19 students in class, 18 participated (1 student walked out and came back when the activity was over). I teach in an ‘active learning classroom’ with desks set up in groups. 4 groups of students working the problem together and individually.
3 different models for the pattern were given to me. I had already created my own model, and the 3 in class were all different from mine.
Each group explained their thinking. Most had not generalized their approach into algebra.
I helped them put it into algebra. I also showed, that we could simplify the algebra for each approach and end up with the same thing!

Comments on the lesson
1. I was surprised how long it took them to come up with the pattern. I had to ‘nudge’ one table along and point out an error in their thinking at another table. The process of coming up with 88 tiles for the 20th iteration took over 10 minutes.
2. I was happy to see so many different approaches to solve the problem. Things also ‘felt different’ in the classroom. Working to figure out a problem like this is much different than working on the process of completing a square. Different people were taking the lead and speaking up to help their group mates. It was great!
3. I was able to use this task as a reference right away. For example, when introducing the idea about the domain of a sequence, which normally is very confusing, I referenced the iteration # in this task and the students seemed more able to make the connection.
4. I didn’t realize the importance of telling the students the handout referred to the first, second, third and (draw-in) fourth iteration until after I handed it out. I had to go to each table numerous times and show them which one was #1, #2, etc… In the future, I would label the handout first.

This activity took about 20 minutes from start to finish. I definitely think it was time well spent!
Thanks!

When it comes to technology, I don’t see the questions in math software programs to be much different than the ones you asked in your own concept checklist tests: http://www.mrmeyer.com/blog/wp-content/uploads/070830_4.pdf . Granted, these aren’t modeling, but they are part of math class. I guess I see software as automating your concept checklist. However, I wholeheartedly agree with your point about software that constrains students’ input too much, and my intent here isn’t to get dragged into a pro/con of software. Lots of software out there stinks, and none of it is what we wish it were.

What interests me is that this particular lesson make-over just feels different, like I could adapt its general strategy to teach anything, even lessons on formalism rather than modeling. Kind of like the difference between Sam Shah’s stuff and yours. With 3-Acts, among other things, I have a sense of what the strategy to overcome disengagement looks like. But I’m still really stumped on the strategy for overcoming confusion with math formalism. For example, how could the basic flow of this makeover be used to teach Precalc students that in the expression (x+5)/x , you can’t cancel the x’s? And then how/when would you interleave drill practice with the meaning-making? Not asking you to respond to this particular dilemma of mine, just showing you where my head is right now.

I’ve understood this to mean that Dan looks at instruction primarily through the lens of motivation, not cognition (the biggest question isn’t whether their brains can understand the lesson, but whether they care enough to turn their brains on).

I blog a lot about motivation and engagement because I find it to be the tougher of the two lenses to focus. I tend to think teachers need less help determining a sequence of worked examples to show their students for a given topic than they do how to get students to care about that topic. I don’t think one is more important than the other. They’re both co-requisite. But one is harder.

Here it seems Dan views the level of difficulty, which I think is mostly a cognitive factor, to be the primary mediator of engagement, at least in this task.

I don’t know if it’s primary, but it’s a big one in these tasks. Three-act math is premised on the idea that the first act is really easy to jump into with intuition and guesswork while the sequels develop the skills to a much higher degree. This isn’t a new feature here, though maybe I haven’t cast it as a focus on cognitive development like I should have.

A well-designed software program can individualize that difficulty level for students better than any one teacher can do it for a whole class. Granted, when you do this with software, you’re mostly doing it with drill-practice questions rather than modeling tasks. But doesn’t the same principle apply?

In theory, sure. But “individualization” in math education technology to this point has only been possible by reducing “math education” down to computational exercises. I’d be happy to be shown an example of mathematical modeling (for instance) that’s personalized, that keeps asking developmentally appropriate questions without constraining the student’s answers to a limited input set. It doesn’t exist, AFAIK.

Since then, I’ve assumed that either you didn’t care much about the cognitive side or that you and I simply don’t agree on what the research says about cognitive factors.

