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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Pathways Blog Available

Our friends at the Carnegie Foundation for the Advancement of Teaching have launched a blog for their ‘pathways’ work.

The blog is at Math Pathways, and I encourage you to take a look; you can also join the community on that page to receive updates.

I like the fact that the current post at the Pathways blog deals with quantitative literacy; we would be better off if we focused on quantitative literacy instead of ‘developmental mathematics’.  Of course, I would rather it be called ‘mathematical literacy’ (to focus on the scientific aspects of mathematics, not just the tools) — but that is a relatively minor point.

I hope you take a look!

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Instant Feedback Lowers Learning

Online homework systems are “cool”.  We like them as faculty (in spite of our frustrations), students generally like them, and we believe that instant feedback is a good thing.

Learning is a different process than connecting a stimulus with the proper response (“conditioning”).  The effect of instant feedback might help conditioning, but can definitely interfere with learning in humans.  Schooler and Anderson published an article entitled “The disruptive potential of immediate feedback” (see http://act-r.psy.cmu.edu/publications/pubinfo.php?id=313 ).  The logic for being disruptive is that the instant feedback competes with the learning content for resources in the working memory.  Paying attention to feedback means that there is less attention available for the concepts and procedures.

 Related to ‘instant feedback’ is the general property of being FAST!   When learners complete activities quickly, research shows that the entire process tends to stay in working memory … never making the transition to long-term memory.  See O’Reilly (page 153), Leron and Hazzon “The Rationality Debate: Application of Cognitive Psychology to Mathematics Education”  (see http://edu.technion.ac.il/Faculty/uril/Papers/Leron&Hazzan_Rationality_ESM_24.3.05.pdf#search=%22co and Kahneman “Maps Of Bounded Rationality: A Perspective On Intuitive Judgment And Choice” (see  http://nobelprize.org/nobel_prizes/economics/laureates/2002/kahnemannlecture.pdf#search=%22Maps%20of%20bounded%20rationality%3A%20A%20perspective%20on%20int) and O’Reilly’s chapter “The Division of Labor Between the Neocortex and Hippocampus” in Connectionist Models in Cognitive Psychology (edited by Houghton, George).

There is a point of view that advocates learning within a gaming environment, which might seem to contradict these statements.  One distinction that might help understand the contrast is that of ‘awareness of learning’ — in many games, the learning takes placed without direct attention to the learning, meaning that the learner has less ability to explain (and transfer) that learning.  We would hope that mathematical learning needs to be transferable, and we like to have learners who can explain what they have learned.  I do believe that ‘instant feedback’ and ‘quick learning’ lowers the overall learning.

Why do I think this is important?  Much of the current ‘movement’ in developmental mathematics involves intensive uses of online homework systems for their instant feedback and quickness.  From a learning theory perspective, this is not a good thing.  My prediction would be that students using these systems have even more surface processing and lack of transfer (of knowledge) than our old-fashion textbooks. 

How should we design instruction for better learning?  Just because feedback can be ‘instant’ does not mean that it’s best; learning support systems (homework) should design the speed of feedback based on parameters from research studies to facilitate deeper processing in the brain.  These systems should also consider breaking up sets of problems to include other activities; a student who quickly completes 30 homework problems without a break might be processing only at the surface level … other learning processes within these sets can give the brain an opportunity to reconcile the new material with prior knowledge (a key step).    As instructors, we can monitor the time on homework to encourage students to slow down, to even take short breaks in the middle.

Given that students may tend to be less patient than in prior periods, we need to pay deliberate attention to slowing things down.  Part of this would be direct and honest statements to our students about how they can improve their learning and success in mathematics.

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Math Wars or Math Together

Change.

Change happens.

Change is happening faster.

Progress is different from change.

Now that I have said the obvious, what does it MEAN?  How do we promote progress, and not just change without progress?  To remind us of the basic meaning, ‘progress’ means movement towards a goal.  In the context of mathematics (including developmental mathematics), our goals reflect shared understandings and values.

And, I think that is part of the problem with ‘math wars’ … we do not focus on our shared understandings and values, and we do not articulate the core goals of our work.  I’ve been thinking about this after reading a ‘math wars’ type article (see http://betrayed-whyeducationisfailing.blogspot.com/2011/05/why-i-quit-teaching-math-at-sfcc.html) … this particular article (possibly not accurately) describes major disagreements at a community college, where the situation resulted in a faculty member resigning their position rather than ‘change’.

Too often, we leave the question of goals as a ‘given’ or a factor not requiring direct attention.  Bad idea!  If we want particular skills for our students, this implies some methodologies would be more appropriate than others.  If we want our students to experience situations like a mathematician, then different methodologies would be more appropriate.  If we want flexible problem solving (involving elements of both prior goals) for our students, an intelligent mix of methodologies would be more appropriate. 

My own guess is, as a community, we would answer “We want all of these things” but to varying degrees.  Within the framework of two courses, or perhaps three, I suspect that the capacity to reach multiple goals like this is just not there … between the resources that we can apply and the resources that our students can apply.   The New Life model, overall, tends to favor the ‘mathematician’ and problem solving goals with less on basic skills.  Other models, including the traditional framework and the redesign models (like emporium) tend to focus on basic skills (with little or no ‘mathematician’). 

We are in this together, so we should get math together.  Our conversations should start with, and focus on, the broad goals for our courses.  Too often, we begin our conversations with “Do you include factoring in beginning algebra?”  Topics are often not the end goal; topics are often secondary to the larger goals in a discipline.

Before launching a redesign project, your department (program, or whatever) should get its math together.  As a profession, we need many more conversations about the core goals; we can have areas of agreement, which will lead to shared work … and shared work can lead to progress.  Merely changing the delivery system is definitely not progress.  Sure, we want higher pass rates; however, higher pass rates just means that there are more people at the end of step n  … we would never accept an explanation that had a good conclusion without examining the quality of the steps preceding it, and this applies to our curriculum as well.   Do steps 1 to n have any connection to our core goals?   That’s my question for you.

Math wars helps nobody.  Math together can lead to progress.  Let us get our math together!!

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