A r c h i v e d  I n f o r m a t i o n

Teaching--A Complex Set of Tasks

Though Gaea Leinhardt, who spearheads these classroom-based studies, does not look directly at the cognitive connection between concepts and procedures, much in her work relates to that connection. In examining the cognitively complex processes of teaching, for example, she has found that expert teachers build into their lessons several components that enable their students to build accurate mental representations of mathematical concepts and behaviors. In effect, these teachers find ways to guide learners, at least implicitly, through the layers of reasoning that Resnick has identified. They do so by a "layering" process of their own. Leinhardt explains, "a lesson given by an effective teacher who has been teaching for many years essentially contains layers of accumulated knowledge about the topic and how to teach it."

Each year, in each class, a good teacher adds increasingly rich strata of effective methods, terminologies, examples, and explanations to her teaching of a topic. The cognitive activity involved is at least as complex as that of a learner, but its goal is to build comprehension in other minds than the teacher's own. To accomplish that goal, an expert teacher develops strong metacognitive--skills those that allow her to monitor each lesson's progress and to make adjustments and repairs as necessary. She also learns how to manage a classroom full of children with minimum disruption, how to keep her instruction logically clear and easy to follow, how to remind students of old material and orient them to new, and how to signal transitions from one part of a lesson to another.

Leinhardt groups these skills and devices of teachers into categories, each of which represents a subcomponent of a lesson. Routines, for example, are behavioral steps that move lessons along without significant interruption. They become familiar to students early on, so that once a teacher signals a routine, it proceeds almost automatically. There are four kinds of routines: management routines take care of the physical movement of people or materials, permitting students to stand in line, pass out paper, share manipulatives, or hand in homework efficiently. Support routines are orchestrated for instructional purposes and consist of the rules by which students approach the blackboard, assemble into groups for discussion or problem solving, exchange papers for correction, or turn to appropriate sections of their texts. Exchange routines are the rules of verbal communication, especially in the service of the topic being discussed. According to Leinhardt, exchange routines not only let students know the usual procedures of communication but also signal changes in those procedures. An expert teacher, for example, will prepare her students for something out of the ordinary by saying, "Now we are going to do something a little different," or "We usually do things this way, but now I want you to try a new method." Finally, learning routines are those that keep students attention on the topic. Leinhardt believes these need to be studied more systematically, but she characterizes them as questions and cues that elicit appropriate subject-matter insights from students. For example, while demonstrating a procedure at the blackboard, a teacher might ask students to describe out loud the steps in her calculation or to identify and explain her deliberate errors.

Classroom routines are supported by instructional scripts that both cover the topical material to be taught and move the management and support routines smoothly along. Teachers constantly revise and update their scripts on topics that they cover frequently. They also form, for each segment of a lesson, content-based agendas that reflect their instructional goals and plans. Agendas are usually not written down but exist in the form of mental notes about which part of a topic the students need to learn, what might cause them problems, how the teacher can assess their comprehension, and what actions she will combine to get the material across. The final teaching component that Leinhardt has studied, and in some ways the most important, is explanations, which she has observed can either be directly expository or can indirectly encourage students own discovery and insight.

Leinhardt has examined lessons, routines, scripts, agendas, and explanations in both mathematics and history classrooms, comparing expert and novice teachers in order to reveal, through differences in their approaches, just what it is about the experts instruction that works. Leinhardt characterizes the overarching goal of her research as an understanding of "the way in which knowledge can be acquired and built under the guidance of a teacher."

Leinhardt usually elects to study a topic that represents a pivotal moment in learning. (Fractions, ratios, and proportionality, for example, mark a point at which students move into an important expansion of the number system and can either gain considerable understanding or become deeply bewildered.) While researching the topic and learning its instructional requirements and pitfalls, Leinhardt also identifies expert teachers who conduct classes in the relevant subject area. Experts are defined as teachers who have had success with students for many years and who are also regarded by their administrators as extraordinarily effective. Novices are usually preservice teachers.

Once Leinhardt understands the topic and has recruited teachers--which may take a year--she begins recording each teacher's instruction, gathering from five to 90 days worth of lessons on videotape and then coding, analyzing, and interpreting their content. In addition, her research team conducts in-depth interviews with the teachers about their professional development, career choices, and subject-matter knowledge.

