This article describes and critiques a phenomenon that we identify as content agnosticism in the “construction genre” of educational software. Our thesis is that the content agnostic position – the assumption that any technology which supports constructionist learning theories must act as a blank slate or empty container – has been erroneously presented as the single trajectory towards the development and implementation of constructionist technologies. We survey some of the disadvantages to the content agnostic position, ranging from violent video game formats to ways in which consumption practices perform a corporate colonization of childhood. As an alternative framework we reconsider the constructionism – instructionism continuum as just one of two orthogonal dimensions; here both content agnostic and content aware constructionist positions are possible. We review some successful content aware design in math and computing education, and finally provide a detailed case study of Culturally Situated Design Tools (CSDTs), as an example of how constructionism can be combined with a content aware approach to math and computing software design.
Since the introduction of LOGO in the 1960s, educators and technologists have been creating open-ended, “sandbox” style computational media; what Mizuko Ito has labeled the “construction genre” of educational technology. 1 Examples include MIT’s Scratch, CMU’s Alice, UC-Boulder’s AgentSheets, and a wide variety of similar systems that are designed to teach math and computational thinking. Advocates of this approach, in particular Seymour Papert, frame these technologies as a means to foster student- and child-centered learning. 2 We are generally in agreement with Papert and his colleagues: our empirical studies, detailed below, suggest strong advantages offered by this bottom-up, “constructionist” framework, in contrast with the usual top-down, “instructionist” approach of most “edutainment” products on the market. Our analysis also supports their contention that constructionism resonates with broader themes of self-empowerment, critical thinking, and democratic politics. 3 However, we find that this emphasis on student- and child-centered learning is often accompanied by an unacknowledged assumption that any technology which supports constructionist learning theories must effectively act as a blank slate or empty container; and that its designers must, throughout the software development process, remain “agnostic” as to the content that learners will create. A critique and examination of the content agnostic position sheds light on how the design and development of media authoring software in the construction genre shapes the media produced by users. 4
We contend that the content agnostic position has the potential to result in disadvantages that are detrimental to students’ development in both computational thinking and social growth. Through a review of several case studies, we illustrate the possibility of alternatives that combine a “content aware” medium with a similar bottom-up, user-centered learning approach, yet explicitly designed within the constraints of community contexts and values. Our goal is not to eliminate the content agnostic position, but rather to provide an opportunity for educators, designers, researchers, and theorists to reflect on its otherwise invisible presence within the construction genre, and to consider alternative approaches when appropriate.
The visual programming environment Scratch (scratch.mit.edu), in which children snap together blocks rather than writing code, serves as a powerful exemplar of educational software that is both constructionist and content agnostic by design. Though constructionist software that is designed at a distance from curricular content can make classroom implementation difficult, 5 teachers can and do use Scratch to make class assignments that are not content agnostic by directing students toward particular themes. In addition, ordinary users in the online Scratch community can group themselves together in “studios” that have some specific content in common; what James Gee calls “affinity groups.” 6 The designers of Scratch accommodate this wide usage by making the software itself as general and flexible as possible, and to impose as little influence as possible in terms of content produced.
We recognize that the content agnostic position emerges out of positive goals: to nurture a sense of being a “producer rather than a consumer”; 7 to encourage more genuine, intrinsic motivation for learning rather than extrinsic rewards such as game points; to give youth a voice from the bottom-up rather than instruction from the top-down; and to help foster the general sense of free expression that is fundamental to an open society; and to make computer programming a common 21st century skill. Yet, we find that unreflective use can turn a blind eye towards the pervasive influence of social forces that can limit children’s cultural development and silence alternative material and perspectives. By “blind eye” we mean not only failing to see negatives, but also the fact that one cannot investigate an absence. If energy research was limited to fossil fuels, we might not only “naturalize” oil spills, petro-dictatorships and global warming as inevitable side-effects, but also fail to envision alternatives such as solar or wind energy – what is often referred to as “undone science.” 8
We begin with an examination of potential disadvantages to the content agnostic position. Using case studies of particular Scratch projects, we will review four categories of potential disadvantages; in particular the strong influence of consumption practices and commercial products, which calls into question the stated Scratch goal of making students “producers rather than consumers.” As a way of mapping possible alternative frameworks, we suggest an analytic separation between the two dimensions of interest: the spectrum between content agnostic and content awareness is on one axis; the spectrum between top-down instructionism and bottom-up constructionism is on the other. Since these can be treated as orthogonal axes, the impression that one can only choose between a content agnostic approach and an instructionist system is a false dichotomy.
We then review possible alternatives, with a final focus on our experiences using Culturally Situated Design Tools (CSDTs: www.community.csdt.rpi.edu/applications). While CSDTs also offer a bottom-up, constructionist approach to math and computing education, they highlight the cultural capital of indigenous and vernacular knowledge (Native American beadwork, urban graffiti, etc.) and foster exploration of these artistic forms as a computational medium. In contrast the content agnostic approach, in which cultural references, if any, are typically restricted to the presence of a character’s ethnic “costume” or artifact, the CSDT’s “ethnocomputing” approach locates the computational thinking in the cultural practices themselves (for example the algorithms used by traditional native bead workers). Under what circumstances that may or may not be more effective in teaching computing and raising interest in computing careers is a more complex set of questions. While we touch on some promising evaluations, our purpose here is merely to “decenter” the content agnostic approach, which is often presented as the only possibility for constructionist learning, and show that there are viable alternatives that offer advantages which are rarely considered in the literature. We conclude that it is possible to maintain the tenets of authorship in constructionism – the valuable sense of free agency in knowledge construction and artifact building – while also supporting cultural connections that are often not readily available to students, and may hold particular importance for underrepresented groups. 9
2. Potential Disadvantages to the Content Agnostic Position: Four Categories
A team in 2003 at the MIT Media Lab, led by Papert’s student Mitchel Resnick, released the visual programming environment, Scratch, with the goal of an authoring tool that would avoid the frustrations of syntax and typing errors by allowing users to snap together different color-coded blocks to create scripts for games, animations, graphic designs, and so on. Scratch was originally designed to couple with Intel’s wide-scale after-school program, The Computer Clubhouse, 10 but it is used in schools as well. As long-time members of the online Scratch community and active participants in the educational implementation of constructionist technologies in the United States, we have experienced firsthand the tendency for young users to gravitate towards the reproduction of commercial content in their design of games, graphic designs, animations, and more.
Searching for commercial content on the Scratch community site reveals a wide variety of brands and products embedded in user projects. There are for example over 28,000 search hits for Dragon Ball Z and 69,700 for Halo. We then sorted the results into “cautionary categories” that emerged when examining the implications for youth. We do not claim that these are exhaustive or totally representative; merely the patterns that emerged in our brief surveys of the online Scratch community and general experiences implementing constructionist software in- and after-school. There is some resemblance here to grounded theory, 11 but as Udo Kelle notes, to say that grounded theory produces analytic categories out of the pure data itself is somewhat misleading; 12 below we explain some of the social theory influences (as well as our discussions talking to students, teachers and parents) that led us to consider these categories. Thus our analysis – for example regarding the significance of gun violence – may only be applicable to US youth; readers should be cautioned that both the commercial products and analytic implications might be very different for Scratch in other nations.
