By Alessandra King, Posted December 12, 2014 – 

How many jelly beans are in the jar? This looks like a typical Fermi problem.

Fermi questions, such as found on http://www.nctm.org/publications/article.aspx?id=20696, are open-ended questions that require the solver to use common knowledge and back-of-the-envelope calculations to estimate quantities about which he or she knows very little. They are one of my favorite types of problem because they foster problem solving, estimation, and number sense. (I will discuss them more in my next blog post.) However, this type of prediction problem also gives us the chance to do a “social” experiment: What happens when we take into consideration the results of the “team,” the entire group of students who participate in the challenge? Now the Jelly Bean Jar problem (and similar Fermi questions) can become a paradigm for studying teamwork and collaboration. 

The NCTM Standards and the Common Core Standards, as well as other frameworks for teaching and learning (for example the Partnership for 21st Century Skills), mention collaboration and communication as well as problem solving as critical skills that students should master to function well and succeed in the twenty-first century. The issues they will have to face are complex: The simple problems have been solved; they have been left with the difficult ones. Matters such as freshwater scarcity, gender equality, and space exploration are not problems that can be solved by any one individual. Therefore, the value of teamwork becomes ever greater. It is well known that a team or a collection of people often makes more accurate predictions than the individuals in it. This phenomenon, called the wisdom of crowds, depends both on talent and on diversity. In fact “the collective accuracy of the crowd depends in equal measures on the accuracy of its members and on their diversity,“ as shown by the Diversity Prediction theorem. A diverse crowd will always be more accurate than its average member and sometimes more than any member in the crowd, as Scott E. Page explains in this video. Diversity here is intended as cognitive diversity, referring to the differences in how we think—the categories and models we use to encode problems, the tools we employ, and the diverse perspectives we apply to solve them. Cognitive diversity is connected with the process engendered by different training, experiences, interests, and outlooks.

So how did my middle school students fare at predicting the number of jelly beans in the jar? Well, there were about 80 individual guesses, ranging from 168 to 8,000, based on a large variety of assumptions and approaches. The average of the team was closer to the real number of jelly beans (1,817) than 85 percent of the individual estimates.

Although often the “crowd” does very well indeed, we do need to be aware of the limitations of group decision making for teamwork to be as powerful and effective as it can be. Fostering diversity and independent contributions and avoiding “group think” and overconfidence can lead to better decision making. Jelly beans provide lots of food for thought!

Alessandra King, Alessandra.king@holton-arms.edu, studies mathematics with her students at the Holton-Arms School in Bethesda, Maryland. She has taught mathematics and physics at the middle school and high school levels and is interested in creative problem solving, critical thinking, and quantitative reasoning.

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