What Sane Person Writes a Novel about Fractions?

fractions“Mathematics is boring. Why do we have to study it anyway?”

“Do I REALLY have to finish all my Math homework? It’s just the same stuff over and over, and it makes no sense to me.”

“I hate doing these algebra exercises. They are SOOOOO boring.”

“I don’t understand this. It’s stupid. Why can’t I do something useful with my time?”

If you are a parent or teacher, then you have probably heard it all. The whining. The complaining. The angry outbursts. For a logical and emotionless subject, Mathematics has an uncanny ability to draw passionate responses from young people. It’s seldom a “YAY, I have Math homework” kind of response. No, it’s more like “ARGHHHH, I HATE Math!”

So why would anyone choose Mathematics as a starting point for a youth novel? And note that we are not talking about some mystical and captivating mathematical subject like String Theory or Equations of Relativity. No, sirree! We like a challenge, don’t we? Out with the exciting stuff, so we can sink our teeth into a common, garden-variety subject: fractions. Yes, you read that correctly. FRACTIONS. Not eye-popping fractal mathematics, mind you. Just regular fractions with numerators and denominators: those little number beasts you encountered way back in grade three of four.

Fractions in all their simple glory were the starting point for “Fractonia”. So is this a story about fractions? (Are you yawning and shaking your head in disbelief?) Yes, but probably not in the way you think. When I was at school and fractions were introduced to the class, the teacher talked about picking apples from a tree. (That was in the days when children actually went outside and climbed trees, so students could relate to the image of apples hanging from a tree.) More recently, while doing research for a new project, I took a look at some junior school materials focused on fractions. The apples were gone from the chapters about fractions. In their place, the reference to pizza slices appeared more often than anything else. (It seems the “an apple a day” phrase has been replaced with “a slice of pizza a day“.) Imagine this pizza being cut into pieces. Your friend eats one slice. You eat five slices. What percentage of the pizza is left? Is this stimulating your imagination and encouraging learning, or is it just making you think that you are hungry? Is there a different way to visualize fractions?

I set out to create a story that would give readers an entirely new perspective on fractions. Why? So they could better understand fractions? No – so they could know it is possible for something as “boring” as fractions to become interesting just by changing our perspective. I wanted to paint an imaginative picture over those sad pizza slices with no story to tell – a picture so unexpected that it would encourage readers to create their own imaginative ways to view subjects they found “boring”. In my experience, an interested student is more likely to learn and overcome learning difficulties than a student who is bored with the topic they are studying. In her article entitled, “How the Power of Interest Drives Learning“, Annie Murphy Paul says the following: “When we’re interested in what we’re learning, we pay closer attention; we process the information more efficiently; we employ more effective learning strategies, such as engaging in critical thinking, making connections between old and new knowledge, and attending to deep structure instead of surface features. When we’re interested in a task, we work harder and persist longer, bringing more of our self-regulatory skills into play.” If we learn better when we are interested, why not find a way to make what we have to learn interesting and engaging? It seems like a simple, common-sense way to ensure we learn more and enjoy doing it.

At its core, “Fractonia” is less about Mathematics and more about attitude. Yes, fractions are part of the story, but no, the story is about exploration, discovery, and possibility. It’s about taking responsibility for our own learning. Our learning is not our parents’ or our teachers’ responsibility – it is OURS: yours and mine. We don’t have to wait for our teacher to make the subject exciting or justify why we should study it. We don’t need to be entertained before we can learn something. No, the process of learning is an adventure that can be created and hosted inside your very own imagination. Go on the adventure, or stay home and mope about how boring everything is.

If you have never had a teacher show you HOW to create your very own learning adventure, sit down and read. But don’t read to be entertained – read to discover. Read so the book can become your teacher and show you how to create your own learning adventure. “Fractonia” is my adventure with something as simple as fractions. Other authors will take you on different adventures. You may not be ready to write your adventure in a book, but you are ready to have an adventure. There are no age limits on learning – we never outgrow a good adventure. What will your next adventure be?


Engaging the reflective mind

The image of a good problem solver is one of an intelligent person. But years of teaching Physics to intelligent young people has convinced me that intelligence isn’t the only criterion for a successful problem solver. Common sense is perhaps more critical than many have recognized. Without it, an intelligent individual may have a lot of knowledge  and the capacity to make complex connections, but may simply lack the practical wisdom to apply it appropriately. Unfortunately, recognition of the fact that common sense is critical to effective problem solving is where most people stop. Possibly this resistance to dig deeper is due to an underlying belief that if you don’t have a whole lot of common sense to begin with, you are never going to get more.

