Worked Examples in Math Ed: Theoretical and Empirical Support

WrongAnswer-isoIn preparing our Small Business Innovation Research (SBIR) proposal on the topic of developmental math games aimed at community college students, we’ve partnered with, and sought feedback from, several experts in math education, notably Dr. Michelle Chamberlin of the University of Wyoming. Dr. Chamberlin volunteered to write a short briefing on the theoretical and empirical support for worked example problems (click to see our first post on the topic).

What follows is Dr. Chamberlin’s sketch on the background of worked example problems, which she has kindly agreed to let us publish here.

Worked Example Problems: Theoretical and Empirical Support

by Michelle Chamberlin, Ph.D., Associate Professor, Department of Mathematics, University of Wyoming

Worked examples have a strong theoretical foundation in the field of cognitive science Continue reading

Threshold Concepts: Education as Rite of Passage

Initiates of ancient mystery religions would learn the meaning of symbols like this one. Image believed to be in the public domain; originally published 1896. From Wikipedia.

Initiates of ancient mystery religions would learn the meaning of symbols like this one. Image believed to be in the public domain; originally published 1896. From Wikipedia.

By dark of night the master leads you, the robed initiate, beyond the columns of the marble portico and into the candle-lit court of the temple. According to custom, the master has overseen your days of fasting, study, and preparation leading up to this, the day you would be ready to cross the threshold. Now, as the cock crows thrice, you are led into the place where you have never been allowed before, to see what you have never seen before, and to know what has been heretofore unknown…You enter into the inner portal of the holy of holies and behold, in the sacred light of a golden candelabrum… the solution to your chemistry homework.

A movement exists in education to view educational standards in the anthropological terms of the rite of passage. As elaborated by Victor Turner [Wikipedia] and other anthropologists, a rite of passage consists of an individual beginning in one social category, crossing a limen (Latin: threshold), and entering into another social category. Examples include rites of passage by which a boy becomes recognized as a man (e.g bar mitzvah), or by which a girl becomes a woman (e.g. bat mitzvah), or certain religious initiations, such as Greco-Roman mystery cults [Wikipedia] or, to use more familiar, contemporary examples, baptism, first communion, or confirmation.

The analogy is this: a student passes a conceptual limen when he or she comes to really understand any threshold concept. There is a metamorphosis from ignorant to understanding. Like a participant faced with the revelation of divine knowledge at Eleusis, the student confronts quadratic equations, the ideal gas law, the principles of engineering, the grammar of a well-composed sentence…whatever the concept, and that student is changed. Once a student masters a threshold concept, he or she doesn’t simply revert to ignorance. A threshold concept is of a class of concept such that it changes the student’s view of the course subject, and, perhaps, whether in a small way or in a profound way, the student’s entire worldview. Continue reading