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My elementary school years are a blur of memories for me; months at a time went by that I remember very little about, and then there were moments that stand out vividly in my mind, even 35 years later. For some of those childhood years I struggle to remember what my teachers looked like, or what their names were. Other years I remember the teachers as though I was in their classroom last week.

I remember my sixth grade teacher like that. Mr. Frank Avey. I remember that before we hit sixth grade, we were all scared to death of him, and we didn't want to be in his class. The rumor was that he was an army drill sergeant before he became a school teacher, and he ran his class like he was still in the army. It turns out, he really was in the army before he became a school teacher (though I have no idea if he was a drill sergeant), and he did (sort of) run his classroom like the army. He could be fierce, and he was certainly strict.

But once I got into his classroom, and had spent a day or two there, I knew that I wasn't going to be afraid of Mr. Avey, and I wasn't going to hate being in his class.

What do I remember? I remember that every day at 10 minutes to noon, a current events news program would come on, so at 15 minutes to noon, Mrs. Avey's class (yes, his wife also taught sixth grade at the same school; she was the sixth grade teacher everyone *wanted *to get!) would come into our classroom (because we had the television) and watch the news together with our class. Then, when the news was over, we'd all have to take a current events quiz over what we'd just watched. It was stressful. Of all the quizzes I ever took, those were the hardest. But I learned a lot about the world that year.

I remember watching Mr. and Mrs. Avey drive up to school in their metallic green Volkswagen bug, and thinking what a cool car that was (I know, it's silly, but I'd never ridden in one of those crazy looking cars).

I remember the classroom getting too loud, and Mr. Avey making everyone put their heads down on their desks for 5 minutes. Oh, what torturous, tedious minutes! I remember my glasses fogging up while my head was down (what weird things I remember!) and I remember being annoyed that I had to suffer this indignity. After all, I wasn't the one who was too loud.

I remember finishing my work early one day, and not having anything to do. Mr. Avey found some work I could do to help him. I felt quite important and helpful sitting at the desk next to his and stapling papers. And then the stack of papers turned into a bin filled with extra things for me to do. I think he labeled that bin "Assistant" or something like that. Oh, didn't I feel special then! He could have labeled the bin "Teacher's Pet" and I probably wouldn't have cared.

I remember that when I graduated from high school, even though Mr. and Mrs. Avey had moved out of state, I got a graduation card from them.

I remember all of these things. And I remember - most vividly - the day I got to ride in the metallic green Volkswagen bug. Here's the story.

Mr. and Mrs. Avey were staunch Democrats. My family was all staunch Republicans. But this was back in the era of American history when people weren't utterly stupid about politics. I know it's hard for people to remember, but there was once a time when people who were Democrats and people who were Republicans actually knew how to talk to each other and get along with each other. Republicans had friends who were Democrats, and therefore weren't about to refer to Democrats as "snowflakes" and "lazy welfare cheats." Democrats had friends who were Republicans and therefore weren't about to refer to Republicans as "bigots" and "uneducated buffoons." Those were the days when people were allowed to have different opinions from one another and still consider each other worth talking to and listening to.

Ah...the good ol' days.

Oh, sorry - where did that come from? Anyway, back to my story.

Mr. and Mrs. Avey contacted my parents one day and said, "We'd like to do something special for Doug. We want to take him to a political rally." My parents knew perfectly well that this was a "liberal" rally, yet they didn't object (insert another good-ol-days comment here)! So one afternoon, I got to climb into that green bug and go on a one-student field trip. I was astonished at how little room there was in the back seat. It was a good thing I was only a sixth grader; my legs were already almost too long to fit comfortably!

We drove to Portland, where I went to dinner with Mr. and Mrs. Avey, and then had the privilege of hearing Walter Mondale speak at a campaign rally. He was, at the time, vice president of the United States under Jimmy Carter, and was running for re-election.

Do I remember anything he said? I do not. Not a word. I remember the excitement, the bright colors, the noise, the crowds. And I remember the strange mix of excitement and nervous tension when the rally was over and Mr. Avey said, "If we go stand over here, maybe you'll be able to get his autograph."* The vice president of the United States!* I'd never met anyone that powerful! (and don't start telling me that the vice president isn't actually all that powerful; I don't want to hear it!).

My hands were sweaty and trembling, my knees were almost certainly knocking. What an exciting moment it was when I thrust a little slip of paper in front of him, and he scrawled on it something that looked more like "B. Monoah" than "W. Mondale", and then shook my damp little hand before moving on to the next person. The moment lasted less than five seconds, but I haven't forgotten it.

