Academic Journal
A Computational Model of Basic Addition Solving.
| Τίτλος: | A Computational Model of Basic Addition Solving. |
|---|---|
| Συγγραφείς: | Chouteau S; Université Grenoble Alpes, CNRS, LPNC., Mazens K; Université Grenoble Alpes, CNRS, LPNC., Thevenot C; Institute of Psychology, University of Lausanne., Lemaire B; Université Grenoble Alpes, CNRS, LPNC. |
| Πηγή: | Cognitive science [Cogn Sci] 2026 Apr; Vol. 50 (4), pp. e70207. |
| Τύπος έκδοσης: | Journal Article |
| Γλώσσα: | English |
| Στοιχεία περιοδικού: | Publisher: Wiley-Blackwell Country of Publication: United States NLM ID: 7708195 Publication Model: Print Cited Medium: Internet ISSN: 1551-6709 (Electronic) Linking ISSN: 03640213 NLM ISO Abbreviation: Cogn Sci Subsets: MEDLINE |
| Imprint Name(s): | Publication: 2009-: Hoboken, N.J. : Wiley-Blackwell Original Publication: Norwood, N. J., Ablex Pub. Corp. |
| Ιατρικοί όροι (MeSH): | Problem Solving*/physiology , Learning*/physiology , Mental Recall*/physiology , Computer Simulation* , Mathematics* , Models, Psychological*, Memory/physiology ; Humans |
| Περίληψη: | This article presents a computational learning model in which procedural execution and memory retrieval codevelop, using simple arithmetic, which provides a particularly well-controlled domain for investigating this issue. In single-digit addition learning, strategies employed initially rely on counting due to the lack of stored answers in memory. Over time, associations between problems and their solutions are strengthened. The model accounts for this learning process by dynamically selecting between counting and memory retrieval, based on their expected duration. It also introduces a mechanism for accelerating counting throuSUPPLEMgh repeated practice along the mental sequence. The model was first tested on data collected from adults learning to solve alphabet arithmetic problems over a 3-week experiment. It successfully replicated the empirical finding that larger problems are memorized earlier than smaller ones. A second simulation was conducted using data from an experiment manipulating problem structure: participants were trained on either contiguous (A+…, B+…, C+…) or noncontiguous (A+…, C+…, E+…) sequences. This variation affected the transition between strategies: participants in the noncontiguous condition showed a greater tendency to rely on retrieval, as the practice of moving from one letter to the next differed. The model also reproduced this pattern. Overall, the results suggest that no single strategy dominates at the end of learning; rather, counting and retrieval coexist, depending on problem size and structure. This model is, to our knowledge, the only one to incorporate a counting acceleration mechanism in line with the automated counting theory and memory retrieval. (© 2026 Cognitive Science Society LLC.) |
| References: | Anderson, J. R. (1982). Acquisition of cognitive skill. Psychological Review, 89(4), 369–406. https://doi.org/10.1037/0033‐295X.89.4.369. Ashcraft, M. H. (1982). The development of mental arithmetic−A chronometric approach. Developmental Review, 2(3), 213–236. https://doi.org/10.1016/0273‐2297(82)90012‐0. Ashcraft, M. H. (1992). Cognitive arithmetic: A review of data and theory. Cognition, 44(1–2), 75–106. https://doi.org/10.1016/0010‐0277(92)90051‐I. Ashcraft, M. H., & Battaglia, J. (1978). Cognitive arithmetic: Evidence for retrieval and decision processes in mental addition. Journal of Experimental Psychology: Human Learning and Memory, 4, 527–538. Retrieved from https://psycnet.apa.org/doi/10.1037/0278‐7393.4.5.527. 