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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Everything I Needed to Know About Math … Learned in 8th Grade?

Various organizations have been issuing reports critical of developmental education, and developmental mathematics in particular.  Those ‘studies’ tend to be repeated and quoted and cited … until everybody assumes that the conclusions are true.

Well, a similar thing happens with the content.  We had an ‘algebra II for everybody’ flurry, and we are starting to see the ‘algebra II for a lot less’ movement.  One recent report is being used to say that the math that students need in life is generally taught in 8th grade — fractions, rates, proportions, and simple equations http://chronicle.com/article/High-Schools-Set-Up/139105/ ).  Of course, if we read the original source for this article (see http://www.ncee.org/wp-content/uploads/2013/05/NCEE_MathReport_May20131.pdf ) the conclusions are much more subtle; the source actually says that students need conceptually understanding in general, and list ‘functions’ as a needed topic — and mentions complex measurement ideas and geometric visualizations.

In many ways, the actual source (at the National Center on Education and the Economy) is very consistent with what we found in the New Life project.  The ingredients of Math Lit (MLCS) are based on a very similar list of quantitative needs in occupations and client disciplines at the basic level.  I encourage you to read their math report (link given above at NCEE).

The worry, however, is that people will remember the articles talking about the study; that people will see that story line repeated enough that we begin to believe that it has to be true.

We need to keep our voices in the public conversation so that policy makers hear a more informed point of view, one based on professional expertise and information about what students really need in college for mathematics.

No, Virginia, students do not just need 8th grade math in life.  Many college programs will involve courses which depend on other quantitative abilities, and many occupations involve more than just 8th grade math.

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Math Lit/Applications for Living: Seeing the Power

Both the Math Literacy course and the Applications for Living course deal with two common models — linear and exponential.  I’m finding it interesting to watch how different and similar the experience is.

For both students, they have not seen exponential models in their college (developmental) courses; none of the current Applications for Living students had the Math Lit course previously.  (That will change as some Math Lit students take Applications for Living.)  In both cases, we explore models from numeric and symbolic forms; the Applications for Living course includes more variety, and also requires active graphing of exponential models.

In both courses, students have a difficult time leaving the linear world of adding and subtracting.  There is confusion about the role of slope in an adding world; during the exploring process, we take the time to show repeated adding as a multiplying, and identify the number as the slope.  When we work in exponential situations, the linear view seems to dominate.  During the exploring process, we show repeated multiplying as an exponent and learn about the role of the multiplier.  The performance learning outcomes are not what we would want; there are some differences between numeric and symbolic problems.

For example, the final exam in the Math Lit course had a doubling problem for which students needed to write the model.  Something like:

At the start, 25 people knew about the latest i-product; this number is going to double every day.  Write the exponential model for N (the number  who know) based on t (days since the start).

Another problem for the Math Lit final was a growth pattern from a numeric standpoint:

The cost of a machine is $400, and this is expected to grow by 10% per year.  Complete the following table of values.  [The table shows years 1 to 5, where the value for each year needs to be completed.]

In Applications for Living, the corresponding problems were this symbolic one:

The value of an investment is expected to grow by 6% per year.  Write the exponential model for the value in terms of the number of years.

And, this numeric one:

At 3pm, 20 mg of a drug were in the body.  At 4pm, 15 mg were in the body.  Complete the following table of values.  [The table shows hours 1 to 5, where the amount of drug needs to be completed.]

Almost half of the Applications for Living students treated the last problem as a linear one: They showed values of 10, 5, 0 and 0 (sometimes with a puzzled comment about having zero as the amount).  In class, we had done drugs in both half-life and percent decrease models; we had calculated the multiplier as well.  They did a little better on the symbolic form; part of this is the fact that this course also does work with finance formula, and one of those formulae is basically the answer for this problem.

The Math Lit students did well on the numeric problem; part of that success was the remediation we did earlier when most students had difficulty on all things exponential.  Few of the Math Lit students wrote a correct exponential model, which is noteworthy since the problem is a slight variation of a situation we used to introduce exponential models.  Most of the incorrect answers were variations on y =mx + b.

Clearly, this assessment feedback is indicating a need for an adjustment to the instructional cycles.

However, I also think that the results reflect a math curriculum that tends to treat topics in isolation.  How often do students need to deal with both linear and exponential models in one assessment?  Also, do we use the word “always” with students?  As in: “Compare the y-values; the difference always tells you what the slope is.”  Or, “If you can see how to get the next value in a table, you can always use this to complete a table.”  Or, “In a function, you can always get the next function value by adding or subtracting.”

During the instructional cycles in both courses, I can see the resistance to leaving the linear model.  It’s a bit like distributing, where students become fixated on one process.  I want students to see the power of understanding exponential models; students want the comfort of one model for all situations.

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