Mathematical Sophistication: What is it? In Seaman & Szydlik (2007) we defined a construct we call mathematical sophistication in terms of nine values and behaviors that give students power to solve problems and create mathematics. We say that an individual is mathematically sophisticated if she has taken as her own the ways of knowing or the mathematical community based on nine interwoven traits: Mathematicians seek to understand patterns. Mathematicians make analogies by finding the same essential structure in different objects. Mathematicians make and test conjectures about mathematical objects and structures. Mathematicians create mental (and physical) models for, and examples and non-examples of, objects. Mathematicians value and use precise definitions of objects. Mathematicains value an understanding of why relationships make sense. Mathematicians value and use logical arguments and counterexamples as our sources of conviction. Mathematicians value and use precise language and have fine distinctions about language. Mathematicians value and use symbolic representations of, and notation for, objects and ideas.

Project Background and Summary While many researchers have measured mathematics students' beliefs, procedural skills, conceptual understanding, or attitudes, few have sought to quantify mathematical behavior. Perhaps the most closely related work is Schoenfeld's 1992 study of the differing behaviors of college students and mathematics faculty in problem solving. He found that students often made quick decisions about an approach to take and pursued that approach even if it was unsuccessful. In contrast, faculty members spent significant time understanding the problem, analyzing and exploring. We assert that which Schoenfeld identified as the mathematician's "expert executive skills" is akin to the construct of mathematical sophistication. In previous research (Seaman & Szydlik, 2007) we argued that students in an elementary education program who demonstrated a deficiency in basic mathematics also showed an inability or unwillingness to engage in mathematically sophisticated behaviors. In the future we hope to find effective ways to foster mathematical sophistication in our students. However the first step is to figure out how to measure it.

Methods of Analysis In Phase 1 (Development) six mathematics faculty at the University of Wisconsin Oshkosh served as expert consultants and six elementary education majors participated as novice consultants in the development of the Mathematical Sophistication Instrument (MSI). In Phase 2 (Pilot Study), a sample of elementary education students in their mathematics content courses completed the MSI. Twelve of these students were then selected as participants for interviews in order for us to study whether sophisticated thinking led to sophisticated responses on the MSI and unsophisticated thinking led to unsophisticated responses. Phase 3 (WTS Study) Based on the results of the previous phases, in Fall 2008 we again revised the instrument and administered it to 56 elementary education majors in their mathematics content courses. Assessing Validity:The faculty teaching those content courses rated each of their students on a 5-point linear scale based on their expert opinion of each student's mathematical sophistication. Faculty ratings were compared to MSI scores to determine the degree to which the instrument can predict mathematical sophistication. Specifically, we tested to see whether there is a significant difference in mean score on the MSI between students at each sophistication level. Assessing Reliability: Reliability was assessed using the Cronbach alpha which indicates the degree to which an instrument measures a single construct.

Sample Items Below we present a sample of MSI items along with the percent of students who selected each response option. The most sophisticated response is shown in bold. Each of these items has an item-test correlation of approximately 0.5. _____________________________________________________ 1) You are a student learning about using lines to model data and, after the lesson, a student raises her hand and makes a guess about how other types of functions could be used to model data. Which option best reflects your view? A) I would probably just want to know whether she was correct or not. (16 percent) B) I would probably want to spend time exploring her guess myself. (48 percent) C) I would probably prefer to focus only on the material that was part of the real lesson. (7 percent) D) I would probably want the instructor to figure it out and explain it to me. (29 percent) _____________________________________________________ 2) Consider the following statement: There are at most six ducks in the pond. Assuming this statement is true, which of the following statements must also be true? A) There is at least one duck in the pond. (20 percent) B) There are six ducks in the pond. (9 percent) C) Both of the above statements must be true. (24 percent) D) None of the above statements must be true. (47 percent) ______________________________________________________________ 3) Consider the following statement: When you multiply any two even numbers together, you always get an even product. Which of these arguments convinces you that the statement is true? A) 6 x 2 = 12, 4 x 8 = 32, 12 x 2 = 24 and so you can see by doing examples that the products will all be even. (4 percent) B) Even numbers have two as a factor. If two is a factor of both numbers then two must also be a factor of their product. (49 percent) C) Both of the above are convincing. (38 percent) D) None of the above is convincing. (9 percent)

Annotated List of Helpful Resources & References Lester, F. K. (2007) Second Handbook of Research on Mathematics Teaching and Learning. National Council of Teachers of Mathematics. This is a synthesis of the research in mathematics education. It contains 31 chapters covering topics such as 'Mathematics Teachers' Beliefs and Affect" and "The Role of Culture in Teaching and Learning Mathematics" and "Research on Statistics Learning and Reasoning." This is the place to begin any literature review for a mathematics education study. Journal for Research in Mathematics Education (JRME). JRME is the flagship journal in mathematics education. Search their database for the most rigorously reviewed research. School Science and Mathematics and the Journal of Mathematics Teacher Education (JMTE) These are both good places to read about (and to submit) mathematics education research studies regarding the preparation of teachers. References Schoenfeld, A. H. (1992). Learning to think mathematically: problem solving, metacognition, and sense making in mathematics. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan Publishing Company. Seaman, C. E. and Szydlik, J. E. (2007). Mathematical sophistication among preservice elementary teachers. Journal of Mathematics Teacher Education, 10, p. 167-182.

Results & Implications The bar graph below shows the distribution of scores on the MSI. The mean score is 11.7 and the standard deviation is 4.65. The Cronbach alpha for the instrument is 0.78 suggesting that the MSI reliably measures a single construct.

The table below shows mean scores on the MSI for the students in each sophistication level as rated by their instructors. Level 1 indicates a rating of "highly unsophisticated;" Level 2 indicates a rating of "fairly unsophisticated;" Level 3 means "neutral sophistication;" and Level 4 indicates a rating of "fairly sophisicated." Only one student was rated at Level 5 ("highly sophisticated"). All of the differences in means are statistically significant at the 0.05 level with the exception of the difference in Levels 2 and 3, suggesting that the MSI is a reasonably valid measure of mathematical sophistication. The Level 5 student scored a 15 on the MSI. The instructors did not feel confident in rating all the students; a total of 43 were rated out of the 56 students who completed the assessment.

Instructor Ratings of Students' Mathematical Sophistication and Mean Score of each Rating Category on the MSI. N = 43.

Career Relevance & Impact This project represents a step in an ongoing program of SoTL research with my colleagues Dr. Carol Seaman and Dr. Eric Kuennen. We hope that the Measuring Mathematical Sophistication Instrument will provide the mathematics education community with a tool to measure an important aspect of mathematical knowledge, and instructors a means for assessing pedagogies designed to teach students to think mathematically. In the Fall 2009, Dr. Eric Kuennen, Dr. Carol Seaman and I plan a large sample trial of the MSI involving a variety of mathematics students at several universities. If you are interested in particiapating in this project, please contact me at szydlik@uwosh.edu. The WTS Program has provided a wonderful context for this work. I thank the University of Wisconsin System and the University of Wisconsin Oshkosh Faculty Development Program for their support. I also thank Dr. Joan Hart, Dr. Stephen Szydlik, Dr. Robert Earles, and Dr. John Beam for their assistance in the development of the instrument.