Sunday, January 24, 2016

My Favorite (Teaching): Improvisation

Sign in data. Most of the variety is size.
This is one of those things that's both a strength and a weakness of my teaching. I have a lot of ideas about things to try, but that is not especially professional. When people talk about the profession of teaching, we always seem to compare to doctors. You don't want your doctor making up treatments on the fly. Research! That's on what doctors should base treatment. Maybe it's okay because the stakes are less high - one topic in a math class vs. your health and well being, or because in teaching we are the researchers, too. Or maybe medical doctors are not the right comparison.

Regardless, I like to improvise! This is the story of two of those moments, in the same class period.

The characters, preservice elementary math teachers; the content, learning quadrilaterals with a focus on reasoning with properties; the setting, they've gone from describing quadrilaterals to thinking about their properties. Day 1 was spent describing quadrilaterals on geoboards to make, and thinking about a variety of different possibilities that still fit a type. They took home geoboard dot paper to make their own quadrilaterals, one of 11 types. (For us: square, rectangle, rhombus, parallelogram, isosceles trapezoid, right trapezoid, trapezoid, kite, chevron, convex, concave.) In general my teaching here is guided by the Van Hiele levels, in particular activities that give students reasons to transition from visual to analysis, from analysis to informal reasoning and then informal to formal, depending on the level. This is K-5 focused, so we don't push at formal reasoning too hardly.

I have an old set of quadrilateral cards that has a lot of visual confusion. Looks like a rhombus but is a parallelogram, etc. They've been good for me in the past, but the longer I teach, the more I want the students involved in the manufacture of math materials. So this time, they made the cards for homework. I made a couple of extra sets in case someone hadn't had a chance.

The first activity was the same as I've done in the past: Quadrilateral Concentration. Players shuffled up their cards (I had them put initials or a symbol on their own so they could get them back, but style would have been enough for most), and dealt them out into a grid. Turn over two cards and if there's a match - two quadrilaterals of the same type - you can pick them up. Most people knew the game already. The conversations were amazing. First question, "I turned over a square and a rectangle, can I pick them up?" The table ruled no, and I supported. The best arguments are over type, though. Either while playing, "hold on, I don't think that is a rhombus," or at the end... "Why don't these have matches?"

My fair quadrilateral
To summarize, I brought up to the document camera the ones that provoked the most discussion. We also used them to talk about variety again - as there were MANY congruent examples. And we discussed the most ambiguous case, which is probably now my favorite geoboard quadrilateral. People thought it was a kite, people thought it was a right trapezoid, people argued about the length of sides, people argued about the size of angles, people compared it to a square... loverly. It was especially nice because people kept cycling back to earlier claims, which seems to prove what I was suggesting about the power of our visual processing.

Another game I've played with the old cards is quadrilateral Go Fish.  We played with the same rules as concentration, using the most specific names possible. Suddenly it occurred to me that we could play concentration as we had, but switch the Go Fish rules to allow for more mathematical subtlety and strategy. Not everyone had played Go Fish, but enough had to make the rules explaining go smoothly in each group. To play a match, they had to be the most specific type. But when your opponent asked you for a type, you could give any that fit the characteristics. BAM! This was great, and almost instantly a better game than the original Go Fish. There was the start of some strategizing, where some people weren't asking for exactly what they wanted, and the conversations were spot on. This is a trapezoid, are you sure you don't have anything that fits, etc.

That's an improvisation that paid off. Better than what I had before.

The other idea on the spot was to go farther into combining properties. I wanted to make it natural to think about what if a quadrilateral was a this and a that. So spur of the moment, sidebar into a weird movie and TV discussion. I asked, what was an adjective that described a show or movie that you liked to watch. Then I shared how my wife's favorite genre was funny + scary. "Like Krampus?" (Side discussion on Krampus, which we recommend. But only one person had seen it, so...) I wrote down the 'equation' funny+scary=Ghostbusters. (Best example is probably Buffy, though.) Then they discussed at their table until each person had one to put on the board. I was worried about = abuse, so I did mention that what we're really doing is finding examples in the intersection.

And one table really got into trying to do adjective arithmetic. We talked about the examples & shows for a bit and then I transitioned to the purpose: what if we combine the quadrilateral types this way? Each table I wanted to come up with one quadrilateral equation. Got some good ones, and I shared about the role of conjecture in mathematics. To their list of four conjectures, I added some questions.

I connected this to the homework, which is to try the very challenging problem of a Venn diagram for all the quadrilateral types. We'll discuss those and the conjectures next class.

Passed it around again, and got much more
variety of property and orientation.

