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Power in the child as a problem solver: making mathematical people and the limits of standardization.

In a document entitled Everybody Counts: Report to the Nation on the Future of Mathematics Education (National Research Council, 1989) it was presumed that, "to participate fully in the world of the future, America must tap the power of mathematics" (p. 1). This power of mathematics emerged as a key focus in the reorganization of math education in the U.S. during the 1980s. With this emphasis, mathematical power was given as a characteristic to bring out of all people through a reformed math curriculum. In this way, the reorganization of the math curriculum intersected with cultural notions of identity, equality, and difference to order who was/was not part of everybody.

In this article, I attend to how the logic of equality embedded in the notion of mathematics for everybody (all) organized certain ways to see and think about children during the Standards-Based reform of math education in the United States. I examine this logic as a way of reasoning about and establishing what is equal, equivalent, and different through the use of the equal sign (=) in the curriculum. Focusing on the principles and practices of equality that organized the Curriculum and Evaluation Standards for Mathematics (National Council of Teachers of Mathematics (NCTM) 1989), I argue that the Standards curriculum was about much more than normalizing expectations for math pedagogy. The reorganization of curriculum intersected with principles of identity and difference to order cultural standards by which to see children as people with mathematical powers who could solve problems of the future.

Expressed in the language of psychology, this ordering of identity, equivalence, and difference made it reasonable to think that standards would "ensure that all students benefit equally from the opportunities provided by mathematics" (NRC, p. 89, my italics). Yet, the commonsense of all students is a fabrication. In other words, it is given materiality through the math curriculum and 'invents' children as certain kinds of people (Hacking, 2002). As a way to understand the cultural production of identity and difference that represent who the child is and should be, I give attention to how Standard mathematics was translated through principles of psychology. This process of translation, what Popkewitz (2004) has called "alchemy", illuminates how rules of psychology and mathematics intersect within a broader system of reasoning about equality to invent cultural representations of children as certain kinds of people.

On one level, then, I analyze the equal sign as a fundamental element of the Standards-based curriculum. As a symbol of mathematical logic, the equal sign ordered how children were to think of relationships of equivalence and non-equivalence. On another, I examine the use of the equal sign as a cultural practice that embodies ways of organizing equivalence and difference between children. At both levels, the analysis investigates how seemingly distinct 'mathematical' and 'cultural' equality related to (re)organize mathematics education--for all.

(Re)Setting the Standards: The Power of Problem Solving for All People

Marking a shift from a focus on individual differences in math learning that characterized the Back to Basics movement (Diaz, 2014), the Standards period of reform was to be organized according to a general notion of everybody. This shift implied a cultural way of reasoning about the equal benefit of math for all. Alongside the National Research Council's public expression for the need for math reform to emphasize the mathematical powers of all people, the Curriculum and Evaluation Standards (NCTM, 1989) also suggested that, "if students are exposed to the kinds of experiences outlined in the Standards, they will gain mathematical power" (ibid, p.5). In line with this goal, the Standards provided expectations for what that power was and how it was to be utilized.

Mathematical power was situated as a "capacity of mind of increasing value," and would become visible as "mathematical modes of thought" (NRC, 1989, p. 32). Mathematical thinking was taken as something that everybody did. This can be seen in discussions of logical-mathematical kinds of intelligence that were thought to frame one of many, yet universal, ways of thinking about and seeing the world (Gardner, 1983). By placing a natural capability to think logically and mathematically in the mind, everybody could be seen as having in them the power to see and reason in particularly mathematical ways.

Yet, mathematical modes of thought were not simply about mathematics. The emphasis given to the 'mathematical' capacities of mind were related to notions of problem solving. Articulated quite clearly in The World Book of Math Power (Johnson, 1983) mathematics was thought of as a "how-to-science, a friend and ally, a way of solving problems of daily life" (ibid, p. 492). To use math as a way to solve these everyday problems, it seemed that, "students need to view themselves as capable of growing mathematical power to make sense of new problem situations in the world around them" (NCTM, 1989, p. 6). Growing this mathematical power seemed imperative to understanding, interpreting, and solving problems in the world.

