What affects strategy selection in arithmetic? The example of parity and five effects on product verification
Tóm tắt
The parity effect in arithmetic problem verification tasks refers to faster and more accurate judgments for false equations when the odd/even status of the proposed answer mismatches that of the correct answer. In two experiments, we examined whether the proportion of incorrect answers that violated parity or the number of even operands in the problem affected the magnitude of these effects. Experiment 1 showed larger parity effects for problems with two even operands and larger parity effects during the second half of the experiment. Experiment 2 replicated the results of Experiment 1 and varied the proportion of problems violating parity. Larger parity effects were obtained when more of the false problems violated parity. Moreover, all three effects combined to show the greatest parity effects in conditions with a high proportion of parity violations in problems containing two even operands that were solved during the second half of the experiment. Experiment 3 generalized the findings to the case of five rule (i.e., checking whether a false product ends in 5 or 0), another procedure for solving and verifying multiplication problems quickly. These results (1) delineate further constraints for inclusion in models of arithmetic processing when thinking about how people select among verification strategies, (2) show combined effects of variables that traditionally have been shown to have separate effects on people’s strategy selection, and (3) are consistent with a view of strategy selection that suggests a bias either in the allocation of cognitive resources in the execution of strategies or in the order of execution of these strategies; they argue against a simple, unbiased competition among strategies.
Tài liệu tham khảo
Ashcraft, M. H. (1992). Cognitive arithmetic: A review of data and theory.Cognition,44, 75–106.
Ashcraft, M. H. (1995). Cognitive psychology and simple arithmetic: A review and summary of new directions.Mathematical Cognition,1, 3–34.
Ashcraft, M. H., &Battaglia, J. (1978). Evidence for retrieval and decision processes in mental addition.Journal of Experimental Psychology: Human Learning & Memory,4, 527–538.
Ashcraft, M. H., &Christy, K. S. (1995). The frequency of arithmetic facts in elementary texts: Addition and multiplication in grades 1–6.Journal for Research in Mathematics Education,26, 396–421.
Campbell, J. I. D. (1994). Architectures for numerical cognition.Cognition,53, 1–44.
Campbell, J. I. D., &Graham, D. J. (1985). Mental multiplication skill: Structure, process, and acquisition.Canadian Journal of Psychology,39, 338–366.
Campbell, J. I. D., &Oliphant, M. (1992). Representation and retrieval of arithmetic facts: A network-interference model and simulation. In J. I. D. Campbell (Ed.),The nature and origins of mathematical skills (pp. 331–364). Amsterdam: Elsevier.
Cohen, J., MacWhinney, B., Flatt, M., &Provost, J. (1993). PsyScope: An interactive graphic system for designing and controlling experiments in the psychology laboratory using Macintosh computers.Behavior Research Methods, Instruments, & Computers,25, 257–271.
Collins, A. M. (1978). Fragments of a theory of human plausible reasoning. In D. L. Waltz (Ed.),Theoretical issues in natural language processing. Urbana-Champaign: University of Illinois Press.
Cooney, J. B., Swanson, H. L., &Ladd, S. F. (1988). Acquisition of mental multiplication skill: Evidence for the transition between counting and retrieval strategies.Cognition & Instruction,5, 323–345.
Dehaene, S., Bossini, S., &Giraux, P. (1993). The mental representation of parity and number magnitude.Journal of Experimental Psychology: General,122, 371–396.
Donley, R. D., &Ashcraft, M. H. (1992). The methodology of testing naive beliefs in the physics classroom.Memory & Cognition,20, 381–391.
Geary, D. C., &Brown, S. C. (1991). Cognitive addition: Strategy choice and speed-of-processing differences in gifted, normal, and mathematically disabled children.Developmental Psychology,27, 398–406.
Geary, D. C., &Burlingham-Dubree, M. (1989). External validation of the strategy choice model for addition.Journal of Experimental Child Psychology,47, 175–192.
Geary, D. C., Frensch, P. A., &Wiley, J. G. (1993). Simple and complex mental subtraction: Strategy choice and speed-of-processing differences in young and elderly adults.Psychology & Aging,8, 242–256.
Geary, D. C., Widaman, K. F., &Little, T. D. (1986). Cognitive addition and multiplication: Evidence for a single memory network.Memory & Cognition,14, 478–487.
Glucksberg, S., &McCloskey, M. (1981). Decision about ignorance: Knowing that you don’t know.Journal of Experimental Psychology: Human Learning & Memory,7, 311–325.
Hasher, L., &Zachs, R. T. (1979). Automatic and effortful processes in memory.Journal of Experimental Psychology: General,108, 356–388.
Hines, T. M. (1990). An odd effect: Lengthened reaction times for judgments about odd digits.Memory & Cognition,18, 40–46.
Jacoby, L. L., &Dallas, M. (1981). On the relationship between autobiographical memory and perceptual learning.Journal of Experimental Psychology: General,110, 306–340.
Krueger, L. E. (1986). Why 2 + 2 = 5 looks so wrong: On the odd—even rule in product verification.Memory & Cognition,14, 141–149.
Krueger, L. E., &Hallford, E. W. (1984). Why 2 + 2 = 5 looks so wrong: On the odd-even rule in sum verification.Memory & Cognition,12, 171–180.
LeFevre, J., Bisanz, J., Kaley, K. E., Buffone, L., &Sadesky, G. S. (1996). Multiple routes to solution of single-digit multiplication problems.Journal of Experimental Psychology: General,125, 284–306.
LeFevre, J., Sadesky, G. S., &Bisanz, J. (1996). Selection of procedures in mental addition: Reassessing the problem-size effect in adults.Journal of Experimental Psychology: Learning, Memory, & Cognition,22, 216–230.
