In this particular world-view, there are “mathy” people and everyone else. The mathy ones learn math (we can discuss what exactly that means to them, but I’d say it is precisely the mathematics they’ve mastered. Anything they haven’t learned, don’t know about, struggled with in school, etc., doesn’t matter), and the rest are. . . unworthy. So why bother to think about different ways of teaching? After all, there’s a nifty little syllogism that involves some very self-serving premises: Whatever I know comprises what’s worth learning. I learned it in the way I learned it. Those who cannot learn what I’ve learned the way I learned it are. . .
well, you finish it, Bob. After all, it’s what keep you convinced you have something valuable to contribute to the world of education.

Unfortunately, that world-view leaves a rather sizable number of folks thrown off the bus of mathematics very early on in the game, and with little or no opportunity to get back on. People like me, who slept through most of high school math thanks to teachers who thought much as Bob does, were clearly far too “other” to be invited to rethink their inevitable dislike of what they’d been misled into thinking was the entire mathematical story.

Luckily, I knew some pretty bright people, read some interesting articles and books, and became motivated to find a more inviting bus to get on. It still contained real mathematics. It simply wasn’t being run by elitists.

It’s remarkable to see how threatened some people are by that, and by people who are working to make the next generation of buses much more likely to keep a lot more riders than the line Bob operates.

Erik, you make a crucial point: we teach students. Not every student is at the same place as every other one at a given point in his/her development. Those who are quite content to see those who aren’t at some arbitrary point at some arbitrary age left behind for good (see the so-called Common Core State Standards for the latest idiotic blueprint) have no business teaching or telling others how to teach.

Then, the questions becomes “how do you get kids ‘to get’ math?” Understanding abstractions is key to this. And, more importantly, we do not teach algebra; we teach students, and making abstractions and the process of concretizing/abstracting explicit is paramount to teaching students to think mathematically.

Ah, yes, “algebra avoidance,” Wayne Bishop’s favorite epithet for any approach to teaching algebra that represents something more than what is dreamt of in his philosophy.”

If you’ve got to channel a mathematician, try George Polya. The guy who wrote “Let Us Teach Guessing.” He knew a good deal more about both mathematics and its teaching than Wayne or you ever will.

Having recently had the pleasure of viewing in its entirety the film of Polya’s legendary mid-’60s class at Stanford in which for some strange reason this world-class mathematician starts by asking to students to guess how many regions of three-space are formed by 5 planes, I’m confident that he’d have been very comfortable with the direction Dan and others are taking things these days. And that he would have looked at the Bishop-Hansen model of instruction as “not very interesting.” Not in 1965 and not now.

It took me a while, but I’ve finally figured out that Dan does not advocate skipping the problem sets and the higher level problems. What he’s doing is encouraging people to build stronger beginnings that will help the students find context in the problem sets they’re doing later.

Example: Graphing Linear Equations

Approach 1: Linear Equations can take the form y = mx + b, or ax + by = c, or y = m(x – x1) + y1. Let’s calculate a slope using m = (y2 – y1)/(x2 – x1). Here are your grids, let’s plot the y-intercept and determine the next point using the slope, etc.

Approach 2: Present a picture of a mountain. Pose the questions: what’s the steepest part of the mountain? Which side looks the safest to climb? Assist by segmenting the mountain profile and having them guess. How can we prove who is right? What would we need? Can I create a point of reference? What now?

Overlay a grid. Give the parts of the mountain coordinates. Ask how to find slope. Determine the winner.

Present them with the same problem set you were going to do in Approach 1.

In the end, the same Algebra gets done. But which version is more engaging? That’s the focus. We all know how to hand out problem sets.

Whether you avoid higher levels by staying at the bottom of the ladder or by memorizing formulas, the result (or lack of) is the same.

No one is advocating staying at the bottom of the ladder. But algebra teachers who can manipulate algebraic abstractions are a dime a dozen. Teachers who can motivate and explain those abstractions are less common. My goal is to help develop those teachers.

Historically, you’ve offered only one suggestion towards that goal: students who don’t like how you prefer to teach algebraic abstraction (ie. as much of it as possible as soon as possible) have no business taking algebra. It never occurs to you to reconsider your preference. That kind of determinism has no place in the education of children and I find it pretty boring around here. Please find a new line or a new place to peddle the old one.

1. Students will be less “frustrated and disengaged” with a task.

Yes, that is why teachers do this. I get that.

2. It will prepare students “to work at higher levels of abstraction later,” with that same task.

The only way to work at higher levels is to WORK AT HIGHER LEVELS. I do not see that here. In fact, that is my main complaint. This is in fact avoiding higher levels, and it is done because of point 1.

3. Students will have more opportunities to excel at crucial mathematical skills, besides for memorizing/using formulas.

Whether you avoid higher levels by staying at the bottom of the ladder or by memorizing formulas, the result (or lack of) is the same. If your point is that some students should be given more time with arithmetic reasoning because it is better to really get arithmetic reasoning than to continue forward into algebra and get nothing (including arithmetic reasoning), then I whole heartedly agree.

4. Students will be more adept at moving up and down the LOA on their own.

Once you are successful with mathematics (you get it) there is no moving up and down the ladder. All levels are in play at the same time.

It seems to me that the main gist here is to avoid teaching algebra. Instead of all of this algebra avoidance (and that is indeed what this is) why not speak out more against pushing students into algebra before they are ready? In three years of watching this thread and all of its claims of teaching algebra better, I have not seen any actual algebra. At what point do you get past the guessing and foreplay and get on with algebra?

Without this focus, students often go into “guess, guess better, guess all day” mode without doing much thinking about the guesses, or only thinking about the guesses (100, that’s too much, 70 that’s too little, 85 that’s too much…). This especially happens if “guess and check” has been taught as a way to solve problems. Guessing is good, estimation is good, abstracting is better. In the classroom I got around this by frequently giving problems that didn’t work out nicely, and the abstractors solved them more quickly than the guessers.

Care must be taken when putting student work side by side to abstract: good student work on the same problem can look a lot different. In the problem Dan gives, one student may operate on the number of kids while another operates on the number of adults while another operates on the budget, etc., etc. It’s not a tough thing to work around, and actually benefits students even more as they can then compare and contrast methods and recognize that there is more than one good way to solve a problem.