In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.
I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).
This student submission again comes from my former student Michael Dixon. His topic, from Geometry: distinguishing between axioms, postulates, theorems, and corollaries.
A2. How can you create a project for your students?
A project that I would have my students do to show that they know what the differences between these four logical terms are to ask them to write a story to model each one. There are several subtleties between these terms that require defining. Axioms and postulates are very similar, both are terms to describe something that is held to be true, and neither require any proof. The general idea is that these are supposed to be “obvious”statement that require no argument. Theorems are ideas that are heavily proven to be true, following the axiomatic method. Corollaries, however, generally follow directly as a result of a theorem, usually requiring only very short proofs.
As an example of what the students could come up with, they could write about two different doctors, who happen to be brothers. The first is a successful general physician in a remote village. He studied for many years to become the man in his village that takes care of all the illness and injuries that the villagers suffer from time to time. He is able to take care of almost anything that requires medicine or general care. But occasionally, the physician decides that a villager needs extra care or surgery that he cannot provide, so he sends them to his brother. His brother is just as successful a doctor, but instead of studying general medicine, this brother focused only on learning how to perform any kind of surgery. When the physician sends a villager to the surgeon, the surgeon figures out what needs to be done and then operates on the villager. Between the two of them, the village hasn’t suffered a death due to sickness or injury in several years.
In this example, the physician would model an axiom, and the surgeon would represent a postulate. Both of them are known by everyone to be excellent in their functions, modeling that they are known to be true. But axioms are held to be true in general, across many categories and sciences. A postulate, however, is known to be true, but is specific to one particular field.
C3. How has this appeared in the news?
If I ask you, “who is the most famous mathematician?”what would you say? Its probably not a question that can safely be answered without causing an argument among mathematicians. But to the layman, the best answer would most likely be Albert Einstein. He is famously known for his General Theory of Relativity. After publishing this work in 1905, Einstein steadily rose to fame, for this work and later for his work on the Manhattan Project and his work in quantum mechanics. And even still today, Einstein’s work still influences the scientific community. Recently it has been reported on PBS that a previously unknown theory that Einstein was working on has surfaced that leads to the idea that he might have supported the idea of a steady-state universe. Pioneered by Fred Hoyle, steady-state theory states that the universe is constantly expanding, but not becoming less dense, hence it remains steady throughout time. Einstein even used equations from general relativity to support his theorem. The article states that Hoyle did not know of Einstein’s support, and though Hoyle’s theorem was mathematically sound, it did not become universally accepted. With Einstein’s support, that result could have turned out differently.
D2. How was this adopted by the mathematical community?
When speaking of the axiomatic method and the history of proofs of this nature, naturally the conversation takes a turn towards the ancient Greeks. Most famously, Euclid developed his geometry using postulates, axioms, theorems, and corollaries. No history would be complete without mentioning these facts. In fact, it was Euclid’s Elements and the parallel postulate that led to a focusing on deductive reasoning and a general application of the axiomatic method in the early 19th century, after the discovery of non-Euclidean geometry. When it is assumed that the negation of parallel postulate is true, an entirely different geometry than we are used to comes into being. Logically it can be reasoned and soundly proven using exactly the same method of logic as Euclidean geometry. This led to a mathematical revolution of sorts, where mathematicians began trying to formalize axiomatically all of mathematics into a system. This led to all kinds of interesting paradoxes, including the incompleteness theorem, among others.