I have just finished reading an article titled “Conceptual Understanding, Problem Solving, Communication and Assessment meet at the Board” by James A Vicich in the February 2007 issue of Mathematics Teacher.
In this article, the author describes a methodology of classroom practice that he has implemented in his mathematics classes that, he asserts, creates a community of students actively engaged in solving problems whilst concurrently learning basic skills and concepts.
Paired board work involves pairs of students working on a problem at the board. Students alternate between taking on the role of ‘teacher’ and doing the problem. The student as ‘teacher’ facilitates the learning of the other student by giving ‘elaborated instruction’ – providing explanations that lead their peers to solve problems for themselves. “In explaining ideas to another, the helper must clarify, organise and possibly re-organise the material. In such a process, the helper may identify misunderstandings or gaps in his/her own knowledge and subsequently resolve those inconsistencies”
How often have we said that we didn’t really understand something until we had to explain it to someone else? This peer-tutoring idea is not new but building it into an instructional procedure that becomes a classroom routine is a wonderful idea. These ‘oral rehearsals’ consolidate and strengthen what students know and provide feedback on the degree of students’ understanding. By compelling students to explain the ‘why’ and not just show a method of solution or providing an answer, students perceive the understanding of the mathematics – the process – to be the primary goal instead of the product. This teacher mentions in the same article that he regularly asks students to come to the board to explain how they went about doing particular homework problems; surely a much better use of class time than a teacher moving through a class and checking that students have ‘done’ the homework. The doing of the homework should not be the primary goal; the understanding developed by exploring the problems set is surely a much better goal.
I regularly do a ‘quiz’ at the start of each lesson. This comprises 3 or 4 problems on the board that revise concepts from the previous class and make connections to either the ideas I am about to explore or to those we have looked at earlier. In addition, I like to include at least one question that extends students’ thinking about a concept being studied. ‘Connect, extend, challenge’ are my guidelines when preparing these questions. Students work on these questions individually or in groups. I then ask students to come to the board to explain their approaches to the rest of the class; students who say “I can’t do it” are not ‘let off’ – they are required to provide something that reflects what they are thinking then I ask other students to ‘lead them through’ the rest of the problem. Even students who are not natural ‘risk-takers’ come to accept this as common practice and I take pains to ensure students know that I don’t care if what they do is ‘wrong’ – their thinking is what I care about seeing. Students treat each other’s work with respect and a pretty good learning community is established.
On another note, we started the second term of the year here in Victoria last week. The first day of the term was a designated staff-only Faculty Day. One of the sessions, in which the mathematics faculty at my school was engaged, involved the work and findings in the doctoral thesis of Dr Gaye Williams – a mathematics lecturer at Deakin University. Her thesis investigated the question: ‘What factors are necessary in mathematics classrooms that lead to students becoming more creative thinkers?’
Some of her findings were quite scary:
• Even in the classes of teachers who had been identified as ‘good teachers’ by their peers, their students and the school community in general, deep analytical thinking wasn’t evident.
• Curricula that are dependent on texts lead to non-connective learning of mathematical ideas as textbooks, in general, separate ideas into chunks and don’t link them.
• Assessment tasks that were designed to elicit higher-level thinking in students didn’t always live up to this expectation as many teachers ‘gave’ students too much and thus took away the opportunities students had to engage in deep thinking.
After many hours of classroom observations, Gaye came to the conclusion that the primary factor that contributed to students being able to think creatively in mathematics was resilience (something I have touched on in a very early post this year). She defined resilience as the ability of students to see ‘failures’ as temporary, externally based and not something they took internal responsibility for. In addition, ‘successes’ were seen as permanent, pervasive and something over which the students had control.
It’s all to do with how students see themselves as learners in a mathematics classroom.
So…how do I want my students to see themselves as learners? How do I want to see myself as a teacher of mathematics? What ‘ways of seeing’ will determine a positive outcome for all involved?