# To Motivate STEM Students, Ask Them Better Questions

What do these four questions have in common?

1. Can all the children of Lake Wobegon be above average?

2. On average, do your friends on Facebook have more friends than you do?

3. Do credit cards make you gain weight?

4. How do I estimate distances to nearby stars?

They are all math questions for high school students. They differ from typical school math problems in that they are phenomenon-based and stated as real-world problems, not as math exercises. More and more educators these days are pushing for such phenomenon-based problems to engage the students, to get them excited about STEM, to advance their critical thinking skills, and to make math and science more fun.

A recent research summary report from Toronto summarizes well the arguments in favor of phenomenon-based learning. According to the report, “a large majority of students find mathematics ‘boring, mostly irrelevant and unrewarding.’'” Phenomenon-motivated problems engage students more directly, offering relevance, opportunity for critical and careful thinking, and even joy. Many students who find math boring view it as merely a sequence of formulas and solution algorithms to memorize. Read a homework problem, recognize the pattern, match it to a recently learned procedure, and then mechanically “plug and chug.”

But, as articulated in the Toronto report, mathematics is not plug and chug:

Mathematics involves learning to problem-solve, investigate, represent, and communicate mathematical concepts and ideas, and making connections to everyday life. … Problem solving is a foundational building block for learning mathematics.

Let’s go to Lake Wobegon, that fictional Minnesota town. Is it possible or impossible for all the students of Lake Wobegon to be above average? Imagine an active classroom, where the teacher becomes mentor and coach rather than lecturer. First, Mike writes on the blackboard the mathematical definition of average, a sum of N numbers divided N. There is active discussion. Virtually everyone is saying, “No, it’s impossible.” But how do we demonstrate that?

Eventually, Susan exclaims, “I’ve got an idea. Let’s 10 of us stand in front of the blackboard and arrange ourselves in a line from shortest to tallest. We find the average and demonstrate that we can’t all be taller than average!” The class cheers as they do the exercise and successfully show the result. But then the teacher breaks into the class: “News Flash from the Lake Wobegon Chronicle, ‘Every member of the senior class of Lake Wobegon High School has been accepted into an Ivy League university!’” Now, is it possible for all the seniors of Lake Wobegon to be above average? I leave it for the reader to imagine how the class responds.

A plug-and-chug approach to computing averages would not have engaged class so actively. Here, “the answer” — what we want students to discover — is not simply the answer to a problem, but rather the process itself, the process of investigation and creative critical thinking. Math is not simply inputting data into a computational algorithm and recording the numerical result.

Returning to the Toronto report, in genuine problem solving the “solver is working on a question where the solver does not know a direct path to achieve the goal.” It “is all about coming up with original thoughts, not about practicing drills or algorithms.” Once a student graduates from formal schooling, she or he will likely work at a job requiring STEM skills. Each day will present new challenges. Rarely if ever will the next day’s challenge be fitted to the pattern found in chapter 3.2 of some textbook. The young professional will need skills to frame, formulate, and eventually solve that challenge. This requires knowing not only how to recognize patterns and use the “algorithms,” but also, more fundamentally, the core ideas behind the algorithms. And the solution process may turn out to be an interdisciplinary one, perhaps involving — in addition to math — physics, chemistry, or even the humanities.

Next challenge: “On average, do your friends on Facebook have more friends than you do?” Wow, that’s a puzzler! Perhaps the teacher would suggest dividing students into small teams and creating “friendship networks” within each team. The networks would have to reflect the randomness of actual friendship networks. We don’t want everyone to be friends with everyone else — too boring! The students work out a way to do this. Then they draw a picture of their group’s friendship network. Then they have to decide exactly what the term “on average” means in this instance. To their surprise, they will discover, if they formulate the problem properly, that each and every friendship network in the class will have this counterintuitive property. We have experimental verification, but how do we prove it mathematically? That’s a central question, and the process of discovery takes the students though a bit of graph theory and senior-level mathematics.

All four of these phenomenon-based questions involve active, inquiry-based learning. Two weeks later, or even two years later, students will remember these learning exercises far better than if the same material were delivered to them in a routine lecture.

Once more, the Toronto report:

The classroom needs to be a place of investigation by supporting unusual ideas and responses by students. … [It] should feel like a community where ideas can be discussed, developed, debated and understood. Students should feel that all ideas are welcome in the classroom, even those that are unconventional.

*Richard C. Larson, a member of the National Academy of Engineering, is Professor of Data, Systems, and Society at MIT. He serves as PI of MIT BLOSSOMS, an OER (Open Educational Resources) project that makes freely available, phenomenon-based, interactive video lessons for high school STEM classes. The four lessons discussed above (plus hundreds more) are available in interactive video format at **http://blossoms.mit.edu**.*