Too Many Math Courses!!

Have you got course(s) in … basic math … arithmetic … pre-algebra … fraction modules … decimal modules … etc??  Although some colleges and a few states have eliminated courses at this level, the vast majority of colleges still have one or more.  One is too many!  #acceleration #FinancialAid

Federal financial aid guidelines prohibit a course to count for enrollment levels unless it is at least at the high school level; courses below high school (K to 8) can not be used.  See https://ifap.ed.gov/fsahandbook/attachments/1415Vol1Ch1.pdf

My institution is currently going through a process (intensely so) dealing with our single course at that level (pre-algebra).  Our department has been asked (about every 10 years) to classify developmental courses as ‘high school’ or ‘below high school’.  The most recent request resulted in our best answer at this time: pre-algebra is at the K-8 level.  That was mostly true 10 years ago, and the answer is even clearer when the Common Core standards are considered.

Does your college follow this rule?  You might know already, but be aware that all institutions can be subject to a financial aid audit; violations can result in financial penalties up to and including loss of all federal financial aid money.  Fines are the most common penalty, from what I’ve heard.

Do you have 3 or more courses below intermediate algebra?  Two of these are likely to be ‘elementary level’ (non-federal financial aid), and one ‘high school’ (beginning algebra).  Three courses at that level creates a practical problem for students (completion), a financial problem for your institution (financial aid audit), and a moral problem as well.

The Math Literacy course is designed to have a minimal prerequisite (basic numeracy).  Some colleges use Math Literacy with a lower placement cutoff than beginning algebra; some offer Math Literacy without any math prerequisite at all.  To me, this is a situation where co-requisite remediation makes a lot of sense.  The prerequisite knowledge is a fairly small set, and the range of ‘gaps’ is therefore more limited than it would be in a higher-level course.

For some of us, ‘arithmetic’ is the most common placement level for new students in our college.  I’ve heard up to 36% in that placement, with my college’s rate a little lower.  In all of the research I’ve seen over the years, one thing has been consistent:  courses before beginning algebra do not benefit students in terms of passing subsequent math courses (in general).  Our instruction and learning is often the least sophisticated at this level, and student motivation tends to be pretty low.

Let’s agree that eliminating courses below Math Literacy (and beginning algebra) is a really, really good goal.  The problems have other solutions within our reach, and our students deserve better than they are getting now.  The federal financial aid rules provide an added incentive; however, we have sufficient rationale from other considerations.

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Co-Requisite Remediation in Tennessee

Has Complete College America (CCA) collaborated with the Tennessee Board of Regents (TBR) to create a great solution … or, have they inflicted an invalid model on the students of the state?  I suggest that “data” will not answer this question. #CCA #CorequisiteRemdiation

To summarize some key features of the Tennessee plan in mathematics, implemented state-wide this fall (2015):

• All students are placed in to college-credit mathematics
• If the ACT Math score is below 19, that college level math course will be statistics or quantitative reasoning
• If the ACT Math score is below 19, the student is required to enroll in a co-requisite ‘support’ course
• This co-requisite support course involves all developmental math learning outcomes

These elements are taken from a TBR memo (http://www.pstcc.edu/curriculum/_files/pdf/cdc/1415/Features%20of%20Corequisite%20Remediation%20-%20Memo.pdf)

From what I can see, actual practice is pretty close to this plan … learning support classes are paired with a QR course and an intro statistics course (but not college algebra or pre-calculus).  The learning support courses list topics from arithmetic, algebra, geometry and statistics.  I noticed that the QR courses tend to be more of a liberal arts math course — set theory, finance, voting, etc (the course is called ‘contemporary mathematics’).  In the 4-year college setting, this type of liberal arts math course is usually offered without any math prerequisite.

The initial data from the Tennessee pilot look very good; in fact, my provost is smitten with the Tennessee program, and wants us to consider doing the same thing.  I think the plan will “work” fairly well in Tennessee because of the non-symbolic nature of their QR course (intro statistics is notoriously non-symbolic, in the algebraic sense) … and the fact that they block students from STEM.  [They also had an inappropriate prerequisite on the non-STEM courses; see below.]

In Michigan, we have tried to establish 3 paths in math … college algebra/pre-calculus, statistics, and QR.  For statistics and QR, we have established a ‘beginning algebra level’ prerequisite (algebra or math literacy).  This level maps roughly to an ACT math score of 17, and we require more algebra in my QR course than in the Tennessee course. When the Tennessee plan ‘works well’, part of that is due to the fact that students never needed any remediation for stat or QR if their ACT math was 16 to 18.

In other words, the good results from the co-requisite pilot is due, in significant part, to the math prerequisite for college level courses.  ACT math = 19 (the Tennessee cutoff)  is a bit low for college algebra, but it is too high for statistics and QR (even if the QR is more algebraic, like mine).  Tennessee could have achieved the benefits for about 30 to 40% of their students by changing the prerequisite on two courses to be more realistic; they had established a ‘intermediate algebra’ prerequisite for all college math when that is not appropriate.  Changing the prerequisite would have helped many of the students without requiring them to take another class.

The problem we face is not that there are ‘bad ideas’ being used; the problem is that policy makers are evaluating ideas at a global level only, when the meaning of any statistical study is derived from analysis done at a fine-grain level.  Aggregated data is either useless or dangerous, and ‘aggregated’ is all policy makers consider.

CCA says “the results are in”.  Nope, not at all … we have some preliminary data about some efforts, which are not necessarily showing what the aggregated data suggests.

