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.

Tuesday, September 1, 2015

A Sorted Beginning

First day of Geometry and Measurement for K-8 Teachers today.  I did some improvising that turned out well, and wanted to think about that a bit. So, quick blogpost.

We were starting with Piece of Me, an activity I've stolen or modified from David Coffey. (I called it Piece of Mind today. I have a pun problem.)  The idea is that instead of the instructor droning on, not looking at you, students find out what they're interested in. One modification I do sometimes is to have them start in their groups. Develop two questions for your tablemates, then ask the person on your right. When they were done, I asked for the questions. I often write down student responses on the whiteboard, just from the principle that it helps them feel listened to. If I'm doing it, I write down them all. Just on impulse, I decided to sort them as they came in.
Now what? I said that I had sorted them. Each group should come up one more question for each list. A few minutes to discuss, then everyone stands up. After you give another question you can sit down. I don't call on people, just first to speak. The only rule was that we had to have one for each column before another one in a used column. When one was suggested, I just asked the class "Agree or disagree?" and we put it where the majority agreed. Sign one of a good semester: no one asked me if they were right! Actually that's sign two. Sign one was that they started on questions without a single person asking me what the columns were. Not that there's anything wrong with that, but the willingness to just give it a go is great.

Then they picked one question to all answer at their table. After all this (about 25 minutes) I asked: were we doing math when we did this? Some yes and no, but when a yes argued that we were noticng, sorting and analyzing by characteristics, plus looking for patterns she crushed the opposition.  I made a point that I want class to be free for people to speak, even if they are the only ones with an opinion.

Finally we did the teacher piece. Lots of why am I teaching, why I am a prof questions, with too long of stories from me. Questions about the course were about working with students, how are they being assessed, what does homework look like.

The next activity is one of my favorites for attributes, and I have used this with all ages.

Game: In or Out?

Set up: draw a circle-ish shape, or lay down a rope, or divide the room in half... two regions is what we're looking for. Players standing around a circle works best in my experience.

One player comes up with a rule that can be determined to be true or false for each player. True, they're in, false they're out. Starting with the player to the rule maker's left, they guess if they're in or out. If they are correct, they can try to guess the rule.

If you need winners, coming up with a rule that no one guesses is a win or a point.

I started with are you wearing sandals. 5 or 6 and they got it. We had rules about shorts, shirts, hair color. One fellow who's rule was at least as tall as me when he was the tallest in the class. I had an every other rule, that went about 15 deep. One great rule was whether you were standing in the shade or not. The rule maker was just at the edge, so we had 15 no's, then yes's until someone got it. We talked about the math we were doing, and I was able to connect their comments to the importance of non-examples. I also talked about the activity being accessible to many different learners and free for different kinds of participation.

When we came back inside, they talked about these questions in their groups:
1)    What were some of the rules used? 
2)    Was there a rule that was easy to guess?  Why?
3)    Was there a rule that was difficult to guess?  Why?
4)    What is a rule that would divide our class into two groups of roughly the same size?
5)    What are two rules that would divide our class into 4 groups of roughly the same size?
6)    Why do two rules divide a population into 4 groups?  Give an example.
Extension: Into how many groups would 5 rules divide a population? N Rules?

To reflect we tried for number 6 as a class. They came up with three ways to visualize in the classroom:
  • hand up, stand up. One question you stand for a yes, the other you raise your hand for a yes. (New to me!) It was neat to be able to look at an individual and interpret, but not a good display to get a sense of the group.
  • end, middle, end, double no's elsewhere. Worked okay, but not as well as ...
  • four corners for four groups. Once we tried it, they divided by a Cartesian scheme, with one direction the first question and the othe question the perpendicular direction. This they liked, and appreciated how there was a dividing line for each question.
We tried:
  • sibling & more than 1 sibling: It turned out we all had siblings, and someone noticed that even if someone didn't, they wouldn't be divided on the other question.
  • wearing shorts & wearing denim: four groups, but it's hot so not many non-shorts.
  • curly hair & shoulder length or longer hair: not many short and curly. This prompted the question - do they have to be linked? No? Well, then...
  • pet & eat a good breakfast: yes fish count as pets. Not many non pets, but good division on breakfast. I mean bad because people, it is the most important meal of the day.
  • TV in the bedroom & eat a good breakfast: computer in the bedroom doesn't count, even if you watch TV on it. (Hmmm.) This was almost perfect, 6,6,5,5.
They jotted down their take aways, then shared in group. Good stuff about nature of the activity, how much of the problem solving they did,  how engaging the sorting questions being about themselves were. I shared how a teacher had students write down sorting ideas, which she could then screen for sensitivity.

