Week 3 of a Great Semester

I am still trying to squeeze out time to work on the book (Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers), so I seldom have the time to blog these days. But I just now had so much fun in my Pre-Calculus class, I have to write about it.

I've been noticing that I'm enjoying all four of my classes this semester. I usually have a favorite, and I am often struggling with a few disengaged students in at least one class. Somehow that hasn't materialized this semester. (One high school student, R, was being goofy the first day, and I called him on it in a puzzled sort of way. Turns out he is a great math student. Yay!)


Pre-Calculus
In pre-calc, the first unit I do comes from the parts of the review chapter that I thought were worth focusing on: Lines, Circles, and Inequalities. (I have them look in the first six sections of the text for problems that would get them stuck, and we work a bit on those, but I don't lecture on all those details.) Although my three topics seem unrelated, I find small ways in which they connect.

We are starting on inequalities, and I was explaining interval notation: "For x ≥ 4, we write [4, ∞). The bracket means we include the 4, and the parenthesis means we do not include infinity, which we never do, because infinity is not a number." R felt that infinity is a number, and explained why, using the phrase 'infinity principle'. I'm not sure what he meant by that, but it actually helped another student think about infinity as "a principle of numbers, not a number itself". I lent R The Cat in Numberland after class.

Yesterday I had used part of Kate's lesson to help them see absolute value as distance. Today I described | x - 5 | < 3 as meaning the distance between our number and 5 is less than 3. A student asked if that was the same as |x| - 5 < 3. I said "Great question. What do you think?" The class was divided. I asked if anyone could give a reason for why it might be the same or different. Someone said that absolute value is a grouping symbol. I agreed and asked how that made the two inequalities different. No one had an answer to that. S then said that the first one is always positive (on the left side), and the second one can be negative (if x=2, for example). I told her after class that that's called a counterexample, and is used often when we're trying to prove something isn't true.

At that point I said I was falling in love with this class, and called them mathematicians. I'm sure they think I'm a bit nuts, but hopefully "in a good way".

Getting back to the task at hand, we picked numbers on the number line for which our first inequality was true, and I got them to tell me I could make it solid (coloring in all the points between 2 and 8). After we did this very concrete process, I showed them the algebraic way to "solve it". I told them we read the solution, 2 < x < 8 as 2 is less than x, which is less than 8.

We then looked at | x - 5 | ≥ 3, and I got them to tell me points that worked first, and then walked them through the steps to get the solution of x ≤ 2 or x ≥ 8. Someone asked if it was ok to write 2 ≥ x ≥ 8. I replied that this says 2 ≥ 8, so it doesn't work.

I won't know until the next quiz how much of this is really making sense to them. It seems great right now, but I am often terribly disappointed once test time comes. The downfall of a good lecture is that it looks and sounds better than it really is.


Linear Algebra
Today I was 'covering' linear independence. The book gives a definition, and I wanted the students to see a need for the definition before I put it up. We had previously seen an example of a vector that was a linear combination of two other vectors. I used a similar example - two of the vectors would make a plane, while the third vector (a linear combination of the first two) would contribute nothing new. So we call this a linearly dependent set of vectors. (And, by our textbook's definition, this happens when there is at least one non-zero c_i in the equation c₁a₁+c₂a₂+...+c_na_n = 0.) Naturally, linear independence is defined to be the opposite situation. If c₁a₁+c₂a₂+...+c_na_n = 0 has only the trivial solution (all the c's = 0), then the set {a₁, a₂, ... a_n } is linearly independent.

That may not sound exciting, but I love how the various concepts in linear algebra all weave together. I couldn't stop myself from mentioning dimension today, even though the book doesn't get to that until the next chapter.


Calculus
Last night, we figured out a few derivatives (which I like to also call the slope function, to help the students keep their eyes on the meaning) using the definition. I keep asking and they keep telling me - it's just change in y over change in x, but I'll only know whether or not they really see that after the first test.

During the first two weeks, they were very confused. It's beginning to come together for them, I think.


I feel very lucky to be teaching students who are willing to play around with math. I also am seeing how my work with math circles, my work on the book, and my blogging have all contributed to my enthusiasm and my steadily increasing skills, even after 25 years of teaching.