Pathways Summer Institute

This week, over 100 faculty (along with over 20 administrators and 20 institutional researchers) met in Palo Alto (California) for the Quantway™ and Statway™ “Summer Institute”.

These “pathways” are being developed under the coordination of the Carnegie Foundation for the Advancement of Teaching (see http://www.carnegiefoundation.org/developmental-math for more details).  AMATYC is a partner in this work.

I see two exciting parts of this work.  First of all, the Carnegie Foundation thinks it is important to have official AMATYC involvement in this work; this allows two of us (Julie Phelps of Florida, and me … Michigan) to be engaged at a deep level in the work.  Both of us were part of the Summer Institute, and have been involved with the planning for it as well, along with work on the actual instructional material.

The instructional materials are the other exciting part of this work.  This is due to multiple factors … the materials are being developed by the Dana Center (see http://www.utdanacenter.org/), known for its quality work in this area.  Another factor is the fantastic fact that the materials will be open resources in 2012, under a “Creative Commons License” which allows general use with source credits.  The materials will include an online homework system of high sophistication.

We also have considerable synergy between the work of the Pathways and the New Life project (AMATYC Developmental Mathematics Committee).  This synergy can be seen in the high degree of agreement between the learning outcomes used in both projects, as well as the multiple people engaged with both projects.

The Pathways are a very sophisticated solution which addresses several needs and problems in developmental mathematics.  I encourage you to become familiar with the work of the Pathways … we live in the best time for developmental mathematics in recent memory.

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The Math Dance

A step forward, step to side, bring feet together … a step backward, step to side, bring feet together.  Dance the waltz enough, and a person can do this sequence without any difficulty.  Many can become experts at the dance, and all can be included. 

Lost is why these are used as the dance steps. 

Of course, the ‘why’ does not really matter — it’s just a dance!!

I have taught a lot of students to dance.  The majority have been able to do dances like the waltz and two-step.  Sadly, the math dance has no particular value if a person does not know why the steps are done like this.  To understand means that a person can improvise; a little understanding allows helpful flexibility, and much understanding allows an artist’s rainbow of insight, logic, and problem solving.

Mathematics has become a dance, one that can be taught as remembered moves to particular musical themes.  There are some experts who assert that this the only possible outcome when society decrees that ALL persons must complete a subject, that mandatory always translates into a lowering of the value of this learning.  The evidence for this view seems abundant, and it is easy to accept this result (especially with the bright and blinding light of accountability shining on education).

We must not give up on mathematics so easily.

Mathematics has much to offer every student, our society, and the future.  Not the math dance — the real mathematics, science of relationships between quantities. 

We can create sound mathematics appropriate for all learners.  All students can learn, given the proper resources and conditions.  I might grant that the more extreme learning disabilities might present obstacles too large for a very small minority; this group is at least 2 standard deviations below the mean.

I encourage you to avoid the current rush of methods that might be more efficient at teaching the math dance.  We have seen these types of improvements before, which provide change but not progress.

I invite you to work with me to imagine a better mathematics program for all of our students, a program that shows the practicality and beauty of mathematics.  We do not need to make mathematicians of all students, just like we do not need to create math dancers … however, I believe that we can create a program that inspires more students to seek out more mathematics.

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Contextualized Mathematics

Should mathematics be learned within the context of a situation that creates (or at least, shows) the need for the mathematics?  Do students learn better when we do?  What do we want our students to gain?

Some of my recent posts might leave a person with a belief that I think mathematics should be highly contextualized.  However, I definitely do not think this is true.  My reasons deal with two discrete issues that connect within our classrooms — the impact of high context on learning, and identifying the goals (what we want students to gain).

The research on contextualized learning is not particularly strong at this point.  Certainly, the general researchers in learning & cognition conclude that context can actually interfere with learning; this is simply a corollary to the principle that learning is improved by making the target (the thing to be learned) as visible as possible.  Context can hide the big ideas.  For a good summary, see this article about cognitive psychology in mathematics http://act-r.psy.cmu.edu/papers/misapplied.html  — one of the best single sources I’ve ever seen.

A second component of the learning dimension is language and culture.  As soon as we present a context, we make demands of our students about other knowledge … sometimes fairly unrealistic.  One example I saw recently involves ‘shooting free throws’, rebounds, and ‘dunks’; another dealt with a baseball infield.  Sports are not uniformly followed by our students.  The same difficulty arises when we talk about projects around the house (whether it is sewing or woodworking).  These language and cultural factors affect both native speakers and those who learn English as a second language, and the problems cut across economic standing as well.  

