In the process of teaching the content I began to discover the meanings behind many short-cuts that I had memorized throughout the years, and to my surprise, was enjoying it quite a bit.

That’s one of the joys of teaching they don’t tell you about right there. The common understanding of math teaching is that once you max out your understanding of secondary math, you teach from your own frozen, calcified body of knowledge. Not so in your case. Not so in my case. So nice to have a job that offers so many opportunities for continued growth.

Also as a statistics fanatic, I agree that z-scores can be a very confusing idea, but when it comes down to it, it is only confusing because of the Standard Deviation. Something few people can probably explain correctly. I think that if you can explain to a student what the standard deviation means well enough, then it becomes natural to try and measure how many deviations away from the average a value is.

Example: you have a set of data, say goldfish weights. Some numbers large, some small.
Q1: What’s “typical” here? (Review measures of center – kids mostly land on mean being “typical”)
Q2: If I wanted to show how typical or atypical these numbers were, how could I do that? (Many students suggest finding how far each number is from the mean, by subtracting the mean from each number)
Q2a: What do positive/negative/large/small numbers MEAN in this data set?
Q3: Suppose I had another data set that had different data (say, the weights of polar bears), but I wanted to make an apples-to-apples comparison between, say, the heaviest goldfish and the heaviest polar bear, how could I do that? (It’s at this point that discussion relating to variability – how many standard deviations from the mean is it? – leads to the formula for z organically)

For normal distribution, I like Amy’s idea.

Very few of my students ever forget WHY they’re doing the calculations they’re doing.

There are increasing numbers of marvelous videos available online, generally free, that give treatments of various college math subjects that are more insightful and user-friendly. And even if they weren’t, if the lectures are sound, you can watch them multiple times for clarification. Sometimes looking at a couple of lectures from different professors/teachers on the same topic will do the trick. And there are lots of places to post questions (Quora has some very good math people answering inquiries, for example). Don’t despair.

I feel as though this is similar to how a lot of students feel when asked to explain the concept of something like a z-score or derivative. Since it is so easy for teachers to move past the explanation quickly, as it makes sense to us, students feel lost very easily and are often embarrassed to ask for help. This then snowballs as the teaching continues and they feel even more lost, so they just check out and memorize how to solve the problem. It’s a very unfortunate issue that is also very common.

In my teacher life before Baxter Academy I taught with Core-Plus Mathematics, an integrated program with an extremely strong and well-developed statistics & probability strand. It was only while learning with Core-Plus that I truly understood the meaning of a z-score. When I observed students learning through these materials, I was astounded that the approach helped students to develop their own formula by asking the simple contextual question, “How many standard deviations away from the mean …”

A more interesting question for me, however, is why standardize in the first place? We have technology that will give us the p-value without standardizing, so why even bother? I wrote about the way this question played out in the classroom in April 2015. (https://rawsonmath.wordpress.com/2015/04/)