Monthly Archives: December 2015

Looking For Our Classrooms

A retiring colleague of mine gave the graduation speech last year.  He is someone who I looked up to a lot and always appreciated my interactions with him.    One point he made during his speech that struck me was when he spoke of how thankful he was to have spent his life in such a “spiritually challenging profession”.  He talked about how it made him a better person, and then a better teacher.  I understood myself to know what he meant because much of my own classroom clarity came from a similar realization.  There is a beautiful quote from a book I love by Robert Pirzig called Zen And The Art Of Motorcycle Maintenance which describes this spiritual challenge perfectly, albeit not from the perspective of a teacher:

“The application of this knife, the division of the world into parts and the building of this structure, is something everybody does.  All the time we are aware of millions of things around us – these changing shapes, these burning hills, the sound of the engine, the feel of the throttle, each rock and weed and fence post and piece of debris beside the road – aware of these things but not really conscious of them unless there is something unusual or unless they reflect something we are predisposed to see.  We could not possibly be conscious of these things and remember all of them because our mind would be so full of useless details we would be unable to think.  From all this awareness we must select, and what we select and call consciousness is never the same as the awareness because the process of selection mutates it.  We take a handful of sand from the endless landscape of awareness around us and call that handful of sand the world.”

It is so hard to know if you are really seeing your classroom for what it is – rather than what you want it to be or what bugs you about it.  Or as simply a collection of small events that deviated from your predictions of how you thought it would go, or heard it would go, or read it would go.  We share stories of our classrooms and those stories are a collection of things that stood out.  Little pieces of awareness that we brought into consciousness.  But who knows what we missed or how our consciousness mutated the actual events in the room.  They are by definition incomplete stories and maybe at the end of the day the best teachers are the ones who can reconstruct the clearest and most accurate picture of their class as a whole.

How do we thrive in a spiritually challenging profession?  These bullet points are my two cents:

Along the way I’ve realized that every time I felt out of control, or stressed, it could be alleviated by improved lessons, or new structures.  No doubt about that.  But somewhere below all of those great teaching strategies that we know work – there are still personal insecurities that need to be understood.  Teaching will make you confront those.  That’s the spiritual part.  I thank my profession and my students for exposing me to my own.   It made me a better person…  who then became a better teacher.




Letter To Myself – 2nd Semester Goals

This letter really is for me – I probably won’t even spell check it.  I’m certainly not going to tweet it.  I’m writing it because I feel I have to.  Even if only discretely – I want to go on the record somewhere as a challenge to myself.

My main goal for second semester is to increase the amount of thinking students do.

I want to decrease the amount of time they spend reproducing and practicing any particular procedure.

I want more of their procedural fluency to come from interesting questions, engaging tasks, or a conceptual understanding.

I need to be more purposeful with my questions.  When assigning tasks I want to have two good questions predetermined for each.

I want to keep track of each activity – specifically whether or not they had these three pillars of a great activity present in them:

  • open middle
  • low entry, high ceiling
  • solution method not immediately clear but solution seems attainable with effort.

Homework to be more review oriented.  1 review, 1 conceptual, 1 new.  Something like that.  Write it out a week in advance.

Continue to rely heavily on vertical non-permenaent surfaces and visible random groupings.

Give students time alone for quite contemplation before having them begin group tasks.

Focus on MP.4 and MP.3.

MP.4 Modeling:  What does it mean to model with mathematics?  Unanswerable Questions type openers.

MP.3 Viable Arguments.  – How often do you get students to compare and contrast approaches?  I want to maximize the effectiveness of this both at whiteboards and at their desks.

  • Meaningful discourse versus call and response, or low level transmitting of process.  An example of the level of communication I want to avoid:  I ask students to share with their partner I get this sort of dialogue:
    • Student 1: “I did this, and then this”
    • Student 2: “ok, I did this and this”
    • Student 1:  “ok”
    • Me: “Did every one get a chance to share with their partners?”
    • Student 1 and 2:  “yes”

Keep having fun and knowing that it’s impossible to reach every student.

