I think there are two dramatically different perceptions
about what is supposed to happen when students are doing mathematics, be it in
class or when they are doing their homework. I think this difference colors the
way mathematics learning and teaching is perceived and causes considerable
confusion when students, teachers and parents discuss progress. One point of
view is that the goal is to get the correct answer to the problems posed,
basically that the problem is a “test” to see if the solver can get the correct
answer. The other belief is that the problems have been posed so that the
solver will learn something by working on the problem, and that working on the
problem will make the solver better at doing math.

I was observing a first grade mathematics lesson last week
and students were learning about how to use “near doubles.” Essentially this was a formal lesson that
encouraged students to do problems like 13+14 by thinking that twice 13 is 26
so 13 +14 =27, or one more than 26. Or they could reason that twice 14 was 28
and so the sum would be one less than 28. As I observed, I thought about how
wonderful it was that these students were learning about the structure of
addition, as well as a handy method for finding sums that many adults use
regularly, rather than just memorizing a rule for adding two digit numbers.
This has to empower them as they learn more and more mathematics.

I walked around and observed that virtually all students got
the correct answers to the addition problem. I also observed that many were
doing it backwards, they were writing down the sum and then the doubles fact
that would help them get that sum. As in
the example above, the student solves 13+14 = 27 first, and then the student solves
it backwards, using 13+13 = 26 (a well-known doubles fact) and then 26+1 = 27. Some had the correct sum but a doubles fact
that did not make sense to me, or was not a doubles fact but some other fact of
addition. (For example, they might have 10+10=20 as their doubles fact. While
this can be used to determine the sum and uses doubles, it shows me that the
student has not grasped the concept of near doubles)

My take away from this observation is that often students
are not focused on anything other than getting the right answer, one way or
another, while the point of the lesson was to make the computation easier as
well as more meaningful by learning about the structure of addition instead of
just memorizing a rule. In fact, using doubles makes a boring addition problem
into an interesting challenge. It then
struck me that there is a dichotomy in the world of mathematics education that
has serious consequences throughout the learning experience of an individual. I
would like to think that the learner was always focused on what the point of
the lesson was, on what the creator of the lesson wanted the user of the lesson
to learn and understand at the end of the lesson. I suspect that the learner is
often not even considering that but is focused on answer getting as opposed to
finding a really great way to get to that answer. I think many parents also
have that same belief, even without realizing it. If their child gets right
answers on tests, they are happy. And what could be wrong with that?

For one thing, the student who only has one way to approach
many problems may find it boring, and may not realize that there are techniques
needed later that benefit greatly from thinking about a concept in many
different ways. An example in later mathematics that comes to mind is the many
different ways to create a graph of a line, or solve a system of equations. The
student who has mastered only one way is severely handicapped if the situation
does not fit comfortably into the solution method they have memorized. For another, the student who learns one way to
do it may find math to be dull and uninteresting, while the student who
explores many different ways of looking at a problem will, I believe, find
mathematics as an outlet for creativity and inventive thought. I believe that
students who enjoy what they are learning, understand that they have a certain
amount of control over how they proceed, and know that there is a utility to
what they learn, will work much harder and learn more than those who just do it
on order to get the right answer and therefore
a good grade.

This explains something that has troubled me for some time.
I frequently hear from parents that their child is not challenged by the
curriculum that is being offered in their school. When I have looked at the
curriculum I find that it is often very rich and full of many challenges.
Perhaps the reason for this disconnect is that when the student looks at the material, the only thing the student is
thinking about is how can I get this done and get a right answer, while I am
looking at how many different things a student can learn from the variety of
approaches taken.

The problem is, that if I am right about this, it does not
give me an immediate plan of action to correct it and we educators believe that
it is our job to correct things that are not working. Perhaps that is food for
another blog, but at the least I am interested in you comments about my thoughts.