10 Facilitating complex thinking

A few years ago one of us (Kelvin) had the privilege of co-teaching with an experienced first grade teacher, Carolyn Eaton. As part of a research project, Ms Eaton allowed some of her reading lessons to be observed. Here is what Kelvin saw when Ms Eaton was having a conference with Joey. They are reading a book “together”, except that Ms Eaton wants Joey to do as much reading as possible himself. Joey’s comments are capitalized, and Ms Eaton’s are in lowercase.

Can you read the whole book?

SURE!

OK, then you start this time.

[Joey looks at first page, alternately at the picture and at the words.]

“IN THE GREAT GREEN ROOM THERE WAS A TELEPHONE”.

[Actual text: “In the great green room, there was a telephone”,]

“AND THERE WAS A RED BALLOON”,

[Actual text: “…and a red balloon”,]

“AND A PICTURE OF THE COW JUMPING OVER THE MOON”.

[Actual text: “…and a picture of the cow jumping over the moon”.]

“AND THERE WERE…” THREE BEARS?… “LITTLE BEARS SITTING ON CHAIRS”.

[Actual text: “And there were three little bears, sitting on chairs,…”]

Could you read this book with your eyes closed?

SURE; WANT TO SEE ME DO IT?!

Well, not right now; maybe another time. Could you read it without the pictures, just looking at the words? That’s how I do best—when I see the words instead of the pictures.

[Joey pauses to consider this.] MAYBE, BUT NOT QUITE SO WELL.

Let’s try it. [Ms Eaton proceeds to copy the words on a large sheet for Joey to “read” later.]

As Carolyn Eaton’s behavior suggests, there are decisions to make “on the fly”, even during the very act of teaching. Ms Eaton wonders when to challenge Joey, and when to support him. She also wonders when to pause and ask Joey to take stock of what he has read, and when to move him on ahead—when to consolidate a student’s learning, and when to nudge the student forward. These are questions about instructional strategies which facilitate complex learning, either directly or indirectly. In this chapter we review as many strategies as space allows, in order to give a sense of the major instructional options and of their effects. We concentrate especially on two broad categories of instruction, which we call direct instruction and student-centered instruction. As we hope that you will see, each approach to teaching is useful for certain purposes. We begin, though, by looking at the ways students think, or at least how teachers would like students to think. What does it mean for students to think critically (astutely or logically)? Or to think creatively? Or to be skillful problem solvers? Forms of thinking lead to choices among instructional strategies.

Forms of thinking associated with classroom learning

Although instructional strategies differ in their details, they each encourage particular forms of learning and thinking. The forms have distinctive educational purposes, even though they sometimes overlap, in the sense that one form may contribute to success with another form. Consider three somewhat complex forms of thinking that are commonly pursued in classroom learning: (1) critical thinking, (2) creative thinking, and (3) problem-solving.

Critical thinking

Critical thinking requires skill at analyzing the reliability and validity of information, as well as the attitude or disposition to do so. The skill and attitude may be displayed with regard to a particular subject matter or topic, but in principle it can occur in any realm of knowledge (Halpern, 2003; Williams, Oliver, & Stockade, 2004). A critical thinker does not necessarily have a negative attitude in the everyday sense of constantly criticizing someone or something. Instead, he or she can be thought of as astute: the critical thinker asks key questions, evaluates the evidence for ideas, reasons for problems both logically and objectively, and expresses ideas and conclusions clearly and precisely. Last (but not least), the critical thinker can apply these habits of mind in more than one realm of life or knowledge.

With such a broad definition, it is not surprising that educators have suggested a variety of specific cognitive skills as contributing to critical thinking. In one study, for example, the researcher found how critical thinking can be reflected in regard to a published article was stimulated by annotation—writing questions and comments in the margins of the article (Liu, 2006). In this study, students were initially instructed in ways of annotating reading materials. Later, when the students completed additional readings for assignments, it was found that some students in fact used their annotation skills much more than others—some simply underlined passages, for example, with a highlighting pen. When essays written about the readings were later analyzed, the ones written by the annotators were found to be more well reasoned—more critically astute—than the essays written by the other students.

In another study, on the other hand, a researcher found that critical thinking can also involve oral discussion of personal issues or dilemmas (Hawkins, 2006). In this study, students were asked to verbally describe a recent, personal incident that disturbed them. Classmates then discussed the incident together in order to identify the precise reasons why the incident was disturbing, as well as the assumptions that the student made in describing the incident. The original student—the one who had first told the story—then used the results of the group discussion to frame a topic for a research essay. In one story of a troubling incident, a student told of a time when a store clerk has snubbed or rejected the student during a recent shopping errand. Through discussion, classmates decided that an assumption underlying the student’s disturbance was her suspicion that she had been a victim of racial profiling based on her skin color. The student then used this idea as the basis for a research essay on the topic of “racial profiling in retail stores”. The oral discussion thus stimulated critical thinking in the student and the classmates, but it also relied on their prior critical thinking skills at the same time.

Notice that in both of these research studies, as in others like them, what made the thinking “critical” was students’ use of metacognition—strategies for thinking about thinking and for monitoring the success and quality of one’s own thinking. This concept was discussed in Chapter 3 as a feature of constructivist views about learning. There we pointed out that when students acquire experience in building their own knowledge, they also become skilled both at knowing how they learn, and at knowing whether they have learned something well. These are two defining qualities of metacognition, but they are part of critical thinking as well. In fostering critical thinking, a teacher is really fostering a student’s ability to construct or control his or her own thinking and to avoid being controlled by ideas unreflectively.

