Lesson Planning in the United States

In the United States (as, no doubt, throughout the world), teachers must write daily lesson plans that spell out in detail what they intend to do. Generally, they are taught how to do this early in their preparation, often before they have encountered any real students, and they engage in this activity throughout their training. Towards the end of their preparation during student teaching -- where, perhaps for the first time, they meet the intended beneficiaries of the plans they have been perfecting -- they are required to write lesson plans before each class they teach and to discuss their plans in conferences with supervisors. New teachers are expected to continue the practice of preparing written plans for each class every day, and in many schools all teachers are required to turn in weekly sets of lesson plans -- often in a prescribed format -- to a principal or department chair. Although this might be a pro forma exercise for experienced teachers, it can have important consequences; teachers frequently are evaluated on the quality, or lack of quality, and on the appropriateness of the plans which they submit.

Given that lesson planning is so ubiquitous and so highly valued, it is surprising that there is no general agreement about what a lesson plan should look like. Most authors suggest a variety of alternative formats, perhaps indicating a preferred format, and then suggest that each teacher develop a personal system which works best for that teacher. Other authors opt for a single format that they justify by common sense arguments bolstered by references to empirical research. Occasionally, one of these formats, has gained widespread acceptance and, partly as a consequence, widespread criticism. In the ensuing controversies, no single style for writing lesson plans has won the day.

Nevertheless, most descriptions of what constitutes a suitable lesson plan contain the same basic elements -- aims, content, procedures, and evaluation -- albeit with different emphases and in different orders. Teachers who are accustomed to one format can easily adjust when they move to another school or a different part of the country where a different format is demanded. Based on a survey of commonly used texts and school district guidelines (e.g., Clark and Starr, 1991; Johnson and Johnson, 1991; Kim and Kellough, 1991; or Callahan et al, 1992), a generic format which would be generally acceptable is outlined in Figure 1. Some formats require more detail, some require less, and some prescribe particular teaching methods. Still a teacher using this outline would produce a plan which most supervisors would recognize as being in an acceptable form, though there might well be differences about the substance of the plan.

This outline should serve to give the reader the flavor of what is generally expected. Keep in mind that it is intended to characterize what teachers are told to write; i.e., what is in the texts that they are expected to read, what the prescriptive educational literature suggests, and therefore what school administrators require or will accept. It does not reveal what teachers actually do -- how they plan, how they think, or how they make decisions. Research on these issues (e.g., Peterson et al, 1978; Clark and Peterson, 1986; Calderhead, 1984, 1987, 1988; Lowyck, 1989; or Day et al, 1993) has had remarkably little impact on what teacher preparation programs and schools expect teachers to put in their lesson plans.
 
 

Even a cursory look at sample lesson plans would show the heavy influence of technical and linear descriptions of the planning process, a perspective that has dominated the literature on lesson planning for many years. (See Taba, 1956; Popham and Baker, 1970; or Gagné et al, 1988.) Invariably, the recommended starting point for constructing a daily lesson plan on any topic is a list of objectives that identify the aim, or aims, of the lesson. The rationale to support this starting point seems quite straightforward: without a clear purpose, how can one begin to make a coherent plan? Consequently, prospective teachers spend a great deal of time learning to formulate objectives and to state them in terms of student outcomes. Of course, individual lessons are always part of some larger instructional context -- a unit, a chapter in the text, or a yearís work -- so it is expected that objectives eventually will be connected to this larger context; but establishing this connection is often a secondary concern. Whether objectives are to be written in behavioral terms, as advocated by Mager (1984), or in some less specific format, most books and teacher preparation courses devote considerable attention -- up to twenty percent of the space or time allotted to planning -- on the selection and statement of objectives for daily lessons.

