Week 4

This was the week when I looked out over the class and realized I had lost them—they were not responding to lesson prompts, their body language spoke of anywhere-else-but-here.

It’s the fourth week of the quarter and for this course it may be the longest week of the quarter. We’re covering a chapter on sequences and proof by induction and the start-up material is vocabulary intensive, with multiple, short topics that need to be stitched together into theorems that are then proved by mathematical induction. Not everyone’s idea of fun.

Mathematical induction, by the way, is a deductive, not inductive, method of proof (in the natural science meaning of induction and deduction).   The induction in mathematical induction refers to the inductive step in the proof method which shows that the assumption of the truth of, P(k), k an integer, induces, or leads to, the truth of P(k+1); P(n) is a predicate that falls true or false based on the integer n.

This week I was pleased to have students, in groups of two and three, ask several questions on difficult homework problems. This was good for several reasons: some students are clearly working on homework before the weekend, and, more importantly, small communities are being formed where students are discovering, together, both answers and dead-ends to mathematical questions. A little not-so-good was my discovery in last week’s homework that several students had the same, exact, almost-correct answer, to one problem—it seemed to be a transcription from a common earlier source.  Apparently teaching now requires some skill in exegesis and hermeneutics. I wrote on each homework paper, the same, exact, almost-correct comment, to which I have been awaiting questions on interpretation.

One embarrassing moment was my forgetting a key insight needed to clarify one of the homework problems (a proof); it’s embarrassing since I solved all the homework problems before the quarter started. After looking at my notes at home, the insight returned, and it really wasn’t too difficult. It may or may not be true for everyone, but I find that I have a problem-solving state of mind that can only to be entered in quiet isolation. In that state of mind solutions to problems develop and outside that state of mind the solutions fade away. I’ve had the same experience with developing software. An algorithm can be constructed in a state of quiet concentration, and luckily it is written down and tested since it has to be used, but, as time goes by, the essence of the reasoning behind the algorithm fades and returning to it at a later date, for example to answer a colleague’s question about it, also creates embarrassing moments.

Next week we continue with induction, strong induction and the well-ordering principle.  And then we move to recursion.  Recursion is challenging and useful and can encode very terse, yet information-dense, definitions, theorems and software.  I’m working on some small-group activities to help introduce recursion.  Next week will tell if they work to make the week shorter than this one.


The third week

This week we began our study of mathematical proof.  We covered several major topics,

  • Proof by example (proving an existential statement true).
  • Proof by counterexample (proving a universal statement false).
  • Proof by exhaustion (proving an existential statement false or a universal statement true).
  • Proof by generalizing from a generic particular (instantiating a generic, arbitrarily chosen element of a set and deducing that it must have a certain property using rules of logic, definitions, and accepted rules of mathematics).
  • Direct proof of a universal conditional statement (in the simplest case, instantiating a generic, arbitrarily chosen, element of the indicated universal set; assuming the hypothesis of the conditional statement is true for the instantiated element; using rules of logic, definitions, and accepted rules of mathematics to deduce the truth of the conclusion).
  • Proof by contradiction (negating the statement to be proved and deducing a contradiction in the enclosing system of mathematics; rejecting the contradiction, not the system, and therefore rejecting the negated statement, and asserting the original statement)
  • Proof by contraposition (instead of directly proving a universal conditional statement is true, directly prove that its equivalent contrapositive is true).

To provide material for practice we established several mathematical definitions and theorems,

  • definitions of even integers, odd integers, prime integers, composite integers, rational numbers, divisibility, standard factored form, div, mod, absolute value, floor and ceiling
  • theorems of factoring integers into primes, properties of even and odd numbers, divisibility, absolute value, floor and ceiling

I presented examples of the above types of proof using the mathematical definitions and theorems that were developed, as well as rules of logic, rules of algebra, etc.  I used a two-column format for the proofs: the left column listing formal mathematical statements; the right column giving reasons that the mathematical statements were validly deduced from what came before.

All of this was done in five hours of lecture.  The students (at least some) followed the lectures closely since they caught my mistakes and challenged the reasoning behind some of the deductions.  I prepared notes for the lectures last weekend (four hours) and reviewed and updated the notes before each class (five hours).  The notes are handwritten so I don’t publish them for the students–I think it is better for students to see the information presented live, record what they think is important, and read the text for the ‘written’ view of the same material.

There was a quiz at the start of the week on the material from week 2.  The scores were again in a bimodal distribution, with most of the students doing very well.  The homework scores were high for those who did the homework.  I spent approximately six hours correcting quizzes and homework (I am teaching 61 students in two sections).

It was important to establish an agreed level of detail for the proofs.  As students learn to question their knowledge of what is mathematically true and false, and why they believe it, they can begin to doubt that they understand much at all about mathematics.  That’s an important awakening, but it can lead to excessive denial of conventions and excessive need for detail.  I offered my examples of proofs in the lectures as templates for the level of detail that they should strive for in the class–a little more detail than they will need in the future as their understanding matures.  From past experience, some will think it too much work that belabors the obvious but others will see the value of exposing more layers of the chains of reasoning that bind our mathematical foundations.

Week Two

For all students in my class there exists a chance to learn.  True or false?  This week the topic was universal and existential statements.  How to prove them true; how to prove them false.  How to negate them.  How common they are in advanced mathematics.  I also returned two assessments: one quiz and one homework–the results were bimodal: students doing well or not well at all–this bimodal distribution of assessment seems to have replaced the normal curve–if it ever existed in graded assessment.

In informal assessment I found the normal distribution, or I imagined it.  From responses to my questions during lectures a small number of students appeared to be fully engaged.  From posture and distraction another small number of students seemed bored.  The  majority of students seemed to be listening and watching and taking notes–I don’t know if they were transcribing or interpreting–it might be nice to have an observer in the room.

In preparing the lesson I personally gained a deeper understanding of the universal quantifier as a short-hand for a conjunction of multiple predicates and the existential quantifier as a short-hand for a disjunction of multiple predicates.  It may be possible to do more with this understanding in my next round of teaching discrete.

Universal instantiation and existential instantiation–the foundational ideas of general rules and particular instances.  These concepts seemed excessively formal when I reviewed them, but as I saw them on the board they gave the impression of great stones of support for the mathematics we use every day–solid and necessary, but best left submerged from view.

Several students dropped my class this week–the drop-without-grade date is Sunday.  They sent no messages to say why.  Almost all of them took the first two quizzes and submitted homework.  Most did well enough.  Perhaps the amount of work required may have discouraged them–I collect and grade homework weekly; I quiz weekly.  Did some of them dislike my method of teaching?  I had to force myself to write that last sentence.

Some students came to office hours to have me solve their harder homework problems. There is a tension in teaching between scaffolding and challenge.  Some faculty see too few failures as low standards.  Some faculty see too many failures as improper support.  My current thought is that a teaching moment is an experiment that has no control.  All lessons learned about teaching are anecdotal.  If there are universal rules can they ever really be true for all particulars?

The first week

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.