Article: Knowing and teaching elementary language arts: a math lesson for English teachers.(

Article: Knowing and teaching elementary language arts: a math lesson for English teachers.(Report)
Article from: The Western Journal of Black Studies
Article date: December 22, 2009
Author: Bowe, Greg





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Introduction
When math educators in the United States talk about "the algebra problem," they aren't puzzling over a formula or an equation. They are struggling with the lingering effects of a pedagogical tradition dating to the establishment of widespread public education in America in the 19th century. Mandatory schooling became the law of the land in the 1850s, and its primary mandate was a utilitarian one: give us graduates who are literate enough to go straight from school to work. The majority of the jobs that needed filling were in a new sector of the economy, commerce, which served a burgeoning middle class. Shopkeepers and staff would need to be able to read competently and to add, subtract, multiply, and divide with speed and accuracy. This concern was reflected in the required school curriculum, in much the same way that "technology" is built in to curriculum today. That is, there may have been very broad variation from one state to another, but all 19th century states and school boards wanted their graduates to be able to survive in the world of 19th century work.
The curricular focus on functional literacy and "shopkeeper arithmetic" was quite successful. By 1907, the lovers in the popular song, "School Days" were able to gaze wistfully back on those "Golden Rule days" of "readin' and 'ritin' and 'rithmetic." Students studied arithmetic for eight years in primary school, honing their skills at computation. Only a very select few, however (somewhere between three and five per cent), pursued schooling beyond the eighth grade (as cited in Kaput et. al, 2008). The other ninety-five per cent never took algebra, geometry, or calculus in their entire lives. The problem for today's math educators is that, while we are living in the beginning of the 21st century, our schools are still teaching math as though it were the 19th century. Is this really a problem though? And what does it have to do with teaching English?
Times Tables and Prime Numbers
In the fall of 2005, I went to a community center in Miami to see some high school students from Bob Moses' Algebra Project run a Flagway tournament. I had no idea what Flagway was, but I did know that the students, all of whom had started out as low-performing students in a low-performing school, had been energized by the Project's ethic: don't remediate, accelerate. I was there because Lisa Delpit and Joan Wynne of the Center for Urban Education and Innovation at Florida International University had introduced me to Bob to see if I could work with the Algebra Project students on their language arts skills (at the time, I was the Director of FIU's first-year composition program).
I am not just an English teacher by virtue of my training. I also hate math of any description, and had always regarded my inability to grasp number concepts as a perverse badge of honor. In high school, I was stunned by the failure of my Algebra I teacher to understand my particular kind of verbal intelligence. He was followed by my Geometry teacher and finally by the Algebra II teacher. Clearly these math types didn't appreciate my skills, and I took my revenge by never taking a math course of any description in college.
On my way into the community center, I had to pick my way through dozens of third, fourth, and fifth graders. The very first child I spoke to, a fourth grader, was preparing for the game by reciting numbers to himself. I had done a little preparation for the day, so I actually did understand what he was doing, but I decided to see if he could explain it to me.
"2, 3, 5, 7, 11, 13..."
"Excuse me, but you skipped some numbers."
"Say what?"
"You skipped over some numbers there: 4, and 6, and then 8 and 9 and 10. What happened to those numbers?"
"No, I'm doin' the prime numbers."
"What are prime numbers?"
"Man, you don' know nothin'!"

I was impressed with the young man's response to
my disingenuous teacher's question. He went on to
not only define prime numbers, and explain--very
patiently, I would add--why l is not a prime number,
but also to show me how primes could be used
to factor larger numbers. And then he explained
the Flagway game to me, which, as it turns out, is
a competitive event, with lots of physical activity,
that depends on a ready knowledge of prime numbers,
the ability to determine the prime factors of
numbers, and a willingness to test hypotheses about
relations between and among numbers.
Watching my new math teacher and his friends run around the gym for the next two hours affected me deeply and in ways I never expected. I knew my arithmetic all right. Straight As in grammar school and 10, 15, or 20% of any amount from my days as a waiter, but these children knew more about numbers and math concepts in the 4th grade than I ever would. The next time I had a chance, I asked Bob Moses why he was teaching little kids about prime numbers. He explained that if a student hasn't got the times tables down by the third or fourth grade, he never will. You could do something much more interesting anyway: teach him about prime numbers. But, I wanted to know, why is it better to understand the prime numbers than to know your times tables?
The times tables are undoubtedly a useful device for internalizing an array of common arithmetic calculations and the surface features of multiplication. Through memorization and practice, students develop fluency in simple calculations. The times tables are what people are talking about when they urge us to 'get back to basics' in math class. Once that frame is drilled into students' heads, they can build bigger and more complex structures to support more involved math ideas. So, for seven or eight generations, Americans have learned the times tables in elementary school and then gone on to conquer space travel, develop nuclear energy, and invent the TomTom Go.
What the times tables don't do, however, is help students learn why or even how multiplication really works. Bob Moses, LiPing Ma and other math educators pressing for change believe that learning a list of all the products of numbers between 1 and 12 actually gets in the way of a student's understanding of the concept of multiplication (Ma 1999, Moses 2001, Kaput 2008). No matter how quickly a student can run up and down the scales of the times tables, there are important things he will miss. Here are a few examples of what English teachers, for example, might learn from middle-schoolers who took the prime numbers route instead.
* Students can learn the times tables forwards and backwards without ever addressing the commutative property of multiplication (5 x 6 is the same as 6 x 5) and without ever relating it to the commutative property of addition (2 x 6 is not only the same as 6 x 2, but both are also the same as 2 + 2 + 2 + 2 + 2 + 2).
* The times tables don't teach some interesting and useful numbers simply because they are not the products of two positive numbers between 1 and 12, despite those interesting numbers (46, or 51, or 69, for example) falling within the range of the products covered by the times tables from 1 x 1 to 12 x 12.
* They don't teach everything about even those numbers that do fall within the times tables' range. For example, students don't learn how to factor 36 from studying the times tables.
On the other hand, teaching young students about prime numbers directly addresses the mechanics and dynamics of multiplication.
* Students learn that 6 x 6 = 36, but as they look for the prime factors of 36, they also learn that 4 x 9 = 36, and that 2 x 18 = 36, and finally that 2 x 3 x 2 x 3 = 36 (as does 2 x 2 x 3 x 3 and 2 x 3 x 3 x 2).
* They learn that 46 (2 x 23) and 51 (3 x 17) share the fact that both are the product of two prime numbers.
* They learn that 46 (2 x 23) and 69 (3 x 23) would live next door to one another if there were a times table for 23.
Repeating the question from above--Is this really important? Mathematicians and math educators think it is. They are unhappy in a general sense with the quality of math education in the United States, and have been so for over thirty years (Klein, 2007, Moses and Cobb 2001). That internal disciplinary discussion has been consistently driven by outside forces, from the space race of the 1950s and 60s to the current comparison of student test scores from other countries. In 1999, a Chinese-born and educated math doctoral student at Stanford University submitted as her dissertation a comparative study of math education in China and the United States, and later published it as very accessible book (from whose title I have baldly stolen), Knowing and Teaching Elementary …