The exam consists of two parts. Two hours is allowed for each part. Unused time from one part may not be spent on the other part. In all cases, support arguments with citations of key research.
Part I. (2 hours). Answer all.
1. Select three established learning theories. Briefly describe each one. Discuss the use of these three theories in mathematics education -- explicitly link to the existing body of literature in mathematics education. Compare and contrast these three theories with your own view of what it means to learn and know mathematics. [Hint: theories to consider include: Associationism, Behaviorism, Gestaltism, Piagetian constructivism, Vygotskian constructivism, Information Processing, Social cognitive theory, Enactivism, Didactique, Embodied cognition.)
a. Describe an (actual or imagined) appropriate use of t-test.
b. Describe an (actual or imagined) inappropriate use of t-test.
c. Refer to the attached excerpts from the National Center for Educational Statistics report High school seniors' instructional experiences in science and mathematics (1996). Consider the data and the conclusions drawn about it. Are the authors justified in their conclusions? Carefully explain why/why not.
3. Select two established qualitative research designs. Calling upon your existing knowledge of the literature, give an example of each of the designs in use. Compare and contrast the two research designs while addressing the appropriateness of making changes in teaching practice that are solely based on qualitative research results.
4. Given the attached excerpts from the NCTM Principles and Standards for School Mathematics, and drawing on your knowledge of the literature, discuss at least two aspects of affect and how research on these aspects appears to inform the Principles and Standards document.
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Part II. (2 hours) Answer two of the four. No additional credit will be awarded if more than two are answered; only two will be graded.
This part focuses on the five research articles given out in April, 2002: Carter&Norwood, 1997;Chapman, 1997; Kieren et al., 1996; Tarte&Fennema, 1995; Yackel and Cobb, 1996, and the essay by Schoenfeld (2000). In all cases, support arguments with citations from these, and other key literature, as appropriate.
1. Consider the research study reported in the article by Olive Chapman (1997). Discuss the learning theory(ies) that appear to have influenced the author's choice of research design. Justify your answer.
2. Consider the papers by Carter and Norwood (1997) and Tarte and Fennema (1995).
a. Carefully discuss and analyze the reliability and validity of the research methods used in each paper.
b. Compare the rigor and specificity of the research reported in the two articles.
c. Describe a rigorous research study, informed by the work reported by Carter and Norwood as well as Tarte and Fennema, that would connect the research in the two papers.
3. Discuss the evolution of cultural norms, socio-mathematical norms, attitudes, and beliefs as reported in the three papers by Chapman (1997), Kieren et al. (1996), and Yackel and Cobb (1996). Cite support from other literature with which you are familiar, as appropriate.
4. Choose two of the five research articles (Carter&Norwood, 1997;Chapman, 1997; Kieren et al., 1996; Tarte&Fennema, 1995; Yackel and Cobb, 1996) and one other research report with which you are familiar. In the context of Schoenfeld's (2000) framework, analyze and compare the justifications for changes in practice offered in the three articles.