Effectiveness of lesson planning: factor analysis.

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 3 9 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.

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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.

Figure 1. Four stages of lesson planning.

OBJECTIVES
formulated in terms of students' observable
behavior

HOMEWORK
matches the objectives

DEVELOPMENTAL ACTIVITIES
reflect the objectives
advance development and learning

MENTAL MATHEMATICS
activates prior knowledge, prepares students for
the acquisition of new concepts

ELEMENTS OF INSTRUCTION

Instructional Environment

* Inquiry-Based Instruction
* Expository/Direct Teaching
* Labs and Projects

Instructional Approaches based on

* Problem Solving
* Multiple Representations
* Critical Thinking
* Communication
* Connections

Class Arrangements
* Individual
* Group Work
* Pair Work

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. Schon (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 (Briggs, 1977; Merrill, 1971 ; Wong & Raulerson, 1974) in a written form of lesson planning. Teachers' records of their lesson plans are "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 Four 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 (1) 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.

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

Lesson Plan Evaluation Rubric

Teacher Number:                Date of Observation:

FOUR STAGES OF LESSON PLANNING

 1.1 OBJECTIVES
      4   All objectives are stated in terms of student observable
          behavior and specified skills and knowledge
      3   All objectives are stated in terms of students observable
          behavior, but do not specify skills and knowledge
      2   Some objectives are stated in terms of student observable
          behavior
      1   Objectives are stated, but none are in terms of student
          observable behavior
      0   No objectives are stated
 2.1 HOMEWORK: Linked to Objectives
      3   Homework problems are listed, match the objectives, and
          contain some problems that reinforce students' prior
          knowledge
      2   Homework problems are listed and match the objectives
      1   Homework problems are listed, but do not relate to the
          objectives
      0   No homework is addressed in the lesson plan
 2.2 Worked-Out Problems--Homework
      2   All homework problems have been worked out
      1   Some key homework problems have been worked out
      0   None of the homework Problems have been worked out
3.1a Worked-Out Problems--Mental Math
      2   Every (or nearly every) mental math problem have been worked
          out
      1   Some (particularly some key) mental math problems have been
          worked out
      0   None (or very few of the mental math problems have been
          worked out
3.1b Worked-Out Problems--Development (Teacher's Presentation)
      3   All problems have been worked out
      2   Key problems critical to the development have been worked out
      1   Some problems have been worked out, but not at least one
          problem critical to the development
      0   None (or very few) of the problems have been worked out
3.1c Worked-Out Problems--Classroom Exercises (Students' Work)
      1   At least key problems for the students' work have been
          worked out
      0   None (or very few) of the problems for the students' work
          have been worked out
 3.2 Student Grouping
      1   Student grouping for activities (seatwork, pairs, group
          work, etc.) is indicated
      0   Student grouping for activities is not indicated
 4.1 MENTAL MATH please circle the elements you find in the lesson
      3   The opening activity relates to the objectives, focuses on
          subskills needed for the lesson, and surfaces student prior
          knowledge
      2   The opening activity has two of the above characteristics
      1   The opening activity has one of the above characteristics
      0   The opening activity has none of the above characteristics
          or there is no opening activity
 5.1 PHASES OF THE LESSON please circle the elements you find in the
     lesson
      3   The lesson contains at least four phases: Opening Activity,
          Teacher Exposition of Material, Student Activities (group
          work, seat work, etc.), Closing Summary with Preview of
          Homework
      2   Three of the four phases are present in the lesson plan
      1   Two of the four hoses are present in the lesson plan
      0   One or none of the four hoses are resent in the lesson tan
 5.2      Logical Flow of the Lesson Through the Phases
      3   The exposition and activities clearly provide logical links
          between students' prior knowledge and the skills need to
          complete the homework
      2   The logical flow of the lesson is unclear, but related to the
          objectives
      1   The logical flow of the lesson is not related to the
          objectives
      0   There are no phase indicated in the lesson plan
 5.3      Embedded Assessment
      3   The embedded assessment is clearly identified in each hose of
          the lesson
      2   The embedded assessment is clearly identified or some phases
          of the lesson
      1   The embedded assessment is not clear but can be inferred for
          at least one phase of the lesson
      0   There is no evidence of embedded assessment
 5.4      Time Guide
      1   There are at least four phases with time guides that are
          appropriate and total time agrees with the period length
      0   There are few or no time guides

