Discussion Document for the Twelfth ICMI Study The Future of the Teaching and Learning of Algebra

algebra, linear algebra, and number theory. Some are similar to the problems of secondary algebra: students’ difficulties with abstraction, concerns of relevance, what to do with computing technology, etc. Other problems such as proof-making or seeing the objects of calculus as algebraic objects seem particular to the tertiary level. The questions below are concerned with these issues of learning and teaching and also with the specific question of education for prospective teachers. • What are the contributions of tertiary algebra courses to the education of prospective secondary mathematics teachers? How do secondary teachers perceive the value of their tertiary algebra courses to their teaching experience? • Secondary algebra has been well researched, and specific obstacles have been found in making the transition from arithmetic thinking to algebraic thinking. Do tertiary level students similarly experience obstacles in making the transition from secondarylevel algebraic thinking to that required for the tertiary level? • Why are certain types of definitions difficult for students? For example, why are definitions given in terms of properties to be satisfied (for example, subspaces and group automorphisms) so difficult for students? How can this problem be addressed? • There are specific questions about specific aspects of specific courses in algebra; for example, why do students who seem competent in R^n have difficulty with more concrete questions in R^2 and R^3? How can such questions be resolved? • How does symbolic logic (through statements, connectives, quantifiers, qualified statements, and arguments) affect students' proof-making and their view of the value of proof-making? • Secondary school algebra seems to lead more directly to applied mathematical modelling at the tertiary level, rather than to abstract algebra. What is going on here? • Should secondary students learn more about algebraic structure?