My course in discrete mathematics has begun. The syllabus is complete. Adds and drops are in process. I want my students to question what they are learning this quarter, not just become proficient in the mechanics of notation. I want them to read definitions for understanding, to pick apart sentences and paragraphs in the text to see why words and phrases such as ‘exactly’ and “at least one element” are important. I want them to learn to better understand results by paying particular attention to details of methods of formal proof. This is a course where they must not only follow proofs, but must produce them as well. For some of my students this will be the first time they need to consciously examine their own methods of deduction, their own chains of reasoning.
I want the students to question me and the text. In every text there are errors, some simple, some subtle. I encourage my students to look for errors and ask about them in class. I make mistakes at the board–I don’t even have to plan them. I encourage my students to catch my mistakes, and quickly, without worrying that the mistakes they see may merely be misunderstandings. Already, in my first class, I’ve dropped an equal sign from an inequality–a clear mistake in transcription–and presented a conditional that was vacuously true–not a mistake but hard to accept. In both cases students challenged, respectfully, what I was doing. Hopefully my answers affirmed the challengers–at least that was my intention.
I want the students to become a mathematical community. I want them to work together in study groups. I want them to share frustration and discovery. But I also want them to mature mathematically as individuals and that requires that they experience some frustration/discovery on their own–they cannot rely entirely on a group for their breakthrough moments. I will assess their progress using homework, quizzes and tests–the homework allowing for more group support, the quizzes and tests requiring more individual demonstration. But assessment to me is still an art that I am mastering and I hope to gain new insight this quarter into the value of my methods.
The topic this week is propositional logic. We’ll develop an algebra for statements using logical connectors. We’ll look at argument forms. We’ll look at the representation of digital logic circuits as Boolean expressions. We’ll have a first quiz to test prerequisite knowledge. Homework will first be due next week.