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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Why Does Co-Requisite Remediation “Work”?

Our academic leaders and policy makers continue to get strongly worded messages about the great results using co-requisite remediation.  Led by Complete College America (CCA), the originators of such messages suggest that this method avoids the failures of developmental mathematics.   [For example, see http://completecollege.org/spanningthedivide/#remediation-as-a-corequisite-not-a-prerequisite] Those of us in the field need to understand why intelligent people with the best of intentions continue to suggest this uni-directional ‘fix’ for a complex problem.  #CCA #CorequisiteRemediation

I want to focus on the educational component of the situation — not the political or fiscal.  In particular, I want to explore why the co-requisite remediation results have been so encouraging to these influencers.

One of the steps in my process was a nice conversation with Myra Snell.  I’ve known Myra for a while now, and she was involved with the New Life Project as well as the Carnegie Foundation’s Statway work.   What I got from this conversation is that Myra believes that there is a structural cause for the increased ‘throughput’ in the co-requisite models.  “Throughput” refers to the rate at which students complete their college math requirement.  Considerable data exists on the throughput using a traditional developmental math model (pre-algebra, beginning algebra, then intermediate algebra); these rates usually are from 7% to 15% for the larger studies.  In each of the co-requisite systems, the throughput is usually about 60%.  Since the curriculum varies across these implementations, Myra’s conclusion is that the cause is structural … the structures of co-requisite remediation.

The conclusion is logical, although it is difficult to determine if it is reasonable.  Scientific research in education is very rare, and the data used for the remediation results is very simplistic.  However, there can be no question that the target of increased throughput is an appropriate and good target.  In order for me to conclude that the structure is the cause for the increased results, I need to see patterns in the data suggesting that ‘how well’ a method is done relates to the level of results … well done methods should connect to the best results, less well done methods connect with lower results.  A condition of “all results are equal” does not seem reasonable to me.

Given that different approaches to co-requisite remediation, done to varying degrees of quality, produce similar results indicates some different conclusions to me.

• Introductory statistics might have a very small set of prerequisite skills, perhaps so small a set as to result in ‘no remediation’ being almost equal to co-requisite remediation.
• Some liberal arts math courses might have properties similar to intro statistics with respect to prerequisite skills.
• Some co-requisite remediation models involve increased time-on-task in class for the content of the college course; that increased class time might be the salient variable.
• The prerequisites for college math are likely to have been inappropriate, especially for statistics and liberal arts math/quantitative reasoning.
• Assessments used for placement are more likely to give false ‘remediation’ signals than they are false ‘college level’ signals.

Three of these points relate to prerequisite issues for the college math courses used in co-requisite remediation.  Briefly stated, I think the co-requisite results are strong indictments of how we have set prerequisites … far too often, a higher-than-necessary prerequisite has been used for inappropriate purposes (such as course transfer or state policy).  In the New Life model, we list one course prior to statistics or quantitative reasoning.  I think it is reasonable to achieve similar results with the MLCS model; if 60% of incoming students place directly in the college course … and 40% into MLCS, the predicted throughput is between 55% and 60%.  [This assumes a 70% pass rate in both courses, which is reasonable in my view.]  That throughput with a prerequisite course compares favorably to the co-requisite results.

The other point in my list (time-on-task) is a structural issue that would make sense:  If we add class time where help is available for the college math course, more students would be able to complete the course.  The states using co-requisite remediation have provided funds to support this extra class time; will they be willing to continue this investment in the long term?  That issue is not a matter of science, but of politics (both state and institution); my view of the history of our work is that extra class time is usually an unstable condition.

Overall, I think the ‘success’ seen with corequisite remediation is due to the very small sets of prerequisite skills present for the courses involved along with the benefits of additional time-on-task.   I  do not think we will see quite the same levels of results for the methods over time; a slide into the 50% to 55% throughput rate seems likely, as the systems become the new normal.

It is my view that we can achieve a stable system with comparable results (throughput) by using Math Literacy as the prerequisite course … without having to fail 40% of the students as is seen in the corequisite systems.

 
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Co-Requisite Remediation and CCA (Saving Mathematics, Part V)

Complete College America (CCA) released a new report on co-requisite remediation this week.  Actually, that statement is not true … the CCA released a web site which shows some data on co-requisite remediation, with some user interaction.  What’s missing?  Anything that would help a practitioner judge whether they should consider co-requisite remediation!  #CCA #Corequisite #SaveMath

Many of us are dealing with policy makers in our states or institutions who see co-requisite remediation as the solution to the “developmental math problem”.  There are, in fact, serious problems in developmental mathematics; there are also serious problems with how ‘college math’ has been defined, and how policy makers are defining a problem away instead of solving it.

Within developmental mathematics, we have been working hard teaching the wrong stuff to our students, frequently using less-than-ideal methods to help them learn.  Our curriculum has too many courses, and the combination is lethal … not many students reach their dream.  When students proceed from developmental math to college algebra or pre-calculus, they often find that the gap in expectations between the two levels is very difficult to deal with.