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College Algebra is Not Pre-Calculus, and Neither is Pre-calc

“Everybody knows what college algebra is!”  This was said by a math chair from a university in my state, as we worked though our state’s new transfer requirement for mathematics.  Of course, he was wrong … though he has a lot of company.  Today’s main question is this:  Is college algebra a subset of pre-calculus?

The original college algebra course developed in the 19th century at the universities of the day (Harvard, Yale, Bowdain, etc), with a focus on meeting a math requirement for their degree.  Of course, those times were very different … the Yale Catalog listed every student, and every student had the same default schedule.  College algebra was everybody’s math course as a freshman; those ‘desiring’ calculus took it as a Junior.  See http://elischolar.library.yale.edu/cgi/viewcontent.cgi?article=1069&context=yale_catalogue

That tradition carries forward to the present day, in the work of the MAA.  The MAA College Algebra guidelines remain a narrative for a general education class, not a pre-calculus course.  See http://www.maa.org/sites/default/files/pdf/CUPM/crafty/CRAFTY-Coll-Alg-Guidelines.pdf

The use of the name ‘college algebra’ for a calculus prerequisite appears to be a regional variation.  In states use ‘college algebra’ as a prerequisite for pre-calculus; other states use college algebra as the first semester of pre-calculus … or as their one-semester pre-calculus (as in “college algebra and trig”).

Our situation has become illogical and disfunctional.

When publishers market their textbooks, sometimes the key difference between college algebra and pre-calculus is this: pre-calc emphasizes a unit circle for trig functions, while college algebra uses right triangles.  Other than that, the pre-calculus book has more complicated problems, but no substantive differences.  Both courses trace their ancestry back to the 19th century mathematics course later known as ‘college algebra’.  [Search for Jeff Suzuki’s talk on college algebra.]

Neither course is really pre-calculus.

Of course, I don’t mean “students can not take these prior to calculus”; they do, though the benefits are small and accidental.  A pre-calculus course would be designed to prepare students for the work of a calculus course.  We make the fatal mistake of equating the ability to solve complicated symbolic problems with the capacity to reason with those objects.

A good preparation for calculus begins much earlier for many students.  “Developmental” mathematics is being re-formed to focus on understanding and reasoning, with a de-emphasis on artificially complex symbolic work.  A mathematical literacy course is a better preparation for calculus than the traditional algebra course.

More importantly, Algebraic Literacy is where we can begin the serious work of preparation for calculus.  Intermediate algebra is a documented failure as preparation for college mathematics; algebraic literacy is designed deliberately for these purposes.  The Algebraic Literacy course has learning outcomes backward-designed to meet the needs of calculus preparation … to be followed by a well-designed course at the next level which completes that preparation.

Here are some conditions necessary for good calculus preparation, based on the available information:

• diversity of content (algebra, geometry, trig as minimum)
• non-trivial reasoning about mathematical objects
• concrete (context) and abstract situations
• properties of functions, and relationships between types
• reasoning and visualization involving related quantities (2, 3, or 4 at a time)
• procedural expertise and flexibility

I do not intend for this list to be exhaustive.  The intent is to focus on key outcomes so we can determine when we have a real pre-calculus experience that will work for our students.  It is my belief that the great majority (>99%) of our current courses used as a calculus prerequisite are not reasonable preparations for the demands of such a course.

Some of our colleagues are beginning the work of correcting the curriculum; we need to support that work when possible.  If you’d like to explore what the new curriculum would look like, the Algebraic Literacy course provides a good starting point;  I’ll be doing a session at the AMATYC conference (Nov 21, 11:55am) in New Orleans.

We can solve this problem, together.

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STEM Path: Your Time has Come!!

The reform work, at the college level in mathematics, has focused on the needs of non-STEM students.  That work has been effective in creating a large, long-term impact on what students experience … as long as they do not need calculus.  Now, we can turn our attention to the needs of students on the STEM path.  #mathpaths #NewLifeMath #Mathways #Precalc

We need to understand, first, that our current STEM path is a weak design with a track record of poor results.  For example:

• In “The Pitfalls of Pre-calculus” David Bressoud shares some sophisticated research on the benefits of pre-calculus to different types of students; less-prepared students were not harmed by pre-calculus (not helped much either), and well-prepared students were actually harmed by pre-calculus.
• Most publicized research on remediation, which focuses on the transition from developmental to college level (intermediate algebra to college algebra/pre-calculus); the standard result in this research is “no benefit” for a significant portion of the population.
• We lack any body of research showing that our courses work above the developmental level (pre-calculus and calculus in particular).

The STEM path is too important to leave it alone, to accept the current sloppy curriculum as “good enough”.  The research is compelling, and we have other research providing guidance for better solutions.  Take a look at the work in co-variational reasoning, as well as the MAA Calculus Readiness test (those use related concepts).  Take a look at the book “Mathematical Sciences in 2025”.

Some curricular models to improve the STEM path are being developed.  The AMATYC New Life Project outlines content for Algebraic Literacy, which is designed to be an effective preparation for college-level mathematics like pre-calculus; the content is engineered for this purpose (unlike ‘intermediate algebra’).  I’ll be doing a session at the 2015 AMATYC conference on Algebraic Literacy (S149, Saturday Nov 21).

The Dana Center is developing its STEM path, with the Reasoning with Functions courses; that work is informed by the sources cited above, with these courses designed to follow a beginning algebra level course (Math Lit, “FMR”, etc).  In other words, their STEM path places pre-calculus immediately after beginning algebra.  The Dana Center will be doing a symposium at the AMATYC conference next month.

You have opportunities to be a part of this work, to make changes at your institution to help students on the STEM path.  Please consider taking advantage of these opportunities.

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