Does everyone understand?

How many times have you asked your students that during a lesson?  I know I asked my students that question too many times to count.  But I never thought about what I meant by the word understand.

Richard Skemp talked about two different types of understanding back in 1976.   The distinction is still important today.  The first type is relational understanding: knowing what to do and why.  The other is instrumental understanding: knowing a rule and how to use it but not understanding why the rule works.

You can probably think of many examples of instrumental understanding in a math class.  Students may know to “flip and multiply” when dividing fractions and may be proficient at computing the answers but they have no idea why the rule works.

Why might it be problematic to only develop instrumental understanding in students?

Skemp explored this by providing a great analogy of visiting a new town.  He talked about learning the route between two points.  He could get from A to B by following step-by-step directions that were given to him.  Or he could explore the town with the goal of developing a mental map of the town.   In the first case, if he makes a mistake, he will be lost.  He needs to rely on outside guidance to help him get back on track.  In the second case, he can find his way around without having to follow step-by-step directions and can choose from multiple paths.  If he turns the wrong way, he can correct his mistake without any outside help and perhaps learn from the mistake.

How does this relate to math class?

If our students only develop instrumental understanding, they need to remember a series of different step-by-step procedures and when to use them.  They are dependent on the teacher for what to do if they make a mistake.  On the other hand, if they develop relational understanding they can use the connections they have developed to begin to correct their own mistakes and use multiple methods to solve a problem.  Skemp further argues that relational understanding is longer lasting and more adaptable to new tasks.

Thinking about what type of understanding about mathematics we want to foster in our students can help us thoughtfully make choices in our classrooms.  Does everyone understand?

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