2.1 Use of Inappropriate Material
We regard the use of what is commonly called “inappropriate material” by teachers – swearing, sexualized images, etc. – as essentially a solved problem, insofar as anyone allowing children the freedom to write an essay, paint a picture or author a website would be faced with similar instances in which they have to weigh freedom of expression against offense to other people. Even the strongest advocate for a content agnostic approach recognizes that some adult supervision and intervention is occasionally needed. The designers of Scratch have been admirable pioneers in developing scalable systems that allow peers, mentors, and/or teachers to “flag” inappropriate material, which often obviates the need for more intense forms of intervention and supervision.
2.2 Tendency to Gravitate Towards Violent Video Game Formats
We recognize that there are a wide variety of perspectives on the controversial topic of violence and media. Perhaps the best empirically supported research has been by Olson et al., an extensive study in the Harvard Medical School’s psychiatric program, which concluded that there is little evidence for any increase in real, physical violence as the causal outcome of exposure to violent video games. 13 We are less concerned with risk of violence, and more concerned with the phenomenon of “gravitate.” In the parlance of nonlinear dynamics, violent video games form a basin of attraction, which draws users towards their commercial material, and whose behavior is in turn used to fine-tune products. There is a distinction between healthy play that may legitimately include violent elements as part of a “free-flow” creative repertoire, 14 and the commodified forms of violence that commercial content, especially in the US, encourages, which may take the place of creative acts that could make a better contribution to the users’ personal, cultural, and intellectual development.
Figure 1 shows an image from a Scratch user. 15 While the icon of a realistic gun image is no doubt inappropriate for a young user, this falls under category 2.1 above: use that will likely be eventually flagged as inappropriate by the community or by a Scratch system administrator. As shown to the right of that icon, the actual games this user has created are based on stick figures and other low-barrier animation components; they are so abstracted that it would be difficult to claim any strong resemblance to commercial video games. Despite the visual reference to a real gun and to first-person shooters much of the activity resembles the ordinary “sociodramatic play” that is often cited as a psychologically healthy activity. 16
Fig. 1. First person shooter projects in Scratch
At the other extreme, Figure 2 shows a series of games created by another user. These projects consist of graphic images imported from the commercial video game Halo. The difference in comments received by Scratch users with self-created images, versus the utilization of commercial first-person shooter images from games such as Halo (particularly in the more realistic images such as the “Halo assault rifle” below), is striking. Responses such as “Nice work bro – its beast” attest to the ways that both the Halo images and the language of response are recognizable tokens for a specific sense of masculinity that rewards these commodified forms of violent fantasy. 17 Although such networks of users may be only a small percentage of the total, we hypothesize that in such cases Scratch has essentially become “colonized” by the varieties of commercial video games that reinforce hegemonic masculinity, and thus may suppress healthier, alternative forms of masculinity. 18 We now move to consideration of commercial “colonization” more generally.
Fig. 2. Additional projects from Scratch users
2.3 Tendency to Gravitate Towards Commercial Content in General
The user in Figure 2 produces just a few of the hundreds of Halo-inspired Scratch projects; a brief search turns up dozens of users creating these works: “Halo Zombie Defense 2,” “Halo reach firefight,” “Halo FPS shotgun,” etc. Other users have up to thirty-seven Halo-related projects. These projects should not be seen as entirely negative: the appropriation and remixing of commercial content creates learning opportunities for computational skills and concepts, and the network of like-minded users offers motivation, camaraderie, and a social learning context. Those working at the intersection of media literacy and fan studies recognized the importance of using popular culture and commercial content to develop essential 21st century digital production skills for both children and adults. 19 New media literacy organizations, such as the Digital Youth Network, are founded on work to formalize these skills in- and after-school. 20 Such efforts adhere to the importance of including youth-oriented, popular commercial culture in 21st century education.
On the other hand, many Halo inspired projects also show generic conventions — gender stereotypes, excessive militarism, etc. – that stem from youth-oriented commercial culture. Beyond commercial gaming products, Scratch projects, studios, and forums yield a high amount of other media franchised brand name traffic: A search for Barbie on the Scratch community site leads to 2,960 results for animations, art, games, music, and stories all featuring the easily recognizable characters of Barbie and friends. Add to that 6,530 results for McDonalds; 4,600 for Disney Princess; 8,210 for Transformers; 17,400 results for Call of Duty; and numerous others such as Bratz, American Girl, Strawberry Shortcake, Power Rangers, Care Bears, My Little Pony, Adidas – let us not overlook over 3 million search hits for Pokemon – and one begins to see the problem with the Scratch website’s motto, “We turn children from consumers into producers.”
The reality is that these commercialized products, advertisements, and media commodities stand in strong contrast to Papert’s idealistic vision: that constructionist software will allow ideas by children-designers to be born out of their own unique creative imagination, to come “from wherever fancy is bred.” 21 Surely he did not have in mind that it would be bred in a corporate boardroom. In category 2.2 we noted that the problematic aspect of violent video games may not be violence itself, but rather its presence as a basin of attraction; the ways in which marketing schemes use children as both target and means of consumption practices. Similarly we recommend caution against a hasty conclusion that Scratch projects featuring highly commodified objects will always convey psychologically damaging or negative properties. Scratch projects do sometimes use irony or humor with these objects, and several scholars have documented the ways in which self-conscious semiotic play can subvert gender roles and passive conceptions of readers/viewers. 22 But the majority of the Scratch projects focused on toys, fast food, violent video games, and other marketing schemes appear to be simple celebrations of the commodities, and even those that offer an ironic take are still caught in the basin of attraction that brings so many projects to focus on commodified cultural forms, at the cost of cultural alternatives that might better contribute to children’s social, cultural, and psychological growth.
Sociologist Beryl Langer refers to the products of global corporations such as Mattel, Hasbro, and Disney as “commoditoys” – objects that propagate their consumption meme across cartoons, movies, fast-food outlets, clothing, and an endless array of add-ons, accessories, and media. 23 Ferguson (2006, 2) documents the ways in which Langer’s commoditoys “implicate children in a collective trance, inspiring or strengthening a subconscious belief in the mythic powers of capitalism.” 24 One need not be against the concept of free enterprise to find something objectionable in the commodification of childhood. While “turning kids from media consumers into media producers” might be true at one level, that characterization ignores a secondary level of consumption: they are now “producers” of what are often essentially commercials for commodities. It is this secondary level of consumption that is made invisible by the content agnostic framework.
2.4 Differential of Computational Complexity between Commercially and Non-Commercially Engaged Projects
We enthusiastically acknowledge that there are many non-commercial, culturally admirable projects on the Scratch community site as well; for example we get 67 search hits for “black history.” However there is a strong tendency to have lower computational complexity for projects that engage with heritage or social critique, in contrast to commodified content. For example, a typical Black History project is a sort of automated slide show that narrates the story of Brown vs. Board, Rosa Parks, C.J. Walker, etc. Most have 2-3 scripts with a small number of blocks that simply switch the image periodically. In comparison, the stick figure games in Figure 1 use up to 15 separate scripts with a vast intricacy of control loops, conditionals and other features. This seems to be a fairly consistent pattern in our brief survey: as if technical proficiency and culturally significant content are in a zero-sum game. But of course they are not in any inherent sense; we contend the symptom is one produced by the content agnostic framework.