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(5), 396–421. https://doi.org/10.2307/749430. Ashcraft, M. H., & Guillaume, M. M. (2009). Mathematical cognition and the problem size effect. The psychology of learning and motivation, 51, 121–151. Ashcraft, M. H., & Stazyk, E. H. (1981). Mental addition: A set of three verification models. Memory & Cognition, 9(2), 185–196. https://doi.org/10.3758/BF03202334. Bagnoud, J., Dewi, J., Castel, C., Mathieu, R., & Thevenot, C. (2021). Developmental changes in size effects for simple tie and non‐tie addition problems in 6‐ to 12‐year‐old children and adults. Journal of Experimental Child Psychology, 201, Article 104987. https://doi.org/10.1016/j.jecp.2020.104987. Bagnoud, J., Poletti, C., Krenger, M., Mahendrathas, M., Dewi, J., & Thevenot, C. (2025). Unraveling the small tie problem mystery: Size effects from finger counting to mental strategies in addition. Journal of Experimental Child Psychology, 252, 106154. https://doi.org/10.1016/j.jecp.2024.106154. Baroody, A. J. (1983). The development of procedural knowledge: An alternative explanation for chronometric trends of mental arithmetic. Developmental Review, 3(2), 225–230. https://doi.org/10.1016/0273‐2297(83)90031‐X. Baroody, A. J. (1994). An evaluation of evidence supporting fact‐retrieval models. Learning and Individual Differences, 6(1), 1–36. https://doi.org/10.1016/1041‐6080(94)90013‐2. Baroody, A. J. (2018). A commentary on Chen and Campbell (2017): Is there a clear case for addition fact recall? Psychonomic Bulletin and Review 25, 2398–2405. https://doi.org/10.3758/s13423‐018‐1440‐y. Barrouillet, P., & Thevenot, C. (2013). On the problem size effect in small addition: Can we really discard any counting‐based account? Cognition, 128(1), 35–44. https://doi.org/10.1016/j.cognition.2013.02.018. Booth, J. R., Perfetti, C., & Macwhinney, B. (1999). Quick, automatic, and general activation of orthographic and phonological representations in young readers. Developmental Psychology, 35(1), 3–19. Campbell, J. I. D. (1995). Mechanisms of single addition and multiplication: A modified network‐interference theory and simulation. Mathematical Cognition, 1(2), 121–164. Chen, Y., & Campbell, J. I. D. (2018). “Compacted” procedures for adults' simple addition: A review and critique of the evidence. Psychonomic Bulletin & Review, 25, 739–753. https://doi.org/10.3758/s13423‐017‐1328‐2. Chouteau, S., Lemaire, B., Thevenot, C., Dewi, J., & Mazens, K. (2024). Learning basic arithmetic: A comparison between rote and procedural learning based on an artificial sequence. Journal of Experimental Psychology: Learning, Memory, and Cognition, 50(3), 418–434. https://doi.org/10.1037/xlm0001241. Chouteau, S., Lemaire, B., Thevenot, C., & Mazens, K. (2025). How learning material shapes learning strategies in an alphabet arithmetic task. Journal of Cognitive Psychology, 37(3), 249–266. https://doi.org/10.1080/20445911.2025.2464600. Chouteau, S., Mazens, K., Thevenot, C., Dewi, J., & Lemaire, B. (2021). A computational model of counting along a mental line. In T. Fitch, C. Lamm, H. Leder, & K. Teßmar‐Raible (Eds.), Proceedings of the 43rd Annual Conference of the Cognitive Science Society (pp. 2010–2016). Dias‐Barriga Yanez, A., Couderc, A., Longo, L., Merchie, A., Chesnokova, H., Langlois, E., Thevenot, C., & Prado, J. (2020). Learning to run the number line: The development of attentional shifts during single‐digit arithmetic. Annals of the New York Academy of Sciences, 1477(1), 79–90. https://doi.org/10.1111/nyas.14464. Dickinson, A. (1985). Actions and habits: The development of behavioural autonomy. Philosophical Transactions of the Royal Society of London, 308(1135), 67–78. https://doi.org/10.1098/rstb.1985.0010. Elofsson, J., Gustafson, S., Samuelsson, J., & Träff, U. (2016). Playing number board games supports 5‐year‐old children's early mathematical development. Journal of Mathematical Behavior, 43, 134–147. https://doi.org/10.1016/j.jmathb.2016.07.003. Farrington‐Flint, L. (2015). Uncovering strategy profiles in young children's reading & spelling. Learning and Individual Differences, 42, 64–69. https://doi.org/10.1016/j.lindif.2015.08.001. Farrington‐Flint, L., Coyne, E., Stiller, J., & Heath, E. (2008). Variability in children's early reading strategies. Educational Psychology, 28(6), 643–661. https://doi.org/10.1080/01443410802140958. Fayol, M., & Thevenot, C. (2012). The