Effectiveness of Lesson Planning: Factor Analysis

By Panasuk, Regina M; Todd, Jeffrey

The paper presents the conceptual framework that guided the development of the Lesson Plan Evaluation Rubric (LPER) instrument derived from the Four Stages of Lesson Planning (FSLP) strategy and the empirical results that provide the insight into the elements of lesson planning. Teachers from urban low-performing middle schools in one of the New England states received training and ongoing coaching in the FSLP strategy. Two hundred sixty one lesson plans from 39 teachers were collected during one school year of the two- year study to conduct factor analysis of the Rubric’s 17 items. The resulting four factors are discussed in this paper. The research shows that the lessons plans developed with the reference to the FSLP strategy revealed a higher degree of lesson coherence.

The research study conducted during the Middle School Mathematics Initiative (MSMI) project provided the opportunity for in-depth investigation of mathematics lesson planning. The Four Stages of Lesson Planning (FSLP) strategy (Panasuk, 1999, see Figure 1 ) was one of the interventions that aimed to assist middle school teachers in the designing of their lesson plans. During the project, we developed and validated Lesson Plan Evaluation Rubric (LPER) instrument (see Figure 2) derived from the lesson planning and delivery evaluation models (Panasuk & Sullivan, 1998). The rubric’s seventeen items, with scores ranging from zero to 37, provided further details about and helped to make explicit the underlying principles of the FSLP strategy. The LPER instrument was used to analyze written lesson plans of the teachers who received training in the FSLP strategy.

The paper presents the conceptual framework for the rubric as it relates to the FSLP strategy, and empirical results that provide insight into the elements of lesson planning.

The Background

According to Clark & Dunn (1991), planning is a psychological process of envisioning the future, and considering goals and ways of achieving them. Lesson planning can be defined as a systematic development of instructional requirements, arrangement, conditions, and materials and activities, as well as testing and evaluation of teaching and learning. It involves analysis of the learning needs and the development of a delivery structure to meet those needs. Schn (1983) described lesson planning as pre-active decision-making that takes place before instruction. Clark and Dunn (1991) stated that, consciously and unconsciously, teachers make decisions that affect their behavior and that of their students. Planning a lesson involves teachers’purposeful efforts in developing a coherent system of activities that facilitates the evolution of students’ cognitive structures. The quality of those decisions and efforts depends on the creativity of teachers and on their ability to apply learning and instructional theories.

Stigler and Hiebert (1999) indicated that, “many teachers in the United States do not even prepare lesson plans, at least not around student learning goals” (p. 151). Kennedy (1994) and Reiser (1994) suggested that experienced teachers do not use Instructional Design features (Rriggs, 1977;Merrill. 1971; Wong & Raulcrson, 1974) in a written form of lesson planning. Teachers’ records of their lesson plans arc “sketchy” (Reiser, 1994. p. 15), “quite brief (Reiser & Mory, 1991, p. 77).or”cryptic shorthand” (Kagan &Tippins, 1992. p. 478). Silver (1998) referred to the results of the Third international Mathematics and Science Study (TIMSS. NCES. 1999). which show that in far too many classrooms, mathematics instruction includes review of the previous lesson’s homework assignment, quick delivery of a set of rules and procedures by the teacher, and the rest of the lesson, if there is any time left, is filled out with a set of exercises for practice. The National Council of Teachers of Mathematics Standards call for more attention to lesson planning and analysis and stress that teachers are responsible for creating an intellectual environment in the classroom where engagement in mathematical thinking is the norm (NCTM, 2000).

Lesson Planning Strategy and its Underlying Principles

To guide teachers’ decision making in planning certain types of lessons, Panasuk (1999) introduced Pour Stages of Lesson Planning (FSLP) strategy (Figure 1), which represents one way to plan instruction. Its purpose is to shape and structure the complex process of mathematics lesson planning to ensure embedded assessment and consistency in student learning. The sequence of the planning steps’ is unique and is the key ingredient of the FSLP strategy. The philosophy of the FSLP is based on the perspectives that emphasize creation of the conditions that optimize learning and the relation of specified events of instruction to learning processes and learning outcomes.

Figure 1. Four stages of lesson planning.

The Four Stages of Lesson Planning strategy and its operational counterpart. Lesson Plan Evaluation Rubric, emerged from and are based on Gagne’s (1962, 2001) instructional theory that emphasized task analysis, Ausubel’s (1968) model of advance organizers, Shulman’s (1987) idea of pedagogical content knowledge, and Tabachneck-Schijf, Leonardo, and Simon’s ( 1997) ideas of multiple representations. The following is a parallel description of the FSLP strategy and LPER.

Stage I. In the first stage of planning (Figure 1 and Figure 2, item 1.1), cognitive objectives are developed and stated in terms of students’ observable behavior that specifies the knowledge structure that produces the behavior (Mager, 1984). Dick and Carey ( 1996) argued that, “perhaps the best-known component of the instructional design models is the writing of performance objectives, or, as they more commonly called, behavioral objectives” (p. 97). The purpose of the specific cognitive instructional objectives is to guide the lesson-planning process. They provide the basis for designing the instructional package and developing evaluation and assessment strategies. Formulating cognitive objectives, teachers convert a set of learning needs to a set of learning objectives that indicate performance. Bruner (1966) suggested that the major goal of objectives is to explicitly describe the skills to be learned, and Mager (1984) argued that cognitive objectives in lesson plans must describe the intended outcomes of learning. Current standards-based instruction (NCTM, 2000) calls for observable, measurable curriculum objectives couched in outcome language such as what students will know and be able to do.

