Studies in Students’ Reasoning and Problem Solving

Author(s):

João-Pedro Da Ponte(submitting)Susana Carreira(presenting)Joana Mata-Pereira(presenting)Renata Carvalho Carrapiço(presenting)Joana Brocardo(presenting)Pauline Vos

Conference:

ECER 2014

Network:

24. Mathematics Education Research

Format:

Round Table

Session Information

24 SES 14, Studies in Students’ Reasoning and Problem Solving

Round Table

Time:

2014-09-05

15:30-17:00

Room:

B113 Sala de Aulas

Chair:

Lurdes Serrazina

Contribution

This Round Table is composed of four different short research reports plus a discussion. The presentations aim to present current studies in students’ reasoning and problem solving in Portugal. Given the nature of the theme (research in Portugal, the host country of the congress), the presenters are all Portuguese, and there is discussant from another country.

1. Logical-deductive reasoning. Logical-deductive reasoning is the basis of much of mathematical reasoning, as is usually associated with a complex cognitive activity (Epp, 2003). Determining the truth or falsity of a mathematical proposition, even in simple cases, may represent a difficult task even for highly educated adults. Conditional reasoning, a central form of logical-deductive reasoning, has attracted research efforts for many decades (Hoyles & Küchmann, 2002). The cognitive psychology of deduction has been dwelling between two main theories (“mental logic” and “mental models”) each accounting for some experimental results on individuals’ difficulties in answering logical problems.

In mathematics education most studies on deductive reasoning have focused on mathematical proof and very few have attempted to look at deductive reasoning in relation to problem solving (Bakó, 2002; Epp, 2003). One particular type of logic problems involves reasoning deductively from a set of instructions, rules or principles that describe relationships between people, things or events. They require the ability to consider a set of facts and rules and to determine, based on logical principles, which may or must be true. Such are analytical reasoning problems, which Cox and Brna (1995) examined in university students and English (1998) focused on reasoning processes of elementary school students (9-12 years old).

In this study, the focus is on young students’ conditional reasoning (aged 10-12) within the context of analytical reasoning problem solving. Our aim is to identify students’ models of conditional reasoning involved in logically connecting statements in an analytical reasoning problem.

2. Students’ mathematical reasoning in an introduction to 2^nd degree equations. This study addresses grade 9 students’ reasoning processes while dealing with 2^nd degree equations for the first time. As reasoning is formulating justified inferences from information available, we consider generalization and justification as key mathematical reasoning processes, but we also pay attention to their relationship with representations and sense making (Lannin, Ellis & Elliot, 2011; Mata-Pereira & Ponte, 2012). The data presented in this communication is from the first lessons about 2nd degree equations, in a grade 9 class, and we focus on whole class discussion moments.

3. Students’ reasoning in mental computation with percent. The aim of this study is to understand, in the context of a teaching experiment emphasizing regular practice and reflection, grade 6 students’ reasoning in mental computation with percent. The theoretical framework addresses the development of mental computation strategies, based on learning theories (e.g., Parker & Leinhardt (1995) that includes the use of numerical relationships with rational numbers, numerical facts and memorized rules in students’ relational thinking (Empson & Carpenter, 2010) and in mental models (Johnson-Laird, 1990) as a support.

4. Mathematical tasks to develop flexible calculation

The project ‘Numerical thinking and flexible calculation: critical issues’ has three major objectives: to identify students’ conceptual knowledge of numbers and operations; to analyze how this knowledge can facilitate adaptive thinking and flexible calculation; to study the implications to the design of mathematical tasks and to mathematical teacher education. We follow the idea that flexibility refers to the ability to manipulate numbers as mathematical objects which can decomposed and recomposed in different ways using different symbolisms for the same objet (Gravemeijer, 2004; Gray &Tall, 1994;). We will present results centered in part of the work done at the level of tier one.

Method

1. The empirical data for this study consist of all the solutions delivered via email to one problem proposed in the course of an online mathematical problem solving competition. The data are therefore exclusively digital documents. The solutions sent by the competitors may just be presented in an email window or come as attachments, such as Word, PowerPoint, or Excel files, and also digital scans of hand-written documents. The data analysis is performed through a descriptive and interpretive combination based on the coding of 334 correct solutions out of a total of 384. The categories refer to the conditional reasoning models exhibited and to the language used in students’ expression of deductive thinking. 2. The methodology is qualitative with participant observation (Bogdan & Biklen, 1982) and data collection includes video-recorded observations of lessons. The participants are from a class of grade 9 students. 3. The study is qualitative with a design research approach (Cobb et al., 2003). It is based on a teaching experiment with mental computation tasks that provide opportunity for work in mental computation in the classroom emphasizing the discussion of students’ reasoning. This study was developed in three phases. In the preparatory phase, in 2010, was developed a preliminary study in grade 5 and planned the teaching experiment. In the experimental phase, two experimental cycles were implemented in 2012 and 2013 involving two teachers and two grade 6 classes and the first author as a participant observer. Data were collected through video and audio recordings of the working moments in the classroom, the researcher’s notes, audio recordings of the preparatory and reflections meetings with the teachers involved and teachers and students interviews. 4. The project plan is based on a three-tiered teaching experiment design research (Lesh, Kelly & Yoon, 2008). The design and reformulation of mathematical tasks is an important part of the project.

