Pupils’ ability in proportional reasoning is essential for their mathematical development. This reasoning is fundamental to solve daily problems and also for learning advanced mathematical topics as well as other fields of study, including natural and social sciences (Post, Behr & Lesh, 1988). Students’ difficulties in this aspect of mathematical reasoning are well known (Bowers, Nickerson & Kenehan, 2002; Van Dooren, De Bock, Hessels, Janssens & Verschaffel, 2005). Furthermore, as Lesh, Post and Behr (1988) note, there are many people that solve direct proportion problems without using proportional reasoning. Silvestre and Ponte (2009) consider that the proportional reasoning ability involves three aspects: (i) distinguishing direct proportion relationship from those that are not, (ii) understanding the multiplicative nature of direct proportion relationships, and (iii) solving various types of problems, demonstrating flexibility to use different approaches, without being affected by the numerical data, context, and representation. These aspects aim at operationalizing the notion of proportional reasoning, indicating the different aspects to consider in its development.

This paper analyses grade 6 students’ mathematical processes and difficulties in missing value problems after the formal teaching of this topic, using an exploratory teaching unit.

The teaching unit  (Wittmann, 1998) has an exploratory nature, seeking to engage students in non-routine tasks. Students need to mobilize their intuitive knowledge to solve them. The theory of teaching and learning behind it assumes that students develop their proportional reasoning when they: explore the multiplicative nature of direct proportion relationships, enhancing their knowledge about the covariation of measures and invariance relationship under certain conditions; solve problems involving direct proportion relationships (missing value and comparison problems) and solve pseudoproportional problems; and work with different representations at the same time (tables, graphs, ratios represented as fractions, ratios represented as divisions).

The study is a teaching experiment, a form of design research (Confrey, 2006) which allows to know the influence of the teaching unit described in the previous section on students’ proportional reasoning ability. The teaching unit was developed in two classes in a school on the outskirts of Lisbon. Two teachers of this school worked collaboratively with the author of this paper in planning and reflecting on classes. All teaching unit classes were videotaped. Written records of students in the different tasks were collected and analysed. Classes were given two tests, a diagnostic and a final test. Considering the nature of the study, the analysis of data is essentially descriptive.

The results show the teaching unit based on a teaching-learning conjecture that values an exploratory approach, promotes the use of multiples representations and encourages students´ discussion and reflection on their own activity to developed students’ proportional reasoning ability. Particularly, students transform gradually theirs additive building down/building strategies in to multiplicative ones.  The results show that students tend to use multiplicative scalar and functional strategies in missing value problems The use of one them depends of students´ numbers sense, particularly the facility to recognize multiplicative relationships.

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