But this email sketches out what typically happens in my classes and private tutoring sessions, so I decided to post it. (But in this post I added a few things to the email to illustrate more of what we had actually done in class, instead of reporting only some of the highlights.)

Students:

Today we started class by looking at one reason why we need to learn to graph linear inequalities (which topic we covered last week): so we can graph, evaluate, and criticize the graphs we use and find in statistics. Graphs of inequalities play an important role in statistics. Then we worked an inequality together: p. 434 #22.

Then instead of doing more inequalities, we started working on graphing linear systems, to make sure we’d have time to cover that topic first. We started out with some motivation: knowing how to find points of intersection is a critical part of understanding how some people navigate, or locate a position on the earth’s surface, using LORAN. LORAN works by finding the intersection of two hyperbolas, as we saw in class. We are not yet ready for working with hyperbolas or systems of hyperbolas, of course; we need to work with lines first. We will build up to hyperbolas one step at a time. (In looking at the LORAN example, we were able to introduce some classic properties of hyperbolas, ellipses, and circles, so we had an idea how a hyperbola was generated and how it was different from other conic sections. And we were able to see how LORAN depends on the basic idea D = RT.)

We read some of the introductory material on p. 453 of our book, then worked three examples of solving linear systems by graphing:
1.  the system 3x – y = 5 and y = x + 1, which has a solution of (3, 4);
2.  the system 3x + 4y = 12 and 6x = 18 – 8y, which has no solution;
3.  the system y = (1/3)x – 1 and -9x + 3y = -3, which has a solution of (0, -1) — no work involved since that point is immediately seen to be the y-intercept of both lines once you have both equations in slope-intercept form. (THINK! We don’t have to waste time actually graphing this one if we take into account our background knowledge of algebra and graphing!)

I pointed out how solving linear systems builds on graphing single lines and solving algebraic equations — mathematical skills build on old knowledge and skills, just as in martial arts and dancing. We can do things like this — math, dancing, martial arts — because we are different from the rest of the animals: we are conceptual beings.

Then, to wrap up, we read and discussed p. 456, covering the three comprehensive and mutually exclusive situations we could have with two linear algebraic equations: one point of intersection; no points of intersection; overlapping lines.

We were then able to do more review work: to take time to work another inequality (p. 434 #23), and to work some direct and inverse variations (p. 439 #10 and p. 441 #32, 34, 28). The variations were good, real-life, practical exercises: wage, pressure-volume, pump speed-time, etc.

Everyone understood and was satisfied, so we stopped there.

Assignments for the Week

Day 1:
Read, study and take notes on “Systems of Equations in Two Variables,” pp. 453-456.  Do p. 457 #1-17 odd.
Do “Cumulative Review: Chapters 1-6,” pp. 449 #1-10 all.

Day 2:
Do p. 457 #2-18 even.
Do “Cumulative Review: Chapters 1-6,” pp. 449 #11-20 all.

Day 3:
Reread and study “Systems of Equations in Two Variables,” pp. 453-456.   Do p. 458 #19-33 odd.
Do “Cumulative Review: Chapters 1-6,” pp. 449 #21-30 all.

Day 4:
Reread and study “Systems of Equations in Two Variables,” pp. 453-456. Do p. 458 #20-34 even.
Do “Cumulative Review: Chapters 1-6,” pp. 450 #31-40.
Read, study, and take notes on Section 7.2, “The Substitution Method.”

Day 5:
Do “Cumulative Review: Chapters 1-6,” pp. 450 #41-62.
Read, study, and take notes on Section 7.3, “The Elimination Method.”

Remember: THINK!! REASON!! MAKE CONNECTIONS!!

See you all next Tuesday at 11:30 at the Library!!

Sincerely,

Michael Gold
Phone: 281-770-2276
Website: www.mgtutoring.com
Blog: www.mgtutoring.com/blog/

A parent responded to this email a few hours after I sent it: “I just have to say it again – Ryan REALLY enjoyed this class today. ”

Ryan had said out loud after class (to paraphrase): “I understand all this much better than I had before class.” (He had completed some pre-algebra and algebra before I started working with him.)

I had to send a quick, short letter of thanks to Dr. Campbell for his “The Classroom Without Reason.” (See my post of 5-2-09.)