The significance of Leinhardt's studies and methodology are especially well illustrated in her comparisons of expert and novice teachers' conduct of lessons in fractions. For this work, she studied generalists (teachers who teach all elementary school subjects) in public school classrooms with 25 to 30 students each, an environment Leinhardt regards as "the acid test" for assessing instructional methods and performance. She found, in general, that the expert teachers were far more skillful than novices at all aspects of their job, from classroom management to the construction of clear and thorough explanations. As Leinhardt writes in a paper entitled "Expertise in Instructional Lessons: An Example from Fractions," expert teachers

keep lessons flowing and are aware of and in tune with what their students are seatwork, learning. The teachers manage homework, seatwork, demonstrations, games, discovery projects, discussions, and drill with fluidity and consistency. Time is always treated as a valuable resource and is not squandered in getting set up and in Expert making multiple unintended false starts. . . . Expert teachers also teach very well. They give detailed, complete explanations and demonstrations, and provide rich mathematical experiences for their students.

For example, just the comparison of teachers' agendas for lessons in fractions revealed the experts' deeper understanding of what their students needed from instruction and how the teacher might provide it. Researchers interviewed both experts and novices about lessons they were planning to teach and found that, although their responses were of generally equal length, the content differed dramatically. Experts specified the instructional moves they intended to make and the actions they would require from the students. They explicitly designed their lessons to flow smoothly and logically, and they identified points at which they planned to check the lesson's effectiveness and, if necessary, adjust its logical flow.

Novice teachers' agendas were never so specific or so detailed. An expert might say precisely how she planned to introduce a new topic (such as adding fractions), relate it to a recent one (such as equivalent fractions), make use of manipulatives or drawings that could clarify crucial concepts, and assess her students' comprehension by having them work on problems at the blackboard. A novice, on the other hand, might say of a similar lesson, "I'm going to go over yesterday's homework on the board. It would probably be good for this class. I don't know how many I am going to go over. There are 36 problems. But I m going to see how it goes. If they're all getting them very quickly, then we'll move on."

This novice, like others studied by Leinhardt's group, expressed no strong mental representation of her own lesson or the topic she intended to cover. As Leinhardt points out in her paper, "The novice had taught a lesson on reducing fractions which had failed, had retaught the same lesson, and had assigned homework. Her entire set of activities for the day was slated for going over the homework. Even though she stated she might go on, she had no idea of what [she] would go on to."

What is implied by the expert teachers' richer, more flexible, and more varied approaches and lesson components is that these teachers have succeeded in forging a solid link between their own conceptual and procedural forms of knowledge. They not only have a deeper understanding than novices about the subject matter and the cognitive problems it can present but a far stronger ability to anticipate and surmount those problems, monitor their own effectiveness, and orchestrate the myriad small activities that surround classroom teaching and learning. In fact, as Leinhardt has documented in many classrooms, these expert teachers are often led by their conceptual understanding of their subject and their craft to incorporate many of the techniques Resnick's work identifies as fruitful: discussion, discovery, collaboration, manipulatives, and guidance toward a grasp of concepts. Leinhardt, pointing out that the experts she studies nevertheless do teach in very traditional settings, recommends that future research investigate expert teaching as it takes place in more inquiry-oriented classrooms; that it analyze when and how to teach procedural as well as conceptual knowledge; and that it study the effects of traditional and discovery-based teaching styles on different topics in math and other subjects. She urges, "Let us find out how much problem solving and inquiry we can get out of straight didactic teaching, how much computational fluidity and accuracy out of an inquiry approach, and then figure out when to use which kinds of approaches."

Expertise in teaching, regardless of the setting in which it occurs, is indispensable to any efforts to raise educational standards and expectations in this country. Though some highly exceptional teachers are always around to ignite students' curiosity and determination, widespread education reform cannot depend on their genius alone. If only a relatively few teachers teach well, only their students and a few gifted others will learn well. In order that excellent teaching--and therefore excellent learning--might take place in more classrooms, Leinhardt has also devoted much of her time to questions of teacher training, assessment, and professionalism. For example, she has collaborated in the Carnegie Forum's Teacher Assessment Project at Stanford University, an effort to develop improved principles and procedures for measuring the skills and effectiveness of practicing teachers. This work gave rise to a project with the Connecticut State Department of Education on the design and development of a scoring system for a semi-structured interview to be used as a performance-based assessment of teachers seeking licenses in that state. Leinhardt has also been a key participant in a collaborative project at NRCSL that has brought researchers and practitioners together to build teacher professionalism and leadership in mathematics instruction.
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[What Do Children Know About Numbers? What Do They Need To Be Taught?]

[Collaborative Projects Foster Teacher Professionalism]