The Future of Education

Seven to ten-year children were able to solve arithmetic problems in dailylife contexts by drawing spatial arrays of correspondences and comparisons, without resorting to symbolic algorithms.  These children had previously experienced a learning environment (LE) we designed; there, arithmetic operations are presented as actions organizing quantities of tangibles on a surface table, with the goal of finding a quantity-unknown.  Control-group children who had not such experience, instead, did not show these solutions and even showed difficulties using algorithms. Upon examining the drawings of children who had reached solutions, we consider that such drawings may be representations of those arrays of tangibles they could observe as the outcome of their performing along the experimentations. Key aspect here is the way of proceeding to arrange quantities, both on a table and on a paper sheet. In these procedural ways, we have observed patterns in the location of known quantities as either one-to-one or one-to-several correspondences, as well as in the comparison of quantities as quantity-parts and quantity-whole. These actions arranging quantities of tangibles are in close proximity to actions/events from children’s dailylife surroundings (f. i. one-to-several correspondence setting table for dinner, a dish that breaks and its pieces are glued together…). Thus, in our LE children follow the orderly sequence in those actions that could be familiar to them, but now experimenting with quantity-numbers. As a result, children seem to become (implicitly) aware that if they arrange quantities in such a way, it will always lead them to visualize the quantity-unknown on the arranged frame they just set/drew. Onwards, this successful procedure could then be repeated to achieve such purpose (schema of action). The child’s raison-d’etre, the signification of this way of performing would then be the outcome achieved by it, in the way of a means-goal frame-of-meaning. Therefore, when facing a new problem in dailylife contexts, she will be in condition to anticipate (assimilation, inference) that she can achieve such purpose by performing an already known action (schema). We will show samples of drawings illustrating both experimentations in the LE, as well as reaching solutions.

Keywords: Mathematics education; Active teaching/learning; Learning environments;