Greater Than

Previously: planning and coaching on inequalities.

The origin of this game was trying to think about a game that helped give students enough experience to intuit the rules for how operations with signed numbers affect inequalities. I think this is doubly hard for students because they don't have much intuition for signed numbers, and learn both integer operation and inequality rules by memorization rather than understanding.

I love playing cards as a material for signed numbers because the red and black make such a nice positive and negative visual. (For more integer games, see this collection elsewhere on the blog.) I thought about students somehow constructing expressions, but I couldn't think of anything not clunky. I knew we wanted comparison, and I like War as a comparison structure. That made me think of the exciting moment of war, playing down extra cards on a tie.

When I played around with the cards, though, it didn't get at inequalities because each player was changing the value of only one side of the comparison. A big no-no in the context we want. So the played cards had to effect both players' values. It seemed confusing to have both players reveal at the same time, so I decided that - at least to start - players should take turns, even though that makes the optimal strategy pretty determinable. (If it's not a word it should be.)



After students get the strategy, go to the variation where both players flip at the same time or the dealer has one less card. If playing with middle schoolers for integer operations practice, try the flipping off the top variation.

I think this is a good educational game, but only close to a good strategy game. I can't quite figure out what's missing, so if you have a variation or adaptation to try, please let me know.

Evaluating this on my game design framework:

1. Goal(s). Gain experience with effect of integer operations on inequalities. Works well. Also good for gaining experience with integer computation.
2. Structure. The game generates a lot of these situations and the cards guarantee a mix of operations and values.
3. Strategy. Pretty simple. On variation becomes a pretty nice bluffing game, but not intensely strategic.
4. Interaction. What you do completely depends on opponent.
5. Surprise. Hidden information and opponent plus randomization of cards helps here.
6. Catch-Up. Victory is almost always possible.
   
7. Inertia. Might be too simple for loads of play, but good enough for the objective.
8. Rules. The idea of applying your card to both is non-intuitive, and remembering suits-operations connections is hard. You might want specialty cards
9. Context. No context, but all the variations generated engagement from the preservice teachers.

One thing I want from this experience is a deck with operations for suits. Not sure if it should have all 4 operations, or multiplication and addition with positive and negative numbers or some other variation. What do you think?

Photo credit: Abulic Monkey @ Flickr

Put Me in Coach

I took my new inequality game (that was the fruit of my planning last post) to Dave Coffey's and Hope Gerson's student teacher assistant seminar. The idea was to teach a lesson, and then get coached as a model for the TAs.  GVSU has a great education program where our student teachers have a full semester of being in the schools all morning before a more traditional student teaching semester. They get supervisors from the College of Education and, for secondary, from their major content area for both semesters.

Inspired by the Learning Network and the Cognitive Coaching model, as well as the Instructional Coaching model from Jim Knight et al, we've moved away from an evaluative assessment model for observations to a collaborative improvement model.  Dave has written a few excellent posts on coaching. One of the things we try to do for the TAs is a coaching demonstration, where they can see what this process is like. Very understandably, they are nervous about being observed. This semester, I've came in and taught a lesson as if to my preservice secondary teachers, and then we debriefed for the demonstration.

We ask the students for whatever they're using for teaching notes (as opposed to expecting a lesson plan) and for them to fill out an action plan. My lesson plan is pretty bare bones... many student teachers wanted to know if that was okay. Yup. The principle is do what is helpful to you. What I brought:

Greater Than Lesson Plan
Math 229, 9/26/12

Objective: understand nature of inequalities and their interaction with operations; apply to measurement

Agenda
5 Start Up, DOS, share HW
45 Greater Than Game

SA: Fill in blanks: 3 ___ 10; -3 ___ 2; -1 ____ -3;
F: explain game rules, play a round vs whole class
A: they play; pose question “what effect did operations have on inequalities?”
R: (10 min) discuss
•    what effect did operations have on inequalities?
•    strategies in game
•    suggestions for improving the game

50 Error analysis investigation

SA: Measuring a line with +/-
F: overview of measuring stations; introduce contest
A:
-they measure volumes and areas
-class-wide table of results
R: explain how did they compute errors for their measurements

5 Contest reveal; debate winning conditions if there’s an opportunity
5 Concerns/ Wrap Up

To Do:
Observation Journal WS
101qs.com how many/how much investigation

Materials:
playing cards
rulers/measuring tape
contest jar

I realized I omitted the lesson objective of analyzing an activity as a teacher, and put that in for the teaching of the lesson 'for real.'  We only did a representation of the first part of the lesson for the coaching. Here's the action plan:

Dave took notes while I was teaching. (He may produce a sharper version of this.) When we're taking notes we often make note of other things to talk or think about aside from the action plan, but don't necessarily bring them up unless it is something that could greatly help the observed.