Lemaire, P., Abdi, H., &Fayol, M. (1996). The role of working memory resources in simple cognitive arithmetic.European Journal of Cognitive Psychology,8, 73–103.
Lemaire, P., Barrett, S. E., Fayol, M., &Abdi, H. (1994). Automatic activation of addition and multiplication facts in elementary school children.Journal of Experimental Child Psychology,57, 224–258.
Lemaire, P., &Fayol, M. (1995). When plausibility judgments supersede fact retrieval: The example of the odd-even effect on product verification.Memory & Cognition,23, 34–48.
Lemaire, P., Fayol, M., &Abdi, H. (1991). Associative confusion effect in cognitive arithmetic: Evidence for partially autonomous processes.CPC: European Bulletin of Psychology,5, 587–604.
Lemaire, P., &Siegler, R. S. (1995). Four aspects of strategic change: Contributions to children’s learning of multiplication.Journal of Experimental Psychology: General,124, 83–97.
Logan, G. D., &Zbrodoff, N. J. (1979). When it helps to be misled: Facilitative effects of increasing the frequency of conflicting stimuli in a Stroop-like task.Memory & Cognition,7, 166–174.
Lovett, M. C., &Anderson, J. R. (1996). History of success and current context in problem solving: Combined influences on operator selection.Cognitive Psychology,31, 168–217.
Mandler, G. (1980). The judgment of previous occurrence.Psychological Review,87, 252–271.
McCloskey, M., Harley, W., &Sokol, S. C. (1991). Models of arithmetic fact retrieval: An evaluation in light of findings from normal and brain-damaged subjects.Journal of Experimental Psychology: Learning, Memory, & Cognition,17, 377–397.
Miller, K. F., Perlmutter, M., &Keating, D. (1984). Cognitive arithmetic: Comparison of operations.Journal of Experimental Psychology: Learning, Memory, & Cognition,10, 46–60.
Norman, D. A. (1973). Memory, knowledge, and the answering of questions. In R. L. Solso (Ed.),Contemporary issues in cognitive psychology: The Loyola Symposium. Washington, DC: Winston.
Reder, L. M. (1982). Plausibility judgments versus fact retrieval: Alternative strategies for sentence verification.Psychological Review,89, 250–280.
Reder, L. M. (1987). Strategy selection in question answering.Cognitive Psychology,19, 90–137.
Reder, L. M. (1988). Strategic control of retrieval strategies. In G. H. Bower (Ed.),The psychology of learning and motivation (Vol. 22, pp. 227–259). New York: Academic Press.
Reder, L. M., &Anderson, J. R. (1980). A partial resolution of the paradox of interference: The role of integrating knowledge.Cognitive Psychology,12, 447–472.
Reder, L. M., &Ritter, F. (1992). What determines initial feeling of knowing? Familiarity with question terms, not with the answer.Journal of Experimental Psychology: Learning, Memory, & Cognition,18, 435–451.
Reder, L. M., &Schunn, C. D. (1996). Metacognition does not imply awareness: Strategy choice is governed by implicit learning and memory. In L. M. Reder (Ed.),Implicit memory and metacognition (pp. 45–77). Hillsdale, NJ: Erlbaum.
Reder, L. M., & Schunn, C. D. (in press). Bringing together the psychometric and strategy worlds: Predicting adaptivity in a dynamic task. In D. Gopher & A. Koriat (Eds.),Attention and performance XVII: Cognitive regulation of performance: Interaction of theory and application. Cambridge, MA: MIT Press.
Schunn, C. D., Reder, L. M., Nhouyvanisvong, A., Richards, D. R., &Stroffolino, P. J. (1997). To calculate or not calculate: A source activation confusion model of problem familiarity’s role in strategy selection.Journal of Experimental Psychology: Learning, Memory, & Cognition,23, 3–29.
Siegler, R. S. (1988). Strategy choice procedures and the development of multiplication skills.Journal of Experimental Psychology: General,117, 258–275.
Siegler, R. S., Adolph, K., &Lemaire, P. (1996). Strategy choices across life span. In L. Reder (Ed.),Implicit memory and metacognition (pp. 79–122). Mahwah, NJ: Erlbaum.
Siegler, R. S., &Lemaire, P. (1997). Older and younger adults’ strategy choices in multiplication: Testing predictions of ASCM via the choice/no-choice method.Journal of Experimental Psychology: General,126, 71–92.
Siegler, R. S., &Shrager, J. (1984). Strategy choices in addition and subtraction: How do children know what to do? In C. Sophian (Ed.),Origins of cognitive skills (pp. 229–93). Hillsdale, NJ: Erlbaum.
Smith, E. E., Shoben, E. J., &Rips, L. J. (1974). Structure and process in semantic memory: A featural model for semantic decisions.Psychological Review,81, 214–241.
Stone, G. O., &Van Orden, G. (1993). Strategic control of processing in word recognition.Journal of Experimental Psychology: Human Perception & Performance,19, 744–774.
Widaman, K. F., Geary, D. C., Cormier, P., &Little, T. D. (1989). A componential model for mental addition.Journal of Experimental Psychology: Learning, Memory, & Cognition,15, 898–919.
Zbrodoff, N. J. (1995). Why is 9 + 7 harder than 2 + 3? Strength and interference as explanations of the problem-size effect.Memory & Cognition,23, 689–700.
Zbrodoff, N. J., &Logan, G. D. (1990). On the relation between production and verification tasks in the psychology of simple arithmetic.Journal of Experimental Psychology: Learning, Memory, & Cognition,16, 83–97.