math education

[cellio]

Being of a certain age, I learned arithmetic the conventional way and neatly dodged New Math. I knew things had changed since then -- at least in the ability of high-school graduates to do arithmetic unassisted -- but I didn't realize just how strange things had gotten. amergina posted a link to this (longish) news story broadcast: math education: an inconvenient truth. Sigh.

To summarize, some (apparently-big-name) published curricula are now skipping conventional methods to teach new ways of doing arithmetic. Some are different ways of breaking down the problems; others are primarily notational differences. All of them seem, on average, slower and more error-prone.

Now granted, I sometimes do arithmetic by the "reason through it" process the reporter dislikes (what did they call that, clusters?), but it's kind of specialized. For example, a 15% tip reduces to a 10% tip and half again; that's fast and easy. If I'm multiplying by a number ending in 9 or 1, it's often easier to reduce to another problem and then deal with the leftovers. If I need the square root of 4862 (I just pulled that number out of thin air), I can't tell you exactly what it is but I know it's a bit less than 70. Sometimes I think in patterns like that. I think this is a fine thing to teach people after they have mastered conventional write-it-down-and-work-it-out methods. Not before, and certainly not instead of. (And I think it's better if you can give them an educational environment in which they figure out these "tricks" for themselves, like I did.)

I assume these new teaching methods (which include "use calculators") are largely responsible for many people being unable to get order of magnitude right. Those of the previous generation undoubtedly said that about the move away from slide-rules, but I never used a slide-rule (except as a novelty) and I can approximate... I once had a calculator-armed teenage clerk at a produce stand insist that my bag of vegetables came to over $200. Even if he had no instincts about what vegetables cost, he should have been able to tell that the price codes he'd read off the list didn't add up to that and maybe he'd mistyped something.

(When shopping I tend to keep rough a mental tally, so when I get to the check-out I know approximately what the total should be. I gather that this is unusual. It's just the way I learned to shop, probably from a time when you had to make sure you didn't exceed cash on hand. Now I use plastic for everything, but the habit remained.)

Well, I guess I can take comfort in one thing: if what they say about mental exercise is correct, I should be pretty close to immune to Alzheimer's. :-)

Comments

[kayre]

Wow, that takes me right back to Benton Harbor. Our kids there were being taught similar things even for addition and subtraction. At third and fourth grade, they still didn't know simple single digit addition securely; only one of the half dozen fourth graders I worked with was starting to work on multi-digit multiplication. The younger teachers would send us textbooks and beg us to help the kids with the concepts as they were taught there-- but the older teachers would tell us to teach the methods WE knew, and help the kids get secure on basic facts. And if those younger teachers were actually being TOLD that 'many children' couldn't master the old methods..... grrrrrrr. Words fail me.

[ichur72.livejournal.com]

What strikes me about this situation is that similar things have happened with reading. I don't know what schools are doing right now, but over the last 10 years or so I've heard much talk about methods for teaching reading that pretty much skip phonics and spelling altogether. Granted, I'm feel like I'm kind of on shaky ground when it comes to saying what works since I was so young when I taught myself to read that I don't even remember learning how to do it, but I do get uneasy when I see how many elementary-school kids of my acquaintance have had trouble with reading and spelling.

[ealdthryth.livejournal.com]

Well, I guess I can take comfort in one thing: if what they say about mental exercise is correct, I should be pretty close to immune to Alzheimer's. :-)

Good plan! I often do math in my head to keep some small level of skill. :-) I often do a rough tally of groceries in my head, but not always.

I work for one of those companies...

[siderea]

Gosh, that's horrible, that must be why after adopting one of those curricula, Boston Public Schools math scores dramatically.... rose.

Oops. Looks like someone's premise is incorrect.

Gary Larson drew a Far Side about "math terror" before either of those curricula existed; people have been complaining about the mathematical inadequacy of high school graduates long before either of those curricula existed. The practice of teaching grade-school "math" as the memorization of arithmetic answers and rote methods -- and the complete avoidance of reasoning skills until pre-algebra -- is precisely why for generations, students hit the wall in 7th and 8th grades when they first encountered, you know, actual math. These things aren't the product of "new math", they're the precipitator of it.

