AAAS Conference on Developing Textbooks That Promote Science Literacy

February 27-March 2, 2001
American Association for the Advancement of Science
Washington, D.C.

Strategic Approaches to Achieving Science Learning Goals

Edward L. Smith
Michigan State University

Overview

In this paper I introduce and illustrate what I refer to as strategic approaches for teaching to achieve specific learning goals. I discuss the role of the Project 2061 Curriculum Analysis Procedures (CAP) in guiding the design of and describing an example strategic approach. Finally, I discuss the nature of a professional knowledge base for teaching specific learning goals.

Throughout my research career, I have specialized in research on the teaching and learning of specific science topics. Since my own classroom teaching and graduate school days, I have been interested in finding ways to teach various topics so that the learners really understand. From the early work of Piaget (1935, "The Child's conception of the World") and my own dissertation (1969), I have been interested in the conceptions that students develop about the world around them. Cognitive science and conceptual change theory helped me to appreciate their significance. In a series of classroom studies, my colleagues and I examined what happened as teachers taught particular topics. We examined the ideas children started out with, the curriculum materials the teachers used, how those materials were used in planning, what transpired in the classroom, and how these elements seemed to account for the usually disappointing learning results we observed.

We developed alternative teaching approaches to some topics. While certain approaches were moderately successful (much more so than the conventional practice), others were not. In some cases, iterations of analysis and revision led to improved success. In this paper, I attempt to capture some of what I think we have learned from these efforts in a way that may support an on-going research and development program.

Summary of My Argument

Well formulated standards can serve to focus a broad range of activities and resources on the achievement of common goals. Often, most attention gets paid to the assessment and accountability aspects of standards. However, the setting of standards and assessment of the extent to which they are achieved is not sufficient to bring about improvement in that achievement. Teachers need knowledge and resources for teaching of the standards for which they are responsible. That is, they need to know how to teach those things. An important contribution of standards is to lay out a research and development agenda for the development of a professional knowledge and resource base to support teachers.

An important set of issues has to do with the adequacy of the standards. Do they capture what is worth knowing? Is the total set of standards feasible to achieve? While these are critical issues, I will not address them in this paper. I simply argue that the current national science education standards (NSES and Benchmarks for Science Literacy) are sufficient to move the enterprise forward in substantial ways, and that improvements in the standards themselves will come from taking them seriously into account in research and development work. Given this assumption, what can be said about the knowledge base needed to support effective teaching of the standards?

My primary purpose in this paper is to introduce the idea of a strategic approach to teaching a related set of learning goals and to illustrate this idea with an example. By a teaching approach, I refer to a generally sequential pattern of activities in which students become engaged over a period of time. A strategic approach is more than a collection of activities, all of which are related to the topic, interesting and doable. It is more than a logical order of presentation of information and more than a logical sequence of activities. An approach becomes strategic when each activity is selected and sequenced to serve particular purposes in moving students from where they are toward the intended learning. Of course, a strategic approach doesn't necessarily work.

My basic thesis is that specific approaches can be developed for teaching particular sets of specific learning goals and that these approaches can be improved until they work reliably for a significant range of teachers and students. Such approaches are not mechanical or rigidly linear, and there is no such thing as a teacher-proof approach. Any successful approach requires a knowledgeable teacher who understands and implements it intentionally and wisely. Providing the systems to support teachers in acquiring this knowledge is an other major challenge. However, if teachers do not know effective approaches for achieving the goals for which they are responsible, it is unlikely those goals will be achieved. I maintain that curriculum materials in which effective strategic approaches are embedded can be a critical, if not essential, resource in improving the achievement of learning goals on a large scale.

Assuming that effective strategic approaches are feasible and desirable, what makes one effective? I would argue that the Project 2061 Curriculum Analysis Procedures (CAP) capture many important features of effective strategic approaches. They can serve as design criteria or as a way to evaluate an approach that has already been devised. However, another important function of this framework is to provide a common way of describing an approach. The CAP criteria and the constructs in terms of which they are described define a set of categories of knowledge. These categories can be viewed as fleshing out Shulman's "pedagogical content knowledge." (Shulman, 1987; Wilson, Shulman and Richert, 1987) The instances of these categories-representations, phenomena, naive conceptions, etc.-constitute a repertoire from which a particular approach can be crafted. A shared knowledge base is essential for a professional community. CAP is based on theoretical perspectives with broad support in the science education research community. They perform an important function in providing a common language and conceptual categories.

