What is math?

My capstone class started today, Math 495: the Nature of Mathematics. We take a historical look at math, and the first assignment is to blog shortly on what is math and 5 milestones that they know
already.

Since, for some reason, I am trying to blog everyday this month, I thought I'd join them.

Mathematics to me is noun and verb.

Noun: it's hard to do better than Eugenia Cheng's description, "the logical study of logical things." Quantities, their properties, operations on them, their properties, treat the operations as the new object and repeat. Pretty soon you have cohomological sheaves and categories dancing with your sugar plum fairies. "Mathematics is beautiful."

Verb: reasoning about, playing with, representing or describing mathematical objects or posing or solving the problems and questions that you have about them. "Hey, watch out, I'm doing mathematics over here."

Students first responses:

Milestones:

1. Invention of number systems to record quantities, especially place value systems (Babylonians, Mayans, Indians)
2. First application problems connected with measurement. (Egyptian and Babylonian stand out to me.)
3. Mathematization: the study of the mathematics itself. Greeks, of course. Brahmagupta, Liu Hui, also.
4. The Persian synthesis of algebra and geometry. Al Khwarizmi in the House! (House of Wisdom that is.)
5. Descartes and the invention of analytic geometry. 

Very hard to pick five. Not having Gauss or Euler is painful. But I'm big on synthesis. My favorite modern mathematics is the Atiyah-Singer Index Theorem.

p.s. If you're interested in the course, here's the Google doc of what we did the last time around.
p.p.s. Found this movie poster on this page. Haven't ever heard of this movie, but I'll be looking for it!
p.p.p.s Erin found the movie as released, which lost the cool x+y title.

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.

What is Math?

As a preassessment for my capstone course in mathematics, I asked these soon to graduate math majors what is math, and what are the big developments in math. This goes along with both my belief in preassessment, and wanting to ask questions about what I really want to know. Here are some of their responses. (See all their blogs in a urlist.)

Roz Chast
http://rozchast.com

What is math?

• Lots of "math is more than numbers" or "it is not about computation"
• Relationships
• Sara: "math is patterns."
• Jennifer: "Math is more of a way of thinking.  It is logical reasoning.  It is looking at patterns and relationships.  It is problem solving and explaining phenomena that seem unexplainable."
• Kate: "I think math is thinking logically, understanding facts and finding relationships between different mathematical concepts."
• Alex: "Math is the study of relationships that many people take for granted."
• Annette: "math is an explanation or an attempt to explain relationships we find in nature or ones that we create."
• Kristine: "math is how we can apply numerical values to how the world works."
• Basis of science
• Becky: "Math can be used to problem solve, find patterns, make predictions, provide reasoning, and much more which is all in the tool box of math – the resource for information."
• Bryce: "Math is the science of solving problems."
• Josh: "Math is the one science that every other subject has in common."
• Kenton: "The reason that I call math a "science" is because without math, no scientific studies would be able to be quantified, all studies would have to be qualitative and therefore much less precise."
• Logical system
• Kerry: "Mathematics is a fantastically broad, beautifully intricate, complexly connected concept of numbers and symbols."
• Emily: "math is thinking critically about different kinds of systems. These systems can range from the number system in algebra and calculus, to a system of shapes, such as geometry, to a system of rings as we saw in modern algebra."
• Andy: "Mathematics is a method for conveying logical principles. A proven mathematical theorem is a reality about logic; it is some organizing principle inherent in the human mind."
• Language. 
• Duncan: "mathematics is the most beautiful language in all of the universe."
• Final answers:
• Biz: "For me, math has been a source of intrigue, education, and frustration."
• Matt: "To ask what is mathematics is like asking what is life. There is no definitive answer."

Top 5
I also asked what are the biggest moments/discoveries in history of math. (Top 5 or milestones, etc.)

• Famous Stuff
• Pythagorean theorem 9
• Fibonacci Sequence 9
• Pi 4
• Fundamental Theorem of Calculus 3
• Natural logarithms and e 2
• Kepler’s laws of planetary motions
• Any of the Fundamental Theorems 
• Law of Sines
• Quadratic Formula
• I Ching
• Fractals
• ϕ
• Culture
• Numbers 9
• Calculators/Computers 8
• Abacus 3
• Discovery of zero 3
• Understanding of different formulas
• Development of measuring units
• Ancient Geometry
• Math in astronomy.
• Concepts
• Unit circle
• Function
• Pattern
• Combinations (in counting)
• Infinity
• Axioms
• Parallel postulate
• the need for complex numbers
• prime numbers
• Fields
• Calculus 7
• Algebra 4
• Calculus 3
• Non-Euclidean Geometry 3
• Geometry
• Differential Equations
• People
• Euclid 7
• Euler
• Newton
• Archimedes
• Gauss
• Karl Pearson (new to me!)

Not too different than I might expect, although the diversity of responses was pretty interesting here.

One more interesting comment:

• Danielle: "The founders of mathematics struggled and devoted their whole lives to the theorems we now take for granted when studying in our classes. I can’t imagine devoting my whole life to proving what we consider now a simple concept." 

If you have a moment, I would love to hear your responses to these prompts in the comments!

Image: I saw this at Peter Liljedahl's very worthwhile site. Even Google couldn't help me find the original. Doesn't it look like it's from the New Yorker? EDIT: Tweeps nailed this one: Roz Chast, a frequent New Yorker contributor.