Preservice teachers enter a mathematics methods course with the expectation they will learn how to teach mathematics but instead often feel like they are presented a menagerie of theories and instructional techniques (Sowder, 2007). For preservice teachers, the seemingly discrete and disconnected information promotes the view that mathematics methods courses are theoretical and impractical (Sowder, 2007; Tatto, 1998).  One approach for bridging the gap between theory and practice, college instruction, and teaching in the K-12 setting is to train preservice teachers to apply a theory of instruction. "A theory of instruction, in short, is concerned with how what one wishes to teach can best be learned, with improving rather than describing learning" (Brunner, 1966, p. 40). 

In order to improve pre-service teachers' understanding and perceptions of constructivist-based teaching, a theory of instruction, Concept-Focused Instruction (CFI), has been developed and integrated in a secondary and middle level mathematics methods course. The theory meets the criteria for a theory of instruction (Bruner, 1966) and addresses the recommended features of K-12 teacher preparation (Sowder, 2007).  This session presents CFI and the impact it had on preservice teachers' perceptions of mathematics and their ability to plan and teach constructivist lessons.  The main research question informing the design and implementation was 'How does a theory of instruction (CFI) impact preservice teachers conception of mathematics and mathematics teaching?' In particular 1) How does CFI impact preservice teacher's perception of mathematics and mathematical knowledge? 2) How does CFI impact preservice teacher's understanding of how students learn mathematics? 3) How does CFI impact preservice teacher's ability to plan and implement constructivist-based lessons?

Theoretical framework

 Concept-Focused Instruction (CFI) is a theory of instruction founded on three core principles. The first principle is that in order for individuals to successfully learn and teach mathematics they must develop the perspective of mathematics as a conceptualizing process.  Mathematics educators have described varying perceptions of mathematics (Dossey, 1992) and how that impacts mathematics teaching and learning (Cooney, 1985; Philipp, 2007; Thompson, 1992).  One such view is mathematics as an interconnected series of concepts that have been constructed through experimentation, observation, and experience.  Based on research on mathematics teachers' beliefs, it can be inferred preservice mathematics teachers who adopt this perspective will be more inclined to use constructivist approaches.

The second principle of Concept-Focused Instruction (CFI) is when individuals engage in a conceptualizing process, they employ and integrate three attributes of a concept: the macroscopic, model, and symbolic.  The macroscopic attribute is a mental visual of the concept.  The model attribute is the tangible representation of the macroscopic attribute, and the symbolic embodies the definitions and formulas associated with the concept.  In the context of CFI individuals who describe macroscopic observations, create models, and apply the appropriate symbolic terms demonstrate conceptual understanding.  The third principle of CFI is in order to effectively teach concepts, teachers need to provide instruction that addresses the macroscopic, model and symbolic attributes of mathematics concepts.

Concept-Focused Instruction (CFI) can be considered a viable theory of instruction because it identifies with the experiences conducive to learning, describes an effective instructional sequence, explains the structure and form of knowledge within a discipline and provides guidelines for improving student motivation.  (Bruner, 1966)  In summary, Principles #2 and #3 address the experiences conducive to learning and the sequence of effective instruction.  Principle #1 addresses the structure and form of the knowledge. Finally, the guidelines for improving student motivation are inherent in Principles #2 and #3.  Students are more motivated when they are engaged in developing their intellectual skills.  This occurs when instruction is adapted to students' knowledge, understanding, and personal experience; when students have opportunities for exploration and experimentation; and when errors and mistakes are treated as a normal part of the learning process (Stipek, 1996).  CFI embeds these practices by emphasizing the need to begin with a macroscopic experience that allows students the opportunity to create, explore, develop and reflect on their models for targeted concepts.

Method

Concept-Focused Instruction (CFI) was used as the framework for a secondary and middle level mathematics methods course in a southeast mid-sized university.  Ten preservice teachers were enrolled in the course in Fall 2010. The methods course met once a week over a 16-week semester. During class meetings the preservice teachers participated in tasks designed to highlight the principles of CFI and applied them to create lectures, demonstrations, and problem-based learning experiences.  In addition to coursework, the preservice teachers were immersed in fieldwork for a period of one and two weeks.  

A collective case study (Stake, 1994) design was used. Each case included relevant course artifacts, documentation from internship (lesson plans, observations, and evaluations), and a transcribed audio recording of at least one interview.  To complete the data set, transcriptions of class meetings, along with the researcher's notebook documenting each class' agenda and any outcomes/thoughts/ideas related to CFI.  Also included were thoughts that conceptualize a collective case study: descriptions of the phenomena/artifacts selected; any patterns observed; decisions for triangulating key observations and interpretations; alternative interpretations to pursue; and any assertions/generalization about the case and/or collective case.

Findings

Eight out of ten preservice teachers' perspectives on mathematics and mathematical knowledge obtained views indicative of mathematics being a dynamic and a problem-driven discipline.  Similarly CFI impacted preservice teacher's understanding of how students learn mathematics and their ability to plan and implement constructivist-based lessons.  All the preservice teachers improved in terms of the level of inquiry in their planning and implementation as well as their use of process skills/standards.

Contribution to Teaching & Learning of Mathematics

This presentation presents a potentially effective approach for training preservice teachers to plan and implement constructivist-based lessons. Preservice teachers find themselves in a figurative sea of educational ideas, lesson plans, and activities from which they must select appropriate instructional materials. The task of selecting, planning, and implementing effective classroom instruction can be daunting.  Providing preservice mathematics teachers with a theory of instruction, i.e. Concept-Focused Instruction, improves their planning and teaching because it simplifies the instructional decision-making process.