All four of the categories above – inappropriate material, violent video games, commodified content, and the reduction in complexity – are admittedly the result of the users’ environment and complex social influences, not an inherent property of Scratch itself. Given this fact, we imagine that an advocate for the content agnostic position might reply that the goal of Scratch is simply to teach computing, and that it provides an unbiased platform in which users must bring their own personal goals and cultural judgments to bear on their development of computational thinking. Such arguments are essentially making an analogy to the concept of free speech in a political system: it is true that no thoughtful advocate for democracy would start placing restrictions on free speech with the misguided goal of improving its content. But in our view, a better political analogy would be the issue of campaign finance reform in the United States.
Since its origins in 1867, many US legislators have proposed that we limit the amount of funding that the wealthy can use to influence elections. Such legislation is controversial; there are always complaints that restrictions on campaign donations will violate free speech. The 2010 Supreme Court decision Citizens United was a landmark victory for the content agnostic position in election funding: since then corporate influence has become so strong that more money for Senate races comes from outside rather than inside the state in which the election occurs. 25 The strong influence of marketing and other dominant forces on Scratch is analogous to the influence of wealth on an election. The problem is not that free speech should be restricted, but rather the question of how “freely” ordinary citizens can be speaking, if the other side is empowered by the enormous resources of corporations: wealth, focus groups, advertising campaigns, market penetration, branding, and so on.
Thus we need alternatives to the content agnostic position, one that can “level the playing field.” As software developers we cannot control external social influences, but we can modify attributes of the learning medium, as discussed below.
3. Constructionism and Contextualism as Orthogonal Dimensions
Our thesis is that constructionist learning theory and the content agnostic position have been erroneously presented as a single phenomenon, resulting in only two possible endpoints along a single spectrum. Once we separate these into orthogonal dimensions, new options arise. Figure 3 shows these separated dimensions using examples in math and computing education.
Fig. 3. Two orthogonal dimensions
At opposite ends of the vertical axis we have the constructionist vs. instructionist distinction that has been the focus of Papert and his colleagues. 26 On the horizontal axis we introduce the spectrum from content agnostic to content aware. Thus there are four possible categories, not just the binary choice of top-down or bottom-up. Still, it is important to ask: where did these two categories emerge from the broader history of the construction genre of educational technology?
From the development of the LOGO programming language in the 1960s and its evolution into “turtle geometry,” 27 to popular 21st century constructionist design tools, 28 information technologies have been utilized as avenues to offer a digital sandbox for young children to develop mathematical and computational thinking. Much of the conceptual basis for this approach originates in the research of Jean Piaget, who framed the development of cognition in terms of four main stages that build towards increasing levels of abstraction. Piaget’s theory of cognitive development describes learning in terms of assimilation of new information into current cognitive structures, followed by the eventual development of those structures as they accommodate from the repeated assimilations.29
For Piaget, children must move linearly through developmental stages. However, Papert and others have taken less dogmatic approaches to these stages, which is fortunate since they did not hold up in further investigations. 30 We will follow the popular convention of educational technologists in referring to Piaget’s original theory as “constructivism” and Papert’s more pluralistic conception of child development as “constructionism.”
According to Piaget, by the age of six children are able to think concretely about objects and things in their world, and to problem solve through processes of trial and error, but it is not until the age of twelve that the abstractions of formal thinking are possible. 31 Concerned with “epistemological pluralism” – validating multiple ways of learning and knowing – in the classroom, Sherry Turkle and Seymour Papert subvert Piaget’s operational hierarchy of development. They posit a more flexible understanding of “concrete” 32 versus “formal” thinking as not fixed developmental stages but rather alternative and equally legitimate ways of knowing. 33 They described the work of K-12 and college students in programming and IT, showing that preferences for less formal programming styles were not necessarily barriers to technical success. In this way, Papert alters Piaget’s biologically fixed developmental trajectory in favor of a more branched multiplicity of paths, with the implication that we will have more students travel farther by including these multiple pathways.
Simultaneous with this emphasis on legitimizing a more concrete cognitive style came an emphasis on discovery learning. From the point of view of Papert and others, these are not two separate domains, because the essential meaning of “discovery” implies starting from a prior set of skills and knowledge, which would likely be more concrete, embodied or experiential. However it is not the case, from Papert’s point of view, that starting with concrete learning automatically creates the conditions for substantial bottom-up discovery, as he describes in this passage:
What can be done to involve the mathematically alienated child? It is absurd to think this can be done by using the geometry to survey the school grounds instead of doing it on paper. Most children will enjoy running about in the bright sun. But most alienated children will remain alienated. One reason I want to emphasize here is that surveying the school grounds is not a good research project on which one can work for a long enough time to accumulate results and become involved in their development. There is a simple trick, which the child sees or does not see. If he sees it he succeeds in measuring the grounds and goes back to class the next day to work on something quite different.34
In other words, measuring the school grounds may be more concrete, but it lacks “recursive depth”; 35 the possibilities for self-generative expansion of learning experiences. Papert contrasts this simple measurement of school grounds with learning from pattern generation on the computer screen via LOGO. Thus the computational medium becomes, in a recursive fashion, its own learning context. He makes this explicit in the introduction to his 1980 book Mindstorms: Children Computers, and Powerful Ideas:
But to say that intellectual structures are built by the learner… does not mean they are built from nothing. On the contrary: Like other builders, children appropriate to their own use materials they find about them, most saliently the models and metaphors suggested by the surrounding culture… When I speak here of “our” culture… I am not trying to contrast New York with Chad. I am interested in the difference between pre-computer cultures (whether in American cities or African tribes), and the “computer cultures” that may develop everywhere in the next few decades.36
Thus Papert’s computational constructionism did not ignore cultural influence, but it did imply that African villages, the American inner city, or similar environments where computers were unlikely to be found in the 1970s were lacking potential for children’s self-development of computational thinking. In contrast, any place with computers (no matter where geographically) could best support children exploring the endless possibilities by which simple patterns can elaborate upon themselves. The computer, in this view, is not merely a medium by which math and computing become exploratory tools, but also a purification device 37 by which we can “siphon off” the irrelevant aspects of African or American culture and allow children to directly experience the raw stuff of science; a direct yet playful engagement with nature’s laws of math and computation. Educators in the global south who are aware of the destruction of indigenous cultural and ecological features due to colonialism and post-colonialism have detailed the ways in which constructivism (usually directed towards Freire and Piaget but it maps equally well to Papert’s approach) can perform a similar role, as both are based on Western assumptions about linear and industrial progress. 38 Constructionist learning theory also insists that children’s whimsical individual interests trump local connections to intergenerational learning and communal traditions. As we describe in the next two sections the constructionist view of computing ultimately ignores the role of computational thinking and design in everyday life; and in doing so, ignoring the long history of computational knowledge that is embedded in non-computer material cultures. 39
As an editor for Socialist Review in the 1950s, Papert was not naive about the power of capitalism to colonize our lives, but his political commentary in “Perestroika and Epistemological Politics” indicated that by the 1990s he saw the authoritarian tendencies of centralized government control to be at least as much a problem, and (rightfully) took pride in the resonance between bottom-up constructionist learning media and anti-authoritarian political critique. 40 This critique is a powerful means to champion humanitarian principles such as the free speech guarantees of the First Amendment of the United States constitution, but difficult to apply to problems such as the imbalance between the speech of individuals and that of giant media corporations.
