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Think Math! materials are designed to have a “low threshold” and a “high ceiling.” In other words, students can approach any Think Math! activity from where they are at the moment and still succeed, learn, and be challenged.
My first impression was that this was not a good thing. It seemed to indicate this program was not organized in a cumulative, logical format that would require a mastery of the previous lesson in order to succeed in the subsequent lesson. In other words, “just jump in any time, because the program skips around topics.” If a student doesn’t fully learn the material (low threshold), that’s fine. On the other hand, if a student completely masters the lesson (high ceiling), well, all the better. Meanwhile, the gap between the slower learners and the faster learners will continue to widen.
And, for assessment purposes, I would imagine that clearing the “low threshold” constitutes “meeting standards”. This would ensure that no child is left behind.
I also found this description.
Problems that can be adapted for multiple ability levels provide a way to engage every student in a class. These problems are sometimes referred to as "low threshold, high ceiling" problems because all students can understand the problem and solve some part of it (low threshold), but even the highest-ability students in the class will not easily complete it (high ceiling). Rich problems can also be extended to allow students to explore more mathematics.
It seems all about heterogeneous grouping, engaging the students at all costs and low expectations. What’s the good part about this that I’m missing?
11 comments:
This just sounds wrong, wrong, wrong to me.
I'm open to the possibility that it could work, but what instantly presents itself to me is the likelihood that everyone will be frozen in place for all eternity.
I find it hard to see how the slowest kids are going to progrss to more difficult material if they're always going to be doing the simplest possible problems, or variants of the problem.
I also fear the fastest kids are going to be ongoingly clobbered by too-difficult material in order to keep them slowed down.
Having now lived through two years of the Phase 4 course here, I've seen how easy it is to take fast, motivated learners and set them up for failure.
I had a great email from Rudbeckia Hirta about the proper structure of an algebra course. I'll quote (then ask permission!):
He [her father] says that ideally a course should include some unfamiliar problems, but that it's completely unrealistic to expect all of the students to get all of the unfamiliar problems right all of the time. And there shouldn't be so many of these as to get the kids to hate math when they can't do them. (I think that if there was a "math pyramid" like the old "food pyramid" that these would be the equivalent of the fat-and-sugar group at the top.) He also says that before introducing unfamiliar problems that the kids need to be able to use the basic techniques really well.
I assume that "unfamiliar" problems is the same thing as "nonroutine" problems....what concerns me about this description is that one of two things will happen to the fast kids:
* they'll be given ALL nonroutine problems and they'll burn out, decide they hate math, etc.
* they'll simply be given the same exact problems but with more steps; the "challenge" will be a challenge to working memory and organization, not mathematical learning & reasoning
Here's what I really don't like, and what makes me feel a lack of confidence: there's no indication that the curriculum developers have made this decision in order to advance everyone's education as rapidly as possible.
I have to assume, given the ideology of ed schools, that the goal is simply to avoid ability grouping on ideological grounds.
That said, I do like the approach Dolciani takes in all of her books, which is to divide each problem set into 3 tiers: easy, medium, and hard.
And in fact, if the curriculum designers put it that way I might feel more trust....
iirc, Mathematics 6 also has tiered problem sets (could be wrong, but I think so....)
*sorry, had to delete and redo because the typos were screaming at me!
I found mention of the high ceiling, low threshold in an article in the Fall 1998 issue of Education entitled Ten ways to mess up a mathematics curriculum: Suggestions for avoiding common mistakes. Check out "mistake #7:
Mistake number 7: Delay higher order thinking until basic skills are learned
One of the age old questions in education is how do you move the class on to new, advanced topics when the most basic facts have yet to be mastered. Many teachers feel it is unwise to move forward to higher order thinking when some of the students seem to be stuck on very concrete low level operations, when in reality, higher order thinking skills and basic skills are actually very compatible. In fact, many times when the students are delayed in the processing of basic skills, it is a direct result of the absence of a meaningful higher order connection. Yes, basic skills are needed to efficiently solve more complex mathematical problems, but this does not preclude the use of higher order thinking in relatively simple activities. It is incumbent on teachers to provide activities that have a low threshold for the students that struggle as well as a high ceiling for those more advanced students so that the process of thinking through mathematics is as much a part of learning the mathematics as the procedure itself.
I actually recommend reading all ten "mistakes". The fact that the article was written immediately following the NCTM's Standards makes it particularly interesting considering it almost seems to foretell the many issues with mathematics education today.
Link: http://findarticles.com/p/articles/mi_qa3673/is_199810/ai_n8819863/pg_1
July 1, 2007 4:26 PM
"Low threshold, high ceiling" is what I was talking about in another thread. This is a technique that (supposedly) allows teaching a mixed-ability group of kids together with the hope that eveyone will get something out of the lesson.
The examples I've been told started in Kindergarten. My son's teacher pointed to an easel with a couple of paragraphs on it. She said that the advanced kids (all kids are sitting on the floor!) could read everything. Some kids could pick out individual words, and some kids could just listen to the teacher read and point to the words. That's it, and the teacher thought my wife and I would think that's all the explanation we needed.
I've heard other examples. They were all just as lame. The goal is full-inclusion, not academics. They're in dreamland if they think this provides a proper education.
Obviously, there is a fundamental difference of opinion over what constitutes a proper education in the lower grades.
I also fear the fastest kids are going to be ongoingly clobbered by too-difficult material in order to keep them slowed down.
But, success for all is promoted. If a kid finds the high ceiling too challenging to learn on his own (since I don’t imagine there’s much TEACHING going on in this scenario), he can drop down to a lower level and SUCCEED. At the end of the day, everyone feels good about their accomplishments. Including the teacher.
Sheesh, I’m so cynical today.
I have to assume, given the ideology of ed schools, that the goal is simply to avoid ability grouping on ideological grounds.
I agree. They prioritize diversity grouping over other considerations about what works best for most students.
" ...the goal is simply to avoid ability grouping on ideological grounds."
For the lower grades, this is their core belief. Everything follows from that; constructivism, spiraling, differentiated instruction, a lack of focus on mastery of content and skills. Never mind that this ideology directly conflicts with their main purpose of academics.
It's interesting to see schools struggle with trying to give more to the advanced students, but they won't give them what they really need. Acceleration.
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