More time and space -- in most cases the majority -- are devoted to a consideration of the appropriate procedures for organizing a lesson. The most prescriptive approaches present a single pattern of instruction that, if used daily and in combination with other weekly or monthly routines, is supposed to result in increased student performance. The procedures proposed are usually justified by plausible arguments or, sometimes, on the basis of empirical research done in classrooms; and they are intended to be used to organize instruction in a wide range of circumstances, even when the research was necessarily done in a more limited context. The seven-step lessons espoused by Madeline Hunter (Hunter, 1982; 1991) or the Missouri Mathematics Program (Good et al, 1983) are examples of this approach. A second, less prescriptive approach admits that there is no single best procedure for organizing a lesson and therefore focuses on general skills of teaching or classroom management (e.g., motivating, questioning, explaining, listening, reviewing, monitoring seatwork, correcting inappropriate behavior, ... ) which various methods require. Here the emphasis is on what the teacher needs to do to organize, monitor, and control the classroom environment, and lesson plans are expected to reflect this concern. A third approach presents a wide range of appropriate methods. These authors try to identify generic methods (e.g., recitation, lecture, small or large group discussion, role-playing, independent projects, ...) to be used across subject areas and/or age levels. The main features as well as the strengths and weaknesses of each method are discussed in detail, and teachers are encouraged to develop a repertoire of methods to call upon. Regardless of which approach is adopted, a conspicuous feature of many proposals is the extent to which they have been influenced by psychological theories and selected results from experimental research done in laboratory settings. (See, for example, Gagné et al, 1988; or Borich, 1988.)

Psychological theories also influence the discussions of evaluation, a third common element in descriptions of the planning process. Most authors distinguish two aspects of evaluation, assessment of what students have learned and reflection on the progress of the lesson. Assessment of student learning is generally driven by the previously established objectives for the lesson; rarely does an author urge teachers to attend carefully to what actually has been learned some of which -- for better or worse -- was not intended. It is assumed that clearly specified objectives are the best measure for determining what students actually have learned. So tools for assessment (such as planned observations of student behavior, rating scales, checklists, criterion referenced tests, ...) become a major focus of attention. Techniques and criteria for systematic reflection and self-appraisal usually receive considerably less attention.

With all this attention to objectives, to methods, to teaching skills, to classroom management, and to techniques for assessment and evaluation the content -- that which is to be taught -- gets pushed to the background. It is treated as something fixed whose main features are generally known. The teacher must master this content, make selections from it, and then transmit it to students. The following quote constitutes the entire discussion about the subject matter in a chapter on lesson planning in a frequently used textbook on teaching methods.

It might be assumed that once teachers have learned techniques for transmitting knowledge they are then in a position to choose techniques appropriate for particular subject matters and circumstances. However, most authors are silent about how to adapt general planning principles to different subject matters or -- equally important -- to various understandings of particular subject matters. They prefer to keep the treatment general, in part because the texts so far discussed are intended to be suitable for -- and salable to -- a general audience. This silence has an important consequence: teachers frequently do not see the full range of possibilities in the subjects they teach. The content of many lessons -- even the sample lessons in the texts for teachers -- seems to be nothing more than what is represented in the textbook. There is seldom even a hint that a chemistry lesson, for example, might be an inquiry into particular sorts of phenomena, or a way of gaining partial control over the physical world, or the systematic use of particular methods, or perhaps the story of the inquiries of particular chemists. Teachers need a wide range of views of what they teach in order to have alternatives from which to choose when developing their lesson plans.

Varieties of Subject Matter

Unfortunately, most teachers in the United States have not been exposed to the alternatives available from more dynamic conceptions of subject matter. Because their prior subject matter education has not prepared them adequately, they have a limited view of the meanings and methods of the subjects they teach. Nor does their pedagogical training help. Though it includes increasing amount of subject matter preparation (Ball and McDiarmid, 1990), this preparation typically separated from preparation focused on methods of teaching, thereby creating a disjunction which is difficult to bridge. In the United States, the dominant paradigm of teacher education has created a curious separation of content from pedagogy. (Doyle, 1993)

If one looks at the more general curriculum literature in the United States -- instead of the prescriptive material written for prospective teachers -- one can find occasions when an analysis of the subject to be taught plays a significant rôle. The attempts at national curriculum reform sponsored by the National Science Foundation 30 to 40 years ago provide an interesting case in point. Spurred on by the sense that students in the United States were ill prepared in mathematics and science, university professors of biology, physics, chemistry, and mathematics became involved with others in a variety of curriculum development projects. Central to this movement was the idea that curricula needed to be reconstructed so that they more adequately represented the structure of the discipline being taught. The importance of teaching the structure of a discipline, as opposed to teaching its specific applications, was articulated by Jerome Bruner in his influential report on the 1959 Woods Hole Conference (Bruner, 1960). Subsequently, it was generally agreed -- at least by all of the nationally organized curriculum projects -- that school subjects ought to teach "the structure" of an academic discipline.