use of procedural knowledge in simple addition and subtraction problems. Cognition, 123(3), 392–403. https://doi.org/10.1016/j.cognition.2012.02.008. Ferrand, L., & Grainger, J. (1992). Phonology and orthography in visual word recognition: Evidence from masked non‐word priming. Quarterly Journal of Experimental Psychology. A, Human Experimental Psychology, 45(3), 353–372. https://doi.org/10.1080/02724989208250619. Fuchs, L. S., Powell, S. R., Seethaler, P. M., Fuchs, D., Hamlett, C. L., Cirino, P. T., & Fletcher, J. M. (2010). A framework for remediating number combination deficits. Exceptional Children, 76(2), 135–156. https://doi.org/10.1177/001440291007600201. Graybiel, A. M. (2008). Habits, rituals, and the evaluative brain. Annual Review of Neuroscience, 31, 359–387. https://doi.org/10.1146/annurev.neuro.29.051605.112851. Groen, G. J., & Parkman, J. M. (1972). A chronometric analysis of simple addition. Psychological Review, 79(4), 329–343. https://doi.org/10.1037/h0032950. Hamman, M. S., & Ashcraft, M. H. (1986). Textbook presentations of the basic addition facts. Cognition and Instruction, 3(3), 173–192. https://doi.org/10.1207/s1532690xci0303_2. Hasher, L., & Zacks, R. (1979). Automatic and effortful processes in memory. Journal of Experimental Psychology: General, 108, 356–388. Retrieved from https://psycnet.apa.org/doi/10.1037/0096‐3445.108.3.356. Kang, I., & Ratcliff, R. (2020). Modeling the interaction of numerosity and perceptual variables with the diffusion model. Cognitive Psychology, 120, Article 101288. https://doi.org/10.1016/j.cogpsych.2020.101288. LeFevre, J., Shanahan, T., & DeStefano, D. (2004). The tie effect in simple arithmetic: An access‐based account. Memory & Cognition, 32, 1019–1031. https://doi.org/10.3758/BF03196878. Logan, G. D. (1988). Toward an instance theory of automatization. Psychological Review, 95(4), 492–527. https://doi.org/10.1037/0033‐295X.95.4.492. Logan, G. D., & Klapp, S. T. (1991). Automatizing alphabet arithmetic: I. Is extended practice necessary to produce automaticity? Journal of Experimental Psychology: Learning, Memory, and Cognition, 17(2), 179–195. https://doi.org/10.1037/0278‐7393.17.2.179. Mathieu, R., Gourjon, A., Couderc, A., Thevenot, C., & Prado, J. (2016). Running the number line: Rapid shifts of attention in single‐digit arithmetic. Cognition, 146, 229–239. https://doi.org/10.1016/j.cognition.2015.10.002. Pashler, H. (1994). Dual‐task interference in simple tasks: Data and theory. Psychological Bulletin, 116(2), 220–244. https://doi.org/10.1037/0033‐2909.116.2.220. Perfetti, C. A., Bell, L. C., & Delaney, S. M. (1988). Automatic (prelexical) phonetic activation in silent word reading: Evidence from backward masking. Journal of Memory and Language, 27(1), 59–70. https://doi.org/10.1016/0749‐596X(88)90048‐4. Poletti, C., Díaz‐Barriga Yáñez, A., Prado, J., & Thevenot, C. (2023). The development of simple addition problem solving in children: Reliance on automatized counting or memory retrieval depends on both expertise and problem size. Journal of Experimental Child Psychology, 234, 105710. https://doi.org/10.1016/j.jecp.2023.105710. Poletti, C., Krenger, M., Létang, M., Hennequin, B., & Thevenot, C. (2025). Finger counting training enhances addition performance in kindergarteners. Child Development, 96(1), 251–268. https://doi.org/10.1111/cdev.14146. Rickard, T. C. (1997). Bending the power law: A CMPL theory of strategy shifts and the automatization of cognitive skills. Journal of Experimental Psychology: General, 126(3), 288–311. https://doi.org/10.1037/0096‐3445.126.3.288. Rickard, T. C. (1999). A CMPL alternative account of practice effects in numerosity judgment tasks. Journal of Experimental Psychology: Learning, Memory, and Cognition, 25(2), 532–542. https://doi.org/10.1037/0278‐7393.25.2.532. Rickard, T. C. (2004). Strategy execution in cognitive skill learning: An item‐level test of candidate models. Journal of Experimental Psychology: