Math Education: An Inconvenient Truth

[WraithETC]

I'm a partial victim of these systems. Growing up in WA we did use some of those types of books. The alternative methods however were usually just side items. By moving up into "honours" math those books were eliminated.

I see the point in the first method but I eventually learned that anyways just by doing math on a regular basis.

 

[Fenixgoon]

Originally posted by: desteffy
education in the US schools is in big trouble...

fixed

 

[mugs]

Originally posted by: bonkers325

Originally posted by: mugs

Originally posted by: bonkers325
what a cumbersome approach to simple math. imagine doing something like that in calculus

Seems to me that they're trying to teach people to use math more effectively in their daily life. Most people don't use calculus on a regular basis, and you highlighted exactly why this wouldn't apply to calculus - this is a method of doing simple math.

my point is that they're using asinine techniques to break down *simple* math. you get taught arithmetic in elementary school - algebra and trig in junior high school and high school - perhaps calculus in high school and maybe in college. how are you going to digest complex problems with those kind of mathematical analysis techniques in a reasonable amount of time? additionally, how are you going to survive tests that forbids the use of calculators?

I don't see how teaching someone a different way to do basic math would hinder their ability to do more complex math, as long as they can still do the basic math...

But like I said above, I think alternate methods are good in addition to the normal algorithm, because the normal algorithm works better on paper.

Originally posted by: WraithETC
I see the point in the first method but I eventually learned that anyways just by doing math on a regular basis.

I think the problem is that most people don't figure it out on their own, which leaves them unable to do mental math later in life when they will need to be able to do it.

 

[Atheus]

Originally posted by: MrsJello
I'm in a class right now about teaching math

...

I haven't seen Everyday Math books above first grade yet, but it seems that their focus is more on real life skills. When was the last time that you had to use pencil and paper to figure out a problem?

When doing certain types of programming, like physical simulations, I go through reams and reams of paper. I also use paper to do geometry when I'm building some kind of structure or machine, or any other time I want to represent information clearly and right now.

Yea maybe _you_ don't use paper, but maybe those kids will want to do technical things?

Using calculators isn't awful if the kids understand what's happening when you do arithmetic.

Does it help that understanding though? Does doing fractions (for example) on a calculator improve your understanding of them?

 

[SSP]

Originally posted by: Minjin
Did anyone learn how to figure out square roots by hand? I've seen it done a few times but we never actively learned it in school. I get the impression that it was phased out not long before I started school.

You can't be serious!

 

[Minjin]

Originally posted by: SSP

Originally posted by: Minjin
Did anyone learn how to figure out square roots by hand? I've seen it done a few times but we never actively learned it in school. I get the impression that it was phased out not long before I started school.

You can't be serious!

Here's what I'm talking about:
Text
Text

 

[mugs]

Originally posted by: SSP

Originally posted by: Minjin
Did anyone learn how to figure out square roots by hand? I've seen it done a few times but we never actively learned it in school. I get the impression that it was phased out not long before I started school.

You can't be serious!

Huh? I never learned how to figure a square root by hand in school... perfect squares are easy, but calculating the square root of a number that is not a perfect square without a calculator was not taught when I was in school. (I suppose I could pull out enough factors to get it down to the square root of a small number like 2 or 3 times some other constant, and I had the square roots of small numbers memorized...)

 

[SSP]

Originally posted by: mugs

Originally posted by: SSP

Originally posted by: Minjin
Did anyone learn how to figure out square roots by hand? I've seen it done a few times but we never actively learned it in school. I get the impression that it was phased out not long before I started school.

You can't be serious!

Huh? I never learned how to figure a square root by hand in school... perfect squares are easy, but calculating the square root of a number that is not a perfect square without a calculator was not taught when I was in school. ( suppose I could pull out enough factors to get it down to the square root of a small number like 2 or 3 times some other constant, and I had the square roots of small numbers memorized...)

