Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Applying Levels of Abstraction to Mathematics Word Problems

Abstract

In many discussions of the ways in which abstraction is applied in computer science (CS), researchers and advocates of CS education argue that CS students should be taught to consciously and explicitly move among levels of abstraction (Armoni Journal of Computers in Mathematics and Science Teaching, 32(3), 265–284, 2013; Kramer Communications of the ACM, 50(4), 37–42, 2007; Wing Communications of the ACM, 49(3), 33–35, 2006). In this paper, we describe one way that attention to levels of abstraction could also support learning in mathematics. Specifically, we propose a framework for using abstraction in elementary mathematics based on Armoni’s (2013) framework for teaching computational abstraction. We propose that such a framework could address an enduring challenge in mathematics for helping elementary students solve word problems with attention to context. In a discussion of implications, we propose that future research using the framework for instruction and teacher education could also explore ways that attention to levels of abstraction in elementary school mathematics may support later learning of mathematics and computer science.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3

References

  1. Aho, A. V., & Ullman, J. D. (1995). Foundations of computer science. New York: W. H. Freeman and Company.

  2. Armoni, M. (2013). On teaching abstraction in computer science to novices. Journal of Computers in Mathematics and Science Teaching, 32(3), 265–284.

  3. Carpenter, T. P., Lindquist, M. M., Matthews, W., & Silver, E. A. (1983). Results of the third NAEP mathematics assessment: Secondary school. The Mathematics Teacher, 76(9), 652–659.

  4. Code.org. (2016). Computer Science Fundamentals for elementary school. Retrieved from https://code.org/educate/curriculum/elementary-school

  5. College Board. (2017). AP Computer Science Principles course and exam description. Retrieved from https://apcentral.collegeboard.org/courses/ap-computerscience-principles/course

  6. CSK-12.org. (2016). K-12 Computer Science Framework. Retrieved from https://k12cs.org/

  7. Greer, B. (1997). Modelling reality in mathematics classrooms: The case of word problems. Learning and Instruction, 7(4), 293–307.

  8. Grover, S., & Pea, R. (2013). Computational thinking in K–12: a review of the state of the field. Educational Researcher, 42(1), 38–43. https://doi.org/10.3102/0013189X12463051.

  9. Hazzan, O. (2003). How students attempt to reduce abstraction in the learning of mathematics and in the learning of computer science. Computer Science Education, 13(2), 95–122.

  10. Hazzan, O. (2008). Reflections on teaching abstraction and other soft ideas. ACM SIGCSE Bulletin, 40(2), 40–43.

  11. Hazzan, O., & Zazkis, R. (2005). Reducing abstraction: the case of school mathematics. Educational Studies in Mathematics, 58(1), 101–119.

  12. Hillis, W. D. (1998). The pattern on the stone: The simple ideas that make computers work. New York: Basic Books.

  13. Kramer, J. (2007). Is abstraction the key to computing? Communications of the ACM, 50(4), 37–42.

  14. Muller, O., & Haberman, B. (2008). Supporting abstraction processes in problem solving through pattern-oriented instruction. Computer Science Education, 18(3), 187–212.

  15. Palm, T. (2008). Impact of authenticity on sense making in word problem solving. Educational Studies in Mathematics, 67(1), 37–58.

  16. Perez, A. (2018). A framework for computational thinking dispositions in mathematics education. Journal for Research in Mathematics Education, 49(4), 424–461.

  17. Perrenet, J., Groote, J. F., & Kaasenbrood, E. (2005). Exploring students’ understanding of the concept of algorithm: Levels of abstraction. In Annual joint conference integrating technology into computer science education (pp. 64–68). New York: ACM.

  18. Reusser, K., & Stebler, R. (1997). Every word problem has a solution -- the social rationality of mathematical modeling in schools. Learning and Instruction, 7(4), 309–327.

  19. Rich, K. M., Yadav, A., & Schwarz, C. V. (2019a). Computational thinking, mathematics, and science: elementary teachers’ perspectives on integration. Journal of Technology and Teacher Education, 27(2), 165–205.

  20. Rich, K. M., Yadav, A., & Zhu, M. (2019b). Levels of abstraction in students’ mathematics strategies: what can applying computer science ideas about abstraction bring to elementary mathematics ? Journal of Computers in Mathematics and Science Teaching, 38(3), 267–298.

  21. Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.

  22. Silver, E. A., Shapiro, L. J., & Deutsch, A. (1993). Sense making and the solution of division problems involving remainders: an examination of middle school students’ solution processes and their interpretations of solutions. Journal for Research in Mathematics Education, 24(2), 117–135.

  23. Statter, D., & Armoni, M. (2016). Teaching abstract thinking in introduction to computer science for 7th graders. In Proceedings of the 11th Workshop in Primary and Secondary Computing Education - WiPSCE ‘16 (pp. 80–83).

  24. Statter, D., & Armoni, M. (2017). Learning abstraction in computer science: A gender perspective. In E. Barendsen & P. Hubwieser (Eds.), Proceedings of the 12th workshop on primary and secondary computing education (pp. 5–14). New York: ACM.

  25. Verschaffel, L., & De Corte, E. (1997). Teaching realistic mathematical modeling in the elementary school: a teaching experiment with fifth graders. Journal for Research in Mathematics Education, 28(5), 577–601.

  26. Verschaffel, L., De Corte, E., & Lasure, S. (1994). Realistic considerations in mathematical modeling of school arithmetic word problems. Learning and Instruction, 4(4), 273–294.

  27. Weyns, A., Van Dooren, W., Dewolf, T., & Verschaffel, L. (2017). The effect of emphasising the realistic modelling complexity in the text or picture on pupils’ realistic solutions of P-items. Educational Psychology, 37(10), 1173–1185.

  28. White House. (2016). Computer science for all [Blog post dated January 30, 2016]. Retrieved from https://obamawhitehouse.archives.gov/blog/2016/01/30/computer-science-all

  29. Wilensky, U. (1991). Abstract meditations on the concrete and concrete implications for mathematical education. In I. Harel & S. Papert (Eds.), Constructionism: Research reports and essays, 1985–1990 (pp. 193–203). Boston: Epistemology & Learning Research Group.

  30. Wing, J. M. (2006). Computational thinking. Communications of the ACM, 49(3), 33–35.

  31. Yadav, A., Stephenson, C., & Hong, H. (2017). Computational thinking for teacher education. Communications of the ACM, 60(4), 55–62. https://doi.org/10.1145/2994591.

Download references

Acknowledgements

This work was supported by the National Science Foundation under Grant number 1738677. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Author information

Correspondence to Kathryn M. Rich.

Ethics declarations

Potential Conflicts of Interest

The authors have no conflict of interest to report.

Research Involving Human Participants

All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards.

Informed Consent

Informed consent was obtained from all individual participants included in the study.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Rich, K.M., Yadav, A. Applying Levels of Abstraction to Mathematics Word Problems. TechTrends (2020). https://doi.org/10.1007/s11528-020-00479-3

Download citation

Keywords

  • Mathematics education
  • Computer science education
  • Abstraction
  • Problem solving
  • Elementary education