AN EXAMINATION OF THE LEVELS OF COGITIVE DEMAND REQUIRED BY PROBABILITY TASKS IN MIDDLE GRADES MATHEMATICS TEXTBOOKS

AN EXAMINATION OF THE LEVELS OF COGITIVE

DEMAND REQUIRED BY PROBABILITY TASKS IN MIDDLE

GRADES MATHEMATICS TEXTBOOKS

DUSTIN L. JONES

Sam Houston State University

dljones@shsu.edu

JAMES E. TARR

University of Missouri – Columbia

tarrj@missouri.edu

ABSTRACT

We analyze probability content within middle grades (6, 7, and 8) mathematics textbooks from a historical perspective. Two series, one popular and the other alternative, from four recent eras of mathematics education (New Math, Back to Basics, Problem Solving, and Standards) were analyzed using the Mathematical Tasks Framework (Stein, Smith, Henningsen, & Silver, 2000). Standards-era textbook series devoted significantly more attention to probability than other series; more than half of all tasks analyzed were located in Standards-era textbooks. More than 85% of tasks for six series required low levels of cognitive demand, whereas the majority of tasks in the alternative series from the Standards era required high levels of cognitive demand. Recommendations for future research are offered.

Keywords: Probability; Curriculum; Mathematics textbook content analysis; Mathematical tasks; Cognitive demands; Middle grades mathematics

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Statistics Education Research Journal, 6(2), 4-27, http://www.stat.auckland.ac.nz/serj

Ó International Association for Statistical Education (IASE/ISI), November, 2007

REFERENCES

Abbott, J. S., & Wells, D. W. (1985a). Mathematics today: Level 6. Orlando, FL: Harcourt Brace Jovanovich.

Abbott, J. S., & Wells, D. W. (1985b). Mathematics today: Level 7. Orlando, FL: Harcourt Brace Jovanovich.

Abbott, J. S., & Wells, D. W. (1985c). Mathematics today: Level 8. Orlando, FL: Harcourt Brace Jovanovich.

Adams, L., Tung, K. K., Warfield, V. M., Knaub, K., Mudavanhu, B., & Yong, D. (2000). Middle school mathematics comparisons for Singapore Mathematics, Connected Mathematics Program, and Mathematics in Context (including comparisons with the NCTM Principles and Standards 2000). A report to NSF, November 2, 2000. Unpublished manuscript, Seattle, WA.

American Association for the Advancement of Science: Project 2061. (2000). Middle grades mathematics textbooks: A benchmarks-based evaluation. Washington, DC: Author.

Arbaugh, F., Lannin, J., Jones, D. L., & Park-Rogers, M. (2006). Examining instructional practices in Core-Plus lessons: Implications for professional development. Journal of Mathematics Teacher Education, 9(6), 517-550.

Batanero, C., Henry, M., & Parzysz, B. (2005).The nature of chance and probability. In G. A. Jones (Ed.) Exploring probability in school: Challenges for teaching and learning, (pp. 15-37). Netherlands: Kluwer Academic Publishers.

Borovcnik, M., Bentz, H. J., & Kapadia, R. (1991). A probabilistic perspective. In R. Kapadia & M. Borovcnik (Eds.), Chance encounters: Probability in education (pp. 27-71). Boston: Kluwer Academic Publishers.

Braswell, J. S., Lutkus, A. D., Grigg, W. S., Santapau, S. L., Tay-Lim, B., & Johnson, M. (2001). The nation's report card: Mathematics 2000. Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement.

Carpenter, T. P., Corbitt, M. K., Kepner, H. S., Jr., Lindquist, M. M., & Reys, R. E. (1981). Results from the second mathematics assesment of the National Assessment of Educational Progress. Reston, VA: National Council of Teachers of Mathematics.

Clopton, P., McKeown, E., McKeown, M., & Clopton, J. (1999a). Mathematically correct fifth grade mathematics review.

[Online: http://mathematicallycorrect.com/books5.htm]

Clopton, P., McKeown, E., McKeown, M., & Clopton, J. (1999b). Mathematically correct second grade mathematics review.

[Online: http://mathematicallycorrect.com/books2.htm]

Clopton, P., McKeown, E., McKeown, M., & Clopton, J. (1999c). Mathematically correct seventh grade mathematics review.

