Don’t Do Your Homework!!

Within days, close to a million students will attend their first class in some developmental math course; some have already started.  The vast majority of their teachers want every student to succeed, though this may not be the students’ perception.  Therefore, one phrase is likely to be heard by close to a million students in a short period of time:  “Do your homework!”

I do not tell my students to do their homework.  Why?  Well, think of this metaphor:

My cousin Alfred walks in to his doctor’s office; being astute, the doctor notices that Alfred is obese (the only question is ‘how obese’).
After letting Alfred share his health concerns, the doctor makes two statements to him:
1) Do you think that you are overweight or obese?
    Alfred’s answer: Well, yes … that is kind of obvious.
2) Okay, your first step is to eat right and exercise?
    Alfred’s response: Yes, I’d like that.

What do you think Alfred is going to do?  Will he eat right?  Will he exercise?

Our students, especially in developmental math courses, do not know how to operationalize “do your homework”.  The vast majority of students believe that ‘doing homework’ means completing the assigned exercises, whether online or on paper.  We sometimes reinforce this perception by “collecting homework”, where we make sure that the student has ‘done it’.  However, the basic purpose of homework is to learn the most possible for that content for that student. 

Instead of telling my students to ‘do homework’, I tell them to follow the learning cycles.  These ‘learning cycles’ are simply stated components of doing homework with a focus on the purpose (learning).  Here are the phrases I am using for the 3 cycles I talk about with my students:

1. Study and Learn
   Read the explanations and information, study the examples, re-work the examples.
2. Practice
   Try every problem assigned, and check your answer.  Look at what is going well for you, and look for areas that you did not understand yet; figure those out.
3. Get Help
   After you examine areas you did not understand, get help on anything you still don’t get. 

You can probably come up with different phrases and a different set of ‘cycles’.  I like to use the word ‘cycles’ because of the implications that the process is repeated and that cycles are related.  My intent is to create an impression that learning involves deliberate work, as well as an impression that answers (right or wrong) are just a step in the process.  It is likely that my students do not see most of what I am trying to say — though indications are that listing these learning cycles helps most students do a better job.

I never collect ‘homework’, because homework is something that happens in the brain while doing the learning.  I would love to be able to directly measure all aspects of learning at the biological level; the world is probably a better place since I can not do so.  Instead, I use assessments in class (like a simple quiz on about half of the class days) along with discussions with students.

Students should not ‘do homework’.  Students should learn math, which involves discrete activities that work together to help that student do the best they can do.

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How much practice is enough?

Do you see repetition as the enemy of a good math class?  Or, do you see practice is the single biggest factor in learning?  More practice might be better … it might be worse; however, repetition is not trivial in the learning process.

One reason I am thinking about repetition is the current emphasis with online homework systems, whether as part of redesigns like emporium or with modules or with ‘regular’ classes.  Sometimes, these systems are marketed with an appeal to a high ‘mastery’ level (percent correct … not the same thing at all).  To understand the impact of various practice arrangements, we need to review some cognitive psychology.

First, a lack of repetition normally places a high work load on short term memory; without repetition, the long-term memory (playing the role of ‘knowledge’ in this case) is anecdotal, like remembering the last web site you visited before leaving home.  Without repetition, new knowledge does not become integrated with related knowledge.  In the extreme, a contextualized math course has almost no repetition; each problem is a novel experience.   In the science of cognition, this type of knowledge is called ‘declarative’ knowledge.

Second, the quality of the practice is a critical factor in how the information is stored.  Much research has been done on factors that raise the quality of practice; in particular, ‘blocked’ (one type at a time) and ‘unblocked’ (mixed) both contribute to better learning.  In my view, this is one of the major drawbacks of both online homework systems and modules … one objective at a time, practice on that, test and move on.  (In cognitive science, ‘blocked’ is used strictly … same steps and knowledge used each time.) 

Third, there is a connection between effective practice and math anxiety.  As accuracy is established via repetition, anxiety can be lowered.  [I am not claiming that practice, by itself, will lower anxiety.  I am claiming that a lack of practice will reinforce the existing anxiety level.]

In the learning sciences, research talks about “automaticity” and “performance time”.  Higher levels of automaticity are associated with faster performance time; both are factors in the brain’s efforts to organize information and ‘chunk’ material for easier recall.

Whatever class you are teaching, keep your practice consistent with your course goals.  If you want students to organize knowledge, apply it to new situations, and improve attitudes, you should consider sufficient quantity and quality of practice.

Here are some references:

Cognitive Psychology and Instruction, 4th edition 2003 Bruning, Roger; Schraw, Gregory; Norby, Monica; Ronning, Royce  (Pearson)

Beyond the Learning Curve: The Construction of the mind 2005   Speelman, Craig P and  Kirsner, Ki    (Oxford University Press)

Automaticity and the ACT* theory   Anderson, John   1992 Available at  http://act-r.psy.cmu.edu/publications/pubinfo.php?id=91

Radical Constructivism and Cognitive Psychology   Anderson, John;  Reder, Lynne;  Simon, Herbert  1998 Available online at http://actr.psy.cmu.edu/~reder/98_jra_lmr_has.pdf

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Practical Math — or Not

Last week, I spent several days with faculty who are working with the Carnegie Foundation’s Pathways — Statway and Quantway, at their National Forum (summer institute).  I continue to be impressed with the quality of these professionals; Carnegie is fortunate to have them involved.  One comment from a faculty member has been stuck in my thinking.  In the context of Quantway, this faculty member said:

Everything in this course has to be practical.  The math students see has to be practical.

