Algebra for Everybody, and Algebra for Nobody

Algebra is so basic that all students need to have a good grasp, and adults without this capability will be limited in their economic and social choices.  Algebra is so esoteric compared to daily life and work that only those in STEM careers need to bother learning it.  Both of these statements can be true; the apparent inconsistency is based on what is meant by the word ‘algebra’. 

As such things normally happen, there is another article suggesting that nobody needs algebra (see Andrew Hacker’s article at http://www.huffingtonpost.com/2012/07/30/in-new-york-times-op-ed-c_n_1719947.html) and a response by Borwein & Bailey (see http://www.huffingtonpost.com/david-h-bailey/algebra-is-essential-in-a_b_1724338.html).  Reflecting our society in general, we tend to view issues as a binary choice — if it is good, all people must; if it is not good, nobody should.  The Bailey & Borwein response is well written, and reflects a balanced point of view.

As a mathematician, I view algebra as a language system used to describe and manipulate features (whether known or not) of the physical world based on arithmetic operators .  Basic literacy in this language system is essential in both academia and ‘real life’; translations into and out of algebra are the most basic literacies, followed by different representations (symbolic, numeric, graphic).  

Unfortunately, the algebra of mass education tends to focus on procedures and complexity of limited value to anybody combined with a focus on solving algebraic puzzles, as if completion of a crossword puzzle is a basic skill for a language.    True to our current binary approach, people who agree with a literacy approach will invest great effort to avoid all procedures, complexities, and puzzles.  The truth is that we undertake these problems ourselves just for fun, and this is one element in our transition to being mathematicians; how are we to capture the attention of potential STEM students if we avoid the fun stuff? 

As a language system, basic literacy in algebra means that a person can read the meaning of statements; transformations to simpler forms is based on that meaning.  I failed to help my students in the algebra class this summer … I know that because students could distribute correctly in a product but failed routinely with a quotient.  [In case you are wondering, the problems involve a 3rd degree binomial and a 1st degree monomial; in the division case, many students ‘combined’ the unlike terms in the binomial instead of distributing.  This is a basic literacy error; very upsetting!]

Hey, I know … nobody needs to distribute algebraic expressions on their ‘job’ (except us!).  That type of  reason is enough for me to conclude that we do not need to cover additional of rational expressions (prior to college algebra/pre-calculus); that process is complex, and is based on a higher level of understanding of the language.  Distributing is a first-order application of algebraic literacy; avoiding that topic means that we present an incomplete picture of algebra as a language.  A pre-college mathematical experience needs to provide sound mathematical literacy — including algebra.

Everybody needs algebraic literacy, as part of basic mathematical literacy.  We can design courses that provide the needed mathematical literacy as a single experience — no need for a numeric literacy course (‘arithmetic’) and an algebra course and a geometry course; all of that, plus some statistical literacy, can be combined into one course.  This is the approach of the New Life model, and is imbedded within the Quantway & Statway (Carnegie), and in the New Pathways (Dana Center).  I encouage us all to include some transformations (‘simplifications’) in the algebraic language.

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The Magic Solution to Learning in Developmental Mathematics

Contextualize … discovery learning … group work … experiments … homework systems … calculators … modules … learning communities … clickers … tutoring … and smiles.

What was that a list of?  To some extent, that was a list of ‘magic solutions’ offered by somebody to improve (often ‘dramatically improve’, according to that person) the learning in our developmental mathematics classrooms.  Every single advocate of these solutions has some ‘data’ (often labeled ‘research’) to support their answer to our problems; if they don’t have this data themselves, then they are a convert or follower — often a person at a foundation or policy group.

These are not solutions, let alone magic solutions.  Solutions deal with problems; solutions make sense.  Solutions fit within the surrounding systems to enable both long-term maintenance and ‘scalability’. 

Here is the magic solution to learning in developmental mathematics:

Offer sound mathematics with academic value, supported by skilled professional educators who can help every student learn by employing a diverse set of tools, focusing on cognitive growth in students.

