Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-5cf477f64f-2wr7h Total loading time: 0 Render date: 2025-04-04T11:15:12.103Z Has data issue: false hasContentIssue false

5 - Cognitive Models of Task Performance for Mathematical Reasoning

Published online by Cambridge University Press:  05 June 2012

Jacqueline P. Leighton  and
Mark J. Gierl
Jacqueline P. Leighton
Affiliation:
University of Alberta
Mark J. Gierl
Affiliation:
University of Alberta
Get access

Summary

It is an understatement to say that mathematical knowledge and skill are valuable for the jobs and careers of the twenty-first century. In fact, they are essential for individuals who want to have the widest array of career options available and a high quality of life. Learners who shun mathematics shut themselves off from many lucrative career paths. According to a recently published article in the Wall Street Journal (Needleman, January 26, 2009), “Doing the Math to Find Good Jobs,” the best occupations in America all required advanced mathematics. According to data compiled by the U.S. Bureau of Labor Statistics and Census, the top five jobs in a list of two hundred included mathematician, actuary, statistician, biologist, and software engineer. These jobs were rated highest because they combined large salaries with desirable working conditions, namely, indoor office environments, unadulterated air, absence of heavy lifting and physical hardship, and conveniences such as controlling one's work schedule. The worst jobs were lumberjack, dairy farmer, taxi driver, seaman, and emergency medical technician. Most of the jobs at the lower end of the list did not require advanced mathematics.

The importance of mathematics for maximizing the likelihood of obtaining a desirable job in the future would make one think that students, desirous of having an edge for a future career, would be clamoring to learn and perform as well as possible in mathematics. Yet this is not the case.

Type
Chapter
Information
The Learning Sciences in Educational Assessment
The Role of Cognitive Models
 , pp. 156 - 196
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

,Alberta Education (2009). Information bulletin: Pure mathematics 30. Edmonton, Alberta. Author.Google Scholar
Alibali, M.W. (1999). How children change their minds: Strategy change can be gradual or abrupt. Developmental Psychology, 35, 127–145.CrossRefGoogle ScholarPubMed
Alibali, M., Knuth, E., Hattikudur, S., McNeil, N., & Stephens, A. (2007). A longitudinal examination of middle school students' understandings of the equal sign and performance solving equivalent equations. Mathematical Thinking and Learning, 9, 221–247.CrossRefGoogle Scholar
Anderson, J.R. (1996). ACT: A simple theory of complex cognition. American Psychologist, 51, 355–365.CrossRefGoogle Scholar
Anderson, J.R. (1990). The adaptive character of thought. Hillsdale, NJ: Erlbaum.Google Scholar
Anderson, J.R., Bothell, D., Byrne, M.D., Douglass, S., Lebière, C., & Qin, Y. (2004). An integrated theory of the mind. Psychological Review, 111, 1036–1060.CrossRefGoogle ScholarPubMed
Anderson, J.R. & Bower, G.H. (1973). Human associative memory. Washington, DC: Winston and Sons.Google Scholar
Anderson, J.R., Boyle, C.F., Corbett, A.T., & Lewis, M.W. (1990). Cognitive modeling and intelligent tutoring. Artificial Intelligence, 42, 7–49.CrossRefGoogle Scholar
Anderson, J.R., Boyle, C.F., Farrell, R., & Reiser, B.J. (1987). Cognitive principles in the design of computer tutors. In Morris, P. (Ed.), Modelling cognition (pp. 93–133). Chichester, U.K.: Wiley.Google Scholar
Anderson, J.R., Matessa, M. & Douglass, S. (1995). The ACT-R theory of visual attention. In Proceedings of the Seventeenth Annual Cognitive Science Society, 61–65.
