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Teacher Candidates' Growth in Designing Mathematical Tasks As Exhibited in Their Lesson Planning

Posted on: Tuesday, 14 March 2006, 03:03 CST

By Anhalt, Cynthia O; Ward, Robin A; Vinson, Kevin D

Abstract

In order to gain insight into preservice teachers' beliefs about planning for mathematics instruction, a study was carried out involving K-8 teacher candidates enrolled in an elementary mathematics methods course. Doyle's (1992) notion of academic task and the research on pedagogical content knowledge served as the theoretical framework for this study. The teacher candidates submitted lesson plans at three intervals during a semester-long methods course; the lesson plans were then coded based on candidates' planned uses of academic tasks. Analyses of the data revealed trends in these teacher candidates' design of academic tasks over the course of the semester. Recommendations and implications are presented highlighting the benefits of incorporating the knowledge base on academic task into a mathematics methods course as a means to contribute to teacher candidates' developing pedagogical content knowledge via their designing of academic tasks in lesson planning.

As part of the preparation process of becoming an elementary school teacher, preservice teacher candidates complete a variety of methods courses designed to develop and enhance their pedagogical content knowledge. During these courses, preservice teachers gain knowledge of and experience with designing, and sometimes implementing, effective and thoughtful activities and lessons for elementary students. We argue that as preservice teachers design lessons, their choices for the tasks they plan to assign to students, and the complexity and sophistication of these tasks, provide insights into candidates' developing understanding for pedagogy. Furthermore, as teacher candidates progress through a methods course designed to further their pedagogical content knowledge, the academic tasks the candidates articulate for their students in lesson plans provide potentially meaningful learning opportunities. Concurrently, the teacher candidates gain experience with designing and implementing a variety of strategies and approaches to effectively teach mathematics to young children.

As a means of gaining insight into their developing pedagogical content knowledge, we conducted a study that examined candidates' K- 8 mathematics lesson plans submitted at three intervals over the course of one semester. Serving as a lens for the coding and analyses of these lesson plans were Doyle's (1992) notion of academic task and Shulman's (1986) conception of pedagogical content knowledge. Analyses of the lesson plans yielded interesting shifts and trends in these teacher candidates' design of academic tasks. The results explore the utility of incorporating the study of academic tasks into mathematics methods courses as a means to promote pedagogical content knowledge in preservice educators.

Perspectives and Guiding Frameworks

Research on Academic Tasks

According to Doyle (1992), "task" refers to "the way in which work, and thus cognition, is organized and structured in a particular setting" and provides "situational instructions for thinking and acting" (p. 503). Doyle (1986) asserted, "the curriculum exists in classrooms in the form of academic tasks that teachers assign for students to accomplish with subject matter" (p. 365). According to Doyle (1986), a task consists of four key elements: (a) an end product, such as a completed worksheet; (b) operations to produce the product, such as applying a rule; (c) resources that can be used to reach the end goal, such as consulting a textbook; and (d) a significance or weight of the task (e.g., the assignment is worth 15% of one's grade).

Doyle (1986) contended that "these elements provide students with essential information about what they are to do with the content of the curriculum and, from this perspective, tasks thus communicate what the curriculum is to students and, therefore, shape their learning in fundamental ways" (p. 366).

Doyle (1992) distinguished between academic tasks involving the following:

(1) memory or the reproduction or recognition of information previously encountered (e.g., spelling tests); (2) routines or algorithms that reliably generate answers (e.g., arithmetic or grammar exercises); (3) opinion or the expression of a disposition toward content (e.g., reactions to a poem); and (4) understanding, including recognizing transformed versions of text, selecting appropriate procedures to solve complex problems, and drawing inferences or making predictions from information given (e.g., solving word problems in mathematics, performing science experiments, or reading a new passage with comprehension). (p. 506)

