My first year of teaching, I was the entirety of the middle school math department. I was the only person in the building who knew very much or cared very much about middle school math. That situation led me to find my online community of math teachers, but it was very lonely on a personal level.

Then, the school hired Dennis Gallagher to teach a couple of courses for our school's most mathematically inclined students. For the first time, I had a colleague and a mentor.

Dennis passed away yesterday. I want to tell you more about him.

Dennis had the best introduction to Algebra 1 that I've seen. He started the first day with the handshake problem. Students had to determine the total amount of handshakes that occur if everyone shakes everyone else's hand in the room, and then write a formula for a room of n people. At the end of that day, he would say "What you just did is notice a pattern, extend it to a larger number, and then generalize it into an algebraic expression. You just did algebra."

The next day, the students find the amount of diagonals in a polygon. Again, they'd find the amount for a rectangle, pentagon, hexagon, and finally a polygon with n sides. The formula, which is a variation on the formula from the previous day, helps make the connection clear: If you arrange a room of people in a circle, every handshake can be drawn as a line segment connecting two people around that circle.

So again he would end class by pointing out that these two totally different problems were actually closely related from a mathematical perspective. "You took ideas and skills from yesterday and applied them in a new and surprising situation. Now you're **really** doing Algebra. And that's what we'll be doing in this class, over and over again."

I remember stepping back from the table as he explained this sequence of lessons to me, both of us almost giddy with excitement. Yep, that's how you start a math class.

I loved Dennis as a teacher and colleague, but more than that, I loved him as a partner in doing math. He taught a geometry class, which he knew was my favorite. Occasionally he would pop into my room holding a sheet of paper: "Ben and Louisa and I have been working on this problem for 20 minutes and we can't figure it out. Want to give it a crack?"

I'd work on it all during my lunch break and at home, ignoring emails and piles of ungraded paper all the while. Then I'd come in the next morning to show them what I'd got: "I still don't know how to solve it, but I did figure out that these two triangles are similar..." and they'd be off to the races.

In those moments I didn't feel like a teacher. I felt like a member of a team, working alongside other people to figure out this tough problem. It didn't matter that one of those team members was around 60 years old and the others were 13 or 14 (or, in one remarkable instance, 8). We were all trying to crack the same problem. That radical parity felt natural to Dennis, and over time it began to feel natural to me.

Dennis was never an answer key kind of guy. He would rather experience the math right alongside our students, because in the end the thing that brought him the most joy as a teacher was simply doing math. And he was the first teacher I ever saw who filled a room with his enthusiasm for the subject. Six years later, I am still trying to show my own students a fraction of the love of math that Dennis simply exuded.

Dennis had a long and fascinating life that I know the tiniest portion about. But I will forever remember the part of it that he shared with me.