Modules in Developemtnal Mathematics — pro and con

I am hearing about colleges either adopting or considering modules in their developmental mathematics program.  Sometimes, this is done as part of an ’emporium model’; however, other designs make use of modules.  Perhaps it would help to have a brief exploration of the pros and cons of modules.

The word ‘modules’ does not have a uniform meaning for us.  In general, a ‘module’ could be another name for a ‘chapter’ — each being a sub-unit within a larger organization of material.  However, most uses of the word ‘module’ refer to one of two approaches to content — uniform sequence of modules or customized sequence.

The difference between the two uses can be subtle, such as a case where the customized exit point is the end of a ‘course’ — some modular programs designate ‘modules 5 to 8’ as a course, and that is where the exit point is.  Customizing is done by either changing the ending module within a course or changing the entry point (starting module) within the course.  Conceptually the contrast for the two designs is important due to the fact that a customized program prevents a summative assessment common for all students.

Over the past several years, I have had discussions with faculty involved in a type of modular program.  Via this obviously non-scientific method, I have developed some pros and cons for modularization.  Most of these apply to either type (uniform or customized).

A modularized approach is usually based on an assumption that the delivery mode is a major source of problems, sometimes stated “we can’t teach this to them the same way they saw it the first time”.   I have not seen any evidence of this being true; it’s not that I want to teach them “the same way” (whatever that means) … it’s that this assumption about the delivery mode often precludes examination of larger issues about the curriculum.  Modularized tends to reinforce notions that ‘mathematics’ is about knowing the procedures to obtain correct answers to problems (often contrived and overly complex).  Our professional standards (such as the AMATYC Beyond Crossroads  … see http://beyondcrossroads.amatyc.org/) begin the discussion about mathematics by describing quantitative literacy.  This aspect — of modularization tending to limit the mathematics considered — is the largest factor in seeing this approach as being weak and temporary.

The other major area of concern, suggested somewhat in the pros and cons, is the professional status of faculty in developmental mathematics.  Administrators and policy makers often do not understand the professional demands of being developmental mathematics faculty; in the modularized approaches, faculty tend to look a lot like tutors.  This similarity then suggests to some that faculty are not necessary, and we can provide a larger pool of tutors.  Our professional standards call for us to see the work in math classrooms as being rigorous in both mathematics and education.  This aspect — the professionalism of faculty — is the most common concern reported by faculty engaged in a modularized program.

Summary:
The attractiveness of modular approaches is easy to understand.  However, the typical implementation of modular approaches will reinforce a traditional content with a weaker assessment system combined with a generally lower faculty professionalism.  When implemented, modular programs will tend to be temporary solutions.  The emerging models — New Life, Carnegie Pathways, Dana Center Mathways — provide a clear alternative to address the problems based on professional standards to create long-term solutions.

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PEMDAS and other lies :)

We use ‘correct answers’ as a visible indication of knowledge.  If the learning environment focuses on homework systems, correct answers may be the only measure used.  However, even when we ‘look at the work’, we may confuse following a procedure with knowing what to do.

PEMDAS may be the most commonly used tool in the teaching of mathematics experienced by our students.  I have seen PEMDAS written on work papers and notebooks; I have heard students say PEMDAS when explaining how to ‘do a problem’ … and I’ve heard instructors say that PEMDAS describes what to do with an expression.

The problem is that PEMDAS is a lie.  PEMDAS only provides a memory tool (a mnemonic) for steps that might apply to some expressions in some situations.  Previously, I have written about the issues with the “P” (parentheses) component of this tool (see https://www.devmathrevival.net/?p=301).  Today, I am thinking about some of the ways in which PEMDAS is false or incomplete.

Take a simple expression like -4².  PEMDAS does not give any interpretation of this expression.  The issue here is that the memory aid only deals with exponents and the 4 binary operations; the negation (opposite) involved here is outside of the rule.  If we established mathematical truth based on an agreement among students passing a course, the truth would be at risk on this expression — whether “16” or “-16” would win a majority would vary by semester.

