What the Hell Did I Learn?

[LostAngeles]

I'm kinda-sorta T.A.-ing in a little prep class for pre-algebra. We're going over a lot of basic math stuff, like the operations and integers and decimals. I went and made up a multiplication table for the students to practice on because I remember that having to do a whole bunch of those it what helped me learn mutliplication...

...twenty years ago in second grade.

That's not a diss on these students at all. I'm fully aware that what's easy for me to get is not easy for others and vice versa. Here's the thing:

I remember learning my times tables in second grade. In seventh grade, I went into pre-algebra and was introduced to my good, long-time friend, x.

What the hell was I learning for four years in elementary schools? Was it fractions and decimals and percents? Was I just repeating the same things over and over again?

Seriously, I can't remember. I can remember up to second grade and everything seventh and beyond, but I have no idea what the hell was in the middle there. This leads me to wonder if my time was being wasted and pre-algebra could have been introduced earlier and to more kids or if there was something really going on there.

 

[stup_id]

Well.. im not sure but I think that algebra education is provided until abstract reasoning is possible for the average person, according to the psichological development theories in vogue...

Anyway it is really quite unfair because theres a large percentage of students that as you feel, they might be able to have abstract reasoning much more sooner

 

[fuelair]

I'm kinda-sorta T.A.-ing in a little prep class for pre-algebra. We're going over a lot of basic math stuff, like the operations and integers and decimals. I went and made up a multiplication table for the students to practice on because I remember that having to do a whole bunch of those it what helped me learn mutliplication...

...twenty years ago in second grade.

That's not a diss on these students at all. I'm fully aware that what's easy for me to get is not easy for others and vice versa. Here's the thing:

I remember learning my times tables in second grade. In seventh grade, I went into pre-algebra and was introduced to my good, long-time friend, x.

What the hell was I learning for four years in elementary schools? Was it fractions and decimals and percents? Was I just repeating the same things over and over again?

Seriously, I can't remember. I can remember up to second grade and everything seventh and beyond, but I have no idea what the hell was in the middle there. This leads me to wonder if my time was being wasted and pre-algebra could have been introduced earlier and to more kids or if there was something really going on there.

Based on when you did times tables, you got geometric figures and formulas (just the basic ones), more and longer numbered number problems and word problems of increasing difficulty. At some point you got number problems that set you up for algebra by using blanks instead of letters or symbols for unknowns. You were later than me ('52-'58 for elementary) as we did not have Pre-Algebra.

 

[Jorghnassen]

Let me think... Actually didn't do multiplication tables until third grade (no biggie). We would regularly do "series" then (start at a number, go by increments of another number, 10 times). That's about all I recall from third grade. I don't recall anything from the fourth because I skipped it, but I do recall having to catch up on long division then. Fifth grade was more of the same I guess. Sixth grade had all those negative numbers, decimals, fractions. There must have been some geometry involved as well through all those years. And a bit of cartesian coords in 6th grade.

Secondary 1 (seventh grade) was not much new except the notion of natural numbers, integers, fractions and the real line (more formalized than before). Then 8th grade had baby algebra (by which I mean symbolic manipulation, not baby algebra: intro to group theory and ring theory...), some more geometry etc. 9th grade was a complete waste of time (all review except for the concept of homothetic transformation). Sec 4 had finally new things (and lots of it, finally the kids were split into the "strong" and "weak" math streams): intro to logic (no numbers or shapes, that's math?), linear equations, factoring polynomials, really basic trig. Sec 5 was pre-calc (functions of all kinds, logs and exponentials, parabolae, circles, ellipses, hyperbolae, and actual trig).

And then cegep 1 science (equivalent of 12th grade) had calculus 1 and 2, up to Taylor series...

 

[mommyrex]

Sixth grade had all those negative numbers, decimals, fractions. There must have been some geometry involved as well through all those years. And a bit of cartesian coords in 6th grade.

Exponents and roots, too, maybe?

 

[Jorghnassen]

Exponents and roots, too, maybe?

Not in sixth grade, no. Maybe a little bit in 7th.

 

[TobiasTheViking]

i have never been taught multiplication tables.

Hell, when learning german in school i told my mother that i was bad at it, and she wanted me to say my <some word i can't remember, but is basicly a verse to remember the grammar rules> like "ich bin du bist er sie es ist". Was never taught those either. She didn't believe me when i said i didn't know them. When a friend from school said i wasn't lying she believed it though.

 

[Cuddles]

I don't know how the American system works, but in Britain it was pretty much repeating the same stuff over and over again. For about 10 years. Didn't really see anything new until GCSE and A-level.

 

[Senor_Pointy]

From what I remember of elementary and middle school, (at least where I live) you do multiplication tables starting in 3rd or 2nd grade, long division in 4th, and then repeat ad nauseam until 7th grade. Then you do algebra in 7th (solving equations, the quadratic formula, etc), Geometry in 8th (With postulates/axioms and formal proofs) and advanced algebra in 9th (matrices, factoring, derivatives), and Pre-calc in tenth (Trig, more matrices, identities, logarithms), and then Calculus and Stats in 11th or 12th.