Figure 2. Lesson Plan Evaluation Rubric

Figure 2. Lesson Plan Evaluation Rubric

Figure 2. Lesson Plan Evaluation Rubric

Different mathematical tasks require different levels of thinking, and the objectives must reflect those cognitive levels in terms of measurable indicators (Bloom, 1956). According to Bloom, learning outcomes for lower level tasks that can be described by recalling, reproducing, reciting (a rule), using (the formula to calculate), or naming (elements of a sequence) would demonstrate students’ knowledge, the first level in his taxonomy. If the students can classify, describe, restate, translate, or recognize, they demonstrate comprehension. When students are able to categorize, differentiate, compare, contrast, examine, experiment, test, compose, summarize, or set up a rule, they exercise higher order thinking such as reasoning and problem solving, and demonstrate their ability to apply, analyze, synthesize, and evaluate. Panasuk, Stone, and Todd (2002) found that while clearly stated objectives at the beginning of planning helps teachers purposely and consciously navigate through the planning process, statements such as “the students will learn” or “the students will understand or know” are vague, lead nowhere, and do not help when teachers or researcher evaluate the very process of learning, knowing, and understanding.

Branch (1994) reported that teachers rarely discuss objectives or lesson plans with other teachers or supervisors in the school. In a study of Canadian teachers, Kennedy (1994) found that most teachers “lacked even rudimentary knowledge to implement an instructional development approach. It seems likely that the respondents, all highly certified teachers with lengthy experience, were reluctant to admit their lack of knowledge and expertise in an area they felt they should know about” (p. 20). Only one-eighth of them were able to develop and classify behaviorally stated instructional objectives. Kennedy testified that some of the most highly educated teachers believed that the use of behaviorally stated instructional objectives was “dehumanizing and restrictive” (p. 20). The LPER contains two items (1.1 and 6.1) that explicitly encourage teachers to formulate cognitive objectives and align them with the establ ished state and national curriculum standards to ensure specific expectations. These standards provide the basis for and guidance in making educational decisions.

Stage II. Designing homework is a critical feature, and its occurrence as the second step in planning a lesson is unique to the FSLP strategy (Figure 1 and Figure 2, items 2.1 and 2.2). It reflects the recognition that all components of instruction must be aligned in order to create coherence from specific cognitive objectives to anticipated learning outcomes. The strategy emphasize\s that planning homework involves working through the assignments to ensure they incorporate the skills specified by the stated objectives. Worked out problems provide teachers with insight into the nature and the details of the problems that the students are expected to do independently, and ensure that selected classroom activities are consistent with the objectives, focused toward outcomes, and linked to both.

We examined the instructional design strategies suggested by Cruickshank, Bainer, and Metcalf (1999), Briggs, (1977), Gagn and Briggs ( 1979), Merrill ( 1971 ), Wong and Raulerson (1974), and Orlich, et al. (1990), and found that they differ considerably in their treatment of homework. Many do not address homework at all. Others seem to treat homework as afterthoughts to planning the developmental activities. Gill and Schlossman (2000) observed that, “homework remains a peripheral concern in teacher training institutions; there is only limited professional interest in translating the consensus for more homework into valuable educational experiences for students” (p. 176).

While little is written that addresses the actual planning of homework, there are some positive research results regarding teachers’ attention to homework planning. Gates and Skinner (2000) determined that students are more likely to complete the homework assignments that have been tailored to their interests. Namboordiri, Corwin, and Dorsten (1993) found that student achievement improved when teachers integrated homework into the summary portion of the lesson. Spadano (1996) demonstrated that when high school students regularly and independently complete meaningful homework assignments, they become autonomous learners and improve their self- control, self-discipline, and self-regulation. Panasuk (2002) asserted that the alignment of objectives and homework provides a foundation for the selection of classroom activities that are consistent with both the objectives and homework. When teachers build alignment of the objectives, learning outcomes, homework, and classroom activities in their planning process, it is likely that instruction based on such planning would facilitate students’ perception of the coherence of the information and would optimize learning (Panasuk, Stone, & Todd, 2002). Class activities would have more impact because the homework directly connects to the activities. Students perceive that the class activities prepare them to complete the homework assignment and that the entire lesson is coherent and integrated.

Panasuk and Todd (2002) found that planning lessons is improved when teachers regularly and carefully analyzed all homework problems before assigning them. By working through homework problems, the teachers scrutinize and determine the features and subtleties of the problems to foresee students’ possible difficulty. Having the homework problems worked out in a manner similar to what the students are expected to, the teachers are better prepared to proactively comment in class on troublesome homework problems as they are assigned, providing students with support necessary to complete homework independently.

Stage HI. The FSLP strategy suggests planningthedevelopmental activities afterthe objectives and homework are drafted. Such a sequence of planning steps offers a basis for strong bonds and consistency between the objectives, the means for meeting the objectives, and the homework as a form of assessment. Planning classroom activities that are developmental (advancing the development and learning) involves selection of materials and format to create an environment that promotes meaningful learning and all levels of thinking. The acquisition of different types of knowledge, skill, and levels of thinking (Bloom, 1956) requires different conditions of learning (Merrill, 1971) that in turn call for different methods of teaching to produce efficient and effective instruction. It is not a matter of preference what teaching and learning strategies to use to meet a particular set of objectives, but it is a matter of making informed pedagogical choices.