For generations, it was socially OK for nobody but a small elite to make it past (or even through) algebra. Unless you were going into a specialist field, all you needed to get by was your sums and your times tables. And then came WWII, and then the Russians put a man in space, and then the Internet happened. As a society, we think it's no longer OK for people to stop with sums and multiplication tables. We no longer think it's adequate to have a small elite of "numerate" people.

So various people who studied how kids learn looked at just why so many people opt out of math around the 7th grade. And what they found was that the method that that pretty, well-spoken meterologist advocated, whereby one is drilled in a synthetic method and the memorization of facts was pretty much directly analogous to teaching reading by having kids memorize vocabulary for six or seven years without ever telling them what the words meant or showing them how they were used, and then beginning teaching reading comprehension. As one might expect, some kids managed to figure out how math worked either on their own or with the help of parents or other supports... and other kids didn't. Rather a lot didn't.

And I remember vividly tutoring my peers in the 7th grade. And let me be clear, by peers I mean the advanced (honors) math track. I remember how many, even most, of my classmates did not know that multiplication was repeted addition; that AxB==BxA (oh, they'd memorized such a rule, and could recite it, but if you posed them a problem which required using that fact, they wouldn't know if "swapping them was allowed"); that a fraction represented division, and vice versa; etc. But they were very good at sums; that's how they got in the advanced class, after all. I remember thinking, incredulously, "I think someone needs to teach these kids arithmetic all over again, because they don't understand anything about it."

New math is an attempt to do just that. It's an attempt to bring the sort of mathematical reasoning that permeats real math down into the lower grades, so that mathematical reasoning is learned young; it is an attempt to teach math comprehension instead of just math memorization, the same way we demand reading comprehension instead of just vocabulary memorization.

Does it succeed? That's a good question. A lot of people think so, especially people who are specialists in studying how children learn. Let me be clear: the product isn't cheap. It's never anyone's lowest bidder. Nobody gets on this bandwagon for any reason other than they think it does the job really well. And the reason there's so much noise about it is that so many people -- teachers, schools, districts -- are opting for it... and other people really hate it. And make things like that video.

[dvarin.livejournal.com]

This seems kind of related to a talk we had at the Amazon developer conference day thingy a while ago. It was on how to utilize intuitive thinking, but it started off with just the seems-obvious statement that novices rely more on rules, and experts rely more on intuition, and as you go up the scale of training and experience from novice to expert you stop relying on rules and start relying on intuition. This is why not all experts are good teachers, because they've been using intution so long they don't think about rules for how to get a result, so they can't explain it to a novice.

Looking at these methods for doing problems, it seems that they're trying to teach intution, sometimes without teaching rules first. (So.. I'm pretty sure that basic addition and subtraction have to be memorized. It seems hard to break 4+7 down into some simpler problem.) The question here that isn't getting mentioned is, how is the teacher teaching it? If they don't understand how it works and can't express that to the students, then it's pretty much equivalent to just teaching them rules, only it's a more complex than necessary set of rules. If it's rules that can generate intution with less requisite experience then they;'re good, but the anecdotal non-evidence suggests that it's hit-or-miss.

Cluster problems... ugh. This sort of backward hole-filling reasoning is really dependent on intuition to go anywhere, and if a particular student can work it, it's good, and if they can't, it's dead. It reminds me of teaching imperative programming proofs when I was a TA--I had no problems coming up with loop invariants, but some of the students just couldn't. The way the professors taught it was the way I thought about it, that the loop invariant was some expression which satisfied a few conditions and enabled proving the conclusion, and as for how you determine what it is...they said nothing. I had no problems with it--stare at a loop, figure what it was supposed to do, think about the shape of the logical hole that needed to be filled... flash! Loop invariant! But several students didn't get it so well, and I'm wishing now that I could have come up with rules for it to tell them until they could get their own intuition. Hole-filling (by which I mean, given the start state and the end state come up with the middle) in some cases may just inherently be a heuristic search problem rather than one susceptible to deterministic rules, which means it's totally intution, which means you just need a lot of experience with seeing all the pieces together. Bleh. It's late, I'm babbling. :)

[starmessenger.livejournal.com]

I think the woman in this video is ignoring the richness of the new-math curriculum in favor of a narrow interpretation of what it means to be proficient in math.

