Full opinion text
GARWOOD, Circuit Judge: The Denison Independent School District [DISD] appeals from certain orders of the district court entered in this school desegregation litigation. I. Introduction. Denison, Texas is a small city located some seventy-five miles north of Dallas. The urban portion of the school district is likewise small, measuring approximately five miles by five miles, though the district also extends several miles beyond the city limits, principally to the north and south, and includes largely rural areas. Residential patterns are such that the black population is principally concentrated in a relatively compact segment of the northeast quadrant of the city, while the western half of Denison, which contains the newer centers of population, is almost exclusively white. During the 1979-80 school year, the DISD provided education to some 5,200 student's, of whom approximately 12 percent were black. II. Background of this Appeal. Until the spring of 1963, the DISD operated a school system that was completely segregated by race, with a dual set of attendance zones for black and white students. On June 24, 1963, the school board, the district’s governing body, voluntarily passed a resolution instituting a “freedom of choice” plan of desegregation, to be phased in on a one-grade-a-year “stair-step” basis. That plan was challenged in a suit brought by a class of black school children and their parents who sought desegregation of all grades in the Denison public schools by the 1964-65 school year. The school board’s plan was approved by the district court, but this Court found the nine years required to fully implement the plan to be excessive in light of the command of Brown v. Board of Education (Brown I), 347 U.S. 483, 74 S.Ct. 686, 98 L.Ed. 873 (1954). Price v. Denison Independent School Distinct, 348 F.2d 1010 (5th Cir.1965). Upon remand, the district court entered an order, dated July 19, 1965, which established a “freedom of choice” plan effective the ensuing school year for all grades in the district. There was no appeal from this order. Thereafter and until the revival of this litigation in 1979, the lawsuit remained essentially quiescent. In 1968 the DISD commenced operating only one high school for all students, and has continued to do so since that time. In 1969 the DISD voluntarily and unilaterally abandoned the “freedom of choice” scheme for grade levels below high school in favor of a neighborhood attendance zone plan. The attendance zones in that plan appear to have remained in effect without substantial change until the 1979-80 school year. In the summer of 1979, the DISD promulgated, effective that fall, a new neighborhood attendance scheme involving the closing of three elementary schools, the conversion of a middle school to an elementary and the drawing of new attendance zone boundaries, principally for the elementary schools. It was the adoption of this new scheme which prompted institution of the present action. On July 31, 1979, shortly after the announcement of this new plan, the plaintiffs and a group of plaintiff-intervenors filed a joint motion for substitution of parties, intervention, and an amended complaint claiming that the 1979-80 attendance plan continued the vestiges of historical segregation in the DISD and seeking relief from the allegedly unconstitutional elementary and junior high student assignment plans and unequal junior high school educational conditions, as well as purported deficiencies respecting teachers, assistant principals and principals. Plaintiffs specifically complained of the closing of the three elementary schools, alleging, among other things, that “white students from the less wealthy socio-economic groups have had their neighborhood schools closed and attendance lines redrawn where they can be counted to cut the minority black percentage down at Terrell [elementary]” and that some 41 black students “would be bussed from North Denison to an elementary school [Mayes] in West Denison.” A temporary restraining order was also sought (unsuccessfully) against the closing of the three elementary schools. It is evident that the contemplated closing of these schools was the motivating factor in the revival of the lawsuit. Hearings were held on August 11-12, 1980. On December 17, 1980, the district court found that the combined differences, in educational conditions and racial composition, between Denison’s two junior high schools, Hughes and McDaniel, violated the Equal Protection Clause of the Fourteenth Amendment and ordered unitization of the seventh, eighth and ninth grades served by those schools. Pursuant to the DISD’s request, the district court, in March 1981, amended this order to include the sixth grade, placing the sixth and seventh grades at Hughes, the eighth and ninth at McDaniel. The junior high school unitization was to go into effect with the 1981-82 school year. On June 10, 1981, the district court entered an order and memorandum opinion which (1) reiterated its findings and order as to the junior high schools; (2) found the 1979-80 elementary school student assignment plan to leave some schools racially identifiable and in violation of the Constitution; and, (3) found that the failure of the DISD to currently have a black principal in any of its schools presented a constitutional violation. The school board was required to submit to the court, within thirty days, an elementary assignment plan which would “assure that the population of each elementary school will be at least six percent black by the beginning of the 1981-1982 school year”; and, to employ a black individual in the next available principalship vacancy. On July 6, 1981, the DISD filed notice of appeal from the June 10 order. On August 12, 1981, the DISD submitted its proposed elementary assignment plan for the 1981-82 school year. The district court rejected that plan in an order of August 14, 1981. The court found that although the DISD’s proposed plan was “in formal compliance with the minimum standards set forth in the June 10 order,” it nevertheless “violate[d] the spirit of the order” because it insufficiently reduced (from 33.8 percent to 29.3 percent) the percentage of black students at Terrell elementary. The -August 14 order required the DISD to submit an “elementary assignment plan which would assure that the population of each elementary school will be no less than six percent black and no more than seventeen percent black by the beginning of the 1981-1982 school year.” On September 1,1981, a hearing was held on a motion for stay of the August 14 order filed by the DISD. At this hearing the parties introduced a “joint exhibit” reflecting school attendance zones that, on the basis of rough calculation, would create elementary school populations in approximate compliance with the August 14 order, both parties agreeing that these attendance zones would be submitted to the school board for approval. On September 8, the school board rejected the proposed plan. On September 18, 1981, the district court entered its final order in this suit. The court (1) denied the motion for stay pending appeal; (2) vacated its order of August 14; and, (3) ordered that the elementary school student assignment plan reflected by the referenced joint exhibit, with some relatively minor modifications, be implemented in January 1982. III. The Issues. In this appeal, the DISD raises diverse questions as to every aspect of the district