The adequacy of CAP in defining necessary and sufficient features of curriculum materials and the teaching approaches embedded in them is something to be determined. Research-based arguments might be made for additional or alternative criteria. Ultimately, empirical evidence that use of highly rated materials tends to result in higher levels of achievement would be important evidence. Of course, adequate professional development and ongoing support are essential, and research will need to sort out the role of teacher implementation and other factors in explaining the results. However, I maintain that having effective strategic approaches as a fundamental component for a professional knowledge base and building those approaches into curriculum materials is a powerful, if not essential component of standards-based science education reform.

Example Approach-Explaining Changes in the Night Sky: Building and Using a Model of the Solar System

My purpose here is to portray an approach developed to help middle school students learn about the solar system. In particular, we want the students to understand the relative positions and motion of the earth, sun, stars, and planets and to be able to use that understanding to explain observed changes in the positions of star constellations and planets in the night sky. The portion of the sequence discussed here does not include the planets.

The sequence is designed to address several benchmark ideas including the following, which I will consider in this discussion:

The patterns of stars in the sky stay the same, although they appear to move across the sky nightly. [4A(3-5)1]
The earth rotating once every 24 hours makes it seem as though the…moon, planets and stars are orbiting the earth once a day. [4B(3-5)2] (American Association for the Advancement of Science [AAAS], 1993)

The approach assumes that the students are aware that the sun appears to move across the sky and that this is a result of the earth turning daily. However, it also assumes that the students are not aware that the same mechanism causes a similar apparent motion of the objects in the night sky. In fact it assumes that many of the students will not have well developed ideas about the changes in appearance of the night sky over time, but will entertain a variety of naive ideas. These might include the idea that the stars are just "up there" and show up at night but disappear in daylight because the sun is too bright. Other ideas might allow for the stars to move about, changing their positions individually or as clusters seen as constellations.

The intent of the approach is to help the students revise their ideas and construct a basic understanding of what the apparent motion is like and how it relate to movements of the earth. We intend that the students learn to use these ideas to predict qualitative changes in particular constellations over a few hours or months.

Beginning: The overnight southern sky constellation prediction. We begin the unit by posing the question about a constellation viewed out of a south-facing window: What would the sky look like if we came back and looked again each hour for three hours, both later that night and again at the same time the next night? Why would it look like that? The students are encouraged to think about, write about, and discuss their ideas about these questions. Their various ideas are noted and recorded, along with any revisions they make in the course of the discussion.

Testing the southern sky constellation predictions. We next discuss how these ideas might be tested and end up with a plan for observing and recording what we observe. These plans are then carried out, resulting in drawings, photographs, or possibly other forms of records. (Accompanying examples can be used if live observation is not possible.) These records are carefully reviewed and compared to the predictions. The results are then analyzed and the students agree upon a description of the results: The stars viewed in the southern sky kept their same pattern while moving from east to west. The next night, the stars appeared in about the same positions as the previous night.

Modeling and observing the southern sky With this result clearly posted, the question of why the positions of the stars changed as they did is considered. Some students are likely to bring up the idea of the earth's turning on its axis. The students are encouraged to discuss this and any other ideas (a few may suggest that the stars are actually moving), and what support they have for those ideas. Some of the students are likely to make an analogy to the apparent motion of the sun and its cause as support for their ideas. If a student does not suggest building a demonstration or model to show their ideas, the teacher does so. A classroom model is built using an object for the sun and a swivel chair for the earth. A person sitting in the chair is the earth observer who experiences a cycle of night and day as the chair is turned. Several students should be the observer while others describe the changing conditions and identify where it is night and day, and so on.

The students are next challenged to add the constellation to the model. A poster of the constellation is prepared, and the students discuss where it should be placed and why. A position opposite the sun will likely be agreed upon, although there may be disagreements about exactly where. After the poster is placed, the students enact the daily rotation with the observer reporting the appearance of the constellation. All the students should have the opportunity to experience the rotation personally. The class discusses ways in which the model is accurate and ways that it is inaccurate.

Introducing a computer simulation. A computer program is then introduced that simulates the appearance of the sky from any location at any time. The students are asked how the simulation could be tested. They will probably suggest checking it with their observations. They make these comparisons and also watch as the simulation runs speeded up to make the changing positions more apparent. With this evidence of agreement, the teacher suggests that they use the simulation program to test further predictions that they will make.

