Silver et al's article "Teaching Mathematics for Understanding: An Analysis of Lessons Submitted by Teachers Seeking NBPTS Certification" presents an analysis of a random collection of 32 portfolios submitted by mathematics teachers of upper elementary and middle grades who wish to be certified by the National Board for Professional Teaching Standards (NBPTS) in 1998. The authors argue that although there was a variety of mathematics topics covered within the portfolios, there was a lack of cognitively demanding tasks for both assessing and developing understanding. Moreover, the authors found that fewer than half of the portfolios analyzed included student generated explanations. These findings were significant in that compared to previous research, significantly more of the teachers in this study included cognitively demanding mathematical tasks within their lessons. Although the inclusion of such tasks signifies good teaching practices, one must remember that these teachers were most likely submitting what they consider some of their best work to receive the certification. Thus, particularly with the authors' generous definition of "demanding task" in mind, the fact that just under half of the proposals contained no cognitively demanding tasks could be considered disappointing.
One of the most significant points in this paper for me was the authors' claim that teachers need "additional support to learn to solicit mathematical explanations as a tool in developing and assessing students' mathematical understanding." Although the teachers excelled in all of their pedagogical practices, student generated explanations seemed to be lacking. The authors suspect that this may be due to different definitions of explanation, but with mathematical justification being a central feature of the NCTM standards, this is still a surprising finding. Collaborative work and student centered learning was at the forefront for many, if not most, of these teachers. How is it then, that student explanations and justifications fall to the wayside? Why is this so much "harder" to do than integrating technology into the class?
It is interesting to read this, as I also feel like the explanations we receive in university math classes are lacking (if existent). I wonder how much assessment factors in? I know in giant classes, the final answer and one or two steps can be enough to get full credit. Logistically anything more is difficult. Even in small classes, parsing poorly-put-together-super-drawn-out paragraphs for some sort of train of thought is exhausting. So what do we do? Target the elementary classes and wait for these new students to arrive? I think we can all help. I worked in a third year class where the instructor made the students write a one page report once a week. As the sole marker, it was not very much fun for very many people (if anyone). That said: in most cases, the reports did get better as the semester progressed. I cannot help but think of Freire's comments we read last term (that went something along the lines of): you cannot write if you do not practice.
ReplyDeleteI'd like to touch upon both of your questions.
ReplyDelete1) I believe that student justification/explanations often are not required by teachers because either a) it is not explicitly written in the IRP's/PLO's or b) it may be written but everyone who reads it, takes away their own definition. To one person an explanation might simply be "show your steps", while to another it would be "show your thought process". These may sound very similar but I believe they are vastly different. To show one's steps, they simply must write down the steps of simplifying an equation, for example. However, to show one's thinking, they have to show their steps and describe why they made those decisions.
I have always found it difficult to have students show me their work while also showing me their explanation/justification. If too much time is spent getting one's thoughts down on paper, the math process may be compromised. I have seen a solution to this problem though! (see #2)
2) While it does seem that technology is being integrated into classrooms at a relentless pace, I don't believe that it's being done so thoughtfully. I think that technology can actually help us in understanding student thought processes/justifications. There are some fantastic apps that allow the student to solve a problem while recording their voice. A teacher can then watch a video of the student talking through the problem-solving in real time. You get to hear all of the ideas that are consequently rejected or accepted, and you also see things being written down that are later erased, which you wouldn't see on a piece of paper. This also allows a student to discuss their reasoning while solving the problem, provided they are comfortable with speaking their thoughts out loud. One of the obvious drawbacks is that a tablet device is necessary, and you are limited to the size of the screen. I see it as providing 'snapshots' of student understanding, and not to be used all of the time.