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Language and Math: Thoughts from Piraha Studies Regarding Language, Comfort, and Recursion

Back when I was teaching Computer Science, I ran across an anthropology article (Science News, The Piraha Challenge) about a research team who tried to teach the Amazon Piraha tribe math and reading. Two particular aspects seemed particularly relevant to me both in teaching computer science and later engineering and STEM.

image by mimwickett (Miriam Wickett), via RGBstock.com

THE GIST OF THE PROJECT

The Everett's, Daniel and Keren, worked with the Piraha in the late 70s to learn more about their language. In the early 80's the Piraha asked them to teach them to count and to read. Eventually, though, both lessons ceased. 

The article explains the discussion among anthropologists and linguists about language and culture roles in the situation. With math, the fact that the Piraha language had only words that roughly meant one, two, and many made simple arithmetic difficult for them. With both lessons, it seemed that practices required to read and do math conflicted with their "cultural conviction on how to converse."

CULTURE AND COMFORT

Most notable was the belief that one should only talk from one's own personal experience. This eliminated language around "abstract concepts or ... distant places and times." In fact, trying to do so may have caused discomfort.

Think about it from a STEM point of view. The most basic mathematics problem is: "If I have two apples, and you give me another, how many will I have?" This concept requires thinking about something that does not yet exist in the real world--and in math, this is just the beginning of abstraction. If your student's culture feels this practice is wrong, success in the field will be difficult.

As a teacher, observing the student was just as important to me as communicating concepts and practices. When a student struggles, I always have to ask myself (or the student, if the right rapport is set up), "Why is this hard?" My job as a teacher is to then use this insight to make the subject make sense to the student. This is why teaching is an art, and really good teachers cannot be replaced by simple (or even complex) programs. 

This idea that a culture may actually feel that the typical practices of the subjects I taught would border immoral was fascinating to me. And perplexing. I don't think there is one, catch-all solution to it, but I share it here so other good teachers may at least consider it as they develop their lessons and teach their students.

RECURSION

One of the aspects the Everetts note is that the Piraha have no recursion in their language. There was some debate about this, but to me, the more interesting aspect is that recursion occurs in language.

My experience with recursion has always been in mathematics and computers. In fact, I was teaching a lesson on recursion the week I read this article, so I brought it up with my math majors, some of whom had already been introduced to it in their discrete mathematics course.

In case you don't know, in math, recursion is a process or definition that is really very simple: It has a general procedure that calls itself, using the next item in the series of numbers it will be applied to. It also has a termination condition. It's probably clearer with an example. Factorial is the most common example of a recursion function: When calculating the factorial of n, a number greater than 0:

  • See if n is 1. If it is, then the answer is 1.
  • If it isn't, then multiply n by the factorial of n-1
This means that 
  • the factorial of 1 is 1
  • the factorial of 2 is 2 times the factorial of 1 (which we see from above is 1) so it is 2 times 1 or 2
  • the factorial of 3 is 3 times the factorial of 2 (which we see from above is 2) so it is 3 time 2 or 6
  • etc.
In computer science, this concept is related to the stack data structure, and is one of the first standard algorithms taught. It was a very power concept in the early days of programming when memory was expensive because the code was tight, but the range was largely infinite.

But students always struggle with it. If you're a standard programmer (not a computer science major), you may be able to go through your whole life without ever using it. I've come up with a few techniques to explain how to design and decipher recursion functions, but there was always a struggle with the first grasp of the concept. If students don't feel that they have an inkling of the idea at the start, the rest becomes a struggle to memorize the bits and pieces they do get.

One of the key elements in recursion (and when implementing it in the computer) is that some information must be "held" until the next step is completed. In the case of the factorial, we know that factorial of 100 is 100 times the factorial of 99, but we have to hold that idea until we find out the factorial of 99, which requires us to wait until we figure out the factorial of 98, etc. Basically, we don't know the answer until we get the problem to be the factorial of 1, and then we can go back to all the multiplications we "held on to" until we got the answer.

This sounds daunting to a student. So I brought up this idea of recursion in the language. They were intrigued, and for some, it was a more gentle introduction to the nut graph of recursion.

Recursion in the English language can be easily seen in two places: 
  • Clauses: Consider the sentence "When I'm doing eating, we'll go to the store." You have to "hold" the data of "When I'm done eating" before you get the the main idea which is that we're going to the store. This makes me wonder if the Germans are particularly good at recursion because the active verb doesn't usually appear until the end of the sentence. Used to drive me bonkers. But in a way, it makes me realize that the action verb is very important to my understanding of the sentence. Perhaps the German mind soaks up more from all those other words and the verb is just a final concept to the mix.
  • Relatives: Consider the fact that the mother of your mother (a recursive relationship) is your grandmother. The mother of your grandmother is your great-grandmother, and then, it becomes just a matter of how many "great"s you need: great-great-grandmother, great-great-great-grandmother. The Piraha have a word for mother and then refer to their grandmother by given name. The language doesn't have this recursive relationship and therefore, it may influence the models and relationships for other ideas. 
Other studies seem to show strong links of language to STEM skills. Many studies I've been reading regarding spatial skills seem to indicate the importance of using spatial words early and often with young children. Even popular writer, Malcolm Gladwell, brought up language and math in his book Outliers:  The Chinese way of counting has a distinct pattern that correlates to the base 10 concept. Compare this to the English way of counting: The translated phrase for 11 is ten-one (not eleven), 12 is ten-two (not twelve), 20 is two-ten (not twenty) and 21 is two-ten-one (not twenty one), etc. This consistency of the language pattern may be helpful when learning numeracy.

LANGUAGE AND STEM, NOT LANGUAGE OR STEM

Too often, I hear from colleagues that there is a battle between the literacy folks in education and the STEM ones. Research like this show that it's not an either-or, but a tight partnership. The next advances in STEM education need to happen hand-in-hand with the literacy education. Both are important, and they help each other. So why not work together?

~ Until next time, Yvonne

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Yvonne NgSTEM or STEAM school, early childhood, elementary school, girls and STEM, middle school, teacher tips