Building the Proper Ecology

The beginning of semester poses many challenges–new classes to teach, names to learn, and class sizes to manage. No challenge is greater than building the correct atmosphere in the classroom while balancing the students’ preconceived notions about you, your class, and mathematics. (Hopefully not all three impressions are poor).

Students talk. They let their friends know how your class runs. This is all the more reason to set the proper atmosphere, because a poor semester can follow you like a plague for semesters to come. I would like to propose that there is more to an effective class than the atmosphere. (This is a duh moment). In fact, as teachers we need to embrace a terminology shift.

Atmosphere is defined as “the air in a particular place.” Taken metaphorically, the classroom atmosphere refers to the set of standards (both academically and inter-personally) that the students and teacher adhere to. Traditionally, an effective “atmosphere” was a silent one, filled with mutual respect; The atmosphere describes the state of the room, and ignores the learning state of its constituents.

A more accurate pursuit would be that for a rich mathematical ecology. Ecology is defined as “the study of the interaction of people with their environment.” An effective classroom not only creates a healthy atmosphere (or environment) but creates opportunities for mathematical interactions within it. The math class should be a place where mathematical discourse is encouraged between all of its members. Students do not learn in an atmosphere, they learn within an ecology–as contributors not consumers.

I have found no better ecology builder than games. There are enough high-quality games to keep them new, fresh, and curious. Games are the perfect ecology-builder because they contain an inherent strategy. The beautiful thing is that good games contain many strategies each with mathematical merit. Games also appeal to our human curiosity. A simple game with a counter-intuitive conclusion urges us forward to discover “why”. Most importantly, games are active and contain interactions with others. Through the organized chaos, an elaborate web of connections is forming. Students are simultaneously drawing from and contributing to the mathematical ecology of the room.

To promote mathematical ecology during the game, I do two things:

• Change Partners
  • A fresh perspective or strategy means constant adaptations to their perfect strategy. It is also a great chance to break down social boundaries within the classroom walls.
• Change Parameters
  • As a game is mastered, students will suggest they know the “trick”. Slightly altering a facet of the game should cause a renewed vigor toward the solution. It also models effective problem posing.

In the first week of this semester, I have used 3 different games. Below are the details of how I set them up, facilitated them, and extended them to encourage mathematical discourse.

1. Manifest

I used this game with a group of grade nines. We were doing place-value at the time, so the strategy held curricular consequences. This was a nice bonus. The game is played with a set of cards each containing a digit 0-9. The ten cards are then placed face-down on the playing surface in a pyramid shape. (One card in the first row, two in the second, three in the third, and four in the fourth.) Players first turn over the number created with the top row. The highest number takes one point. They continue this process with the second row. The largest two-digit number created (Base-10) wins two points. The third row is worth 3 points, and the fourth row is worth four points. Equal numbers result in points for both sides. The winner is the player with the highest total after all 10 digits are revealed.

This game has some very interesting strategies and extensions. I set it up in a 32 team, single elimination bracket. Once a person was eliminated, they became a follower of the person who beat them. In the end, the class was divided into two camps. Sixteen people arranging one set, and sixteen arranging another. (For those interested, I lost in the semis…disappointing result).

Possible extensions include:

• What if we revealed the 4-digit number first?
• What if the rounds were each worth one point?
• What if the tiles were randomly chosen and placed?
• What is the base was changed from 10 to 5?
• What if the lowest number took the points from each round?

The last question was a very interesting conversation with my students.

     2.    Three Way Duel

This game is from James Tanton’s, Solve This. Essentially, three combatants engage in a duel. Alex (A) hits target 1/3 of time, Bob (B) hits 2/3, and Carlos (C) is a perfect shot. If they shoot in order of accuracy starting with Alex, who wins the duel?

I gave the students a persona and a dice. They got into groups of three and simulated the duel for 15 minutes. This was a very lively time. Certain groups began making “human” decisions and allowing grudges to creep into their mathematical decisions. Overall, the results began to skew and certain characters began to complain. I asked them how to change the game in their favour. Altering the parameters invented new games. When others protested, I asked them why. Rich probability arguments emerged. One student pleaded, “I always die, they are teaming up on me!”

Possible extensions include:

• What if the duel started with Bob? Carlos?
• What if Alex got 2 shots in a row to begin?
• What if you had to be hit twice to be eliminated?
• What if the order of shot was random?
• What is the best strategy for each player?

When we tabulated the results, I moved the class into large groups of Alex, Bob, and Carlos. They then each developed their winning strategy.

    3.      Fifteen

A simple, yet popular, game. It is traditionally played with 2 people. The goal is to sum to fifteen; the winner is the player who says the number “15”. Each alternating turn, a player adds a “1” or “2” to the current sum until 15 is reached. As the trials continue, students begin to deduce the winning strategy. In fact, if played right, the player who goes first always loses. I keep playing with confident players until they begin to turn the tables on me.

Possible extensions include:

• What if the game is played with 3 people?
• What if you could add 1,2, or 3?
• What if the target sum changed?
• What if each player got to alternate 2 guesses in a row? (like a tennis tie-breaker)

A simple strategy can be tested with a particularly confident student.

Mathematics carries with it a heavy feeling for most students. If we are going to break down the attitude that there is one path to one answer, teachers need to create more than a classroom where the atmosphere is mathematical. An effective class creates an ecology that encourages uninhibited mathematical interactions between its members.

NatBanting