From the assumption that everybody has mathematical powers that help solve real problems, it would follow that everybody is a mathematician and does mathematics somehow consciously (NRC, 1989; NCTM, 1989). School mathematics, then, was to "endow all students with a realization that doing mathematics is a common human activity" (NCTM, 1989, p. 6). As a capacity of mind framing day-to-day activities, the naturalness and rationality given to mathematical thinking served as a standard way to establish similarity among all humans.

Taken as a normal way of living in the world, mathematical thinking and problem solving were presumed to be common human practices. Making sense of the world through 'mathematical' sensibilities seemed to be universal for people no matter where or when they lived. For the ancient Greeks, math and its powers served as a logical way to see, think about, and order a complex world (Johnson, 1983, p. 19). Expressed with this example of the ancient logic, the abstraction of mathematical power and problem solving was represented as a universal way of ordering the world, according to predictable patterns.

Consider "a pattern such as the following:

9 + 5 = 5 + 9

27 + 58 = 58 + 27

132 + 6 = 6 + 132" (ibid, p. 20).

This pattern, represented as a series of whole number sums, seemed to be generalizable as a logical rule. "For counting numbers, called n and m here, it is always true that n + m = m + n" (ibid., p. 21). So, no matter the ordering, two discrete things can be added to establish an equal identity. This symbolic expression of equivalence assumes an "If ... then ..." rule: If 2 + 3 = 5 and 5 = 2 + 3, then 2 + 3 = 3 + 2. Accepted as a truth of mathematical logic, this expression of equivalence and identity inscribed a seemingly universal law of equality.

In relation to the discussion of mathematical power and thinking, this use of the equal sign (=) reinscribed a truth by which to establish equality, as a form of universal logic. At the same time, it intersected with a cultural standard by which to identify what is presumed as the power to think logically and mathematically. This power relied upon what was considered logical thought, ordered by the ability to see patterns and generalize relationships between objects in the world.

More than a representation of how people sort out and reduce the complexity of the world around them, mathematical thought also served as a cultural norm by which to judge animal rationality. It would seem that an animal's problem solving process could be made visible by identifying certain behaviors as 'mathematical'. In having the ability to see people as discriminate objects from a bird's eye view, crows appear to count and solve problems based on the mathematical information taken from the environment (Johnson, 1983). If birds could think and reason mathematically to problem solve, certainly all humans could.

However, there appeared to be differences in the crow's mathematical powers as compared to people. This distinction was grounded on the assumption that "since crows cannot use language, they have developed the ability to judge larger quantities by sight" (ibid, p. 14). So, although the crows could see the world from a mathematical perspective and solve problems based upon that information, they were not able to communicate or expand their mathematical thinking. This key distinction signified language as an important factor identifying the power of mathematics and how it was to be used.

The assumption that language represented a distinctly human mode of mathematical thought was not new to the Standards period of reform in terms of organizing how children were to learn mathematics. However, during the reorganization of math education in the 1980s, the association between language and the representation of mathematical thinking framed the norms by which to speak and think mathematically. This, in relation to the given to educational psychology as an arbiter of truth about the child, organized cultural norms by which to order learning as a personalized, yet social, endeavor. These cultural norms not only organized the Standards by which to evaluate what was given as mathematical power, they also gave new meaning to how the child was to be seen as a specific kind of person, called the problem solver.

Math as a Language and Tool for Becoming a Mathematical Problem Solver

In Speaking Mathematically (Pimm, 1987), the "intellectual powers of children evidenced in the mastery of their first language" served to deem all children capable of expressing an inherent mathematical power (p. 197-198). Math was described as a natural power, like the ability to learn language. If all children had both intuitive abilities to learn language and inherent mathematical-logical frames of thought, then it would follow that they could all be taught to "grow in their ability to communicate mathematically and use higher-level thinking processes," (NCTM, 1989, p. 23). Communication in general, and the use of a particular language, appeared to nurture this growth. In the use of psychology as a tool to translate mathematics and its presumed power into math education, language became a lens through which to see this mathematical kind of person.