How best to teach critical thinking remains a matter of debate. One issue is whether to infuse critical skills into existing courses or to teach them through separate, free-standing units or courses. The first approach has the potential advantage of integrating critical thinking into students’ entire educations. But it risks diluting students’ understanding and use of critical thinking simply because critical thinking takes on a different form in each learning context. Its details and appearance vary among courses and teachers. The free-standing approach has the opposite qualities: it stands a better chance of being understood clearly and coherently, but at the cost of obscuring how it is related to other courses, tasks, and activities. This dilemma is the issue—again—of transfer, discussed in Chapter 3. Unfortunately, research to compare the different strategies for teaching critical thinking does not settle the matter. The research suggests simply that either infusion or free-standing approaches can work as long as it is implemented thoroughly and teachers are committed to the value of critical thinking (Halpern, 2003).

A related issue about teaching critical thinking is about deciding who needs to learn critical thinking skills the most. Should it be all students, or only some of them? Teaching all students seems the more democratic alternative and thus appropriate for educators. Surveys have found, however, that teachers sometimes favor teaching of critical thinking only to high-advantage students—the ones who already achieve well, who come from relatively high- income families, or (for high school students) who take courses intended for university entrance (Warburton & Torff, 2005). Presumably the rationale for this bias is that high-advantage students can benefit and/or understand and use critical thinking better than other students. Yet, there is little research evidence to support this idea, even if it were not ethically questionable. The study by Hawkins (2006) described above, for example, is that critical thinking was fostered even with students considered low-advantage.

Creative thinking

Creativity is the ability to make or do something new that is also useful or valued by others (Gardner, 1993). The “something” can be an object (like an essay or painting), a skill (like playing an instrument), or an action (like using a familiar tool in a new way). To be creative, the object, skill, or action cannot simply be bizarre or strange; it cannot be new without also being useful or valued, and not simply be the result of accident. If a person types letters at random that form a poem by chance, the result may be beautiful, but it would not be creative by the definition above. Viewed this way, creativity includes a wide range of human experience that many people, if not everyone, have had at some time or other (Kaufman & Baer, 2006). The experience is not restricted to a few geniuses, nor exclusive to specific fields or activities like art or the composing of music.

Especially important for teachers are two facts. The first is that an important form of creativity is creative thinking, the generation of ideas that are new as well as useful, productive, and appropriate. The second is that creative thinking can be stimulated by teachers’ efforts. Teachers can, for example, encourage students’ divergent thinking—ideas that are open-ended and that lead in many directions (Torrance, 1992; Kim, 2006). Divergent thinking is stimulated by open-ended questions—questions with many possible answers, such as the following:

• • • How many uses can you think of for a cup?
    • Draw a picture that somehow incorporates all of these words: cat, fire engine, and banana.
    • What is the most unusual use you can think of for a shoe?

Note that answering these questions creatively depends partly on having already acquired knowledge about the objects to which the questions refer. In this sense divergent thinking depends partly on its converse, convergent thinking, which is focused, logical reasoning about ideas and experiences that lead to specific answers. Up to a point, then, developing students’ convergent thinking—as schoolwork often does by emphasizing mastery of content—facilitates students’ divergent thinking indirectly, and hence also their creativity (Sternberg, 2003; Runco, 2004; Cropley, 2006). But carried to extremes, excessive emphasis on convergent thinking may discourage creativity.

Whether in school or out, creativity seems to flourish best when the creative activity is its own intrinsic reward, and a person is relatively unconcerned with what others think of the results. Whatever the activity—composing a song, writing an essay, organizing a party, or whatever—it is more likely to be creative if the creator focuses on and enjoys the activity in itself, and thinks relatively little about how others may evaluate the activity (Brophy, 2004). Unfortunately, encouraging students to ignore others’ responses can sometimes pose a challenge for teachers. Not only is it the teachers’ job to evaluate students’ learning of particular ideas or skills, but also they have to do so within restricted time limits of a course or a school year. In spite of these constraints, though, creativity still can be encouraged in classrooms at least some of the time (Claxton, Edwards, & Scale-Constantinou, 2006). Suppose, for example, that students have to be assessed on their understanding and use of particular vocabulary. Testing their understanding may limit creative thinking; students will understandably focus their energies on learning “right” answers for the tests. But assessment does not have to happen constantly. There can also be times to encourage experimentation with vocabulary through writing poems, making word games, or in other thought-provoking ways. These activities are all potentially creative. To some extent, therefore, learning content and experimenting or playing with content can both find a place—in fact one of these activities can often support the other. We return to this point later in this chapter, when we discuss student-centered strategies of instruction, such as cooperative learning and play as a learning medium.

Problem-solving

Somewhat less open-ended than creative thinking is problem solving, the analysis and solution of tasks or situations that are complex or ambiguous and that pose difficulties or obstacles of some kind (Mayer & Wittrock, 2006). Problem solving is needed, for example, when a physician analyzes a chest X-ray: a photograph of the chest is far from clear and requires skill, experience, and resourcefulness to decide which foggy-looking blobs to ignore, and which to interpret as real physical structures (and therefore real medical concerns). Problem solving is also needed when a grocery store manager has to decide how to improve the sales of a product: should she put it on sale at a lower price, or increase publicity for it, or both? Will these actions actually increase sales enough to pay for their costs?

Problem solving in the classroom

Problem solving happens in classrooms when teachers present tasks or challenges that are deliberately complex and for which finding a solution is not straightforward or obvious. The responses of students to such problems, as well as the strategies for assisting them, show the key features of problem solving. Consider this example, and students’ responses to it. We have numbered and named the paragraphs to make it easier to comment about them individually:

Scene #1: a problem to be solved

A teacher gave these instructions: “Can you connect all of the dots below using only four straight lines?” She drew the following display on the chalkboard:

Exhibit 10: The teacher gave these instructions: “Can you connect these dots with only four lines.

The problem itself and the procedure for solving it seemed very clear: simply experiment with different arrangements of four lines. But two volunteers tried doing it at the board, but were unsuccessful. Several others worked at it at their seats, but also without success.

Scene #2: coaxing students to re-frame the problem

When no one seemed to be getting it, the teacher asked, “Think about how you’ve set up the problem in your mind—about what you believe the problem is about. For instance, have you made any assumptions about how long the lines ought to be? Don’t stay stuck on one approach if it’s not working!”

