Here’s my problem with what you’re saying, and (apologies) but I’m having trouble seeing how you’re clearly answering my objections.

(1) *“If you are working with kids in that category and think they are, as a group, genuinely interested and capable in abstractions, then you are fooling yourself.”*

You haven’t defined abstractions well. All of math is an abstraction of some sort or another. Multiplication is an abstraction. Fractions are an abstraction. Simple equations are abstractions. Which abstractions do you believe low ability kids to be capable of, and which do you not?

(2) I think that you’ve contradicted yourself. You write that low ability kids can’t handle modeling for the sake of modeling’s sake. You also write that teachers introduced modeling for the sake of modeling because they found that their kids couldn’t handle finding answers (“Then we started pretending that the process was more important than the product…”). So, which is it?

(3) I think that you’ve tripped up in another way. You write that you’d just as well see low ability kids out of school. That means that school doesn’t have much to offer them. Then you also write that it’s more important to be able to find answers for most people in math. Now, if you’re just taking a stand on what you think dumb people are interested in, fine. If you are saying that it’s more important for people to know how to find answers than to understand the process, then you’ve tripped up. Because *nothing* ultimately matters about the education you’re providing these kids, as far as content goes. You’re just trying to stretch their minds, not trying to teach them anything that lasts. So who cares what’s “more important”? Nothing is really important.

Thanks for your patience in explaining your positions.

]]>You were comparing your kids to doctors and lawyers. With due respect, you weren’t working with low level kids if that were true. So I’ll keep it simple: low ability kids have IQs between 85 and 100. If you are working with kids in that category and think they are, as a group, genuinely interested and capable in abstractions, then you are fooling yourself. This really isn’t a matter of opinion; the link between IQ and ability to think abstractly is well established. See the Flynn op ed here in this earlier post: https://educationrealist.wordpress.com/2012/10/06/teaching-students-with-utilitarian-spectacles/

I would argue that even low mid-level ability kids (IQs from, say, 95-105) have difficulty with modeling and abstractions and benefit from more concrete presentations, with modeling presented as a way to make abstract equations concrete situations, but that’s at least debatable.

*Is the problem with teaching modeling (for the sake of modeling) to low-ability kids that they won’t be interested in it? Or is it that it’s a lowering of standards, because the product is (in fact) more important than the process?*

There is no problem with teaching modeling. There is a problem with pretending that creating a model is more important than finding the answer for MOST people, particularly low ability people.

As I’ve said before, it may be possible to teach low ability kids abstractions in connection with specific skills, but we haven’t tried that yet, and it’s certainly not what math class is about.

]]>Because low to mid ability kids are far more utilitarian. They aren’t interested in problems that don’t have answers.

You’re wrong, in my experience. I teach very weak kids, and they are very interested in problems that don’t have answers. But at least we agree: we should teach kids the math that they will find challenging and interesting. That’s pretty significant, and I’m glad we agree.

Still, I don’t think that what you’re saying now jives with what you wrote in your post:

Getting the answer used to be enough for math teachers, too, until kids stopped getting the answer with any reliability. Then we started pretending that the process was more important than the product.

Is the problem with teaching modeling (for the sake of modeling) to low-ability kids that they won’t be interested in it? Or is it that it’s a lowering of standards, because the product is (in fact) more important than the process?

]]>We aren’t debating whether to teach modeling. You have already agreed that I do teach modeling, which for low ability kids is often just a matter of having them visualize what would actually happen if the events in the word description took place. I teach them to model the problems BECAUSE it helps them to get the right answer. Because low to mid ability kids are far more utilitarian. They aren’t interested in problems that don’t have answers.

Remember, this all began because you thought I was wrong to quibble about the article, which said that the kids got more right answers. In fact, the treatment did not result in more right answers, simply more accurate models–that were then solved incorrectly. But you think the more accurate models (which in this case, were equations) is a “right answer” in and of itself. I’m simply saying it’s not true. There may be kids for whom an interesting model is more important than the right answer, but they aren’t kids who struggle to pass algebra. They need the right answer.

]]>This makes a lot of sense to me:

As long as a kid is in school, I want to be giving that kid intellectual problems that spur their interest in thinking about “Why”, “What”, and “How” to the best ability they can. That strikes me as the purpose of school.

But I don’t see how that’s consistent with your post. You also said,

Remember, though, we’re talking about the lowest ability kids here. Do they need models, or do they need to know how to find the right answer?

If we’re just trying to spur their interest then why do kids need to know how to find the right answer? That’s what I’m having trouble putting together about your position. Why not teach modeling, if that will interest children and spur their interest?

