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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Math – Applications for Living VIII

In our class (Math – Applications for Living) we are investigating linear and exponential functions.  One of the assessments in the class is a quiz which covers linear functions connected to contexts.  This quiz had 3 items, which gave almost all students quite a bit of difficulty.  Every student in class has passed our introductory algebra class where we make a big deal of slope, graphing with slope, the linear function form, finding equations of lines, and a few related outcomes.  Perhaps the difficulty was due to ‘normal’ forgetting … I am more inclined to attribute the difficulty to a surface-level knowledge that inhibits transfer to new situations.

The first item was not too bad: 

The cost of renting a car is a flat $26, plus an additional 23 cents per mile that you drive.  Write the linear function for this situation.

The class day before the quiz, we had done quite a bit of work with y=mx + b in context like this.  The majority got this one right (though there were some truly strange answers).

The second item got a little tougher:

At 1pm, 2.5 inches of snow had fallen.  At 5pm, 3.5 inches had fallen.  Find the slope.

This item involved two related (connected!) ideas — the independent variable (input) ‘results’ in the change in the dependent (output) variable, and slope is the change in dependent divided by the change in the independent.  This ‘sieve of knowledge’ filtered out about 2/3 of the class — a third missed the fact that time is usually an independent variable, and another third lost the idea of slope as a division.  This is the outcome that I was most concerned by.

The last item was the ‘capstone’:

A child was 40 inches tall at age 8, and 54 inches tall at age 10.  Write a linear function to find the height based on the age.

A quick read of this problem might make one think that it is problem 2 (two issues) with a third issue piled on.  However, the problem said which variable was independent (age); the intent was to combine the ‘what is slope’ issue with knowing  how to find a y-intercept.  Essentially, nobody got this problem correct.  Some missed the independent variable stated in the problem … some could not find slope … the majority found some slope-type number but had no clue what to do with the problem from there.  If we strip the problem of context, it becomes this classic exercise:

Find the equation of the line through the points (8, 40) and (10, 54).  Write the answer in slope-intercept form.

Every student in class had survived doing at least a dozen of these problems in the prior math class; this item is pretty common on all of our tests in introductory algebra … and often is on the final exam for that course.

We spent about 10 minutes going over this one problem after the quiz. The questions from class were really good — and indicated how weak their knowledge was (I can only hope that their knowledge is getting deeper!).  Some students found a ‘slope’ and just used that (“y = 7x”); several felt compelled to use one of the given values in the equation (y=7x + 40 and y=7x +54 both were seen).  One common theme that came out was that students forgot that the ‘b’ in the function was the y-intercept; however, it was more than that … they were mystified by my statement that we could find the y-intercept from the given information.  I showed the symbolic method; not much luck with that.  I showed a graphical method … that helped a little more.  On this one item, I am guessing that we went from about 25% correct knowledge to about 60% knowledge.

Behind all of this difficulty is the manner of learning normally seen in a basic algebra class — 40 topics (sections), containing a few types of problems each, lots of repetition but few real problems (as opposed to exercises), and almost no connections between topics.  The mental map resulting from this is ‘not pretty’; an open-ended and unusual problem like on my quiz shows a number of gaps and misunderstandings.

In a separate post, I have called for “depth and breadth” (mile wide and mile deep!).  If we need to error on one of these two dimensions, let us error on the side of depth … wide exposure without depth is often worse than no exposure at all.  My students are having a difficult time unlearning what they ‘learned’ before; it is easier to extend good knowledge to a new area.

We all have these experiences — where we see the basic problems with student’s knowledge of basic mathematical objects like linear functions.   It helps to know that we share this process.  Perhaps together we can build a mathematics curriculum that does a much better job of building mathematical proficiency.

 
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