Home Mathematics Modeling mathematical ideas: developing strategic competence in elementary and middle school

# Organizing a table tennis tournament (Unstructured problem from PRIMAS)

You have the job of organizing a table tennis league. Gather the information you need to organize the match. Plan how to organize the league, so that the tournament will take the shortest possible time. Put all the information on a poster so that the players can easily understand what to do.

I. Pose the Problem Statement: Pose questions. Is it real world and does it require math modeling? What mathematical questions come to mind?

• 1. How many players will take part?
• 2. Are these single or double matches?
• 3. What is the rule of who plays whom?
• 4. How many tables are there?
• 5. How long will each game be?
• 6. What time will the first match start?

II. Make Assumptions, Define, and Simplify: What assumptions do you make? What are the constraints that help you define and simplify the problem?

III. Consider the Variables: What variables will you consider? What data/information is necessary to answer your question?

Seven players will take part. All matches are singles.

Every player has to play each of the other players once.

There are four tables at the club. Games will take up to half an hour.

The first match will start at 1.00 p.m.

IV. Build Solutions: Generate solutions.

Formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables

V. Analyze and Validate Conclusions: Does your solutions make sense? Now, take your solution and apply it to the real world scenario. How does it fit? What do you want to revise?

VI. Present and Justify the Reasoning for Your Solution.

In our professional development for teachers, we gave them the unstructured task above of organizing a table tennis tournament. One of the teachers shared his work stating, “Before starting the table tennis problem, I realized that it was asking for a problem-solving process identical to the one needed for a common issue of mine: creating a sequence/schedule of assigned seats throughout the school year so that all students are able to work with and/or sit next to each of their classmates at least once.” It was interesting to see how the teacher was recognizing that he uses mathematical

Figure 3.5 Teacher working through an unstructured mathematical modeling task. Source: Authors.

modeling in his work as a teacher. The teacher represented his thinking using a diagram and a table (see Figure 3.5).

Another structured way this problem could have been posed is featured below.

You have the job of organizing a table tennis tournament. Seven players will take part. “All matches are singles.” Every player has to play each of the other players once. Call the players A, B, C, D, E, F, and G. Complete the list below to show all the matches that need to be played.

A v B, B v C, ..., A v C, B v D ...

There are four tables at the club, and each game takes half an hour. The first match will start at 1.00p.m. Copy and complete the poster below to show the order of play, so that the tournament takes the shortest possible time. Remember that a player cannot be in two places at once! You may not need to use every row and column in the table!

Take this same task and consider this setup above to the more unstructured version of this problem presented before. What would be the benefits and challenges of each version of the problem?

Many of the teachers appreciated the nature of the open-ended task stating that it was easier to solve this problem in an unstructured way without being constrained by someone else’s approach. They stated that

“I loved that I was not forced to use a specific strategy and that I could model my math thinking the way that works best for me!”; “Multiple approaches adds to everyone’s understanding of the concepts.”; “If we want our students to become flexible thinkers, we need to allow them opportunities to do this.”

As they reflected on solving this problem, teachers saw the value that this type of modeling has allowed students to think flexibly, create their own problem-solving strategies, and become better thinkers.

How does considering these different yet important ways we model math ideas help develop “Strategic Competence” for students and teachers? Strategic competence has been defined as one of the important strands of mathematical proficiency, as the “ability to formulate, represent, and solve mathematical problems” (National Research Council, 2001, p. 116). This strand includes problem solving and problem formulation, which requires students to solve a problem by representing it mathematically: numerically, mentally, symbolically, verbally, or graphically. The key attribute for someone who has strategic competency is flexibility in their problem-solving process and strategies.