Anderson, J.R. & Schunn, C.D. (2000). Implications of the ACT-R learning theory: No magic bullets. In Glaser, R. (Ed.), Advances in instructional psychology: Educational design and cognitive science, Vol. 5 (pp. 1–33). Mahwah, NJ: Erlbaum.Google Scholar
Ashcraft, M.H. & Kirk, E.P. (2001). The relationships among working memory, math anxiety, and performance. Journal of Experimental Psychology: General, 130, 224–237.CrossRefGoogle ScholarPubMed
Ashcraft, M.H. & Krause, J.A. (2007). Working memory, math performance, and math anxiety. Psychonomic Bulletin and Review, 14, 243–248.CrossRefGoogle ScholarPubMed
Baldi, S., Jin, Y., Skemer, M., Green, P.J., & Herget, D. (2007). Highlights from PISA 2006: Performance of U.S. 15-year-old students in science and mathematics literacy in an international context (NCES 2008–016). National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education. Washington, DC.Google Scholar
Bendixen, L.D. & Feucht, F.S. (2010). (Eds.). Personal epistemology in the classroom. Cambridge, MA: Cambridge University Press.CrossRef
Bonnet, G. (2002). Reflections in a critical eye: on the pitfalls of international assessment. Assessment in Education: Principles, Policy, and Practice, 9, 387–399.CrossRefGoogle Scholar
Brannon, E.M. (2005). What animals know abut numbers. In Campbell, J.I.D. (Ed.), Handbook of mathematical cognition (pp. 85–107). New York: Psychology Press.Google Scholar
Bransford, J.D., Brown, A.L., & Cocking, R.R. (Eds.). (1999). How people learn: Brain, mind, experience, and school. Washington, DC: National Academy Press.
Briggs, D.C., Alonzo, A.C., Schwab, C., & Wilson, M. (2006). Diagnostic assessment with ordered multiple-choice items. Educational Assessment, 11, 33–63.CrossRefGoogle Scholar
Brown, J.S. & Burton, R.R. (1978). Diagnostic models for procedural bugs in basic mathematical skills. Cognitive Science, 2, 155–192.CrossRefGoogle Scholar
Cameron, J. & Pierce, W.D. (2002). Rewards and intrinsic motivation: Resolving the controversy. Westport, CT: Bergin & Garvey.Google Scholar
Clements, D.H. & Sarama, J. (2007). Early childhood mathematics learning. In Lester, F.K. Jr. (Ed.), National Council of Teachers of Mathematics: Second handbook of research on mathematics teaching and learning (pp. 461–555). Charlotte, NC: Information Age Publishing.Google Scholar
Corbett, A.T., Anderson, J.R., Carver, V.H., & Brancolini, S.A. (1994). Individual differences and predictive validity in student modeling. In Ram, A. & Eiselt, K. (Ed.), Proceedings of the Sixteenth Annual Conference of the Cognitive Science Society. Hillsdale, NJ: Erlbaum.Google Scholar
Davis, S.H. (2007). Bridging the gap between research and practice. Phi Delta Kappan, 88, 568–578.CrossRef
Deci, E.L., Koestner, R., & Ryan, R.M. (1999). A meta-analytic review of experiments examining the effects of extrinsic rewards on intrinsic motivation. Psychological Bulletin, 125, 625–668.CrossRefGoogle ScholarPubMed
Dehaene, S., Bossini, S., & Giraux, P. (1993). The mental representation of parity and numerical magnitude. Journal of Experimental Psychology: General, 122, 371–396.CrossRefGoogle Scholar
Lange, J. (2007). Large-scale assessment of mathematics education. In Lester, F.K. Jr. (Ed.), National Council of Teachers of Mathematics: Second handbook of research on mathematics teaching and learning (pp. 1111–1142). Charlotte, NC: Information Age Publishing.Google Scholar
Dorman, C. (1991). Exceptional calendar calculation ability after early left hemispherectomy. Brain & Cognition, 15, 26–36.CrossRefGoogle ScholarPubMed
Fias, W. & Fischer, M.H. (2005). Spatial representations of numbers. In Campbell, J.I.D. (Ed.), Handbook of mathematical cognition (pp. 43–54). New York: Psychology Press.Google Scholar