The worked example research is useful but I struggle with its ecological validity. It’s a dangerous game, generalizing these one-off lab experiments to 180 days in the classroom. Sweller himself presumes motivation in his research. It only works if the students are already motivated. And there’s nothing quite so de-motivating, from my experience, than repeated daily doses of worked examples. That’s why I fix on motivation more than cognition.

But then in this last comment, Dan says, “Instead I’ve tried to start the task easy enough to bring in a lot of students and end hard enough to be worth their time, with or without any context.” Here it seems Dan views the level of difficulty, which I think is mostly a cognitive factor, to be the primary mediator of engagement, at least in this task. I actually agree that carefully calibrating the difficulty of a sequence of questions can lead to engagement without much context.

But this is exactly why Dan’s tech contrarianism seems way too pessimistic to me. A well-designed software program can individualize that difficulty level for students better than any one teacher can do it for a whole class. Granted, when you do this with software, you’re mostly doing it with drill-practice questions rather than modeling tasks. But doesn’t the same principle apply? Can’t a sequence of drill questions easy enough to draw students in and hard enough to make them feel they’re learning something produce engagement? The sense I’ve gotten from Dan is not just that no software currently on the market is good at this, but that computers are fundamentally unsuited for that kind of practice.

In fact, RAND recently released results of an efficacy study of Carnegie Learning Algebra 1, which found an 8 percentile increase in scores for high school students using that blended curriculum, though no impact for middle school students. The study was extremely large–$6 million of federal funding with 18,700 high school students and 6,800 middle school students as subjects. You can read the results here:

http://www.rand.org/content/dam/rand/pubs/working_papers/WR900/WR984/RAND_WR984.pdf

In any case, I’ve sometimes grown as a teacher by having a dialogue with Dan in my head, so I’m genuinely curious: is it that you think cognitive factors are really important, but you’ve chosen to specialize mainly in motivational factors? Or do you see level of difficulty as motivational, not cognitive? Or is my whole cognitive/motivational dichotomy false in your opinion? What got me thinking about this was when we disagreed about the significance of the worked example effect a long time ago. Since then, I’ve assumed that either you didn’t care much about the cognitive side or that you and I simply don’t agree on what the research says about cognitive factors.

My own feeling is that (some) students are not asking for the context to be relevant to them, so much as asking for the context to shown to be relevant to somebody in the real world (other than mathematics buffs.)

Dan, you’ve repeatedly shown that making a real-life problem tractable, but not trivial, takes a huge amount of creativity.

When you ask “What does the real world buy us here?” I am not clear if you are now backing off slightly. Or if you are just accepting additional constraints specifically for the Makeovers (e.g. it must fit on a piece of paper that can be handed out.)

I’m compelled by Andrew Busch’s vocational math class, a class which is premised around math that is useful for professionals. I suppose there the math had better be.

In general, though, I don’t find students so eager to know someone with a job uses their math as much as eager to not feel stupid, to not feel small, to feel intrigued and surprised instead.

I’m asking, How does the cafeteria context do any of that? How could any context do that when the rest of the problem is rather intractable?

On another note, while “low-literacy demands” help give EL students (and many others) easier access to problems, these students need to be immersed in language in math classes in order to help them learn the English. Schmoker and Marzano have been pushing for the reading of more text across the curriculum, and now the CCSS is calling for 70% of student reading to be reading informational text from other content areas. The only place for students to learn how to read technical material is in math and science class. With that said, I can say as the father of a dyslexic child, that offering the border problem visually would engage my son more readily than a garden context. He would want to see how close his guess was, rather than solve someone else’s contrived problem. At the same time, I hope there will still be ample opportunity in his math class to develop his impaired reading skills.

If we give them the first four “iterations” they will simply count to get the sequence: 12, 16, 20, & 24 blue squares. That is low entry, which is good, but won’t almost all students see it as an “increase by 4” number pattern at this point? Is that what you want? It seems like this is directing students towards a non-visual approach for a visual problem (not necessarily bad) and away from creative thinking on a problem that is inviting creativity.