Intermediate Algebra is NOT College Math ! :(

I actually spend a fair amount of time looking at other colleges math courses, partly from my interest in seeing how many colleges are doing New Life Project courses (Mathematical Literacy, Algebraic Literacy).  From that work, it is clear that the landscape is changing in both beginning algebra and general education mathematics.  However, two patterns are still present:

  1. We continue to offer one or more courses in arithmetic focusing on procedures.  The presence of these courses is a tragedy on our campuses, since they negatively impact exactly the student groups we want to help (minority, poor).  I’ve posted on these issues earlier this year.
  2. We frequently classify intermediate algebra as a college course, and commonly use it as a general education requirement.  Using a course which mimics a high school course in this way is professional embarrassment.  That’s the topic of this post.

We all know that “intermediate algebra” varies considerably between colleges, states, and regions.  In some cases, the intermediate algebra course has content at the level of the Common Core Mathematics (see http://www.corestandards.org/Math/ ) within the algebra and functions categories.  In most cases, however, our intermediate algebra courses fall below those expectations.

Intermediate algebra is a remedial course!!

The primary distinction between K-12 algebra and intermediate algebra is assessment — the college intermediate algebra course most likely requires a higher level of performance by the student in order to earn a passing grade.   It’s like “So, you were supposed to have learned this stuff in high school but NOW you are going to have to REALLY know that stuff.”

However, in many ways, our intermediate algebra (IA) courses are inferior copies of the K-12 curriculum.  Our IA courses are still descendants of copies of Algebra II from the 1970’s; much emphasis on procedures and correct answers … not much dealing with reasoning.  Given that we don’t deal with most of the discipline issues that occupy a K-12 teacher’s time, we should to better.    The K-12 content has responded to a series of standards (NCTM, Common Core) while our intermediate algebra has been standing still.

The Algebraic Literacy (AL) course is a modern system to help students get ready for college mathematics.  However, AL is still “not college math”, even though AL raises the expectations for students.

Entire states use intermediate algebra (IA) as an associate degree requirement.  In Michigan, which lacks a central governing body for community colleges, most colleges use that as one option for degrees.

We can, and must, do better.  If students do not need a course like Pre-Calculus, then we should use quantitative reasoning (QR) or statistics for their degree requirement … or even a course like ‘finite mathematics’.

Personally, I think intermediate algebra must die (and soon).  The issue in this post is whether a K-12 level standard course should be used for associate degree requirements.  Beyond the criteria of ‘expediency’, there is no rationale for that use.  IA is remedial, not college level.

Let’s MOVE ON!!

 
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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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