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Costs and Benefits of Telling Children the Quantitative Meaning of Manipulatives

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Costs and Benefits of Telling Children the Quantitative Meaning of Manipulatives

Adrien, Emmanuelle ORCID: https://orcid.org/0000-0003-4363-9755 (2020) Costs and Benefits of Telling Children the Quantitative Meaning of Manipulatives. PhD thesis, Concordia University.

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Abstract

The objective of the present study was to identify the costs and benefits of directly telling students the quantitative referents for manipulatives compared to allowing them to construct meaning for the manipulatives in more open and exploratory learning environments. Sixty-five (N = 65) first graders were randomly assigned to one of three conditions that differed in the type of encoding instruction they received: direct instruction (DI), guided exploration (GE), or control. The overarching research question was: How do the ways in which children assign a quantitative referent to a target manipulative (DI vs. GE vs. control) influence their (a) learning, (b) near-transfer abilities, (c) symbolic flexibility and symbolic fluency through far-transfer tasks, and (e) problem-solving accuracy?

Results indicated that direct instruction seemed to be most beneficial for children’s learning. In terms of the learning assessment, children from the DI condition benefitted relative to children in the GE condition, in that they needed fewer items and less time before using the target manipulative in the prescribed way. Evidence suggested that children in the DI condition also outperformed their counterparts in the GE condition on a near-transfer task when looking at their initial responses, but when both initial and post-prompt responses were considered, the performance of children in the GE condition was not significantly different from the performance of children in the DI condition. In contrast, students who learned through guided exploration seemed to be more flexible in their use and interpretation of the manipulatives in the context of the far-transfer tasks than those who were told explicitly what the objects represented. The greater flexibility demonstrated by children in the GE condition also conferred an advantage on their accuracy when solving word problems with the manipulatives compared to children in the DI condition.

This study contributes to the existing literature in that it offers a nuanced view of the use of manipulatives in classroom contexts. Results suggest that teachers may wish to tailor their instructional methods to the learning objectives (e.g., learning, near transfer, far transfer) they have set for their students when using concrete representations with them.

Divisions:Concordia University > Faculty of Arts and Science > Education
Item Type:Thesis (PhD)
Authors:Adrien, Emmanuelle
Institution:Concordia University
Degree Name:Ph. D.
Program:Education
Date:February 2020
Thesis Supervisor(s):Osana, Helena P.
Keywords:Elementary mathematics; Instruction; Manipulatives
ID Code:986616
Deposited By: EMMANUELLE ADRIEN
Deposited On:25 Jun 2020 17:54
Last Modified:25 Jun 2020 17:54

References:

Abrahamson, D., & Kapur, M. (2018). Reinventing discovery learning: a field-wide research program. Instructional Science, 46(1), 1-10. https://doi.org/10.1007/s11251-017-9444-y

Acevedo Nistal, A., Van Dooren, W., Clarebout, J. E., & Verschaffel, L. (2009). Conceptualising, investigating and stimulating representational flexibility in mathematical problem solving and learning: a critical review. ZDM Mathematics Education, 41, 627-636. https://doi.org/10.1007/s11858-009-0189-1

Adrien, E., Osana, H. P., Uttal, D., & Orsini, A. (2019, March). Instructional environment influences how first-graders interact with math manipulatives [Poster presentation]. Society for Research in Child Development biennial meeting, Baltimore, MD.

Ainsworth, S. (1999). The functions of multiple representations. Computers & Education, 33(2-3), 131-152. https://doi.org/10.1016/S0360-1315(99)00029-9

Alfieri, L., Brooks, P. J., Aldrich, N. J., & Tenenbaum, H. R. (2011). Does discovery-based instruction enhance learning? Journal of Educational Psychology, 103(1), 1-18. https://doi.org/10.1037/a0021017

Alibali, M. W., & Nathan, M. J. (2012) Embodiment in mathematics teaching and learning: Evidence from learners’ and teachers’ gestures. Journal of the Learning Sciences, 21(2), 247-286. https://doi.org/10.1080/10508406.2011.611446

Astle, A., Kamawar, D., Vendetti, C., & Podjarny, G. (2013). When this means that: The role of working memory and inhibitory control in children’s understanding of representations. Journal of Experimental Child Psychology, 116(2), 169-185. https://doi.org/10.1016/j.jecp.2013.05.003

Baddeley, A. (1990). Human memory. Erlbaum.

Ball, D. L. (1992). Magical hopes: Manipulatives and the reform of math education. American Educator: The Professional Journal of the American Federation of Teachers, 16(2).

Ball, D. L. (1993). Halves, pieces, and twoths: Constructing and using representational contexts in teaching fractions. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 157-196). Erlbaum.

Baroody, A. J., Purpura, D. J., Eiland, M. D., & Reid, E. E. (2015). The impact of highly and minimally guided discovery instruction on promoting the learning of reasoning strategies for basic add-1 and doubles combinations. Early Childhood Research Quarterly, 30, 93-105. http://dx.doi.org/10.1016/j.ecresq.2014.09.003

Barsalou, L. W. (2008). Grounded cognition. Annual Review Psychology, 59, 617-645. https://doi.org/10.1146/annurev.psych.59.103006.093639

Behr, M. J., Lesh, R., Post, T. R., & Silver, E. A. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91-126). Academic Press.