In this way, "everybody counts" was not merely a catch phrase. It invoked a commonsense way of thinking about who everybody is, how they think, and what they do to express a seemingly inherent mathematical thought process. This reasoning about everybody set the Standard-based expectations for "what children should do when they learn mathematics," (ibid.). Clearly, all children would be expected to use mathematics as a language and mode of reasoning to interpret, translate, and communicate about every day and unfamiliar problems. This is important in how 'mathematical' forms of language and reasoning framed the characteristics that appeared to belong to everybody, as a certain kind of 'mathematical' people.

This way of organizing the equivalence given to children is explored in and through the use of the equal sign (=) in the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards (1989) math problems. In this, the use of the equal sign (=) in Standard math problems organized what children were expected to know and do as mathematical problem solvers.

As the practices of problem solving intersected with the rules of psychology that situated learning within a social relationship, a new way of seeing children as mathematical kinds of people emerged. In this, the mathematical power and problem solving seemed to provide an opportunity to "get inside students' heads" (Schoenfeld, 1987) and give meaning to what it meant to be a person who could see, think, and speak mathematically. Identifying children as social beings who learn through interactions with others and the world, the tools of psychology assembled with the logic of mathematics to create a commonsense about who the mathematical problem solver was and should be. Importantly, the abstraction of mathematical power (and the person who had it) was not a given. Rather, it was created as a set of cultural principles that gave new sensibilities to mathematics - made visible in the curriculum that was, itself, part of the alchemy that remade mathematics.

Modeling with Math:

Representing Problems and Metacognitive Awareness

Standard #2 of the K-4 curriculum, "Mathematics as Communication" includes an activity in which children were to use language and counters to represent the problem 14 - 5 = []. Here, the emphasis was not for children to find the correct answer. Rather, the modeling was so that they could "see how problems that appear to be different in fact share the same underlying structure," (NCTM, 1989, p. 27). In other words, children were supposed to recognize that 14 - 5 = [] is a common way of representing problems in relation to individuals and how they should think about everyday life, such as Maria and some pencils, Eddie and several balloons, and Nina and Pedro's seashells.

Yet, "seeing" everyday life as a particular system of ordering and classifying was not necessarily so obvious, or easy to "see". The shared structure to the problems was to become visible through a mathematical discussion of how they were different--as a way to see how they were alike. In many ways, this modeling with both the counters and the mathematical expression was related to an importance in the curriculum placed on "translating a problem or idea in a new form" (ibid, p. 26). This translation involved: decoding the text, communicating to establish a consensus about what the problem is, modeling the problem with counters, and reorganizing the model into a symbolic representation. Therefore, the focus of using mathematics as communication was to establish a common way for children to see and communicate about problems in their world--given their differences. So, in spite of the various contexts in which the problems were situated, children were to talk through the distinctions, reduce them, and agree to represent their equivalence as a general problem of difference. In other words, the problems were all alike in that the relationship between 14, 5, and [] in each situation was to be organized by the rules of subtraction and given as 14 - 5 = [].

The mathematical problems of equivalence and difference used as the Standard were related to the psychological problem of identifying who the problem solver is and what this child does. In fact, as the Standards were in development, Mathematical Problem Solving (Schoenfeld, 1985) was a topic of research emerging to examine the "spectrum of behaviors that comprise people's mathematical thinking" and "to explain what goes on in their heads as they engage in mathematical tasks" (p. 5). This spectrum was bound by the limits of language and symbolic expressions to model a form of thinking that could be recognized as drawing upon and having control over relevant resources, knowledge, and strategies.

Mathematics learning is not "purely" about numbers and relations but part of a cultural set of principles that operate in fabricating the child. As a mode of establishing equivalence between people as "good problems solvers", the psychological explanation of the social behaviors that were presumed to represent internal thoughts provided a cultural norm by which to identify mathematical power in people. As one of many behaviors given on this "spectrum", the person that exhibited mathematical power was expected to self-monitor--so as to assess and control the problem solving process. This evaluation was based upon an assumption that good problem solvers were equivalent in their ability to know "what to do with the facts at hand" (ibid, p. 33). In other words, "the issue is not what the students know, it is how they use that knowledge" that made them successful in solving non-routine problems in mathematics (ibid, p. 32).

