Consider this problem. How many squares are on
a standard 8x8 chessboard? The immediate answer
is 64. There are 8 rows, each containing 8 squares.
But look again. The board also contains squares
that are made from 4 smaller squares. It contains
3x3 squares, 4x4 squares, and so on, all the
way up to the single 8x8 square. that is formed
by the entire board. The answer is therefore
1 squared plus 2 squared plus 3 squared plus
dot dot dot up to plus 8 squared, which is 204.
Now, our first answer was not foolish, it just
simply was not complete. Finding the larger answer
required us to remain with the problem after
the obvious answer appeared. We had to notice
what we overlooked. Then we had to revise our
interpretation and begin again. This is productive
struggle. It is the uncertainty that we experience
while learning something that is within reach
but not quite yet within sight. Productive struggle
may involve wrong answers, it may involve frustration,
and the uncomfortable feeling of not knowing
what to do next. But it is not evidence that
we are incapable of mathematics. Very often it
is the evidence that learning is happening. But
we should distinguish struggle from suffering.
A student wrestling with a difficult idea is
not in the same position as a student who is
humiliated or excluded or overwhelmed or left
without needed support. Difficulty can strengthen
learning. but shame and abandonment usually does
not. So we should be careful not to call every
painful experience, quote, good for us, simply
because something might eventually be learned
from it. Probably the important question is not
how can I eliminate every difficult feeling,
and it's probably not how much pain can I endure.
Maybe a better question is what kind of difficulty
is this? And what response does it require? Now,
sometimes the answer is persistence. Let's try
another example, or maybe draw a picture, or
maybe return to a definition. But sometimes the
answer is assistance. Let's ask someone to explain
it, or maybe offer a hint, or maybe someone can
work besides us. And sometimes the answer is
simply rest. Wisdom probably lies partly in learning
the difference. Now the chessboard did not become
clearer because we stared at it miserably. It
became clearer because we changed the way we
looked. So think of something that you are currently
struggling to understand. What have you unsuccessful
attempts already taught you? And maybe what small
hint would help you continue without taking the
discovery away? Difficulty is not always a barrier
to learning. but sometimes it is the place where
learning begins.