Monday, February 20, 2012

Monty in Math Ed

Before you continue reading, This is not a post about the Monty Hall problem. Not because I don't love the problem, it just isn't. That ship has sailed.

This post was born from a conversation I had with a teaching friend after a games night. Along the way, we were reminiscing some past experiences and he mentioned the famous clip from Monty Python and the Holy Grail where Sir Bedevere handles a crowd's plea to burn a woman suspected of witchcraft. When I got home that evening, I searched the clip and watched it for the first time in years. My lens has obviously changed, because I found myself not interested solely in the logic and hilarity of the piece, but on Sir Bedevere's teaching methods. As you refresh your memory below, I want you to do so with your "educator lens". Does Sir Bedevere run an effective lesson?

It is remarkable how much information can be garnered with a trained eye. There are several observations that can be pulled from the clip:
  • The clip opens with a very enthusiastic group of 'students' coming to 'class'. I tried to remember the last time my students came triumphantly into my room as the bell rang. Those experiences are few and far between, but the Red Card, Blue Card problem did get that response.
  • Notice that the teacher is physically raised and is very obviously the source of knowledge in the 'class'. Students are looking for the correct answer from the authority.
  • Teacher opens the class with a broad question ("How do you know she is a witch?"), and gets a shallow answer ("Because she looks like one").
  • Teacher asks for verifying criteria, and further specifies the opening question.
  • Teacher then informs the class that there is a specific set of criteria that they need to know to diagnose their problem. ("There are ways of telling whether she is a witch")
  • Teacher leads the class along a problem solving trail with multiple pauses for student thought and involvement. Unfortunately, questions are always focused on a select few students. (The three keeners that sit front-and-center)
  • Students begin to critique each other's responses. ("More witches!" ; "Shhhh")
  • Teacher takes a student response and forces them to think deeper. ("Build a bridge out of her."; "Can you not also make bridges out of stone?")
  • When the class becomes stumped, the teacher asks a semi-rhetorical question to re-spark thinking. (Does wood sink in water?")
  • The entire class is a struggle between two wills: The class wants to arrive at an answer, while the teacher is moving toward a foreseen objective. (Medieval curriculum...possibly?)
  • Teacher encourages a brainstorming session, but quickly disapproves when students go off track. ("What also floats in water?"; "Bread", "apples", "very small rocks"...)
  • The teacher reviews the problem solving with the class before deciding how to test their new-found hypothesis.
Obviously, there are advantages to this setting as a classroom. Math teachers will complain that there is a high level of inherent motivation involved in the class so it is easy to loosen the control a little. Students would much rather burn witches than factor polynomials. Also, the teacher was equipped with the necessary resources and was not being pushed with curriculum or testing restraints. How did Sir Bedevere go about assessing the students? Did he contact parents about those students who seem disengaged in the witch craze?

I think there are good and bad things to draw from the "lesson". It demonstrates that a teacher can lead students through discovery while still standing at the front of the room. It also demonstrates how poor feedback can quickly derail a great lesson. I am not nominating 'The Flying Circus' for teacher of the year, but do think there is plenty of food for thought contained in the clip.

I think it would be a very interesting activity to get teachers to assess the learning of the three "students" or take the lens of Intern Supervisor and evaluate the lesson.

If this demonstrates nothing else, it proves that my class is a lot like a flying circus in more ways than one.


Saturday, February 11, 2012

Continuum Assessment

Yesterday I took part in a multi-division professional development day on assessment and critical thinking. My division has been enamoured with Assessment for Learning for the longest time, but I have not been able to effectively transfer that knowledge into effective summative assessment in my math courses. I have, for the most part, stuck with the traditional assessment methods.

My foray into Project Based Learning necessitates a shift, and that shift was finally solidified through my activity with peers at the sessions. I scratched down a form of project assessment, and labeled it "Continuum Assessment". I called it this for three reasons:
  1. Placing something on a continuum requires a large amount of critical thinking.
  2. A continuum implies that there is room for growth. This "growth mindset" aligns perfectly with my department's priority this year.
  3. It avoids an quantitative scale when reflecting.
The process goes as follows:
  • I begin the assessment process by asking the students to tabulate a list of characteristics of a good project.
  • Initially, each suggestion is placed on the board. I will suggest topics if they are overshadowed, and even bargain to include some aspects.
  • Every characteristic that makes the final cut is assigned a continuum on the assessment. The continuum contains no marks or labels.
  • Below each continuum, there are two spaces. One is for evidence of their assessment, and the other is for improvements on their performance.
  • Examples of projects are presented to the students. They fill out mock assessments with good, written evidence and improvements. The assessments are discussed and the characteristics are altered, added, or deleted (if needed).
  • Each student completes an assessment on their own project individually, and each group is assigned a partner group to complete an assessment on.
The process takes time, but it is time well spent. The goal of the class is to get students self-monitoring their mathematical progress. The project environment provides a large amount of autonomy, and the assessment continues this feeling. The absence of numbers (ie. from 1-10) requires students to further uncover the evidence of a job well done. My goal is to sit down with students at midterms and look through their assessments and chart their progress.

I plan on assigning their blogs a continuum on the assessment. The use of this reflective tool throughout will become yet another tool to build interconnectedness. Some students have taken to their blog right away; others need to get use to the fact of documenting their thoughts and struggles. The posts do not need to be long to show me that they are thinking about the class, the work, and their strategies.

For example, one student documents the slow implementation of the technology here:
While another student details her excitement to begin her first project. A description and initial strategy are included:

As the class continues, and freedom increases, the reflective blogs will become a very important part of the assessment.

My growth as a teacher has been largely centered around instruction. I attempted to build my lessons with a focus on deep student thinking. I just assumed that critical thinking ended when the task was done and assessment began; I imagine the years of mathematics instruction I received solidified that fact. Developing the continuum model for my projects has finally illuminated what it means to continue to learn throughout assessment. Assessment is as much as an ecology builder as instruction. An effective assessment model not only documents whats been accomplished, it links seamlessly into what's to come.


Saturday, February 4, 2012

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.