Math is...

Our standard (non-thesis) capstone is a course called The Nature of Modern Mathematics. For me, this is a math history course. 

Our essential questions:

• what is math?
• what is its nature? (Is it invented or discovered? Is it completable? Is it beautiful?)
• what are the important ideas of math?
• how do I do math?
• what is the history of math?
• who made/discovered math?
• what are the important milestones?

• what do mathematicians do now?
• who are they?
• what are the big open questions?

I love teaching this course. 

The first assignment is a pre-assessment of sorts, asking them to start blogging with a short post on what math is and what are the milestones they know about.  Given their responses, I think we can see that this is going to be a good semester. What have college majors learned about math? We have about a third future elementary teachers, a third secondary teachers, and a third going on for graduate school or the corporate world. You might be able to see a stong influence of calculus courses, geometry and discrete mathematics. 

The amazing Ben Orlin

This blogpost is in case you would find what they think about math interesting, or if it might start you thinking about what your students think about math. I sorted their responses by my own weird classifications.

Here is the list of all their blogs. If you read just one, try Brandon's.

Math is... 

(patterns)

• patterns
• about trying to find universal patterns that we can apply to infinite situations or problems.
• a way of thinking about patterns throughout the universe. Math is interpreting and studying these patterns to find more patterns.

• about pattern recognition

• the study of patterns in the world and in our minds and how they connect to each other.

(tools)

• a tool
• all the computational things we learn throughout life, but it is also a tool and language humans use to make sense of the world around us.
• a collection of tools that we use to quantify and describe the world around us. We use mathematics very similarly to how we use language. Using language, we can identify objects, convey ideas, and argue. Math can be used in the exact same way when communicating scientific ideas, defining mathematical objects, and proving theorems. The most interesting relationship between language and mathematics is that both can be utilized to describe events and objects that do not exist in the physical universe.

(science)

• logical science
• a framework we use to understand, and like science, it is not reality itself
• the study of everything around us. It is how we quantify structures. It's a science that deals with logic. It is a measurement of the physical space around us. It is so much more then just a simple discipline or school subject.
• a logical way of explaining everything in the world and you can find math everywhere you go
• a quantifiable way to explain physical phenomenon but also includes ways to predict imaginary situations.
• a numeric and logical explanation of the world around us.
• our human desire to give order and regularity to the world.

(language)

• a language
• a language used to study and discuss patterns found in nature.

(system)

• using logical and analytical thinking to derive solutions to the problems we see from all directions
• the use of objects that have been given accepted values and meanings to help us to quantify the world around us.

Things We Forget

(hmmmm…)

• context.  Math gives us a common ground from which to clearly and accurately communicate with the world.  Math transcends language.
• much more than just numbers, it can be used theoretically to answer some of worlds most unexplainable phenomenon. We are in the age of information where researchers and engineers are making breakthroughs everyday using advanced computes powered by mathematical formulas and theories.
• a way of explaining what happens around us in a logical and numerical way, but there is also so much more to math than just numbers and logic.  New discoveries in mathematics are occurring all the time to describe anything and everything about the world, and with these the definition of math is growing as well.  So for me, the best way I could define math is by likening it to an infinite series, how mathy of me.  Just like with the next term in the series, each new discovery broadens the scope of mathematics and as a result the definition becomes that much different than before.
• literally everything

The brilliant as usual
Grant Snider

Name 5 Milestones...

(concepts)

• x 3 Number
• x2 counting
• Egyptian numeration
• zero as a number
• the acceptance of i as a number
• the acceptance of irrationals as numbers
• x2 e
• x2 pi
• x3 Measurement
• Quantifying time and number systems in Egyptian times
• a definite monetary system
• x4 number operations (+, –, x, ÷)
• proportional reasoning
• functions
• The coordinate plane
• x2 the discovery of infinity

(system)

• x2 Proof
• when mathematical concepts could be argued and verified through what we all now recognize as a proof.
• the first math proofs for example the geometry proofs by the Greek mathematicians
• x2 the power of communication
• symbols
• how to communicate what we know to others outside the math world
• The movement into abstraction.

(fields)

• x7 geometry
• x2 pyramids
• x3 non-Euclidean
• x3 algebra
• x2 to predict, plan, and control the environment
• ballistics
• x2 trigonometry
• x5 calculus
• the computer age of statistics

Usually he says "practice"!
(Sydney Harris)

(people)

• Pythagoras and his theorem
• x7 Euclid
• x4 Elements

• way to prove concepts and communicate mathematically

• Al Khwarizmi
• Galileo
• Descartes
• Newton and his Laws
• Leibniz
• Blaise Pascal's invention of the mechanical calculator

(Theorems)

• x4 The Pythagorean theorem
• the realization that the Earth was round and not flat
• x3 Euler’s Identity
• (I swear this is the closest thing the real world has to magic.)
• The Nine Point Circle
• The Seven Bridges of Konigsberg
• Euler’s Method

If you want to answer those questions in the comments, I'd be fascinated. Or if you want to share what you notice about their responses.