Although there was agreement on the importance of structure, there was no common understanding of what was meant by the term. It was assumed that the mathematicians, biologists, physicists, historians and other academics who were involved in curricular reconstruction would know the structure of their own disciplines. Since most groups quickly came to their own understanding of what "the" structure was, this assumption was seldom questioned. Furthermore, once the central concepts and principles for organizing instruction were chosen, they typically were assumed to be the only reasonable choices that could have been made. It was only later, after the curriculum materials were developed and in use, that scientists and mathematicians began to question these assumptions. René Thom, a highly regarded mathematician, argued that the new curricula did not pay sufficient attention to the logic of invention and discovery and suggested that more attention be given to the development of mathematical intuition and informal methods of reasoning. Morris Kline, another mathematician, argued that the new curricula did not fully represent the mathematical tradition, calling their dedication to rigor "rigor-mortis." (Kline, 1974) Both authors questioned the prevailing notions of what constituted "the" structure of mathematics.

One voice expressed similar concerns early on, not just about mathematics but about the assumption that every discipline has just one structure that is best known by scholars working in the field. Joseph Schwab, a scientist with a systematic knowledge of philosophical and educational matters, developed his ideas in several papers written in the early 60ís (Schwab, 1962; 1964b; 1964c; 1978). At a symposium convened to analyze the concept of knowledge structure and its relevance for education, he laid out the arguments for his three major points and for their educational significance. (Schwab, 1964a) First, the organization of knowledge into disciplines -- e.g., into geography, history, physics, biology, and so on -- is not a settled matter. Not only are there fuzzy boundaries, we can use a number of different criteria to distinguish disciplines. Moreover, once disciplines have been identified and have been used to determine school subjects, we can arrange them into different hierarchies and therefore into different pedagogical relationships. Second, a discipline will inevitably have a number of different substantive structures. Each structure bounds and analyzes the subject in a different way by determining the questions to be asked, the data to be sought, the experiments or other activities to be performed, and the way the data obtained are to be interpreted. Third, each discipline will have more than one syntactic structure -- i.e., more than one "pathway by which it moves from its raw data to its conclusions" (1964a: 11) In any field, scholars have differing views on how knowledge should be verified, on the proper rôle of discovery and of proof, on the way the quality of data should be determined, and, in general, on the appropriate canons of evidence.

A central theme of this body of work is that curriculum developers are constantly confronted with choices arising from the inevitable complexity of any subject matter. This complexity generates conflicting meanings, alternative methods, fuzzy boundaries, and thus complementary possibilities for teaching. Schwab recognized that subject matter as a curricular commonplace was a neglected field of study having been "taken as familiar, fixed, and at hand when wanted." (1964a: 4) He sought to convince us that it is a variable term and, therefore, that detailed analyses of particular subject matters are curricular resources in ways not typically imagined in schools. To be convincing, Schwab had to furnish examples, not only of analyses of particular subject matters but also of their curricular implications. Some of these were provided through his teaching and deliberation with others on teaching at the University of Chicago. (See Westbury and Wilkof, 1978). Others were published as papers (1958; 1960), and a book, College Curriculum and Student Protest (1969a). As chairman of the teacher preparation committee of BSCS (Biological Sciences Study Committee), Schwab had the opportunity to influence the development of three different versions of a biology curriculum, each based on a different set of principles yet each in some sense covering the same content. A number of his students wrote dissertations (Herron, 1970; Siegel, 1975; Aron, 1975; Fox, 1975) which provide additional examples.

In all of this work, Schwab and his students used a topical method of analysis which provided tools for determining the range of choices available in particular subject matters. This mode of reasoning -- part of a tradition whose roots lie in the work of Bacon and Aristotle -- looks for commonplaces (topoi) which locate those aspects of a subject which require attention. By themselves, they are empty categories; but when these categories are filled by the details derived through the analysis of the subject under consideration, they generate a diversity of viewpoints on the subject. "An adequate set of commonplaces, then, provides a map on which each member of a plurality can be located relative to its fellow members. It not only permits [us to know] some part of the whole, it also enables [us] to know -- to some degree, at any rate -- what part of the whole [we] will see." (Schwab, 1971: 513) Schwab frequently made productive use of the set of commonplaces proposed by Aristotle often referred to as the four causes (namely the original, formal, efficient, and final causes). For example, he used these terms to discuss the structure of disciplines (1978), to distinguish the theoretical from the practical (1969b), and to delineate different approaches to science teaching (1974). In the same vein, the same commonplaces have been used to describe different approaches to mathematics and to computer science (Pereira, 1990). In the next section, we will use the same set of commonplaces to develop a framework for analyzing mathematics lessons.