Daniel Willingham, in a recent article, raises the question of whether common sense can indeed be taught. Willingham debates this question from a psychological perspective and eloquently references psychologist Keith Stanovich who, in his new book What Intelligence Tests Miss, offers a way to understand the difference between intelligence and common sense.   Stanovich sticks to a more traditional definition of intelligence that focuses on the ability to solve problems and make effective decisions. Stanovich suggests that there are three components to the cognitive system that handles these functions: the autonomous mind (which engages in quick thinking based on simple associations and past experiences), the algorithmic mind (which processes information by making comparisons and combining concepts), and the reflective mind (which interprets goals and beliefs and determines appropriate actions to achieve those goals). What most people don’t realize is that typical intelligence tests measure the efficiency of the algorithmic mind, but fail to consider the moderating effect of the reflective mind.

To problem solve effectively, you don’t only need to decide which facts should be combined to generate a solution. You have to test and adapt that selection (made by the algorithmic mind) to the situation at hand. In other words, the solution needs to fit into the environment of the problem, or the solution will never be practical. And this is the job of the reflective mind. According to Willingham, “You need to see your environment for what it is, you need to set realistic goals, and you need to select actions that move you towards those goals.” Intelligent people (categorized this way by typical intelligence tests) don’t always successful engage their reflective minds (the source of common sense) to determine the appropriateness of their solution. The result? Intelligent people are not always naturally good problem solvers. But could they become good problem solvers? To the critical question, “can common sense be taught?” Willingham’s response is “To some extent, yes. With sufficient practice, people can come to recognize the types of errors the reflective mind makes, and learn to avoid them.”


Common sense – the stepping stone to successful problem solving

“My kids seem to have no common sense.  What do I do?”

 Before we consider whether common sense is something that can be acquired through exercise or practice, a curious reader may well ask, “But how do we know if we (or our children or students) have enough common sense?”  How much is enough?  Was I born to struggle with issues that require common sense?  Is my lack of common sense just the result of my genetic coding?  And why is problem solving hampered by the absence of common sense?  Can’t I find a way to become good at problem solving without growing my common sense?

These questions introduce complex topics that promise to weigh down the most athletic mind.  It’s easy (and extremely informative) to get caught up in the theories and debates that psychologists and educators invest themselves in.  My experience, however, is that most parents and teachers need practical solutions that will make learning easier for the children, and not a bunch of theoretical textbook quotations.  So here, we will rather focus on the practical issues, and how to overcome real-life hurdles that keep students from succeeding.  Parents and teachers may find they identify a little better with their children and students if they first challenge themselves to a fun, common sense test (an example is found at http://www.kathimitchell.com/commons.htm).  The score doesn’t matter nearly as much as the insight this test will offer us into recognizing why common sense is so very important in the problem solving process.


Is it possible to put common sense back where it belongs?

For many students, something very fundamental is missing from the problem solving process. Many of my students simply couldn’t see the “obvious” as it glared at them from the question paper in front of them. Because they missed the simple sign posts that point the way to the solution in a problem solving activity, they quickly became hopelessly lost, and almost all would give up the moment the hopelessness attacked.

It took a while to realize that many of the students who struggled with the challenges of science or mathematics were not tripped up by a lack of knowledge of the subject. They knew the facts – they just didn’t know how to make the facts evolve into a solution to a fact-related problem. There are a number of reasons for this happening, but from my observation, the most common problem is that the students simply missed the “obvious”. It’s not that the students were rebelling against “common sense” just for the sake of rebelling. Most students simply had no idea that they lacked that vital ingredient to successful problem solving, that simple human quality which previous generations called good, old “common sense”. Sadly, it has become apparent that “common sense” is no longer common.

If common sense is missing, is it possible to put it back where it belongs? As we explore the process of learning, we will try to answer this question.


Where is the gap in my education?

As a teacher of Physics, I have spent years searching for ways to make complex concepts simple to grasp. I have looked for ways to make the learning process easier. And I have studied my students, listened to them, watched them, and experimented with different ideas to see which will enhance their understanding. In this process, one thing has never ceased to amaze me. In speaking to colleagues in similar study fields, I discovered that I was not the only one to notice this strange “phenomenon”. What astounded me was the gaping hole in the education of my students, and the frightening thing about it was that most parents and students didn’t seem the least bit concerned about it. What was missing? Common sense. Common sense? Surely I am mistaken? Everyone has common sense – it comes with the being human, right?