## Only Fractions

My love and I saw Only Yesterday recently. It’s a 1991 Studio Ghibli film, directed by Isao Takahata. It hasn’t had a United States release before, which is a pity; it’s quite good. The movie is about a woman, Taeko, reflecting on her childhood as she considers changing her life. One of the many wonderfully-realized scenes is about ten-year-old Taeko’s struggles with arithmetic. You probably guessed that, as otherwise the movie would seem outside the remit of this blog.

In the scene Taeko has had a disastrous arithmetic test. Her older sister is trying to coach her through how to divide fractions. It goes lousy. Her older sister insists it’s just a matter of inverting and multiplying. This is a useful tip if you understand how to divide fractions and need to keep straight what you’re doing. If you don’t understand, then it’s whatever the modern equivalent is for instructions on how to set a VCR.

Taeko tries to understand one problem, $\frac{2}{3} \div \frac{1}{4}$. [Edit, 25 April 2021: for years now I had the problem as $\frac{2}{3} \div \frac{4}{1}$ which is wrong.  And several people wrote to tell me of this and I somehow did not parse what they were writing about.  I apologize for all this confusion.]. She pictures it as an apple and draws a circle, blacking out a third of it. She cuts the rest into four equally-sized pieces and concludes that you could fit six slices into the original apple. Her sister stammers over this and fumes. She declares “that’s multiplication!”. She complains her sister isn’t doing the right thing, she’s not inverting and multiplying. I recognize her sister’s panic. It’s the bluster of someone trying to explain something not actually understood, on watching someone going far off the script.

The scene’s filled with irony. Taeko has a better understanding of what she’s doing than her sister has, but never knows it. Her sister understands a procedure but not what fractions dividing signifies. She can’t say why one wants to invert anything or multiply something. Taeko knows what the question she’s asked means, but not how to relate that to what she’s asked to do.

I don’t want to undervalue learning procedures. They’re worth knowing. They are, once you master them, efficient ways to compute. But there are many ways to master a procedure. I can’t believe there is one way to learn anything that works for everyone. One of many challenges teachers face is exploring the different ways their students best learn. Another is getting close enough to how they best learn that most of the students can understand something. It’s a pity when real people akin to Taeko can’t get that little bridge to connect their drawings of an apple to the page of fractions to be worked out.