John Walkup

I love listening to Jennifer Elkins and Gerlinde Olvera banter. Jennifer and Gerlinde form two of the Team Leaders in science and mathematics at the company, which has decided that providing researchers plenty of free time to discuss complex educational issues in detail is greatly fruitful. For the past two weeks, they have been discussing (and arguing) over some of the finer points of cognitive rigor, a framework for analyzing student work. Sometimes I jump in. One of our recent discussions centered on a subtle element of cognition that greatly affects teaching, a (sometimes) undesirable behavior we call <i>cognitive subversion</i>. Jennifer and Gerlinde have a detailed account of it in their own blog, but I saw a real-life account of it in a Southern California classroom.

A teacher was asking her students to find the area of a right triangle. All that was known was the length of the hypotenuse and that one of the angles measured 30 degrees. This is a complex activity, requiring that students first recognize that placing two of these triangles next to each other forms an equilateral triangle. There are quite a few steps that follow, and I won’t go into them in detail. But I will mention that this activity aligns to a depth-of-knowledge (DOK) level of 3.

However, the teacher began providing the students hints. The first hint she mentioned was, “Think about placing two of the triangles next to each other.” By providing hints to each step, the teacher was replacing a DOK-3 level activity with a succession of DOK-1 (recall) level steps.

Hints are often helpful, especially to weaker students. But hints can also subvert the cognitive rigor of an assignment, transforming a higher-order thinking activity into a series of low-level steps. We call this “cognitive subversion.”

There is a time and place for providing hints. And as objective observers, we are mindful that the strategies used by teachers can often be difficult to judge as beneficial without understanding the classroom environment. But teachers need to be aware of <i>why </i>they do the things they do (a process called metacognition). By being well-informed of what constitutes Bloom’s taxonomy and depth-of-knowledge, teachers can provide hints as part of a well-informed strategy, rather than based only on gut feelings.