Understanding Inquiry, an Inquiry into Understanding: a conception of Inquiry Based Learning in Mathematics

IBL (Inquiry Based Learning) is a group of educational approaches centered on the student and aiming at developing higher-level thinking, as well as an adequate set of Knowledge, Skills, and Attitudes (KSA). IBL is at the center of recent educational research and practice, and is expanding quickly outside of schools: in this research we propose such forms of instruction as Guided Self-Study, Guided Problem Solving, Inquiry Based Homeschooling, IB e-learning, and particularly a mixed (Inquiry-Expository) form of lecturing, named IBLecturing. The research comprises a thorough review of previous research in IBL; it clarifies what is and what is not Inquiry Based Learning, and the distinctions between its various forms: Inquiry Learning, Discovery Learning, Case Study, Problem Based Learning, Project Based Learning, Experiential Learning, etc. There is a continuum between Pure Inquiry and Pure Expository approaches, and the extreme forms are very infrequently encountered. A new cognitive taxonomy adapted to the needs of higher-level thinking and its promotion in the study of mathematics is also presented. This research comprises an illustration of the modeling by an expert (teacher, trainer, etc.) of the heuristics and of the cognitive and metacognitive strategies employed by mathematicians for solving problems and building proofs. A challenging problem has been administered to a group of gifted students from secondary school, in order to get more information about the possibility of implementing Guided Problem Solving. Various opportunities for further research are indicated, for example applying the recent advances of cognitive psychology on the role of Working Memory (WM) in higher-level thinking