We currently do not have sound mathematics in the majority of our courses; there are emerging models that provide some specific alternatives (New Life, Carnegie Pathways, Dana Center Mathways).   Many of our colleagues (perhaps the vast majority in some places) have limited skills further hampered by a limited conceptualization of their profession; organizations such as AMATYC and its state affiliates provide professional development to supplement the internal opportunities.  As a profession, we have not articulated a standard set of tools necessary for faculty to meet the needs of our students.  And, far too often, we look at surface outcomes of success (completion, passing) instead of looking at measures of meaningful growth in our students.

As you can see, there is nothing simple or quick about this magic solution.  I still call it ‘magic’, because this solution creates a qualitative shift in our profession — instead of ‘avoidance’, we have a positive target; instead of a discouraged and sometimes desperate people, we can be inspired and proud (both as mathematicians and as educators).  I admit that this magic solution is not quick, nor is it easy; however, it is a real solution, not a temporary distraction like the items listed at the start of this post.  [Those items are possible tools to use, not solutions.]

Many of us are currently involved with projects that are not really a solution, whether this consists of modules or mastery learning or a temporary redesign such as emporium.  Do not worry about this work; it is part of the process … not the end.  Whether it takes 2 years or 5 years, the incomplete solution will be identified as such, and the next stage will be started.  THAT (the next stage) is what you should be concerned about. 

Behind this basic change is a more developed and refined use of research.  Much of the ‘data’ used in our profession (internally or externally) is just a little better than the charts in USA Today — they are not statistically sound, and do not fit into a body of research for our profession.  Most of this data is better left ‘ignored’.  Our work should be informed by theory and research that develops over time; fads are a distraction from basic change.

I hope that you can focus on the larger picture, on what is a ‘magic solution’; perhaps you can look at the emerging models for inspiration or encouragement.  Our success in this endeavor called developmental mathematics depends more on our internal visions of solutions than on a temporary distraction or ‘data’.

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What is Our Plan?

In the classic problem solving methodology, the intense effort is placed in two early stages — understand the problem, and make a plan.  In the case of mathematics (especially developmental mathematics), we have seen much hand-wringing and gnashing of teeth, often accompanied by saber-rattling, about ‘the problem’.  However, a problem can only be defined by comparing the current condition to the desired condition.  Looking at data is a first step, but can often lead to short-sighted efforts that do not solve any significant parts of the problem.

Here is one overview of a plan for mathematics in community colleges (focusing on developmental mathematics, though not restricted to that):

• All math courses must provide good mathematics (appropriate and powerful concepts to deal with quantitative situations).
• All math courses must prepare students for mathematical needs that they will encounter in college.
• Community college mathematics is not a repeat of school mathematics.
• Community college mathematics is compatible with, and supportive of, university mathematics.
• Reasoning and problem solving are central goals of mathematics as part of a general education.
• Remediation is needed for some students, ideally limited to one course or a fast-track experience for most of those students.
• Any student might be inspired to higher goals, and many are capable of additional mathematics in a reasonable amount of time.

If a “solution”, whether modules or online homework or emporium model, only deals with the patterns of the data, then the solution will not solve anything important.  In some cases, the ‘pass rates’ might rise temporarily or even long-term; however, there is still likely to exist a substantial gap between a larger plan for mathematics and what is actually delivered to students.  If the traditional mathematics does not contribute to a larger plan (which is my view), then a solution plan involves much more than the delivery system and much more than course organization.

In the case of developmental mathematics, we have a historical artifact which is based on a premise that we need to provide the same mathematics that students should have learned in high school.  Such an approach is arbitrary, unrelated to mathematical needs, and dooms our courses … and dooms our students … at the system level.  Having a sequence of 4 courses in developmental mathematics guarantees that less than 20% of the students will reach college work, based on an unreasonably high 80% pass rate and 80% retention rate.  The response, based on ‘the data’, is to get students to their exit point in this ‘school mathematics’ as quickly as possible (modules); is our plan for mathematics that students should be shoved off the train as soon as possible … or do we want to have an opportunity to inspire students?

The emerging models — AMATYC New Life, Carnegie Pathways, and Dana Center Mathways — are based on a larger plan.  However, many of us are looking at them as responses to ‘the data’.  For these models to work well, the faculty and colleges involved need to have a deeper understanding of a plan for mathematics.   Hopefully, you will see much in the plan outlined above that you can agree with.  One of our basic problems is that policy makers do not have this larger plan for mathematics in mind; they, naturally, focus on the data.  We, and our professional organizations, need to articulate a larger plan so that we can better serve our students.

