Answer Standards Fight Mathematics

“Negative exponents are not allowed in answers.”

My intermediate algebra class has not had negative exponents in the material so far.  However, here is what most students remember:

I can not have negatives, so write the reciprocal.  No exceptions.

Given the ‘answer desperation’ of most students, any ‘rule’ about the answer gets added emphasis in learning.  As teachers, many of us try to make it easy for students … so we add our emphasis.  The result is that our standards (often sensible but arbitrary) fight mathematical knowledge.  Students focus on the form of the answer and our rules about that, and have less understanding of the mathematics involved in the situation.

In the case of negative exponents, the rationale for ‘no negative exponents’ is marginal at best.  True, in some cases, positive exponents are simpler; however, for the majority of situations, negative exponents are simpler — they often avoid the need to write a fraction.  The evolution of exponential notation and meaning is based partially on the idea that negative exponents are a simpler way to show division … and fractional exponents are a simpler way to show roots.

For my class, the previous emphasis on ‘no negative exponents’ distracts them from understanding simple division problems.  We do more polynomial arithmetic than is really needed, but these division problems are just dividing a binomial or trinomial by a monomial.  The student answer desperation and the negative exponent prejudice combine to distract them from the basic ideas of division.

A related issue came up in my beginning algebra classes … students wondered if they should finish a fraction problem by changing it to a mixed number.  The context was solving a linear equation with one variable; the form of the answer is a trivial matter compared to the mathematics.  Overall, one of the most common questions I ever hear is “how do we need to write that answer”.  Sometimes, this deals with algebraic concepts and the question is valuable; many times the question is a distraction.  I often tell students that I don’t care what form they give an answer as long as it clearly communicates a correct result.

Perhaps we should take a step back from all of our simplistic statements about ‘form of the answer’.  Many of them are conveniences for grading work, nothing more.

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Webinar Recording Available — Reform in Dev Math (Treisman & Rotman)

On June 6, 2012, an AMATYC webinar was held on Issues in Implementing Reform in Developmental and Gateway Mathematics Uri Treisman and Jack Rotman; the web page http://www.amatyc.org/publications/webinars/index.html has links to view the 54 minute recording or to download it.

This webinar presented an overview of dimensions of reform (5 areas), some background on those, presented some choices in implementation, and reviewed some current reform efforts in mathematics (especially developmental).  Also included in the recording is the question and answer period based on participants’ questions.

The webinar was limited to AMATYC members, in terms of registration.  At this point, the recording is available as a professional resource to non-members as well.

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Emporium Model … A Good 3-Year Solution to the Wrong Problem

Many of us are involved with emporium-type projects; these changes are often in the popular media … such as the article about college algebra at the University of Texas – Arlington (http://www.nytimes.com/2012/09/07/us/ut-arlington-adopts-new-way-to-tackle-algebra.html).  I refer to emporium-type projects as ‘3-year solutions’ because they do not address basic problems … the main focus is a symptom of a problem (low pass rates) and a generalized threat to higher education (costs). 

A particular project might in the emporium style might last longer than 3 years, but mostly due to the investment the institution has made in space and infrastructure.  And, most emporium-type projects will produce better data (the symptom) especially in the first two years … the motivational effects of ‘something new’ can help faculty and students, at least for a while.  Over time, the data will tend to degrade from the early improvements.  The cost savings will be difficult to maintain in the long term; since the primary savings comes from limiting of faculty time, administrators will face pressures to increase faculty assistance for students.   One of the forces that supports ‘survival’ of the project (investment in space, especially) will eventually turn out to be a challenge for survival, as other demands develop in the institution.

Like ‘module solutions’, emporium-type solutions tend to avoid curricular problems.  Procedural techniques in the absence of understanding mathematical concepts, with a pronounced lack of applications to useful situations (in academia and in life), within a context of “you have to pass this math course in order to do what you want” … these are some of the basic curricular problems we face.  In theory, an emporium-type model COULD address these issues; however, doing so is likely to be more difficult (perhaps much more) than in other designs which place faculty in a more active role as facilitator of learning.

Part of these difficulties are a result of using technology as a foundational component in the learning process.  Most of the technology used is mass-market software focusing on the ‘greatest common factor’ (often mis-spoken as ‘least common denominator’) of math faculty — these technology solutions deal primarily with problem types that most faculty can agree to … in other words, procedural techniques and routine applications that can be done repeatedly without understanding.  Addressing the curricular problems with technology would involve large investments of resources in development, which no single institution can afford … and publishers are reluctant to provide it without the ‘market’.

Let’s put it this way:  The curricular problems must be addressed first, before we can identify or build appropriate instructional systems.  The problem solving of solving the curricular issues is the strength of faculty, which means that the solutions will tend to be very faculty and classroom-based for quite a while.   Eventually, we might be able to use a model like the emporium in a long-term solution; right now, this is just not possible.  Technology follows curricular change … curricular change CAN be inspired by technology, but this is unusual.  (As an example, the curricular changes due to graphing calculators turned out to be less substantial than many thought or some feared — even though there have been changes, the long-term effect was mediated by the curricular problems.)

