Once upon a time, so the story goes, there was a Wise Old Master who, despite his wisdom, seemed uncharacteristically glum one morning.  When asked why, he responded: “It was a clear beautiful night last night, with the most spectacular full moon.  So I decided to take the disciples outside for a wordless sermon. Standing among them in a field, I simply pointed at the moon.”
When asked why this dejected him, he replied “all the disciples knelt and worshipped the pointing finger.”
A couple of times recently, we’ve visited the idea of authentic student understanding, and I’d like to pull into that station again this week, for a couple of reasons.  As I was thinking about this, I remembered the story above, and perhaps I’m stretching the analogy, but it seems like we – both in education and in as a public – might sometimes get ourselves in the position of the Wise Master’s disciples.
In our early March visit, I had been discussing my experiences with the concept of division in fourth grade. (https://larryncampbell.com/index.php/2021/04/26/short-division-student-understanding/). I mentioned that I was a whiz at ‘naked division problems’ themselves, while simultaneously spending time wondering why an answer like 5 R 3 (five, remainder 3) couldn’t just be ‘shortcut-ed’ to an answer of 8 by adding the 5 & 3. 
I got an interesting response from Bob Egbert, frequent reader and former colleague in the MSU sciences: “As a student, back in the old days, I think I was able to perform without actually learning the material, especially in math.  I could figure out what sort of problems would likely be on an exam (three to five problems none too involved given the time allotted for the exam) and off I’d go.   I think that it wasn’t until some time later that I would say that I actually learned and really understood the material.”
Bob’s comment stuck with me, not only because of its relevance and its broad implications, but also because of his last sentence, which got me thinking.
I suspect we all have similar stories.  These stories may come from any discipline, of course, but I think they tend to occur more often in my own field of mathematics (and arithmetic) because understanding of mathematics and what it is/isn’t often gets sidetracked with a metaphorical focus on pointing fingers.
Consider the Wise Master and his disciples.  Note that his pointing finger was in fact an important tool used to help acknowledge the goal: the beauty of the moon. But the tool was not the goal, and the disciples stopped short.
Likewise, just for one limited example, skill tests in mathematics (standardized or otherwise) can be an important tool – among many others – in helping assess student understanding.  But they are not the final goal, and we shouldn’t stop too soon, thinking that they are.  It’s a common hurdle in the difficult pilgrimage toward assessing authentic understanding.
Bob’s last sentence above highlights another factor to consider.  How often do we all note that the real learning/understanding of a concept often comes slowly and in fragments?  Don’t we all have stories of ‘learning’ something in school, but not ‘understanding’ it until later?
I’m wondering if that isn’t part of the natural process.  If so, it creates an even more delicate balance in both teaching and assessing.  How do we determine how a student is progressing and still allow time for that ‘maturation’ process of fully understanding?  How, when, and how often do we assess?
It’s just another indicator of how complex the education process is and how well our schools manage to consistently achieve it.