-5
Some school districts in the bay area are doing better than San Francisco.
In the Los Gatos school district my daughter completed her first year of college level calculous and some second year calculous by the time she graduated from high school. This was better than my education as I only had access to pre calculus in my high school.
As an aside - giving kids math puzzles is a much better preparation for math than memorization. The only thing I had to memorize was the multiplication table, everything else I could derive. I could always derive a formula faster than memorizing it even at a young age. I went to school in the era of "new math" where we were introduced to functions and variables in 2nd grade.
I don't think anyone is suggesting that more advanced high school level math should be transformed into an exercise in memorization. They're talking about kids that are 7 or 8 years old memorizing their multiplication tables.
We might be saying the same thing since the only thing you need to memorize is the multiplication tables. ;-) How long can that possibly take?
My grandsons are between 3 and 8 - so they are in this age group. They are amazing at solving problems when it leads to something they want like winning a game against their brother. Puzzles are great for teaching math to this age group.
It's a relative comparison. Kids are far better at memorizing stuff and dealing with things that have been made "concrete" to them, compared to abstract thinking - because effective abstraction requires a sort of overall maturity of thinking that very few kids can be expected to develop at an early age! So it's better to just teach the memorization and concrete steps first, and focus on the more abstract stuff later on.
It would be good to find a way (an alternate path) that helps the kids who aren't great at memorizing. But don't do it in a way that makes things worse for the kids that are good at memorization. Don't "equalize" by making the best worse.
The key to memorization is repetition. Also, paying attention.
If I had to design a system that helps everyone with repetition, it would go like this:
- each concept must be explained and understood first (never progress to repetition unless you are sure the child groks it);
- afterwards, add the concept to your "spaced repetition" software (conveniently provided by the teaching institution);
- give kids some time at school when they can do the spaced repetition exercises (so it doesn't become an extra homework).
The first step avoids the kind of stupid memorization that people do when they don't have enough time, and that gives them problem (and even greater time pressure) in the future. This makes the entire thing more meaningful that current education.
The second step avoids the situations like "I felt pretty sure I understood it yesterday, but now I have no idea where to begin". This makes the entire thing more efficient.
> But don't do it in a way that makes things worse for the kids that are good at memorization.
Uhm, if you complete the spaced repetition exercise and there is still some time left, you can go play or read something. Or perhaps take a bonus lesson that is not a part of the standard curriculum.
Kolmogorov had to go against a strong political pressure to convince the government to allow him to create a school for mathematically gifted kids.
Also, a large part of Soviet mathematical output came from people who were politically forbidden from getting an official scientific job (and then the Academy of Science was filled with people like Lysenko), and sometimes had to get a manual job. So they did math in their free time, corresponding with each other. Which, paradoxically, probably provided more freedom than they would get in an academic institution: no pressure to "publish or perish", freedom to spend years solving some obscure problem with no impressive intermediate results, true cooperation between the outcasts instead of fighting for a limited number of academic job.
A large part of Soviet mathematical progress was made in opposition to the regime. Or perhaps, to put it more mildly, the system was so hopeless that people didn't waste their time, and organized better mathematical education outside the system. Then they created things like the International Mathematical Olympiad (the first twenty years, it was an Eastern Bloc thing), various correspondence seminars, etc. But this all happened outside the official educational system.
My understanding of "strong political pressure" for the years of Kolmogorov in the USSR meant that ended up in a salt mine in Siberia and dead in a year or so. But apparently somehow Kolmogorov was smart enough to avoid that end.
Yup, suspicions confirmed, all around the world, no matter what, bureaucracies have a lot in common: If have something good and want to kill it off, then set up a bureaucracy to help it!!!!! That's partly a black joke; likely the US NSF and NIH and lots of private efforts in math, physical science, medical science do terrifically good things. And, no joke that the LHC is a big bureaucracy and found the Higgs Boson, a darned good thing.
Still there can be some truth to the joke. E.g., in the US I've seen some of what you said about publish or perish: While I never really wanted such an academic research career (I just wanted a JOB and to support a family and looked at my math Ph.D. as vocational education), eventually it dawned on me that actually in some fields the publication standards sufficient for publish or perish are low and that a good researcher should be able to toss out dozens of such papers, if only as side efforts, while otherwise doing the important stuff.
Bluntly when I was considering academic math research, the main obstacle, and it was SEVERE, was that I didn't quite yet have access to a computer that could run Knuth's TeX math word processing software. Bluntly the word processing for typing in the math research was MUCH more challenging than doing the math research!
There's another relevant rule: In social, political, bureaucratic, etc. systems, when want to do better, measure it, and, for that, create, design, implement, and setup measures.
Then the system responds, does well on the measures, just the measures -- games, hacks, manipulates, tricks, fools, takes advantage of weaknesses in the measures -- with harm to what was being measured! Partly another black joke but, still, with some sad truth.
With bad measures, the Soviets were not the only ones with the problem of "They pretend to pay us, and we pretend to work.".
Then we get reminded that democracy and free enterprise are the worst systems ever invented except for all the other systems that have ever been tried. Another black joke; sometimes democracy and free enterprise do really well. E.g., in a democracy, we can elect liars, crooks, incompetents, etc. but eventually the disaster becomes obvious to enough voters, and then we get some better candidates and an ELECTION to throw the rascals out -- been known to happen, sometimes slowly, but, still, happens.
I know next to nothing about the Soviet system, but from what little I know it looks like it was an astounding juxtaposition of just horrible suffering and lost opportunities, waste, e.g., on military efforts, lots of just horrible inefficiencies, yet at times some, maybe not enough, maybe with the movie remark "But do you know what it cost?", astoundingly good things -- Kolmogorov, Oistrakh in violin, Giles in piano, Garanca in voice (grew up in one of the Baltic states when they were part of the USSR), the Mariinsky Ballet, etc.
I'm trying to get my startup live on the Internet; there are some ways it could help civilization. Otherwise I can't solve all the problems. But democracy and free enterprise when they work well -- in the US from 1929 to 1942 they didn't work so well -- can be amazingly good things.
Something feels off about this whole debate. I took algebra in 7th grade, so I naturally lean towards the "push it earlier, not later" camp. But the more I think about it, the more I'm convinced that the way we group mathematical concepts together in grade school is fundamentally boring for students. It seems like nit-picking to fall hard on one side of this "8th vs 9th" dichotomy when I actually suspect the whole math cirriculum could use a revamp from the ground up. Maybe we should consider basic discrete math in elementary school. Maybe basic calc in 6th grade. How can we get substantive, critical thinking skills earlier into the curriculum so we don't have to just assure students: "trust us, math will get interesting if you can just stick with it through calculus and linear algebra".
I agree. I hated math up until 8th grade. My 7th grade math teacher suggested I take algebra, which was the "advanced" option for 8th grade math. I have no idea why, because my math grades were very average, but he must have seen something or had a hunch.
It was like a light bulb went on. Math went from being rote drudgery to something that I could see had some useful purpose. I enjoyed math from that point on.
What I hated was doing 50 homework problems for every type of whatever it was we were learning that day. Even if the answers to have odd-numbered ones were in the back of the book.
Once an equation or soomething 'clicked', I got it. There was no point of doing 50 of them, each slightly different than the last.
I'd argue that you are in the minority here. Most people (especially children) require repetition and depth of work to master a skill / understand a strategy.
