## Which is easier to teach and understand – fractions or negative numbers?

Erlina Ronda is a Math Ed specialist at the University of the Philippines who seems seriously interested in teaching for understanding and her latest blog post asks Which is easier to teach and understand – fractions or negative numbers?

Interesting question!  But I suspect that it doesn’t have an answer as some people may be better conditioned (either by make-up or experience) to handle one, and others may find it easier to deal with the other.

With regard to fractions I think one of the main barriers, in addition to the fact that the notation is more complex (each fraction being made up out of two other numbers), is the non-uniqueness of representation – by which I mean the fact that two different fractions (eg 4/6 and 2/3 ) can represent the same value.

And with regard to signed numbers I think it is partly (as Erlina noted) breaking away from the concept of numbers as representing just some kind of quantity or size, and letting them also have a direction.  In fact the signed numbers (despite having the “advantage” of a more unique symbol) are better thought of as relationships (translations) than values (positions on the line).  So +5 is an increase of 5 units   and -3 is a decrease of 3 units (each of which corresponds to a whole family of differences just as each rational number corresponds to a whole family of divisions)

This approach makes it easier to understand the commerce-based rules for signed multiplication that were laid out by Brahmagupta in 7th century India. It may not be so obvious what was meant by the “product” in what is often translated as  “The product of two debts is a fortune” , but read as “the loss of a loss is a gain” it becomes much more obvious.

I guess I could say that signed numbers and fractions are both about relationships between pairs of magnitudes  – or more properly families of pairs that are all related in the same way (and when people come back to elementary arithmetic in advanced math classes that is really how they do it).

The problem with fractions then is mainly that this is more explicit in the notation (so the complication is visible) and the problem with negatives being that it is not (so something important is hidden)!

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