Leveraging Interactive Technologies to Accelerate Math Learning in Urban Schools

Leveraging Interactive Technologies to ­Accelerate Math Learning in Urban Schools

 

By Roland Lucas

CUNY Graduate Center

Urban Education Department

Mathematics and Technology

Abstract

Worldwide technological capacity is growing exponentially, and in doing so it increases human memory sense perception, data search, retrieval and processing powers. Our collective human power of analysis and synthesis supersedes that of any previous era due to technological advances. Transnational businesses with local reach are ever employing these leading edge technology tools, and are increasingly requiring that their workforce, even low skilled workers, have competencies for using them. Students can hardly keep up with this exponential growth of data processing speed and knowledge production. Certainly our public schools in urban areas fall far short overall in preparing our youth to meet these demands, due in large part to outdated teaching methods and insufficient resources. If there are advances, in teaching methods and resources in public schools, these advances tend to have a linear growth rate, whereas what is needed is an exponential or radical growth rate to match the exponential demands of a modern workforce. One means of helping students to adapt is to fight fire with fire; use technology to help them keep up with technological advances. Students attending our urban public schools have already immersed themselves into technology to varying degrees, in their activities outside of formal school settings. Leveraging this social and knowledge capital in more formal educational public school settings is one means of enhancing their academic learning experiences and narrowing the achievement gaps they face.

What can youth teach us about learning through their practices on friendship-driven on-line network spaces?

In 2006 the MacArthur Foundation sponsored a study to help determine how digital media are changing the way young people learn, play, socialize, and participate in civic life. In this study, researchers identified three broad patterns of behavior practiced by youth while using on-line public networks. These behavior patterns are categorized in the study as “hanging-out”, “messing around”, and “geeking out”. They convey varying levels and intensities of cultural capital production and exchange. The findings of this study have much to teach us about how to leverage student tendencies and competencies, habitus, enacted in informal fields of activity, to enhance learning in more formal educational settings.

“Hanging-out”

Youth these days hang out with each other not only face-to-face but on social networks. We all know that the social bonding of hanging out outside of the school setting can cross over the school boundary and significantly affect the cultural climate, hence learning, in schools. The hanging out that youth do on social networks is similar in many ways to how its done face-to-face; however; there are many new affordances to on-line hanging out that adult educators would be wise to recognize. Referring to on-line hanging out, the white paper from the study posits that, “while hanging out with their friends, youth develop and discuss their taste in music, their knowledge of television and movies, and their expertise in gaming. They also engage in a variety of new media practices, such as looking around online or playing games, when they are together with friends” (Living and Learning with New Media, 2008). Later in this paper I will discuss some of what educators can learn from students play with on-line games, using the theoretical lens of James Gee.

“Messing Around”

The “Living and Learning with New Media” white paper gave the following distinctions of the messing around on-line activity by youth: “Unlike hanging out, in which the desire is to maintain social connections to friends, messing around represents the beginning of a more intense, media-centric form of engagement. When messing around, young people begin to take an interest in and focus on the workings and con­tent of the technology and media themselves, tinkering, exploring, and extending their under­standing. Some activities that we identify as messing around include looking around, searching for information online, and experimentation and play with gaming and digital media production” (Living and Learning with New Media, 2008).

Below I list some salient aspects of students’ on-line messing around from a social cultural perspective, that I think are important to leverage in public school teaching practices. I will give more descriptions of my social cultural frames of reference following this discussion of student on-line activity.

 

1)    The messing around by youth in virtual spaces is a form of social production, or enactment, where learning and identity formation is continuously happening to varying levels. Participants exchange information, ideas, and give each other mutual support. They sanction each other when activity does not agree with the common interests that fundamentally bind the group. They positively reinforce each other when actions do support the group, thus strengthening group bonds. Participants of these spaces are inscribed by the ideologies, consciously and unconsciously of others in the shared space.

2)    Distance between participants is not necessarily an issue. Participants can still connect in on-line spaces despite never having met in person, and influence each other’s learning and identity formation.

3)    Youth members are the main directors and authors of the cultural production that occurs. The activity is primarily self-directed, with adults on the periphery. Even though adults may have designed and continuously maintain elements of the shared space, the youth are in control of how resources are used within the given constraints, generating new cultural artifacts and new cultural inscriptions of each other.

4)    Time is not as great a factor either. Associates are able to access each other practically at anytime of day, synchronously or asynchronously. Dialogue is constrained only by their access to a networked electronic device.

 

 

“Geeking out”

Youth on-line geeking-out has all of the aspects of hanging out and messing around, but with higher intensity levels of activity that also entail more specialized knowledge and associated competencies. The white paper says of student on-line geeking out that …

When young people geek out, they are delving into areas of interest that exceed common knowledge; this generally involves seeking expert knowledge networks outside of given friendship-driven networks. Rather than simply messing around with local friends, geeking out involves developing an identity and pride as an expert and seeking fellow experts in far-flung networks. Geeking-out is usually supported by interest-based groups, either local or online, or some hybrid of the two where fellow geeks will both produce and exchange knowledge on their subjects of interest. Rather than purely “consuming” knowledge produced by authorita­tive sources, geeking-out engagement involves “I am the Greater God of video edit­ing” (Living and Learning with New Media, 2008).

The following are some salient aspects of youth on-line geeking out that I think are important to leverage in teaching practices.

 

1)    Youth will usually demonstrate highly skilled competencies in producing the artifacts that signify the learning and enculturation that has taken place.

