In Twice As Less: Black English and the Performance of Black Students in Mathematics and Science, Eleanor Wilson Orr suggests that differences between black English vernacular and standard English usage limit some black students’ understanding of mathematical and scientific concepts.
Ms. Orr, a founder and teacher at the Hawthorne School in Washington, concludes that "[i]f these students are to be successful ..., they must acquire the verbal features that are essential to the expression in standard English of mathematical and scientific ideas.”
“Making the work easier,” she contends, will not help. Rather, “the habit of learning by pattern must be broken; the dependence on the replicable as the only guarantee of being correct must be replaced by the habit of thinking.”
In the following excerpts from her study, Ms. Orr analyzes examples of comprehension problems arising from nonstandard uses of prepositions and conjunctions.
There was explicit evidence that these students were using one kind of function word, prepositions, in a manner different from other students; their misuses were different even from the misuses with which we were familiar. Although there was no direct evidence yet that their usage of relative pronouns and conjunctions was also different from that of other students, the fact that these students tended to see the ideas of main and subordinate clauses as interchangeable, while other students did not, suggested that the signals picked up by these students from these function words might be different from those picked up by other students--or perhaps that these students picked up no signals at all from these words. ...
[These examples of typical errors] suggest the possibility that when words, or symbols, are used as instruments with which to think, the use in one language of a single symbol in contexts where a second language requires two or more can lead a speaker of the first language to arrive at a different mental construct of some given information from that arrived at by a speaker of the second language. ...
I want to stress how much of what a student is actually thinking can remain hidden because one tends to assume one knows what a student means by what he or she says in spite of the way he or she says it. The tendency is to see the student’s words in relation to what one is thinking oneself, and to assume as well that as long as one knows the standard English equivalent of a nonstandard expression there is no problem: One “knows” what the speaker or writer “means” by the expression.
This is simply not the case. ...
I had known that the students’ written and oral work was replete with references to distances as locations and with cases of locations being treated as distances, but I looked upon these as simply the result of carelessness or of a lack of practice in writing about such situations.
It had never occurred to me that nonstandard perceptions could accompany such nonstandard usage--perceptions that might lie at the root of some of the difficulties the students were having in geometry. ...
In English one identifies a distance by the names of the two locations that determine it: These names are held in relation to one another and to the rest of the sentence by certain prepositions. The prepositions most commonly used are from, to, and between, as in the distance from Washington to New York and the distance between Washington and New York. In my experience, students who are in the habit of using these prepositions according to the conventions of standard English do not confuse location and distance; and students who confuse location and distance do not use these prepositions according to those conventions.
The combinations from ... to ... and between ... and ... can trigger the need to name the two locations that determine a distance; in a sense each combination sets up two spaces that need to be filled. Without such a “space holder” the structure collapses: The spaces are not there, and what might otherwise be separated is not. For example, a student writes, Aurora Ohio is equal to the distance from Lakeland to Wash. Here two cities that determine a distance have collapsed into one, in the sense that one of the cities has itself become the distance: Aurora has become the distance from Washington to Aurora.
Whole phrases and even clauses can collapse for students who are required to speak or write about a number of distances when they are not yet in the habit of using the standard English distance prepositions. And the collapse in sentence structure is often accompanied by a collapse in idea. ...
Unfortunately, useful as the term collapse is, it could be understood to indicate that, for example, the entire from ... to ... structure exists for the student before the collapse, in the same sense as cannot exists for a speaker of English who then may or may not choose the contracted form can’t. That is not what I wish to convey. ...
[T]he problem with these collapsed sentences is that when they are read in context, what the students have in mind can seem obvious; the assumption is that the standard equivalents of these sentences are what the students “mean,” or are what the students would say if they had available to them the expressions of standard English. But in fact the students often think exactly as the words in their own sentences indicate.
When they speak of locations as being equal, they are not necessarily just using the word location when they actually “mean” distance; ... they may actually be thinking of locations as having magnitude in the same sense as distances have magnitude and of these magnitudes as being equal when the distances between these locations are equal. And when they speak of two distances as being “equally apart from one another,” they are not necessarily just using the word distance when they mean location; they may actually be thinking of distances as being separated by distances in the same sense as cities are separated by distances. ...
Thus, lack of familiarity with the standard English distance prepositions can lead to collapsed sentences, and the standard English literal meaning of a collapsed sentence, rather than its standard English equivalent, can indicate what a student is actually thinking. ...
In addition to being prepositions of distance, from and between are also prepositions of subtraction, as in subtract from and the difference between. In fact, a metaphor of distance is used to depict subtraction: Subtraction determines the distance between two locations on the number line. ...
Like many of these students’ verbal expressions of distance, many of their verbal expressions of subtraction are, to a speaker of standard English, both unconventional and confusing. Furthermore, these students frequently misinterpret problems involving subtraction, and they do so in a way that, from the perspective of standard English, matches their verbal expressions. Many of these students also reason according to an unconventional perception of subtraction, which again, from the perspective of standard English, matches their verbal expressions. And this perception of subtraction, especially when combined with an unclear image of distance, interferes with the students’ understanding of negative numbers. ...
[J]ust as these students’ lack of practice in producing sentences that use from and between as distance prepositions often results in collapsed sentences, so too does their lack of practice in producing sentences that use these prepositions as subtraction prepositions. ... When there is no distinction between the possible directions of a subtraction, negative numbers make no sense. Subtraction without direction is equivalent to thinking only in terms of absolute value. ...
To me, it is clear that the perceptions these students have of certain quantitative comparisons are shaped by the language they use.
They come to school without an as ... as structure in the language they speak. Before acquiring a multiplicative mode of expression, they are required to talk and write about comparisons that are multiplicative and partitive as well as additive and subtractive. Using the -er ... than structure, they produce additive and subtractive expressions of the comparisons they need to discuss. They see and hear the as ... as structure but understand it as additive. They begin to use it as additive, sometimes with a vector adjective (twice as larger as), sometimes merging it with the -er ... than structure (twice as large than, twice as larger than). Then, picking up the multiplicative meaning of the as ... as structure, they employ as ... as in subtractive expressions of division, using marked adjectives to indicate decrease and multipliers to indicate the amount of decrease, as in twice as less. In spite of knowing the algorithms of subtraction and division, they then interpret algebraic expressions and think through quantitative relationships according to the standard English, subtractive meaning of their own expressions.