Scene #3: Alicia abandons a fixed response

After the teacher said this, Alicia indeed continued to think about how she saw the problem. “The lines need to be no longer than the distance across the square,” she said to herself. So she tried several more solutions, but none of them worked either.

The teacher walked by Alicia’s desk and saw what Alicia was doing. She repeated her earlier comment: “Have you assumed anything about how long the lines ought to be?”

Alicia stared at the teacher blankly, but then smiled and said, “Hmm! You didn’t actually say that the lines could be no longer than the matrix! Why not make them longer?” So she experimented again using oversized lines and soon discovered a solution:

Exhibit 11: Alicia’s solution

Scene #4: Willem’s and Rachel’s alternative strategies

Meanwhile, Willem worked on the problem. As it happened, Willem loved puzzles of all kinds, and had ample experience with them. He had not, however, seen this particular problem. “It must be a trick,” he said to himself, because he knew from experience that problems posed in this way often were not what they first appeared to be. He mused to himself: “Think outside the box, they always tell you…” And that was just the hint he needed: he drew lines outside the box by making them longer than the matrix and soon came up with this solution:

Exhibit 12: Willem’s and Rachel’s solution

When Rachel went to work, she took one look at the problem and knew the answer immediately: she had seen this problem before, though she could not remember where. She had also seen other drawing-related puzzles, and knew that their solution always depended on making the lines longer, shorter, or differently angled than first expected. After staring at the dots briefly, she drew a solution faster than Alicia or even Willem. Her solution looked exactly like Willem’s.

This story illustrates two common features of problem solving: the effect of degree of structure or constraint on problem solving, and the effect of mental obstacles to solving problems. The next sections discuss each of these features, and then looks at common techniques for solving problems.

The effect of constraints: well-structured versus ill-structured problems

Problems vary in how much information they provide for solving a problem, as well as in how many rules or procedures are needed for a solution. A well-structured problem provides much of the information needed and can in principle be solved using relatively few clearly understood rules. Classic examples are the word problems often taught in math lessons or classes: everything you need to know is contained within the stated problem and the solution procedures are relatively clear and precise. An ill-structured problem has the converse qualities: the information is not necessarily within the problem, solution procedures are potentially quite numerous, and a multiple solutions are likely (Voss, 2006). Extreme examples are problems like “How can the world achieve lasting peace?” or “How can teachers insure that students learn?”

By these definitions, the nine-dot problem is relatively well-structured—though not completely. Most of the information needed for a solution is provided in Scene #1: there are nine dots shown and instructions given to draw four lines. But not all necessary information was given: students needed to consider lines that were longer than implied in the original statement of the problem. Students had to “think outside the box”, as Willem said—in this case, literally.

When a problem is well-structured, so are its solution procedures likely to be as well. A well-defined procedure for solving a particular kind of problem is often called an algorithm; examples are the procedures for multiplying or dividing two numbers or the instructions for using a computer (Leiserson, et al., 2001). Algorithms are only effective when a problem is very well-structured and there is no question about whether the algorithm is an appropriate choice for the problem. In that situation it pretty much guarantees a correct solution. They do not work well, however, with ill-structured problems, where they are ambiguities and questions about how to proceed or even about precisely what the problem is about. In those cases it is more effective to use heuristics, which are general strategies—“rules of thumb”, so to speak—that do not always work, but often do, or that provide at least partial solutions. When beginning research for a term paper, for example, a useful heuristic is to scan the library catalogue for titles that look relevant. There is no guarantee that this strategy will yield the books most needed for the paper, but the strategy works enough of the time to make it worth trying.

In the nine-dot problem, most students began in Scene #1 with a simple algorithm that can be stated like this: “Draw one line, then draw another, and another, and another”. Unfortunately this simple procedure did not produce a solution, so they had to find other strategies for a solution. Three alternatives are described in Scenes #3 (for Alicia) and 4 (for Willem and Rachel). Of these, Willem’s response resembled a heuristic the most: he knew from experience that a good general strategy that often worked for such problems was to suspect a deception or trick in how the problem was originally stated. So he set out to question what the teacher had meant by the word line, and came up with an acceptable solution as a result.

Common obstacles to solving problems

The example also illustrates two common problems that sometimes happen during problem solving. One of these is functional fixedness: a tendency to regard the functions of objects and ideas as fixed (German & Barrett, 2005). Over time, we get so used to one particular purpose for an object that we overlook other uses. We may think of a dictionary, for example, as necessarily something to verify spellings and definitions, but it also can function as a gift, a doorstop, or a footstool. For students working on the nine-dot matrix described in the last section, the notion of “drawing” a line was also initially fixed; they assumed it to be connecting dots but not extending lines beyond the dots. Functional fixedness sometimes is also called response set, the tendency for a person to frame or think about each problem in a series in the same way as the previous problem, even when doing so is not appropriate to later problems. In the example of the nine-dot matrix described above, students often tried one solution after another, but each solution was constrained by a set response not to extend any line beyond the matrix.

Functional fixedness and the response set are obstacles in problem representation, the way that a person understands and organizes information provided in a problem. If information is misunderstood or used inappropriately, then mistakes are likely—if indeed the problem can be solved at all. With the nine-dot matrix problem, for example, construing the instruction to draw four lines as meaning “draw four lines entirely within the matrix” means that the problem simply could not be solved. For another, consider this problem: “The number of water lilies on a lake doubles each day. Each water lily covers exactly one square foot. If it takes 100 days for the lilies to cover the lake exactly, how many days does it take for the lilies to cover exactly half of the lake?” If you think that the size of the lilies affects the solution to this problem, you have not represented the problem correctly. Information about lily size is not relevant to the solution, and only serves to distract from the truly crucial information, the fact that the lilies double their coverage each day. (The answer, incidentally, is that the lake is half covered in 99 days; can you think why?)

