Wednesday, November 11, 2015

Counting Circles Brainstorm

Let it be known that Sadie Estrella is a Hawaiian treasure.

She made her way north for SUM2015 in Saskatoon and I got the opportunity to learn from her about counting circles (as well as share an eventful dinner). 

It is probably good to understand her work on counting circles before reading a couple of ideas I had during her session. 

I went to her blog and searched for #countingcircle, and the results can be read here

Use this time to read Sadie's work

A couple things struck me while she was talking:

  1. She is so honestly passionate. You can tell that she cares when she talks. I immediately felt comfortable as the "student". 
  2. She had such a precision of language. At one point, she noticed herself saying "anything else?" and purposefully corrected it to "other ideas?" Such a little detail that makes the idea generation flow. 

As she was talking, I started to think where this routine would fit in my room. I teach high school, so most of my students are past counting 10s, but still struggle with numerical flexibility. I had two ideas that I would love to try in my room. One is a structure, and the other is a conversation.


Sadie had us counting by 11s and dissecting our thinking. She modelled similar counting circles with decimals, fractions, qualitative sections of time (i.e. quarter of), etc. One thing that I thought would be interesting is attaching a function to the circle. So as the students were counting inputs (0, 1, 2, 3...) they were audibly giving the output of the function. 

If we started with linear, my hope is that students would see the pattern of outputs quickly. Then we could have them skip input numbers (possibly for functions like y = 1/2x +3). Why is it easier to skip? As the group gained fluency, it would lead into a profitable conversation...


Certain counting gaps are simpler than others, and that may result in an increase in speed around the circle. This analysis of speed is a fantastic conversation. Once students see the pattern, the counting will speed up. Stopping the flow and asking, "Why can we do this so fast?" might be an effective stem to get kids talking about the patterns they've noticed. 

Keeping track of the circle speed would be a great assessment tool for pattern recognition. The danger here is equating speed with standardization and getting away from the good conversations that occur around the circle. The goal shouldn't be to go fast, but going fast is a symptom of some great mathematics. 

If you cheated and didn't read Sadie's work, this makes no sense to you. I did that on purpose. Go read it now. 

Now don't you wish you had done that to begin with?

The routine is simple; it can embed into your room as a daily or weekly activity. It makes thinking visible and connects learners. Win-Win. 


Sunday, November 8, 2015

Clothesline Series

I joined a middle years math community organized by my school division. I have a growing interest in the transition of students from middle school to high school because many of the tasks I use or create get at middle years content. I'm wondering what knowledge students come to my room with and what atmosphere it was learned in. Both have huge impacts on how students operate in my room.

I was surprised to hear that middle years teachers lamented that students could not use number lines. I use number lines as a support in my high school classes because I (ignorantly) assumed that this was an accessible tool from their elementary days. As it turns out, what I thought was making things easier for kids to conceptualize, probably was causing cold sweats and night terrors.

Typical day in the office for me.

When Andrew Stadel blogged about clotheslines I had to investigate further.

It wasn't long until my intern and I had retractable clotheslines attached to our walls.

We started with Grade 9 fractions and decimals. We wanted to include a plethora of reasoning situations:
Like denominators
Unlike Denominators
Reducible fractions
Negative and positive
Terminating and repeating decimals
Equivalent fractions and decimals

I refined the resource creation process by using PowerPoint and then saving as a .pdf with two slides per page. I then brainstormed several situations where number lines would facilitate useful mathematical reasoning and discourse. They range from early grades to pre-calculus (with the ordering of radians and degrees). 

The complete "series" of slide sets is below. They include a brief instruction slide. My hope is that they are ready to print, cut, and use. 


Algebraic Thinking
Domain and Range
Improper Fractions
Fractions and Decimals
Radians and Degrees

How I used them:

I installed clotheslines onto my walls, but writing the line on the board or tacking a string across a wall would work as well. Sometimes I started by placing a few referents, but as they got better, I allowed students to begin conversations by placing the referents. I then handed each student a number one-by-one and asked them to talk through their reasoning as they pinned it to the line. Students were free to interact, but not interrupt. 

The key is the attention to reasoning and stress on mathematical communication. If I give more than one student a value at a time, it becomes a collection of independent (and silent) reasoners. I want students to argue about issues like sequence, scale, inverses, distance, etc. 

I ran two clotheslines simultaneously with a teacher asking questions alongside each.


Print extra blank values, and have students pair up and write a value for their partner to place. 

As an exit ticket, give students a value and have them write to you where they would place it and why. (I use journalling and portfolio work regularly in class, but a stand alone activity would be great also).

Change the scale by moving a couple referents. Ask the students what would happen and to adjust the values.

Any other curricular topics, classroom structures, or extension ideas are welcome. I've found that the impermanent nature of the placement encourages sense making, mathematical talk, and risk taking. Any activity that checks those three boxes is great at any level. 


** Check out Mary Bourassa's Log Clothesline here **