Encouraging critical thinking in Maths

Day 2 of multiplication with Year 7s. After completing two examples on the board, showing exactly how students should do their working out, I asked “Does this make sense? Is anyone confused about any of the steps I have done on the board here?” With nods and general silence I set the class to continuing their work – short and long multiplication questions.

At this stage of the lesson, I move away from the board and start walking in between the rows of students, stopping to check their work and see how they’re going.  By doing so, I get to know the students a little better, have a bit of a chat with them and take note of the pace of their work (are they at level with the rest of the class, doing challenge problems, or doing some more practice on the basics?).  Approaching one student, Lisa, I take a quick look at her work: she has left out the 0 at the start of the second line of working in about five long multiplication questions. Putting in this 0 is something I mentioned during last lesson and re-emphasised this lesson. So why did Lisa continue to miss this basic step?
As David Wetzel has noted in a recent article, “Encouraging students to use critical thinking is more than an extension activity in science and math lessons, it is the basis of true learning.”  Critical thinking moves students beyond passive learning to active learning. By having students watch me explain multiplication problems on the board, they are passively taking in what I am saying. While Lisa may have comprehended every step, she was not actively engaging with the problem. It is perhaps unsurprising, therefore, that she went on to make mistakes.
A mentor of mine who happened to be watching that class, suggested that I try different methods for checking for understanding among the students. In particular, he noted that it could be useful to get students to articulate the meaning behind steps in the working out and the solution. So the next day I tried something different.  Instead of putting up a question and showing how the working should be done, I wrote up a couple of questions with their worked solutions, and each containing a mistake.  I asked students to have a think about these problems, then work in pairs and figure out exactly where I had gone wrong. As a final step, I got a few students to come up to the board and explain their solutions to the class. Rather than silence and blank stares, there was what I like to think of as “productive noise”.  Students busily working out the problems and asking one another questions.
What I’ve learned from this class, I will work to apply to others.  It’s easy to just ask, “Do you understand?”, but to get students more actively involved in their learning, higher level cognitive questioning is needed.
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