Diversion, Advancement, and Math

Developmental education continues to be a subject of much research, and even more discussion.  A few studies on remediation have become the drum beat for some foundations and projects that call remediation a failure.  As is normal in scientific research on a human activity, the landscape is much less clear … and much more attractive … than the ‘failure’ studies imply.

A recent report authored by Judith Scott-Clayton and Olga Rodriguez for the National Bureau of Economic Research describes efforts to study a broader range of possible outcomes.  The report is titled Development, Discouragement, Or Diversion? New Evidence On The Effects Of College Remediation and available at http://www.nber.org/papers/w18328.  Statistically, this report uses a discontinuity (regression-discontinuity, or RD) approach  as did several other studies; they expanded the research by using high school records and other college records. 

The major findings of the study are:

We find that remediation does little to develop students’ skills.

We also find relatively little evidence that it discourages either initial enrollment or persistence, except for a subgroup we identify as potentially mis-assigned to remediation.

The primary effect of remediation appears to be diversionary: students simply take remedial courses instead of college-level courses.

The first finding is consistent with other RD reports.  My own explanation for this discouraging finding is that developmental courses focus more on ‘facts’ than ‘skills’ or understanding; our students experience a long sequence of tasks related to what they need, but we do not generally provide a cohesive treatment that addresses the need directly.  The second finding shows some differences with other studies, which suggest that developmental courses might lower persistence.

The third finding is the one that raises concerns for me.  In their report, the authors suggest that diverting students from college-level academics might be a valid approach for a group of students with a lowered probability of success.  While I agree that there are some parts of the ‘developmental population’ with such extreme learning challenges that diversion is best, I do not agree in general that diversion is healthy or desirable. 

Is developmental education about diversion, or are we about advancement?  For us as professionals, this is one of those fundamental questions that determines our classroom behavior and expectations of students.  I see myself as an ‘advancement fanatic’; not only do I see advancement as the fundamental goal of developmental education (even mathematics), I believe in the advancement goal for every single student in spite of the predominant temporary evidence that this might not be reasonable.

One of my students said “You believed in me when nobody else would”.  I do not know if this student will reach her goal of becoming an elementary teacher; I hope she does … children deserve to have a teacher like her.  Even if she does not make it there, I believe that she can.  Quite a few of our students are in our courses because other people gave up on them.  I will not give up.

Diversion for developmental math students is the last resort for us.  Advancement is the thing.

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Research Trends in Developmental Mathematics

If you teach basic statistics in any form, you have probably dealt with the sharp contrast between ‘statistics’ and ‘statistical study’; in other words, there is a large difference between statistical data and the practice of statistics.  Having data does not mean there has been a statistical study.  In a similar way, having data does not mean that there has been research.

Research is an abstraction of this ‘data => statistical study’ to a higher level; research involves a prolonged effort to answer meaningful questions in a field of study, usually involving multiple researchers.  Research, in this meaning, is rare in developmental mathematics — we have lots of data, quite a few studies, but not that much research.  Research strives to provide richer and more subtle answers, and deals with a common core of issues.

One of my friends (thanks, Laura!) recently passed along a link to an article on research in developmental mathematics; this article is by Peter Bahr, whom I had read a few years ago (he’s been busy!).  The current article is called A Case for Deconstructive Research on Community College Students and Their Outcomes, and is available online at http://cepa.stanford.edu/sites/default/files/Bahr%203_26_12.pdf 

This article places research on developmental math within a larger framework of research in community colleges, focusing on student progression.  Which factors in a progression make a difference in the eventual outcome?  One of the conclusions Dr. Bahr reaches is that beginning algebra is a critical course; not passing this course on the first attempt raises the risk that a student will not complete — even if they persist to try the course again.  The article has several other points with practical implications for us, and for policy makers.

Instead of saying that remedial math is part of a ‘bridge to nowhere’ (the mistaken message of Complete College America), research into developmental mathematics takes a more intelligent (and difficult) approach of identifying specific features that have positive or negative impacts on student outcomes.   This research is too specialized for policy makers to understand, even if they understand research as opposed to statistics; part of our responsibility is to articulate what this research means in a manner that policy makers can understand.

I hope that you will use research like the Bahr article to suggest basic changes in your developmental math program.

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Variables Less Understood

In a traditional beginning algebra course (like one I am teaching), we spend much of the time working with variables in expressions and equations … and functions.  The course first shows variables within simple expressions with a progression towards complexity and equations.  One problem (and the student errors on it) really caught my attention this semester; it’s actually a simple problem, not requiring any procedural steps.

