__________
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 09 (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 neverending recap of my CMCSouth session. This can lead to metaconjectures 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 counterexample 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.
Avery, such great stuff. I am noticing the power of teaching children language like, "'I have a conjecture that...' and 'Respectfully, I think I have a counterexample to ________'s conjecture.'" I totally agree with you that "It's a great way to build a community of mathematics for the rest of the year." I hope you share this presentation multiple times in multiple venues in the near future. I believe many many math teachers, grades ?12 will gain a great deal of insight about teaching proof from your ideas.
ReplyDeleteJust last Tuesday I played a 3digit version with my 6th graders. (aka Pico Fermi Bagels, and an online version http://goo.gl/nByZ2e)
ReplyDeleteI've never played the 14 you described but will definitely try it.
Mine is: I have a secret 3digit number, make your guess, and I'll tell you how many digits are correct and how many of those digits are placed correctly.
Thanks for sharing, Avery.
Here's a version of Mastermind on Scratch. Could probably easily remix it to change the colors to digits. Might only need to change the sprites. https://scratch.mit.edu/projects/12243923/
ReplyDeleteMiloMac  We bring latest mac tools which you can easily get online. You will find best tools of market on our website. Get mac applications via torrents.
ReplyDeleteMacVies  Get latest updates on your software free of cost. It is very difficult to find latest tools which can help you in daily life.
Machacx
ReplyDeleteGet unlimited free tools and cracks for mac os. You can download all premium mac and windows files https://www.machacx.com/
ReplyDeleteThis was really helpful and gave me far more ideas.
Ghost Mannequin services
A very awesome blog post. We are really grateful for your blog post, Reading nice blogs on instagram with site instagram viewer
ReplyDeletenice
ReplyDeleteThe Top Apps
Bu web sitesini hızlı bir şekilde ziyaret etmek ve tüm meslektaşların görüşlerini okumak harika.
ReplyDeleteBu yazının konusu hakkında, aynı zamanda aşinalık kazanmaya da hevesliyim.
https://www.totosafeguide.com
We are professionals with all Main Florida Real Estate Agent Orlando Whether you have a Florida Deluxe home, Orlando investment home or a luxury Florida vacation home. We will use our experience to market your house both online as well as in print. You may ask yourself just how much is my residence worth? We will certainly utilize the Florida mls to locate comparative homes in your neighborhood. We utilize this to market our multimillion buck estate homes and also it will benefit your Orlando house to buy. Orlando Realtors
ReplyDeleteSuzuki WagonR Parts
ReplyDeleteThat doesn’t mean that JackpotCity Ontario is a flawed platform, however the system is fully automated, and errors can still occur, not simply right here however wherever on the internet. The casino is licensed by in Malta by the Malta Gaming Authority and has the authority merely acceptto simply accept} gamers from round the} world. Jackpot City 빅카지노 Casino has been around for over 20 years now, so its administration positively is aware of} the way totips on how to} run a webbased casino and to do it properly enough to outlive for decades. And as you level up, you will gain more access to helpful loyalty rewards, like daily specials, unique access to new games, invitations to inperson VIP occasions, and more.
ReplyDelete