PEMDAS is incomplete about operations in general, such as the negation above … or absolute value.  Given the visual similarity with parentheses, most students see that the ‘inside’ of an absolute value is simplified first.  However, what to do with an expression like  3|x – 2|?  Is there a choice to distribute?  As we know, and students are confused about, the order of operations provides one possible procedure … properties of numbers and expressions completes the story, and these properties are more important in mathematics.  Getting the correct answer to 8 + 5(2) in a pre-algebra course has nothing to do with being ready to succeed in algebra, or math in general.  Basic expressions like 8 + 5x are a challenge for many students, partially due to how strong the PEMDAS link is.

Another example:  what does PEMDAS tell us about mixed numbers?  This is a special case of the ‘parentheses problem’, where there is no symbol of grouping.  Fractions, in general, are an area of weakness.  We tell students that “you need a common denominator” or “cross multiply” — both of which appear to violate PEMDAS (we would divide left to right).  Properties are the important thing here as well; adding requires similar objects.  We focus so much on correct answers and perhaps ‘correct steps’ that we miss opportunities to address the mathematics behind the visible work.

The meaning of an expression with mixed operations is based on the priority of each operation; mathematically, the level of abstraction of an operation determines the priority.  Multiplying is abstracted from the concept of repeated adding, so multiplying carries a higher priority; exponentiation is abstracted from the concept of repeated multiplying, and has a higher priority.  Lowest abstractions are the basic concepts — add, subtract, negation.  For those of you involved with programming, this approach should sound familiar — computing environments are based on a detailed list of these levels of abstractions.  In mathematics, our world is defined by properties which provide necessary choices for types of expressions where equivalent forms can be created without using the prioritization.

The big lie in PEMDAS is that those 6 words say something important about mathematics.  Those 6 words do not say anything important about mathematics, only about an oversimplification that produces some correct answers to some expressions without understanding the mathematics.  Properties and relationships are the important building blocks of mathematics; a student starting from PEMDAS has to unlearn that material before understanding mathematics.   If our goal is to have students compute correct answers for any expression, then we would never use PEMDAS — it is woefully incomplete, and we would need the prioritization list like a computer program uses.  If our goal is to have students understand mathematics, we would deal with the concepts that determine the order along with the properties that provide choices; a focus would be on the correct reading and interpretation of expressions.

Do your students a favor; avoid using PEMDAS.  Use mathematics instead.

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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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Towards Effective Remediation

Do we have a vision of effective remediation … a model which minimizes the pre-college level work for students, in total, while providing an opportunity for all adults to be included in the process of completing credentials leading to better employment and quality of life?  Based on some 39 years in developmental education, what would I suggest?

I have been thinking, as hard as I can, lately on the problems caused by policy makers looking for a simple solution.  Often, the policy makers’ interest in remediation has been prompted by reports issued by groups like Complete College America (CCA); the CCA “Bridge to Nowhere” report is excellent use of rhetorical tools, but is not a good foundation for building policies in support of effective remediation.  The simple solutions involved are usually crafted by groups that do not include people with expertise in developmental education.  Somehow, the viewpoint that we present, as experts, is difficult to understand by non-experts; perhaps some policy makers are worried that experts will only want to preserve the current system, or that we will suggest that even more courses be provided in our field.

Effective remediation involves providing the appropriate learning opportunities for each learner so that the learner reaches college courses with adequate preparation.  Traditionally, we establish remediation in discrete content areas (reading, writing, math), with an independent decision in each area based on a placement test.  Some promising practices have evolved recently with efforts to link developmental content courses, and efforts to include learning skills.   Especially within mathematics, considerable effort has been invested in creating a modularized approach; modularization is a topic of its own.  However, two observations might help us:

1. Each student is considered for 0 to 4 developmental courses in each of the 3 content areas, usually based on one placement test in each area.
2. The content is the developmental courses is often severely constrained by the historical roots of the system; especially in mathematics (though still true in reading and writing), the focus is on mechanics and procedure, with less emphasis on reasoning and analysis.

