At the Altar of Alignment

The answer to all questions is “42” (see Hitchhikers Guide).  The solution to all problems is “alignment”.  Academic leaders, government officers, and policy makers are using the word “alignment” in attempts to address many perceived failures in academia.  Alignment is not even a necessary property, and is certainly not sufficient, for an academic system to be successful.

At the micro-level, people tell us to align course outcomes.  If course A is a prerequisite to course B, then the outcomes should be “aligned”. In cases where our goals are strictly operational (just the doing, not the understanding nor the reasoning), we can align courses.  I’d suggest that this is a very weak methodology for a mathematics curriculum, since aligning outcomes directs our attention to the fine levels of granularity as opposed to the basic story line of a course.  A stronger design is to focus on mathematical abilities being developed over time … both within a course as well as across courses.  Alignment is often counter-productive in mathematics.

At the mid-level, we are told to align the mathematics required with the needs of the student’s program.  In other words, if the primary quantitative need of an occupation is the consumption of statistics, then the mathematics required for the program should be a statistics course.  As attractive as this alignment might be … the practice is based on two unfounded assumptions — (1) that a student KNOWS what they plan to become when they begin college, and (2) that this plan is relatively stable over time for each student.  Unless we plan to return society to pre-global, pre-fluid periods for occupations, alignment is a dis-service to many students.   Instead of alignment, we’d be better served by offering a good mixture of valuable mathematics, not specialized.

At the macro-level, we try to align K-12 mathematics with college mathematics (or, vice versa).  The unfounded presumption here is that K-12 mathematics exists primarily to prepare students for college mathematics. And, there is an assumption that this ‘alignment’ (whatever it means in this context) will make a significant difference.  Like aligning course outcomes, aligning levels of education tends to push our attention down to small details —  in other words, alignment is based on focusing on insignificant details while ignoring larger concerns.  For this level alignment, think about what would be more powerful:

• Students have mastered skills A1 to A5, B1 to B7, C1 to C4, and D1 to D8 which logically can be followed by A6 to A9, B8 to B12, C5 to C10, and D9 to D11.OR
• Students develop learning and academic skills (including mathematics) to develop reasonable proficiency as well as an ability to learn in a variety of situations using different tools.

We spend time at the altar of alignment, working on ‘solutions’ which have little chance of helping students.  Education is much more than the sum of a finite series of detailed objectives … education is much more than learning just the mathematics needed for an expected occupation … education is more than a series of steps which present a surface logic but lack power in a person’s life.

Our time would be better spent in seeking a vision and some wisdom on educating students, educating them for capacities and success.  The checklist success of alignment is worthless compared to the benefits of education done well.

 Join Dev Math Revival on Facebook:

Talking About Equity as an Avoidance

My department has begun a process which will (hopefully) lead to meaningful and sustained improvements in our equity picture.  Current, and historical, data makes it clear that our program is not serving all groups adequately.  Black students (aka “african american”) almost always have a pass rate significantly lower than other groups, after accounting for their level of preparation.

I am very pleased with my colleagues and their willingness to spend time working on a problem which involves some discomfort … it’s not always easy to talk about race and equity.  Much of our initial discussion focused on our point of view and problems that make sense to us … phrases like “student skills”, “role models”, and “tutoring” we very common.  “Compassion” and “empathy” were also used.  These are all good thoughts, but tend to focus on the surface and symptoms.

However, I am sure that our conversation will need to progress to deeper levels of understanding.  One reason to believe this is that this conversation has occurred hundreds of times in other institutions and organizations without producing an accepted basket of ‘best practices’ for eliminating the inequity as we generally would like.

One perspective that might help our profession actually make progress on this comes from Danny Martin (University of Illinois at Chicago).  Dr. Martin delivered a talk entitled “The Collective Black and Principles to Actions” (available at http://ed-osprey.gsu.edu/ojs/index.php/JUME/article/view/270/169) .  The ‘Principles to Actions” part of the title refers to the 2014 publication by NCTM of that name.  The “collective black” in the title refers to a way to understand a social structure in the United States.

A quote from near the end of that article is:

Does this document represent, symbolically and in spirit, the kind of disruptive violence to the

status quo that can move the last to first?  Can it truly help in improving the collective conditions

— not isolated examples of success — of African American, Latin@, Indigenous, and poor

students? By success, I do not mean slow growth and incremental gains.

The “disruptive violence” in this quote might bother some readers.  Remember that Dr. Martin is speaking of social institutions, not a personal philosophy of political change.

