Modularized mathematics is a common curricular strategy in our era, with a common justification and design strategy being the identification of what math students need. Separately, I have posted about the use of modules (and I will have more to say on them); today, this is about the use of ‘identifying the math they need’.
Here is a short story, a parable, with your indulgence:
Felicia and Ashley have been managing a service-oriented hardware store in their town for five years, and they finally have enough capital accumulated to remodel their store. In their planning process, they realize that it is important to make sure that they effectively meet the needs of their customers. With the help of a PR company, Felicia and Ashley design a web survey form that the customers can use to identify the items and categories of need. Naturally, the items and categories are based on what the store has already been selling. Many customers complete the survey, with a surprising consistency in the general results. Based on the results of the survey, a remodeled store opens with the merchandise reflecting the survey … items needed by many are in-stock and visible in an attractive display; items needed by a few are done as a special order.
After two months, it becomes clear that the new store is far less profitable than the old. A new survey is done to determine the problem, including areas for general comments. The results of this survey show that there were two causes of the problem. First, it turns out that the ‘items needed by a few’ were significant as a group … many items “less needed” accumulated over many customers creates a large change; the special order process did not meet the needs. Second, and mentioned on every comment, is the fact that there were four areas of emerging need in hardware that were not listed on the original survey; since they were not even listed, customers could not report this need. These emerging needs reflect both the newest do-it-yourself projects and the maintenance of the newest homes.
When we design modules or courses based on a content survey, we are beginning with the assumption that “what is needed” is within the existing content. This survey approach is commonly used for module designs, as well as research on the mathematics needed in various occupations. If we run a hardware store, there is an implied responsibility to understand ‘hardware-ology’ deeply to understand the needs of the patron even before they know what they need.
We run a mathematics learning enterprise. We carry a responsibility to deeply understand the mathematical needs of our students. Our situation is, in fact, far worse than the hardware store in the parable; the hardware store was successful before the change to ‘needed’ items. Mathematics programs …developmental or college credit … are definitely not successful currently. If a particular math program was already successful, there would not be much motivation to ‘modularize’ or to identify math needs; the fact that a program is modularized is a direct statement of non-success.
I suggest that you consider the more basic question:
Is it possible that our mathematics programs are not meeting the mathematical needs of our students and that this is a major factor in the program not successful?
Existing developmental math content is based on an archaic set of school mathematics content; it does not reflect the changes in schools since 1965. Existing introductory college mathematics is based on a curriculum extended from that archaic foundation. As you know, extrapolating from a model to a new set of domain is a risky process; not only do we assume the validity of the original model (not justified in math), we assume that the extrapolation is valid. We have significant curricular studies that conclude that the extrapolation is not well founded; see the work of MAA ‘CRAFTY’ (http://www.maa.org/cupm/crafty/.
Here is what we need instead of ‘need based on current content’:
We need to identify the basic mathematical knowledge needed for our students to be prepared well for the mathematical needs of their college academic work as well as societal needs.
A friend of mine is a somewhat famous economics educator in community colleges. Current economics work is very advanced mathematically; however, introductory economics (micro, macro) are taught qualitatively with very small doses of quantitative work. The reason? It’s not that economics educators don’t want or need quantitative methods at the introductory level … the reason is that their students are woefully prepared for quantitative work, even after algebra courses. We ‘cover’ slope, but not rate of change in general (for math courses most students take); what we do cover is done in a way that inhibits transfer of learning to a new setting (economics). I’ve had this same conversation with science faculty, with the same result; I expect that much of the same story would be found in some social sciences.
The use of modules in curricular design raises issues about learning mathematics. The use of ‘what students need’, when based on existing content, reinforces an archaic model of mathematics. It is our responsibility to understand our students’ mathematical needs at a deep level, to the depth that we can identify content that is outside of the current curricula. If we can not judge this need, nobody else will.
The New Life model was based on exactly this type of work; we identified the needs based on a professional understanding of the quantitative demands of current students, especially those in community colleges. Some of this work is now imbedded in the Carnegie Pathways, and has a similar development in the Dana Center’s “New Mathways Project”. The curricular design in these efforts seeks to begin meeting students quantitative needs, starting on the first day of their first math course (developmental or not).
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