2)    The cultural enactment and learning that takes place can push boundaries of existing knowledge because the processes of knowledge creation are not completely bounded by traditional rules, conventions, values and sanctions of what constitutes knowledge and valid forms of expression. These spaces then create forms of knowledge and competencies that are divergent from mainstream or dominant forms of knowledge. This is a basis for innovation. Confirming this point in it’s reference to on-line collaborative gamming spaces the white paper posits, “In all of these cases, players are engaging in a complex social organization that operates under different sets of hierarchies and politics than those that occupy them in the offline world” (Living and Learning with New Media, 2008).

3)    Youth take pride in their self-authored products and unique competencies. They take pride in the recognition received by members of their group for producing specialized knowledge and artifacts.

4)    The positive reinforcement received is a motivator for continued and deeper incursions into the specialized field of knowledge and cultural production.

5)    These competencies can often transfer into other fields that have resemblances to the spaces where they were initially developed in. Youth evidence higher ordered learning by being able to transfer and apply their specialized knowledge and competencies to novel situations. Adults may be involved with the learning and production that occur, but they are not leading or directing it. Adult educators may find themselves in the reversed role of being a student of youth who are leading the way.

Through on-line hanging out and geeking out, youth demonstrate highly engaged modes of cultural enactment and learning that inextricably involve their identity development as well as competency development. They demonstrate adaptability to using on-line networks for production and exchange of knowledge and media artifacts.

As a math teacher who uses technology in the curriculum, I consistently observe students demonstrating a readiness to transfer their developing on-line competencies into the classroom. Students continuously search for ways to express their evolving identities through their competencies in school and out. The fields of their private social life have porous boundaries that are always in interchange with their school life. Students bring their acquired self-directed competencies, experiences, and tendencies to the school table, whether this fact is recognized and leveraged by school educators or not. Teachers should ask, what can the on-line self directed activities of students teach us about how to teach youth today? Teachers and other stakeholders can leverage the cultural and knowledge capital produced and created through student on-line activities to enhance and accelerate student learning. Teachers can evolve effective strategies of teaching that plug into this on-line habitus of students, and leverage it to reach more formal educational goals. This means mobilizing the same or similar resources out of school online network spaces that can produce the same or similar products in school.

Interests and Positive Sanctions

1)     Student on-line activity out of school is primarily self-directed towards self-interested goals. In my experience with students in public high school in math courses, they often don’t recognize how the study of math aligns with their self-interested goals. Teachers need to recognize the value of continuously negotiating with students how the skills taught can align with student interests, developing ideologies and cultural identities. Teachers must allow students freedom to explore, experiment, and exercise a degree of autonomy without penalizing them for straying from predefined requirements or convention. Meaning making must be recognized as a joint and negotiated venture between students and teachers. Not a one-way dictate from teachers. Furthermore, the products they produce should be validated so long as they relate in a significant way to the learning objectives at hand. Even when student authored products don’t seem to do so, they may express a serendipitous insight by the student that should also be validated. “Among fellow creators and community members, the context is one of peer-based reciprocity, where participants can gain status and reputation but do not hold evaluative au­thority over one another” (Living and Learning with New Media, 2008).

Students are motivated not just to get a grade by teachers, but also to gain acceptance and positive reinforcement by their peers and teachers, which can occur face to face or through an online activity.

Solidarity

Another important lesson to learn from student self-directed on-line activity is the importance of solidarity for youth who inhabit these spaces. Fostering collective social bonds should be recognized as essential in the context of formal student classes as well. Students in self-directed context are willing to show each other the ropes of the specialized field of activity. Teachers should foster this solidarity among students; this understanding that everything that happens in the course affects everyone, and that their collective goals can be reached more effectively if everyone supports each other. The learning experience then transcends the individual and extends out through a ripple effect to all members of the group, and even to the wider community. Students learn in effect citizenry through their specialized contributions.

As a teacher of mathematics who is keen on integrating technology into the curriculum, I see high value in leveraging self-directed student social / cultural capital and competencies in the classroom to enhance their meaning making potentials and success with mathematics. I think students are prepared to meet us more than half way in doing so.

Theoretical Frameworks

I use Bakhtin’s theory on dialogical discourse and “ideological becoming” to envision processes of positive collective identity development via collaborative discourses and knowledge construction that can take place in culturally empowering learning spaces. I also borrow heavily from Bourdieu, Tobin, and Turner, for their theories and analyses on multi level social \ cultural structures, and how these theories can support a vision of social transformation. Any project addressing the transformation of educational praxis must be grounded in a socio-cultural theory of action that accounts for the structures of the school environment as being embedded in larger cultural structures. It must also account for the motives of the groups focused upon, and relational fields social change that they inhabit.

Bakhtin’s Dialogical Discourse and Identity Development

Agency involves an actor using available tools, structures and resources to carry out actions to obtain a goal. My unit of analysis is not the agency of African American or other minority students as individuals or even as a group onto themselves achieving their isolated goals. Too often this kind of focus produces deficit theories, finding the problem of under achievement by African American students in their own bodies, minds and ethnic culture. It is not the African American student in isolation, even while connected to advanced tutorial software applications that I consider. I consider activities of groups of students in collaboration through interactive technologies for the purpose of increasing their agency to offset much of the limiting structures of racism and classism as manifested in their local communities. I consider how through their collective activities they can proactively solve problems that are relevant to the motives of their group and to their communities. I consider how having access to timely, relevant, and up-to-date knowledge capital and conduits afforded by interactive technologies can serve as vital resources to accomplish collective motives.

Furthermore, collaborations in networked learning spaces involving a community of students tend to make the construction of knowledge less centered on the adult teacher as the only source of knowledge that students consider. Knowledge construction becomes less an exercise of reproducing established knowledge and power structures. It would be less authoritative, monological and passive, as Bakhtin (1986, 2004) describes it. Dialogue would rather be more in keeping with Bakhtin’s concept of inner persuasive discourse where participants actively scrutinize, challenge, change, reject and argue over existing knowledge to suit their needs and evolving understandings. Multiple “utterances” or dialogues are then considered, synthesized or otherwise reshaped as needed (polyphony). New products that students appropriate and generate can be acted upon to support their agency and identity formation. Existing knowledge and relational structures will be reproduced only if they support the motives of the collective. If not, these structures should be targeted for transformation, thus providing for student agency.