Content as a Starting Point for Lesson Planning

Though in his published writings Schwab never addressed the micro-level of daily lessons, he was particularly concerned with this issue. In all of his work, Schwab was driven "inward to this classroom, to a careful analysis of the students he had this semester, and always to a concern for the here-and-now of the next class, in this course, in this program." (Westbury and Wilkof, 1978; 35) In seminars for prospective teachers at the University of Chicago, he discussed at length the possibilities inherent in various subject matters and worked with students to develop classroom applications developed from the diversities that were uncovered. He did similar work in a series of seminars for teacher educators at Michigan State University in order to demonstrate the uses of diverse understandings of subject matter.

If not overcome, static conceptions of subject matter are a severe handicap for teachers. The central curricular issue always involves bringing a subject matter and a learner together. When we see the meanings and methods of the subject matter as static -- embalmed, as it were, between the covers of the textbook -- our options for promoting this interaction are severely limited. If, for example, we say that children should learn literature, we must ask ourselves which set of literary concepts and principles we mean and in what senses exposure to this version of literature is educational. Among literary critics, there is some variety of views as to what constitutes a reading of a text; and these different views implicate contrasting sets of reading skills, sensitivities, and strategies for teaching. As a result, educators have choices about the educational goods that could be achieved. Such choices are obscured when literature, or any subject matter, has imposed upon it a fixed image of its shape and educational possibilities.

The typical American approach to lesson planning does not help teachers to uncover these alternatives. By starting with aims, particularly with highly specific objectives, attention is diverted away from the content and its various possibilities. Objectives become ends in themselves, disconnected from the content which gives them meaning and also from the learners who are expected to accomplish them. The heavy emphasis on general procedures for teaching adds to the problem by encouraging teachers to force the subject matter into a structure that may not accommodate all of its complexity. Locating oneís starting point for lesson planning within the content to be taught can help to reveal new potentials that one might otherwise miss. Only oneís starting point, however; aims, objectives, and procedures should develop as one thinks about how to exploit some of the more promising possibilities.

As an example, suppose a geography teacher cannot get his students to grasp "the basics" (by which he means map reading skills, the location of principal cities, the names of important rivers, the chief imports and exports of countries, and so on). It might be helpful for him to clarify his aims although these seem fairly well specified by his interpretation of what is ëbasicí. It would be more helpful for him to revise his procedures, but starting there would focus him on how to manipulate events instead of on possible fruitful connections between geography and the learners. It would be even more helpful for him to start by reconsidering his understanding of geography, particularly what he means by ëthe basicsí, for what is basic in his conception of geography may be less essential detail in anotherís understanding of geography. Rather than seeing geography as a collection of skills and facts which later get put to use, he might choose to see it as an inquiry into particular sorts of questions or as a way of looking at the world around us or as the application of geographic methods. Each choice suggests alternatives that he could use to serve the particular educational needs of his students.

Starting from a consideration of the content of a lesson, what aspects of the subject demand consideration? Or, to put the question another way, from the perspective of content what might be taken to be the commonplaces of lesson planning? As a start, we can locate four central questions:

Though they provide some guidance, these questions still are quite general and so relatively devoid of meaning. Phrases like "mathematical objects" or "mathematical habits of mind" may convey something to readers familiar with the literature on mathematics education, but they need further specification. This will be done by first describing and then analyzing a lesson on perimeter and area.

