# What do you mean…a ‘model of students’ mathematics’?

As a middle school mathematics teacher who has stepped into a world here at the AIMS Center that is bigger than my experience, I am often overwhelmed by the scope of all I need to keep in mind. Still, many of the ideas are accessible, and allow me to reflect on my practice as an educator. Thoughts like,

*knowledge is not a transferable commodity; *

and

*communication is not a conveyance*,

have caused me to see knowledge as a tool of adaptation generated by a learner in a responsive effort to meet to the goals and demands of his or her experience within an environment.

As an educator, I can no longer assume that by communicating in a learning experience I am building a student’s knowledge. As a matter of fact, for a physical demonstration (much less a verbal explanation) to significantly impact a child’s understanding, brain research shows the child must plan to do what it is the educator is doing (or speaking of doing). For an educator to say that a student has knowledge of a “thing” would require evidence of the student spontaneously using that knowledge in the service of achieving another goal.

This raises an issue. How can I teach more responsively when I can’t know what’s happening in a student’s brain? Educational researchers have developed a way to think about teaching students from diverse experiential backgrounds, with mathematical notions based on well-developed models (or progressions) of how actual students have constructed those ideas in the past. These psychological models allow an educator to predictively anticipate and respond to how an individual student operates in real-time, the first-time. They have the potential to transform static attempts at ‘one size fits all’ mathematics instruction into dynamic and tailored mathematical exchanges that provide powerful opportunity for each student to leverage and advance their conceptions. Well-developed, second-order models called ‘children’s mathematics’ can allow educators to purposefully provide opportunities for each student to both construct and spontaneously use their mathematical knowledge.

These models can serve as organizing principles to help educators explain and predict recurrent patterns in student mathematical activity. Though they are not to be understood as what “really” goes on in the head of the student, models serve in showing the progress actual students have been observed to make under the influence of constructivist teaching. These are typically in a format general enough to account for the progress of other students and specific enough to orient an educator’s experience of an individual student in a particular mathematical situation.

In a future blog, I hope to talk in more detail about how you might begin utilizing generalized models of students’ mathematics to predict and explain student thinking in your educational setting.