The professor’s blaming of videogames and Facebook for the purported decline in mathematics skills strikes me as particularly eye-rolling…
Heh, exactly. And presumably, you learnt far more about logarithms by playing around trying to figure out what that button did, how it reacted to changes in inputs, its relation to other logarithms and other functions, trying to determine its base, and so on, then you ever would by merely being drilled on how to mindlessly run the same code agonizingly slowly by hand, something you will never have to do in your life. Plus, the former is just that much more fun.
I think I have not made clear that what I’m griping about is what’s going on in my state, which is probably more abysmal than where the rest of you are.
The “cranky professor” is speaking from the experience of his own kids’ lousy math education, in addition to his teaching experiences. He’s actually no cranky professor; he’s a prominent scientist here whose position has been backed up by hundreds of the UW science faculty, as well as much of the high school math communities. The elementary schools here do no drills anymore and spend most of the time doing blocks, strips, meaningless charts, and logic problems (If-John-is-wearing-a-red-shirt-and-drinks-coke-and-Mary-is-sitting-two-people-away,-then-what-is-Jane-drinking- types of problems.) If a kid can’t tell you how much 7x6 is without computing it in their head every single time, it’s going to hinder them. My kids would get 2 or 3 math problems for each topic. The system here is characterized by being a mile wide and an inch deep. Did any of you look at the math test linked to from the blog (pdf’s)? Whether or not you consider the math test to be an “inauthentic exam”, the fact is that the students are going to need to know this math for SATs, ACTs and in college, and are going to be stuck in remedial math courses. If they can teach what’s needed in remedial math, why couldn’t they have just taught it in the first place?
EVERY high school math teacher I’ve talked with here is distressed with the math illiteracy of the kids they get. Yet EVERY elementary school teacher has vigorously defended the discovery-based system they’re using as being best for the kids. There’s a real disconnect going on here.
You guys all have very good points about what is important in math and what is not, which I don’t dispute. I’m not arguing that using calculators is wrong, just that the understanding and facility has to be there first, and that the understanding and facility is not happening here. You can’t imagine how poor the instruction is here. And you also are assuming that most students are like yourselves: top-tier intelligence people for whom math understanding comes easily, and who will learn regardless of the method taught. Most kids aren’t.
I don’t know why the discovery-based math is not working. It seems logical to me. But in practice it seems to work much better for adults than for kids. It appears that the discovery-based portion is emphasized with not enough of the practical built in.
I’m not proposing unguided “discovery”, mind you. I’m hardly proposing anything; I don’t know how to teach young children. I just don’t see the value in arithmetic drills, or even arithmetic skills.
The SATs and ACTs both allow calculators on the mathematics sections, incidentally. I agree, students should know things like what 64[sup]1/2[/sup] means. That’s something guided instruction complemented by access to calculators can help with.
But homework assignments with a bunch of problems like 7*6 = ? don’t do anything for understanding, either. That’s what’s making kids dependent on calculators, except that it’s making them be the calculators themselves. The calculator I have in my pocket can do all those calculations without breaking a sweat, but I’d never say that it understands a lick of math. For it, it’s all just a matter of pushing electrons around, just like long division is just a person pushing a pencil around.
As for the question of why constructivist teaching techniques don’t work in elementary classrooms, I would guess that the primary reason they don’t work is that they aren’t used.
If I remember rightly, five is pretty much the upper limit for humans, with three or four more common. You can do better, of course, if there is an obvious shape – a two-by-three rectangle, for example, for six. But a straight line, or a random distribution, seems to cut off at five.
However, a very few people in history, all of them, I believe, lightning-calculator prodigies, seem to have been able to get up to about twenty.
I seriously have to defend why it’s valuable to know 7x6 off the top of your head rather than have to pull out the calculator?
I’m with you.
I can do a lot of relatively simple calculations, and even some more complex ones, in my head quicker than I can pull out a calculator or smartphone. I usually know the proper change before a cashier can even ring it up. You can avoid being ripped off this way (unintentionally or otherwise) without having to publicly imply that you don’t trust the cashier by pulling out some device to doublecheck them. I also like being able to figure out a tip without pulling out a calculator, although in that case time and public appearance are admittedly not as important.