court’s orders. Its principal challenges are to the district court’s determinations of constitutional violations in (1) the elementary school student assignments; (2) the junior high schools; and, (3) the absence of a black principal. The DISD further contends that the court’s findings and conclusions do not support its determination of constitutional violations. IY. The Elementary Schools. A. Background. For the school year 1979-80, the DISD operated seven elementary schools for approximately 2,800 students. The seven facilities were relatively evenly dispersed throughout the city, with Terrell elementary in the northeast, Lamar in the east, Golden Rule in the south, Hyde Park in the southwest, Mayes in the west, Layne in the northwest, and Houston elementary located in the center of Denison. The previous school year, 1978-79, Denison had nine elementary schools. The plan which prompted this litigation and went into effect for the 1979-80 year closed three of these elementary schools, Central (then 55.3 or 56.9 percent black), Peabody (.9 percent black) and Raynal (9.7 percent black), and converted Terrell, then a grade six, seven and eight middle school (24.2 percent black), to an elementary. The former Peabody attendance zone was divided between Houston, Golden Rule and Lamar. Part of the Central attendance zone was assigned to Mayes, the remainder to Terrell, which is located some five blocks north and .three east of Central. Raynal and Lamar, each located somewhat east of Terrell, had previously served students in grades five and below, with sixth graders from these zones attending Terrell middle school. Under the 1979-80 scheme, all of the Raynal attendance zone was assigned to Terrell. Lamar sixth graders continued to attend Terrell. Each of the seven elementary zones under the 1979-80 plan was geographically a wholly contiguous unit, as was also the case with the previous nine-zone plan. The evidence indicated that the principal purposes of the 1979-80 plan were to effect some economies and instructional improvement by reducing the number of elementary schools and to eliminate the then majority black elementary school, Central. The United States Department of Health, Education and Welfare had notified the DISD that the black concentration at Central should not exceed 32 percent. A competing plan advanced by some of the plaintiffs was rejected by the board because it did not reduce the number of schools and insufficiently reduced the concentration of blacks, which would have been over 40 percent at Central. The following table shows the elementary school statistics for the 1980-81 school year (the statistics which the district court principally focused on), as well as the black student percentages for the years 1978-79 and 1979-80: 1980-81 1980-81 1979-80 1978-79 School Black All Other Total Black % Black % Black % Golden Rule 12 318 330 3.6 1.7 2.4 Houston 35 348 383 9.1 13.1 12.2 Hyde Park 5 362 367 1.4 .8 .9 Lamar 25 322 347 7.2 10.0 9.5 Layne 2 339 341 .6 .8 .3 Mayes 38 390 428 8.9 8.4 .3 1980-81 1980-81 1979-80 1978-79 School Black All Other Total Black % Black % Black % Terrell 211 414 625 33.8 32.6 24.2 Central - Closed Closed 56.9 Raynal - Closed Closed 9.7 Peabody - Closed Closed .9 328 2,493 2,821 11.6 During the 1980-81 year, some 59 percent of the white elementary students attended schools having more than 7 percent black students, and some 64 percent of the black students attended Terrell. During the period of dual racial attendance zones, Central was an all-white elementary school, and Terrell was an entirely black school apparently serving all grade levels. Terrell was entirely closed at the end of the 1967-68 school year. It reopened for the 1969-70 year as, in the words of the joint pretrial order, “a desegregated middle school serving grades 6-8.” It then had a 23.9 percent minority enrollment. Terrell remained as a middle school through the 1978-79 year, its highest minority enrollment being 29.05 percent in 1975-76 and its lowest 22.5 percent in 1970-71 and 1971-72. Central’s minority enrollment was less than 2.5 percent until the 1966-67 year when it became 25.5 percent. In 1967-68, it became 53.9 percent minority. Thereafter, minority enrollment varied from a low of 49.4 percent in 1969-70 to a high of 63.5 percent in 1976-77, falling somewhat in 1977-78 to 56.7 percent. Layne, Hyde Park, Houston and Lamar apparently were all-white schools during the years of dual racial attendance zones. In the years 1966-67 through 1977-78, Houston’s minority enrollment varied from a low of 8.4 percent in 1966-67 to a high of 16.2 percent in 1976-77. Hyde Park’s low was .75 percent in 1967-68, its high being 2.4 percent in 1971-72. Layne had no minority students until 1971-72, and thereafter the percentage varied between 1 percent and 1.4 percent. During this period Lamar’s minority percentage increased from 15.3 percent in 1966-67 to 22.6 percent in 1971-72. It was closed for renovation in 1973-74, and reopened (apparently without a sixth grade) in 1974-75 with an 11.2 percent minority enrollment, thereafter reaching a high of 12.54 percent in 1977-78. There is no evidence concerning elementary school closing or construction prior to the 1962-63 year. Until 1979-80, the only school closings, other than Terrell, were the all-black Langston and Dalton schools, apparently elementary, which were closed just after the 1964^65 year. Apparently only two elementary schools have been added. Golden Rule, in the south central part of the district, previously served a small adjoining rural school district which merged into the DISD commencing with the 1964-65 year. Its minority enrollment that year was .3 percent. It had no minority students the following year. Thereafter, its minority enrollment varied from a low of .9 percent in 1969-70 to a high of 6.1 percent in 1976-77. Mayes, located in west Denison, was opened in 1968-69 and had no minority students until 1971-72, when the minority enrollment was 1.7 percent. Thereafter and prior to 1979-80, minority enrollment at Mayes varied from a low of 1 percent to a high of 4.11 percent. In 1979-80 it became 8.4 percent. The DISD’s elementary schools were, and apparently always had been, essentially “neighborhood” schools, in that the great majority of students attending each school lived in its immediate vicinity. Students living more than two miles distant, principally those residing in the rural areas, were bused to school. Prior to the 1979-80 year, some 10 to 12 percent of the elementary students were transported in this manner. Apparently, these students were disproportionately white, reflecting the racial composition of the rural students. Under the 1979-80 zones, the percentage of elementary students bused increased to about 24 percent. About 9 percent of those bused were black (some 19 percent of all black elementary students). Most of the increase in busing of black elementary students was necessitated by the northeastern extension of the Mayes attendance zone to encompass a predominantly black area in the western part of the former Central zone. Two desegregation experts, Dr. Edward Tapscott and Mr. James Williams, were appointed by the court in December 1979 to study and report on the DISD. Neither recommended any material change in the DISD’s elementary school attendance scheme put in effect in 1979-80. Williams described it as “an appropriate plan, particularly evidencing good faith,” and Tapscott