Wednesday, June 25, 2014

Problem Posing with Pills

My class always welcomes conjectures. This is made explicit on the very first day of the semester. This goes for everything from grade nine to grade twelve. As the grades advance, the topics have us venturing into increasingly abstract concepts, but conjectures are always honoured. 

Certain class structures promote conjecturing more than others. Students offer questions during lectures, but they are often of a surface variety. They notice a pattern that has occurred in three straight examples, or think they have discovered a short-cut. I don't like using tricks, but if they are "discovered" or "re-invented" (to borrow a term from Piaget and genetic epistemology), then we use them. 

Several posts on this blog have been born out of conjectures offered in class. Many task ideas come from shifts in tasks that we were working on. By far the most profitable, conjecture-rich structure has been that of large whiteboards. There is something about the organization mixed with the rich problems and communication obligations that opens student minds. Regardless of class structure for that day, if a conjecture arises, I get everyone's attention:

"I need your attention please. [Insert name here] has a conjecture to make."

After the conjecture is made, we have a conversation around its feasibility and even vote on its validity. Some groups (or individuals) devote themselves to refutation, while others remain in whole-hearted affirmation. It creates an interesting (yet non-competitive) dynamic. Some students even like having their conjectures disproven and get right back to modifying them to make them stand. 

When I feel like I have loosened the curricular pressure, I take my students through explicit problem posing (conjecturing) exercises. We use the process documented explicitly in Brown and Walter's book. I complete this exercise with a task that is solvable in 5-10 minutes and has many discrete attributes that can be changed. That leaves plenty of time to pose new problems, exchange, attempt solutions, and discuss. 

Side note: The Art of Problem Posing is a must read for any teacher of mathematics at any level. It's algorithmic encapsulation of the fluid process of posing problems jives well with beginners and is extended easily for experts. Like seriously, stop reading this post and get that book. L-I-F-E C-H-A-N-G-E-R.

On this particular day, I gave my grade 9s the "Poison Pills" problem from Stella's Stunners. (The initial solution offers avenues into factors and multiples)
The problem has nice places for students to create conjectures. For example, students quickly realize that two poison pills can ever be adjacent. They can use these certainties to build on their solution. 

After we solved the problem and discussed, I asked them to change attributes and exchange new problems. These are the new problems they created:

  • We split pills in half and put in two containers. Take pill from #1, place in top of #2 and then eat from bottom of #2.
  • Two players. We used three containers. Take one pill from bottom of all three. Ingest one and place the other two back in whichever column you like. Goal is to be last one alive.
  • Switch the bottom and top pill. Then take two from the bottom and put them on top. Eat the next pill.
  • Instead of taking two pills, you take three. Eat the third and place the first two back. 
  • Take three pills and eat the second. Return the other pills if you are alive to do so. Don't take third pill if you die when you eat second. 
  • Split the poison pills into six half-pills of poison. You need to eat two of them to die. 
  • Have 16 pills (12 good, 4 poison) and 12 prisoners. Look if pattern still exists.
  • Odd prisoners eat first pill and pass the second one back to the top. Even prisoners pass the first pill and eat the second. 
  • Add one antedate pill that can make one prisoner invincible to poison.
We managed to solve a few of them, but had to leave some for them to work on in their problem journals. 

There are three large benefits to encouraging problem posing in class:

Mathematical Intuition
Students were able to recognize and reason why new problems were too easy to even spend time on them. This is a great thing to see as a teacher. Four months ago, these same kids would have happily accepted work that resulted in no cognitive struggle, now they ditch their new problems because they see quick solutions. 

Mathematical Complexity
Students quickly discover that some of the hardest problems come from a simple switch. Their instinct was to change almost everything about the problem until the original was a distant cousin and fleeting memory. Some groups found that shifting one small attribute can create a difficult problem. The beauty and intricacy of mathematics shows itself. 

Mathematical Ownership
Students would much rather work on problems they authored. It makes me think of a quote from a recent presentation: "You know what students are interested in? Their own thinking". Students took these problems home and created new ones from them. I even had one student bring back an idea for a card game based on his problem. This ownership can be authored into problems in subtle or extreme ways. 

I remind my students that a good mathematician will try and keep problems alive. We are so used to math being about killing the problem--problem solving. While this is a noble pursuit, I am more interested in resuscitating problems and extending their mathematical lives--problem posing. What can we do to revive or extend this curiosity. A good class, in my books, leaves with more problems than it started. 


Sunday, June 15, 2014

Garbage Can Task

The following task happened by accident:

I was about to introduce a problem to my Math 9 Enriched class that we were going to complete with group whiteboards. Before I could introduce, life got in the way. Students wanted to know about their most recent examination. As I launched into a speech on their performance, a student got up to sharpen their pencil. She walked right in front of me. I made a comment, and she replied that the garbage can should be in the back corner where it would be more convenient. 

I told her that having it by my desk was most convenient for me. Then another student said:

"Why don't we put it in the middle of the room? Wouldn't that be the most convenient?"

In this class, we call this "breaking the math". Students are always welcome to stop our class and make a conjecture. When this leads us into further problems, we joke that the conjecturer broke the math. 

I then flipped the question (to many groans from students) and asked where we would place the can so that if every student had to travel directly to it, we would travel the least amount of distance collectively. 

