To C or not to C

Probably the hardest decision to make as a teacher is when to pass a student that is just on cusp of passing, but is not quite there. I just had such a student this semester, and ultimately decided not to pass them, but it was not an easy decision.

The stakes could be very high, either way. Perhaps I'm overestimating my influence, but I can see not passing them as affecting their future ability to get financial aid, discouraging them from trying again, or sending out a flag to some sponsoring agency to change their sponsorship. I almost never get to know the repercussions of these decisions.

On the other hand, I'm not completely comfortable even with some of the Cs I did hand out in this class. I'm not sure how prepared these students are for the next class. Did I just hand off trouble to the next instructor? Again, I'll probably never know the answer to this, either.

In the end, I think the homework points were a big part of this decision. We use an on-line homework system. Out of 40 possible points, the student only earned 11. Most of the sections were not even opened. I don't know if this is proper logic or not, but I'd have been much more likely to award the C for a similar score if 30 points had been earned.

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Geometry proofs are backwards

While I have not been technically on hiatus, I have been playing a lot of Civilization IV lately, rather than blogging. Yes, I realize the latest release is Civilization V. However, by staying one step behind the curve, I get several benefits. I can buy a complete set at a single time, instead of buying all the expansions separately, and the price is much reduced that way.

Meanwhile, I'm still teaching Elementary Geometry on Saturday mornings. One of the most important aspects of the course is teaching proofs. While there is a subset that will use the geometry itself, there is a larger group that will require training in the type of thinking that these proofs use. Anyone working in the legal field, for example, will need to know how to put together a proof. Yet, the traditional way of writing proofs seems backwards to me.

This is how I typically present on proof on a test:

Notice that the statements are on the left, while the reasons for those statements is on the right. That is backwards to me. Americans read left-to-right, and that puts the statements as something that comes before the reasons when you read a proof. However, the reasons are the connections between that line of the proof and the lines that precede it. Formally, you take the prior information and the reason, put them together using a law of logic or two, and produce the statement.

Of course, the design is traditional. One of my retirement plans is to write a good textbook for teaching Geometry at community colleges (they have good texts for Pre-algebra, Algebra I, and Algebra, but Geometry seems to be confined to high-school texts. In today's world, that would mean creating video segments and an on-line interactive homework system (our college uses MathXL, but they have no Geometry test listed) as well as writing the book with a little less flash and a fewer pictures of kids. When I do, I think I will write the proofs with reasons on the left, and maybe moved up a half-line.

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Jerry Sloan -- a fond farewell

I haven't had time to write the post I planned to be next. I'm teaching Calculus for the first time, and I am putting a huge amount of effort into it, compared to previous classes I have taught. It's one of those situations where I'm not sure I can measure up to what the class needs. Still, if they let me, I'll be teaching it again. Mental growth comes from mental struggles.

However, I heard yesterday that Jerry Sloan resigned. From what I can tell, he just doesn’t have the energy it takes to constantly herd young players into trusting in his system anymore, and that's not really anyone's fault. I first started rooting for the Jazz in the mid 90s, so it's been quite a time. I'll briefly ramble on about it below the fold. Next post, I'll get back to the topic of the difference between deductive and inductive reasoning.

Sloan has always had arguments with players, some much more publically, and just as prominent within the franchise, as Williams. Karl Malone certainly comes to mind, although their disagreements were not as well-documented as those with Ostertag. Sloan seems to have always relied on personal drive, energy, and force of will to keep players playing the system. For better or worse, these things fade as you get older.

Great players tend to have big personalities, and Williams is no exception. You can't expect a man like that, at that age, to just take direction. There will be battles of wills, unless the coach has no authority at all, or just lets the players do whatever they want. Ten years ago, Sloan probably would have been more than up for those battles. In that time, he's been widowed, remarried, and has a knee replaced. Rather than just hang on (ala Don Nelson), Sloan has decided that if he can't be the coach he wants to be, he'd rather not coach at all. I respect that.

I will miss Sloan. I'm going to keep watching, though. I was watching last night, when Kirilenko showed just how good he was by getting injured. I don't know if the Millers will let Corbin coach for 20 years, but it wouldn't surprise me. I'm curious to see if Corbin takes up Karl Malone on his offer to be an assistant coach (I'm sure he'd have a thing or two to show Jefferson). I'm still a Jazz fan, just a sadder one, at least for this season.
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A discussion of asymptotes

One of the objectives in the Intermediate Algebra class is identifying some basic properties of a relation from looking at a graph (domain, range, is it a function). So, on a test I put in a graph that could roughtly correspond to -log (-x). This graph has the y-axis as an asymptote.

