As a relatively new teacher, it's hard for me to really know what to believe. In my math methods class, I was taught the socio-constructivist philosophy. I accepted it not blindly, but because I had already been teaching for a couple of years (I took afternoon/evening classes over a period of 2 years while teaching on my intern credential), and the ideas really resonated with me, based on what I saw from my students. One of the units I developed in the class was the slope unit that I mentioned in an earlier post, and when I implemented that unit, it worked much better with my students than anything I had previously tried. We read a lot of research and several case studies, and there seemed to be a lot of evidence supporting the benefits of this approach. Plus, it just made a lot of sense to me.
This philosophy is different than what has been described as the purely constructivist, or inquiry-based learning that many people seem to abhor. And it is quite different than the behaviorist (teacher as source of all knowledge) philosohpy.
I have read many posts and articles from mathematicians who are opposed to the NCTM and its beliefs. But here is a quote from their standards document pertaining to high school level:
Because students' interests and aspirations may change during and after high school, their mathematics education should guarantee access to a broad spectrum of career and educational options. They should experience the interplay of algebra, geometry, statistics, probability, and discrete mathematics. They need to understand the fundamental mathematical concepts of function and relation, invariance, and transformation. They should be adept at visualizing, describing, and analyzing situations in mathematical terms. And they need to be able to justify and prove mathematically based ideas.
This seems reasonable to me, and I wonder if those of you who take issue with the NCTM's beliefs could comment specifically on what the concerns are. Is it the content of their standards, or the way they are implemented, or something else?
I am bound by the CA content standards and STAR testing. I feel that most of the standards are things that a student taking algebra, for example, should know how to do. But there seems to be very little emphasis on problem solving, critical thinking, and application. And the STAR tests themselves do not really assess these things, only the fundamental "tools" of algebra. As teachers know, if it is not assessed, it will not get done - especially when teachers and schools are under the gun to raise test scores. So is the plan that students will master the tools of math in high school, and then will somehow be able to become problem solvers in college? And what about the students who don't go to college?
What thoughts do people have on the CA standards (or, if you are familiar with another state's standards)? Are they a subset of a good math education? Do they mesh with the NCTM standards at all? Are the state standards counterproductive? (As for the STAR testing, that will probably be a good topic for a later posting).
And finally, why is there such hostility over these issues? What I've read seems more like political partisanship, and less like people trying to collaboratively build a consensus as to how best to teach our country's students.
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2 comments:
California's standards are pretty darned rigorous.
As for your statement that "if it isn't tested, it doesn't get done", that's a reflection more of poor teaching than on the standards themselves.
If you read the CA Framework in addition to the standards (the standards come in a stand-alone book or as Chapter 2, I think, in the Framework) your concerns about application and higher order skills will be assuaged.
I agree that the CA standards are rigorous. I also think that the skills outlined in the standards are critical to being able to solve more complex problems in each subject area. I've read through a lot of the framework, and I think it is well written and quite reasonable. I haven't found anything in the framework that I object to.
However, if something that is not tested is not taught, I don't think it is correct to necessarily blame the teacher. The standards for each grade level are written with the assumption that the standards from the previous grade levels have been adequately mastered. This is clearly not the case for the majority of students (otherwise, there would not be such high levels of failure on the CA exit exam, which barely even touches high school level math).
Therefore, as a 9th grade Algebra 1 teacher, I am faced with a daunting list of standards, which I would love for my students to learn - and they, I, and my school will be judged based on how well they learn them. I have the problem, though, that I must spend a huge amount of time teaching standards from lower grades that students have never learned. This eats away from the time needed to teach the current standards at just a procedural level, let alone a conceptual level.
The framework emphasizes that three components of math education are critically integrated: procedural knowledge, conceptual understanding, and problem solving skills. However, the standardized assessment that students take is almost entirely composed of procedural level questions.
In our algebra 1 classes, we used to do a Problem of the Week assignment to really develop students' thinking and problem solving skills, but we had to drop this. They were so weak in their problem solving abilities, that the amount of time going into developing them (though I think this was time very well spent) was making it impossible to make any headway into the list of content standards. We still work on traditional word problems, but these are pretty formulaic and don't really push the boundaries of critical thought. We also have to spend a lot of time developing pre-algebra concepts (this is in addition to the separate numeracy class that students concurrently take to develop their basic arithmetic skills). Add to that the fact that the STAR tests occur when there is still over a month of class left, and yet they test the entire year's worth of standards.
Given all of these constraints, we've had to strand out the concepts vertically among classes. Some things we have just accepted we will not teach in Algebra 1 (i.e. rational expressions, absolute value equations, fractional exponents), and will instead push to Algebra 2.
And, in the high stakes testing climate that we now have, we do look at the blueprint (the document that shows how many questions relating to each standard that will be on the test) to make decisions. For example, here is standard 21 from algebra 2:
Students apply the method of mathematical induction to prove general statements about the positive integers.
According to the blueprint, there is one question every three years on the STAR test relating to this standard. I think mathematical induction is fascinating and really develops critical thinking. But to really teach students mathematical induction takes a lot of time. How can I justify spending that time, when so many other standards are tested much more heavily?
In summary, I don't think there is a problem with the skills that are listed in the standards. I think there is a larger systematic problem that the standards are a part of.
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