[Online: http://mathematicallycorrect.com/books7.htm]

Collins, W., Dristas, L., Frey-Mason, P., Howard, A. C., McClain, K., Molina, D. D., et al. (1998a). Mathematics: Applications and connections, course 1. New York: Glencoe/McGraw Hill.

Collins, W., Dristas, L., Frey-Mason, P., Howard, A. C., McClain, K., Molina, D. D., et al. (1998b). Mathematics: Applications and connections, course 2. New York: Glencoe/McGraw Hill.

Collins, W., Dristas, L., Frey-Mason, P., Howard, A. C., McClain, K., Molina, D. D., et al. (1998c). Mathematics: Applications and connections, course 3. New York: Glencoe/McGraw Hill.

Conference Board of the Mathematical Sciences. (2001). The mathematical education of teachers. Washington, DC: Mathematical Association of America.

delMas, R. C., & Bart, W. M. (1989). The role of an evaluation exercise in the resolution of misconceptions of probability. Focus on Learning Problems in Mathematics, 11, 39-53.

DeVault, M. V., & Weaver, J. F. (1970). Forces and issues related to curriculum and instruction, K-6. In P. S. Jones & A. F. Coxford, Jr. (Eds.), A history of mathematics education in the United States and Canada: Thirty-second yearbook (pp. 90-152). Washington, DC: National Council of Teachers of Mathematics.

Dolciani, M. P., Beckenbach, E. F., Wooten, W., Chinn, W. G., & Markert, W. (1967a). Modern school mathematics: Structure and method 7. Boston: Houghton Mifflin.

Dolciani, M. P., Beckenbach, E. F., Wooten, W., Chinn, W. G., & Markert, W. (1967b). Modern school mathematics: Structure and method 8. Boston: Houghton Mifflin.

Doyle, W. (1983). Academic work. Review of Educational Research, 53(2), 159-199.

Doyle, W. (1988). Work in mathematics classes: The context of students' thinking during instruction. Educational Psychologist, 23(2), 167-180.

Duncan, E. R., Capps, L. R., Dolciani, M. P., Quast, W. G., & Zweng, M. J. (1967). Modern school mathematics: Structure and use 6. Boston: Houghton Mifflin.

Fey, J. T., & Graeber, A. O. (2003). From the new math to the Agenda for Action. In G. M. A. Stanic & J. Kilpatrick (Eds.), A history of school mathematics (Vol. 1, pp. 521-558). Reston, VA: National Council of Teachers of Mathematics.

Garfield, J., & Ahlgren, A. (1988). Difficulties in learning basic concepts in probability and statistics: Implications for research. Journal for Research in Mathematics Education, 19(1), 44-63.

Grouws, D. A., & Smith, M. S. (2000). Findings from NAEP on the preparation and practices of mathematics teachers. In E. A. Silver & P. A. Kenney (Eds.), Results from the Seventh Mathematics Assessment of the National Assessment of Education Progress (pp. 107-141). Reston, VA: National Council of Teachers of Mathematics.

Hake, S., & Saxon, J. (1985). Math 76: An incremental development. Norman, OK: Saxon Publishers, Inc.

Hake, S., & Saxon, J. (1987). Math 65: An incremental development. Norman, OK: Saxon Publishers, Inc.

Hake, S., & Saxon, J. (1991). Math 87: An incremental development. Norman, OK: Saxon Publishers, Inc.

Hinkle, D. E., Wiersma, W., & Jurs, S. G. (1988). Applied statistics for the behavioral sciences (2nd ed.). Boston: Houghton Mifflin Company.

Hogg, R. V., & Tanis, E. A. (1993). Probability and statistical inference (4th ed.). New York: Macmillan Publishing Company.

Konold, C. (1983). Conceptions about probability: Reality between a rock and a hard place. (Doctoral dissertation, University of Massachusetts, 1983). Dissertation Abstracts International, 43, 4179b.

Konold, C. (1989). Informal conceptions of probability. Cognition and Instruction, 6, 59-98.

Konold, C. (1991). Understanding students’ beliefs about probability. Cognition and Instruction, 6, 59-98.

Konold, C., Pollatsek, A., Well, A., Lohmeier, J., & Lipson, A. (1993). Inconsistencies in students' reasoning about probability. Journal for Research in Mathematics Education, 24(5), 392-414.

