To Piaget, learning is predominately situated in on the individual-cognitive end of the spectrum, where knowledge is constructed by the recombination of what is known and what is newly presented. Papert, too presents learning on the level of the individual, but with respect to his/her place in a continuous world of interconnected people and objects. Where Piaget presents knowledge construction as the development of mental mastery over objects outside the individual, Papert speaks of a world where internal mind and external object become one through personally meaningful, constructive acts. This existential melding suggests not only the enhanced learning that occurs when outer and internal realities reinforce one another, but when learning is situated in a social context.

Encompassing of this entire spectrum is Paulo Freire. Freire is present in Papert when he suggests that constructionist technologies in the hands of children can provide the means of emancipation from the structural elements that frame society and interact, critically, with learning. In Freire we find the theoretical framework for viewing social structures and paradigms as systems that also engage in learning, albeit with people and ideas as the constructionist building blocks. Like agents in a NetLogo model, these substructural elements autonomously act on and interact with the entrapping paradigms of the encompassing system. Unlike NetLogo, people and ideas serve as the agents of thought and operation by which the system “learns,” suggesting the unending potential of this conflict to reconstruct dominant paradigms. Freire suggests that this happens when the oppressed take ownership of their own learning while educators simultaneously cede their patronizing roles to allow for expressive inquiry, an act that requires perfect trust. Analogous to Papert, the learning moment occurs when people and practice become existentially one.

Where Freire presents theory, Cavallo and Blikstein demonstrate realizations of this approach in various workshops and pilot studies in Brazil. By training Constructionist educators, then releasing them upon their schools to “infect” other faculty and students through expression and consumption of ideas, a critical mass is ultimately reached when structural paradigms start shifting, and learning happens on all levels.

The design of the Mathematical Black Box is theoretically grounded in a pedagogical approach that seeks to combat unfortunate, recent trends in education:

Undistributed Cognition
Mathematical education over the past half-century has moved toward the use of manipulatives, calculators, computers, and even extensive use of pencil and paper to distribute the cognitive act of mathematical reasoning across minds, bodies, and artifacts. As a result, the ability of students to manipulate numbers and produce computational results has been weakened. The Black Box, on the other hand, forces exclusive use of auditory senses and abstract reasoning to produce results.

Desocialized Learning
During this same time period, education in general has increasingly encouraged socialization in learning via collaboration and communication. This, too, has contributed to the distribution of cognition beyond the capacity of the individual mind. The Black Box counters this approach by removing the student from the influence of peers and teachers, effectively forcing her/him to solve problems on their own. This will prepare them to be strong, discerning individuals in their lives beyond school.

Decontextualized Learning
Finally, education in general has experienced significant pressure from academics to contextualize learning (i.e. pose problems within the context of meaningful tasks with respect to student experiences and preferences). This, too, weakens the computational power of the student with respect to potential performance in novel settings. The Black Box, on the other hand, fosters reasoning that is multipurpose. Thus, when faced with any mathematical reasoning task in their future lives, students need only transfer the skills they have developed in the Box to solve their new problem.

We conceived of students learning about scheduling by way of observing and attempting to model typical scenarios at their local intersection. Furthermore, we wished to provide a mechanism by which students could implement scheduling algorithms to govern the intersection as they deemed appropriate. NetLogo appeared to be a perfect technology for integrating the configuration of testable traffic scenarios, implementing scheduling algorithms, and providing a visual representation that would encourage exploration and experimentation. By encouraging interaction with such a model, we hope to engender not only an understanding of scheduling, but, more importantly, the realization that values and preferences can be (and are) encoded in perfunctory yet powerful pieces of technology (i.e. the traffic light).

