Mathematical Literacy WITHOUT a Prerequisite

Starting this Fall (August 2016) my department will begin offering a second version of our Mathematical Literacy course.  Our original Math Lit course has a prerequisite similar to beginning algebra (it’s just a little lower).  The new course will have NO math prerequisites.

So, here is the story: Last year, we were asked to classify each math course as “remedial, secondary level”  or “remedial, elementary level” or neither.  This request originates with the financial aid office, which is charged with implementing federal regulations which use those classifications.  Our answer was that our pre-algebra course was “remedial, elementary level” because the overwhelming majority of the content corresponded to the middle of the elementary range (K-8).  We used the Common Core and the state curriculum standards for this determination, though the result would be the same with any reference standard.

Since students can not count “remedial, elementary level” for their financial aid enrollment status, our decision had a sequence of consequences.  One of those results was that our pre-algebra course was eliminated; our last students to ever take pre-algebra at my college finished the course this week.

We could not, of course, leave the situation like that — we would have no option for students who could not qualify for our original Math Literacy course (hundreds of students per year).  Originally, we proposed a zero credit replacement course.  That course was not approved.

Our original Math Literacy course is Math105.  We (quickly!) developed a second version … Math106 “Mathematical Literacy with REVIEW”.  Math106 has no math prerequisite at all.  (It’s actually got a maximum, not a minimum … students who qualify for beginning algebra can not register for Math106.)  The only prerequisites for Math106 are language skills — college level reading (approximately) and minimal writing skills.

Currently, we are designing the curriculum to be delivered in Math106.  We are starting with some ‘extra’ class time (6 hours per week instead of 4) and hope to have tutors in the classroom.  Don’t ask how the class is going because it has not started yet.  I can tell you that we are essentially implementing the MLCS course with coverage of the prerequisite skills, based on the New Life Project course goals & outcomes.

We do hope to do a presentation at our state affiliate conference (MichMATYC, at Delta College on October 15).  I would have submitted a presentation proposal for AMATYC, but all of the work on Math106 occurred well after the deadline of Feb 1.

One of the reasons I am posting this is to say: I am very proud of my math colleagues here at LCC who are showing their commitment to students with courage and creativity.  We will deliver a course starting August 25 which did not exist anywhere on February 1 of this year.

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Algebra in General Education, or “What good is THAT?”

One of the questions I’ve heard for decades is “Is (or should) intermediate algebra be considered developmental?”  Sometimes, people ask this just to know which office or committee is appropriate for some work.  However, the question is fundamental to a few current issues in community colleges.

Surprising to some, one of the current issues is general education.  Most colleges require some mathematics for associate degrees, as part of their general education program.  Here is a definition from AACU (Association of American Colleges and Universities):

General education, invented to help college students gain the knowledge and collaborative capacities they need to navigate a complex world, is today and should remain an essential part of a high-quality college education.  [https://www.aacu.org/publications/general-education-transformed, preface]

What is a common (perhaps the most common) general education mathematics course in the country?  In community colleges, it’s likely to be intermediate algebra.  This is a ‘fail’ in a variety of ways.

  1. Algebra is seldom taught as a search for knowledge — the emphasis is almost always on procedures and ‘correct answers’.
  2. The content of intermediate algebra seldom maps onto the complex world.  [When was the last time you represented a situation by a rational expression containing polynomials?  Do we need cube roots of variable expressions to ‘navigate’ a complex world?]
  3. Intermediate algebra is a re-mix of high school courses, and is not ‘college education’.
  4. Intermediate algebra is used as preparation for pre-calculus; using it for general education places conflicting purposes which are almost impossible to reconcile.

We have entire states which have codified the intermediate algebra as general education ‘lie’.  There were good reasons why this was done (sometimes decades ago … sometimes recently).  Is it really our professional judgment as mathematicians that intermediate algebra is a good general education course?  I doubt that very much; the rationale for doing so is almost always rooted in practicality — the system determines that ‘anything higher’ is not realistic.

Of course, that connects to the ‘pathways movement’.  The initial uses of our New Life Project were for the purpose of getting students in to a statistics or quantitative reasoning course, where these courses were alternatives in the general education requirements.  In practice, these pathways were often marketed as “not algebra” which continues to bother me.

Algebra, even symbolic algebra, can be very useful in navigating a complex world.

If we see this statement as having a basic truth, then our general education requirements should reflect that judgment.  Yes, understanding basic statistics will help students navigate a complex world; of course!  However, so does algebra (and trigonometry & geometry).  The word “general” means “not specialized” … how can we justify a math course in one domain as being a ‘good general education course’?