This improvisation was okay. Don't think I did much harm, it was a moment of high engagement, but not necessarily in mathematics. Well, it was mathematics, but not our quadrilateral content.  The disappointing thing is that the conversation about the shows - reasonably analytical - didn't carry over to the conversation about the quadrilaterals.

I'm okay with this, however, because even a bad result is going to happen sometimes. The same activity that is a gas burner with every class that has ever tried it can crash and burn. So the improvisation increases my store of supplies, keeps my interest, and gives me things to think about for student thinking.

Tuesday, January 12, 2016

Similar Triangles

So now I'm going to blog about something that I'm just starting to think about.

For two days, I've had a tab open with a neat Futility Closet post. (So many clever bits of mathematics and reasoning there.) It has this image:

A pleasing fact from David Wells’ Archimedes Mathematics Education Newsletter
I immediately made it up in GeoGebra, but being the start of a new semester, hadn't really thought about it yet. I didn't see what parallel lines had to do with it, nor being equilateral. About to close the tab finally, I shared it on Twitter. Boom!

Matt and John jumped in. And then Henrí... 

I love the cycle of generalization in math! Get rid of this restriction, and that restriction.

Get rid of the 2nd line.
Get rid of the shared vertex.

And then the what ifs. What if we restricted a vertex in the preimage?


Lines and circles to lines and circles... must be complex. But John had already gotten there!

Then Simon found circles another way!

Now I'll go find someone to talk with locally about the complex transformations here. I could do it alone, but I prefer math in dialogue! I think I want to see someone else get excited the math, too.  I also enjoyed this coming up so soon after the post on Willingham's 4C's of story. Great illustration of the causality and complications inherent in an interesting mathematics situation.

It's available in GeoGebra if you want to play, too.

Sunday, January 10, 2016

Story Teaching

Quick post thinking about Dan Willingham's post on The Privileged Status of Story, which I got to via Dan Meyer's post Study: Implicit Instruction Rated More Interesting Than Explicit Instruction

What constitutes a story?
"The first C is Causality. Events in stories are related because one event causes or initiates another. For example, "The King died and then the Queen died" presents two events chronologically, but "The King died and the Queen died of grief" links the events with causal information. The second C is Conflict. In every story, a central character has a goal and obstacles that prevent the goal from being met. "Scarlett O'Hara loved Ashley Wilkes, so she married him" has causality, but it's not much of story (and would make a five-minute movie). A story moves forward as the character takes action to remove the obstacle. In Gone With the Wind, the first obstacle Scarlett faces is that Ashley doesn't love her. The third C is Complications. If a story were just a series of episodes in which the character hammers away at her goal, it would be dull. Rather, the character's efforts to remove the obstacle typically create complications—new problems that she must try to solve. When Scarlett learns that Ashley doesn't love her, she tries to make him jealous by agreeing to marry Charles Hamilton, an action that, indeed, poses new complications for her. The fourth C is Character."

At the end, Willingham challenges us to incorporate these C's into lessons. In particular, the most important C, Conflict. "Teachers might consider using 10 or 15 minutes of class time to generate interest in a problem (i.e., conflict), the solution of which is the material to be learned."

I think this is compatible with several MTBoS approaches, in particular & obviouly, 3-Act lessons.

Character - my biggest question after my first read was who are the characters? Not in a heavy handed Life of Fred way, but in the story. I think it must be teacher and students for us. We resolve the conflict, after all.  Probably one of the inherent advantages of inquiry teaching is making the students the central characters. Not that we teachers can't be involved - I think we have to be ready to jump in, too. But we can't be Deus Ex Machina everytime, and let the students know there's always an out.

Math lessons are well set up for storytelling otherwise, I think.

Causality - why does this work is a great basis for an investigation. Add up the digits - if that's divisible by three the original number was, too. What? How could that work? Look - these three centers of a triangle are always on the same line. Why on earth...? Of course, if we make it out that knowing the fact is more important, we're killing the story. This is historically a great spark for mathematical developments as well. While I was writing this Sam Shah posted this image which got my mind wandering, making me go off and do some GeoGebra.

Conflict - I have no idea if this is unusual, but I try to get good math arguments going every chance I get. I usually refuse to be the authority. ("Is this right?" What do you think? "I think so..." Well, let's ask the class!) Plus anytime I ask for an answer, I always ask if there are any other answers. And when the students propose answers, there's a chance for a math argument. It also makes me think of Chris Luzniak and his Math Debates.  Even whether a particular topic is math can be a great argument. There's a course I start with Sudoku, and the last question is, were we doing mathematics? I have never had a class agree on this answer.

