Reducing Costs of Developmental Math

The ‘cost’ of developmental mathematics is one of the major issues being faced by states and institutions.  Although this is commonly stated as a financial cost, an equally important cost is present — the cost to our students (time, credits).  There is also a risk involved, given that most studies of developmental education seem to report that students placed into developmental courses have a lower chance of completing programs.

Is there a solution?  Is there a simple solution?

In a recent post (https://www.devmathrevival.net/?p=756) I talked about what a reasonable prerequisite to beginning algebra could be.  That post hinted at some solutions which could be implemented to reduce costs.

Here is a simpler solution that can be done right away, and may not have the kinds of problems you might predict:  Place students into beginning algebra, even if their placement test suggests something before that.

I admit that this is a strange suggestion.  However, think about how ‘strange’ our current system can be … at many institutions, students who start in pre-algebra have about a 20% ‘chance’ of completing their college level math requirement.  Are we helping that 20% so much that this process is worth the risk to the other 80%?

Before you jump up and down, screaming “THIS IS NOT GOING TO WORK” … look at some potential numbers.  If we assume that 70% of the students placed into pre-algebra pass that course, and that 50% of those who proceed to beginning algebra pass that second course, we have a net 35% who complete beginning algebra in the second semester.  This 35% assumes that ALL students will pass pre-algebra continue to beginning algebra; this is not reasonable.  Based on estimates from my data work at my college, from 70% to 80% actually go on to the second course.  Applying the highest rate (80%) to the 35% value gives us a realistic net of 28% … about 28% of students who start in pre-algebra complete the beginning algebra course the second semester.

What would we expect to happen to students who go directly to the beginning algebra course?  Would they be half as likely to pass that course, compared to having taken pre-algebra?  This “half” seems like a reasonable estimate (and may be too low).  Half of 50% … is 25%.  Since 25% is generally not statistically different from 28%, there is a good chance that placing all students in to beginning algebra would not create any additional risk to the student — and would save a semester of credits.

There is actually evidence that suggests this 25% ‘direct’ rate is too low.  A study (http://ccrc.tc.columbia.edu/Publication.asp?UID=1030) shows the predicted pass rates for students above and below the cutoff on a placement test (Accuplacer in this case); the predicted values for rates of C or better are above 30% for all placement test scores.  If this is accurate, then it would actually help students to never place them in to pre-algebra.

Based on years of talking with students struggling in beginning algebra, there is another reason why ‘skipping’ pre-algebra might help quite a few students: of the students who pass pre-algebra, quite a few of them were not challenged by the material … in fact, many do not study … and still pass.  This “no study, and pass” experience is exactly the opposite of what most students need; students need to know that working hard and continuing are critical for academic success.  As long as a pre-algebra course is primarily procedural, with a focus on correct answers, it will not contribute to habits that help students in later courses.

Think of that … a simple solution that saves a lot (money & credits for students, costs and resources for colleges), with either no risk or even some significant benefits.  Let’s agree to not place any student into pre-algebra (or whatever your course is called); if their placement test suggests that they don’t have enough ‘basic skills’, we would be better off placing them into beginning algebra anyway, perhaps with a sheet of references for refreshing those skills.

 
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Pre-Algebra … an Oxymoron?

One of the issues our department is looking at is the course prior to beginning algebra … we call it Pre-Algebra.   The current incarnation of this course includes early work with variables, expressions and equations — based on the thought that this will help prepare students for algebra.  The results are not what we would hope for.

Is “Pre-Algebra” an oxymoron?  The word oxymoron (apparently) literally means “sharp dull”, and the implication of the word is that the two concepts are contradictory.  What really comes before algebra (‘pre’)??

I’ve commented previously that arithmetic, as an academic topic, is more difficult than algebra and therefore it is illogical to use arithmetic as a prerequisite to algebra.  If we look at ‘tests’ of readiness for algebra (a typical one is http://www.algebra-class.com/algebra-readiness-test.html) the content is a mixture of arithmetic of whole numbers and fractions, order of operations, ‘simple’ algebraic expressions and equations, and perhaps some basic geometry knowledge.  Our own pre-algebra course looks a lot like this. 

