I was flipping through TV channels recently when I came upon The Defenders, a courtroom drama. The show was new to me, but one courtroom scene—and one line in particular—resonated with my experience as an educator.
In this scene, the defense attorney (and star of the show) gave a very short opening statement to the jury. When he returned to the table, his client questioned the shortness of his remarks. The defense attorney’s reply was, “Opening arguments never sway a jury. It is the closing arguments, supported by the evidence, that make for favorable verdicts.”
It struck me how this could be analogous to the classroom. For years, we educators have opened mathematics lessons with our entire defense of the mathematics concepts, asking students to accept and understand the concepts based only on our say-so. We then wonder why the members of the jury—our students—aren’t convinced by the lack of evidence.
An example that comes to mind is the inverse relationship of exponential and logarithmic functions. We usually start the lesson with the definition of a logarithm: if x = am, then log a x = m. I recently worked with a group of teachers who presented this relationship with a mnemonic, BEN.
BE=N (exponential form) means log B N = E (logarithmic form)
Students are asked to use the BEN formulas to solve for x in problems such as log x 8 = 3 or 5x = 100. The teacher then introduces the properties of exponents and logarithms, i.e. log a xy = log a x + log a y, then asks students to accept the formula as fact and use it to solve problems.
How different—and more impactful— would it be to have students complete a table for values of f(x) = 10x and its inverse, as follows?
We could then present the students with a table to complete with decimal forms of logarithms, then ask them to explore relationships.
As students look for pairs of values that add to a third value, for example log 2 + log 3, they are examining evidence that the log a x + log a y = log a xy.
Now that the students have explored the concepts, we can then make closing arguments that summarize the evidence—like any good defense attorney would.
Two of the eight Standards for Mathematical Practice ask us to have students examine evidence:
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
As we move toward the teaching envisioned in the Common Core State Standards, we may have to reconsider the power of closing summaries, which refer to and make sense of the evidence, over the futility of opening statements. Otherwise, we may continue to leave our students unconvinced.