The dialogues following an observation are some of my favorite times at work. Discussing teaching with another professional based on something that we both just saw happen is amazing. Exciting as a teacher and satisfying and growing as a colleague.  We've tried to think of a way to encourage this kind of teacher to teacher interaction at the university, but it hasn't happened yet.


The video of this coaching isn't exactly like what it is when not for demonstration, but what is here is pretty authentic. (For one thing, the discussion is usually 30 min to an hour.) I did make some of the changes that were suggested here or in the discussion afterward with TAs participating in the coaching. They recognized the value of positive feedback, seemed less intimidated by the prospect, and were interested in how the person being coached does more of the talking.

What: the game does work for generating experiences to consider inequalities. I think it also works on the level of an activity for novice teachers to evaluate for use in class. It raises the issues of materials, effort to implement, and when a game might be good in class.  (Some of this is based on my use of the game with my PSTs.) I did hear a couple things that made me think about how do I make sure more of the connections that I'm thinking of get shared in class, for example the connection to War.  I do like the structure of the game.

So what: I want to consider the idea of generating a good demonstration game vs authentically playing a game to model.  I feel hesitant to stage a game, and I'm not sure why.  I want to think more about when should a game be well-determined and cleanly set, and when should a game be something that you kind of unroll over a few days.
        The idea of being clearer with preservice teachers about the difference between how things work in our college classroom and how it would be with K-12 students is definitely a valuable one. 

Now what: I did clean up the instructions as they suggested. I think the game is good as is for high school. The choice of cards vs flipping off the top helps emphasize that the only thing that flips the inequality is multiplying by a negative. The flipping is a nice variation for younger students who are focused more on operations with signed number. As a game I have to think more about the full strategy version. Is there a way to make it a real game?
        It's a constant danger that PSTs will decide our experiences are irrelevant when confronted with the schools as they are. So thinking about this transition and connection piece from teacher education to teacher reality should always be considered.  I also want to continue to be sensitive to the idea of big shifts and subtle shifts. Subtle shift - questioning, medium - finding a game online, big - designing your own game or writing your own curriculum.

Note: I'll shortly have a post up that's just about the game with the current version.

Image credit: By Sgt. Robert Adams [Public domain], via Wikimedia Commons

Think Aloud: Planning Inequality

From the always fun Zappa Blamma

In my preassessment for preservice teacher (PST) high school mathematics course, I had a new topic grab their attention - inequalities.  In previous times teaching this course, that hasn't been an issue to which we've given much attention. My bad, as I've seen plenty of struggles with it in secondary classrooms. It's clearly an error where we retreat to instrumental understanding (using Skemp's terminology); rules, rules and more rules.

So what are the key ideas? It's an extension of the algebraic methods we're already using. So I want to focus more are the conceptual components of inequalities... which I haven't ever really deeply considered before.

The heart of inequalities is comparison, which is a key component of number sense. A key and often omitted component. Usually we introduce numbers (multi-digit, fractions, decimals, signed) and then are lucky if we even get representation work before we get to computation rules and rote. So that makes me think of a couple of my favorite games - Fraction Catch and Decimal Pickle. Probably not appropriate for a high school class, though I use them in the preservice middle school class to good effect. What would a high school version look like?

Hmm. It's the rules for changing the sign that get people all bothered. What if the players/teams started with a number, then multiplied or added to both players numbers... the goal could be to get the larger quantity in the end.  You could make special cards... or (I love being able to use regular dice, cards, dominoes, etc.) have the suits imply the operations. I'd have to think of how many turns, how many cards to have, some of the game mechanics. But that idea might be worth the work of developing something new.

