Do students really understand adding fractions?

I asked a year 7 student last year if he could add the fractions 1/5 and 1/4. He showed perfect working: 1/5 = 4/20, 1/4 = 5/20, 1/5+1/4 = 9/20. Since he was supposed to be teaching the class how to do this, I asked him why he found a common denominator? He thought for a while but could not answer the question.

It´s interesting that when I was growing up, I was taught a standard algorithm to add fractions and had no idea why the algorithm worked. It appears on the surface that this year 7 student has a better understanding of adding fractions than I did, but once I started to peel back the layers he was simply using the same algorithm presented in a different way.

It was only after drawing two congruent rectangles (one with 1/4 shaded and one with 1/5 shaded) that he saw that you couldn´t add the fractions until each rectangle was split up into the same number of pieces (the common denominator).

This suggests that he was either taught this topic in an algorithmic fashion or that he was taught to promote conceptual understanding but had simply forgotten it.

This will definately have implications for how I teach adding/subtracting fractions in the future.

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