Abrahamson, D. (2009). Embodied design: Construction means for constructing meaning. Educational Studies in Mathematics, 70(1), 27–47. https://doi.org/10.1007/s10649-008-9137-1
Ainsworth, S., Prain, V., & Tytler, R. (2011). Drawing to learn in science. Science, 333(6046), 1096–1097. https://doi.org/10.1126/science.1204153
Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215–241. https://doi.org/10.1023/A:1024312321077
Barnett, S. M., & Ceci, S. J. (2002). When and where do we apply what we learn?: A taxonomy for far transfer. Psychological Bulletin, 128(4), 612. https://doi.org/10.1037//0033-2909.128.4.612
Barsalou, L. W. (1999). Perceptions of perceptual symbols. Behavioral and Brain Sciences, 22(4), 637–660. https://doi.org/10.1017/S0140525X99532147
Brenner, M. E., Mayer, R. E., Moseley, B., Brar, T., Durán, R., Reed, B. S., & Webb, D. (1997). Learning by understanding: The role of multiple representations in learning algebra. American Educational Research Journal, 34(4), 663–689. https://doi.org/10.3102/00028312034004663
Carpenter, T., Franke, M., Jacobs, V., Fennema, E., & Empson, S. (1998). A longitudinal study of invention and understanding in children’s multidigit addition and subtraction. Journal for Research in Mathematics Education, 29(1), 3–20. https://doi.org/10.2307/749715
Catrambone, R. (1998). The subgoal learning model: Creating better examples so that students can solve novel problems. Journal of Experimental Psychology: General, 127(4), 355–376. https://doi.org/10.1037/0096-34220.127.116.115
Chen, Z., & Klahr, D. (2003). All other things being equal: Acquisition and transfer of the control of variables strategy. Child Development, 70(5), 1098–1120. https://doi.org/10.1111/1467-8624.00081
Cooper, M. M., Stieff, M., & DeSutter, D. (2017). Sketching the invisible to predict the visible: From drawing to modeling in chemistry. Topics in Cognitive Science, 9(4), 902–920. https://doi.org/10.1111/tops.12285
Court, J. E. (1993). Free-body diagrams. The Physics Teacher, 31(2), 104–108. https://doi.org/10.1119/1.2343674
Day, S. B., & Goldstone, R. L. (2012). The import of knowledge export: Connecting findings and theories of transfer of learning. Educational Psychologist, 47(3), 153–176. https://doi.org/10.1080/00461520.2012.696438
Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers, grades 6–10. Heinemann.
Ellis, A. B. (2007). A taxonomy for categorizing generalizations: Generalizing actions and reflection generalizations. The Journal of the Learning Sciences, 16(2), 221–262. https://doi.org/10.1080/10508400701193705
Fuson, K. C., Wearne, D., Hiebert, J. C., Murray, H. G., Human, P. G., Olivier, A. I., Carpenter, T. P., & Fennema, E. (1997). Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction. Journal for Research in Mathematics Education, 28(2), 130–162. https://doi.org/10.2307/749759
Fyfe, E. R., McNeil, N. M., Son, J. Y., & Goldstone, R. L. (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
Gagnier, K. M., Atit, K., Ormand, C. J., & Shipley, T. F. (2017). Comprehending 3D diagrams: Sketching to support spatial reasoning. Topics in Cognitive Science, 9(4), 883–901. https://doi.org/10.1111/tops.12233
Garnier, B., Chang, M., Ormand, C., Matlen, B., Tikoff, B., & Shipley, T. F. (2017). Promoting sketching in introductory geoscience courses: CogSketch geoscience worksheets. Topics in Cognitive Science, 9(4), 943–969. https://doi.org/10.1111/tops.12291
Gentner, D., & Markman, A. B. (1997). Structure mapping in analogy and similarity. American Psychologist, 52(1), 45. https://doi.org/10.1037/0003-066X.52.1.45
Gick, M. L., & Holyoak, K. J. (1980). Analogical problem solving. Cognitive Psychology, 12(3), 306–355. https://doi.org/10.1016/0010-0285(80)90013-4
Gick, M. L., & Holyoak, K. J. (1983). Schema induction and analogical transfer. Cognitive Psychology, 15(1), 1–38. https://doi.org/10.1016/0010-0285(83)90002-6
Glenberg, A. M. (1997). What memory is for. Behavioral and Brain Sciences, 20(1), 1–19. https://doi.org/10.1017/s0140525x97000010
Gobert, J. D., & Clement, J. J. (1999). Effects of student-generated diagrams versus student-generated summaries on conceptual understanding of causal and dynamic knowledge in plate tectonics. Journal of Research in Science Teaching, 36(1), 39–53. https://doi.org/10.1002/(SICI)1098-2736(199901)36:1%3C39::AID-TEA4%3E3.0.CO;2-I
Goldstone, R. L., & Son, J. Y. (2005). The transfer of scientific principles using concrete and idealized simulations. The Journal of the Learning Sciences, 14(1), 69–110. https://doi.org/10.1207/s15327809jls1401_4
Hatano, G., & Inagaki, K. (1986). Two courses of expertise. In H. Stevenson, H. Azuma, & K. Hakuta (Eds.), Child development and education in Japan (pp. 262–272). Freeman.