Within the concerns about standardizing problem solving methods in math, the consciousness of one's thoughts and organization of one's actions seemed to establish a cultural norm about the kinds of people who make good problem solvers--or mathematically powerful people. This model of thought resembled how children were expected to see similarity in the problems of difference and represent their common structure as 14 - 5 = []. Embedded within the rules for establishing equality between things that did not seem to be alike but shared a similar organization was a particular way of ordering equivalence as well as a way to regulate difference. This was not just in the mathematics but transmogrified into cultural modes of reasoning about how equivalences could be organized as standards for children and their expected behaviors.

Managed by the rules of psychology, the use of mathematical knowledge would become identifiable as particular actions. Specifically, practices of self-monitoring, self-regulating, and evaluating cognitive activity were all associated with a general problem-solving process (Silver, in Schoenfeld 1987, p. 49). More than components of mathematical thinking, these were psychological traits that ascribed a specific identity to people who were aware of, able to see, and could think about their own thinking. Entangled in a standard "process model" of problem solving, metacognitive awareness emerged as a psychological characteristic to identify this person. It was presumed that "awareness of one's intellectual behavior is a prerequisite for learning to change it" (Schoenfeld, 1987, p. 199). Operating below the surface, at a "metalevel", were processes of planning, monitoring, and evaluating that it seemed one should be aware of in order to change intellectual behaviors, engage in problem solving, and learn.

Evaluating children in relation to the "Mathematics as Communication" Standard introduced the possibility of seeing some as unable to change their thoughts into mathematical language. In this assessment, however, it was acknowledged that it might not be the mathematical understanding that is wrong, but rather an "inability" to translate and communicate the thought process that represented it (NCTM, 1989, p. 27). The difference between children, while similar in their innate capacity for language and mathematics, was expressed as an inability to interpret what was going on in their head in an ordered way. So, the neutrality given to language that appeared to give equivalence to all children also inserted a notion of difference in who was/ was not a problem solver.

So, while finding the "right" answer did not appear to be the focus of the activity, there was a "right" way to think about 14 - 5 = [] as the representation of equivalence in the language of mathematics. The presumed universality given to math as a language was to provide children with a way to express 'mathematical power'--inscribed as cultural traits of a new kind of person who could think about, model, communicate, and change mathematical modes of thought.

Mathematics as Reasoning: Strategies for Making Sense and Building Confidence

As a rule, learning mathematics was not purely a matter of cognition. Math learning was also contingent upon and entangled with what were considered to be affective dimensions. These affective dimensions gave sensibility to the idea that "a major goal of mathematics instruction is to help children develop the belief that they have the power to do mathematics and that they have control over their own success or failure" (NCTM, 1989, p. 29). Articulated in the discussion of Standard #3 "Mathematics as Reasoning", children's math power was to emerge not only through learning math as a language, but also through an internalization of the belief that mathematics makes sense and that they can make sense of it. Therefore, the power to benefit from mathematics seemed to rely upon building knowledge of one's own reliability, certainty, and assured sense of direction with math.

Particular problems were thought to "[help] children see that mathematics makes sense," (ibid). In solving 35 - 19 = [], the emphasis was, again, not on children computing the correct answer. Instead, this problem was posed in relation to the idea that "children need to know that being able to explain and justify their thinking is important" (ibid). In that, while children might be able to identify 16 as the correct response, the goal was for them to have a strategic and analytical way to think about the problem that was more than simply filling in the box.

A series of questions posed by the teacher were thought to illuminate an inherent logic organizing how children should see and think about mathematical equivalence as reasonable, but not necessarily self-evident:

"Do you think it would help to know that 35 - 20 = 15?"

"How would it help to think of 19 as 15 + 4?"