Complications - is there anything more mathematical than this? Oh, that worked. What if we added this? Could we do it still if we didn't know that? Messing around with conditions is prime mathematical behavior. And if the problem is problematic enough, this happens by itself. I could solve it if I knew that, now how do I find that out? Or you're trying all the cases and get to one where the freak out lives. Or you're practicing the very mathematical habit of mind of trying to find counter-examples to your own idea.

Where I think these C's might be helpful to me is in being more intentional about the type of math the students are working on, and using this structure to help design how I'm going to try to get my lead characters to find the problem.

For my first math for elementary teachers tomorrow, I want to create the conflict for my students between what they know about elementary mathematics and what they need to know. I loved Graham Fletcher's progression of multiplication, so I'm going to try to use that in contrast with their native ideas about teaching multiplication. (Also such a nice synthesis of understanding to model for them.) In the past I've mirrored mathematics development in children and schools, starting with number concept and building up. This will essentially be going in reverse, but will hopefully be a more obvious need to know that will motivate the deconstruction on supposedly simpler topics to follow. Wish me luck!

Monday, December 21, 2015

Book Club Fall 15

Here's my senior mathematics students' discussion about the great selection of books they read. No more than four are allowed per book, and they can choose from (or add to) this list. This page is our in class discussion. The links on the students' names go to their reviews.

How to Bake π, Eugenia Cheng: Sarah Park, Nick Karavas, Rob Wilson, Kate Vandenberg
We all liked it, with mixed feelings about the end. Good book for everybody, especially the first half. Games and stuff to do that really increases interest, and you can apply to every class we’ve had to take. Really gave examples that made sense; like about logic, “cookies don’t obey logic.” Made me laugh out loud. She talks about baking while drunk, having her heart broken and then having mathematical thoughts about these kind of things.

Journey through Genius, by William Dunham: Lindsay Czap, Kevin Forster, Adam Keefer, Joe Young
A great read, follows history. All the guys he covered contributed. Newton could focus so hard he wouldn’t sleep for 4 days. Leibniz & Bernoulli sent him a problem they worked on for a year, but he solved in 12 hours. Everyone seemed curmodgeonly, but Euler was well rounded and produced the most. Author does a good job relating unsolved problems, too. Newton exemplified perseverance. A lot of the chapters have historical background and fairly dense proofs, but you can skim the proofs if you want. The last two chapters were Advanced Calculus, basically. But if you don’t like proofs, we might not recommend it. The historical background is accessible and useful to teachers, though.
(This is the default text when other people teach this course, for good reason. Almost always enjoyed and appreciated.)

Euler: The Master of Us All
, William Dunham: Brennan Kulfan, Brandon Piotrzkowski
The title might be true. He seemed like a cool guy the Journey through Genius people said, but this book focuses on the breadth of his mathematics. Actual proofs, includes the still open questions. It was a little boring the way it was organized; every chapter was before Euler, Euler, after Euler. But I really appreciated his crazy methods, and his reliance on logarithms and infinite series.

How Not to Be Wrong, Jordan Ellenberg: Holli McAlpine, Natalie Van Dorn, Lauren Noyes, Schmitty
Mathematics can give power to our common sense. The book falls apart from there. The book tries to reach back to that, seemingly not knowing who the audience is. Overexplain simple ideas, use super-complex statistics. Big data has power, and there were cool examples. It was a good read, and parts were captivating, but it will lose you at some point. I liked it a lot more; the examples were really good. Like the connection between reading entrails and advanced statistics. It did include applications to teaching. The second half is the in depth math parts, which could get confusing. But even then you can get the big point of the chapters. There was no single thesis to tie it all together. There were a lot of statistics, and thinking about how they are misused logically. Important, but doesn’t tie together.

Love and Math, Edward Frenkel: Kali Orenstein, Brian Hurner, Khadijah Shaaf
We all enjoyed it. It’s the story of his life, and how he went from physics to math, and the connection between the two. Goes back and forth between his life and mathematics. About halfway through the book the math gets really deep; what are lie algebras? Unified theory of mathematics: number theory and curves over finite fields, etc. using symmetry.  He is definitely writing for a general audience, so he simplifies everything he can. If you don’t understand, skip it, and I’ll explain it later. The problem of his Jewish last name in Russia is fascinating and troubling. Amazing to think about that going on so recently. Starts at 17, invited to Harvard at 21 even before his bachelor’s.