In my experience (having watched a LOT of my friends go through varying degrees of good/bad public schools in NJ) most kids really do need the repetition. Without it they don't hammer in why something like adding 2x to each side of an equation cancels out -2x on one side, even if the last problem had you balancing an equation by adding 4x+5 to each side. These concepts might seem absolutely trivial to us now, but as a person first approaching them I think they can be pretty complex.
As someone else who was in the same situation: In a way that additional problems would help, almost never. All my mistakes at that point were the equivalent of typos, things like dropping a sign from one line to the next.
Yes - this discussion is kind of ridiculous. If San Francisco had announced that they were removing elective CS from the High School curriculum, Hacker News would be saying "How can anyone expect to succeed in a STEM world without High School CS?"
I was one of the advanced math kids and took calculus in 11th grade. Then I went on to Calc 2 in college, and it was a different league entirely. I would have been better off building a stronger foundation throughout High School and then doing Calc 1 in college.
...which is exactly what the new curriculum aims to do. I have a 6th grade daughter in SF, and the strength of the new curriculum is that they spend a lot more time making math more intuitive. Instead of learning one approach to long division, they learn multiple approaches. In theory at least, this pays off in the long run for a lot of kids.
What American schools need isn't more acceleration (Algebra at age 14 instead of 15) - it's a better understanding of what mathematics actually is and why it matters.
What American schools need isn't more acceleration (Algebra at age 14 instead of 15) - it's a better understanding of what mathematics actually is and why it matters.
While I agree with this sentiment, my problem is with the one-size-fits-all approach taken in San Fran. Lumping gifted children in a class with everybody else does them a huge disservice...
My personal experience in 7th grade (pre-algebra) was horrible. My school decided to experiment by placing gifted students in math with the rest of the kids, with the idea being that we'd magically bring up the average performance. Instead what happened was the nerds sat in the back bored out of our minds and lost a year of math education (and this was with an extra teaching aide in the class - the two teachers simply couldn't keep the non-gifted kids on track AND provide us any extra attention). This left us all behind when we entered Algebra in 8th grade.
I hate to be "that parent" - but gifted kids have different needs than normal students and deserve the opportunity to excel without waiting around for everybody else to figure out 2+2.
Edit - I don't care if the advanced math offering in 8th grade is called Algebra or something else, as long as there is an advanced offering. The linked article made it sound like there was not such a class.
I think a solution could be to have classes based on age groups..giving kids 5 years to finish 5 years worth of curriculum so they can finish at their own pace and then grade them at the end of the five year term.
This way, if something is easy or interesting...or tough or boring, kids can choose how they want to tackle it. Having many different ages in the same class without the pressure to finish everything crammed within one year should help. Here is where maybe bright kids can teach kids that need help..or older kids can help younger ones.
Example: ages 5-10 study together in huge single room schoolhouses ..each one on whatever they want to learn. Ages 11-15 is another group etc. amongst them, they can be in sub groups according to interest or ability. A class can have 4-5 teachers who can tackle all of the subjects. Volunteer parents.
Test and grade them at the end of five years. Only test them every year or semester.
They should eventually be able to form groups...and work as peers. Is there really a difference between a 12y/o and a 13y/o...as adults, we don’t always consort with those born within the same solar year. Why wouldn’t it work for kids?
> Is there really a difference between a 12y/o and a 13y/o
There can be huge differences in social development there, and not even strictly tied to age. The variance is rather large.
But even if you just look at averages, as far as I can tell the average 13 y/o girl and the average 12 y/o boy are quite far apart in terms of their social interactions, starting with basics like "are you thinking about dating yet?"
All that said, getting 12- and 13- year olds to work as peers is a much simpler problem than the originally posed one of getting 5- and 10- year olds to work as peers. In _that_ context there are a bunch of problems, ranging from difference in attention spans to the basic issue that most 5-year-olds don't know how to read yet, and most 10-year-olds do, and so conveying information to both together in a way that's not frustrating to one or the other can be quite difficulty.
Now if we want to group students by ability (5-year-old who can read, great, let them work with that 8-year-old if they have an interest in common) instead of age, that might work much better than any sort of age-based grouping. Of course that can exacerbate the social aspects, but there are in fact ways of making this work well. Having the older student partially teach the younger one, for example, is a good way for the older student to significantly improver their own understanding of the material.
Maybe they should learn a diverse group of children. Girls and boys are ‘different’ but we don’t segregate them anymore...why should we segregate them on the basis of one year age gap?
Except all the places we do (single-gender schools do exist, and even in non-single-gender schools physical education and health are often taught separately).
Also, maybe we should do more of this; there is some evidence in the literature that at some ages educational outcomes would be better with gender segregation if we did things right in terms of keeping equality of resources. Which is of course the sticking point.
> why should we segregate them on the basis of one year age gap
In case it wasn't clear from my answer above, I don't think we should. I think we should group kids by interests and current level (i.e. by what they will be trying to learn) a lot more than we do now.
I am saying that there are some studies showing gender segregation at certain ages may improve educational outcomes, largely depending on how it affects teacher behavior. Whether that means we should do it rather depends on whether we can get those teacher behavior effects in other ways and whether gender segregation would cause other issues (e.g. unequal resource allocation).
For some specifics that are pretty easy to find, see http://econweb.umd.edu/~turner/Lee_Turner_Gender.pdf for recent evidence that boys do may do better in all-boy schools, at least in some cultural contexts. There were a bunch of studies in the '90s that claimed girls do better with no boys in the class, due to teachers actually noticing them, but that effect seems to have more or less disappeared over the last 20-25 years.
In general, as in all things to do with kids and education the answer is almost certainly "it depends". Some children do better in a gender-segregated environment. Some do better in a gender-integrated one. Some don't particularly care. Hence all the caveats above about "some" and "may" and so forth. The hard part is figuring out when gender-segregated education is appropriate and when it's harmful, on a student-by-student basis. Unfortunately, public education is too cookie-cutter for such details.
5 year age difference may be a bit much but beyond that, montessori is doing exactly that. they group children in ranges of 3 years: 0-3,3-6,6-9,9-12,12-15,15-18
we can probably argue about the exact age ranges (maybe follow the schools age-ranges: preschool/kindergarten, primary school, middle school, high school all sound like good age groupings).
but the concept is good. younger kids can learn from older ones. older kids internalize the material by helping younger ones.
and it turns out that this model is already proven too: it's applied in montessori, and it's practiced in scouting as well.
America doesn't need more acceleration, but nor does it need more deceleration, which is what this policy does. What they got rid of was: some kids are ready for Algebra in 8th grade, and they can take it. New policy is: everyone takes the same class, regardless of skill or preparation.
Everyone gets the same 'x' regardless of individual circumstances. Easy policy to prescribe if one holds certain political beliefs.
Unintended consequences: Wealthy and middle class kids will still have access to the same quality of education as they did before, those that rely solely on public education will have access to lower quality.
Inevitably, students from CA will be less prepared for college entrance exams. CA will have to institute a 'statewide college entrance examination' and, maybe they'll pass a law prohibiting using any other entrance exam in an admission process (because they're racist or something or other).
100% agree - the real intent here is to reduce the achievement gap, and the real effect is to increase it. High achieving wealthy parents make sure their kids get math outside of school. The kids who lose are the smart, poorer kids who would have thrived and advanced given the challenge.