Learning, Memory, and Cognition, 30(1), 65–82. https://doi.org/10.1037/0278‐7393.30.1.65. Rittle‐Johnson, B., & Siegler, R. S. (1999). Learning to spell: Variability, choice, and change in children's strategy use. Child Development, 70(2), 332–348. Rosenbaum, D. A. (1980). Human movement initiation: Specification of arm, direction, and extent. Journal of Experimental Psychology: General, 109(4), 444–474. https://doi.org/10.1037/0096‐3445.109.4.444. Sherman, J., & Bisanz, J. (2007). Evidence for use of mathematical inversion by three‐year‐old children. Journal of Cognition and Development, 8(3), 333–344. https://doi.org/10.1080/15248370701446798. Shiffrin, R., & Schneider, W. (1977). Controlled and automatic human information processing: II. Perceptual learning, automatic attending and a general theory. Psychological Review, 84, 127–190. Retrieved from https://psycnet.apa.org/doi/10.1037/0033‐295X.84.2.127. Siegler, R. S. (1988). Individual‐differences in strategy choices–Good students, not‐so‐good students, and perfectionists. Child Development, 59(4), 833–851. https://doi.org/10.2307/1130252. Siegler, R. S., & Ramani, G. B. (2008). Playing linear numerical board games promotes low‐income children's numerical development. Developmental Science, 11(5), 655–661. https://doi.org/10.1111/j.1467‐7687.2008.00714.x. Siegler, R. S., & Shrager, J. (1984). Strategic choices in addition and subtraction: How do children know what to do? In C. Sophian (Ed.), Origins of cognitive skills (pp. 229–293). Lawrence Erlbaum. Szameitat, A. J., Vanloo, A., & Müller, H. J. (2016). Central as well as peripheral attentional bottlenecks in dual‐task performance activate lateral prefrontal cortices. Frontiers in Human Neuroscience, 10, 119. https://doi.org/10.3389/fnhum.2016.00119. Thevenot, C., & Barrouillet, P. (2020). Are small additions solved by direct retrieval from memory or automated counting procedures? A rejoinder to Chen and Campbell (2018). Psychonomic Bulletin & Review, 27(6), 1416–1418. https://doi.org/10.3758/s13423‐020‐01818‐4. Thevenot, C., Barrouillet, P., & Fayol, M. (2001). Algorithmic solution of arithmetic problems and operands: Answer associations in long‐term memory. Quarterly Journal of Experimental Psychology A: Human Experimental Psychology, 54A(2), 599–611. https://doi.org/10.1080/02724980042000291. Thevenot, C., Dewi, J. D. M., Bagnoud, J., Uittenhove, K., & Castel, C. (2020). Scrutinizing patterns of solution times in alphabet‐arithmetic tasks favors counting over retrieval models. Cognition 200, 104272. https://doi.org/10.1016/j.cognition.2020.104272. Thorndike, E. L. (1922). The psychology of arithmetic. Macmillan Company. Towse, J. N., & Hitch, G. J. (1995). Is there a relationship between task demand and storage space in tests of working memory capacity? Quarterly Journal of Experimental Psychology, 48A(1), 108–124. https://doi.org/10.1080/14640749508401379. Towse, J. N., Hitch, G. J., & Hutton, U. (1998). A reevaluation of working memory capacity in children. Journal of Memory and Language, 39(2), 195–217. https://doi.org/10.1006/jmla.1998.2574. Uittenhove, K., Thevenot, C., & Barrouillet, P. (2016). Fast automated counting procedures in addition problem solving: When are they used and why are they mistaken for retrieval? Cognition, 146, 289–303. https://doi.org/10.1016/j.cognition.2015.10.008. Widaman, K. F., Geary, D. C., Cormier, P., & Little, T. D. (1989). A componential model for mental addition. Journal of Experimental Psychology: Learning, Memory, and Cognition, 15(5), 898–919. https://doi.org/10.1037/0278‐7393.15.5.898. |
| Grant Information: | Université Grenoble Alpes |
| Contributed Indexing: | Keywords: Alphabet; Arithmetic; Computational modeling; Learning; Mathematical cognition |
| Entry Date(s): | Date Created: 20260422 Date Completed: 20260715 Latest Revision: 20260715 |
| Update Code: | 20260716 |
| DOI: | 10.1111/cogs.70207 |
| PMID: | 42015605 |
| Βάση Δεδομένων: | MEDLINE |
καταχωρήστε σχόλιο πρώτοι!