In the context of mathematics, tasks can be distinguished and identified when an observer closely monitors what students are doing and how students are interacting with the content. The first type of academic task (memory, reproduction, or recognition of information previously encountered) can be illustrated in an example of students doing a timed test on basic facts in any of the four operations. In this example, students are already familiar with the content and need to respond with information from memory and produce-and in some cases reproduce-the solutions. The second type of academic task (routine, algorithmic, or procedural) can be illustrated with an example of students finding the area of a quadrilateral by using a given formula, such as A = s^sup 2^ or A = 1 x w. The third type of academic task (opinion or the expression of a disposition toward content) can be illustrated with an example of students expressing their opinions, feelings, or dispositions toward doing certain kinds of mathematics-such as solving for an unknown variable in an algebraic equation. The fourth type of academic task (understanding, problem-solving, and other high-level cognitively demanding tasks) can be illustrated with an example of students recognizing and determining patterns in a given set of items, such as a growth pattern (1, 3, 5, 7, . . .), determining the next few numbers, and generalizing that the growth pattern to produce odd numbers is 2x + 1.

Doyle (1990) asserted that teachers engage students in classroom activities and simultaneously carry them through a curriculum by designing academic work and engaging them in the intellectual processes required to understand and do that work. Further, the kinds of tasks teachers plan to use reveal a great deal about teachers' theories of curriculum enactment and teaching and learning mathematics; Doyle (1986), therefore, suggested using academic task as an analytical tool for examining subject matter as a classroom process (see also Stein, Smith, Henningsen, & Silver, 2000).

Research on Pedagogical Content Knowledge

Many researchers have documented the various types of knowledge needed by effective teachers (Anderson, 1989; Ball, Lubienski, & Mewborn, 2001; Ball & McDiarmid, 1990; Kennedy, 1998; Ma, 1999; Shulman, 1986, 1987). These efforts have repeatedly identified pedagogical content knowledge as one of the most relevant and significant types of knowledge. Shulman (1986) defined pedagogical content knowledge as the ability to represent ideas in ways that are understandable to particular groups of students. Shulman further described pedagogical content knowledge as the capacity "to transform the content knowledge he or she [the teacher] possesses into forms that are pedagogically powerful and yet adaptive to the variations in ability and background presented by the students" (p. 15). Carter (1990) refined this conceptualization to present pedagogical content knowledge as an attempt to determine what teachers know about their subject matter and how they translate that knowledge into classroom curricular events. Doyle (1992) contended that this ability distinguishes a teacher from a non-teaching specialist; for example, "Knowing biology is necessary, but certainly not sufficient, to know how to represent biological content to students in a teaching situation" (p. 498). Doyle (1990) further argued that one of the key components of teachers' pedagogical content knowledge is the ability to represent subject matter to students. Similarly, Shulman argued that "What is also needed is knowledge of the most useful forms of representations of those ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations, [in other words,] the ways of representing and formulating the subject that make it comprehensible to others" (p. 9).

Connecting Academic Task to Pedagogical Content Knowledge Considering Doyle's work on academic tasks and using, in particular, Shulman's definition of pedagogical content knowledge, we contend that preservice teachers' pedagogical content knowledge is demonstrated, in part, by their choices of the academic tasks they assign to students. Furthermore, we hypothesize that as teacher candidates progress through their methods courses, their pedagogical content knowledge is developing and expanding. By examining the academic tasks that preservice teachers plan to use in the teaching of mathematics at successive time intervals in their methods courses, insight can be gained into these teacher candidates' developing pedagogical content knowledge.

Methodology

Participants

The study involved 31 elementary educatio\n majors (2 males and 29 females) enrolled in a field-based mathematics methods course at a large southwestern university. All of these individuals had successfully completed a mathematics content course, a prerequisite to the methods course. During the methods course, the teacher candidates concurrently participated in the course's field component, where they observed, assisted, and taught in an elementary classroom daily. Additionally, throughout the semester, the teacher candidates were taught in a constructivist manner (using manipulatives, technology, engaging in problem solving, hands-on exploration, writing, discourse, making real-world connections, etc.) consistent with reform standards (National Council of Teachers of Mathematics [NCTM], 1991, 2000; see also Anhalt, Ward, & Vinson, 2004).

