I'm finishing up my first year as a high school math teacher, and even with the occasional day or two when I thought I might pull out my hair in exasperation, I love it!
There's just one thing that has haunted me from the first week right through this closing part of our spring semester: Although I always enjoyed math and did well in it, I sometimes wonder if I am truly 'mathematically minded' or if I'm just 'book smart'. The way it shows up is when I observe some other teachers in our school and see how they can take a student's wrong answer and find a way to relate it to the concept and meld it into a step in the right direction.
But for me the big thing is coming up with examples off the top of my head that have nice neat answers. At first I thought, Wow these people have a lot memorized, but then I realized that, No, they are simply remarkably flexible in their math capabilities. Just the other day, I sat in on a review lesson for a test. After going over some systems of equations the teacher had typed up for them on a prepared sheet, a student asked the teacher if he could show one more. I assumed the teacher would open the textbook, but instead he just paused for a second and then wrote one out on the board. He definitely appeared to make up in the spot, and it worked out neatly, with no remainders. Now I could probably do that in the privacy of my home, but to do it while an entire class of students is staring at your back makes everything harder.
If I am not that quick and I never become such a flexible problem solver, won't that pop up whenever a gifted student or a parent questions me about something a little deeper that the material I've mastered to teach high school? And wouldn't the students be better served by a true mathematical genius rather than just an ordinary person who happens to like math and teaching?
Name and Address Withheld, For Obvious Reasons,
Miss. LaV _ _ _ _
Dear Miss LaV _ __ _ _ ,
You have absolutely nothing to fear!
For reasons I will soon get to, I can tell you that you have everything you need to make a fine high school math teacher -- except, of course, a little more experience. And that will accumulate quicker than you might expect, as long as you do NOT quit. DO NOT!
The world needs more math teachers like you; "humble" and "real" are words that come to mind, but to me the litmus is that you are almost finished with your first year and you STILL feel positively about teaching math. When I ran a department, the two most important ingredients I would look for when hiring were: 1) an authentic desire to work with students/adolescents/teenagers, or whatever level was relevant, and 2) An open-minded intelligence that is associated with a love of learning, and which is contagious. So, below college level, specific knowledge of the subject matter is NOT one of those two top conditions. (That knowledge will come as necessary). The world has a lot of talented mathematicians, but fewer talented math teachers who can not only illuminate the material to make sense of it but convey a positive feeling about the fun, relevance, beauty, and elegance of mathematics.
SO, know this:
Stephen W. Hawking, the famous Lucasian Professor of Mathematics at Cambridge University in England, openly discusses the fact that he had little formal training in mathematics and for the first few years of his professorship stayed only one chapter ahead of the class!
Yes, some teachers may be more flexible in their thinking than you currently are, but don't worry about it. Worrying is a waste of time and energy that could be used for other things.
And, finally, there are techniques ("tricks") that I myself have found helpful in situations like the one you mention above about 'systems of equations'. Why? Because I have a bad memory and so must derive everything constantly from scratch at the board while the students are staring at my back; fortunately, I've developed a knack for doing this quickly, so it appears that I have replaced recall with an odd ability to figure things out in a few seconds when I need them.
Since you mentioned making up systems of equations 'on the spot', which can indeed be tricky with lots of eyes staring at you, here's what I do:
Use numbers that mean something to you and attach them to variables that are connected to those numbers by something personally meaningful.
Let's say I have a brother who is three years older than I.
Therefore, when I was 2, he was 5. I remember the number "2" because it rhymes with true, and this is a true story.
Next, I think of his name and then use his first initial as my variable. So, secretly, in my head I think: J= 5. Next, my first initial, M, of course, = 2.
J= 5 and M= 2
(BUT, since students soon catch on to my ways of thinking, I might want to twist it slightly and use my middle initial, which is L.)
So: J=5 and L=2.
From there, with a little practice, it is surprisingly simple to set up a system of equations on the chalkboard in thirty seconds or less.
5J = 7L +11
4J – 3L = 14
Let's check it:
5(5) = 7(2) + 11
25 = 14 + 11
25 = 25
YES for equation number one!!
4(5) – 3(2) = 14
20 – 6 = 14
YES for equation number two!!
Here's one more example of the type you can make up at the board without too much stress:
Let's say the address of your home has the house # (or apartment #) 116.
And let's say that your previous home had the house or apartment number 8.
So, secretly, you say to yourself:
H (for house) = 116
P (for "previous house") = 8
Now you do whatever simple math you need to to produce two equations which both use your H and your P.
H – 56 = 60... and then you change the 56 into (7x8) and come up with this:
H- 7(P) = 60
The other equation could be 120 – H = P/2
H- 7(P) = 60
116 – 7(8) = 60
116 -56 = 60
60 = 60
YES, it works!
120-H = P / 2
120 – (116) = 8/2
4 = 4
Yes, it works!
Hope this helps!