OTHER ELEMENTS OF LESSON PLANNING

 6.1 Aligned with the state Math Curriculum Standards
      2   The objectives are adequately referenced to the state MCS
          using strand and standard number
      1   The objectives incorrectly reference the state MCS using
          strand and standard number
      0   The objectives do not reference the state MCS using strand
          and standard number
 6.2 Multiple Representations please circle the elements you find in
     the lesson
      2   Classroom examples incorporate all three forms
          (pictorial/concrete, verbal, symbolic) of representation
      1   Classroom examples incorporate two of the three forms of
          representations
      0   Classroom examples incorporate only one form of
          representation
 6.3 Adaptations for Special Needs
      1   The lesson has been adapted/extended to meet individual
          students' needs
      0   There is no indication that the lesson has been adapted/
          extended to meet individual students' needs
 6.4 Address Student Misconceptions
      2   There is an indication of at least one possible student
          misconception and a plan to address the misconception(s)
      1   There is an indication of at least one possible student
          misconception(s)
      0   There is no indication of possible student misconceptions
 6.5 Mathematics Content
      1   The lesson plan contains no mathematical errors, or only
          minor mathematical errors
      0   The lesson plan contains several mathematical errors, or
          serious mathematical errors

Scorer:              Date of Scoring:               Total Score:

Teacher Number:                Date of Observation:

LESSON PLAN EVALUATION RUBRIC (LPER)

 1.1 OBJECTIVES
      4   All objectives are stated in terms of student observable
          behavior and specified skills and knowledge
      3   All objectives are stated in terms of students observable
          behavior, but do not specify skills and knowledge
      2   Some objectives are stated in terms of student observable
          behavior
      1   Objectives are stated, but none are in terms of student
          observable behavior
      0   No objectives are stated
 2.1 HOMEWORK: Linked to Objectives
      3   Homework problems are listed, match the objectives, and
          contain some problems that reinforce students' prior
          knowledge
      2   Homework problems are listed and match the objectives
      1   Homework problems are listed, but do not relate to the
          objectives
      0   No homework is addressed in the lesson plan
 2.2 Worked-Out Problem Homework
      2   All homework problems have been worked out
      1   Some key homework problems have been worked out
      0   None of the homework problems have been worked out
3.1A Worked-Out Problems Mental Math
      2   Every (or nearly every) mental math problem has been
          worked out
      1   Some (particularly the key) mental math problems have been
          work out
      0   None or few of the metal math problems have been worked out
3.1B Worked-Out Problems Development (Teacher's Presentation)
      3   All problems have been worked out
      2   Key problems critical to the development have been worked
          out
      1   Some problems have been worked out, but some critical to the
          development are not
      0   None or few of the problems critical to the development have
          been worked out
3.1C Worked-Out Problems Classroom Exercises (Students' Work)
      1   At least key problems for students have been worked out
      0   None or few of the problems for students have been worked out
 3.2 Student Grouping
      1   Student grouping for activities (seatwork, pairs, group work,
          etc.) is indicated
      0   Student grouping for activities is not indicated
 4.1 MENTAL MATH please circle the elements you find in the lesson
      3   The opening activity relates to the objectives, focuses on
          sub-skills needed for the lesson, and surfaces student prior
          knowledge
      2   The opening activity has two of the above characteristics
      1   The opening activity has one of the above characteristics
      0   The opening activity has none of the above characteristics
          or there is no opening activity
 5.1 PHASES OF THE LESSON please circle the elements you find in the
     lesson
      3   The lesson contains at least four phases: Opening Activity,
          Teacher Exposition of Material, Student Activities (group
          work, seat work, etc.), Closing Summary with Preview of
          Homework
      2   Three of the four hoses are resent in the lesson plan
      1   Two of the four hoses are resent in the lesson plan
      0   One or none of the four hoses are resent in the lesson plan
 5.2      Logical Flow of the Lesson Through the Phases
      3   The exposition and activities provide logical links between
          students' prior knowledge and the skills need to complete the
          homework
      2   The logical flow of the lesson is unclear, but related to the
          objectives
      1   The logical flow of the lesson is not related to the
          objectives
      0   There are no hoses of the lesson
 5.3      Embedded Assessment
      3   The embedded assessment is clean identified in each hose of
          the lesson plan
      2   The embedded assessment is dean identified for some hoses of
          the lesson plan
      1   The embedded assessment is not dear but can be inferred for
          at least one hose of the lesson plan
      0   There is no evidence of embedded assessment
 5.4      Time Guides
      1   There are at least four phases with time guides that are
          appropriate and total time agrees with the period length
      0   There are few or no time guides

