In our Mathematical Literacy course, we are working through concepts from a numeric point of view with less emphasis on algebraic (symbolic) statements. This weeks’ content dealt with ratios, scaling rates up or down, linear rate of change and exponential rate of change. Our work might indicate what our students are capable of, in a general way.
This course is ‘at the same level’ as beginning algebra, which means that we share prerequisite settings for math, reading, and writing; the students are similar, in many ways, to a typical beginning algebra class. The Math Lit class also has a few students who did not meet all three prerequisites (due to some system problems at the college).
It’s true that students struggled at times in class. One of those struggles dealt with language processing; we are using nutrition labels as a context for working on rates and scaling. When students needed to read specific questions and then extract information from the label, most students did not see what they should do. This is not a matter of mathematical ability or skills; in fact, students who have passed our beginning algebra class often exhibit the same pattern when I see them in the applications course (Math – Applications for Living). A few students are having trouble with the scaling ideas, which is a non-standard approach; however, since they usually know an alternate method this is not a big issue.
Although I have not done an individual assessment yet, students did not seem to have any trouble with the concepts of linear and of exponential change. We did numeric examples in two settings, and I observed groups and individuals — no issues spotted. Most students are having difficulty connecting a situation to a symbolic model — both linear and exponential. In the case of linear, we did “the salary is increased by 5%” … all of them could calculate the result for a given salary, but few of them could make the transition to the symbolic model (new = 1.05S). The same kind of thing happened with exponential models. Since we are not emphasizing symbolic work (yet!), this gap is not a big problem (yet!).
I’ve dealt with exactly the same issue in the Applications course (symbolic models for linear and exponential change), and observed the same proportion of students having difficulty. The traditional beginning algebra course has an insignificant impact on students’ abilities to write symbolic models for situations — except when the correct key words are used in the problem. If the problem is stated in a way that “normal” people talk every day, students can not make the connection to symbolic forms (in general).
In some ways, this was a discouraging week. The difficulty with language is very frustrating; my judgment is that students (and people in general) are far less skilled with the written word than in prior decades. Basic verbal skills like parsing and paraphrasing are not normally seen. The transition to symbolic forms seems like such a small step, so that difficulty is troubling to a mathematician. Our course is designed to build these skills over the course by visiting similar ideas from different points of view; I can hope it gets better! However, I find it encouraging that these students — even the ones who lack all the prerequisites — are having no more difficulty than students who passed our beginning algebra course.
This Math Lit course is a good class for a mathematician to teach; we deal with basic ideas in detail and work on transfer of knowledge, with an emphasis on problem solving (as opposed to exercises and repetition). In that work in-depth, we can see where students really do not get the idea and work on creating better mathematical knowledge.
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