When explaining a process of providing instruction Why is it important to separate information into three buckets?

• Atkinson, R. K., Derry, S. J., Renkl, A., & Wortham, D. (2000). Learning from examples: Instructional principles from the worked examples research. Review of Educational Research, 70(2), 181–214.

  Article  Google Scholar 

• Atkinson, R. K., Renkl, A., & Merrill, M. M. (2003). Transitioning from studying examples to solving problems: Effects of self-explanation prompts and fading worked-out steps. Journal of Educational Psychology, 95(4), 774.

  Article  Google Scholar 

• Barbieri, C., & Booth, J. L. (2016). Support for struggling students in algebra: Contributions of incorrect worked examples. Learning and Individual Differences, 48, 36–44.

  Article  Google Scholar 

• Begolli, K. N., & Richland, L. E. (2016). Teaching mathematics by comparison: Analog visibility as a double-edged sword. Journal of Educational Psychology, 108(2), 194–213. https://doi.org/10.1037/edu0000056

  Article  Google Scholar 

• Begolli, K. N., & Richland, L. E. (2017). Bridging cognitive science and real classrooms: A video methodology for experimental research. The Journal of Experimental Education. https://doi.org/10.1080/00220973.2017.1347775

  Article  Google Scholar 

• Begolli, K. N., Richland, L. E., Jaeggi, S. M., Lyons, E. M., Klostermann, E. C., & Matlen, B. J. (2018). Executive function in learning mathematics by comparing representations: Incorporating everyday classrooms into the science of learning. Thinking & Reasoning. https://doi.org/10.1080/13546783.2018.1429306

  Article  Google Scholar 

• Berliner, D. C. (2002). Comment: Educational research: The hardest science of all. Educational Researcher, 31(8), 18–20.

  Article  Google Scholar 

• Billings, E. M. H. (2001). Problems that encourage proportion sense. Mathematics Teaching in the Middle School, 7(1), 10.

  Article  Google Scholar 

• Booth, J. L., Cooper, L. A., Donovan, M. S., Huyghe, A., Koedinger, K. R., & Paré-Blagoev, E. J. (2015a). Design-based research within the constraints of practice: AlgebraByExample. Journal of Education for Students Placed at Risk, 20(1–2), 79–100. https://doi.org/10.1080/10824669.2014.986674

  Article  Google Scholar 

• Booth, J. L., Lange, K. E., Koedinger, K. R., & Newton, K. J. (2013). Using example problems to improve student learning in algebra: Differentiating between correct and incorrect examples. Learning and Instruction, 25, 24–34. https://doi.org/10.1016/j.learninstruc.2012.11.002

  Article  Google Scholar 

• Booth, J. L., McGinn, K. M., Barbieri, C., Begolli, K. N., Chang, B., Miller-Cotto, D., ... & Davenport, J. L. (2017). Evidence for cognitive science principles that impact learning in mathematics. In Acquisition of complex arithmetic skills and higher-order mathematics concepts (pp. 297–325). Academic Press.

• Booth, J. L., McGinn, K. M., Young, L. K., & Barbieri, C. (2015b). Simple practice doesn’t always make perfect: Evidence from the worked example effect. Policy Insights from the Behavioral and Brain Sciences, 2(1), 24–32. https://doi.org/10.1177/2372732215601691

  Article  Google Scholar 

• Booth, J. L., & Newton, K. J. (2012). Fractions: Could they really be the gatekeeper’s doorman? Contemporary Educational Psychology, 37(4), 247–253.

  Article  Google Scholar 

• Booth, J. L., Newton, K. J., & Twiss-Garrity, L. K. (2014). The impact of fraction magnitude knowledge on algebra performance and learning. Journal of Experimental Child Psychology, 118(1), 110–118. https://doi.org/10.1016/j.jecp.2013.09.001

  Article  Google Scholar 

• Booth, J. L., Oyer, M. H., Paré-Blagoev, E. J., Elliot, A. J., Barbieri, C., Augustine, A., & Koedinger, K. R. (2015c). Learning algebra by example in real-world classrooms. Journal of Research on Educational Effectiveness, 8(4), 530–551. https://doi.org/10.1080/19345747.2015.1055636

  Article  Google Scholar 

• Boyer, T. W., & Levine, S. C. (2015). Prompting Children to Reason Proportionally: Processing Discrete Units as Continuous Amounts., 51(5), 615–620.

