Special Topics in Mathematics – Patterns and Algebra
Instructor: Lisa Fontaine-Rainen
Mondays at 12:00 pm Eastern, 10 weeks, starting August 26
Mathematics courses often teach students how to solve problems, use algorithms, and number crunch.
Instead, in Special Topics, we play with how to pose problems, develop algorithms, explore ideas, prove (both formally and informally) their methods and ideas work, and propose next steps. Students can use the skills learned in these classes to stretch their regular math curriculum, challenge their assumptions about mathematics, and truly think like a mathematician.
In Special Topics we focus deeply on problem posing and exploring those ideas generated by students. Homework will focus on exploring (but not necessarily answering) questions they pose. We will focus on learning to explore, and thus also learning how to stick with a challenging problem and make progress even in the face of failure. We will modify the syllabus as we go to follow the paths students suggest. We will take considerable time to reflect on their questions and ideas rather than having our course dictated by the topics listed below. As such, this is a course learners could opt to take more than once – perhaps as a learner beginning to work with variables, then later to explore more advanced algebraic concepts related to patterns. We will weave representing patterns algebraically throughout the course.
This also means that the content in the course will have a large range of ways to respond – students who need to approach things on a simpler level will have plenty of opportunities to engage, as will those who need to go deeper, tackle more challenging questions, and even approach concepts of proof. Students should approach all work with the attitude of “what from this grabs me, what questions can I ask, where do I want to go” and be careful about comparison, given that a large range of ability may potentially be engaging in the material.
There is some overlap between this course and the course previously titled “Special Topics I” – but little enough that anyone who wishes to take this after having taken that should be able to find new avenues to explore.
Day 1: Introduction to mathematical thinking, billiard table investigation.
*The premise for this investigation is taken from Mathematics: A Human Endeavor, though this investigation will go much further than that text.
Day 2: Billiard Table investigation – generating mathematical questions, how to construct an exploration.
Day 3: Sharing billiard table investigations, discussion of further possible investigations, application of algebra. Introduction to using tables to express patterns and find algebraic representations.
Day 4: Patterns in tables
Day 5: Sharing patterns in tables investigations, discussion of further possible investigations, introduction to series.
Day 6: Geometric patterns and the Tower of Hanoi, continued applications of series.
Day 7: Sharing geometric patterns investigations, exploring next steps, posing new questions. Patterns in Pascal’s Triangle.
Day 8: Sharing investigations on patterns in Pascal’s Triangle. Introduction to Fibonnaci sequences, algebraic representations.
Day 9: Sharing Fibonnaci investigations, misleading patterns.
Day 10: Final sharing of investigation projects.
Supplemental materials:
These materials are NOT required for the course, but could provide additional enrichment for those who wish to go further.
Mathematics: A Human Endeavor by Harrold Jacobs (our first investigation is based on the first few sections of chapter 1 of this book).
What’s Next? – a series of books with mathematical patterns. Some patterns in this class are taken from these books.