Students would no longer fall into serving the intentions of dominant groups, nor channel the words, ideologies, goals, and problems of dominant groups. Students become critical thinkers who analyze data and problems in service of their communities. Without this directed critical facility, they would simply implicate themselves in sustaining the reproduction of their own subordination in society through a mis-educational system. They would “recite by heart” other people’s voices or structural rules. They would, in the Bakhtinian sense, parrot authoritative discourses, rather than retell their stories in their own words. Learning to privilege one’s own critical voice is what Bakhtin (1981), refers to as “ideologically becoming”. This is education proper and is sorely lacking in public schools where minority students predominate. A culturally sensitive approach to education that respects difference and builds upon the life experiences of students would encourage this kind of critical thinking and development of “voice” or ideological self. It would furthermore facilitate sharing with and building upon ideas of others who have common liberating motives. It would very much be involved with building solidarity for the purpose of helping students realize their collective motives and individual educational goals.

Extending James Gee’s Ideas On What Video Games Teaches Us About Learning

It is my understanding as an educator and an African American who has gone through the public education system in New York, that African American students in public education view themselves and are treated externally, as a Diaspora or “affinity group” that has a different set of challenges, consequently, a different set of immediate goals than other student groups. James Gee, in his book “What video games have to teach us about learning”, describes “semiotic domain” as “any set of practices that recruits one or more modalities (e.g., oral written language, images, equations, symbols, sounds, gestures, graphs, artifacts, etc.), to communicate distinctive types of meaning.” (Gee 2007) I will extend the definition of semiotic domain to refer to the situation of minority students in the mathematics semiotic domains. For Gee an “affinity group” is simply any group associated with a particular “semiotic domain”. I will refer to the subset of minority students, particularly African American students, situated in public schools and who generally struggle with math, as an affinity group that I am principally targeting in this discussion.

The legacy of institutionalized racism as manifested in the public education of African Americans specifically, and in society generally, has contributed to the formation of the African American student affinity group that is set apart from normalized groups. As I say this, a memory comes to mind of seeing all Blacks in the cafeteria at the college I attended, Westfield State University in Massachusetts, always sitting together. The university was 95% White and 2% Black. I will note here that every individual and group have unique measures of achievement. I see achievement as the attainment of knowledge and the development of competencies or habitus, necessary to produce structures that will enable an individual to successfully reach goals that uplift not just the individual, but the collectives that the individual identify with (Gee’s identity groups).

It is worth investigating how African American students located in economically depressed urban areas see themselves in the context of the public education system. Though this inquiry is germane to the topic of this paper and touches on Gee’s “identity principle”, I will not venture to present a review of attitudes that shape the African American student self perception here. I do agree with Boykin’s “triple quandary” and Dubois’ “double consciousness” as concepts that illuminate general dispositions of African Americans in the context of the public education system. I will focus on the issue of how this affinity group can utilize current technologies effectively to close the knowledge and achievement gaps they face. Tied to this, is the inquiry into practices that can promote the positive “projective identity” of weaker African American students in their study of math and science, which I will discuss.

Affinity Groups and Math Identity

Gee talks about how members of an affinity group can enter a semiotic domain with performance weakness, therefore needing a “psychosocial moratorium”, which he describes as “a learning space in which the learner can take risks where real-world consequences are lowered” (Gee 2007). Applying this concept to the subset of African American students that are weaker in math and science, teachers of these students must not only be sensitive to the academic weaknesses they bring to these classes, but effectively use methods that can mediate the complexities of problems that often defeat weaker students. This is where the use of technology in the classroom can be a vital asset. But first there needs to be an understanding that education in large part has to do with identity construction and the building up of self esteem or self efficacy. With African American students, who may have developed oppositional attitudes to public education due to their experience of various forms of racism in society, it is vital to address these attitudes in positive ways, starting with acknowledging the prevailing structures of racism experienced by students. The teacher must encourage the development of a constructive and engaged projective identity capable of transcending student conditioned responses to racism, and the structures of racism themselves in the students’ environment. Gee says, “Without such an identity commitment, no deep learning can occur. The student will not invest the time, effort, and committed engagement that active, critical learning requires.” (Gee 2007). Once the student is on the path of developing a healthy projective identity, believing that he/she can learn the subject at hand, it is then vital to reinforce this identity formation by scaffolding methods.

Mathematics Identity Building In My High School Classes

In preparing my algebra 2 classes to take the New Jersey state standardized high school proficiency exam in mathematics, I make heavy use of graphing utilities, like a graphing calculator and Maple 15. Though I teach my students how to handle challenging questions about functions and their graphs analytically or “by hand” I frequently reinforce the concept at hand by using the graphing utility. Sometimes I may even present the concept with the graphing tool first. The reasons are several and critical. My students tend to be more receptive to visual representations of a problem and its solution as opposed to simple text representations. In my view, teachers should embrace multiple approaches to teaching students in need of remediation. Gee refers to this as the “multimodal principle”, where “meaning and knowledge are built up through various modalities (images, texts, symbols, interactions, abstract design, sound etc.), not just words” (Gee 2007). This is not to imply that African American students or any other group can’t master textual representations as well.

Since my students do tend to have weaknesses in analytical problem solving, the graphing utility scaffolds around those weaknesses, thus providing the immediate satisfaction of understanding and solving the problem at hand. We can then spend more time talking about the solution and its potential value or relevance to them, than not. Once they are aware of the solution and its possible value to them, it is then easier for me to “sell” to them the value of knowing how to handle the same problem analytically. They also trust that I will successfully guide them to the solution “by hand” as I did by the graphing tool.