Six-month prediction investigation. The next problem is to predict what would be seen if they were to look again in six months. The students work with the classroom model to develop their predictions. They will probably predict that they will not be able to see the same constellation because it would then be on the same side of the earth as the sun. Other constellations-those on the side of the earth away from the sun- would then be visible.. This prediction is successfully tested against the simulation program. This leads into small group projects in which each groups investigates a particular date or constellation, preparing and putting up a poster in the appropriate location. Students check each other's predictions, writing up reports to be shared with those who added the date and constellation they check. The final arrangements are reviewed by the class as the model is enacted to show what would happen in several new, in- between cases. Each case is then checked against the simulation.

Polar star investigation. Until this point, all the observations have been of the southern sky, as limited by the original problem of looking out the south-facing window. Now students consider the question of how a star over the pole would appear. After students write about and discuss their ideas and reasons, a "star" is placed directly over the swivel chair. Students take turns as the observer and compare notes as to what was observed. They will probably emerge with the idea that a polar star will appear stationary. They may disagree about where it would appear, or they may all assume that it will be "overhead" in the sky. If students suggest it, a globe can be used instead of the swivel chair. After their predictions and reasons are discussed, they are tested against the simulation by finding Polaris in the northern sky at several different times during a night. Different times of the year are checked. The result will probably be agreed upon as "it stays in about the same spot."

The question of where students would look outside to see Polaris is raised and discussed. Due to a naive idea of an absolute up and down based on the local perspective, the two dimensional display of the simulation program, and the lack of a "tilt" in the swivel chair model, there is likely to be disagreement as to whether Polaris will be directly overhead or lower in the sky. A globe is brought into the model. The class should be able to figure out that if Polaris is over the north pole, it is not directly overhead at our location on the globe. This should then be related to where it would appear in our local sky. Live observations are used to test their ideas. A carefully planned visit to a planetarium might be a poor substitute for this, but better than nothing.

Northern sky constellation investigation. The last step in this part of the sequence is to have small groups of students investigate how constellations in the northern sky would appear if observed at different times during the night. (Running the simulation for the northern sky over time should be avoided in the polar star investigation). They should use the classroom model, but not the simulation, in developing their prediction and placing their constellation where they think it should go. Each group should write up their predictions and explanations. When all of the constellations have been placed, the class can take turns observing from the swivel chair. They will probably reach agreement that the northern sky constellations will rotate in unison around the northern star. An evening observing session would be the best way to test these ideas. They can also watch the simulation run at this point. All students should feel competent to use the model to predict when particular constellations would be visible and how they would appear if observed throughout the night. They should all have the opportunity to do this, perhaps with visiting parents or other guests.

A strategic approach

I call what I have just described a "strategic approach" to addressing the learning goals for this sequence. Each step is planned for specific reasons to promote the intended learning, taking into account what we know about learning and the ideas that the students are likely to bring with them to the study of this topic. This not just a plan for a logical sequence or organization for the presentation of content. Nor is it a set of relevant activities put into a logical sequence or a set of activities on the topic that don't necessarily focus on the learning goals. Rather, each step is chosen to serve particular purposes in helping students to move from where they are at the outset to where we hope they will end up by the end of the sequence-that is, achieving the specific learning goals addressed. While there are many considerations in planning such an approach, I will unpack this particular example by discussing four steps in sequence:

engaging students in a problem involving relevant phenomena,
addressing commonly held student ideas,
introducing the new scientific idea(s) with appropriate representations, and
providing practice with using the new idea(s).

Engaging students in a problem involving relevant phenomena

The "southern night sky" example of a strategic approach described above was begun in a very intentional way:

A question was posed.
The question was understandable before the students have learned the new ideas.
The question involved relevant phenomena in a realistic setting.
The students had the opportunity to engage the question as individuals in small groups, and in the class as a whole.

The problem prepares the way for subsequent introduction of the scientific ideas that will help answer it; these ideas are the specific learning goals for the sequence. The rationale for this way of beginning derives in part from a cognitive-conceptual change theoretical perspective (cite How People Learn, Strike and Posner revisit, Smith, 1991). Briefly, understanding involves relating new information to existing knowledge. However, scientific ideas are often counter to everyday ideas. Unless the learner "activates" their personal knowledge structures and revises and reconciles them with the new scientific ones, they are likely to learn the scientific ones in isolation, memorizing to meet the demands of school, while continuing to rely on their everyday ideas.

From this perspective, establishing a problem at the outset accomplishes several important functions:

Students begin actively thinking.
Students bring up relevant existing ideas.
Students realize there are differences of opinion.
The teacher becomes aware of the students' ideas.
A sense of purpose and direction is created for further activity.