The FSLP strategy adheres to the idea of multiple perspectives on learning and teaching (Shiffman, 1995). Theories that contribute to our knowledge about learning and teaching are essential, and offer scientifically-based approaches to the process of lesson design in general, and selection of the teaching models in particular. Teaching and learning models based on these theories, represent the basis for scientifically warranted, pedagogically sound lesson plans.

Behaviorism, cogniti vism, and constructivism provide a general explanation of the nature of knowledge and how people learn. Behaviorism is based on observable changes in behavior and focuses on a new behavioral pattern being repeated until it becomes automatic. Cognitivism helps to understand the thought process that is manifested through behavior, which is an observable indicator of what is happening inside the learners’ minds. Constructivism, based on the premise that the learners construct their own perspective of the world through individual experiences and schema, suggests that learning is an active search for and construction of meaning. We support Ertmer and Nevvby “s ( 1993) position to advocate no one single theory to draw instructional strategies from, and suggest correlating different theories with the needs of the learners, the content to be learned, and the environment to be created. Approaches based on the behavioral theory would help facilitate mastery of mathematics content through careful and detailed identification of the objectives. The application of cognitive theory principles would guide the process of incorporating problem solving approaches and heuristics to be applied in new or unfamiliar situations. The teaching methods based on the constructivist theories would promote students’ active involvement and facilitation of knowledge development rather than transmission of information. We believe that instructional approaches go beyond one particular theory and must be based on the integration of different theories and models. Various strategies allow the teacher to make the best use of all available practical applications of the different learning and instructional theories. With this approach the teacher is able to draw from a large number of strategies to meet a variety of learning situations.

Figure 1 displays universal elements of instruction that have been examined and described in multiple research studies incorporated into the FSLP strategy. The instructional approaches, referenced as five process standards (NCTM, 2000), are aimed at creating an intellectual environment that engages students in mathematics thinking and utilizes various activities that meet the general purpose and specific objectives of the lesson. While the content, the student needs, and abilities are the primary issues, the form in which the content is presented and the student needs are met (i.e. class arrangement. Figure 2, item 3.2), is secondary and should not prevail over the main focus of planning.

The LPER reflects the idea that having worked out problems (Figure 2, items 3.1a, 3.1b, 3.Ic) is central to effective planning. Panasuk (2005b) asserts that working through a problem helps to see its structure and provides teachers with the basis for making conscious decision when selecting and sequencing the activities.

StageIV. Planning mental mathematics, the final stage (Figure 1 and Figure 2, items 3.1a, 4.1) of lesson design is based on and integrates all three previous stages. Constructing mental mathematics activities, teachers create brief and fast-paced problems that are basic elements of student prior knowledge as well as prerequisites of the new learning (Panasuk & Cutler, 2001; Panasuk, 2002). Mental mathematics, as it is regarded in the FSLP, is similar in some way to Ausubel’s (1968) concept of advanced organizers. As Ausubcl suggested, “The principle function of the organizer is to bridge the gap between what the learner already knows and what he needs to know before he can successfully learn the task at hand” (p. 148). The organizers should be formulated in language and concepts familiar to the students (p. 331). The principle functions of mental mathematics are to surface and connect learners’ prior knowledge to new information, to precipitate new material, and to provide a framework for new knowledge, and review previous lesson homework efficiently (Panasuk, 2005a).

Concept and Task Analysis

Pertinent to each stage of planning is the notion of concept and task analysis that is based on Gagn’s (1965) hierarchy of principles and the notion of the organized knowledge structure. Many behaviors and reasoning skills in which mathematics students are engaged are quite complex. Performing operations with numbers, or solving equations, or applying the Pythagorean theorem, students execute a set of distinct steps in a particular order, which shows evidence of certain reasoning skills. The purpose of concept and/or task analysis or decomposition is to gain insight into the nature of a given concept or task and to identify subtasks and their underlying sub-concepts (Panasuk, 2005b). For example, the task of solving linear equations (i.e. (2(x-3)-3(2-4x) = 12), when using formal procedure, involves several subtasks such as application of the distributive property, collecting like terms, and solving one step equations. In turn, each of the subtasks requires knowledge and skills of the concepts of operations with positive and negative numbers and operations with fractions. Each of these sub-concepts can be further broken down into subordinate concepts that build up the mathematical system related to solving linear equations.

Concept and task analysis is a cornerstone of planning mathematics lessons. Mathematical concepts cannot be understood in isolation and would ma\ke sense only as a part of a system in which meanings have been established. Through concept and task analysis, teachers develop a detailed picture of the structure of the concept/ task to be learned and its constituent parts, and are better prepared to create a classroom environment that would facilitate students’ meaningful learning. Concept and task analysis helps in identifying students ‘ prerequisite knowledge needed for learning new material. Turning these prerequisites into a series of mental mathematics exercises foruse at the beginning of the class, teachers would activate students’ prior knowledge and give them a sense of how the day’s lesson is similar to and different from their existing knowledge base.

While designing the developmental activities, concept and task analysis helps teachers to plan a gradual progression from one level of representation to another (Tabachneck-Schijf, Leonardo, & Simon, 1997). [For example, from working and operating with real objects or geometrical shapes, to mental imagery of pictures or diagrams, to symbolic representations of formulas.] In addition, as teachers perform concept and task analysis during lesson planning, they have an opportunity to predict the kinds of misconceptions that students may have. Through planning examples that address misconceptions, teachers can establish conditions for students to rethink and consider their alternative conceptions.

