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Articles

Learning Fractions by Splitting: Using Learning Analytics to Illuminate the Development of Mathematical Understanding

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REFERENCES

  • Aghababyan, A., Symanzik, J., & Martin, T. (2013, August). Visualization of “states” in online educational games. Paper presented at the World Statistics Congress, Hong Kong, China.
  • Ainsworth, S. (1999). The functions of multiple representations. Computers & Education, 33(2–3), 131–152. doi:10.1016/S0360-1315(99)00029-9
  • Ainsworth, S. (2006). DeFT: A conceptual framework for considering learning with multiple representations. Learning and Instruction, 16(3), 183–198. doi:10.1016/j.learninstruc.2006.03.001
  • Amershi, S., & Conati, C. (2009). Combining unsupervised and supervised classification to build user models for exploratory learning environments. Journal of Educational Data Mining, 1(1), 1–54.
  • Baker, R. S. J. D., D’Mello, S. K., Rodrigo, M. M. T., & Graesser, A. C. (2010). Better to be frustrated than bored: The incidence, persistence, and impact of learners’ cognitive–affective states during interactions with three different computer-based learning environments. International Journal of Human-Computer Studies, 68(4), 223–241. doi:10.1016/j.ijhcs.2009.12.003
  • Baker, R. S., Hershkovitz, A., Rossi, L. M., Goldstein, A. B., & Gowda, S. M. (2013). Predicting robust learning with the visual form of the moment-by-moment learning curve. Journal of the Learning Sciences, 22, 639–666. doi:10.1080/10508406.2013.836653
  • Baker, R., & Siemens, G. (2014). Educational data mining and learning analytics. In K. Sawyer (Ed.), Cambridge Handbook of the Learning Sciences (2nd Ed., pp. 253–274). New York, NY: Cambridge University Press.
  • Baker, R. S. J. D., & Yacef, K. (2009). The state of educational data mining in 2009: A review and future visions. Journal of Educational Data Mining, 1(1), 3–16.
  • Ball, D. L. (1988). Unlearning to teach mathematics. For the Learning of Mathematics, 8(1), 40–48.
  • Ball, D. L. (1993). Halves, pieces, and twoths: Constructing representational contexts in teaching fractions. In T. Carpenter, E. Fennema, & T. Romberg (Eds.), Rational numbers: An integration of research (pp. 157–196). Hillsdale, NJ: Erlbaum.
  • Barlow, W. (1998). Modeling of categorical agreement. In P. Armitage & T. Colton (Eds.), The encyclopedia of biostatistics (pp. 541–545). New York, NY: Wiley.
  • Beal, C., & Cohen, P. (2010). Evaluation of AnimalWatch: An intelligent tutoring system for arithmetic and fractions. Journal of Interactive Online Learning, 9(1), 64–77.
  • Behr, M., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio, proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–333). New York, NY: Macmillan.
  • Behr, M., Lesh, R., Post, T., & Silver, E. (1983). Rational number concepts. In R. Lesh & M. Landau (Eds.), Acquisition of mathematics concepts and processes (pp. 91–125). New York, NY: Academic Press.
  • Behr, M., Wachsmuth, I., Post, T., & Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323–341. doi:10.2307/748423
  • Bishop, Y. M. M., Fienberg, S. E., & Holland, P. W. (1975). Discrete multivariate analysis: Theory and practice. Cambridge, MA: MIT Press.
  • Blikstein, P., Worsley, M., Piech, C., Sahami, M., Cooper, S., & Koller, D. (2014). Programming pluralism: Using learning analytics to detect patterns in the learning of computer programming. Journal of the Learning Sciences, 23, 561–599. doi:10.1080/10508406.2014.954750
  • Bouchet, F., Harley, J., Trevors, G., & Azevedo, R. (2013). Clustering and profiling students according to their interactions with an intelligent tutoring system fostering self-regulated learning. Journal of Educational Data Mining, 5(1), 104–146.
  • Bowers, A. J. (2010). Analyzing the longitudinal K-12 grading histories of entire cohorts of students: Grades, data driven decision making, dropping out and hierarchical cluster analysis. Practical Assessment Research and Evaluation, 15(7), 1–18.
