by Barry Garelick
I attended Mumford High School in Detroit, from the fall of 1964 through June of 1967, the end of a period known to some as the golden age of education, and to others as an utter failure. For the record I am in the former camp, a product of an era which in my opinion well-prepared me to major in mathematics. I am soon retiring from a career in environmental protection and will be entering the teaching profession where I will teach math in a manner that has served many others well over many years and which I hope will be tolerated by the people who hire me.
I was in 10th grade, taking Algebra 2. In the study hall period that followed my algebra class I worked the 20 or so homework problems at a double desk which I shared with Raymond, a black student. He would watch me do the day’s homework problems which I worked with the ease and alacrity of an expert pinball player.
While I worked, he would ask questions about what I was doing, and I would explain as best I could, after which he would always say “Pretty good, pretty good”—which served both as an expression of appreciation and a signal that he didn’t really know much about algebra but wanted to find out more. He said he had taken a class in it. In one assignment the page of my book was open to a diagram entitled “Four ways to express a function”. The first was a box with a statement: “To find average blood pressure, add 10 to your age and divide by 2.” The second was an equation P = (A+10)/2. The third was a table of values, and the last was a graph. Raymond asked me why you needed different ways to say what was in the box. I wasn’t entirely sure myself, but explained that the different ways enabled you to see the how things like blood pressure changed with respect to age. Sometimes a graph was better than a table to see this; sometimes it wasn’t. Not a very good explanation, I realized, and over the years I would come back to that question—and Raymond’s curiosity about it—as I would analyze equations, graphs, and tables of values.
The study hall was presided over by a high school counselor whose office was in the corner of the great room. The day came when we were to sign up for next semester’s courses, and she called out the names of the students in her custody. Raymond went in for his appointment. There was some discussion going on inside and suddenly the counselor, not given to sensitivity nor controlling the volume of her speech, blared out in unrelenting anger for all the study hall to hear: “You want to take algebra 2? You didn’t finish Algebra 1, and you got a “D” in the part you did finish! You will take ‘General Arithmetic’, young man!”
Raymond walked back and took his seat next to me. I said I was sorry about what happened, but he didn’t look in my direction. I was 15 and didn’t quite know what to do, so I didn’t do or say anything else to him. He didn’t speak to me for the remainder of the semester, and when I worked my algebra problems he found something else to do.
I never saw him again after that semester though I do know he graduated when I did. I suspect that he didn’t take any more math classes, arithmetic or otherwise. I don’t know whether his interest in math was based on my making it look easy, or whether he would have made a serious effort to get up to speed. I also don’t know whether his poor performance in the algebra course he took was because of poor teaching, lack of ability, or because he was a victim of neglect who had been passed on and promoted to the next grade as many students had been. I doubt his counselor knew either. Whether his counselor would have reacted differently had Raymond been white is also something I don’t know. What I do know is that his interest was strong enough to want to enroll in an algebra course, and the question “Why do you need to learn that stuff?” did not seem to be on his mind.
The Exchange of One Inequity for Another
I have written previously and extensively about math as it was “traditionally taught” because I feel strongly about it and it is what I know best. Two of the main criticisms about traditional math – which have also been levied against education in general for the era — are that 1) it relied on memorization and rote problem solving, and 2) it failed thousands of students.
This last criticism refers to the low numbers of students taking algebra and other math classes in the 50’s and 60’s and is taken as evidence that the techniques of traditional math—drills, memorization and word problems that were not necessarily related to the “real world”—worked only for bright students who learned math no matter how it was taught. Another side to this argument, however, is that the low numbers of students who took algebra and other math classes during this period was because of the tracking practices that were in force at the time.
The history of tracking students in public education goes back to the early part of the 1900′s. By the 20′s and 30′s, curricula in high schools had evolved into four different types: college-preparatory, vocational (e.g., plumbing, metal work, electrical, auto), trade-oriented (e.g., accounting, secretarial), and general. Students were tracked into the various curricula based on IQ and other standardized test scores as well as other criteria. By the mid-60’s, Mirel (1993) documents that most of the predominantly black high schools in Detroit had become “general track” institutions that consisted of watered down curricula and “needs based” courses that catered to student interests and life relevance. Social promotion had become the norm within the general track, in which the philosophy was to demand as little as possible of the students. The educational system in the U.S. pitted many groups against each other— skin color was not the only determinant. Children from farms rather than from cities, and children of immigrants, for example, were often assumed to be inferior in cognitive ability and treated accordingly.