Hegarty, M., & Kozhevnikov, M. (1999). Types of visual–spatial representations and mathematical problem solving. Journal of Educational Psychology, 91(4), 684–689. https://doi.org/10.1037/0022-0618.104.22.1684
Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., Olivier, A., & Wearne, D. (1996). Problem solving as a basis for reform in curriculum and instruction: The case of mathematics. Educational Researcher, 25(4), 12–21. https://doi.org/10.3102/0013189X025004012
Hiebert, J., & Wearne, D. (1996). Instruction, understanding, and skill in multidigit addition and subtraction. Cognition and Instruction, 14(3), 251–283. https://doi.org/10.1207/s1532690xci1403_1
Hutchins, N. M., Biswas, G., Maróti, M., Lédeczi, Á., Grover, S., Wolf, R., Blair, K. P., Chin, D., Conlin, L., Basu, S., & McElhaney, K. (2020). C2STEM: A system for synergistic learning of physics and computational thinking. Journal of Science Education and Technology, 29(1), 83–100. https://doi.org/10.1007/s10956-019-09804-9
Jaeger, A. J., Velazquez, M. N., Dawdanow, A., & Shipley, T. F. (2018). Sketching and summarizing to reduce memory for seductive details in science text. Journal of Educational Psychology, 110(7), 899–916. https://doi.org/10.1037/edu0000254
Jitendra, A. K., Petersen-Brown, S., Lein, A. E., Zaslofsky, A. F., Kunkel, A. K., Jung, P. G., & Egan, A. M. (2015). Teaching mathematical word problem solving: The quality of evidence for strategy instruction priming the problem structure. Journal of Learning Disabilities, 48(1), 51–72. https://doi.org/10.1177/0022219413487408
Kamii, C., & Dominick, A. (1998). The harmful effects of algorithms in grades 1–4. In L. J. Morrow & M. J. Kenney (Eds.), The teaching and learning of algorithms in school mathematics (pp. 130–140). Reston, VA: National Council of Teachers of Mathematics.
Kaminski, J., Sloutsky, V., & Heckler, A. (2009). Concrete instantiations of mathematics: A double-edged sword. Journal for Research in Mathematics Education, 40(2), 90–93. https://doi.org/10.2307/40539326
Kapur, M. (2008). Productive failure. Cognition and Instruction, 26(3), 379–424. https://doi.org/10.1080/07370000802212669
Kapur, M. (2014). Productive failure in learning math. Cognitive Science, 38(5), 1008–1022. https://doi.org/10.1111/cogs.12107
Kieran, C. (1992). The learning and teaching of school algebra. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning: A project of the national council of teachers of mathematics (pp. 390–419). Macmillan.
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(2), 75–86. https://doi.org/10.1207/s15326985ep4102_1
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
Knuth, E. J., Stephens, A. C., McNeil, N. M., & Alibali, M. W. (2006). Does understanding the equal sign matter? Evidence from solving equations. Journal for Research in Mathematics Education, 37(4), 297–312. https://doi.org/10.2307/30034852
Koellner, K., Jacobs, J., Borko, H., Schneider, C., Pittman, M. E., Eiteljorg, E., Bunning, K., & Frykholm, J. (2007). The problem-solving cycle: A model to support the development of teachers’ professional knowledge. Mathematical Thinking and Learning, 9(3), 273–303. https://doi.org/10.1080/10986060701360944
Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. Basic Books.
Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words. Cognitive Science, 11(1), 65–100. https://doi.org/10.1016/S0364-0213(87)80026-5
Lehrer, R., & Schauble, L. (2004). Modeling natural variation through distribution. American Educational Research Journal, 41(3), 635–679. https://doi.org/10.3102/00028312041003635
Lehrer, R., Schauble, L., Carpenter, S., & Penner, D. (2000). The inter-related development of inscriptions and conceptual understanding. In P. Cobb, E. Yackel, & K. McClain (Eds.), Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional design (1st ed., pp. 325–360). London: Routledge.
Leopold, C., & Leutner, D. (2012). Science text comprehension: Drawing, main idea selection, and summarizing as learning strategies. Learning and Instruction, 22(1), 16–26. https://doi.org/10.1016/j.learninstruc.2011.05.005
Loyens, S. M. M., Kirschner, P. A., & Paas, F. (2011). Problem-based learning. In S. Graham, A. Bus, S. Major, & L. Swanson (Eds.), APA educational psychology handbook: Application to learning and teaching (Vol. 3, pp. 403–425). American Psychological Association.
