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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