Math – Applications for Living IX

In our Math119 course, we are studying models — linear (repeated adding) and exponential (repeated multiplying).  Although some of the details we are including are not very practical, some are practical … and helpful in understanding everyday numbers like ‘inflation’.

Here is a situation we looked at:

If prices increase at a monthly rate of 1.5%, by what percentage do they increase in a year?

Much of our work in class has been on translating from a “percent change” statement to a “multiplying statement”.  Most students saw that this 1.5% increase meant that the multiplier was 1.015.  To answer this question, we just evaluated

We did have a little struggle about using the resulting value (1.1956 …); with a little nudging, we agreed that the annual increase was 19.6%.  Even though we have done quite a few finance applications, this result was a little surprising … students thought we would multiply 0.015 by 12 (18%).

While we were working on models, we also introduced using a calculator procedure to find answers to ‘difficult’ questions [meaning that we used a numeric approach to solving exponential equations].  Take a look at this problem:

Fifty mg of a drug are administered at 2pm, and 20% of the drug is eliminated each hour.  When will it reach 10 mg in the body (the minimum effective level)?

We’ve got that percent change going on; students are generally getting that — this is a multiplier of 0.80.  [This problem is much tougher when I give them drug levels for consecutive 1 hour intervals … like after 3 hours and after 4 hours.].  We set up this equation

To solve this problem, we used a graphing calculator ‘intersect’ process … placing this function on ‘y1’ and the output we needed (10) on ‘y2’.   Our solution (about 7.2 hours)  is useful in understanding the frequency for some prescriptions (3 times per day in this case).  In class, we also approach this same problem as a ‘half-life’ situation; conceptually, that is more complex … and specialized, so we do not emphasize the half-life method.  [Half-life is mostly there to help students if they take a science course which uses half-life concepts.]

We also point out that the intersect process used here is very flexible; it may be one of the most practical things they get out of the course.

 
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Redesign: The “Basic Skills” Phrase of Today

Let me  say the most important thing first:  Redesign of developmental mathematics is not what is needed; we need to implement new models of meeting the mathematical needs of college students.   Okay, so that is the primary point … here is some background.

You are wondering about this redesign stuff … what does it mean?  How does redesign help students in developmental mathematics courses?  The word itself (“redesign”) has multiple meanings, essentially captured in this definition:

Redesign:  to revise in appearance, function, or content   (from Merriam-Webster dictionary)

A redesign might be referring to just the appearance, like having a 3-color cover for a textbook instead of 2 colors.  Most faculty would be looking for a redesign which looked at function or content (or both), with little concern for appearance.

A redesign is a revision to an existing course or curriculum which results in an altered functioning or content.  I suggest to you that we do not need redesign of developmental math courses; we need something more basic than revision.  Developmental mathematics has not (previously) had a deliberate model for identifying and addressing college student needs for pre-collegiate mathematics.  No, we have not had a model to revise … we have had a history, in fact a long legacy, consisting of loosely connected skills in polynomial arithmetic in service of a mythical calculus preparation.

Beginning a redesign effort assumes (or is based on evidence that) our current system is essentially sound, that it only takes some amount of revision to be good enough.  Think of it with this parable:

In the 1970’s, car companies realized that they would need to produce vehicles with improved fuel efficiency.  Their initial responses were based on the redesign — they took an existing model car, made the body smaller and made the engine as small as possible; with a few cosmetic changes, cars like the Ford Pinto were born.  Although these ‘redesigned’ cars sold reasonably well, the car companies were essentially basing their work on the same designs.  Meanwhile, other car companies (such as Toyota) created cars based on a totally different design — designs in which the better fuel efficiency was just part of a larger vision.  Eventually, the American car companies realized that a new vision of fuel efficient cars was needed … resulting in vehicles that offer a package of benefits including fuel efficiency.

If we redesign our existing developmental mathematics courses, we are putting a GPS unit on a 1973 Ford Pinto.  Now, I’ve got nothing against Ford; it’s a good company, and they have come out with some really nice vehicles.  However, the point is that redesign of developmental mathematics is reinforcing the current vision of the curriculum; this vision is not based on a coherent analysis of student needs and curriculum process … we have historical artifacts which have been given the look & feel of a curriculum.

