If you believe the theories of mathematics presented by a university researcher/teacher who proves and uses those theories every day in his/her/their work, and which are presented to you for your learning in a university mathematics course, why would you not believe the theories of learning presented by a university researcher/teacher who proves and uses those theories every day in his/her/their work, which are presented to you for your learning in a university mathematics education course?

When you were in university mathematics courses, which mathematics theories did you disregard and say were wrong, were false? Of the mathematicians who taught you in those university courses, which of them did you say were lying to you by presenting mathematics based on false theories? I expect you believed everything those instructors wrote on the blackboard for your learning (clearly transposition mistakes aside).

Admittedly, a few theories, such as Fermat�s last theorem were unproven for centuries, but were regarded as believable given the incredible number of examples that showed this theorem worked, and the absence of a counter example to the theorem. They were accepted, worked with, and further studied with a disposition that it is likely a likely true and valid theory. But those are the exception; let�s focus on the theories that are proven. For example, does it matter how long a theory has been in existence, that is, has been known and been proven? If yes, then the example of Fermat�s Last Theorem used above is a counter example. Are we to ignore the proof just defined in 1994 by Dr. Andrew Wiles because it was within a few decades, or within this century? So perhaps it does not matter how long a theory is in existence, as long as it has been proven.

The burden of appropriate proof is left for those who are experts in the field, the peers of the one making the conjecture and offering the proof. It is not, for example, up to someone like me, who, although I have an undergraduate mathematics degree, would not presume to be able to judge the accuracy, logic, and matheamtics of Dr. Wiles� proof, even though I use the related Pythagorean Theorem and some of the number theory he employed in his proof almost daily, definitely frequently, when I teach secondary school mathematics. He is the expert in his field, I am not an expert in his field. I trust him and his peers for mathematics knowledge.

In the field of mathematics education, more accurately, the field of teacher education for the teaching and learning of secondary school mathematics, the theories used by mathematics educators are psychological theories, education theories, and the combinations of these theories with mathematics, such as theories of teacher efficacy, student efficacy, social constructivism, intelligence, learning disabilities, belonging, behaviour, collaborative learning environments, manipulatives, or technology. These are theoretically and empirically proven theories. Why then would these theories be dismissed by some of the preservice teachers enrolled in a mathematics education university program? When a pedagogical strategy for the secondary school mathematics classroom and the associated classroom teacher and student artifacts are brought to preservice teachers� attention, why is scepticism, or blatant dismissal with �No, that won�t work in a classroom!� a common response? Which of the mathematics education theories that are being used within that teaching and learning strategy are being accepted or dismissed, and why are they being accepted or dismissed?

As a secondary school mathematics teacher, I do not recall my fellow mathematics teachers dismissing a mathematics theory that has been thought about, proven, or applied to other contexts and situations (we could call these �application questions�, or �chapter problems�, or �the exercises in section C of the textbook�). In fact, I don�t recall such a dismissal of the mathematics in a curriculum by preservice teachers in my mathematics education courses either. The order and selection of particular mathematics for a curriculum� possibly, but not the mathematics identified in the curriculum expectation itself. However, the teaching and learning strategies for such mathematics, based on proven theories of learning, can be readily and quickly dismissed by preservice teachers. There is a savage corollary, the theories supporting teacher education pedagogical practice can be dismissed and ignored too. Which of the proven theories of teacher education are being dismissed and ignored when a preservice teacher won�t do the work, won�t have the experience of learning in a particular way, or does something else with the materials rather than the activity that was designed� based on teacher education theory and practice?

It is well known in the post-secondary research environment that theory and practice are closely intertwined. This is also a commonly held perspective specifically in the field of education. The argument of theory versus practice is much like the chicken and the egg argument � which came first? At this point in evolution it is perhaps not necessary, except for philosophical debate and discussion purposes, to worry about what really came first. As for theory and practice, which comes first? Often a theory inspires practice, or practice inspires a theoretical idea that is then tested and shown to be a theory. Often emergent theories are based on a combination of proven theories and practice. (There is an edge of discontinuity, some might say hypocrisy, for the practice to be privileged in the secondary school mathematics classroom, but not in the tertiary school mathematics education classroom. Or the theories of mathematics to be privileged but not the theories of the teaching and learning of mathematics.)

At this point in time, again, does it matter whether it was theory or practice that came first when we talk about the theories and the practice that we have accepted in mathematics and built a field of knowledge upon? A field of knowledge upon which other fields have flourished, such as engineering, computer science, medicine, architecture, and law for a few examples. It is a field that is continually changing; a field of knowledge that is continually learning and improving itself, as evidenced by Dr. Wiles� recent mathematics proof. The same should be said for the field of mathematics education or teacher education. They are fields of knowledge for which research simultaneously exists with practice, and it is reassuring that these fields of education are known to be continually improving and learning. What matters is the acceptance of the knowledge of the field, and the expertise of those who are the knowledge creators (researchers) and knowledge mobilisers (university teachers).  

As a side note of something related�and perhaps what inspired my writing�

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Jamie Pyper

Jamie Pyper

mathematics educator and researcher | mathematics teacher (OCT) | and craft beer enthusiast! | he/him | #MSTE Coordinator, #PBLmathEd #CMESG #NCTM | mathperceptionproject.ca @314_per
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