This way of beginning contrasts with several common alternatives. For example, starting off by simply presenting content accomplishes none of these functions. While identifying that "we are going to learn about the solar system" does provide some sense of direction, note that it requires that students already understand a technical term. It does not address the other functions. Announcing that we are going to learn about how the stars move, or even posing the same question used above in a rhetorical way, is more informative about the purpose, but fails to accomplish the other functions. The question must be engaged by the students to fulfill these functions.

Notice that in the "night sky" example, at least three opportunities were provided to express ideas about the question. First, as individuals, all students could write about or illustrate their ideas. Then, small group context allowed for several students to talk simultaneously. Finally, the whole class context allowed the teacher and other students to push for clarity and elaboration, and to demonstrate interest and respect for the ideas by formally acknowledgment.

A common approach among teachers is to first introduce the vocabulary that will be used to read a section of text. This is not likely to accomplish any of the necessary functions and fosters the idea that science is mostly about memorizing a bunch of strange new words. In contrast, beginning by establishing a problem that centers on relevant phenomena fosters the fundamental idea that science involves asking questions about the world around us.

In addition to the general rationale for beginning a successful teaching approach with a problem, there is a specific rationale for the "southern sky" example and how it is framed. First, it is a problem that can be empirically investigated and whose answer directly supports one of the learning goals of the unit: "The patterns of stars in the sky stay the same, although they appear to move across the sky nightly." The explanation developed for this result is based on another of the learning goal ideas. "The rotation of the earth on its axis every twenty four hours…makes it seem as though the stars are orbiting the earth once a day." Finally, by posing the problem as looking out of a south-facing window, the initial data collection is constrained so that a relatively simple pattern is observed. This pattern is similar to that observed for the sun and has a similar explanation. The more complex and less familiar patterns for the northern sky constellations and the north star are observed and explained later, after the students have become familiar with the model.

Addressing commonly held student ideas

I have made references above to "conceptual change" and students' "naive conceptions." These ideas grow out of theoretical and empirical research, much of which was inspired by Piaget's early work. The body of research on children's "misconceptions," "alternative frameworks," and "naive conceptions" also drew heavily on cognitive psychology. The conceptual change model proposed by Posner and Strike and their colleagues (Posner, et al 1982; Strike & Posner, 1992) was heavily influenced by philosophers of science, particularly Stephen Toulmin (1972), who pointed out the "problem of conceptual change." Briefly, Toulmin observes that we understand new information by relating it to our current knowledge. He then asks how we come to understand when our current ideas are in conflict with the new idea to be understood (as often occurs in science). Susan Carey (1986) made much the same point with her "paradox."

This conceptual change work is often oversimplified when applied to students' everyday ideas which are often viewed as wrong answers that just need to be corrected or somehow avoided in the first place. However, as explained by Toulmin and cognitive psychology, our own current knowledge provides the basis for thought and represents the tools we have to work with. And since these tools work well most of the time, we do not change our thinking easily.

The thrust of the conceptual change perspective is that teaching needs to recognize the nature and importance of students' current knowledge, including their "naive conceptions," and use teaching approaches that guide students' through a process of conceptual change. There is an element of having to convince students of the value of the new ideas. Do what we will, if the students are not convinced, they will not be understanding, but will simply be doing whatever they can to meet the demands of school.

The teaching approach described above is designed to take into account that students will likely have ideas that are incompatible with those we are addressing and anticipates what some of their naive conceptions might be. A key part of the process of developing and improving on this kind of teaching approach, involves ongoing analysis of the conceptions students' seem to be using. In our research, my colleagues and I found it useful to construct charts that summarize the contrasts we identify between the goal conceptions we define for a unit or topic and the naive conceptions we identify. These contrasts are labeled as "issues" that can have more than one position or perspective. These have become a standard required part of the units planned by the students and interns in the Michigan State University teacher preparation program. The chart below summarizes some of my current thinking about students' thinking relevant to the solar system unit. I have organized my discussion of how the approach addresses students' naive conceptions around the two issues in the chart.

Goal/Naive Conception Chart for Solar System Unit

Issue

Goal Conception

Naive Conception(s)

Stars' appearance over time

The patterns of stars in the sky stay the same…

they appear to move across the sky nightly…

…different stars can be seen in different seasons.

