Whom Do we Follow?

At the college level, most introductory mathematics has been materials-based … meaning that the curriculum experienced by our students was largely determined by the textbook used.  The vast majority of textbooks considered for a given course would be very similar in ‘objectives’, presentation, and terminology.  Authors would bring minor variations in explanations, and more substantive variations in the range of problems included.

Along comes the ‘redesign-modules-web homework’ push.  The content becomes atomized, with most content described by the algorithms used to generate the problems.  Many of us are delivering courses which emphasize students “doing problems” as a way to get them to “do mathematics”; I would argue that these are not the same, and the differences have weakened our curriculum.

Another push would be the Pathways of the Carnegie Foundation for the Advancement of Teaching (Statway™ and Quantway™), developed in partnership with the Dana Center (Univ Texas- Austin), with assistance from AMATYC.  Rather than driving the curriculum by algorithm, these materials seek to remain true to the mathematical description of the content … the homework system is much more difficult to develop (though they have the best people working on it, and they are succeeding).

So, the question for our profession is this:  Whom do we follow?  Do we follow the atomized content with algorithms defining the outcomes, or do we follow new voices that seek to deliver mathematics needed by our students? 

Perhaps the question is not fair, as some of us are not following anybody … some of us are being told that we WILL walk a certain path, with the atomized content and algorithms; frequently this is due to administrators reaching a critical point in the process, and there is no more patience for a faculty-led process.

As long as we are professionals, we should continue to advocate for our responsibilities relative to our curriculum.  When administrators push, we need to look for all avenues to push back — not to avoid change, not to deny the existence of a problem; no, we need to push back so that the responsibilities remain ours as faculty. 

I hope you, and all of us, will consider our responsibilities … whether we are able to chose whom to follow, or whether somebody attempts to tell us whom to follow.  We have our own professional standards, and our future will depend upon how well we are true to those standards.

 
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Mastery Learning is …

I have heard many faculty speak in favor of mastery learning … and almost as many speak in opposition.

The heart of this set of opposing viewpoints is an incomplete notion of what mastery learning IS.  Many equate mastery learning with basic skills … with repetition … with homework systems.  These are not definitions, nor even descriptions, of ‘mastery learning’.

The origins of ‘mastery learning’ were centered in a philosophical base which claimed that almost all students could learn any particular content to the level of ‘masters’ (usually defined to be a 4.0 or A student) given the correct conditions … with a primary condition to vary being ‘time on task’.  In a classic view of higher education, all students are imbedded within a learning environment so they experience similar conditions; those who perform at a high level are rewarded with 4.0/3.5/A/B grades and encouraged to pursue more learning … those who failed to perform within these constant conditions were told that they needed to make an alternate choice of activity (as in, some other class … some other major … or not in college at all).

Those of us who adopted a mastery learning model turned this conception on its head.  We were not here to sort students; we were here to create the conditions for all students to have the opportunity to become masters of the content.  Our content was not changed, only the conditions for learning.  Our assessments did not reflect lowered expectations, but they did create positive conditions for additional learning.

The current misconception of mastery learning is based on the technology that is often used to deliver ‘content’.  Offering modules, online homework, and requiring ‘80%’ before moving on … these have little to do with mastery learning.  These learning environments focus on basic skills primarily because that is easier for mass-produced homework systems (though it also reflects a bias among many colleagues). 

In essence, mastery learning is only limited by our capacity to design instruction and assessment.  If applications … transfer … problem solving … creativity are important elements in your ideas about mathematics, mastery learning can be designed to support them.

Mastery learning, in 2011, is more about the economics of publishing and grants than it is about the flexibility (and power) of mastery learning.  I have spent many years in a program that had mastery learning as a founding principle, and I understand the complexities of creating a mastery learning model that includes ‘more than basic skills’.  I would suggest that most of these difficulties are present regardless of whether mastery learning is involved. 

Mastery learning does not determine the nature of the mathematics faced by our students.  No, what determines the mathematics that our students experience is our own conceptions of mathematics.  We should, as a community of professionals, have honest discussions about what it means to “learn mathematics”.

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Michigan Student Success Summit

The Michigan Center for Student Success hosted a summit this week, and I gave a short presentation on the New Life model.

Here is the link to that presentation (PDF): http://jackrotman.devmathrevival.net/Prominent%20efforts%20to%20redesign%20developmental%20mathematics%20Stu%20Success%20Summit%20Sept2011%20JackRotman.pdf

 
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Excuse Aunt Sally … part II

Sometimes, an experience in class illustrates a concern.

In today’s beginning algebra class, we were doing order of operations with signed numbers … the first problem for students to do had a quotient and then a product.  MANY students in class were convinced that they needed to multiply first; their rationale was “PEMDAS” — where multiply clearly comes before divide. The problem looked like -16 ÷ (-4) ·2; not very complex … and more than half the class insisted on multiplying -4 and 2 before dividing.

Now, it is true that the type of problem involved is not that important; it’s not needed to model any situation, does not find any reasonable answer, and does not support future learning (outside of order of operations).  The correct answer for this particular problem is not very valuable.

However, our students should be developing a coherent system of knowledge.  In an earlier post, I suggested that “Dear Aunt Sally be Excused” from all math classes; my rationale was that PEMDAS directly causes confusion in algebraic reasoning.  This post is further suggesting that PEMDAS is not very functional even within the original domain of use (order of operations, no variables).

I am becoming more convinced that “PEMDAS” should be avoided in all mathematics classes … whether it is school mathematics or college mathematics.  PEMDAS is short-sighted and misleading; PEMDAS does not support an organized system of understanding.  PEMDAS harms students in the long term, and somewhat in the short-term.

 
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