## Placement in Math … What’s the GOAL?

So, I’ve been thinking quite a bit about college placement.  People are advocating ‘multiple measures’ (often done as ‘alternative measures’); Accuplacer is announcing “The Next Generation Accuplacer” during their summer conference (https://accuplacer.collegeboard.org/professionals/national-conference/concurrent-session-highlights) .  There are people who believe that all placement tests are ‘evil’ (including some professional friends).  The placement tests are a tool used in a process … do we agree on the goal of that process?

At the highest level of analysis, there are 3 competing goals for math placement:

1. Which course best reflects the student’s current skills?
2. What is the highest course in which the student has a reasonable chance of success?
3. What skills does a student have now, or lack now?

We often make the mistake of equating these 3 goals; they are separate ‘hypotheses’, and the outcome for a given student can be quite different.

My perception is that we, in the mathematics community, have tended to fixate on the first and last goal (skills oriented placement).  To consider how different the 3 goals are, consider this student prototype:

Cassie can solve basic equations (linear, quadratic by factoring), with sound understanding of polynomials basics.  Cassie knows some function basics, including graphing; however, using slope in a flexible way is lacking.  The main weakness Cassie has is in fractions both arithmetic and algebraic — Cassie has some memorized rules that help in some cases, but produce wrong answers in general.

Goal #1 tends to find a ‘mean skill level’; since Cassie lacks some arithmetic, this mean is likely to be beginning algebra.  Goal #2 tends to find a ‘floor’ level of functioning, a minimum for success in the maximum course; depending on her program, Cassie would end up in Algebraic Literacy/Intermediate Algebra or a College Level course.  Goal #3 tends to push students down to the lowest level of gaps in skills [goal #3 is used in modular and emporium designs]; Cassie would end up in an arithmetic or pre-algebra class.

Many of us are drawn to the two skill-based goals, and that is a problem.  The first goal with it’s focus on ‘mean skill level’ tends to under-place students (as seen by outsiders); more importantly, students are always starting in a course with less challenge and more review than needed.  The third goal creates the longest sequence of learning for every student.  Both of the skill-based goals operate from the assumption that reviewing skills produces the intended results; in 40+ years in the business, I have only seen modest effect sizes for review courses in general.

The second goal has more potential to both serve our students well and provide faculty with reasonable teaching challenges.  This goal operates from a viewpoint that skills are general indicators of capabilities, and it is these capabilities that allow a student to pass math courses.  The prototype student, Cassie, is based on a student I had in beginning algebra … she is currently taking a test in my intermediate algebra class; Cassie has been very pleasant about the two courses, but I think she’s wasted a semester of math.  [My department tries to do goal #2, but our operational system produces #1 in most cases.]

Of course, there are many basic connections between the placement system goals and the design of our curriculum.  If we already had Algebraic Literacy, my student (Cassie) could have succeeded in that course in her first semester.  Our traditional curriculum (basic math, pre-algebra, beginning algebra, etc) tends to be consistent with goal #3 (the maximum number of developmental math courses).

One of the steps we often take when setting cutoffs for placement is that we ask other colleges what they use … or we see a reference online or in manuals.  It’s very rare for the goal of placement to be shared when we get this information; for years, all assumed that we shared one goal (one of the skill-based goals).

We live in a period of change in both our math courses and in the placement system.  Our developmental math courses are trending towards Math Literacy & Algebraic Literacy; the placement tests are undergoing similar shifts.  [The new Accuplacer tests will be less skill-based, more diverse, and include some reasoning.]  These changes provide us with an opportunity to re-focus our goal in placing students, and I am hoping we can develop a professional consensus on a goal of placement:

What is the highest course in which the student has a reasonable chance of success?

I am hoping that the changing tests (Accuplacer Next-Gen) will allow my institution to produce results consistent with this goal.  My draft of a Mathematical Literacy Placement Assessment is written with this goal in mind (http://www.devmathrevival.net/?p=2480).

The big question, though, is what our profession does!