During the 60’s and 70’s, radical critics of schools such as Jonathan Kozol, brought accusations of sadistic and racist teachers, said to be hostile to children and who lacked innovation in pedagogy. “Traditional” schooling was seen as an instrument of oppression and schools were recast in a new, “hipper” interpretation of what progressivism was supposed to be about. In moving away from the way things were, the education establishment’s goal was to restore equity to students rather than maintaining the tracking that created dividing lines between social class and race. The end product however was a merging of general track with college prep with the result that college prep was becoming student-centered and needs-based with lower standards, and less homework assigned. Classes such as Film Making and Cooking for Singles were offered, and requirements for English and History courses were reduced if not dropped. Social class and race was no longer a barrier for such classes as evidenced by the increasing numbers of white students began taking them.
By the early 80’s, the “Back to Basics” movement formed to turn back the educational fads and extremes of the late 60’s and the 70’s and reinstitute traditional subjects and curricula. The underlying ideas of the progressives did not go away, however, and the watchword has continued to be equal education for all. While such a goal is laudable, the attempt to bring equity to education by eliminating tracking had the unintended consequence of replacing it with another form of inequity: the elimination of grouping of students according to ability. Thus, students who were poor at reading were placed in classes with students who were advanced readers; students who were not proficient in basic arithmetic were placed in algebra classes. Ability grouping was viewed as a vestige of tracking and many in the education establishment consider the two concepts to be synonymous.
The elimination of ability grouping occurs mostly in the lower grades but also extends to early courses in high school. The practice of such full inclusion is now so commonplace that theories have emerged to justify its practice and to address the problems it brings. “Learning styles” and “multiple intelligences” are now commonplace terms that are taught in schools of education, along with the technique known as “differentiated instruction” to address how to teach students with diverse backgrounds and ability in the subject matter. Teachers are expected to “differentiate instruction” to each student, and to keep whole-group instruction to a minimum. To do this, the teacher gives a “mini-lesson” that lasts 10 to 15 minutes; then students work in small groups and told to work together. The prevailing belief is that by forcing students to solve problems in groups, to rely on each other rather than the teacher, the techniques and concepts needed to solve the problem will emerge through discovery, and students will be forced to learn what is needed in a “just in time” basis This amounts to giving students easy problems, but with hard and sometimes impossible approaches since they have been given little to no effective instruction to the mathematics that results in effective mathematics problem solvers.
The limitations of differentiated instruction work hand-in-hand with other aspects of the educational beliefs that shun “traditional” modes of instruction. (Beals (2009) describes the current trends in very accurate detail.) It is not unusual to hear parents concerned over art-based projects in English classes that call for students writing book reports in the form of a book jacket or poster—in which the artistic merits of the poster or book jacket may count as much as the actual composition. Exercises in grammar have declined to the point that they are almost extinct. Essays now are “student-centered” which is to say that students write about how they feel about certain events that occur in a story, relating it to themselves—this extends to history classes as well. They may be asked how Hester Prynne would write a profile about herself on Facebook, or George Washington on the eve of battle. Objective analysis, along with grammatical drill, sentence and paragraph structure and other tenets of a basic education are considered passé and not in keeping with the current watchword of 21st century education.
Brighter students are seated with students of lower ability in the belief that the brighter students will teach the slower ones what is needed. And frequently this occurs, though the fact that the brighter students are often obtaining their knowledge via parents, tutors or learning centers is an inconvenient truth that is rarely if ever acknowledged. The result is that brighter students are bored, and slower students are either lost, or seek explanations from those students in the know. Another inconvenient truth is that in lower income communities, there are unlikely to be students who have obtained their knowledge through outside sources; they are entirely dependent on their schools.
Students forced to endure this form of education do not progress as rapidly and do not master the essentials necessary to be successful in high school math courses. Even many of the classes for gifted and talented students are conducted in this manner. For example, one gifted class for seventh graders designed a new playground to a budget. The parent who told me this was quite proud to add that the playground was built with minor changes. The project took the bulk of the semester and there wasn’t much else in the way of geometry, proportions, rates, and pre-algebra concepts such as negative numbers, exponents and radicals—unless such concepts were visited on a “just in time” basis as discussed above.
In lower grade math classes, teaching mathematical procedures and algorithms has given way to more pictorial explanations, using alternative methods of adding, subtracting, multiplying and dividing in the name of providing students with “deep understanding”. Process trumps content. The results are that such students are passed on into algebra courses in high school with little to no mastery of the arithmetic procedures that are essential to move on to more abstract versions of the same. As such, they do not qualify for the honors track courses, nor—ultimately—AP calculus.