OTHER ELEMENTS OF LESSON PLANNING

 6.1 Aligned with the state Math Curriculum Standards (MCS)
      2   The objectives are adequately referenced to the state MCS
          using strand and standard number
      1   The objectives incorrectly reference the state MCS using
          strand and standard number
      0   The objectives do not reference the state MCS using strand
          and standard number

 6.2 Multiple Representations please circle the elements you find in
     the lesson
      2   Classroom examples incorporate all three forms (pictorial/
          concrete, verbal, symbolic) of representation
      1   Classroom examples incorporate two of the three forms of
          representations
      0   Classroom examples incorporate only one form of
          representation
 6.3 Adaptations for Special Needs
      1   The lesson has been adapted/extended to meet individual
          students' needs
      0   There is no indication that the lesson has been adapted/
          extended to meet individual students' needs
 6.4 Addressing Student Misconceptions
      2   There is an indication of at least one possible student
          misconception and a plan to address the misconception(s)
      1   There is an indication of at least one possible student
          misconception(s)
      0   There is no indication of possible student misconceptions
 6.5 Mathematics Content
      1   The lesson plan contains no mathematical errors
      0   The lesson plan contains several mathematical errors

Scorer:              Date of Scoring:               Total Score:

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 established 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 emphasizes 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), Gagne 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. Cates 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 III. The FSLP strategy suggests planning the developmental activities after the 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, cognitivism, 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 Newby'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 in formation. 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, 3.1c) 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.

Stage IV. Planning mental mathematics, the final stage (Figure l and Figure 2, items 3.1 a, 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 Ausubel 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 Gagne'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 make 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 for use 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 summative 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 summative 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 summative 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 whether the students are ready for meaningful 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, Gagne, 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 Gagne'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-concepts 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) (2) for observing the delivery of the planned lesson.

In the second year, 39 teachers volunteered their participation in the project. Among 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 [right arrow] the specialists [right arrow] the teachers, and the research team [right arrow] 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 necessary 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 pre- and 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 < x < 0.4) in italics. We named the factors to demonstrate the essence and to communicate the nature of the underlying constructs in each factor (Kachigan, 1986). The following section discusses the meaning of underlying the LPER's factors as they relate to the FSLP strategy.

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 (3.1a), problems that teacher would use during instruction (3.1b), and problems that the students would complete during the class (3.1c).

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 of 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 could 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 are 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 mathematics 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 well-specified 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 flow 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 OMITTED]

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<-2, or making a decision whether x = 2 is a solution of the equation 5(x-3)=10. While designing mental mathematics, teachers analyze the sub-skills and sub-concepts that need to be emphasized by means of mental mathematics activities, thus they think about different forms of representations, which, in turn, affect their ability to produce more refined objectives.

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 are 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 FSLP strategy 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.

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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.

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: [e-mail]

Table 1
Principle Components Analysis of LPER Items (Varimax Rotation)

     Factors
                                           1       2      3       4

     LPER Item     Percent of Variance    15.2%  14.1%   10.3%   10.3%

1.1  Objectives                           0.058  0.096   0.095   0.580
2.1  Homework linked to objectives        0.097  0.106   0.804   0.096
2.2  Homework worked out                  0.602  0.230   0.187   0.085
3.1A Mental math worked out               0.646  0.110   0.074   0.157
3.1B Teacher problems worked out          0.784  0.121   0.144   0.043
3.1C Student problems worked out          0.811  0.008  -0.017   0.170
3.2  Student grouping                    -0.007  0.683  -0.067   0.175
4.1  Mental math criteria                 0.160  0.102   0.557   0.363
5.1  Phases of the lesson                 0.171  0.713   0.276   0.170
5.2  Logical flow of the phases           0.370  0.345   0.506   0.097
5.3  Embedded Assessment                  0.177  0.571   0.293   0.246
5.4  Time guides                          0.157  0.694   0.130  -0.111
6.1  Aligned to the state Math
       Curriculum Framework               0.149  0.420  -0.188   0.503
6.2  Multiple representations             0.015  0.062   0.270   0.581
6.3  Special need adaptations             0.200  0.250  -0.231   0.318
6.4  Student misconceptions               0.247  0.010   0.151   0.524