Lappan, G., Fey, J., Fitzgerald, W. M., Friel, S. N., & Phillips, E. (1996). Getting to know Connected Mathematics: A guide to the Connected Mathematics curriculum. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998a). Accentuate the negative. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998b). Bits and pieces I. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998c). Bits and pieces II. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998d). Clever counting. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998e). Comparing and scaling. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998f). Covering and surrounding. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998g). Data about us. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998h). Data around us. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998i). Filling and wrapping. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998j). Frogs, fleas, and painted cubes. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998k). Growing, growing, growing. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998l). How likely is it? Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998m). Hubcaps, kaleidoscopes, and mirrors. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998n). Looking for Pythagoras. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998o). Moving straight ahead. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998p). Prime time. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998q). Ruins of Montarek. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998r). Samples and populations. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998s). Say it with symbols. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998t). Shapes and designs. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998u). Stretching and shrinking. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998v). Thinking with mathematical models. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998w). Variables and patterns. Palo Alto, CA: Dale Seymour Publications.

Lappan, G., Fey, J. T., Fitzgerald, W. M., Friel, S. N., & Phillips, E. D. (1998x). What do you expect? Palo Alto, CA: Dale Seymour Publications.

Lecoutre, M. (1992). Cognitive models and problem spaces in “purely random” situations. Educational Studies in Mathematics, 23, 557-568.

Lloyd, G. M. (1999). Two teachers’ conceptions of a reform-oriented curriculum: Implications for mathematics teacher development. Journal of Mathematics Teacher Education, 2, 227-252.

Lloyd, G. M., & Behm, S. L. (2005). Preservice elementary teachers’ analysis of mathematics instructional materials. Action in Teacher Education, 26(4), 48-62.

Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook (2nd ed.). Thousand Oaks, CA: Sage Publications.

National Council of Teachers of Mathematics. (1980). An agenda for action: Recommendations for school mathematics of the 1980s. Reston, VA: Author.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author.

National Research Council. (2004). On evaluating curricular effectiveness: Judging the quality of K-12 mathematics evaluations. Washington, DC: National Academy Press.

Newman, C. M., Obremski, T. E., & Scheaffer, R. L. (1987). Exploring probability. Palo Alto, CA: Dale Seymour.

Nichols, E. D., Anderson, P. A., Dwight, L. A., Flourney, F., Kalin, R., Schluep, J., et al. (1974a). Holt school mathematics: Grade 6. New York: Holt, Rinehart and Winston.

Nichols, E. D., Anderson, P. A., Dwight, L. A., Flourney, F., Kalin, R., Schluep, J., et al. (1974b). Holt school mathematics: Grade 7. New York: Holt, Rinehart and Winston.

Nichols, E. D., Anderson, P. A., Dwight, L. A., Flourney, F., Kalin, R., Schluep, J., et al. (1974c). Holt school mathematics: Grade 8. New York: Holt, Rinehart and Winston.

Osborne, A. R., & Crosswhite, F. J. (1970). Forces and issues related to curriculum and instruction, 7-12. In P. S. Jones & A. F. Coxford, Jr. (Eds.), A history of mathematics education in the United States and Canada: Thirty-second yearbook (pp. 153-297). Washington, DC: National Council of Teachers of Mathematics.

Payne, J. N. (2003). The new math and its aftermath, grades K-8. In G. M. A. Stanic & J. Kilpatrick (Eds.), A history of school mathematics (Vol. 1, pp. 559-598). Reston, VA: National Council of Teachers of Mathematics.

Phillips, E., Lappan, G., Winter, M. J., & Fitzgerald, W. (1986). Probability. Menlo Park, CA: Addison-Wesley.

Porter, A. C. (2006). Curriculum assessment. In J. Green, G. Camilli & P. Elmore (Eds.), Handbook of complementary methods in education research (pp. 141-159). Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Pratt, D. (2000). Making sense of the total of two dice. Journal of Research in Mathematics Education, 31, 602-625.

Pratt, D. (2005). How do teachers foster students’ understanding of probability? In G. A. Jones (Ed.) Exploring probability in school: Challenges for teaching and learning, (pp. 171-189). Netherlands: Kluwer Academic Publishers.

Remillard, J. T., & Bryans, M. B. (2004). Teachers’ orientations toward mathematics curriculum materials: Implications for teacher learning. Journal for Research in Mathematics Education, 35, 352-388.