I think Dr. Martin’s point, perhaps shared by Dr. Martin Luther King as well, is that incremental change and “stuff around the edges” will not produce meaningful changes at the level necessary.  Our  problems are too well established in the existing structures, and even in the vocabulary we use to describe ‘the problem’.  For example, millions of white people have had “compassion” and “empathy” for a wide variety of students (including the group ‘black students’ my department is focused on).

Here is a point … Perhaps “white people” only support working on “equity” when this work does not involve any change in the white power relationships and social structure.  Are we willing to share power and authority to reach the lofty goals we seek?

Perhaps we will find that reaching equity in our department depends upon fundamental changes in the  local community.  The urban schools have old buildings, few resources, and other significant challenges; this district is heavily ‘minority’ (black students in particular) … because our state allows “school of choice’, where THOSE WITH RESOURCES can take their students to a ‘better’ school in the suburbs.   Can ‘separate and sort of equal’ ever allow us to achieve equity in higher education?  [The local condition amounts to sanctioned segregation of schools, especially at the high school level.]

We are likely to encounter large-size problems in our work to eliminate inequity in our courses.  We have only begun the conversation, and I’m proud that my colleagues are willing to begin this journey.  Our success will likely involve changes that would have been difficult to imagine prior to beginning the process.  So … I appreciate your “moral support”.

Is your department ready to face the challenges of doing effective work to reduce inequity?

 
Join Dev Math Revival on Facebook:

Why Does Co-Requisite Remediation “Work”?

Our academic leaders and policy makers continue to get strongly worded messages about the great results using co-requisite remediation.  Led by Complete College America (CCA), the originators of such messages suggest that this method avoids the failures of developmental mathematics.   [For example, see http://completecollege.org/spanningthedivide/#remediation-as-a-corequisite-not-a-prerequisite] Those of us in the field need to understand why intelligent people with the best of intentions continue to suggest this uni-directional ‘fix’ for a complex problem.  #CCA #CorequisiteRemediation

I want to focus on the educational component of the situation — not the political or fiscal.  In particular, I want to explore why the co-requisite remediation results have been so encouraging to these influencers.

One of the steps in my process was a nice conversation with Myra Snell.  I’ve known Myra for a while now, and she was involved with the New Life Project as well as the Carnegie Foundation’s Statway work.   What I got from this conversation is that Myra believes that there is a structural cause for the increased ‘throughput’ in the co-requisite models.  “Throughput” refers to the rate at which students complete their college math requirement.  Considerable data exists on the throughput using a traditional developmental math model (pre-algebra, beginning algebra, then intermediate algebra); these rates usually are from 7% to 15% for the larger studies.  In each of the co-requisite systems, the throughput is usually about 60%.  Since the curriculum varies across these implementations, Myra’s conclusion is that the cause is structural … the structures of co-requisite remediation.

The conclusion is logical, although it is difficult to determine if it is reasonable.  Scientific research in education is very rare, and the data used for the remediation results is very simplistic.  However, there can be no question that the target of increased throughput is an appropriate and good target.  In order for me to conclude that the structure is the cause for the increased results, I need to see patterns in the data suggesting that ‘how well’ a method is done relates to the level of results … well done methods should connect to the best results, less well done methods connect with lower results.  A condition of “all results are equal” does not seem reasonable to me.

Given that different approaches to co-requisite remediation, done to varying degrees of quality, produce similar results indicates some different conclusions to me.

• Introductory statistics might have a very small set of prerequisite skills, perhaps so small a set as to result in ‘no remediation’ being almost equal to co-requisite remediation.
• Some liberal arts math courses might have properties similar to intro statistics with respect to prerequisite skills.
• Some co-requisite remediation models involve increased time-on-task in class for the content of the college course; that increased class time might be the salient variable.
• The prerequisites for college math are likely to have been inappropriate, especially for statistics and liberal arts math/quantitative reasoning.
• Assessments used for placement are more likely to give false ‘remediation’ signals than they are false ‘college level’ signals.

Three of these points relate to prerequisite issues for the college math courses used in co-requisite remediation.  Briefly stated, I think the co-requisite results are strong indictments of how we have set prerequisites … far too often, a higher-than-necessary prerequisite has been used for inappropriate purposes (such as course transfer or state policy).  In the New Life model, we list one course prior to statistics or quantitative reasoning.  I think it is reasonable to achieve similar results with the MLCS model; if 60% of incoming students place directly in the college course … and 40% into MLCS, the predicted throughput is between 55% and 60%.  [This assumes a 70% pass rate in both courses, which is reasonable in my view.]  That throughput with a prerequisite course compares favorably to the co-requisite results.