Other Elements of Lesson Planning

Embedded assessment (Figure 2, item 5.3) and phases of lessons (Figure 2, item 5.1 ) are built-in to the FSLP strategy. Formative and sumrnative forms of student assessment and evaluation are equally important and should be incorporated into lesson planning consistently. Branch and Gustafson (1998) define formative evaluation as “identifying needed revisions to the instruction” and sumrnative evaluation as “being directed to assessing the degree to which the objectives have been achieved” (p. 5). Homework can be viewed as a form of sumrnative assessment when considered in the context of a daily lesson. It is the indicator of students’ ability to meet the instructional objectives when they work independently without the teachers’ assistance and guidance.

Assessment has a formative role when it is ingrained in planning and implementation. Classroom practice should be designed to explicitly and implicitly provide the sources from which teachers and students are able to make informed decisions about progress towards the day’s objectives. Airasian ( 1994) and Stiggins (2001) suggest that student questioning is an integral aspect and the most common form of teacher/student interaction and formative evaluation. Planning clear questions in advance that probe for reasoning, not just for facts and information, are important to understand students’ progress toward the instructional objectives and are central to teacher/student interaction and assessment. The questions should encourage students to recall facts, to analyze those facts, to synthesize or discover new information based on the facts, or to evaluate knowledge. It takes skill and practice to pose questions that go beyond short and low-level response and to balance both high and low level questions.

In addition to its function of surfacing prior knowledge, the use of mental mathematics is an example of formative assessment (Panasuk, 2002), as it informs the teacher whetherthe students are ready formeaningful participation in the new lesson.

Phases of the lesson (Figure 2, item 5.1) are discrete yet necessarily connected components of the planning and instruction. The FSLP strategy implicitly defines mental mathematics and the developmental activities phases of the lesson. Within the developmental phase, there might be the segments of direct teaching, student activities, guided inquiry, individual, or group work. A final phase of the lesson is homework orientation and guidance when the teacher summarizes the lesson and refers to the homework assignment, noting its relationship to the problems solved in class and indicating nuances and possible hurdles.

Integrating Pedagogical Content Knowledge into Lesson Planning

Effective planning requires an integration of knowledge of pedagogy, content, and instructional design. Shulman ( 1987) defined pedagogical content knowledge (PCK) as “that special amalgam of content and pedagogy that is uniquely the province of teachers, their own special form of professional understanding” (p. 8). Mathematics teachers, reveal strong pedagogical content knowledge when they show an understanding of the associations between general pedagogical principles and mathematics content.

The view of pedagogical content knowledge accepted in this paper is based on works of Piaget, Ausubel, Gagn, and Simon and associates.

Piaget’s (1963, 1970) theory helps to explain the development of operational structures in school-aged students. David Ausubel’s (1968) idea of meaningful learning promotes the concepts of student active involvement and his model of advanced organizers emphasizes connecting current learning to prior knowledge. The works Larkin and Simon ( 1987), Simon ( 1992), and Tabachneck-Schijf, Leonardo, and Simon’s ( 1997) help to understand the various forms of representations and their interrelations. Together with the Gagn’s (1965, 2001) theory of instructional design, the principles developed by Piaget, Ausubel, and Simon and associates form the basis for pedagogical content knowledge for mathematics teaching that is pertinent to the FSLP strategy. Among others, these concepts constitute a cognitive perspective of classroom learning: (a) students are viewed as active learners; (b) the conditions for meaningful learning are enhanced when student prior knowledge is activated; (c) the use of multiple representations offers a framework for teachers to present the mathematics concepts in more than one modality (visual, verbal, or symbolic) and provides the students with the opportunity to develop their cognitive operational structure by accommodating various forms of representation; teachers collect the evidence of the students’ progress as they demonstrate newly learned ideas in more than one form of representation; (d) concept and task analysis helps teachers reveal underlying sub- eoncepts and skills to plan a gradual progression from one level of representation to another. This view of learning and teaching, together with the established national (NCTM, 2000) and local content standards for mathematics provides the professional knowledge base for mathematics teachers. The conception of pedagogical content knowledge combined with findings from instructional design research, forms the framework for Four Stages of Lesson Planning strategy and the Lesson Plan Evaluation Rubric. While the strategy provides a means by which mathematics teachers they can apply professional knowledge in their classroom practice, written lesson plans provide a trail of evidence that can be used to gain insight into teachers’ pedagogical content knowledge.

The Project and the Research

The Middle School Mathematics Initiative (MSMI) professional development program implemented the FSLP strategy to affect the instructional core of teaching, which includes lesson planning (Elmore, 2000). The purpose of the two-year Middle School Mathematics Initiative (MSMI) project was to assist underperforming middle schools, as identified by the statewide standardized test scores, in improving student achievement in mathematics. Fifty teachers volunteered for the program in the first year of implementation. They came from 14 middle schools in 8 districts and were served by six mathematics specialists selected by the state department of education through an interview process. The specialists were expert mathematics teachers with advanced knowledge in mathematics content and pedagogy, had been teaching in the public schools for ten or more years, and had been identified as educational leaders in their schools and districts. They received training in the use of the FSLP strategy and were assigned to coach the participating teachers in the use of the strategy and monitor the quality of their lessons. They did not carry a teaching load and, therefore had the opportunity to work on a daily basis with the project teachers individually and collectively. They used the Lesson Plan Evaluation Rubric (LPER) for scoring the teachers’lesson plans and the Lesson Observation Guide (LOG)^sup 2^ for observing the delivery of the planned lesson.