said it was “a good-faith effort to try to keep neighborhood schools and yet at the same time draw an attendance zone which would move minority students into some of the other elementary attendance zones.” Integrative aspects of the 1979-80 system which the experts commended were the northeastern expansion of the Mayes zone to incorporate a mostly black area formerly in the Central elementary zone, having the Lamar sixth graders attend the new Terrell elementary, and the closing of Raynal, with the concommitant assignment of its heavily white, northeast rural students to the new Terrell elementary. The experts refused to fault the district for not moving the Mayes boundary still farther east, since the 1979-80 move had already placed its line only a few blocks west of the Terrell facility. Because of Layne’s size and condition, Tapscott “would be hesitant of reassigning large numbers of students to it.” He also would not condemn the elementary system because its only pairing involved the Lamar zone sixth graders attending Terrell. Though certain other plans which had been suggested resulted in substantially greater racial “balance” in the elementary schools by extensive “pairing” and transportation, Tapscott and Williams were unwilling to recommend any of them in preference to the 1979-80 system or to say that they were materially superior or other than “about equal” to it. Respecting the Peabody closing, Tapscott testified that among the elementary schools Golden Rule, Houston and Peabody, which were located relatively close to each other, it made sense to close one, and Peabody was a logical choice as it “was the closest to the downtown area and would probably be able to serve the community for the least amount of time.” The downtown area was losing population, particularly of school age. Further, Tapscott considered that the board “made a proper decision” in closing Raynal, which was “too small,” and “reassigning those students to Terrell.” None of the experts criticized the closing of Central or the conversion of Terrell to an elementary school. The experts were concerned with the disproportionately high percentage of black students at Terrell elementary, located in the heart of an overwhelmingly black residential neighborhood. However, this appeared to be more a fear for the future than a criticism of the present. The concern principally related to the possibility of future “white flight,” which would take place if the “tip ratio” usually experienced in Texas of “approximately 35% black” were reached. As Dr. Tapscott explained the “tip ratio,” “Once [a school’s black population] reaches that 35%, then it seems to jump very quickly to 50 to 60 and move very fast to become a minority school .... The patrons find some way to another elementary zone . However, the experts felt that this did not require a present change in the Terrell attendance zone, but rather indicated that the board should carefully monitor the situation there. Superintendent Alexander testified that the board was committed to such monitoring, and would take steps if Terrell became over 35 percent black. Testimony also disclosed that the district’s honors program for grades four, five and six was located at Terrell, and intra-district elementary transfers were limited to that program. Similarly, all elementary transfers from without the district were assigned to Terrell. Dr. Tapscott, in describing the 1979-80 plan, said it placed at Terrell a larger number of white students “than would normally be there with the elementary attendance boundaries as they would normally be drawn.” The experts were high in their praise of Terrell. Dr. Tapscott described it as “probably the best educational program that I observed going on in the elementary schools of Denison” and as “the best elementary school in the district.” Williams “wholeheartedly” concurred, and testified Terrell represented “the best use of a formerly all-black school” he “had ever seen.” While Tapscott and Williams were generally commendatory of the board’s 1979-80 elementary program, they were somewhat critical of the district’s performance in past years. The criticism, however, appears to have been essentially that the board had waited too long to do what it did in 1979. Thus, Dr. Tapscott addressed the prior situation where Central had been over 50 percent minority and most of the other elementary schools (generally five out of eight) had minority enrollments several percentage points under the minority elementary percentage of the district as a whole, stating: “I would have to conclude that they should have responded earlier, that they didn’t and, therefore, I would have to assume that they wanted to keep it that way for some reason, and they should have responded much earlier than they did.” Mr. Williams also was critical of the board’s prior inactivity, though he would not ascribe it to segregative intent. He answered in the affirmative when asked, “whatever their intention,” whether the building of Mayes was a cause of the Annex and Central having become “resegregated.” However, Williams seemed to regard the previous high concentrations of blacks at Central (and at McDaniel junior high) to be largely the result of demographic factors unrelated to school board actions or educational considerations. He testified in that regard: “[W]hy and how did this city get to seem to be a town that had the east half of town and the west half of town with an awful lot of railroad yards and ponds and lakes and all sorts of things in between, but ... it really boils down to Old East Denison and New West Denison. “One of the explanations ... was Perrin Air Force Base. A tremendous housing development in the late fifties and, primarily, the sixties all, of course, to the west, typical Texas pattern, and with the closing of Perrin Air Force Base, it seemed to me that the district and the town went under a considerable condition of economic stress, ... and, consequently, it seemed to me that even though a district may have had the best of intents at any given moment, it did not have the best tax base.... “[I]t was a poor economic base, a disruptive closing of an Air Force Base, and an apparent pattern established by realtors, none of whom, you know, are responsible to the school, of putting all the new folks out west and leaving an interior city, the old City of Denison, on pretty shaky ground when it comes to who is left there. “Quite often you have elderly people, retired railroad people, people with no children. We found this a unique and disturbing type of condition ....” Other evidence showed that the district suffered an approximately 20 percent enrollment decline from 1969-70 to 1979-80. While the black student percentage in the district as a whole remained in the 10 to 12 percent range during this period, the testimony of Mr. Williams suggests that the Perrin-sparked westward population movement, largely composed of whites of an age to have younger children, raised the percentage of black elementary students residing in the areas of “old” or “east” Denison. Williams also testified: “Q. Do you see anything in the past actions of the district as it relates to school closings and attendance zones and that sort of thing that has caused a greater concentration of black people in the Terrell area? “A. No, I did not.” Elementary classes in the DISD were generally composed of 25 students. While some black residents testified that they preferred an