Concept Before Shortcut

 This year, I chose to do something different.  Students arrive in high school knowing so many shortcuts but they don’t know why the shortcut works.  For instance, when students divide one fraction by another, they are told not to divide but flip the second fraction and turn it into a multiplication problem, yet I’ve seldom seen students shown why it works this way.  

Unfortunately, shortcuts tend to make it faster to do certain problems but it doesn’t help students learn about the associated topic.  In fact, I would say, shortcuts emphasize procedure rather than understanding the concept they are working with.  I will be the first to admit that I have been teaching shortcuts because I learned math that way and my teachers program didn’t discourage their use.  

In addition, learning a shortcut often contributes to students solving problems by rote rather than seeing the concept that can be applied to a variety of situations.  When students learn shortcuts such as the turn the division of fractions into a multiplication problem where you are multiplying by the reciprocal, they often flip the wrong fraction.  

At the community college, they used the saying “Flip the right one, not the wrong one” to remember it was always the fraction on the right that was flipped.  If you had a problem where one fraction was over the other one, the saying didn’t work and students often set it up incorrectly.  Now I go through the whole process if we are dividing fractions or algebraic fractions so they see why one of the fractions is inverted.

Unfortunately, the use of shortcuts discourages the conceptual understanding of the topic which makes it harder for students to apply their knowledge to the math in later math classes.  When they learn the butterfly method for arithmetic fractions, they don’t know how it relates to algebraic fractions and if they try to apply it, they often end up with a mess.

I have nothing against shortcuts but I do believe students need to master the concept before they are introduced to the shortcut so they know what is going on.  When shortcuts are taught before they’ve learned the concept, most students never get the concept because they are focused on remembering the shortcut itself. Thus when they hit a more complex example of the concept, they may not understand how to solve it because they do not see how to apply the shortcut.

In Algebra I, I hit the chapter that shows the formula for multiplying perfect trinomials squared and the (x +8)(x-8).  I showed students via multiplying the two binomials why the book said (x +c)(x-c) = x^2 - c^2). When they saw the math, it made sense and without the mathematical explanation, I don’t think they would have connected the reason for no middle term.

So it’s important to show the whole process before teaching them the shortcut.  Let me know what you think, I’d love to hear.  Have a great day.