Thus the rise of a content agnostic position can be attributed to at least three influences. One is a psychological theory of computationally creative media as one in which production is essentially individualistic and self-determined, such that nature’s universal laws of math and computing can be playfully elaborated and scaffolded upon itself without external influence. Another is the political atmosphere in which violations of human rights by centralized governments made anti-authoritarian critiques available as a kind of “proof of concept” for the individualist psychological framework. Finally, there is the fact that Scratch exists in a media ecology that offers its users easy access to using and re-using the signifiers of commodified youth-culture, an access not foreseen in Papert’s original vision.
4. Content Aware Learning: Examples from Culture-Based Math Education
In the case of Scratch, learning can take place without bias towards specific content because the scripting platform it provides is teaching computational thinking regardless of which images are used, which narratives are explicitly or implicitly supplied, etc. Such content agnostic software is possible for many educational disciplines. Software such as Geometer’s Sketchpad and GeoGebra can create any geometric form; Algodoo is similarly a general-purpose physics “sandbox.” In contrast to these content agnostic forms, “content aware” educational software is designed to provide students with a kind of “value added” orientation or valence.
The idea of content aware STEM (science, technology, engineering, and mathematics) education has become increasingly important in the US at national, state, and local levels. For example, the Next Generation Science Standards (NGSS), created through a partnership from the AAAS, NRC, and the National Science Teachers Association has recommended that rather than teach STEM education strictly through abstract universal principles, science education should include contextual connections, with “the goal that all students should learn about the relationships among science, technology, and society.” 41 In particular they highlight the approach of González et al., who demonstrated improvements in science education practices by incorporating local knowledge from home and community. 42 The integration of content on sustainability with STEM education has also been a growing movement, as can be seen for example in the National Association of Biology Teachers’ position statement. 43 The National Council of Teachers of Mathematics standards has emphasized that the “opportunity to experience mathematics in context is important” and thus “students should connect mathematical concepts to their daily lives, as well as to situations from science, the social sciences, medicine, and commerce.” 44 The Computer Science Teachers Association has similarly supported content aware approaches to computing. 45
The use of cultural, social, and environmental context in STEM education is thus an important category for content aware learning media. As implied by Figure 3, it is possible to take both instructionist and constructionist approaches to content aware media. A useful illustration of this distinction can be found in ethnomathematics. Although initially defined in the context of “indigenous” cultures, 46 it quickly expanded to include ancient non-Western state societies (Egyptian hieroglyphics for example), mathematics in vernacular culture (e.g. calculations by push-cart vendors; quilters, etc.), and even historical and sociological investigations of professional mathematicians.
One class of instructionist versions of ethnomathematics pedagogy can be classified as “number base systems.” Different cultures use different bases for counting; for example Mayan hieroglyphics show a base-20 system with a sub-base of 4, and this is easily converted directly into lesson plans. 47 That is not to say that it’s impossible to develop a constructionist approach using this material, but a quick Internet search will show a number of “Mayan math” worksheets available to teachers in which students are simply translating between our decimal (base-10) system and the Mayan vigesimal (base-20) system. A similar category is that of culture-based word problems. As in the case of using number systems from different cultures, using word problems from different cultural scenarios also runs the risk of trivialization; replacing Dick and Jane counting marbles with Tatuk and Esteban counting coconuts. 48 The presence of these instructionist approaches to culturally specific content is surprisingly common; the ubiquity may be partially due to the fact that it requires very little modification of standard math lessons, and little distraction or additional labor for the already-overburdened math teacher, to add on this kind of shallow “cultural” veneer.
This shallow gesture towards “multicultural math” bears little resemblance to scholarly ethnomathematics research. For example, Ascher’s description of the American Indian game of Dish goes beyond a mere cultural example of probability. 49 In the Cayuga version of the game there are six peach stones that are blackened on one side. These are tossed in the air and the numbers landing black side or brown side are recorded as an outcome. The Cayuga point scores for each outcome is, to the nearest integer, inversely proportionate to the probability: it appears as if they were calculated. Ascher does not claim that is the case; rather she notes that this accuracy can emerge over time as a consequence of how the game is embedded in community ceremonies, spiritual beliefs, and healing rituals; specifically through the concept of communal playing in which winnings are attributed to the group rather than the individual player. 50 Thus ethnomathematics research offers a correspondence with Western knowledge of probability while situating it in relation to a rich set of cultural practices, rather than reducing it to a decontextualized algebra or word problem.
But translating this research into classroom lessons is a difficult challenge. If a student is already alienated from classroom work, expecting them to read through a complex text to gain an understanding of these deeper connections is unlikely to succeed. Thus one advantage of constructionist approaches to ethnomathematics is the possibility of using the creative, discovery learning elements that Papert and his colleagues have rightfully emphasized to allow for this deeper engagement with underlying cultural and technical meanings. Lipka et al. for example developed a set of culture-based lessons for native Alaskan students, and found that pencil and paper constructionist-like exercises showed statistically significant improvement using content aware approaches rather than content agnostic. 51 Children’s familiarity with local fish rack construction not only held their interest better, but also became the basis for successive activities and explorations (for example finding out how elders used hand lengths and other body parts as measurement units).
It is worth comparing this to the earlier quote from Papert regarding a similar example of measuring the dimensions of artifacts in local surroundings, which he criticized: “surveying the school grounds is not a good research project on which one can work for a long enough time to accumulate results and become involved in their development.” 52 The phrase “involved in” gets to the heart of the matter. If measuring the school grounds is insufficient for this deeper sense of involvement, then what is? There is no doubt that commodified content can create a sense of deep involvement: video game enthusiasts show their life-long brand loyalty with tattoos for Grand Theft Auto. “Shopping haul” videos in which consumers showcase their bargains have been one of the fastest growing trends on YouTube. 53 But there is little resemblance between this kind of “obsessive passion” in contrast to the “harmonious passion” which can best foster the social, ethical, and intellectual strengths of a developing child. 54 Creative exploration of the rich set of fish rack connections – how many fish can the local ecosystem sustainably yield; its relation to native cosmology; bringing elders into a mathematics classroom, etc. – lends the power of constructionism to communities rather than corporations.
The power of a content aware approach to constructionist education is that it can aid young people’s cognitive and emotional development of community and self; including a healthy understanding of ethnicity. Healthy self-conceptions can vary greatly; from students who see ethnicity as central to those who do not. There is however substantial evidence to suggest that negative racial and ethnic stereotypes have significant impact on student performance, 55 and that students can develop creative strategies for resistance which include simultaneously embracing both ethnic identity and academic prowess. 56 A dynamic view of racial and ethnic identity – one in which children can fully explore its myriad connections – is thus better for allowing this healthy developmental trajectory.
