“Ours is not to reason why, just invert and multiply”-- Week 8

One of the most common methods of teaching students how to divide fractions is to use the saying: “Ours is not to reason why, just invert and multiply”.  I find this troubling because throughout my undergrad courses, I have constantly been taught that as an educator, it is my job to ensure students are able to make sense of the concepts being taught.  However, this saying contradicts everything I learned because it discourages students from reasoning and understanding mathematical processes.  Many students struggle with fractions, so when you bring in a rule about dividing factions without working with students to build an understanding of the method used, it discourages making sense of the mathematics.  As future educators, it’s important for us to encourage students to practice ‘knowing’ rather than simply ‘doing’.  i.e. we need to teach students in such a way that promotes understanding of the concepts rather than rely mostly on the rules and procedures to get to an answer. 

I think there are many mathematics rules which we use often that are similar the inversion example, in that we do not know how to explain why we use them if a student were to ask.  I think teachers often answer questions posed by students about why we use certain rules, and why they work with statements like: “That’s just how it works” or “Just do what I taught you to do” because they do not know how to explain these procedures.   For example, when asked to explain why we invert fractions when dividing them, many of my colleagues, including myself, found is difficult to do so.  I found this interesting because we all continued onto post-secondary education in mathematics, however, when it came to explaining a concept taught in grade 7 mathematics, we struggled.  This made me think of how as a student, I was good at following the procedures in math, however I may not have taken the time to make sense of many of the rules being presented to me.  As a future educator, I do not want to teach students to use rules and methods simply because I told them it works.  Rather, I would like my students to practice active learning and thoroughly understand the math processes.  For this reason, I think it’s important for math educators to start reflecting on what they do not know, and figure out how to explain and understand these processes in student friendly terms. 

In the example of inverting fractions and multiplying in order to solve a division problem with fractions, one explanation would be to have students first look at how .  I would than 

remind students that  .   Two other important concepts to .  So if we were to divide , which could be written as:   Form the previous example, we know that a number multiplied by its reciprocal is 1.  For this reason, we will than multiply   (which we know is equal to one, so it does not change the equation).  This equation than becomes , and this is why we invert and multiply when dividing by fractions.