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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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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