Elementary students’ fraction reasoning: a measurement approach to fractions in a dynamic environment

References

• Abrahamson, D., & Abdu, R. (2021). Towards an ecological-dynamics design framework for embodied-interaction conceptual learning: The case of dynamic mathematics environments. Educational Technology Research and Development, 69(4), 1889–1923. https://doi.org/10.1007/s11423-020-09805-1
   Web of Science ®Google Scholar
• Abrahamson, D. (2002). When “the same” is the same as different differences: Aliya reconciles her perceptual judgment of proportional equivalence with her additive computation skills. In D. Mewborn, P. Sztajn, E. White, H. Wiegel, R. Bryant, and K. Nooney (Eds.), Proceedings of the Twenty Fourth Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 1658–1661). Eric Clearinghouse for Science, Mathematics, and Environmental Education, Columbus, OH.
   Google Scholar
• Abrahamson, D., Lee, R. G., Negrete, A. G., & Gutiérrez, J. F. (2014). Coordinating visualizations of polysemous action: Values added for grounding proportion. ZDM Mathematics Education, 46(1), 79–93. https://doi.org/10.1007/s11858-013-0521-7
   Google Scholar
• Anderson-Pence, K. L. (2017). Techno-mathematical discourse: A conceptual framework for analyzing classroom discussions. Education Sciences, 7(1), 40. https://doi.org/10.3390/educsci7010040
   Web of Science ®Google Scholar
• Artigue, M. (2002). Learning mathematics in a CAS environment: The genesis of a reflection about instrumentation and the dialectics between technical and conceptual work. International Journal of Computers for Mathematical Learning, 7(3), 245–274. https://doi.org/10.1023/A:1022103903080
   Google Scholar
• Bartolini Bussi, M. G., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom: Artefacts and signs after a Vygotskian perspective. In L. English, M. B. Bussi, G. Jones, R. Lesh, & D. Tirosh (Eds.), Handbook of international research in mathematics education (2nd ed., pp. 746–783). Lawrence Erlbaum.
   Google Scholar
• Battista, M. T. (2012). Cognition-based assessment and teaching of place value: Building on students’ reasoning. Heinemann.
   Google Scholar
• Behr, M. J., Harel, G., Post, T. R., & Lesh, R. (1992). Rational number, ratio, and proportion. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 296–332). Macmillan Publishing Company.
   Google Scholar
• Brousseau, G., Brousseau, N., & Warfield, G. (2014). Teaching fractions through situations: A fundamental experiment. Springer.
   Google Scholar
• Charalambous, C. Y., & Pitta-Pantazi, D. (2007). Drawing on a theoretical model to study students’ understandings of fractions. Educational Studies in Mathematics, 64(3), 293–316. https://doi.org/10.1007/s10649-006-9036-2
   Google Scholar
• Chase, K., & Abrahamson, D. (2015). Reverse-scaffolding algebra: Empirical evaluation of design architecture. ZDM Mathematics Education, 47(7), 1195–1209. https://doi.org/10.1007/s11858-015-0710-7
   Google Scholar
• Chval, K. B., Lannin, J. K., & Jones, D. (2013). Putting essential understanding of multiplication and division into practice in Grades 3–5. National Council of Teachers of Mathematics.
   Google Scholar
• Clark-Wilson, A., Robutti, O., & Thomas, M. (2020). Teaching with digital technology. ZDM Mathematics Education, 52, 1223–1242. doi:10.1007/s11858-020-01196-0
   Google Scholar
• Clarke, D. M., & Roche, A. (2009). Students’ fraction comparison strategies as a window into robust understanding and possible pointers for instruction. Educational Studies in Mathematics, 72(1), 127–138. https://doi.org/10.1007/s10649-009-9198-9
   Web of Science ®Google Scholar
• Clements, D. H., Sarama, J., Yelland, N. J., & Glass, B. (2008). Learning and teaching geometry with computers in the elementary and middle school. In M. K. Heid & G. W. Blume (Eds.), Research on technology and the teaching and learning of mathematics: Research syntheses (Vol. 1, pp. 109–154). Information Age Publishing.
   Google Scholar
• Confrey, G.et al (1994). Splitting, similarity, and rate of change: A new approach to multiplication and exponential. In G. Harel, and G. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 291–330). Albany, NY: State University of New York Press.
   Google Scholar
• Corbin, J., & Strauss, A. (2014). Basics of qualitative research: Techniques and procedures for developing grounded theory. Sage.
   Google Scholar
• Davydov, V. V., & Tsvetkovich, Z. H. (1991). The object sources of the concept of fraction. In V. V. Davydov (Soviet Edition Editor) & L. P. Steffe (English Language Editor) (Eds.), Soviet studies in mathematics education: Psychological abilities of primary school children in learning mathematics (pp. 86–147). National Council of Teachers of Mathematics.