The Math Book, Clifford Pickover: Brooke Ramsey, Dakota Doster
Very short one page summaries, great images, great overview of all of math history. Some of them I wished went more in depth or I had time to dig in more deeply. But anyone high school and up with any interest in math or history could benefit from it.

The Number Mysteries, Marcus du Sautoy: Josie Whitsel, Katie Tizedes
Primes, shapes, uncrackable code, the future… I liked how it jumped around. Seemed random, but always fresh. What has not been answered at all is the end. A few were deep enough you might not understand. Like the rock, paper, scissors section with real life lizards that live that way or the world champion RPS player. Strongly recommend it. Primes, for example, we talk about a lot, and we know so much about it, how can we not know these basic things about them; then the David Beckham . Every chapter has something like that.

Visions of Infinity, Ian Stewart: Anthony Pecoraro, This book looks at some of the hardest problems mathematicians have faced and why solving these problems have been so important. Along with a glimpse into what the future has in store for mathematicians. Pretty dense book with a lot of abstract math. Some of the greatest math was discovered along the way to proving something else. The connections were cool. The people who were wrong section was interesting. Pretty hard book to read. Explained sigma summation, but assumes knowledge about elliptical curves.

Joy of X, Steven Strogatz:
Paige Melick, Joey Montney, Abby Fatum. Guided tour from 1 to infinity; it really does go from numbers, to algebra, uses lots of examples. Do not have to be a math person at all. This could convince you of why you learn math even if you don’t like it. The notes at the back give ideas for deeper math content and proof. Recommended for teachers, negative times negative. Not as much depth because it was an easy read; reading stuff we already know. Original examples. Could read some of it to a fifth grader… doesn’t really teach you math. Mentions the topic and why it’s important.

The Calculus of Friendship, Steven Strogatz: Molly Carter
Tuesdays with Morrie, with a lot of proofs in it. My takeaway was more about the relationship than the proofs. Since the proofs were in correspondence, it was not as precise as it could be. All these years of letters were about math, not about the personal stuff, but the relationship was the important part.

The only assignment beyond the discussion is a one page-ish review and a chance to see their annotations or notes. What follows this is a book swap, supplemented with a few of mine, and I ask them to skim the second book. I love how we get references from different people's books as we progress through the history of math. Next week is Euler week, for example, so there will be lots of connections.

Saturday, October 17, 2015

Angle of Coincidence

Quick idea for a math game on angles, hopefully I get to try it this week.

Materials: deck of angle vocabulary cards, blank paper, ruler, pens, protractor.

Set up: (make if necessary and) shuffle angle vocabulary cards.

Draw phase: teams take turns
  • add a point, and 
  • connect to one, two or three other points from your new point.
  • each team adds right angle mark or congruent length if that's their intent
  • both teams make 5 points.
An example:
Play phase: on your team's turn
  • roll a die (that's this turn's points)
  • flip a card. Claim an angle or a set of angles that fit the condition. You can only claim unused angles.
  • score that many points for each angle you claimed that fits the condition.
  • check: if you can't find one or find one mistakenly, the other team can catch you for 2 points per angle.
  • game is to 20 points, run out of cards, or all angles are claimed.
 Example: red scored an acute, a right and a pair of vertical angles. Green scored a pair of congruent angles and a set of supplementary angles.

Design reflection:
Could use a context, but the only thing that comes to mind is shooting metaphors. Maybe bird watching? You know how the kids love bird watching!

Foxtrot, of course, has angle games covered.
They even get triggy with it.
Lots of nice bits here, I hope. Constructing the board, using notation, eventually even making the cards. Some classic interaction (catch the opponents out in a mistake), but could be more. The thing I like the best is how the game will change in between playing. What angles were you unable to find, what combinations can you make, etc.

Possible starting card set:
  • an acute angle
  • a right angle
  • an obtuse angle
  • a pair of congruent angles
  • a set of congruent angles that are not right angles
  • a pair of complementary angles
  • a pair of adjacent angles
  • a pair of cute adjacent angles
  • a pair of acute obtuse angles
  • a pair of vertical angles 
  • a set of angles that add to 180 degrees
  • a set of angles that 
  • a pair of corresponding angles
  • a set of interior angles
  • a pair of congruent exterior angles
  • a pair of angles that add to 180 degrees
  • a straight angle
What else would you add? I'd want a set of cards a playing field to start, then introduce the making aspects when the students know how to play. Warning: only roughing out playtesting so far.

What do you think?

Sunday, October 4, 2015

Six Sides to Every Story

I finally got a chance to teach the Hexagons.

