I often wonder how effective we can be as teachers if we
don’t first get an indication of what students misunderstand. I did a lot of
work with Dynamic Classroom Assessment,
George Bright and Jeane Joyner’s work and they talk about four ways to think
about students’ written responses. They differentiate among “Communicating an Understanding”,
“Communicating a Misunderstanding”, “Miscommunicating an Understanding”, and “Miscommunicating
a Misunderstanding”. If we think
about student work in terms of these categories then I believe it also means we
need to have multiple representations for which we can determine whether there
is understanding or not, and if it is computational or procedural, conceptual
or representational.
Think about the students who submitted the following work (I
am recreating it here but have not changed any of the students’ notations):
Which of the following procedures results in a larger
numerical answer?
1 ½ ÷ ¾
or 1 ½ x ¾
Student A: They are
the same because division is just the opposite of multiplication.
Student B: 1 ½ x ¾
because multiplication always gives a bigger answer.
Student C: They are
the same because in division you just change the sign and multiply.
What was interesting is the fact that none of the students
actually computed the quotient or product to allow them to check their
thinking.
In talking with these students I asked them what it means to
divide. None of the three students was able to articulate that division is used
to determine how many equivalent groups into which a dividend can be arranged,
nor given a number of groups what makes a fair share for each of the groups.
I am convinced that conceptual understanding must be
developed rather than giving students “cutesy” sayings like “keep, change,
flip”. How might we, as teachers,
develop a stronger conceptual model to deepen student understanding of division
and then division by a fraction? Please
post your thoughts and I will be reading them and posting a follow-up to this
in the next couple of weeks.
Anne
