Practical Tools, Habits of Mind, and A More Satisfying Life: Three Reasons to Learn and Do Math

By: Jeff Heyck-Williams

This year I have a wonderful opportunity.  I get to teach math.  Specifically, I teach middle school students in an algebra class and elementary and middle school teachers during professional development in an effort to build their mathematical capacity.  I honestly love my job.  

While these adolescents and adults are fairly different in their temperaments and instructional needs, they both have challenged me with the essential question: Why do we need to learn this stuff?

    While you might not think that this question should give me pause...  it has.  In particular, I feel that too often the question of why we should have a rich depth of mathematics knowledge and skills boils down to a single answer: math is a great tool for doing a bunch of other disciplines that will solve the world’s problems and make you rich.  Don’t get me wrong, math has been and is an essential tool for fields as different as finance and physics.  However, math is much more, and this argument not only ignores the other reasons for learning math, it cheapens the discipline.  I would argue that understanding both a depth and a breadth of mathematical concepts is important for an educated society for three reasons.  Only through understanding, appreciating, and sharing all three will we be able to get the most out of our math education.

REASON # 1: Mathematics Is a Practical Tool

I’ll start with the common assumption.   Mathematics is a practical tool for many of the tasks we have to complete in our daily lives as well as in most if not all jobs.  Mathematics is relevant in our daily lives when we calculate the tip and how to split the bill when we eat out with a group of friends, when we figure out how many gallons of paint to buy to paint that room, or when we ignore the odds and go ahead and buy that lottery ticket anyway.  Math is useful.  

  

In addition, we are experiencing a resurgence of calls for improvement in science, technology, engineering, and math education (STEM as it is called).  Much of the rationale for this resurgence is couched in economic terms.  As this argument goes, we live in what Thomas Friedman has coined a flattening world, and if students in the United States are going to be able to continue to compete in the global marketplace they need math.  The types of white collar technical jobs that will pay a living wage require a high degree of mathematical proficiency because math is a practical tool.  Jobs in fields such as finance, accounting, economics, statistics, computer science, engineering, chemistry, and physics require a high level of mathematical proficiency.  In addition, as information technology advances, jobs in every other sector will require increasing levels of mathematical know-how.     

All of this is true.  However, this reasoning only views mathematics as a tool.  From this perspective, math is no better than a very versatile hammer.  It is a means to an end.  The problem with only viewing math from this lens is that for many of us it means that we could learn some elementary computation, a bit of measurement and geometry, and possibly a bit of probability for parlor tricks on the weekend, and be done with it.  Many of my students rightly ask, “when will I use all of this analysis of linear equations in my daily life?”  And if the only view of mathematics is as a tool, I have to admit that those who don’t go into a math-related field won’t ever need to find the slope of a linear pattern.  However, I would argue that it is still important for them to study how to generalize and analyze linear patterns.  Here’s why.  

REASON # 2: Mathematics Cultivates Habits of Mind

Regardless of the mathematical content you learn, the study of mathematics cultivates habits of mind or thinking skills that are powerful disciplines for negotiating our world.  At the heart of good mathematics instruction is problem solving.  To do math effectively, students need opportunities to learn how to struggle through and solve ill-defined problems.  This process teaches perseverance, how to communicate an idea precisely, how to build a reasoned argument, how to think flexibly, and how to be metacognitive, or to monitor their thinking and problem solving process.  These skills are not unique to mathematics, but are skills that are relevant to the process of solving any problem in any area.  So, whether a person is destined to use calculus in their daily life or work, the study of calculus can build in them the skills to tackle whatever problems they face.  Mathematics is uniquely positioned as a part of the core curriculum of our schools to help build these skills.  

However, this aspect of mathematics might only be useful to a person during their school years, and could easily be left behind.  If you don’t particularly like doing math and find it difficult during your school years, you can easily choose a career path that isn’t dependent on mathematics.  You will still have gained the benefits of cultivating habits of mind for solving problems, and you will go on to apply them in other areas.  Which brings me to my third reason for learning and doing mathematics.

REASON # 3: Mathematics Leads to a More Satisfying Life      

Mathematics as a system is one of the most beautiful and enduring creations of the human mind.  Developing understanding of mathematics is learning to appreciate and contribute to this work of collective human ingenuity.  Over the course of human civilization, mathematics has developed as a tool for organizing our complex social lives as well as understanding our universe.  Specifically, as the science of patterns, mathematics finds connections between often disparate phenomena and explains how those phenomena behave in surprisingly similar ways because of mathematical patterns.  Furthermore the whole of the mathematical world that describes these patterns was not invented or discovered by a single individual but by the collective work of millions of mathematicians striving for understanding.  Knowing this provides inspiration for my own explorations in the world of math.  While there have been many great individuals who have made monumental contributions from Euclid and Archimedes to Newton and Gauss, mathematics as a whole is the work of many.  Each of us has the potential to contribute to this work.    

With this realization and a little mathematical knowledge, everyone can open doors to understanding our world that are shut without mathematics, but don’t take my word for it.  Watch the short video Nature by Numbers created by Cristobal Vila for Eterea Studios.  
Nature by Numbers from Jeff DeMeglio on Vimeo.

Vila’s video highlights connections between a nautilus shell, a sunflower, and a dragonfly’s wings that are inaccessible without math.  What is even more interesting is that the math behind this video has been known for centuries and has been used in artwork from the architecture of the Parthenon to the paintings of Leonardo da Vinci.  The ratio featured in the video (The Golden Ratio) and the sequence of numbers here (the Fibonacci sequence) are but two examples of how math amazingly describes our world by capturing patterns at the heart of our physical reality.  There are many more examples of how mathematics creates surprising, elegant forms that are aesthetically pleasing and yet somehow illuminate connections in the natural world.