Through scaffolding students are able to, for example, bypass having the adeptness of graphing functions “by hand” and analytically solving for unknowns, by simply entering the function definitions into a graphing utility and displaying the results. The various attributes of the function can be readily known by visual inspection (such as, critical points, local maxima/minima, asymptotes, tail end behavior, domain, range, undefined points, inflection points, roots, intersections with other functions to solve for an unknown etc.) with relative ease, compared to discovering them analytically. Shielding students from undue complexities affords them more time to engage in higher ordered thinking, and the thrill of solving more complex problems, thus boosting their sense of self efficacy. This speaks to Gee’s concept of “psychosocial moratorium”, where success is not dependent on managing all the complexities of a problem all at once.

The above approach exposes students to the sphere of higher ordered thinking as it relates to the math semiotic domain. Student weaknesses can then be addressed from a position of gleaning the big picture, increased engagement, self-confidence, and self-motivation, all translating into progressive achievement. Gee sums up the positive effects of applying good gaming principles to education when he says “they situate meaning in a multimodal space through embodied experiences to solve problems and reflect on the intricacies of the design of imagined worlds and the design of both real and imagined social relationships and identities in the modern world.” (Gee 2007). I take this to mean, as it relates to the unique learning needs of African American students in math and science, that we must empower them to be active doers of math in these creative learning spaces. I will add that for this to be accomplished, weaker students must engage in math that is relevant to the unique challenges they face in both the education realm, and the larger society. These learning spaces must actively prepare them to be critical thinkers and to solve the unique problems they experience. Throughout the process we must be consciously aware that education is certainly deeply involved with identity construction. With the above approach, I have witnessed increased student engagement and willingness to work through math problems.

Let’s CHAT

The CHAT, cultural historical activity theory, also has promise in conceptualizing transformative, agentic educational practice that can further the educational goals of the African American collective. The unit of analysis of this theory reaches beyond the individual personal activity and subjective reality, in isolation from the context of the wider fields of social interaction. According to this cultural theory, individual activity, including learning in the classroom, is in symbiotic or dialectic relation with the wider community within which individual activity is embedded. “CHAT leads to changes in the location of representing what is educationally relevant: Its inherently dialectical unit of analysis allows for an embodied mind, itself an aspect of the material world, stretching across social and material environments” (Roth and Lee, 2007). Hence, individual learning goals, must be considered in context of the wider goals of the community within which the individual is situated. This is precisely the thrust of my approach on urban educational practices. I see collaborative technologies as tools that enhance the agentic activity of individual | collective African American student achievement. They can facilitate knowledge construction in iterative and accumulating developments, not unlike the multiplier effect of money in a closed economic system, thereby magnifying the capital wealth (knowledge capital) across the entire community of stakeholders.

Multiplier Effect of Interactive technologies

Interactive technologies foster the acceleration of the interactive transforming exchanges needed to accomplish the progressive development of African American communities, and those of other minority groups. Interactive technologies can facilitate a multiplier effect of knowledge capital that can reverse the knowledge and achievement gaps faced by African American students. An increase in the number of persons (nodes) distributing information (capital) in a closed networked system will increase the frequency that each person will receive a form of capital, in the Bourdieusian sense. Each exchange of capital is the equivalent of an injection of new capital in the system, thus representing a multiplier effect. That is, the net gain in capital of each person in the system is increased many more times than if a person was acting alone or with a very few other persons (nodes). The goal then in urban schools is to utilize tools, processes, and approaches that will increase the frequency that students in a networked system (extended classroom) can share information with stakeholders. In this paradigm students appropriate interactive technology tools that facilitate sharing of timely information. The more knowledge capital that is circulated the better the overall quality and effect of produced products that have embedded in them the knowledge. What promotes effective meaningful discourse around problem scenarios engaged by students, also promotes an increase in meaning potentials for effectively solving those problems. Through the improvement of the means of transacting forms of capital in the Bourdieusian sense, comes a multiplier effect of net capital accumulation, including knowledge capital.Modeling With Interactive Learning Spaces

An important aspect of collaboration is not only the facility to distribute knowledge across networks, but also those of constructing knowledge collaboratively, leveraging the gains from modeling, scaffolding and feedback. The products of collaboration can then be distributed across networks to stakeholders who can further build on these products.

Modeling

Using powerful Graphical User Interface (GUI) tools and forth generation computer languages (4GL), students can construct a model of a problem along with a possible solution set and submit it for collaborative review. A teacher can in the same sense as a text document, markup the model or otherwise point out to the student areas where the model may be enhanced for a better solution. Students can then explore the suggested solution path. I am currently exploring a power math tool called Maple 15 that does this kind of modeling combined seamlessly with student co-authored rich text.

The NRC report “How People Learn” also offers critical insights into how the use of technology, backed by sound pedagogical principles, can enhance the learning of African American students and help close the knowledge and achievement gaps they face. One method it promotes is using technology as a modeling tool. Modeling problems visually through technology, in my experience has tremendous advantages for students with math weaknesses. There is a strong trend in modern societies of visually modeling or simulating a problem set using advanced technologies such as CAD (Computer Aided Design). It is popular in current gaming design and many fields, such as medicine, meteorology, architecture, film, and aviation to name just a few. There is no reason why this approach should not be taken full advantage of to aid students in tackling complexities in math and science in the classroom, particularly for those who struggle in these fields.

