Unveiling the Hidden Embodied Mathematical Knowledge

I start this blog with a story of May – she was a recent immigrant student from the Philippines and lived in an urban city of Japan. 

Because of the immigration policy, May could not be granted permanent residency or citizenship. This political context placed many children of migrant families including May in stressful and uncertain circumstances. May could not picture where she would be living in the next year and she said she would not feel that she fully belonged anywhere. 

One day, I noticed May was doing something with her hands, but this was under her desk. She was hiding something as she solved a mathematics word problem. 

I became curious. I asked her what she was doing and she told me that it is something she learned from her parents. I then asked her if she can teach me. I also met her parents to teach me how it works. 

What May was hiding was the finger multiplication algorithm. This method was used for single digit multiplication of numbers between 6 and 9. 

Historically, there is mention of this method, in a documented published in 1888 (Ball, 1888 – A Short Account of the History of Mathematics). According to this classic text of history of mathematics, this method originated from Florence Italy and known as regula iqnavi

For example, May explained to me she would use it for 8 x 9. Using two hands, each hand represents a factor. Five is represented by the closed hand and any number above five is represented by the number of open fingers (e.g., eight is three fingers open). In the case of 8 x 9, another hand represents 9 (so, four open fingers). Then, you counted the number of open fingers in both hands and multiply this number by ten (product A) (e.g., in the case of 8 x 9, 7 times 10). Then, you multiply numbers of closed fingers in each hand (product B) (e.g., in the case of 8 x 9, one closed finger in one hand and one closed finger in another hand, 2x 1). The final multiplication product was calculated by adding products A and B (70 +2=72). 

This finger multiplication method can be also extended to the multiplication of numbers between 11 and 15 (with a variation), as I learned in the community. 

When parents taught children this finger multiplication method, it was taught as a secret. All parents I talked to emphasized that children should exhibit the mainstream ways taught at the school.

And even when children know this finger multiplication method, they tended to hide that in school and they were asked to use the mainstream method to memorize multiplication table. May was aware of what is considered to be legitimate at school –so she was hiding this method, under her desk.

After learning this finger multiplication method from May and her parents, I designed workshops to further explore this method mathematically that they were hiding, with other (im)migrant parents and children.

In the workshop, we explored why this method work, using algebraic expression (let x be one factor represented with one hand and let y be another factor represented with the other hand):

(5+x)(5+y)=(5-x)(5-y)+10(x+y)

(x-5) × 10 + (y-5) × 10 + (10-x) × (10-y)=xy 

We also discussed the decimal system underlying the finger multiplication method.

A scene sketch from the workshop…with children who were initially reluctant to try this method (because it is not a mainstream method taught in the school).

While watching the child participants trying out the method, Irene (parent) was nodding. After a while, Ryan (Irene’s child) all of the sudden shouted “Ah! You’re right. … This is amazing.” I asked participants to try 8×9. Another child participant, Brian said, “Oh wow.” Ryan looked at Irene and said, “I got it.” Irene clapped her hands for Ryan. 

Based on: Takeuchi, M. A. (2018). Power and identity in immigrant parents’ involvement in early years mathematics learning. Educational Studies in Mathematics, 97 (1), 39-53. doi: 10.1007/s10649-017-9781-4

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