Fu, W.T. & Anderson, J. (2008). Dual learning processes in interactive skill acquisition. Journal of Experimental Psychology, Applied, 14, 179–191.CrossRefGoogle ScholarPubMed
Geary, D.C., Hoard, M.K., Byrd-Craven, J., Nugent, L., and Numtee, C. (2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Development, 78(4), 1343–1359.CrossRefGoogle ScholarPubMed
Gentner, D., Loewenstein, J., & Thompson, L. (2003). Learning and transfer: A general role for analogical encoding. Journal of Educational Psychology, 95, 393–408.CrossRefGoogle Scholar
Gentner, D., Rattermann, M.J., Markman, A., & Kotovsky, L. (1995). Two forces in the development of relational similarity. In Simon, T. & Halford, G. (Eds.), Developing cognitive competence: New approaches to process modeling (pp. 263–313), Hillsdale, N.J.: Erlbaum.Google Scholar
Greeno, J.G., Pearson, P.D., & Schoenfeld, A.H. (1997). Implications for the NAEP of research on learning and cognition. In Linn, R., Glaser, R., & Bohrnstedt, G. (Eds.), Assessment in transition: Monitoring the nation's educational progress (pp. 151–215). Stanford, CA: National Academy of Education.Google Scholar
Halford, G.S., Smith, S.B., Dickson, J.C., Maybery, M.T., Kelly, M.E., Bain, J.D. & Stewart, J.E.M. (1995). Modeling the development of reasoning strategies: The roles of analogy, knowledge, and capacity. In Simon, T. & Halford, G. (Eds.), Developing cognitive competence: New approaches to process modeling (pp. 77–156). Hillsdale, NJ: Erlbaum.Google Scholar
Hatano, G. (1988). Social and motivational bases for mathematical understanding. New Directions for Child Development, 41, 55– 70.CrossRefGoogle Scholar
Hidi, S. & Harackiewicz, J.M. (2000). Motivating the academically unmotivated: A critical issue for the 21st century. Review of Educational Research, 70, 151–179.CrossRefGoogle Scholar
Hiebert, J., Carpenter, T.P., Fennema, E., Fuson, K.C., Wearne, D., Murray, H., Olivier, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.Google Scholar
Hill, H.C., Sleep, L., Lewis, J.M., & Ball, D.B. (2007). Assessing teachers' mathematical knowledge: What knowledge matters and what evidence counts. In Lester, F.K. Jr. (Ed.), National Council of Teachers of Mathematics: Second handbook of research on mathematics teaching and learning (pp. 111–155). Charlotte, NC: Information Age Publishing.Google Scholar
Hofer, B.K. & Pintrich, P.R. (1997). The development of epistemological theories: Beliefs about knowledge and knowing and their relation to learning. Review of Educational Research, 67, 88–140.CrossRefGoogle Scholar
Hope, J. (1985). Unravelling the mysteries of expert mental calculation. Educational Studies in Mathematics, 16, 355–374.CrossRefGoogle Scholar
Jordan, J.A., Mulhern, G., & Wylie, J. (2009). Individual differences in trajectories of arithmetical development in typically achieving 5- to 7-year-olds. Journal of Experimental Child Psychology, 103, 455–468.CrossRefGoogle Scholar
Kalaman, D.A. & LeFevre, J. (2007). Working memory demands of exact and approximate addition. European Journal of Cognitive Psychology, 19, 187–212.CrossRefGoogle Scholar
Kaminski, J.A., Sloutsky, V.M., & Heckler, A.F. (2009). Concrete Instantiations of mathematics: A double-edged sword. Journal for Research in Mathematics Education, 40, 90–93.Google Scholar
Kieran, C. (2007). Learning and teaching of algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In Lester, F.K. Jr. (Ed.), National Council of Teachers of Mathematics: Second handbook of research on mathematics teaching and learning (pp. 707–762). Charlotte, NC: Information Age Publishing.Google Scholar
Kilpatrick, J., Swafford, J. & Findell, B. (Ed.). (2001). Adding It Up: Helping Children Learn Mathematics. Washington, DC, USA: National Academies Press.