  • Bruce, C., Chang, D., Flynn, T., & Yearley, S. (2013). Foundations to learning and teaching fractions: Addition and subtraction. Retrieved from http://www.edugains.ca/resources/ProfessionalLearning/FoundationstoLearningandTeachingFractions.pdf
  • Bulgar, S. (2009). A longitudinal study of students’ representations for division of fractions. Montana Mathematics Enthusiast, 6(1), 165–200.
  • Calvo, R. A., & D’Mello, S. (2010). Affect detection: An interdisciplinary review of models, methods, and their applications. IEEE Transactions on Affective Computing, 1(1), 18–37. doi:10.1109/T-AFFC.2010.1
  • Carpenter, T., 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, 24(5), 428–441. doi:10.2307/749152
  • Center for Game Science. (2013). Refraction. Retrieved from http://centerforgamescience.org/portfolio/refraction/
  • Charles, K., & Nason, R. (2000). Young children’s partitioning strategies. Educational Studies in Mathematics, 43, 191–221. doi:10.1023/A:1017513716026
  • Confrey, J. (1991). The concept of exponential functions: A student’s perspective. In L. P. Steffe (Ed.), Epistemological foundations of mathematical experience (pp. 124–159). New York, NY: Springer.
  • Confrey, J. (1994). Splitting, similarity, and rate of change: A new approach to multiplication and exponential functions. In G. Harel & J. Confrey ( Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 291–330). Albany, NY: SUNY.
  • Confrey, J. (1995). Student voice in examining “splitting” as an approach to ratio, proportions and fractions. In L. Meira & D. Carraher (Eds.), Proceedings of the 19th International Conference for the Psychology of Mathematics Education (Vol. 1, pp. 3–29). Recife, Brazil: Universidade Federal de Pernambuco.
  • Confrey, J. (2007, February). Tracing the evolution of mathematics content standards in the United States: Looking back and projecting forward. Paper presented at the Conference on K-12 Mathematics Curriculum Standards, Arlington, VA.
  • Confrey, J. (2012). Better measurement of higher-cognitive processes through learning trajectories and diagnostic assessments in mathematics: The challenge in adolescence. In V. Reyna, M. Dougherty, S. Chapman, & J. Confrey (Eds.), The adolescent brain: Learning, reasoning, and decision making (pp. 155–182). Washington, DC: American Psychological Association.
  • Cramer, K., Post, T., & delMas, R. (2002). Initial fraction learning by fourth- and fifth-grade students: A comparison of the effects of using commercial curricula with the effects of using the rational number project curriculum. Journal for Research in Mathematics Education, 33(2), 111–144. doi:10.2307/749646
  • Dai, X., Martin, T., & Aghababyan, A. (2014). Supervised learning methods for predicting student success in a digital game environment. Manuscript in preparation.
  • Dhar, V. (2014). Big data and predictive analytics in health care. Big Data, 2(3), 113–116. doi:10.1089/big.2014.1525
  • Dumbill, E. (2013). Making sense of big data. Big Data, 1(1), 1–2. doi:10.1089/big.2012.1503
  • Fazio, L. K., & Siegler, R. S. (2013). Microgenetic learning analysis: A distinction without a difference. Human Development, 56(1), 52–58. doi:10.1159/000345542
  • Gould, P., Outhred, L. N., & Mitchelmore, M. C. (2006). One-third is three-quarters of one-half. In P. Grootenboer, R. Zevenbergen, & M. Chinnappan (Eds.), Identities, cultures and learning spaces: Proceedings of the 29th Annual Conference of the Mathematics Education Research Group of Australasia (Vol. 1, pp. 262–269). Adelaide, Australia: Mathematics Education Research Group of Australasia.
  • Hackenberg, A. (2007). Units coordination and the construction of improper fractions: A revision of the splitting hypothesis. Journal of Mathematical Behavior, 26, 27–47. doi:10.1016/j.jmathb.2007.03.002
  • Hair, J. F., Anderson, R. E., Tatham, R. L., & Black, W. C. (1992). Multivariate data analysis (3rd ed.). New York, NY: Macmillan.
  • Han, J., Kamber, M., & Pei, J. (2011). Data mining: Concepts and techniques (3rd ed.). Waltham, MA: Morgan Kaufman.
