Using Mastermind to Model The Life Cycle of a Problem

The following is part 3 of a recap for a workshop I led at the CMC-South (California Math Council) conference in Palm Springs, CA making the claim that educators and mathematicians should expand the definitions of proof in order to make proof accessible to elementary and middle school students.
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Mathematics the verb is cyclic, an idea I outline in part 2 of my recap on the life cycle of a math problem  and explain to my sixth graders using the game of Mastermind.

If you're unfamiliar with the game, here are the basic rules .
If you want to play a few games against the computer, we have that too.

I use digits instead of colors, which means this can be played using an iPad drawing tool instead of having to go and fork out twenty bucks for a physical game. I suppose you could also use pencil and paper. I also keep track of "correct digit, correct place" and "correct digit, wrong place" instead of using white and black pegs (to be honest, I can never remember which one is which). Even using the digits 0-9 (equivalent to 10 colors), a game goes pretty quickly.

We start with some Avery vs. Class action, with some specific language used throughout.

Before the first guess: " _____, would you like to make a wild guess?"

After each guess: "Does anyone have a conjecture about this game?" We've used this language before, but you could introduce this language here. If so, the student states their conjecture.
"Do you have a proof for your conjecture?" After the student's proof, a concept that should be broadened, I ask the class: "Skeptical peers, does anyone have any questions or concerns about this proof?" If so, we try and resolve it by altering the proof or abandoning the conjecture. If not, we write it up as a theorem.

One important note: If students come up with incorrect conjectures that do not get detected, I still write it up as a theorem. I actually hope for this to happen at least once to reiterate to the class that they are the skeptical peers, not me. And I'm not a prophet (by any means) that will let them know whether they're right or wrong. They need to work at this. That said, this is a great medium for allowing incorrect theorems to live for a bit. After all, it's just a game of Mastermind and mistakes will quickly become apparent.

I then ask for more conjectures. If there aren't any, I call on another student for an educated guess. Rinse, lather, repeat. I always get great conjectures. "There cannot be a 2 in this puzzle." "There must be a 1 in the first or second place." These conjectures get more specific and more involved over time. In the end, this almost always naturally leads to students making conjectures about the solution.

We also talk about "the problem space" or the set of truths/axioms and how different Mastermind games will have similarities in their problem space (the rules of Mastermind), but also differences (different solutions). More on this in part 4 of my never-ending recap of my CMC-South session. This can lead to meta-conjectures and theorems about Mastermind such as:

The following information implies that none of the guessed numbers are in the solution.

GUESS	CORRECT DIGIT, CORRECT PLACE	CORRECT DIGIT, WRONG PLACE
? – ? – ? – ?	0	0

 The following information is impossible.

GUESS	CORRECT DIGIT, CORRECT PLACE	CORRECT DIGIT, WRONG PLACE
? – ? – ? – ?	3	1

We play as a class. We play in small groups. It's pretty amazing hearing middle schoolers say things like "I have a conjecture that..." and "Respectfully, I think I have a counter-example to ________'s conjecture." It's a great way to build a community of mathematics for the rest of the year.

Oh, and I also have some Mastermind puzzles. Below are a few examples.

Find the secret code with the following information if you can use the numbers 1, 2, 3, and 4.

GUESS	CORRECT DIGIT, CORRECT PLACE	CORRECT DIGIT, WRONG PLACE
4 – 1 – 4 – 1	0	2
4 – 3 – 3 – 1	0	4

Find the secret code with the following information if you can use the numbers 1, 2, 3, and 4.

GUESS	CORRECT DIGIT, CORRECT PLACE	CORRECT DIGIT, WRONG PLACE
3 – 3 – 2 – 1	0	2
4 – 4 – 1 –2	2	0
4 – 1 – 1 –3	1	1

Using this first clue, what is the maximum number of guesses you will have to make before finding the secret code if you can use the numbers 1, 2, 3, and 4.  Why?

GUESS	CORRECT DIGIT, CORRECT PLACE	CORRECT DIGIT,  WRONG PLACE
1 – 3 – 2 – 4	0	4

This last puzzle is also a great problem to talk about lower and upper bounds, an important mathematical habit of mind. If you share solutions to the puzzle, please post a spoiler alert. If you have insights, suggestions, or opinions please post an insight, suggestion, or opinion alert.

Teaching Problem Solving, Part 1: Starting with a Good Problem

I'm helping lead a PD on problem solving tomorrow.  We'll start by working on some problems that I think do an a'right job of passing my "Characteristics of a Good Problem" test.  I blogged about this a while ago, but here's a shiny new version:

Characteristics of a Good Math Problem

1.     The problem is accessible. It minimizes vocabulary and notation (and vocabulary and notation that does exist should simplify, not complicate).  It should only be as precise as necessary.  The problem should have multiple entry points, and include ways to collect data of some sort. It should have multiple methods that promote different learning styles and celebrate different ways of being smart.  It may have multiple valid solutions.

2.     The problem is deep. It is rich enough to spend hours, days, weeks, months, or years working on variations, generalizations, and extensions. It leads to and connects different aspects of mathematics. The problem motivates developing procedures, vocabulary, notation, and mathematical concepts.

3.     The problem is captivating. This does not mean that it has to be a “real world” problem and pseudocontext (from Jo Boaler's What's Math Got To Do With It?) should be avoided. A captivating problem may lead to a surprising result. It may feel like a puzzle waiting to be solved. It may be necessary to solve a different, interesting problem (which is not the same as “you’ll need to know this next year”). The problem may be posed by students. The problem consists of benchmarks along the way where one is re-energized by the feeling of success.

4.     The problem scales sideways. This characteristic may be more applicable to school mathematics, but sideways scaling allows for practice and quality assessment (beyond just solving exercises measuring the ability to follow procedures).

5.     The problem is mathematical. Problem solving skills and/or the language of mathematics help make progress in defining, simplifying, quantifying, dividing and/or solving the problem. Exploring the problem promotes mathematical habits of mind.