Belenky, D. M., & Nokes, T. J. (2009). Examining the role of manipulatives and metacognition on engagement, learning, and transfer. The Journal of Problem Solving, 2(2), 6. https://doi.org/10.7771/1932-6246.1061

Belenky, D. M., & Schalk, L. (2014). The effects of idealized and grounded materials on learning, transfer, and interest: An organizing framework for categorizing external knowledge representations. Educational Psychology Review, 26(1), 27-50. https://doi.org/10.1007/s10648-014-9251-9

Berch, D. B., Krikorian, R., & Huha, E. M. (1998). The Corsi block-tapping task: Methodological and theoretical considerations. Brain and Cognition, 38(3), 317-338. https://doi.org/10.1006/brcg.1998.1039

Bialystok, E. (1992a). The emergence of symbolic thought: Introduction. Cognitive Development, 7, 269-272. https://doi.org/10.1016/0885-2014(92)90015-J

Bialystok, E. (1992b). Symbolic representation of letters and numbers. Cognitive Development, 7, 301-316. https://doi.org/10.1016/0885-2014(92)90018-M

Bialystok, E. (2000). Symbolic representation across domains in preschool children. Journal of Experimental Child Psychology, 76(3), 173-189. https://doi.org/10.1006/jecp.1999.2548

Bialystok, E., & Codd, J. (1996). Developing representations of quantity. Canadian Journal of Behavioural Science/Revue canadienne des sciences du comportement, 28(4), 281-291. https://doi.org/10.1037/0008-400X.28.4.281

Bieda, K. N., & Nathan, M. J. (2009). Representational disfluency in algebra: Evidence from student gestures and speech. ZDM Mathematics Education, 41(5), 637-650. https://doi.org/10.1007/s10648-014-9259-1

Bonawitz, E., Shafto, P., Gweon, H., Goodman, N. D., Spelke, E., & Schulz, L. (2011). The double-edged sword of pedagogy: Instruction limits spontaneous exploration and discovery. Cognition, 120(3), 322-330. https://doi.org/10.1016/j.cognition.2010.10.001

Boone, H. N., Jr., & Boone, D. A. (2012). Analyzing Likert data. Journal of Extension, 50(2), 1-5.

Brown, M. C., McNeil, N. M., Glenberg, A. M. (2009). Using concreteness in education: Real problems, potential solutions. Child Development Perspectives, 3(3), 160-164. https://doi.org/10.1111/j.1750-8606.2009.00098.x

Burns, B. A., & Hamm, E. M. (2011). A comparison of concrete a virtual manipulative use in third- and fourth-grade mathematics. School Science and Mathematics, 111(6), 256-261. https://doi.org/10.1111/j.1949-8594.2011.00086.x

Callaghan, T., & Corbit, J. (2015). The development of symbolic representation. In R. M. Lerner, L. S. Liben, & U. Mueller (Eds.). Handbook of child psychology and developmental science, Volume 2, Cognitive Processes, 7th edition (pp. 1-46). Wiley.

Carbonneau, K. J., & Marley, S. C. (2015). Instructional guidance and realism of manipulatives influence preschool children's mathematics learning. The Journal of Experimental Education, 83(4), 495-513. https://doi.org/10.1080/00220973.2014.989306

Carbonneau, K. J., Marley, S. C., & Selig, J. P. (2013). A meta-analysis of the efficacy of teaching mathematics with concrete manipulatives. Journal of Educational Psychology, 105(2), 380-400. https://doi.org/10.1037/a0031084

Carleton University Math Lab. (2017). Go/No Go (Version 1.1.01) [iPad software]. https://carleton.ca/cacr/math-lab/apps/gono-go-app/

Carlson, S. M., & Beck, D. M. (2009). Symbols as tools in the development of executive function. In A. Winsler, C. Fernyhough, & I. Montero (Eds.), Private speech, executive functioning, and the development of verbal self-regulation (pp. 163-175). Cambridge University Press.

Carpenter, T. P., Ansell, E., Franke, M. L., Fennema, E., & Weisbeck, L. (1993). Models of problem solving: A study of kindergarten children's problem-solving processes. Journal for Research in Mathematics Education, 428-441. https://doi.org/10.2307/749152

Carraher, T. N., Carraher, D. W., & Schliemann, A. D. (1985). Mathematics in the street and in schools. British Journal of Developmental Psychology, 3, 21-29. https://doi.org/10.1111/j.2044-835X.1985.tb00951.x

Ceuppens, S., Deprez, J., Dehaene, W., & De Cock, M. (2018). Design and validation of a test for representational fluency of 9th grade students in physics and mathematics: The case of linear functions. Physical Review Physics Education Research, 14(2), 1-19. https://doi.org/10.1103/PhysRevPhysEducRes.14.020105

Chao, S. J., Stigler, J. W., & Woodward, J. A. (2000). The effects of physical materials on kindergartners’ learning of number concepts. Cognition and Instruction, 18(3), 285-316. https://doi.org/10.1207/S1532690XCI1803_1

Chase, K., & Abrahamson, D. (2018). Searching for buried treasure: uncovering discovery in discovery-based learning. Instructional Science, 46(1), 11-33. https://doi.org/10.1007/s11251-017-9433-1

Chi, M. (2000). Self-explaining expository texts: The dual processes of generating inferences and repairing mental models. In R. Glaser (Ed.), Advances in instructional psychology, Volume 5 (pp. 161-238). Erlbaum.

Clark, R. E. (2009). How much and what type of guidance is optimal for learning from instruction. In Tobias, S., & Duffy, T. M. (Eds.), Constructivist instruction: Success or failure (pp. 158-183). Routledge.

