Schedule: This is a 10-week course that meets three times per week, one hour each session. (30 total hours, or more.) Contact us for other scheduling options.
Format: we will use a combination of lecture, interactive discussion, examples, exercises, Q&A, and in-class work.
Cost: $600 per person for a group class of 4 or more students; $2250 for one-on-one tutoring.
Payment options: Payments can be made via PayPal, Venmo, Zelle, cash, or check.
Materials: Pencil and paper.
For more details or to schedule a class, contact Michael by phone at 281-770-2276 or by email at michaelgold@goldams.com.
Suggested text
Plane Geometry by Welchons and Krickeberger. Used copies on sale on Amazon, but it is out of copyright so free copies are available on the Internet.
1. Plane Geometry on the Internet Archive
2. Teachers’ Key:
i. Teachers’ Key 1
ii. Teacher’s Key 2
3. Answers, Answers to Separate Achievement Tests
Class Description and Rationale
Abraham Lincoln was struggling in his 40s, struggling professionally, personally, and politically. Stephen Douglas was wining out in politics.
So how did Lincoln turn things around? And could you learn from it?
Could you deeply improve your own life and work? Imagine the benefits that would result.
Lincoln was interdisciplinary before “interdisciplinary thinking” was cool.
It’s in the historical record that, of all the things this wise man could have done, the one thing he did was study and memorize the proofs in Euclid’s Elements, the foundation for the geometry we learn in school (the Elements includes what we learn, and lots more).
As Drew McCoy says in “An ‘Old-Fashioned’ Nationalism: Lincoln, Jefferson, and the Classical Tradition:”
“[B]y the late 1830s he was already well-known for the logical precision of his political speeches, in which he characteristically stated propositions and proceeded to prove them, both by adducing documentary or empirical evidence and by deducing them from axioms or self-evident truths. Lincoln then turned even more seriously to Euclid. … As mental exercise, Lincoln’s long hours with Euclid doubtless made him a better “close reasoner,” to use Herndon’s term, and hence a more effective lawyer, which was surely his conscious purpose. But they also helped prepare him in ways he could not have known.”
A mathematician and a lawyer, Dan Van Haften and David Hirsch, have even written a recent book, Abraham Lincoln and the Structure of Reason, about precisely how geometry helped Lincoln be a better thinker, writer, speaker, lawyer, and reasoner — and helped him become President and change our country and our world for the better. He is an inspiration for us all, one were have a lot to learn from.
So, in this course, we will study some geometric proofs and draw lessons from it. We will be heavy into both. Lincoln wanted to understand how to demonstrate something — naturally enough, because he wanted to be an ethical, effective lawyer — so he turned to geometry to learn how to do so, and drew lessons from what he learned that he applied in thinking, writing, speaking, presenting legal cases, and arguing.
You can, too.
Lincoln got much more out of geometry in his 40s than he ever could have pre-18. He had much more mental prowess, life experience, practiced wisdom, and financial and professional need. He had a context beyond what he did at 18, and could make cognitive connections beyond what he could at 18.
You can, too.
Note: throughout this course, I will bring into play quotes, thoughts, and ideas about good thinking and its importance. This will be a deep, powerful, multidisciplinary course.
Class Content
Hour 1
Introduction and beginnings.
What we will do in this course and what we will get out of it.
How it helped Lincoln, Aristotle, and others.
Learn and practice basic terms and principles of geometry.
Discuss some philosophy of mathematics.
Discuss the importance of philosophy — i.e., of a general, big-picture outlook — to thinking about any specific topic or field.
Process and discuss.
Hour 2
Review the basic terms of geometry and what we will get out of this course.
Prove and practice the Vertical Angle Theorem.
Learn parallel lines and angles around a transversal. Practice with them.
Hour 3
Review the VAT and angles around a transversal.
Prove that the angles of a triangle add to 180.
Prove the exterior angle theorem.
Learn basic concepts of circles and angles in circles.
Prove the Inscribed Angle Theorem.
Hour 4
Review the VAT, angles around a transversal, the EAT, and the IAT.
Learn hierarchy (structure) diagrams: a means of connecting ideas back to the evidence of the senses.
Discuss a hierarchy diagram of “all planets orbit the sun in elliptical orbits with the sun at one focus” so we start to get an idea of how the diagrams work and their importance for your work and life.
Discuss a hierarchy diagram of Newton, Kepler, Galileo, the Greeks, the Egyptians, and the Mesopotamians, again, for an example.
Process and discuss what we learned and why it matters.
Hour 5
Review the VAT, angles around a transversal, the EAT, and the IAT.
