Mathematical Problems That Optimize Learning for Academically Advanced Students in Grades K-6

Abstract

Several decades ago, V. A. Krutetskii conducted a multiyear study to investigate the various types of thinking that academically advanced, or as he called them, gifted mathematicians used. Following an in-depth look at Krutetskii's nine ways of thinking, a model is proposed that will provide direction for teachers in selecting problems. The model is comprised of four levels of mathematical tasks. Level 1 is mathematical exercises, Level 2 is word or story problems, Level 3 is mathematical problems, and Level 4 is authentic mathematical problem-solving tasks. Subsequently, an elaboration of high- and low-level tasks is applied to the four-level model. Consistent with Krutetskii's theory, the suggestion is then made that approximately of the curricula for students of advanced intellect in mathematics should be comprised of Levels 1 and 2 tasks, should be comprised of Level 3 tasks, and should be comprised of Level 4 tasks. Three implications are offered for teachers and four are offered for researchers. The first implication is that teachers must carefully scrutinize their curriculum to see that it meets the needs of all students, including academically advanced students. The second implication is that conceptual (deep) understanding of algorithms can be attained through the use of mathematical problems and authentically challenging tasks. The third implication is that teachers are not likely to have a database of problems that represents all levels if they use only the provided textbook. Researchers and educators should be reminded that additional time and effort is necessary to empirically research the proposed theory. Moreover, authentically challenging tasks, such as Model-Eliciting Activities, should be used with students, and they could be used for assessment.