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.

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!