The mathematics K-12 teachers need to know

From a tweet by @dcox21:

…then they will realize how misguided it really is to teach the addition of fractions using the LCM of the denominators. p62 http://bit.ly/eLhu0y

The link is to a 70-page article by H. Wu (2008) called The Mathematics K-12 Teachers Need to Know.  I’m indebted to David Cox for pointing it out, and my skimming of it reveals much to be read and thought about as time goes on.  I was particularly interested in the alleged misguidedness of using the LCD (lowest common denominator) to add fractions.  Seems like a great method to me!  Anyway, on p. 62, Wu gives the example

$\frac{x^2}{3x^4+x+2} + \frac{6}{x^2+5}$

and notices that the addition is carried out using the formula

$\frac{k}{l} + \frac{m}{n} = \frac{kn+ml}{ln}$

Well, my senior students don’t have any great problem with fractions like that, and they use LCDs routinely to simplify algebraic fractions.  What’s more, they’ve never seen a formula like the $klmn$ one above.  Adding fractions is a first-principle process of finding a common denominator (lowest or otherwise), modifying each fraction to have that denominator, then combining.  For hairy algebraic ones like

$\frac{5}{x^2-4} - \frac{3}{x^2-4x+4}$

it’s a relief to find a common denominator because it makes the overall working and answer simpler.

I’m not seeing why I shouldn’t teach Year 7 students to add numeric fractions by finding a common denominator, nor teach Year 8-11 students to add algebraic fractions in the same way.  It seems to me it’s a great opportunity to reinforce understanding with every example given.

One thought that occurs is that textbooks in NSW emphasise the addition of complex algebraic fractions with worthwhile LCDs, but perhaps textbooks elsewhere do not.  Or perhaps Mr Wu doesn’t think such emphasis is warranted.  (I could be pressed into agreeing with that…)  But as it stands, for building algebraic skills and reinforcing fractional understanding, I fail to see what I’m doing wrong.

Reminder: I see much of interest in the 70 pages Wu offers, including other aspects of the teaching of fractions.  I just happen to disagree with this point.  At the moment.