PERCENTAGE AS RELATIONAL SCHEME: PERCENTAGE CALCULATIONS LEARNING IN ELEMENTARY SCHOOL A.F. Díaz-Cárdenas, H.A. Díaz-Furlong, A. Díaz-Furlong, M.R. Sankey-García & G. Zago-Portillo Universidad Autónoma de Puebla (MEXICO)

Percentage calculations require the development and comprehension of a variety of mathematical schemes. As we conceptualize knowledge organization as a social construction process the main objective is to make thinking patterns visible to students with the aim of improving achievement through the analysis of mistakes and misconceptions. Mathematical concepts, such as percentage, can be conceptualise as cognitive schemes. Particularly, percentage can be characterise as a relational scheme. We defined a relational schema as one in which the central component outlined what constitutes a relationship between elements integrated in it.

Assessment procedures at school must be guided by a clear definition of what it is you want to measure. Assessment of competencies in mathematics is related to define the cognitive structures, such as schemes, that underlie the ability to recognize meaningful patterns of information. It is important to create evaluation procedures to assess mathematical competencies such as the meaningful use of mathematical language, modelling and problem solving skills

In elementary school it is necessary to encourage the comprehension of terms that explicitly communicate the mathematical thinking that allow the appropriate use of concepts and relationships. We have adopted a microgenetic approach as a means for studying percentage comprehension development. The microgenetic method requires that the observed behavior is analysed and assessed intensively with the goal of inferring the processes that gave rise to both quantitative and qualitative aspects of the cognitive change. In elementary school it is necessary to encourage the comprehension of terms that explicitly communicate the mathematical thinking that allow the appropriate use of concepts and relationships.

Fifty five children (9-10 years old) attending one private elementary school and one public elementary school in Puebla city (Mexico), participated in this study. Two groups of fifth-grade elementary school from each institution participated in our research. One group participated as a control group and this group only participated in the assessment tests.

Measures Every student solved the assessment items both before and after that the percentage learning group have participated in learning sessions concerning percentage calculations. The classroom percentage problems and the related assessment items developed were based on relational thinking and were designed in order to assess the following competencies: Identification and characterization of the essential elements involved in percentage calculations; analysis of the relationships that play a role in those calculations; construction of a mathematical relation between percentage and its fractional and decimal representations.

Data Analysis Items responses were analyzed with an item analysis program designed by our research group. We designed an applet that can be used with Microsoft Excel application. With this tool, the teacher can perform basic item analysis in relation to a parameter of the previous academic performance. We get as a result the item response curve, a difficulty parameter, and a discrimination parameter related to mathematical academic grade point average as an ability parameter. Parameters are calculated according to a logistic function (Díaz-Furlong, Héctor Adrián, et al,).

As can be seen in the Figure the percentage learning group showed a clear improvement related to question before sessions How much is 0.25 × 40? In this case, students with high academic grades develop greater increments.

Similarly; the percentage learning group demonstrated better performance as a group regarding the item: What is 25% of 40? This result can be seen in the Figure.

Finally, the following Figure shows the results to the question: How much is one-fourth of 40? As a group, children answered correctly this item from the first assessment. That is, children know how much is one-forth of a number.

CONCLUSIONS We presented percentage as equivalent to fractions of 100. For instance, 25% is equivalent to a fourth of 100, therefore students could compute 25% of any number as the fourth part of it. The percentage learning group exhibited an improvement related to percentage calculations compare to control group. Essentially, the performance shows improvements related to the mathematical expression of percentage. Students with high academic grades showed greater increments. Percentage calculations require the development and comprehension of a variety of mathematical relational schemes. Essentially, percentage was presented as a relational scheme, i.e., as a relation between two numbers or quantities, not as a simple formula.