chapter_1

When introducing the course, I stressed from the beginning that it is all about translation between, and building-up of skills within, three modes of representations (of mathematical ideas and reasoning): 1) Algebraic 2) Graphical 3) Tabular Like a foreign language, sometimes the hardest part of learning this material is that we've got to translate back and forth between modes (e.g., understanding what a vertical shift looks like on a graph and in an equation).

Example 3 from Section 1.2: Which of the points A, B, C, D, E, and F, represent ordered pairs that satisfy the equation ? (followed by a linear graph with labeled points, some of which are on the line)
 * This is a good example of translation (between graphical and algebraic) that seems to cause students problems. The red text ("points") seems to suggest a graphical reading of the question (i.e., it suggests that the reader will be able to think about this question within a graphical mode of reasoning). Yet, later on in the sentence, the purple text ("satisfy the equation") is a rather abrupt shift into a more algebraic mode of reasoning. So, perhaps not surprisingly, students reach this point and become confused -- apparently, they assume they need to answer the question within a graphical mode of reasoning, and they are not sure what "satisfy the equation" means in that context.
 * What I found was that by addressing these modes of reasoning and helping students to see how these translations between modes are //implied// by the language of the problems (and they are usually not adept at recognizing this language), they become a bit less tense about asking questions and feeling so helpless. In fact, even knowing only that you're having a translation problem can help you figure out how to proceed. In the above example, we talked about how, given that there is no equation given in the problem (let alone any numbers), there needs to be some kind of connection between what "satisfy the equation" and "points" on a graph mean. Seeing those two modes of reasoning as separate but translatable seems to have been useful for some.