Ban Intermediate Algebra!?

Sometimes, there is a fine line between ‘reasonable interpretation of reality’ and ‘bad idea’.  Should we ban intermediate algebra in colleges?  Would it hurt anybody … help anybody … would anybody notice?

My current ‘reality’ includes teaching an intermediate algebra course that is quite traditional, except for us using an ebook (to save students money, and provide equal access).  This course has the usual combination of topics — functions, absolute value statements, polynomials, factoring (lots), rational expressions, rational equations, rational exponents, radicals, radical equations, quadratic equation methods, and quadratic functions (along with a variety of word problems, which are mostly puzzles).

In case you did not know, I have been teaching for quite a while (something like 39 years).  Originally, intermediate algebra was taken primarily by those who needed pre-calculus … and most of them needed calculus.  For a variety of reasons, the vast majority of my current students are not in this category; for them, intermediate algebra is part of their general education process.  [At my college, intermediate algebra is the MOST commonly used course to meet a gen ed requirement.]

Outside of the small minority of my students who actually need calculus (a group which should be larger), most students are not well served by an intermediate algebra course.  The traditional course does little to enhance their mathematical literacy or reasoning, with its focus on symbolic procedures; the traditional course does not contribute to the GENERAL education of students, since it is fairly specialized (polynomial arithmetic and related symbolic procedures).

For many of my students, intermediate algebra is where their dreams and aspirations wither and die under the negative influence of a curriculum which does not serve their needs.  Even for those who need pre-calculus, the traditional intermediate algebra course does not signficantly increase their mathematical proficiencies.  [The procedures learned are soon forgotten, and not much else was learned in the first place.]

Let’s ban intermediate algebra.  In its place, we should offer a version of the New Life “Transitions” course.  The Transitions course learning outcomes focus on providing mathematical preparation as part of a general education, especially if the student will take science courses (biology, chemistry, etc).

If you do not know about the Transitions course, take a look at the learning outcomes listed at https://dm-live.wikispaces.com/TransitionsCourse.   This course focuses on concepts and connections between concepts, so that students gain more than procedures.  The particular outcomes were chosen to be part of the general education of students needing science courses; some ‘STEM enabling’ outcomes are listed as an option for a course preparing them for pre-calculus.  The “Instant Presentations” page here has a presentation on the Transitions course; see https://www.devmathrevival.net/?page_id=116 

Of the two New Life courses, the first course “Mathematical Literacy for College Students” (MLCS) has generated more interest as an alternative to a traditional beginning algebra course.  I find this interesting, since we could argue that intermediate algebra is a worse match to student needs.  Curiously, the Transitions course is somewhat similar to some materials that are already on the market … which means that implementing Transitions avoids some of the challenges faced by those working on MLCS.

Some of you have been thinking “hey, we are required to use intermediate algebra as the prerequisite for all college-credit math courses”.  Well, I know … our profession needs to work on that problem.  Presently, the Developmental Mathematics Committee (DMC) in AMATYC is working on a position statement related to this problem; see http://groups.google.com/group/amatyc-dmc 

Obviously, I do not really expect us to ‘ban intermediate algebra’ (though I can dream!!).  Perhaps some of us can help our students by using the Transitions course as an alternative for those students who to not need pre-calculus.

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Evidence Based Decisions

How do we prevent ‘evidence based decisions’ from becoming ‘evidence constrained decisions’?

First, let’s get clear on what ‘evidence based decision making’ is about.  Primarily, the idea is to apply evidence from the scientific method to decision making.  This is the definition given at http://en.wikipedia.org/wiki/Evidence-based_medicine ; much of the current push to use this method in education comes from improvements in medicine due to using scientific evidence as a basic methodology.

The idea is to base decisions on the evidence, when appropriate evidence is available.  Remember that we are talking about scientific evidence — which is a stronger standard than ‘data’.  The scientific evidence provides a connection between a practice or treatment with the outcomes (usually stated as a probability or odds).  Sounds good, doesn’t it?