"Would it help to count on from 19 to 35?"

(ibid, p. 30).

The above questions implied strategies for seeing the sense built into the related expressions of equality. To the first question, children were to respond, "Yes". Knowing that 35 - 20 = 15 is a valid expression was supposed to help them think strategically about the problem 35 - 19 = []. By comparing the 19 to 20 as 1 less, children should reason that the 15 compared to the [] would be 1 more. Reframed as a different, yet related problem, 35 - 20 = 15 was to provide a sound way of seeing and rationalizing a mathematical relationship of equivalence between 35, 19 and [].

This way of making sense of the expression appealed to the identity given to whole numbers that ordered the relationships between them. In other words, the meaning ascribed to numbers meant that they could be seen as identical (such as 35 and 35) and different (like 20 and 19). As a series of ordered and predictable patterns, the identity given to numbers formed the relationship by which to determine such equivalence and difference. In the Standards, the ability to distinguish between numbers, understand how they were related, and predict their patterns was referred to as "number sense" (ibid, p. 38). This sense of number entangled with the equal sign (=) was the basis on which children should be able to recognize "valid arguments" about mathematical equivalence and non-equivalence.

However, this way of assigning identity of discrete objects and the relationship between them (mediated by the equal sign =) was not only mathematical. The logic that ordered the validity of numbers and their relationships resembled cultural principles of identity and equality that organized how children were to build valid arguments as a way to build confidence and a belief in their abilities to learn mathematics.

As mentioned earlier, the questions about 35 - 19 = [] were to help children see that "mathematical reasoning cannot develop in isolation ... the ability to reason is a process that grows out of many experiences that convince children that mathematics makes sense" (NCTM, 1989, p. 31). This convincing was about showing children that math made sense so they would believe in and appreciate it. Not just for the sake of liking math, it was thought that if children believed in and saw the power of mathematics in many situations, they could become more certain they had the power in them to know and do it.

"Number sense" was not only a way of thinking about numbers but was related to psychological notions of how that sense was made. Not necessarily distinct from matters of cognition, it was thought that the feelings, perceptions, and beliefs of children were intricately related to how they conducted themselves as problem solvers. That is, "the performance of most intellectual tasks takes place within the context established by one's perspective regarding the nature of those tasks. Belief systems shape cognition," (Schoenfeld, 1985, p. 35). This statement embodies a cultural norm, expressed in psychological language, which identifies a link between how one's views of the world shape how one thinks and acts.

While this was historically not a new association, the affective and cognitive considerations in learning mathematics were re-signified during the Standards reform movement. Marking a shift in emphasis from all children as self-motivated individuals, the Standards reform ordered and maintained that all children could learn math as strategic and confident problem solvers. Motivation was still taken as an important psychological factor in learning math, but it became reassembled as the psychological concept of interest, fueled by the certainty that math could be done. Whether or not children wanted to learn math entangled with psychological considerations of confidence --ordering how children should see and believe in the value given to math as a way to build certainty in their problem solving as mathematically powerful kinds of people.

This confidence framed a standard approach to all students learning math since it was thought that the confidence could be built from the validity granted to math itself. In fact, the Standards maintained that the assessment of mathematical power, "should examine students' disposition toward mathematics, in particular their confidence in doing mathematics and the extent to which they value mathematics," (NCTM, 1989, p. 205). Taken as psychological dispositions of people who engaged in effective problem solving, confidence and an intentional appreciation of math were associated with particular learning behaviors and therefore, possible to evaluate. The "good problem solver" could use a variety of strategies and approaches to test the validity of arguments, check in with his/her thought process, and persevere to logical and confident conclusions (Schoenfeld, 1987). Even when unsure, children could work through what they knew and plan what to do next. The shaping and reshaping of mathematical thought, then, was tied to a reorganization of the child's value system as well as their value as a person.

