The Sport of Thinking
by Mary Meier
Middle school students feel they are unique beings. They want to be perceived as unlike the other members of their families, yet identical to their friends. Students gain much of their identity by trying to fit in with their peers. Their conformity is seen in the same clothes they wear, the same music they enjoy, and their ever evolving (but identical) hair styles. They conform not only to belong but to avoid standing out. This trait extends to their experiences in the classroom. Many middle school students are shy about asking or answering questions during class fearing they will be viewed by their peers as "dumb" or, sadly, "smart."
Math classes tend to exacerbate this condition. Mathematics is a science based upon asking and answering questions. Students are seldom given the opportunity to recite information they have learned; rather, they are continually asked to use information to solve problems. As social as middle school students are, they can become quiet in a mathematics classroom. They would rather agree with everything the "Math Whiz" of the class has to say rather than offer an explanation, opinion, or question. To help our students overcome their silence, our challenge as teachers is to help students evolve into "independent thinkers."
Alverno College, a small liberal arts school in Milwaukee, has become known for producing graduates with the power to think independently. The curriculum extends beyond textbooks and focuses on developing eight important abilities, including problem solving, social interaction within cooperative groups, and effective communication skills. As an Alverno graduate, I was expected to express my opinions, work with others to solve problems, and ask and answer complicated questions.
Before I completed a project, paper, or assessment, I was required to fill out a self-evaluation to reflect on the abilities I used and the steps I had taken along the way. I received written critiques of my work, instead of a traditional letter grade. These critiques would often fill an entire page. I was then encouraged to resubmit my work, or simply take note of the abilities which helped to create success.
I feel my accomplishments as a teacher have been enhanced by being educated in an environment which focused on learning and creative thinking. I know first hand how my interest level and critical thinking skills soared when given an opportunity to become an active learner. I see the same benefits when my students work with others, solve "real life" problems, evaluate performances, write, speak, and use manipulatives to think about math and its relationship to the world.
The five "W" approach
This transformation does not occur overnight but develops quickly once students realize the only way they can escape a problem is by solving it! Unfortunately, many students arrive in my classroom with good problem solving strategies, but an apprehension to share them in front of their peers. I use a "five W" questioning approach to help all my students engage in problem solving discussions while instilling self-confidence.
The best question we can ask our students is simply, "HOW did you solve this problem?" Often, students are simply asked to state an answer, ignoring all of the mathematical concepts and effort that went into the solution. When students are required to communicate their thinking process, they take pride in the steps of the process as well as the solution.
Because each problem may be solved in multiple ways, it is important to also ask, "WHO was able to solve this problem differently?" Students are given the opportunity to share their ideas and to see different ways a problem may be solved. This question gives me tremendous insight into the thinking patterns of my middle school mathematicians. It also allows me to quickly spot unconventional and creative thinkers.
Next I ask, "WHY did each approach yield the same results?" For instance, my students recently shared with me many different strategies they used to calculate the perimeter of a rectangle. One student told the class his group "added together all of the sides." Another group stated that since multiplication is repeated addition, they multiplied the length by two, multiplied the width by two, then added the products together. A third group found an alternate method by adding the length and the width together first, and then multiplying the sum by two. Through our discussions, my students were able to tell me the last two groups found identical perimeters because of the Distributive Property of Multiplication over Addition.
Unfortunately, not every group will be successful with each problem solving opportunity. It is then important to help students learn from their mistakes by asking "WHERE did our slip ups occur?" Students who "almost got it" are encouraged to share their partial successes. Most students find they made a common mistake. They find comfort in knowing that they were not the only mice to be caught in a trap!
Finally, I have students evaluate their problem solving approach. I ask students "WHAT technique will you use next time you are faced with a similar problem?" Students evaluate and compare their methods to find the most effective method. Students find the most efficient way to solve the problem and mentally file it for future use.
When middle school students feel comfortable expressing their problem solving ideas, they will take over the teachers role and ask questions of themselves and others. More importantly, they will begin to see math as a "sport of thinking" rather than merely a "numbers game." Albert Einstein once wrote, "The important thing is to not stop questioning." Encourage your students to follow his lead in search of new meaning and answers.
- Mary Meier teaches math at Oak Creek East Middle School in Oak Creek, Wisconsin 53154