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Assessing Mathematical Proficiency

Alan H. Schoenfeld, editor
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Christine Latulippe
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Assessing Mathematical Proficiency, edited by Alan Schoenfeld, presents the results of the discussions at a Mathematical Sciences Research Institute conference on mathematics assessment. The chapters are written by a diverse collection of notable stakeholders in the assessment and mathematics arenas, including Schoenfeld, Deborah Lowenberg Ball, and Pendred Noyce. Probably due to the fact that just over half of the authors are based in California  (as is the Mathematical Sciences Research Institute), the book has somewhat of a California slant in topics and examples.

Overall, I think the book has the potential to be useful to anyone who’s ever written or graded a math test, but it is primarily appropriate for faculty who are interested in assessment issues in K–12 education. It would be a good choice of text for a masters level assessment course, with nice overviews of national and classroom assessment, No Child Left Behind, and considerations of assessment myths and goals. Somewhat of a bonus feature for graduate students is the listing in the epilogue of twenty-four topics or questions that the group were left with at the end of the conference as a sort of “research agenda.” A few chapters might be applicable for new mathematics graduate teaching assistants (for example, Chapter 8 regarding mathematics for citizenship), or even for pre-service teachers or student teachers (Chapters 6, 19, 20).

Hugh Burkhardt’s Chapter 6 (“Mathematical Proficiency: What is Important? How Can It Be Measured?”) is a great collection of open-ended assessment items utilized to address some important questions surrounding assessment. The realistic and almost conversational tone of this chapter leads me to believe it could be used to discuss student mathematical capabilities at different ages or grade levels; advantages and disadvantages of open-ended questions vs. writing prompts (“write a recommendation to the mayor based on your findings”) vs. scaffolded questions; as well as ways that teachers can give hints or break down a problem without taking away the richness of the situation it is set in.

While reading this chapter, I enjoyed solving the test items, and also reconsidering my own goals and purposes when creating assessments for undergraduate courses. How do my own assessments align with the competencies that I value as an instructor and want students to leave my courses with? Have I ever really sat down to consider what I want students to gain in my course and then align every step of the course with those valued competencies?

Although there’s an embarrassing blunder regarding percentages in the footnote on page 333, Chapter 19 saved the final section of the book for me. Lilly Wong Fillmore explains through use of an example the difficulties related to assessing mathematics capabilities of someone whose native language is not English. In this chapter, the California-specific data about English language learners is relevant to new teachers in various parts of the country. There is a very clear explanation and illustration of the difference between no longer being labeled as an “English learner” by the school system and still having little understanding of academic language in either the native language or English. It’s an eye- opening discussion that could be paralleled with one about visible and invisible disabilities and their influences on assessment.

Assessing Mathematical Proficiency’s cover states that “a special feature is an interview with a student about his knowledge of fractions, demonstrating what interviews (versus standardized tests) can reveal.” My first thought was that this had been done before by Stanley Erlwanger in 1973 when he interviewed Benny, a 6th grader (Journal of Children’s Mathematical Behavior 7-26) about his understandings of fraction algorithms. However, Deborah Lowenberg Ball’s interview with Brandon, also a 6th grader, is impressive. Not only does this young boy hold up for over an hour and a half answering questions in front of an audience of adults, Lowenberg Ball’s questioning styles and interviewing techniques are masterful. Reading this interview transcript has value for teachers and qualitative researchers alike.

There are a limited number of typos in the text; just enough to be noticeable, but not enough to be distracting. In multiple places, the examples of student work or sample questions that must have been scanned in were quite fuzzy. It’s wonderful to have real student work within the text, but some models were difficult to read (not because of children’s handwriting but because of the technology utilized).

I felt that the book was not tied together very coherently, especially in the second half. Schoenfeld does a nice job in the preface, laying out the entire book, and each section also has an overview or introduction of the chapters within. In the beginning these are strong and the chapters within the section tie together quite nicely (for example, Section 2 consists of two chapters with different perspectives on the question of “what is mathematical proficiency?”). But by the final section (“The Importance of Societal Context”), the umbrella is larger and the chapters within don’t mesh together as well. The lack of cohesion is also visible in the authors’ references to one another: some don’t mention each other, some say “see X, this volume”, and still others say “see Y, chapter #.” With such a large collection of essays consistent cross-referencing could help the flow of the book for the reader.

Some chapters were weaker than others but taken as a whole I found Assessing Mathematical Proficiency to be informative, easy enough to sit down and read, but also a good “on-the-shelf” resource for additional assessment materials which were cited throughout.

Christine Latulippe is Assistant Professor of Mathematics Education at California State Polytechnic University at Pomona. She is a Project NExT Fellow (Sun Dot ’07), and is interested in professional development issues for preservice and inservice K–12 teachers as well as for math GTAs. As a new professor, she doesn’t have much time for hobbies, but Christine recommends that you make time to donate blood this year at your local American Red Cross or community blood bank. One of the top reasons that people give for not donating blood is “no one ever asked me… I didn’t realize my blood was needed”. Consider yourself asked!


Preface Alan H. Schoenfeld; Part I. The Big Picture: 1. Issues and tensions in the assessment of mathematical proficiency Alan H. Schoenfeld; 2. Crucial contemporary social, political, and cultural issues in mathematical assessment in the United States Judith Ramaley; 3. Crucial contemporary social, political, and cultural issues in mathematical assessment in the United States Susan Sclafani; Part II. Perspectives on Mathematical Proficiency: 4. What is mathematical proficiency? R. James Milgram; 5. What is mathematical proficiency (and how can it be assessed)? Alan H. Schoenfeld; Part III. What Does Assessment Assess? Issues and Examples: 6. Assessing mathematical proficiency: what is important? Hugh Burkhardt; 7. Aspects of the art of assessment design Jan de Lange; 8. Mathematical proficiency for citizenship Bernard Madison; 9. Learning from assessment Richard Askey; 10. Using assessment to design professional development David Foster; Part IV. The Case of Algebra: 11. Context and learning: an assessment of 'real world' mathematics tasks Ann Shannon; 12. Making meaning in algebra: examining students’ understandings and misconceptions David Foster; 13. Assessing the strands of student proficiency in elementary algebra William McCallum; Part V. What Do Different Assessments Assess?: 14. Learning about fractions from assessment Linda Fisher; 15. Brandon interview and commentary, plus CD of interview Deborah Ball; Part VI. The Importance of Context: 16. Assessment of mathematics learning in France Michele Artigue; 17. Assessment to improve learning in mathematics: the BEAR assessment system Mark Wilson and Claus Carstensen; 18. English learners and math learning: language issues for the math educators to consider Lily W. Fillmore; 19. Beyond words to mathematical content: assessing English language learners in the mathematics classroom Judit Moschkovich; 20. Assessment in the real world: the case of New York city Elizabeth Taleporos; 21. Perspectives on state assessments in California Elizabeth Stage; Part VII. What Do We Need To Know?: 22. Research agenda emerging from the conference.