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.