Feedback And Scaffolding

Another advantage that using technology to aid learning has is that it gives students and teachers more opportunities for feedback. Think of the power that we may currently take for granted of writing a paper electronically and submitting a draft for review to peers and a teacher who are sometimes a half a world away. The reviewer can offer critique by marking up the document electronically next to the original text, without changing the original. The student author can then accept or reject some or all of the changes. The student can explore what the results would be (see how it reads) if the changes were accepted. This also applies to the math and science models. Students can construct a model of a problem along with a possible solution set and submit it for review. A teacher can in the same sense as a text document, markup the model or otherwise point out to the student areas where the model may be enhanced for a better solution. Students can then explore the suggested solution path. This aspect of scaffolding is critical for students who are weaker in a given semiotic domain and who are struggling to navigate it. It also enables students to become more reflective and aware of successful strategies in navigating the semiotic domain. As stated in “How People Learn”, “technology creates opportunities to incorporate into curricula a meta-cognitive approach to instruction by using an inquiry cycle that helps students see where they are in the inquiry process” (How People Learn, 2007). The end result is that students learn the processes of becoming adept at a particular semiotic domain. They develop self-efficacy as authors / designers who not only adequately function in the semiotic domain, but also advance the semiotic domain to wider frontiers though unique authoring / designing contributions. The sound pedagogical use of technology by teachers affords promising opportunities to mediate academic weaknesses in math and science that African American students tend to have in typical urban schools.

Collaboratories – Networked Learning Spaces

One other concept discussed in the “How People Learn” report that I will extend to address the learning needs of African American students is that current technologies can provide opportunities for these students to collaborate with peers, teachers, experts, and anyone associated with a particular semiotic domain of interest, via virtual learning spaces or what the NRC report refers to as “collaboratories”. In my experience, students in urban public schools already have adeptness in using collaborative technologies, such as cell-phones and Facebook on the web. The problem is that this collaboration tends to focus not on academic and collective socio-eco-political problems, but rather on non-academic, social circle concerns. Educators should leverage the familiarity of urban school students with the later usage of technology and apply it to the more frequent usage for the former type to address the knowledge and achievement gaps they face.

The knowledge and achievement gaps in math and science faced by urban school students have deep historical and macro causes. The digital and knowledge divide are macro national problems requiring macro national solutions. The beauty of the Internet and collaborative technologies with respect to these problems is that it is designed to solve problems that are distributed over dispersed geographical domains as easily as problems that are locally situated. Furthermore, the technology can be used to roll-up or incorporate solutions of local problems, into models that treat the same problem from a macro or global perspective. The “How People Learn” report used the example of students collecting and analyzing local data related to global warming and then through collaborative technologies, uploading their local findings to a centralized global model of the phenomena for shared use. In the same way, urban school students can roll-up their solutions to local problems into models that treat the same problem from a macro or global perspective, through the vehicle of collaborative learning spaces. Specifically, interactive technologies in education offer powerful tools for addressing field trip and meeting constraints, access to experts in a given discipline with video-based problems, computer simulations, electronic communications systems that connect classrooms with communities of practitioners and experts in science, mathematics, and other fields. This all allows students to collaborate in wider collaborative learning communities. In these spaces, students can use shared collaborative and visual tools and see how their local data fits into a larger model (e.g., local environmental studies of environmental pollution that causes disproportionate occurrences of asthma in urban communities). Integrating this approach into the curriculum results in positive student attitude towards and engagement with complex problems.

Students can be engaged in online learning communities for creating, sharing, and mastering knowledge: exchanging real-time data, deliberating alternative interpretations of that information, using collaboration tools to discuss the meaning of findings, and collectively evolving new conceptual frameworks. Knowledge and meaning is obtained through the synthesis of multiple dialogues and points of view, where each “utterance” in the Bakhtin sense, is predicated on those that came before. In an interactive virtual learning space, these utterances can be contributions to a threaded discussion on a discussion board. To accomplish this I have for example, used the Moodle collaborative software as a collaborative tool in my high school math classes. The outcomes of this collaboration benefit not just the individual student, but can become a repository of relevant knowledge capital, by and for the community at large. It becomes a collective competency, directed at meaning making that is meaningful to both the individual and the community the individual comes from. It will remain a collective competence so long as the discourses and knowledge produce remains directly relevant or synchronous to the common problems faced by the collective. Once this golden rule is violated, then the environment has been compromised. An accumulation of compromises past a critical point will render the collaborative environment ineffectual in the uplift of both the individual and collective. This possibility has to be vigilantly mitigated by both the participants and designers of the learning space.

Scaffolding and Modeling with Web-Based Tools

One step forward to affording urban math students with tools needed to accelerate the meaning making potential within the domain of high school math classes are with on-line math programs that can more clearly reveal math relationships and solutions through powerful GUI interfaces. There is a plethora of accessible web-based tools that can help enhance learning for students in urban schools. I maintain that the best of these tools would allow students to generate visual models of math concepts they learn in their courses. These models would allow students to inspect critical aspects of the problem they are modeling, without getting bogged down with the mechanics of revealing those critical aspects. I further maintain that the best of these modeling tools would allow students to interact dynamically with the model, changing parameters and presentations as desired, so as to further expose nuances of their problem. With the better tools, students would be able to share their models with others to get feedback, and they would be able to update the model based on this feedback, and publish their findings. There is yet another dynamic feature that I think would make for a better on-line tool, as that is the ability of students to couch their mathematics within the context of a real-world example. The tool would allow a student to define a real-life problem. It would make available constructs that facilitate establishing the parameters of the problem. These constructs can be applicable math task, formulas, figures and shapes, typical formulas, and preexisting models of a similar kind. Students would in essence be able to construct full-blown virtual representations of their scenario that are realistic and maps well to the real-life problem.