Oh, I thought he wasn't taught how to break it down. Obviously finding the exact root of some numbers is challenging and a calculator is necessary.

 

[fierydemise]

I'm going to ditto what everyone here has said, the TERC method is great for mental math but tedious for pencil and paper, however I think it is something that should be taught because many people have a very difficult time with mental math. The everyday math method I like, again a little tedious but the concept of place value can be hard for some. The lattice method is interesting but I think its a little confusing and definitely time consuming. I can see value in teaching the TERC and Everyday Math methods however the standard algorithm should be taught first, then the others used to help illustrate concepts (Everyday Math) or teach mental math (TERC).

As a current high school student I've gotten hit by all manner of POS math curriculum, this year I finally got out of the IMP (Interactive Mathematics Program) and into real math. The IMP program suffers from two major flaws, first it tries to teach "real world" math but it does it in long units that end up being incredibly far from real world. My favorite example of the this is a unit from last year, a man is being dropped off a moving ferris wheel into a moving bucket of water, when does the man have to be dropped for him to land in the bucket of water? Second it creates its own names for basic mathematics principles and never tells us what they are actually called for example factoring is referred to as mystery lots which wouldn't be a problem if they eventually dropped the name but instead you have high school seniors who don't know what factoring is knowing it only as mystery lots which is an incredibly slow and very limited factoring method.

 

[chuckywang]

Originally posted by: Minjin
Did anyone learn how to figure out square roots by hand? I've seen it done a few times but we never actively learned it in school. I get the impression that it was phased out not long before I started school.

There is an algorithm you can perform by hand. It's long and complicated though.

 

[Minjin]

Originally posted by: chuckywang

Originally posted by: Minjin
Did anyone learn how to figure out square roots by hand? I've seen it done a few times but we never actively learned it in school. I get the impression that it was phased out not long before I started school.

There is an algorithm you can perform by hand. It's long and complicated though.

I know, I linked it above. I wasn't actually asking if anyone knew how to do it, I was wondering who all had it taught to them. That one link says that it was phased out of schools around WW2 although I think it held on for longer than that.

 

[daniel1113]

Originally posted by: mwtgg

Originally posted by: BigJ
Looking at the first example (only 3:20) that's similar to how a lot of people do it in their heads. 26 x 31 = (20 x 31) + (6 x 31) = 620 + 186 = 806

Right, but there's no reason to do that when you don't have to do it in your head.

Exactly.

During the summers, I work on construction job sites, mainly doing layout and control. Since all of our equipment was in base 10 decimals (32.4427 feet), and all of the subs and laborers only worked in feet and inches (32 feet 5 5/16 inches), I would have to do all the conversions in my head. I can do them extremely fast, because I break the decimal into parts and other shortcuts. However, there is no way I could use these shortcuts if I didn't have a solid understanding of the fundamental methods, even for division and multiplication. It was quite shocking that many of the other engineers that did not have a solid mathematics background couldn't do it without running for a calculator. Perhaps they learned the lattice method in grade school?

 

[waggy]

This topic scares me.


My kids start school soon. I really hope they do not use the crappy methods. A friend of mines child was forced to use the Everyday math until 4th grade then something weird in 5-7th (not sure what it was).

but it was stupid. it was not about finding the the answere to the problem. but trying to reason why they are asking such a problem.

the father did his best to teach his kid the correct way to find the asnwere. BUT the teacher would not give him credit for it and even threatened to have the kid put in the "special" class because he would not fallow what the teacher was teaching. the poor kid is so screwed up (so is the class). i really hope none of them want to go to college. they will fail pretty bad.

 

[DrPizza]

I didn't watch the video, but I don't see how 26 x 31 = (20 x 31) + (6 x 31) = 620 + 186 = 806 is any different from the traditional algorithm taught in US schools?
31
26
------
186
620
----
806

The only thing that differs is a horizontal arrangement vs. vertical arrangement, and the 10's column is multiplied before the digits column. Hwoever, it should be noted that the traditional method is *just* an algorithm. There is virtually no understanding involved, except a "how" to do something. That's why so many US students are pathetic in mathematics - they have no understanding of what they're doing. At least the 26 x 31 = (20 x 31) + (6 x 31) = 620 + 186 = 806 order of doing it lends itself a little better toward comprehension.

