Artículos
THE QUADRATIC GLASS CEILING AND ITS CONSEQUENCES
O TETO QUADRÁTICO DE VIDRO E SUAS CONSEQUÊNCIAS
EL TECHO DE CRISTAL CUADRADO Y SUS CONSECUENCIAS
William Crombie
(The Algebra Project, USA)
billcrombie@gmail.com
Recibido: 10/07/2023
Aprobado: 10/07/2023
abstract
Over the past thirty years, the Algebra Project has worked with a diversity of stakeholders (parents, university partners, administrators, teachers, and their students) to raise the floor of math literacy for the most underserved students in US school systems. In the course of this work, we have come to recognize a glass ceiling that the most underserved students and their teachers must contend with. It has blocked access to topics that have historically been considered advanced, but which actually are accessible to students at the levels of Algebra I and Geometry. It has fuelled community debates around how much math is too much math for some students and not enough for others. In this article, we will examine a few examples from Algebra I and Geometry, which clearly show this accessibility. These examples demonstrate that an appropriate floor for math literacy in the 21st century needs to be reconstructed to account for the gap between what could be taught and what is taught in secondary mathematics.
Keywords: math literacy. accessibility. algebra. geometry. calculus.
resumo
Nos últimos trinta anos, o Projecto Álgebra trabalhou com uma diversidade de partes interessadas (pais, parceiros universitários, administradores, professores e seus alunos) para elevar o nível de alfabetização matemática para os alunos menos atendidos nos sistemas escolares dos EUA. No decorrer deste trabalho, chegamos a reconhecer um teto de vidro com o qual os alunos mais carentes e seus professores têm de lidar. Ele bloqueou o acesso a tópicos historicamente considerados avançados, mas que na verdade são acessíveis aos alunos dos níveis de Álgebra I e Geometria. Ele alimentou debates na comunidade sobre o quanto a matemática é matemática demais para alguns alunos e insuficiente para outros. Neste artigo, examinaremos alguns exemplos de Álgebra I e Geometria que mostram claramente essa acessibilidade. Esses exemplos demonstram que um piso apropriado para a alfabetização matemática no século 21° precisa ser reconstruído para dar conta da lacuna entre o que poderia ser ensinado e o que é ensinado na matemática secundária.
Palabraschave: alfabetização matemática. acessibilidade. álgebra. geometria. cálculo.
resumen
Durante los últimos treinta años, el Proyecto Álgebra ha trabajado con una diversidad de partes interesadas (padres, socios universitarios, administradores, maestros y sus estudiantes) para elevar el nivel de alfabetización matemática para los estudiantes más desatendidos en los sistemas escolares de EE. UU. En el curso de este trabajo, hemos llegado a reconocer un techo de cristal con el que deben lidiar los estudiantes más desatendidos y sus maestros. Ha bloqueado el acceso a temas que históricamente se han considerado avanzados, pero que en realidad son accesibles para los estudiantes en los niveles de Álgebra I y Geometría. Ha alimentado debates comunitarios sobre cuántas matemáticas son demasiadas para algunos estudiantes e insuficientes para otros. En este artículo, examinaremos algunos ejemplos de Álgebra I y Geometría, que muestran claramente esta accesibilidad. Estos ejemplos demuestran que es necesario reconstruir un piso apropiado para la alfabetización matemática en el siglo XXI para tener en cuenta la brecha entre lo que se podría enseñar y lo que se enseña en matemáticas secundarias.
Palabras clave: alfabetización matemática. accesibilidad. álgebra. geometría. cálculo.
Introduction
In a posthumously published editorial Bob Moses (2021), the founder of the Algebra Project, wrote:
Amidst the planetwide transformation we are undergoing, from industrial to informationage economies and culture, math performance has emerged as a critical measure of equal opportunity. We can see the collateral damage of inequities in math education in the way that students are tracked into deadend math courses and how that tracking is then used to deny them other opportunities because they cannot demonstrate the required math competencies on standardized tests. Simply look at how the failure to complete math requirements is strongly correlated with not completing either high school or postsecondary education.
These inequalities in math education and their collateral damage affect the vast majority of American students, but their most severe impact is upon communities of color. While the American meritocratic perspective sees these inequalities as evidence that the equality of opportunity does not imply an equality of results, the literacy paradigm that the Algebra Project operates under views the inequality of results as implying an inequality of opportunities to learn. The purpose of this paper is to reveal one source of these inequalities affecting all students in the United States.
Within the Algebra Project, we have seen that these inequalities stand in plain sight but are not easily recognized because their origins lay deeply embedded in the presuppositions of our standards, policies, and practices. These presuppositions determine what we think is advanced mathematics versus elementary mathematics, what we think certain students can or cannot learn, and consequently, what we think certain students should or should not learn (Rosenstein, 2017; Thompson, 2008).
Presently the US math education research community has a major focus on examining how we teach and how students learn. Pedagogical content knowledge, PCK (Shulman, 1986), and mathematical knowledge for teaching, MKT (Ball, 1990) are two of the most prominent perspectives for how we teach. Constructivist research is focused on the mental construction of knowledge and understanding by students. This broad research agenda extends from cognitive constructivism (Clements, 1990) and neoPiagetians such as APOS Theory (Dubinsky, 2001) to social constructivist (Van de Veer, 1994; Bransford, Brown & Cocking, 1990: Steffe, & D’Ambrosio, 1995). But these research frameworks take the underlying mathematics as given. They have not considered whether the mathematical content itself is problematic. Not only is
