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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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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