Strategies to assist problem solving

Just as there are cognitive obstacles to problem solving, there are also general strategies that help the process be successful, regardless of the specific content of a problem (Thagard, 2005). One helpful strategy is problem analysis—identifying the parts of the problem and working on each part separately. Analysis is especially useful when a problem is ill-structured. Consider this problem, for example: “Devise a plan to improve bicycle transportation in the city.” Solving this problem is easier if you identify its parts or component subproblems, such as (1) installing bicycle lanes on busy streets, (2) educating cyclists and motorists to ride safely, (3) fixing potholes on streets used by cyclists, and (4) revising traffic laws that interfere with cycling. Each separate subproblem is more manageable than the original, general problem. The solution of each subproblem contributes the solution of the whole, though of course is not equivalent to a whole solution.

Another helpful strategy is working backward from a final solution to the originally stated problem. This approach is especially helpful when a problem is well-structured but also has elements that are distracting or misleading when approached in a forward, normal direction. The water lily problem described above is a good example: starting with the day when all the lake is covered (Day 100), ask what day would it therefore be half covered (by the terms of the problem, it would have to be the day before, or Day 99). Working backward in this case encourages reframing the extra information in the problem (i. e. the size of each water lily) as merely distracting, not as crucial to a solution.

A third helpful strategy is analogical thinking—using knowledge or experiences with similar features or structures to help solve the problem at hand (Bassok, 2003). In devising a plan to improve bicycling in the city, for example, an analogy of cars with bicycles is helpful in thinking of solutions: improving conditions for both vehicles requires many of the same measures (improving the roadways, educating drivers). Even solving simpler, more basic problems is helped by considering analogies. A first grade student can partially decode unfamiliar printed words by analogy to words he or she has learned already. If the child cannot yet read the word screen, for example, he can note that part of this word looks similar to words he may already know, such as seen or green, and from this observation derive a clue about how to read the word screen. Teachers can assist this process, as you might expect, by suggesting reasonable, helpful analogies for students to consider.

Broad instructional strategies that stimulate complex thinking

Because the forms of thinking just described—critical thinking, creativity and problem solving—are broad and important educationally, it is not surprising that educators have identified strategies to encourage their development. Some of the possibilities are shown in Table 24 and group several instructional strategies along two dimensions: how much the strategy is student-centered and how much a strategy depends on group interaction. It should be emphasized that the two-way classification in Table 24 is not very precise, but it gives a useful framework for understanding the options available for planning and implementing instruction. The more important of the two dimensions in the table is the first one—the extent to which an instructional strategy is either directed by the teacher or initiated by students. We take a closer look at this dimension in the next part of this chapter, followed by discussion of group-oriented teaching strategies.

Table 24: Major instructional strategies grouped by level of teacher direction and student focus

As the name implies, teacher-directed instruction includes any strategies initiated and guided primarily by the teacher. A classic example is exposition or lecturing (simply telling or explaining important information to students) combined with assigning reading from texts. But teacher-directed instruction also includes strategies that involve more active response from students, such as encouraging students to elaborate on new knowledge or to explain how new information relates to prior knowledge. Whatever their form, teacher-directed instructional methods normally include the organizing of information on behalf of students, even if teachers also expect students to organize it further on their own. Sometimes, therefore, teacher-directed methods are thought of as transmitting knowledge from teacher to student as clearly and efficiently as possible, even if they also require mental work on the part of the student.

Lectures and readings

Lectures and readings are traditional staples of educators, particularly with older students (including university students). At their best, they pre-organize information so that (at least in theory) the student only has to remember what was said in the lecture or written in the text in order to begin understanding it (Exley & Dennick, 2004). Their limitation is the ambiguity of the responses they require: listening and reading are by nature quiet and stationary, and do not in themselves indicate whether a student is comprehending or even attending to the material. Educators sometimes complain that “students are too passive” during lectures or when reading. But physical quietness is intrinsic to these activities, not to the students who do them. A book just sits still, after all, unless a student makes an effort to read it, and a lecture may not be heard unless a student makes the effort to listen to it.

Advance organizers

In spite of these problems, there are strategies for making lectures and readings effective. A teacher can be especially careful about organizing information for students, and she can turn part of the mental work over to students themselves. An example of the first approach is the use of advance organizers—brief overviews or introductions to new material before the material itself is presented (Ausubel, 1978). Textbook authors (including ourselves) often try deliberately to insert periodic advance organizers to introduce new sections or chapters in the text. When used in a lecture, advance organizers are usually statements in the form of brief introductory remarks, though sometimes diagrams showing relationships among key ideas can also serve the same purpose (Robinson, et al., 2003). Whatever their form, advance organizers partially organize the material on behalf of the students, so that they know where to put it all, so to speak, as they learn them in more detail.

Recalling and relating prior knowledge

Another strategy for improving teacher-directed instruction is to encourage students to relate the new material to prior familiar knowledge. When one of us (Kelvin) first learned a foreign language (in his case French), for example, he often noticed similarities between French and English vocabulary. A French word for picture, for example, was image, spelled exactly as it is in English. The French word for splendid was splendide, spelled almost the same as in English, though not quite. Relating the French vocabulary to English vocabulary helped in learning and remembering the French.

As children and youth become more experienced in their academics, they tend to relate new information to previously learned information more frequently and automatically (Goodwin, 1999; Oakhill, Hartt, & Samols, 2005). But teachers can also facilitate students’ use of this strategy. When presenting new concepts or ideas, the teacher can relate them to previously learned ideas deliberately—essentially modeling a memory strategy that students learn to use for themselves. In a science class, for example, she can say, “This is another example of…, which we studied before”; in social studies she can say, “Remember what we found out last time about the growth of the railroads? We saw that…”

If students are relatively young or are struggling academically, it is especially important to remind them of their prior knowledge. Teachers can periodically ask questions like “What do you already know about this topic?” or “How will your new knowledge about this topic change what you know already?” Whatever the age of students, connecting new with prior knowledge is easier with help from someone more knowledgeable, such as the teacher. When learning algorithms for multiplication, for example, students may not at first see how multiplication is related to addition processes which they probably learned previously (Burns, 2001). But if a teacher takes time to explain the relationship and to give students time to explore it, then the new skill of multiplication may be learned more easily.