In the second year, 39 teachers volunteered their participation in the project. Amont; them were 24 teachers were from the previous year. The teachers came from 12 middle schools in 7 districts and were served by the same six mathematics specialists. Each specialist served six to eight teachers in one or two schools. For both years, the project provided the teachers with a fund for student-usable classroom materials and the option for taking a graduate level content course focused on middle school mathematics. The total of 44 teachers took the courses during the project.

Training of the Specialists and the Teachers

Specialists: The training was provided on a multilevel basis: the research team * the specialists * the teachers, and the research team * the teachers. The goals of the training were (a) to deepen the specialists ‘ understanding of the Four Stages of Lesson Planning strategy, (b) to establish the reliability of the instruments used to analyze lesson plans and classroom observations, and (c) to advance in the specialists the skills necessaiy to provide feedback to the teachers on their lessons. The specialists participated in a six-day session of formal training and were also engaged in four sessions based on collaborative lesson observations of middle school mathematics teachers who had been involved in the project. In addition, to assist the specialists in conducting lesson \analysis, the project research director accompanied them on a visit to each teacher to observe lessons together. They held joint conferences with the teachers after each observation. Through the course of the formal training sessions, the collaborative observations, and joint classrooms visits, the research team and the specialists worked on the development of a common language for the analysis of lessons, developed inter-rater reliability, and validated the LOG and LPER instruments.

Teachers: All participating teachers attended after-school workshop at the beginning of the school year lead by the project research director. The purpose of the workshops was to describe the Four Stages of Lesson Planning strategy and set up the expectations for lesson development. The teachers viewed and discussed videotaped lessons produced by the research team for training purposes. In addition, the specialists regularly met with their teachers to provide ongoing training.

The specialists regularly conducted preand post-observation conferences with each teacher. During the pre observation conference, the specialist and the teacher reviewed the lesson plan, and after the lesson both engaged in the analysis of teaching.

Data collection

Since “there is big leap from preparing to do something to actually doing it” (Hall & Hord, 2001, p. 36), the data were collected consistently during the second year of the project allowing the teachers time for implementation of the FSLP strategy. We gathered and examined 261 lesson plans generated by 39 participating teachers. Each specialist collected from five to eight lesson observation packets. The packets included a lesson plan from the teachers with a fully presented homework assignment (not only the problems assigned by numbers from a textbook), a Lesson Plan Evaluation Rubric (LPER) completed by the specialist, field notes of the classroom observation written by the specialist, a Lesson Observation Guide (LOG) completed by the specialist, and student work samples (class work or homework).

We also scored each lesson plan using LPER, and together with the specialists reviewed all discrepancies that occurred when applying the rubric to the lesson plans to achieve total agreement on all items.

Analysis of the LPER Data

We focused our investigation on the patterns and relationships among the LPER items as they associate with the FSLP strategy, on detecting the nature of the clusters of items, and on the verification of the conceptualization of the FSLP construct. We posed two questions: (a) Why are certain factors are grouped together empirically? (b) What are the underlying principles in the development of lesson plans that result in the factors that include different items of the LPER? To analyze interrelationships among a large number of variables produced by LPER items and to explain these variables in terms of their common underlying dimensions, we chose the method of Principal Components with Varimax rotation (SPSS, 2002).

While we postulated the connection between the stages and the items in the LPER, the empirically obtained evidence helped to surmise the interrelation between the individual items within the components. Analyzing the data, we continuously reflected on how well the hypothesized components explain the data and observed how the components correspond to the meaningful relationships between the LPER items, FSLP strategy, and their underlying theoretical constructs. The parallel analysis helped to demonstrate high internal consistency of the LPER and the FSLP strategy.

The LPER items clustered into four factors accounted for 49.9% of variance (Table 1). The strong correlations (> 0.4) for each LPER item are highlighted in bold with a weak correlation (0.3

Factor 1 : Worked-Out Problems. Four items are strongly correlated in this factor. These items stand for having all types of problems in the lesson plan worked out, namely, homework problems (2.2), mental mathematics problems (3. Ia), problems that teacher would use during instruction (3. Ib), and problems that the students would complete during the class (3.Ic).

One of the underlying principles of the FSLP strategy is to build the alignment between homework, classroom activities and mental mathematics to better facilitate students’ perception of the coherence of the studied concepts and tasks. The purpose of solving the problems is not to merely obtain the answers, but to scrutinize each task and its underlying concepts, to comprehend the nature ol’the concepts, and to delineate them by developing a hierarchy of prerequisite knowledge. By examining the solutions of all problems, teachers have a better picture of the scope of the concepts and sub- concepts that students will be learning. This makes them better prepared for making conscious decision when selecting and sequencing the activities by ordering them from simple to complex and more inclusive.

Also, the correlation with the mental mathematics problems involved in this factor indicates that their selection was not random. One of the FSLP tenets, is to create mental mathematics that is closely connected to the homework and classroom problems to ensure consistency and coherency of the information presented.

There is a weak correlation with the item 5.2, logical flow of the lesson through the phases. Such an association is consistent with the principles of having the problems from the various phases (mental mathematics, homework, and developmental activities) worked out. Having worked through problems allow a teacher to make better decision about the relevance of the selected problems across the phases, provide better inclusion of the concepts, thus ensure a better flow of the lesson. Association between all types of classroom problems (and/or exercises) provides the foundation for building connections among mathematics concepts, helps to avoid unnecessary repetition and drill that do not lead to understanding, and to present the concepts through a variety of contexts to substantiate meaningful learning.