Plenty of extremely poor Chinese immigrants scrimp and save every penny to pay for top notch tutoring for their children. This isn't really about wealth, it's about culture and ability.
I half agree. Chinese immigrants strongly value education, and on average, have more inherent ability. So many overcome the disadvantage of being poor. That doesn't mean that the average poor kid isn't at a disadvantage, and that the smart poor kid is hurt by removing advanced math courses. That smart poor kid may not be getting the support at home like a Chinese immigrant, so support at school is all they are going to get.
Why not send them to private school then? Or is top-notch tutoring only available outside of a traditional school? In which case, why have a traditional school model at all?
It is not only that, if you parent is a successful engineer, businessman or whatever. Not only they will hire tutors if needed they can often teach kids themselves.
I learned more math from my dad who has PhD in physics then from all of my teachers and tutors combined.
One of the reason home schooling works so well. It is much easier to teach few kids you care about then whole classroom of the ones you don’t.
From the school district’s page justifying the new course sequence:
“Historically, rigor meant doing higher grade-level material at earlier grades, and equity meant providing all students equal access. The CCSS-M require a shift to seeing rigor as depth of understanding and the ability to communicate this understanding, and to seeing equity as providing all students equal success.”
If the goal is equal success for everyone, you have to hold back the high-achievers so the rest of the group can keep up.
The problem is hinted at here: "equity meant providing all students equal access"
What they are saying is that they refused to keep unqualified students out of the advanced classes. Keeping those students out would have exposed the schools to all sorts of accusations of discrimination, so they didn't do it. Parents insist that low-performing students be in the advanced classes, and the school doesn't say "no", so the class becomes a mess of failure.
When you try to understand math education, remember that it’s basically what would happen if foreign language classes were designed and taught by people who only spoke English and had no training in formal language theory.
Of course it’s largely a disaster: it’s taught by people who brag about hating the subject!
Algebra education can (and does) safely begin in first grade, with the introduction of workbooks: “4 + [ ] = 7” is a common exercise, and involves the implicit solving of a basic linear equation, where you solve for the value which goes in the box.
That you can teach algebra to first graders, but math education is so abysmal, only speaks to the complete disregard mathematics is given in education.
But what else would you expect when you leave it up to people who hate math?
Can confirm! Have a first grader and she actually found learning basic adding/subtraction, and now multiplication more fun by learning it through games played with implicit basic algebra. “I am thinking of a mysetry number, can you guess it? X marks the soot where the number goes... I’ll give you a clue 2 plus x equals 4. Or 9 times x equals 18.” Etc. She’ll laugh and play this game all day, and make up algebra puzzles for us too. Great way to learn her basic math and implicitly algebra too through play rather than by rote, or the plodding pace of the grade school system.
Heh, this reminds me of a fond story. My eighth grade algebra teacher (who did seem to genuinely love math) once said that mathematics was a game invented by man. I didn’t appreciate this at the time but I wish this was the mentality.
IMO the US math curriculum now is designed to train kids to become pre-PC-era clerks (who only need basic, manual arithmetic) rather than engineers.
I remember that, and hated it. Why? Because I had real trouble memorizing sums and products. I still have to stop and think to do simple sums in my head. I sometimes still count on my fingers when I add two numbers. And these types of problems were taught as memorization. Four plus what equals seven? You either knew it or you didn't.
If they had taught us how to solve these problems, I think I would have enjoyed it more. "You can take away the four from both sides, and there's the answer" somehow that would have been easier for me to work with.
I was frustrated in this way during my math education, but lately my opinion about memorization has softened a bit.
The trick is that it has to be approached in the same way as in sports: Demonstrate a technique, explain the principles, attempt in live play, then return to drill muscle memory(now that the student has discovered how bad they are at it). Music is very similar - you can play a song badly, then drill scales and arpeggios a while, come back and suddenly you play the song better.
The point of memorizing is, in the end, to make the knowledge closer and more available to you. But there are several ways in which this cycle drops the ball during education and succumbs to rote learning as the sole factor:
* The teacher themselves doesn't understand the principle, and thus is poor both at explaining concepts and grading results. "You get a zero," they will shrug, when the student has misunderstood something and turns in a problem set with wrong answers.
* The principle isn't connected to "live play", making the technique unrelated to existing knowledge. It's the way in which education systemically fails most frequently, and it starts with having classes specialized per subject and limiting the crossover between them. All too often, all that happens is that you do some problem sets, get tested a little later, and that's it - and so all your focus as a student is on passing, not on learning.
* The drill focuses overly much on tricks and "gotchas" and not on developing confidence and long-term retention, making the student uncertain about how to generalize the technique to the tricks. In comparison, when I took judo, we drilled all techniques on one side only, for the entire semester. Is it useful to be able to mirror the techniques? Yes, but that doesn't mean that any study time needs to be allocated to it.
This was true for me. I absolutely hated math in middle and high school. I never studied for it and got mediocre grades. I had a few teachers who really had no enthusiasm or drive to make it interesting. Fortunately, I started taking some community college classes my last couple of years, where I really fell in love with linear algebra. I wound up being a math major in undergrad and a geophysics PhD.
This development is part of a ground up movement that aims to change education from an activity in which learning is done to an activity in which the correct opinions are formed. Math, not being an opinion-based matter, gets dumbed down so that other topics can be put forth to the students.
It’s the kill zone competitive strategy applied to education (competitors to your children) instead of big tech. Make sure nothing can develop that threatens your family’s market position.
I was bored out of my mind at that point so tuned out. 7 years later I was struggling through the math portions of my CS degree and now with 10 years in the workforce I'm genuinely frustrated about not being allowed to do the shiny stuff (namely ML). The NZ school system at the time basically fucked me when it comes to basic mathematics and now in my early 30s I'm trying to fix this using Khan Academy. It's not easy and I know I'm going to struggle a lot more being significantly older. Maybe things have changed in the intervening years but I still hold genuine scorn towards whoever designed that math curriculum. This stuff is actually important.
It doesn't need to be harder because you are older. Many young minds lack developed abstract thinking. I recall hearing that many don't develop this until their mid twenties. My first pass in 9th grade of geometry and 11th grade trigonometry were rough. I later self-studied and found the material to be crazy approachable. If you are still struggling, I advise going further back in your studying or getting a good tutor who will help fill holes in your knowledge.
IMO do top down ML stuff and learn the math as-needed. This is recommended by Rachel Thomas and Jeremey Howard of fast.ai as well. There are a lot of great resources for the specific kinds of math you'd need for ML. I personally am self-proclaimed terrible at math ( also had very been schooling ) and was able to ML with not much extra work after their courses. Good luck!
>How can we get substantive, critical thinking skills earlier into the curriculum so we don't have to just assure students: "trust us, math will get interesting if you can just stick with it through calculus and linear algebra".
Don't teach math to elementary school students. Seriously. The current system mostly just teaches children to hate mathematics. Talk to some random adults if you doubt me - if you like math, you're in a large minority.
There's a good chunk of math that is nearly impossible to grasp until you've developed sufficiently abstract thought, at which point they're pretty obvious. I remember reading something like "sixth graders who enter with zero prior math exposure are merely a year behind fully educated peers at the end of the year". People are extremely good at picking up the mathematical concepts they need that are within their conceptual grasp. Hell, I know someone who wasn't exposed to algebra until they were taking calculus courses for their mechanical engineering major - they had to put in a ton of work and get help from their greek peers, but they passed their courses.