Well, in education, there are difficulties in getting scientific evidence.  We have tons of data, which are raw measurements organized in some manner; however, this has little to do with scientific evidence.  Most commonly, we have either before and after data relative to some change; sometimes, we have data from two groups under different treatments … data on the outcomes, without data on other variables that we suspect have an impact on the outcomes. 

Scientific evidence does not come from one set of data.  After one set of data suggests, scientifically, that we have reason to believe that this treatment results in a change in the outcomes, this hypotheses gets tested by replication — done by different practitioners.  The idea of scientific evidence is that we achieve something close to an empirical proof that we have a cause and effect relationship — not just a one-time correlation.

I can not resist bringing up one of my favorite oxymorons — “data based decision making”.  Data is simply organized measurements; no decisions can be made based on data, because data is not evidence of anything.  I use brand X gasoline one week, and the next week I use brand Y — and get 10% better mileage … which means nothing; this data just means that I get slightly different outcomes, nothing else.  I normally find the phrase ‘data based decisions’ to be used as a cover for a hidden agenda.

Back to evidence based decisions … as mathematicians, we are all scientists; we understand the power of research — and it’s limitations.  The presence of evidence (in the scientific sense) suggests better courses of action (decisions) to the extent that the probable outcomes are ‘likely’.  The presence of evidence does not determine the best decision … wise people still need to evaluate the current situation and apply their understanding of the evidence.

What do we do when there is no scientific evidence relevent to our decision?  Are we constrained by the evidence available?  Even in medicine, with its superior collection of evidence, decisions are not constrained by evidence.  We should be guided by the evidence we have, and use our wisdom combined with our understanding of the outcomes desired to determine the best available decision.

Relative to mathematics education in colleges, I would present these observations:  We have large bodies of evidence about learning which can (and are) being applied to our courses.  We often mistake data for being evidence, and mistake reporting data for research, and this has led to some dramatic failures (and some less dramatic).   When we do remember the distinction between data and research, we tend to skip the step of ‘replication’ before announcing a conclusion; this has led to cynical colleagues and a skeptical public.

If we do not understand what the word ‘evidence’ means, who will?  Certainly not external forces such as politicians.  We need to be much better at articulating what we are basing a decision on, and clearer at describing results.  We need to focus on our shared values, and use them to describe the desired outcomes.  We need to focus on our wisdom, to provide guidance in the absence of evidence.

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Modules in Developmental Mathematics

Are modules a good thing in developmental mathematics?  They are certainly popular these days, with quite a few colleges … and some entire state systems … putting their entire developmental math program in modules.

The word ‘modules’ overlaps in meaning with some more generic labels such as ‘chapters’ or ‘units’.  However, the word module implies a planned independence — you install only the modules needed, or only repair the one that is ‘broken’.  In the case of developmental mathematics, one of the benefits attributed to modules is that students only have to study what they ‘need’ (based on some assessment).

The design implies that there are discrete skills necessary, that developmental mathematics is all about students demonstrating accurate procedures in the modules prescribed for them, and that we should not expect any other ‘value added’ for our students.  Briefly, here are arguments against each of these implications.

A number of studies have looked at the actual mathematics used in various occupations.  This list often includes arithmetic skills (especially with percents), some formula work, and a few other items.  Rather than discrete skills, the occupational need is for the problem solving and ‘STEM-like skill set’ (see http://www.insidehighered.com/news/2011/10/20/study-analyzes-science-work-force-through-different-lens for example).  Our 2010 developmental mathematics courses emphasize discrete skills because of history, not because that is what students need.  By constructing a series of independent modules, we are creating a wider gap between our curriculum and the needs of our students. 

Much of the technology behind the current module ‘frenzy’ is heavily procedural, with the main thing being correct answers.  There are two significant shortcomings of this approach.  First, getting a correct answer has only an indirect connection to knowing something; we have all seen students get a ‘correct’ answer with either multiple errors or no comprehension of what they are doing.  Second, procedural details are notorious for being forgotten — the old ‘use it or lose it’ syndrome; if we focus on procedures, we are essentially saying that developmental mathematics has no lasting benefit for the students.  If this is true, we would be more professional to take the student’s money and give them their grade without them going through the game of producing the correct answers for us.