I think some of you are missing the big picture: when students get to algebra, they will pretty much already know algebra. They will see 26 times (x + 10) to be nothing different from what they've already done (etc.) Perhaps before you decide this method is bad because it's not the algorithms you originally used, you might want to take a look at countries that are FAR superior to the US in mathematics, or more specifically, what they're doing differently.

How many of you, who have learned the traditional algorithms used in the US, can explain 2/3 divided by 4/7? (I'll guess that unless you google, there are very few of you who can explain "why" you take the reciprocal of 4/7 and then multiply; in fact, I'd estimate no more than 2% of you can explain why.) Guess what. That's not the algorithm used in Singapore for division of fractions. (uhh, where do they rank in math? 1st??). They use common denominators when dividing.

What goes beyond what most US students ever learn is the interconnectedness of math concepts. Everything seems to be taught as a bunch of disconnected algorithms.
-----
However, the biggest problem isn't which algorithm that's used. The biggest problem is that - especially at the elementary level - the teachers aren't that competent in mathematics. They don't have the deeper understanding on mathematics that's essential for teaching methods other than the traditional algorithms (the way they were taught.) Since no else has brought up the similarities between that method of multiplication and algebra, I guess you're helping to reinforce my case.

However, take a look back at the countries that are tops in the world in mathematics education... They aren't ahead of the US in the 4th grade. Starting at about the 4th grade though, they take off, leaving the US in their dust. Why? Because the kids are taught with more of an emphasis on understanding vs. learning algorithms. Their kids are taught to understand they "why's", not the "how's". In the US, we have to start from scratch in Algebra, and go back to teaching the distributive property. Back to teaching FOIL, etc. These foreign kids look at a product of two binomials and say, "oh, I get it." Two or three day's worth of lessons aren't needed.

 

[oboeguy]

Originally posted by: Minjin

Originally posted by: SSP

Originally posted by: Minjin
Did anyone learn how to figure out square roots by hand? I've seen it done a few times but we never actively learned it in school. I get the impression that it was phased out not long before I started school.

You can't be serious!

Here's what I'm talking about:
Text
Text

Heh, from the 2nd link:

Note: In computers and hand-held calculators, square root, sine, cosine, and other transcendental functions are calculated with sophisticated functions based on the Newton-Raphson method, sometimes called Newton's method. Here is a more detailed lesson that explains, for instance, possible difficulties in convergence.

I was going to say I'd use Newton's Method if someone asked me to compute a sqrt on paper. (Edit: or Taylor Series hehe)

The video was interesting but over the top. The points made in the last couple of minutes were far more important than the "OMGnewstuffsucks" propaganda at the start. I agree that the box or lattice method is nifty but pretty useless. As for division, I was highly amused by the fact that a condescending tone is used to describe alternatives to the old-skool method, while the trial-and-error aspects of the older stuff are glossed-over. Weak, very weak. Anyhow, it's clear that the "standard" methods scale the best in the sense that they are "local" (dealing with one or place values at a time, essentially) and aren't that hard to understand so there's no reason to not teach them. Some of the more "global" methods are cool but, heh, cumbersome past more than a few digits. OTOH, I see nothing wrong with teaching divide-and-conquer strategies which are, again ironically, used all the time in computer programming.

I also see very little point in "teaching" the use of a calculator in elementary school. It should take maybe one day at most. Seriously, how hard is it to push a few buttons? I understand the utility of fancy TI calculator in HS or even 7th/8th grade but the "why teach _____ when a calculator can do it" or "who uses _____ in every day life" mentality is a pathetic cop-out motivated by who-knows-what (I have no idea).