Data Collection and Method of Analyses

At the beginning of the semester, we compiled a list of eight mathematical topics that would be explored throughout the methods course and which represented typical K-8 mathematics topics the candidates would be expected to teach. These topics included the multiplication of fractions, division of fractions, area of a circle, area of a trapezoid, area of a parallelogram, perimeter of polygons, addition and subtraction of integers, and mean. The topics, along with a corresponding and appropriate grade level, were written individually on index cards prior to the first class, and one index card was distributed to each teacher candidate during the beginning portion of the semester. The teacher candidates were then instructed to develop and submit a lesson plan 1 week later reflecting what they considered to be an effective way to teach that particular topic; that is, a way to teach that topic so that student understanding would be maximized. The teacher candidates were encouraged, but not required, to seek out resources such as books, the Internet, inservice teachers, and their mentor teachers to assist them in completing this assignment. The teacher candidates were asked to identify the topic and grade level, any materials, the procedure, a closure, and the source (if any) of their ideas when completing their lesson plans. These "initial lesson plans" were reviewed and coded according to the types of academic tasks included within the lesson plan.

After completing and submitting their individual lesson plans, the teacher candidates were placed in groups with their classmates sharing the same topic and asked to reach a group consensus on how to best teach their shared topic. The candidates were encouraged to share individual ideas described in their initial lesson plans and to justify to the group why they believed their approach and strategies to teaching this topic were effective. The conclusion of the group activity was the generation of a consensus method for best instructional practice for the academic content by submitting a "group lesson plan."

During the semester, the preservice teachers did not receive explicit instruction regarding categorization of the mathematical academic tasks. Presenting the theoretical framework on academic tasks during the course of the study would have interfered with the validity of the study. However, the preservice teachers were encouraged to reflect on and discuss with their peers the tasks or the work that they had designed in which their students would interact with the mathematical content, especially during the group work at the middle interval.

Near the end of the semester, the teacher candidates were asked individually to submit one final lesson plan, again using the previously described format, but this time detailing how to best teach a K-8 mathematical topic of their choice. Topics chosen by the teacher candidates for these "final lesson plans" included a wide range of concepts at different grade levels.

Coding the academic tasks. Upon receiving all three of each teacher candidate's lesson plans, we classified the academic tasks consistent with Doyle's framework. Only three of the four types of Doyle's academic tasks were used in the coding process as categories, (a) memory or the reproduction or recognition of information previously encountered; (b) routines, procedures, or algorithms that reliably generate answers; and (c) understanding, including recognizing transformed versions of the content, selecting appropriate procedures to solve complex problems, and drawing inferences or making predictions from information. We chose to exclude Doyle's third academic task-the opinion or the expression of a disposition toward the content-as a category in the coding process because we felt it was not possible to capture this type of task within the limitations of this study. The three tasks used in the coding process will be referred to hereinafter as memory/ reproduction/recognition tasks, routine/algorithmic/procedural tasks, and understanding/problem-solving tasks.

The coding process was both an independent and a collaborative effort by the three researchers whereby each lesson plan was chunked into sections to discretely identify the academic tasks articulated in the lesson plans. As each lesson plan was read by a researcher, the text describing an academic task was identified and coded as follows: (a) memory/reproduction/recognition; (b) routine/ algorithmic/procedural; and (c) understanding/problem-solving academic tasks. After each researcher independently coded each lesson plan according to type of academic task, we compared initial coding categorizations. In the instances in which discrepancies existed regarding the categorization of an academic task, a discussion took place to better understand each other's thinking regarding the coding and then a consensus was reached.

