'Yes, I posted this some time ago. However, the questions are not rhetorical. I really would like an answer from my colleagues in the primary and secondary school system — particularly those in the education union establishment. And I’d like an answer because I really do want to know why I have to do your job as well as mine.
Thanks ahead of time for your response. Here it is:
I don’t want anyone to get the wrong idea. I’ve had many extremely bright students. Many. Some had the necessary knowledge for the class, and some did not; of those who did not, most worked their butts off and did well. I also don’t want you to think smarts are what I care most about.
My favorite students are the ones who aren’t that bright, but work their tails off to do as well as they can. My least favorite students are the ones who are extremely sharp, but don’t work.
Sometimes, however, there’s too much missing knowledge, so much that the best thing the student can do is drop the class. It breaks my heart when I get a student like this.
I had a student some time back I’ll call Mark. Mark was bright, though his high school had cheated him, and he was lost almost from the first day. He worked hard, and came to my office hours. But … well, there was too much missing, as I discovered early in the semester when he was in my office.
I asked him what he wanted me to clarify, and he said he didn’t understand the 68-95-99 Rule. The conversation went something like this:
Me: “In a normal distribution, 68% of the data fall within one standard deviation in either direction of the mean. So here’s our distribution,” I drew a bell curve on the whiteboard, “And here’s our mean,” I drew a dashed line bisecting the curve. “Our mean is 50, and our standard deviation is 2, so 68% of the data fall between 50-2 and 50+2, 48 and 52.” I drew lines and arrows, and a 68% beneath.
Mark: “I don’t understand. Wouldn’t it be 75?”
Me: “Wouldn’t what be 75?”
Mark: “The mean.”
Me: “Why would it be 75?”
Mark: “That’s what you said in class.”
Ah. He was stuck on the example — and being a firm believer in introducing concepts in contexts familiar to students, I introduce basic descriptive stats in terms of grades, since what else are students more familiar with?
Me: “The mean could be 75, sure. In this particular distribution,” I pointed to the curve on the whiteboard, “the mean is 50.” I then erased the 50. “But let’s say the mean is 75,” and I wrote 75 after the x-bar, “then 68% of the data falls between 75-2 and 75+2, 73 and 77.”
Mark: “How do you know if the mean is 50 or 75?”
One of the difficult parts of teaching is diagnosing the problem. Students have questions, but the problem may actually be more fundamental than what they are asking about, as I was beginning to understand here.
Mark was having two basic problems: He didn’t understand what a mean was, and he was having trouble abstracting the idea out of the example. The former is easy to fix; the latter is not.
Me: “Can I erase this?” I pointed to the whiteboard, and he nodded. I erased the curve, and wrote a series of numbers on the board in a vertical column: 90, 85, 70, 65, and 50. “These are test scores,” I said, “How do you calcualate the mean, or average?”
Mark didn’t volunteer an answer.
Me: “Okay, let’s say the whole class takes an exam, and these are the scores. An average, or mean, tells me how well the class did overall. To calculate the average, I add all the scores, then divide by the number of scores. Here, you do it.” I have him the marker.
Mark added the numbers, then stopped.
Me: “How many scores are there?”
Mark: “Five.”
Me: “Okay, divide the total by five.”
Mark complied.
Me: “What’s the mean?”
Mark: “Seventy-two.” He looked at the numbers for a minute, then smiled. “I get it!” he said.
That’s when I realized what I’d suspected: Mark was a university freshman who had not, until just now, understood the concept of an average. I found that disturbing, but Mark was on a roll.
Mark: “So what’s a median?”
Me: “The middle score.” I pointed to the five numbers. “Half of the scores will fall above the median, and half will fall below the median. What’s the median of these scores?”
Mark: “Seventy.”
Mark was in my office three hours. No wonder he’d been lost. He didn’t understand an average. He didn’t understand sampling or distributions. We didn’t get to the 68-95-99 rule that day, because there was too much he didn’t understand.
I worked with him twice every week, and he got a B in the class. He worked harder than nearly any other student I’ve had. But if he had not come to my office every chance he got, he would never have passed.
Mark had no sense of entitlement. He wanted to understand, and he wanted a good grade, and he worked for both. He was bright. The thing is, I pretty much ran him through a high school math program in the office during the course of the semester.
I’m a teacher, so I can ask the obvious question, and some other teacher can’t come back with any of the usual non-answers.
I did it. Why couldn’t you, when Mark was in high school? It’s not money. I got no extra pay for helping Mark. It’s not time. I spent many hours in the office working with him. It’s not his intelligence or ability to learn. He’s smart, and he learned quickly, once we got started.
So I’ll ask again: Why couldn’t you do your job? It wasn’t my job to teach Mark high school math, but I did. Why did I have to? How did Mark get through all the required high school math courses without understanding what an average is? How did Mark get through all the required math courses without ever having seen y=mx+b? How did Mark get through all the required math courses and not understand that each flip of the coin is independent of all others? Most of all, how did you, his teachers, let such a bright, hard-working, motivated student slip through the cracks?
What’s going on there?
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