Category: Discrete Math Project

Discrete Math Project 1.1 – An Introduction to Logic

Section 1.1 covers the basics of logic: Statements, open sentences, truth tables, negation, conjunction, and disjunction. The authors use “Aunt Buosone” in several examples. I’m excited to see what other characters I meet. The math isn’t new to me yet, but the name “Buosone” sure is.

I have never taught logic to this extent. When I taught geometry, we spent quite a bit of time discussing logic and especially reasoning, but we never actually worked with truth tables. In early August each year, I would convince myself that spending time with truth tables and doing a deep dive into logic would be a great learning experience for my students and would pay off in the long run. But each year, I backed off and decided to only spend time on “geometry” logic and not on “discrete math” logic. The payoff for truth tables didn’t seem great enough, and we did a ton of great thinking and reasoning without them. I wonder if many geometry teachers include truth tables in their curriculum. If not, do they show up anywhere in the curriculum? I learned about truth tables in PDM – Precalculus and Discrete Mathematics – using the Chicago series (UCSMP). Given the emphasis on coding, I’m curious where discrete math fits into the curriculum.

Anyway, I like making truth tables. Well, I don’t mind it. I guess it gets tedious after a while, especially once you understand them. I did find the word problems interesting. Here’s an example:

Higher Ed problem

That’s just a good problem, the sort I might have given my geometry students back in the day. Although the problem does not specifically call for a truth table, I made one anyway:

Truth Table

While this made the questions extremely easy to answer, I wonder what value the truth table really has here. I thought about the questions and answered them before making the truth table and used it to confirm my answers. The truth table certainly adds clarity, but in doing so, does it actually remove some of the critical thinking necessary to answer the questions? Or does it seem that way only because I have a fairly strong background in logic?

Curriculum Connections

This chapter included a number of logic puzzles. You know the type – Alice and Bob go to the same school, Bob and Carol both major in history, … Which person studies physics and is under six feet tall? I’ve never done much with this sort of puzzle, but I know students tend to like them. Definitely a good resource to have around to allow for easy differentiation after an activity.

Questions to Ponder

Does constructing truth tables help students develop logical thinking and reasoning skills? Does the process become rote? Is it just another procedure to learn?

How does the knowledge gained through learning about elementary logic help students to learn about mathematical proof? What do we gain by studying logic at this level?

What are the consequences of differences between mathematical language and the vernacular? How can we help students learn how and when to use and, or, and not correctly within a mathematical context? Does the idea of an inclusive or pose a significant barrier to student understanding, and if so, how do we overcome it?

The Discrete Math Project

The Discrete Math Project

I recently bought Discrete Mathematics for Teachers by Ed Wheeler and Jim Brawner. I have a bit of a habit of buying math books. Learning about math – even if it’s unrelated to the curriculum I teach –  makes me a better teacher. While I’ve taken a bunch of education courses over the last few years, it’s been quite some time since I took a math course. I’ve decided to replicate that experience by working through this textbook. I’m excited!

Discrete math is not an unfamiliar subject to me. I learned a fair amount of discrete math in high school, and as part of my math degree, I took a course in discrete math. I’ve specifically taught sequences, combinatorics, and probability, and I’ve also touched on some set theory and logic. I hope to learn some new things and to refresh some knowledge tucked deep away in the furthest recesses of my brain.

My plan is to work through roughly one section each week. I’ll probably do it on Friday or Saturday night. That’s just how my social life works. There are 39 sections, so in theory, this project could take me the better part of the year. As I go along, I’ll certainly explore certain topics in more depth, and I welcome suggestions for avenues that I might pursue. For each section, I hope to put together some sort of blog post – an interesting problem, curriculum connections, etc.

The following quote from the Preface stood out to me:

The goal is to develop teachers who not only know the mathematics they are teaching, but also understand the larger mathematical context in which the mathematics they teach has life.

My content knowledge has enriched my teaching in so many ways. I look forward to continuing to develop it in the coming weeks. I invite you to join me on this journey.