Another reason for using a culture-based approach is the myth of genetic determinism; the false conception that underrepresented students are incapable of performing at the same level as other students because of gene-based differences in neural structure. There is no evidence supporting this, and much to the contrary. For example, after the occupation of Germany at the end of WWII, there was a population of illegitimate children of black and white soldiers who were raised by their white German mothers. Eyferth found no black/white IQ differences in this group: not surprising since they were raised in similar (German) environments. 57 More recently similar results were found for black children raised by white mothers in the US. 58 But the myth is not harmless: studies of “stereotype threat” show that African American students do worse on standardized testing when they believe the test may be reflecting racially determined intelligence. 59 In other words, the myth of genetic determinism is a potent self-fulfilling prophecy. Demonstrating complex mathematical and computational thinking as part of the heritage culture for underrepresented students is a potential counter to this pernicious falsehood.
5. Content Aware Constructionist Software in Computer Science Education
Math and computational connections to cultural heritage are not the only legitimate path to this sense of deep involvement. Jean J. Ryoo et al. and Kimberly A. Scott and Mary Aleta White describe curricula in which underrepresented high school students used a variety of electronic media to investigate – in open-ended, constructionist fashion – issues at the intersection of social structures (the “subject positions” of race, class and gender) and quality of living for groups organized by those structures (health, employment opportunities, etc.). 60 For example, Ryoo et al. noted that in one students’ reflection on her health game she “described her realization that her aunt’s obesity was affected by complex factors beyond diet and exercise, including the intersection of her aunt’s low-pay/low-status job, high crime neighborhood, inaccessibility to healthy food options, and her family’s material needs as contributing factors in her struggle toward a healthy lifestyle.” 61 The impact of this program was due to the “culturally responsive practices (asset building, reflection, and connectedness) embedded within the curriculum” 62; one might say the digital media here merely played a “supporting role.”
It is possible, however, to build a content aware approach directly into the underlying structure of constructionist software: such software would have the potential advantages of easier scaling (like any web-accessible media), facilitating the ability of instructors to add the desired social values, and – even in the case of users without instructors – it could offer a counter to the forces of commodification critiqued earlier. This is the case for the suite of applets we have designed and tested, Culturally Situated Design Tools (CSDTs). Similar to the concept of ethnomathematics, the basis for this and other “ethnocomputing” projects 63 is the idea that computational knowledge exists within cultural designs and practices: iterative patterns in Native American beadwork; recursive 64 applications of transformational geometry in African American cornrow braiding patterns; polar coordinates in urban graffiti, etc. In the case of indigenous knowledge systems there is potential for opposing the myth of genetic determinism and, in both cases, the potential for opposing the myth of cultural determinism.
The process of creating CSDTs begins with interviews with artisans, videos of their practice and “reverse engineering” of their designs; these are used to create a quantitative model that attempts, as closely as possible, to reflect the creative and technical processes of the artisans as well as the indigenous knowledge of their social context. This is important because simply imposing computational thinking externally would not have any impact on the myths of genetic determinism and might also detract from its effects on myths of cultural determinism. In the case of Native American beadwork, for example, we found that the concept of two orthogonal axes embedded in the rows and columns of the bead loom resonated with deeper cultural themes that were also organized by four-fold symmetry: native languages using base four counting; teepees made with four base poles; prayers offered to “the four winds;” etc. Interviews with artisans also revealed the use of iterative patterns; “up one over one” to create a 45 degree angle for example. The resulting model – iterative patterns on a Cartesian grid – is not necessarily something a traditional beadwork artisan would immediately tell you (in fact most artisans begin these discussions with something similar to “it can’t be explained, you just have to learn it with your hands”). But neither is it merely imposing alien math and computing. It is, rather, a sort of “composite picture” of the web of the computational thinking that is embedded in the cultural background and the individual artisan’s thinking and practice.
Fig. 4. Simulations by Navajo students
The next step is in creating a graphical interface in which these indigenous or vernacular knowledge and practices can be easily manipulated to create the traditional patterns. Some compromise has to be made between faithfully reproducing the artisan’s culture-based concepts and behaviors, creating an interface that is easy and intuitive for children, and satisfying the teacher’s need for relevance to the curriculum. One might think that systems such as Scratch are sufficiently generic for this purpose, but we have found that significant coding tasks can be encountered in creating this “sweet spot” for cultural simulations. In the cornrow braiding simulation, for example, we need to use a parameter to control “translation” – the term from the standard school curriculum – to set the distance between each of the Y-shaped “plaits” that make up one braid. But since the plaits are usually scaling in size as you move along the braid, the translation distance has to scale as well. So this CSDT sets translation as “percentage of image width” – that way students do not have to create a separate variable and modify its magnitude, which would be a significant barrier (a “steeper learning curve”) to new users. In our later move from a parametric interface to a more Scratch-like scripting tool, we found that the same technique could be introduced by offering users a “translate by percentage of width or height” block (in contrast to the Scratch “move by x or y” block).
Interestingly, braiding stylists also place the distance between plaits using a visual estimate of plait size ratios, so quantifying this as percentage actually brings it closer to the artisan’s emic, insider point of view. It was striking to see this connection in action when we observed that students with prior experience doing physical braiding often began their simulation using a small plait and scaling up by a percentage greater than 100; whereas students without that experience always scaled down. Since many students did not know that it is possible to have a percentage greater than 100, this was a helpful illustration for how cultural knowledge could be leveraged as mathematical understanding.
In much of the literature on multicultural education (including ethnomathematics), the motivation is described by what we might call the reflection theory: the need for creating lessons in which the student’s cultural identity is reflected in the math or other content. Melissa A. Millerick for example suggests that, “by framing instruction to align with students’ cultures, teachers can use curricula that honor each student’s life experiences.” 65 While that is true in some cases – the above example of prior braiding knowledge for example – we have not found the reflection theory to be a good framework for understanding the variety of learning possibilities and student interests in our experiences with CSDTs. That is not to say CSDTs are ineffective: to the contrary, evaluations have shown statistically significant increases in controlled studies of both math and computing skills as well as interest in computing careers. 66 But given a choice between the CSDTs from a variety of cultural origins – fractals in African architecture, native beadwork, break-dance movements, etc. – students do not show a strong correlation between their heritage culture and the tool they select. How to explain this contradiction?