A Geometry Lesson Described

Suppose that it has been decided to teach about perimeter and area to a group of reasonably well prepared and motivated students of age thirteen or fourteen. Starting from a consideration of the subject matter implied by this topic, a traditional approach to the topic, one indicated in many textbooks, would start by reminding students of the meaning of the terms area and perimeter. (For example, "Area means the amount of space inside a region" and "Perimeter means the distance around the boundary of a region.") This might be followed by a few diagrams drawn by the teacher on the blackboard, which would show various polygonal shapes with the interior shaded in or the boundaries highlighted so that students could get an intuitive understanding of the meanings of the terms. Focusing first on perimeter, the teacher might present a definition of perimeter: for example, "The perimeter of a polygon is the sum of the lengths of its sides." After that, he or she would provide a few sample diagrams with the lengths of the sides given and then ask students to calculate the perimeter; or, given the perimeter and one side, ask for the length of the missing side. To extend this idea, the teacher might leave out some of the given lengths but add information about the characteristics of the polygon. (For example, "This is a rectangle with sides 4 and 6; what is its perimeter?") Many variations on this theme are possible, most of which would be left for homework. The eventual outcome of this phase of the lesson would be for the teacher and students to discover, or somehow arrive at, the formulas for the perimeter of a square (p = 4s) and a rectangle (p = 2l + 2w). These two formulas might be connected by observing that a square is actually a special case of a rectangle. Homework would require students to use these formulas.

Area, though somewhat more difficult, would be treated in a similar way. The area of a rectangle would be defined as the product of its length and width (A = lw). To establish the reason for this definition, the teacher would present a diagram of a rectangle superimposed on a square grid and ask students to calculate its area. Most students could do this quite readily, but it might be necessary for some to count unit squares. These students would be expected to move quickly to the obvious shortcut: instead of counting the rectangular array of squares, simply multiply the number of rows times the number of columns. Once established, the formula would be used to calculate the areas of various rectangles (and squares) with a variety of given information Again homework would have students apply these formulas to a wider variety of figures that could be decomposed into rectangles.

For students of the age and experience envisioned, this could well be the end of a dayís lesson. Still it would be helpful to see how this teacher would continue the next day or so. Having established the formulas for the area of a square and a rectangle, it is natural to ask how to find the area of less rectilinear figures, such as triangles, parallelograms, and trapezoids. Formulas for these figures can be found in a series of logical steps. similar to one that you may have encountered when you were in school. First, show that a right triangle is one-half of a rectangle and thus conclude that the area of a right triangle is half the area of the rectangle (A = ab/2). Next show that any triangle can be split into two right triangles and so with the help of a little algebra, that the area of a triangle is given by the formula:  A = bh/2. A parallelogram can always be divided into two congruent triangles and, therefore, into two triangles of equal area; so we can conclude that its area is twice the area of the triangle (A = bh). Finally a trapezoid can be split into a parallelogram and a triangle; adding the now known areas of these two pieces (and also with a little algebra) we arrive at the desired formula: A = (b + c)h/2. Having derived some or all of these formulas, the teacher would ask students to apply them to specific diagrams of gradually increasing complexity. Homework would consist of further application of the formulas.

The Geometry Lesson Analyzed

What can we say about the way mathematics is portrayed in this lesson? The framework outlined in Figure 2 will be used to organize this analysis. Although the illustrative questions do not exhaust the questions that could be asked, they should suffice to indicate what is implicated when we ask the apparently simple main questions. Remember, we are trying to open teachersí minds to the richness and variety of the subjects that they teach in order to help them choose from the range of possibilities that arise when we think of content in more flexible ways.

1. What is this lesson about? To many teachers the answer to this question seems so obvious ("Perimeter and area, of course!") that the question hardly seems worth asking. Even so, there is more to it than may first appear, and teachers should be encouraged to dig beneath their first, and probably superficial, reaction. A more thoughtful response would seek answers to the following sorts of questions.

Two circumstances make this an especially important question. First, the smooth logic of the lesson is more apparent than real. Many issues have been swept under the rug. Two examples may show you the sorts of difficulties I have in mind. At one point in the description of the lesson it is said, "Any triangle can be split into two right triangles." But is this really true for any triangle? How do we know? What if one of the angles is obtuse? A bit of thought may convince you that the argument is a bit more complex than it appears. A more serious difficulty occurs right at the beginning of the treatment of area when the square grid was superimposed on a rectangle. Did you assume that the grid could be made to fit exactly on top of the rectangle? Most people do, but suppose that a little bit of the rectangle was left over. That is, suppose for instance that the rectangle measured 3.21 by 4.85. What do we do with this extra material? It no longer is so "obvious" that we can count the squares by multiplying.