Elaborating information

Elaborating new information means asking questions about the new material, inferring ideas and relationships among the new concepts. Such strategies are closely related to the strategy of recalling prior knowledge as discussed above: elaboration enriches the new information and connects it to other knowledge. In this sense elaboration makes the new learning more meaningful and less arbitrary.

A teacher can help students use elaboration by modeling this behavior. The teacher can interrupt his or her explanation of an idea, for example, by asking how it relates to other ideas, or by speculating about where the new concept or idea may lead. He or she can also encourage students to do the same, and even give students questions to guide their thinking. When giving examples of a concept, for example, a teacher can hold back from offering all of the examples, and instead ask students to think of additional examples themselves. The same tactic can work with assigned readings; if the reading includes examples, the teacher can instruct students to find or make up additional examples of their own.

Organizing new information

There are many ways to organize new information that are especially well-suited to teacher-directed instruction. A common way is simply to ask students to outline information read in a text or heard in a lecture. Outlining works especially well when the information is already organized somewhat hierarchically into a series of main topics, each with supporting subtopics or subpoints. Outlining is basically a form of the more general strategy of taking notes, or writing down key ideas and terms from a reading or lecture. Research studies find that that the precise style or content of notes is less important that the quantity of notes taken: more detail is usually better than less (Ward & Tatsukawa, 2003). Written notes insure that a student thinks about the material not only while writing it down, but also when reading the notes later. These benefits are especially helpful when students are relatively inexperienced at school learning in general (as in the earlier grade levels), or relatively inexperienced about a specific topic or content in particular. Not surprisingly, such students may also need more guidance than usual about what and how to write notes. It can be helpful for the teacher to provide a note-taking guide, like the ones shown in Exhibit 11.

Notes on Science Experiment

Guide to Notes About Tale of Two Cities

Exhibit 13: Two note taking guides

In learning expository material, another helpful strategy—one that is more visually oriented—is to make concept maps, or diagrams of the connections among concepts or ideas. Exhibit 10 shows concept maps made by two individuals that graphically depict how a key idea, child development, relates to learning and education. One of the maps was drawn by a classroom teacher and the other by a university professor of psychology (Seifert, 1991). They suggest possible differences in how the two individuals think about children and their development. Not surprisingly, the teacher gave more prominence to practical concerns (for example, classroom learning and child abuse), and the professor gave more prominence to theoretical ones (for example, Erik Erikson and Piaget). The differences suggest that these two people may have something different in mind when they use the same term, child development. The differences have the potential to create misunderstandings between them (Seifert, 1999; Super & Harkness, 2003). By the same token, the two maps also suggest what each person might need to learn in order to achieve better understanding of the other person’s thinking and ideas.

Exhibit 14: Maps of personal definitions of “child development”

Mastery learning

This term refers to an instructional approach in which all students learn material to an identically high level, even if some students require more time than others to do so (Gentile, 2004). In mastery learning, the teacher directs learning, though sometimes only in the sense of finding, writing, and orchestrating specific modules or units for students to learn. In one typical mastery learning program, the teacher introduces a few new concepts or topics through a brief lecture or teacher-led demonstration. Then she gives an ungraded assignment or test immediately in order to assess how well students have learned the material, and which ones still need help. The students who have already learned the unit are given enrichment activities. Those needing more help are provided individual tutoring or additional self-guiding materials that clarify the initial content; they work until they have in fact mastered the content (hence the name mastery learning). At that point students take another test or do another assignment to show that they have in fact learned the material to the expected high standard. When the system is working well, all students end up with high scores or grades, although usually some take longer to do so than others.

As you might suspect, mastery learning poses two challenges. The first is ethical: is it really fair to give enrichment only to faster students and remediation only to slower students? This practice could deteriorate into continually providing the fast with an interesting education, while continually providing the slow only with boring, repetitious material. In using the approach, therefore, it is important to make all materials interesting, whether enrichment or remedial. It is also important to make sure that the basic learning goals of each unit are truly important—even crucial—for everyone to learn, so that even slower individuals spend their time well.

The other challenge of mastery learning is more practical: the approach makes strong demands for detailed, highly organized curriculum. If the approach is to work, the teacher must either locate such a curriculum, write one herself, or assemble a suitable mixture of published and self-authored materials. However the curriculum is created, the end result has to be a program filled with small units of study as well as ample enrichment and remedial materials. Sometimes providing these practical requirements can be challenging. But not always: some subjects (like mathematics) lend themselves to detailed, sequential organization especially well. In many cases, too, commercial publishers have produced curricula already organized for use in mastery learning programs (Fox, 2004).

Direct instruction

Although the term direct instruction is sometimes a synonym for teacher-directed instruction, more often it refers to a version of mastery learning that is highly scripted, meaning that it not only organizes the curriculum into small modules or units as described above, but also dictates how teachers should teach and sometimes even the words they should speak (Adams & Engelmann, 1996; Magliaro, Lockee, & Burton, 2005). Direct instruction programs are usually based on a mix of ideas from behaviorism and cognitive theories of learning. In keeping with behaviorism, the teacher is supposed to praise students immediately and explicitly when they give a correct answer. In keeping with cognitive theory, she is supposed to state learning objectives in advance of teaching them (providing a sort of mini-advance organizer), provide frequent reviews of materials, and check deliberately on how well students are learning. Direct instruction usually also introduces material in small, logical steps, and calls for plenty of time for students to practice.