The majority of discrepancies, although few in number, arose between the categories of routine/algorithmic/procedural academic tasks and understanding/problem-solving academic tasks. This may have been caused by the fact that in some instances there seemed to be a potential for the task to become more of a problem-solving, meaning-based task as it initially appeared in the lesson plan. Shortly after the description of the task, however, there were statements in the lesson plans regarding the teacher providing a procedure or a formula for the students to use. No discrepancies were noted in the coding and categorizing of the memory/ reproduction/recognition academic tasks.

In looking closely at the coding process, specifically at the interrater reliability of the total number of tasks that were coded at each interval, the number of tasks that were initially coded differently was low relative to the total number of tasks that were found. In the initial lesson plans, out of the 66 total tasks there were 7 tasks in which we had initially disagreed on the coding. Out of the 36 total tasks that were coded in the group lesson plans, we had 4 tasks in which we initially disagreed on the coding. In the final lesson plans, out of the total 106 tasks that were found, we had 12 tasks in which we initially disagreed on the coding. In all cases we eventually came to a unanimous coding decision.

Examples of coded academic tasks. Consider the following portion of a lesson plan designed by one of the teacher candidates. Addressing the area of a trapezoid, this lesson plan was submitted at the initial interval and as articulated below, contained examples of each of the three types of academic tasks.

The lesson began by asking students to recall or reproduce properties of a trapezoid, an activity coded as a memory/ reproduction/recognition academic task:

As students enter the classroom, they will find on the board a large drawn trapezoid. They will be asked to write down anything that they recognized about the shape including any properties or other notable aspects. The teacher will then ask students to share some of their observations about the shape and compile a list on the board. If the name is not mentioned, the teacher will define the shape as a trapezoid.

As the lesson continued, students were asked to compute the area of a trapezoid, an activity coded as a routine/algorithmic/ procedural academic task:

After the groups have their trapezoids formed, each group will get several sheets of graph paper and at least two colored pencils. The teacher will ask the groups to decide on a unit size (the teacher will suggest the graph paper squares) and to trace and find the area of their trapezoid. The teacher may want to ask the students to quickly define perimeter and area as a quick review. Each group will then be given the opportunity to discuss how they found the area. The teacher will ask, "Did any group have a method where they did not count the graph paper squares?"

Later in the lesson, the teacher candidate described an understanding/problem-solving academic task in which students were to engage:

The class will then be broken into groups or pairs and given a set of tangrams including . . . five right triangles, a square, and a parallelogram per group. The groups are then asked to make a trapezoid out of their tangrams. Each group should then share how they made their trapezoid and what they considered while constructing it or what difficulties they may have encountered.

This activity was coded as an understanding/problem-solving academic task as students were applying and extending their knowledge about the shape of a trapezoid while engaged in an exploratory experiment.

Results

The lesson plans submitted by the teacher candidates in this study represent hypothetical teaching acts, as these lesson plans were not implemented in an actual classroom setting. Thus, the teaching approaches and ideas presented in the lesson plans provide us with insights into these teacher candidates' beliefs about how to plan for and teach mathematics. Additionally, we argue that these teacher candidates' chosen strategies for planning and teaching mathematics provide p\artial insight into their developing pedagogical content knowledge.

Trends in the Number of Academic Tasks Planned

Table 1 displays the number, as well as the percentage, of each type of academic task described in the lesson plans submitted at three points in the semester. As shown in the table, relatively large differences surfaced in the number of occurrences of specific tasks planned at each of the three intervals. For example, in the initial lesson plans and group lesson plans, the routine/ algorithmic/procedural academic tasks were by far the most planned academic tasks with a modest representation of memory/reproduction/ recognition academic tasks. In the final lesson plans, a different pattern was observed, where memory/reproduction/recognition academic tasks were the most frequently planned and understanding/problem- solving academic tasks were far more common than in the prior two intervals.

