Instant Presentations now available!

There is a new page in DevMathRevival.net — called “Instant Presentations”.

The idea here is simple: provide an on-demand presentation that people can share and view.  Similar to a webinar, and more direct.

Initially, the page has 3 presentations — the Mission of Developmental Mathematics; What is Now; the New Life Vision.  Each of these presentations is under 5 minutes, with video and audio.  [The format is “Flash video”, which means that browsers will handle the files for you when you click on them.]

Give it a try!! The direct link is https://www.devmathrevival.net/?page_id=116

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Standards Based Reforms — What Research Says

The United States has seen a number of ‘standards based reforms’ over the past 20 years or so. Many of these deal with school mathematics, though a few of us in the college environment have worked towards a similar process. The most recent effort is the “Common Core Standards” (http://www.corestandards.org/ ), which is the highest profile effort yet.

The Rand Corporation published a report, in 2008, called Standards-Based Reform in the United States: History, Research, and Future Directions (online at http://www.rand.org/content/dam/rand/pubs/reprints/2009/RAND_RP1384.pdf).  I was impressed by some of their observations.

First, “Standards Based Reform” is usually implemented as “Test Based Reform”.  The point here is that content and pedagogy reflects a testing emphasis such that the actual standards are secondary — the tests (such as those used for No Child Left Behind, NCLB) take on the primary importance.  Behind this is a tension you will understand: Standards, by themselves, produce very little change.  “Aligning” testing to the standards is very common, and very understandable, as a method to create change.  Change is not always progress, however.

Second, high-stakes testing with sanctions ‘distorts’ teaching practice; as you’d expect, teachers focus more on preparing for the test when there are sanctions involved.  In general, most of the current testing involves sanctions of some kind such as NCLB or state-level impacts.  Since tests must, by design, address small subsets of the larger domain of knowledge described by the standards, the result tends to produce students who can perform better on the tests connected to the sanctions compared to other measures of their knowledge.  Specifically, they do not do relatively as well on our college placement exams. 

Third, the report goes back to a critical document that describes 4 categories of standards … and also analyzes the track record of some specific efforts.  A shot blog post is not an appropriate venue to report on these comments (I don’t want to inflict a journal-length article on you 🙂 ).

Although community colleges have not faced the standards based reforms and tests with sanctions directly, we deal with the consequences of these efforts.  Some policy makers assume that the “developmental math problem” will go away once the standards are implemented (like the Common Core).  The Rand analysis provides some insight into why the problem is not that simple; we should assume that our problem might change in the next 10 years … not that it will go away.

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The Sum of All Shortcuts

As I work with my beginning algebra students, and think about how they can learn this ‘stuff’ for use later, it occurs to me that we have developed a curriculum based on shortcuts.

Okay, so what do I mean?  “Shortcuts” are the separate rules that are provided to describe WHAT to do with a type of problem.  For example, this week we had negative exponents.  Our textbook, like almost all others, says that negative exponents show the reciprocal … students know that x^(-2) should be written as 1/(x^2).  Is this really how students should see this idea?  I do not think so.  For this particular notation, the origins come from needing to show a division … x^(-2) means dividing by x^2; this division meaning provides a nice connection to positive exponents and to place value, in addition to being more accurate.  In spite of these advantages, why do we so often show the reciprocal meaning?

The ‘shortcut’ property of this is not isolated.  Open any book, listen to any of us talk in class, and you will see (hear) shortcuts.  When we add two fractions, we need a common denominator; we can add like terms.  Do we connect these ideas (they are the same principle)?  To solve an equation, we ‘do the same to both sides’ … it’s a balanced scale; do you realize how many students have a visual map of this that is strictly positional — not even dependent upon having an equality statement?  (Just show them  ‘3x + 5 + x + 4’ and see how many subtract 4 or x.) 

More?  How about “is over of” … ‘circle groups of 3 numbers inside a cube root’ … ‘Y1, Y2, intersect — answer is x’.  ‘PEMDAS’. Is there anything substantial in our curriculum, or is our curriculum the sum of all shortcuts? 

