Heuristic Trees as a Digital Tool to Foster Compression and Decompression in Problem-Solving

Springer Science and Business Media LLC - 2022
Rogier Bos1, Theo van den Bogaart2
1Freudenthal Institute, Utrecht University, Utrecht, The Netherlands
2Archimedes Institute, HU University of Applied Sciences Utrecht, Utrecht, the Netherlands

Tóm tắt

AbstractThis design-based study addresses the issue of how to digitally support students’ problem-solving by providing heuristics, in the absence of the teacher. The problem is that, so far, digital tutoring systems lack the ability to diagnose students’ needs in open problem situations. Our approach is based on students’ ability to self-diagnose and find help. To this purpose, we introduce a new type of digital, interactive, help-seeking tool called a heuristic tree. Students’ use of this tool is supported by a help-seeking flowchart. The design of heuristic trees is based on our reinterpretation of the notion of heuristic in terms of terms of compression. Our research question is: How do heuristic trees and the help-seeking flowchart influence students’ problem-solving behaviour? This question was studied in the context of a number theory course for in-service mathematics teachers. During five weeks, fifty students worked on fifty-five problems supported by heuristic trees. Our data consists of video observations of two small groups of students, a teacher log, interviews with these two groups, and a survey filled in by twenty-three students. The main results are that the support by heuristic trees and the help-seeking flowchart allows students to work in the absence of a teacher and to engage strongly with problems, maintaining ownership of the solution methods. Moreover, as intended by the tree structure, students learned to focus not just on the small steps of the solutions, but also on the general heuristic techniques, theorems, and concepts that should be learned in the process of finding those solutions.

Từ khóa


Tài liệu tham khảo

Aleven, V., McLaren, B., Roll, I., & Koedinger, K. (2006). Toward meta-cognitive tutoring: A model of help seeking with a cognitive tutor. International Journal of Artificial Intelligence in Education, 16(2), 101–128.

Aleven, V., Stahl, E., Schworm, S., Fischer, F., & Wallace, R. (2003). Help seeking and help design in interactive learning environments. Review of Educational Research, 73(3), 277–320.

Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Fuentes, S., Trigueros, M., & Weller, K. (2014). APOS theory: A framework for research and curriculum development in mathematics education. Springer.

Bakker, A. (2018). Design research in education: A practical guide for early career researchers. Routledge.

Bos, R. (2017). Structuring hints and heuristics in intelligent tutoring systems. In G. Aldon & J. Trgalová (Eds), Proceedings of the 13th International Conference on Technology in Mathematics Teaching (pp. 436–439). Lyon, France: ICTMT. (https://hal.archives-ouvertes.fr/hal-01632970/document)

Bos, R., & van den Bogaart, T. (submitted). Teachers’ design of heuristic trees.

Hershkowitz, R., Schwarz, B., & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education, 32(2), 195–222.

Lemmink, R. (2019). Improving help-seeking behavior for online mathematical problem-solving lessons Unpublished Master’s thesis. Utrecht, the Netherlands: Utrecht University. (https://dspace.library.uu.nl/handle/1874/382857)

Lester, F. (2013). Thoughts about research on mathematical problem-solving instruction. TheMathematics Enthusiast, 10(1–2), 245–278.

Lester, F., & Cai, J. (2016). Can mathematical problem solving be taught? Preliminary answers from 30 years of research. In P. Felmer, E. Pehkonen, & J. Kilpatrick (Eds.), Posing and solving mathematical problems: Advances and new perspectives (pp. 117–135). Springer.

Miles, M., & Huberman, M. (1994). Qualitative data analysis: An expanded sourcebook (2nd ed.). Sage Publications.

Pol, H., Harskamp, E., Suhre, C., & Goedhart, M. (2008). The effect of hints and model answers in a student-controlled problem-solving program for secondary physics education. Journal of Science Education and Technology, 17(4), 410–425.

Pol, H., Harskamp, E., Suhre, C., & Goedhart, M. (2009). How indirect supportive digital help during and after solving physics problems can improve problem-solving abilities. Computers and Education, 53(1), 34–50.

Pólya, G. (1945). How to solve it: A new aspect of mathematical method. Princeton University Press.

Reiss, K., Heinze, A., Renkl, A., & Groß, C. (2008). Reasoning and proof in geometry: Effects of a learning environment based on heuristic worked-out examples. ZDM: The International Journal on Mathematics Education, 40(3), 455–467.

Roll, I., Aleven, V., McLaren, B., & Koedinger, K. (2011). Improving students’ help-seeking skills using metacognitive feedback in an intelligent tutoring system. Learning and Instruction, 21(2), 267–280.

Roll, I., Baker, R., Aleven, V., & Koedinger, K. (2014). On the benefits of seeking (and avoiding) help in online problem-solving environments. Journal of the Learning Sciences, 23(4), 537–560.

Schoenfeld, A. (1985). Mathematical problem solving. Academic Press.

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), 1–36.

Sfard, A. (2008). Thinking as communicating: Human development, the growth of discourses, and mathematizing. Cambridge University Press.

Tall, D. (2013). How humans learn to think mathematically: Exploring the three worlds of mathematics. Cambridge University Press.

Thurston, W. (1990). Mathematical Education. Notices of the AMS, 37(7), 844–850.

van Hiele, P. (1986). Structure and insight: A theory of mathematics education. Academic Press.

White, P., & Mitchelmore, M. (2010). Teaching for abstraction: A model. Mathematical Thinking and Learning, 12(3), 205–226.

Thông tin
Thông tin xuất bản

Springer Science and Business Media LLC

Thông tin tác giả