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 I 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.
A few years later, I gave a test on counting and probability that involved finding the number of ways a pizza could be made, given the ingredients on the menu. This was, sadly, a question with a single correct answer (64: the curious reader can likely figure out how many ingredients were on the menu given this answer). One of my top thinkers wrote 32, but because he had not shown his work, I could not know where he got the answer, so gave him no points on this problem. As was the policy in that class, students could earn back half of the points they lost by doing “test corrections”—where they revisited the question, explained their mistake, and redid the problem. This student came to me certain he was correct. He explained his method, which was truly elegant, and far better than the approach I and all my other students had used. It made sense, but he was getting the wrong answer, so I was determined something had to be wrong with the method. It turns out the only thing wrong was his calculation: 26 simply does not equal 32. But his method was so much better than what I had been doing, he received a substantial change in his test score. His method could lead to more interesting thinking, and by comparing our two methods, we could think about how to simplify other problems. This is where the beauty begins to happen.
Both examples do have a single right answer. But these questions are most useful when they lead to other questions or when the focus is on exploring the method or multiple methods, rather than a straight answer. Single right-answer-questions, known as exercises, are useful for practicing skills. Automaticity with things like multiplication and fractions makes it far easier to make connections in mathematics. These do have their place, but most certainly not in the center of mathematical learning.
To deepen this kind of thinking and connecting, create problems that don’t have a question. Set up a situation, such as playing the game of SET, and ask students what questions we can ask. This may require some modeling to help them see what kinds of questions are interesting, particularly as most students are used to questions that have a single right answer. The real trick is figuring out how our answers lead us to more questions, to unlearn the habit of being “done” simply because you have an answer. We need to see an answer as the genesis of new ideas.
The most important thing a teacher models in mathematics is curiosity and excitement. Modeling how to pose problems is important when students show they aren’t sure how to do so, but I have found that most gifted learners pose problems pretty naturally. While they often take pride in getting the “right” answer, they also tend to want to ask the next question. So often, teaching mathematics to gifted learners is about giving them the space and the culture to explore rather than answer. To seek the beauty they often already know exists but have been denied. To get to finally think like a mathematician.
Lisa Fontaine-Rainen has been teaching gifted children in the public school system and working with homeschoolers for 15 years. She earned her MA in Gifted Education from William and Mary in 2001, and has been lucky to be able to work with these amazing young people ever since. Check out her online math classes through GHF Online.