This dude is strong. No question about it. He can lift very heavy things. Every day he hits the gym, spends an hour or so on the bench, probably another hour doing lat pull-downs. He may vary it up, doing chest/back on even days, arms/shoulders/legs on odd. Whatever his routine, clearly it's working. And maybe that's great. Maybe there's a certain utility in being able to curl 300 pounds today, in order that you can curl 305 tomorrow.

But are all strengths the same? Indeed, could we even say that this guy is strong in a real-world way? The dictionary definition of "strength" is the *capacity for exertion or endurance*, which suggests an application, some activity for which you're using all this strength. So is he a rock-climber? A boxer? Is he a firefighter responsible for dragging hoses into burning buildings? Put another way: what's the point? Beyond body building competitions, is there some reason why he's lifting all this weight?

I read an article in yesterday's New York Times about Jump Math, a math curriculum designed by John Mighton, hailed by some as a "kind of math miracle worker." Teachers in England and Canada praise Jump Math's effect on student performance, including dramatic gains in standardized test scores. Curious what a miracle looks like, I downloaded the student workbook for Grade 8, and was surprised by what I found:

It seems very traditional, your run-of-the-mill skills-based approach to math instruction. In this sense, it's not unlike most curricula that students see every day, in that it takes math, breaks it into a bunch of smaller skills, and addresses those individually. As described by the article, a key difference is that Jump Math takes this even farther, "break[ing] things down into minute steps and assess[ing] each student’s understanding at each micro-level before moving on." LCM today. GCF tomorrow. Then prime factorization, order of operations, identifying fractions, etc., etc. *Bench press. Flat bench fly. Incline press. Decline flies.... *It's hyper-focused--every muscle group is covered!--and, at least judging by the test scores, very effective.

But that's the crux, right? What are we measuring? Of course, standardized tests aren't going anywhere. Like it or not, they're a fundamental part of what it means to be a teacher these days, and bemoaning this reality prompts the *Don't hate the player, hate the game* truism. Still, in the same way that being big and being strong are different, knowing how to do skills and knowing how to do math are not the same. Skills are a subset of math, just like grammar is a subset of language. Necessary? Yes. Sufficient? Hardly.

Which gets to the question that arguably matters most: what's the purpose of a teacher? What's the point? Does a teacher train for training's sake, or does he/she prepare students for what comes next: the game of life? Breaking math into discrete skills is good...until it's not. You can't build a house from a whole tree, so you cut it into boards. Fine. But maybe those boards are still too big, too unwieldy, so you cut them down further. Okay. But if you keep going, and going, and going, then ultimately you're going to be left with sawdust, and that's pretty tough to build with.

Quarry, stone, rock, pebble, sand. If the purpose of math is ultimately to construct something larger and more meaningful--if the purpose of being strong is to apply that strength to some real-world activity--then at what point do we draw a line in the road, lest we end up in the sand?

Teaching is incredibly liberating, but can be a drag when we feel like we're just implementing someone else's agenda, someone else's priorities (especially when those priorities seem to change each year). Again, standardized tests aren't going anywhere, but what if they did? How many of these "miraculous" programs would disappear with them? Look at a textbook, a Mathalicious lesson, your own approach to math. Look at everything there is to look at. And then ask: "If the rules were different, would this still exist?" *You still have to be able to lift heavy things, but now you have to run, too.*

Math is timeless, and trumps legislation. Real teaching is timeless, and doesn't require outsourcing autonomy simply to meet AYP. Yes, there's a game to play, and yes, we have to play it. But let's remember something, something that's always been--something that will always be--true:

Cogs spin. Teachers teach.

It's funny your example was about GCF. I recently wrote a post about the same kind of thing... http://demandeuphoria.blogspot.com/2011/03/burden-of-proof.html

I am a math tutor, and I find that the "breaking down into minute steps" is actually counter-productive when looking at the big picture. Maybe it helps for standardized tests. Maybe. But I find that the kids who "get it" don't need the minute steps. They just get it.

And the kids who don't get it, seem to be very confused by the breaking down. They can't see how it all fits together. I find this especially with the concept of FOILing. They learn to FOIL for the sake of FOILing, but when it comes up in the middle of a problem, they look at me and say "do we need to FOIL?" Like it's a totally separate thing.

I prefer to teach about the big concept first, and encourage the students to do problems their own ways. Rather than prescribing the "one true" method and discounting their own processes.

I like to teach for fluency. A variety of ways to make sense, as many as possible from the students themselves, and then they choose a solution path that best suits the circumstance. Finding an LCM for 9 and 12, for example, is different than finding an LCM for 324 and 381. You can teach one brute force approach that always works, but that is not going to be efficient, typically.

Mathalicious's tree/chop/build run makes the teacher in me feel so warm and sparkly.

And I think it gets at my framework of a valuable education--a teacher makes material accessible so that it can be usable to a student who then actually uses it...who can even go beyond what applications are currently "teachable" to create new possibilities. And that would be good. (And probably compleeettteeely sufficient for most kids to grow up to be rockstars, diplomats, and non-indicted investment bankers.) But--as I think about the spirit of a public education, I wonder if there isn't also more of an obligation.... (And when I say wonder, I do mean "wonder" and invite insights! It's not a mitigation of my hardline thoughts.)

The real-world application is still just using someone else's description/understanding/definition of the "real world" to operate within it..and should we also be concerned with pushing some kids to be the kinds of thinkers who can dig into that working understanding to potentially expand or refine it? To not just use existing knowledge do something new, but to actually learn or infer something actually new in the world of mathiness. Does that start in HS? Or is that none of our business and should be left to quantum labs at universities?

If it is our business, how do we do it well if we aren't now? Short of me getting tipsy with Arne D. (which I hope for often, but is unlikely) and convincing him to invest in non-congressionally authorized federal policy I've dreamed up in yoga class, I surely do not know....but somebody who reads a blog called Mathalicious must have thoughts, yeah?

Karim, you offer thoughtful points about the purpose of math. However, I've used the JUMP curriculum and maintain that it is NOT simply exercising or drill and kill. The JUMP workbook you are referring to is just ONE of many resources used in the program. In fact, it's one of the last things used in a lesson cycle. A lesson cycle in the program actually begins with a Socratic Q and A session involving the teacher and the class. Through guided discovery, conceptual understanding is attained. Once mastery is determined via mini-quizzes, practice is done in workbooks to consolidate gain, through guided discovery and incremental difficulty the concepts being taught. This isn't necessarily a sequential order, but often activities follow or a part of the initial exposure to the concept. In the 8th grade probability lesson, for example, this entails students doing real world research (often through technology) to add relevance to the task. The JUMP program isn't simply chop and drill. For more information, I suggest registering for free on the website and downloading free teacher's manuals. Also, a quick summary of research is available here... http://jumpmath1.org/supporting_research

That's REDICULOUS!!!! I wonder how many pounds those weights r?!?!