even district-wide dispersion (theoretically resulting in three blacks per class), at least one of the plaintiffs’ witnesses stated there should be no less than five blacks per class of 25 or 30. Williams indicated his general concern with only 10 percent blacks in a class, and also stated, “I have never approved of having only 1 or 2 or 3 blacks in an elementary classroom.” Plaintiffs’ expert Dr. Gay testified that eight or nine blacks in a class of 25 (32 to 36 percent) was generally a better situation than two in such a class (8 percent). B. District Court Orders. (i) Violations found and relief granted. In the portion of its June 10, 1981, opinion dealing with the elementary schools, the district court stated: “The primary issue as to the new elementary assignment scheme is whether it allows any particular elementary schools to remain as racially identifiable vestiges of segregation. It is well settled that the existence of racially identifiable schools in a formerly de jure dual school system constitutes prima facie evidence of a constitutional violation, which can be overcome only by a showing that the racial composition of the schools is unrelated to the prior discriminatory educational structure. Swann v. Charlotte-Mecklenburg Board of Education, 402 U.S. 1, 18, 26 [, 91 S.Ct. 1267, 1277, 1281, 28 L.Ed.2d 554] (1971).” (Footnote omitted.) Relying on the language in Swann v. Charlotte-Mecklenburg Board of Education, 402 U.S. 1, 26, 91 S.Ct. 1267, 1281, 28 L.Ed.2d 554 (1971), concerning “a presumption against schools that are substantially disproportionate in their racial composition,” the district court ruled that Layne, Hyde Park and Golden Rule elementary schools were racially identifiable (as white), since the proportion of black students in each varied by more than five percentage points from that of the elementary school population of the district as a whole (11.6 percent). While in regard to Terrell certain aspects of the June 10 opinion are ambiguous, considered as a whole the only fair conclusion is that it did not find Terrell racially identifiable. The court accordingly ordered the board to submit “a plan designed to ensure that each elementary school within the district will contain a minimum Black population of six percent as of the beginning of school in the fall of 1981.” The plan submitted by the school board met the requirements of the June 10 order, and was not objected to by the plaintiffs, but was rejected by the district court in its opinion of August 14. The court there noted that in its opinion of “June 10, 1981, Golden Rule, Hyde Park, Layne and Terrell were found to be racially identifiable.” While the plan submitted by the board reduced the percentage of blacks at Terrell from 33.8 percent to 29.3 percent, the district court held that “Terrell remains a racially identifiable school” since its proportion of black students varied by more than five percentage points from the 11.6 percent district-wide black elementary percentage. For this reason, the court rejected the plan, and ordered the board to submit a plan “which assures that the population of each elementary school will be no less than six percent black, and no more than seventeen percent black.” At a September 1 hearing on the district’s motion for stay, a potential compromise elementary plan was developed which was later rejected by the board. On September 14, at a resumption of the stay hearing, the plaintiffs requested the court to order a modification of such plan into effect, and the court did so, effective January 1982, by its September 18 order. Although this order vacated the August 14 order and purported to enforce that of June 10, nevertheless, under the plan ordered implemented the projected elementary attendance figures would meet the requirements of the August 14 order that no school’s proportion of black students be less than 6 percent or more than 17 percent. This was largely achieved through noncontiguous zones and increased busing. (ii) Determination of racial identifiability- It is in some respects difficult to determine whether the district court treated the issue of racial identifiability as essentially one of law, ultimately determined by the general statistical criterion of a variation as large as five percentage points in the school’s proportion of black students from that of the district as a whole, or as a question of fact to be determined for each individual school and district in the light of all the relevant circumstances. The June 10 opinion in part evidences something of the latter approach. Its analysis in this regard commences by stating “an institution is racially identifiable when its racial composition differs significantly from that in the district as a whole.” It then cites authorities for the proposition that, in districts with black student proportions similar to the DISD’s, racially identifiable schools “have been held to be those whose Black population deviate by more than five percentage points” from the district-wide percentage. Nevertheless, the June opinion then sets out the following qualification; “Of course, it must be emphasized that such a statistical measure is only a starting point, albeit a valuable one, in determining the presence of racially identifiable schools. Other factors may militate against a conclusion of racial identifiability and counteract the import of the statistics.” However, the “only a starting point” approach seems to have been applied chiefly, if not entirely, to Terrell, with the result that, in the June 10 opinion, Terrell was not found racially identifiable. The determination in the June 10 opinion that Golden Rule, Hyde Park and Layne were racially identifiable appears to be essentially a statistical one, coupled with the fact that these schools had been all white in the time of dual racial attendance zones. Moreover, the district court’s subsequent orders herein reflect that it treated the question of racial identifiability essentially as one of law, to be ultimately determined strictly on the basis of a generally applicable statistical formula. This approach is evident in the court’s order of July 17 (denying motion for stay) which states that any school in the district is “racially identifiable” if its black population is “greater than 17.6 percent, or less than 6.6 percent.” Similarly, in the order of August 14, in which it rejected the board’s plan submitted pursuant to the June 10 order, the court stated, “Terrell school is racially identifiable, because black students are relatively over-represented in that school.” (Emphasis added.) No new or different factual analysis is made of Terrell, nor are any different facts stated or conclusions drawn, as compared with the June 10 order, with the sole exception of the ultimate statistical determination of racial identifiability. The August 14 order states that “a racially identifiable school is one in which the racial composition of the school varies more than five percentage points, plus or minus, from the racial composition of the school district,” then recites the DISD percentages, and concludes by stating, “any school with more than 16.6 percent black population is a racially identifiable school.” (Emphasis added.) In its final order of September 18 the district court again reviewed the methodology it had followed in its determinations of racial identifiability. This order reflects that the district court did not consider the issue of racial