Stars (and constellations) in the southern sky appear to move from east to west like the sun.*

Stars in the northern sky appear to rotate around a point. The North Star, near this point, appears to remain nearly still.*

a) The pattern of stars in the sky stay the same, but they all appear to move from east to west across the sky like the sun.

b) Constellations appear to move about as clusters and can get closer together or farther apart.

c) Stars appear to move about individually, forming different patterns/constellations at different times.

d) The pattern and apparent positions of stars in the sky stay the same. They are visible at some times and not others.

Cause of changes (or lack thereof) in stars' appearance

The earth rotating once every 24 hours makes it seem as though the sun, moon, planets, and stars are orbiting the earth once a day.

The earth orbiting the sun once a year explains how different stars and constellations can be seen in different seasons.

Stars stay still while we turn by them as the earth rotates on its axis.
Constellations are star clusters that move around in space.
Stars move around in space individually and form different patterns at different times.
Stars are just "up there" and we are "down here." We can't see them in the day because the sun is too bright.

* These ideas are not explicit in the standards. They are included as learning goals in the unit to avoid confusion and encouragement of a misconception and to support students' ability to apply the model to all parts of the sky.

Stars' appearance over time: What would we see if we looked up at the same place in the night sky at different times? It is difficult for someone who is familiar with the observable patterns and the explanatory models to appreciate the complexity involved in just looking up into the night sky now and then, occasionally seeing a recognizable pattern, but otherwise just a bunch of "stars." Most students will probably not have very well developed ideas about the observable changes in the appearance of the stars, but as shown in the table, there are several naive ideas that they might entertain.

These and/or other ideas are elicited in the initial activity in which the problem of how an observed pattern of stars would appear later in the same night and the next night at the same time. The teacher should be alert for and record the various ideas that students express. Some may be abandoned by the students as they hear others' ideas and their explanations. The teacher should help keep track of these revisions of thinking (e.g., crossing out abandoned ideas, encouraging expression of their reasons). The teacher should help students clarify how they think other stars appearing near the constellation would appear later and whether other constellations might come nearer to the one observed. In each case, their reasons for suggested ideas should be sought. The record of the ideas students continue to consider should be kept and referred to later when the observations are planned and when the results of the observations are being discussed. Those results will be a major basis for testing the students' ideas.

Although some students will probably use rotation of the earth as an explanation for their prediction of motion, it is important to continue to focus on testing the empirical issue before pursuing the explanation per se. If the students are not clear about this issue, they are likely to be confused about the discussion of the explanation. Even if they are later convinced by the explanation and/or the simulation, they will be less likely to see the observations as evidence supporting the model. This relation between empirical data and models is another issue to be addressed in teaching, but is beyond the scope of the present paper.

Different ideas about whether all the stars appear to move together or not are taken into account in planning the data collection. Records should include other noticed stars in addition to those considered part of the constellation. The original problem specified the south-facing window. This orientation should be maintained at this time to limit the complexity of the observed patterns of movement. The results should indicate that all of the observed stars appeared to move in the same direction. After discussing the results, most of the students will probably agree with this finding. Laterwork with the explanation will help it make sense.

Cause of changes (or lack thereof) in stars' appearance: What would explain why the stars might appear like this? Some of the students are likely to bring up the rotation of the earth explanation to support their prediction of what would be observed out the window. It may well be brought up again in different discussions. At those times, the explanations should be treated as reasons for their predictions. However, after the observed results have been established, the question of why the stars appear to move across the sky during the night and reappear in about the same place the next night is posed directly for consideration by all students. This prepares the way for consideration of the rotating earth explanation as well as others that students may come up with. (The actual movement of the stars is a possible explanation, although we have not seen it expressed yet in our cooperating classrooms.)

Some of the students are likely to refer to the similarity of the observed motion of the stars to that of the sun. They may argue that the rotation of the earth explains both. They may even suggest using objects to demonstrate their ideas. If not, the teacher should ask about the similarity and suggest building a classroom model. Even though most or all of the students may associate their observations with rotation of the earth, there are still problematic issues that may limit their understanding and ability to apply this idea. One of the main problem issues is that the appearance of northern sky constellations over time is quite different than that of the southern sky ones. (This is written from the perspective of the northern hemisphere.) While the southern sky constellations appear to move across the sky from east to west, as does the sun, the northern sky constellations appear to rotate around a point. The "north star" is located very near that point. Further difficulty is encountered due to a naive idea of an absolute up and down based on the local perspective, the two dimensional display of the simulation program, and the lack of a "tilt" in the swivel chair model. Because of these complexities, it is important to undertake more extensive work with the rotating-earth explanation, even if most students seem to agree with that explanation of the southern-sky observations.