The intersection of focus is that of Central Expressway and Rengstorff Avenue in Mountain View. In addition to the typical four directions of traffic, a railway line crosses Rengstorff just to the West of the intersection. The train adds an additional layer of complexity to the scheduling behavior of the traffic light, one which we wanted the students to both model and account for in their algorithms. Below is a screenshot of the model interface next to a satellite-based rendering of the actual intersection:

The model consists of a simplified, normalized version of the Rengstorff x Central Expressway intersection. Although the number of lanes is reduced to one in each direction, additional characteristics remain, including the train and tracks, pedestrians and crosswalks, cars, trucks, emergency vehicles, traffic lights, railroad crossing guards, traffic sensors, and emergency vehicle sensors. The following aspects of the model are configurable:

traffic light timing [green/orange/red light duration] train frequency [0 - 10% of the time] traffic density [percentage of traffic in each direction] truck and emergency vehicle frequency [0-10%] and [0-5%] respectively. percentage of traffic turning left and right (not fully implemented) pedestrian density [percentage of pedestrians in each direction]

Charts of maximum, minimum, and average wait time for vehicles and pedestrians are also provided:

The vehicles and pedestrians in the model are created with random speeds (within a range). They attempt to cross the intersection and exit the opposite side of the screen, but are constrained by rail-road crossing guards (which prevent crossing the tracks when a train is on screen), the tracks themselves if another vehicle is stopped directly on the other side, red traffic lights, the intersection itself if another vehicle is stopped directly on the other side, and vehicles directly in front of them. Packaged with the model is a simplistic round-robin traffic light scheduling algorithm. This algorithm cycles through the three traffic directions (North-South, East-West, and Pedestrian), giving them each an equal time slice as configured by the user.

There are two complementary behaviors encouraged by the model. The first is to explore the ramifications of the configurable parameters with respect to the scheduling algorithm. The hope here is to encourage the discovery of emergent phenomena within this system of independent actors. The second is to write more intelligent scheduling algorithms to maximize vehicle and pedestrian throughput (and safety). Presumably, these algorithms would avail of the various sensors provided, such as vehicle sensors at the edge of intersections, emergency vehicle sensors further down the road, pedestrian crosswalk sensors, and train sensors.

In the spirit of exploring emergent behavior, I ran two rounds of BehaviorSpace automated testing to explore the effect of changing just the green traffic light duration in a standard simulation with a high train frequency. The results were surprising. Indeed, I do not fully understand the results (the patterns of which were consistent across test runs). Here are charts of the average and maximum wait times for these sessions:

The dips in maximum wait time for 200 and 400 tick green light durations, relative to test runs where the durations are 100, 300, and 500 suggest either emergent behavior or an idiosyncrasy of my model that is exposed by this level of testing. Although I can’t explain the pattern, the fact that BehaviorSpace can be used to rapidly expose these results will be instrumental in both my understanding and further refinement of the code.

On Monday, May 3, I engaged in a more qualitative form of testing by introducing the model to three sections of eighth grade computer science at the Girls’ Middle School (in the context of the unit described above). The reception ranged from neutral to quite positive. Some students wondered how long it took to make, while others commented on how “cute” the pedestrian schoolchildren and the train were. Students found (or were prompted by me to find) the 3-D rendering, and became further excited. The 3-D view gives a feel for individual interactions while the top-level view reveals system dynamics. Both seemed to be of interest to the students. I asked them for suggestions for improvements and received the following (mostly from Carly): control over pedestrian speed, smaller train guard window, control over pedestrian light time, sports cars, and the ability to make cars crash. I encouraged them to play with the model over the next week, and consider how they might design a smarter traffic light. In the meantime, I will update the model to accommodate some of their wishes, and prepare instructions to scaffold their code-based scheduling algorithm additions.

