Parallel Thinking

In my most recent post, I discussed Marian Small's book Good Questions. I really wasn't a huge fan of the book, but there were some redeeming features that I did enjoy. Her view of parallel tasks is one that deserves some mention.

What do we mean by a "parallel task"? The idea of a parallel task is taking a question that a student might be facing and "scaling it back" to a question that is more familiar to the student. I tend to do this a lot in precalculus, especially when working with rational expressions. Let me give you some examples.

Suppose we have to add 1/(x-2) + 1/(x-3). We might write:

1/(x-2) + 1/(x-3)
= (x-3)/(x-2)(x-3) + (x-2)/(x-2)(x-3)  [find LCM]
= (2x-5)/(x-2)(x-3)                          [add the numerators]

To motivate this, we can think of the simpler problem of 1/2 + 1/3.

1/2 + 1/3 = 3/6 + 2/6 [find LCM]
                = 5/6         [add the numerators]

Sometimes the rational expressions get trickier with common factors like

1/(x^2 - 25) + 1/(x^2 + 8x +15)
= 1/(x+5)(x-5) + 1/(x+5)(x+3)                            [factor]
= (x+3)/(x+5)(x-5)(x+3) + (x-5)/(x+5)(x-5)(x+3)  [find LCM]
= (2x-2)/(x+5)(x-5)(x+3)                                     [add numerators]

To motivate this, we can think of the simpler problem of 1/6 + 1/9 (since 6 and 9 share a common factor, but do not share their other factors).

1/6 + 1/9 = 1/(2)(3) + 1/(3)(3)          [prime factor]
                = 3/(2)(3)(3) + 2/(2)(3)(3) [find LCM]
                = 5/(2)(3)(3)                    [add numerators]
                = 5/18

So this is the idea of parallel tasking. Where my students tend to have troubles is with the fact that they are not strong with fractions. Perhaps teachers are not showing the importance of using the LCM when finding common denominators (ie. 1/6 + 1/9 = 9/54 + 6/54... WHY WOULD YOU TEACH THIS?!), or not dedicating enough time to prime factorization (this second fault might be due to the awful curriculum we have that forces "mental math" strategies that only work in contrived situations).

In my opinion, parallel tasking is something that an excellent teacher will do and use effectively without thinking. However, it requires a lot of knowledge in the subject matter, and the ability to see how everything is interconnected (something most teachers do not attain from education faculties). But that is another post for another day.

Have you had any success with parallel tasks?