[It might be argued that the learning goals themselves do not say anything about the difference between northern and southern sky movement nor the direction of the stars' apparent movement. Why, therefore, should we go into all that? Our concern is that northern sky observations are so unlike the sun's apparent motion that the students' association of the rotating earth with apparent stellar movement will be shaken and leave them confused. A fuller understanding also seems implied here in the standards because these issues are not addressed at this level of sophistication at the higher levels. (ref. Benchmarks and NSES)]

Introducing the new scientific idea(s) with appropriate representations

If students are to develop understanding of ideas, they need to encounter, interpret, use, and create representations of those ideas. Representations can be verbal statements, diagrams, animations, analogies, or role-playing, or can take other forms. Finding representations that are accessible and effective in communicating important aspects of the idea is an important and challenging part of curriculum materials development. All representations have their limitations, and more than one is frequently needed. Since most sequences or units address more than one idea, representations for each of them must be considered.

The first idea addressed in our example-the patterns of stars in the sky stay the same-is an empirical generalization. The first planned introduction of this idea is in the analysis of the observation records. Inspection of the drawings indicates that the position of the constellation is different on the different drawings. The analysis should lead to the interpretation that this is movement from east to west. This serves as supporting evidence for the generalization. Next, a representation of this idea is presented in the form of a set of views for the same times viewed on the simulation program. Running the simulation at a speeded up rate provides a vivid portrayal. The simulation does not provide direct evidence, but rather agreement from another kind of model. Finally, the students experience a representation of the stars' apparent movement across the sky as they are turned in the swivel chair in the classroom model. This representation, also not direct evidence, has the advantage of the students realizing that although the poster stars are appearing to move, it is they who are actually moving.

The second idea-that the stars appear to move across the sky nightly-differs from the first in that the students do not get direct evidence for it. It is an abstract idea. They develop a representation of it in the classroom model as the swivel-chair earth and its passenger-observer. They watch others rotate and experience themselves rotate in a context which supports them in thinking of themselves as on the earth (in relation to the sun and stars). Exchanging the globe for the swivel chair earth provides another representation as the globe is rotated. The twenty-four hour period for one rotation is represented in the review of the day-night cycle.

The third idea-that the earth rotating…makes it seem as though the…moon, planets and stars are orbiting the earth once a day-is an explanatory relationship between the first two. The classroom model is the primary representation of this idea. Again, it has the advantage of allowing the students to both observe the relationship from an outside perspective and experience it as a part of the model.

While the classroom model is a vivid representation, there should also be a clear verbal representation for this idea, and the other two as well. An example might be: "It looks like the stars are moving across the sky, but it is the earth turning that makes it seem that way." These representations should be developed in class discussions, recorded in a deliberate way and acknowledged as ideas that have been agreed upon, at least by most of the class, the teacher, and by any other sources that may have been consulted.

Providing practice with using the new idea(s)

One of the main things that makes learning something worthwhile is being able to use that knowledge. The central use of the goal ideas addressed by our solar system sequence is in predicting changes in the observed positions of constellations in the night sky and explaining why predicted or observed changes take place. The use of an idea is something that typically has to be learned and practiced. Thus, in our approach, the students are given many opportunities to predict, observe and explain changes in positions of stars and constellations.

We have found limited guidance in how to address the use of knowledge from the conceptual change perspective. Our treatment of the activity aspect of understanding (MDE, 1991) has been especially informed by a social psychology perspective (Brown, et al, 1989) and, in particular, by the "cognitive apprenticeship" model described by Collins, et al, (1989). "Application" is often viewed as something that follows more or less automatically after an idea has been learned, at least if it has been understood. In contrast, the cognitive apprenticeship model suggests that use is part of understanding and that ideas be learned in the context of their use.

As the name implies, the cognitive apprenticeship model portrays the relationship between a novice and a more expert teacher as they engage together in a meaningful activity. The expert "models" and "coaches" the novice as the novice takes more and more of the responsibility for the activity. Active guidance and feedback are important elements of the modeling and coaching. While initial examples of the activity are relatively simple, they are none-the less meaningful instances of the full activity. Using the analogy with craft apprenticeships, a silversmith might start out with a simple cup as the first object for which the novice will take an active role. More complex shapes and ornamentation would come later.

In addressing the practice aspect of the approach, I tried to figure out what it was that the students were practicing. This flowed out of our "objectives" pretty well, since they are to specify what it is the students are su