Organized by psychological principles identifying effective problem solving strategies, children were seen as people who could make sense of, plan, communicate, rethink, and represent a well-organized, logical, and justifiable mathematical thought process. This assumption could be seen in how children were expected to discuss the problem 35 - 19 = []. As a Standard by which to evaluate mathematical power, confidence in mathematical reasoning was to be represented by participation in math discussions aimed to uncover an inherent logic of what they are doing, how they are thinking, and why (Schoenfeld, 1985, 1987; Johnson, 1983). As such, the mathematical notions of identity and equivalence embodied in expressions like 35 - 19 = [] interacted with cultural ways of representing the child as capable of using math as a language and mode of reasoning - to communicate a belief in and control over the logic of one's "own" conclusions.

This mode of characterizing a cultural standard of problem solving gave equivalence to all children - as having equal opportunities to think and speak mathematically so as to benefit from math education. In this, children's participation was to be organized around the assumption that they can contribute to knowledge production and decision-making in the classroom. Yet, the benefit of this liberal form of participation was based upon the ability to contribute to mathematical problem solving in ways that were already determined by the Standards. In that, this mode of mathematical reasoning characterized children and their learning around the abilities to plan, strategize, assess, rethink, and, therefore, feel confident about their powers to problem solve.

While the discussion and questioning about 35 - 19 = [] can be read as an interaction between a teacher and students, it can also be considered in relation to the process of alchemy whereby mathematics becomes math education. It is in this translation that certain normative values about a mathematically powerful problem solver make visible a cultural thesis about who this child is. In ways that seem almost natural, the cultural theories about what children should know and be able to do interact with the subjects of school math to make the abstraction of mathematical power 'real', as a measurable goal of the Standards.

To this point, I have examined the use of the equal sign (=) in the organization of the curriculum around particular Standards. This has been to provoke a way of thinking about how the rules for establishing identity and equivalence are visible in the Standard expectations for knowing, seeing, and representing equality as a truth of mathematical logic. In this way, the principles organizing the problems in the Standards have been enacted as norms for how kids should speak and think mathematically in schools.

Within this standard approach to developing and evaluating what was expressed as mathematical power, school mathematics also carried a normative and cultural set of values that are about more than mathematics. In more ways than one, the mathematical was linked to the cultural inscription of everyday people living an everyday life. The activities that framed what children should know and be able to do aimed to emphasize "connections between mathematics and the real world and [encourage] children to recognize and use a variety of situations and problem structures," (NCTM, 1989, p. 12). This instantiation of the "real world" brings with it an assumption of what was considered real in the world as well as what constitutes a problem in it.

Problems in the "Real World": Mathematical Illiteracy

Given that "math power is a term important for everyone" (Johnson, 1983, p. 8), it served to provide a general expectation for everyone and frame the common sense about how all children should know and do mathematics. Yet, as the expression of equivalence between all people based upon an inherent capacity to learn math, the instantiation of math power also embodied a notion of difference. More directly, "math power can mean the difference between confidence and insecurity; vitality and boredom; goals and regrets," (ibid). On one hand, the expressions of confidence, vitality, and goals can be related to the inscriptions of identity presumed to represent the normal kind of child as a person who could think and speak mathematically. On another, the notion of difference becomes visible as representations of insecurity, boredom, and regret. These distinctions (expressed as problems of difference) become clear in discussions of "innumeracy".

Taken as "an inability to cope with common quantitative tasks," innumeracy and its consequences were presumed to be "magnified by the very insecurity that it creates" (Steen, 1990, p. 216). In naming this problem, "innumeracy" was associated with psychological notions of apprehension, fear, and uncertainty (ibid). Even more, it assembled with concerns over a culturally and historically specific form of "math anxiety" that was visible as expressions of helplessness, guilt, unwillingness, and boredom (Chavez and Widmer, 1982; Cockcroft, 1982; Stodolsky, 1985). As a problem in the pedagogy of mathematics, the differences that defined the innumerate as insecure, disengaged, ignorant, and unacceptable were conceived within and emerged alongside the language of the alchemy that produced literacy - of a mathematical kind.

