A study of children learning multicolumn addition with microcomputer software support

Three computer-aided tutoring procedures were devised to teach multicolumn addition according to the standard school algorithm, one procedure to each of three groups of 2nd-grade children. The key differences between groups were the demands placed on short term memory and the amount of conceptual understanding the procedures attempted to teach. Each child solved a sequence of two-digit problems on a computer screen by touching each digit with a light pen in the correct sequence. The control group did not receive on-screen number-fact assistance. One treatment ( assisted ) group did receive on-screen number-fact assistance, testing the hypothesis that the algorithm is learned more effectively when learned first as a sequence of procedural steps alone, without subjects\u27 need to recall number-facts. A second treatment ( simulation ) group received the same on-screen assistance along with an additional display of simulated blocks which, like concrete manipulative materials, represented symbol manipulations. The simulation group tested a second hypothesis that a concurrent display of the meaning of procedural steps contributes to even more effective algorithmic learning. T-tests (one-tailed, 5% level) applied pair-wise to pretest/posttest difference scores indicated support for the first hypothesis but not for the second, an indication that 2nd-grade children learn the addition algorithm more effectively if demand on short term memory is temporarily lifted. A descriptive framework called superposition of frames is proposed to account for anomalies in findings and for the rich diversity of errors generally manifested by children in multidigit addition. Drawing on current concepts in cognitive psychology and mathematics education, this description suggests that children\u27s mathematical knowledge is fragmented into isolated, unstable, and sometimes entrenched frames of knowledge. When a child finds appropriate correspondences between frames and initiates a superposition of frames, the child\u27s procedural and conceptual knowledge, previously in disarray, may then become integrated. Implications for elementary mathematics instruction are discussed