  • Hare, J. (2014). Bring it on, big data: Beyond the hype. Big Data, 2(2), 73–75. doi:10.1089/big.2014.1520
  • Hershkovitz, A., de Baker, R. S. J., Gobert, J., Wixon, M., & Sao Pedro, M. (2013). Discovery with models: A case study on carelessness in computer-based science inquiry. American Behavioral Scientist, 57(10), 1480–1499. doi:10.1177/0002764213479365
  • Hiebert, J. (1988). A theory of developing competence with written mathematical symbols. Educational Studies in Mathematics, 19(3), 333–355. doi:10.1007/BF00312451
  • Jigyel, K., & Afamasaga-Fuata’i, K. (2007). Students’ conceptions of models of fractions and equivalence. The Australian Mathematics Teacher, 63, 17–25.
  • Kapur, M., & Bielaczyc, K. (2012). Designing for productive failure. Journal of the Learning Sciences, 21(1), 45–83. doi:10.1080/10508406.2011.591717
  • Kaput, J. J. (1987). Toward a theory of symbol use in mathematics. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 159–195). Hillsdale, NJ: Erlbaum.
  • Kaput, J. (1988, November). Truth and meaning in representation situations: Comments on the Greeno contribution. Remarks prepared for the meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, De Kalb, IL.
  • Kerslake, D. (1986). Fractions: Children’s strategies and errors—A report of the strategies and errors in secondary mathematics project. Berkshire, England: NFER-NELSON.
  • Kieren, T. E. (1988). Personal knowledge of rational numbers: Its intuitive and formal development. In J. Hiebert & M. J. Behr (Eds.), Number concepts and operations in the middle grades (pp. 162–181). Hillsdale, NJ: Erlbaum.
  • Kirschner, P., 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. doi:10.1207/s15326985ep4102_1
  • Kuhn, D. (1995). Microgenetic study of change: What has it told us? Psychological Science, 6(3), 133–139. doi:10.1111/j.1467-9280.1995.tb00322.x
  • Lampert, M. (1986). Knowing, doing, and teaching multiplication. Cognition and Instruction, 3(4), 305–342. doi:10.1207/s1532690xci0304_1
  • Lampert, M. (1989). Choosing and using mathematical tools in classroom discourse. In J. Brophy (Ed.), Advances in research on teaching (Vol. 1, pp. 223–264). Greenwich, CT: JAI Press.
  • Lesh, R., Behr, M., & Post, T. (1987). Rational number relations and proportions. In C. Janvier (Ed.), Problems of representation in teaching and learning mathematics (pp. 41–58). Hillsdale, NJ: Erlbaum.
  • Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics (pp. 33–40). Hillsdale, NJ: Erlbaum.
  • Lipkus, I. M., Samsa, G., & Rimer, B. K. (2001). General performance on a numeracy scale among highly educated samples. Medical Decision Making, 21(1), 37–44. doi:10.1177/0272989X0102100105
  • Liu, Y., Mandel, T., Butler, E., Andersen, E., O’Rourke, E., Brunskill, E., & Popovic, Z. (2013). Predicting player moves in an educational game: A hybrid approach. In S. D’Mello, R. Calvo, & A. Olney (Eds.), Proceedings of the 6th International Conference on Educational Data Mining (pp. 106–113), Memphis, TN: International Education Data Mining Society.
  • Lorr, M. (1983). Cluster analysis for social scientists. San Francisco, CA: Jossey-Bass.
  • Marshall, S. P. (1993). Assessment of rational number understanding: A schema-based approach. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 261–288). Hillsdale, NJ: Erlbaum.
  • Martin, T., Petrick, C., Andersen, E., Liu, Y., & Popovic, Z. (2012, April). Refraction time: Making “split” decisions in an online fraction game. Paper presented at the meeting of the American Educational Research Association, Vancouver, British Columbia, Canada.
  • Martin, T., & Sherin, B. (2013). Learning analytics and computational techniques for detecting and evaluating patterns in learning: An introduction to the special issue. Journal of the Learning Sciences, 22(4), 511–520. doi:10.1080/10508406.2013.840466
  • Martinie, S. L., & Bay-Williams, J. M. (2003). Investigating students’ conceptual understanding of decimal fractions using multiple representations. Mathematics Teaching in the Middle School, 8(5), 244–247.
  • McDiarmid, G. W., Ball, D. L., & Anderson, C. W. (1989). Why staying one chapter ahead doesn’t really work: Subject-specific pedagogy. In M. C. Reynolds (Ed.), Knowledge base for the beginning teacher (pp. 193–205). New York, NY: Pergamon Press.
  • Mislevy, R. J., Behrens, J. T., Dicerbo, K. E., & Levy, R. (2012). Design and discovery in educational assessment: Evidence-centered design, psychometrics, and educational data mining. Journal of Educational Data Mining, 4(1), 11–48.