One problem with the reflection theory is that it frames each student’s identity as singular, unified and static. This is a poor model, as identity is constantly in a process of being constructed, especially in youth. Pierre Bourdieu provides a better framework with the concept of habitus, which allows for multiple proclivities, habits, expectations, etc. that constitute identities as dynamic and multidimensional. 67 An individual’s habitus, in Bourdieu’s formulation, is still strongly influenced by social structures – working class kids learn to value blue-collar ways of being – and the social structures are in turn created by those individuals. In this aspect it is similar to Willis’s classic on “how working class kids get working class jobs.” 68 But Bourdieu also provides a theory of social change: “the systematic exploration of the unthought categories of thought that delimit the thinkable and predetermine the thought.” 69 Thus habitus leaves open the possibilities for agency, interpretation and contestation: what is sometimes referred to as “reflexivity.” 70 This combination of habitus and reflexivity holds up well in empirical studies of youth identity. 71
Stressing reflexive freedom too much makes cultural identity seem trivial or too easily malleable. Thus the content agnostic position implies that the identity of “consumer” is easily transformed to that of “producer” despite the evidence that some of the children’s construction is filled with commodified forms and practices. At the same time, stressing deterministic, unitary models of identity – as we have found in some of the literature on multicultural education – also leads to poor predictions. Viewing identity as more multidimensional and flexible – a constant negotiation between creative invention and structural influence – helps to explain why underrepresented youth might respond positively to a broader array of cultural connections. A website that makes the case for sophisticated math and computing concepts from African, American Indian, Latino, and urban vernacular practices is in itself a kind of symbolic representation of an anti-racist stance: we have experienced students expressing joy in simply knowing that someone took the time to think about the mathematical ideas in their culture. 72 As a set of design tools, it offers the attraction of what Pollock terms “everyday anti-racism”: not the heroic role model of Dr. Martin Luther King, Jr. but rather a supportive environment in which mixtures of agency and identity can be creatively explored and developed. 73
Bringing together the “pluralist” constructionism of Papert with Bourdieu’s concept of habitus also helps to illuminate how a content aware constructionist medium like CSDTs can facilitate underrepresented students’ capabilities in bringing together computational elements, cultural hybrids and technological appropriations. For example, Navajo Rug Weaver is a CSDT that draws on the “embedded” geometry and algorithmic processes found in the weaving and design of Navajo rugs. 74 Like Scratch, students develop computational skills as they move through cycles of design and testing. However, the content agnostic position emphasizes the accrual of this capital to the individual: if there is something to be discovered, it is because of the ingenuity of the computer user. In the case of Navajo students discovering the computational aspects of traditional Navajo rugs, they are essentially reverse-engineering grandma’s algorithm, an accrual of computational capital to the collective identity (and perhaps to grandma as an individual as well). Helping students become aware of these “heritage algorithms” – capital that they already possess, but which lacked a “translation” to knowledge as represented in classrooms or computing machines – is an important difference between the content agnostic and content aware approaches.
Navajo Rug Weaver is designed to be appropriable by students beyond the imitation of traditional designs. Like the rest of the CSDTs, a software development and pedagogic goal is to emphasize its flexible design aspect in ways that “allow students to utilize a synthesis of math, computing, and culture in creative expression.” 75 Some of the students’ narratives about their designs reflected contemporary concerns; others focused more on creative use of cultural tradition. One students’ description reads, “I wanted to get the four colors of the directions so once I got them I added the four hills. I added the purple background to represent my clan, ‘water flows together.’” Clearly, this student is able to build on the CSDT platform in such a way that reinforces the cultural capital of the Navajo people (four sacred mountains) while bringing out mathematical ideas (four-fold symmetry) that foster computational thinking. Other student designs reveal the appropriation of the tool for interests that are explicitly not part of Navajo culture. Without this ability, the critique of essentialism (Navajo youth will learn best with Navajo artifacts) would render CSDTs as culturally deterministic.
For example, an African American student of Caribbean heritage used a CSDT based on the native American bead loom to simulate the Jamaican flag. One of the Navajo students spotted this, and created a simulated rug design titled “Jamaican Summer Nights;” it featured the flag colors and shapes but using proportions closer to traditional Navajo design.
Fig. 5 Jamaican Summer Nights
These examples indicate a synthesis between the guidance of situated content and the openness of the construction genre. The fact that the student above chose to apply the colors of the Jamaican flag shows that it is sufficiently open-ended; the fact that it is the Jamaican flag and not, say, the colors of a McDonald’s sign suggests that there is a kind of alternative basin of attraction around culture, just as Scratch can have a basin of attraction around commercial content.
Thus the explorations are not limited to the situated content itself. Students continue to design in ways that meet their own creative ends and goals. For example, in an exercise in which students used CSDTs to explore the logarithmic curves of African Adinkra symbols, one student added in the corporate logo of the Nike swoosh. But this was illuminating a connection – who knew that both African artisans and Nike designers were using the same mathematical object? – rather than simply wallowing in commercial content. The negotiation enables a dynamic view of culture as something that is continually changing and being worked on at the everyday level, and fosters a sense of “design agency” in open-ended play between the social, the natural, and the technical. 76
Without a doubt the rise in the construction genre of educational software expands and legitimizes epistemological pluralism in and out of the classroom. Scratch and other platforms provide students with open-ended design tools that motivate authorial engagement with math and computational thinking. Yet, the coupling of constructionism with a content agnostic position can result in a basin of attraction towards commercial content. The enormous amount of content involving violent video games, fast food outlets, and other items in the “commoditoy” network – generated top-down in terms of the billion dollar corporations that benefit – can be viewed as obstacles to Papert’s original goals of bottom-up, discovery learning.
This is not to say that content agnosticism is always negative, only that the gravitational pull of commercial culture needs to be recognized as a problem in the lives of children. Educators, technologists, designers, and others should not give up the authorial and democratic learning goals that surround the construction genre. This is especially important in the US during the age of No Child Left Behind and Race to the Top, which reinforces a shift toward high-stakes testing that can result in homogeneous styles of teaching and learning. But it is critical to understand the ways in which attention to content for culturally rich and relevant lessons for students can be partnered with this technological scaffolding of democratic politics in the classroom.
Balancing content aware design with open-ended constructionist goals, as we have seen in CSDTs, invites students to creatively engage the connections between computational thinking and cultural knowledge, so as to make their own cultural capital more available to them as “things to think with,” and thus develop new ways of extending their own cultural capital into science, technology, mathematics, and engineering fields. CSDTs are just one of many possible ways of approaching the future of constructionism, such that it enables more inclusive educational practices. It is important that constructionism, in both theory and practice, continues to reinvent itself as much in its aspirations for positive social impact as in its technological sophistication.
The authors would like to acknowledge NSF grant DGE-0947980 in support of this work.