Expected Outcomes

1. Two main approaches were identified in the participants’ solutions, each focusing in one of the two semantic dimensions of the rules and conditions given in the problem, one focusing on the dimension “who lied” and another on “who spilled the popcorn”. Most participants (89%) used hypothesis testing for the dimension “who lied”, choosing the approach that requires more effort to represent all the available possibilities. The use of conditional reasoning was a constant in all participants’ answers, following three types of reasoning models. The modus ponens syllogistic form was very clear in students’ reasoning and that no errors have been observed in this type of inferential reasoning. 2. While making generalizations students follow mostly an inductive approach, generalizing not only particular cases, but also known properties to a larger class of objects. Some of these reasoning processes also reveal abductive approaches. Students were able to make justifications based on previous knowledge or on counterexamples that refute a statement. However, students’ justifications mainly emerge from teacher’s questioning, as students rarely justify their generalizations and statements spontaneously. 3. The results suggest that with percent, students’ reasoning is based on numerical relationships. A common strategy is the change of representation from percent to fraction or decimal. When the computation involves benchmarks (50% or 75%) students prefer halving and when it involves computing of 20% or 25%, they prefer dividing by 5 and 4 respectively. Students’ reasoning in mental computation with percent emphasizes properties of proportional relationships, part-whole relationships, and ratios. 4. Findings suggest that conventional arithmetic tasks are not appropriate to develop flexibility and indicate that students tend not to think about the quantitative relations of the problems. They also do not envisage alternative ways of thinking nor use the support that different representations and ‘friendly’ numbers can give to them.

References

Bogdan, R., & Biklen, S. K. (1982). Qualitative research for education: An introduction to theory and methods. Boston: Allyn & Bacon. Cobb, P., Confrey, J., diSessa, A., Lehere, R., & Schauble, L. (2003). Design experiments in education research. Educational Researcher, 32(1), 9–13. Empson, S., Levi, L., & Carpenter, T. (2010). The algebraic nature of fraction: Developing relational thinking in elementary school. In J. Cai & E. Knuth (Eds.), Early algebraization: A global dialogue from multiple perspectives (pp. 409-428). Heidelberg: Springer. Gravemeijer, K. P. E. (2004). Local instructional theories as means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6(2), 105-128. Gray, E. M., & Tall, D. O. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115-141. Johnson-Laird, P. N. (1990). Mental models. Cambridge, UK: Cambridge University Press. (trabalho originalmente publicado em 1983) Lannin, J., Ellis, A. B., & Elliot, R. (2011). Developing essential understanding of mathematical reasoning: Pre-K-Grade 8. Reston, VA: NCTM. Lesh, R., Kelly, A. E., & Yoon, C. (2008). Multitier design experiments in mathematics, science and technology education. In A. E. Kelly, R. Lesh, and J. Baek (Eds.), Handbook of design research in education: Innovations in science, technology, mathematics and engineering. New York: Routledge. Mata-Pereira, J., & Ponte, J. P. (2012). Raciocínio matemático em conjuntos numéricos: Uma investigação no 3.º ciclo. Quadrante, 21(2), 81-110. Parker, M., & Leinhardt, G. (1995). Percent: A privileged proportion. Review of Educational Research, 65(4), 421-481. Threlfall, J. (2009). Strategies and flexibility in mental calculation. ZDM Mathematics Education, 41(5), 541-555. Bakó, M. (2002). Why we need to teach logic and how can we teach it? International Journal for Mathematics Teaching and Learning, (October 17th). (Retrieved from: http://www.cimt.plymouth.ac.uk/journal/bakom.pdf). Cox, R. & Brna, P. (1995). Supporting the Use of External Representations in Problem Solving: The Need for Flexible Learning Environments. Journal of Artificial Intelligence in Education, 6(2-3), 239-302. English, L. (1998): Children’s Reasoning in Solving Relational Problems of Deduction. Thinking & Reasoning, 4(3), 249-281. Epp, S. (2003). The Role of Logic in Teaching Proof. American Mathematical Monthly, 110, 886-899. Hoyles, C. & Küchemann, D. (2002). Students’ understandings of logical implication. Educational Studies in Mathematics, 51, 193-223. Newstead, S. E., Bradon, P., Handley S. J., Dennis, I., & Evans, J. (2006): Predicting the difficulty of complex logical reasoning problems. Thinking & Reasoning, 12(1), 62-90.

Author Information

João-Pedro Da Ponte (submitting)

Instituto de Educação da Universidade de Lisboa, Portugal

Susana Carreira (presenting)

University of Algarve

Faculty of Sciences and Technology

Albufeira

Joana Mata-Pereira (presenting)

Instituto de Educação da Universidade de Lisboa

Lisboa

Renata Carvalho Carrapiço (presenting)

Instituto de Educação da Universidade de Lisboa

Torres Vedras

Joana Brocardo (presenting)

Escola Superior de Educação Politécnico Setúbal

Barreiro

Pauline Vos

University of Agder

Faculty of Engineering and Science

Kristiansand

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