#### Virtuous Democracy

#### (Gordon Hamilton, 2005)

Many top quality creative mathematics classrooms get all students to make conjectures about their mathematics explorations. True conjectures are easy to get excited about, but the mark of a strong mathematics classroom is that all students feel comfortable enough to make a *false* conjectures. Alison Hansel’s (twitter: @ms_hansel) class is particularly excellent.

We can think of conjectures as commentary on the output of a puzzle. The complement of conjectures are constraints. Sometimes students should explore adding new constraints to existing puzzles. If the constraints are too harsh, the puzzle may have no solution. That’s good for students to discover.

Not all constraints are created equal. Creative students should try to develop an aesthetic sense about what makes a beautiful constraint. There is no strong rule for this, but I’ll give you two examples and let you draw your own conclusions…

Example of a constraint which sounds ugly: “Numbers 2, 13, 14 and 17 must be Yellow.”

Example of a constraint which sounds beautiful: “All the yellow numbers must be consecutive.”

I invite you to explore virtuous democracy – both through making conjectures and adding your own constraints.

**Standards for ****Mathematical Practice**

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

##### MP1 Toughen up!

This is problem solving where our students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

##### MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

##### MP6 Be precise!

This is where our students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

##### MP7 Be observant!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.