Lincoln, Gettysburg, and Geometry

381px-Abraham_Lincoln_O-77_by_Gardner,_1863In my last blog, I showed a connection between Lincoln’s Cooper Union speech and Euclid’s Elements. There is also a connection between the fundamental postulates in a mathematical geometry and the fundamental postulates in a system of government or political geometry.

All mathematical geometries have a core set of postulates from which all statements or theorems about the relationships between the objects in the geometry are derived. Different sets of postulates result in different sets of geometric theorems. For example: in the geometry of a flat surface, Euclidean geometry, there is a parallel line postulate from which we can deduce that the sum of three angles in any flat surface triangle equals 180 degrees. In the geometry of a sphere, all lines intersect and therefore there is no parallel line postulate. As a result, the sum of the three angles in any spherical triangle is greater than 180 degrees and less than 540 degrees.

All political geometries, likewise, have a core set of postulates from which the laws of the political geometry are derived. The opening lines of the second paragraph of the Declaration of Independence stated the fundamental postulate of the new American democracy, “We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.” This new American political geometry was radically different from previous political geometries.

The meaning of the word “all” in the fundamental postulate of American democracy eventually led to the Civil War. Initially, Lincoln viewed the Civil War as a struggle to preserve the Union. Only later did he see it as a struggle to make the fundamental postulate of American democracy a reality for all Americans. Students of mathematics come to fully appreciate and understand terms like “all,” “each,” and “every” only after they have acquired a sufficient level of mathematical maturity. Lincoln had acquired sufficient political maturity by the time he gave his Gettysburg Address on November 19, 1863.

Lincoln began his Gettysburg Address by reminding his audience of the fundamental postulate of American democracy. The Civil War was about making this postulate a reality for all Americans. He then tells Americans that we, the living, have the responsibility to finish the work of the living and dead Americans who so nobly fought to advance the fundamental postulate of American democracy. The struggle goes on today.

All human enterprises are connected. Whether the enterprise involves art, music, literature, dance, theater, mathematics, physics, chemistry, biology, political science, computer science, history, psychology, religion, sports, etc., there is always some connection to be found. When I find a connection that I have not realized before, I experience one of life’s special moments, a moment of epiphany.

Have you read anything, or experienced anything in your life that prompted you to think about math concepts? What math epiphanies have you experienced?

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Photo credit: “Abraham Lincoln O-77 by Gardner, 1863” by Alexander Gardner – Library of Congress. Taken on November 8, 1863, just 11 days before Lincoln’s address at Gettysburg.



What Links Abraham Lincoln to Euclid’s Elements?

Lincolnx400On February 27, 1860 Abraham Lincoln gave a speech at the Cooper Union Institute in New York City. The vast majority of the audience of approximately 1,300 were members of the Republican party. Lincoln, like many in the audience, was well aware that this speech was his one chance to show the party’s movers and shakers that the prairie lawyer from Illinois was cut from presidential timber. Lincoln’s physical appearance, poorly tailored suit, awkward gait, and frontier twang caused many in the audience to form the initial opinion that he would not be a suitable Republican candidate for President.

That evening, Lincoln’s speech was sensational, evaporating any doubts about his suitability as a Republican candidate for President. The audience was awed and dazzled by his command of historical facts and the airtight logic of his speech. Three months later Lincoln received the Republican nomination for President, and the Cooper Union speech effectively became the Republican Party platform for the 1860 presidential election.

In his book, Lincoln at Cooper Union, Harold Holzer brilliantly captures that magical evening and gives the reader a wonderful description of the events leading up to Cooper Union and the events afterward that made Lincoln President.

After reading and studying Holzer’s book several times, I realized that Lincoln’s Cooper Union speech was, astonishingly, modeled after a classic Euclidean geometric proof. This made sense. Both Lincoln and Thomas Jefferson admired and read Euclid’s Elements as a way to improve the mind and promote an individual’s ability to think and reason logically. For me, a former high school geometry teacher and Civil War buff, this was a thrilling realization, a moment of epiphany.


Lincoln organized his Cooper Union speech into three sections. The objective of the first section was to demonstrate or prove that Senator Steven Douglas’ principle of popular sovereignty was a false doctrine. The principle of popular sovereignty claimed that only the states had the power to decide whether or not to allow slavery into the territory of a new state, and the Federal government had no control over the spread of slavery. Lincoln believed that the Federal government had the right and duty to control the spread of slavery.