But I’d make the caveat that whether the irrationalists are of the kind he identifies, or are of the kind we find in the Middle Ages (some of which type are still around today), they are enemies of reason and to be opposed. The Scholastics, for example, were “followers” of Aristotle, but they did not even get the essence of Aristotle: follow reason and the evidence of the senses. A Scholastic was anti-Aristotle and anti-reason to the extent that he/she violated the authority and kingship of reality for the sake of subjugating himself or herself to printed word and authority — and was just as anti-reason as the people Dr. Campbell identified in his article. Irrationalism can come disguised as Christianity just as it can come disguised as Marxism. The Salem witch trials, to take a clear example, were not executed by Marxists.

As Jefferson pointed out, it is every form of tyranny over the mind of man (i.e., reason) that must be opposed.

I just wanted to thank Mr. Campbell as a fellow “fighter” for a rational education and against indoctrinating students or numbing or destroying their minds through years of cognitive neglect or abuse. I work really hard in teaching mathematics to get students to think for themselves, to be independent, to know and live first-hand, to develop the methods of mind that will allow them to achieve their own goals and live their own lives — in short, to train students to reason — so I had to tell Mr. Campbell thanks for being another rational teacher out there (going by only the evidence of his article), and I wanted to let him know there there are more teachers facing the challenges and opposition he is facing while striving to develop the conceptual awareness, the “conceptual eyes,” of students.

Dr. Campbell:

I enjoyed your essay “The Classroom Without Reason.” I have seen the same thing, and fight the same fights: against irrationality and for reason.

Thanks for your comments on Becoming a Critically Reflective Teacher by Stephen D. Brookfield, and Courage to Teach: Exploring the Inner Landscape of a Teacher’s Life by Parker J. Palmer. I saw exactly that type of thing and those types of books when I was getting my teacher’s credentials from the University of Houston in the 90s. And in reading what is going on in modern education, I have learned about more of the same — for example, William Ayers’  “social justice.” In getting my credentials, I kept quiet mostly, so I could get my credentials and get out. I was “threatened” in writing once (a professor saying he thought I should not be a teacher, and if I contested the low grade he gave me, he’d grill me and speak against me) and in getting my B.S in Math and B.A. in Philosophy at UT, Austin, I had a few professors threaten to fail me for speaking against them (‘do not oppose me, or you will be failed’).

Too many teachers I taught with through the years were more collectivist than individualist. (One teacher I taught with told me, a few years after I left the school we had taught at, that he was criticized for wearing a shirt with an American flag on it to work during the Elian Gonzalez fiasco!!) And they bought into many bogus theories of education — well, some people followed those things because their jobs and well-being were on the line.

As you do, I focus on teaching students to reason and think independently. I don’t censor what they say — I just let them know that they damn well better be able to back up their statements with facts and good reasoning. I teach/tutor math, but the way I teach, we of necessity get into some science, writing, logic, philosophy, art, history, and literature. I could not and would not teach if I could not get into diagramming sentences so they understand proofs in geometry and word problems in algebra; if I could not relate learning proofs to learning law (and logic and reasoning) by discussing Lincoln’s study of Euclid’s Elements to become a better reasoner (as presented in Drew McCoy’s “An ‘Old-Fashioned’ Nationalism: Lincoln, Jefferson, and the Classical Tradition”); if I could not show how geometry and algebra are important and useful by discussing painting and sculpture and movies; if I could not show how mastering proofs in geometry helps one master writing; if I could not show how geometry and trigonometry were critical to the development of modern science.

Sincerely,

Michael Gold

Update (8:45 PM): Made “I had to send…” its own paragraph and added the “I just wanted to thank” paragraph.

I kind of like that….Lone Wolf Tutoring – Setting the Gold Standard in Education 

…

Meant in an entirely positive, Paleo way.  Strong in mind and body, individual, not following the pack, wise, self-sufficient, hunter…… you get the idea.  Quite fitting for you, I think.

I like it. There’s a Lone Wolf Productions film company…so why not Lone Wolf Tutoring?

Update (5-7-09, 2:10 PM):  I am only being rhetorical. I like the name when looked at as above, when looked at as a dignified appelation, but I don’t think the name would go over and be practical in today’s context, where “lone wolf” is a pejorative.