### A Golf Ball Mathematics Inquiry

This series of lessons started with a simple premise, but with a deep desire to get my students thinking more mathematically; and reflecting on their thinking.  I have been trying to get more Meta-Cognition (or Meta-Learning, as I frame it with my students) into the classroom.  This was something that I tried to weave into the activity, and not necessarily tack it on at the end as something to think about.

As for the subject of golf, there was no particular reason to choose it.  I am not interested in golf.  My students are not interested in golf.  Their parents are not interested in golf.  The reason I choose it was because I found a bucket of golf balls in our school’s multi-purpose room, and it sparked a question in my head (which I explain below).  Here, I am trying to model inquiry in daily life for my students.  These things make me curious, what makes you curious?

The actual work that was done in this series of lessons took place over several days, but for the sake of distilling it down to the main essence, I will describe it as one continuous lesson (which I wanted to do originally, but too many external forces kept pulling me away from it).

Step One – Building the Environment

I brought in a small bucket of golf balls and let them all touch, feel, and make observations about the balls.  We made a list of characteristics that golf balls have, and tried to get some new vocabulary to use in the problem.  I asked them questions about golf balls and asked them to defend their answers; How many golf balls do you think the average golfer uses in a year?  Imagine a golf hole with a little lake, some dense trees and a gully.  Where would you find a lot of balls if you looked?  How many golf balls do you think a pro golfer would lose in a year?  A lifetime?

The point here is not to use the skills we are going to practice, but rather to embed ourselves in the environment, and to engage with the subject, even if we have no interest in it!  Golf is not something that kids are all that passionate about, but it was quite fun to step into the world of golf and imagine what it would be like.

Also, there are some hidden skills that will be necessary later on; things like making assumptions and backing them up, estimating with reasons, and multiplying out.

Step Two – Setting the Problem

Here I presented the materials we would use, and I set up the problem.

“Here are your tools, an empty shoe box, a bucket of golf balls, and a pile of rulers, paper, and pencils.  Using only these tools, we are going to try and figure out how many golf balls can fit into this classroom, assuming all the furniture has been taken out.  You will be in teams of three, and you may not ask the teacher any questions.  You may, however, ask the other teams in the class as many questions as you wish, but they can only have yes/no answers.  This is a race.  Alright?  Get to it.”

There are a couple of things to note here in the set-up of the activity.  First, the materials were selected to be a clue to one method of doing this problem.  They are meant to offer some sort of support, and a spark to get ideas flowing.  There are multiple methods to solve this problem, and as teachers we should encourage kids to use all the possible methods they can think of.

Second, there are some rules in here that seem odd, but they have a distinct purpose.  Brent Davis at the University of Calgary is a great researcher and educator who is bringing complexity science/thinking to the field of education.  He calls these Enabling Constraints, rules that prohibit students from doing one thing, but in turn, open more possibilities in a different direction.  They purpose of my rules is to prevent children from relying on me (hence, no questions to the teacher) and instead, communicating with each other.  However, I don’t want them to just copy the kids who they think will get it, so by specifying questions that have yes/no answers, I am forcing them to think before they ask, to be specific about their questions, and to focus the point of their questions.

Lastly, the concept of competition in not something I do often in my class, so the addition of a race in something that I experimented with.  My ultimate goal when I put competitive elements into the classroom is to have the students so engrossed in what they are doing that they forget this is a contest, because knowing the answer is more important than winning or losing.  Another reason for competition is too have the students under a certain amount of stress or pressure.  Too much stress negatively impacts performance, and too little stress has the same effect.  Trying to find that balance is what I am looking for, and it something I have not yet figured out in regards to my own practice.  This is just my own experimentation.  I will this say that when I did this project with my class, the idea of contest disappeared and was forgotten.

Step Three – Meta Learning

Before we get to our work, we are going to understand what we are doing, and what skills will be required to accomplish the task.  I always give them this template before we start, and either I fill it in beforehand, or we do it as a group.  The skills and attitudes we focused on during our investigation were:

• Resilience

• Teamwork

• Organization

• Communication

These are the areas that the students recognized as important to the problem.  I usually try and keep the list down to 2-4 skills that we directly work on or talk about.

After we have set our skills, have the kids talk to their group mates about what these skills look like, and how as a team, they can support each other and help each other improve in these areas.  Share any insightful ideas with the class.

Step Four – The Work

Here is where the students start to investigate and figure out the puzzle.  Some of them will pick up right away what the purpose of the shoebox is.  Others will need some time and will need to watch to get the ideas down.  From the teacher’s perspective, this is a great time to start wandering around and supporting students in their use of meta-learning.  Have them reflect on how they are communicating, what strategies they are using to organize their work, and how they are dealing with adversity (or whatever skills you choose to highlight).  Make notes for the carpet discussion later.

Step Five - Communication

Now that most groups have figured out the methods and are on their way to the answer, tell them that as an added element of math communication, they will be required to design a poster showing all their work and thinking.  It must be so clear, that any stranger could look at and know exactly what the problem was, how we did, and the solution.

Step Six – Math Reflection

Put all the completed posters up somewhere in the class and have a silent gallery walk.  The kids will walk around and take notice of the strategies that other groups used in silent concentration.  After everyone has had a chance to think about each poster, have them sit with a partner (from a different team) and debate methods.  This is where we reflect on our use of mathematics and our mathematical organization.   Which poster was more effective?  Why?  What is easy to understand about it?  What is difficult to understand? What would you do differently next time?  Switch partners as many times as needed.

Step Seven – Meta-Learning Reflection

This would be our final reflection for the activity, relating it back to the skills we discussed at the beginning of the lessons.  Bring the students back to those skills and attitudes and have them with partners talk about how they did.  I also have a large group discussion on the skills where everyone can make observations about themselves or about others.

Finally, the last step is a written reflection.  There are a ton of templates and ideas out there, but one that I tired with this activity was a PMI (Plus, Minus, Idea).  Remember these are related to Meta-Learning skills, not the math work.

Plus – What did I do well?  What am I proud of?

Minus – Where was I not successful?  Where did I struggle?

Idea – What will I do different next time?