  Google Scholar 

• Bramwell, R., West, H., & Salmon, P. (2006). Health professionals’ and service users’ interpretation of screening test results: Experimental study. BMJ British Medical Journal, 333(7562), 284. https://doi.org/10.1136/bmj.38884.663102.AE

  Article  Google Scholar 

• Bryant, P., & Nunes, T. (2012). Children’s understanding of probability: A literature review (full report). Londres: The Nuffield Foundation.

• Butler, A. C., Marsh, E. J., Slavinsky, J. P., & Baraniuk, R. G. (2014). Integrating cognitive science and technology improves learning in a STEM classroom. Educational Psychology Review, 26(2), 331–340. https://doi.org/10.1007/s10648-014-9256-4

  Article  Google Scholar 

• 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.

  Article  Google Scholar 

• Chen, O., Kalyuga, S., & Sweller, J. (2015). The worked example effect, the generation effect, and element interactivity. Journal of Educational Psychology, 107(3), 689.

  Article  Google Scholar 

• Chen, O., Kalyuga, S., & Sweller, J. (2017). The expertise reversal effect is a variant of the more general element interactivity effect. Educational Psychology Review, 29(2), 393–405.

• Chi, M. T., De Leeuw, N., Chiu, M.-H., & LaVancher, C. (1994). Eliciting self-explanations improves understanding. Cognitive Science, 18(3), 439–477.

  Google Scholar 

• Chi, M. T. H., Bassok, M., Lewis, M. W., Reimann, P., & Glaser, R. (1989). Self-explanations: How students study and use examples in learning to solve problems. Cognitive Science, 13(2), 145–182.

  Article  Google Scholar 

• Cronbach, L. J. (1951). Coefficient alpha and the internal structure of tests. Psychometrika, 16(3), 297–334.

  Article  Google Scholar 

• Daniel, D. B. (2012). Promising principles: Translating the science of learning to educational practice. Journal of Applied Research in Memory and Cognition, 1(4), 251–253. https://doi.org/10.1016/j.jarmac.2012.10.004

  Article  Google Scholar 

• DeWolf, M., Bassok, M., & Holyoak, K. J. (2015). From rational numbers to algebra: Separable contributions of decimal magnitude and relational understanding of fractions. Journal of Experimental Child Psychology, 133, 72–84. https://doi.org/10.1016/j.jecp.2015.01.013

  Article  Google Scholar 

• DeWolf, M., & Vosniadou, S. (2013). The representation of fraction magnitudes and the whole number bias reconsidered. Learning and Instruction. https://doi.org/10.1016/j.learninstruc.2014.07.002

  Article  Google Scholar 

• DiStefano, C., Zhu, M., & Mîndrilă, D. (2009). Understanding and using factor scores: Considerations for the applied researcher. Practical Assessment, Research & Evaluation, 14(20), 1–11.

  Google Scholar 

• Dunlosky, J., & Rawson, K. A. (2012). Despite their promise, there’s still a lot to learn about techniques that support durable learning. Journal of Applied Research in Memory and Cognition, 1(4), 254–256. https://doi.org/10.1016/j.jarmac.2012.10.003

  Article  Google Scholar 

• Dunlosky, J., Rawson, K. A., Marsh, E. J., Nathan, M. J., & Willingham, D. T. (2013). Improving students’ learning with effective learning techniques: promising directions from cognitive and educational psychology. Psychological Science in the Public Interest, 14(1), 4–58. https://doi.org/10.1177/1529100612453266

  Article  Google Scholar 

• Falk, R., & Wilkening, F. (1998). Children’s construction of fair chances: Adjusting probabilities. Developmental Psychology, 34(6), 1340–1340.

