Workshop: Secondary Students' Understanding of Distributive Property

@ Singapore Polytechnic on 5 March 2008 (3-5 pm)
by Dr Ida Mok Ah Chee 莫亚慈
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From the beginning, Dr Mok already 'tuned' our expectation - she emphasized that what's she going to share is on a study where she used an instrument to diagnose the 'cause', and not to propose any solution to the problem identifed. The purpose is to surface symptoms so as to help one to draw out more information about a problem. She also shared that, from the researcher's point of view, we learn from when the 'subjects' made more mistakes - there's where we could find out how one thinks.

Was wondering - what's distributive rule - to me, it's just a procedure that we perform, when need to 'open' brackets - ie. doing expansion. It's amazing, especially when we go beyond 2 terms within each factor. Things can get so complicated and become out of hands!

In fact, it goes all the way back to our understanding of the 'letters' used in mathematics - its meaning changes when put in different context. Here are some commonly made mistakes when dealing with distributive law.

Example 1: (t+1)²
=t² + 1²
= (t² + 1)

Example 2: 5 (x²)(2xy)
=(5 x x² )(5 x 2xy)
= (5 x² )(10 xy)

Example 3: cos 90° - cos 30°
= cos(90° - 30° )
= cos 60°

At one point, it's mentioned that Distributive property is a systemic rule that links multiplication and addition - and in fact, division "is" multiplication while addition "is" subtraction. Well, sounds abstract??? Not really, it brings us back to the fundamentals...

• 5 - 2 = 5 + (-2)
• 5 ÷ 4 = 5 x ¼

Dr Mok shared with us the 4 questions that were used in the study; and all participants had the chance to attempt (of course, we were reminded to think - how our pupils will respond to those questions! Hey, that sounds familiar, isn't it?) True enough, a range of strategies (to solve the problems) were shared - some were the 'long-forgotten' methods like substituting numbers to test! Here are some: