## Exponential Attrition in Mathematics

One of the motivations behind current reform efforts — especially in developmental mathematics — is the negative impact of long sequences of courses, regardless of individual pass rates within the sequence.  This negative impact is often summarized by the concept of exponential attrition, and is a cousin to basic probability:

The probability for a sequence of (relatively independent) events is the product of the probabilities for each event.

The probability concepts provide a more subtle way of looking at the problem.  Let’s take the simplest possible sequence — two courses.  There are three events involved:

1. Course A
2. Transition to Course B
3. Course B

Clearly, there is an event (or multiple events) prior to Course A.  However, those factors deal with systematic factors generally outside of the mathematics curriculum.  Event 2 is a retention or continuation measure, subject to impacts from within the mathematics curriculum.  However, this transition is an event with a probability less than 1.

 Event Nature of probability 1. Course A Pass rate for course A 2. Transition to Course B Continuation rate 3. Course B Conditional probability: Given ‘course A’, what is pass rate in Course B

For a two-course sequence at my college, the approximate values for the probabilities are:  .68, .75, and .55.  The product of these probabilities is about .28; approximately 28% of students starting in course A will pass course B .  In this case, the conditional probability in course B hurts; however, even if the probability in course B is equal to the pass rate of that course, the result is only a little higher — 33% in our case.

For students placed one level lower, they have a 3-course sequence with 5 probabilities:

 Event Nature of probability 1. Course A Pass rate for course A 2. Transition to Course B Continuation rate 3. Course B Conditional probability: Given ‘course A’, what is pass rate in Course B 4. Transition to Course C Continuation rate 5. Course C Conditional probability: Given ‘courses A & B’, what is pass rate in Course C?

For a three-course sequence at my college, the approximate values for the probabilities are: .65, .80, .58, .70, and .64, which have a product of about .15 — approximately 15% of students starting in this course A will pass course C.  [The ‘course A’ in this sequence is not the same as ‘course A’ in the prior sequence.]

When our department did a 3-year study following students in a 3-course sequence, we came up with a net rate of 18% (compared to the theoretical value of 15%).  The difference was caused by some additional students who repeated and passed one or more of the 3 courses.

Clearly, the primary method to reduce this net probability — the negative impact of exponential attrition — is to eliminate events in the sequence.  Some acceleration models seek to eliminate transition events — two classes combined into one semester; in some designs, this truly does produce a unitary value for the transition event (100% move from course A to course B).  However, the majority of students probably can not manage a doubling-up like this where they have 6 or 8 (or even 10) credits of math in one semester; this combination model also creates challenges for math departments — small and large.

Another approach is to eliminate the need for a given student to take course A.  In some cases, this is done by state mandate.  More professionally valid solutions involve early testing and intervention programs like El Paso Community College (see http://achievingthedream.org/college_profile/el_paso_community_college ) or boot camps.  Some of these models eliminate both course A and the transition event; most eliminate course A and still have the transition event to course B.  Some other models are described at the California Acceleration Project (see http://cap.3csn.org/ )

The New Life model seeks to eliminate courses from the general sequence and from a given student’s sequence.  A ‘typical’ student faces a 3-course sequence such as beginning algebra, intermediate algebra and then a college-credit math class.  In the New Life model, this 3-course sequence would often be a 2-course sequence (saving 2 events in the probabilities).

For more information on the New Life model, take a look at the Instant Presentations page (http://www.devmathrevival.net/?page_id=116)

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