Mitsuo's Magic

Before after-school tutoring commenced yesterday afternoon, I found freshman Mitsuo at my door. I had left it open to let the cool December breeze in. He hesitated for a moment when he realized I was chatting with Ashley, one of my talented sophomores. I quickly sensed that he was anxious to tell me something. Before he had the chance to excuse himself and turn away, I asked him to come in.

Although Mitsuo, who's still taking Algebra 1, is not a student of mine, he has warmed to me. A few months earlier, while I was on after-school duty, the boy, then a stranger to me, approached and impressed me with his magic card tricks. He left a really good impression when he thanked me profusely for indulging in his magic.

When I realized that his concern was about a math idea he had conjured up, I motioned him to approach the white board. He grabbed a marker and wrote the following:

[90^(1-3) + 80] /4

I initially didn't understand the part that appeared to be the exponent of 90, but I assumed that it had something to do with the first, second and third six-week periods. Earlier that afternoon I had discussed semester grades with some of my students. The semester grade is the average of the three six-week period grades and the semester exam.

I also wondered why he didn't multiply 90 by 3.

"It looks like you want to determine your semester grade." He said yes.

As if he read my mind, he said, "I'm sorry, Sir, I know I didn't write it right, but I want to show that this 90 is the average for the three six-week periods."

I decided to not call his attention yet to his failure to multiply 90 by 3.

"You see, Sir, I was told that, to get the average..." He stopped. Suddenly I had a feeling that he had shown his idea to other people and it was dismissed. "But I was explaining that the way it's done is harder for me, because it's harder when you multiply and then divide. Right?"

"What do you mean?"

I thought that he probably just didn't like to work with bigger numbers.

"I have an easier way to get the semester grade." He now looked more determined. "All I do is get the difference between 90 and 80, which is 10. Then I divide 10 by 4, which I then multiply by 3."

I felt relief when he mentioned multiplying by 3!

He wrote 10/4 x 3 = 7.5.

"It's right, right?" I nodded. I now realized that the boy was on to something.

Although I already knew that 7.5 must be added to 80 to get the correct average of 87.5, I had to ask how he's going to use it.

His last step turned out to be wrong, but I did not point it out to him. Not the right time.

I knew in my heart that, despite the careless mistake at the end of his solution, Mitsuo deserved to be congratulated for bravely pursuing his unconventional yet intelligent method of determining the semester grade. He made a careless mistake, but his investigative mathematics is the sort that we hope to see in our students. Clearly, he's been empowered!

To assure him that his mathematics was right, I asked him to demonstrate the conventional solution, i.e. [90x3 + 80]/4, and then pointed out to him where I see his 10, the 10/4,and the (10/4)x3 in said solution.

It was only then that he realized his last step was wrong!

But even to him, the little mistake didn't matter. He was just too happy to know that, after being dismissed for his unconventional method, he got validated.

I promptly attended to my older students when Mitsuo left. There were seven of them already who needed some assistance with their circumcenter constructions.  It didn't matter that I was really tired and hungry. Mitsuo got me under his math spell!