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Are We Multiplying or Dividing?

Being Explicit in Teaching Mathematics

Ana Vaisenstein

It is common to hear that students with disabilities need explicit teaching.
Usually explicit teaching is understood as telling the student what to do, such as
explaining what procedure to follow to solve a multiplication problem. The be-
lief underlying this approach is that because of their disability, and in some cases
cultural background, students cannot come up with their own strategies for solv-
ing problems and/or do not have language to explain their thinking. Therefore,
the teacher needs to be the one who talks and explicitly gives these students the
necessary information to solve the problem.
Over the last four years, I have worked with students who struggle in math.
Some of them have Individualized Education Programs (IEPs), and some of them
have not yet been formally identified as students with special needs. I have ob-
served many students who have been taught procedures to solve the four basic op-
erations but who have not been exposed to the concepts underlying these proce-
dures. They are often not able to reason through the procedures or explain the
mathematical meaning of what they are doing.
Let me share an example from a student I had last spring. This is how Davel
solved 18 12:

18

 12

836

What procedure do you think Davel used?
This is what he explained to me: “2 times 8 is 16. Write the 6 and carry the

1. 2 times 1 is 2, plus the 1 I carried is 3. 1 times 8 is 8—836.” Davel had forgot-
   ten or maybe misunderstood the steps of the procedure he was taught, yet he com-
   pletely trusted the procedure and did not even wonder whether his answer made
   sense. When his peers commented that the answer could not be in the 800s, he