Robinson, E., & Robinson, M. (1996). A guide to standards-based instructional materials in secondary mathematics. Unpublished manuscript.

Robitaille, D. F., & Travers, K. J. (1992). International studies of achievement in mathematics. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 687-723). Reston, VA: National Council of Teachers of Mathematics.

Saxon, J. (1980). Algebra 1/2. Norman, OK: Saxon Publishers, Inc.

School Mathematics Study Group. (1961a). Mathematics for junior high school (Vol. I). New Haven, CT: Yale University Press.

School Mathematics Study Group. (1961b). Mathematics for junior high school (Vol. II). New Haven, CT: Yale University Press.

School Mathematics Study Group. (1962). Mathematics for the elementary school: Grade 6. New Haven, CT: Yale University Press.

Shaughnessy, J. M. (1992). Research in probability and statistics: Reflections and directions. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 465-494). Reston, VA: National Council of Teachers of Mathematics.

Shaughnessy, J. M. (2003). Research on students' understandings of probability. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.), A research companion to Principles and Standards for School Mathematics (pp. 216-226). Reston, VA: National Council of Teachers of Mathematics.

Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. Mathematics Teaching in the Middle School, 3, 344-350.

Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455-488.

Stein, M. K., & Smith, M. S. (1998). Mathematical tasks as a framework for reflection. Mathematics Teaching in the Middle School, 3, 268-275.

Stein, M. K., Smith, M. S., Henningsen, M., & Silver, E. A. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York: Teachers College Press.

Stohl, H., & Tarr, J. E. (2002). Developing notions of inference with probability simulation tools. Journal of Mathematical Behavior 21, 319-337.

Tyson-Bernstein, H., & Woodward, A. (1991). Ninteenth century policies for twenty-first century practice: The textbook reform dilemma. In P. G. Altbach, G. P. Kelly, H. G. Petrie & L. Weis (Eds.), Textbooks in American society: Politics, policy, and pedagogy. Albany, NY: State University of New York Press.

Usiskin, Z. (1979). Algebra through applications with probability and statistics. Reston, VA: National Council of Teachers of Mathematics.

Usiskin, Z. (1985). We need another revolution in secondary school mathematics. In C. R. Hirsch & M. J. Zweng (Eds.), The secondary school mathematics curriculum: 1985 yearbook of the National Council of Teachers of Mathematics (pp. 1-21). Reston, VA: National Council of Teachers of Mathematics.

Usiskin, Z., Feldman, C. H., Flanders, J., Polonsky, L., Porter, S., & Viktora, S. S. (1995). Transition mathematics. Glenview, IL: Scott, Foresman.

Valverde, G. A., Bianchi, L. J., Wolfe, R. G., Schmidt, W. H., & Houang, R. T. (2002). According to the book: Using TIMSS to investigate the translation of policy into practice through the world of textbooks. Boston: Kluwer Academic Publishers.

Weiss, I. R. (1978). Report of the 1977 National Survey of Science, Mathematics, and Social Studies Education. Research Triangle Park, NC: Center for Educational Research and Evaluation.

Weiss, I. R. (1987). Report of the 1985-1986 National Survey of Science and Mathematics Education. Research Triangle Park, NC: Research Triangle Institute.

Weiss, I. R., Banilower, E. R., McMahon, K. C., & Smith, P. S. (2001). Report of the 2000 national survey of science and mathematics education. Chapel Hill, NC: Horizon Research, Inc.

Willoughby, D., Bereiter, C., Hilton, P., & Rubenstein, J. H. (1981). Real math: Level 6.
La Salle, IL: Open Court.

Willoughby, D., Bereiter, C., Hilton, P., & Rubenstein, J. H. (1985a). Real math: Level 7.
La Salle, IL: Open Court.

Willoughby, D., Bereiter, C., Hilton, P., & Rubenstein, J. H. (1985b). Real math: Level 8.
La Salle, IL: Open Court.

Yáñez, G. C. (2002). Some challenges for the use of computer simulations for solving conditional probability problems. In D. S. Mewborn et al. (Eds.), Proceedings of the 24th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 1255-1266). Athens, GA: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

DUSTIN L. JONES

Sam Houston State University

Mathematics & Statistics

Box 2206

Huntsville, TX 77341

USA