Transition from Secondary to Tertiary Mathematics: Culture Shock – Mathematical Symbols, Language, and Reasoning

Our education is marked by large-magnitude discontinuities called transitions, during which significant changes—which require more than just academic reconstruction—occur over a relatively short period of time. The passage from high school to university, i.e., the secondary-to-tertiary transition, is the subject of this chapter. A key ingredient in helping students transition successfully lies in the two-way communication between high school teachers and university instructors. The case studies we discuss illustrate what the topics of these conversations could be. For instance, Ontario high school mathematics curriculum expectations do not adequately address mathematics language and logical reasoning. However, university mathematics instructors assume that their students have experience in working with definitions, universal and existential quantifiers, in constructing simple implications, or providing counterexamples. Surprisingly, standard university textbooks that review high school material do not even have hints or guidelines about understanding mathematics language and “mathematics culture,” nor do they provide examples illustrating rules of logical deduction. In another case study we investigate difficulties that students face as they navigate trough a myriad of mathematical symbols, and work with their changing, content-dependent meanings. Case studies presented in this chapter could be included into high school teachers’ horizon knowledge. An ability to see and understand how mathematical ideas and reasoning develop over a longer time scale can inform teaching, and thus better prepare students for their transition to tertiary mathematics. For exactly the same reasons, these case studies should find their way into university teaching.