I happily teach all three sections of advanced geometry this year at Juneau-Douglas High School. The door of my classroom serves as the neck of a funnel, as students straight from primarily Dzantik'i Heeni and Floyd Dryden flow into my classroom. It should, though, be duly noted that each of my classes has a few clever sophomores who add just the right amount of spice to the mix.

I finished teaching 15 minutes early the other day, gave out my first book assignment and let my students start their work in class. They all immediately opened their books and started to work. Fabulous. Then the hands started to pop up everywhere, and the more exuberant kids called out, "Where do we write the heading? Do we need to write out the problem?" Then the classic, "Do we need to show our work?" One more: "Do we need to skip lines between problems?" Hmm, someone has trained them well. I answer their questions and they continue to work.

Then the hands start to fly again, "Is this right? Where is the graph paper? What, you mean you just want me to sketch a graph without graph paper?" It's hard adjusting to my style. It's not that I am opposed to highly structured learning environments - they obviously work - it's just that I am not a highly structured human being and I think there are benefits to having all kinds of teachers.

After eight years of managing my own classroom, I've slowly developed a set of guidelines that work for me. When they don't work, I change them. I'm always up for a good solid argument for change. Last year, my advanced geometry students certainly changed me and, to some degree, trained me. I now better understand their needs, how they think, their likes and dislikes and what makes them tick. They are certainly a diverse group of young people.

I've learned about half of what I know about being a teacher through experience, which basically means trying it in the classroom. The other half I learned from my mentor teacher, Susan Joling. After working in Susan's classroom during my postgraduate-study year, I walked away thinking "critically" about all aspects of education. It was an essential experience to becoming an observant teacher. I find it strange though, that the opportunity to learn from other teachers by observing them at work rarely happens after you graduate, at least at the high school level.

One thing I learned from Susan that I have taken to heart is the belief that you have got to sneak some fun into your year whenever you can. So in the interest of fun, and because I never did get to see the legendary Mary Borthwick teach before she retired last spring, I gave her a jingle.

My request was simple, "Will you come to my advanced geometry classes on the first day of school and help play a round of tangrams with us?"

About half of my current students had Borthwick last year for algebra I, and I knew that - whether my former Borthwick students loved her methods or not - they all respected her tremendously. I wanted to get their attention on that first day, and perhaps what I really wanted to say, through the juxtaposition of the two of us, is that algebra I and geometry are very different beasts, just as Mary and I are very different teachers. And it's all going to be OK.

Borthwick's algebra I is a hard act to follow. I really wish I could have seen her teach. I'm quite sure that the classy 35-year legend could teach this nine-year gal a thing or two.

So what happened? Mary declined my offer. Thus, I met my students alone on that first day, as usual. I did mention to them that I had invited Borthwick to class. A few backs straightened. Several heads snapped toward the door, expecting her to walk in. I chuckled to myself.

I was tempted to ask someone to stand and recite the perfect squares and cubes from one to 20, but I restrained myself. The amazing thing is that they could probably all do it, and I am so glad. Thanks to all the teachers who have trained them well, I'll take it from here.

• Mary-Lou Gervais is a math teacher at Juneau-Douglas High School.

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