   Google Scholar
• DeWolf, M., & Vosniadou, S. (2015). The representation of fraction magnitudes and the whole number bias reconsidered. Learning and Instruction, 37, 39–49. https://doi.org/10.1016/j.learninstruc.2014.07.002
   Web of Science ®Google Scholar
• Dougherty, B. J. (2008). Measure up: A quantitative view of early algebra. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 389–412). Lawrence Erlbaum Associates.
   Google Scholar
• Drijvers, P., Godino, J. D., Font, V., & Trouche, L. (2013). One episode, two lenses. Educational Studies in Mathematics, 82(1), 23–49. https://doi.org/10.1007/s10649-012-9416-8
   Web of Science ®Google Scholar
• Empson, S. B. (2003). Low-performing students and teaching fractions for understanding: An interactional analysis. Journal for Research in Mathematics Education, 34(4), 305–343. https://doi.org/10.2307/30034786
   Web of Science ®Google Scholar
• Flood, V. J., Shvarts, A., & Abrahamson, D. (2020). Teaching with embodied learning technologies for mathematics: Responsive teaching for embodied learning. ZDM Mathematics Education, 52(7), 1307–1331. https://doi.org/10.1007/s11858-020-01165-7
   Google Scholar
• Fuchs, L. S., Schumacher, R. F., Long, J., Namkung, J., Hamlett, C. L., Cirino, P. T., Jordan, N. C., Siegler, R., Gersten, R., & Changas, P. (2013). Improving at-risk learners’ understanding of fractions. Journal of Educational Psychology, 105(3), 683–700. https://doi.org/10.1037/a0032446
   Web of Science ®Google Scholar
• Ginsburg, H. (1997). Entering the child’s mind: The clinical interview in psychological research and practice. Cambridge University Press.
   Google Scholar
• Goldin, G. (2000). A scientific perspective on structures, task-based interviews in mathematics education research. In A. E. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 517–545). Lawrence Erlbaum.
   Google Scholar
• Götze, D., & Baiker, A. (2021). Language-responsive support for multiplicative thinking as unitizing: Results of an intervention study in the second grade. ZDM Mathematics Education, 53(2), 263–275. https://doi.org/10.1007/s11858-020-01206-1
   Google Scholar
• Guest, G., Bunce, A., & Johnson, L. (2006). How many interviews are enough? An experiment with data saturation and variability. Field Methods, 18(1), 59–82. https://doi.org/10.1177/1525822X05279903
   Web of Science ®Google Scholar
• Hackenberg, A. J., Norton, A., & Wright, R. J. (2016). Developing fractions knowledge. Sage.
   Google Scholar
• Heid, M. K., & Blume, G. W. (Eds.). (2008). Research on technology in the teaching and learning of mathematics: Research syntheses (Vol. 1). Information Age Publishing.
   Google Scholar
• Hollebrands, K. F. (2007). The role of a dynamic software program for geometry in the strategies high school mathematics students employ. Journal for Research in Mathematics Education, 38 (2) , 164–192. https://doi.org/10.2307/30034955
   Web of Science ®Google Scholar
• Hoyles, C., & Noss, R. (2003). What can digital technologies take from and bring to research in mathematics education? In A. J. Bishop, M. A. Clements, C. Keitel, J. Kilpatrick, & F. Leung (Eds.), Second international handbook of mathematics education (Vol. 1, pp. 323–349). Kluwer Academic Publishers.
   Google Scholar
• Hwang, S., Yeo, S., & Son, T. (2021). A comparative analysis of fraction addition and subtraction contents in the mathematics textbooks in the US and South Korea. International Electronic Journal of Elementary Education, 13 (4) , 511–521. https://doi.org/10.26822/iejee.2021.208
   Google Scholar
• Jackiw, N., & Sinclair, N. (2009). Sounds and pictures: Dynamism and dualism in dynamic geometry. ZDM Mathematics Education, 41(4), 413–426. https://doi.org/10.1007/s11858-009-0196-2
   Google Scholar
• Kaput, J. J. (1992). Technology and mathematics education. In D. A. Grouws (Ed.), Handbook of teaching and learning mathematics (pp. 515–556). Macmillan.
   Google Scholar
• Konold, C., Harradine, A., & Kazak, S. (2007). Understanding distributions by modeling them. International Journal of Computers for Mathematical Learning, 12(3), 217–230. https://doi.org/10.1007/s10758-007-9123-1
   Google Scholar
• Laborde, J. M. (2016). Technology-enhanced teaching/learning at a new type with dynamic mathematics as implemented in the new Cabri. In M. Bates & Z. Usiskin (Eds.), Digital curricula in school mathematics (pp. 53–74). Information Age Publishing.
   Google Scholar
• Lamon, S. J. (2012). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers (3rd ed.). Routledge.