Factor 2: By-products of the FSLP. This factor consists of five items that are strongly related; student grouping (item 3.2), the presence of distinct and specific phases of the lesson aligned to the FSLP strategy (item 5.1), embedded assessment in each phase (item 5.3), time guides for each phase (item 5.4), and alignment to the state mathematics framework (item 6.1). These items are logical by-products of the FSLP. They illustrate an important underlying organizational principle of the strategy. This lesson planning strategy results in a lesson that is structured in phases: an opening activity (such as mental mathematics, do-now exercises, etc), developmental activities that cou Id include a teacher- directed phase, student activities in pairs, groups, or individually, and a phase to explicitly link the homework to the lesson’s activities.

Three items, students grouping (item 3.2), phases of the lesson (item 5.1), and time guides (item 5.4) are closely connected; such association seems logical. Activities of different types arc more effective when they are coordinated with the most appropriate class arrangement. To create effective classroom setting and to provide students with different learning experiences, the teachers need to make decisions whether to set up a pair or group work, or to address the whole class. Changing the classroom environment would associate with phases of the lesson for example, from mental mathematics with a whole class to pair work on a problem that requires exploration, and then back to the whole class setting for summary and conclusion. Time guides help to treat the time allowed for each phase as a valuable resource. The idea of time guides does not contradict to the belief that plans should be considered tentative and be flexible. Estimating time for a certain phase of the lesson is important to prevent a common shortcoming of many lesson plans: they are overwhelmed with the concepts to be learned and problems to be solved. The data from TIMSS (NCES, 1999) show that over 90% of mathematics class time in the United States 8th-grade classrooms is spent on practicing routine procedures, with the remaining time generally used to apply procedures in new situations. Virtually, no time is given to inventing new procedures and analyzing unfamiliar situations. Such lesson “design” is a result of minimal attention to planning, in general, and no attention to proper classroom time treatment in particular. Leinhardt (1993) found that common to many lessons is the following, “I will go over yesterday’s homework on the board, but I don’t know how many I am going to go over, because there are 25 problems. I will see how it goes. If the students are getting them quickly, we’ll move on” (p. 12). Perhaps the appropriate metaphor that would portray such lesson is a trip that ran out of time and was not completed. That is why the FSLP strategy fosters and encourages time guides to make realistic estimation of each phase of the lesson.

To achieve the alignment required by the state, we asked teachers to match their objectives with the state mathematics standards, which are formulated in terms of observable indicators including all levels of thinking. Because most of the objectives formulated by the teachers were identical to the standards, item 6.1 will be referred as objectives. The correlation between item 6.1 and embedded assessment (item 5.3) supports another fundamental principle of the FSLP strategy, the significance of formative assessment. The objectives (in this component aligned to the state curriculum framework) provide explicit context \for both the teacher and the students to make informed decisions about the status of the learning outcomes. They offer the basis for planning instructional and assessment strategies, thus allowing the teacher to make adjustments to lesson and facilitate the progress toward those objectives. The strategy encourages continuous recognition of the needs in revising instruction and assessing the degree to which the objectives have been achieved.

Factor 3: Lesson Coherence. Three items are strongly correlated in this factor; homework linked to instructional objectives (item 2.1), effective use of mental mathcmatics in light of the objectives, students’ sub-skills, and prior knowledge (item 4.1), and the logical flow of the lesson through the phases (item 5.2). Two other items correlate weakly with the factor, distinct phases of the lesson (item 5.1 ), and embedded assessment (item 5.3). The correlation of these items reveals the consistency of the lesson plan. The core proposition of the FSLP strategy is that well- structured lessons flow from wellspecified objectives that are closely connected to homework-uniquely placed as a second stage of planning in the FSLP strategy. Item 2.2, homework related to the objectives, received the highest scores in this factor. The association of the mental mathematics (item 4.1 ) with this factor supports the FSLP principle of the strong connections between all stages of planning. Designing mental mathematics when the objectives, homework, and all major activities are drafted completes the planning cycle and ensures coherence and the logical (low through the phases (item 5.2) of the lesson from its very beginning.

Factor Four: Representations. Four items are strongly correlated with this factor, the specification of objectives (item 1.1), the use of multiple representations (item 6.2), the alignment with the state curriculum framework (item 6.1), and student misconceptions (item 6.4). The association of the items 1.1 and 6.1 has been already elucidated in factor three.

Interesting is the association of the multiple representations (items 6.2) with both items that relate to objectives. Such association shows that the lesson plans that contained well-stated cognitive objectives formulated in terms of observable behavior, also incorporated varied forms of representations. This seems essential to lesson planning that is based on the FSLP strategy. Multiple representations are distinct verbal, visual, and symbolic means of communicating information through external representations. Objectives exemplify the types of skills the students are expected to exhibit and involve the descriptors of the various treatment of representations such as organizing, recording, recognizing, drawing a picture, explaining, interpreting a graph, selecting, applying mathematical ideas, using symbolic (formal) mathematical language, and translating among mathematical representations to solve problems. Thus, to measure learning outcomes and to assess students’ progress toward meeting the objectives, the teachers must plan activities that involve visual representations (diagrams, pictures, graphs, tables), verbal representations (words), and symbolic representations (variables, expressions, operations, equations) for students to convey ideas and make the connections between them. When students have the opportunity to use, compare, and contrast different forms of representations (pictorial, verbal, symbolic), it is likely that they develop a capacity for and expand their operational structures.

