Return with me briefly to my fourth grade school year (and bring your own dinosaur jokes with you). We’re headed to Mrs. Dice’s classroom at the end of the hall, where there’s a sharp right turn (both physical and metaphorical) into the 5^th and 6^th grade classrooms down that wing.

As we sneak into the back of the classroom, we observe that we were venturing into this new area called ‘short division’.  (Did you encounter it?)   Exciting new stuff!  If we could learn these shortcuts, we would no longer have to multiply by the divisor, subtract the result from the original number, and repeat the process with the new number. (Yawn) 

I now have to tell you that I was a whiz at arithmetic procedures that year.  It you needed a times-tables or long-division answer, I was your man (or boy, I guess).   But this is not to brag, as you’ll see.

I soon got good at this new procedure too.  If I encountered a problem like 7 ‘goezinto’ 38, I could get the answer as 5, with a remainder of 3, using the shortcut procedures we were taught.  We wrote this answer as 5 R 3. 

Now here’s the problem.  I could get that fine.  But there was a period of two weeks or so that for the life of me, I didn’t understand why we needed two numbers in the answer!  I mean, we have 5 R 3.  Why isn’t the answer just 8?  It was very puzzling to me. (And I didn’t have the courage to just ask.)

In summary, I could do short division problems lickety-split and get usually-perfect scores.  But I didn’t ‘get’ that 38 could be split into 5 full groups of 7, with 3 left over that won’t fit into equal groups.

So, here’s the question:  Given that limited scenario, would we say that I understood the concept of division?  The quizzes (and perhaps the standardized tests) would say that I did. All correct answers, after all.  But I’m pretty sure I know the dark secret that I really didn’t understand the concept during that time.

You may have deduced that I’m still pondering one of our topics from last time, namely the idea of student understanding, what it really means, and education’s surprisingly difficult task of evaluating when it is achieved – individually or as a group.

The example is from the field of arithmetic, and that’s probably because that’s buried in my ‘academic area’.  Perhaps that’s why I remember my frustration so well?  But we don’t have to limit our discussion to that subject.

After the events of the past two months (some would say the past 4 months – or even years), there’s been a lot of talk that we need to beef up the content in our school classes in history and civics, etc.  But let’s think about that.  I remember learning a lot of dates in history classes, studying a lot of maps, and learning a lot of facts (good and useful ones, I might add!  And I remember many of them).  But, did we really learn to understand the lessons of history or civics?  Have we only recently come to learn and appreciate (the hard way) how valuable history is in our present times and how incredibly tough it is to be a good citizen, understand the Constitution, and to protect a democracy?

So, we return to the beginning.  We toss around the term student understanding as if we knew what we mean and automatically knew how to evaluate it.  But do we?  And will skills tests enlighten us?