identifiability as, in any meaningful sense, a factual or evidentiary question, dependent on local circumstances or history, but rather viewed it as purely a question of law to be ultimately determined by a generally applicable statistical formula. The court labeled as “mistaken” the contention that selection of an appropriate statistical variation for application in a given case was an “essentially ‘evidentiary’ ” process, and stated that among the DISD’s “many misapprehensions” was the “assumption that the determination of racial identifiability is a factual issue.” Instead, the court ruled that “a standard for determining racial identifiability is a legal matter.” It explained that its choice of a five percentage point deviation factor as the relevant standard for the DISD, though it represented an application of the statistical formula espoused by Dr. Gordon Foster as reported in the opinions in Penick and Higgins (see note 30 supra), was nevertheless a “legal conclusion” amounting to the adoption by the district court of this statistical formula as the broadly applicable legal standard by which racial identifiability in all cases of this general nature, not just in the DISD’s case, would be determined. The September 18 opinion then addressed the analysis of racial identifiability as made in the June 10 opinion. Preliminarily, it again observed that “racial identifiability” is “a legal conclusion,” and that when such a legal determination is made respecting “a school” in a former de jure system, such “determination establishes a prima facie case that the school system violates the Constitution.” This prima facie case, however, can be rebutted by evidence that the racial identifiability of the school in question “is unrelated to the prior history of legally mandated segregation.” The court then turned to the specific language in its June 10 opinion that “a statistical measure is only a starting point, albeit a valuable one, in determining the presence of racially identifiable schools.” The court observed that such phraseology contained “an unfortunate verbal confusion,” as the term “racially identifiable” was used in the quoted sentence of the June 10 opinion to mean “a finding of constitutional dimension, rather than the prima facie conclusion.” As we understand it, then, the court’s September 18 opinion holds that a school’s “racial identifiability” is determined finally by a purely statistical analysis, and that for purposes of such determination the statistics are not “only a starting point.” Moreover, as we read this portion of the September 18 opinion, the status of racial identifiability (“the prima facie conclusion”), determined statistically and as a matter of law, gives rise, again as a matter of law, to a prima facie case of unconstitutionality. The school district cannot prevent the prima facie case from arising by presenting evidence of “other factors” which “militate against a conclusion of racial identifiability and counteract the import of the statistics.” But the case is only prima facie, and may be rebutted by showing that the racial identifiability is unrelated to past (or present) de jure segregation; however, if the prima facie case is not rebutted, then the racial identifiability is conclusively presumed to result from de jure segregation and hence constitutes a condition the school board is constitutionally required to eliminate (“a finding of constitutional dimension”). In line with this analysis, the September opinion finds that Terrell, on the basis of the statistical formula, is “racially identifiable as black,” although it concedes that “unquestionably, Terrell school is predominantly white, since the student population is two-thirds white.” Accordingly, the court holds that the “DISD is charged with the responsibility of establishing that this characteristic [racial identifiability as black] is not traceable to a history of racial discrimination.” At this point, however, the path of the September 18 opinion becomes less plain. It states “as the June 10 order makes clear, DISD singularly failed to carry this burden.” We are unable to so construe the June 10 opinion. Not only does it conspicuously fail to find Terrell racially identifiable (see note 25 supra), it likewise contains no indication that the DISD failed to carry any burden respecting Terrell. The September opinion continues by stating, however, that other evidence “militated against the conclusion of racial identifiability” of Terrell, citing its educational quality. This seems to suggest, contrary to the above-noted implications of the September opinion’s discussion of the “unfortunate verbal confusion” concerning “racial identifiability,” that statistical racial identifiability may be rebutted by evidence, such as quality of education, not directly related to whether the statistical condition results from present or past intentionally segregative acts by the educational authorities. In any event, in the remaining portion of the September opinion, the district court does not expressly address the effect, factually or legally, of the evidence which it found to “militate against the conclusion of racial identifiability” of Terrell. Nor does it expressly address the presence or absence of any other rebuttal as to either Terrell’s racial identifiability or any prima facie case arising therefrom. The September opinion does refer to “the conclusions of the June 10 order ... that the racial identifiability of Terrell establishes a presumption of unconstitutionality.” However, as we have noted, the June 10 order makes no such determination. The district court in its September opinion placed heavy reliance, in its racial identifiability analysis, on the following from this Court’s opinion in Lemon v. Bossier Parish School Board, 566 F.2d 985, 987 (5th Cir.1978): “In the conversion from dual school systems based on race to unitary school systems, the continued existence of all-black or virtually all-black schools is unacceptable where reasonable alternatives exist.” The district court then concluded that school desegregation law had become sufficiently “refined” so that it was appropriate to substitute the concept of “racially identifiable” as white or black (presumably as determined statistically under the Foster formula) for the concept of “all-black or virtually all-black” as set forth in Lemon. Accordingly, though the matter is not entirely free from doubt, considering all the district court’s opinions herein dealing with the elementary schools, we conclude that it finally treated the issue of whether a school was racially identifiable as a question of law ultimately determined by a legally established statistical formula of broad general application, calling by its terms in this instance for a maximum deviation of five percentage points, and not as a discrete question of fact depending on, or subject to rebuttal by evidence of, the particular history or circumstances of the school or district in question. (iii) Determination of constitutional violation. Having determined the existence of racially identifiable elementary schools in a former de jure district, the district court noted that such schools, absent a showing that their racial composition is unrelated to racially segregative action by the school authorities, must be “found to be violative of the Constitution.” We observe at this point that the ultimate constitutional violation, the existence of which may be presumed