Math Applications — Taking it Personally

Contextualized math is a current theme in our profession; some courses are taught strictly working from context — the story is the thing, and only the mathematics that relates to that story is developed.  Other courses emphasize context, while many of us take a moderated approach in which we blend context and abstraction.  Whatever the mix, these contexts are experienced as ‘applications’ or word problems by our students.  Do applications (or context) present issues of equity in our developmental math classes?

I would like to tell you about what two of my students (beginning algebra) are experiencing this semester.  First, a student to be called Mary.

Mary: (looking at a typical ‘distance’  problem about two cars)  I don’t know how to write the algebra for this, but I can figure out the answer.
Instructor: Okay, so tell me more about that.  How do you figure it out?
Mary: Well, the problem says that one car is going 10 miles per hour faster, so I put myself in that situation; I know that the speed limit is 70 miles per hour, so that must be the faster car.  The other car must have been going 60.
Instructor: I see.  What part of the problem told you that the cars were on a highway with a speed limit of 70?
Mary: The problem did not say that, but the only way I can understand the problem is to put myself in to it.

The second student will be called John (whose native language is Arabic).

John: (looking at a problem about a tree and a flag pole dealing with their heights)  This problem is really hard.
Instructor: What makes it hard?
John: Everything in the problem … I need to translate it into my language; it does not make sense to me.
Instructor: Are you talking about the individual words?
John: Yes, yes … they are confusing.

The prognosis for Mary is not as good as the prognosis for John.  They are both taking the applications personally; the difference is that Mary thinks that she has to see herself in the problem for it to make sense, while John thinks that he will understand the problem once he knows all of the words.

This experience made me think of some research I saw a few years ago dealing with how word problems in mathematics might raise issues of equity.  The research suggested that students from a ‘lower class’ (this was British research) get distracted by the details of the applications as they relate to their personal life.  My student, Mary, was doing exactly that.  Her learning skills, and her life experiences, provided a limited view of applications; some problems dealt with objects or situations with which she had no experience, and she did not know what to do … other problems activated related but not worthwhile information (like the car problem).   Clearly, we will need to work together (Mary, the class, and me) to help broaden the view and provide more resources.

Taking an application personally can create difficulties in forming a solution strategy; taking it personally highlights information (which might be trivial) and causes us to possibly ignore other information critical to a solution.  This situation deals with perception and motivation.  For those of us who are using high-context classrooms, I wonder if you are finding that the approach is equally accessible to all of your students.

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Reform Models in Developmental Mathematics

For many years, our developmental mathematics programs were based on a remedial image — filling in the ‘swiss cheese’ of student’s knowledge of school mathematics, with the school mathematics based on an archaic content (circa 1965).  Now, for the first time, we have an opportunity to explore a model of developmental mathematics that is based on mathematical needs of students — designed especially for community colleges.

During the June 6 (2012) webinar, Uri Treisman presented some general concepts to guide our work in reforming our curriculum; my component of the webinar dealt with applying these concepts in our departments.  In this post, I want to share two possible structures for reform of developmental mathematics as presented that day.  [The recording of the webinar will be available later this summer.]

One approach to reform is to target reform for particular groups of students.  You might identify students who need an intro statistics course, or those who need a quantitative reasoning course, and design a prerequisite course just for these students.  In this approach, the existing developmental mathematics curriculum is left undisturbed … at least for now.  The resulting curricular model looks something like this:

This ‘targetted’ approach is reflected in the Statway and Quantway work, for example.  However, this is not the only … nor necessarily best … approach.  Since our content is heavily influenced by archaic high school content, the mathematical needs of students — especially in reasoning and transfer of learning — would be better served by a total reform.

A reform for all students (total reform) has a goal of replacing existing courses.  In this model, the beginning algebra course is replaced by mathematical literacy course (which is also part of the target reform model); the intermediate algebra course is replaced by a reform algebra course … which some students would not have to take to meet their math needs. 

This reform for all students model creates this visual:

The reform algebra course (“B” in this visual) might be the one described as “Transitions” in the New Life model; see http://dm-live.wikispaces.com/TransitionsCourse.  Some colleges might consider a combined beginning & intermediate algebra course for course B; this is not a reform course (as the content is the traditional … and archaic … material).  Another option in this total reform model is to create a faster path in pre-calculus — blend ‘course B’ (reform algebra) and pre-calculus in to a 2 semester sequence for those students. 

Reform in developmental mathematics is needed.  However, reform in developmental mathematics is not sufficient; we also need to reform the introductory college mathematics courses to reflect current needs and professional knowledge.  Our students deserve the best mathematics we can provide, both in developmental and college-level courses.

 
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Modules for the Developmental Mathematics Not Needed

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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Webinar Registration is open — Treisman & Rotman, June 6 (AMATYC)

Registration is now open for AMATYC’s next webinar: Issues in Implementing Reform in Developmental and Gateway Mathematics.  Details appear below; registration is currently open to AMATYC members and there is no cost to register for the webinar.

Webinar Details:

Presenters: Uri Treisman and Jack Rotman

Date: Wednesday, June 6, 2012

Time: 4:00pm EDT / 3:00pm CDT / 2:00pm MDT / 1:00pm PDT

Description: Uri Treisman and Jack Rotman will discuss issues that should be considered in implementing reform efforts in early college mathematics starting with a comprehensive theoretical framework to guide the work and narrowing down to key principles for the work on-the-ground.

Sponsoring Committee: Developmental Mathematics

To Register, Click Here

Registration is limited to AMATYC members; however, the webinar will be recorded and posted for general viewing later.  I’ll post a notice when that recording is ready.

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