By Steven Klee, Seattle University
It persistently rises to the surface of your memory – that afternoon when you fell in love with a person or a place or a mood … when you discovered some great truth about the world, when an indelible brand was seared into your heart, which is, of course, a finite space with limited room for searing.
Arthur Phillips, Prague
It was my senior year of high school. I had spent the first half of my day taking the AIME exam. At the end of the exam, there was one problem that really intrigued me. I couldn’t stop thinking about it! It lingered in the back of my mind through lunch and gym class. When I got to my history class, I had an idea to start looking at small examples: what if there were only two houses on the street? Or three? Or four? Then I had an “a-ha” moment, which let me see a recursive pattern and ultimately led to the solution of the problem.
The joy I experienced at solving this problem was profound, and it still stands out in my mind, almost 20 years later, as a significant moment in my mathematical journey. I had had this insight that was completely new (at least it was new to me), and led me to solve a problem that was unlike anything I had ever seen before. It was exciting! It didn’t count towards my grade anywhere, but that didn’t matter. I had discovered something new, and mathematics had left an indelible brand on my heart.
My goal in this article is to examine this experience more carefully, along with the experiences of other mathematicians and scientists, to try to understand the “a-ha” moments that can be so powerful for our students. To gather data, I asked a large group of people, including high schoolers, academics, and people in industry to reflect on the following question:
Tell me about one of the first times you ever experienced joy or excitement at solving (or not solving) a math problem. When did this happen? Do you remember the problem? What made this experience so memorable?
In what follows, I will reflect on general themes that surfaced in the responses I received in the hopes that they can help us more deeply reflect on our own teaching. I am grateful to my friends and colleagues who shared their stories. Each one was exciting and inspiring in its own way, and I regret that I was not able to include an excerpt from each of them. I would love to hear about your stories of joy and mathematical discovery in the comments section below.
Theme #1: Venturing into the unknown
Several people commented that their moments of inspiration came from venturing into the unknown of the mathematical landscape. Sara Billey (University of Washington) reflected on the joy of solving her first research problem:
One day after studying almost everything known about the problem, I decided to close the books, put away the previously published papers, and pull out a clean sheet of paper. I asked myself “What could I prove that was not written?” I wrote down a formula that combined a fact I knew from the literature with the problem I was trying to solve. I asked if that formula could also be true. I sat there for a while and the proof came to me. I wrote it up in my notebook, and declared that a successful day. About two weeks later, I showed this formula to another student, who got inspired to write down another, related formula. He came back a few days later and said he could prove the conjecture if a third formula was true. Well, I had the feeling I could prove the third formula by putting a bunch of things together. Sure enough, my rather intricate proof worked! It was a very exciting time, and I got to be a part of it because I forced myself to close the textbooks and ask myself a question beyond what was already written.
Matthias Beck (San Francisco State University) echoed these sentiments, writing:
I vividly remember the first original research problem I solved. I knew the literature well enough by that point that I was pretty sure that my theorem was novel, and that caused a certain sense of excitement: the thought that at this point in time nobody else had ever scribbled down what was written on my pieces of paper.
The power of making one’s mark on the mathematical landscape by discovering some fact that was previously unknown to the world is no doubt significant. The feeling of accomplishment that comes with a new research discovery has affected researchers at all levels, from undergraduate REU participants to established researchers.
On the other hand, this venture into the unknown need not be predicated on a research experience. A problem does not need to be new to the world in order for its solution to be meaningful; it just needs to be new to the student. Another respondent recalled her first memorable problem:
The problem was as follows: a pencil costs X, an eraser costs Y, and a pen costs Z. Can you buy these items in such a way that the total cost is M? The point of the solution was that X, Y, and Z were divisible by 7, but M was not. I was eleven at the time. It took me a few hours, but then it finally hit me how to solve it. I felt so excited when I finally got that “a-ha” moment.
Theme #2: Owning the problem
In many cases students were moved because they had a sense of ownership of the problem and its solution. It is easy to feel a sense of ownership in research where we write papers with our names on them and other people refer to our results, but this same feeling can be fostered in the classroom. Dylan Helliwell (Seattle University) reflected on proving that the bisectors of a linear pair are perpendicular in his high school geometry class:
I couldn’t immediately put my finger on it, but this problem felt different than the others. I realized that I wasn’t solving for the measure of an angle or showing two things were congruent. I was establishing a new general fact! I was creating new mathematics! (Well, not really. Presumably the author of the textbook knew it was true, too.)
He went on to reflect more about the nature of this problem:
The statement wasn’t immediately obvious. I had to review the precise definitions and draw some examples before I believed it. Then I had to figure out the actual steps to prove it. We were using a “two-column” structure for our proofs and my proof took 31 lines! This was so much more than any of the other problems, and in the end I knew it was correct because I had proved it!
As with many research problems, this experience was significant because the student was challenged to do more than he had been asked to do before. The discovery was genuine to him; was new to him. His 31-line proof was his proof, and the work was meaningful because he had to think of how he could most meaningfully convey the information in those 31 lines. Tim Chartier (Davidson College) reflected on a similar experience in proving that there are infinitely many primes:
We were asked, prior to seeing the proof, to make an argument as to why we might and then why we might not have infinitely many primes. Could we run out of primes? Or, if we have some finite set of primes, is some integer large enough such that we need some new prime to form its prime factorization? Even today, I remember where I sat on campus as I pondered these thoughts. That evening, we worked on a proof of infinitely many primes in preparation for the next day’s class. In class, we developed the short proof. It was like a haiku of mathematics – elegant and focused.