   Google Scholar
• Mack, N. K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267–295. doi:10.2307/749828
   Web of Science ®Google Scholar
• Maschietto, M., & Soury-Lavergne, S. (2013). Designing a duo of material and digital artifacts: The pascaline and Cabri Elem e-books in primary school mathematics. ZDM Mathematics Education, 45(7), 959–971. https://doi.org/10.1007/s11858-013-0533-3
   Google Scholar
• Moon, J., & Ke, F. (2020). In-game actions to promote game-based math learning engagement. Journal of Educational Computing Research, 58(4), 863–885. https://doi.org/10.1177/0735633119878611
   Web of Science ®Google Scholar
• Moreno-Armella, L., Hegedus, S. J., & Kaput, J. J. (2008). From static to dynamic mathematics: Historical and representational perspectives. Educational Studies in Mathematics, 68(2), 99–111. https://doi.org/10.1007/s10649-008-9116-6
   Google Scholar
• Morris, A. K. (2000). A teaching experiment: Introducing fourth graders to fractions from the viewpoint of measuring quantities using Davydov’s mathematics curriculum. Focus on Learning Problems in Mathematics, 22(2), 33–84.
   Google Scholar
• Moyer-Packenham, P. S., Ulmer, L. A., & Anderson, K. L. (2012). Examining pictorial models and virtual manipulatives for third-grade fraction instruction. Journal of Interactive Online Learning, 11(3), 103–120. http://www.ncolr.org/jiol/issues/pdf/11.3.2.pdf
   Google Scholar
• Ng, O. L., & Sinclair, N. (2015). Young children reasoning about symmetry in a dynamic geometry environment. ZDM Mathematics Education, 47(3), 421–434. https://doi.org/10.1007/s11858-014-0660-5
   Web of Science ®Google Scholar
• Ni, Y., & Zhou, Y.-D. (2005). Teaching and learning fraction and rational numbers: The origins and implications of whole number bias. Educational Psychologist, 40(1), 27–52. https://doi.org/10.1207/s15326985ep4001_3
   Web of Science ®Google Scholar
• Noss, R., Hoyles, C., Mavrikis, M., Geraniou, E., Gutierrez-Santos, S., & Pearce, D. (2009). Broadening the sense of ‘dynamic’: A microworld to support students’ mathematical generalisation. ZDM Mathematics Education, 41(4), 493–503. https://doi.org/10.1007/s11858-009-0182-8
   Google Scholar
• Olive, J., & Lobato, J. (2008). The learning of rational number concepts using technology. In M. K. Heid & G. W. Blume (Eds.), Research on technology and the teaching and learning of mathematics: Research syntheses (pp. 1–54). Information Age and the National Council of Teachers of Mathematics.
   Google Scholar
• Pitkethly, A., & Hunting, R. (1996). A review of recent research in the area of initial fraction concepts. Educational Studies in Mathematics, 30(1), 5–38. https://doi.org/10.1007/BF00163751
   Google Scholar
• Rabardel, P., & Bourmaud, G. (2003). From computer to instrument system: A developmental perspective. Interacting With Computers, 15(5), 665–691. https://doi.org/10.1016/S0953-5438(03)00058-4
   Web of Science ®Google Scholar
• Rabardel, P. (1995). Les hommes et les technologies. Une approche cognitive des instruments contemporains [People and technology: A cognitive approach to contemporary instruments]. Armand Collin.
   Google Scholar
• Roschelle, J., Kaput, J., & Stroup, W. (2000). SimCalc: Accelerating students’ engagement with the mathematics of change. In M. J. Jacobson & R. Kozma (Eds.), Innovations in science and mathematics education: Advanced designs for technologies of learning (pp. 46–76). Lawrence Erlbaum Associates.
   Google Scholar
• Roschelle, J., Noss, R., Blikstein, P., & Jackiw, N. (2017). Technology for learning mathematics. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 273–296). National Council of Teachers of Mathematics.
   Google Scholar
• Saldaña, J. (2015). The coding manual for qualitative researchers (3rd ed.). Sage.