The last implication, that we should not expect any other value-added for our students, goes beyond the prior concern.  Are students in developmental mathematics classes so limited intellectually that we should give up on their capacity to learn mathematics in the academic sense?  By setting the standard so low, we are not only limiting our students — we are actively reinforcing every bad attitude about mathematics.  Many of these attitudes are based on perceptions of mathematics as dealing only with specific procedures and correct answers, coupled with a belief that normal people are not capable of understanding mathematics. 

You may be wondering if I somehow believe that the existing developmental mathematics curriculum is better than what I describe; in general, it is not much better.  There might be a better problem solving component, and a little bit of conceptual understanding.  However, we have inherited a curriculum that has been fixated on algebraic procedures (and a particular collection of procedures). 

No, it is not that modules are breaking something good.  Rather, modules are giving the illusion that we are fixing something because we can point to a change.  Change is not progress, not unless the change results in achieving what the community of professionals sees as something of basic and intrinsic value.  When I have conversations with mathematicians, I do not hear people describe the outcomes of modules as being of value … we value concepts and connections, thinking and problem solving.

Modules are an easy ‘solution’ to the wrong problem, and I suggest to you that modules create additional problems.  Let us not get distracted by the technological appeal of modules; instead, let us look critically at the mathematics we deliver … and how we can actually help our students.

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Roots and the Mythology of Mathematics

We teach ‘mathematics’.  We believe ‘mathematics’ to be valuable.  What is this ‘mathematics’?

Our students firmly believe that ‘mathematics’ is a difficult mystery hidden from normal people.  Why do they have to ‘learn’ this ‘mathematics’?  When will they use it?  Does anybody really care (outside of a math class)?

Whether we are in a developmental classroom, or pre-calculus, or some other ‘math class’, we do not generally deliver an honest presentation of our subject.  How can I possibly make that statement?  Well, I’ve been thinking for years … and reading other peoples’ informed judgments … and conclude that the core property of mathematics is “the science of quantitative relationships”.  Mathematics is a science, not an abstract play ground; neither is mathematics a complex set of occasionally connected manipulations on various symbols and statements.

Mathematics enjoys a privileged position in American society, a position based more on the mythology of of mathematics than any reality.  Decision makers think ‘more mathematics’ is a good thing, and they can find statistical data that supports that position.  Our skeptics (and there are a few) can present better statistical studies that show that it is actually not the mathematics that makes the difference — there is a common underlying cause.

One of my students said this week (as she asked another question) “How can you stand to teach something that everybody hates so much?”  This was a spontaneous comment, and shows the type of mythology that I speak of.  If ‘mathematics’ was valuable, as we teach it, students would (to varying degrees) understand the benefits and gain motivation for working hard.

Instead, ‘mathematics’ is normally experienced as that complex set of occasionally connected manipulations on various symbols and statements.  We have students ‘simplify variable expressions’, but we have no clue that they realize we are talking about representations of quantities in their lives.  They ‘solve equations’, with no clue of how equations state conditions that people, objects, and properties must meet in specific ways.  We make students ‘graph functions’, without either making sure that they know how functions express the central relationships of quantities important to them or letting them in to the powerful tools of ‘rate of change’.

The roots of mathematics are in the rich intersection of practicality and science.  We have lost our roots, and cover neither side of this intersection.  We survive only because of the mythology surrounding ‘mathematics’; this mythology is not correct, and is offensive to a mathematician (in my view).  We teach mythology instead of mathematics.

Get up!  Look back at our roots as a practical science.  Do all you can to dispell the myths held by people concerning mathematics.  A central part of this work is to build a curricular structure that emphasizes actual mathematics.  You can begin this process by looking at the New Life model for developmental mathematics, as one model based on mathematics not mythology.

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