The American math curriculum is embarassing to even explain to most europeans. An “advanced” high school senior in the US might be taking Calculus 1. Most students graduate barely able to explain what a linear function or a quadratic equation is.
What American math curriculum? Every American state has its own curriculum and every school district has very broad autonomy, right? I’m no expert but most states don’t even have state exams at the end of high school that are exactly comparable. There are many states that have more internal variety in their education system than most European nations.
The closest the US comes to having a national curriculum is the College Board Exams.
This has changed with Common Core and the pushing of standardized testing to the lower levels (elementary and middle school), though high schools still tend to have more variation.
I grew up in a European education system (Scotland) and subsequently emigrated to the US. I have two kids in the school here. Although your sentiment was my initial expectation too, it wasn't borne out by some time I spent studying the two curricula (present-day Scotland vs the classes my kids have). What I found was that they pretty much matched up age for age.
I was taking ap calculus bc in 9th grade (calculus I). There were 2 other students in 9th grade with me. 6 were in 10th grade, many were juniors or seniors.
I left high school soon after 10th grade started so I don’t know what track I would have been on. But there are tons of overachievers taking on as much advanced stuff as possible in high school. So I think it’s a little unfair to say an advantages student only might be taking calculus I as a senior, because my personal experience and observations don’t suggest that.
On the other hand, what you are saying might have been true a long time ago. My wife’s father is a professional economist and he only took Calculus in college and he’s pretty smart. So I think education has improved in the US in that time.
American schooling is centered around standardized testing, and you can get a perfect score on the SAT Math without knowing the quadratic equation, so of course it's just a footnote in Algebra education.
The other problem with Calculus in American schooling is the number of students who took and aced their AP Calculus classes only to completely flunk their college-level multivariable calculus. I remember it always being recommended that, even if you took AP calc in high school, you should re-take calculus in highschool if you were going into Physics, Engineering, or Computer Science. Is that still the case
Americans seem to make a really massive deal out of calculus. I wonder if that's counter-productive and puts people off.
You give it a special name, you talk about other topics purely in relation to calculus ('pre-calculus') as if calculus is the central big thing, and people talk about dreading it at college.
In the UK we never used the term 'calculus' when we learned it at school - we were just introduced to differentiation one day without any fanfare as part of an ongoing maths course, and then integration later. You didn't get a chance to get apprehensive about it and build up a mental block because you didn't know it was coming and it was no big deal.
FWIW, prior exposure to calculus is really helpful for tackling introductory undergrad engineering classes. If you can't do derivatives and some basic integrations without thinking, then you will really struggle in the subject-specific engineering classes (e.g. Circuits, Statics, etc.) that you start to take in your 2nd year. Not sure if pre-college calculus experience is as helpful for other fields, though.
Our high school physics teacher pushed for alignment of math schedule with physics lessons, as basic mechanics is much more intuitive with the understanding of derivatives, and derivatives get a clear illustration (of the principles, and also of the reasons why one might care about derivatives) in these physics lessons, so it makes sense to teach these topics hand-in-hand.
In the UK, differentiation and integration aren't taught for GCSE maths (to ~16 year olds, last year of compulsory schooling) [1] but are for AS-level maths (to ~17 year olds) [2]
However, students select which AS-levels and A-levels they want to study; students can drop math entirely if they so wish. And some CS departments will accept such students, putting them through a high-speed remedial math course.
> They explained derivatives to us on the first year of UK university CS degree
That's done partly as a refresher for those who didn't do maths at A-level (so would be 2 years out of not doing maths at all) and to take into account some systems that don't teach it.
At least that was the case for my UK university CS degree.
Yes this was my experience in Australia as well we did not have separate "Algebra", "Geometry", "Calculus" etc classes it was all just 'Mathematics'.
From memory I think concepts from Calculus were first introduced in year 10/11 via geometry (plotting curves and finding points of inflection) from there derivatives just made a lot of sense - slopes as a rate of change and all that.
OK so I started calculus at 15, IIRC, but I was in the advanced class at the young end for my year. Most people who are going to do it start it at 16-17, if doing maths at A-level.
Everyone else drops maths at 16, never having encountered calculus.
But your college courses seems to be so high quality. How does a student in the US go from what sounds like quite a limited education in maths at high-school to doing so well in maths at college? What connects the two up?
Individual school districts and occasionally schools have enormous autonomy compared to the norm in Europe. So two schools in the same state can each have a class called Algebra II, with literally zero overlap in the material covered. Yesterday I read about a high school math teacher who decided to teach partial differential equations, normally, I believe what Americans call Calculus II in university, as an elective. There are good schools, but the system is very far from uniform. Partly this is because it wasn’t designed from the ground up to teach nationalism with education fit in around that goal. Puritan New England was the first society with mass literacy. Schools were locally funded, run and organised and that organisation of local rather than state administration persists, possibly in every state, certainly in most. Education came before nationalism so there was never a state system designed from the top down to turn everyone into Americans, nationwide, though many reformers gave it a good try.
No, partial differential equations would be the fourth semester.
Calculus I is differentiation. Calculus II is integration. Calculus III is vector calculus, with stuff like curvature in 3 dimensions. Differential Equations would be the next class.
The AP test covers differentiation and, optionally, integration. It's the first semester or two. This is what a good high school student will do unless the school itself is really bad or really small.
I remember Calc I (1st semester) being limits, differentiation and integration. And Clac II (2nd semester) being partial differential equations. This was in the engineering school though... it may have been different for other schools in the university.
This is the original poster's point. There is no consistency. Even in university. Some schools use quarters, some semesters, some trimesters. Some have letter grades, some percentages. Some are pass/fail freshman year, but are graded in subsequent years (e.g. MIT). What makes up "Calc I" at university varies tremendously.
One thing to keep in mind is that although there are a lot of elementary and high schools that do a bad job teaching math in the US, there's also a lot that do a good job. Kids that struggle in math in high school and/or come from high schools with poor programs mostly don't even try doing it in college. One other observation I've made is that people who do poorly in what I would think of as engineering math (calc 1-3, linear algebra, differential equations, probability and statistics, CS theory classes) often have more trouble doing the algebra error free than doing the "complicated" parts of problems. It's possible spending more time on algebra is actually beneficial to doing well in college level math.
My high school math teacher had spent years teaching remedial math at a university level. We thought she was being a curmudgeon when she complained that universities even offered remedial classes.
Nowadays, I think she was right: university students should be ready for university-level work. Many people aren’t at that level the moment they graduate high school. The US really doesn’t have a place for those people, aside from remedial classes.
But, anyhow, for many people, there’s an intermediate step that doesn’t get talked about.
they went to private schools or, alternatively, they went to the wealthier public schools which teach these sort of courses, although usually on a 'tracking' system which segregates the 'smart' from the 'not so smart' at around 12/13.
for the US university (at least the more elite ones):
the above, in combination of the fact that a big chunk of the of the students are international students.
An advanced high school senior in the US will take Calculus 1. An advanced high school senior in the US might take Calculus 2. (Yes, I am intentionally implying that taking calculus is what defines a student as advanced.)
My high school don't go beyond Calculus 1 (edit to add: this was two semesters and covered differentiation and integration).
That was years ago (pre-internet) and if a student wanted to take other college-level math they were released to take it at the local university campus. Today, I guess online classes are an option.