We could probably come up with lots of instances where you would prefer to do the math surreptitiously, and there’s also the obvious case of lacking a calculating device due to a dead battery or something.
I love calculators. I write software for a living and I love computers, which is really what a calculator is, but I also like the convenience of not constantly having to pull out a machine (or even have one on hand) for mundane day to day transactions.
Also, I tend to do at least a sanity check (right order of magnitude, etc.) in my head when using a calculator because, as perfect as they are, the operator is not perfect and can hit the wrong keys, misplace a decimal, etc.
With me, I absolutely believe it’s important to know the basic addition and multiplication facts. My goal is for folks to have flexible and powerful strategies for solving problems, and if you have to figure out 7*6 from scratch every time you solve a problem, then you won’t ever be very powerful.
However, if I taught multiplication, I’d teach 76 by asking kids to use previously-solved equations. 76 is the same thing as 75+7, right? And 75 is easy. Or use 66+7 to figure it out, or 77-7.
Your arguments, needscoffee, sound exactly wrong to me, I’m sorry. I teach this way precisely because not everyone finds math easy. My traditional math education was pretty easy for me, and I was able to figure out a lot of the “why” on my own. I’ve got a very good memory for numbers; I can memorize tables of equations. Not everyone can. That’s why you teach from the ground up, why you prove to kids that something works and gradually move from concrete examples to abstract representations of the math. Kids who find abstract math difficult can fall back on the concrete.
As for constructivism teaching a mile wide and an inch deep, your own example contradicts that. Your kids get two problems a night, right? Are they asked to prove their work? This is based on the Japanese model of education, and is a major shift from the way things used to work here: math educators looked at why Japanese kids were doing so much better in math, and discovered that instead of being given a list of 30 problems to solve, they were given a couple of rich problems but then being required to prove their answer (or show their work, depending on how you want to phrase it–I say “prove,” because it cuts shorts the smart-alecks who want to tell me they just knew the answer and did no work to get it). 30 problems a night is a mile wide and an inch deep. One problem that you must prove the answer to is in-depth education.
If your district is seriously not doing any work on the basic facts, then yeah–they’re too far in one direction. But it really sounds to me like what you’re proposing is way too far in the other.
Point of order: I often encountered this situation from the student side, and I assure you I wasn’t doing it to be a smart-aleck. I knew how to get the answer but explaining it was often difficult, either because the steps seemed intuitive to me, or I didn’t have the necessary language to describe them.
Once it was explained to me that the point was not just to get the answer but to demonstrate mastery of the process, there were many fewer problems. (Although to me, it must have seemed that getting the right answer every time was sufficient evidence of mastery!) It’s amazing, though, how many teachers don’t have the patience to make that simple explanation.
That said, your approach ought to already head off most of these problems, as you seem less concerned that the proper method be used than that a reliable method be used. But the students who still balk seem more likely to be struggling to express a method that seems intuitive to them, not defiantly refusing out of spite or malice.
Powers &8^]
And forcing them to be able to express that method will help them understand it even better. It’ll help them, too, they just might not realize it at the time.
That’s exactly it. And that’s why I challenge them as I do (Sorry, Powers, to imply you were a smart-aleck: I’m just projecting on why I did it as a kid, I guess 
). There’s a common conversation in my classroom that goes something like this:
Me: Okay, so Jim’s told us that the equation we’re trying to solve is 27+5. How are we going to solve it? Fran?
Fran: 32!
Me: That’s your answer. How did you solve it?
Fran: I just knew it!
Me: I think you’re wrong. I think the answer is 35. Prove to me that you’re right.
Fran:…
At which point I wait while she thinks about it, and usually she’ll come up with a way to prove it: she’ll use a number line, or she’ll count up from 27, or she’ll break the problem into 20+(7+3)+2, or she’ll use some other method that gets her to the answer. Then I’ll look astonished, admit I was wrong, and thank her for the explanation.
The same thing happens on class work and tests: “I just knew it” isn’t an acceptable answer, except for very particular things like addition facts.
For folks who are skeptical of the blocks-and-strips method, here’s a real-life example from today.
Kids were playing a game designed to help them memorize (through familiarity) addition facts with a sum of 10. It was like Go Fish, only they’d ask the partner for a card such that they’d have a pair of cards with a sum of 10. That is, if I had a 7 in my hand, I might ask for a 3. If I got one, I’d record the resulting equation.
