Every great advance in science has issued from a new audacity of imagination.

~John Dewey, The Quest for Certainty, 1929

Catherine Twomey Fosnot

Most people assume that learning results from teachers transmitting knowledge: clearly explaining concepts, procedures to be practiced, and facts to be memorized; then testing to assess retention and application, with subsequent feedback. Yet this could not be further from the truth. Mathematicians and scientists alike describe the processes by which they learn quite differently from the ones characteristically employed in our schools. The renowned Dutch mathematician, Hans Freudenthal (1991), frustrated with the state of affairs in education once said:

Cognition does not start with concepts, but rather the other way around: concepts are the results of cognitive processes… How often haven’t I been disappointed by [those] who narrowed mathematizing to its vertical component [teaching one skill or abstract idea upon another]. Mathematics should be thought of as a human activity of “mathematizing”—not as a discipline of structures to be transmitted, discovered, or even constructed—but as schematizing, structuring, and modeling the world mathematically.

His point was that the starting place for learning mathematics is not explanation of the abstractions constructed by previous mathematicians, but context, realistic (or at least imaginable) situations from a learner’s world. Working with mathematicians, scientists, and educators from the University of Utrecht over a 20-year period, he established a learning program now commonly known around the world as Realistic Mathematics Education (RME). Rather than using explanation of concepts and procedures as the starting point of instruction about a topic, the RME group begins by inviting learners to find ways to informally mathematize open-ended problems with many starting points. Subsequent discussions on the various productions and strategies used, as well as working through sequences of further carefully crafted problems designed with constraints and potentially realizable solutions, support learners over time to develop, progressively, more formal solutions and generalizable ideas. From this perspective mathematics is generated from problem solving, rather than previously taught procedures being applied to problem solving.

Many scientists from around the world have argued for a similar problem-based instruction for science education. In 2004, the U.S National Science Teachers Association (NSTA) produced a standards document emphasizing scientific inquiry,

Scientific inquiry describes the diverse ways in which scientists study the natural world and propose explanations based on the evidence derived from their work. Scientific inquiry also refers to the activities through which students develop knowledge and understanding of scientific ideas…. it is at the heart of how students learn. From a very early age, children interact with their environment, ask questions, and seek ways to answer those questions. Understanding science content is significantly enhanced when ideas are anchored to inquiry experiences. Scientific inquiry is a powerful way of understanding science content. Students learn how to ask questions and use evidence to answer them. In the process of learning the strategies of scientific inquiry, students learn to conduct an investigation and collect evidence from a variety of sources, develop an explanation from the data, and communicate and defend their conclusions. The National Science Teachers Association (NSTA) recommends that all K–16 teachers embrace scientific inquiry and is committed to helping educators make it the centerpiece of the science classroom. The use of scientific inquiry will help ensure that students develop a deep understanding of science and scientific inquiry (National Science Education Standards (NSES p. 23).

Regarding students’ abilities to do scientific inquiry, NSTA recommends that teachers help students to:

• Learn how to identify and ask appropriate questions that can be answered through scientific investigations.
• Design and conduct investigations to collect the evidence needed to answer a variety of questions.
• Use appropriate equipment and tools to interpret and analyze data.
• Learn how to draw conclusions and think critically and logically to create explanations based on their evidence.
• Communicate and defend their results to their peers and others.

Regarding students’ understanding about scientific inquiry, NSTA recommends that teachers help students understand:

• That science involves asking questions about the world and then developing scientific investigations to answer their questions.
• That there is no fixed sequence of steps that all scientific investigations follow. Different kinds of questions suggest different kinds of scientific investigations.
• That scientific inquiry is central to the learning of science and reflects how science is done.
• The importance of gathering empirical data using appropriate tools and instruments.
• That the evidence they collect can change their perceptions about the world and increase their scientific knowledge.
• The importance of being sceptical when they assess their own work and the work of others.
• That the scientific community, in the end, seeks explanations that are empirically based and logically consistent.

Yet even with such examples and mandates, the teaching of math and science in our schools often still remains entrenched in misconceived behaviors related to explanation, practice and feedback, and outdated notions of thinking and learning. The Space Design Competition (SDC) offers a powerful learning experience as a contrast.

The Active Nature of Thinking and Learning

The Cognitive and Biological Foundation

Cognitive psychologists have known for years that real learning is much more complex than just assimilating information, as it requires the development of a neural network of synapses and pathways; it requires cognitive reorganization. It has an emotional component, and is more about the active nature of the posing and solving of problems within social communities of discourse than it is about the passive taking in of information. Rather than behaviors or skills as the goal of instruction, cognitive development and deep understanding are the foci. Genuine learning is now understood as constructions of active learner reorganization. Rather than viewing learning as a linear process of the accumulation and mastery of one skill after another, it is understood to be a complex network of big ideas, strategies, and models and fundamentally non-linear in nature.

This new view of learning, commonly termed constructivism, stems from the burgeoning field of cognitive science, particularly the later work of Jean Piaget just prior to his death in 1980; the socio-historical work of Lev Vygotsky and his followers; and the work of Jerome Bruner, Howard Gardner and Nelson Goodman, among others, who have studied the role of representation in learning. It also has roots in biology and evolution; the logical outcome of the work of contemporary scientists such as Ilya Prigogine, Humberto Maturana, Francisco Varela, Ernst Mayr, Murray Gell-Man, Wolfgang Krumbein, Betsy Dyer, Lynn Margulis, Stuart Kauffman, and Per Bak (among others), as attempts were made to unify physics with biology.[1]

Knowing and cognitive processes are rooted in our biological structure. The mechanisms by which life evolved, from chemical beginnings to cognizing human beings, are central to understanding the psychological basis of learning. We are the product of an evolutionary process and it is the mechanisms inherent in this process that offer the most probable explanations to how we think and learn.

