First, the usual description of “traditional unproductive.” Why, why is the “traditional” way of doing things always deemed “unproductive”? It’s incredibly unscientific to start that way. Can we not learn from the Evil Traditional Way?
The slide itself just called ’em unproductive vs. productive… first unproductive one being “Students can learn to apply mathematics only after they have mastered the basic skills.” Well, we’ll see.
…. I’ve seen, and I really like. They left half an hour or so for questions… and there’s a real emphasis on building the basics and directly teaching how to think about math. Now, all of it was attacking problems at GED level — but there was even mention that something could be used with basic numeracy as well as GED level, so they aren’t of that group who can’t fathom anything below “GED.”
An answer to a question included introducing algebra by giving groups of students a jigsaw puzzle with a piece missing, and recognizing that they could put the puzzle together even without all the pieces, and that variables in algebra were those “missing pieces,” and that you could figure out all kinds of things about it from pieces you had.
I’ll be back for those folks. (Now, I’m not saying I like the GED test… just that I liked what these folks were saying about math…)
Resource Room Dot Net Blog 
howardat58
April 28, 2016
re the last paragraph:
“..and so long as the first missing piece is twice the size of the second missing piece, everything will work out fine.”
xiousgeonz
April 28, 2016
There was only one missing piece in the puzzles 🙂 However, one of their examples was about ten factories, with a string of qualifications like “the second one made twice as much as the first, the third made 40 more than the second…” with the last one making nothing because the weather was bad…
I rarely hear people talking about the “value” of having students struggle, and then actually explaining how you *teach* how to “struggle.” Most of ’em seem to be about telling the student to “think about it.”
howardat58
April 29, 2016
“Struggle” is one of the new buzz words, like “grit”, but it does make one think of a failing Houdini imitator. “Think about it” is fairly useless advice, better to try more specific advice a la Polya, and one which I don’t ever see, which is”can you think of a simpler problem?” or “Why not try it with easy numbers?”.
xiousgeonz
April 29, 2016
Yes! I found Polya had quite a few of my “secret tricks” — like “change the numbers and make them simpler” (which, granted, requires you to understand enough to maintain the relationship of the problem)… or to ask “what do you *wish* the problem was? Is there a way you could make it that way?”
howardat58
April 29, 2016
This one is classic ! From a person who shall be nameless, as an example of a straightforward “task” (direct instruction here).
Bill mows the lawn in 20 mins, Bob is slower, it takes him 30 mins. How long will it take them if they work together? His solution was to work out how much lawn each could do in one minute, (1/20th and 1/30th) add the fractions, and solve a linear equation.
I thought ” that’s a heavy handed way to go”.
In one hour Bill can mow 3 lawns, Bob can mow 2 lawns, together in 1 hour they can mow 5 lawns, so one fifth of an hour to mow one lawn.
But now it’s not an example of application of “what we have just studied” !!!
xiousgeonz
April 29, 2016
Yes — that is a classic example of something that we “don’t have time” to teach past the Magic Formula. IMHO it would be far superior to guide students through the connections between rate and time and fractions — and I’d prob’ly have stuff that looked like direct instruction in the mix.
I have *just* had a fellow figure out why the square root of a fraction is bigger than its square, based on figuring out that 50 divided by 2/1 is the same as 50 times 1/2…