Calc 1 (Calc AB) is basic derivatives and integration without too many frills. Calc 2 (Calc BC) adds advanced integration (by parts, partial fractions, etc), L’Hopital’s rule and indeterminate forms, coordinate transforms, and basic Taylor Series.
There's 300M people there, you're bound to find 6 who are pretty good at it. And there's definitely resources somewhere to teach them stuff, it's the richest big country in the world.
The question is more how does the system work for average kids.
The US not only separates smart kids into different classes, eg AP and IB programs and early admission to community colleges, they separate the elites into whole other schools, eg private boarding schools, magnet schools, and technical schools.
This is ignoring the huge baseline variance in quality, due to funding and cultural variations.
Yes, I recall that very basic sets topics (e.g. unions and intersections),might have been introduced in 3rd or 4th grade? Definitely was in there somewhere before middle school.
The entire concept of grades is what's broken; as well as isolating and refining everything to the point that it looses meaning and ties to anything useful in the real world.
It would be better if there were projects that everyone were mentored on. Projects that had sub-assignments related to doctrines of focus and mentors (teachers) ready to help and to upgrade to high levels of detail and quality.
Even more ideally the work would be anything that currently qualifies as a government or civil need (double checking work by full time employees), re-enacting historical work with period engineering constraints (sort of steam-punk-ish) to teach live history, or otherwise maintaining the commons. (Infrastructure projects in software, analysis of actual civil infrastructure, conducting studies and tabulating results; with the interesting ones actually checked in more depth.)
At my school they started introducing algebra in 5th grade with legit algebra classes in 6th grade. I think that is a good time to start, although at least half the kids could have started a year earlier even.
Calculus was an absolute mind-fuck for my algebra centered mind when I got to college. Took me probably two full months before it clicked and I basically settled on "this is still math, but it's completely different rules and the old rules don't apply at all."
> Took me probably two full months before it clicked
That doesn't actually sound like very long. I can easily image it taking more than two months to learn a new programming language, or how to build your own radio. Pretty much any nerd activity will take some time to get into.
As someone who graduated a couple years ago this policy change feels ridiculous to me. The difference between someone entering college with either one or two years of calculus and someone entering with precalc completed is huge, and any policy that moves away from that is a bad idea.
The benefits imo are mostly around allowing more flexibility in scheduling for taking classes that require prereqs of calculus, which you're blocked from the first year otherwise. This can be compounded if you come in with many of your school's general requirements met, as this can make it hard to find classes to take that will be helpful towards your desired major. Additionally, some programs have (poorly designed) hard pipeline dependencies that mean you're set back from a couple quarters to a year if you enter without calculus completed.
This is just another nail in the coffin of public schooling. We've watered down the high school diploma so much that 20% of high school graduates are functionally illiterate. Based on the increasing requirements for bachelor's degrees where it doesn't seem to make sense on the surface, it seems employers have caught on to how meaningless it is.
Who is “we”? Public schools at the local level in the US are very diverse. If you have the money you can find public school systems as good as Ivy League colleges, with a self-selected peer population of students with highly engaged parents.
> The difference between someone entering college
> with either one or two years of calculus and
> someone entering with precalc completed is huge...
I am not so sure about that.
Too many students enter college with a half-assed grasp of math because they were rushed and pressured through the subject matter before they were ready to handle it. Combine that with grade-inflation and the fact that so many students just can't focus on one thing at a time, leads to a really sub-optimal outcome IMHO.
Students who go through private schools with mastery-based curriculum are far better prepared for rigorous college math, not because they take 2 years of calculus, but because they've actually mastered the subject matter that they've studied.
There should be no shame in slowing things down but keeping the course work rigorous for college-bound students.
The only sensible reason to not teach algebra in middle school is because you've taught it in primary school.
Moving it to high school sounds crazy to me. What do you do between doing your times tables and fractions in maybe 3rd or 4th grade and 9th grade? Geometry?
I'm any case it's quite fortunate there's online resources like Khan Academy where kids can learn on their own.
> What do you do between doing your times tables and fractions in maybe 3rd or 4th grade and 9th grade? Geometry?
Other countries do a sort of "pre-algebra", mostly focusing on non-trivial word problems which are then solved by setting up the equivalent to an algebraic equation, but in natural language arguments as opposed to symbols. AIUI, this sort of onboarding towards algebra is lacking in the U.S. system, and much of the difficulty that folks seem to experience w/ algebra and higher-level math might have something to do with that. (Of course, Geometry could also play a role in filling that gap. Euclidean geometry has traditionally been studied quite early, as a general introduction to mathematical rigor.)
Edit: Discussed in Andrei Toom, Word Problems in Russia and America [pdf] http://www.de.ufpe.br/~toom/travel/sweden05/WP-SWEDEN-NEW.pd... The reference to Russia is a bit misleading - of course Russian math education is traditionally among the strongest and most effective, but the same approach to using word problems as an introduction to algebra is found in plenty of other places.
Sure, but what do they do in grade 4 and 5, after basic arithmetic and fractions? I'd be surprised if the answer didn't involve quite a bit of proto-algebraic "word problems" of the sort described in my comment, and in the attached reference.
In the US states I've lived in students start first grade at age 6 or 7, sometimes depending on the month they were born in, but also depending on readiness, but also depending on what age they started kindergarten or preschool if they did. Kindergarten is nearly universal in the US, hence the K-12 system, and preschool (the grade before kindergarten) is what is optional, and daycare before that. I was 12 myself in 6th grade. So though he seems to be in another country (Estonia), his system seems to me to be similar to the US as regards grade numbering. I am not aware of 5 yr olds in the US commonly being in 1st grade. May I ask what country you are in where first grade begins at age 5 on average?
Pre-algebra was standard 8th grade in my school district in the US, with algebra in highschool. Advanced kids were able to take algebra in 8th, though typically only if they did pre-algebra in 7th grade.
Yes, but what US schools call "pre-algebra" is largely review of earlier material, with a few random concepts tacked on such as how to factor natural numbers. It's nothing like the comprehensive introduction to algebraic thinking that the reference above talks about, that, to reiterate, is extremely widespread outside the US (barring a few places where the negative influence of US- Ed School fads was especially strong - the reference talks about this, as well).
I think we need to distinguish between "Algebra" the field in mathematics and "Algebra" the class taught in US schools. The two are not the same. For example the "Algebra I" class my kids had was essentially analytic geometry : Cartesian plane, linear equations, y = mx + b. They were taught the basic concepts of Algebra (variables, commutative, associative rules, reciprocal) way earlier in I think 5th grade.
This confusion with class titles carries on through the years. For example "Geometry" includes matrices.
It seems that an excellent predictor of policy in SF is: will it increase inequality and the advantage of the very rich while being presented with liberal rhetoric? Then do it. This is a great way to keep smart but poor kids from competing with private school kids.
This is also across the board in most California districts. Not just SF. (This article is dated 2016)
I posted this because in our Bay Area school district, algebra and geometry was an accelerated course with just 35 students. (In just one school of the district) Selected by lottery.
It costs 7500/year but they want to get rid of it. Teachers are also planning to strike. To me, it looks like funding in CA public schools is no longer about education but about the care and feeding of public sector unionized employees.
Parents are horrified but in the same neighborhood thread, teachers are using it to ask the parents to support them. Students are given extra credit if they show their support for work the rule(only school work hours and no more clubs and extra classes like this) and strikes and walk outs.