 

[drkitten]

From what I remember of elementary and middle school, (at least where I live) you do multiplication tables starting in 3rd or 2nd grade, long division in 4th, and then repeat ad nauseam until 7th grade. Then you do algebra in 7th (solving equations, the quadratic formula, etc), Geometry in 8th (With postulates/axioms and formal proofs) and advanced algebra in 9th (matrices, factoring, derivatives), and Pre-calc in tenth (Trig, more matrices, identities, logarithms), and then Calculus and Stats in 11th or 12th.

It varies a bit from district to district, but that's about right.

What you're not seeing, Lost, is that there are a lot of "hidden" milestones that are important between mere exposure to the multiplication table and an adult ability to multiply/divide any two numbers whatsoever.

As an example, the Arizona state standards (citations available upon request, or simply Google for them) demands that second graders be able to multiply by 2s, 5s, and 10s, and count by multiples of 3s. Third graders are expected to be able to "state" (but not necessarily to apply, I guess) multiplication facts through 9s. Fourth graders are expected to state through 12s and to be able to multiply multi-digit numbers by two-digit numbers. They are only expected to be able to divide with one-digit divisors, however. By fifth grade, students are expected to be able to multiply and divide arbitrary whole numbers and decimals, but not fractions and mixed numbers-- that comes in 6th grade. "Integers" (i.e. negative numbers) are introduced in 7th grade, and only at 8th grade are students really expected to have full adult-level mastery of multiplication and division.

 

[GreedyAlgorithm]

full adult-level mastery of multiplication and division.

Is it sad that this makes me scoff? It probably shouldn't be. At one point anyone "educated" was supposed to know much more than someone today "educated" knows, but back then they didn't try to educate as many people.

 

[roger]

Lots of repetition. Plus factoring, finding primes, word problems, etc. I remember being excuiatingly bored with math class, though I loved it. Because we were doing arithmetic, not math. This week: multiply 3 digit numbers together. Next week: multiply 4 digit number. The next week? Why, multiply 5 digit numbers!!!

ETA: reading DrKitten's post reminds of extra stuff. For example, for a long time division had "remainers' 7/3 = 2 remainder 1. After awhile we learned decimals.

From that, you learn converting fractions into decimal, and the harder decimal into fractions.

I think we did some work in different bases: base 8, base 7, etc.

I think we learned modulo arithmetic.

Greatest common divisor, least common multiple.

Exponents.

Scientific notation.

General equations dealing with algebraic notation, operator precedence, etc:
(8 -3)
------
(9 - 4)*3

I guess a bunch of stuff leading up to the beginnings of algebra. But all I clearly remember doing is endless exercises of muliplying and dividing.

 

[LostAngeles]

It varies a bit from district to district, but that's about right.

What you're not seeing, Lost, is that there are a lot of "hidden" milestones that are important between mere exposure to the multiplication table and an adult ability to multiply/divide any two numbers whatsoever.

As an example, the Arizona state standards (citations available upon request, or simply Google for them) demands that second graders be able to multiply by 2s, 5s, and 10s, and count by multiples of 3s. Third graders are expected to be able to "state" (but not necessarily to apply, I guess) multiplication facts through 9s. Fourth graders are expected to state through 12s and to be able to multiply multi-digit numbers by two-digit numbers. They are only expected to be able to divide with one-digit divisors, however. By fifth grade, students are expected to be able to multiply and divide arbitrary whole numbers and decimals, but not fractions and mixed numbers-- that comes in 6th grade. "Integers" (i.e. negative numbers) are introduced in 7th grade, and only at 8th grade are students really expected to have full adult-level mastery of multiplication and division.

I suspected that's what I wasn't seeing. In second grade, I (so I thought at least) was expected to know my tables up through 12 and so I did.

Makes sense. Thanks.

I'm guessing then, that there isn't an expectation for elementary age children to be able to do the abstract reasoning of algebra because most of them have not been exposed to such a thing or because generally, the brain isn't developed enough for that yet?

 

[drkitten]

Is it sad that this makes me scoff?

It is, because it indicates that you have bought into several myths about history and the history of eductaion in general.

At one point anyone "educated" was supposed to know much more than someone today "educated" knows, but back then they didn't try to educate as many people.

Er, no. As recently as the 1940s and 50s, "calculator" and "computer" weren't equipment, but jobs; relatively well-paying white collar jobs for people who could accurately perform arithmetic. The ability to add, subtract, multiply and divide was considered to be a valuable skill, precisely because so many people had not "mastered" it. There's this continuing myth about this "golden age" of education when "educated people" knew so much more than now... a myth that unfortunately seems to evaporate if you look at the actual performance levels at any given time.

 

[wybili]

I noticed a sort of progression in the development of math skills; maybe this doesn't apply more generally, but for me it went something like this: first there's the stage where you first learn how to do some computation; then after repeating that a million times doing the computation becomes second nature, and you just do it instead of thinking about how to do it. Then after doing it about a million more times, you don't even bother to do the computation any more and you use a calculator or something (or accept that you could do it if you wanted to ). If you keep doing maths long enough the same kind of progression happens with, for example, integration.

Anyhow I think that process takes a few years, so it makes sense that there's a four year gap between first learning multiplication and getting into algebra.

 

[RSLancastr]

hat the hell was I learning for four years in elementary schools?

Long division.

Hell, I still haven't finished some of those problems...