Redesign: Not an Either/Or Situation

NOTE: This post comes from Kathy Almy, who has worked on the New Life project, and is very involved with the profession.

There are two emerging approaches in developmental math redesign:  the emporium model and non-STEM pathways like Quantway and Statway.  These approaches are very different in their makeup so it seems that schools have to make a choice.  That doesn’t have to be the case.  Both can live together in one department and actually, that model may serve students and faculty better.

The emporium model uses online software to help students fill gaps in their understanding.  It can work for a student who needs a brush-up on skills.  It also works well for the student who plans to take college algebra or precalculus but doesn’t quite place into those courses.  For the student who is motivated and just shy of where they need to be on the college track, it can shorten the time for completing developmental coursework.  But what about the student that places into beginning algebra or below and has many more issues than just filling skill gaps? 

For these students the New Life course Mathematical Literacy for College Students (also in the Carnegie Quantway path), can be a less time-consuming model and one that serves them well.  MLCS works extremely well for the student whose college level math class will be statistics or general education math.  In one semester, students gain the mathematical maturity and college readiness they need to be successful in one or both of these college level courses.  The course integrates algebra with numeracy, functions, proportional reasoning, geometry, and statistics.  It does so using an integrated approach, touching each of these 6 topics in each unit.

How can using both approaches serve students and faculty?  The reality is that most new initiatives bring controversy and potentially resistance.  These two redesign models are no exception.  Faculty often feel very strongly, pro or con, about one or both of them.  Because of that, using one model across the board at a school may bring more challenges.  Using a variety of models helps faculty and students work and learn in ways that make sense for them.  People feel respected in terms of how they work and what they need in their future.  That’s an important facet of a successful redesign.  Because when redesign is imposed instead of invited, the effects will be short-lived and potentially less than they could have been.

Ultimately, both approaches have the same goal:  change the current one-size-fits-all mindset of developmental math to serve students better.  How we go about doing that need not be one size either.

Kathleen Almy is a math professor at Rock Valley College in Rockford, IL.  She has worked with more than 30 colleges nationwide to assist with their developmental math redesigns.  She is also implementing a pilot of the MLCS course in fall 2011.   For more information, see her blog at almydoesmath.blogspot.com. 

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An Emporium Story

Once upon a time, a community college noticed that their developmental math classes were not very effective … too few students were completing the course, and the cost of delivering the courses was higher than the results warranted.  An obvious solution was a methodology now called the emporium model; however, this was 1970 so that phrase was not available.  Students spent their time working problems, focusing on what they needed to learn — no lectures, just work.

The college was my institution (Lansing CC), and the emporium model was called the LCC Math Lab.  I began working in this Lab in 1973, when we had strong faculty leadership to make it more than isolated skills taught in modules.  The work was not easy, but we were able to provide improved instruction and results (though we did not worry as much about saving a lot of money). 

Fast forward to 2010 … the College closed the Math Lab at LCC because the results showed the method was not very effective and the cost of delivering it was too high. 

After working in this ’emporium’ methodology for 37 years, I can tell you that it does not take any outstanding wisdom to predict that the emporium model will work on a limited basis for a limited period of time.  The student results depend greatly on the institution’s support and planning, and the cost savings is grounded in administrative procedures — not the method itself.  Our program went from using 80% of the standard cost to using 190% of the standard cost, due to administrative changes.

Unless we want to return to a painful change process in a few years, we should look further than these “ISO” type redesign methodologies … the improvements are not universal, the curriculum is not up-to-date, and the cost savings are administrative (and perhaps temporary).  Using emporium-buffet-etc redesign is like installing a GPS unit on a 1973 Pinto — yes, we get better data and we feel ‘with it’, but it’s still a 1973 Pinto.  We will not save the planet by driving a Pinto, nor will we save our profession by emporium models; we need a top-to-bottom new vision of our work, whether this is the New Life vision or some other model.

ps — ‘ISO’ is ‘International Organization for Standardization’, as in “ISO 9001” a set of management criteria (see http://www.iso.org/iso/home.html)… focusing efficiency from a management point of view; I see emporium and related models as being ‘ISO applied to developmental mathematics’.

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Visual Learners and Designing Instruction

“I am a visual learner … I have to see it, and then I can understand.”

You’ve heard this, and you may have said this.  What does it mean?  Does it mean that graphical representations are best?  Are they required?

The situation is not that simple, and is misleading at times.  The issues deal with what a person actually sees in a graph, and how the brain processes information.  In a clever study, Gagatsis and Elia conducted a study that showed two things … first, that multiple representations by themselves do not improve mathematical understanding; second, that the different modes of representations are not ‘equal’ in learning.   (See http://dipmat.math.unipa.it/~grim/quad14_gagatsis.pdf)

Behind this, I believe, is how a human brain processes information.  Experts seem to feel that there are separate ‘channels’ for phonological (word-like) and visual information.  Some evidence indicates that the phonological channel has a higher priority in the processing; perhaps this is simply due to the native difference in the information — after all, a visual image is global when first processed while verbal information tends to have higher level of details.  I would point out that an expert sees a very different set of information in a graph compared to a novice.  See the ‘cognitive psychology’ books by Roger Bruning (2003, page 122 ), and by Bruce Goldstein (2005, pages 166 to 168).

A recent study by Aleven and others looked at the effectiveness of various graphical representations with fractions.  They found that multiple graphical representations of the same concept AND switching between graphical and symbolic forms produced the best learning.  If a student only sees one ‘view’ graphically, it is not as good; different views help the learner ‘see’ more information, similar to a ‘compare & contrast’ writing assignment.  Learning was also improved when there was a routine switch from graphical to symbolic forms.  (See http://www.learnlab.org/research/wiki/index.php/Sequencing_learning_with_multiple_representations_of_rational_numbers_(Aleven,_Rummel,_%26_Rau)

How should we use ‘visual’ displays with our students?  Students like them (at least, compared to the symbolic forms).  The bottom line is that students do not see the same visual information that we see … when they understand well, they will see much of what we see.  When you see a graph of an exponential function over a small domain, you see the parts that are not shown — even though the shape of the graph does not necessarily imply this information.  Students need to see the relationship in multiple graphical forms, as well as symbolic and numeric forms, to become more ‘expert’ — to have the understanding that is needed.   When you see a graph of any relationship, you understanding the ‘approximating’ nature of the graph, and know to look at the numeric and symbolic forms for additional accuracy; again, students need scaffolding along this path of understanding. 

In addition to graphs, we also tend to use a variety of visual aids with our students — whether it is a triangle image to represent part-whole work with fractions or percents or a table to summarize information.  You may have noticed that some students are helped, and others confused, by these displays.  To become mathematically literate, our students would be able to translate from these and to state some basic limitations of such a visualization; a tool used without knowing the limitations is a ticket for future difficulties.

It’s not enough to say that “graphical representations are good”; we need to design a learning experience that brings the student to a more complete understanding of all representations.   Visual images might help students in the short-term, or might confuse; neither condition is good for the long-term.  Having “different representations” is not just a “been there, done that” approach … we need to attend to the transitions between representations and to the native limitations of each.  Good instruction makes explicit the connections between representations in an ‘iterative’ process.

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Equity, Opportunity and Developmental Mathematics

Our friends at the Developmental Education Initiative (DEI) kindly offered an opportunity to write a guest post for their blog Accelerating Achievement … my post on Equity is up at their blog http://deionline.blogspot.com/2011/05/guest-post-equity-opportunity-and.html

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