Prove
i. Angles supplementary to congruent angles (or the same angle) are congruent.
ii. Angles complementary to congruent angles (or the same angle) are congruent.
Learn the triangle congruence theorem.
What proof is.
Why it matters at work and in life. How to use it.
Hour 6
Review past material.
Justify some triangle congruence theorems.
Learn the concepts of axiom, postulate, theorem, and corollary.
Discuss some basics of logic so you can think better, form better sequences of thought and argument, and do better at work and in life.
Hour 7
Practice applying the triangle congruence theorems.
Start proofs involving triangle congruence theorems.
Hour 8
Practice proofs involving triangle congruence theorems.
How to do proofs.
Struggles some have.
Why it matters.
How to apply principles of geometric proofs to thought in general and to business.
Hour 9
Practice proofs involving triangle congruence theorems.
How to do proofs. (And what to learn from them.)
Thinking as asking and answering questions.
Using questions in our own thinking, at work, and in life.
Hour 10
Practice proofs involving triangle congruence theorems.
How to do proofs. (And what to learn from them.)
Using questions in our own thinking, at work, and in life.
The Five Whys: finding root causes.
Process and discuss what we learned and why it matters.
Hour 11
Practice harder proofs involving triangle congruence theorems.
Proofs and sub-proofs, arguments and sub-arguments.
Sequences of thought.
Hour 12
Practice harder proofs involving triangle congruence theorems.
Hour 13
Martin Luther King, Jr., on the essence of education.
Practice harder proofs involving triangle congruence theorems.
Hour 14
Thales, triangles, and ancient Egyptian pyramids: thinking in principles.
Practice harder proofs involving triangle congruence theorems.
Hour 15
How Aristotle generalized from geometric proof to discover logic.
Aristotle vs. Plato, implications of their thought, and some effects in history.
How it applies to your own thinking, to our world, and to working more efficiently and effectively.
Hour 16
Review Aristotle and logic.
What logic is and is not.
Ptolemy vs. Kepler: a case study in bad thinking vs. good.
Galileo and Newton: what they learned from Aristotle, and why they were great thinkers (and hence great scientists).
Some bad scientific and logical ideas to be aware of today.
Lessons learned.
Hour 17
Logic and hierarchy (structure) diagram practice: deriving the area formulas — so continue to get an idea of how the diagrams work and their importance for your work and life.
Lessons learned.
Process and discuss what we learned and why it matters.
Hour 18
Logic and hierarchy diagram practice: deriving the volume formulas — so we continue to get an idea of how the diagrams work and their importance for your work and life.
Lessons learned.
Process and discuss what we learned and why it matters.
Hour 19
Practice harder proofs involving triangle congruence theorems.
Hour 20
Abraham Lincoln, proof, and reasoning.
How do we get the concept of logic? A hierarchy diagram.
Lessons learned.
Process and discuss what we learned and why it matters.
Hour 21
Use triangle congruence theorems to prove properties of quadrilaterals.
Hour 22
Use triangle congruence theorems to prove properties of quadrilaterals.
Practice using properties of quadrilaterals.
Hour 23
Use triangle congruence theorems to prove properties of quadrilaterals.
Practice using properties of quadrilaterals.
Hour 24
The importance of definitions, examples, classification.
Review basic terms of circles.
Use triangle congruence theorems to prove theorems involving circles.
Hour 25
Use triangle congruence theorems to prove theorems involving circles.
Practice using circle theorems.
Hour 26
Use triangle congruence theorems to prove theorems involving circles.
Practice using circle theorems.
Process and discuss what we learned and why it matters.
Hour 27
Use triangle congruence theorems to prove theorems involving circles.
Practice using circle theorems.
Hour 28
Proving the Pythagorean Theorem in a few ways.
Process and discuss what we learned and why it matters.
Hour 29
Putting together evidence, ideas, generalizations, and deductions into a process of reasoning.
Process and discuss what we learned and why it matters.
Hour 30
Review and conclude.
Process and discuss applications to:
-training programs
-teaching your team
-writing better
-thinking better
-judging people
-assessing employees and new hires
-problem-solving
Objectives
1. To use geometric proof to learn how to reason better.
2. To master the process of geometric proof.
3. To understand how Aristotle discovered logic from geometric proof.
4. To understand basic principles of logic.
5. To understand how Abraham Lincoln applied abstract ideas learned in geometric proof to law, speaking, writing, debating, thinking, and reasoning.
6. To analyze arguments and processes of reasoning.
7. To understand the hierarchy (structure) of knowledge.
8. To understand how to make “hierarchy diagrams.”
9. To understand how to formulate better training programs at work.
10. To understand how to better train subordinates.
11. To make better, wiser decisions at work and in life.