Clements, D. H., & Joswick, C. (2018). Broadening the horizons of research on discovery-based learning. Instructional Science, 46(1), 155-167. https://doi.org/10.1007/s11251-018-9449-1

Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal for Research in Mathematics Education, 2-33. https://doi.org/10.2307/749161

Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers.

DeCaro, M. S., & Rittle-Johnson, B. (2012). Exploring mathematics problems prepares children to learn from instruction. Journal of Experimental Child Psychology, 113(4), 552-568. http://dx.doi.org/10.1016/j.jecp.2012.06.009

Deliyianni, E., Gagatsis, A., Elia, I., & Panaoura, A. (2016). Representational flexibility and problem-solving ability in fraction and decimal number addition: A structural model. International Journal of Science and Mathematics Education, 14(2), 397-417. https://doi.org/10.1007/s10763-015-9625-6

DeLoache, J. S. (1987). Rapid change in the symbolic functioning of very young children. Science, 238(4833), 1556-1557. https://doi.org/10.1126/science.2446392

DeLoache, J. S. (1991). Symbolic functioning in very young children: Understanding of pictures and models. Child Development, 62, 736-752. https://doi.org/10.1111/j.1467-8624.1991.tb01566.x

DeLoache, J. S. (1995). Early understanding and use of symbols: The Model model. Current Directions in Psychological Science, 4(4), 109-113. https://doi.org/10.1111/1467-8721.ep10772408

DeLoache, J. S. (2000). Dual representation and young children’s use of scale models. Child Development, 71(2), 329-338. https://doi.org/10.1111/1467-8624.00148

DeLoache, J. S., de Mendoza, O. A. P., & Anderson, K. N. (1999). Multiple factors in early symbol use: Instructions, similarity, and age in understanding a symbol-referent relation. Cognitive Development, 14(2), 299-312. https://doi.org/10.1016/S0885-2014(99)00006-4

DeLoache, J. S., & Marzolf, D. P. (1992). When a picture is not worth a thousand words: Young children's understanding of pictures and models. Cognitive Development, 7(3), 317-329. https://doi.org/10.1016/0885-2014(92)90019-N

DeLoache, J. S., Uttal, D. H., & Pierroutsakos, S. L. (1998). The development of early symbolization: Educational implications. Learning and Instruction, 8(4), 325-339. https://doi.org/10.1016/S0959-4752(97)00025-X

diSessa A. A., & Sherin, B. L. (2000). Meta-representation: An introduction. The Journal of Mathematical Behavior, 19(4), 385-398. https://doi.org/10.1016/S0732-3123(01)00051-7

Donovan, A. M., & Fyfe, E. (2019, June 21). Connecting concrete objects and abstract symbols promotes children's mathematics learning. https://doi.org/10.31234/osf.io/ye2j6

Dufour-Janvier, B., Bednarz, N., & Belanger, M. (1987). Pedagogical considerations concerning the problem of representation. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 109-122). Erlbaum.

Durik, A. M., & Harackiewicz, J. M. (2007). Different strokes for different folks: How individual interest moderates the effects of situational factors on task interest. Journal of Educational Psychology, 99(3), 597-610. https://doi.org/10.1037/0022-0663.99.3.597

Empson, S. B. (1999). Equal sharing and shared meaning: The development of fraction concepts in a first-grade classroom. Cognition and Instruction, 17(3), 283-342. https://doi.org/10.1207/S1532690XCI1703_3

Empson, S. B., Levi, L., & Carpenter, T. P. (2011). The algebraic nature of fractions: Developing relational thinking in elementary school. In J. Cai, & E. Knuth (Eds.), Early algebraization (pp. 409-428). Springer.

English, L. D. (2004). Mathematical and analogical reasoning. In L. English (Ed.), Mathematical and analogical reasoning of young learners (pp. 1–22). Erlbaum.

Freeman, S., Eddy, S. L., McDonough, M., Smith, M. K., Okoroafor, N., Jordt, H., & Wenderoth, M. P. (2014). Active learning increases student performance in science, engineering, and mathematics. Proceedings of the National Academy of Sciences, 111(23), 8410-8415. https://doi.org/10.1073/pnas.1319030111

Fugate, J. M., Macrine, S. L., & Cipriano, C. (2019). The role of embodied cognition for transforming learning. International Journal of School & Educational Psychology, 7(4), 274-288. https://doi.org/10.1080/21683603.2018.1443856

Furner, J. M., & Worrell, N. L. (2017). The importance of using manipulatives in teaching math today. Transformations, 3(1), 2.

Fuson, K. C. (2009). Avoiding misinterpretations of Piaget and Vygotsky: Mathematical teaching without learning, learning without teaching, or helpful learning-path teaching? Cognitive Development, 24(4), 343-361. https://doi.org/10.1016/
j.cogdev.2009.09.009

Fuson, K. C., Fraivillig, J. L., & Burghardt, B. H. (1992). Relationships children construct among English number words, multiunit base-ten blocks, and written multidigit addition. In J. I. Campbell (Ed.), Advances in psychology (Vol. 91, pp. 39-112). North-Holland.

