Developmental: Skills or Capabilities?

At a recent conference (MDEC, the Michigan affiliate of NADE), we were having a conversation with Hunter Boylan about developmental education. One of the participants commented that a major concern was that students sometimes leave developmental courses as developmental students.

What did they mean by ‘developmental students’?  I think the basic concern is that students were leaving our courses without the capabilities (not abilities) to handle college academic work.  One of my colleagues who is a ‘reading’ faculty commented that it seems like the developmental course was a collection of discrete skills which did not add up to any additional capabilities.

There is a somewhat different point of view for professionals engaged with NADE or the National Center for Developmental Education (which is directed by Hunter Boylan).  Their framework specifically includes ‘personal growth’, referring to a collection of cognitive and affective factors … which I categorize as ‘capabilities’.    [The “NADE-type” definition of developmental implies that it is not a nicer name for remedial; most of us in the mathematics community equate the two phrases.   As implemented, most developmental math programs are ‘remedial’; I wish they were not.]

In reading, for example, parsing a phrase … vocabulary … decoding … these are groups of skills; however, without additional capabilities, students remain developmental in their functioning — resulting in a higher risk of failure in college courses.  That is, basic literacy skills are not sufficient in a good developmental reading program.

How does a typical developmental math program compare?  Sadly, I think we are the epitome of skill courses that do not impact the capabilities of our students.  A beginning algebra course usually has 8 to 10 chapters of material, with a preponderance of … parsing phrases … vocabulary … procedures; our ‘applications’ are mostly stylized puzzle problems which avoid the need to think deeply about relationships.  In fact, we sometimes take pride in providing rules or tools to cope with word problems so students do not have to analyze them. 

A basic reason behind the New Life project is this:  serving up skills with symbols does not change the capabilities of our students.  Dealing with basic concepts, connections, transfer, analysis … this process changes the capabilities of our students.  It is our belief that good preparation for college work is based on an emphasis on deeper academic work in ‘developmental’ courses.

As you look at the learning outcomes for New Life (or the New Mathways), keep in mind that the model is making a serious attempt to build student capabilities.  Since there is not a linear sequence of basic skills, you will have to work harder to understand what the curriculum is trying to accomplish for our students.

Any course — ‘developmental’ or not — that only seeks to add skills to a student, without a larger focus on capabilities, is a missed opportunity.  When that course is used in a gate-keeper fashion (like mathematics is), we need to move towards a design that truly helps our students.

 
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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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Education as Transformation

Much is made these days of ‘value-added’, including the use of student ‘gains’ on standardized tests in the evaluation of teachers.  In colleges, we have defined courses in terms of student learning outcomes … which might reflect a comparable view of higher education (similar to K-12 and emphasis on ‘skills’).

“It must be remembered that the purpose of education is not to fill the minds of students with facts…it is to teach them to think.”  [Robert M. Hutchins]

What is the primary mission of colleges?  We all want our students to get better jobs, and would also like them to have a better quality of life.  Can these goals be achieved by the accumulation of discrete skills and learning outcomes?

Education is what remains after one has forgotten what one has learned in school. [Albert Einstein]  

Community colleges tend to serve the less-empowered segments of society.  People often cite mathematics as a key enabler of upward mobility, with some demographic studies to support this position.  These correlational studies produce a false impression of the processes involved.   The motto is not ‘algebra for all’ … the motto is ‘building capacity to learn and function’.

Education… has produced a vast population able to read but unable to distinguish what is worth reading. [G.M. Trevelyan]

Education should be a transformative experience.  Independent thinking, reasoning with a variety of methodologies (including quantitative), and clear communication should be evidence of this transformation.  In a community college, we can not strive for the same level of transformation as a university or liberal arts college education; however, we stand in the critical first steps for students along this path.

Education is the ability to listen to almost anything without losing your temper or your self-confidence. [Robert Frost]

In developmental mathematics, we have too often been content to provide little snippets of essentially useless knowledge — procedures to deal with a variety of calculations.  Even though it is not easy, and there is always a discomfort involved, our students are capable of much more.  Without reasoning and clear communication, these procedures will not benefit students (beyond a data bit that says they ‘passed math’).

Education is not filling a pail but the lighting of a fire. [William Butler Yeats]

As we work together to build a better model for developmental mathematics, we need to appreciate our place in the education of our students.  A good mathematics course produces a qualitative change in students. We can measure some aspects of this process by examining the reasoning and communication processes that students use.  However, there is no sure-fire and objective measure that says  a student has made progress.  We will develop better tools for this — including some focused on quantitative literacy and reasoning.  The challenges of measurement should prevent us from keeping our proper focus; we need to work to make the important measurable.

Education is the key to unlock the golden door of freedom. [George Washington Carver]

The pre-algebra/introductory algebra/intermediate algebra model of developmental mathematics needs to be re-made into a valid curriculum.  We can include mathematics that is practical, and that is an improvement — however, it  is not sufficient.  A central goal of developmental mathematics needs to be the improvement of quantitative reasoning and communication … a contribution which will enable our students to be educated, free people in a world facing diverse challenges.

 
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Student Success — New Life at Grayson (TX)

Some of the best work in the profession is being done at smaller colleges.  Grayson County College (Denison, TX) exemplifies this in their good work in developmental mathematics.

This year, Grayson is running pilot sections of both New Life courses — MLCS and Transitions — as part of their plan to completely replace the traditional developmental math courses next year.  Like most of us, the Grayson math department is primarily adjunct faculty; three of the full-time faculty — Stanley Henderson, Shawn Eagleton, and Sherre Mercer — have  been willing to share some of their ideas and tools with us.

Here are some comments from Sherre Mercer:

We are using a “recipe for success” in our new courses.  The document was developed by Stanley Henderson, one of the professors in the department, and is based partially based on his first day of class activities over the years. Students are asked to grade themselves on their recipe for success and encouraged throughout the semester to make improvements in their study/life habits with regard to the four areas on the recipe.

The recipe for success is this document RECIPE FOR SUCCESS SPRING 2012 Grayson County College

Also, Sherre goes on to say:

The students are required to write verbal explanations frequently.  As part of the focus on conceptual understanding, my class was required to complete writing assignments before and after each exam.  These documents were developed by Terra Diehl, one of the presenters at NADE last year.   The students are asked to complete the pretest page before the exam.  They complete the post-test page after the graded exam is returned. 

 The math department at Grayson is doing a very good job, and showing generosity in sharing both the ideas they use and some of the tools that have helped students succeed.  Thanks, Grayson County College!!

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