True Story 10.13
Hurdles to teaching “half” to children of the Kpelle tribe
The setting for this story
For two years, John Gay and Michael Cole worked among the Kpelle people of Liberia, carrying out research aimed at improving the teaching of mathematics in Africa. Below they recount one of the hurdles encountered by Western teachers who were trying to teach Kpelle youngsters about arithmetical concepts that you and I regard as so simple that we sometimes say they’re “intuitive.”
A story of misaligned minds16
The Kpelle children with whom we were working knew almost no standard English when they began school. What English they did know was Liberian English, the local Liberian pidgin, with elements drawn from standard English as well as tribal languages. The children had become familiar with Liberian English before they entered school, where they identified it – inaccurately – with the standard English they were being taught. Confusion was inevitable.
An example is the word “half,” which means two things to a Kpelle pupil: (1) an ill-defined part of a whole or (2) a strange term used in school. Ordinarily, youngsters use “half” to refer to a part that’s as little as one-tenth of its whole or as much as nine-tenths of its whole.
If you ask a Kpelle schoolchild who’s been studying arithmetic, “One-half of what is eight?” they will be unable to answer correctly. They are willing to memorize “One-half of sixteen is eight” or “One-half of fourteen is seven,” and they often do. But they seem unable to grasp such facts at the level of general rules or abstractions.* In other words, after memorizing a series of statements using “half,” they still have no answer for a new question such as “One-half of what is ten?”
John and Michael’s question
Why is the abstract concept “half” almost impossible for Kpelle children to master and apply?
Critique of story 10.13
In Liberian English, two words could be translated as “half.” One is kpua, which can mean one of two or more portions, which need not be equal. The other word is hâvu; but again, the two hâvu portions need not be equal. In Kpelle daily life, rarely do any two things need to be equal. So using either of these Liberian English terms during arithmetic lessons is problematic.
Another reason concerns the Kpelle’s mindset. The closest they come in daily life to using precise measurement terms is in relation to rice, their staple food and a commodity constantly harvested and traded. For example, “two and a half (hâvu) sacks of rice” is understood as you and I understand it. But for them, that phrase refers to two and a half sacks of rice – only sacks of rice – which to them is one whole thing. Based on that phrase, “two and a half” cannot be repurposed to refer to anything else, e.g., two and a half onions, or even two and a half sacks of onions. So an arithmetic teacher cannot refer to “two and a half sacks of rice,” then point to hâvu to teach the meaning of “half” as an abstract concept.
Almost never do Kpelle youngsters witness adults separately referencing the color, shape, size, weight, pattern, or other quality or characteristic of anything. For them, each fact is indivisible, complete, and accepted on the authority of tradition. The mindset of the Kpelle is holistic or relational, 180° different from the analytic mindset that, to youngsters in WEIRD societies, becomes highly familiar because (a) they constantly hear adults using an analytic mindset; (b) they are taught to perceive and think analytically at a tender age by adults using toys designed for that purpose; (c) many of their other toys have multiple parts intended to be manipulated (building sets, doll houses, model trains); and (d) analysis is implicitly and explicitly taught in Western-style schools.
Early life for young Kpelle includes few opportunities to have their attention drawn to the parts of wholes or to notice that wholes or parts have abstract qualities such as “half” that can be applied to a wide range of things – half a sack of rice, half a fish, half a meter, half an acre, and so forth.
For thought
Why do you and I regard a range of arithmetic concepts – half, average, triangle, digit, etc. – as basic and simple? Do we arrive in this world with these concepts ready for use in our brain? Few would agree with that notion. So if such concepts are not present at birth, they are learned. In our modern industrialized societies, children learn them – or better, absorb them – by being surrounded by people and ideas that make use of them. For example, as a youngster, you gradually became aware that, whenever your parents spoke the word “half,” they were referring to two things of roughly equal size and/or weight. Several years later, when “half” was taught in your primary school, the formal definition made sense to you. It seemed “intuitive” because, at least subconsciously, you’d become familiar with it before you entered school.
Meanwhile in Liberia, Kpelle children – at least during the era when Gay and Cole’s research occurred – had no comparable experiences. They were growing up in another perceptual world, which they figured out how to understand and successfully navigate in ways that differ from ours.
Related stories
Story 10.03 relates a similar challenge: teaching the meaning of “1” to children in the Carolina Piedmont. Story 10.16is also drawn from John Gay and Michael Cole’s research with the Kpelle.
* Abstraction refers to a process whereby a general concept or rule is derived from concrete examples. Consider polar bears of various ages, sizes, etc.; the concept white can be derived from them all and applied to other things, e.g., snow. Consider “one half of eight is four,” “one half of six is three,” etc.; the concept half can be derived from these examples and applied to other statements such as “one half of ten is what?” Abstraction also refers to the resulting generalization or rule; half is the abstraction. People raised to think analytically learn to do this; people raised holistically ordinarily do not. See also the note at the end of Story 10.16.
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Endnotes:
16 Gay & Cole, 31–35, 41, 94.
Full citations are available at misalignedminds.info/References.