The other dimension of my concern deals with our goals … what do we want our students to gain?  Some people bring in contextualized learning so that students can experience ‘doing mathematics’ like a mathematician does.  Other people use high-context because it motivates students.  To me, both of these goals are important … however, they are not the whole story.  A major goal of any math class should be to provide general tools that can be used, especially in future classes that the student needs to take, and this suggests a need to be able to transfer learning to other situations.  This transfer is inhibited when a learner has not practiced a skill or process repeatedly; meeting this threshold is very difficult if we contextualize most problems.  See http://jackrotman.devmathrevival.net/sabbatical2006/3%20Life%20in%20The%20Grey%20Zone.pdf, which deals with ‘how much practice is enough’.

I am especially concerned about preparing students to cope with the quantitative needs in their science and technology classes.  These needs vary from the very specific context to quite broad conceptualizations, and we seldom know which mixture of needs a particular student will have.  Developmental mathematics needs to deal primarily with broad sets of needs.  I do not think we can limit the mathematics to that which there are contexts that the student will understand.

It would be simple to say that we need ‘balance’ in our curriculum, and this would be true.  However, we should talk about what students should gain.  Some of their future classes are actually after some very specific skills (such as equivalence of different forms, or dimensional analysis); others are general … almost theoretical (such as behavior of types of functions for biology).  For a particular college, the needs might shift the ‘balance’ more strongly in one direction or the other.

The New Life reference material (at http://dm-live.wikispaces.com/) was developed to summarize many of these needs.  I encourage you to look at the sound mathematics described, most of which can be well served by combining learning about the process along with dealing with various contexts.  Needs exist for which contexts may not exist; some needs deal with theory, where context is a temporary step off the trail.

Yes, there are “what works” studies that conclude a high-context approach improves math classes.  These are not proofs of a result.  Like other ‘what works’ studies, there are many factors involved and only a few measured for inclusion in analysis.  From a learning standpoint, the best to be expected is “no significant difference” with high context … and this would reflect a great deal of effort to avoid the known difficulties with high context.

We are preparing students for success, and their success involves a multitude of needs.  Our math classes should focus primarily on general concepts, with a limited role for contextualized learning.

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All of The Above

Somehow, I am on my congressperson’s email list; fortunately, he does not produce too much spam.  However, sometimes the emails sent from his office provide an interesting thought.

A recent email focused on energy policy, and touted an “all of the above” approach (exploration, federal lands, nuclear, coal, green, etc).  I started thinking … policy is based on a goal (or problem) and reflects the understanding of how we can achieve a desired outcome (also known as progress).  In the context of energy policy, ‘all of the above’ is a non-policy.  It strikes me more as a desperate person thrashing wildly in the water to avoid drowning.

In developmental mathematics, though, we have been practicing “all of the above” for many years.  We work from where we are, and we add ideas that sound productive.  I think we appear to many others to be ‘thrashing wildly in the water’, and I know that some of us actually think this is accurate. 

Is there an alternative?  Yes, of course … though it requires going back to an empty page in many ways.  We have a small set of basic questions that should guide (and somewhat determine) all of our work: What is important mathematics for all students?  What does it mean to ‘learn’ this mathematics?  How can we determine if a student has achieved this learning?

Our understanding of these questions is critical, for we know that various methodologies have different strengths … their impact on learning is different, and each is better suited to particular learning goals.  For example, we generally give broad support to ‘problem solving using mathematics’, and this means capacity to transfer learning in our domain; research has determined how this outcome can be enhanced, and which methodologies are likely to be more effective.  One specific point: Contrary to popular mythology, ‘drill’ is not ‘kill’ — repetition of skills forms a critical basis for development of problem solving; the problem is not ‘drill’ … the problem is ‘only drill’.

Think about this ‘all of the above’ idea.  Take a look at totally new models (like New Life), and consider your own ‘answers’ to the basis questions.  Our professional standards (Beyond Crossroads, http://beyondcrossroads.amatyc.org/) suggest that our work be focused on achieving our shared goals.  “Thrashing wildly” in an “All of the Above” mode does not appear in the improvement cycle.

As long as we continue an “All of the Above” approach to our work, we actually achieve “None of These”.

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