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.

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