Today I searched through a giant file of papers from my elementary school years to find this scrap of paper. It took a long time to unearth it. Why today? Because this morning's newspaper had the following obituary headline:

## Francis L. Avey Jr. (1936 - 2017)

Mr. Avey passed away last week after a "courageous battle with cancer" at age 80.

It's funny, I have thought of that autograph many times over the years, but I've never felt compelled to try to find it until today. And as I studied it over this evening, I realized something important. All those times I thought about Walter Mondale's signature, the sequence was never that I thought of Walter Mondale, and it reminded me of Mr. Avey; the sequence always was, thinking of Mr. Avey reminded me of Walter Mondale.

And so it seems that in the life of one child, a sixth grade teacher truly was more powerful than the vice president.

I hope and pray that I too will have that kind of power in the life of a child. Thank you Mr. Avey, and all the others who have had this power in my life. May God bless you all.

After school one day last week, one of my students wanted a little help with some problems she was working on. One of the problems was a system of two equations in two unknowns:

x^{2} + xy = 30

xy + y^{2} = 6

I said to her, "I like problems like this. I don't know right off the top of my head the best way to start it, and that means it's an interesting problem!" I asked her what she had tried.

She said, "I multiplied the second equation by -1, and then I added it to the first equation, to get rid of the xy term."

x^{2} + xy = 30

-xy - y^{2} = -6

------------------

x^{2} - y^{2} = 24

"That's good," I said. "So what next?"

"I don't know," she said, "that didn't seem to get me anywhere."

I said, "Okay, well, let's leave that there; we may want it later." So I put a box around it, to make sure I wouldn't erase it from the board. "Did you try anything else?"

"I couldn't think of anything else to try."

"Did you try adding the equations?"

"But nothing will cancel if I do that..."

"Try it anyway. I think something interesting will happen. I'm not sure it'll be USEFUL, but it'll at least be interesting," I said.

We did the equation addition:

x^{2} + xy = 30

xy + y^{2} = 6

------------------

x^{2} + 2xy + y^{2} = 36

It didn't take her long to notice that the left side could be factored:

(x + y)^{2} = 36

"And what does that tell us?" I asked.

"x plus y is either 6 or -6."

"That's great," I said, "you've just turned this into two problems!" So I drew a chart showing the two choices:

Choice One: x + y = 6

Choice Two: x + y = -6

"Now, I said," let's go back to your first equation that you came up with. Can you think of anything you can do with it?"

She realized she could factor the left hand side:

x^{2} - y^{2} = 24 ⇒ (x - y)(x + y) = 24

At this point she was getting a little excited about the problem, because it was starting to fall apart quite nicely. After all, if x + y = 6, then (x - y)(x + y) = 24 is equivalent to (x - y)(6) = 24, or x - y = 4. Similarly, if x + y = -6, then (x - y)(-6) = 24, or x - y = -4.

Now we find that we have two systems of equations:

Choice One: x + y = 6 and x - y = 4

Choice Two: x + y = -6 and x - y = -4

Both of these systems can be quickly solved (my student did them both in her head), to obtain the solutions (5, 1) and (-5, -1).

And the best part of my teaching day was hearing these words: "That was FUN!"

Then I showed her how the Teacher's Manual suggested doing the problem. Solve one of the equations for a variable:

x = (6 - y^{2})/y, and then substitute that into the second equation. I showed her the full solution. It was UGLY.

"I like our method better! Ours was fun!"

I understand the reasons for teaching the method that's in the text book; it's straightforward (even though it's ugly), and will work on a variety of problems. In addition, using that method gives students practice on a few different topics such as fractions and squaring binomials, etc.

We had no guarantee, when we first started the problem, that our ideas of subtracting and adding the equations were going to get us anywhere useful. As I said to the student, "We might try a bunch of things before we find something that works, but sooner or later we'll hopefully stumble on something useful."

But that's what problem solving is. Problem solving is not just using a rote method that a textbook shows you. It's looking at a problem and not knowing where to go, but being willing to try things to see what will happen. It's adding two equations together because you notice that adding them creates a perfect square, and that might be useful. It's subtracting two equations because you notice that it eliminates an xy, and *that *might be useful too.

It's a labyrinth of possibilities, with no guarantee that the things you try will be useful, but with the hope that somewhere there's a path that leads to a solution.

And, in the end, it's the delight of watching a system of quadratic equations disintegrate into pieces that you can solve in your head. It's the spark of excitement when the solution starts to dawn in a student's mind.

And it can be FUN!