Fyfe, E. R., DeCaro, M. S., & Rittle-Johnson, B. (2014). An alternative time for telling: When conceptual instruction prior to problem solving improves mathematical knowledge. British Journal of Educational Psychology, 84(3), 502-519. https://doi.org/10.1111/bjep.12035

Fyfe, E. R., McNeil, N. M., & Son, J. Y. (2014). Concreteness fading in mathematics and science instruction: A systematic review. Educational Psychology Review, 26(1), 9-25. https://doi.org/10.1007/s10648-014-9249-3

Fyfe, E. R., & Nathan, M. J. (2019). Making “concreteness fading” more concrete as a theory of instruction for promoting transfer. Educational Review, 71(4), 403-422. https://doi.org/10.1080/00131911.2018.1424116

Fyfe, E. R., & Rittle-Johnson, B. (2012). The effects of feedback during exploratory mathematics problem solving: Prior knowledge matters. Journal of Educational Psychology, 104(4), 1094. https://doi.org/10.1037/a0028389

Fyfe, E. R., & Rittle-Johnson, B. (2016). Feedback both helps and hinders learning: The causal role of prior knowledge. Journal of Educational Psychology, 108(1), 82-97. https://doi.org/10.1037/edu0000053

Gelman, S. A., & Ebeling, K. S. (1998). Shape and representational status in children's early naming. Cognition, 66(2), B35-B47. https://doi.org/10.1016/S0010-0277(98)00022-5

Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38(4), 293-304. https://doi.org/10.1177/00222194050380040301

Gibson, J. J. (1977). The theory of affordances. In R. Shaw & J. Bransford (Eds.), Perceiving, acting, and knowing: Toward an ecological psychology (pp. 67-82). Erlbaum.

Glenberg, A. M. (2008). Embodiment for education. In P. Calvo, & T. Gomila (Eds.), Handbook of cognitive science (pp. 355-372). Elsevier.

Gravemeijer, K. P. (2002). Preamble: From models to modeling. In K. P. Gravemeijer, R. Lehrer, B. van Oers, & L. Verschaffel (Eds.), Symbolizing, modeling and tool use in mathematics education (pp. 7-22). Kluwer Academic.

Greeno, J. G. (1994). Gibson's affordances. Psychological Review, 101(2), 336-342. https://doi.org/10.1037/0033-295X.101.2.336

Gromko, J. E. (1998). Young children's symbol use: Common principles and cognitive processes. Update: Applications of Research in Music Education, 16(2), 3-7.

Hatano, G. (2003). Foreword. In A. J. Baroody, & A. Dowker (Eds.), The development of arithmetic concepts and skills: Constructing adaptive expertise (pp. xi-xiii). Erlbaum.

Hiebert, J. (1984). Children’s mathematics learning: The struggle to link form and understanding. Elementary School Journal, 84, 497-513. https://doi.org/10.1086/461380

Hiebert, J. (1988). A theory of developing competence with written mathematical symbols. Educational Studies in Mathematics, 19, 333-355. https://doi.org/10.1007/BF00312451

Hiebert, J. (1992). Mathematical, cognitive, and instructional analyses of decimal fractions. In G. Leinhardt, R. Putnam, & R. A. Hattrup (Eds.) Analysis of arithmetic for mathematics Teaching (pp. 283–322). Erlbaum.

Hiebert, J., Carpenter, T. R., Fennema, E., Fuson, K. C., Wearne, D., Murray, H., & Olivier, A., & Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. Heinemann.

Hiebert, J., & Grouws, D. A. (2007). The effects of classroom mathematics teaching on students’ learning. In F. K. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 371-404). Information Age.

Hmelo-Silver, C. E. (2004). Problem-based learning: What and how do students learn? Educational Psychology Review, 16(3), 235-266. https://doi.org/10.1023/B:EDPR.0000034022.16470.f3

Hmelo-Silver, C. E., Duncan, R. G., & Chinn, C. A. (2007). Scaffolding and achievement in problem-based and inquiry learning: a response to Kirschner, Sweller, and. Educational Psychologist, 42(2), 99-107. https://doi.org/10.1080/00461520701263368

Holmes, J., & Adams, J. W. (2006). Working memory and children’s mathematical skills: Implications for mathematical development and mathematics curricula. Educational Psychology, 26(3), 339-366. https://doi.org/10.1080/01443410500341056

Honomichl, R. D., & Chen, Z. (2012). The role of guidance in children's discovery learning. Wiley Interdisciplinary Reviews: Cognitive Science, 3(6), 615-622. https://doi.org/10.1002/wcs.1199

Horan, E., & Carr, M. (2018a). A review of guidance and structure in elementary school mathematics instruction. Review of Science, Mathematics and ICT Education, 12(2), 41-60. https://doi.org/10.26220/rev.2889

Horan, E., & Carr, M. (2018b). How much guidance do students need? An intervention study on kindergarten mathematics with manipulatives. International Journal of Educational Psychology, 7(3), 286-316. https://doi.org/10.17583/ijep.2018.3672

Hughes, M. (1986). Children and number. Basil Blackwell.

Hume, T. (2015). PathSpan (Version 2.0.1) [iPad software]. https://hume.ca/ix/pathspan

Hushman, C. J., & Marley, S. C. (2015). Guided instruction improves elementary student learning and self-efficacy in science. The Journal of Educational Research, 108(5), 371-381. https://doi.org/10.1080/00220671.2014.899958

Jacobs, V. R., & Kusiak, J. (2006). Got Tools? Exploring Children's Use of Mathematics Tools during Problem Solving. Teaching Children Mathematics, 12(9), 470-477.