   Google Scholar
• Saxe, G. B., Taylor, E. V., McIntosh, C., & Gearhart, M. (2005). Representing fractions with standard notation: A developmental analysis. Journal for Research in Mathematics Education, 36 (2) , 137–157. https://doi.org/10.2307/30034828
   Web of Science ®Google Scholar
• Schmittau, J., & Morris, A. (2004). The development of algebra in the elementary mathematics curriculum of VV Davydov. The Mathematics Educator, 8 (1) , 60–87. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.452.3117&rep=rep1&type=pdf
   Google Scholar
• Shvarts, A., Alberto, R., Bakker, A., Doorman, M., & Drijvers, P. (2021). Embodied instrumentation in learning mathematics as the genesis of a body-artifact functional system. Educational Studies in Mathematics, 107(3), 447–469. https://doi.org/10.1007/s10649-021-10053-0
   Web of Science ®Google Scholar
• Siegler, R. S., & Pyke, A. A. (2013). Developmental and individual differences in understanding of fractions. Developmental Psychology, 49(10), 1994–2004. https://doi.org/10.1037/a0031200
   PubMed Web of Science ®Google Scholar
• Simon, M. A., Placa, N., Avitzur, A., & Kara, M. (2018). Promoting a concept of fraction-as-measure: A study of the learning through activity research program. The Journal of Mathematical Behavior, 52, 122–133. https://doi.org/10.1016/j.jmathb.2018.03.004
   Web of Science ®Google Scholar
• Sinclair, N., & Crespo, S. (2006). Learning mathematics in dynamic computer environments. Teaching Children Mathematics, 12 (9) , 436–444. https://doi.org/10.2307/41198823
   Google Scholar
• Smith, J. P., III., & Barrett, J. E. (2017). Learning and teaching measurement: Coordinating quantity and number. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 355–385). National Council of Teachers of Mathematics.
   Google Scholar
• Soni, M., & Okamoto, Y. (2020). Improving children’s fraction understanding through the use of number lines. Mathematical Thinking and Learning, 22(3), 233–243. https://doi.org/10.1080/10986065.2020.1709254
   Web of Science ®Google Scholar
• Steffe, L. P., & Olive, J. (2002). Design and use of computer tools for interactive mathematical activity (TIMA). Journal of Educational Computing Research, 27(1), 55–76. https://doi.org/10.2190/GXQ4-60W8-CJY4-GKE1
   Google Scholar
• Steffe, L. P., & Olive, J. (2010). Children’s fractional knowledge. Springer.
   Google Scholar
• Suh, J., Moyer, P. S., & Heo, H. J. (2005). Examining technology uses in the classroom: Developing fraction sense using virtual manipulative concept tutorials. Journal of Interactive Online Learning, 3(4), 1–21. http://www.ncolr.org/jiol/issues/pdf/3.4.2.pdf
   Google Scholar
• Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A research companion to principles and standards for school mathematics (pp. 95–113). The National Council of Teachers of Mathematics.
   Google Scholar
• Tzur, R. (1999). An integrated study of children’s construction of improper fractions and the teacher’s role in promoting that learning. Journal for Research in Mathematics Education, 30(4), 390–416. https://doi.org/10.2307/749707
   Web of Science ®Google Scholar
• Venenciano, L., & Heck, R. (2016). Proposing and testing a model to explain traits of algebra preparedness. Educational Studies in Mathematics, 92(1), 21–35. https://doi.org/10.1007/s10649-015-9672-5
   Web of Science ®Google Scholar
• Vergnaud, G. (1996). Au fond de l’apprentissage, la conceptualisation. [At the basis of learning, the conceptualization.]. In R. Noirfalise & M.-J. Perrin (Eds.), Actes de l’école d’été de didactique des mathématiques [Proceedings of the summer school on didactics of mathematics.] (pp. 174–185). IREM.
   Google Scholar
• Vygotsky, L. S. (1978). Mind in society. Harvard University Press.
   Google Scholar
• Vysotskaya, E., Lobanova, A., Rekhtman, I., & Yanishevskaya, M. (2021). The challenge of proportion: Does it require rethinking of the measurement paradigm? Educational Studies in Mathematics, 106(3), 429–446. https://doi.org/10.1007/s10649-020-09987-8
   Web of Science ®Google Scholar
• Webel, C., & DeLeeuw, W. W. (2016). Meaning for fraction multiplication: Thematic analysis of mathematical talk in three fifth grade classes. The Journal of Mathematical Behavior, 41, 123–140. https://doi.org/10.1016/j.jmathb.2015.12.003
   Web of Science ®Google Scholar
• Wilkins, J. L., & Norton, A. (2018). Learning progression toward a measurement concept of fractions. International Journal of STEM Education, 5(1), 1–11. https://doi.org/10.1186/s40594-018-0119-2
   PubMedGoogle Scholar
• Yeo, S. (2020). Integrating digital technology into elementary mathematics: Three theoretical perspectives. Research in Mathematical Education, 23(3), 165–179. https://doi.org/10.7468/JKSMED.2020.23.3.165
   Google Scholar
• Yeo, S. (2021). Semiotic mediation through technology: The case of fraction reasoning. The Mathematical Education, 60(1), 1–19. https://doi.org/10.7468/MATHEDU.2021.60.1.1
   Google Scholar
• Young, J., Gorumek, F., & Hamilton, C. (2018). Technology effectiveness in the mathematics classroom: A systematic review of meta-analytic research. Journal of Computers in Education, 5(2), 133–148. https://doi.org/10.1007/s40692-018-0104-2
   Web of Science ®Google Scholar