Mastery Learning is …

I have heard many faculty speak in favor of mastery learning … and almost as many speak in opposition.

The heart of this set of opposing viewpoints is an incomplete notion of what mastery learning IS.  Many equate mastery learning with basic skills … with repetition … with homework systems.  These are not definitions, nor even descriptions, of ‘mastery learning’.

The origins of ‘mastery learning’ were centered in a philosophical base which claimed that almost all students could learn any particular content to the level of ‘masters’ (usually defined to be a 4.0 or A student) given the correct conditions … with a primary condition to vary being ‘time on task’.  In a classic view of higher education, all students are imbedded within a learning environment so they experience similar conditions; those who perform at a high level are rewarded with 4.0/3.5/A/B grades and encouraged to pursue more learning … those who failed to perform within these constant conditions were told that they needed to make an alternate choice of activity (as in, some other class … some other major … or not in college at all).

Those of us who adopted a mastery learning model turned this conception on its head.  We were not here to sort students; we were here to create the conditions for all students to have the opportunity to become masters of the content.  Our content was not changed, only the conditions for learning.  Our assessments did not reflect lowered expectations, but they did create positive conditions for additional learning.

The current misconception of mastery learning is based on the technology that is often used to deliver ‘content’.  Offering modules, online homework, and requiring ‘80%’ before moving on … these have little to do with mastery learning.  These learning environments focus on basic skills primarily because that is easier for mass-produced homework systems (though it also reflects a bias among many colleagues). 

In essence, mastery learning is only limited by our capacity to design instruction and assessment.  If applications … transfer … problem solving … creativity are important elements in your ideas about mathematics, mastery learning can be designed to support them.

Mastery learning, in 2011, is more about the economics of publishing and grants than it is about the flexibility (and power) of mastery learning.  I have spent many years in a program that had mastery learning as a founding principle, and I understand the complexities of creating a mastery learning model that includes ‘more than basic skills’.  I would suggest that most of these difficulties are present regardless of whether mastery learning is involved. 

Mastery learning does not determine the nature of the mathematics faced by our students.  No, what determines the mathematics that our students experience is our own conceptions of mathematics.  We should, as a community of professionals, have honest discussions about what it means to “learn mathematics”.

Join Dev Math Revival on Facebook:

The End of Learning Styles

Although I did not hear this particular report, NPR (National Public Radio) aired a report on the scientific research related to “learning styles”; see http://www.npr.org/blogs/health/2011/08/29/139973743/think-youre-an-auditory-or-visual-learner-scientists-say-its-unlikely 

If this is the end of ‘learning styles’, what does that mean?  Which ‘learning styles’?

I think the end of learning styles can only be a good thing, for teachers and our students.  The basic constructs of ‘learning styles’ involve vague descriptions of sensory processing, skewed to favor one or more categories of input (auditory, visual, kinetic, etc), without regard to the research of cognitive scientists.  Categorizing students within these skewed categories creates dangers, and real damage, to our students.  We all have had students who have been told “I am a very hands-on learner; if I can not touch it and move it, I will never understand it” … and similar statements of limitations for other ‘styles’.

Ed Laughbaum, a long time friend, said in a recent post that ‘basic brain function is the same in all normal brains’; he does not say this lightly, and has good scientific reasons for that statement.  My own humble reading of current research and theories of cognition certainly supports that statement.   Unless the student has a temporary (drug induced, for example) or chronic (birth defect, closed brain injury) biological issue, the learning needs are quite similar across all students with comparable current learning. 

The constructs of learning styles have not worked, and they conflict with science.  Too often, we have accepted “proof by parable” or even “proof by rhyming” … what does “drill & kill” mean?  An “inch wide and a mile deep”?  “She is a visual learner.”  “Our students need manipulatives.”  “Sage on stage … Guide on side.”  I am afraid that our profession, and teaching in general, has been guided more by the appeal of the words in statements rather than by known properties of learning.

It is true that very few of us, and teachers, will be able to study the actual work of cognitive scientists.  We will depend upon others to translate and summarize this work so that we can use it.  If these resources are not available, we must avoid the pop-psychology notions that might seem to have some truth in them.   

If you would like a source, here is the best one-stop summary I have seen: http://act-r.psy.cmu.edu/papers/misapplied.html  , an article called “Applications and Misapplications of Cognitive Psychology to Mathematics Education”.

Join Dev Math Revival on Facebook:

Dear Aunt Sally … Please be Excused (from math)

In developmental mathematics classes, as in school mathematics, Aunt Sally seems to be everybody’s friend.  As in “Please Excuse My Dear Aunt Sally” as a memory aid for order of operations (aka “PEMDAS”).  I would, indeed, like to excuse Aunt Sally from ever being in my math class. 

In another post, I talked about the “Sum of all Shortcuts”; in this post, the issue is mnemonic aids.  You can improve your students’ “learning” if you minimize the use of these ‘easy to remember’ tricks. 

This may sound counter-intuitive … isn’t it a good thing if students can remember something?  Well, it CAN be a good thing; the issue is what exactly do they remember?  In the case of PEMDAS, they remember ‘do inside parentheses first’ or ‘do parentheses first’.  Fine, to a point — students can evaluate  “8 + 10 ÷ 2” and “(8 + 10) ÷ 2”.

The student then sees these ideas, which do not follow PEMDAS:

• 8x + 2x   (can add before multiply)
• 8x + 2y    (can not add)
• 8(x + 2y)   (can multiply ‘first’)
• (2x²y)³     (can ‘power’ first)
• f(x)=8x + 2    (what does that x mean?)
• f(-3) for that function    (what do we do with the -3 on the left side)

The “P” in PEMDAS is especially worrisome.  Parentheses have multiple purposes in mathematics, and only some of them relate to the order of operations.  We also use other symbols of grouping, some of which are another operation (radicals, fractions, absolute value, etc).

Now, we actually make students do too much with expressions of extra complexity just to see if they can follow the order of operations.  We create our own need for an easy-to-remember tool (PEMDAS) which then results in students having to unlearn later when we do other work in ‘simplifying’.  This is a bit like designing a tool to require disposable parts, in order to keep a business active; I would suggest that our artificial level of difficulty with numeric expressions serves no purpose, not even our own.

It’s important, however, for our students to be literate and comfortable with the basic meanings of expressions and forms.  As I talk with my students, I am impressed by how many of them remember ‘PEMDAS’ years later and by continuing difficulty in doing work that does not involve applying ‘PEMDAS’.  We are not doing our students any favor by giving them an easy thing to remember which does not transfer to future work.

Some readers are likely upset by my suggestion; yes, I know PEMDAS has helped millions of students in their math classes; yes, I am aware of research showing mnemonics help learning disabled students in particular.  However, the benefits for most students do not seem that great to me; the long-term result may be more negative than positive.  [Many of my most struggling students have learning problems, and survived by using tools like PEMDAS; they have difficulty in the situations listed above that do not follow the PEMDAS priorities.]

We know that PEMDAS does not cover most expressions involving variables.  I am suggesting that PEMDAS directly interferes with the algebraic literacy of our students; quite a few students suffer needless discouragement when their algebraic difficulties increase as they painfully discover the real limits of PEMDAS.

Let’s send Dear Aunt Sally on a much needed vacation; she has been used for many years, and perhaps is ready to retire.  Instead, let us focus on basic literacy dealing with reasonable objects of valuable mathematics.

Join Dev Math Revival on Facebook:

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.

Join Dev Math Revival on Facebook: