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This is easily the most challenging chapter in the text. The ideas found here are not at all easy for students to develop, or even for teachers with experience. My personal belief is that these ideas are at the very heart of teaching developmentally - teaching from the perspective of a child who must develop his or her own ideas and understanding. To that end, I make a serious attempt in my classes to help students work through the main ideas found here. In the spirit of a constructivist approach, I have no problem with the notion that these ideas will be poorly formed or not totally integrated at the outset of the course. My intent is to lay some groundwork upon which students can reflect as the basic ideas found here are utilized throughout the course. Constructivism is the approach to learning that is currently espoused by most mathematics educators. Because the ideas are difficult, does not, in my opinion, mean that the pre-service teacher should not be exposed to a comprehensive framework for learning, within which effective instructional activities can be designed. To do less leaves the teacher without any significant basis for making instructional decisions other than intuition. That seems inadequate. And so I generally invest about a week in the ideas of this chapter and capitalize on every possible opportunity throughout the semester to reflect back on what we worked on in the beginning.

You may decide that, for your undergraduate students, this chapter is a bit too heavy. I have that feeling recurrently, although I persist. If there is a single key idea for the practitioner, it is that students must be mentally engaged, not passive. That is, teachers must find ways for students to be reflective. Building on the theory of Vygotsky, the social atmosphere of a mathematical community of learners is also significant. The need for reflective thought on the part of every learner at every age and for all content is the principal rationale for the suggestions for teaching found in Chapter 4.

However you choose to help your pre-service teachers with this chapter, be sure to consciously use these ideas throughout the course. If we teach theory and don't use it, we are guilty of the worst form of "educationese" that there is. I think you will find that an emphasis on reflective thinking will be valuable.

In classes, in-service, and in talks, I have found the "blue dot metaphor" depicted in Figure 3.1 and transparency T18 to be extremely effective. Students often talk about their lessons in terms of trying to activate students' "blue dots" or of finding out "what blue dots kids have" so that they can build on them.

If understanding is placed on a continuum (T-20) at one extreme is what I have called instrumental understanding, the possession of knowledge with no connections or at least very few and very weak connections. It is usually procedural knowledge that is learned without connections and is the most susceptible to instrumental understanding. (Note that many students confuse instrumental understanding with procedural knowledge. This is a serious error.) As more and more connections are made, the understanding improves. The connected end of this continuum of understanding I have called relational understanding. As noted in the text, these terms are taken from Richard Skemp.

What I find important here for my students is the realization that understanding is not an all or nothing proposition. Understanding grows with time and reflective experiences. Related to this idea is the personal nature of that understanding that is so highly mediated by the ideas that a child brings to the subject. You can only make connections with what is there.

The idea of model has been expanded to include all representations of an idea. In this sense, Dick Lesh's idea of five different representations (Figure 3.9, p. 30 and T-21) is also useful. New ideas can be tested in each of these domains and if the more the idea remains the same across representations, the more likely it is that students understand.