Table 1

Number and Percentages of Mathematical Academic Tasks Found in Lesson Plans

Trends in the Number of Lesson Plans Articulating Academic Tasks

Table 2 lists the number of lesson plans, per interval, in which each of the different tasks appeared. In the initial interval, 31 lesson plans were submitted (one from each of the teacher candidates), and 8 group lesson plans were submitted during the middle interval. Only 29 individual lesson plans were submitted in the final interval, as 2 teacher candidates failed to submit their final lesson plans.

As before, the data indicated a preference for the routine/ algorithmic/procedural and understanding/problems-solving academic tasks in the initial and group lesson plans. However, in the final lesson plans, a larger percentage of lessons represented the understanding/problem-solving tasks.

In order to explore trends in the planned uses of the three academic tasks, the observations listed in Table 1 were used to compare the academic tasks planned in the initial and final lesson plans. The results of a chi-square test confirmed the patterns observed were significant, χ^sup 2^(2) = 22.55, p < .001, demonstrating a statistically significant difference between the academic tasks planned in the initial and final lesson plans. We assert these results support that the teacher candidates demonstrated growth in planning effective academic tasks during the semester.

Table 2

Number and Percentages of Lesson Plans in Which Mathematical Academic Tasks Were Found

Discussion

Doyle's (1986) definition of academic tasks refers to the academic activities that take place while students work with the material that is to be learned. The fact that the total number of planned academic tasks increased substantially from 66 to 106 occurrences (see Table 1) indicates that these prospective teachers were planning mathematics instruction in ways that included increased opportunities for students to interact with the curriculum through academic tasks. Doyle (1983, 1988) and Schoenfeld (1985) have underscored that learning opportunities are enhanced when students do most of the mathematics work during instruction. Therefore, the prospective teachers' final lesson plans held greater potential for learning opportunities by exposing students to more mathematical tasks.

The increase in the number of understanding/problem-solving academic tasks from the initial to the final lesson plans and the increase in the number of lesson plans containing understanding/ problem-solving academic tasks, indicates that the understanding/ problem-solving type of tasks were becoming more prevalent over time. Because of the nature of the understanding/problem-solving type of tasks, students would potentially be given more opportunities to do the required thinking to participate in the tasks, and therefore, their learning would be enhanced. According to NCTM's (2000) Principles and Standards for School Mathematics, problem-solving tasks offer students opportunities to draw on their knowledge, and through this process, enable students to develop new mathematical understandings. In addition, NCTM (2000) emphasized that solving problems is not only a goal of learning mathematics but also a major means of doing so, advocating that students "have frequent opportunities to formulate, grapple with, and solve complex problems that require a significant amount of effort" (p. 52).

We support Doyle's (1986) contention that the kinds of tasks that teachers plan to use reveal a great deal about their theories of curriculum enactment and teaching and learning mathematics. As shown in Table 1, during the initial and middle intervals the vast majority of the tasks assigned were memory/reproduction/recognition academic tasks and routine/algorithmic/procedural tasks; very few understanding/problem-solving type of tasks were assigned during these intervals (6% and 3%, respectively). Because these prospective teachers' thoughts about teaching and learning mathematics is partially revealed through the teachers' planning, a more evenly balanced approach to the types of academic tasks assigned to students in the lesson plans submitted during the final interval is evidence of growth. The closer distribution of planned academic tasks designed in the final interval, as well as the increase in the number of tasks planned, may indicate that these prospective teachers were attaching nearly equal importance to the various academic tasks at the end of the mathematics methods course.

At the end of the methods course, a questionnaire that described the notion of academic tasks was administered to the 31 teacher candidates. The candidates were asked to comment on how a knowledge base of academic tasks might assist them when planning K-8 mathematics lessons. Several of the teacher candidates acknowledged that possessing the knowledge base of the different ways students can interact with the material to be learned when they are offered different academic tasks was "extremely valuable" and "crucial." Many of the teacher candidates also acknowledged an obligation to think about using all three types of academic tasks in their future planning. Although none of the teacher candidates specifically indicated that one type of task was superior to the others, a majority indicated the "importance" and "worthiness" of including many understanding/ problem-solving tasks for students to complete, as these types of tasks "get students to really think" and "help students solve real world problems."