I recognize that there is a high probability of head nodding and agreement with this sentiment among people reading this post.  Can we … is it reasonable or desirable … to shift from a ‘nothing in this course is practical’ to ‘everything in this course is practical’ position?

First of all, we need to recognize that ‘practical’ is a matter of perception, communication, and culture.  Our students will not see the same ‘practicality’ that we do.  For example, if we have a series of material looking at the cost of buying a car including operating and finance, many students will definitely not see this as practical.  The majority of my students are not able to consider this situation in their real life now, nor for several years; for some, they can not even imagine having a real choice to make about a car.  What we often mean is that math needs to be contextualized, not practical — context is a simpler matter to establish than practical.

Secondly, the ‘practical’ or ‘contextual’ emphasis reminds me of the old school approach to low-performing math students:  If a student was not doing well in math, put them in an applied math course (business math, shop math, personal finance), as a way of being polite about lowered expectations.  I realize that many of our students are initially happy with the lowered expectations of ‘practical math’; however, this approach does not honor their real intelligence, nor does it recognize the capacities in our students to understand good mathematics just because it is enjoyable to do so.

More important than these two points is the learning implications of ‘practical math’.  I’ve been reading theories of learning and research testing these theories … for close to 40 years now.  Nothing in the theory suggests that learning in a practical context is better than learning without the context; without deliberate steps to decontextualize the learning, the practical approach often inhibits general understanding and transfer of learning to new situations.  I do not believe that ‘all is practical’ is a desirable approach to learning mathematics.

However, context and practicality can be very motivating.  Motivation is the most elemental problem in developmental mathematics.  Therefore, it is reasonable to provide considerably more context for students than the traditional developmental math courses with its ‘train problems’.  I also would add that most students are motivated by learning mathematics with understanding when they can see the connections; true, our students need some extra support for this process, and it conflicts with the approach emphasized with them in the past (primarily memorization without understanding).

I have summarized my view on the ‘practical’ issue with this statement:

I will always include some useless and beautiful mathematics in all of my math classes.

Education is about expanding potentials and creating new capacities; practical learning is the domain of ‘training’ (which is also critical … but it is not education).  I encourage all of us to help our students learn mathematics in different ways: sometimes practical, sometimes in a context, sometimes imaginative, and sometimes logical extensions.  The mix of these ingredients might reasonably shift as a student progresses; developmental math courses might be more practical than pre-calculus.  Diverse learning is better than limited learning.  Diverse learning respects the intelligence of our students and maintains high expectations for all students.

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Variable Concepts … Variable Notation

Let’s talk variables.  Do we want students to develop an understanding of variable concepts (whether in a developmental math course or not)?  Is accurate use of variable notation enough?  If students can model applications and accurately determine solutions … is that enough?  Is there a role for linguistic literacy in mathematics?

A few years ago, I was able to use a sabbatical leave to explore a number of issues related to learning in developmental mathematics; the primary product of this leave was a series of short reports intended for my department though appropriate for faculty at other colleges; one of the reports dealt with variable concepts — see http://jackrotman.devmathrevival.net/sabbatical2006/1%20Variable%20Understanding%20and%20Procedural%20Skills.pdf.  The other mini reports from that sabbatical are available at http://jackrotman.devmathrevival.net/sabbatical2006/index.htm

One of the issues we face in college is dealing (or not) with prior learning.  Without intervention, prior learning (even when inaccurate) survives — often surviving in the face of conflicting information in the current learning environment.  Visualize the prior learning as being as a stable mass of ‘knowledge’ (even though it has gaps and errors); as students go through a class as adults, information that connects positively with the old reinforces the old.  When new information does not connect or conflicts with the old, the low-energy (natural) response is to build new storage … resulting in that solid core being supplemented by weak veneers of new knowledge.  This, of course, is an incomplete visualization for the actual processes in the human brain.  The suggestion is that students approach a math class with an attitude that supports old information and minimizes cognitive effort for dealing with new or incompatible information.

In my beginning algebra class this week, we did the test on exponents and polynomials.  Although the test includes some artificially difficult problems with negative exponents, most of the items deal with important ideas.  One of the most basic items on the test was this:

Evaluate a² + (3b)² for a = -3 and b = 2

Several students made this mistake with the first term:

-3² = -9

A smaller group of students made this mistake with the second term:

(32)²

Now, this is a good class — all students are actually doing homework and attending class almost every day.  We had dealt with the first situation at the start of the semester.  How could these errors survive to this point?

Both errors are based on variable as a symbol to be replaced by a number, which is not complete.  They might represent a visual approach, not verbal.  Variables represent quantities involved in sums and products, where products with variables are implied … and more than this.  Simplifying expressions might — or might not — uncover the incomplete understanding.  What can I do to help students with this?

I am planning on incorporating some linguistic activities around variables in the first week of the semester.  Some of the ideas are from a old book called “English Skills for Algebra” from the Center for Applied Linguistics (Joann Crandall, et al); I believe this book is out of print.  The authors wrote this book from the viewpoint of helping students with ‘limited English proficiency’, which might just apply to many of our developmental students.  Some of their activities involve listening to somebody read mathematical statements and the student writing them down.  I think I will mostly activities that deal with written statements — identifying translations and paraphrasing (both to algebra and from algebra).

I do know that just saying “that was wrong … this is right” will not help these students develop a more complete understanding.  I need to create situations where they get uncomfortable and really dig into the concepts related to variables.  Some energy needs to be created so that we don’t just place a veneer on top of that mass of prior knowledge; parts of that prior knowledge need to be broken up and put back together.  Without that process, many of these students will be limited in their mathematics and blocked from many occupations.

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