I've been impressed by (and even jealous of) my daughter's new math education. New math got her ready and eager to take advanced geometry in ninth grade.

My kid and her fourth-and-fifth-grade classmates learned all the techniques lamented in the video. They also learned the standard algorithms, plus visualization techniques (plots, charts, graphs, arrays, maps) that I didn't get to in the old-math regime until ninth-grade algebra. The algorithm-lady sneers at the textbook chapter on visualizations, but visual support for what she was doing with numbers was crucial to my kid's understanding and enjoyment of math.

Example: when Gwen was in fifth grade, she showed me how she'd learned about squared and cubed numbers by drawing me a picture of a "number square" and a "number cube", and showing me how the visualization and annotation were related. Dang, we didn't even get to the *number line* until I was in middle school! Under new math, kids are doing the number line, and learning visually about negative numbers, in third and fourth grade.

Her elementary-school curriculum made math so interesting to my daughter that she routinely shared what she was learning with me. Sometimes she needed help with mechanics, but just as often what she was learning sparked relationships in her mind, and she wanted to talk about them. New math gave me a way to talk to ten-year olds about hypercubes -- fun for them, fun for me, and they've been chomping at the bit for physics ever since.

Compare: in fifth grade, my classmates and I were having our minds numbed with rote memorization and practice of the standard algorithms for multiplication and division. We spent *years* doing nothing but multiplying and dividing columns of numbers. The algorithms are efficient, but what do you have when you're done? Bored, math-loathing fifth-graders who can efficiently multiply and divide long numbers.

Death to old math, I say, and good riddance.

Oh my, you've put your foot in it...

[larryo.livejournal.com]

Valorie and I have spent a HUGE amount of time talking about this (she's known as the "Math Fairy" at school).

Daniel (age 15) uses the Lattice method (Everyday Math) almost exclusively for simple multiplication. Why? Because it's faster and more accurate than the "standard algorithm".

The reality is that the group that made this slick video (and other related groups) are faux advocacy groups that appear to be focused primarily on the idea that "because it's not what I learned, it's bad". This doesn't reflect the fact that different kids learn differently.


From Valorie (in an email sent on the discussion we had at work about this):
Another thing to remember is that one of the reasons we are so pathetic in math these days is that so many of the *parents* believe they are awful in math. Parents believe themselves competent to help their child write/edit a paper but many parents (in my experience) believed the math was too hard by third grade. Once students start two digit multiplication or long division, parents started to drift away. Fractions typically got NO support from home, and don't get me started on units of measure! If the parents convey to the kids that "Math is hard. I don't understand it so I don't expect you to understand it," then kids believe they can't learn math. I had some kids doing fourth grade work by the end of second grade, but by the middle of the third grade standard math curriculum, they'd decided that math was too hard even though the math was easier than what they'd been doing in second grade.

This is why I had to start teaching the first graders "algebra" - their parents needed to feel competent to help their child and every parent I've met thus far doesn't think first grade math is too hard. Once I started to get the parents feeling competent, the number of kids who struggled with math in 4-6th grades decreased dramatically. Of course, this is relatively anecdotal and a very small sample set, but it makes sense to me. How many times do you hear someone say something along the lines of "Let me get out my calculator so I can figure out the tip. I'm lousy at math"? A lot fewer times than I hear someone say something along the lines of "You read it to me. I'm a lousy reader."

Having a curriculum that changes the way students think about math makes this innumeracy worse, but most of these progressive math curriculums support a lot of parent teaching. Everyday Math includes materials for school-wide math nights, games parents can play with their kids and letters to send home which include "How-to" examples of the methods covered in class. It isn't great, but parents have to be willing to ask questions if they want their kids to be able to ask questions.