A few of the teacher candidates described their anxiety and frustration over wanting to include understanding/problem-solving tasks in their future planning but felt "underprepared" or "unable" to do so, expressing concerns of "not getting the big picture" or "not knowing enough" to answer potential questions posed by students. Also, 3 teacher candidates alluded to understanding/ problem-solving academic tasks as taking "too much time" to design and for students to complete, which the 3 candidates indicated as a reason for not wanting to include these tasks in their lesson planning. Interestingly, several of the preservice teachers noted that it was "easier" to plan and implement memory/reproduction/ recognition mathematical tasks because these tasks can be accomplished through games or a review of the content, such as having the students recall basic facts.

Implications

Doyle (1986) argued that the curriculum exists in the form of academic tasks that teachers assign for students to accomplish with subject matter; therefore, academic tasks "communicate what the curriculum is to students and, thus, shape their learning in fundamental ways" (p. 366). Students are engaged in the intellectual process of understanding the material through the academic tasks teachers design. Therefore, the type of academic tasks in which students engage affect what mathematics they learn, how they learn it, and how they can apply their learning in various contexts. Although Doyle (1992) did not suggest the existence of a hierarchy in terms of the difficulty or complexity of the various types of academic tasks, we contend that when designed and implemented in the spirit of the NCTM Standards (1991), understanding/problem-solving tasks are more cognitively demanding undertakings in comparison to both memory/reproduction/recognition academic tasks and routine/ algorithmic/procedural tasks; thus, understanding/problem-solving tasks provides more fruitful opportunities for students to engage in the learning of mathematics.

We argue that a primary goal of a methods course is to develop and enhance teacher candidates' pedagogical content knowledge by providing prospective teachers with opportunities, experiences, approaches, and strategies to teach subject matter in effective and engaging ways that maximize student understanding. We recommend the explicit teaching of and practice with designing academic tasks in mathematics methods courses as a means to enhance teacher candidates' developing pedagogical knowledge by assisting candidates in making careful pedagogical choices when designing academic tasks for their future students. It is our recommendation, based on our findings, that the theoretical framework(s) on academic tasks (Doyle, 1992; Stein et al., 2000) be introduced to elementary preservice teachers early on in their mathematics education courses. This recommendation is to emphasize the importance of giving preservice teachers the opportunity to reflect and critically analyze the type of academic tasks they design, based on how they intend their students to interact with the mathematics content they are to learn. Our conjecture is that by discussing, asking for clarification, and questioning each other regarding their designs of mathematical academic tasks, teacher candidates will improve their knowledge of how to purposefully design rich academic tasks for their students and thereby improve student learning of mathematics. We assert that it is imperative f\or preservice teachers to become knowledgeable of and selective in choosing and designing cognitively demanding tasks when planning for mathematics instruction.

Furthermore, methods instructors should consider using the theoretical framework of academic tasks as an analytical tool in ascertaining the types of academic tasks about which their preservice teachers are thinking when designing lesson plans. Many of the discussions that can take place among preservice teachers in a methods course regarding planning for instruction may also be analyzed in terms of the types of academic tasks preservice teachers view as worthwhile and the reasons behind their thinking. The data from this study indicate that as these preservice teachers progressed through their mathematics methods course and were given opportunities to design lesson plans both independently and collaboratively, not only did they include a greater number of academic tasks, but they also provided more opportunities for students to engage in richer and more meaningful mathematical tasks.

References

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Cynthia O. Anhalt

Mathematics, University of Arizona

Robin A. Ward and Kevin D. Vinson

Teaching and Teacher Education, University of Arizona

Copyright Ball State University Teachers College Winter 2006

Source: Teacher Educator, The

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