Co-requisite remediation steps in to this complex problem domain, and declares that all will be fine if we just put students into college math with some support.  The most common (and sometimes the ONLY) co-requisite remediation done is in Intro Statistics and Quantitative Reasoning [QR] (or Liberal Arts Math).  The history, frequently, is that students had to pass intermediate algebra prior to these courses … even though that background has nothing to do with the learning; the requirement was to establish “college level”.

So, the CCA and allies declare that students can take Stat or QR instead of developmental math.  Of course this is ‘successful’; the old prerequisite was unreasonable, and the co-requisite method puts students directly in to courses they are relatively ready for, and also provides extra support (in some cases).  Many colleges, including mine, had already lowered the prerequisite for Stat and QR years ago; our results from both Stat and QR are better than what the CCA states for their co-requisite model.

The co-requisite ‘movement’ is an illusion.  The work succeeds (almost totally) because students are placed in to math courses that have minimal needs for algebra.  I get better results by just changing the prerequisite to Stat and QR.

We also face a risk to mathematics in this illusion:  students with dreams that involve STEM are frequently told that this dream is being shelved in favor of co-requisite remediation, that they will take either Stat or QR.  The path to calculus is either not available or involves work that is not articulated well to students.  Policy makers are treating math as a barrier to cope with, a problem to solve with the least remediation.  The need for mid- and high-skill STEM workers is well documented, but the co-requisite ‘solution’ often blocks the largest pool of students from those fields … the minorities, the poor, the students served by under-performing schools.

Society needs our work to succeed for all students.  We can not accept a solution which reduces upward mobility; a solution which does not provide ‘2nd chances’ is a risk to both mathematics and to a democratic society.

Don’t get me wrong — Stat and QR have a major role to play in our curriculum, and these courses might be the most common math courses students should take in college.  My main message is that we need to question the illusion called ‘co-requisite remediation’, AND we need to articulate a vision of our curriculum which enables ALL students to consider STEM and STEM-like careers.   [The New Life Project provides a vision of such a curriculum.]

If you really want to read the CCA “Report”, go to http://completecollege.org/spanningthedivide/#the-bridge-builders

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Saving Mathematics, Part IV: This is College, Right?

For whatever reason, we in the mathematics community have an obsession with high school … we define college mathematics by assigning a prerequisite that suggests a level above high school (often the prerequisite is intermediate algebra).  We also accept the notions about people not being good at mathematics, which results in the contradictory policy of allowing intermediate algebra to meet a degree requirement in college.  What’s up with THAT?  #remediation #FinAid

All of our institutions (with very few exceptions) must comply with financial aid laws and regulations.  Those regulations make a distinction between remediation at the high school level and remediation at the elementary level (K-8); courses at the elementary level (like arithmetic, pre-algebra) can not be used to determine eligibility for financial aid.  Courses at the high school level can be used for financial aid, though there is a limitation on the total remediation.  (see https://ifap.ed.gov/fsahandbook/attachments/1415Vol1Ch1.pdf)

The K-12 professional standards of the past 25 years (NCTM) and the Common Core provide a way to judge the level of our courses.  Most of our intermediate algebra courses map to 9th and 10th grades in those standards; even prior to that, intermediate algebra was considered 10th or 11th grade level.  Overall, 57% of our enrollments (community-college-type) is in pre-college mathematics … 32% of that enrollment is in remediation at the elementary level.  [CBMS 2010 data; http://www.ams.org/profession/data/cbms-survey/cbms2010-Report.pdf]

My position is that these high numbers in remediation are the result of artificial parameters for ‘college level’ and our obsession with high school.  Many of us accept the position that the mathematics actually needed for college work (whether STEM-path or not) is not delivered by our basic-math > pre-algebra >beginning-algebra > intermediate-algebra filtering system.  Our curriculum in those courses is often inferior to what our K-12 colleagues are using.

• Remediation does not mean high school mathematics

We need to throw out our traditional developmental courses (as well as most college-algebra-level courses).  Convenient copying of courses does not help students.

The question is:

• What does a COLLEGE student need prior to a college math course?

The needs do vary somewhat depending on the particular college math course.  We need to show our integrity by offering courses designed to serve the purpose for which we use them:

• Only require students to take courses with validity for the purpose!

This is not a quick process, but it is something we can do together … and even be inspired by.

In the meantime, let’s show our professionalism by doing the following:

1. Always classify arithmetic and pre-algebra as “elementary level” remedial courses
2. Always classify beginning algebra and intermediate algebra as “high school level” remedial courses, which have no role meeting a college degree requirement
3. Identifying appropriate college-level math courses required for each degree

Complete College America says much that I disagree with; quite a bit of their communication is rhetoric to support pre-determined solutions.  However, one thing from CCA I really agree with:

College students come to campus for college, not more high school. Let’s honor their intentions — and refocus our own good intentions to build a new road to student success.
http://www.completecollege.org/docs/CCA-Remediation-final.pdf

To get started on a path to replace the traditional developmental math courses, take a look at the New Life Project courses (Mathematical Literacy and Algebraic Literacy).  I hope that you will join me and hundreds of other professionals working to create better models to serve our students and communities.

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