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).

Join Dev Math Revival on Facebook:

Is this the end of beginning algebra?

Sometimes, we are so accustomed to seeing the world in the same ways that we can not completely process new information.

When people look at the curriculum described in the New Life model (http://dm-live.wikispaces.com) there is no label that looks like a beginning algebra course.  Does this mean that algebra is not a valid domain for developmental mathematics?

The issue here is “What are the powerful ideas of mathematics” … ideas that will help students prepare for a variety of opportunities in college and in life.  If we focus on the good stuff — ideas with power — we will find ourselves renewed and our students rewarded.

In the case of ‘beginning algebra’, we normally sort through a topic list that has phrases like ‘expressions’, ‘linear equations’, ‘graphing equations in two variables’, ‘systems of equations’, ‘exponents and polynomials’.  These phrases are so entrenched in developmental mathematics that our textbooks are almost required to use this list to create their table of contents.  If these phrases are not used, we wonder if there is any algebra involved.

Instead, think about two fundamental ideas of algebra (just to start) — proportional quantities and variables.  Proportionality is fundamental to many concepts that students either need to use or will encounter in future courses.   This includes unit conversions and dimension analysis — work that exceeds the simple mechanical conversions we often limit ourselves to.  Proportionality can be studied in various representations; the numeric level is the applied level, but we should abstract this information to the graphical level and the symbolic level.

When we look at proportionality from a symbolic viewpoint, there is an opportunity to develop a deeper understanding of variables and constants.  Students leave our existing courses convinced that any letter is something to be solved for; the power of ‘variable’  focuses on the ability to represent information and relationships in a way that communicates well.

Rather than the demise of beginning algebra, we are actually looking at the revival of algebra.  Algebra is a set of powerful ideas; the reformulated curriculum focuses on these ideas … and avoids the artificial ‘procedural’ content that has been performing an impersonation of ‘algebra’ in our curriculum.