This story inspires two important lessons. First, the students were not told to prove there are infinitely many primes. Instead, they were presented with the question of “are there infinitely many primes?” and asked to explore the meaning of that question. Second, the students first came up with their own proofs, and in the next class they were presented with what Erdös would call the “Book Proof” – the elegant proof that cuts to the core of mathematics. However, there was pedagogical value in this struggle against mathematics and in coming up with a proof, even if it was not the proof, because it was their proof. The students had ownership of the experience.
Theme #3: The joy of play and exploration
Many people who found inspiration in proving a theorem or solving a hard problem echoed a sentiment of joy in the realization that there was more to mathematics than rote calculation. José Samper (University of Miami) said
The first problem I remember enjoying was during a math competition in 8th grade. I remember it well: There are 100 people on an island, some always lie, the rest always tell the truth, the islanders all know who lies and who tells the truth. A reporter comes to the island, lines everybody up, and asks the N-th person if there are at least N liars. Everybody answers “yes.” How many liars are there?
This problem made me realize that math could be more than a bunch of dull computations.
I was surprised to learn that several people had deep learning experiences as a result of rote computation. Rachel Chasier (University of Puget Sound) recalled learning her times tables:
After computing the multiples of 9 by hand, I quickly devised my own algorithm: to compute 9*N, put N-1 in the tens place and 9-(N-1) in the ones place so that the digits sum to 9. I tried explaining this to my friend, but it only made them more confused. This was one of the first times I realized I was thinking about math differently than other people and that I had a mathematical mind.
Similarly, Luke Wolcott (Edifecs Software) recalled
In early elementary school we learned about long division, and this set off a competition with me and a friend to divide the biggest numbers we could manage. I remember the passion with which I filled a 8.5 x 11 sheet, the long way, with a really big number, then drew the division bar over it and to the left, and came up with a (shorter) number to put on a piece of paper to its left. I remember the joy I felt when I realized that a list of the first nine multiples of the divisor would be very helpful, and reduced this enormous long division problem into repeated comparison and subtraction.
And finally, Lucas Van Meter (University of Washington) added
When I was in 8th grade I wrote down all the squares and took their differences. I was surprised to find they were all odd. Then I took the differences of the differences and was amazed to find they were all two. Then I decided to do the same thing with cubes and finally found the differences of the differences of the differences were all equal to six. What makes this memory stick is that it was one of the first times I made a mathematical discovery on my own with no outside intervention. It felt like a personal discovery of my own.
I was surprised by these three reflections because we tend to hear that students dislike mathematics because it seems like a bunch of rote, boring computations, while these stories all seemed to stem from that rote computation. But perhaps this shouldn’t be so surprising. The important takeaway seems to be that the inspiration stemmed from discovering something new as a result of playing with all the mathematics they had at their disposal.
Theme #4: Elegance, beauty, and simplicity
Finally, a number of people recalled the feeling of being struck by the simple elegance of a solution to a problem they had failed to solve. Jonathan Ke (Kamiak High School) recalled:
One day, my dad showed me a book filled with mathematical puzzles and questions, one of which was to add up the numbers from 1 to 100. I found a calculator, plugged in as many numbers as I could, got bored, and gave up. Then my dad showed me a video of how Gauss found the sum. I was amazed at the trickery he used and the mathematical explanation of why it worked. I realized that math is far more than just bunch of formulas that I choose to plug-and-chug and get an answer. It is far more complicated and beautiful.
Similarly, a colleague who works in industry wrote:
It was actually a simple problem if you knew trigonometry, but at that time I didn’t (I was in seventh grade). The obvious way to find the angles didn’t work, and I had no clue how to solve it despite a lot of effort. It turned out to be a proof by picture – just a picture – but once the teacher drew it, it was like a bright light at noon right after a pitch black midnight. The discovery was meaningful because it was the result of suddenly and deeply understanding something that you couldn’t understand before…there was joy in transitioning from being hopelessly clueless to knowing. It was one of the first times I saw that you could understand something, but first you had to make it more complicated.
In these stories, we see that failing to solve a problem can also lead to a meaningful experience. Again, the important aspect of these stories seems to be that the students had time to play with the problems first. They devoted considerable efforts to solving the problem, which led to a deeper appreciation of why the ultimate solution was so elegant.
What should these stories mean to us as teachers? On the one hand, many of them contain ideas that are prevalent in leading teaching philosophies:
- We should work to make it clear that mathematics is more than a set of arbitrary rules that govern mindless computations.
- We should create an environment in which students are encouraged to explore and share their ideas, ranging from observations about multiplication tables to new ideas about unsolved problems.
- Students need time to explore and struggle with ideas on their own before they see an elegant and perfectly rigorous solution to a problem.
So how do we do this? Some of these issues have been addressed in this blog and in other places, while others present ongoing issues to be overcome:
- How do we create a grading system in which students can be rewarded for working on a difficult problem as opposed to getting the “right answer”?
- How do we empower students to view themselves as creative problem solvers as opposed to human calculators?
- How do we assign interesting, substantive problems whose solutions cannot be found through a simple Google search?
- In many instances, people were inspired by mathematical explorations of their own design, not by problems that had been assigned to them. How can we foster this type of exploration in the classroom?