Keep learning more and more and more about your students as people.

Read the standards you are being asked to teach.

Can we represent this differently?

I want to say CONVINCE ME a lot.

Focus on process versus answer getting.

Be more honest with myself when I am giving problems that have only one solution method.

Use Michael’s student teaching as an opportunity to study the effectiveness of lessons without having to facilitate the lesson.

Managing “flow” and allowing for the existence of “productive struggle”.

Allow the last 5 minutes of class for debrief / exit tickets.

Visit more classrooms.

Finish my presentation on modeling for TMC16.

Determine what a conceptual understanding of each topic would look like, and how would we use that to build procedural fluency?

Share everything with collegues.







Capture / Recapture – Proportional Reasoning Lab

This is a popular activity – here is a link to a post from Dan’s blog back in 2008 about the same lesson.  I’m writing about it now because I love it and want to remind people of it if they had forgot.

The lesson is great because…

  1.  It’s “real enough world
  2.  It’s a true mathematical model so
    1.  Students get to question the assumptions its based on
    2.  Students get to see what effect those assumptions have on the model
    3. Students get to validate the model through experimentation.
  3. Allows students to estimate the solution.  (Based on these results would we perceive the forest to have a high or low population of deer?)
  4. Has a high ceiling as part B above is a fairly high level evaluation of the model.

The lesson has a few drawbacks

  1.  It doesn’t have an open middle.  The only variation in methods comes from how they set up the proportion.

Brief overview of Capture / Recapture:   It’s a technique used to estimate populations.  For example, let’s assume we wanted to estimate the population of fish in a lake.  We would first capture and tag a set of fish.  After releasing them and giving them time to disperse back into the lake we would go back out and capture another set of fish and see how many of these recaptured fish had a tag from the first capture.  We would then assume that the ratio of tagged fish in our second capture, to the size of our second capture, is exactly the same as the ratio of fish we tagged in our first capture to the total population of the lake.

Is that model valid?  Turns out to be surprisingly accurate based on some interesting assumptions that the worksheet at the end will take the students through.

I use regular and pink goldfish to take students through a simulation.  In groups of two, each student gets a lake with a lot of fish in it, a lake with not very many fish in it, and a bag of pink fish.

Big and Small Populations


Then I have them capture 10 fish from each lake

(first capture)

(first capture)

Now they take each captured fish (by replacing them with a pink fish)

(tagging all the fish in the first capture)

(tagging all the fish in the first capture)

Each fish goes back into the lake (bag) and are given sufficient time to reintegrate into fish society (shake the bag).  Then go back out to the lake and again capture 10 fish.  Record how many of the fish in this second capture have a tag from the first capture.

(take a look at how many of the fish from the first capture are in the second)

(take a look at how many of the fish from the first capture are in the second)

Now setup your proportion and solve.  Math says that the proportion of tagged fish to non-tagged fish in the second capture is exactly the same as the proportion of fish captured in the first capture, to the total population.  That those little piles of fish up there are a model of the entire fish population.

Of course we want to verify our results so we can test the accuracy of our mathematical model.  After all we are trying to be the “least wrong” here, and ultimately accepting that our model doesn’t give the exact answer…  but it is close enough to be acceptable to us?  Let’s see, now I have them count to see exactly how many fish were in each lake:

(highly populated lake)

(lake with lots of fish)

(lightly populated lake)

(lake with a few fish)

Then the next day give them these two slides (also from Dan)

“What if this were the results of the second capture.  Would we predict the total fish population was large or small?” (red fish are tagged, blue are not tagged)

Screen Shot 2015-12-18 at 12.08.49 PM

“A lot or a little?”

Screen Shot 2015-12-18 at 12.09.07 PM

Ok that’s it.  Other than that I just recommend doing some research on capture / recapture because there are a lot of interesting research articles and youtube videos on the topic.

The Goods

Here is the worksheet that I give the students after completing the lab.  I’m pretty sure this originally came from Dan Meyer’s blog, I can’t find it on there now so…  yeah:




– B