The Quadratic Glass Ceiling we propose to uncover is the product of gaps, Blind Spots, in the mathematics that is taught for the first two years of high school in the US, namely in Algebra I and Geometry. The difficulty in recognizing these gaps is captured in the EinsteinWertheimer Correspondence on Geometric Proofs and Mathematical Puzzles (Luchins, 1990). Albert Einstein suggests:
Concepts that have proved useful for ordering things easily assume so great an authority over us, that we forget their terrestrial origin and accept them as unalterable facts. They then become labeled as ‘conceptual necessities,’ ‘apriori solutions,’ etc. The road of scientific progress is frequently blocked for long periods by such errors. It is therefore not just an idle game to exercise our ability to analyse familiar concepts, and to demonstrate the conditions on which this justification and usefulness depend.
Are these Blind Spots, these conceptual necessities, easily seen for what they are? No, not at all. For if they were, we wouldn’t label them as Blind Spots. But we can perceive them if we first develop a picture of what should occupy these spots and then look again to see that these spots are empty. The Blind Spots creating the Quadratic Ceiling run through standardsbased curricula, teacher mathematical and instructional practice, learning progressions and their associated tasks and products. In a discussion of learning progressions Maloney (2022) asserts
… learning trajectories provide detailed descriptions of instructionallygrounded development of mathematical concepts and reasoning from the perspective of student learning, and, overall, building on decades of accumulated experience in mathematics education research. However, their greater importance may lie in their potential as frameworks that contribute an unprecedented coherence across classroom instruction, professional development, standards, and assessment, by focusing squarely on conceptual understanding and reasoning instead of assessmentdriven procedural knowledge. This potential was sufficiently compelling as an organizing framework to have been cited as a basis for the Common Core mathematics standards, the new mathematics learning expectations that are consistent across most of the United States.
It, therefore, seems that the consequences of these Blind Spots and their associated Quadratic Glass Ceiling would follow a similar trajectory with parallel but detrimental impacts. Indeed, these gaps and ceilings produce an unprecedented incoherence across curricula, classroom instruction, professional development, learning progressions, and assessments, as the following examples will show.
An Example of a Blind Spot in High School Algebra I
The following two questions have been used in professional learning sessions with teams of high school mathematics teachers. The typical responses are given below.
the line of symmetry is givenby
These responses are consistent with standard Algebra I texts. In the conventional formulation of Algebra I, the meaning of polynomial coefficients is never fully developed. The literature recognizes that this omission leads to certain misconceptions but has no notion of either a clear interpretation of all the coefficients or how to address the issue of their interpretation effectively. Ellis and Grinstead (2008) note that the equation for a parabola
In our work with teachers and their students, we use the following exercise to resolve this indeterminacy. In addition to identifying the geometric meaning of each coefficient by observing the salient geometric feature affected by varying the coefficient’s value, students are given the task of determining the effect of adding the nextorder term starting with a constant polynomial and going up to a quadratic polynomial. The salient geometric feature of the constant monomial is its height above or below the
In general, students find that the highestorder coefficient describes a global property of the graph. Adding the nextorder term causes the previous global feature to become a local feature at the
What is the meaning of the parameters
What is the meaning of the parameters
The linear portion of the quadratic polynomial at the
This example is the first instance of the Quadratic Glass Ceiling. The notion of the tangent line is an easily accessible concept within the framework of Algebra I. Its avoidance, as noted above, leads to several misconceptions on the part of students in their attempt to see some coherence and relationship among the mathematical concepts they are being taught.
An Example of a Blind Spot in High School Geometry
The following three questions have been used in professional learning sessions with teams of high school mathematics teachers. The typical responses are given below.
How is the area from
a horizontal line given by the equation,
a straight line given by the equation,
a parabola given by the equation,
The conventional wisdom regarding the area under a parabola is typically framed along similar lines. We find in
Suppose we want to find the area of the shaded region
Despite this pronouncement of what cannot be done, the following exercise is a problem in high school Geometry requiring little more than the set of transformations that the Common Core Standards target as an essential component for learning Geometry. In this exercise, students are asked to complete a proof given a sequence of pictures. The third column is initially blank. In the third column, students are asked to explain how each picture relates to the following picture and how the shaded areas are connected. The problem is to determine the area of the parabolic triangle,