           

           Truly, as mathematician Jerry P. King wrote in his book The Art of Mathematics “... one’s intellectual and aesthetic life cannot be complete unless it includes an appreciation of the power and beauty of mathematics.”  If we are serious about educating our children to be active participants in their own education and responsible and compassionate members of society, then they must have access to a mathematics education that not only teaches the powerful tools that mathematics has to offer, but also opens them to the possibilities of beauty and wonder that only mathematics affords.

Questioning: The Roadmap to Learning

By: Elaine Hou

When you think back to your first years in the world, what kinds of questions did you ask? Were they narrow questions with one set answer, or big questions with endless possibilities? Children at a young age naturally bring a big curiosity about the world around them and inside them. They don’t usually ask “What color is the sky?” but rather, “Why is the sky blue?” They first ask, “How did we get here?”, only to later let the fear of not having an answer make their questions smaller. “How did we get here?” becomes “It is because it must be so. How do I replicate what I’ve been told must be so?”

Now think about the types of questions you asked and encountered in your experiences, especially in the subject of mathematics. Was your math story characterized mostly by broad questions such as “Why do we have numbers?” and “What is addition?”, or by narrow questions such as “What number is this?” and “What is 3+2?” The type of questions we ask ourselves as learners shapes the types of questions we ask our children. The power of asking bigger questions determines the richness of the worlds that we help them discover around and within them.

There is a lovely children’s book titled “Toby Shoots for Infinity,” that takes us to the heart of questioning in math and in life. In the story, Toby, a young boy brimming with curiosity, ventures on a journey to discover what is at the end of infinity. Toby’s father, aptly named Professor Copernicus, tells him that infinity is a kind of magic number in mathematics, used by engineers to build dams, make cake pans, and send rocket ships into space. While Toby is confused that an idea in mathematics could be used to create actual things, he is equally determined to find out just what infinity is. He sets out with a group of close friends, vigilantly named “The Conquerors of Infinity,” to pave a way to the end of infinity using numbers. They take turns passing the baton of counting higher and higher, until they get tired and come to a realization that “infinity is too big to fit in our heads.” Toby then has a mini-existentialist crisis at the age of 8 when he makes the following lament: “Yesterday, I learned that there was no fence at the end of the time. Today, I learned that there was no answer at the end of my questions.”

At one time in our learning journeys, we had a small Toby inside of us, not afraid to ask big questions. If we were lucky enough, we had someone in our lives, whether it was a teacher or family member, who saw the spark in our questions and celebrated the asking itself more than correct answering. Unfortunately, many of our learning stories, especially the way we learned math in schools, involved a series of rote tasks that had pre-determined answers rather than an exploration of the history and beauty of the subject itself. Our textbooks were our teachers’ teaching bibles, dimming the bright world of mathematics, science, literature, history, and even the arts to only a shadow of what it could be with endless practice problem drills and basic recall questions. When the Tobys inside of us tried to ask why and how, our first timid steps into infinite possibilities were stopped and redirected toward memorizing formulas, procedures, and right vs. wrong answers.

With our own Tobys never fully cultivated, it is easy to become the same replicators of a soul-less learning story for our own students and children. And rather than traveling toward an infinity of possibilities and new discoveries, we find ourselves in a sad cycle of hating a certain subject because we never got the “right answers,” or loving that subject only because we never took risks and experienced failure that could actually lead to deeper learning. To limit ourselves to these two dichotomies prevents us from really understanding what learning actually is and could be. In fact, true learning means being equipped to find our own answers to the larger questions of the 21st century.

How do we break this cycle as educators and parents? We start with our own learning again, whether it’s developing our own math capacities or re-discovering a familiar work of art using a new perspective. In the grown-up season of our lives, we begin to think like young children again and find the Toby within us. We learn to ask bigger questions again. Where did the concept of infinity come from? What mathematical patterns can we see in nature? How do formulas describe the beauty and order of the world around us?

At the end of the story, Toby’s mom brings hope back into his math story. While she does not have an iconic name like Professor Copernicus, she does work as a nurse in an emergency ward where “people arrive in bits and pieces.” Toby’s mom nurses Toby’s fragmented state of mind back to health by reminding him that his questions are the beginning of a beautiful journey. While the world with its infinite possibilities is both scary and exciting, Toby’s mom comforts him with the idea that the moon is really just one big nightlight. As we help our children search for the answers to bigger questions, we bring light to a much bigger world they can be active participants in, and help them pave their own ways to understanding.

We each have a Toby, a Professor Copernicus, and a Life Nurse in all of us. As parents, educators, and learners, let’s ponder this simple yet wise dialogue as we embark on learning journeys with our own children:

Toby: “Dad, I’m tired of living in a world where no one knows anything.”

Dad: “If we knew everything, the world would be boring.”

Here is a list of beautiful children’s books that inspire rich questioning. Please feel free to suggest others!

Small Stories that Inspire Big Questions

Toby Shoots for Infinity by Jean Lemieux

The Three Questions by John Muth

The Curious Child by Donyell Floyd

Wilfrid Gordon McDonald Partridge by Mem Fox

The Kids’ Book of Questions by Gregory Stock

Not a Box by Antoinette Portis