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. . 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.

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## Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.
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to Joseph:

In the film (*), the problem assigned is to divide 2/3 by 1/4. Takeo is using her drawing of an apple to visualize 2/3 divided by 4. That’s 8/12 divided by 4, or 2/12 = 1/6, Takeo’s answer.

I rarely “invert and multiply”. Instead, I change to fractions with common denominators; in this case 2/3 to 8/12 an 1/4 to 3/12. How many times does 3/12 go into 8/12? More than twice but less than three times; 2 and 2/3 times.

I really enjoyed your post, especially the juxtaposition of “invert and multiply” and “whatever the modern equivalent is for instructions on how to set a VCR”. That made me laugh–gallows humor, though.

Roger Purves

(*) The site blog.wowzers.com (September 11, 2013) has the relevant clip.

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I think moving things to common denominators should help. At least it makes it possible to take the denominators out, and what’s left looks much more like regular old division. But that does also add extra steps to the process, and there is this drive to not do unnecessary steps. But I am biased toward taking extra steps when learning how to do problems.

Thanks so for the kind words.

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I love this movie, here is my two cents writing as an Asian growing up in Asia:

(First, , as the person above says, the actual math problem is 2/3 ÷ 1/4 = 1/6 (!?)

I didn’t have Takeo’s problem….I just didn’t question my teacher, who was an authority in our society. The true issue for me was calculus: it took me a long time to understand differention. Both my textbook and my teacher took the approach of directly cutting to calculation; there was very little explanation on what was done–and it was placed in a blue box in smaller text, signifying “outside syllabus and not included in the standardized test”! My teacher skipped it altogether. And it didn’t help thst the textbook was written in poor English–the same level of language proficiency you find in math major international students.

And that was my Takeo moment…. I flunked it. Luckily I finished my study in the US, where I got to take calculus again, this time fully knowing what to expect. And finally in the US did somebody filled me in on what exactly the ____ I have been calculating all those years :)

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I appreciate the perspective! I have a sliver of understanding of a Southeast Asian perspective, from having taught in Singapore for a half-decade, but that’s not a good way to learn what it’s like to grow up learning subjects a particular way.

It seems to me like many people who are comfortable learning

`these are the rules to calculate by'' crash when they get into calculus. It seems weird. After all, you can lay out maybe a half-dozen important rules for differentiation. Combine that with knowing some special cases like`

the derivative of the arctangent function” and it seems like someone should have all the mechanical operations they need. But it isn’t that straightforward somehow. The easy thing to say is that calculus just has too many abstractions and too many things to do for older techniques to work. But trigonometry and algebra are fields that are far more open, and abstract, than the classes that come before, and fewer people crash there.I was always fortunate, as a native-English-speaker, that there were plenty of mathematics books written in a language I should have been able to understand. But, then, the first time I took Real Analysis, I was not having much luck with any of it, until I took from the library a textbook written in French. It didn’t help me to mastery of the subject, but it did keep me from completely failing. I think it’s that I am not a fluent reader in French. I could piece my way through sentences in the book. But I had to put a lot of time and thought into each of them, and at least pieces of that took.

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