Moving Away from Results-Oriented Math Instruction

by John R. Walkup, Ph,D.

Each year, the mathematics education community talks more and more about the need to focus on process, not results. Back in the day, if students obtained correct answers on their assignments, teachers reasoned that the students must have learned all they needed to know about the topic.

Not so today.

Now, we focus on process because, among other things, it provides students more flexibility in applying their knowledge to problems lying outside the traditional experience.

There are a number of things teachers can do to shift the focus onto process, not results.

1. For each lesson, teach the entire Knowledge Dimension of Bloom's Revised Taxonomy.  The Knowledge Dimension includes factual knowledge, procedural knowledge, conceptual knowledge, and metacognition. I have also added relevance knowledge, deep knowledge, and collaborative knowledge.

Factual and procedural knowledge are already incorporated into traditional mathematics instruction. However, conceptual knowledge is often found lacking. For example, instruction associated with solving absolute value equations often focuses on using an algorithm to generate the correct answer. however, most simple absolute value equations can be solved conceptually as well since the absolute value indicates the distance between two points on the number line.*

For example, |x - 7| = 5 can be reasoned just as easily as calculated: "Two points, one at x and the other at 7, are separated by a distance of 5... hmmm... 2 and 12 are a distance of 5 from 7, so the solutions must be x = 2 and x = 12."  (Note that this also teaches students why two answers exist for x.)

For more on this revised Knowledge Dimension, see my other blog post, which focuses on an art education lesson as a clarifying example.

2. Focus on evidence of instruction. Student work should display the thought processes students used to solve the problem. This extends beyond simply "show your work." Students should routinely comment on their problem-solving approach to the extent that their work can be used as a learning tool later. When grading, teachers should also provide meaningful feedback on the process used by the student. If the grading burden is too high, then teachers can walk through the solutions, then ask students to write down their own process feedback.

Notice that in the sample student work shown above, the student has provided little meaningful feedback to the teacher on his thought processes. While each step is shown, the work provides no mention of the concepts or decision-making employed by the student to solve the problems.

Essentially, one could ask:  If my students blacked out their answers on homework assignments, what would remain for me to examine? What would they have in their possession that would help them understand what they learned?

Note that evidence of instruction carries with it a huge metacognitive aspect, one of the seven domains in the Knowledge Dimension of Bloom's Taxonomy.

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Seeking training at your school or district centered on Cognitive Rigor or Depth of Knowledge?  Call me at (559) 903-4014 or email me at jwalkup@standardsco.com. 

We will discuss ways in which I can help your teachers boost student engagement and deep thinking in their classrooms. I offer workshops, follow-up classroom observation/coaching, and curriculum analysis to anywhere in the country (and even internationally).

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