Many who make it to the honors have received the instruction and knowledge they need through tutoring/learning centers or their parents. In some cases, there are schools whose gifted and talented program consists of a traditional approach for math and other subjects. Thus, students who qualify for such programs are exempted from the one-size-fits-all, student-centered classes. In either case, students entering high school have been unintentionally split into groups of students, some of whom will qualify for honors classes and those who will not. Depending on the high school, the non-honors courses may be watered down versions often by necessity. These students are passed on through the system in some schools; in others they receive failing grades. Students, through circumstances beyond their control, may end up “tracked” in sub-standard courses and will be ill-prepared to take math courses in college, thus shutting out possibilities of a career in the sciences or engineering.
A recent study by William Schmidt of Michigan State University (Schmidt, et al., 2011) also observes the differences in learning opportunities and concludes that the differences are a function of the education system structure. Thus, there are differences in content depending on the area of the school district and that there is a fundamental relationship between content coverage and achievement. Schmidt states that with respect to mathematics, if the districts examined in his study were to hold generally for the U.S. then “any student can be disadvantaged simply due to differences in the rigor of the mathematics taught in the district in which they happen to attend school.” While a variety of factors contribute to the disadvantaging of students as discussed above, eliminating ability grouping is a big one. Through the efforts and philosophies of otherwise well-meaning individuals, full inclusion and equality for all has served as a form of tracking.
Not Good Enough for Traditional
Critics of the traditional model of education–particularly math–argue that traditional methods worked only for the gifted kids (for whom it is assumed they will learn what they need to know no matter how it is taught). And the corollary to such thinking is that students not gifted are not good enough for the traditional method. The move to homogenize skill levels in the classrooms has been entrenched now for several decades. It has come to the point now that students who have been forced through circumstances into non-honors tracks, and judged to not be able to handle the “traditional mode” of education and are thus “protected” from it. And in being protected from learning they are therefore not presented with the choice to work hard—and many happily comply in a system that caters to it.
Which raises the question of whether higher expectations and more teacher-centered instruction yield better results. Vern Williams is a middle school math teacher in Virginia, who teaches gifted students and served as a member of the President’s National Math Advisory Panel. He relates a story about how he was recently assigned a tutorial class made up of students who had failed and barely passed Virginia’s sixth grade math exams. When he first started teaching the class they wanted to play games, but Williams challenged them and included material that he was teaching to his seventh grade (gifted) algebra classes. He reports “Many of the students wanted to ditch their regular ‘baby’ classes and just attend mine. They viewed my class as not only interesting but serious.”
But students who have been put on the protection-from-learning track fulfill the low expectations that have been conferred upon them. The education establishment’s view of this situation is a shrug, and—despite their justifications for the inquiry-based and student-centered approach that brings out all children’s’ “innate” knowledge of math—respond with “Maybe your child just isn’t good in math”. The admonition carries to subjects beyond math and is extended to “Maybe your child isn’t college material.” And while it is true that a “college for all” goal is unrealistic, the view that so many students somehow are lacking in cognitive ability raises serious questions. Simply put, you no longer have to be a minority to be told you may not have cognitive ability. As Schmidt (2011) states in his paper: “To attribute achievement differences solely to differences in student efforts and abilities is grossly unfair and simpleminded and ignores the fundamental relationship between content coverage and achievement.”
There is now an in-bred resistance against ability grouping using explicit instruction. That such approaches may result in higher achievement, with more students qualifying for gifted and honors programs, is something that the education establishment has come to deny by default. What they have chosen instead is an inherent and insidious tracking system that leaves many students behind. They have eliminated the achievement gap by eliminating achievement. And many of those left behind disdain and despise education and the people who managed to achieve what they could not—just as I imagine Raymond must have many years ago.
Beals, Katharine. 2009. “Raising a Left-Brain Child in a Right-Brain World”. Trumpeter. (Chapter 3).
Mirel, Jeffrey; David L. Angus. Equality, Curriculum, and the Decline of the Academic Ideal: Detroit, 1930-68; History of Education Quarterly, Vol. 33, No. 2 (Summer, 1993), pp. 177-207
Schmidt, W., et al. 2011. Content Coverage Differences across Districts/States: A Persisting Challenge for U.S. Education Policy, American Journal of Education, Vol. 117, No. 3; University of Chicago Press; (May 2011), p. 422