# Gifted Children and Mathematical Thinking

I have a few favorite mathematicians: Sophie Germain, Paul Erdös, and Euclid. But when it comes to talking about gifted children and learning mathematics, my favorite story is of Carl Gauss.

Gauss was clearly a gifted youngster. As the story goes, his primary school teacher assigned the class to add all the numbers from 1 to 100. While most of the pupils were still busy writing down the question, Gauss gave the answer: 5,050.

This can be, of course, arduously completed by adding 1, 2, 3, etc. until reaching 100. Had Gauss actually done all of that addition in his head, that would have been remarkable indeed, but it would not have been as interesting. What he did instead was a much faster, simpler approach. (I won’t spoil it, for those of you who’d like to ponder it.) What was truly remarkable was that he thought beyond the expected algorithm, made connections his teacher did not expect, and crafted something that can be used give many more answers easily, can lead to many more questions, and is truly elegant—perhaps even beautiful.

Typical mathematics education lacks this beauty.

When we teach English, we hope to inspire readers and writers, encouraging students to see themselves as authors, to love the beauty of the written word, word play, imagery. Grammar is part of it, but a small part that helps us understand structure and communicate clearly.

Increasingly, other subjects are taught the same way—teaching in the way experts in the field engage. Students do science experiments and record results and draw conclusions, critique studies, examine data, and learn not just what the answer are but how to ask the questions. History students examine primary source documents from historical events and try to piece together different points of view to understand what actually happened, to see the stakeholders in different situations, to watch the interplay of different points of view rather than just wallow in the stories of the victors who write the history.

Yet mathematics instruction remains the transmission of a set of skills, tools to be employed to solve carefully contrived problems written solely to assess whether the student can use those tools. Why are students not encouraged to do as mathematicians do?

Perhaps there is a sense that mathematics is only a tool, rather than something worthy on its own. Perhaps we believe that mathematicians are simply presented well-formed problems to solve and hand back. But this is not what mathematicians do. This is not how mathematicians think. Mathematics is not a tool, but a way of thinking, a structure that allows us to explore ideas. Without the exploration, mathematics becomes nothing more than spelling and grammar, losing all its beauty.

Little Carl Gauss thought something vastly different than the rest of the students.

Instead of employing mathematics as a tool, he posed a problem to himself: Not simply “What is the sum of all of the numbers from 1 to 100?,” but “Is there a way to do this quickly and easily?”

This is what mathematicians do. They are not problem solvers, they are problem posers. Solving the problem is exciting and useful, but not the end, as a solution can and should lead to more problems to be posed. Scholars of mathematics who have gone so far as to earn their PhD in mathematics have to prove something previously unproven to earn those letters after their name (though that alone is not sufficient). We have unsolved problems, like Goldbach’s Conjecture, because someone originally posed the problem. Without the big questions, we would have no big ideas. Mathematics isn’t about finding the right answer, it is about posing problems, exploring the connections and patterns and ideas that form our thinking, that model our world, that define our understanding of so much.

How should a teacher support this kind of mathematical thinking, where students develop tools through asking questions, then weave the ideas together into a tapestry of mathematics that is elegant and beautiful? The first step is to ask problems that don’t have a single right answer. In mathematics, this sounds revolutionary! Whatever will we do with the answer key? Or worse, what if I, the teacher, don’t know what all the right answers are?

This is part of why such things so rarely happen. The fear of math has become so ingrained in our society that people feel justified saying “I don’t do math,” even though no one would ever feel so comfortable saying “I don’t do reading.” Teachers aren’t immune to that fear, particularly teachers, such as elementary teachers, who are expected to tackle every subject. The first step teachers have to take for themselves is to let go of the need to be right. You may even have to let go of knowing how to get to the right answer.

In my own teaching, I’ve succeeded and failed in exactly this. The first, in my second year of teaching in a fifth-grade gifted classroom, I introduced my students to the game SET, and we had begun posing problems about the game. One problem posed was “How many distinct sets exist?” We worked on this problem in our “free time” for multiple days and did not solve it. A year later, teaching in a new district to students two years older, I realized what the solution was, and it was elegant and easy, but with that fifth-grade class did not know how to solve it. We never came to a solution, but the process held a tremendous amount of value: Students knew they could not come to me and say, “Is this right?” I certainly did not know if it was. They would know it was right because they were sure of it, not because they checked it with me and I confirmed.