STEM or What? What Trajectory?

In community colleges, ‘developmental mathematics’ courses are the highest enrollment math courses.  These courses are based on a ‘calculus’ track, in most cases; I explore this issue in one of the Instant Presentations.  Most of our students are not required to take calculus, or even pre-calculus, for their program.  What is their trajectory?

Many of my students are on a trajectory to meet a general education requirement in mathematics.  In some cases, this is the requirement of my college (a course after beginning algebra); in other cases, it is the requirement of a transfer institution … for those who transfer.  What is the trajectory of general education in mathematics?

The “SIGMAA-QL” (quantitative literacy special interest group of the MAA) conducted a study in 2009 in an attempt to determine any commonality in our general education requirements.  The results are available in the report http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.158.6128&rep=rep1&type=pdf , which I have been studying this week.

This “QL” survey was intended for both 2-year and 4-year institutions, and they tried.  They sent the survey to every MAA-liaison, and this included over 400 at community colleges.  The return?  About 45 out of those CC.  Given the small number, the results could not be summarized.   The response rate was about 25% for 4-year colleges (275 returned, I believe), so the report deals with the 4-year situation.

Within the 4-year environment, the survey sound a surprising amount of ‘diversity’ in the general education requirement across an institution.  However, most of these courses fall into two categories — part of the pre-calculus sequence, or statistics.  What is trajectory we are designing here?  Are we saying that all students should attempt but fail to complete a sequence towards calculus?  Are we saying that ‘getting ready for calculus’ is equivalent to general education? Are we saying that statistics is the only exception for students — no other branch of mathematics has validity for all students?

Given the direct connection between developmental mathematics and these ‘general education’ requirements, we need to do some critical thinking relative to the trajectories we create.  What does it mean to be ‘quantitatively literate’, and how does this differ from ‘quantitative reasoning’?  Dealing with these questions will form a strong foundation for building trajectories which we impose on our students … this imposition needs to be based on a sound argument for their benefit.  (We need an argument stronger than “math is good for you”.)

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Instant Presentations now available!

There is a new page in DevMathRevival.net — called “Instant Presentations”.

The idea here is simple: provide an on-demand presentation that people can share and view.  Similar to a webinar, and more direct.

Initially, the page has 3 presentations — the Mission of Developmental Mathematics; What is Now; the New Life Vision.  Each of these presentations is under 5 minutes, with video and audio.  [The format is “Flash video”, which means that browsers will handle the files for you when you click on them.]

Give it a try!! The direct link is https://www.devmathrevival.net/?page_id=116

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Standards Based Reforms — What Research Says

The United States has seen a number of ‘standards based reforms’ over the past 20 years or so. Many of these deal with school mathematics, though a few of us in the college environment have worked towards a similar process. The most recent effort is the “Common Core Standards” (http://www.corestandards.org/ ), which is the highest profile effort yet.

The Rand Corporation published a report, in 2008, called Standards-Based Reform in the United States: History, Research, and Future Directions (online at http://www.rand.org/content/dam/rand/pubs/reprints/2009/RAND_RP1384.pdf).  I was impressed by some of their observations.

First, “Standards Based Reform” is usually implemented as “Test Based Reform”.  The point here is that content and pedagogy reflects a testing emphasis such that the actual standards are secondary — the tests (such as those used for No Child Left Behind, NCLB) take on the primary importance.  Behind this is a tension you will understand: Standards, by themselves, produce very little change.  “Aligning” testing to the standards is very common, and very understandable, as a method to create change.  Change is not always progress, however.

Second, high-stakes testing with sanctions ‘distorts’ teaching practice; as you’d expect, teachers focus more on preparing for the test when there are sanctions involved.  In general, most of the current testing involves sanctions of some kind such as NCLB or state-level impacts.  Since tests must, by design, address small subsets of the larger domain of knowledge described by the standards, the result tends to produce students who can perform better on the tests connected to the sanctions compared to other measures of their knowledge.  Specifically, they do not do relatively as well on our college placement exams. 

Third, the report goes back to a critical document that describes 4 categories of standards … and also analyzes the track record of some specific efforts.  A shot blog post is not an appropriate venue to report on these comments (I don’t want to inflict a journal-length article on you 🙂 ).

Although community colleges have not faced the standards based reforms and tests with sanctions directly, we deal with the consequences of these efforts.  Some policy makers assume that the “developmental math problem” will go away once the standards are implemented (like the Common Core).  The Rand analysis provides some insight into why the problem is not that simple; we should assume that our problem might change in the next 10 years … not that it will go away.

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The Sum of All Shortcuts

As I work with my beginning algebra students, and think about how they can learn this ‘stuff’ for use later, it occurs to me that we have developed a curriculum based on shortcuts.

Okay, so what do I mean?  “Shortcuts” are the separate rules that are provided to describe WHAT to do with a type of problem.  For example, this week we had negative exponents.  Our textbook, like almost all others, says that negative exponents show the reciprocal … students know that x^(-2) should be written as 1/(x^2).  Is this really how students should see this idea?  I do not think so.  For this particular notation, the origins come from needing to show a division … x^(-2) means dividing by x^2; this division meaning provides a nice connection to positive exponents and to place value, in addition to being more accurate.  In spite of these advantages, why do we so often show the reciprocal meaning?

The ‘shortcut’ property of this is not isolated.  Open any book, listen to any of us talk in class, and you will see (hear) shortcuts.  When we add two fractions, we need a common denominator; we can add like terms.  Do we connect these ideas (they are the same principle)?  To solve an equation, we ‘do the same to both sides’ … it’s a balanced scale; do you realize how many students have a visual map of this that is strictly positional — not even dependent upon having an equality statement?  (Just show them  ‘3x + 5 + x + 4’ and see how many subtract 4 or x.) 

More?  How about “is over of” … ‘circle groups of 3 numbers inside a cube root’ … ‘Y1, Y2, intersect — answer is x’.  ‘PEMDAS’. Is there anything substantial in our curriculum, or is our curriculum the sum of all shortcuts? 

Most shortcuts developed as an effective device to help students remember what to do, so they could arrive at more accurate answers.  If you have some old textbooks around, check out this theory.  I believe that textbooks evolve as they are published in new editions, and new ones mimic the newer ones, so that the content is often examples and shortcuts.  In the name of simplicity and ease for students, we take out the substantive narration around the shortcuts; the back-story is lost, and students think that the tricks they see are the real mathematics.  This is not doing a favor to our students.

One of the reasons to revitalize the curriculum is to give us a fresh start.  We can go back to the mathematics, the back-story, the connections.  In theory, we could take out the shortcuts and ‘fix’ what we have.  Unfortunately, our instructional practices are so wrapped up in the shortcuts that I suspect we will not identify even a majority of the shortcuts.  As mathematicians, we value understanding connections, applying concepts, and problem solving … shortcuts present a clear and present danger to these values.  The prevalence of shortcuts is not limited to developmental math classes; I see a number of them at the next level as well (whether it is college algebra or pre-calculus).  However, I have to say that we in developmental mathematics use shortcuts to a much greater extent.

It’s not that I do not want students to get correct answers.  This is about transferring knowledge — dis-connected knowledge (shortcuts) has little chance of being used in any other context.  This is about students remembering what they ‘learned’ — unstructured knowledge (shortcuts) forms stories to be remembered, and need to be indexed and accessed in the same manner.  This is about an education, which is more than the sum of shortcuts (or facts).

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