  Article  Google Scholar 

• Falk, R., Yudilevich-Assouline, P., & Elstein, A. (2012). Children’s concept of probability as inferred from their binary choices-revisited. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-012-9402-1

  Article  Google Scholar 

• Fergusson-Hessler, M., & de Jong, T. (1990). Studying physics texts: Differences in study processes between good and poor performers. Cognition and Instruction, 7, 41–54.

  Article  Google Scholar 

• Fischbein, E., & Gazit, A. (1984). Does the teaching of probability improve probabilistic intuitions? Educational Studies in Mathematics, 15(1), 1–24.

  Article  Google Scholar 

• Fryer, R. G., Jr., & Levitt, S. D. (2004). Understanding the black-white test score gap in the first two years of school. The Review of Economics and Statistics, 86(2), 447–464.

  Article  Google Scholar 

• Garfield, J., & Ben-Zvi, D. (2007). How students learn statistics revisited: A current review of research on teaching and learning statistics. International Statistical Review, 75(3), 372–396. https://doi.org/10.1111/j.1751-5823.2007.00029.x

  Article  Google Scholar 

• Gigerenzer, G., Gaissmaier, W., Kurz-Milcke, E., Schwartz, L. M., & Woloshin, S. (2009). Knowing your chances. Scientific American Mind, 20(2), 44–51.

  Article  Google Scholar 

• Gilovich, T., Vallone, R., & Tversky, A. (1985). The hot hand in basketball: On the misperception of random sequences. Cognitive Psychology, 17(3), 295–314.

  Article  Google Scholar 

• Große, C. S., & Renkl, A. (2007). Finding and fixing errors in worked examples: Can this foster learning outcomes? Brain Research Reviews, 17(6), 173–196. https://doi.org/10.1177/0192513X12437708

  Article  Google Scholar 

• Hanushek, E. A., & Woessmann, L. (2012). Do better schools lead to more growth? Cognitive skills, economic outcomes, and causation. Journal of Economic Growth, 17(4), 267–321. https://doi.org/10.1007/s10887-012-9081-x

  Article  Google Scholar 

• Hoffrage, U., & Gigerenzer, G. (1998). Using natural frequencies to improve diagnostic inferences. Academic Medicine: Journal of the Association of American Medical Colleges, 73(5), 538–540.

  Article  Google Scholar 

• Kahneman, D., & Tversky, A. (1972). Subjective probability: A judgment of representativeness. Cognitive Psychology, 3(3), 430–454.

  Article  Google Scholar 

• Kaiser, H. F. (1974). An index of factorial simplicity. Psychometrika, 39(1), 31–36.

  Article  Google Scholar 

• Kalyuga, S. (2007). Expertise reversal effect and its implications for learner-tailored instruction. Educational Psychology Review, 19(4), 509–539.

  Article  Google Scholar 

• Kalyuga, S., Chandler, P., & Sweller, J. (2001a). Learner experience and efficiency of instructional guidance. Educational Psychology, 21(1), 5–23.

  Article  Google Scholar 

• Kalyuga, S., Chandler, P., Tuovinen, J., & Sweller, J. (2001b). When problem solving is superior to studying worked examples. Journal of Educational Psychology, 93(3), 579.

  Article  Google Scholar 

• Kalyuga, S., & Renkl, A. (2010). Expertise reversal effect and its instructional implications: Introduction to the special issue. Instructional Science, 38(3), 209–215.

  Article  Google Scholar 

• Kapur, M., & Bielczyz, K. (2011). Classroom-based experiments in productive failure. In Proceedings of the annual meeting of the cognitive science society (Vol. 33, No. 33).