[benzado.livejournal.com]

Having spent a year studying Mathematics Education, this whole thing is depressing to me. What you basically have here are scientists who do good research and find a way to teach mathematics so that children will actually understand it. So they publish their research and somebody else attempts to put it in the schools. But then you have teachers that have been teaching for 30 years and don't care for new ideas or really poor training programs that don't help the teachers understand how to teach them.

Usually the program requires some fundamental changes (like, let's make sure the kids learn this smaller set of topics than crash through this huge set of topics) which are only half-heartedly supported by the school (the state's standardized tests demand you cover all of these things by January). So the program is kind of implemented half-assedly, and parents complain to the school board, which is made up of elected people with no educational training whatsoever, who deride these newfangled ideas as stupid hogwash and wind up vilified on TV like above, and then everybody says "let's go back to basics" and all the research that the scientists did that might one day lead the way to some gender equity in mathematics is thrown down the drain for another ten years.

[baron-steffan.livejournal.com]

This is a huge "button" for me. I should mention that I'm a pharmacist, and that pharmaceutical calculation (of which subject my professor wrote the bible) is highly dependent on plugging figures into set formulas. Ratios are our life, doncha know.

Well, I'm here to tell you that set formulas are, or can be, death. No, really, I mean that literally. Because you have to understand the logic of what you're calculating. We got zero credit for "decimal point errors" in pharmacy school, thank goodness! "Sorry: your patient is dead."

An Illustrative Anecdote
Every semester, in one class or another, we got a version of the same problem. You get a prescription for one pint of potassium permanganate solution, such that, when the patient takes a tablespoonful and
dilutes it to a gallon, he gets a 1:5000 concentration. How much potassium permanganate do you need? It's the sort of problem most students figure you "need to know for the test" becasue you'll never see it in real life.

Guess again.

When I worked at the hospital, my director was an incompetent Captain Queeg from "the Caine Mutiny". And one day, we got this problem. Everyone recognized it, and Queeg shut down the department while he set everyone working on it. I mean, really, c'mon, it's just a basic ratio problem but NOOOoooo, it was that ratio problem! Work on it and show your work.

Well, what I figured was this. I needed to know how much KMnO4 had to be in that tablespoon. And I needed to know how many tablespoons I had to make. So I told him the answer, and when he asked for my work I told him what I just said. Wouldn't buy it. Where's your work, where's your formula? I just told you. Nope, unacceptable. And how come you didn't use the metric system, you know we use the metric system here.

That's where dependence on formulas can lead you. What you need to realize is that disciplines reduce to simpler disciplines. Medicine is biology. Biology is chemistry. Chemistry is physics. Physics is math. And math is logic. What kids need to be taught is how to solve math problems with logic. And from what I see, that isn't being done any more.

[http://users.livejournal.com/merle_/]

I actually really like the "non-conventional" methods, like approximating tips and the like. That sort of thing can be quite useful. My mom was astounded when I came up for the "knock a digit off, then add half again" method for computing 15% tips, because she could do that, and even thinking about the multiplication would cause her to freeze up. Personally, I look for nearby integers divisble by 6 or 7: the tip on $27 is about $4 since 28/7=4.

However, what you say is certainly correct: you must know how to do the basics before any of these techniques becomes useful. If you do not know your multiplication tables, you do not stand a chance of getting a correct answer. And letting kids use calculators is boneheaded. One may as well allow them dictionaries for spelling tests, or Clif notes for book reports.

Where the methods of estimation are really useful, though, is for people who have some skills but cannot come to appreciate mathematics because it was drilled into them that it was hard, boring, and useless. I may not be able to come up with 20% of $69.34 easily, but 20% of $70 is a snap. And I know enough about transitivity that I could come up with a first-order error correction (say, adjusting my answer by 20% of $.65).

I am told that I learned math in the age of "new math". I don't think so, because, like you, I had to come up with those tricks myself. And that's a useful learning experience in and of itself. But many people are unable to discover those tricks.