Eleanor Wilson Orr, a teacher of math and science, was with her husband a director of the Hawthorne School for 25 years. The genesis of her recently published book "Twice as Less: Black English and the Performance of Black Students in Mathematics and Science" was her work with D.C. public school students who attended Hawthorne under a cooperative program with the District of Columbia Board of Education. Concerned because the students were failing an alarming percentage of their math and science courses, Orr found the roots of the problem lay in the grammatical and syntactical differences between black English and standard English. In this article she summarizes her thesis, illustrating how the failure to understand distinctions common to standard English can impede a student's ability to grasp certain mathematical concepts. "The result is nonunderstanding and misunderstanding," Orr says, "both of which remain beneath the surface as long as the work students do in their mathematics and science courses requires no more than the execution of well-learned algorithms."

In one of my high-school chemistry classes several year ago, a student said that the volume of a gas would be half more than it was if the pressure was doubled with the temperature remaining constant. When I asked if she meant that the volume gets larger, she said, "No, smaller." When I then explained that half more than would mean larger, one and a half times larger or half again as much, indicating the increase with my hands, she said she meant twice and with her hands indicated a decrease. When I then said, "But twice means larger, two times larger," again indicating the increase with my hands, she said, "I guess I mean half less than. It always confuses me."

By the time this exchange occurred, I at last understood enough to know what was happening. I had heard many black students -- intelligent and hard-working students -- lose their way as they carefully pursued a line of reasoning. I had begun to see that their language -- the tool they counted on to reason with -- was interfering. I knew that my chemistry student did not have automatically available to her the as . . . as structure of standard English (twice as many as; half as many as). I knew that she instead depended on than, and I knew something about the kinds of to page 26, 27 understanding that can go hand in hand with this dependence on than. Above all, I had learned how hugely important it is not to assume I know what a student means but instead to ask a number of questions to dig out the meaning.The Role of Black English TWENTY YEARS ago linguists made it clear that Black English is a rule-governed language as systematic and well-structured as any other language, that it is not just "bad English." Knowing that all children have mastered and internalized a language by the time they are 5 or 6, these linguists spelled out for educators how certain differences between Black English and standard English can interfere with a black child's learning to read. In 1979 a federal judge in Michigan found that Black English was a barrier to equal participation in educational opportunities, and thus found the School District of Ann Arbor in violation of the Equal Educational Opportunities Act of 1974: school officials had failed to take "appropriate action" to remove a "language barrier" that impeded equal participation. And yet today when school systems are at last making public the embarrassing gaps between the scores of white students and black students and are at last declaring their first priority to be the closing of these gaps, one hears and reads nothing about the role Black English may play in the poor academic performance of blacks. Too many preconceived notions have intervened.

What does a teacher do when an obviously intelligent student says that 50 miles is half as much as 25 miles? When another says that 25 miles is half more than 50 miles and another that 50 miles is half more than 25 miles? Another, that if you subtract 50 from twenty-five the remainder is 25? And another, that if you divide 25 by 50, the answer is two?

Such statements are precisely what one is apt to hear and read when students whose out-of-school language is Black English cannot rely on familiar numerical patterns in their mathematics and science courses but must instead reason with words. Most important, underlying such statements can be conceptual problems that are crippling in mathematics and science by the time one reaches high school.The Roots of Confusion THE SOURCE of the problem is that certain prepositions, conjunctions and relative pronouns -- essential in English to the expression of certain quantitative ideas -- are not used in Black English in the same ways as they are in standard English. For example, in standard English:

As is for comparisons made by multiplication or division (five times as many as; one-fifth as many as);

And than is for those comparisons made by addition or subtraction (five more than; five less than).

In addition:

Only a comparison by multiplication can, when the multiplier is greater than one, without negation, be expressed with either as or than (five times as many as; five times more than);

When a comparision is by addition or subtraction, it cannot be expressed with as (five as many as; five as few as);

And when a comparison is by division, it cannot be expressed with than (one-fifth more than; one-fifth less than).

Of course, speakers of standard English are not necessarily conscious of the above distinctions. The distinctions are maintained automatically. But a student whose first language does not include as . . . as cannot automatically maintain the distinctions.

Thus my chemistry student relies on half more than or half less than (instead of half as much as) for a comparison that is actually by division. But students also understand half more than as indicating twice. For them 50 miles is half more than 25 miles because 25 miles is one of the halves of 50 miles and the other half is the amount more 50 miles is than 25 miles.

Ingeniously, what is conventionally understood as multiplicative/partitive is, by means of than, thought through consistently as additive/subtractive.

As the student then begins to acquire the standard as . . . as structure, he or she understands it at first as just another form of the than structure. So half as much as, like half more than, also identifies the relationship of 50 miles to 25 miles rather than of 25 to 50 miles. As does not at first trigger a perception of multiplication or division.

Since the word half is employed to identify what is twice, students turn to using twice to identify what is half in expressions like twice as smaller than. Typically my algebra students would write, "So the car traveling twice as faster will take twice as less hours." Or "The hours of the faster car are less by two times than those of the slower car." And the same students said that 8 -- 2x is twice as less than 8 -- x.

Thus, just as they use half more than to identify what is twice as if it were additive, so they use twice as less to identify what is half as if it were subtractive. So when half more than didn't work for my chemistry student, she tried twice, and then resorted to half less than. Some students will reason as if to divide by two is to subtract two; some as if to divide by two is to subtract twice as much from a quantity in one instance as in another; some as if to divide by two is to subtract twice the quantity from the quantity.Consequences Of Misunderstanding SO THE TERMS in which the student thinks end up being the inverse of what he sees and hears in the classroom. And at a time when he or she is in the process of acquiring the language of school, the student will hear the teacher, and perhaps other students, say that 25 miles is half as much as 50 miles while the student, using the same expression, is thinking of 50 miles as half as much as 25 miles. The student is thus saddled with the burden of translating back and forth between the languages of school and his or her own inverted expressions. When this occurs in algebra where all those xs and ys must also be dealt with, the confusion and discouragement can understandably leave the student with no choice but to memorize any apparently correct patterns he or she can spot. And instead of the confidence that can come with understanding, thinking is replaced with a dependence on rote learning. Quantitative thinking appears to offer little reward and certainly no enjoyment.

The alternative, which sadly is too often the case, is to have the work required of students be performable by means of replicable model examples. School officials will feel better because scores on standardized tests will for a while be higher; students will temporarily feel they are doing well; and the potential misunderstandings will simply remain untouched beneath the surface of mechanical manipulations. But the trouble will surface at the high-school level as is evidenced annually by the drop in test scores reported for 11th graders.

The problem can be solved. But it can be solved only by facing it, by acknowledging it. The current preoccupation with test scores deals only with the cosmetic. Instead of teaching test-taking skills and dwelling on repeatable mechanical operations, teaching materials and classroom strategies must be designed to lead students into thinking on their own, into developing the confidence to trust their own minds. For a while test scores will not go up, but the results will show up later where it really counts. More black students will continue in science and mathematics beyond the high-school level. And most important, fewer bright black students will be relegated to second-class status in a job market where employability is going to depend increasingly on one's background in mathematics and science. ::