My Discovery-Based Math Elective: The Purpose of the Elective (Part 2)

This is part 2, of a 15-part series of posts detailing how I developed and piloted a discovery-based high school math elective. The first, introductory, blog post for this series can be found here [Introductions]. The goal of this post is to describe the purpose of the course, as well as the fundamental curricular and pedagogical design philosophies of the course.

• This is a one year high school math course, surveying five topics selected from pure mathematics. At my school we broke the course into two unrelated semesters, where students could take either/both semesters. We had to give the semesters two different names, neither of which fit the content correctly, but we were working with a constrained district catalog). 
• Number Theory (we called it Discrete Mathematics)
• Modular Arithmetic (8 weeks)
• Alternative Number Bases (5 weeks)
• Discrete Mathematics (we called it Advanced Quantitative Reasoning)
• Combinatorics (Pascal's Triangle, primarily) (4 weeks)
• Sequences (4 weeks)
• Graph Theory (5 weeks)
• Who is this course for?
• The course is designed to be accessible in grades 9-12. This year I had grades 10-12 in the class, some of whom were 10th graders struggling in their other (primary) math class, others of whom were 12th graders crushing AP Calc at the same time. It also was mostly upperclassmen, as they tended to have more space in their schedules for electives. It was one of the options for a 4th year of math, for students two didn't want to do AP math. The course was eligible for math credit fulfilling my school's math graduation requirement.
• With some work, this course could totally be adapted to meet the readiness of middle schoolers, though it would need someone with a deeper understanding of middle schoolers than I have. Similarly, the course could be modified for post-secondary use.
• The objective of the curriculum advanced in this course is to expose students to topics in mathematics that are typically less covered in traditional K-12 coursework. Below is a pretty cool Map of Mathematics that's floating around, which you can see full-size here (and even buy a poster!) Circled in yellow is my understanding of the extent of math covered in traditional high school mathematics. Circled in blue is the math that is surveyed in this course.

• This focus on non-traditional topics has two major impacts:
• Facilitates differentiation for readiness
• By focusing on topics that are outside the traditional K-12 pathway, this course can provide a differentiated experience for an uncommonly wide range of incoming student readiness levels. The basics are Number Theory are accessible at the elementary level. For example, division with remainder is technically an elementary level standard. But by extending it through modular arithmetic, we can experience meaningful and unfamiliar work all the way up to advanced topics like Linear Diophantine Equations and the Chinese Remainder Theorem.
• Any of the five topics covered could definitely be a year-long elective course, on their own. But by including five different topics, I really only need to include the richest and most accessible 40% of each field. And then I only need to reasonably expect a student to access around half of that. But by having so much math available to be studied, it becomes incredibly easy to provide multiple options for students to find relevant math that feels interesting and possible to them.
• Facilitates re-invitation of previously discouraged students
• Traditional high school math coursework has an unnaturally narrow focus of on Algebra, with guest appearances by Statistics and Geometry. As such, we ask students to retread familiar content over and over. And yes, there is some value to that, since there is much math to be learned as we dive deeper and deeper into math we've already begun to understand. It helps us to construct an understanding of how one topic in math can have unfathomable depth, and surprising applications across many different topics.
• But this is a double-edged sword, as it also sends the message that all of math is basically fundamentals in algebra. Many students have negative experiences in some of their math classes, especially in high school. And these experiences often compound, as strict prerequisites force students to repeat classes that they have not succeeded in before. Even if a student does pass a class, they may very well feel like they're just being pushed to do more of the same math, and so have more of the same negative experiences.
• Certainly, the environment in which a student learns has a greater impact on whether or not they have a positive experience in a given math class. All else being equal, changing the topic alone likely won't radically shift a student's perspective on math. But by providing access to math that feels, looks, and is meaningfully different, we can create an opportunity for student to positively reinvent their feelings about math.
• The objective of the pedagogy advanced in this course is to prioritize authentic mathematical practices, with a focus on exploration, discovery, and proof. This focus on practices facilitates the course's ability to differentiate the course for readiness, by providing multiple means of action and expression.
• It is more natural to hold fair, but different expectations for students to engage in a given practice. If I have 9th graders and 12th graders in my class, it's difficult to ask them BOTH to engage in a problems where they have to prove trig identities, for example, because my 9th graders haven't done enough of the prerequisite work. But I also can't ask them both to solve basic linear equations with integer coefficients, because my 12th graders would (hopefully) be mostly bored.

• Instead, say I give students a class the classic Counting Trains problem, posed here by PlayWithYourMath.com. This is a classic "Low Floor, High Ceiling" problem, and a really incredible example at that. Because there is truly meaningful math to be done at levels accessible to both elementary and high school students, and beyond. The interesting math content introduced by the various parts of the problem may be different for students with different levels of readiness. But at all levels, students are asked to engage in mathematical practices (explore, discover, use repeated reasoning, etc.)