The item related to predicting and treating student misconceptions (6.4), is also strongly correlated to this factor. The examined lesson plans that were carefully elaborated, evidenced teachers’ deeper understanding of the subject matter and the cognitive problems students might experience, thus showed the teachers’ stronger ability to anticipate and treat students’ possible misconception. The correlation of the items 6.2 and 6.4 is very important. It presents the evidence that the lesson plans, incorporating multiple representations, better articulated and suggested different tactics of treating misconceptions. Mathematics misconceptions often result from the use of only one representation of a concept demonstrated to the students and the lack of different opportunities to see the concept by ways that make sense to them. When multiple representations are encouraged by the teacher and new evidence about the concept is presented, for example, treating a common misconception related to the erroneous assumption that the sum of two squares is equal to a square of the sum of two quantities by using diagrams (see Figure 3), the students are empowered with multiple ways of learning mathematics.

Figure 3. A diagram that treats a misconception about the sum of two squares.

While weak, the factor shows a relationship between planning mental mathematics (item 4.1) and incorporating multiple representations (item 6.2). By definition, mental mathematics is a set of problems that can be solved mentally. However, such problems should reflect not only low-level cognitive skills. They must incorporate pictures or diagrams to recognize or to read, as well as problems that require higher order thinking such as explaining the process of transition from -2y > 4 to y

Table 1

Principle Components Analysis of LPER Items (Varimax Rotation)

The LPER item related to mathematical errors (Figure 2, item 6.5), show very low score variance. Only 24 lessons (9%) have serious mathematical errors. The item doesn’t correlate well with the instrument, and the data from the item do not contribute to a better understanding of the relationship between FSLP and LPER.

Concluding Remarks

In conclusion, we stress that lesson plan should be viewed as tentative and flexible composition of lesson elements that are connected by means of logical bonds, which arc rooted in the relations among mathematical concepts. The teachers show they are skillful in planning when they utilize varied approaches and lesson components and focus on lesson coherency. They realize that content and student needs dictate the choice of methods and not vice versa. They forge a solid link between the presented concepts and combine the myriad of small classroom activities into a coherent structure. The lessons plans developed using the FSLPstrategy and its objectives-first, homework second, and opening-activity-last focus, showed a higher degree of lesson coherence.

This article calls for renewing attention to the elements of effective lesson planning.

If we are to change the status of and improve mathematics learning, substantial attention and time must be invested in promoting thorough development of detailed and well-thought-out written lesson plans.

Footnotes

1 The FSLP strategy suggests the sequence of planning not delivery steps.

2 The description of the Lesson Observation Guide (LOG) goes beyond the purpose of the paper.

References

Airasian, P. W. ( 1994). Classroom assessment (2nd ed.). New York: McGraw Hill.

Ausubel, D. P. ( 1968). Educational psychology: A cognitive view. New York: Holt, Rinehard & Winston.

Bloom, B. S. (Ed.). (1956). Taxonomy of educational objectives: The classification of educational objectives by a committee of college and university examiners. Handbook 1 : Cognitive Domain. New York: D. McKay.

Branch, R. C. ( 1994). Common instructional design practices employed by secondary school teachers. Educational Technology. March 1994, 25-33.

Branch, R. M., & Gustafson, K. L. (1998). Re-visioning models of instructional development. Paper presented at the Association for Educational Communications and Technology, St Louis, MO, February, 1998. ERIC document number ED416837.

Briggs, L. J. (1977). Instructional design: Principles and applications. Englewood Cliffs, NJ: Educational Technology Publications.

Bruner, J. (1966). Toward a theory of instruction. Cambridge, MA: Harvard University Press.

Cates, G. L., & Skinner, C. H. (2000). Getting remedial mathematics students to choose homework with 20% and 40% more problems: An investigation of the strength of the interspersing procedure. Psychology in the Schools, 37, 339-347.

Clark, C. M., & Dunn, S. (1991). second-generation research on teachers’ planning, intentions, and routines. In H. C. Warren & H. J. Walberg (Eds.), Effective teaching: Current research (pp. 183- 200). Berkeley, CA: McCatchum Publishing.

Cruiekshank, D. R., Bainer, D. L.. & Metcalf, K. K. (1999). The act of teaching, (2nd ed.). Boston: McGraw-Hill College.

Dick, W. & Carey. L. ( 1996). The systematic design of instruction, (4th ed.). New York: Harper Collins College Publishers.

Elmore, R. F. (2000). Building a new structure for school leadership. Washington, DC: Albert Shanker Institute.

Ertmer, P. A., & Newby, T. J. (1993). Behaviorism, cognitivism, constructivism: Comparing critical features from an instructional design perspective. Performance Improvement Quarterly, 6 (4), 50- 70.

Gagn , R. M. (1962). The acquisition of knowledge. Psychological Review. 69. 355-365.

Gagn, R. M. ( 1965). The conditions of learning. New York: Holt, Rinehart & Winston.

Gagn, R. M. (2001 ). Preparing the learner for new learning. Theory Into Practice 19 (1), 6-9.

Gagn, R. M., & Briggs, L. J. (1979). Principles of instructional design, (2nd ed.). New York: Holt, Reinhart, and Winston.

Gill, B., & Schlossman, S. (2000). The lost cause of homework reform. American Journal of Education, 709(1), 27-63.

Hall, E., & Hord, S. (2001). Implementing change. Boston: Allyn & Bacon.

Kac\higan, S. K. (1986). Statistical analysis: An interdisciplinary approach to univariate and multivariate models. NY: Radius.