The project can be found here

[Coram Bryant]

Piaget’s legacy, with respect to designing for education, is (at least) twofold. The first stems from the claim, as described by Papert, that Piaget was the first to recognize that children do not think as if they were undeveloped adults, but rather as agents of a unique logic and understanding that is peculiar to their condition. The fact that the specific developmental stages he espoused may not be generalized beyond those which he specifically observed (as suggested by cross-cultural studies revealing exceptions) only reinforces the underlying message that an understanding of knowledge frameworks may only be achieved through extensive observation. The countless hours Piaget devoted to interaction with and analysis of children in formal and informal contexts remains the gold-standard for user-centered design. His example challenges us to observe and attempt to understand those about which we might theorize, rather than project our personal beliefs onto our subjects. This message applies to the design of educational technologies not just for children, but for learners in all conditions at all stages of the lifecycle.

Second, Piaget’s Constructivism continues to underlay any voiced alternatives to the still prevalent pedagogical paradigm that views children as empty vessels to be filled with quanta of information. His influence on the more inclusive theories of Vygotsky, Papert, and others allow us to dream of learning environments in which individual conceptual understanding is not uniform (and therefore not measurable by standardized tests), but, rather, the unique manifestation of personal struggles with and accommodations of novel experiences. This approach is likewise fundamental to the design of any educational technologies that seek to provide meaningful, differentiated learning opportunities that satisfy the needs of students to construct their own knowledge.

Our team interviewed JJ in his home on Tuesday evening of this week, and found him to be energetic, confident, and engaging. We entered JJ’s domain with a short list of pre-fabricated questions but a semi-structured mindset, which allowed us to insert relevant queries to subtly direct JJ’s self-directed tour of his play space. We immediately learned that JJ’s room is a treasure trove of playthings, replete with constructionist cars, baseballs, a swing, and a rock-climbing wall emerging from a room-wide, naturalistic mural! Even the bunk bed served as an object on which to climb and frolic. Our tour continued in the living room, where tubs of Legos were put to constant use, and a cabinet of board games lay open. It was at this time that JJ revealed his enjoyment of multi-tiered Tic-Tac-Toe and subtraction-based math games. When asked why he liked the latter, he indicated that it was because subtraction was “hard.” Finally, we asked JJ to draw and describe three ideal toys, to which he readily agreed. The evening finished with our being invited to eat sandwiches and cupcakes at a tea party hosted by his siblings and him (and his mother), which served as a pleasant ending to the process.

The set of questions, variants of which we employed iteratively throughout the process, was as follows:

• What’s your favorite subject in school?
• What’s your favorite animal?
• What’s your favorite game?
• Can you show us your favorite toys? What do you like about them? How do you play with them?
• Do you have any special toys you play with at school?
• Do you like to play games by yourself or with friends/your brother and sister?
• Can you give us a tour of your room?
• How could we make a toy special to you?

Our key insights into JJ’s needs and interests are that he values the following:

• Gifted toys (Hanukkah, birthday) are special to him
• “Hidden” toys
• constructed toys
• lasers/axes
• hybrid toys (R2D2 + Scooby Doo)
• Math games, specifically subtraction-based
• remote control toys
• motion-based toys
• Challenges – things that are “hard”
• red and yellow as favorite colors
• a parakeet as his next pet
• playing, outside so he can run around
• his family

Below are the JJ’s sketches of ideal toys. The first is a Scooby Doo + R2D2 hybrid with lasers, axes, and a chainsaw tail. The second is a challenging mathematics games with numerical buckes. The last is a toy embedded in concentric shields protected by lasers.

R2D2-Scooby Doo Hybrid

Challenging Math Game

Apocryphal Laser-protected Toy

And here are JJ’s descriptions of his creations:

JJ Describing Design #1

JJ Describing Designs #2 and #3

Armed with this information, we engaged in a prototyping session in which we evolved from a rubix-cube-like object that when properly assembled would light a laser-like LED, to a series of concentric R2D2 sculptures that require mathematical reasoning to unlock, to our final prototype: a set of transparent, magnetic cubes that when arranged properly in a super-cube, light an internal LED. The outer faces (i.e. those that are showing when the cube is arranged properly) depict images of import to JJ, namely members of his family and favorite toys (e.g. R2D2 and Scooby Doo). The internal faces of the composite cubes depict numbers and mathematical operators, are abstractly visible in the super-cube. When the super-cube is disassembled, the mathematical faces may be used to create linear equations for JJ to solve. Ideally, the faces of some of the cubes would be blank and white-board-maker-erasable, so he could input his own answers and create his own mathematical equations. Finally, an important aspect of JJ’s favorite toys is that they are gifted. Consequently, we plan to create an accompanying scrapbook of the creation process to augment the sentimental value of the final toy.