If "numeracy is to mathematics as literacy is to language" (Steen, 1990, p. 211) then it would seem reasonable that, "numeracy and literacy are the entwined complements of human communication" (ibid, p. 212). In its various forms, "functional literacy," "cultural literacy," "scientific literacy," "environmental literacy," and "quantitative literacy" set the historical and cultural backdrop for the demand of a specific form of "mathematical literacy" as the equivalent expression of numeracy (ibid). The rising cultural expectations of numeracy and literacy emerged in the Standards to further define what this mathematical form of literacy would look like. In this, it became reasonable that "the intent of these goals is that students will become mathematically literate," (NCTM, 1989, p. 6). As an expression of mathematical power, mathematical literacy was taken as an integral part of learning how to draw upon what had been granted as a power to see, think, and communicate 'mathematically' in and about the world.

Becoming mathematically literate and realizing the ideal of mathematical power, however, had little to do with learning mathematics. This was clear in how it seemed self-evident, that "mathematical literacy is essential as a foundation for democracy in a technological age" (NRC, 1989, p. 8). This link between democracy and the role the mathematically literate were to have in maintaining its foundation was made reasonable in relation to numeracy, as the mode of thinking and acting about the world in ways that seemed 'mathematical'. The sense of democracy entangled with the production of the mathematically literate was clear in cultural statements regarding how, "with numeracy comes increased confidence for individuals to gain control over their lives and jobs. Numeracy provides the ability to plan, to challenge, and to predict," from a series of choices and alternatives (Steen, 1990). Gaining a sense of control and confidence would seem to provide individuals with the tools of thought and action by which to plan their future through making rational decisions in the present. Improving mathematical literacy, or numeracy, inscribed a mathematical interpretation of the world that was thought to make a better kind of person--and, in turn, somehow produce a better kind of worker.

The production of "mathematically literate workers" within the Standards based curriculum was in line with the goal of "economic survival and growth" (NCTM, 1980, p. 3). Emerging in a web of technological, democratic, and economic considerations, the mathematical kind of person was presumed capable of contributing to the "world of work in the twenty-first century" (NRC, 1989, p. 11). This world was presumed to be, "less manual but more mental; less mechanical but more electronic; less routine but more verbal; and less static but more varied" (ibid). As such, certain kinds of people were thought to be best "prepared to cope confidently with quantitative, scientific, and technological issues" (ibid, p. 12). Those people were seen as 'mathematical' --able to see, think, represent, and speak about the world in logical, ordered, planned, and methodical yet flexible ways in the face of uncertainty and unpredictability.

To return to the equal sign (=), while its use in the curriculum ordered standard notions of identity and equivalence, its organizing principles gave a particular nuance to how children were to be seen as mathematically literate, and powerful. In other words, the math kind of person was not born as such, but was to be made through the cultural inscription of traits in the math curriculum that assembled to create a commonsense of who that person is and should be. Carrying a cultural standard about mathematics as the "certain" view of the world, the Standard-based math problems also operated to give civic meaning and normative value to those that were seen as mathematically literate. With mathematical literacy as the common way of framing the democratic identity of the nation, its citizens, its workforce, and its schoolchildren, a notion of mathematical illiteracy emerged as the distinction from this norm.

This difference was expressed in relation to the production of cultural distinctions that seemed to threaten a democratic sense of equality and opportunity. That is, "apart from economics, the social and political consequences of mathematical illiteracy provide alarming signals for the survival of democracy in America" (ibid, p. 13, my italics). In Innumeracy: Mathematical Illiteracy and Its Consequences (Paulos, 1988), the social and political consequences were given as the embodiment of various inadequacies, insufficiencies, and an overall lack of ability or awareness of how to function as a productive member of democratic society.

The mathematically illiterate was characterized as the kind of person that had "a lack of numerical perspective, an exaggerated appreciation for meaningless coincidence, a credulous acceptance of pseudosciences, and an inability to recognize social trade-offs," (ibid, p. 5-6). Apparently, this person could not appreciate the sense of certainty a mathematical perspective would lend to one's approach to living and problem solving in the social world. To use another language, they were an insecure and passive group who left life to chance, could not access benefits or assess risk, and were otherwise unplanned, unpredictable, and unreliable. These kinds of people based upon their presumed inabilities to think and speak mathematically were seen as less productive and overall lacking the power to act in intentional, rational, and organized ways.