Kagan, D. M., & Tippins, D. J. (1992). The evolution of functional lesson plans among twelve elementary and secondary student teachers. Elementary School Journal, 92 (4), 477-489.

Kennedy. M. F. (1994). Instructional design or personal heuristics in classroom instructional planning. Educational Technology, March 1994, 17-24. “

Larkin, J. H., & Simon. H. A. ( 1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11 (1), 65- 100.

Leinhardt, G. (1993). Instructional explanations in history and mathematics. In W. Kintsch (S.A.),Proceedings of the Fifteenth Annual Conference of the Cognitive Science Society (pp. 5-16).

Hillsdale, NJ: Lawrence Erlbaum Associates. Mager, R. F. (1984). Preparing instructional objectives (2nd ed.). Belmont. CA: Lake Publishing.

Merrill, M. D. (1971). Instructional design: Readings. Englcwood Cliffs, NJ: Prentice Hall.

Namboodiri K., Corwin R. G., & Dorsten L. E. ( 1993). Analyzing distributions of school effects research: An empirical illustration. Sociology of Education. 66 (4), 278-294.

National Center for Education Statistics. ( 1999). TlMSS videotape classroom study: Methods and findings from an exploratory research project on eighth-grade mathematics instruction in Germany, Japan, and the United States [NCES 199-074]. Washington, DC: Government Printing Office.

National Council of Teachers of Mathematics. (2000). Principle and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.

Orlich, D. C., Harder, R. J., Callahan, R. C., Kauchak, D. P., Pendergrass, R. A., Keogh, A. J., & Gibson, H. (1990). Teaching strategies: A guide to better instruction, (3rd ed.). Lexington, MA: D. C. Heath.

Panasuk, R. M. (1999). Collaborative investigations: enhancing elementary and middle school teachers pedagogical content knowledge and increasing mathematics student achievement. Unpublished Manuscript. University of Massachusetts Lowell.

Panasuk, R. M. (2002). Connected mental mathematics: Systematic planning. Illinois Mathematics Teacher Journal, 53 (4), 3-9.

Panasuk, R. M. (2005a). Planning mathematics instruction with a strateg\. Manuscript submitted for publication.

Panasuk, R. M. (2005b). Concept ami task analysis: Another look at planning instruction Manuscript submitted for publication.

Panasuk. R. M., & Cutler, J. (2001). Promoting number sense. Mathematics Educational Leadership Journal, 5(1), 1-10.

Panasuk. R. M., Stone, W. E., & Todd. J. W. (2002). Lesson planning strategy for effective mathematics teaching. Education, 122 (4), 808-826.

Panasuk, R. M., & Sullivan,M. (1998). Need for lesson analysis in effective lesson planning. Education, 118, 330-345.

Panasuk, R. M., & Todd, J. W. (2002). Final report on the second year of the Middle School Mathematics Initiative. Unpublished manuscript. University of Massachusetts Lowell.

Piaget, J. ( 1963). The origins of intelligence in children. New York: Norton.

Piaget, J. ( 1970). Science of education and the psychology of the child. New York: Orion Press.

Reiser, R. A. (1994). Examining the planning practices of teachers: Reflections on three years of research. Educational Technology, March 1994, 11-16.

Reiser, R. A., & Mory, E. H. (1991). An examination of the systematic planning techniques of two experienced teachers. Educational Technology Research and Development 39 (3), 71 -82.

Schiffman, S. S. ( 1995). Instructional systems design: Five views of the field. In GJ. Anglin (Ed.), Instructional technology: Past, present and future (2nd ed., pp. 131-142). Englewood, CO: Libraries Unlimited, Inc.

Schn, D. ( 1983). The reflective practitioner. New York, N. Y.: Basic Books.

Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57(1), 1-22.

Silver, E. (1998). Improving mathematics in middle school: Lessons from TlMSS and related research. U.S. Department of Education, Office of Educational Research and Improvement. ERIC document number ED417956.

Simon, H.A. ( 1992). Alternative representations for cognition: Search and reasoning. In H.L. Pick. Jr., P. van den Broek, and D.C. Knill (Eds.), Cognition: Conceptual and methodological issues (pp. 121-142). Washington, DC: American Psychological Association.

Spadano, J. W. (1996). Examining a homework model us a means of advancing ownership of understanding and responsibility in secondary mathematics education. Unpublished doctoral dissertation: University of Massachusetts Lowell.

SPSS. (2002). SPSS 10.0 for the Macintosh, Version 10.0.7α. Chicago: SPSS.

Stiggins, R. J. (2001). Student-involved classroom assessment (3rd ed.). Upper Saddle River, NJ: Merrill Prentice Hall.

Stigler. J., & Hiebert, J. ( 1999). The teaching gap. NY: The Free Press

Tabachneck-Schijf, H. J. M., Leonardo, A. M., & Simon, H. A. (1997). CaMeRa: computational model of multiple representations. Cognitive Science, 21 (3), 305-350.

Wong, M. R., & Raulerson. J. D. (1974). A guide to systematic instructional design. Englcwood Cliffs, NJ: Educational Technology Publications.

Regina M. Panasuk, Ph.D., Professor of Mathematics Education, University of Massachusetts Lowell. Jeffrey Todd, Graduate School of Education, University of Massachusetts Lowell.

Correspondence concerning this article should be addressed to Dr. Regina M. Panasuk, Professor of Mathematics Education. University of Massachusetts Lowell, 61 Wilder Street, Lowell, MA 0185; Email: [email protected]

Copyright Journal of Instructional Psychology Sep 2005