A redesign of the current courses may provide some temporary relief, just as the ‘small’ cars of the 1970s.  However, we must recognize this basic fact:

We do not have a coherent model of developmental mathematics.

We work hard, we help quite a few students, they work hard … it’s impressive what we have accomplished without a model for our work.  Can you imagine what we are capable of, if we have a model for our work?  Guided and inspired by a vision for a model which meets real students’ needs with solid mathematics, our courses can become places where students realize their dreams and ambitions … where mathematics provides an on-ramp for college success.

So … do NOT redesign.  Get inspired by a new model; take a look at New Life … at Pathways … at Mathways.

 
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Workshop at AMATYC 2012

At the AMATYC 2012 conference (Jacksonville, FL — November 8 to 12) I will be making a general presentation on New Life … AND a workshop on the two courses in the New Life model (Mathematical Literacy for College Students – MLCS, and Transitions).  For general conference information, see http://www.amatyc.org/Events/conferences/2012Jacksonville/index.html 

I am thrilled to be able to provide both the general session and the workshop on the courses.  We are collecting ideas for the workshop over on the wiki for New Life (see http://dm-live.wikispaces.com/AMATYC+2012).  If you are not a member there yet, just follow the directions for joining the wiki; it is fairly easy to join, and membership is open to anybody with an interest.

Tentatively, the general New Life session is scheduled for November 8 (Thursday) from 9:00 to 9:50; the Workshop on New Life courses (MLCS and Transitions) is scheduled for November 9 (Friday) from 1:45 to 3:45.  Both sessions are being held in a larger room — feel free to pass along this information!

Hope to see you there.

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Ignore Common Core?

Can college math faculty ignore the Common Core?  Specifically, can those of us working in developmental mathematics ignore the Common Core?

If you need to read more about the Common Core Math Standards, take a look here http://www.corestandards.org/the-standards/mathematics. The Standards are listed for each grade K to 8, and then high school by area of mathematics.

As you might know, a primary motivation for the Common Core was that of alignment … getting K-12 outcomes to align with expectations, especially for college readiness.  This alignment is connected to the standardized tests used for ‘No Child Left Behind’ (NCLB) as well as some teacher evaluations.  [A current theme in teacher evaluations is the use of ‘value added models’ (VAM), which is a statistical methodology to estimate the impact of individual teachers; I may address VAM in a future post.]

A logical approach might be to think that … if a student places in to developmental mathematics … there is no reason that we need to be especially aware of the Common Core.  If this placement is accurate, we might conclude that the Common Core ‘did not work’ for whatever reason, so our work is independent.

Look at the situation in a different ‘frame’:  Because the Common Core is closely tied to standardized testing and NCLB, the mathematics assessed is often discrete skills with a focus on procedures and simple applications.  This emphasis in K-12 will, therefore, tend to produce students in college — whether ‘developmental’ or not — who have a less complex package of mathematical proficiency.   

I have been suspecting something like this happening in the last few years (even before Common Core, though the Common Core will expand the impact) … students obtain about the same average scores on placement test even though their functioning, mathematically, is more limited.  Solving a linear inequality might go okay for them, and then difficulty emerges when there is a discussion about how to represent the solutions in a different way.  Finding slope from two ordered pairs might be okay, and then confusion appears when slope needs to be interpreted in words or a context.

Recently, I did a post on “Lockhart’s Lament”; in that essay, an observation is that a sure way to ruin a subject is to require all students to ‘take it’.  With the Common Core, we have a movement to make all students take the same subject for almost all of their K-12 experience.  Since this ‘subject’ is almost always tied to standardized tests and sometimes to teacher evaluations, the forces operate on the subject to reduce all topics to operational steps.  (I’m reminded of the “paint by numbers” analogy in Lockhart’s Lament.)

Policy makers are often looking for simple solutions, which makes the Common Core look very attractive as well as standardized tests.  If only we could present ‘understanding and reasoning’ as simple solutions for the mathematical needs of K-12 students.  Are not those the central enablers of success for students  in our college courses?

We ignore the Common Core at our own peril.  Some college faculty actively support the use of the Common Core mathematics standards, and there is a real danger that this wish will be granted.  There is no single mathematical standard in the Common Core that I object to; the tragedy is that the summation (or integration in the mathematical sense, if you will) of the Common Core is a worsening of the mathematics problem in colleges … starting with developmental, but including all college mathematics in the first two years.
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