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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.

A Constructivist View of Learning

The big idea here is that all learners use the ideas they currently have to form or create new ones. Knowledge is never gained passively. Children must be mentally active for learning to take place. Teachers tend to confuse constructivism with a way of teaching. It is worth pointing out that constructivism is a theory about how the human mind learns. If the theory is correct, and presently most believe it is, children will learn this way regardless of how we teach. You can't turn on constructivism some days and turn it off other days. This is not a choice. Therefore, if we treat children as passive receptors, they will either reflect on the ideas on their own or they will likely not learn.

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 knowledge is the possession of an idea, then understanding is a measure of how well this idea is integrated with or connected to other existing ideas in the learner's cognitive framework. Therefore it is reasonable for several children to each have constructed an idea - to possess a bit of knowledge - but each understand it differently and to different degrees. Transparency T-19 illustrates this idea with the blue dots. In that figure, the idea is that several students may each own the same idea (the dot at the center). It may or may not be connected to related existing ideas or the child may not have useful ideas with which to connect it.

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.

Knowledge of Mathematics: Conceptual vs. Procedural

It seems important to understand that the ideas of mathematics have no physical exemplars "out there" in the real world. This idea is taken directly from Piaget's categorization of knowledge based on its source. That a mathematical concept is a relationship is somewhat profound when one realizes that you cannot show a child a relationship and thus you cannot show him a mathematical idea. Relationships are logical constructs. Our minds must invent them. It is the task of the teacher to promote that invention. Therefore, when we show students a physical manipulative, we are not showing a concept but only a physical object.

Conceptual vs. Procedural Knowledge

The distinction between conceptual and procedural knowledge has been made in many places in the literature. Most importantly I would refer you to Hiebert's book, Conceptual and Procedural Knowledge: The Case of Mathematics and also the chapter by Hiebert & Carpenter, "Learning and Teaching with Understanding," in Grouws' Handbook of Research on Mathematics Teaching and Learning. As these authors point out in the latter work, it is not important to argue over which is the most important or even to draw clear lines between these types of knowledge. Rather, it is important to consider how concepts and procedures are related. To this end, it is useful to help your students see that conceptual knowledge and procedural knowledge actually are distinctly different and that students can and do acquire one without the other. Unfortunately, the most common scenario is the acquisition of procedural knowledge without a conceptual connection or basis. This, at least is one idea with which my students have little difficulty - they all possess an array of procedures that they have learned instrumentally without any conceptual foundation. They recognize this and are willing to talk freely and openly about it. I assure them that this is not at all their fault but more likely a result of instruction. I generally pronounce a golden rule for all school mathematics:

No child should ever be asked to learn any mathematics of any kind without a conceptual understanding! There are no exceptions to this rule.

The Value of a Community of Learners

The view I've tried to develop in this chapter is that reflective thought is the key to learning. A strict look at constructivist theory often ignores the value of classroom interactions and the way that students learn from each other. Shared meanings develop and useful language emerges. Ideas developed and shared by one student might be challenged, adapted, or provide the inspiration for new ideas. This social view of learning is generally built on the theories of Lev Vygotsky. As presented here, this is not a competing theory for the constructivist view but rather a companion view that compliments that theory. Social learning occurs in students zone of proximal development.

The Role of Models

For some time now I have chosen to use the word model instead of manipulative. I have found that teachers actually believe in some way that color, or attractiveness, or the tactile nature of a manipulative is what makes a manipulative an effective teaching tool. A model is generally a manipulative but it can be a drawing or a picture or a calculator just as well. More importantly, the mathematics does not come from the model, but is used to test out, confirm, and refine the relationships that the learner is developing. The habitual question for the teacher to ask is this: Is the model being used as a thinker toy or a tester toy and not just used in imitation of how they were shown to use it? This is another phrase that gets repeated a lot in my class - models as thinker toys. As we formulate a new logical relationship, we need to test it, to see if it fits and makes sense. To the extent that the relationship can be imposed on the model without dissonance, then the model is providing us with some feedback as we construct the new idea. (This last idea is a bit more than my students are able to wrestle with.) So we do not get ideas from models but rather think with them. What I do work hard at is for my students to distinguish between the models they use and the concepts they are developing. Constance Kamii warns continually that children can learn to push blocks around just as mindlessly as they push pencils. Her warning is well taken.

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.

Strategies for Effective Teaching

The text lists six foundational ideas of teaching developmentally: children construct their own ideas and understandings; each learner's ideas are unique; reflective thought is the single most important key to effective learning; the sociocultural environment of the classroom has a significant effect on student learning; models for mathematical ideas can help students explore new or emerging concepts; and effective teaching is child centered. It follows that the key idea for teachers is to conduct activities that cause children to be reflective, to think about and to construct the desired ideas. It is not the use of manipulatives that makes teaching effective. Rather, it is causing children to be reflective that is the most important consideration for teachers.

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