Each student in the secondary teaching concentration will complete a capstone project (MAT 5520). The preliminary work involved in selecting a topic, learning about human subject research, and building a literature review should be completed during enrollment in Investigations in Teaching Mathematics (MAT 5910). The project must be carried out in a high school mathematics classroom, with the active participation of a practicing teacher if the student is not currently teaching.

## Goals and Guiding Principles

The Capstone Project for is designed to be a culminating experience for students in the MA in Mathematics, Secondary Education program, integrating the knowledge and expertise gained over the course of the program with their work in the classroom. The teacher will apply their knowledge of mathematics, of teaching, and of mathematics education research to the systematic investigation of their classroom practice. Each teacher will work with a capstone advisor from the Department of Mathematics Sciences as they develop, implement, and summarize the capstone project. The goals of the capstone project are for the teacher to:

- develop familiarity with research on mathematics teaching and learning along with techniques for finding, understanding, critically analyzing, and applying that research;
- develop techniques for sustainably assessing, understanding, and improving one's own practice in teaching mathematics.

These goals will be met through a *research-based* investigative project in which the teacher investigates an aspect of his or her mathematics teaching practice. The investigation will be *research-based* in the sense that it will be informed and inspired by high-quality mathematics research. However, it is not meant to imitate the formal research procedures of a thesis. Rather, emphasis shall be placed on systematically analyzing one's own practice in a manner that allows the teacher to develop a deeper understanding of teaching and learning, ultimately leading to improved student learning.

## Possible Formats

The capstone project may take a variety of forms, some of which are discussed at the end of this document. However, all projects must:

- focus on the mathematics classroom and student learning of mathematics
- include an examination of relevant literature and provide evidence of understanding the research methodologies that influence the design of the project.
- include both a paper and an oral presentation/defense of the product in an open forum of university faculty, student peer, and/or professional practitioners

The specific capstone project is to be developed in by the student beginning in MAT 5910 and in consultation with the student's capstone advisor. The following are recommended formats, although other formats, based in or originating from high-quality mathematics education research, are possible. Any format should be based on a formal research methodology, of which the student is expected to convey understanding.

## Thesis Option

At the student's discretion, and in consultation with the graduate coordinator and capstone advisor, he or she may choose to develop the capstone project into a Master's Thesis. The thesis will satisfy the requirements of the capstone project but should embody high-quality mathematics education research. The goals and aims of the project are similar, but a thesis option student assumes greater responsibility for defining a conceptual basis for their project, identifying epistemological assumptions and their consequences, and for examining and utilizing relevant literature in order to justify and precisely define their methodology. In short, whereas a capstone project is described as *informal* research, a thesis should follow the principles of* formal* mathematics education research.

## Relevant Standards

The capstone project or thesis will be used as evidence in the student's Product of Learning. It will demonstrate evidence of North Carolina Professional Teaching Standards, Standard 3: "Teachers know the content they teach", particularly those elements of the standard that indicate knowledge of "content appropriate to their teaching specialty, recognition of "the interconnectedness of content areas/disciplines, and making "instruction relevant to students". It will also demonstrate evidence of Standard 4: "Teachers facilitate learning for their students". The capstone project or thesis may also serve as evidence in the Product of Learning of other standards, depending on the particular topic and investigation.

## Paper and Oral Presentation

Every capstone project will include both a paper and an oral presentation. Because formats of the project may differ, the specific format of these products may differ somewhat from student to student. However, each should communicate the following:

- Familiarity with the relevant research that serves as a basis for the project, communicating important results or frameworks and their relationship to the work undertaken in the capstone project. The paper, in particular, should articulate how this research has led to the decisions made in the capstone project.
- A discussion of the design of the project, justifying methodological choices.
- A communication of the results and implications, with emphasis on what was learned about student learning and how the project will impact classroom practice.
- A plan for extending the project for continued growth in teaching practice.

## Formats

### Informal Action Research

Action research is often used as a catch-all term for teachers investigating their own practice, but it does have a well-defined theoretical basis in literature. In one of the earliest descriptions of the process, Corey (1954) introduces the term as follows: "*Action research* in education is research undertaken by practitioners in order that they may improve their practices. The people who actually teach children or supervise teachers or administer school systems attempt to solve their practical problems by using the methods of science" (p. 375). The important defining characteristics of action research then, are the definition of a practical problem related to improving professional teaching practice and the systematic collection of data to address the problem and improve practice (Parsons & Brown, 2002). A student engaged in a capstone project utilizing informal action research should begin by defining an issue they would like to address, then investigate relevant sources in the research literature in order to design a process for collecting and analyzing data – quantitative, qualitative, or both – that will offer insight into the problem at hand.

