Month: October 2013

Geometry Instruction [MTBoS #2]

Lately I’ve been reflecting on feedback from my students- it’s the end of quarter 1 so a natural opportunity arose. Many asked for me to have “more examples”.

I’m not sure if “more examples” means:

  1. More lessons that are traditional “I do one example while students furiously scribble into notebooks, we do one example where I cold call from the class, you do 20 of these silently and independently” OR
  2. More examples/experiences on how/where to apply concepts we are learning (this one’s a bit more nebulous for me)

If I’m not doing enough of 2, then I’m failing as a teacher.

I lean towards pretty much flat out refusing to do much more of #1 because I never want my students to think that math is about mimicry and that I can somehow magically model every geometry problem they would/could ever encounter. Also, geometry in TN is not a state-tested subject, so there is not a lot of pressure towards an end-of-course performance test. This allows me with a lot of flexibility to make my curriculum a little more… shall we say progressive? I resonate with and align my core teaching values with many of the values that I’ve observed in the rest of the MTBoS re: quality math tasks.

Enter Dan Meyer.

No one I respect thinks students should discover all of geometry deductively. … To motivate a proof, students need to experience that “Wait. What?!” moment of perplexity, the moment where the seemingly-true has revealed itself, a perturbing moment experienced by so many mathematicians before them.

Which has forced me to reevaluate and define the difference between meaningful struggle (what Dan would refer to as perplexity) and frustrating struggle [it sounds so obvious written out- I guess that’s why I should blog more often?]. If I have to present my students with something in order to allow us to engage in more meaningful struggle, I am not robbing my students of the opportunities to make deductive conclusions. I am supporting them by preventing frustrating struggle that leads to disengagement and shutdown.

Enter Chris Danielson and the Van Hiele levels of Geometry Instruction. These are not something that I’ve paid any attention to when planning my lessons, which should probably will definitely change. Like Chris said, high school students come in, or maybe should come in?, around a level 1.

Frustrating struggle [current]: My questions/examples are aimed towards getting students to remind themselves of properties (which maybe 1 person in a class can currently provide for us).

Meaningful struggle [goal]: I need to purposefully and consistently provide the opportunity for students to make deductions about properties and shape relationships based on experiences (data collection -> observations) AND for students to learn how to defend those relationships (whether presented or deduced) through quality math talk.

Now, to bring it back to Dan’s comment. How do you decide what is appropriate for students to discover deductively? Is a traditional “I do, we do, you do” lesson ever appropriate? If so, when?

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Transformations and Chalk [MTBoS Mission #1]

The very first activity I did with my kids to start congruence was to go outside and get our hands dirty (literally).

Our medium: sidewalk chalk.

This was less of a problem and more of an activity, but it definitely sparked some great conversation. I started out with an activity for each rigid motion (reflection, rotation, translation) and then had discussion.

 

Reflections

Photo Aug 29, 9 01 18 AM

Directions:

Have a line drawn on the parking lot. Each student draws and labels their own point. “Walk directly to the reflection line and count the number of steps it took you to get there. Now, keep walking the same number of steps and draw your new point. Label it with the same letter and an apostrophe- this is read letter prime”.

Questions:

What kinds of paths did we take?

What about the distance from the line?

Are the two points we drew the same? Why or why not?

 

Rotations

Photo Aug 29, 9 01 09 AM Photo Aug 29, 9 01 22 AMDirections:

Make a point where you are standing. Now draw a circle around you and your point. Where you are facing, make a little mark on the circle and write 0. Now turn halfway around on your point. Where you are facing the circle, make another mark and write 180. Now turn halfway between your two marks. Make another mark labeled 90. Now, turn halfway around the circle, mark that 270.

Now stand on the line of your circle. Walk on the line until you get to the 90 mark.

Questions:

Did we move from our point? What is the point that we are standing on?

What do these numbers represent?

What angle got us to halfway around? What angle will get us all the way around?

Whose 90 is to the left of their 0? to the right? What are these directions called in this context?

Is this still rotation? How can we know?

Where is our reference point?

Can I rotate a negative angle?

 

Translations

Directions:

Simon says move (scholars ask move where?)

Sorry, Simon meant to say move to the right (how much?!)

Sorrrrrry, Simon really meant to say move to the right 3 steps.

Simon says move backwards 5 steps.

Questions:

What clarifications did you need to follow my directions?

Did you end up the same distance from the people around you? Would you have expected to?

Why didn’t you?

 

The discussions were really awesome and we ended up with this sweet set of notes. We were even able to come up with perpendicular paths for reflections (a little prompting from me) and it was great to be able to refer back to the activity throughout our congruence unit.

Photo Oct 14, 10 32 00 AM

 

I would love to hear your feedback- are there questions I should have asked but didn’t?