If you are involved with an emporium-type project, I would say that you should enjoy it as much as possible … and do not count on the project to last past a few years.  In the meantime, become familiar with the emerging models for developmental mathematics (AMATYC New Life, Dana Center Mathways, Carnegie Pathways) and the reform work in college mathematics (MAA CRAFTY, AMATYC Right Stuff).  You will need to be prepared for the day when your institution decides that the emporium-type project is not good enough.  Perhaps you will even be involved in convincing your institution that there is a better path forward, a path focusing on solutions that address basic problems. 

Emporium does not mean forever.

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Are Modules the Answer for Developmental Mathematics?

The number of institutions implementing modules in developmental mathematics continues to increase, which I expect to continue for another year or two.  Over the next 5 years, I expect most of these institutions to shift to other models and solutions for their developmental mathematics programs.  Perhaps you can think of some reasons why colleges would try modules now and then replace modules.

Our context for this problem is complex, with multiple expectations for developmental mathematics and multiple measures of current problems.  Modules are appealing because of the clear connection between a modular design and some measures of current problems — low pass rates and low completion rates in particular.  For a change to survive in a longer term, the methodology needs to address enough of the basic problems to be sustainable.

When we started the New Life project in the Developmental Mathematics Committee of AMATYC, we asked a set of national leaders in the field to identify the basic problems they saw.  In analyzing that input (done via email, primarily), the problems could be clustered in a few basic categories:

The content of developmental mathematics courses is not appropriate for the majority of students.

The typical sequence of courses has too many steps for students to complete in a reasonable amount of time.

The learning methods emphasized in most programs were not effective, and do not reflect the accumulated wisdom about learning and cognition.

Faculty, especially in developmental mathematics, were professionally inactive and they tended to be isolated.

Faculty were not using professional development opportunities, both due to lack of information and due to lack of institutional support.

Modules are often selected based on rationale of content and sequence.  However, when we look deeper at the content problem, the issue is a very basic one: the typical developmental mathematics sequence emphasizes symbolic procedures presented in isolation from both applications and other mathematics.  In other words, completing a developmental math course typically does not result in a significant increase in the mathematical capabilities of students … the learning was of the type that is quickly forgotten.

One reason, then, that modules will tend to be a short-term process is that the design does not generally address basic content problems.   A modular program makes it easier for students to complete; a consequence of this is that the content is deliberately compartmentalized and isolated.  Module 4 is independent of Module 3; the learning is not connected, nor is there (normally) a cumulative assessment at the end of a sequence (like a final exam). 

I am hoping that you are thinking … “Wait a minute, modules can do more — the learning can be connected, and we can have a cumulative assessment”.  Great, good job thinking critically.  However, every single modular implementation I am reading about focuses on the independence of the modules, and none have a ‘final exam’.  Some colleges will eventually try to address this problem.  The challenge is that doing so is fairly difficult, and will tend to increase cost.  [You might have noticed that cost was not a general problem as identified by leaders in the profession.]

The learning methods are also a problem in the typical modular design.  Modules have a high probability of using online homework systems; these systems tend to be limited to symbolic procedures.  More fundamentally, though, I see modular programs as missing the learning power of groups and language.  Modular programs tend to be individual-based; social settings, such as small group work, are either difficult to manage or just plain impossible.  Language (meaning speaking and writing) are often quite limited; as in traditional developmental programs, modules tend to emphasize the correctness of answers as a measure of learning … as opposed to quality of work, written explanations, or spoken explanations.  Therefore, I generally expect that modular programs will result in levels of learning that are statistically equal to the programs they replace; this (if true) is enough of reason for colleges to leave the module design in a few years.

Some modular designs have addressed some of the problems related to faculty … at the point of implementation, and in limited areas.  Not enough for long-term viability.  We, the faculty in developmental mathematics, have much to do.  The overwhelming majority of us are not engaged in any professional activity (beyond a few hours of work per year at our own campus); we generally do not attend conferences, we don’t join AMATYC and state affiliates; we don’t read professional journals, let alone publish in them.  We need to develop a deeper understanding of our profession; in particular, we need to be proficient in analyzing learning mathematics as a matter of mathematics and of cognition.  We need both deeper toolsets and the knowledge about best uses for those tools.  None of the modular designs I read about have a long-term strategy for supporting faculty.

The designs I often call “the emerging models” all deal with multiple problem areas, resulting in long-term viability.  The emerging models (AMATYC New Life, Carnegie Pathways, Dana Center Mathways) address content, sequence, learning, and faculty issues.  Over the next few years, you will begin hearing of institutions who had implemented modules switching to one of these emerging models.  We all are committed to helping our students, and these models provide a better solution.

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