The best educators in my life were my music teachers, and I try to adapt their techniques to mathematics. A piano teacher doesn't sit on the bench and show the student how things are done. Rather, the student is on the bench with fingers on the keys. A music teacher knows that doing scales (what we'd call "rote" in other subjects) is important, not for knowledge, but for familiarity with one's tools. A music teacher also knows that the best parts of instruction are the creative bits.
There's nothing that stops us from importing these techniques into math instruction. I try to have my students hold the pencil and I encourage them to do lots of "just plain" problems; these are the "scales" of math. But there are lots of fun or at least interesting, things to do in every mathematical subject at every level. Almost every idea and problem in math is related to almost every other idea and problem in some fashion. Finding those connections is not only the fun part of math, but it also gives real mathematical insight. Coaxing a student toward finding these connections is the best part of teaching and tutoring.
If a student is struggling, it's because there's a concept that is bigger than he can handle at the time. There are two solutions to this problem. One is to break the big concept down into smaller bits, so that the student can understand them. While this is a valid view, and often necessary, I think it is far over-used. The other solution is to make the student stronger, so that he can handle the big concept; and this facet of education is almost entirely neglected. Students do become stronger, but it's only by accident most of the time. In my tutoring and teaching, I deliberately do things to make the students better students. Mostly, I show them that they are not helpless to answer many of their own questions. When they say, "Is this right?" I try to show them how they could have checked the answer themselves. The end result of my efforts is that soon, the student no longer needs a tutor. Maybe that's not a good business plan, but at least it's altruistic.
Another trait of my music teachers (this seems universal among music teachers) is that when I did well, they said, "Good!" And when I did not do so well, they said, "Good! Good! Good!...now let's...." This sort of encouragement is not so universal among math teachers, but maybe it ought to be.
I've been a math professor since 1990. Before that I was a graduate student teaching 6 courses per year. I've taught Intermediate Algebra, College Algebra, Finite Math, Calculus, Vector Calculus, Differential Equations, Linear Algebra, Number Theory (I love number theory,) Discrete Math, Complex Variables, Statistics, Geometry, and a number of graduate courses. I put myself through graduate school by tutoring (and, yeah, painting. Lots and lots of painting.)
I've received my best student evaluations in the remedial and freshman courses. Struggling students and I seem to get along well. On the other hand, I also seem to have a knack for directing undergrad research projects, so I've mentored quite a few budding math whizzes. I like math and I like talking about math at any level.
My hobbies are: I'm learning to play the pipe organ for my church. I'm good enough now that I can play a hymn while the real organist goes up for Communion. I'm a fanatic homebrewer. I started in 1993 "before it was cool." I'm a member of the Austin Saengerrunde, which is a German singing club. I'm not a native Texan, but I've learned to smoke brisket, ribs and pork butt so that you couldn't tell.
I've been married since 1980 and we have two adult children, who have been quite stubborn about providing me with the grandchildren I require. The photo of me is about 10 years old and was taken as a joke. (The joke is a long, boring inside story.) But good pictures of me are hard to come by.
Here is a cute paper I wrote with a rising star named Jeremy Rouse. Most of it is elementary, until the elliptic curves sneak in: https://arxiv.org/pdf/1607.06681.pdf
Here's a nice proof of the Pythagorean Theorem. If you have a right triangle with legs $a$ and $b$ and with hypotenuse $c$, then you can form a square of side $a+b$ like this (actually, you're probably a better artist than I am. This picture is not to scale or even to proportion.)
On each side, where the segments meet we can mark a point. If we join those four points with segments, we form a square of side $c$. Now let's compute the area of the big square in two different ways.
First, since the side is $a+b$, the area of the big square is $(a+b) ^2$ . Second, the big square is cut into $4$ triangles and one smaller square. The area of the smaller square is $c^2$. The area of each triangle is $ab/2$. So the total area is $c ^2 + 4(ab/2)$.
This gives us the equation:
$(a+b) ^2 = c ^2 + 4(ab/2) $ which we expand to
$a ^2+2ab+b ^2 = c ^2+2ab.$ Cancel the $2ab$ from both sides and we have: $a ^2 + b ^2 = c ^2.$
Pythagoras would be proud.
Stand by for the best explanation of the product rule ever...
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The fee can be less or more, depending mostly on travel. The number of minutes can also vary. Everyone has different needs and goals, and I'm pretty flexible. I operate above board all the time, so you are free to deduct my fees from your taxes (if that's possible.) If you need a 1099, that's not a problem.
North-central Austin, near the intersection of Braker and Dessau. map
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