Here are some images of stages in this process, as well as our final prototype:

Brainstorming...

Process!

Prototype View 2

Prototype View 1

It is into this landscape that Abrahamson and Howison have introduced the Mathematics Image Trainer (MIT), an ambidextrous interactive set of sensors and display that scaffolds a proportional reasoning task consisting of covariant movement of two tennis balls in vertical relation to one another. The impetus for creating such a device is grounded in the dualistic conjecture that perfunctory activities do not provide occasions for the physical experience of proportionality, and that cognition consists not of syntactical propositions, but of simulated dynamic imagery grounded in life experiences. Thus, like an abacus or a pendulum, the MIT is intended to provide an imagistic vocabulary for abstract mathematical reasoning (Abrahamson & Howison, 2010).

In terms of both theoretical underpinnings and empirical evidence, the MIT offers a promising mechanism for encouraging the embodiment of proportionality. Even within the span of a short intervention, students expressed that they subsequently knew that “it’s ok to be different,” indicating an expansion of a prior fixed-difference mathematical mentality (Abrahamson & Howison, 2010). This is in accord with Dewey’s claim that “an ounce of experience is better than a ton of theory simply because it is only in experience that any theory has vital and verifiable significance” (Dewey, 1919, p. 6). But, in what world is this experience taking place?

The question remains whether pre-fabricated laboratory learning better serves students than authentic, situated tasks. As Dewey points out, there is another aspect of the mechanical pedagogy: when “exercises are given for their own sake” (Dewey, 1919, p. 5). What the MIT lacks, in a manner consistent with much of traditional education, is the value of proportional reasoning. It may be that these values may only be learned in real-world activities, of which many abound in the realm of proportion, despite the authors’ claims. Scaling quantities for cooking and scientific experiments, or constructing both physical and virtual models within the context of an enterprise application might likewise engender proportional reasoning, and, further, ascribe value to such knowledge. The researcher-initiated followup discussions of hot-air balloon ascents and comparative modes of transportation speaks to the need for situated learning to both solidify and legitimize understanding.

One final aspect of situated instruction that is essential to learning is the socially-based construction of knowledge. The suggested future support for students to develop and negotiate an understanding of proportional reasoning, in tandem, looks to be a promising and exciting step in this direction. Again, looking for ways to authentically situate the experience would likely elevate and legitimize the learning even further.

Abrahamson, D., & Howison, M. (2010). Embodied artifacts: coordinated action as an object-to-think-with. In Embodied and Enactive Approaches to Instruction: Implications and Innovations. AERA 2010, Denver, May 3, 2010.

Dewey, J. (1916). Democracy and education, Chapter 11: Experience and thinking. New York: Macmillan (pp. 152-179 in the original).

On the other hand, I find myself increasingly aware of the limitations of computational technologies to address the indubitable challenge facing formal education. Papert implicitly describes this challenge as overcoming the paradigm in which students are forced to “discover” conclusions from a known corpus that has been fabricated by a paternalistic system. Kay’s clown fish simulation, although engaging and informative, still falls victim to this paradox. Although students are provided freedom of exploration within the realm of the simulation, they are unavoidably constrained by the constructs of the prefabricated environment, not to mention by the subject matter and content. That they are permitted to explore these materials in novel and relatively liberating ways does not constitute escape from the pedagogical dilemma. In comparing the simulated experience with a process of authentic scientific inquiry, we see that the simulation, as programmed by an engineer in a far-off lab, fills in numerous gaps in the structure, shape, and interactions of the actors. The scientist, alternatively, would necessarily engage in lengthy observations, and in so doing, generate new questions for subsequent exploration. Unfortunately, the computer simulation is far more akin to Kay’s description of digital representations of great works of visual art: sufficiently appealing to engender a false sense of experience, thereby effectively truncating further inquiry. Even constructionist technologies fall victim to this contradiction, given that they predetermine the shapes and potentials of their virtual building blocks.