Picture 1 

We want to find the area of the region between the parabola, 
Picture 2 

The shaded figure is the same size and shape as the parabolic triangle in Picture 1. It is derived from the original parabolic triangle by a reflection about a vertical line at the base’s midpoint. 
Picture 3 

The shaded figure is the same size and shape as the original parabolic triangle. It is derived from the parabolic triangle in Picture 2 by a reflection about a horizontal line located at the midpoint of the vertical side of the square. 
Picture 4 

The area of the square ( 
Picture 5 

Since the area of the gap depends only upon the difference between the upper boundary (UB) and the lower boundary (LB), A(UB – LB) = A(UB) – A(LB), the area of the hill is equal to the area of the gap ( 
Picture 6 

The hill is horizontally translated to the left by half the length of the square’s base. The area of the shaded figure is equal to half the area of the gap. 



Picture 7 

Horizontally scaling the figure in Picture 6 by a fact of 2 results in a figure with an area twice that of half the gap. The area of the shaded figure is simply 
Picture 8  2 
Vertically scaling the figure in Picture 7 by a factor of 2 results in a figure with twice the gap’s area. The area of the shaded figure is 2 
Picture 9  This figure is a reflection of the figure in picture 4, about a vertical line located at the midpoint of the base. Since the area of the previous figure was equal to twice the area of the gap (2 