Mayer, R. E. (2004). Should there be a three-strikes rule against pure discovery learning? American Psychologist, 59(1), 14. https://doi.org/10.1037/0003-0066X.59.1.14
McDaniel, M. A., Cahill, M. J., Michael, J., Robbins, M., & Wiener, C. (2014). Individual differences in learning and transfer: Stable tendencies for learning exemplars versus abstracting rules. Journal of Experimental Psychology: General, 143(2), 668–693. https://doi.org/10.1037/a0032963
McNeil, N. M., & Alibali, M. W. (2004). You’ll see what you mean: Students encode equations based on their knowledge of arithmetic. Cognitive Science, 28(3), 451–466. https://doi.org/10.1207/s15516709cog2803_7
McNeil, N. M., & Alibali, M. W. (2005). Why won’t you change your mind? Knowledge of operational patterns hinders learning and performance on equations. Child Development, 76(4), 883–899. https://doi.org/10.1111/j.1467-8624.2005.00884.x
Montague, M. (1998). Strategy instruction and mathematical problem solving. Journal of Reading, Writing, & Learning Disabilities International, 4(4), 275–290. https://doi.org/10.1080/0748763880040405
Montessori, M. (1965). Dr. Montessori’s own handbook. Schocken.
Paas, F., Renkl, A., & Sweller, J. (2003). Cognitive load theory and instructional design: Recent developments. Educational Psychologist, 38(1), 1–4. https://doi.org/10.1207/S15326985EP3801_1
Piaget, J. (1973). To understand is to invent: The future of education. New York: Viking.
Reed, S. K. (1993). A schema-based theory of transfer. In D. K. Detterman & R. J. Sternberg (Eds.), Transfer on trial: Intelligence, cognition, and instruction (pp. 39–67). Ablex Publishing.
Richland, L. E., Stigler, J. W., & Holyoak, K. J. (2012). Teaching the conceptual structure of mathematics. Educational Psychologist, 47(3), 189–203. https://doi.org/10.1080/00461520.2012.667065
Rittle-Johnson, B., & Star, J. R. (2007). Does comparing solution methods facilitate conceptual and procedural knowledge? An experimental study on learning to solve equations. Journal of Educational Psychology, 99(3), 561–574. https://doi.org/10.1037/0022-0622.214.171.1241
Ross, B. H. (1987). This is like that: The use of earlier problems and the separation of similarity effects. Journal of Experimental Psychology: Learning, Memory, and Cognition, 13(4), 629–639. https://doi.org/10.1037/0278-73126.96.36.1999
Ruchti, W. P., & Bennett, C. A. (2013). Develop reasoning through pictorial representations. Mathematics Teaching in the Middle School, 19(1), 30–36. https://doi.org/10.5951/mathteacmiddscho.19.1.0030
Schwartz, D. L., & Bransford, J. D. (1998). A time for telling. Cognition and Instruction, 16(4), 475–523. https://doi.org/10.1207/s1532690xci1604_4
Schwartz, D. L., Bransford, J. D., & Sears, D. (2005a). Efficiency and innovation in transfer. In J. Mestre (Ed.), Transfer of learning from a modern multidisciplinary perspective (pp. 1–51). IAP.
Schwartz, D. L., Chase, C. C., & Bransford, J. D. (2012). Resisting overzealous transfer: Coordinating previously successful routines with needs for new learning. Educational Psychologist, 47(3), 204–214. https://doi.org/10.1080/00461520.2012.696317
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), 759–775. https://doi.org/10.1037/a0025140
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
Schwartz, D. L., Martin, T., & Pfaffman, J. (2005b). How mathematics propels the development of physical knowledge. Journal of Cognition and Development, 6(1), 65–88. https://doi.org/10.1207/s15327647jcd0601_5
Sheredos, B., & Bechtel, W. (2017). Sketching biological phenomena and mechanisms. Topics in Cognitive Science, 9(4), 970–985. https://doi.org/10.1111/tops.12290
Sherman, J., & Bisanz, J. (2009). Equivalence in symbolic and nonsymbolic contexts: Benefits of solving problems with manipulatives. Journal of Educational Psychology, 101(1), 88. https://doi.org/10.1037/a0013156
Singley, M. K., & Anderson, J. R. (1989). The transfer of cognitive skill. Harvard University Press.
Star, J. R., & Newton, K. J. (2009). The nature and development of experts’ strategy flexibility for solving equations. ZDM, 41(5), 557–567. https://doi.org/10.1007/s11858-009-0185-5
Stigler, J. W., & Hiebert, J. (2004). Improving mathematics teaching. Educational Leadership, 61(5), 12–17.
Sweller, J. (1988). Cognitive load during problem solving: Effects on learning. Cognitive Science, 12(2), 257–285. https://doi.org/10.1207/s15516709cog1202_4
Van Garderen, D. (2006). Spatial visualization, visual imagery, and mathematical problem solving of students with varying abilities. Journal of Learning Disabilities, 39(6), 496–506. https://doi.org/10.1177/00222194060390060201
Van Heuvelen, A. (1991). Overview, case study physics. American Journal of Physics, 59(10), 898–907. https://doi.org/10.1119/1.16668
Van Meter, P. (2001). Drawing construction as a strategy for learning from text. Journal of Educational Psychology, 93(1), 129–140. https://doi.org/10.1037/0022-06188.8.131.52
Wylie, G., & Allport, A. (2000). Task switching and the measurement of “switch costs.” Psychological Research Psychologische Forschung, 63(3–4), 212–233. https://doi.org/10.1007/s004269900003