Kintsch, W. (1998). Comprehension: A paradigm for cognition. Cambridge, UK: Cambridge University Press.Google Scholar
Klahr, D. & Dunbar, K. (1988). Dual search space during scientific reasoning. Cognitive Science, 12, 1–48.CrossRefGoogle Scholar
Kloosterman, P. (2010). Mathematics skills of 17-year-olds in the United States: 1978 to 2004. Journal for Research in Mathematics Education, 41, 20–51.Google Scholar
Koedinger, K.R. & Anderson, J.R. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. Cognitive Science, 14, 511–550.CrossRefGoogle Scholar
Koedinger, K.R. & Anderson, J.R. (1991). Interaction of deductive and inductive reasoning strategies in geometry novices. In Proceedings of the Thirteenth Annual Conference of the Cognitive Science Society, Hillsdale, NJ: Erlbaum.Google Scholar
Koedinger, K.R. & Anderson, J.R. (1993). Effective use of intelligent software in high school math classrooms. InProceedings of the Sixth World Conference on Artificial Intelligence in Education (pp. 241–248). Charlottesville, VA: Association for the Advancement of Computing in Education.Google Scholar
Koedinger, K.R., Anderson, J.R., Hadley, W.H., & Mark, M. (1997). Intelligent tutoring goes to school in the big city. InternationalJournal of Artificial Intelligence in Education, 8, 30–43.Google Scholar
Kuhn, D. (2005). Education for thinking. Cambridge, MA: Harvard University Press.Google ScholarPubMed
Larkin, J. & Simon, H. (1987). Why a diagram is (sometimes) worth 10,000 words. Cognitive Science, 11, 65–99.CrossRefGoogle Scholar
Lesh, R. & Zawojewski, J.S. (2007). Problem solving and modeling. In Lester, F.K. Jr. (Ed.), National Council of Teachers of Mathematics: Second handbook of research on mathematics teaching and learning (pp. 763–804). Charlotte, NC: Information Age Publishing.Google Scholar
LesterJr., F.K. (Ed.). (2007). National Council of Teachers of Mathematics: Second handbook of research on mathematics teaching and learning. Charlotte, NC: Information Age Publishing.
LesterJr., F.K. & Kehle, P.E, (2003). From problem solving to modeling: The evolution of thinking about research on complex mathematical activity. In Lesh, R. A. & Doerr, H. M. (Eds.), Beyond constructivism: Models and modeling perspectives on mathematics problem solving, learning, and teaching (pp. 501–518). Mahwah, NJ: Lawrence Erlbaum Associates.Google Scholar
LesterJr., F., Garofalo, J., & Kroll, D. (1989). Self-confidence, interest, beliefs, and metacognition: Key influences in problem-solving behaviour. In McLeod, D.B. & Adams, V.M. (Eds.), Affect and mathematical problem solving: A new perspective (pp. 75–88). New York: Springer-Verlag.Google Scholar
Mabbott, D.J. & Bisanz, J. (2008). Computational skills, working memory, and conceptual knowledge in older children with mathematics learning disabilities. Journal of Learning Disabilities, 41, 15–28.CrossRefGoogle ScholarPubMed
Maher, C.A. & Martino, A.M. (1996). The development of the idea of mathematical proof: A 5-year case study. Journal for Research in Mathematics Education, 27, 194–214.CrossRefGoogle Scholar
Mayer, R.E. (2008) Learning and instruction. Columbus, OH: Pearson Merrill Prentice Hall.Google ScholarPubMed
Mayer, R.E. (2003). Mathematical problem solving. In Royer, J.M. (Ed.), Mathematical cognition (pp. 69–92). Greenwich, CT: Information Age Publishing.Google Scholar