He began his Euclidian proof by stating a postulate that all parties in the popular sovereignty debate could agree with:  “Our fathers, when they framed the Government under which we live, understood this question just as well, and even better, than we do now.” “Question” refers, of course, to the spread of slavery, and the postulate was a direct quote from a speech given by Senator Douglas.

The manner in which Lincoln constructs and develops the statements in his proof is brilliant. It’s like watching a creative master teacher presenting a proof of a theorem to a class of geometry students. Each statement in his proof is a logical consequence of previous statements, all supported by facts in the historical record. During the course of his speech, Lincoln referred to the postulate no less than 15 times.

In section three, Lincoln’s goal is to rally the Republican Party around the principles that slavery is wrong and should not be allowed to spread beyond where it already exists. Republicans have a duty to publicly declare these principles and should not compromise them in order to appease Southern slaveholders. To prove his point, Lincoln used proof by contradiction, a sophisticated technique commonly used by mathematicians to prove a theorem.  The proof begins with the assumption that the negation of the statement is true. At Cooper Union, the negation of the statement Lincoln was attempting to prove was, “Slavery is right.” From the assumed statement, one draws logically correct conclusions that eventually lead to a statement that’s false. Therefore the negation of the statement must be false and the original statement must be true.

To fully appreciate and understand the Cooper Union speech, read Holzer’s book. Even if you normally don’t enjoy history books, I think you’ll find Lincoln at Cooper Union a great read. If Lincoln had failed at Cooper Union, he would not have become President, and the course of world history would have taken a far different path.

Have you read anything, or experienced anything in your life that prompted you to think about math concepts? What math epiphanies have your experienced?

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What my professors at Western Illinois University taught me about how to teach

Mathematics backgroundAfter serving in the U.S. Navy, I enrolled at Western Illinois University to get certification to teach high school math. In addition to taking the required education courses, I took a few pure math courses to see if I could still do high level math. The math professors at Western were exceptional math educators, and professors Joseph Stepanowich and James Calhoun were especially influential.

Joe Stepanowich was very friendly, the most down-to-earth person you could meet. On first meeting Joe, you wouldn’t guess he was a legend in math education circles and with former students. Joe taught me some number theory. I can still hear him saying, “9 bundles of x-squared minus 4 bundles of x-squared equals 5 bundles of x-squared. Bundles of x-squared are not the same as bundles of x- cubed.” What a wonderful way to explain to kids how like terms in an expression are combined! I was dumbstruck after Joe showed our class the method of finite differences, which is an algorithm for finding a formula for the nth term of a sequence when the nth term is a polynomial. When I later taught finite differences to my advanced math students in high school, some of them told me they found finite differences to be fun and easy. A fundamental activity of mathematicians and scientists is to find a set of equations that express relationships between two or more variables, the rules of nature. The finite difference algorithm is just one of many pattern or rule finding tools.

James Calhoun taught me the development of the real numbers. Jim was not one of those professors who used proof by intimidation to prove a theorem. If a concept was subtle, he explained the concept from a variety of viewpoints. I remember his discussion of the concepts of equivalence classes and a well-defined operation. Students could easily see that he was deeply committed to helping all his students gain a clear understanding of these important concepts. Jim’s explanation of equivalence classes and well-defined operations served me well in other graduate level math courses. I was so impressed with his teaching style, I tried to adopt it as my own.

Experienced math teachers know that about every ten years a new method of teaching math to kids comes about. The new method is supposed to be the grand elixir. But there is no grand elixir! If there was, we would have discovered it many years ago. Only hard work by students and creative teachers will move math education forward.

More of What Zalman Usiskin Taught Me

equation-transform-pic1In my previous blog, I wrote about how Zalman Usiskin, Director of the University of Chicago School Mathematics Project, showed me a better way to teach slope of a line and linear relationships. Usiskin also demonstrated a better way to teach the equation transformation rules. Over the years, many of my math teachers have given excellent presentations on a variety of math concepts. However, Usiskin’s presentation was sensational. If you have ever attended one of his presentations, you know what I mean.