  • Moss, J. (1997). Developing children’s rational number sense: A new approach and an experimental program ( Unpublished master’s thesis). University of Toronto, Toronto, Ontario, Canada.
  • Moss, J., & Case, R. (1999). Developing children’s understanding of the rational numbers: A new model and an experimental curriculum. Journal for Research in Mathematics Education, 30(2), 122–147. doi:10.2307/749607
  • Myers, M., Confrey, C., Nguyen, K., & Mojica, G. (2009). Equipartitioning a continuous whole among three people: Students’ attempts to create fair shares. In S. Swars, D. Stinson, & S. Lemons-Smith (Eds.), Proceedings of the 31st Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education ( Vol. 5, pp. 792–799). Atlanta: Georgia State University.
  • National Council of Teachers of Mathematics. (2007). Second handbook of research on mathematics teaching and learning. Washington, DC: Author.
  • National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education.
  • National Research Council. (2005). How students learn: History, mathematics, and science in the classroom. Washington, DC: National Academies Press.
  • Norton, A., & Hackenberg, A. J. (2010). Continuing research on students’ fraction schemes. In L. P. Steffe & J. Olive (Eds.), Children’s fractional knowledge (pp. 341–352). New York, NY: Springer.
  • Olive, J., & Steffe, L. (2002). The construction of an iterative fractional scheme: The case of Joe. Journal of Mathematical Behavior, 20, 413–437.
  • Pardos, Z. A., Baker, R. S., San Pedro, M., Gowda, S. M., & Gowda, S. M. (2014). Affective states and state tests: Investigating how affect and engagement during the school year predict end-of-year learning outcomes. Journal of Learning Analytics, 1(1), 107–128.
  • Pearn, C., & Stephens, M. (2004). Why you have to probe to discover what Year 8 students really think about fractions. In I. Putt, R. Faragher, & M. McLean (Eds.), Proceedings of the 27th Annual Conference for the Mathematics Education Research Group of Australasia (pp. 430–437). Sydney, Australia: Mathematics Education Research Group of Australasia (MERGA).
  • Pescosolido, R. (2010). Developing effective assessment for learning trajectories: The case of equipartitioning ( Unpublished dissertation). North Carolina State University, Raleigh, NC.
  • Post, T. R., Cramer, K. A., Behr, M., Lesh, R., & Harel, G. (1993). Curriculum implications of research on the learning, teaching and assessing of rational number concepts. In T. P. Carpenter, E. Fennema, & T. A. Romberg (Eds.), Rational numbers: An integration of research (pp. 327–362). Hillsdale, NJ: Erlbaum.
  • Pothier, Y., & Sawada, D. (1983). Partitioning: The emergence of rational number ideas in young children. Journal for Research in Mathematics Education, 14(5), 307–317. doi:10.2307/748675
  • Rau, M. A. (2013). Conceptual learning with multiple graphical representations: Intelligent tutoring systems support for sense-making and fluency-building processes ( Unpublished doctoral dissertation). Carnegie Mellon University, Pittsburgh, PA.
  • Rencher, A. C. (2002). Methods in multivariate analysis (2nd ed.). Hoboken, NJ: Wiley.
  • Reschly, A. L., & Christenson, S. L. (2012). Jingle, jangle, and conceptual haziness: Evolution and future directions of the engagement construct. In S. Christenson, A. L. Reschly, & C. Wylie (Eds.), Handbook of research on student engagement (pp. 3–20). New York, NY: Springer.
  • Reyna, V. F., & Brainerd, C. J. (2007). The importance of mathematics in health and human judgment: Numeracy, risk communication, and medical decision making. Learning and Individual Differences, 17(2), 147–159. doi:10.1016/j.lindif.2007.03.010
  • Romesburg, H. C. (1984). Cluster analysis for researchers. Belmont, CA: Lifetime Learning.
  • San Pedro, M. O. Z., Baker, R. S., Bowers, A. J., & Heffernan, N. T. (2013). Predicting college enrollment from student interaction with an intelligent tutoring system in middle school. In S. D’Mello, R. Calvo, & A. Olney (Eds.), Proceedings of the 6th International Conference on Educational Data Minin (pp. 177–184). Memphis, TN: International Educational Data Mining Society.
