Desarrollo de conocimientos matemáticos informales a través de resoluciones de problemas aritméticos verbales en primer curso de educación primaria

Abstract

The overall objective of this research is to study the development of informal knowledge on base 10 grouping and place value. This is done through the study of the strategies used by children in solving arithmetic word problems and the analysis of representations of discrete quantities used in their procedures, and the description of the evolution of strategies and representations along one academic year. Fifty-four students have participated in the research. They were studying first grade of primary education in a public school in the northwest of Madrid (Spain). We have designed an arithmetic problem-solving workshop composed of 25 sessions, one per week, developed over a school year. The workshop posed problems of multiplicative structure, of equal groups, with 10 groups, of multiplication and division; other problems of equal groups, without groups of ten; and additive structure problems with two digit numbers. The problems were based on stories previously read in the classroom. We offered to the students various manipulatives (structured and unstructured), without instruction on its use, among which they could choose freely. In the workshops, there was a phase of individual work followed by sharing strategies, and writing a letter explaining the problem solving process. Data collection has been done through individual interviews videotaped in the classroom. We have also taken notes on record sheets and photographs of the resolution process, while students used manipulatives. Finally, we have collected the students' worksheets and their written letters. For the analysis of strategies, we start with a categorization from previous studies. Direct modeling strategies have been analyzed according to the representation of the quantities and the mode to carry out counting. This circumstance coupled with the freedom given in the selection and use of materials, has led to the detection of great diversity of modalities for implementing strategies, not described in previous studies. Some of them are transition strategies from direct modeling to counting strategies and other strategies involving the use of number facts, promoted by the use of rekenrek and hundred chart. It also shows, with greater detail than previous studies, the development of direct modeling strategies, since the lack of representation of numbers in groups of 10, to the representation of the amounts separated in tens and units, using non structured manipulatives as cartons of ten eggs and ten bars constructed by children with interlocking cubes. This has allowed the description of the evolution from informal direct modeling strategies to formal strategies, as well as developing an understanding of tens, for which we describe transitions between levels of understanding identified in previous studies. After categorization of strategies, in a second analysis, we describe strategies as successions of capacities and represent them, in a diagram, as possible learning paths to solve a problem. When analyzing representations of discrete quantities, we consider each quantity as composed by a number and an object type, and we identify each component as iconic, symbolic, or a mixture of both, which results in a classification scheme. Representations have been classified depending on the phase in which they are produced: the process of solving the problem, the moment of annotation of the solution, or the stage of writing the letter in which the process and the answer are communicated. Children have most often used iconic representations to solve problems. The frequency of symbolic representations has been increasing throughout the workshop, because of the use of hundred charts and the application of algorithms. At the stage of communication of the written solution, and in the elaboration of the letter, they use more formal representations, as writing the number and the name of the object in figures or words. Among the conclusions of the investigation, we noted that first grade children applied preferably, throughout the entire course, direct modeling strategies that reflect an informal knowledge. This has occurred in a learning situation governed by rules that allowed free choice of strategies and manipulatives, and despite formal mathematical knowledge, which students were learning in their daily math classes. As an implication for teaching, we propose to include tasks in the classroom that promote the use of informal knowledge, by providing experiences that enable children building ideas on mathematical concepts, prior to their formal teaching. For example, problems of multiplicative structure may be included in first grade of primary education. This supposes a change of teaching approach, in which we think about problem solving as a way of construction of mathematical content, to overcome an applicationist approach. The use of manipulatives also must undergo reflection, paying more attention to thinking that children develop using manipulatives of their own choice, that to the structure of the manipulatives. Students’ performance in the workshop has evidenced characteristics of learning with understanding, as the connection between informal and formal strategies, children's knowledge of the applicability of the algorithms of addition and subtraction, or the use of different strategies for the same problem. The introduction, in primary education, of a teaching approach as described in the workshop, inspired in cognitively guided instruction, can promote learning with understanding. As implication for theory, I propose to use the teaching-learning trajectories and the learning paths for a task, as complementary tools for curriculum design and classroom planning. Strategies can subdivided into sequences of capacities, which constitutes the highest level of concretion of student learning expectations and, from a broader perspective, this analysis identifies the necessary capacities to move from a level of understanding to the following. These tools serve also to articulate informal and formal knowledge, establishing connections between them.