Christopher Danielson has a distaste for boring old quadrilaterals, so he came up with teaching hexagons. (Blog posts: one, two and three.) Dylan Kane wrote briefly about his experiences with them at NCTM:NOLA, and Bredeen Pickford-Murray has a three post series on them.

For me, the idea of type in geometry is pretty complicated. You need to have a large variety of the thing that will have types, you need experience with that variety, you need to be doing something with them - often describing them. (Boy, you're needy.) Types are the way we sort, which means attention to properties and attributes. Definitions come after types, as the more we work with something the more we need precision. Making a good definition is a great task for forcing you to realize some of your assumptions about the objects. 

With a group of preservice elementary teachers, quadrilaterals offer a lot of these opportunities.   Mostly they are at the visual geometry level, and debating whether a square is a rectangle is a good discussion. (Most have been told that, but don't really believe it.) This class I'm teaching is all math majors who have had a proof writing course and many of whom have had a college geometry course. Their quadrilateral training is not complete, but they are strong in the fours.


For me this made the conditions right for trying hexagons. I was up front with the idea that part of this was to put them in their learner's shoes, about trying to make sense of this material seeing it for the first time. So we started with variety and description. 

The first day we played guess my shape on GeoBoards. One person describes and another or a group tries to make what they're describing. For more challenge, the describer can't see the guesser's geoboard either.  I limited us to hexagons. I also tried to bar instructions; you weren't trying to tell them how to make it, but rather describe it so well they could make it.  

After playing in their groups, we tried one person describing while all of us tried to make it. When she was done, several people presented their guess. That gave us a sense of the holes in her description. But on the reveal, there was appreciation for what she was getting at, and several good suggestions about more description that would have helped. Then we looked for what made descriptions helpful.

With that experience, we brainstormed some possible types. The homework was to draw two examples of each possible type, and come up with reasons that is should or shouldn't be a type. Also they read Rubinstein and Crain's old MT article (93!) about teaching the quadrilateral hierarchy. 

The next day I gave them some time to make classes of hexagons within their table, with the idea that we'd come together to decide on class types. We went totally democratic: each group would pitch a type to the class, and then we voted yea or nay. The one type we started with was a regular hexagon. Mostly I was quiet in the discussions, and I abstained from the votes unless I felt strongly about it. Once we had a pretty good list of types, we did class suggestions for names. I shared how historically names are often either for properties or what they look like. The table that proposed the shape had veto authority over the name. But Channing Tatum almost became a type. Several people reflected that this was a pretty powerful and engaging experience.

Homework was to make a set of 7 hexagons. At least one is exactly three types, three are at least two types, two are exactly one type and one of your choice.

The third day we opened with one of my favorite activities: Circle the Polygons.

We do rounds of finding out how many polygons people or groups have circled, and then they can ask about one of them. Then they recallibrate. If students have no experience with polygons (2nd or 3rd grade) I will start with some examples and non-examples. Once we're agreed with what a mathematician would say, I asked them to define 'polygon.' Closed and straight edges are pretty quick, and someone usually thinks about 2-dimensional. (Although I usually wonder if they have a counter example for that in mind.) The last quality is a struggle. After we have a couple student descriptions, I share that mathematicians also had trouble expressing non-intersecting, no overlap, no shared sides and just made a new word: simple. Then I asked a spur of the moment question about what did they see as the purpose of definitions and, wow, did they have great ideas.

So we worked on definitions of the hexagon types, sorted them, and looked at each others' sets. Super nice variety. People brought up a few that were tough to type, and they caused nice discussions on which properties were possible together and some good informal arguments on connections. The note to the side was a discussion of Dan Meyer's "If this is the aspirin, what's the headache?" question for definitions. We agreed that most lessons give the aspirin well in advance of any headache they might prevent.

To sum up the experience, I asked them what they noticed about what they had done, especially in connection with the Common Core standards for shape, and what advice they would give to teachers who were asked to teach these things.

To finish up, we did one drawing Venn problem together. Come up with labels, and draw a shape in each region that fits or give a reason why you can't.

So thanks to Christopher and this great group of students for a great three days of mathematical work.

P.S. Student reaction: several of these preservice teachers chose to write about the hexagon experience for their second blogpost. If you want to hear about it from their perspective...
  • Jenny thinks "It is time for teachers to get away from the cute activities that are fun for the students and get into the real meat of teaching these shapes." 
  • Chelsea is thinking about adapting it to quadrilaterals. 
  • Heather wrote about the inquiry aspect.

Monday, September 7, 2015

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
  • 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.

  • 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.

  • 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.

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

  • 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

  • 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...
  • 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

  • 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.

  • 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)
  • 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

  • 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.