My siblings and I recently held our annual reunion.  At these reunions, one of the obligatory conversational paths is the one that heads down memory lane to our childhood days.  Stories of brotherly bullying (I am oldest – I plead the fifth), sisterly spoiling (the youngest was the first and only female!) and other growing-up memories are shared and exaggerated greatly, which is, of course, part of the fun.

At this recent gathering, one of my brothers told this story on me. I’d never heard it before and have absolutely no memory of it.  Decades have blurred his memory of details and timeline a little, as you’ll see.  Yet, I don’t think he was spinning a yarn.  I find the story interesting in numerous ways, especially all these decades later, and after my own career in mathematics education.  

In his story, I came home one afternoon from a Calculus test I had worried about. (I lived at home during my first two years in college.)  He asked how it went, and I lamented that I had forgotten how to work a particular problem.  Having remembered studying the concept, however, I was able to remember the page number of the text where the method was found, and even where on the page to find it.  (It’s my brother’s story and I’m stickin’ to it.)  So, I listed those facts on the test, and I was upset that I didn’t get (at least partial) credit for that.

(It’s not clear how I knew I didn’t get credit.  Perhaps the story takes place a day or two later when the tests were returned?)

On the surface, I’m not here to defend my take on that grievance at that time.  I suspect if I could now somehow visit with my alter-ego back there in his era, my knee-jerk reaction might be to say “Shape up, kid.  You can’t carry a Calculus book everywhere you go.”  Besides, as a later teacher myself (and of Calculus at that), I’ve had similar interesting takes shared with me by sly students.

On the other hand, the story has me thinking.  What’s the educational perspective here?  Let’s try to leave the story itself back in the 60’s (ouch!) where it belongs, and let’s examine this situation today.

Aren’t we suddenly right back in the middle of the basic questions that underlie so many of these columns?  What is it that we want our ‘educated’ students today to be able to do, especially in these digital days of instant answers on phones and laptops?

I haven’t needed or used any really esoteric Calculus computations in decades.  If I suddenly need one of them (shudder), and I’m rusty, am I cheating to pull out an old Calc text (especially the one named Google) to review?

If I encounter a simple arithmetic problem involving a 4^th-grade skill, do I necessarily need to remember old paper/pencil algorithms? Isn’t the basic skill here the ability to spot what operation is called for and how to proceed?  If I can’t do that, not even a calculator will help me, let alone paper/pencil. 

I’m sometimes shaky on subtle shades of meanings for words I think I want to use. (Happened twice writing this column!) Am I less educated if I double-check the meanings online?   

This is obviously over-simplified, so I say it very carefully.  Should we be preparing our students to be ‘knowers of facts and procedures’ or confident problem-solvers who know how to find tools they need and use them?