Direct instruction programs share one of the challenges of other mastery learning approaches: because they hold all students to the same high standard of achievement, they must deal with differences in how long students require to reach the standard. But direct instruction has an additional challenge, in that they often rely on small-group interaction more heavily than other mastery learning programs, and use self-guiding materials less. This difference has the benefit that direct instruction works especially well with younger students (especially kindergarten through third grade), who may have limited skills at working alone for extended periods. The challenge is that reliance on small-group interaction can make it impractical to use direct instruction with an entire class or for an entire school day. In spite of these limits, however, research has found direct instruction to be very effective in teaching basic skills such as early reading and arithmetic (Adams & Engelmann, 1996).

Madeline Hunter’s effective teaching model

A number of direct instruction strategies have been combined by Madeline Hunter into a single, relatively comprehensive approach that she calls mastery teaching (not to be confused with the related term mastery learning) or the effective teaching model (M. Hunter, 1982; R. Hunter, 2004). Important features of the model are summarized in Table 25. As you can see, the features span all phases of contact with students—before, during, and after lessons.

Table 25: Madeline Hunter’s “Effective Teaching Model”

What happens even before a lesson begins? Like many forms of teacher-directed instruction, the effective teaching model requires curricula and learning goals that are tightly organized and divisible into small parts, ideas, or skills. In teaching about photosynthesis, for example, the teacher (or at least her curriculum) needs to identify the basic elements that contribute to this process, and how they relate to each other. With photosynthesis, the elements include the sun, plants, animals, chlorophyll, oxygen produced by plants and consumed by animals, and carbon dioxide that produced by animals and consumed by plants. The roles of these elements need to be identified and expressed at a level appropriate for the students. With advanced science students, oxygen, chlorophyll, and carbon dioxide may be expressed as part of complex chemical reactions; with first-grade students, though, they may be expressed simply as parts of a process akin to breathing or respiration.

Once this analysis of the curriculum has been done, the Hunter’s effective teaching model requires making the most of the lesson time by creating an anticipatory set, which is an activity that focuses or orients the attention of students to the upcoming content. Creating an anticipatory set may consist, for example, of posing one or more questions about students’ everyday knowledge or knowledge of prior lessons. In teaching about differences between fruits and vegetables, the teacher could start by asking: “If you are making a salad strictly of fruit, which of these would be OK to use: apple, tomato, cucumber, or orange?” As the lesson proceeds, information needs to be offered in short, logical pieces, using language as familiar as possible to the students. Examples should be plentiful and varied: if the purpose is to define and distinguish fruits and vegetables, for example, then features defining each group should be presented singularly or at most just a few at a time, with clear-cut examples presented of each feature. Sometimes models or analogies also help to explain examples. A teacher can say: “Think of a fruit as a sort of ‘decoration’ on the plant, because if you pick it, the plant will go on living.” But models can also mislead students if they are not used thoughtfully, since they may contain features that differ from the original concepts. In likening a fruit to a decoration, for example, students may overlook the essential role of fruit in plant reproduction, or think that lettuce qualifies as a fruit, since picking a few lettuce leaves does not usually kill a lettuce plant.

Throughout a lesson, the teacher repeatedly checks for understanding by asking questions that call for active thinking on the part of students. One way is to require all students to respond somehow, either with an actual choral response (speaking in unison together), another way with a non-verbal signal like raising hands to indicate answers to questions. In teaching about fruits and vegetables, for example, a teacher can ask, “Here’s a list of fruits and vegetables. As I point to each one, raise your hand if it’s a fruit, but not if it’s a vegetable.” Or she can ask: “Here’s a list of fruits and vegetables. Say together what each on is as I point to it; you say ‘fruit’ or ‘vegetable’, whichever applies.” Even though some students may hide their ignorance by letting more knowledgeable classmates do the responding, the general level or quality of response can still give a rough idea of how well students are understanding. These checks can be supplemented, of course, with questions addressed to individuals, or with questions to which individuals must respond briefly in writing. A teacher can ask everyone, “Give me an example of one fruit and one vegetable”, and then call on individuals to answer. She can also say: “I want everyone to make a list with two columns, one listing all the fruits you can think of and the other listing all the vegetables you can think of.”

As a lesson draws to a close, the teacher arranges for students to have further independent practice. The point of the practice is not to explore new material or ideas, but to consolidate or strengthen the recent learning. At the end of a lesson about long division, for example, the teacher can make a transition to independent practice by providing a set of additional problems similar to the ones she explained during the lesson. After working one or two with students, she can turn the rest of the task over to the students to practice on their own. But note that even though the practice is supposedly “independent”, students’ understanding still has be checked frequently. A long set of practice problems therefore needs to be broken up into small subsets of problems, and written or oral feedback offered periodically.

What are the limits of teacher-directed instruction?

Whatever the grade level, most subjects taught in schools have at least some features, skills, or topics that benefit from direct instruction. Even subjects usually considered “creative” can benefit from a direct approach at times: to draw, sing, or write a poem, for example, requires skills that may be easier to learn if presented sequentially in small units with frequent feedback from a teacher. Research supports the usefulness of teacher- directed instruction for a variety of educational contexts when it is designed well and implemented as intended (Rosenshine & Mesister,1995; Good & Brophy, 2004). Teachers themselves also tend to support the approach in principle (Demant & Yates, 2003).

But there are limits to its usefulness. Some are the practical ones are pointed out above. Teacher-directed instruction, whatever the form, requires well-organized units of instruction in advance of when students are to learn. Such units may not always be available, and it may not be realistic to expect busy teachers to devise their own. Other limits of direct instruction have more to do with the very nature of learning. Some critics argue that organizing material on behalf of the students encourages students to be passive—an ironic and undesirable result if true (Kohn, 2000, 2006). According to this criticism, the mere fact that a curriculum or unit of study is constructed by a teacher (or other authority) makes some students think that they should not bother seeking information actively on their own, but wait for it to arrive of its own accord. In support of this argument, critics point to the fact that direct instruction approaches sometimes contradict their own premises by requiring students to do a bit of cognitive organizational work of their own. This happens, for example, when a mastery learning program provides enrichment material to faster students to work on independently; in that case the teacher may be involved in the enrichment activities only minimally.