Kaminski, J. A., Sloutsky, V. M., Heckler, A. (2009). Transfer of mathematical knowledge: The portability of generic instantiations. Child Development Perspectives, 3(3), 151-155. https://doi.org/10.1111/j.1750-8606.2009.00096.x

Kapur, M. (2014). Productive failure in learning math. Cognitive Science, 38(5), 1008-1022. https://doi.org/10.1111/cogs.12107

Kaput, J. J. (1987a). Representation systems and mathematics. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 19-26). Erlbaum.

Kaput, J. J. (1987b). Toward a theory of symbol use in mathematics. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 159-196). Erlbaum.

Karmiloff-Smith, A. (1991). Beyond modularity: Innate constraints and developmental change. In S. Carey, & R. Gelman (Eds.), The epigenesis of mind: Essays on biology and cognition (pp. 171-198). Erlbaum.

Kirschner, P. A., Sweller, J., & Clark, R. E. (2006). Why minimal guidance during instruction does not work: An analysis of the failure of constructivist, discovery, problem-based, experiential, and inquiry-based teaching. Educational Psychologist, 41, 75–86. https://doi.org/10.1207/s15326985ep4102_1

Klahr, D. (2009). “To every thing there is a season, and a time to every purpose under the heavens”: What about direct instruction? In S. Tobias, & T. M. Duffy (Eds.), Constructivist instruction (pp. 291-310). Routledge.

Klahr, D. (2010). Coming up for air: But is it oxygen or phlogiston? A response to Taber’s review of constructivist instruction: Success or failure. Education Review, 13(13), 1-6.

Klahr, D., & Nigam, M. (2004). The equivalence of learning paths in early science instruction: Effects of direct instruction and discovery learning. Psychological Science, 15(10), 661-667. https://doi.org/10.1111/j.0956-7976.2004.00737.x

Koedinger, K. R., & Nathan, M. J. (2004). The real story behind story problems: Effects of representations on quantitative reasoning. Journal of the Learning Sciences, 13(2), 129-164. https://doi.org/10.1207/s15327809jls1302_1

Kuhn, D. (2007). Is direct instruction an answer to the right question? Educational Psychologist, 42(2), 109-113. https://doi.org/10.1080/00461520701263376

Laski, E. V., Jor’dan, J. R., Daoust, C., & Murray, A. K. (2015). What makes mathematics manipulatives effective? Lessons from cognitive science and Montessori education. SAGE Open, 5(2), 1-8. https://doi.org/10.1177/2158244015589588

Latour, B. (1990). Drawing things together. In M. Lynch & S. Woolgar (Eds.), Representation in scientific practice (pp. 19-68). MIT Press.

Lazonder, A. W., & Harmsen, R. (2016). Meta-analysis of inquiry-based learning: Effects of guidance. Review of Educational Research, 86(3), 681-718. https://doi.org/10.3102/0034654315627366

Lee, H. S., & Anderson, J. R. (2013). Student learning: What has instruction got to do with it? Annual Review of Psychology, 64, 445-469. https://doi.org/10.1146/annurev-psych-113011-143833

Lee, J. S., & Ginsburg, H. P. (2009). Early childhood teachers’ misconceptions about mathematics education for young children in the United States. Australian Journal of Early Childhood, 34(4), 37-45. https://doi.org/10.1177/183693910903400406

LeFevre, J. A., Fast, L., Skwarchuk, S. L., Smith-Chant, B. L., Bisanz, J., Kamawar, D., & Penner-Wilger, M. (2010). Pathways to mathematics: Longitudinal predictors of performance. Child Development, 81(6), 1753-1767. https://doi.org/10.1111/j.1467-8624.2010.01508.x

Lesh, R. (1999). The development of representational abilities in middle school mathematics. In I. E. Sigel (Ed.), Development of mental representation: Theories and application (pp. 323-350). Erlbaum.

Lesh, R. (2000). What mathematical abilities are most needed for success beyond school in a technology based age of information. In M.O.J. Thomas (Ed.), Proceedings of TIME 2000: An international conference on technology in mathematics education (pp. 72– 82). Auckland University of Technology.

Lesh, R., Behr, M., & Post, T. (1987) Rational number relations and proportions. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 41-58). Erlbaum.

Lobato, J., Clarke, D., & Ellis, A. B. (2005). Initiating and eliciting in teaching: A reformulation of telling. Journal for Research in Mathematics Education, 36(2), 101-136. https://doi.org/10.2307/30034827

Loehr, A. M., Fyfe, E. R., & Rittle-Johnson, B. (2014). Wait for it... Delaying instruction improves mathematics problem solving: A classroom study. The Journal of Problem Solving, 7(1), 5. http://dx.doi.org/10.7771/1932-6246.1166

Manches, A., & O'Malley, C. (2016). The effects of physical manipulatives on children's numerical strategies. Cognition and Instruction, 34(1), 27-50. https://doi.org/10.1080/07370008.2015.1124882

Marley, S. C., & Carbonneau, K. J. (2014a). Future directions for theory and research with instructional manipulatives: Commentary on the special issue papers. Educational Psychology Review, 26(1), 91-100. https://doi.org/10.1007/s10648-014-9259-1

Marley, S. C., & Carbonneau, K. J. (2014b). Theoretical perspectives and empirical evidence relevant to classroom instruction with manipulatives. Educational Psychology Review, 26(1), 1-7. https://doi.org/10.1007/s10648-014-9257-3

Martí, E., Garcia-Mila, M., & Teberosky, A. (2005). Notational strategies for problem solving in 5- to 7-year olds. European Journal of Developmental Psychology, 2(4), 364-384. https://doi.org/10.1080/17405620500317607

Martí, E., Scheuer, N., & de la Cruz, M. (2013). Symbolic use of quantitative representations in young children. In B. M. Brizuela & B. E. Gravel (Eds.), “Show me what you know”: Exploring student representations across STEM disciplines (pp. 7-21). Teachers College Press.