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- Mizuko Ito, Engineering Play: A Cultural History of Children’s Software (Cambridge: The MIT Press, 2009), 143-186. ↩
- For an overview of Papert’s and colleagues’ math and computing educational interventions see: Seymour Papert, Mindstorms: Children, Computers, and Powerful Ideas (New York: Basic Books, 1980).; Cynthia Solomon, Computer Environments for Children: A Reflection on Theories of Learning and Education (Cambridge: The MIT Press, 1986).; Idit Harel and Seymour Papert, Constructionism (New York: Ablex Publishing, 1991).; Yasmin Kafai, Minds in Play: Computer Game Design as a Context for Children’s Learning (Norwood: Lawrence Erlbaum Associates, 1995).; Mitchel Resnick, Turtles, Termites, and Traffic Jams: Explorations in Massively Parallel Microworlds (Cambridge: The MIT Press, 1997).; Mitchel Resnick, Amy Bruckman, and Fred Martin, “Constructional Design: Creating New Construction Kits for Kids.” In The Design of Children’s Technology, ed. Allison Druin, (Burlington: Morgan Kaufmann Publishers, 1999), 149-168.; Andrea A. DiSessa, Changing Minds: Computers, Learning, and Literacy (Cambridge: The MIT Press, 2001).; Sylvia Libow Martinez and Gary Stager. Invent to Learn: Making, Tinkering, and Engineering in the Classroom (Torrance: Constructing Modern Knowledge Press, 2013).; Yasmin B. Kafai and Quinn Burke. Connected Code: Why Children Need to Learn Programming (Cambridge: The MIT Press, 2014). ↩
- Seymour Papert, “Perestroika and Epistemological Politics.” In Constructionism, ed. Idit Harel and Seymour Papert (New York: Ablex Publishing, 1991), 13-28.; Matt Ratto, “Critical Making: Conceptual and Material Studies in Technology and Social Life,” The Information Society 27, no. 4 (2011): 252-260. ↩
- Lev Manovich, “Inside Photoshop,” Computational Culture 1, no. 1 (2011): 1-12. ↩
- Angelos Agalianos, Geoff Whitty, and Richard Noss. “The Social Shaping of Logo.” Social Studies of Science 36, no. 2 (2006): 241-267.; Tamara Lemerise, “Can we Integrate Logo into the Regular Mathematics Curriculum and Sill Persevere the Logo Spirit?” For the Learning of Mathematics 10, no. 3 (1990): 17-19. ↩
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- Idit Harel, Children Designers: Interdisciplinary Constructions for Learning and Knowing Mathematics in a Computer-Rich School, (New York: Ablex Publishing, 1991). ↩
- David J. Hess, “Undone Science and Social Movements,” in Routledge International Handbook of Ignorance Studies, ed. Matthias Gross and Linsey McGoey, (New York: Routledge, 2015), 141-154. ↩
- Underrepresentation can be defined by economic class, gender, and other identifiers, but for the sake of simplicity we will focus most of our comments here on ethnicity. In the US context these would be African American, Native American (including Pacific Islander and Native Alaskan) and Latino students. Outside the US these could be groups such as “tribals” in India, Roma in Spain, etc. Rather than minorities within a nation, one could also apply the demographic analysis internationally; for example Africans in general are underrepresented within the international computing profession for reasons that are primarily the legacy of colonialism; see G. Pascal Zachary, “Fertile Ground in Africa for Computer Science to Take Root,” The New York Times, accessed November 5, 2015, http://www.nytimes.com/2011/12/06/science/fertile-ground-in-africa-for-computer-science-to-take-root.html. ↩
- Yasmin B. Kafai, Kylie A. Peppler, and Robbin N. Chapman, The Computer Clubhouse: Constructionism and Creativity in Youth Communities, (New York: Teacher College Press, 2009). ↩
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- We have changed names and modified the images to preserve anonymity. ↩
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- For an introduction to the intersections of media literacy and fan studies see: Rebecca Black, Adolescents and Online Fan Fiction, (New York: Peter Lang, 2008).; Henry Jenkins et al., Confronting the Challenges of Participatory Culture: Media Education for the 21st Century, (Cambridge: The MIT Press, 2009).; Paul Booth, “Fandom in the Classroom: A Pedagogy of Fan Studies,” in Fan Culture: Theory and Practice, ed. Katherine Larsen and Lynn Zubernis (Newcastle upon Tyne: Cambridge Scholars Publishing, 2012), 174-187.; Michael Lachney, “Students as Fans: Student Fandom as a Means to Facilitate New Media Literacy in Public Middle Schools,” in Fan Culture: Theory and Practice, ed. Katherine Larsen and Lynn Zubernis (Newcastle upon Tyne: Cambridge Scholars Publishing, 2012), 188-207. ↩
- Barron, Brigid, Kimberley Gomez, Nichole Pinkard, and Catlin K. Martin. The Digital Youth Work: Cultivating Digital Media Citizenship in Urban Communities. Cambridge: MIT Press, 2014. ↩
- Papert, “Perestroika and Epistemological Politics,” 13-28. ↩
- Camille Bacon-Smith, Enterprising Women: Television Fandom and the Creation of Popular Myth, Philadelphia: University of Pennsylvania Press, 1992).; Henry Jenkins, Textual Poachers: Television Fans & Participatory Culture, (New York: Routledge, 1992).; Constance Penley, NASA/Trek: Popular Science and Sex in America, (New York: Verso, 1997.) ↩
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- Seymour Papert, The Children’s Machine: Rethinking School in the Age of the Computer, (New York: Basic Books, 1993). ↩
- To explore the educational application of turtle geometry see Papert, Mindstorms.; Harold Abelson, Harold, and Andrea A DiSessa, Turtle Geometry: The Computer as a Medium for Exploring Mathematics, (Cambridge: MIT Press, 1981). ↩
- Uri Wilensky, “Making Sense of Probability Through Paradox and Programming,”in Constructionism in Practice: Designing, Thinking, and Learning in a Digital World, ed. Yasmin Kafai and Mitchel Resnick, (New York: Ablex Publishing, 1996), 269-296.; Mark Gura, Getting Started with LEGO Robotics: A Guide for K-12 Educators, (Eugene: International Society for Technology in Education, 2011).; Brennan, Karen, and Mitchel Resnick. “New frameworks for Studying and Assessing the Development of Computational Thinking” (proceedings of the 2012 Annual Meeting of the American Educational Research Association, Vancouver, Canada, April 13–17, 2012). ↩
- Jean Piaget, The Construction of Reality in the Child. (New York: Ballantine Books, 1955). ↩
- Susan A. Rose and Marion Blank, “The Potency of Context in Children’s Cognition: An Illustration Through Conservation.” Child Development 45, no. 2 (1974): 499-502. ↩
- Jean Piaget and Barbel Inhelder. The Growth of Logical Thinking from Childhood to Adolescence. (New York: Basic Books, 1958). ↩
- Their terminology of concrete/abstract is perhaps misleading: in the case of different programming styles, both are pure code and in that sense equally “digital abstractions.” A better characterization might be bottom-up – structuring code organically by iterative experiments – versus top-down; planning out a hierarchical decomposition of components. ↩
- Sherry Turkle and Seymour Papert, “Epistemological Pluralism and the Revaluation of the Concrete,” in Constructionism, ed. Idit Harel and Seymour Papert, (New York: Ablex Publishing, 1991), 161-192. ↩
- Seymour Papert, “Teaching Children Thinking.” Contemporary Issues in Technology and Teacher Education 5, no. 3 (2005): 353-365. ↩