from the unrebutted prima facie case, is not merely racial imbalance in the schools, Milliken v. Bradley, 433 U.S. 267, 280-81, 97 S.Ct. 2749, 2757, 53 L.Ed.2d 745 (1977), but is rather “a current condition of segregation resulting from intentional state action.” Washington v. Davis, 426 U.S. 229, 240, 96 S.Ct. 2040, 2047, 48 L.Ed.2d 597 (1976). Accordingly, assessment of whether the prima facie case has been rebutted involves a determination of whether the current racial composition of the schools does not result from past or present intentionally segregative action by the educational authorities. Obviously such action includes that which is covertly, as well as avowedly, for a segregative purpose. The current racial composition of the schools will, of course, result from intentionally segregative action if the present system of student assignment is adopted or maintained with a segregative purpose. Where that is not the case, the current racial composition of schools may nonetheless reflect, in whole or in part, the present effect of past intentional educational segregation. This may be the ease where the effects of such past educational segregation are reflected in current housing patterns, or school locations, to the extent that such conditions are in turn reflected in the complained of current racial composition of the schools. See United States v. Texas Education Agency (Lubbock I.S.D.), 600 F.2d 518, 525 (5th Cir.1979). The district court in its June 10 opinion expressly found that the “DISD has failed to fulfill its burden of proving that the current racial identifiability of Layne, Hyde Park and Golden Rule is unrelated to past discriminatory action.” It seems apparent that this determination was not to any extent based on a conclusion that the DISD had failed to demonstrate that its actions, either in adopting the 1979-80 plan or at any time prior thereto and after the dismantling of the dual racial attendance zones some fifteen or sixteen years previously, were taken without segregative purpose or intent. There was certainly evidence reflecting both present and past good faith on the part of the DISD. In its June 10 opinion the district court stated: “Certainly, some progressive efforts have been made in the past by DISD, and the plan instituted in 1979 represented an improvement over the previous assignment pattern.” And in its order of July 17, denying the district’s request for stay of the June 10 order, the court further addressed the question of intent, as follows: “[M]any of the measures taken [by the DISD] in the past fifteen years were conscientious efforts to ameliorate the disturbing legacy of segregation that was inherited from a time when separate and unequal schooling for students in the Denison school district was enshrined in law and custom.... However, the command of the law is clear — the good intentions of officials in the school system cannot compensate for actual failures of any plans ultimately implemented. If the plans do not work, then they must be rejected .... Thus the laudable past efforts of DISD to diminish the vestiges of racial discrimination may not now be used as a foil to the continuing affirmative duty, incumbent on DISD, ultimately to eliminate all vestiges and establish a truly unitary system of public education.” (Emphasis added.) Moreover, we note that just before reconvening following the mid-afternoon recess on the first day of trial, while the plaintiffs’ case-in-chief was still being presented, the court, sua sponte, advised the parties as follows: “THE COURT: All right. It’s my sense that we’re wasting a lot of time in this case talking about intent. The Supreme Court held in Dayton Board of Education against Brinkman in 1978 at 443 U.S. 526 [99 S.Ct. 2971, 61 L.Ed.2d 720] — the decision came down July 2, 1979 — says, ‘The measure of the post Brown I conduct of a school board under an unsatisfied duty to liquidate a school system is the effectiveness, not the purpose, of the actions in decreasing or increasing the segregation caused by a school system.’ “So, we don’t have here a question of intent; we’ve got a question of effectiveness, so I suggest that all of this testimony about intent is totally irrelevant, so let’s go forward with the evidence that is relevant.” The above statements not only expressly reflect the court’s view that intent was irrelevant and should not be further explored, but also, by implication, its view that there was no issue for further evidentiary development, or for factual determination, as to whether the current racial compositions of the schools resulted from past illegal segregation by the educational authorities. The district court did not expressly address the question of whether the residen-. tial concentration of blacks in a segment of the northeast quadrant of Denison was shown not to have been caused by intentionally segregative acts of the educational authorities. It merely stated in this connection: “The school district cannot meet the presumptive burden raised by the presence of the racially identifiable schools with the contention that the present level of segregation results from the neighborhood school policy, in combination with segregated housing patterns.” No evaluation was made of the testimony of Mr. Williams, the court-appointed expert, as to some of the causes for Denison’s racially concentrated housing. It seems apparent, then, that the district court did not make any kind of factual determination or analysis as to the presence or extent of any relationship between the current racial compositions of the respective elementary schools and intentionally segregative conduct, past or present, on the part of the educational authorities. Rather, the court simply stated that the presumption from statistical imbalance was unrebutted, and hence the statistical imbalance was automatically equated to a constitutional violation. C. Review of the District Court Orders. As the district court aptly noted, Denison’s case presents “uncharted terrain” in this field of jurisprudence and “is unusual among the gamut of southern desegregation litigation, in that the city’s school system has a relatively small percentage of Black students.” Approximately a decade and a half has elapsed since all overt segregation in the DISD ceased, and in that time it is unquestioned that considerable progress has been made without judicial intervention. There are no all-black or virtually all-black schools. There are no majority black schools. In regard to its several elementary schools, the DISD follows, in apparent good faith, a neighborhood school policy, and there is no suggestion that the attendance lines have been segregatively gerrymandered. Because of the small number and residentially concentrated nature of the black elementary students, it is plain that, even if somewhat integratively gerrymandered, normal geographically continuous neighborhood attendance zones would nevertheless still result in a disproportionately large number of blacks in one of the zones and a disproportionately large number of whites in others. The disproportionately black elementary school was nonetheless two-thirds white, and was described by impartial experts as the best in the district. Transfers to it were encouraged. It had had large white majorities for over ten years. It is described as being a “desegregated” school as long ago as 1969-70. There was evidence that the residential concentration of blacks