• Klahr, D., & Li, J. (2005). Cognitive research and elementary science instruction: From the laboratory, to the classroom, and back. Journal of Science Education and Technology, 14(2), 217–238. https://doi.org/10.1007/s10956-005-4423-5

  Article  Google Scholar 

• Kornell, N., Rabelo, V. C., & Klein, P. J. (2012). Tests enhance learning-Compared to what? Journal of Applied Research in Memory and Cognition, 1(4), 257–259. https://doi.org/10.1016/j.jarmac.2012.10.002

  Article  Google Scholar 

• Kuder, G. F., & Richardson, M. W. (1937). The theory of the estimation of test reliability. Psychometrika, 2(3), 151–160.

  Article  Google Scholar 

• Labarge, A. S., McCaffrey, R. J., & Brown, T. A. (2003). Neuropsychologists’ abilities to determine the predictive value of diagnostic tests. Archives of Clinical Neuropsychology, 18(2), 165–175.

  Article  Google Scholar 

• Lappan, G., Fey, J., Fitzgerald, W., Friel, S., & Philips, E. (2006). Connected mathematics two. Prentice Hall (School Division).

• Lemke, M., Sen, A., Pahlke, E., Partelow, L., Miller, D., Williams, T., Kastberg, D., & Jocelyn, L. (2004). International outcomes of learning in mathematics literacy and problem solving: PISA 2003 Results from the US perspective. Highlights. NCES 2005-003. US Department of Education.

• Martignon, L., & Krauss, S. (2009). Hands-on modelling with Wason cards and tinker cubes: First steps in logical and Bayesian reasoning in fourth grade. International Electronic Journal of Mathematics Education, 4(3).

• Mayer, R. E. (2012). Advances in applying the science of learning to education: An historical perspective. Journal of Applied Research in Memory and Cognition, 1(4), 249–250. https://doi.org/10.1016/j.jarmac.2012.10.001

  Article  Google Scholar 

• National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics (Vol. 1). Author.

• National Governors Association. (2010). Common core state standards. Washington, DC.

• National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. In Foundations (Vol. 37, Issue 9). https://doi.org/10.3102/0013189X08329195

• Noelting, G. (1980). The development of proportional reasoning and the ratio concept Part I—Differentiation of stages. Educational Studies in Mathematics, 11(2), 217–253.

  Article  Google Scholar 

• Nunes, T., Bryant, P. E., Evans, D., Gottardis, L., & Terlektsi, M. E. (2014). The cognitive demands of understanding the sample space. ZDM: The International Journal on Mathematics Education, 46(3), 437–448. https://doi.org/10.1007/s11858-014-0581-3

  Article  Google Scholar 

• Ohlsson, S. (1996). Learning from performance errors. Psychological Review, 103(2), 241–262. https://doi.org/10.1037/0033-295X.103.2.241

  Article  Google Scholar 

• Paas, F., & Van Merriënboer, J. J. G. (1994). Variability of worked examples and transfer of geometrical problem-solving skills: A cognitive-load approach. Journal of Educational Psychology, 86(1), 122.

  Article  Google Scholar 

• Pan, S. C., & Rickard, T. C. (2018). Transfer of test-enhanced learning: Meta-analytic review and synthesis. Psychological Bulletin, 144(7), 710.

  Article  Google Scholar 

• Pashler, H., Bain, P. M., Bottge, B. A., Graesser, A. C., Koedinger, K. R., McDaniel, M. A., & Metcalfe, J. (2007). Organizing instruction and study to improve student learning. IES Practice Guide. NCER 2007-2004 (pp. 63–63).

• Peterson, R. A. (2000). A meta-analysis of variance accounted for and factor loadings in exploratory factor analysis. Marketing Letters, 11(3), 261–275.

  Article  Google Scholar 

• Piaget, J., & Inhelder, B. (1975). The origin of the idea of chance in children.(Trans L. Leake, P. Burrell & HD Fishbein). WW Norton.

• Post, T., Behr, M., & Lesh, R. (1988). Proportionality and the development of prealgebra understandings. The Ideas of Algebra, K–12, 78–90.

• Prinz, R., Feufel, M. A., Gigerenzer, G., & Wegwarth, O. (2015). What counselors tell low-risk clients about HIV test performance. Current Hiv Research, 13(5), 369–380.

