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.

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:

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 (https://www.devmathrevival.net/?page_id=116)

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New “Instant Presentations” on the New Life Model

The Instant Presentations page (https://www.devmathrevival.net/?page_id=116) now has a set of new presentations on the New Life model.

New presentations include:

1. Reform — the Big Picture
2. Reform — the New Life Model
3. The Mathematical Literacy Course overview
4. The Algebraic Literacy Course overview
5. New Life at your Institution

Instead of a redesign, or just flipping a classroom, look at ways to provide better mathematics to your students.  We can create shorter paths through math and enable students to learn sound mathematics that means something.

If you have ideas for other quick presentations, let me know!

 
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Getting to Know the New Life Model

We started our work on the New Life Model in 2009.  When I started this blog in 2011, I created some short presentations about different aspects.  Since then, the New Life Model has become used in more colleges … which means that we understand more about the issues.

I am started an updated series of presentations on the “Instant Presentations” page https://www.devmathrevival.net/?page_id=116

Two of the new videos in this series are now available — “Reform, the Big Picture” and “Reform: The New Life Model”.  I hope you and your colleagues find these videos helpful!  If you have suggestions for other presentations, pass them along.  The planned videos at this point include: the Math Lit course, the Algebraic Lit course, and Implementing New Life at your institution.

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Building a Learning Attitude — Hope

Perhaps you have a person like this in your classes.  A person who believes that struggle means that the other person has not done their job.  A person who gets discouraged because problems come up that are not her fault.  A person who believes that 20 years of completion means that they have shown sufficient achievement.

In fact, I am sure that most of us have a member of our math faculty like this (or several).

Yes, the description is for a faculty member.  She has asked me what I do when students will not contribute in class, and what to do with behavior problems; she has shared frustrations with students who don’t do their part.  In some ways, the New Life project exists for faculty like this, as a way to give hope and engage all faculty.

“If you want to make peace with your enemy, you have to work with your enemy. Then he becomes your partner.” (Nelson Mandela)

We want progress.  Progress is not measured by the 50 exceptional programs; progress is measured by what all of our students experience.  We face a challenge much less complex than President Mandela did when apartheid was ended in South Africa; to make progress, he knew that all people must be included in the work.  To exclude a group just meant to exchange places with them, and prevent progress.

“We Have Met The Enemy and He Is Us” (Walt Kelly, “Pogo”)

The only way to make progress is together.  We need the discouraged and cynical faculty, just as they need us. They need our hope for a better future, and we need them so progress is real, and has a chance of lasting.  To exclude people from our work is to reduce our probability of success.  Inclusion means dialogue, even when uncomfortable.  We share values about mathematics, though we may have different perspectives based on our experiences.

Do not give up on other faculty.  Keep up a dialogue.

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