Here it is:

Solve   -6k + 3 = 3 – 6k

I have to admit that I did not emphasize ‘reading the equation to see what general statement it is making’, though we did actually talk about equations where the variable term was equal on both sides.  One of the common errors is shown below:

Every student making this error could some an equation like  ‘2y – 5 = 4 – y’.  What was causing the error?

Sadly, the problem was that many students are learning the ‘algebra dance’: Duplicate these steps, record the result.  Part of the dance is to write the opposite of the variable ‘thing’ on the other side to get one variable in the problem.  Students used this dance to solve a number of equations to produce quite a few correct answers.  For this problem, part of the dance was the ‘get one variable’ — the student knew that -6 + 6 was zero, so we just have the letter.  The variable was less understood than we thought, based on the consistent correct answers to other problems.

It’s very likely that you can list some mistakes that are similar in showing a less understood variable concept.  One of the errors I am seeing is “5 + 2x” becoming “7x”; the numbers and letters become the whole story … the operations are not even being read.

If you are curious, there is a wide body of research on learning variable concepts; for one summary, see http://www.nctm.org/news/content.aspx?id=12332 (an NCTM item).  Some particular research items:  http://www.emis.de/proceedings/PME29/PME29RRPapers/PME29Vol3KoiralaEtAl.pdf and http://www.merga.net.au/documents/Steinle_RP09.pdf  and http://elib.mi.sanu.ac.rs/files/journals/tm/16/tm915.pdf. 

What I am focused on, however, is not the research nor the particular misunderstandings — rather, I am thinking about WHY this happens.  It seems the problem is most likely when students have a higher motivation for ‘correct answers’ compared to their valuing of understanding (which is a combination of desire to understand and confidence in being able to).  In my classes, I often say that I am not that interested in the answer they get; I am more interested in the knowledge you have about that type of problem (the understanding).  Obviously, this statement from me does not change the drivers of student motivation (answers or understanding); I need to create instructional spaces where the understanding is the result being assessed directly.  I suspect that I will be using some type of writing for this purpose; this will be a challenge, given the range of writing abilities in the class.  For one reference on writing in math, see http://www2.ups.edu/community/tofu/lev2/journaling/writemath.htm. 

However, I can count on a basic human trait:  We (meaning our brains) naturally prefer to understand the world around us.  Knowledge organized by understandings is easier to maintain and use, compared to knowledge that is random memories.  The problem with ‘variables less understood’ is that this natural desire to understand has been subverted, perhaps caused by messages about ability … perhaps reinforced by social messages that math is about formulas and answers.

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The Math Bridge

Imagine, if you will, two small towns near a bridge over a large river. One town (Prima Factoid), priding itself on details and being thorough, shared a belief that ‘being ready’ meant having all of the basic skills taught in their local high school.  They spoke of alignment, of mastery, of students’ taking responsibility for their learning.  The neighboring town (Stepped Up), being populated by realists, shared a belief that every body was ready enough … or they were not eligible.  They spoke of evidence, reports, and things not working.

These towns share the bridge that is developmental education, a major part of this structure being called developmental mathematics.  Prima Factoid constructed levels and additional ramps to the bridge; Stepped Up put everybody in vehicles all going the same speed (fast) with some extra handbooks and ‘life line’ calls.  The two towns had a friendly football rivalry, but this hid a deep mistrust between citizens of the two towns.

So here is my motivation:  Complete College America released a report Remediation: Higher Education’s Bridge to Nowhere    (see http://www.completecollege.org/docs/CCA-Remediation-summary.pdf).  I am disappointed in this report … within their goal of fostering a completion agenda, they label remediation as a failure beyond recovery; they suggest that we place all students in college-level courses (as in Stepped Up). 

However, many of us actually live in Prima Factus, and we need to recognize how mismatched this approach is to the needs of college students.  By living via a basic skills mentality, with an honest desire to help students, we present unnecessary barriers and extra courses in front of students without much evidence of this being effective for the majority of students.

For the developmental education bridge to actually work, we need to be much more deliberate and thoughtful in its design.  To think that all students are ready for college courses with support ignores the deep educational needs of a large portion of our students; to think that all students need to pass courses covering basic skills from arithmetic and polynomial algebra is to provide a weak foundation for college work.

We need balance; we need a clear vision … a vision that recognizes that there are many students who just need some extra support to be successful in college courses without taking developmental courses, while there are many other students with academic needs that should be met in a few courses (like 1 or 2 math courses). 

Reports that totally condemn what we are doing do not help us move forward, just as reports that totally defend the current basic-skill oriented models.  We have fundamental work to do so we truly help our students … ALL of our students.

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