### Curriculum Development and Analysis

In a curriculum development project, the teacher-researcher plans and enacts a curriculum in the classroom, analyzing its impact and effectiveness. Curriculum here is defined according to Clements (2007) definition: "a specific set of instructional materials that order content used to support pre-K-grade 12 classroom instruction" (p. 36). The field of curriculum design and assessment is extensive, and a student undertaking a curriculum development project should expect to develop familiarity with important principles before working to design curriculum. Clements (2007) identifies a *Curriculum Research Framework* consisting of three phases:

: In this phase, the subject matter and pedagogical bases of the curriculum are established. The guiding principles and reasons for decisions are well-established.*A priori foundations*: Here, existing research and prior knowledge are consulted in constructing a model for student thinking and learning specific to the particular subject addressed in the curriculum.*Learning model*: In the evaluation phase, curriculum standards, guidelines, standards, and audience factors are considered, and field tests are conducted to refine the curriculum.**Evaluation**

It should be noted that these phases are not distinct but part of an interwoven cycle. For instance, some evaluation will take place even during the foundations stage, and each phase will be revisited multiple times. A student engaging in a curriculum development project might choose a unit or topic for the focus of the project. Limiting the subject matter focus of the project permits a sufficient level of detail and analysis. The project should address all three phases of Clements' (2007) framework, justifying curricular decisions through established research knowledge, developing learning models, and evaluating the curriculum. It is possible that a student may choose to devote more attention to a certain phase if they develop more original or unique approaches to a specific aspect of curriculum development. For instance, if a student develops new models for learning a particular subject instead of drawing on previously-established models, the project may address that phase in more depth than others.

### Informal Teaching Experiment

A teaching experiment is a method for investigating the impact on teaching techniques or approaches on students' learning and understanding of mathematics. Steffe and Thompson (2000) describe the methodology as follows: "It is primarily an exploratory tool, derived from Piaget's clinical interview and aimed at exploring students' mathematics ... the teaching experiment is directed toward understanding the progress students make over extended periods. ... In this, it is a living methodology designed initially for the exploration and explanation of students' mathematical activity" (p. 273).

Teaching experiments are based on the understanding that individual students construct their own conceptions of mathematics independent of the mathematical understandings of teachers and other students (though a students' conceptions are certainly shaped by interactions with others). A teaching experiment, then, is an opportunity to investigate and analyze students' mathematical understanding. It differs from a curriculum development project in focus. A curriculum development project is primarily focused on analyzing the curriculum and instructional approach, whereas a teaching experiment is primarily focused on analyzing student thinking and knowledge. In many cases, this is done by an in-depth analysis of student cognition as students engage in learning mathematics in a single classroom, or by comparing student learning in two classes where different approaches are taken. In either case, attention should be paid to the fact that the lack of randomized assignment of students and other factors contributes to problems of generalizability. Nevertheless, teaching experiments provide valuable information to those who teach mathematics. A capstone project that adopts a teaching experiment approach will identify a domain of mathematical knowledge that is of particular interest and design a method for investigating and analyzing how students come to learn or understand a particular topic or in a particular context.

### Case Study

In-depth analysis of a single, well-bounded case can often provide a level of detail that isn't available with other types of projects. Though there are a variety of ways of conceptualizing case study, a prototypical case study may be defined by characteristics listed by VanWynsberghe and Khan (2007):

- Intensive and in-depth focus on a small sample, such as a single classroom or a very small number of students,
- Highly detailed context, carefully descriptions of the particular instance,
- Natural settings where the researcher does not exert control over organization or behavior. VanWynsberghe & Kahn note that "case study is uniquely suitable for research in complex settings because it advances the concept that complex settings cannot be reduced to simple cause and effect relationships." (p.4)
- Bounded by a specific time frame or particular context.
- Working hypotheses are constructed during data collection and analysis. A case study is not designed to test a hypothesis, but understanding of a phenomenon emerges over the course of the study.
- Drawing from multiple sources of data.

Critics of case study research point to the fact that the reliance on a single case means the conclusions of such a study are not necessarily generalizable. While true, this does not mean that case studies do not produce generalizations. Indeed, much of our individual knowledge is drawn from specific cases, and case studies can be instrumental in generating theories and disproving theories (Flyvbjerg, 2001). See Erlwanger's (1973) case study of a student's fraction arithmetic rules in a self-guided curriculum or Raymond's (1997) case study of a teacher's practice mirroring her own experience as a student instead of her teacher preparation program for two excellent examples of how case studies have served such a role in mathematics education research.

A capstone project adopting a case study approach will identify a phenomenon (perhaps a lesson, a class, a few students, a single student, a single teacher) that is of interest, justifying the focus on that particular case. The project should then seek to collect a variety of data in order to describe the case and generate working hypotheses throughout the process. The final product will include a rich description of the case along with supporting evidence and lessons learned from the research.

## References

Clements, D.H. (2007). Curriculum research: Toward a framework for "research-based curricula". *Journal of Research in Mathematics Education* 38(1), 35-70.

Corey, S.M. (1954). Action research in education. *The Journal of Educational Research*, 47(5), 375-380.

Erwanger, S.H. (1973). Benny's conception of rules and answers in IPI mathematics. *Journal of Children's Mathematical Behavior*, 1(2), 7-26.

Parsons, R.D., & Brown, K.S. (2002). *Teacher as reflective practitioner and action researcher.* Belmont, CA: Wadsworth/Thomason Learning.

Raymond, A.M. (1997). Inconsistency between a beginning elementary school teacher's mathematics beliefs and teaching practice. *Journal for Research in Mathematics Education*, 28(5), 550-576.

Steffe, L.P. & Thompson, P.W. (2000). Teaching experiment methodology: Underlying principals and essential elements. In R. Lesh & A. E. Kelly (Eds.), *Research design in mathematics and science education* (pp.267-307). Hillsdale, NJ: Erlbaum.

Van Wynsberghe, R. & Khan, S. (2007). Redefining case study. *International Journal of Qualitative Methods*, 6(2), Article 6. Retrieved from http://www.ualberta.ca/~iiqm/backissues/6_2/vanwynsberghe.pdf.