*In this multi-part series of blog posts, I’d like to tell the story of the O’Reilly School of Technology and the new Make: Mathematics project. This is also a great opportunity for me to explain and justify our overall mission and plans.*

**Part 2: My most important teaching experience**

Because of my own “ah ha” learning experience in mathematics, I was convinced that everyone could learn math well. l if they had a passionate teacher with the communication skills to get them there. I certainly had the passion, so I set out to transform students into mathematicians.

By all the standard measurements and expectations, I became a great teacher. At least my students thought I was great;nothing feeds the ego like teaching. My night time math review sessions had become so popular that I had to hold them in the largest auditoriums on campus. One of my classes even presented me with a large engraved trophy to thank me for my teaching . They did well on exams, and they were very happy. All was well.

But despite all of this positive reinforcement, my point of view (confidence?) and self-esteem began to suffer a bit after a visit in 1993 to Moscow. I visited the mathematics department at Moscow State University, and stayed with a couple who were both professors there. They had two wonderful daughters, the younger of whom was still in high school. At one point during my stay, she asked me to help her with homework in mathematical mechanics. Of course, I was proud to help, since Mechanics was my research (specialty?). Her homework seemed to me to be rather difficult (advanced) for high school; I had a bit of trouble with it myself. Suspecting that this high level of work might be a strange exceptionin the Russian educational system, I asked her parents if she was in a school for gifted students or some other advanced learning program. They told me that, no, she was actually an average student in an average school. This confused me. Mathematical mechanics is historically a major focus in Russian Academics, but that didn’t explain how their students could be so much more advanced than the students in the United States at this level. After a lot of probing questions, I eventually found out that all of the math exams in Russia are oral exams. Students in Russia have to be able to explain mathematical problems and their solutions out loud.

Intrigued and inspired, I came back to the states, and started giving my students oral exams after their written exams. This was a long and arduous process, because I had to schedule an hour for testing for each student. However, I discovered something that depressed the hell out of me. None of my students knew what they were talking about. Even students who got perfect scores on my written exams didn’t really understand what it was that they were doing.

It became clear that students were simply emulating calculation techniques, without understanding where those techniques came from, or how to create them themselves. Then it became clear to me why my review sessions were so popular. In those sessions, students would ask me to solve every type of problem they could find in the text book. Even though I’d have them try the problems before showing them the solution, they were really preparing a decision matrix for a matching game. If the problem was like this, then they would do this; if it was like that then they’d do that, and so on. I also realized that the problems I was asking them to do, were designed with this system in mind. It seemed that most of the calculation techniques were designed to help students pass tests, but did not illuminate the true nature of the mathematical structures .

In the American system of teaching mathematics, we are actually teaching algorithms for getting answers from synthetically designed problems,but not teaching students the art and science of mathematics. Sure, some students get through school, and become great mathematicians despite this system, but we are losing most students through attrition. Ask your friends and neighbors about their experiences with math education, and most will say they hated math in school. Others might say they were good at it, but hated word problems, which I always thought was a curious thing to say (what those people are really saying is that they were better at the matching game when the thing to match was given in a simple form).

I wanted to quit teaching. The whole thing seemed a bit hopeless. I could certainly try to change how I was doing things, but how? I didn’t know. I thought back to my own “ah ha” experience and realized that the reason it had happened was because the instructor actually didn’t show me how to solve or understand problems directly, but because I wanted to help him figure it out. I was given space and motivation to explore and figure things out for myself. I suppose I could have tried to apply that teaching technique in my class, but I realized in that particular class, I was the only one who was having that special learning experience. The rest were simply waiting on ME to explain the math. Helping just one or two students reach their potential out of fifty wasn’t very appealing to me.

After discussing this with my colleague, Lee Wayand at Ohio State, he told me he’d started working with Jerry Uhl and Bill Davis and their Calculus & Mathematica math reform project. He told me it was a completely different paradigm, where students were really learning the math, and not just learning computational tricks. I was suspicious of using technology to teach. I’d tried some Calculator driven courses that were complete failures, but Lee assured me that this was different. I talked to Bill, and after some coaching, I started teaching Calculus & Mathematica at the Ohio State University in 1994.

These courses were clearly different from their traditional counterparts. Bill asked me not to lecture to the class, but instead to participate as their coach and their peer. This was possible because the course materials were written in Mathematica software files and housed on computers in the computer lab. When I’d walk into the room, the students were already working on the material. When I left, they’d still be there, working on math. I’d never seen anything like it. Students were working, engaged, and talking about math with each other without me leading them. They’d call me over, and ask me questions, but instead of answering the questions directly, I’d ask them leading questions, and give them ideas to explore. Bill once told me that to answer a student’s question outright, is to steal from them the opportunity and thrill of discovering the solution themselves. He was right!

Mathematica is an extraordinary piece of software that is a bit like combining the text editing capabilities of Microsoft Word with the most powerful scientific calculator in the world. The language of Mathematica is, of course, very mathematical and natural, and as you learn that languauge, you are actually learning mathematics, and visa versa. In these courses, Mathematica is never taught directly, but it is learned indirectly, as a consequence of learning Mathematical concepts and combing though patterns.

Suddenly, by going through these materials, and interacting with their instructor in this new way, the students were not only learning, but gaining ownership of mathematical concepts. For the first time , they were seeing Mathematics as a science. They were looking for patterns, finding them, and explaining them. That’s what mathematics is all about. When I gave oral exams to the students in these classes, a lot of them were able to explain the concepts and structures of Calculus.

That first term of teaching Calculus & Mathematica changed my life. I became a disciple and evangelist for this new teaching paradigm, and began a quest to understand more about why it worked, and how to extend and expand this paradigm to change education.

*{To be continued. In the next post: How Jerry Uhl at the University of Illinois, and Bill Davis at Ohio State discovered this new teaching paradigm, it’s relation to Constructivism, and how my quest to spread it’s reach led to the creation of Useractive, inc. which eventually became the O’Reilly School of Technology} *