Mazzocco, M.M. & McCloskey, M. (2005). Math performance in girls with turner or fragile X syndrome. In Campbell, J.I.D. (Ed.), Handbook of mathematical cognition (pp. 269–297). New York: Psychology Press.Google Scholar
McCrink, K. & Wynn, K. (2004). Large-number addition and subtraction in infants. Psychological Science, 15, 776–781.CrossRefGoogle ScholarPubMed
Miura, I.T. & Okamoto, Y. (2003). Language supports for mathematics understanding and performance. In Baroody, A.J. & Dowker, A. (Eds.), The Development of Arithmetic Concepts and Skills – Constructing Adaptive Expertise: Studies in Mathematical Thinking and Learning (pp. 229–242). Hillsdale, NJ: Erlbaum.Google Scholar
Munn, P. (1998). Symbolic function in pre-schoolers. In Donlan, C. (Ed.), The development of mathematics skills: Studies in developmental psychology (pp. 47–71). East Sussex, UK: Psychology Press.Google Scholar
Morris, L.W., Davis, M.A., & Hutchings, C.H. (1981). Cognitive and emotional components of anxiety: Literature review and a revised worry-emotionality scale. Journal of Educational Psychology, 73, 541–555.CrossRefGoogle Scholar
,National Mathematics Advisory Panel. (2008). Foundations for Success: The Final Report of the National Mathematics Advisory Panel. U.S. Department of Education: Washington, DC.Google Scholar
,National Research Council. (2005). How Students Learn: History, Mathematics, and Science in the Classroom. In Donovan, M.S. & Bransford, J.D. (Eds.), Division of Behavioral and Social Sciences and Education. Washington, DC: The National Academies Press.Google Scholar
,National Research Council. (2009). Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity. Committee on Early Childhood Mathematics, Cross, Christopher T., Woods, Taniesha A., and Schweingruber, Heidi, Editors. Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: The National Academies Press.Google Scholar
Needleman, Sarah E. (January 26, 2009). Doing the math to find the good jobs. Wall Street Journal, p. D2.
Newell, A. (1973). You can't play 20 questions with nature and win: Projective comments on the papers of this symposium. In Chase, W. G. (Ed.), Visual information processing (pp. 283–310). New York: Academic Press.Google Scholar
Newell, A. (1990). Unified theories of cognition. Cambridge, MA: Harvard University Press.Google Scholar
Newell, A. & Simon, H.A. (1972). Human problem solving. Oxford, UK: Prentice-Hall.Google Scholar
Niss, M. (2007). Reflections on the state and trends in research on mathematics teaching and learning: From here to utopia. In Lester, F.K. Jr. (Ed.), National Council of Teachers of Mathematics: Second handbook of research on mathematics teaching and learning (pp. 1293–1312). Charlotte, NC: Information Age Publishing.Google Scholar
,Organization for Economic Cooperation and Development. (2004). Learning for tomorrow's world: First results from PISA 2003. Paris, France: Author.Google Scholar
Pesek, D.D. & Kirshner, D. (2000). Interference of instrumental instruction in subsequent relational learning. Journal for Research in Mathematics Education, 31, 524–540.CrossRefGoogle Scholar
Provasnik, S., Gonzales, P., & Miller, D. (2009). U.S. Performance Across International Assessments of Student Achievement: Special Supplement to The Condition of Education 2009 (NCES 2009–083). National Center for Education Statistics, Institute of Education Sciences, U.S. Department of Education. Washington, DC.
