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Comment by texaslonghorn5 | Hacker News Reader

Ultimately the solution is better STEM foundation at the very very early ages. Universal pre school would probably be very useful. Empirically [edit: n = 1 or 2], it seems that drilling basic arithmetic and then multiplication tables early in pre K and earlier elementary will give students a better intuitive math foundation to do algebra very well. That would enable everyone to go into more advanced classes at the same time (earlier) rather than these policies which want everyone to go in at the same time (later).

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As a math PhD with dyscalcula, I'm very skeptical. I was nearly held back as a child because of poor arithmetic performance, and really only started to be above average when we started on algebra. Poor arithmetic isn't that uncommon among the mathematicians I know.

Interesting. I do not know much about dyscalcula. At an early age do word problems make it worse or better?

I don't think they make much difference either way, as it's specifically the processing of numbers.

Poor arithmetic actually drove me to learn how to program calculators and caused me to be interested in being a software engineer

Did that dyscalculia prevent you from learning and attaining familiarity with the standard algorithms? That's the sensible goal of "drill and kill" in early grades, not doing routine arithmetic with high amounts of significant digits.

Nope, its literally recalling the times tables that I choke on. There were timed segments, and I couldn't keep up

I love this.

On a similar note, I have a friend who majored in math at Harvard. He once told me that he came into Harvard being arrogant because in high school he was always at the top of his class in math. He enrolls in his first college level math course thinking he's got this, but he soon realizes that "higher math", which is largely proof-based, is a completely different subject than what he learned in high school. A month in he bombs the first exam. He went to the professor, who is originally from Italy, and explained his situation and how he was a star in high school. He responds in a thick Italian accent "that was not math, that was computation. In this course I teach math".

The math you typically learn in high school is very important, but I wish that we did a better job of explaining to high school students that what they are learning is completely different from what "real mathematicians" study (although I do think that computation is quite important in engineering, for example).

One of the major problems is that most of the high school teachers are themselves not very familiar with math, and are more educated in the very different problem of corralling children.

To be fair to your friend, 90 years ago they probably could've had a productive career in applied math as an expert computer. There was a point when calc tricks could get you a PhD, but those days are gone.

For me it was the opposite, i was a pretty average math student in high school and fell in love with proof based math in college. Proof based math requires much more creativity and thus is much for fun for me at-least, i wish i was taught proof based math in HS.

I wish we did probability with equal gusto as kids. I very occasionally multiply 7 and 6 in my head, but have to reason with probabilities and statistics all day long.

Agree. I think getting really good at addition subtraction and multiplication and then division/fractions is the way to go though. Have really strong math fundamentals and then learn algebra and then everything else is far easier to pick up.

Fair point. My first introduction to stats was at second year in uni though, far too late. By that time I had already done a lot of calculus, which while important, hasn’t exactly been critical in my daily life.

Arithmetics is absolutely the most important topic to learn, since it is the basis for all other quantitative reasoning.

For example, it is really important to understand that 1 / 3 chance is the same thing as 3 / 9 chance. It is obvious to you now since you have done so much arithmetic's, but to someone who never properly learned it they wouldn't be able to properly compare those two and could think that one is a very different number than the other. Without basic understanding about quantities all other quantitative skills become worthless.

Statistics is the only reason I was able to get over my math struggles in college.

>"drilling basic arithmetic and then multiplication tables"

I get the sense that such rote methods are no longer encouraged and a lot of the "new math" in Common Core is aimed at approximation and reckoning so that students won't rely on memorization.

If you want to teach people methods to solve equations, limits, integrals etc. speed with basic algebraic operations is necessary.

Facility with those methods is then necessary to be able to adequately follow important proofs and gain understanding of more advanced concepts.

I don't know how you would teach people important results in their fields (physics, computer science etc., I'm not talking about actual mathematicians) without those skills.

TL;DR: algebra != arithmetic AKA real math doesn't use numbers

I'm only good at _arithmetic_ because of making Warhammer 40k armies (true story bro).

I'm good at _algebra_ because I was taught well, on top of a knack.

Speed with basic algebraic operations was very helpful in many places but speed with arithmetic operations has only been helpful in board games.