One of my colleagues said, back in 2008, that pass rates are the least of our problems in mathematics.  I agree.  We need to have a plan for mathematics, and build new curricula to support that plan.

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What the Experts Say … about Remediation

In our profession (developmental mathematics), the most common phrase this year seems to be “remediation is a failure”; states consider banning all developmental courses, and organizations call remediation a ‘bridge to nowhere’.  What is the validity of these statements?  What is the true status of developmental education in 2012? 

To start with, take a look at a recent article by Hunter Boylan and Alexandros Goudas called “Knee-Jerk Reforms on Remediation”   see      http://www.insidehighered.com/views/2012/06/19/essay-flawed-interpretations-research-remedial-education#ixzz1yG6A5hL2.  Boylan and Goudas review the largest studies that are cited for the ‘failure’ statement, and easily point out the limitations of the research involved.  Some studies employ a discontinuity analysis around the cut score for placement into developmental courses as an estimate of the effects of remediation.  Other studies employ large data sets over a period of time to produce a demographic summary of who is referred to developmental math, who completes developmental, and who completes a college course.  Like other demographic work, these studies can not prove causality.  Neither type of study is a scientific basis for measuring the effect of developmental courses; both are valid estimates to determine the presence of a problem.

Now, I need to address two things … first, why the ‘failure’ message is the default position for so many people inside and outside of the profession; second, what is the true condition of developmental mathematics. 

The failure message is most heard from two sources:  the non-profits advocating for change and a completion agenda, and the foundations funding much of our experimentation.  Neither of these sources is unbiased.  However, sheer repetition from apparently independent sources creates the impression that the failure message is valid.  I think the use of certain metaphors (like ‘bridge to nowhere’) creates an impression of certainty of conclusions, and suggests a cultural acceptance of ‘failure’.  One problem we face is that we have used similar tactics ourselves, as in ‘drill and kill’ and ‘guide on the side’; proof by metaphor …or proof by rhyming … is not scientifically valid.

The true condition of developmental mathematics is much more subtle, which brings with it opportunities and challenges.  A simple ‘failure’ message is easier to interpret and act upon (basically, throw it out!).  The fact is that developmental mathematics delivers some benefits to many students.  The problem is not a total failure of the concept but a lack of an appropriate model to implement the mission and goals.  Developmental mathematics has its roots in remedial mathematics, which was a deliberate repetition of school mathematics; this, in turn, was based on a selective admissions college or university approach.  The vast majority of developmental mathematics is currently carried out in the community college setting, with a diverse population of students; many of these students have an occupational goal … although they may eventually consider a university, their current education is employment based.  Of course, many other students have a university goal.

We have not had a model appropriate for our population of students.  We need to create a deliberate sequence of mathematical experiences to prepare students as quickly as possible for places they will have quantitative needs, whether STEM-bound or not.  Even for STEM students, our existing curriculum is not a deliberate model; the current model presumes that exposure to a topic at a simple level will enable more advanced thinking in a complex setting.  We need a model that emphasizes basic mathematical ideas from the beginning (the ‘good stuff’, as I call it), and let go of making sure that students can produce volumes of correct answers to symbolic questions with fractions and percents … or equations with fractions or radicals.  Mathematical reasoning is far more important than a bag of 100 symbolic tricks and procedures.

The true condition of our profession is that we have become confused by the combination of our own frustrations and these external failure messages.  Ours is a noble calling … if done correctly, developmental mathematics can be part of the process that enables people to be upwardly mobile; instead of the younger generation having a lower standard of living, we can part of the process that creates a better life for the next generation.  Developmental mathematics can also be part of the process of major adjustment for adults who find that their occupation is no longer available.

The true condition of developmental mathematics is an opportunity for the transformative change to sound mathematics to help our students succeed in college and in society (quantitatively).  We face great opportunities; we are not a failure.  We need to look past the external messages to examine our profession with honesty and vision.  Together, we can meet this opportunity with pride and enthusiasm.

 
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