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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Algebra for Everybody, and Algebra for Nobody

Algebra is so basic that all students need to have a good grasp, and adults without this capability will be limited in their economic and social choices.  Algebra is so esoteric compared to daily life and work that only those in STEM careers need to bother learning it.  Both of these statements can be true; the apparent inconsistency is based on what is meant by the word ‘algebra’. 

As such things normally happen, there is another article suggesting that nobody needs algebra (see Andrew Hacker’s article at http://www.huffingtonpost.com/2012/07/30/in-new-york-times-op-ed-c_n_1719947.html) and a response by Borwein & Bailey (see http://www.huffingtonpost.com/david-h-bailey/algebra-is-essential-in-a_b_1724338.html).  Reflecting our society in general, we tend to view issues as a binary choice — if it is good, all people must; if it is not good, nobody should.  The Bailey & Borwein response is well written, and reflects a balanced point of view.

As a mathematician, I view algebra as a language system used to describe and manipulate features (whether known or not) of the physical world based on arithmetic operators .  Basic literacy in this language system is essential in both academia and ‘real life’; translations into and out of algebra are the most basic literacies, followed by different representations (symbolic, numeric, graphic).  

Unfortunately, the algebra of mass education tends to focus on procedures and complexity of limited value to anybody combined with a focus on solving algebraic puzzles, as if completion of a crossword puzzle is a basic skill for a language.    True to our current binary approach, people who agree with a literacy approach will invest great effort to avoid all procedures, complexities, and puzzles.  The truth is that we undertake these problems ourselves just for fun, and this is one element in our transition to being mathematicians; how are we to capture the attention of potential STEM students if we avoid the fun stuff? 

As a language system, basic literacy in algebra means that a person can read the meaning of statements; transformations to simpler forms is based on that meaning.  I failed to help my students in the algebra class this summer … I know that because students could distribute correctly in a product but failed routinely with a quotient.  [In case you are wondering, the problems involve a 3rd degree binomial and a 1st degree monomial; in the division case, many students ‘combined’ the unlike terms in the binomial instead of distributing.  This is a basic literacy error; very upsetting!]

Hey, I know … nobody needs to distribute algebraic expressions on their ‘job’ (except us!).  That type of  reason is enough for me to conclude that we do not need to cover additional of rational expressions (prior to college algebra/pre-calculus); that process is complex, and is based on a higher level of understanding of the language.  Distributing is a first-order application of algebraic literacy; avoiding that topic means that we present an incomplete picture of algebra as a language.  A pre-college mathematical experience needs to provide sound mathematical literacy — including algebra.

Everybody needs algebraic literacy, as part of basic mathematical literacy.  We can design courses that provide the needed mathematical literacy as a single experience — no need for a numeric literacy course (‘arithmetic’) and an algebra course and a geometry course; all of that, plus some statistical literacy, can be combined into one course.  This is the approach of the New Life model, and is imbedded within the Quantway & Statway (Carnegie), and in the New Pathways (Dana Center).  I encouage us all to include some transformations (‘simplifications’) in the algebraic language.

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Placement Tests To Go?

Placement Tests are an important part of the process at the vast majority of community colleges, especially relative to mathematics.  Over on the
MATHEDCC discussion list, Fred asked an honest question about finding an online placement test that was not a commercial test.  Most of the public responses to his query have been critiques of placement tests in general (some would say toilet-emptying, self-affirming statements).  Under the surface, is it possible that we do not need placement tests?

Some readers will have an extreme reaction to a question raising the possibility that placement tests are not good.  Let me clearly state my opinion after working with them for 39 years:  Most placement tests are reasonably good assessments of the content that they were designed to measure.  Given the limitations that users place on them (users being most of us), the tests achieve the best measurements possible.  Of course, these statements don’t tell you if I think placement tests are ‘good’ or not … and that is my point.  Our use of placement tests might be good or not; the tests themselves are just what they are designed to be.

The use of placement tests involves several issues.  The largest issue right now is whether placement test results are the only factor in initial math placements.  The best research I have seen suggests that we should supplement test results with other information, especially high school performance (overall) for recent ‘graduates’ (whether they graduated or not).  Some states have a common data system for K-16 which makes this relatively easy; others (like my state) have significant barriers.

Another issue deals with the content definitions for placement tests.  Some of us see the companies involved as ‘evil’, with a higher priority on money or prestige than on helping students.  I suspect that this point of view is held by people who have not been involved with the companies work.  Although it is true that some of the field representatives of the companies are not helpful academically … the people with actual control at the companies are focused on academic success.  Personally, I fault ‘us’ more than the companies.  We have been telling the companies that the content for the tests needs to identify skills that the student does or does not have; skills are forgotten, and are vulnerable to trivial details.  If we would focus more on comprehension, application, and reasoning … the placement tests would have more meaning for us and our students.

A related issue is the use of placement tests in a deficiency model, such as some modular programs.  We sometimes expect a placement test to indicate whether a student ‘knows percents’ (alternatively, ‘does not need the module on percents’).  We should not use placement test for diagnostic purposes.  We might use well-designed diagnostic tests for this purpose, though I actually have more concerns about diagnostic tests than placement tests.  Diagnostic tests involve the effective ‘waiving’ of instruction; as a profession, I do not think we can support a 20-item diagnostic test as being equivalent to the instructional value of 3 weeks of class.  I digress!

Another issue with our use of placement tests is ironic:  We do not apply number sense to the results of a test.  For measurements of objects, we know that there is no signficant difference between 3.1 meters and 3.2 meters — if the measurements are made with a meter measure.  However, for placement tests having essentially 1 digit of precision, we often make a distinction between a 64 and a 61.  Take a look at the standard errors for your placement tests, and remain humble.  If tests are the only measure used, a ‘line’ needs to be drawn somewhere; this line might separate the 64 from the 61, but that does not mean that they are really different. Too often, we look at placement tests as if they were precision calipers when they are really meter measures.

The title of this post has two meanings (at least).  The obvious would be ‘will placement tests go away?’; I do not think so, and I would not advocate that.  Another meaning is ‘placement tests as ‘take-out’ process’ (like drive-in restaurants) … “would you like fries with that algebra test?”.  In other words, ‘give the student what they want’; over on the MATHEDCC discussion, this was the point for some people — just let students decide on their best math course.  Self-selection has been the subject of research, which I think has generally found less effective than placement based on tests for the entire population of students.

We do not need to get rid of placement tests.  We need to support changes in the content of those tests, and we need to show a better understanding of the measurements resulting from placement tests.  The ‘placement test problem’ is more about us as math faculty than it is about the tests.

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