I don't think anyone here would disagree with your point about algebra, but I think a lot of people such as myself would disagree that pre-K memorization of arithmetic helps with algebra later.

Performing a bunch of calculations for tabletop wargaming is basically the same as learning multiplication tables and solving related problem sets. It should help every time that for example you have to simplify a polynomial involving fractions and similar operations.

As I stated in another post, I don't know when it is neurologically ideal to learn arithmetics, it seems something that would be important to study carefully (personally I learnt before grade school, when I was 3-4 years old, but I didn't learn to read until I was 6, something that is often taught earlier).

Algebra does use numbers. I learned to move the numbers to one side and the variables to the other.

10x+7 = -2x+31

Move 7 to the other side (picking up a minus sign)

10x = -2x + 31 - 7 = -2x + 24

Move -2x to the other side (flipping sign)

12x = 24

Recover 12*2 = 24 via memorization, quick division (though not long division), or whatever method

Therefore, x = 2. Then I drew a square and was done.

>Then I drew a square and was done.

I don't get this part.

You mean to highlight the end result? In that case, it would, most of the time, be a rectangle (or try to be).

I was joking, I meant the QED box.

I know about QED, but I am not familiar with the concept of "QED box".

A cursory web search yielded nothing.

It’s a symbol that goes by many names: the tombstone, end-of-proof, or Q.E.D.

"∎"

Interesting.

I wasn't familiar with this, despite being a graduate. Its use might depend wildly on where.

But strong arithmetic fundamentals are absolutely necessary for strong algebra. I've watched kids struggle with basic algebra because when they don't instantly recognize that 7x8=56, they also don't recognize that 7zx8z=56z(edit: squared).

Edit: thanks to the reply; HN ate my superscript 2. Apparently it doesn't like the unicode multiplication x, either: × ² ?

Good point ... however, 7z*8z=56z^2.

Hmm. Alright, I can see solid arithmetic being good for introducing the concept of a variable...

...but the only time I ever saw significant numbers when actually doing math was in toy problems that deliberately chose weird, big coefficients, where the arithmetic part was by far the least significant.

> Empirically, it seems that drilling basic arithmetic and then multiplication tables early in pre K and earlier elementary will give students a better intuitive math foundation to do algebra very well.

This aspect of the Common Core was about recognizing deficits in conceptual understanding resulting from rote methods of drilling arithmetic.

The empirical evidence is the opposite of OP's assertion, but the end point of giving students a better intuitive foundation for higher level math is indeed the goal!

Signed,

An elementary school math teacher who has studied the 60 years of math reform in America, internationally, and worked very hard to ensure all students have a foundation to succeed in higher level mathematics

Makes sense, the drilling that works for one student probably doesn't generalize all. Thanks for your perspective!

More and more earlier and earlier doesn't comport well with child development and can backfire by making students feel incompetent. We need better teachers, probably by paying more so more talented people join the profession.

Other countries do a lot of "pre-algebra" in the later grades of primary education, when the kids are quite ready for it; "drill and kill" rote methods are generally focused on in very early grades, since they help build familiarity with the sort of rigorous, algorithmic thinking that's required for good math proficiency. This is what Russian Math, French Math, Singapore Math, etc. are built on, and the approach has stood the test of time indeed. The fuzzy "Common Core" approach pushes abstract content way too early, and ends up confusing kids as a result.

I wholeheartedly endorse Singapore Math, Russian Math, Canadian Math (this is a thing, check out John Mighton's fantastic JUMP program). ANYTHING but US math.

I think we could pay more, but only if we could actually get better teachers. So many teachers are bad and not worth even what they get paid (personal experience from a nominally quite good school district). My concern with paying more before figuring out how to weed out bad ones is that it would just be a waste.

Pre-K (4 year olds) seems a bit young for teaching multiplication tables.

They do not require abstraction, I don't know if it is necessary but it is practical to teach them.

What's the point of teaching kids to memorize something that they can't apply? When I was a kid, schools taught multiplication tables in 2nd grade, when most kids are 7 years old. The difference in cognition between a 4 year old and 7 year old is insane.

