Math Circles: A Structured Approach to Problem Solving Paul Carreiro and Kelly Zinck MTA 2010

What are Math Circles? Similar to Literature Circles where each student is assigned a role in the group Flexible and adaptable Use the roles most appropriate for the problem the group is working on. Can be used to solve closed (one answer) or open ended problems (many answers – tell me everything you know about area and perimeter)

The Roles Situation Summarizer Vocabulary Master Idea Person Model Maker Computation Kid Connection or Link Master * complete as a group Path Finder * complete as a group

Why Use Math Circles? Problem Solving and Reasoning Moving through the Five Representations in math Effective Communication Student Engagement

Classroom Use… Have used literature circles Looking for a problem solving approach that would engage, be collaborative, accommodate a range of learners

First steps Initial question: Given a sample, how many jelly beans are in a bag. Initial roles: Situation Summarizer, Model Maker, Computation kid, vocabulary master, Idea person

First discoveries Some roles are hard Collaboration is hit and miss Some give up easily They can’t model; vocabulary is weak It takes a long time They talk a lot and write a little Some interesting, unexpected problem solving

Next steps Added vocabulary section to their notebooks Did some research on modelling; in effect had to model how to model Lots of guided reading of the text Evolved into a once a month activity

This led to… Tried toothpick problems (move the toothpick to make a square Tried problem solving in geometry (make art that rotates, reflects and translates) Discovered MangaHigh (spaceship game) Narrowing of roles: Summarizer, connections maker, idea person, computation kid, model maker

Signs of Progress Increasing engagement Longer, more focussed talk Perseverance

September Fermi problems It’s all about you project Counting hair Ancestors

Where we are and where we are going Four roles: Ideas, Computation, Summarizer, Materials Like to reintroduce model maker Talk is excellent; writing still weak Assumptions and conditions- a new idea Perseverance strong; engagement high

What is a Fermi Problem? A Fermi problem is a multi-step problem that can be solved in a variety of ways, and whose solution requires the estimation of key pieces of information. How many piano tuners are in Chicago?

Setting the Context Fermi problems at first might appear not to have an answer. Your initial response may be “I need more information or there is not enough information.” There are many different ways to solve a Fermi problem. Be creative. Use any necessary tools. You may use a variety of estimation strategies and need to take a few risks. Don’t be afraid to piggyback on someone else’s ideas. You will be working as a team and sharing strategies and ideas, encouraging and supporting each other using social skills that we will reinforce each day.

Fermi Problems How many water balloons will it take to fill the school gymnasium? How many jellybeans fill a one litre bottle? How many hairs do you have on your head? How many golf balls will fit in a suitcase? How fast does human hair grow (in km/hr)? Assuming one Santa Claus visits all children on Christmas, how fast would he have to travel? How many books are in a bookstore? How many hours do you have to work to pay for a car? How long would it take to count to a million? How long a row of pages would be needed to show a million stars? How many CHEERIOS are there in a box?

Let’s try! Please arrange yourselves into groups of 4 and pick a problem to try out math circles. Pens come in packages of 3, 5 and 8. Mrs. M bought 26 pens for her class. How many packages of each type might she have bought? In a pattern, you add two numbers to get the next number. If the 9 th number is 212 and the 10 th number is 343, what is the 1 st number?

Final Thoughts This idea cam from an article published in Mathematics Teaching in the Middle School Clean copies of the role sheets and a copy of the presentation will be available at We would love to hear stories of how you tried Math Circles in your classroom! You can leave a message on the website.