Picture 10  Therefore, the area of the square (2 
So, referring back to Figure 2, there is
The area under the parabola is typically the gateway problem to the Integral Calculus. As with the previous Algebra I exercise, we have successfully used this exercise with high school teachers and their Geometry students. Moreover, we see that not only is the conventional wisdom on the area problem mathematically incorrect but that it needlessly delays the introduction of concepts and techniques that are well within reach of students in a high school Geometry class.
Consequences of the Quadratic Glass Ceiling
The present situation with high school mathematics in the US is reminiscent of what Bob Moses referred to as sharecropper education. This was and still is an arrangement where communities are denied access to literacy levels within the educational system only to have those same levels of literacy used as conditions for blocking or enabling greater participation in the system. The data from the National Assessment of Education Progress (NAEP) Report for 2019, preCOVID, provides a prime example of the effect on communities of color of this type of double bind. (NCES, 2019). The 12^{th}grade NAEP mathematics achievement level results by race/ethnicity have 32% of white 12^{th} graders at or above proficient. NAEP defines proficiency as demonstrating “solid academic performance and competence in challenging subject matter.” Blacks performed at 8%, Hispanics at 11%, and Native Americans at 9%. These results are a consequence of institutional constraints which perpetuate historical inequalities rather than simply the unpreparedness of the students from the communities involved. To expand upon the position stated in the Introduction, not only is
Calculus is not just a system of knowledge and techniques. It is also an institution. (Kaput, 2000). And as with many institutions, Calculus in the US educational system is a marker of privilege. The examples of the Quadratic Glass Ceilings given here demonstrate that there is no mathematical reason for the privileged access to Calculus that we presently see in US public schools. Instead, the reasons for restricting access are historical, social, and ultimately institutional. They serve to maintain an institutional system of privilege even after the original justifications no longer seem to be socially acceptable or politically relevant. This is the critical difference between institutional racism and its earlier explicit virulent variant. Under the aegis of meritocracy, the effects of institutional policies are claimed to be raceneutral. But the presence of institutional racism is found in its differential impact on communities of color as noted in the 2019 NAEP data.
The point of view developed here calls for a reassessment of the relationship between Algebra as the elementary discipline and Calculus as the advanced discipline. Alfred North Whitehead (1929) offers a clarifying perspective. “Algebra is the intellectual instrument for rendering clear the quantitative aspects of the world.” Along these lines, Algebra can be understood in terms of its defining or foundational problems. Algebra addresses four fundamental problems: the problem of dependency, the problem of comparison, the problem of rate of change, and the problem of net change. The dependency problem asks, “How does one quantity depend upon another?” The answer to this question is typically a function, a concept often viewed as the central notion that Algebra should be built around. (Klein, 1932). The comparison problem asks, “When is one quantity or function less than, equal to, or greater than another?” The answer to this question typically involves the solution of equations and inequalities. The rate of change problem asks, “How fast is a quantity or function changing?” In Algebra I, this question is typically addressed by conditions of either constant rates of change or average rates of change. The net change problem asks, “By how much has a quantity or function changed?” And again, these problems are only considered for the cases of a constant rate of change or an average rate of change.
The preceding examples demonstrate that this restriction to constant and average rates of change is not a consequence of the mathematics itself. The notions of varying rates of change, slopes/derivatives, and their corresponding net changes, areas/integrals, are accessible through the study of polynomial functions, and in the first instance, through quadratic functions as early as Algebra I and Geometry. The present restrictions on their study are historically derived. And, as demonstrated above, the reason for their continued use in the restricted sense is an institutional choice, not a mathematical requirement.
At present in the US, Calculus has assumed a position as the capstone course of the high school mathematics sequence. Access to Calculus is presently based on a meritocratic rationale. According to the conventional wisdom, Calculus as an advanced topic requires four or more years of preparation for its acquisition: Algebra I, Geometry, Algebra II, Trigonometry, PreCalculus, and only then Calculus. (Almora Rios, 2023) The previous examples in this article demonstrate that Calculus is not the advanced topic it is presently assumed to be because Calculus is not synonymous with Analysis, the mathematics of approximation and limits. The main results of the Calculus can be derived from the advanced topic of limits or as shown above from an application of basic Algebra and Geometry. We have presented only the first steps in such a derivation here. Because the main concepts and techniques of Calculus fall squarely within the domains of Algebra I and Geometry, the present meritocratic prerequisites serve only to support the privilege of some and retard the advancement of others. In effect, the most practical prerequisite for Calculus with Limits is Calculus itself as a fundamental problem domain of introductory Algebra and Geometry. Calculus, in spite of the way it is presently taught, is nothing more than quantitative reasoning: reasoning in terms of quantity, rate of change, and net change. Or as Whitehead reminded us, “Algebra is the intellectual instrument for rendering clear the quantitative aspects of the world.”
If we can indeed see these Blind Spots and the Quadratic Glass Ceiling they create, an appropriate floor for math literacy in the 21^{st} century needs to be reconsidered and reconstructed to account for the gap between what could and should be taught and what is taught in secondary mathematics to all communities in the US.
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