I'd be surprised if there were any countries where multiplication was formally taught to pre-K students as part of the standard curriculum, but i'd love to be proven wrong.

I don't know if there are countries. I believe that if there actually was a unified accelerated math framework that was really emphasized starting age 4/5 then kids would be absolutely fantastic at math.

> What's the point ... ?

Paraphrasing what I said a comment above, you drill addition and subtraction until everyone is good at it, then you drill multiplication, then you do basic division, then you start introducing basic one variable algebra with "move plus to the other side to get minus" etc. The application is using algebra for word problems; formalism can come later.

In context I meant get really good at addition/subtraction starting pre K and then multiplication once +/- is mastered.

Though empirically, I don't know about age 4 but kindergarten is definitely not too young for learning up to 12*12. And once you figure out multiplication and eventually mental division, it's not too big of a leap to have one variable algebra with "move a plus to the other side to become minus" etc. The formalism can come later but it's fantastic to have some exposure to moving numbers and symbols around from an early age.

Out of curiosity, is that a hunch or are you aware of any schools teaching multiplication tables even in Kindergarten? This used to be done in second grade when I was a kid.

You're right, it's a hunch (n=1.5±0.5).

When I was in kindergarten, I used to do math booklets at home with my mom for fun. I learned basic multiplication sometime around then. 13 years later I majored in engineering.

So I'm not saying it can't be done by any 5 year olds, but it seems young to teach this to the majority of 5 year olds.

I'm not aware of any schools teaching multiplication tables in kindergarten, but I did memorize the 9x9 table when I was in kindergarten because my older siblings' Big Kids Notebooks all had the times table on the back and it formed a rhyme/ditty in the local language. After it was explained to me that multiplication was repeated addition, that made perfect sense.

But don't ask me about division, my siblings/parents/whoever tried to explain it as "the opposite of multiplication", which was complete nonsensical gibberish and I didn't learn division until years later.

My sixth grader and first grader score in the 90+ %ile in mathematics and didn't come close to learning multiplication up to 12 in kindergarten. In fact, the topic isn't even covered until second grade at the earliest.

I think establishing a foundation of addition and subtraction takes far longer for children to master than you're considering, especially since there is evidence that children of this age appear to intuitively view numbers logarithmically rather than linearly [0].

I suppose you could take advantage of this by somehow prioritizing multiplication and division over addition and subtraction, but I think there's too much value in comprehension of linear numbers and addition/subtraction since that is the lion's share of interactions they will experience at that age.

On the other hand, if you're merely talking about abstracting multiplication and division into patterns, then I wholeheartedly agree with you, and there is evidence supporting this [1]. Although pattern identification is already part of kindergarten/1st grade curriculum here.

Ultimately, IMO the most important aspect of education in general is covered in the open letter linked to the OP:

> There cannot be a “one size fits all” approach to K-12 mathematical education.

My children have thrived with their current math curriculum, and I know some of their classmates have struggled in contrast. One size does not fit all in education, nor in many aspects of life.

0: https://news.mit.edu/2012/thinking-logarithmically-1005 1: https://www.tandfonline.com/doi/abs/10.1080/15248372.2012.68...

If you can memorize the alphabet, you can memorize the multiplication table to 12x12.

And you'll start seeing beauty in patters and sequences of numbers. The sooner the better.

Anecdote: my then 5 year old and I would "practice counting by different numbers" on the walk to school. By the end of kindergarten, she could count by everything up to 12s. In 1st grade, we started reversing it and asking how many 4s in 48 and the like, and by the start of second grade, we were firmly in adding and subtracting fractions with different denominators (though, on paper at this point, no longer mental math).

She had (has?) a solid grasp on numeracy. I recall asking her why, around 7th grade, "0.999..." is equal to 1. I was prepared to show some fancy algebra and she one upped me when she said "well, 1/9 is 0.111... so 9/9 is one and 0.999...".

She never liked math though. She spurned calculus.

There is little evidence universal preschool would reduce academic variance later on. On cognitive measures (though not necessarily social/emotional), randomized trials of such programs tend to show fade-out (no difference between control and treatment groups) within several years.

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