Edit: And I agree with Dr. Pizza's points (of course), especially that of teachers in the US not having enough math competence / experience to properly lead the way, so to speak, to future topics. I may get flamed for this, but I'll state it anyway: education degree programs suck, and don't focus enough on content.

 

[Jeff7]

Good post, DrPizza. I remember seeing algebra for the first time - it just looked bizarre, and seemed horrendous. Letters in math? Huh?

How many of you, who have learned the traditional algorithms used in the US, can explain 2/3 divided by 4/7? (I'll guess that unless you google, there are very few of you who can explain "why" you take the reciprocal of 4/7 and then multiply; in fact, I'd estimate no more than 2% of you can explain why.) Guess what. That's not the algorithm used in Singapore for division of fractions. (uhh, where do they rank in math? 1st??). They use common denominators when dividing.

I think you're right there too. (2/3)/(4/7) - first thing I'd do is just put *(7/4) next to (2/3) and multiply through. I know why I do it too - long time ago, the teacher said to do it.
I often wonder why we're stubborn like that - if another country is doing something really well, why not follow their example? I think it's Denmark, that has an excellent levee system, and some of their engineers have offered to help us design strong, safe ways of keeping the ocean out. Their "levee" system even allows for proper intermixing of fresh and salt water, and is designed to withstand a once-in-10,000 years storm. They're built to last. But the government seems to want to go it alone for whatever reason. Same here - Singapore is #1 in math education? Fine, let's follow their example.

I have trouble with calculus now because the only time I see it is in calculus class, and I'm in an engineering major. The calc professors I've had thus far all have Ph.D's in some math field, so I'm confident that they know what they're talking about when they say that calculus is useful all the time in engineering. I have yet to see it though. None of my other classes use calculus, so it just vanishes from my mind in the time between calc classes. Things like physics or statics though, that I see in multiple classes, so that knowledge is quite solidified in my mind. Calc? It seems like the mysterious thing stuck into the curriculum that everyone says is wonderfully important, but it's a big secret as to exactly where it's important.

Originally posted by: Minjin
Did anyone learn how to figure out square roots by hand? I've seen it done a few times but we never actively learned it in school. I get the impression that it was phased out not long before I started school.

I never did. Hell, my middle and high school math classes outright skipped logarithms. I got to CalcII in college and she's doing logarithms on the board, and everyone else seemed to know what was going on. I'd never seen the things before.

Originally posted by: daniel1113
Exactly.

During the summers, I work on construction job sites, mainly doing layout and control. Since all of our equipment was in base 10 decimals (32.4427 feet), and all of the subs and laborers only worked in feet and inches (32 feet 5 5/16 inches), I would have to do all the conversions in my head. I can do them extremely fast, because I break the decimal into parts and other shortcuts. However, there is no way I could use these shortcuts if I didn't have a solid understanding of the fundamental methods, even for division and multiplication. It was quite shocking that many of the other engineers that did not have a solid mathematics background couldn't do it without running for a calculator. Perhaps they learned the lattice method in grade school?

Now see, I'll try stuff like that, but if I have .4427 in my head, then think of 5 5/16, the .4427 just goes away. Even things like 24 x 56. Once I figure out the 6 * 24 part (following the good old "algorithm" method) to get 144, the number 24 is gone from memory. Then I'm left with "56" as a multiplier, and "144" as a partial answer, but I can't remember "24."
I'm wary that this problem started around the time my psychologist started me on Celexa, then Wellbutrin, the latter of which gave me a seizure. I've been off of those things for a few years now, but I'm just hoping they didn't permanently screw up something in my brain.

 

[DrPizza]

lmao! I couldn't for the life of me remember that old method... All I could remember about it was breaking it up into two digit chunks, and how to get the first number. But, that's the first thing that came to mind.. Newton's method for finding a root, and afterward, I thought of using a Taylor or MacLaurin Series.