This evening I attended a seminar at Monash University in Melbourne led by Professor Ron Tzur of Purdue University. Its title was Counter Intuitive Teaching: Proposing a paradigm shift in mathematics teaching and teacher education.
He asserted that there were two pedagogical constants in the work of a teacher: to answer the questions
(1) what will I teach?
(2) how will I teach it?
Ron divided the ‘how’ of teaching mathematics into ‘traditional’ or ‘reform’. I’m sure we’re all familiar with the teaching behavioural aspects of the traditional paradigm – show and tell through examples (or templates as my previous post would probably classify them as) and an emphasis on skill drill. The ‘reform’ model involves students actively solving realistic (or authentic to the discipline) problems, the use of manipulatives, group work in addition to individual work, discussions etc.
Ron’s contention is that, whilst the reform model is more student-centred and therefore intrinsically a more worthwhile learning experience for students, both models actually are underpinned by the same pedagogical approach. This approach he calls intuitive teaching.
Intuitive teaching has at its roots an entrenched presumption about human communication – mainly that the student will learn because the teacher has introduced/explained the new material well….that the new material is an entity on its own. The teacher identifies the concept or idea he/she wants to investigate and then decides on an activity (physical or mental) that addresses this concept to the learner. He believes that the learner is introduced to the unknown too quickly and that it isn’t owned by the student and that this has negative effects on the learner. This happens despite the fact that the teacher may be ‘first-rate’ in terms of their level of caring for their students, their knowledge of the discipline, their meticulous preparation, their desire to see their students learn and succeed, their knowledge of technologies etc.
With the brain research that has taken place in the last ten years, it can be shown that students will not actually create an appropriate cognitive platform for a new concept that allows them to transfer this knowledge into different contexts (and thus demonstrate true understanding) if the new material could not attach itself to a hook that had already been established in the brain (the pathway needs to be constructed between ‘old’, embedded knowledge and the new).
Ron’s paradigm of counter-intuitive teaching involves activities that offer deliberate assimilation of the new knowledge with the old. Time should always be spent establishing what conceptual knowledge students already have and then making connections between this ‘available’ conceptualisation with the new idea teachers want to embed. This, he stressed, isn’t the same as what skills knowledge they already have. “The brain cannot see if it doesn’t have the ways to see” The brain will only take on board what it is able to. Teachers need to build conceptual bridges through their choice of activities. He asserts that it is only through this process that a transformation in extant conceptions will result ie. learning will take place!
He described the process of counter-intuitive teaching as assimilating the new concepts into the learner’s current conceptions through an activity that sets goals for the learner and has opportunities within its execution for the learner to reflect on what they’re doing by compelling them to notice the effects of their actions and thus provide the cognitive bridge between what they already ‘know’ and the new concept. The learner then has seen and created the connecting bridge within their own brain’s framework. The learner ‘owns’ the concept on a biological basis. Prompts from the teacher can help orient the students’ reflections towards the new concept (teacher intervention).
This had a tremendous impact on me – this ‘psychologising’ of mathematics educating has been an interest, indeed a passion, of mine for some time now. To have it articulated and validated was very empowering and encouraging.
To finish, Ron said that he has visited many classrooms and assisted teachers with designing these counter-intuitive tasks and that its major benefits were:
*more often than not it works for students
*it was particularly powerful for hard-to-grasp concepts
*it gave teachers somewhere to look when students didn’t succeed
He also said that:
*it was not a risk-free pedagogy
*it was very demanding on teachers
He also recommended that mathematics curriculum documents be divided into 3 columns: one for the understandings we wish students to have about mathematics, one for the mathematics we wish to ‘cover’ and one for the activities we could use to develop those understandings for our students. As this is what I am currently working towards, this was also a fantastic endorsement for what I am trying to achieve.
A most uplifting, informative and heartening experience.