My reservations regarding technology speak to the subtly insidious, pervading desire to provide students with technologies that make learning “easier.” This mindset has its roots in fluidic theory of education, described by Kay, which suggests that students, as empty vessels, must be filled with bitter quanta of knowledge, a process that may only be enhanced by transporting the information in sugar-coated forms. On the contrary, technology should strive to provide students with opportunities for authentic learning, which invariably involves difficult explorations of thought and practice. As such, it seems that the greatest potential for technology is to empower students to take ownership of their learning, while simultaneously guarding against the danger of viewing technological worlds as ends in themselves. Ultimately, technology should serve as a bridge to authentic discovery in an asymptotic continuum of progressively less constrained realms.

Our project is a sequential compilation of the personal histories of our group members. The individual vignettes also provide hints regarding why or how we arrived at Stanford. As you can see, we had a little fun at the end :).

The first three sections require user interaction to move through the scenes. For the first segment, use the arrow keys to navigate between “music”, “tennis”, “ocean”, and “books”, in order. Movement through the second and third segments is also controlled by the arrow keys, (the third segment also requires the use of the space bar to move between some images). Starting with the final segment, the remainder is fully animated.

Here is a link to the project: Project Source

Enjoy!
Tiffany Tseng, Tony Schloss, Brian Donovan, Coram Bryant

Some of my earliest memories involve my brother and I constructing playworlds with cardboard boxes. Our plans were initially modest, no doubt consisting of a single unit, open on both ends, through which we could crawl in an endless circle, like the Ouroboros consuming its tail. Days turned into years, and our endeavors grew more sophisticated, culminating with what seemed to be an endless maze of connected boxes in our backyard, with forks and forts, through which we crawled for extended periods without encountering one another. Tears were shed the day we were forced to deconstruct our creation. Around this time, as we explored the back alleys of our quiet neighborhood, we discovered, with amazement, that cardboard boxes constituted waste in the adult world. A new period of adventure ensued, in which we scoured the trash bins of neighboring houses and stores, marveling at the pristine treasures that were carelessly disposed by those of lesser vision. We even staked our claim on a refrigerator box that had been placed outside a local business at midday. Like patient hunters, we waited in the weeds for the shop patrons to depart for their evening commute before swooping in to land our biggest treasure to date.

Yet somehow, in the years that followed, our interest in the cardboard box waned, turning toward more complicated objects and constructs in the adolescent and adult worlds, until the cardboard box became what we could never comprehend in youth: an annoyance to be compressed for ignominious recycling. This trend might have continued had it not been for that fateful night in November, when my understanding of the beauty of the cardboard box returned to consciousness. I recognized the spirit of the cardboard box in many of the values I cherished, be it elegance and modularity in engineering, or remembering to acknowledge the endless potential in all I meet. Although no longer a fixture in my everyday consciousness, I was heartened to realize that my appreciation for cardboard, albeit in different forms, lived on in me.

In June of last year, in anticipation of my November epiphany, no doubt :), I was privileged to witness the birth of my first son, and revel is his subsequent development. It should have come as no surprise when, in the early stages of his self-willed physical agency, I observed him repeatedly forgoing opportunities to engage with bright, colorful, specialized childhood toys in order to overturn a small cardboard box for use as a drum, seat, or object to push across the floor.  I can’t wait to see the creative solutions he engineers with this worthy Hall of Famer.