Barry Cherkas and Rachael Welder of Hunter College in their research paper identified three major types of on-line interactive tools to enhance math learning using web-based tools: a) static, “meaning that knowledge-seekers read passive content published on a website similar to material printed in a textbook; b) interactive, “having the ability to provide learner with a richer experience … but they do not actually teach students how to solve problems”; c) dynamic, meaning “that knowledge-seekers interact with web tools that generate fresh customized content based on user input.” (Cherkas & Welder 2012). In other words they are interactive and responsive to user-defined parameters. It is of course the later type of on-line tool that affords greater learning potential for students. As Cherkas states it, “The value of such resources resides in their ability to give learners the opportunity to develop visualization skills, explore mathematical concepts in innovative ways, and obtain solutions to self-selected problems … giving learners “more control of their learning environment of meet their individual cognitive and developmental needs.” (Cherkas & Welder 2012) Part of the best practices of using on-line math tools, as identified by Cherkas and Welder, is that he tools can provide “insight into the development of reasoning behind the mathematical ideas discussed” (Cherkas & Welder 2012). So the tools do not only present solutions, but can aid in exposing the reasons and concepts that support the solutions. The tool helps with first principles of the concept the student is exploring and using to solve problems. These advantages are precisely what students in urban public schools need to accelerate their learning and close achievement gaps.

Using ethnicity data they collected over 5 websites dedicated to math education, Alvan Baranchik and Barry Cherkas were able to show that more ethnic minority students use these sites disproportionately more than Caucasian students. This supports the claim that students who have traditionally had difficulty handling these higher-level courses, and/or were not getting the help they needed in schools, as alluded to in the paper, found high utility with on-line tools to help gain a better understanding of the concepts. I have made use of one such on-line math tool, located at webgraphing.com (Cherkas 2010). This tool allowed my algebra 2, pre-calculus, and calculus students to graph higher ordered functions and to investigate their properties with relative ease. My students appreciated the ease that they were able to quickly find critical aspects of functions and to use this information to answer higher ordered questions about concepts they were learning in class. I think this kind of on-line tool use will become more common in classes such as mine. I think though in the future, these tools will gain in their dynamic capabilities as I described above, allowing students to construct full-blown virtual models of real-life problems, with relative ease, thus affording greater opportunities for making their knowledge construction align more closely with what is relevant to them.

I am currently exploring a power math tool called Maple 15 that does this kind of modeling that combines powerful mathematical modeling seamlessly with student co-authored rich text. The Maple work environment is standalone; however, the worksheets produced can easily be shared through a network. The worksheets can be exported into HTML format so that recipients of these worksheets can view them. I can envision a time when math worksheets will be editable by tools other than Maple 15; just like MS Word and Excel files can be modified by other applications. Think about how word processing programs such as MS Word and Excel spreadsheets allows for knowledge construction and sharing. I see Maple as doing the same, but in aiding constructing math knowledge, as well as fostering dialog and fluency with mathematics. This particular tool has math capabilities ranging from arithmetic through advanced calculus and beyond. As such it can be intimidating for students if the teacher is not careful to help students managing these capabilities. For instance Maple has both “in-line” and menu driven modes for accomplishing most every task. I have students work exclusively with the menu driven options that are context sensitive; so powerful operations are available by clicking on a math expression, equation or graph.

Criteria For Implementing an Interactive Learning Environment

The following are some requirements of an online collaborative learning environment.

  • Teacher can create new content to augment any pre-existing content.
  • Student authors and mashes up content.
  • Organize teacher and student authored content in context of the lesson.
  • Allow for shared objects and models.
  • Allow for organization of authored Asynchronous content
  • Allow for synchronous dialog
  • Allow for controlled multi-user access to content
  • Handle wide array of multimedia objects
  • Ease of use
  • Scalability to Web 2.0 tools
  • Easy setup and access
  • Scalable cost
  • Capture student feedback
  • Allow for simultaneous or near simultaneous control and manipulation of objects within the context of a lesson.

These areas would be assessed in a given locality on a scale of: none existent, recently implemented, low, medium, and high). It would be important to give a locality periodic feedback on how well it is doing with regards to its implementation of a collaborative learning environment across the curricula. The promise of such feedback and subsequent advice would be a selling point to encourage a school to participate in this project initiative.

Assessing for a Web 2.0 Learning Environment

Web 2.0 tools – online platforms that allow nonprogrammers to contribute content to the World Wide Web (O’reilly, 2005), are increasing transforming not only our society, but has the greatest transformative effect on education in the foreseeable future. Having said this, not all teachers are in a position to use web 2.0 tools in their classes to the greatest positive effect. I think it’s safe to say that most educational practitioners are aware of the gulf between having access to computers in the classrooms, and the facility of both teachers and students to use them in effective ways to enhance learning. Even when teachers make use of interactive technologies, there are gradients on how they use them ranging from one-way, teacher centered, and local to multi-directional and collaborative and global. I maintain that it is the later usage that maximizes the meaning potential developed by and for students, hence the higher level learning by students.

Justin Reich, the first author of the article entitled “The State of Wiki US in the U.S. K-12”, identified four general types of Wiki usage by high schools across the U.S. The focus of this study was on Wiki’s because of their collaborative characteristics, they are emblematic of Web 2.0 technologies, and they are easily accessible to schools having a networked environment. Their primary research data set came from a web site that offer’s free wiki usage to educators and students. The study identified four major categories of Wiki usage along with their frequency of usage: (a) trial wikis and teacher resource-sharing sites (40%), (b) teacher content-delivery sites (34%), (c) individual student assignments and portfolios (25%), and (d) collaborative student presentations and workspaces (1%). The study found that wikis created in schools serving low-income students have fewer opportunities for 21st –century skill development and shorter lifetimes than wikis from schools serving affluent students.