What does a real student take away from (gain from) this pre-algebra experience?  Here is a (somewhat cynical) summary:

• Whole numbers — “Okay, got that; thanks for making me go through this for the 20th time”
• Fractions — “Are we done with those yet?  I hope you don’t expect me to remember this; I certainly do not understand fractions.”
• Order of operations — “Look for parentheses; there is also something about multiplying before adding, I just have trouble remembering what it is.”
• Algebraic expressions — “I don’t get why x and y are different … they are both unknown; I remembered the rule for simplifying until I took the test … not a minute longer.”
• Basic equations — “I like finding x; just don’t give me a word problem.  Oh, and I prefer to not show steps for solving … is that okay?”
• Basic geometry — “Don’t give me too many formulas to memorize; I can do the geometry.  What’s the deal with area having a different unit than perimeter?”

Like many of us and our courses, our course is mostly about using procedures to get correct answers … and this shows in what students get out of the class, and painfully shows when the students take an algebra course.  Some parts of a typical beginning algebra course can be done by simple procedures; some topics are ‘procedure challenged’ (like graphing, and systems).  If the list above is an accurate summary of what a student gains from pre-algebra, then it has nothing to do with being ready for algebra … pre-algebra has become an oxymoron.

Once upon a time, pre-algebra did not exist.  We offered some arithmetic courses, and sometimes used arithmetic as a prerequisite to algebra (reinforcing the myth that all mathematics is a linear and dependent sequence of steps).  Some pre-algebra courses are really arithmetic courses with a less remedial name; some are an honest attempt at improving readiness for algebra.  Is there ANY evidence that a prerequisite to algebra exists in a form that could be the basis for a course?

Some research has been done, mostly at the middle school level.  For example:  http://www.mheresearch.com/assets/products/ea5d2f1c4608232e/CA_Algebra_Readiness_Research_Base.pdf  and http://www.rachaelwelder.com/files/vitae/Welder_Prereq_Know_Algebra.pdf  .   The research cited by such articles is often descriptive in nature, which can show (at best) some correlation.  [The latter reference even includes a 1991 presentation I made at an AMATYC conference!   That report might be helpful, in spite of that.]

Some middle school programs might do a better job of algebra readiness than colleges; we, in the college setting, generally do a procedural course with little designed to help students mature their thinking and reasoning.  I do not have a definite answer … I can not say “THIS stuff is the real prerequisite to algebra!”  However, I can tell you that having a student generate 2000 correct answers to a variety of problems has nothing to do with being ready for algebra.  We might be better off using a general test of reasoning as the prerequisite (ie, active writing skills … as opposed to reading).

In the New Life model, we do not have a pre-algebra course or level.  The first course, Mathematical Literacy for College Students (MLCS), stands mostly on its own.  The people involved in designing MLCS have identified some prerequisites to MLCS — which fall in the category of ‘basic numeracy’.  Here is that list:

• Use arithmetic operations to represent real-world operations, such as putting together, comparing, distributing equally, etc.
• use real-number arithmetic to solve stated problems.
• Use graphical representations on a number line to demonstrate fluency in
• interpreting interval notation,
• ordering numbers,
• representing operations (i.e., addition, subtraction, doubling, halving, and averaging)
• representing decimal numbers, including negative numbers.

This list was created by ‘backwards analysis’ — looking at the learning in MLCS, what specifically does a student need to know before then?  This list is quite short, and our dream is that students who need this content can be served WITHOUT taking another course — whether this is done via a boot-camp review, or ‘just in time remediation’ within MLCS.  [The list is from https://dm-live.wikispaces.com/file/view/MLCS_Numeracy_March2010.pdf]

It is possible … really possible … that we could start students directly in MLCS (or the beginning algebra) without a prerequisite; a little bit of support on prerequisite knowledge, and some scaffolding within MLCS or algebra, would be enough for almost all students.