From a logical point of view, these are not trivial difficulties. Rigorous arguments to overcome them can be constructed, but it takes considerable effort to do so -- an effort not well supported by the assumptions and intuitive ideas used in this lesson. Moreover, such difficulties are not unique to this lesson. They are inherent in any geometry lesson that purports to be logical. Unless the teacher uses a highly abstract, purely formal set of assumptions and undefined terms, the logic of the lesson will inevitably contain gaps and flaws. So, wittingly or unwittingly, any teacher of 13 and 14 year olds who wishes to move forward by deductive logical steps will have to make choices and compromises along the way. In such a situation, the teacher must consider where and when to make these compromises and what other kinds of arguments will be considered acceptable.

This brings us to the second circumstance: deductive methods are not the only way in which mathematicians do their work. They are necessary to make intuitively understood ideas more precise and to communicate results to others, but new ideas and new theorems are developed by intuitive and less formal methods. George Polya who, in numerous books and articles has tried to specify and exemplify the less formal methods that mathematicians use, has created the most persuasive argument for non-deductive methods. Mathematical knowledge, Polya argues, is developed by complex processes involving -- in addition to generalization and deduction -- specialization, analogy, observation, plausible reasoning, and even guessing. "In many classrooms 'guessing' is taboo, whereas in mathematical research 'First guess then prove is almost the rule.'" (1981, vol. 2: 157) In this situation, a more complete response to the second question would seek answers to the following sorts of questions.

From what has so far been said, or implied, you can see that the sample lesson has a number of problematic features. These are not faults because in the hands of a reasonably competent teacher the lesson would work quite well for most students. Nevertheless, these features are things the teacher should think about because there are many students and circumstances for whom this lesson would not work. The question about the origins of the lesson raised two of the most troubling issues. First, the reasons (mathematical or otherwise) for engaging in this activity are obscure. The lesson needs some mathematical issue to initiate the activity. Second, there is good reason to believe -- from research as well as from personal experience -- that some students will not have an intuitive grasp of the difference between perimeter and area. Many people -- even those who know the formulas -- misunderstand the relationships between the two. For these students, one would like to have an alternative approach that will engage their attention and give them a better grasp of fundamental concepts. The apparently logical organization of the lesson raises a third troubling issue, though the extent to which it is troubling depends on the nature of the students and what kind of mathematical diet they get in other lessons. Many students do not connect well with a deductive approach but do connect with other forms of mathematical inquiry. Another approach that stressed less formal mathematical methods and habits of mind would be desirable for these students and no doubt would benefit the others as well.

A revised lesson on the topic of area and perimeter starts with the assumption that the concepts of area and perimeter, though known to students, are not clearly differentiated. The teacher would open this lesson by displaying a length of string, tied at the ends and stretched around four pegs approximately in the shape of a square. This would be described as an enclosure, a pasture say, with grass inside and surrounded by a fence. The pegs would then be rearranged to form a rectangle, obviously different in shape from the first; and the string, still tied as before, would be stretched around the pegs in their new location. Students would then be asked, "Is there more grass in the second pasture, less grass in the second pasture, or do both pastures contain the same amount of grass?" In many classes, there would be a difference of opinion at this point, but it is not unusual to find near universal agreement that both pastures contain the same amount of grass (even with students who think know something about area and perimeter).

Either way, the teacher would ask the class to work -- individually or in groups -- to justify their answers using the materials provided (string, pins, thick cardboard, and graph paper) and the following suggested procedure:

Pin a piece of graph paper to the cardboard.

Stick four pins into the graph paper so they approximate a square.

Tie the ends of a piece of string together so it fits snugly around the four pins.

Once it has been established that area can change while the perimeter stays the same -- an idea that troubles some adults -- new questions naturally arise. For instance:

Given a fixed perimeter, which rectangle gives the largest area?

Similarly, which gives the smallest area?

What happens if we fix the area, can the perimeter change?

What shortcuts can we find to calculate (instead of measure) area or perimeter?

• What if we started with a triangular pasture? Does the area change?
• If a square gives the maximum area for a rectangular pasture, what triangle would give the maximum area? How could you check?
• How could we calculate the area of a triangular pasture?
• Why stick to triangles? Why not parallelograms, trapezoids, or kites?
• For a fixed perimeter, which of all possible shapes gives the maximum area?
• What happens in three dimensions?

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