Mathematicians sometimes refer to “motivating” a definition, theorem, etc. This departs from everyday usage, in which the object of the verb “motivating” is a person, by focusing on the mathematical motivation for a particular definition, proof, or topic. In a teaching context, “motivating” generally means using a particular sort of sequence of teaching actions or a segment of a textbook before introducing a definition, theorem, topic, or problem. The authors of the 1916 textbook Elementary Algebra claim to have tried to “motivate the various topics of algebra either through special problematic situations, or through the gradually rising demands of the equation for particular phases of algebraic technique” (p. iii).
In HK4 (034) the teacher creates a situation that makes salient the difference between two equations. [link to this section of the lessons by country doc] Both equations are linear in one unknown, but one equation has a single solution and the other has infinite solutions, where every number is a solution. The difference in these two equations is a motivation for singling out equations with the property that every number is a solution. Equations with this property have a special name: “identity.” Having students solve one equation with a unique solution and one equation that has every number as a solution motivates the concept of identity.
In N2 the teacher motivates the Pythagorean theorem by looking at the particular case of a (3,4,5) triangle drawn on a lined board. In this case it is easy for the students to figure out the area of the square formed by the length of the hypotenuse by computing the area of another square and of four triangles (see the N2 lesson graph). The teacher then extends this heuristic argument to a general right triangle.
Myers, G. & Atwood, G. (1916). Elementary algebra. Chicago: Scott Foresman.
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