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Creativity vs Standardized Tests

Realizing that this is a little outside of the stuff usually discussed here, I wanted to share somethat that one of my students shared with me.

You might already know of his work (I did not), this is a short presentation by Sir Ken Robinson: http://www.youtube.com/watch?v=zDZFcDGpL4U&feature=youtu.be 

One of the points he makes is that the standardized tests that are so much emphasized these days are part of the industrial model of education, which served students well in the early 20th century and probably well enough in the mid-20th century.  However, in the world of today, the industrial model misses the real needs dealing with creativity and collaboration. 

As usual, there are points where I disagree with what he says; however, it’s an enjoyable 11 minutes … complete with a good sense of humor and some nice ‘animation’ (a little bit like a “Prezi”, if you know what that is).

So … enjoy!
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Prerequisites

Prerequisites are placed on courses for various reasons, from convenience to supporting student success.  Few prerequisites are placed on courses based on validation studies … some prerequisites are used based on professional validity, while others have even less of a scientific basis.

Let’s say that you are working for the department of education in a state such as mine (Michigan), and you have been getting more concerned about the possibility that academic standards are not consistent across regions or levels of education.  You look at mathematics, and notice that college courses use a different organizing system than high schools … and you do not want students to get credit for a college math course that is really a high-school level course.  An easy, and somewhat logical, approach is to enact a rule (or law) that says any college level math course needs to have at least an intermediate algebra prerequisite.

What’s so bad about that arrangement?  If intermediate algebra is at the level of 1oth to 11th grade high school, this seems like a pretty low standard for ‘college’; when challenged, you might add that the really logical rule would be that a college level math course needs to have at least a college-algebra prerequisite … and you’d like to do this, if WE would just make up our minds about what ‘college algebra’ really is.

However, the problems with this approach are too basic to be resolved by this framework.  First, it assumes that all mathematics builds on the stuff in an intermediate algebra course; several basic areas, most notably statistics, do not have any relationship to the concepts and skills of intermediate algebra.  By requiring intermediate algebra as the minimal prerequisite, we mislead students and cause them to take unneccessary courses … both problems are non-trivial.

Second, this approach assumes that a subtle concept like ‘rigor’ can be measured by the prerequisite.  This is not one of the valid uses of prerequisites; rigor is measured by properties of the course in question (the content, concepts, assessment and practices) … which do not necessarily change just because we list “IA” (intermediate algebra) as a prerequisite.  If we want sufficient rigor in college level mathematics classes (and I hope we do), we need to measure those courses — not a prerequisite to those courses.

Third, prerequisites tend to disproportionately affect underrepresented groups.  At my institution, it is not unusual to have 30% of a pre-algebra class be minority; the courses which immediately follow intermediate algebra are often 90% majority.  Sadly, our curriculum is still not a pump … more filtering happens, so any unvalidated prerequisite can lead to wasteful reductions in minority completion.

I’m pleased that my own institution has 5 college-credit math classes that do not have an intermediate algebra prerequisite.  Two of these 5 courses transfer to several institutions in the state.  However, students are still advised to take intermediate algebra “just in case” … they don’t really need it, but they might change their mind later.

If this topic is of interest to you, you will want to follow a position statement being worked on in the Developmental Mathematics Committee (DMC) of AMATYC.  The DMC web site is http://groups.google.com/group/amatyc-dmc?hl=en   The motivation for this position statement is to help institutions and states use appropriate prerequisites, based on validation — not prerequisites to enforce an abstract policy on ‘rigor’.

In the long term, we will replace “IA” with a more reasonable course like the New Life Transitions course — at least for the majority of students who do not need a pre-calculus type course later.  We will also replace beginning algebra with something like Mathematical Literacy for College Students, and this kind of course could really serve all students.  Until this change happens, we can work on better prerequisites relative to IA … and for all courses (including Transitions).

 
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