# Teaching Practice: Mathematics

A British teacher, Craig Barton, recently wrote about his transformation as a mathematics teacher in *American Educator*. He poses the classic change dilemma: don’t upset the applecart! If I am a well-liked teacher who gets good results, why am I doing anything wrong and why do I need to seek any further boundaries to explore?

Eric Mazur, a physics teacher at Harvard posed the same dilemma for his own teaching in 1990. He had been teaching for 7 years, got great evaluations from his students, gave brilliant lectures, and his students passed his exams with flying colors. But a simple test of his students’ understanding of the concept of force demonstrated that they held exactly the same misconceptions at the end of his course as they had held at the beginning of his course. The idea that his students were learning was an illusion. Mazur wrote: “The students did well on textbook-style problems. They had a bag of tricks, formulas to apply. But that was solving problems by rote. They *floundered* on the simple word problems, which demanded a real understanding of the concepts behind the formulas.”

So are we teaching mathematics, or are we teaching students how to answer exam questions? Barton writes in his article that “asking and responding to diagnostic questions is the single most important thing I do every lesson… teaching without formative assessment is like painting with your eyes closed.” He calls this “responsive teaching” or checking for understanding in real time and improving teaching and learning before it’s too late.

Where to go as a mathematics teacher? As in any other subject, the teacher first has to be released from the grip of the mark book. If everything is for ‘marks’, the chance that a student will take a risk is close to zero. The mark book also advantages certain kinds of learners – those predisposed to do what they are told to do with little variation. The research on valedictorians shows that great students (as defined by the mark book) are certainly successful in life, but they do little to move our civilization forward. They tend to be traditional and compliant. But understanding doesn’t come from getting something right; it comes from getting something wrong.

I learnt this when I took a 2-man canoeing course. I thought I was doing brilliantly at maneuvering my canoe around the lake. The instructor thought differently. I’m going to leave, he said, if you don’t get wet. Get wet, I asked? Yes, fall in the lake. You won’t understand water, your paddle, and the relationship between the two until you fall in the lake. He was absolutely right. I was a good canoeist before the course. I went to a whole different level of understanding the more mistakes I made allowing me to fully grasp both the intellectual and affective aspects of canoeing. As I lent on my paddle, I gained confidence in the tension I could feel and I could lean my canoe over further and further allowing me to run much more difficult rapids once we were allowed off the lake and into the river.

Mazur accidentally discovered what the teaching solution was. He was so frustrated in trying to get his clever Harvard students to understand what was so obvious that he finally told them to talk about it amongst themselves. In three minutes of peer instruction, understanding of the concept occurred and his lesson could continue. He developed this revelation into what he called “interactive learning”. He discovered that this way of learning erased the gender gap, enabled students to retain knowledge far longer, triples students knowledge gains, and made more students interested in physics. That’s a wow.

Let’s go back to Craig Barton and his understanding of the value of feedback or formative evaluation. He got rid of giving assignments for grades and replaced them with assignments for learning. He made everyone answer every question rather than picking out individuals to answer questions. And he had to relearn how to ask questions. He learnt to ask questions that would give him insight into the reasoning the student was using i.e. when they got it right or wrong, it was evident why that was so. Which led to the final insight that students each get something wrong for different reasons. He calls these questions diagnostic. Barton has written around 3,000 diagnostic questions (all available on his website).

Here are Barton’s criteria for each question:

- It should test a single skill or concept. This is not the time for interleaving, says Barton: “The purpose of a diagnostic question is to home in on the precise area that a student is struggling with and provide information about the precise nature of that struggle.”
- It should be clear and unambiguous. The teacher should be able to accurately infer students’ understanding from their answers.
- Students should be able to answer it in less than 10 seconds.
- The teacher should learn something from each incorrect response without further explanation from the student (that’s because the teacher has chosen the incorrect answers very carefully).
- It cannot be answered correctly while still holding a key misconception. This is the most important characteristic, and the one that makes formulating questions so difficult.

Interestingly and not surprisingly, this fits in with what John Hattie has found in his research. He wrote in 2003: Assessment is primarily concerned with providing teachers and/or students’ feedback information, which they then need to interpret when answering the three feedback questions: Where am I going? How am I going? and Where to next? And feedback is one of the top influences on a student’s success, with an effect size of .7, and proving much more useful than, for example, practice tests (effect size of .57), the teacher-student relationship (effect size of .52), small group learning (effect size of .47), technology in mathematics (effect size of .33), ability grouping (effect size of .3), and reducing class size (effect size of .21).

Barton’s website also has free math support and resources for teachers and students, and his book is available on Amazon!