Criticisms like these have led to additional instructional approaches that rely more fully on students to seek and organize their own learning. In the next section we discuss some of these options. As you will see, student-centered models of learning do solve certain problems of teacher-directed instruction, but they also have problems of their own.

Student-centered models of learning

Student-centered models of learning shift some of the responsibility for directing and organizing learning from the teacher to the student. Being student-centered does not mean, however, that a teacher gives up organizational and leadership responsibilities completely. It only means a relative shift in the teacher’s role, toward one with more emphasis on guiding students’ self-chosen directions. As we explained earlier in this chapter, teacher-directed strategies do not take over responsibility for students’ learning completely; no matter how much a teacher structures or directs learning, the students still have responsibility for working and expending effort to comprehend new material. By the same token, student-centered models of learning do not mean handing over all organizational work of instruction to students. The teacher is still the most knowledgeable member of the class, and still has both the opportunity and the responsibility to guide learning in directions that are productive.

As you might suspect, therefore, teacher-directed and student-centered approaches to instruction may overlap in practice. You can see the overlap clearly, for example, in two instructional strategies commonly thought of as student-centered, independent study and self-reflection. In independent study, as the name implies, a student works alone a good deal of the time, consulting with a teacher only occasionally. Independent study may be student-centered in the sense that the student may be learning a topic or skill—an exotic foreign language, for example—that is personally interesting. But the opposite may also be true: the student may be learning a topic or skill that a teacher or an official school curriculum has directed the student to learn—a basic subject for which the student is missing a credit, for example. Either way, though, the student will probably need guidance, support, and help from a teacher. In this sense even independent study always contain elements of teacher-direction.

Similarly, self-reflection refers to thinking about beliefs and experiences in order to clarify their personal meaning and importance. In school it can be practiced in a number of ways: for example by keeping diaries or logs of learning or reading, or by retelling stories of important experiences or incidents in a student’s life, or by creating concept maps like the ones described earlier in this chapter. Whatever form it takes, self-reflection by definition happens inside a single student’s mind, and in this sense is always directed by the student. Yet most research on self-reflection finds that self-reflection only works well when it involves and generates responses and interaction with other students or with a teacher (Seifert, 1999; Kuit, Reay, & Freeman, 2001). To be fully self-reflective, students need to have access to more than their existing base of knowledge and ideas—more than what they know already. In one study about students’ self-reflections of cultural and racial prejudices (Gay & Kirkland, 2003), for example, the researchers found that students tended to reflect on these problems in relatively shallow ways if they worked on their own. It was not particularly effective to write about prejudice in a journal that no one read except themselves, or to describe beliefs in a class discussion in which neither the teacher nor classmates commented or challenged the beliefs. Much more effective in both cases was for the teacher to respond thoughtfully to students’ reflective comments. In this sense the use of self-reflection, like independent study, required elements of teacher- direction to be successful.

How might a teacher emphasize students’ responsibility for directing and organizing their own learning? The alternatives are numerous, as they are for teacher-directed strategies, so we can only sample some of them here. We concentrate on ones that are relatively well known and used most widely, and especially on two: inquiry learning and cooperative learning.

Inquiry learning

Inquiry learning stands the usual advice about expository (lecture-style) teaching on its head: instead of presenting well-organized knowledge to students, the teacher (or sometimes fellow students) pose thoughtful questions intended to stimulate discussion and investigation by students. The approach has been described, used, and discussed by educators literally for decades, though sometimes under other names, including inquiry method (Postman & Weingartner, 1969), discovery learning (Bruner, 1960/2006), or progressive education (Dewey, 1933; Martin, 2003). For convenience, we will stay with the term inquiry learning.

The questions that begin a cycle of inquiry learning may be posed either by the teacher or by students themselves. Their content depends not only on the general subject area being studied, but also on the interests which students themselves have expressed. In elementary-level science, for example, a question might be “Why do leaves fall off trees when winter comes?” In high school social studies classes, it might be “Why do nations get into conflict?” The teacher avoids answering such questions directly, even if asked to do so. Instead she encourages students to investigate the questions themselves, for example by elaborating on students’ ideas and by asking further questions based on students’ initial comments. Since students’ comments can not be predicted precisely, the approach is by nature flexible. The initial questioning helps students to create and clarify questions which they consider worthy of further investigation. Discussing questions about leaves falling off trees, for example, can prompt students to observe trees in the autumn or to locate books and references that discuss or explain the biology of tress and leaves.

But inquiry is not limited to particular grade levels or topics. If initial questions in a high school social studies class have been about why nations get into conflict, for example, the resulting discussions can lead to investigating the history of past wars and the history of peace-keeping efforts around the world. Whether the topic is high school social studies or elementary school biology, the specific direction of investigations is influenced heavily by students, but with assistance from the teacher to insure that the students’ initiatives are productive. When all goes well, the inquiry and resulting investigations benefit students in two ways. The first is that students (perhaps obviously) learn new knowledge from their investigations. The second is that students practice a constructive, motivating way of learning, one applicable to a variety of problems and tasks, both in school and out.

Cooperative learning

Even though inquiry-oriented discussion and investigation benefits when it involves the teacher, it can also be useful for students to work together somewhat independently, relying on a teacher’s guidance only indirectly. Working with peers is a major feature of cooperative learning (sometimes also called collaborative learning). In this approach, students work on a task in groups and often are rewarded either partially or completely for the success of the group as a whole. Aspects of cooperative learning have been part of education for a long time; some form of cooperation has always been necessary to participate on school sports teams, for example, or to produce a student-run school newspaper. What is a bit newer is using cooperative or collaborative activities systematically to facilitate the learning of a range of educational goals central to the academic curriculum (Prince, 2004).