While I have done this a few times before, this is the first time I've had members of the class question whether it was possible for a graph to be an asymptote. As one student said, "Doesn't it need to either go straight up or curl back?" No, not exactly.

So, to make it easier to understand, I talk about how y=1/x behaves as x increases. I won't bore with the details. I get to end the discussion with one of my favorites things to tell a student: It makes perfect sense, it's just counter-intuitive.

If you know anything at all about mathematics, you should know that.
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My first hour in a Geometry class

This is basically a longer form of the signature quote at the bottom of the page, originally created by a poster named HRG on CARM. My personal attitude toward mathematics is much closer to some combination of formalism and fictionalism than any other position, and I find it useful to remind students that what we are studying is doesn't have to be a perfect representation of the real world in order to be valuable. Indeed, since at the very least the parallel postulate doesn't hold for reality, we are not studying such a system.

So, I start of talking about three criteria that we would like any sort of knowledge to possess. The differ slightly from the usual description of epistemology.

Reality
We prefer knowledge that reflects the world in which we live, one that has an application to things outside of our imagination. Saying that a ball is red, that an action is evil, that 2 + 2 = 4 would ideally say something about the apple, the action, or the numbers themselves.

Demonstrability
We prefer knowledge that we can show, by some sort of objective process, holds to be true. This allows us to convince skeptical people that the knowledge is valid. Saying that a ball is red, and action is evil, or that 2 + 2 = 4 would mean we had accepted means of demonstrations so that every observer who accepted such means as valid would accept the result.

Certainty
We prefer knowledge that won't change with time, that stands firm. when something is true, it should stay true and not become partly true or even false later on. A red ball should always be red (assuming the ball itself does not change), an evil action should always be evil, and 2 + 2 should always equal 4.

The next part discusses what I see as three different systems that focus on knowledge using the desired attributes above. Of course, any truly broad system of inquiry will make us of all three of the systems I discuss below, but generally only one of them at time. Like a three-way combination of oil and vinegar, they don't seem to mix together. In the examples, I try to offer samples where the primary dependence seems to match the system I use, and the other systems are of secondary significance.

Empirical systems
Empirical systems rely on the investigation of the real word and use inductive processes and assumption to decide what is more generally true about that world. They use demonstrable methods such as taking measurements, running controlled experiments, and making predictions to be verified by further observations and experiments. Because the observations are founded in actual investigations, their reality to the world is a natural consequence. However, the use of the inductive method means that the results can not be certain, that all finding are provisional and possibly erroneous. The best that can be hoped for is beyond a good reason to doubt a finding. Examples of empirical systems would include physics and the results of national polls regarding elections.

Formal systems
Formal systems rely on working from a starting point accepted as being true and using an accepted, objective, deductive means of deriving new truths within that system. The method of deriving new truths within the system is the demonstration of that truth, and the process of deriving the truth would be repeatable regardless of the observer. Also, given a specific starting point, the same truth will always be the result (at least, ideally). However, since the starting point is selected rather than derived, there is no guarantee that such a starting point will have any bearing on the world around us. A formal system may be a useful model of reality, but we can never know whether it matches reality. Examples would include philosophy and law (as practiced at the appellate level).

Belief systems
Belief systems use the trusted, revealed knowledge of an authority of some sort to provide as description of the world and its operation. The truths that we get from this authority are accepted because of the level of trust, not because of any independent test that we run. Belief systems do provide a view of reality and knowledge about the state of affairs in the universe. Also, because the source is some consistent authority, they have the feature of certainty, of not changing within the system itself. However, because our acceptance of this knowledge is based upon trust rather than a means to validate them, these truths are generally not demonstrable to others in a significant or meaningful way. Examples include Shintoism and Objectivism.

Then the question that guarantees a minute of silence and ten more of discussion: which one of these is mathematics?
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Easier means harder, at least this time

I teach math at a local community college, as an adjunct. This past school year that mean 13 credit hours each fall and spring, and now 7 this summer. One of the courses is the Liberal Arts math class. I have taught this quite a few times, and really enjoy it. The other course is the one I am struggling with.

I received a Review of Arithmetic course (the high-school equivalent of pre-algebra). Honestly, I had been avoiding this course, but it was all they had left. I feel completely inadequate to teach it. 15-20 years ago, I really struggled to relate to students at this level, and usually came across as condescending and smug (although I never felt that way). This go-round, I've actually been working my way "down" the ladder (College Algebra, than Intermediate Algebra, then Geometry, and Introductory Algebra last summer).

I have never felt so intimidated about teaching a class, though. This summer will tell me a lot about myself, one way or another. Here's to learning, teaching,and learning by teaching!
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