]]>Hey, I missed this comment somehow.

*It seems to me as if you’re suggesting that there isn’t much point of school for low-performing kids. You’d send them out of school if you could, but you can’t, so you might as well give them something that they can handle so that they don’t feel awful about themselves. Is that right? *

To some extent, yes. Our economy has changed; these kids could early have gotten manufacturing or other jobs in which they could achieve (to varying degrees) based on their ability to achieve, not their interest in abstract math well above the needs of the job. So yes, in earlier days, these kids would not have been an educational problem.

Our willingness to accept that our original goal–educating every to be a symbol manipulator, rather than a concrete doer–is completely out of the question will be the primary determinant of our educational success moving forward. Right now, our unwillingness to accept this is the reason we are declaring our schools a failure when they aren’t.

*who cares whether we teach low-performing kids math that involves getting a right answer at the end, or teaching them proof and modeling as ends in themselves.*

As long as a kid is in school, I want to be giving that kid intellectual problems that spur their interest in thinking about “Why”, “What”, and “How” to the best ability they can. That strikes me as the purpose of school. Some kids will be interested in that more than others—regardless of ability. But no one is harmed by stretching their brain. This seems pretty obvious, to me. School has more than one function; it’s not just about creating good industrious workers for society. It’s also about creating good citizens, good thinkers, and people who feel vested in society. If our school system gives kids the sense that we not only care about them, but value their abilities and interests where they *are*, not where we delude ourselves they should be, it strikes me as a good thing.

No.

1) https://educationrealist.wordpress.com/2012/07/01/the-myth-of-they-werent-ever-taught/

2) I have no idea what my kids’ scores are in any objective sense when I first meet them, no access to test scores–at least, I haven’t 3 years out of 4. I give my kids a test the first day of class, a test much easier than the class they are entering (https://educationrealist.wordpress.com/2012/09/10/my-math-classes-are-they-prepared-um-no-so-what/). Most, but not all, of my students match that performance over the year–that is, the students who ace that test continue to ace the class, the students who tank struggle. But it’s not a perfect correlation. Some students finally start to “get it” that particular year. Additionally, a few students who do well on the test, which is highly structured and predictable, struggle in the class. If I were being fooled by the Pygmalion effect, that wouldn’t happen.

3) I am a very good assessor of in-class abilities, and in the first two years of teaching, I continually underestimated the degree to which some kids were cheating. So I’d always be running into the case of a kid who man, I’d thought didn’t have a clue and yet was killing on the tests. In the early days, I trusted the kids’ performance on my tests rather than my own effect–that is, I trusted what I thought was demonstrated performance over Pygmalion. However, when I got it together enough to create 2, 3, or 4 tests to offset cheating (which, in the early days, took me a couple months to put in place), the kids who I thought didn’t know a thing didn’t, in fact, know a thing when they couldn’t cheat. They all had the answers to the wrong test–the answers on the test of their neighbors. A couple years of this left me more confident of my own assessments–now, if I see a kid overperforming my assessments, I covertly check for cheating.

4) Both as a test prep instructor and a teacher, I am notable for the degree to which I defy the Pygmalion effect. I always have kids who had previously done poorly in math because they didn’t do things the way the teacher liked, didn’t follow the rules, didn’t do homework and who do very well in my class. I also have kids who learned for the first time that they weren’t good at math, that they’d done well by following the rules, doing homework, but not gaining understanding.

5) I’m a Pygmalion skeptic, but to the degree it holds, it holds for achievement, not actual ability. And my classes are all about demonstrated ability. You screw around in my class, never do your homework, but nail the tests, you’re getting an A. The Pygmalion effect is about “soft” skills, and the ability to please a teacher–and that’s not a major factor in my classes.

]]>Is it possible that your own beliefs about what the low performing kids are capable of create a ceiling for their capabilities?

]]>So in that sense, giving them interesting problems that challenge their mind appropriately that force them to deal with rules and constraints is as good a way as any and much better than most as a way for them to use their mind while in school.

I assumed that it was better for them to use their minds in school because it would benefit them later in life. Apparently, you didn’t mean that.

It seems to me as if you’re suggesting that there isn’t much point of school for low-performing kids. You’d send them out of school if you could, but you can’t, so you might as well give them something that they can handle so that they don’t feel awful about themselves. Is that right? Or do I have you totally off? (Sorry if I do; trying to read you accurately.)

If that’s true (and I might have misunderstood you), then who cares whether we teach low-performing kids math that involves getting a right answer at the end, or teaching them proof and modeling as ends in themselves. (You said earlier that low-performing kids **need** to learn how to find the right answer. That’s what I’m pushing on.)