Even though teachers usually value cooperation in students, circumstances at school can sometimes reduce students’ incentives to show it. The traditional practice of assessing students individually, for example, can set the stage for competition over grades, and cultural and other forms of diversity can sometimes inhibit individuals from helping each other spontaneously. Strategies exist, however, for reducing such barriers so that students truly benefit from each other’s presence, and are more likely to feel like sharing their skills and knowledge. Here, for example, are several key features that make cooperative learning work well (Johnson & Johnson, 1998; Smith, et al., 2005):

• • • Students need time and a place to talk and work together. This may sound obvious, but it can be overlooked if time in class becomes crowded with other tasks and activities, or with interruptions related to school (like assemblies) but not to the classroom. It is never enough simply to tell students to work together, only to leave them wondering how or when they are to do so.

• • • Students need skills at working together. As an adult, you may feel relatively able to work with a variety of partners on a group task. The same assumption cannot be made, however, about younger individuals, whether teenagers or children. Some students may get along with a variety of partners, but others may not. Many will benefit from advice and coaching about how to focus on the tasks at hand, rather than on the personalities of their partners.

• • • Assessment of activities should hold both the group and the individuals accountable for success. If a final mark for a project goes only to the group as a whole, then freeloading is possible: some members may not do their share of the work and may be rewarded more than they deserve. Others may be rewarded less than they deserve. If, on the other hand, a final grade for a group project goes only to each member’s individual contribution to a group project, then overspecialization can occur: individuals have no real incentive to work together, and cooperative may deteriorate into a set of smaller individual projects (Slavin, 1994).

• • • Students need to believe in the value and necessity of cooperation. Collaboration will not occur if students privately assume that their partners have little to contribute to their personal success. Social prejudices from the wider society—like racial bias or gender sexism, for example—can creep into the operations of cooperative groups, causing some members to be ignored unfairly while others are overvalued. Teachers can help reduce these problems in two ways: first by pointing out and explaining that a diversity of talents is necessary for success on a group project, and second by pointing out to the group how undervalued individuals are contributing to the overall project (Cohen, Brody, & Sapon-Shevin, 2004).

As these comments imply, cooperative learning does not happen automatically, and requires monitoring and support by the teacher. Some activities may not lend themselves to cooperative work, particularly if every member of the group is doing essentially the same task. Giving everyone in a group the same set of arithmetic problems to work on collaboratively, for example, is a formula for cooperative failure: either the most skilled students do the work for others (freeloading) or else members simply divide up the problems among themselves in order to reduce their overall work (overspecialization). A better choice for a cooperative task is one that clearly requires a diversity of skills, what some educators call a rich group work task (Cohen, Brody, & Sapon-Shevin, 2004). Preparing a presentation about medieval castles, for example, might require (a) writing skill to create a report, (b) dramatic skill to put on a skit and (c) artistic talent to create a poster. Although a few students may have all of these skills, more are likely to have only one, and they are therefore likely to need and want their fellow group members’ participation.

Examples of cooperative and collaborative learning

Although this description may make the requirements for cooperative learning sound somewhat precise, there are actually a variety of ways to implement it in practice. Error: Reference source not found summarizes several of them. As you can see, the strategies vary in the number of how many students they involve, the prior organization or planning provided by the teacher, and the amount of class time they normally require.

Table 26: Strategies for encouraging cooperative learning

Instructional strategies: an abundance of choices

Looking broadly at this chapter, you can see that choices among instructional strategies are numerous indeed, and that deciding among them depends on the forms of thinking that you want to encourage, the extent to which ideas or skills need to be organized by you to be understood by students, and the extent to which students need to take responsibility for directing their own learning. Although you may have personal preferences among possible instructional strategies, the choice will also be guided by the uniqueness of each situation of teaching—with its particular students, grade-level, content, and purposes. If you need to develop students’ problem solving skills, for example, there are strategies that are especially well suited for this purpose; we described some (see, “Problem solving strategies” in this chapter). If you need to organize complex information so that students do not become confused by it, there are effective ways of doing so. If you want the students to take as much initiative as possible in organizing their own learning, this too can be done.

Yet having this knowledge is still not enough to teach well. What is still needed are ideas or principles for deciding what to teach. In this chapter we have still not addressed an obvious question: How do I find or devise goals for my teaching and for my students’ learning? And assuming that I can determine the goals, where can I find resources that help students to meet them?

Chapter summary

Teaching involves numerous instructional strategies, which are decisions and actions designed to facilitate learning. The choice of strategies depends partly on the forms of thinking intended for students—whether the goal is for students to think critically, for example, or to think creatively, or to solve problems. A fundamental decision in choosing instructional strategies is how much to emphasize teacher-directed instruction, as compared to student- centered models of learning. Teacher-directed strategies of instruction include lectures and readings (expository teaching), mastery learning, scripted or direct instruction, and complex teacher-directed approaches such as Madeline Hunter’s effective teaching model. Student-centered models of learning include independent study, student self-reflection, inquiry learning, and various forms of cooperative or collaborative learning. Although for some students, curriculum content and learning goals may lend themselves toward one particular type of instruction, teaching is often a matter of combining different strategies appropriately and creatively.

On the Internet

<www.glossary.plasmalink.com/glossary.html> This web page lists over 900 instructional strategies— about ten times as many as in this chapter! The strategies are arranged alphabetically and range from simple to complex. For many strategies there are links to other web pages with more complete explanations and advice for use. This is a good page if you have heard of a strategy but want to find out its definition quickly.

<www.olc.spsd.sk.ca/DE/PD/instr/alpha.html> Like the web page above, this one also describes instructional strategies. It includes fewer (about 200), but they are discussed in more detail and organized according to major categories or types of strategies—a good feature if you have a general idea of what sort of strategy you are looking for, but are not sure of precisely which one.

Key terms

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