Most shortcuts developed as an effective device to help students remember what to do, so they could arrive at more accurate answers.  If you have some old textbooks around, check out this theory.  I believe that textbooks evolve as they are published in new editions, and new ones mimic the newer ones, so that the content is often examples and shortcuts.  In the name of simplicity and ease for students, we take out the substantive narration around the shortcuts; the back-story is lost, and students think that the tricks they see are the real mathematics.  This is not doing a favor to our students.

One of the reasons to revitalize the curriculum is to give us a fresh start.  We can go back to the mathematics, the back-story, the connections.  In theory, we could take out the shortcuts and ‘fix’ what we have.  Unfortunately, our instructional practices are so wrapped up in the shortcuts that I suspect we will not identify even a majority of the shortcuts.  As mathematicians, we value understanding connections, applying concepts, and problem solving … shortcuts present a clear and present danger to these values.  The prevalence of shortcuts is not limited to developmental math classes; I see a number of them at the next level as well (whether it is college algebra or pre-calculus).  However, I have to say that we in developmental mathematics use shortcuts to a much greater extent.

It’s not that I do not want students to get correct answers.  This is about transferring knowledge — dis-connected knowledge (shortcuts) has little chance of being used in any other context.  This is about students remembering what they ‘learned’ — unstructured knowledge (shortcuts) forms stories to be remembered, and need to be indexed and accessed in the same manner.  This is about an education, which is more than the sum of shortcuts (or facts).

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Because the Data Says So!!

A very brief story:  During the portion of a statistics course dealing with inference, the professor presents data on predictors of success in graduate school.  The variables included test scores (GRE, GMAT, or similar), undergraduate transcript data, economic background, and lifestyle (diet, exercise, etc).  The analysis of this data was conducted to see which variables correlated the highest with eventual success in graduate school.  The undergraduates listened with some interest, and expected the professor to announce that it was the grade in an undergraduate statistics course the correlated the best; the class was sad to hear the identity of the winner — eating cooked carrots.  Eating cooked carrots (as opposed to raw, or not eating any) had the strongest correlation to graduate student success.  The statistics professor then stated the obvious conclusion:  We should only accept students who eat cooked carrots, and we should all start eating cooked carrots; only after a uncomfortable minute did the professor challenge us to examine the validity of transitioning from data to a policy decision. 

Last week, the Washington Post published an article on the “Algebra II movement” for all high school students.  (See http://www.washingtonpost.com/business/economy/requiring-algebra-ii-in-high-school-gains-momentum-nationwide/2011/04/01/AF7FBWXC_story.html?)  

Within the movement, the policy makers cite data showing that success in Algebra II is the strongest predictor of success in higher education and the workplace.  Based on this pattern in the data, the ‘obvious’ conclusion is that we need to require all students to complete Algebra II in high school.  As part of the mathematics community, should we accept this support for mathematics even though it is clearly based on faulty reasoning?  Does it matter to college math professors & instructors?

We should definitely care about this … “algebra II” is a self-reinforcing mythology that affects mathematics in the first two years of college.  To some extent, our entrance requirements, our placement tests, and our graduation requirements are all predicated on the “algebra II standard”. Policy makers tend to assume that students who have ‘had algebra II’ should not (as a group) need to repeat it in college — there is an expectation that remediation needs will decline over the next few years. The Common Core Standards (http://www.corestandards.org/the-standards/mathematics) are consistent with the algebra II movement, and many states are adopting these standards with the promise of lessened needs for remediation.

Developmental mathematics is one of the few professions in which the practitioners would like to be in a world where their services were not needed; I know I would be happy beyond description if most students arrived at our doors with sufficient mathematics to enter directly into college mathematics. However, past experience indicates that this outcome for the Common Core & Algebra II are far less than certain.

Of course, we as college math faculty can not (and should not) seek to undermine any standard or policy for school mathematics. However, when possible, we should share our expertise and judgment with policy makers so that they might have a more realistic expectation for the results of a proposed change in school mathematics.
We, along with our professional associations, should seek to remain engaged partners with efforts to improve the mathematics preparation in our schools and colleges.

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