Statistics is necessary, but not sufficient, for general education in college.

All of these ideas then connect to ‘guided pathways’, where the concept is to align the mathematics courses with the student’s program.  This reflects a confusion between general education and program courses; general education is deliberately greater in scope than program courses.  To the extent that we allow or support our colleges using specialized math courses for general education requirements … we contribute to the failure of general education.

In my view, the way to implement general education mathematics in a way that really works is to use a strong quantitative reasoning (QR) design.  My college’s QR course (Math119) is designed this way, with an emphasis on fundamental ideas at a college level:

  • Proportional reasoning in a variety of settings (including geometry)
  • Rate of change (constant and proportional)
  • Statistics
  • Algebraic functions and basic modeling

If a college does not have a strong QR course, meeting the general education vision means requiring two or more college mathematics courses (statistics AND college algebra with modeling, for example).  Students in STEM and STEM-related programs will generally have multiple math courses, but … for everybody else … the multiple math courses for general education will not work.  For one thing, people accept that written and/or oral communication needs two courses in general education … sometimes in science as well; for non-mathematicians, they often see one math course as their ‘compromise’.

We’ve got to stop using high school courses taught in college as a general education option.  We’ve got to advocate for the value of algebra within general education.

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Statistics: No Box-and-Whiskers; A Better Histogram

Many of you know that I have ‘been around’ for a long time.  My first statistics course was around 1970, and I started teaching some statistics in 1973.  I’ve had some concerns about a tool invented about that time (box and whisker plots), and want to propose a replacement graphic.

Here are two box & whisker plots (done in horizontal format, which I prefer):

box-plot-Wait_Times_May2016 box-plot-HDL_May2016

 

 

 

 

 

 

 

 

There are two basic flaws in the box & whisker display:

  1. The display implies information about variation, when the underlying summary does not (quartiles).
  2. The display requires the reader to invert the visual relationship:  A larger ‘box’ means a smaller density, a smaller ‘box’ means a larger density

Here are the underlying data sets, presented in histogram format (which is not perfect, but avoids both of those issues):

Histograma_HDL

 

 

 

 

 

 

 

 

histogram_wait_time

 

 

 

 

 

 

 

 

 

 

 

 

Some of the problems with box plots are well documented; a number of more sophisticated displays have been used.  See http://vita.had.co.nz/papers/boxplots.pdf. These better displays are seldom used, especially in introductory statistics courses.

The main attractions of the box-plot was that it provided an easy visual display of 5 numbers — minimum, first quartile, median, third quartile, maximum.  The problem with creating a visual display of such simple summary data is that it will always imply more information than existed in the summary.  We’ve got a solution at hand, much simpler than the alternatives used (which are based on maintaining the box concept):

Replace basic box-and-whisker plots with a “quartiled histogram”.

A quartiled histogram adds the quartile markers to a normal histogram display.  Here are two examples; compare these to the box plots above:

Quartiled-Historgram-HDL_May2016

 

 

 

 

 

 

 

 

 

Quartiled-Historgram-Wait_Times_May2016

 

 

 

 

 

 

 

 

 

 

 

 

The quartiled histogram combines the basic histogram with a simplified cumulative frequency chart — without losing the independent information of each category.

Perhaps a basic box and whisker plot works when the audience is sophisticated in understanding statistics (researchers, statisticians, etc).  Because of known perceptual weaknesses, I think we would be better served to either not cover box & whisker plots in intro classes — or to cover them briefly with a caution that they are to be avoided in favor of more sophisticated displays.

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TPSE Math … Transforming Post Secondary Ed Mathematics

One of my Michigan colleagues recently reminded me of a national project on transforming post secondary education mathematics “TPSE Math”, which you can find at http://www.tpsemath.org/

This broad-based effort seeks to engage faculty and leadership from all segments of college mathematics, with an impressive leadership team.  I encourage you to check it out.

One of the first things I explored on their site deals with equity; they have a 2016 report on equity indicators (see http://www.pellinstitute.org/downloads/publications-Indicators_of_Higher_Education_Equity_in_the_US_2016_Historical_Trend_Report.pdf)  Interesting reading!

Another part of their web site I want to look at in more detail … “MAG” (Mathematics Advisory Group), which is focused on an ‘action oriented role’.  Take a look at http://www.tpsemath.org/mag

I’m expected that we will all be involved with this TPSE work, to varying degrees.

 
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