Quilici, J.H. & Mayer, R.E. (1996). Role of examples in how students learn to categorize statistics word problems. Journal of Educational Psychology, 88, 144–161.CrossRefGoogle Scholar
Ritter, S., Anderson, J.R., Koedinger, K.R., & Corbett, A. (2007). Cognitive tutor: Applied research in mathematics education. Psychonomic Bulletin and Review, 14, 249–255.CrossRefGoogle ScholarPubMed
Rittle-Johnson, B., Siegler, R.S., & Alibali, M.W. (2001). Developing conceptual understanding and procedural skill in mathematics: An iterative process. Journal of Educational Psychology, 93, 346–362.CrossRefGoogle Scholar
Schoenfeld, A.H. (2007). Method. In Lester, F.K. Jr. (Ed.), National Council of Teachers of Mathematics: Second handbook of research on mathematics teaching and learning (pp. 69–107). Charlotte, NC: Information Age Publishing.Google Scholar
Schoenfeld, A.L. (1992). Learning to think mathematically. Problem solving, metacognition, and sense-making in mathematics. In Grouws, D.A. (Ed.), Handbook of research in mathematics teaching and learning (pp. 334–370). New York: Macmillan.Google Scholar
Schommer, M. (1990). Effects of beliefs about the nature of knowledge on comprehension. Journal of Educational Psychology, 82, 498–504.CrossRefGoogle Scholar
Siegler, R.S. & Araya, R. (2005). A computational model of conscious and unconscious strategy discovery. In Kail, R. V. (Ed.), Advances in child development and behavior, Vol. 33, (pp. 1–42). Oxford, UK: Elsevier.Google Scholar
Siegler, R.S. & Booth, J.L. (2004). Development of numerical estimation in young children. Child Development, 75, 428–444.CrossRefGoogle ScholarPubMed
Siegler, R.S. & Svetina, M. (2006). What leads children to adopt new strategies? A microgenetic/cross sectional study of class inclusion. Child Development, 77, 997–1015.CrossRefGoogle ScholarPubMed
Simon, T.J. & Klahr, D. (1995). A computational theory of children's learning about number conservation. In Simon, T.J. & Halford, G.S. (Eds.), Developing cognitive competence: New approaches to process modeling. Hillsdale, NJ: ErlbaumGoogle Scholar
Shrager, J. & Siegler, R.S. (1998). SCADS: A model of children's strategy choices and strategy discoveries. Psychological Science, 9, 405–410.CrossRefGoogle Scholar
Sophian, C. (2007). The origins of mathematical knowledge in childhood. Lawrence Erlbaum Associates.Google Scholar
Spencer, S.J., Steele, C.M., & Quinn, D.M. (1999). Stereotype threat and women's math performance. Journal of Experimental Social Psychology, 35, 4–28.CrossRefGoogle Scholar
Skwarchuk, S.L. & Anglin, J.M. (2002). Children's acquisition of the English cardinal number words: A special case of vocabulary development. Journal of Educational Psychology, 94, 107–125.CrossRefGoogle Scholar
Stanovich, K.E. (2009). What intelligence tests miss: The psychology of rational thought. New Haven, CT: Yale University Press.Google Scholar
Stewart, S. & Thomas, M.O.J. (2010). Student learning of basis, span and linear independence in linear algebra. International Journal of Mathematics Education, 41, 173–188.Google Scholar
Towse, J. & Saxton, M. (1998). Mathematics across national boundaries: Cultural and linguistic perspectives on numerical competence. In Donlan, C. (Ed.), The development of mathematics skills: Studies in developmental psychology (pp. 129–150). East Sussex, UK: Psychology Press.Google Scholar
Wayne, A.J. & Youngs, P. (2003). Teacher characteristics and student achievement gains: A review. Review of Educational Research, 73, 89–122.CrossRefGoogle Scholar
Wiest, L.R. (2000). Mathematics that whets the appetite: Student-posed projects problems. Mathematics Teaching in the Middle School, 5, 286–291.Google Scholar
Wilson, J. & Clark, D. (2004). Towards the modeling of mathematical metacognition. Mathematics Education Research Journal, 16, 25–48.CrossRefGoogle Scholar
Wu, H. (1999). Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education. American Educator, 14–19, 50– 52.Google Scholar
Wynn, K. (1995). Origins of numerical knowledge. Mathematical Cognition, 1, 35–60.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×