Mathematical literacy and illiteracy served to divide and differentiate kinds of people - who would either support the nation in its growth or risk its economic, political, and cultural demise. Quite directly, "we are at risk of becoming a divided nation in which knowledge of mathematics supports a productive, technologically powerful elite while a dependent, semiliterate majority, disproportionately Hispanic and Black, find economic and political power beyond reach" (NRC, 1989, p. 14). This divide, would be drawn along economic, linguistic, and racial lines and serve as another cultural expression marking the difference between a mathematically il-/semi-literate and the norm of a mathematically literate and powerful person who could represent the world and their place in it in well-ordered and justifiable ways.

In a similar style of comparative thought that validated standards of identity and difference in the mathematical formula for equality, the mode of representing the mathematically literate as equivalent with progress and growth classified the mathematically illiterate as the cultural expression of difference. Generally accepted as unacceptable, mathematical illiteracy emerged as a social problem and provided new ways to see and think about the concern of organizing mathematics for all. The cultural organization of all seemed to be entangled with a concern that "changing demographics have raised the stakes for all Americans. Never before have we been forced to provide true equality in opportunity to learn," (ibid, p. 19).

Reasoning about and providing true equality embodied a particular way of assigning identity and difference to establish equality and formulate how that problem would be solved. In this way, the cultural statement about the expectation of equality bears resemblance to the Standard way of approaching equality as a problem, expressed by the equal sign (=). That is, the particular system of reason in the curriculum that orders the solving of equality inscribes difference as the problem of inequality. In this way, the Standard modes of reasoning about equality, equivalence, and difference embodied in the equal sign (=) were worked into the cultural imperative to provide a "true equality" - wherein the mathematically illiterate, coded by cultural markers of identity, were seen as the problem.

Reforming People and the Limits of Standardizing Equality

As 'mathematical' kinds of people, everyone was presumed to have inherent powers to use the language and reason of mathematics to solve the problems of everyday life. Through an appeal to a democratic notion of math as a universal model of this life, apparently natural modes of communicating and thinking mathematically established equivalences between children as common in their ability to gain mathematical literacy and have a greater sense of control over their present and future life choices, chances, and opportunities. Importantly, this way of seeing and evaluating children emerged as the abstraction of math power and was given visibility as a cultural norm. This could be seen in how the use of the equal sign (=) was linked through the language of psychology to standards framing what all children should know and do to confidently express math powers. As methodical, organized, planned, and reflective, mathematically literate people represented a normalized way of seeing and being in the world.

In a similar style of comparative thought, new ways of organizing who was not part of the everybody emerged--distinguishing people who lack confidence in their powers to formulate, communicate, and rethink the mathematical modes of communication that organized the problems of everyday life. The child who was imagined as a risk to the nation's progress and democratic survival was marked by a difference in ability to read, write, and speak about the world mathematically. Seen as mathematically illiterate, this child was presumed as unreliable, uncertain, and otherwise unproductive in terms of the economic, technological, and social production that was valued in the calculated effort to manage equality.

The image of the child as a mathematical problem solver was and still is integral to the narrative about how math education should be (re)organized for all. As the practices with math pedagogy intersect with the principles of identity and difference expressed in the language of psychology, the standard problems of school math entangle with cultural norms distinguishing what it can mean to be a person who sees, thinks, and speaks mathematically about the everyday world. Yet, the making of this kind of person also embodies the limits and historical conditions whereby the possibility to include all children in the math curriculum reinscribes cultural notions of difference, as the inequality that reforms are to solve.

JENNIFER DIAZ

diazj@augsburg.edu

Education Department,

Augsburg College

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Author:Diaz, Jennifer
Publication:Knowledge Cultures
Article Type:Essay
Geographic Code:1USA
Date:Jul 1, 2016
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