If The Piece Doesn't Fit, You MUST Acquit!

In math class we have started investigating triangles.  My good friend, running coach, and teaching partner, Erika, suggested to give them a bunch of straws, some vocabulary, and let them go off and running.

Who am I to say no to such a wonderful structure?

We reviewed some geometric terms: Acute, right, obtuse, scalene, isosceles, and equilateral.  Then we gave them a challenge: create a triangle with each of the vocabulary terms: one for sides and one for angles.  They realized that they had to create nine total triangles.

Students got started by building an equilateral acute triangle.  It was a solid beginning with students easily conquering that task:

Many thought they'd finish all of them inside of 10 minutes... then they tried the next one.

Feeling confident, they went on to another triangle on their list: equilateral right.  Students used the straws and tried to build it, but no matter how they arranged the straws they just couldn't manage:

If the piece doesn't fit, you must acquit!

Students became frustrated and annoyed. There was some fantastic and frank mathematical discussion amongst themselves. They discussed lots of options.  Some said it was impossible, others argued that can't be the case, but then had second thoughts... Is it possible?  After conferring and discussion, they decided such a shape was not possible to build because "there will always be a little piece of triangle missing on one side, and that side will always be longer."

Why do these teachers constantly try to trick us?

Students continued on to build the other triangles, obtuse scalene, isosceles right, but then  got stumped with 'obtuse equilateral'.  The students went back to their previous thoughts and ideas that were built from 'right equilateral' and concluded that such a triangle could not be built.  Students also made amazing observations:

"...when I built an isosceles triangle, it looks like there are two angles that are always the same too.... So maybe when sides are the same length the angle is the same degrees?"

Those two acute angles look eerily similar

We followed this lesson up with one that used protractors.  Students have begun to confirm similar thoughts and hypotheses, as well as showed that the angles of triangles "always seem to add up to about 180 degrees."

I loved all of the discussion and discovery this lesson gave the students.  Erika and I facilitated discussion, but we never clued them into the 'impossibility' of building a right equilateral triangle.  They came to this conclusion on their own and were successfully able to argue (in a middle school way) why it wasn't possible to build such a shape.

I'm excited to see how they apply this knowledge to the rest of our geometry unit!

Collaboration for the win!

Miyagi or Cobra Kai?

I now take you to your regularly scheduled movie, already in progress:

Daniel: No the problem is, I'm getting my ass kicked every other day, that's the problem.
Miyagi: Hai, because boys have bad attitude. Karate for defense only.
Daniel: That's not what these guys are taught.
Miyagi: Hai - can see. No such thing as bad student, only bad teacher. Teacher say, student do.

And so begins an epic relationship between student and teacher.  When reflecting upon my own teaching, I often find I'm in the Miyagi dojo, as opposed to Cobra Kai.

In the Cobra Kai dojo, there is one master.  The sensei is all knowing and all controlling.  He snaps orders and the students complete the drill in unison.  When he asks a question they answer together, "Yes, sensei!" There is no questioning, no deviation, no discussion.

Students are given very straight forward workouts - practice the jab, practice the round kick.  There is no doubt these students are learning karate. When anyone walks into their dojo they are surrounded by awards, trophies, and other accolades.  The students of this dojo pass all of their tests.  The students win for their sensei and their dojo.

Does your classroom intimidate or inspire?

On the other side of town, Daniel is training with Mr. Miyagi.  Daniel is washing cars, painting fences, sanding floors, and feeling like he's Mr. Miyagi's personal servant... and he tells Miyagi such.  He doesn't see the connection between waxing cars and defeating his enemies.

When he questions Mr. Miyagi about this, Miyagi shows him the connection between painting the fence and defending himself against an attack.  "Show me wax on, wax off." To his surprise, Daniel defends himself against Mr. Miyagi's attack.  He was learning karate the whole time without even knowing it.

Sometimes 'math' lessons don't have to look like math

Curriculum is just that - curriculum.  It is a series of standards or studies that a teacher must impart onto the students. It is a collection of facts.  Pedagogy is how to best implement those standards, how best to teach the lessons. That is the art of education.

In education there are multiple ways to teach. Pre-service teachers are introduced to Gardner, Montessori, Piaget, and others. Yet I would argue that many teachers and schools model the Cobra Kai style of teaching and development. One teacher, the sage on the stage, preaching a skill with students practicing: I do, we do, you do.  Students in this model get really good at drills.  They pass the tests. They win awards for their teachers and school.

But what else do students in this system learn - or more importantly what do they not learn?  Students learn to be uncreative and unimaginative  They learn to not try alternative methods. They learn to say "the teacher told me to do this way, so I must do it this way." Teacher say, student do.

Many websites (such as AACU) talk about how employers are looking for creative thinkers, problem solvers, and other 'outside the box' innovators.  Miyagi helped produce such a student, and hopefully my classroom will as well.

Yes, sensei