For us to develop a vision of effective remediation, we need to understand the deeper problems with the existing system such as those suggested by these observations.  In order to provide appropriate learning opportunities (whether courses, workshops, or other experiences), we need a more advanced conceptualization of remediation itself.  We need to more beyond a simple binary choice independently made in discrete content areas based on one test in each.

I suggest that we consider the following framework:

• Students roughly within a standard error of placing in to college work in a content area be provided just-in-time remediation and register for the college course.
• Students over one standard error away are placed into one of two populations based on other measures (such as high school GPA).  Some might be placed into the ‘just-in-time’ remediation group.
• The low-intensity developmental students are placed into a one-semester ‘get ready for college work’ course in one or more content areas.
• The high-intensity developmental students are provided a year of connected course work which blends reading, writing, math, and learning skills designed to address content and thinking needs.

The first two categories involve a significant portion of our developmental students, who have less intense needs; their remediation can be quicker than we often provide now.  For those who ‘almost place’ into college work, the ‘discontinuity’ research on placement tests suggests that we might be able to avoid any developmental enrollments in that content area.   The low-intensity developmental students are those who are not predicted to succeed in college but have limited needs; within mathematics, this group would include those who can review areas of algebra and quantitative reasoning in one course with minimal support outside of the class.

The high-intensity developmental group would include students with broad needs across multiple content areas.  These are the students who now struggle to complete developmental courses.  However, their educational needs are not limited to the content area skills; reasoning skills and study skills are a problem for many students in this group.  I am envisioning a two-semester package of courses (three or four each semester) with intentional overlap of cognitive skills being addressed … the math course, the reading course, and the writing course would all address issues of inference and concise language use.  This high-intensity group would also have a student-success type course to prepare them for the academic demands of the college course that lie outside of a content area.

Here is an image of this model:

A goal of this model is to make a better match between student needs and the remediation that they receive.  Our traditional system is designed for the ‘low-intensity’ type of student, and I believe that these students are well served on average.  The just-in-time remediation group is the source of our current problems from the policy makers; because these people exist in our current developmental program when the evidence raises questions about this practice, policy makers generalize the conclusion to all developmental students. 

The biggest change, and our largest opportunity, comes from the high-intensity group of students.  In our society, these are often called ‘low-skilled’ adults; they might be functionally literate (or perhaps not), and generally have few options in the economy.  Our traditional developmental program tends to be either limited in helpfulness or a problem for these students.  In a mathematics class, the high-intensity students have difficulty with both the mathematical ideas and the language factors in the work.  We tend to expect some magical cognitive growth in these students, as if working on discrete content areas will generate spontaneous global changes in the brain; I have no doubt that this does, in fact, happen to some students … I have seen it.  However, we do not create conditions for the larger cognitive changes.

Colleges might create a one-semester option for the less intense of the high-intensity group — those who can accomplish the goals with a one-semester package.  Smaller colleges might have difficulty with the logistics of this, while larger colleges would probably benefit from having two categories of high-intensity students.  Part of the rationale for the design for high-intensity need students is that preparation for them, is a more complex challenge.  Some will have had special services in the K-12 schools, and some will have significant learning disabilities.  This is the group most at risk; if community colleges are to serve all adults, then our remediation design needs to provide an appropriate pathway through to college work.  The alternative is to have a significant group of adults who will always be economically and socially vulnerable.  This high-intensity group are the ones that we need to educate policy makers about, so that they can understand the needs better — both the student needs, and our needs if we are to truly help them.

If you would like to do some reading on research related to this model, much of what I am thinking of resides in reports from the Community College Research Center (CCRC) at Columbia (http://ccrc.tc.columbia.edu/). Two specific articles: placement tests in general (http://ccrc.tc.columbia.edu/Publication.asp?uid=1033) and skipping developmental based on discontinuity analysis (http://ccrc.tc.columbia.edu/Publication.asp?uid=1035).    An article of interest by Tom Bailey and others on state policy appears at http://articles.courant.com/2012-05-18/news/hc-op-bailey-college-remedial-education-bill-too-r-20120518_1_remedial-classes-community-college-research-center-remedial-education.

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