Canon Fodder: Beginnings

A beginning (and ending?) point for the search for the root/canon of math education.

Being a mathematician by training and profession, I’m often driven to make things outside of mathematics conform to the patterns of mathematics: abstraction, definitions, theorems, deductive proofs, etc.  I can also be pretty blind to the possibility that this won’t work for everything.  This is my attempt to make some (mathematical) sense of Education Learning Theories.

Theory = Model: A theory is just a model. A model is initially defined by the context in which is designed to work, and the question it is designed to answer.  Much can go into the context, e.g. domain, scale and resolution, and there is a natural give and take between the context and the question (and the assumptions that go into the model).

For example, in a context where we have a set of data (inputs and outputs) that are related, we wish to predict the output value at some other input.  Based on these components,  and some additional assumptions, we construct a linear function, and that linear function is our model, or our theory as to how the outputs are related to the inputs.

Of course, for a given context/question we could come up with different theories, and then we have to consider how to compare and evaluate them (if at all possible).  However, the point for now is that to even have a chance of comparing theories, we have to identify their forming context/question and consider the overlap.

Learning ≠ Physics: Considering the ‘unreasonable effectiveness’ of mathematics in describing and predicting physical phenomenon, it is natural to assume that we can use a mathematical framework for models to describe other phenomenon, like learning, with the same level of success. But, of course, learning is much more complex and there are too many variables outside the range of easy or reasonable measurement, and so, we should expect that the derived theories will focus on a limited part and will be somewhat imprecise.

Learning Here ≠ Learning There: In modeling it is common (or a common mistake) to extend the model beyond the context/questions that originally defined it. Good models are robust enough to take this abuse, but I suspect that unless stated quite broadly, Learning Theories are much more fragile and do not extend easily.

Learning Theory≠ Model: It seems, by transitivity, that I have contradicted myself here, but what I’m really saying is that Learning Theories aren’t really used in the same way that (mathematical) Models are.  Math models are used primarily (and because they can be) to predict and secondarily to understand.  They also, in a small way, form a framework/context for discussing the object being modeled.  In our example above of constructing a linear function, we could think about the general process of replacing a set of data with the parameters that define the model and then talk about different sets of data in terms of just these parameters, e.g. “Is the slope/correlation positive or negative?”. Learning theories are used with reverse emphasis.  They are primarily about creating a context, or perspective or ‘lens’ for discussing learning.  They establish a based language that those ‘here’ and those ‘there’ can use to talk about and evaluate learning.  There is some part of theories related to prediction and understanding, but it is minor.

Canon or Not?:  I don’t think I can put enough structure on learning theories to make them more ‘mathematical’, primarily due to the complexity and variety of contexts involved. I do think that there is value is pursuing more understanding and formalizing the base ideas so that we have some way to discuss, apply and teach learning theories.



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