I am looking forward to a semester with these people!

Sunday, August 23, 2015

Math Circle: MacMahon Squares

When I got an invitation from Judy Wheeler to come lead a math circle activity I jumped at it. I've never been, but have wanted to for so long. Sue Van Hattum's influence, no doubt, with the great math circle stories on her blog and Playing With Math.

Judy asked for my topic during #tiles week in the #MathPhoto15 challenge. That had provoked an in depth discussion on tiling vs tessellation, and whether aperiodic tilings were tessellations, and then somebody mentioned Wang Tiles. Wang Tiles? Oh are those cool. Digging around about those led to finding a very fun series of blogposts from Steve Natusiak on MacMahon tiles. (Here’s the first. One cool sequence. The whole schmegegge.) These tiles were introduced early in the 20th century by Percy MacMahon and are just a lovely construction. MacMahon's idea was squares with colored edges, that you could tile if the matching edges colors matched.

I read up on math circles protocols, and headed for Kalamazoo. (After some printing troubles, which had the Xerox actually spewing curled up sheets of paper into the air like in a sitcom.) I prepared a Google doc, mostly for follow up resources, and some blank tiles (pdf). Judy said they would have scissors and markers.

In the break before my session, LuAnn Murray posed a sticky not problem. How many of the numbers between 1 and 100 can you make with 4 9's  and any operation found on a calculator. (So exponentiation and square roots - despite the implied 2 - are in.)

I showed a couple tiles (P,P,R,G & P,P,G,R), and asked what they noticed. First question: do all three colors need to be present? So I also showed all Green.  The observed the properties and I asked what I was going to ask them. (So meta. But a room full of teachers, so...) Correct: how many tiles? I asked for estimates ahead of time, which ranged from 12 to 1296. They jumped to working right away and then the clarifying questions began. Biggest: if you rotate them and they match are they different? I asked what did they think? Unanimously, they thought those should be the same. What about flipping? Different. "So they're not colored on both sides!" Is no color an option? (No.) Is red, red, green, purple different than red, green, purple, red? (Yes.)

People worked on lists, making diagrams, a few trees and a couple purely combinatorial approaches. A few were actually coloring them out. I let them know that each table would need a set for the next part, which encouraged some more actual coloring.  A couple times I polled the tables for how many they thought, and answers started to converge. When there was agreement but not yet unanimous, I brought them together to share. One teacher jumped up right away: these were all the ones with four red sections, three red sections and two red sections. She didn't do one red, because that would show up in the other tiles. The green, watching out for repeats then purple. Went down by two each time. One person brought up her list with less, and we worked together to figure out what was missing. "It's hard to figure out what's left out!" So how do we do it?

Then I asked: now what? We've got these tiles figured out. What should we do next?

I was really curious to see what kind of problems were posed. Here's what they suggested...

Michael Tanoff takes off when I start...
comes back and he's got the book!
Very cool.
Good extensions! Very representative of usual math teacher extensions. But I wanted play with the tiles we had, so I put on a restriction of using these tiles and the rules we were given. Immediately they posed the rectangle problem - which is what I wanted to get to, and which was MacMahon's original puzzle. I gave them his extra condition, that all the colors match on the outside edge. I want to think more about the kind of extensions we do in math class, because it seems to me we extend to big general ideas versus the kind of closely related problems where mathematicians are more likely to start. As a profession we do more of the 'let's make this harder' extensions than 'here's a parallel problem.' I think.
Now they were playing! 

They had several different approaches to this, as well, but it was much more collaborative in general. Maybe because most tables only had one set of tiles made from the first half. They posed conjectures pretty quickly. They gathered data about how many triangles of each color. They got close and tried small swaps, but also realized that some configurations were a dead end and required starting over.

Good problem.

Nobody was quite done when our time was up. I assured them there was a solution, then stole a couple minutes from working for a reflection and explained why reflection is so important to me. They'd been focused on the SMP, so I asked them to think about SMP1 - especially the perseverance. I asked them to share at table and then just a couple shared with the whole group. They pointed out how I asked more questions than told answers, but encouraged, too. They mentioned how working together helped with perseverance and the problem solving. They appreciated the different methods that people had.

All in all, I was pleased. I think this problem got at the spirit of the math circle, and had plenty of problem solving opportunity. The teachers were great and showed a lot of strong mathematical thinking and practice. And they continued to work on the puzzle while I was leaving.

P.S. Totally an aside & a plug: one of the other benefits of the tiling discussions was that I finally got around to making a sketch for all 17 wallpaper groups in GeoGebra

I'm pretty happy with how it turned out, but am very open to suggestions.