Can ANY Sequence of Math Courses Succeed??

We share a commitment to student success … we work hard to help students reach their goals.  Sometimes, it just seems like that success is very rare.  We’re told by many sources that most students fail to reach their goal, especially if they are placed into the “dead end” (developmental mathematics).  What’s the problem?  #CCA  #pathways  #collegemath

Mathematics in college consists of sequences of courses.  Historically, the primary variable was the ‘exit point’ (the student’s last math class); recent work has created a more subtle solution where the prerequisites are variable … not all college-level courses require intermediate algebra.  We still have sequences.

The extremes of a sequence design are easily seen as failures.  At the one extreme, we might have a sequence of 4 prerequisite courses prior to the ‘one that counts’; even if we have an astounding pass rate (80%) and perfect retention (100% to next course), the net result is 41% start the college course.  The more reasonable pass rate (70%) and retention rate (80%) mean that about 12% start the college course.

At the other extreme, we have no prior courses … it’s a sequence of one course, the college level one.  That’s what the radical “co-requisite remediation” advocates suggest (and some states try to implement).  In this approach, 100% start the college math course, so even if just 30% pass it’s a gain over the long sequence.  Most of us do not support this type of policy.

So, the question is this:

Can ANY Sequence of Math Courses Succeed?

As for many human endeavors, it is far easier to create something that does not work than something that does.  Here are some principles for designing for success.

• Courses copied from another context will not support success in the sequence.

Developmental mathematics is full of course copies … basic math copied from 8th grade, beginning algebra copied from algebra I, intermediate algebra copied from algebra II, etc.  These remedial courses are part of a different tradition: students should be ‘college ready’ so we provide courses to remediate high school.  College algebra, erroneously seen as pre-calculus, is also a copy of a course.  These courses have nothing to do with success in the sequence.  We create some coincidental features for success in the algebra courses, but the entire package is doomed.

• Arithmetic is too difficult for college

Learning arithmetic involves applying properties of real numbers, standardized rates (percents), and solving fractional equations (proportions).  These advanced topics might make sense after a good algebra course, but certainly not before.  Saying that arithmetic is a prerequisite to algebra is like saying that running is a prerequisite to walking.  A course on arithmetic is doomed; either the content is too advanced … or we take all of the arithmetic out and just deal with correct answers.

• One course at the developmental level should be enough for at least 80% of the students.

Too often we think about “what the student does not know”, instead of “what is needed for success”.  We get trapped in to a process that tries to fill in all ‘holes’.  Being ready for success in a college level math course does not involve ‘everything’.

• Capabilities are just as important as skills.

In the traditional sequence, we do exclusively skill work; sure, we include ‘applications’ with the thought that these will improve something (though we are not sure what).  Our courses often delay intense work on reasoning until the calculus I course.  General reasoning is one of the ‘capabilities’ required for success; we might even focus on the 5 strands of mathematical proficiency.  Other capabilities are number sense, proportional reasoning and algebraic reasoning.

• Good mathematics should start from the first day of every class.

The traditional sequence directly says “you are not ready for the good stuff yet; let’s see if you make it though this n-course sequence and then start good mathematics”.  I leave “good mathematics” as an undefined term.  If you look forward to teaching it and are proud of the mathematics, you are probably doing good mathematics.  One trait of good mathematics is likely to be that connections are built for each idea.

The AMATYC New Life Project has advocated a curricular model consistent with these principles.  We’re not alone in doing so; the Dana Center New Mathways work also does a good job.  The old courses (developmental and college algebra) need to be replaced by courses designed to succeed.  The New Life  math courses emphasize important mathematics with a plan to efficiently get students ready:  Mathematical Literacy, and Algebraic Literacy.   [I’m doing a session at this year’s AMATYC conference on the Algebraic Literacy course.]

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1 Comment

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  By schremmer, September 28, 2015 @ 11:02 am

  1. Re. “Learning arithmetic involves applying properties of real numbers”

  Absolutely NOT. I am ready to make the case for this but I don’t want to inflict it.

  2. Re. “One course at the developmental level should be enough for at least 80% of the students.” On the one hand I do agree that the length of any sequence is a major issue (See Effect Of Length Of Sequence) but that the length should be 1 in 80% of the cases is most debatable. Rebuilding people who have been brainwashed into accepting and memorizing whatever “authority” hands down is bound to take more than one course.

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