Many students find the equation transformation rules to be counterintuitive. When we replace every instance of the variable x in an equation with x + 5, most students will guess that the graph is slid 5 units to the right in the positive direction, not 5 units left in the negative direction. When we replace every instance of the variable x in an equation with 2x, most students think that the graph will be stretched horizontally by a factor of 2, not shrunk horizontally by a factor of ½.

Usiskin showed how the solutions of two x-y variable relations compared. The two examples shown below are similar to his. The first shows how we can slide the graph of a circle, and the second shows how we can shrink or stretch the graph of a circle. After he generated a solution of the equation x2 + y2 = 25, Usiskin asked the audience how he generated a solution of the transformed equation. After comparing and checking the solutions of pairs of equations, it quickly became clear why the transformation rules work the way they do. What a cool way to explain the equation transformation rules to my students.

Slide the graph 5 units to the left in the negative direction and up 3 units in the positive direction.

x2 + y2 = 25            (x + 5)2 + (y – 3)2 = 25
(3,4) —————>    (-2, 7)
(-4, 3) ————->    (-9, 6)
(0, -5) ————->    (-5, -2)
(5, 0) ————–>    (0, 3)
(-3, -4) ————>    (-8, -1)

Shrink the graph horizontally by a factor of ½ and stretch the graph vertically by a factor of 3.

x2 + y2 = 25           (2x)2 + (y/3)2 = 25
(3, 4) ————–>    (3/2, 12)
(-4, 3) ————->    (-2, 9)
(0, -5) ————->    (0, -15)
(5, 0) ————–>    (5/2, 0)
(-3, -4) ————>    (-3/2, -12)

Last week I used the above examples to teach the equation transformation rules to my college algebra students. I had a student enter the equations in my Basic Trig Functions program. This allowed me to write on the board and ask/answer questions. While students did not find my presentation sensational, most of them gained a good understanding of how the equation transformation rules work.

You can download my free summary of the Equation Transformation Rules by going to Check out the Basic Trig Functions program by viewing a wide variety of screen shots that demonstrate many features of the program. Equations can be entered as an explicitly defined function of x, an explicitly defined function of y or an implicitly defined relation in the variables x and y. Just enter the equations. The program will figure out the type of equation entered.

What moments of math epiphany have you experienced? What methods have you found effective in teaching math to kids?

Yours in math,
George Johnson

Teaching the Slope of a Line and Linear Relationships: What I Learned from Zalman Usiskin

Blog4PicWhenever I attended conferences where the brilliant math educator, Zalman Usiskin, Director of the University of Chicago School Mathematics Project, was giving a presentation, I always made sure I was near the front of the line to get in. Usiskin had a profound influence on how I viewed the world and approached teaching math.

In one of Usiskin’s presentations, he told about a time when he gave his babysitter a ride home. She explained that she did not understand slope of a line. She just didn’t get it. He asked, “ How much per hour did I pay you?” After a few micro-seconds, she came up with the correct answer. To her surprise, he told her that she had just calculated a slope. In the presentation, as Usiskin went on to explain that all slopes are unit rates, I wondered why I hadn’t thought of that before. The amount that the y-variable increases or decreases when the x-variable increases one unit is the slope of the line. After his talk, I placed much more emphasis on teaching slope as rate, such as cost per part, profit per item sold, weight loss per week, etc.

To help teachers with the concept of a slope of a line and linear relationships, you can download my free Linear Growth and Decay handout by going to The first example in this handout is about predicting gross pay given the number of parts produced in 8 hours. Students seem to have no problem understanding how I derive the equation P(n) = 0.35n + 60. When I ask them why 0.35 in the equation should be no surprise, I’m still amazed that many of them don’t make the connection that the piece work rate stated in the problem was 35 cents per part. But after more coaching, almost all students understand the concept.

Two short examples:

  • The first four digits of the square root of 3 = 1.732. What is special about the year 1732?

Answer: George Washington was born.

  • The first ten digits of the irrational number e = 2.718281828. My son’s significant other remembers e as 2.7 Andrew Jackson Andrew Jackson. I googled Andrew Jackson and found out that he was elected President in 1828 and served from 1829 – 1837. It worked for her.

I’ll have more moments of epiphany for you in my next blog. Feel free to share your stories about how teachers have added to your understanding of math concepts.

All the best,

George Johnson

photo courtesy of Used by permission.

