Fractions are Hard!
2.1 Context for a unifying approach
I work with upper-middle school and high-school students on the topic of fractions. (But haven’t students mastered fractions by this stage of schooling?)
There is often a great deal of emotional angst around the topic of fractions and I work to remove that angst by starting at the very beginning of the topic again, in an emotionally appealing setting. It is my deliberate goal to work towards obviating and feelings of fear and dread and a sense of judgment that “you should already know this.” (This applies as much to my high-school seniors as it does to junior high-school grades.)
I go through this same story for each and every grade level. I do not skip details as students often have new levels of insight seeing it all again after their intellectual might has developed after another year or two.
I bounce off of the ideas students have experienced from grade school: fractions as coming from sharing problems (division problems) and use the imagery of parts per whole. But I point out that no model in mathematics is perfect and will capture all the features of any mathematical system being explored. So our goal here is to identify the key beliefs that seem to be speak to some truth about how we think fractions should behave, list those beliefs, and then see how all the “rules” of fractions that were taught in grade school are actually logical consequences of those fundamental beliefs.
This topic and approach is absolutely appropriate for people who have already gone through the typically grade-school fraction training. Moreover, we will now be conducting the process of real mathematics: Identifying the key beliefs that seem to make the theory hold together and then exploring the logical consequences of those beliefs. (And there is nothing stopping us from doing that in a natural and appealing way!)
Fractions really are an advanced topic worthy of high-school attention after all.
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