Martin, T. (2009). A theory of physically distributed learning: How external environments and internal states interact in mathematics learning. Child Development Perspectives, 3(3), 140-144. https://doi.org/10.1111/j.1750-8606.2009.00094.x

Martin, T., & Schwartz, D. L. (2005). Physically distributed learning: Adapting and reinterpreting physical environments in the development of fraction concepts. Cognitive Science, 29(4), 587-625. https://doi.org/10.1207/s15516709cog0000_15

Marzolf, D. P., & DeLoache, J. S. (1994). Transfer in young children's understanding of spatial representations. Child Development, 65(1), 1-15. https://doi.org/10.1111/j.1467-8624.1994.tb00730.x

Mason, J. (1987). What do symbols represent? In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 73-82). Erlbaum.

Mayer, R. E. (2004). Should there be a three-strikes rule against pure discovery learning? American Psychologist, 59(1), 14. https://doi.org/10.1037/0003-066X.59.1.14

McGraw-Hill Education. (2008). Number knowledge test. McGraw-Hill Education.

McNeil, N. M., & Jarvin, L. (2007). When theories don’t add up: Disentangling the manipulatives debate. Theory Into Practice, 46(4), 309-316. https://doi.org/10.1080/
00405840701593899

McNeil, N. M., Uttal, D. H., Jarvin, L., Sternberg, R. J. (2009). Should you show me the money? Concrete objects both hurt and help performance on mathematics problems. Learning and Instruction, 19, 171-184. https://doi.org/10.1016/j.learninstruc.2008.03.005

Meira, L. (1995). The microevolution of mathematical representations in children's activity. Cognition and Instruction, 13, 269-313. https://doi.org/10.1207/s1532690xci1302_5

Ministère de l’Éducation du Québec. (2001). Québec Education Program. http://www.education.gouv.qc.ca/fileadmin/site_web/documents/PFEQ/educprg2001.pdf

Ministère de l’Éducation et de l’Enseignement supérieur du Québec. (2019). Indices de défavorisation (2017-2018). http://www.education.gouv.qc.ca/fileadmin/site_web/documents/PSG/statistiques_info_decisionnelle/Indices-defavorisation-2017-2018.xlsx

Moch, P. L. (2001). Manipulatives work! The Educational Forum, 66, 81-97. https://doi.org/10.1080/00131720108984802

Moyer, P. S. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in Mathematics, 47(2), 175-197. https://doi.org/10.1023/A:1014596316942

Myers, L. J., & Liben, L. S. (2008). The role of intentionality and iconicity in children’s developing comprehension and production of cartographic symbols. Child Development, 79(3), 668-684. https://doi.org/10.1111/j.1467-8624.2008.01150.x

Nathan, M. J. (2008). An embodied cognition perspective on symbols, gesture, and grounding instruction. In M. de Vega, A. Glenberg, & A. Graesser (Eds.), Symbols and embodiment: Debates on meaning and cognition (pp. 375-396). Oxford University Press.

Nathan, M. J., Stephens, A. C., Masarik, D. K., Alibali, M. W., & Koedinger, K. R. (2002). In Representational fluency in middle school: A classroom study. In D. S. Mewborn, P. Sztajn, D. Y. White, H. G. Wiegel, R. L. Bryant, & K. Nooney (Eds.), Proceedings of the 24th annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 462-472). ERIC Clearinghouse for Science, Mathematics and Environmental Education.

Nathan, M. J., Walkington, C., Boncoddo, R., Pier, E., Williams, C. C., & Alibali, M. W. (2014). Actions speak louder with words: The roles of action and pedagogical language for grounding mathematical proof. Learning and Instruction, 33, 182-193. http://dx.doi.org/10.1016/j.learninstruc.2014.07.001

National Research Council. (2009). Mathematics in early childhood: Learning paths toward excellence and equity. National Academy Press.

Okamoto, Y., & Case, R. (1996). Exploring the microstructure of children’s central conceptual structures in the domain of number. Monographs of the Society for Research in Child Development, 61, 27-58. https://doi.org/10.1111/j.1540-5834.1996.tb00536.x

Osana, H. P., Adrien, E., & Duponsel, N. (2017). Effects of Instructional Guidance and Sequencing of Manipulatives and Written Symbols on Second Graders’ Numeration Knowledge. Education Sciences, 7(2), 52. https://doi.org/10.3390/educsci7020052

Osana, H. P., Blondin, A., Alibali, M. W., & Donovan, A. M. (2018). The affordances of physical manipulatives on second-graders’ learning of number and place value [Paper presentation]. American Educational Research Association, New York, NY.

Osana, H. P., & Pitsolantis, N. (2013). Addressing the struggle to link form and understanding in fractions instruction. British Journal of Educational Psychology, 83, 29-56. https://doi.org/10.1111/j.2044-8279.2011.02053.x

Osana, H. P., & Pitsolantis, N. (2019). Supporting Meaningful Use of Manipulatives in Kindergarten: The Role of Dual Representation in Early Mathematics. In K. M. Robinson, H. P. Osana, & D. Kotsopoulos (Eds.), Mathematical learning and cognition in early childhood (pp. 91-113). Springer.