was caused at least in part by factors unrelated to educational policies or actions. The proportion of blacks in the district’s elementary student body, 11.6 percent, was so small that a completely even dispersion would result in all schools having white percentages so large (over 88 percent, or, with a five percentage point variance, several schools over 90 percent) as to be at a level generally considered “virtually one race.” Nevertheless, there were no 100 percent white schools, and over half the white students attended schools with approximately 6 percent or more black students. As to Layne, Hyde Park, Golden Rule and Terrell, the district court’s judgment is based on an interrelated two-step process. First, these schools were ultimately determined to be “racially identifiable,” as a matter of law, essentially on the sole basis of their former overt “de jure” status combined with the deviation of their respective minority percentages from that of the district as a whole by a number of percentage points greater than that allowed by the Foster formula. This finding of “racial identifiability” was basically arrived at purely as a matter of law, as distinguished from being the result of factual conclusions or determinations with reference to the particular conditions and circumstances of these schools and the DISD. Second, it was concluded, as a legal consequence of the presumption arising from the “racially identifiable” status of these schools, that their condition in this respect resulted from intentionally segregative conduct by the educational authorities. Similarly, this determination was in essence a legal one, and did not rest on an articulated consideration of the particular facts and circumstances of these schools and the DISD or on a drawing of factual inferences or conclusions therefrom. We hold that, under the peculiar circumstances of this case as outlined in the preceding paragraph, the referenced method employed by the district court, to determine whether the racial composition of these elementary schools represented an unconstitutional vestige of “de jure” educational segregation, was erroneous in that such determination was made essentially only as a matter of law and was not supported by any sufficient underlying factual determinations and conclusions. Our consideration of this matter must necessarily begin with, and be guided by our understanding of, the Supreme Court’s opinion in Swann, particularly the following portions thereof: “(1) Racial Balances or Racial Quotas. U “Our objective in dealing with the issues presented by these eases is to see that school authorities exclude no pupil of a racial minority from any school, directly or indirectly, on account of race; ... u “[I]n the opinion and order of the District Court of December 1, 1969, we find that court directing ‘that efforts should be made to reach a 71-29 ratio in the various schools so that there will be no basis for contending that one school is racially different from the others . .., [t]hat no school [should] be operated with an all-black or predominantly black student body, [and] [t]hat pupils of all grades [should] be assigned in such a way that as nearly as practicable the various schools at various grade levels have about the same proportion of black and white students.’ “The District Judge went on to acknowledge that variation ‘from that norm may be unavoidable.’ This contains intimations that the ‘norm’ is a fixed mathematical racial balance reflecting the pupil constituency of the system. If we were to read the holding of the District Court to require, as a matter of substantive constitutional right, any particular degree of racial balance or mixing, that approach would be disapproved and we would be obliged to reverse. The constitutional command to desegregate schools does not mean that every school in every community must always reflect the racial composition of the school system as a whole. a “We see therefore that the use made of mathematical ratios was no more than a starting point in the process of shaping a remedy, rather than an inflexible requirement. From that starting point the District Court proceeded to frame a decree that was within its discretionary powers, as an equitable remedy for the particular circumstances.... Awareness of the ra cial composition of the whole school system is likely to be a useful starting point in shaping a remedy to correct past constitutional violations. In sum, the very limited use made of mathematical ratios was within the equitable remedial discretion of the District Court. “(2) One-race Schools. “... Schools all or predominantly of one race in a district of mixed population will require close scrutiny to determine that school assignments are not part of state-enforced segregation. “In light of the above, it should be clear that the existence of some small number of one-race, or virtually one-race, schools within a district is not in and of itself the mark of a system that still practices segregation by law. The district judge or school authorities should make every effort to achieve the greatest possible degree of actual desegregation and will thus necessarily be concerned with the elimination of one-race schools. No per se rule can adequately embrace all the difficulties of reconciling the competing interests involved; but in a system with a history of segregation the need for remedial criteria of sufficient specificity to assure a school authority’s compliance with its constitutional duty warrants a presumption against schools that are substantially disproportionate in their racial composition. Where the school authority’s proposed plan for conversion from a dual to a unitary system contemplates the continued existence of some schools that are all or predominately of one race, they have the burden of showing that such school assignments are genuinely nondiscriminatory. The court should scrutinize such schools, and the burden upon the school authorities will be to satisfy the court that their racial composition is not the result of present or past discriminatory action on their part.” 402 U.S. at 22-24, 91 S.Ct. at 1279-80 (footnotes omitted; emphasis added, except in headings). In applying the foregoing principles, we turn first to Terrell, which the district court determined to be, as a matter of law, racially identifiable as black, despite its clear white majority, solely on the basis of its prior overtly de jure status and its failure to meet the Foster formula percentage criteria. Under the circumstances of this case, such a mechanical method of determination was legally erroneous. While, as reflected by the above quotation, Swann does mention “a presumption against schools that are substantially disproportionate in their racial composition,” the context in which this language appears strongly suggests that the intended reference is limited to schools that are all or predominantly of one race. The language in question appears in a section of the opinion entitled “One-race Schools.” Earlier in the opinion, the Court has explained that this section will deal with the question of “whether every all-Negro and all-white school must be eliminated as an indispensable part of a remedial process of desegregation.” Id. 402 U.S. at 22, 91 S.Ct. at 1279. In the portion of the section in question preceding the “substantially disproportionate” sentence, there are several references to schools that are “all or predominantly of one