Most kids were playing it well, but one girl was really struggling: she couldn’t figure out what to ask for. I pulled an 8 from her hand and asked her what she needed to ask her opponent for; she couldn’t figure it out. I suggested she use her fingers; she still couldn’t do it.
So I went and snagged 10 blocks and put them in piles of 5 (I’m trying to reinforce the count-in-groups skill while I’m at it), asking her how many there were. She counted them individually and said 10. I split them into 6 and 4 and asked how many there were; she counted them and said 10. I split them again; this time she was able to answer 10 without counting. I split them several more times just to make sure she got the idea that no matter how I split them, there would still be 10.
Then I pulled 8 of them aside and asked her to count only those. “How many do you need to get to 10?” I asked, and pointed to the 2 blocks that remained. It took her a little bit, but she answered correctly. We played this way a few more times, and slowly she caught onto the idea that, if there were 10 total and I’d removed n, what remained was 10-n, and that n + (10-n) = 10. (Obviously she didn’t express it that way).
That’s how this kind of math works. The kids that were having an easy time with the game got a harder version, a version that combined knowing sums of 10 and the memory skills of Concentration, so that they’d have less mental processing power to devote to the sums and they’d need to have a more robust memory. The kids that were struggling with the addition got something concrete to fall back on, so they could see how it works.
I can easily imagine someone who doesn’t understand how to use manipulatives, who substitutes playing with blocks for rigorous teaching of the underlying math: they’re every bit as plausible as someone who substitutes drill sheets for rigorous teaching of the underlying math. But used as part of a strong mathematical curriculum, manipulatives like blocks can be very powerful tools.
Hey, can you point me to some of that evidence?
No, no, no; all answers are 6. Here, just take an arbitrary equation and solve:
1
- * sin x = ?
n
1
- * si[del]n[/del] x = ?
[del]n[/del]
1 * six = ?
six = 6
I don’t think you understand what the Singapore method is.
<Goofed big time. Probably not enough time to edit.>
Nope. That’s the method I’ve seen taught everywhere. Mine is similar to long division.
I guess I can explain for you mathies without hijacking the thread too much. I’ll include a short transcription of the process, and explain in a spoiler.
6 8 6
√47 05 96
36 6*6=36
11 05 96
10 24 [del]9*129=1161[/del] 8*128=1024
81 96
81 96 6*1366=8196
0
[spoiler]Let’s say you have to find the square root of 470,596. You first put everything in groups of two, so the number is 43 03 36. You then pick the next lower perfect square: 6*6=36, so your first digit of your answer is a 6. You then sorta treat it like long division, and subtract the 36 00 00 from the total, leaving you with 1105 96. You multiply your previous digit by 20, and try to find a multiple that is close. But this is where it gets tricky, because you have to add your prospective answer to the previous answer * 20 mentioned earlier.
So you might think you’d use 9, but that gives you 1299=1161, which is greater than 1105. So you try 8, and get 8128=1024, which works. You now know your answer starts with 68. You subtract 1024 00 from the 1105 96, and get 8196. Now you again take your answer and multiply by 20, thus giving you 1360. 7836/1360≈6, so you get 1366*6=8196 which leaves no remainder, giving you the answer: sqrt(470,596) = 686.[/spoiler]
And if I’d’ve read the thread further, I’d have found out I’m not the only one who learned that way. I have not seen it in any of the various modern math books, and no one I grew up with learned that method.
They learned one of the iterative methods. My guess is because even the most basic calculators have square root buttons, and thus doing it by hand is less frequently needed. It’s more useful to learn a method that teaches you other mathematics principles. But I did always beat out anyone who used the other method.
Also, in high school, we had both calculator and no calculator parts to every test. And in other classes I frequently forgot my calculator and had to do stuff manually in other math-based classes. (Only the math teacher had calculators we could borrow.) By the time I was in calculus, I used the solve function on the provided Ti-89 calculator to solve equations that involved more than a couple of steps.
What is it?
ETA: This is it.
Hm, BigT, that method is indeed completely unlike any I’ve seen before. I’d have to play around with that a bit, to see exactly why it works, and what it does with non-perfect-squares, and how it works in other number bases.