  Article  Google Scholar 

• Renkl, A. (1997). Learning from worked-out examples: A study on individual differences. Cognitive Science, 21(1), 1–29.

  Article  Google Scholar 

• Renkl, A. (2014). Toward an instructionally oriented theory of example‐based learning. Cognitive Science, 38(1), 1–37.

• Renkl, A., & Atkinson, R. K. (2003). Structuring the transition from example study to problem solving in cognitive skill acquisition: A cognitive load perspective. Educational Psychologist, 38(1), 15–22.

  Article  Google Scholar 

• Renkl, A., Atkinson, R. K., Maier, U. H., & Staley, R. (2002). From example study to problem solving: Smooth transitions help learning. The Journal of Experimental Education, 70(4), 293–315

• Renkl, A., Stark, R., Gruber, H., & Mandl, H. (1998). Learning from worked-out examples: The effects of example variability and elicited self-explanations. Contemporary Educational Psychology, 23(1), 90–108.

  Article  Google Scholar 

• Richardson, M. W., & Kuder, G. F. (1939). The calculation of test reliability coefficients based on the method of rational equivalence. Journal of Educational Psychology, 30(9), 681.

  Article  Google Scholar 

• 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(2), 346–362. https://doi.org/10.1037/0022-0663.93.2.346

  Article  Google Scholar 

• Rogers, W. (1994). Regression standard errors in clustered samples. Stata Technical Bulletin, 3(13).

• Roy, M., & Chi, M. T. H. (2005). The self-explanation principle in multimedia learning. The Cambridge Handbook of Multimedia Learning, 271–286.

• Siegler, R. S. (2002). Microgenetic studies of self-explanation. Microdevelopment: Transition Processes in Development and Learning, 31–58.

• Siegler, R. S., & Lortie-Forgues, H. (2014). An integrative theory of numerical development. Child Development Perspectives, 8(3), 144–150. https://doi.org/10.1111/cdep.12077

  Article  Google Scholar 

• Stark, R. (2004). Implementing example-based learning and teaching in the context of vocational school education in business administration. Learning Environments Research, 7(2), 143–163.

  Article  Google Scholar 

• Sweller, J., & Chandler, P. (1991). Evidence for cognitive load theory. Cognition and Instruction, 8(4), 351–362.

  Article  Google Scholar 

• Sweller, J., & Cooper, G. A. (1985). The use of worked examples as a substitute for problem solving in learning algebra. Cognition and Instruction, 2(1), 59–89.

  Article  Google Scholar 

• Traub, R. E. (1994). Reliability for the social sciences: Theory and applications (Vol. 3). Sage.

• Vamvakoussi, X., & Vosniadou, S. (2012). Bridging the gap between the dense and the discrete: The number line and the “rubber line” bridging analogy. Mathematical Thinking and Learning, 14, 265–284. https://doi.org/10.1080/10986065.2012.717378

  Article  Google Scholar 

• Van Dooren, W., De Bock, D., Depaepe, F., Janssens, D., & Verschaffel, L. (2003). The illusion of linearity: Expanding the evidence towards probabilistic reasoning. Educational Studies in Mathematics, 53(2), 113–138.

  Article  Google Scholar 

• van Loon-Hillen, N., Van Gog, T., & Brand-Gruwel, S. (2012). Effects of worked examples in a primary school mathematics curriculum. Interactive Learning Environments, 20(1), 89–99.

  Article  Google Scholar 

• Wason, P. C. (1968). Reasoning about a rule. The Quarterly Journal of Experimental Psychology, 20(3), 273–281.

  Article  Google Scholar 

• Widaman, K. F., Helm, J. L., Castro-Schilo, L., Pluess, M., Stallings, M. C., & Belsky, J. (2012). Distinguishing ordinal and disordinal interactions. Psychological Methods, 17(4), 615.

  Article  Google Scholar 

• Zhu, L., & Gigerenzer, G. (2006). Children can solve Bayesian problems: The role of representation in mental computation. Cognition, 98(3), 287–308.

  Article  Google Scholar 

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