race,” or “one-race, or virtually one-race,” or simply “one-race,” which reinforces the view that schools which may fairly be so described constitute the subject matter of this section of the opinion. Moreover, in the sentence immediately following the “substantially disproportionate” language, the Court again refers to “schools that are all or predominantly of one race,” and says that the school authorities “have the burden of showing that such school assignments are genuinely nondiscriminatory.” And in the next sentence, it is said that the “court should scrutinize such schools,” and the school authorities have the burden to show “that their racial composition is not the result of present or past discriminatory action” (emphasis added). It seems clear that in such sentence of the Swann opinion the words “such schools” and “their” (which we have italicized) refer to the phrase “schools that are all or predominately of one race” appearing in the immediately preceding sentence in Swann. Moreover, the sentence in Swann mentioning “a presumption against schools that are substantially disproportionate” does not explain the operation or object of such presumption. The only mention of any burden on the school authorities, or any requirement for a showing by them respecting the effects of current or past discriminatory actions, is contained in the next two sentences. These sentences make clear that what such burden and showing relates to is “schools that are all or predominately of one race.” And, it is those schools which constitute the subject matter of this section of the Swann opinion. Further, it is “in a district of mixed population” that the opinion has said such all or predominantly one-race schools “require close scrutiny.” Accordingly, to the extent the sentence in Swann concerning “substantially disproportionate” may be understood to modify the language in that opinion concerning a burden of proof arising from the continued existence of former de jure “schools that are all or predominately of one race,” we believe that such modification would likely have been conceived of as operating more in the nature of a limiting condition to, rather than an expansion of, the one-race or virtually one-race concept. Finally, we observe that the central issue before the Court in Swann related to the elementary assignment plan, and the Court observed that under the board’s plan, which it held the district court properly rejected, “[m]ore than half of the Negro elementary pupils were left in nine schools that were 86% to 100% Negro”, id., 402 U.S. at 9, 91 S.Ct. at 1272, though the minority population in the entire district was approximately 29 percent. The Supreme Court’s discussion in Swann of “Racial Balances” in schools likewise reinforces our conclusion. In the section of the opinion in which that topic constitutes the subject matter under consideration, there is no indication that mere racial imbalance in the schools of a former de jure district gives rise as a matter of law to a presumption that the imbalance results from intentionally segregative conduct by the educational authorities. Rather, the discussion of racial imbalance is in terms of its relevance to the general nature of the remedy which the district court may frame for a condition it has already found to be the result of intentionally segregative conduct. That, of course, was the context in which the matter was presented to the Supreme Court in Swann. Moreover, even in that context the Court was obviously concerned with limiting the use and significance of mathematical ratios and statistical racial balance. The opinion counsels against using the district-wide racial percentages as a “norm” (even a norm from which variations might be permitted) by which to judge the several schools, and states that remedial decrees should not be framed to require as a substantive constitutional right “any particular degree of racial balance” and that the “command to desegregate” does not require each school to mirror the racial balance of the whole district. Mechanical or “inflexible” use of general statistical ratios is clearly discouraged. On the other hand, where mathematical ratios are used as “no more than a starting point in the process of shaping a remedy ... for the particular circumstances,” then this “very limited use” is within the district court’s “remedial discretion.” Id. 402 U.S. at 24, 25, 91 S.Ct. at 1280. Here the district court employed mathematical ratios primarily to determine the existence of a condition of unlawful segregation. Moreover, its use of such ratios cannot fairly be described as “very limited.” The district court did not use “awareness of the racial composition of the whole school system” as only a “starting point” in the flexible and discrete exercise of its remedial jurisdiction to fashion a remedy for the particular circumstances of this case. Rather, it determined “racial identifiability” by applying the Foster formula, fixing the maximum permissible variation from the district-wide racial composition “norm,” as an essentially inflexible rule of law of general application. Our reading of Swann appears consistent with other decisions of the Supreme Court. None has been cited to us, nor are we aware of any, presuming continued illegal segregation merely because of the existence of a disproportionately large black enrollment in a formerly de jure school having a clear majority of white students. We will mention some of the leading decisions of the Court since Brown II. In Green v. School Board of New Kent County, 391 U.S. 430, 88 S.Ct. 1689, 20 L.Ed.2d 716 (1968), where the Court spoke of the duty to create “a unitary system in which racial discrimination would be eliminated root and branch,” id. 391 U.S. at 438, 88 S.Ct. at 1694, the opinion notes the absence of residential segregation in the district and that the district has only two schools, one of which is “all-Negro” and is attended by “85% of the Negro children in the system.” Id. 88 S.Ct. at 431, 441, 88 S.Ct. at 1691, 1696. In Davis v. Board of School Commissioners of Mobile County, 402 U.S. 33, 91 S.Ct. 1289, 28 L.Ed.2d 577 (1971), a case handed down the same day as Swann and like it a unanimous opinion authored by Chief Justice Burger, the Court states that having “found a violation” there should be “every effort to achieve the greatest possible degree of actual desegregation” and that “the measure of any desegregation plan is its effectiveness.” Id. 402 U.S. at 37, 91 S.Ct. at 1291. The opinion notes that under the plan, which the Court rejects, “64% of all the Negro elementary school pupils” in the district attended schools “over 90% Negro,” and “over half” of the “Negro junior and senior high school students” were attending “all-Negro or nearly all-Negro schools.” Id. Wright v. Council of City of Emporia, 407 U.S. 451, 92 S.Ct. 2196, 33 L.Ed.2d 51 (1972), sustained an injunction against the division of a school district which was attempted two weeks after the issuance of a district-wide court desegregation order. The division would have exacerbated the racial imbalance in the schools, and the district court found that “ ‘in a sense, race was a factor in the city’s decision to secede’ ” (though this was apparently not the primary or dominant purpose). Id. 407 U.S. at 461, 92 S.Ct. at 2202. Justice Stewart’s opinion for the majority was at pains to point out that the Court was not holding that the racial imbalance factor was s