One Math Teacher and My Math Epiphany

chalkboard1An epiphany is a sudden and profound understanding of something. In this blog, I would like to share some of my moments of sudden and deeper understanding of a math concept. A “profound” understanding is probably a stretch.

I was fortunate to have had Vivian Jones as my math teacher at Moline High School in 1959. Vivian could teach math to a post. I will remember some of the things she taught me until the day I die because they made intuitive sense to me.

For example:

• To find the area of a trapezoid, multiply the height by the average of the two bases. Of course, this method works for a rectangle, but it also works for a triangle! A triangle is a trapezoid with the length of one base equal to zero.

• To find the sum of the first n terms of an arithmetic sequence, multiply the average of the first and last term by the number of terms.

• When doing polynomial division, and you are at the subtraction step, change the sign of the term and add. Like all math teachers, Vivian knew that students are much better at adding than subtracting positive and negative numbers.

• To find the area of any polygon when the x-y coordinates of each vertex are known, use the Surveyor’s rule which is a really slick algorithm. Every math team coach should teach the Surveyor’s rule. Shoelace algorithm, shoelace formula and Gauss’ area formula are other names for the Surveyor’s rule procedure.

Vivian could explain concepts by asking simple, penetrating questions that got the point across. For example, she taught me the difference between a rational number and an irrational number by asking 3 simple questions:

• Question 1 – What is the square root of 2? My answer – 1.414
• Question 2 – What kind of number is 1.414? My answer – rational
• Question 3 – What kind of number is the square root of 2? My answer – irrational

Did you have an influential math teacher? What were your moments of math epiphany? Feel free to share your stories about how teachers added to your understanding of math concepts.

Basic Trig Functions, v.3.6, now available!

trig-Graph-1-pdf1As I mentioned in my last blog, I will be offering three main lines of software on my website, Available now is the first program, Basic Trig Functions, version 3.6. As the title indicates, the main emphasis is on teaching trigonometry at the high school and college level. However, teachers of high school algebra, college algebra and pre-calculus will also find many features of the program to be very useful.

Basic Trig Functions has two modes of operation. The first is circle and trig mode. Use it to:

  • explore standard trigonometric angles, radian measure, arc length, sine, cosine, and tangent functions.
  • create a variety of trig-circle diagrams, which teachers can use to create handouts and test questions.
  • find and graph the powers of a complex number in either standard a + bi or polar format.
  • find and graph the roots of a complex number in either standard a + bi or polar format.
  • graph a wide variety of X-Y variable relations. When the user moves the mouse near a point of intersection, min/max point, or x-intercept, the X-Y coordinates of these points can be found with a click of the mouse.
  • graph polar functions. Users can see how polar points are graphed with the click of a mouse.
  • explore the Mandelbrot Set. Users can graph the orbit of a complex number and a mini-graph of the Julia Set of  a + bi with the click of a mouse.

The second mode of Basic Trig Functions allows teachers to easily demonstrate the geometry of the trig functions and special features of the tangent, cotangent, secant and cosecant functions, and investigate special geometric properties of these functions.

I invite you to visit our website to view screen shots that demonstrate the capabilities of the program. Users can copy all program output to the clipboard to be used in the creation of their own materials.

All the best,

George Johnson


New Resource for Math Teachers

George HeadshotWelcome! I’m glad you’ve joined me for this exciting launch of My mathematics teaching career has covered over 40 years. I have taught courses ranging from general mathematics through calculus, and I am currently teaching College Algebra and Elementary Statistics at my local junior college. Over the years, I have developed software programs that have helped me do a better job of teaching algebra, trigonometry, pre-calculus, calculus and statistics. It is my core belief that teachers should help students understand math concepts from both an algebraic and geometric point of view, and these programs are designed to do that.

The tools found on my website fall into two major categories. The first includes three main lines of software. The second category includes free teacher-created handouts. The first offering of handouts are those created by me; my future goal is to add to this inventory of handouts and to invite other math teachers to share their handouts through the website. More information on this will be coming in future blogs.

My software offers many unique features that make it easy for teachers to give dynamic presentations of core concepts in mathematics. Please visit my website to see some of the possibilities of my program, Basic Trig Functions.

Thank you for joining me for this launch. Feel free to contact me with feedback, and I hope you’ll join me for more in the weeks to come.

~George Johnson