Osana, H. P., Przednowek, K., Cooperman, A., & Adrien, E. (2018). Encoding Effects on First-Graders' Use of Manipulatives. The Journal of Experimental Education, 86(2), 154-172. https://doi.org/10.1080/00220973.2017.1341862

Palmer, S. (1977). Fundamental aspects of cognitive representation. In: E. Rosch, & B. B. Lloyd (Eds.), Cognition and categorization (pp. 259-303). Erlbaum.

Petersen, L. A., & McNeil, N. M. (2013). Effects of perceptually rich manipulatives on preschoolers' counting performance: Established knowledge counts. Child Development, 84(3), 1020-1033. https://doi.org/10.1111/cdev.12028

Petit, M. M., Laird, R. E., Marsden, E. L., & Ebby, C. B. (2016). A focus on fractions: Bringing research to the classroom, 2nd ed. Taylor & Francis.

Pouw, W. T., van Gog, T., & Paas, F. (2014). An embedded and embodied cognition review of instructional manipulatives. Educational Psychology Review, 26(1), 51-72. https://doi.org/10.1007/s10648-014-9255-5

Puchner, L., Taylor, A., O’Donnell, B., Fick, K. (2008). Teacher learning and mathematics manipulatives: A collective case study about teacher use of manipulatives in elementary and middle school mathematics lessons. School Science and Mathematics, 108(7), 313-325. https://doi.org/10.1111/j.1949-8594.2008.tb17844.x

Purpura, D. J., Baroody, A. J., Eiland, M. D., & Reid, E. E. (2016). Fostering first graders’ reasoning strategies with basic sums: the value of guided instruction. The Elementary School Journal, 117(1), 72-100. https://doi.org/10.1086/687809

Rasmussen, C., & Bisanz, J. (2005). Representation and working memory in early arithmetic. Journal of Experimental Child Psychology, 91(2), 137-157. https://doi.org/10.1016/j.jecp.2005.01.004

Resnick, L. B., & Omanson, S. F. (1987). Learning to understand arithmetic. In R. Glaser (Ed.), Advances in instructional psychology (Vol. 3, pp. 41-95). Erlbaum.

Rittle-Johnson, B. (2006). Promoting transfer: Effects of self‐explanation and direct instruction. Child Development, 77(1), 1-15. https://doi.org/10.1111/j.1467-8624.2006.00852.x

Rochat, P., & Callaghan, T. C. (2005). What drives symbolic development? The case of pictorial comprehension and production. In L. L. Namy (Ed.), Symbol use and symbolic representation (pp. 25-46). Erlbaum.

Roll, I., Butler, D., Yee, N., Welsh, A., Perez, S., Briseno, A., Perkins, K., & Bonn, D. (2018). Understanding the impact of guiding inquiry: The relationship between directive support, student attributes, and transfer of knowledge, attitudes, and behaviours in inquiry learning. Instructional Science, 46(1), 77-104. https://doi.org/10.1007/s11251-017-9437-x

Rosen, D., Palatnik, A., & Abrahamson, D. (2018). A better story: An embodied-design argument for generic manipulatives. In N. Calder, K. Larkin, & N. Sinclair (Eds.), Using mobile technologies in the teaching and learning of mathematics (pp. 189-211). Springer.

Rosenshine, B. (2009). The Empirical Support for Direct Instruction. In S. Tobias, & T. M. Duffy (Eds.), Constructivist instruction (pp. 201-220). Routledge.

Sarama, J., & Clements, D. H. (2009). “Concrete” computer manipulatives in mathematics education. Child Development Perspectives, 3(3), 145-150. https://doi.org/10.1111/j.1750-8606.2009.00095.x

Schmidt, H. G., Loyens, S. M., Van Gog, T., & Paas, F. (2007). Problem-based learning is compatible with human cognitive architecture: Commentary on Kirschner, Sweller, and Clark (2006). Educational Psychologist, 42(2), 91-97. https://doi.org/10.1080/00461520701263350

Schoenfeld, A. H. (2004). The math wars. Educational Policy, 18(1), 253-286. https://doi.org/10.1177/0895904803260042

Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475-5223. https://doi.org/10.1207/s1532690xci1604_4

Schwartz, D. L., Chase, C. C., Oppezzo, M. A., & Chin, D. B. (2011). Practicing versus inventing with contrasting cases: The effects of telling first on learning and transfer. Journal of Educational Psychology, 103(4), 1-17. https://doi.org/10.1037/a0025140

Schwartz, D. L., Lindgren, R., & Lewis, R. (2009). Constructivism in an age of non-constructivist assessments. In S. Tobias, & T. M. Duffy (Eds.), Constructivist instruction (pp. 34-61). Routledge.

Schwartz, D. L., & Martin, T. (2004). Inventing to prepare for future learning: The hidden efficiency of encouraging original student production in statistics instruction. Cognition and Instruction, 22(2), 129-184. https://doi.org/10.1207/s1532690xci2202_1

Shafrir, U. (1999). Representational competence. In I. E. Sigel (Ed.), Development of mental representation: Theories and applications (pp. 371–389). Erlbaum.