It’s all mind blowing to me(I don’t have kids in the public school system) and I can’t wrap my mind around this. I have been approached by friends who are parents and don’t want to be vocal about their kids children in the community forums as I don’t have the vulnerabilities of parents whose kids are already in the system.
Certain Asian/Indian parents send their kids to tutoring classes which is seen as elitist and reactions have been borderline racist.
I just tell my friends to pull their kids from public school if they can afford it or homeschool them. I am very impressed with khan academy. I looked into it and I find myself learning from their classes.
I have to ask ...what is the point of public education in CA anymore? Should educational methods change and the way we teach subjects change? Conformity is tyranny at this point.
Teachers overload kids with useless assignments and the real learning happens after school in tutoring centers. And then the kids have to learn extra curricular subjects especially as they get closer to applying for universities. This is supposed to make them ‘well rounded’ but it only makes them depressed, stressed and uninterested in education. They become cynical and jaded and entirely unprepared when they go to college. It’s worse for the smart kids as they are bored but still have to go through the motions.
California has made home schooling more difficult than most states. The cynic in me says that it’s done that way on purpose to protect the position of the public school employees.
Since the graduates of SF public schools are expected to leave the city and make room for wealthier folks, the use they will still have for the city is as ambassadors of San Francisco's political views to the rest of the country, where they will be able to afford to live.
I think that’s the most natural model for a virtue-signaling economic sinkhole that can’t sustain a viable population, but is trying to spread its governance ideas anyway: it’s literally analogous to unhealthy cells using false signaling to sinkhole resources while replicating their damaged DNA — a cancer.
It’s not just SF: Seattle, NYC, and basically every major metro that subscribes to particular left-wing ideas.
There are many strong opinions (ITT and elsewhere) about how algebra should obviously be taught in middle school.
The data I've seen does not support those conclusions. Eg [1][2].
This is kind of an interesting article, but it really does not convey the essential facts at hand. Middle school alg harms student achievement. Many/most students end up retaking it anyway (!!!) and there is a simple and effective fix. SFUSD should be lauded for braving the political heat.
Maybe there is a kind of educational system where middle school alg is a good idea. But our system is not that kind of system. If you live/vote in SF please support evidence-based practices and common sense. <3
[1] https://www.edweek.org/media/2018/06/11/v37-35-algebra-chart...
[2] "But in a 2015 study, the University of North Carolina's Thurston Domina and colleagues tracked how California's uneven 8th grade algebra-for-all rollout played out across districts. In a surprise finding, they discovered that higher enrollments in early algebra were linked to a decline in students' scores on a state math test."
The problem is that a country's technological advancement is not usually based on the mean performance of everyone, or the equity of the population in general achieving a minimum level.
A country's advancement is based largely on the outstanding rockstars / top performers and their innovation.
I'm afraid this is trading off wanting to help students who are low achieving, at the expense of the highest performers getting to achieve their potential.
Of course the step that SFUSD took led to fewer students retaking the class later. You make a class easier / later, and of course everyone will pass more successfully. But did it measure how many students were held back from higher achievement? Of course not, there is no good way to quantify that.
China and India, who let far fewer aspects of equality, racial/group concerns, privacy and self-esteem issues interfere with their schooling, are going to eat our lunch.
You really think you can tell who the "outstanding rockstars" of your country are going to be from an 8th-grade Math class? I mean, if we were teaching stuff like "data science" and JavaScript coding in elementary school, you might have a point. But this is elementary math, that everyone should be able to do to a high standard. I really have trouble grokking what your point might be here, other than trying to justify the all-too-common "soft bigotry of low expectations".
No, you don't know exactly who the rockstars are going to be, at any level. That's exactly the point of having at every age, some "reach" opportunities for those who can use it at that stage and show that they're outstanding and benefit from being helped to test/learn to their potential.
Versus what, you suggest that we have mandatory remedial level education for everyone to sit through until the last kid is performing up to grade level?
Maybe there is a kind of educational system where middle school alg is a good idea. But our system is not that kind of system. If you live/vote in SF please support evidence-based practices and common sense. <3
Indeed, your system is not that kind of system. On the other side of the Pacific, Asian kids are routinely prepared for algebra by 7th grade, and their scores on international math tests in later years lead the world. That's Asian "common sense".
One can't help wondering why it is that their cousins, the Asian kids in the SFUSD, are so poorly prepared for algebra even by the 8th grade. The fact that San Francisco-style "common sense" decides that the solution is to declare that even those who are prepared will no longer be allowed to take algebra until high school says a lot about the fraught relationship between San Francisco and common sense.
Non-SF common sense might conclude instead that the fix should involve making elementary school better rather than making middle school slow down to match the failing elementary schools. Maybe expand your "evidence-based" thinking beyond your own failing districts and look at statistics and practices from the rest of the world. What is it that Singapore, Hong Kong, Taiwan, ML China, Korea, and Japan do with primary-age Asian kids that your SFUSD doesn't do with its Asian kids? Why not start with that?
My original comment was really focused on my narrow read of the question in the thread title. My comment is basically "because there is clear A/B test data to support the decision".
I take your comment as a valid answer to a broader question -- something like "thinking about the bigger picture, and envisioning all possibilities, how could and should SFUSD work?" I have a lot of thoughts on that too.
But if I take your comment as a direct rebuttal, on my same topic, I think it's an impractical way to manage the system. It's like you're arguing that middle school alg is a cool feature that shouldn't cause the app to fall over, and we should redeploy the cool feature and reengineer everything. I don't agree that your conclusions flow from your premises and I think the pragmatic approach (given your premises) is to disable the feature while we do the reengineering.
"What is it that ... do with primary-age Asian kids that your SFUSD doesn't do with its Asian kids? Why not start with that?"
Because that's culture. That's respect and reverence for education and teachers. And then there's the after-school and weekend tutoring.
There's always this discussion in comparison with Finland, as well, which takes a very different approach to education than China. Yep, again culture. A commitment to children. In Finland, a commitment to funding schools and social services as a society.
It's way easier to mandate a change in the curriculum. As a mathematics educator in the US, I'll make this claim: the curriculum is almost irrelevant. It's the teachers and the attitude of society toward education that matters the most. If teachers are good & have the freedom and training to teach, they'll adjust the curriculum, whatever it may be.
The average salary for primary education teachers with 15 years experience in Finland is about $37,500, compared to $45,225 in the United States.
SFUSD starting salary for a teacher with no experience is $52,657, average salary is $69,572, $90,000+ compensation is not uncommon. That salary does not exist for teachers in Finland.
Finland does require all teachers to have Masters degrees though. Teachers are also allowed a great deal of autonomy and trust in designing their curriculum and teaching approaches.
What form of achievement are we talking about? I’m seeing schools were anything less than all A’s is a failure. Are we dropping because kids are getting C’S?
I'll recap the links I dropped in case they're hard to digest or whatever. They cover two measurements of achievement:
* Before this change, 40% of SFUSD students had to retake Alg. After this change, it was 10%.
* In a separate study focused on statewide 2008 effects of middle school alg, "enrolling more students in [middle school alg] has negative average effects on [10th-grade standardized test scores]".
This SFUSD plan does not seem dumb from what I can tell, and I'm an informed and interested observer. Change my mind!
