At the front of the class, demonstrations are given and tough problems are solved—but this time it’s the students that are doing the explaining. Steven Williams, a faculty member in the Department of Mathematics Education, has published research on how these classrooms still incorporate traditional teaching methods.
Williams’ publication is all about how the student can benefit from being more involved in the learning process. Along with a colleague from the University of Oregon, Williams studied classroom interactions at schools who had implemented a new approach to teaching math.
Under the direction of university and college professionals in their area, these schools implemented a reformed teaching method where students work towards solutions together and then share their solutions with the class. The teacher makes specific suggestions that guide students toward the correct methods. This concept is referred to as “scaffolding,” where understanding is initiated as the teacher helps to point out connections.
This approach seeks to avoid one drawback of the traditional classroom setup, where students are told the answers in a rigid, formalized lecture.
“When students get used to a teacher telling them everything they need to know, they begin to tune out until they get to what they think is the important part,” Williams said.
So, when do teachers tell? Through his research, Williams found that the dilemma of telling is two-sided. In the reformed approach, how and when teachers choose to tell is still vital to learning. His research shows that teachers were able to scaffold student-led discussion by having clear expectations and keeping a high level of involvement in the classroom.
“We observed that these teachers removed themselves as the single mathematical authority in the classroom and allowed students to explain and justify the mathematics they were doing using their own reasoning,” said Williams. “In order to pull this off, teachers need new skills. The term in the field is teachers that have ‘withitness,’ meaning they are aware of what is going on in the classroom. Good teachers can and should interact. They need a much fuller toolbox than just worksheets and timed tests.”
Williams feels the greatest impact of his research is its potential ability to change the poor reputation that has been attached to this kind of teaching. His publication provides support for an improved system that requires students to meet higher expectations by being independent thinkers. The role of teachers is enlarged, not diminished, through increased interaction. His paper acts as a caution towards extreme implementation of either reform or traditional methods.
“I realize that when you describe this kind of teaching, people say this is just discovery teaching and that didn’t work in the nineties,” he said of the politics involved. “They say it’s like turning over the asylum to the inmates, when really it’s more subtle than that. It’s not a wholesale abandonment of telling. Teachers in our study did break in when necessary. They did in fact tell, and any teacher needs to do that. Neither a pure lecture nor a pure discovery approach will work. Each side has more to give than the other side realizes.”
Though not part of his published findings, Williams found that this blended approach did, in fact, work for the school studied. In the case of 8th grade algebra students, they saw a huge jump in the number of students who were able to test into higher math classes, evidence that they were still performing well on standardized tests.
While many concede that talking about history or literature is helpful, many don’t see how more than one answer in math can actually be useful. Williams said that such a prejudice comes from not fully understanding the subtleties of mathematics.
“Different answers sometimes serve different purposes,” he said. “The ways of thinking about and approaching the problem are more important. If you understand math correctly as being about ideas and making connections, not just memorizing rules and applying them, you can see that [discussion among students] is suited to math.”
With the addition of Williams’ research to the literature, educators can better evaluate how these teaching methods can work together to create a richer education for math students.
For more information: Baxter, J. & Williams, S. (2010). Social and analytical scaffolding in middle school mathematics: Managing the dilemma of telling. Journal of Mathematics Teacher Education, 12(1), 7-26.