As a part of this study, the researchers developed the Wiki Quality Instrument (WQI), as a measuring rubric to ascertain the levels to which schools are using wikis to promote the development of 21st-century skill sets for students. The tool has 5 major categories for assessment, each having several sub-categories. The 5 major categories with their number of sub-categories are: (a) Information Consumption (2 items) (b) Student Participation (4 items), (c) Expert Thinking (5 items), (d) New Media Literacy (6 items), and (e) Complex Communication (7 items). Coders of wiki changes made in a content management system such as Moodle or Blackboard, assess whether students participate in activities that support the development of 21st-century skills as part of their high school curriculum. Their findings after applying this rubric to a sample of 241 schools, is that schools serving more affluent students provide more opportunities for development of 21st-century skills, as revealed through the wiki usage. Furthermore, when teachers do use technology in the classroom, it is more for the purpose of gaining efficiencies with existing practices, such as disseminating teacher generated information, rather than for the purpose of transforming those existing practices such as allowing students to author and share newly created knowledge, of rich multi-media content. According to their student only about 1% of the wiki usage was of a collaborative nature. I maintain that it is precisely collaborative kind of activity, leveraging Web 2.0 technologies that will afford urban students the means to accelerating their learning exponentially, and preparing them to thrive in our modern economies.

Using Interactive Technologies In My High School Math Classes

Over the last 7 years of teaching in urban schools in New Jersey, I’ve engaged students in projects that I hope will develop their identification of themselves as doers of mathematics not just for their own benefit, but to the greater good of the community from which they come. This is in keeping with the unit of analysis that I mentioned already. It is not the student’s activities in isolation, even if it involves advanced technologies. Rather, I consider activities of groups of students in collaboration through interactive technologies for the purpose of increasing their collective meaning making potential and their agency to problem solve on behalf of their communities. This goal was reflected in the project requirements in one way or another, as I attempted to develop this collective identify of my high school math students. The following are some activities I engage my class with to do this:

1)    Access resources on the Moodle placed there either by other teachers, other students or myself. Students can modify copy and then modify contents.

 

Resources include:

 

  1. PowerPoint slides on math lessons
  2. Videos on particular skills
  3. Assessments
  4. Journal entries in forums
  5. Stored files

 

2)    Create shared folders in Gmail where students can share electronic files with each other and I.

3)    Create Web site for students in Google to establish virtual math identity and share files with public (rest of school).

4)    Smart board to facilitate whole class sharing of products

5)    Google survey, Word Press, Geometer’s sketchpad, Maple 15.

6)    Co-teaching and cogens are mechanisms for students to share their perspectives on math.

7)     Maple 15 allows students to create narratives or problem scenarios and seamlessly embed math into them. They can then share this with others by exporting it as an html file.

Micro Discourse Analysis

Coding for positive evaluations of student discourse as they work towards collective and individual goals would need to take place over all the types of products that were collected while students engaged with math content using technology tools, such as the ones I mentioned. The following is a tool for doing discourse analysis on student dialogue and authored products that I adopted and adapted from Lisbeth Amhag and Anders Jakobsson, who used it in their analysis of student dialog while taking an online course. This tool is in keeping with Bakhtin’s ideas of ideological becoming.

Dialogic Level The levels of thematic patterns in the dialogue
Passive and

authoritative

 

• Accepting and confirming

• Passively reproducing knowledge

• Monological and authoritative

• Failure to explicate the possible meaning potential (in the dialogue, and through artifacts as a basis for learning, development, and solving collective problems

 

Meaning potential can be understood as a sample space that is composed of all the possible ways to understand or interpret statements made in a dialogue.

Persuasive and preliminary

negotiation

• Accepting, confirming and questioning

• Elements of passively reproducing knowledge

• Negotiations

• Responses and artifacts create possible meaning potentials

• Failure to use meaning potential as a basis for learning, development, and solving collective problems

Persuasive and co-authorial

negotiation

 

• Accepting, confirming or actively questioning and a desire to develop the discussion

• Few or no elements of passively reproducing knowledge or artifacts

• Others’ statements reworded to own words

• Participants are shareholders and co-authors in the account, negotiations

• Responses create possible meaning potentials

• Use of meaning potential actively as basis for learning, development and solving collective problems

 

I’m in my 6th year of teaching mathematics and using interactive technologies to enhance student learning as I’ve described. I have seen indications of my students developing an ideological stance and identity as a problem solver on behalf of student communities. Using the above dialogical tool will be helpful going forward to determine the level of student “Ideological Becoming”, in the Bakhtinian sense. Another means to ascertain that students in urban classrooms are developing positive identity as doers of math on behalf of their communities, is to gage the level of synchrony and entrainment amongst students and teachers in the classroom in the sense given by Randall Collins in his book, Interaction Ritual Chains (2004):

“As persons the person become more tightly focused on their common activity, more aware of what each other is doing, and feeling, and more aware of each other’s awareness, they experience their shared emotion intensely, as it comes to dominate their awareness”.

This solidarity on a sustained and continuous level, towards a common goal of community uplift is what I hope to engender in my students.

Ken Tobin has done extensive work in this area of microanalysis of synchrony in the classroom and has demonstrated its usefulness.