I went to a parochial grade school and hated it. I begged my parents to take me out, and by the time 7th grade came around, they gave in. Unlike my siblings, I went to public school for 7th and 8th grade. In retrospect, if I ever have children, I would make every sacrifice imaginable to ensure they go to private schools. No amount of complaining from them would ever make me reconsider.
I believe that actively working to prevent top students from making too much progress over the median student is the norm in all public schools.
It is wrong that we primarily divide the thousands of students in a given school district by the year they were born.
I agree with your last point whole-heartedly. We need to start adjusting for ability and proficiency as early as 2nd grade.
The big challenge with all teaching is that you must teach to the "middle" ability child. Instead of splitting by age we should be splitting into groups like the top 45% in one class, the next 35% in the next and the bottom 20% in the last. This lets each child have more or less time based on the aptitude and creates smaller deviance between the "middle" and top/bottom of each group.
Absolutely true. However, everyone knows who the kids are that got an F on the test anyway. In this structure, they get more help and may be more able to get a C- instead. It is even more so with math since everything builds on previous knowledge unlike something like history.
Stigma isn't all bad. It's a motivating factor. Decades ago, the norm was to post grades on the wall for all to see. We lost a lot of motivation when we got rid of that.
I went to public schools and then to Caltech, where I could sleep through freshman chemistry because my public school chem teacher was so great that I already knew the material.
I read this article with a lot of interest a while ago, since I have a kid in middle school in SFUSD.
Having been through the system, I think the whole thing is pretty stupid. People, seriously, the problem with math achievement in SFUSD is not a function of whether we favor concept development or technical approaches. The problems are far deeper. We can only dream of being at the the point where this sort of debate is actually what matters.
I think what happened is that US educators looked at Finland (among other places that have high math scores) and saw that there is more development of concept and less emphasis on technical drill. So they went back and made a curriculum that does, like, the concept thing.
Interestingly, they just don't seem to want to copy that part where math teachers (well, all teachers) are a highly talented and respected group of professionals, drawn from the upper tier of graduates.
I think that the US failed to understand that Finland does things this way because it's how talented teachers who are good at math teach math. You can't put this in a curriculum, hand it off, and get the same results without that talent base. And if you have that talent base with autonomy to do the job properly, well, they'll probably do it regardless of the curriculum.
Until we address this (it's not the only thing we need to address), this whole debate is worthless.
The article restates a presumption that by offering advanced students harder coursework, they are "leaving behind" the rest of the students. However, by the very same logic, if 20% of the kids get ahead by one year, aren't we just leaving behind 80% there, vs 100% in the no-algebra-offering scenario?
In most analyses of STEM, the top echelon of US students is NOT behind the rest of the world's top echelon: it's only in the medians/means, accounting for the vast number of poorly performing Americans, do we see the international STEM education gap.
Therefore, I'd argue this policy is actually detrimental to the one and only group of students we are doing well with in the USA, namely the top tier of students.
I wonder what level the public school 8th grade teachers are at with their Algebra at this point and what that has to do with this. I get the impression there are very few 8th grade math teachers who studied math in college as their first choice and then transitioned to education. There are too many more attractive STEM careers if you are inclined towards math, and the best math teachers in a school district are likely to be teaching the advanced 11th and 12th graders.
I took Algebra in public school in 8th grade in 1990. The public school was pretty bad, I remember how frequently we (the 8th graders) were correcting the teacher, he was that bad he was screwing up the examples on the board almost every class.
I got sent to Catholic School for high school. Everyone in the math department had an advanced math degrees. Calculus was taught by a PhD. The curriculum blended Algebra/Geometry/Trig/Calculus across all 4 years instead of the "one thing at a time approach." I remember being very cognizant of thinking I was probably better at algebra by the end of 9th grade than my public school 8th grade teacher had been. The quality difference in the teachers for math was mind blowing between the two schools.
I'm conflicted on this. Having moved 18-20 times across 7 states before high school, I'm intimately familiar with the inconsistencies between curriculums that existed before common core. I was taught some things 3-4 times while others I missed entirely. I also, through a series of (in hindsight) unfortunate events, ended up at the point of taking calculus my freshman year of high school and had three years of no math before college. While it was convenient as a high schooler, it really bit me when I started college and dropped straight into advanced calculus after a three year hiatus of all things math.
Harmonizing with the common core is fantastic. But school districts should also strive to harmonize with surrounding districts when it comes to "extras" like this, as well.
Lowering educational standards to meet the needs of the 20th percentile student is preternaturally stupid and is why we’ll be moving out of San Francisco before our kids are in school.
>the CCSS Math 8 course that eighth graders are now expected to take includes 60% of the material from the old Algebra I course. This includes linear equations, roots, exponents, and an introduction to functions. The new course also offers students a taste of geometry and statistics—hardly your typical middle school fare.
If you're worried about social justice/equity, I have great news for you: the SFUSD "joke" created dramatic and positive improvements across student demographics, including (and especially) black and Latino students.
In most districts, private school is often better than public school. This is especially true in California for a variety of reasons (first and foremost Prop 13). You may wish to send your kids to private school (makes sense to me) but from the data I've seen, these changes significantly improved SFUSD's math offerings.
Thanks for your reply, and for the links. I followed them and read through them, but I did it quickly, I'll take another look later.
I don't think the data supports the change as an unqualified success. The article you posted (again, I quick read it) doesn't compare calculus enrollment before and after the change. This article from the chronicle provides that data:
"While more students are taking precalculus now, the enrollment in Advanced Placement calculus courses has declined by nearly 13 percent over the past two years. Enrollment in AP Statistics, which requires only Algebra II as a prerequisite, has surged nearly 50 percent."
My understanding is that all courses other than calc that require algebra have seen higher retention and enrollment. So we could view this as a qualified success, a trade-off: better results pre-calc, at the cost of fewer students taking calc. Keep in mind, we don't know how many of the students who are now succeeding in Stats would have been succeeding in Calc, were it not for the change. We are also only given percentage increases/decreases. Without starting populations, we actually don't know if the increase in Stats offsets the decrease in Calc.
Unfortunately, this is precisely the trade-off people were worried about. Fewer of the top students are taking calc. Any claim of success must address this.
FWIW, I do agree with that last bit from the article, about calc in high school not being quite as essential as (some) people think. If an entering college student has very strong algebra, trig, and pre-calc (let's say the ability to solve elaborate algebraic equations involving trig functions, logs, various exponentials, tricky multivariable quotients... and really gets it), then I don't think the lack of formal calc would be all that big a problem.
Then again, calc is a good way to get into those kinds of equations.
I wouldn't call this change an unqualified success either. There are probably ways to improve from this new status quo. Maybe a 13 percent decline in calc enrollment is a sign of overcorrection.
I would call it an improvement, though. That's all I'm on about. Basically what you said about "calc in high school not being quite as essential as (some) people think", together with a dash of "everyone thinks their kid is a genius who's being held back by this change but most of them are wrong" and a heavy dash of "stats is obv better than calc anyway".
calc enrollment goes down by 13%; alg re-dos go down by 30%; seems like most people are winning or breaking even right?
I disagree that “a heavy dash of stats is obv better than calc”. Not saying there’s no case for it, but it isn’t obvious at all.
I also don’t think the line about everyone thinking their kid is a genius is productive for this discussion. I think there’s a reasonable point in there, but I think it’s needlessly caustic and fails to recognize legitimate concerns about the effect this will have on aspiring STEM students in college.
