Session Information
24 SES 04, Exploring Text and Textbooks in Mathematics Teaching and Learning
Paper Session
Contribution
BACKGROUND
There are several studies showing that students perform better on posttests if they have constructed (at least parts of) the solution method compared to students that were given a procedure to follow (Jonsson, et.al, 2014); Kapur, 2011, 2014; Brunstein, et,al, 2009). However, the same studies show that the students creating their own methods do not have the same rate of success during practice as those who followed a procedure to reach the correct answer. That is, the students who follow given instructions how to solve the problem will perform better during practice than the students who have to create their methods (however, not necessary all by themselves), but the latter is shown to be more effective. These circumstances could be one of the reasons why teachers tend to prefer to provide students with instructions and procedures how to solve problem rather than support them to become skilled problem solvers. It is easy to misguidedly regard successful practice during lessons as proof of that the students have reached the target knowledge and will perform well on the posttest. These results are in line with research by for example Schoenfeld (1985), Hiebert & Stigler, J (2004) and Brousseau (1997), shortly described here
TEORETICAL FRAMEWORK
Schoenfeld (1985) stated that in order to learn and to become skilled problem solvers students need to actual solve ‘problems’, i.e. intellectual challenges that no procedures how to solve the problem is known in advance. Hiebert & Grouws (2007) stated that struggling with fundamental mathematics subjects is critical to developing an understanding of mathematical concepts. Allowing a positive type of struggle in the problem-solving process leads students to create their own problem-solving methods rather than imitate instructed procedures. This process can be conducted by implementing Brousseau’s (1997) theory of didactical situations in mathematics. Brousseau suggests that instead of providing students with procedures on how to solve the tasks, teachers should designate part of the responsibility of developing the problem-solving process to the students. In this part of the didactic session, described as an adidactical situation, the students should create their own solutions rather than be guided to the correct answer by their teacher. However, to prevent students from adopting erroneous constructed methods the adidactical situation should involve feedback related to the students’ actions. Brousseau (1997) refers to feedback as “positive or negative sanctions relative to her action, which allows her to adjust this action, to accept or reject a hypothesis” (p. 7).
TEXTBOOKS EXPLOARTIVE PROBLEMS
Textbooks in mathematics, in general, have a strong focus to provide students with instructions, procedures etc to help students solve tasks imitating the given method (Lithner 2003, 2004). Even problems that aims for inviting students to explore and find out mathematical relations on their own tend to be accompanied with guidelines to the students how to explore and what to pay attention to. However these kind of tasks, leaving a small part of the construction to the students have not been studied in the same amount as strict procedural tasks compered to explorative problems.
The present study will look into possible differences regarding learning outcomes when students, are given an explorative problem with guidelines (as provided in textbooks) compared to if no guidelines are given. The following hypotheses will guide the study
HYPOTESES:
1) Students that are provided with guidelines will solve the problem to a greater extent than students who create their own solving method
2) Students that have created their own solution method will perform better on posttests than students who were provided with guidelines
Method
Expected Outcomes
References
Brousseau, G. (1997). Theory of didactical situations in mathematics: didactique des mathématiques (1970-1990). Brunstein, A., Betts, S., & Anderson, J. R. (2009). Practice enables successful learning under minimal guidance. Journal of Educational Psychology, 101(4), 790. Hiebert, J., & Stigler, J. W. (2004). A world of difference. Journal of staff development, 25(4), 10-15. Jonsson, B., Norqvist, M., Liljekvist, Y., & Lithner, J. (2014). Learning mathematics through algorithmic and creative reasoning. Journal of Mathematical Behavior, 36, 20-32. Kapur, M. (2011). A further study of productive failure in mathematical problem solving: Unpacking the design components. Instructional Science, 39(4), 561-579. Kapur, M. (2014). Productive Failure in Learning Math. Cognitive science, 38, 1008-1022. Kapur, M., & Bielaczyc, K. (2012). Designing for productive failure. Journal of the Learning Sciences, 21(1), 45-83. Lithner, J. (2003). Students mathematical reasoning in university textbook exercises. Educational Studies in Mathematics, 52, 29–55. Lithner, J. (2004). Mathematical reasoning in calculus textbook exercises. Journal of Mathematical Behavior, 23, 405–427. Schoenfeld, A. H. (1985). Mathematical problem solving: ERIC.
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