- Ron Eglash and David Banks, “Recursive Depth in Generative Spaces: Democratization in Three Dimensions of Technosocial Self-organization,” Information Society 30, no. 2 (2014): 106-115. ↩
- Papert, Mindstorms, 19-20. ↩
- Bruno Latour, We Have Never Been Modern, (Cambridge: Harvard University Press, 1993). ↩
- C.A. Bowers and Frédérique Apffel-Marglin, Rethinking Freire: Globalization and the Environmental Crisis, (Mahwah: Lawrence Erlbaum Associates Publishers, 2005). ↩
- Ron Eglash, African Fractals: Modern Computing and Indigenous Design, (New Brunswick: Rutgers University Press, 1999). ↩
- Papert, “Perestroika and Epistemological Politics.” ↩
- Next Generation Science Standards, “Next Generation Science Standards.” Next Generation Science Standards, accessed January 21, 2014, http://www.nextgenscience.org/next-generation-science-standards. ↩
- Norma Gonzalez, Luis C Moll, and Cathy Amanti, Funds of Knowledge: Theorizing Practices in Households, Communities, and Classrooms, (New York: Routledge, 2005). ↩
- National Association of Biology Teachers, “Sustainability in Life Science Teaching,” National Association of Biology Teachers, accessed September 19, 2015, http://www.nabt.org/websites/institution/index.php?p=520. ↩
- National Council of Teachers of Mathematics, “Connections,” National Council of Teachers of Mathematics, accessed January 11, 2014, http://www.nctm.org/standards/content.aspx?id=26855. ↩
- Computer Science Teachers Association, “CSTA Voice: Equity,” Computer Science Teachers Association, http://csta.acm.org/Communications/sub/CSTAVoice_Files/csta_voice_01_2013.pdf ↩
- Marcia Ascher, Ethnomathematics: A Multicultural View of Mathematical Ideas, (Boca Raton: CRC Press, 1991), 1. ↩
- Rafael Lara-Alecio, Beverly J Irby, and Leonel Morales-Aldana, “A Mathematics Lesson from the Mayan Civilization,” Teaching Children Mathematics 5, no. 3 (1998): 154-159. ↩
- Marianne M. Jennings, “Rain-Forest Algebra and MTV geometry,” The Textbook Letter November-December, accessed September 14, 2015, http://www.textbookleague.org/75math.htm.; Betina Zolkower, “Math Fictions,” in Technoscience and Cyberculture, ed. by Stanley Aronwitz, Barbara Martinsons and Michael Menser (New York: Routledge, 1996), 57-96. ↩
- Ascher, Ethnomathematics, 87-94. ↩
- Ascher, Ethnomathematics, 93. ↩
- Jerry Lipka et al., “Math in a Cultural Context: Two Case Studies of a Successful Culturally Based Math Project,” Anthropology & Education Quarterly 36, no. 4 (2005): 367-385. ↩
- Papert, “Teaching Children Thinking,” 356. ↩
- Elisabeth Harrison, From Ordinary Shopper To Celebrity, Overnight, NPR, accessed September 17, 2015, http://www.npr.org/2011/01/05/132379365/from-ordinary-shopper-to-celebrity-overnight. ↩
- Sami Timimi, “Children’s Mental Health in the Era of Globalisation: Neo-Liberalism, Commodification, McDonaldisation, and the New Challenges They Pose,” in Essential Notes in Psychiatry, ed. Victor Olisah (Rijeka: Intech, 2012), 413-438. ↩
- Daphna Oyserman, Kathy Harrison, and Deborah Bybee, “Can Racial Identity be Promotive of Academic Efficacy?” International Journal of Behavioral Development 25, no. 4 (2001): 379-385.; Inna Altschul, Daphna Oyserman, and Deborah Bybee, “Racial‐Ethnic Identity in Mid‐Adolescence: Content and Change as Predictors of Academic Achievement,” Child Development 77, no. 5 (2006): 1155-1169. ↩
- David, W. Stinson, “Negotiating the ‘White Male Math Myth’: African American Male Students and Success in School Mathematics,” Journal for Research in Mathematics Education 41, no. 1 (2013): 69-99. ↩
- Klaus Eyferth, “Leistungen Verschiedener Gruppen von Besatzungskindern in Hamburg-Wechsler Intelligenztest fur Kinder (HAWIK),” Archiv für die gesamte Psychologie 113 (1961): 222-241. ↩
- Peter Arcidiacono et al.,“Isolating Mechanisms for the Racial Divide in Education and the Labor Market: Evidence from Interracial Families,” (unpublished Manuscript, Duke University, 2012). ↩
- See Claude M. Steele, Steven J. Spencer, and Joshua Aronson, “Contending with Group Image: The Psychology of Stereotype and Social Identity Threat,” Advances in Experimental Social Psychology 34, (2002): 379-440. for a review of this literature. ↩
- Jean J. Ryoo et al., “Democratizing Computer Science Knowledge: Transforming the Face of Computer Science through Public High School Education.” Learning, Media and Technology 38, no. 2 (2013): 161-181.; Kimberly A. Scott and Mary Aleta White, “COMPUGIRLS’ Standpoint: Culturally Responsive Computing and Its Effect on Girls of Color, ” Urban Education 48, no. 4 (2013): 657-681. ↩
- Ryoo et al., “Democratizing Computer Science Knowledge,” 16. ↩
- Scott and White, “COMPUGIRLS’ Standpoint,” 658. ↩
- For an overview of current and past ethnocomputing projects see: Ron Eglash et al., “Culturally Situated Design Tools: Ethnocomputing from Field Site to Classroom,” American Anthropologist 108, no. 2 (2006): 347-362.; Matti Tedre and Ron Eglash, “Ethnocomputing,” in Software Studies: A Lexicon, ed. Matthew Fuller (Cambridge: The MIT Press, 2008), 92-101.; Yasmin Kafai et al., “Ethnocomputing with Electronic Textiles: Culturally Responsive Open Design to Broaden Participation in Computing in American Indian Youth and Communities” (presentation at the 45th ACM Technical Symposium on Computer Science Education, Atlanta GA, March 05 – 08, 2014).; Kristin A. Searle, and Yasmin B. Kafai. “Boys’ Needlework: Understanding Gendered and Indigenous Perspectives on Computing and Crafting with Electronic Textiles,” (presentation at the Eleventh Annual International Conference on International Computing Education Research, Omaha, NE, August 09 – 13, 2015). ↩
- Here we are using “recursive” in the sense of “circular feedback of information” rather than in the coding sense of calling a procedure that calls itself. Whether or not such circular information flow is implemented as recursion or iteration in the coding structure is beside the point. ↩
- Melissa A. Millerick, “Multicultural Students’ Perspectives on Their Mathematics Education,” (M.A. Thesis, Dominican University of California, 2008), 18. ↩
- Ron Eglash et al., “Fractal Simulations of African Design in Pre-College Computing Education,” ACM Transactions on Computing Education 11, no. 3 (2011): 1-14.; William Babbitt et al., “Adinkra Mathematics: A study of Ethnocomputing in Ghana,” Multidisciplinary Journal of Educational Research 5, no. 2 (2015): 110-135. ↩
- Pierre Bourdieu, Outline of a Theory of Practice, (Cambridge: Cambridge University Press, 1977). ↩
- Paul E. Willis, Learning to Labor: How Working Class Kids Get Working Class Jobs, (New York: Columbia University Press, 1977). ↩
- Pierre Bourdieu and Loïc J. D. Wacquant, An Invitation to Reflexive Sociology, (Chicago: University of Chicago Press, 1992), 40. ↩
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- William Babbitt, Dan Lyles, and Ron Eglash, “From Ethnomathematics to Ethnocomputing,” in Alternative Forms of Knowing (in) Mathematics, ed. Swapna Mukhopadhyay and Wolff-Michael Roth (New York: Springer, 2012), 205-219. ↩
- Mica Pollock, Everyday Antiracism: Getting Real About Race in School, (New York: The New Press, 2008). ↩
- Babbitt et al., “From Ethnomathematics to Ethnocomputing,” 205-219. ↩
- Eglash et al., “Culturally Situated Design Tools, 348. ↩
- Ron Eglash and Audrey Bennett, “Teaching with Hidden Capital: Agency in Children’s Computational Explorations of Cornrow Hairstyles,” Children Youth and Environments 19, no.1 (2009): 58-73. ↩