  • 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. doi:10.1207/s1532690xci2202_1
  • Sherin, B. (2013). A computational study of commonsense science: An exploration in the automated analysis of clinical interview data. Journal of the Learning Sciences, 22(4), 600–638. doi:10.1080/10508406.2013.836654
  • Shute, V. J. (2011). Stealth assessment in computer-based games to support learning. Computer Games and Instruction, 55(2), 503–524.
  • Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., … Wray, J. (2010). Developing effective fractions instruction for kindergarten through 8th grade: A practice guide (NCEE#2010-4039). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/PracticeGuide.aspx?sid=15
  • Siegler, R. S., & Crowley, K. (1991). The microgenetic method: A direct means for studying cognitive development. American Psychologist, 46(6), 606–620. doi:10.1037/0003-066X.46.6.606
  • Siegler, R. S., Thompson, C. A., & Schneider, M. (2011). An integrated theory of whole number and fractions development. Cognitive Psychology, 62(4), 273–296. doi:10.1016/j.cogpsych.2011.03.001
  • Slavin, R. E., & Lake, C. (2008, September). Effective programs in elementary mathematics: A best-evidence synthesis. Review of Educational Research, 78(3), 427–515. doi:10.3102/0034654308317473
  • Slavin, R., & Smith, D. (2008, March). Effects of sample size on effect size in systematic reviews in education. Paper presented at the annual meeting of the Society for Research on Effective Education, Crystal City, VA.
  • Stigler, J., Givvin, K., & Thompson, B. (2010). What community college developmental mathematics students understand about mathematics. MathAMATYC Educator, 1(3), 4–16.
  • Su, Z., Song, W., Lin, M., & Li, J. (2008). Web text clustering for personalized E-learning based on maximal frequent itemsets. In Huaibei Zhou (Chair), 2008 International Conference on Computer Science and Software Engineering (pp. 452–455), Wuhan, Hubei, China, December 12–14. Washington, DC: Institute of Electrical and Electronics Engineers (IEEE). doi:10.1109/CSSE.2008.1639
  • Sweller, J. (2010). Element interactivity and intrinsic, extraneous, and germane cognitive load. Educational Psychology Review, 22(2), 123–138. doi:10.1007/s10648-010-9128-5
  • Sztajn, P., Confrey, J., Wilson, P. H., & Edgington, C. (2012). Learning trajectory based instruction: Toward a theory of teaching. Educational Researcher, 41(5), 147–156. doi:10.3102/0013189X12442801
  • Ulrich, D., & McKelvey, B. (1990). General organizational classification: An empirical test using the United States and Japanese electronics industries. Organization Science, 1(1), 99–118. doi:10.1287/orsc.1.1.99
  • Vellido, A., Castro, F., & Nebot, A. (2011). Clustering educational data. In C. Romero, S. Ventura, M. Pechenizkiy, & R. S. Baker (Eds.), Handbook of educational data mining (pp. 75–93). Boca Raton, FL: CRC Press.
  • Wang, F.-H., & Shao, H.-M. (2004). Effective personalized recommendation based on time-framed navigation clustering and association mining. Expert Systems With Applications, 27, 365–377. doi:10.1016/j.eswa.2004.05.005
  • White, T. (2014). Hadoop: The definitive guide (4th ed.). Sebastopol, CA: O’Reilly Media.
  • Wilson, P., Myers, M., Edgington, C., & Confrey, J. (2012). Fair shares, matey, or walk the plank. Teaching Children Mathematics, 18(8), 482–489. doi:10.5951/teacchilmath.18.8.0482
  • Wilson, S. M. (1988). Understanding historical understanding: Subject matter knowledge and the teaching of American history (Unpublished doctoral dissertation). Stanford University, Palo Alto, CA.
  • Wilson, S. M., Shulman, L. S., & Richert, A. (1987). 150 different ways of knowing: Representations of knowledge in teaching. In J. Calderhead (Ed.), Exploring teachers’ thinking (pp. 104–124). Sussex, England: Holt, Rinehart & Winston.
  • Witten, I. H., Frank, E., & Hall, M. A. (2011). Data mining: Practical machine learning tools and techniques (3rd ed.). Burlington, MA: Morgan Kaufman.
  • Xu, B., Recker, M., Qi, X., Flann, N., & Ye, L. (2013). Clustering educational digital library usage data: A comparison of latent class analysis and k-means algorithms. Journal of Educational Data Mining, 5(2), 38–68.

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