“We show that specific prosodic features in face-to-face encounters—alignment and misalignment—are associated with the production of solidarity and conflict, which in turn are associated with successful and unsuccessful lessons. They are also associated with different degrees of solidarity and emotional energy that participants in science classrooms experienced.” Roth & Tobin (2010)

 

Micro level data, as in video taping of students as they are engaged in the type of collaborative project work that involves community, can be coded for prosody, entrainment or sustained focus, and synchrony. This data can be compared to the same type of micro level data of students engaged in typical teacher centered lessons. Though I have not carried out this micro level analysis and comparison with my students, I have notice higher levels of student engagement, entrainment, and synchrony when students were allowed to author their own math products and were free to present them in the role of class teacher. I would not want to overstate what this micro level analysis can reveal in terms of identity formation. I think that looking at micro level data in isolation from wider cultural structures that students are embedded in, can lead to misinterpretation of events. For instance simple smiles by students in and of themselves can be interpreted as positive emotional engagement, but can actually be an expressions of subversion or carnival (in the Bakhtinian sense) to a teacher’s practice. However this is true for all of the forms of evidence that I have listed thus far. It is only when one form of evidence is held together with all of the other student produced cultural artifacts that we can gain higher confidence levels that this collaborative, student centered approach cultivates in students a positive identity formations as doers of math on the behalf of their communities. Having stated this caution, it is my experience that the levels of solidarity are markedly higher when students are given freedom to work on problems that are relevant to their communities, and allowed to present them, without undue teacher interventions, to their classmates and other stakeholders. This may also translate into higher levels of student achievement in math and science, as well as entry into these professions.

Where does this lead? I think this can lead to a feedback loop of cultural capital forming a habitus, or educational praxis that supports not just individual positive affective identity formation, but positive collective group identity formation, that can be the basis for transforming dominant structures that have to this day, thwarted the educational aspirations of African American students. I refer again to Turner for aid in conceptualizing prospects for transformative agentic action, as the outcome of positive affective identity formation of participants in my project: “the flow of positive sanctions in an encounter tends to circulate among the participants to the encounter, with individuals mutually sanctioning each other in ways that build up local solidarities, although at times this flow of mutual positive sanctioning can work its way up to meso-structures and macrostructures” (Turner 2007). This reminds me again of the multiplier effect of circulating money in a closed system repeatedly before allowing it to leave the system, as a means of accruing wealth exponentially. This then provides a cultural capital accumulation that may reach a critical mass, allowing for transformation structures in the educational arena. The goal of this project is no less than such a transformation of mesostructures and macrostructures that tend to constrain the educational and life possibilities of African American students.

Ripple Effect of Micro-Level Transformative Education On Macro-Level Structures

What becomes essential in the deployment of interactive technologies is not the technology itself, but the meaning making, liberating ideologies, and problem solving that are all directly relevant to the participants acting for their own benefit and that of the wider collective from which they come. I have been involved with student learning in the math domain; however, what I have said about interactive technologies I think applies generally to the various science domains as well. New forms of computer models coupled with increasing ease and power of modifying and sharing these models without regard for distance or time, makes possible a broader, more powerful repertoire of pedagogical strategies that can be pressed into service to accomplish common goals of the collective.

Culturally empowering learning spaces that utilize advanced interactive technologies, coupled with liberating ideologies embedded in the curriculum, have the potential of producing educational experiences for minority students in public schools that are transformative of existing constraining structures in public schools, affording agency for both individuals and collectives. These spaces are not isolated enclaves that locate the problems facing African American students, for example, in their bodies or in their ethnic practices. Rather, there is a recognition that agency of students is interlinked with how students and stakeholders access and manage available resources to construct meaning and knowledge that can be applied to their collective problems and motives. These learning spaces can serve as models for a public education generally, for not only minority students, but for all students. So these learning spaces in particular localities can have a transformative effect on meso and macro level structures of education and society as a whole.

The exponential advances in technology are changing the way we have to address the ongoing issue of racism and its various manifestations of oppression. We have to recognize that these changes are speeding up the movement towards a condition where African Americans collectively will not develop the necessary survival tools to avoid becoming a permanent underclass in society, or a non-existent factor in total. As one who has become intimately involved in the forces of change due to technology in both the corporate world and the educational world, I could not and would not in due conscience not involve myself in researching how to advance the unique educational needs of African American students who push to the extreme abyss by the technological engine of change. Perhaps if educators of good will can bridge the technological learning gaps faced by minorities, we will also be able to bridge economic and social gaps as well thus promoting a greater cosmopolitanism in American society. If we can close the achievement gap faced by lower achieving groups, we would have learned valuable praxis for closing the achievement gap between America and other countries that have surpassed America in student achievement.

References

Bakhtin, M. M. (1981). The dialogic imagination: Four essays by M. M. Bakhtin (C. Emerson, & M. Holquist, Trans.). Austin, Texas: University of Texas Press.

Bakhtin, M. M. (1986). Speech genres and other late essays (V. W. McGee, Trans. Vol. 9). Austin, Texas: University of Texas Press.

Bourdieu, P. (2000). Cultural reproduction and social reproduction. In R. Arum & R. Beattie (Eds.), The structure of schooling: Readings in the sociology of education (pp. 56-68). Mountain View, CA: Mayfield Publishing Company.

Collins, R. (2004). Interaction ritual chains. Princeton, NJ: Princeton University Press.

Gee, J. (2007). What video games have to teach us about learning. New York, NY: Palgrave Macmillan.

Bransford, J.D., Brown, A.L., & Cocking, R.R., Editors (2007). How People Learn. Commission on Behavioral and Social Sciences and Education, Chapter 9. Washington, D.C.: National Academy Press.

Roth, W-M., & Lee, Y-T. (2007). “Vygotsky’s neglected legacy”: Cultural-historical activity theory. Review of Educational Research, 77(2), 186-232.

Roth, W-M., & Tobin, K. (Eds.) (2007). Science, learning, and identity: Sociocultural and cultural-historical perspectives. Rotterdam, NL: Sense Publishing.

The John D. and Catherine T. MacArthur Foundation Reports on Digital Media and Learning | November 2008. Living and Learning with New Media: Summary of Findings from the Digital Youth Project [White paper].

Turner, K. (2007). Human emotions: A sociological theory. (p. 73). New York, NY: Routeledge.

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