Example of Resource Support of Mathematics Text
How could the types of resources (see our eText Typology) support a source text in mathematics?
Consider the text shown in Figure 1. This is taken from a theorem on number theory by Leonhard Euler (Euler, 1773/2007). This is not the ordinary fare of middle or high school algebra classes, but is in fact not as daunting as it might first appear. What resources might be useful for understanding this “obvious” mathematical text?
Figure 1. A Lemma from Euler used in proving that every integer is the sum of four squares
In point of fact, proving the equality in the next to last line is conceptually the same as demonstrating that (x+y)(x-y) = x2 +xy – xy +y2 = x2 + y2 which is a standard topic for every Algebra 1 student studying quadratic equations. The lemma simply requires more bookkeeping in repeated applications of the distributive and associative laws. Listed below are some possible resources in support of this mathematical text:
- Read the entire lemma aloud.
- Translate the text into other languages or representations: Braille, ASL, Spanish, etc.
- Provide definitions and examples for key vocabulary: Lemma, sum of four squares, expressed, cross products, and cancel.
- Provide a graduated set of examples of n-component expression multiplication. E.g. (1+2)(3+4); (x+1)(3+4); (x+y)(3+4); (x+y)(x-y); (x+ y +z)(x-y-z).
- Provide graduated set of examples of the distributive law.
- Provide graduated set of examples of the associative law.
- Provide a list of examples: 5= 22 + 12 + 02 + 02, 52 = 72 + 12 + 12 + 12.
- Provide an explanation for how a2 + b2 + c2 + d2 represents a number expressed as a sum of four squares.
- Summarize the steps of the main proof showing where the lemma is used.
- Provide metacognitive problem solving prompts.
- Provide prompts for readers to construct other examples.
- “Do you believe it?” “Try some examples.” “How could you square the expression for A?”
- Assess students’ understanding of the associative and distributive laws and recommend instructional activities to reinforce or reteach concepts and procedures.
- Provide links to where the lemma is using the proof of the main theorem.
- Provide mathematical text editor capable of creating and manipulating multi-component expressions.
- Change the size, color or arrangement of the text
- Shift the typographic conventions used to identify the three set of variables, especially away from the use of Greek letters.
- The theorem states that it is true for all integers. Can negative integers be the sum of squared numbers?
This list of ideas is not meant to be exhaustive or to consist of good and effective interventions. Even less does it suggest how such resources might be incorporated into an electronic reading system and presented to students in a manner that is useful and accessible, but not distracting. These are empirical questions, any of which could be the subject of research undertaken by MeTRC and its collaborators.