Before being in math ed, I worked in index theory, that includes plenty of analysis. Analysis, as a mathematical field, involves a lot of estimates. Think nitty-gritty epsilon-delta proofs. This is less that that, which is less than or equal to the other, which means the whole thing is less than... I loved that stuff. So is there a way to get students making those kind of estimates?

That makes me think about experimental error. Error analysis is nice because of the connections to measurement. That gets us into volume formulas. Could even do a guess how many kind of competition. Hmm. I took a picture of a neat set up like that at the ND-U of M football game over the weekend, but it came out too blurry. And I don't know the result! (I guessed 1337. [LEET.] The bottom pyramid was 10 balls wide and about 20 balls tall.) Lots of good images for that on 101qs.com. Could use in class or make a good HW assignment out of that, and get the PSTs looking at that great resource. It would also let us talk a bit about measuring technique - important to get in somewhere. Volume ties into some of the higher degree polynomial stuff we've been doing...

I think that would make for good foundational experiences, and then we could do a follow up day to see how the ideas of inequalities we see apply in traditional algebra and algebra 2 problems.

Now to get working on that game...

ps. What I didn't do here that I usually do when planning, is look around what other people have about inequalities. That usually involves a visit to Sam's virtual cabinet, searching reader, raiding Kate's archive, etc. Where do you look for inspiration, adaptation and out and out robbery? (Accredited, of course.)

This story continues in the post on coaching. With video of an actual dialogue!

Slant Wise

I caught a great talk by Eugene Peterson this week. He's a pastor and spiritual writer who gave a talk at Valparaiso University in honor of Walter Wangerin, Jr. (who was also there); the talk was "What are writers good for?" (I found a pdf of a previous iteration of the talk.) There'll be mention of God below, but really, these are my connections from his talk to math teaching. I've written one other post inspired by Peterson, Jonah the Math Teacher.

Peterson's bottom line for writers is that they can reclaim language from debasing use. For religous writers this is particularly important, because we as a society have turned our spiritual words into godtalk that is easy to ignore and not worth our time.  In other words,

Knowledge of speech, but not of silence; Knowledge of words, and ignorance of the Word... Where is the Life we have lost in living? Where is the wisdom we have lost in knowledge? Where is the knowledge we have lost in information? -TS Eliot, Choruses from The Rock Poems and Plays, 96

Important ideas, I think. And amazingly close to the challenges we face in mathematics teaching.

"So what are writers good for? It is our vocation to maintain and practice this core, basic, revelational, personal nature of language, living speech.    In a world in which language has been uprooted from its originating God soil and put to the use of information or propaganda, it is the vocation of writers to represent and practice language as revelation, to re-orient language into the personal world in which men and women actually live—in their families, and neighborhoods and workplaces," says Peterson.

What are math teachers called to do? To recover the debased math, practiced in schools for years, to bring it into the world in which the students live, to share math as it's truly done, to share learning that will make a difference in our students' life.

So how does a writer do it?  For Peterson, the illustrating example is the middle parable section of the gospel of Luke. A lot of Jesus most powerful teaching. Stories with no explicit mention of God, no direct lesson. A story about fertilizing a tree instead of cutting it down. Told in Samaria to people who are not interested in his religion, and not fond of his people. He might as well have been a math teacher.

Not to double up on poetry, but Peterson also went to Dickinson.

Tell all the Truth but tell it slant – Success in Circuit lies Too bright for our infirm Delight The Truth’s superb surprise As Lightning to the Children eased With explanation kind The Truth must dazzle gradually Or every man be blind -Emily Dickinson

(Fantastic art by  David Clemsha. Shows up in my Reader the next morning as a wonderful coincidence.)

How well does that capture the essence of good teaching?  Peterson notes, "A parable keeps the message at a distance, in the shadows, slows down comprehension, blocks automatic prejudicial reactions, dismantles stereotypes. A parable comes up on listener obliquely, on the slant." A writer does this by having the reader come to them, going slow, countering the impatience of the age.

This leaves teachers the challenge of knowing what is best, in a culture that wants what is lesser. When we propose that there is better, they want solutions that are immediate and rushed. How can we convince them that the answer is slow? Real stories, finding the way themselves, experiencing the superb surprise. I think we have to just persist. Write our real lessons. Participate in the community. Share our success stories that will buoy us through a dozen bad days.

"The Truth must dazzle gradually."

It's our vocation to tell it slant.