> Enrollment in AP Statistics, which requires only Algebra II as a prerequisite, has surged nearly 50 percent."
Probably has a lot to do with the intro-to-stats stuff that they introduced to the math curriculum as part of this reshuffling. Hard to see that as a negative - if the one thing we can worry about is "more students taking AP stats, fewer taking calculus", I'd say that this change was a huge success!
Abstraction is really hard. An economics professor of my acquaintance said he started requiring calculus for his game theory class so he could be sure everyone was competent with algebra. Every time you increase the level of abstraction you lose some people. Some of them can be taught it but some can’t, no matter how hard they or their teachers try.
Interesting. That’s probably the best reason I’ve heard for why calculus and computer science prerequisites should be similar, even when the latter doesn’t use the former: if you don’t have the abstraction capacity for calculus, you probably won’t have it for programming.
That's exactly how I started teaching my 2nd grader. They got some problems like that for their homework and on the scratch paper I said, "well for the blanks we can just stick an x in there" and she didn't even blink and went with it. Because well blanks look just odd. They might look like a "-" if it is written too high.
This is an example of them taking something that's basic and making it more complicated and cumbersome by trying to "simplify" it.
Montessori uses clear test tube learning devices with beads. Less abstract and more easy to visualize. This is taught in 2nd grade. In your above example, the blank or variable is an empty test tube, the 4 is a tube with 4 beads, and the 9 is a tube with 9 beads.
That works well for teaching how to solve problems like that, especially in an early age.
However, do you know what's the Montessori approach to learn abstraction as such, so that you can move from the visual/tangible approach (which works well for simpler problems) to solving problems with the abstract symbol manipulation that's needed for all the more complicated tasks including applied math at physics, economics, etc ?
In my experience as a math tutor, they’re untaught natural abilities via poorly constructed (and hence, understandably confusing) “math” lessons taught by humanities majors turned primary school teacher.
I often make the same point, and have occasionally found just rewriting it as “4 + [x] = 9” is enough to help span the conceptual bridge. Sticky notes in white board equations is another way to make this point.
What an interesting article. I ready it for a bit of morning outrage and now find the issue far more complex and subtle than I had thought.
My own child entered the US system in grade 8 as it happens and due to language issues was tracked into algebra, which was mostly stuff he’d had a couple of years before (not because he was an advanced student, simply how his German system had worked). He wasn’t able to get back on track until the end of the year.
So I of course assumed this change would be a disastrous concession. Now I’m not yet sure what to think.
I am curious if you are in CA and if your child was picked because english wasn’t ‘spoken at home’.
CA has a funding formula that takes money from all districts and put it in a common pot. Better well funded schools are no longer assigned according to affluence of neighbourhood.
Funding for schools is then redistributed act to principles of no child left behind..which means that if there are more kids in a school that avail free lunch/meals or if they don’t speak English at home(English not native language), then the school can ask for more $$ from Sacramento.
In my school district, kids fluent in English are often tracked to non English speaking/English as second language streams to tap into these funds.
It’s a bit of a mess because affluent areas don’t get enough funding but teachers need salaries for standard of living costs in said affluent areas...so most of the funding goes to salaries and then to cover unfunded pension liabilities of staff.
Eventually students perform due to parent involvement and extra tutoring classes after school hours. They do end up over worked and stressed out.
I'm a fan of "common pot" but it doesn't work as well as one might like: in the public schools here in Palo Alto the parents routinely hold auctions that raise multiple millions of dollars which goes into paying for things that I think every kid should be getting (art, PE, music, an aide in the classroom etc).
PTA donations got so extreme that the school district had to implement the same "one pot" approach just to spread the money around the schools in the Palo Alto school district.
Palo Alto district is a basic aid district. This means that they get no money from the state...they have parcel tax to fund their schools. Palo Alto is a wealthy enclave in CA. They don’t have to deal with the restrictions of other counties and school districts.
[..]Some districts, known as “basic aid” or “excess tax” districts, fund their revenue limit entirely through property taxes and receive no general purpose state aid. They also retain any excess property taxes within their district. ... Basic aid districts are generally more advantaged than other districts.[..]
That's interesting! Yes California, but it was a private school (chosen mainly because it had a low snooty factor but also because it had both English literature and English language/writing as two different subjects) so the problem was he didn't know any of the vocab when given a placement test so they just threw him in the default classes.
FWIW, students in SFUSD who want to get calculus as a senior are required to “double up” on math one year - for example, taking algebra and geometry simultaneously.
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...unequal access to high quality education is a comparable injustice to unequal access to the ballot box and is "the clearest manifestation of the nation's caste system."
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The real caste system in education is those who can afford to send their kids to a high-performing private school vs. those who cannot. This solution only exacerbates that problem.
> the real caste system in education is those who can afford to send their kids to a high-performing private school
Is the problem here is not with the public school system, but rather with the existence of high performing private schools? Should these schools be banned? Perhaps along with private tutoring? Such policies might well lead to increases in equality of outcome.
I would not advocate for removal of algebra from middle school.
But I would consider advocating removal of add/subtract/multiply/divide facts from grades 1-5, and replace the time with learning of how to reason.
I feel like learning you math facts, rules for multiplication, etc challenge children early on, but could be taught to a naive middle school child within a couple of weeks or less. Thus teaching this to young children seems like waited effort with little transfer benefit. On the other hand, getting children interested in learning, learning how to organize thoughts into an argument, etc...there is clear benefit for that, I wonder.
I hear so much moaning about students with "poor math skills" when it's really poor arithmetic skills. The moaners typically wouldn't recognise a differential equation if it hit them in the face.
What is so abstract about Algebra I? Isn't it pure rote? I mean, it's so rote that I am confident everything in Algebra I can be solved by Mathematica (or Maple) automatically.
They started teaching us algebra in grade 5. It took me until my first year university to really fully understand it. A lot of this was due to my own lack of efforts but also due to lousy teachers. The math teacher I had university was probably one of the best teachers i've ever had for any subject. I pretty much owe my ability to understand math to that teacher.
It's an unfortunate fact that teachers that really enjoy and understand their subjects is a rare treat and most people won't have the chance to encounter one in all subjects till college.
I think there would be a lot of use for teachers to get additional training in whatever field they're teaching so they have a deeper understanding than just what they remember and what's taught by the book. I don't even really blame the teachers so much because they got into the work to teach generally not to teach a specific subject they may get stuck with.
It has more to do with the left-wing ideology of the San Francisco school district, and its loathing for anything that might resemble elitism. There's a reason why private schools are so expensive in SF. Parents are abandoning public schools faster than new private schools can expand to meet demand.
I have observed the curriculum so far in US, at least up until the 2nd grade and it's a hot mess. Even for double digit addition problems they jump right into "cool tricks" and "mathematical thinking" while there are just really a few steps to memorize and, guess what, kids are great at memorizing stuff. Later on it's time to show a few short-cuts and kids even discover those themselves.
Now going back to algebra, I am firm believer that a simplified version of this has to be introduced much earlier. It's a bit like a spiral, do basic arithmetic, geometry, number line manipulation, simple word problems, next year do the same but more complicated, add some algebra in there and so on.
But I guess that's just too boring and teachers, and probably higher level authorities, decide that they need to do "something" so they start making changes just to they can slap something on their resume. Developers do that too some extent, "here let me rewrite all this..." we can certainly understand that, but doesn't mean it's a good thing.