After setting some parameters about the room, we whipped up an idealized model on the board (pictured below). We decided that the can should be on a grid intersection and the distance between each student is one meter. Also, the students travel as the crow flies. I placed dots where the students were sitting around the room. 

A few really cool ideas began to emerge. It should be mentioned that I foresaw the close parallels to the Road Building task. I anticipated that the Pythagorean theorem would need to be used. I didn't let them know this until one group unearthed the massive amount of calculation that was necessary. 

Once this was common knowledge, groups turned their attention to symmetry. They tried to place the can in a spot that created as many congruent triangles as possible. This enabled them to cut down on their calculations. I over heard the verbiage of 2-3 triangle and 4-5 triangle. They began to name the triangles based on the length of the legs. 

One group noticed that any seat in the same row or column with the can didn't require a calculation. They then decided to set their sights on finding the placement that was collinear with the maximum number of students. 

We had a conversation about the meaning of "center". The geometric center of this rectangle may not be acting as the center of the people placed within it. I saw parallels to measures of central tendency, but decided that it was not in the class' best interest to switch to statistics at this point. 

After answers filtered in, students started posing their own problems. Many started to pose problems around designing seating arrangements to meet certain criteria:

Design a classroom that only needs one calculation.
Design a classroom where every student needs their own calculation.
Design a classroom where the center of the room is the best place for the can.
Design a classroom where the corner of the room is the best place. 

I have a lot of curricular freedom with these students, but this problem would be a good one to practice the Pythagorean theorem. I introduce the idea with the simpler Road Building task, and then solidify knowledge with this one. 

One student asked what would happen if the can didn't have to be on the floor. You should have heard the groans as we pursued this latest instance of "broken math". 


Sunday, June 8, 2014

Basketball Golf Task

The other day, a future teacher asked what one piece of advice I would give to a soon-to-be mathematics teacher. I immediately had several. I settled on one that I felt encapsulated my belief both in and out of class:

Honour curiosity

In class, this finds me wandering through student suggestions and constantly posing new problems that create relevant challenges. Curiosity (both student and teacher) keeps a vibrant ecology going, and I would argue that the intellectual tension so often provided through curiosity is necessary for a positive ecology to thrive.

Outside of class, this has me interacting with my curios online and with others. The purpose of this blog was to document and elaborate on my educational (specifically mathematical) creativity. This is such an instance where a simple problem popped into my head and I forced myself to see it through. Who knows, it may become an important piece of a student's learning someday.

For no apparent reason I became curious whether it was easier (mathematically speaking) for a basketball to go through a hoop or a golf ball to fall into the cup. It was an innocent enough question--a starting point.

I shared it with a couple colleagues and we began to discuss strategy. We immediately placed it within our neat boxes of curricular units. I said that it would be a great example of scale. I would find the diameters of the large items (basketball), the diameters of the small item (golf), and then find the scale factors between the balls and holes respectively.

She said it would be a great idea for area and percent. She wanted to find all four areas and then find the percentage of the hole that each respective ball would cover. We both thought this was a great start and took to Google.

My strategy
Basketball - 9.07" diameter
Hoop - 18" diameter

Golf Ball - 1.680" diameter
Hole - 4.25" diameter

SF = Basketball / Hoop = 9.07 / 18 = .50 (two significant digits)
SF = Golf Ball / Hole = 1.680 / 4.25 = .40 (two significant digits)

This told me that the basketball diameter was approximately a one-half scale model of the basketball hoop while the golf ball was approximately a four-tenths scale model. Thus, it is easier to sink a golf ball.

Her strategy
Area Basketball = 64.61 (units omitted)
Area Hoop = 254.47 (units omitted)

Area Golf Ball = 2.22 (units omitted)
Area Hole - 14.19 (units omitted)

Ball / Hoop = 64.61 / 254.47 = .25 = 25%
Ball / Hole = 2.22 / 14.19 = .16 = 16%

This told her that the golf ball took up less of the hole than the basketball did of the hoop.

Regardless of strategy, this question poses some interesting extensions if you are willing to search for them. Enabling this curiosity is the critical piece to effective mathematics teaching. I'm curious about a men's basketball. The stats above are for a female ball, the men's ball is an additional inch in diameter. How much harder is it to sink a guy's ball? 

What if we combined the strategies and took the scale factors of the areas or percentages of the diameters? Would our answers be any different?

Two basketballs will squeeze into a hoop simultaneously How small would the golf hole need to be to create this exact phenomenon? How wide would the hoop have to be to create the same ratio that exists in golf? The PGA is wondering about expanding the golf hole, is this a good idea? why or why not? How wide would a basketball hoop need to be to  match the new 15 inch golf hole? 

I could see this task fitting nicely into a unit on area in the middle years. (I like how the relationship between 1/2 diameter and 1/4 area can be explored). That is beside the point of this post. The goal is to encourage teachers to view themselves as creative beings. Follow your queries and develop them. Don't be embarrassed to share; this blog is filled with posts I am sheepish about.

My favourite teacher once told me that he was having trouble with curricular reform because he wasn't creative. This was coming from one of the most creative men I had ever learned from. I think this is more common than we think. Share, collaborate, critique, and honour your curiosities. They just might make the difference.