The other point in my list (time-on-task) is a structural issue that would make sense:  If we add class time where help is available for the college math course, more students would be able to complete the course.  The states using co-requisite remediation have provided funds to support this extra class time; will they be willing to continue this investment in the long term?  That issue is not a matter of science, but of politics (both state and institution); my view of the history of our work is that extra class time is usually an unstable condition.

Overall, I think the ‘success’ seen with corequisite remediation is due to the very small sets of prerequisite skills present for the courses involved along with the benefits of additional time-on-task.   I  do not think we will see quite the same levels of results for the methods over time; a slide into the 50% to 55% throughput rate seems likely, as the systems become the new normal.

It is my view that we can achieve a stable system with comparable results (throughput) by using Math Literacy as the prerequisite course … without having to fail 40% of the students as is seen in the corequisite systems.

 
Join Dev Math Revival on Facebook:

College Algebra is Not Pre-Calculus, and Neither is Pre-calc

“Everybody knows what college algebra is!”  This was said by a math chair from a university in my state, as we worked though our state’s new transfer requirement for mathematics.  Of course, he was wrong … though he has a lot of company.  Today’s main question is this:  Is college algebra a subset of pre-calculus?

The original college algebra course developed in the 19th century at the universities of the day (Harvard, Yale, Bowdain, etc), with a focus on meeting a math requirement for their degree.  Of course, those times were very different … the Yale Catalog listed every student, and every student had the same default schedule.  College algebra was everybody’s math course as a freshman; those ‘desiring’ calculus took it as a Junior.  See http://elischolar.library.yale.edu/cgi/viewcontent.cgi?article=1069&context=yale_catalogue

That tradition carries forward to the present day, in the work of the MAA.  The MAA College Algebra guidelines remain a narrative for a general education class, not a pre-calculus course.  See http://www.maa.org/sites/default/files/pdf/CUPM/crafty/CRAFTY-Coll-Alg-Guidelines.pdf

The use of the name ‘college algebra’ for a calculus prerequisite appears to be a regional variation.  In states use ‘college algebra’ as a prerequisite for pre-calculus; other states use college algebra as the first semester of pre-calculus … or as their one-semester pre-calculus (as in “college algebra and trig”).

Our situation has become illogical and disfunctional.

When publishers market their textbooks, sometimes the key difference between college algebra and pre-calculus is this: pre-calc emphasizes a unit circle for trig functions, while college algebra uses right triangles.  Other than that, the pre-calculus book has more complicated problems, but no substantive differences.  Both courses trace their ancestry back to the 19th century mathematics course later known as ‘college algebra’.  [Search for Jeff Suzuki’s talk on college algebra.]

Neither course is really pre-calculus.

Of course, I don’t mean “students can not take these prior to calculus”; they do, though the benefits are small and accidental.  A pre-calculus course would be designed to prepare students for the work of a calculus course.  We make the fatal mistake of equating the ability to solve complicated symbolic problems with the capacity to reason with those objects.

A good preparation for calculus begins much earlier for many students.  “Developmental” mathematics is being re-formed to focus on understanding and reasoning, with a de-emphasis on artificially complex symbolic work.  A mathematical literacy course is a better preparation for calculus than the traditional algebra course.

More importantly, Algebraic Literacy is where we can begin the serious work of preparation for calculus.  Intermediate algebra is a documented failure as preparation for college mathematics; algebraic literacy is designed deliberately for these purposes.  The Algebraic Literacy course has learning outcomes backward-designed to meet the needs of calculus preparation … to be followed by a well-designed course at the next level which completes that preparation.

Here are some conditions necessary for good calculus preparation, based on the available information:

• diversity of content (algebra, geometry, trig as minimum)
• non-trivial reasoning about mathematical objects
• concrete (context) and abstract situations
• properties of functions, and relationships between types
• reasoning and visualization involving related quantities (2, 3, or 4 at a time)
• procedural expertise and flexibility

I do not intend for this list to be exhaustive.  The intent is to focus on key outcomes so we can determine when we have a real pre-calculus experience that will work for our students.  It is my belief that the great majority (>99%) of our current courses used as a calculus prerequisite are not reasonable preparations for the demands of such a course.

Some of our colleagues are beginning the work of correcting the curriculum; we need to support that work when possible.  If you’d like to explore what the new curriculum would look like, the Algebraic Literacy course provides a good starting point;  I’ll be doing a session at the AMATYC conference (Nov 21, 11:55am) in New Orleans.

We can solve this problem, together.

 Join Dev Math Revival on Facebook: