Why bother with zscores and Table A?
We use zscores and Table A to help us find area under normal distributions (Pvalue) so that we can make a conclusion to a significance test. But calculators and technology can do all this work for us. Tables of values are archaic. We most certainly aren’t still using one of these to find logarithms:
So why bother with zscores and Table A?
The Most Obvious Answer
Besides helping us to get the Pvalue, zscores allow us to fairly compare values that come from two different distributions. Kyra is a 4th grader who is 49 inches tall and Kirah is a 6th grader who is 56 inches tall. Who is taller within their age group?
But there is a better reason.
The Better Answer: Inferential Thinking
When experienced statistics thinkers are given a scenario, we usually have some idea about whether or not the results are statistically significant. Let’s look at an example:
The school newspaper took a random sample of 400 students at East Kentwood High School and asked them whether or not they approved of building a new parking lot in front of the school. 59% said that they approve. Does this provide convincing evidence that a majority of students approve the new parking lot?
In my mind, I think “for a majority, we need 50% and this result is 9% above that. Moreover, the sample size of 400 is large enough that the variability in the sampling distribution will be pretty small…so I think the probability of getting 59% purely by chance is very low (Pvalue), so our result is statistically significant.”
There are three levels of thinking displayed here:

How far is my value away from what was expected?

How many standard deviations is my value away from what was expected? (zscore)

How likely is it to get my value purely by chance (Pvalue!).
Calculating and understanding zscores (level 2) is the thinking that connects us to the Pvalue (the holy grail). If we simply use a calculator or an applet to find the Pvalue, we have skipped over the inferential thinking. But let’s be careful here. If students are simply calculating the zscore as a memorized process, then their inferential thinking is not being developed. We need to be sure that students are interpreting the zscore (like it’s their job).
In most cases, I will argue for phasing out the old and in with the new technology (see Graphing Calculators are the New Slide Rule), but in this case I stay old school with my students…because ultimately I want them to be able to “feel” statistical significance using their inferential reasoning.