THIS IS A DRAFT. This is an untested outline for a culminating project for a course in probability (or the probability portion of a probability & statistics course).
(Updated: 27 Aug 2011. Acknowledgements: Danny Tan, David H, Nate Straight)
Overview: Games constantly ask players to make decisions that will help them win the game. In many games, deciding what the best move is involves some math, and therefore these games are a good way to exemplify many mathematical ideas and provide venues to practice skills. Through analyzing a game and producing a strategy guide, students will apply course concepts to solve a non-trivial problem. The project also seeks to engage student interest with fun activities (i.e. playing games).
1. Introduction. Early in the semester (first week or even the first day while explaining the syllabus), state that there will be a semester project. The goal of the project is to produce a strategy guide for a game of the students’ choosing. Along the way, students will identify and explore key decisions and apply mathematical concepts from the course to explain their choices for these decisions. Furthermore, students will have the option of using an existing game or creating their own. As we haven’t introduced many of the mathematical concepts that can be used, we can’t go into much detail in that respect right now, but we can introduce a number of games to give students an idea of the games and decisions we’re talking about.
2. Getting started. By about 2-3 weeks in, students should have been introduced to some basic concepts (e.g. events, probability of events, independence, union/intersection/complement events, conditional probability) and several games, including all of the teacher suggested ones below. They probably have 1-2 problem sets under their belt too, and some may have study groups. This should provide them with enough background knowledge to get started on the project.
Students will form groups of 4. Prior to forming groups, poll the students on which games seen so far they have enjoyed the most / are interested in working on. Post the results so students can use it to find people who want to work on the same problem as them. There should also be an option in the poll for people who want to design their own game. However, emphasize to students that they are not pressured to form groups according to the poll results, nor are they required to do the game they specified in the poll, and they are encouraged to form a group with their existing study group.
What are the roles within the group? This is so various parts can be delegated to different team members, who become responsible for them.
Each group should now decide what game they want to analyze (from a list of suggestions or supplying one of their own) or design one of their own. Tell the students that groups that design their own game will do about 2/3rds the analysis of other groups to compensate for the work in designing a game. Furthermore, their games should be able to accommodate 4 or more players and aim to finish in 30-60 minutes (times are for players who are well-versed in the rules; add 50-100% play time for new players). Descriptions of various games will be provided to give students a range of games to choose from and to give ideas to students who are designing their own.
3. Group exchanges. Throughout the semester, groups will meet with one another to show what progress they have made and provide feedback to each other. Groups will need to make at least one meeting in every 2-week period, rotating between different groups. Meetings should take about half an hour, though week 10 may take longer. A suggested format is for groups to split into pairs: one pair will explain their project to members of the other group; the second pair gives feedback to the other group. Here is a tentative schedule for groups that are not designing their own games:
- Week 4: Strategy brainstorm. Play the game, at least partially, so the players get a feel for the game. Discuss strategies and from there what are the key decisions players face.
- Week 6: Knowns and unknowns. Focus on one key decision the group has identified. Discuss what you know and don’t know about the problem. What information is relevant to the decision? What course concepts would you apply to decide upon a choice? If you are stuck on a problem, what knowledge or plan of action might get you unstuck?
- Week 8: 1st key decision. Present as full of an analysis of one key decision as you have. See the project rubric below for what constitutes a full analysis.
- Week 10: Playtest! Play a full game against people in the other group. Does your strategy work?
- Week 12: 2nd key decision
- Week 14: 3rd key decision
And a schedule for those that are designing their own games:
- Week 4: Brainstorm. The group should describe the theme and/or the central mechanic(s) of the game. What would make the game fun and interesting to play? What details can be added/removed? (Note: in game design, it is often very easy to add new things but harder to remove them. Game designers should determine what is essential to the game experience and what is irrelevant/distracting.)
- Week 6: Prototype. The group should have rules and a physical prototype of their game (artwork and such optional) and play it with the other group. Get feedback on what works (i.e. is fun) and what doesn’t.
- Week 8: Knowns & Unknowns.
- Week 10: Playtest! While the group can tweak their game up until the final deadline
- Week 12: 1st key decision
- Week 14: 2nd key decision
After each meeting, both groups should submit a short memo (2 paragraphs) summarizing the feedback they received and the feedback they got. One copy of the summary should go to the other group, one copy goes to the teacher to make sure groups are procrastinating. The summary is graded purely on completion.
The meeting during week 10, the week for playtesting, will probably take longer than usual. However, it should be more fun and less brain intensive than other meetings. This may be a good way to relax after a midterm.
4. Checkpoints. Twice during the semester, the group will turn in a more substantial report in addition to the short summary.
After week 4, students will submit a 1-page project proposal. The proposal will give the game (or a description of the game, if they are making their own) and point out at least one key decision they plan to analyze. It is okay if they cannot fully analyze the decision right now, as much of the course is still to come. The purposes of the project proposal is 1) to make sure students haven’t picked a problem that is too easy or too ambitious; and 2) to give the teacher feedback on which course concepts may be important to various groups, so emphasis can be placed at different parts of the course as needed. The proposal will be mostly graded on completion, with maybe 20% reflecting how much thought the students put into the problem.
About week 9 (by this point, all of the basic ideas should have been introduced, such as conditional expectation, but more advanced concepts like random processes probably have not been broached yet.), groups turn in a progress report. For groups designing their own game, this should describe their game mechanics, including their current rulebook, and preliminary analysis of the two key decisions they plan on studying. Their choice of decisions should follow the description of game mechanics. For other groups, they should identify the three key decisions they plan to analyze and provide as much of their analysis for one key decision as they have. Again, the purpose of the checkpoint is to make sure students have not trivialized the project or are procrastinating. Again, grading will be 80% completion, 20% content.
4. Final product. Groups who designed their own game will deliver the following:
- A copy of their game, including all game components (give suggestions for ways to produce print & play components) and rulebook. This will be returned to the group afterwards.
- A strategy guide analyzing at least 2 key decisions.
Other groups will deliver a strategy guide analyzing at least 3 key decisions. The analysis of these key decisions should be thorough: students should identify what decision the game is asking the player to make, what information is relevant to picking the optimal decision, what course concepts they used to solve the problem, showing one or more examples they solved, and providing strategy advice to players. It is important that these key decisions not be trivial; as a rule of thumb, any probabilities/calculations should take the teacher at least 5 minutes to solve. The project asks the students to apply course concepts to answer substantial questions.
The project rubric will reflect these objectives. For example: 25% each for 3 key decisions, each one broken into components as described above; remainder split between readability/clarity of the strategy guide and oral presentation; for groups designing their own games, replace 1 key decision with an evaluation of their game design (whether their game contains interesting decisions that can be solved with math; rules clarity; component aesthetics).
In place of project presentations, the class can have an end-of-the-semester game convention where groups demo their game, explain their findings, and play other groups’ games.