Sharon, T. (2005). Made to symbolize: Intentionality and children's early understanding of symbols. Journal of Cognition and Development, 6(2), 163-178. https://doi.org/10.1207/s15327647jcd0602_1

Sherin, B. L. (2000). How students invent representations of motion: A genetic account. The Journal of Mathematical Behavior, 19(4), 399-441. https://doi.org/10.1016/S0732-3123(01)00052-9

Sherman, J., & Bisanz, J. (2009). Equivalence in symbolic and nonsymbolic contexts: Benefits of solving problems with manipulatives. Journal of Educational Psychology, 101, 88-100. https://doi.org/10.1037/a0013156

Siegler, R. S., & Lemaire, P. (1997). Older and younger adults' strategy choices in multiplication: Testing predictions of ASCM using the choice/no-choice method. Journal of Experimental Psychology: General, 126(1), 71. https://doi.org/10.1037/0096-3445.126.1.71

Sloutsky, V. M., Kaminsky, J. A., & Heckler, A. F. (2005). The advantage of simple symbols for learning transfer. Psychometric Bulletin & Review, 12(3), 508-513. https://doi.org/10.3758/BF03193796

Stanford Achievement Test (9th ed.). (1995). Psychological Corp.

Steffe, L. P., & Gale, J. E. (Eds.). (1995). Constructivism in education (p. 159). Erlbaum.

Sullivan, G. M., & Artino, A. R., Jr. (2013). Analyzing and interpreting data from Likert-type scales. Journal of Graduate Medical Education, 5(4), 541-542. http://dx.doi.org/10.4300/JGME-5-4-18

Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257-285. https://doi.org/10.1016/0364-0213(88)90023-7

Sweller, J., Kirschner, P. A., & Clark, R. E. (2007). Why minimally guided teaching techniques do not work: A reply to commentaries. Educational Psychologist, 42(2), 115-121. https://doi.org/10.1080/00461520701263426

Tabachneck, H. J. M., Koedinger, K. R., & Nathan, M. J. (1994). Toward a theoretical account of strategy use and sense-making in mathematics problem solving. In A. Ram, & K. Eiselt (Eds.), Proceedings of the 16th Annual Conference of the Cognitive Science Society (pp. 836-841). Erlbaum.

Thomas, M. O. (2008). Conceptual representations and versatile mathematical thinking. In M. Niss (Ed.), Proceedings of the 10th International Congress on Mathematical Education, 4-11 July 2004 (pp. 1-18). Roskilde University.

Thomas, M. O. J. & Hong, Y. Y. (2001) Representations as conceptual tools: Process and structural perspectives. Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 4, 257–264. https://www.math.auckland.ac.nz/~thomas/index/staff/mt/My%20PDFs%20for%20web%20site/PME01YY.pdf

Tran, C., Smith, B., & Buschkuehl, M. (2017). Support of mathematical thinking through embodied cognition: Nondigital and digital approaches. Cognitive Research: Principles and Implications, 2(1), 16. https://doi.org/10.1186/s41235-017-0053-8

Trninic, D. (2018). Instruction, repetition, discovery: Restoring the historical educational role of practice. Instructional Science, 46(1), 133-153. https://doi.org/10.1007/s11251-017-9443-z

Uttal, D. H., Amaya, M., del Rosario Maita, M., Hand, L. L., Cohen, C. A., O’Doherty, K., & DeLoache, J. S. (2013). It works both ways: Transfer difficulties between manipulatives and written subtraction solutions. Child Development Research, 2013, 1-13. https://doi.org/10.1155/2013/216367

Uttal, D. H., Liu L. L., & DeLoache, J. S. (2006). Concreteness and symbolic development. In L. Balter & C. S. Tamis-LeMonda (Eds.), Child psychology: A handbook of contemporary issues (2nd ed.) (pp. 167-184). Psychology Press.

Uttal, D. H., O’Doherty, K., Newland, R., Hand, L. L., & DeLoache, J. (2009). Dual representation and the linking of concrete and symbolic representations. Child Development Perspectives, 3(3), 156-159. https://doi.org/10.1111/j.1750-8606.2009.00097.x

Uttal, D. H., Scudder, K. V., & DeLoache, J. S. (1997). Manipulatives as symbols: A new perspective on the use of concrete objects to teach mathematics. Journal of Applied Developmental Psychology, 18, 37-54. https://doi.org/10.1016/S0193-3973(97)90013-7

Von Glasersfeld, E. (2013). Radical constructivism. Routledge.

Vygotsky, L. S. (1978). Tool and symbol use in child development. In M. Cole, V. John-Steiner, S. Scribner, & E. Souberman (Eds.), Mind in society: The development of higher psychological processes (pp. 19-30). Harvard University Press.

Wechsler, D. (1974). Manual for the Wechsler Intelligence Scale for Children-Revised. Psychological Corporation.

Weisberg, D. S., Hirsh-Pasek, K., & Golinkoff, R. M. (2013). Guided play: Where curricular goals meet a playful pedagogy. Mind, Brain, and Education, 7(2), 104-112. https://doi.org/10.1111/mbe.12015

Wise, A. F., & O’Neill, K. (2009). Beyond more versus less: A reframing of the debate on instructional guidance. In S. Tobias, & T. M. Duffy (Eds.), Constructivist instruction (pp. 94-117). Routledge.

Zacharia, Z. C., & Olympiou, G. (2011). Physical versus virtual manipulative experimentation in physics learning. Learning and Instruction, 21(3), 317-331. https://doi.org/10.1016/j.learninstruc.2010.03.001

Zelazo, P. D. (2006). The Dimensional Change Card Sort (DCCS): A method of assessing executive function in children. Nature Protocols, 1(1), 297-301. https://doi.org/10.1038/nprot.2006.46
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