For cells to live, evolve and flourish, there must be an exchange of matter and energy with the environment around them. The flow of matter and energy through complex systems has been described by the Nobel Laureate, Ilya Prigogine, as a dissipative structure; an open system that maintains itself in a state far from equilibrium. Dissipative structures produce new forms of order, order that arises spontaneously in a complex system when it is far from equilibrium, at ‘the edge of the chaos’. When the flow of energy increases, the increased activity produces instability and a ‘bifurcation’ results. At this bifurcation point, reorganizing occurs; self-organizing that results in the emergence of a new structure for coherence and efficiency. Without dissipative structures, without exchange with the environment, entropy would result. Capra (2002, p. 14) explains the importance of this model:

This spontaneous emergence of order at critical points of instability is one of the most important concepts of the new understanding of life. It is technically known as self-organization and is often referred to simply as “emergence.” It has been recognized as the dynamic origin of development, learning, and evolution. In other words, creativity—the generation of new forms—is a key property of all living systems. And since emergence is an integral part of dynamics of open systems, we reach the important conclusion that open systems develop and evolve. Life constantly reaches out into novelty.

The active nature of thinking and learning, the sequences of contradictions and bifurcations, can be seen in the history of ideas about aspects of the physical world, for example, light. Before Newton’s time, the notion that light was in the form of rays made it possible for people to explain shadows and “images” from pinholes, but it did not provide a mechanism to explain refraction as light passes through a transparent medium. This was not a big problem until lenses began to be used in Galileo’s day. Within one generation the issue was puzzling enough to cause a search for a notion of the nature of light sufficient to handle this inadequacy. Newton suggested that if we thought of light as actual tiny, material particles, spherical in nature, one could explain refraction in terms of a mechanism. He suggested that if you roll actual particles, such as marbles, across a horizontal surface toward a wide ramp sloping down to another horizontal surface you would find that the marbles would approach the ramp traveling in straight lines. At the ramp they will change direction slightly and then change again at the bottom of the ramp resulting in a change in direction of travel much the same way as light is observed to change direction at the interface between two transparent media. For a while all was well, and light was conceived of as particles by scientists until we could make light sources bright enough and well collimated enough to see detail in the edges of the shadows of objects. Thomas Young resolved the problem this new phenomenon created by convincingly putting forth a wave model of light, in contrast to a particle model. Much later the attempts to interpret the photoelectric effect with a wave model created new problems until Planck, and later Einstein, provided data in support of the idea that light was composed of chunks or packets of energy traveling in a similar fashion to billiard balls. When they hit an object they knocked a particle out of the mass of that object, just as a billiard ball hitting another would send it traveling at the same speed as the original ball. This model explained refraction and the photoelectric effect, whereas the wave interpretation had been insufficient, but now a new paradox remained. How could light be packets of energy and yet be waves at the same time?

It is important to note here that the paradoxes in the scientists’ interpretations existed between the abstractions, for example, light as waves versus light as particles. The experimental results were contradictory only insofar as they fit or contradicted the given abstraction, the currently deemed viable explanation. The notions of light as rays, or waves, or packets of energy are all constructed abstractions as humans evolve, solve problems and seek coherency. The data by themselves are not contradictory; they are contradictory only in relation to the meaning that the learner (the scientist, in this case) attributes to them.

As can be seen in the prior example, scientists seek meaning and coherency of the world we live in by generating mathematical equations and scientific models to explain it. They seek beauty and understand the process to be one of imagination and invention. Henri Poincaré (1890, p. 143) stated it well,

A collection of facts is no more a science than a heap of stones is a house. A scientist worthy of his name…experiences in his work the same impression as an artist; his pleasure is as great and of the same nature…. It is by logic that we prove, but by intuition that we discover. To know how to criticize is good, to know how to create is better.

As they create, scientists seek data to confirm or contradict ideas. Contradictions may be in the form of two theories that both seem plausible, and yet are contradictory, or theories that become insufficient given new evidence. This active process of scientific construction is not just seen in the activity of professional adult scientists and mathematicians. Learners of all ages do not just passively take in information and accept ideas as truths; they question and generate theories, too.

Possibilities generated by subjects (4 to 8 years of age) as they sought to understand how to balance a series of blocks on a fulcrum are evidence of this process (Fosnot, Forman, Edwards & Goldhaber, 1988). At first young children tended to plunk blocks randomly on the fulcrum and to push harder or hold them in place if they didn’t balance, almost as if they believed that balancing blocks on a fulcrum had to do with stickiness, or the force of their actions. But soon, as they explored this problem further, they began to investigate moving the blocks back and forth across the fulcrum. These procedures resulted at times in balance (with symmetrical blocks), but did not work for asymmetrical blocks; yet, children persisted with these actions for some time, even correcting in the wrong direction, seemingly constructing a theory that involved finding the midpoint of their back and forth actions! Eventually they began to explore other actions, sometimes correcting in the right direction to restore balance. This successful action generated contradictory data, a negation to their earlier theories. Eventually this contradiction was resolved with the construction of a new theory: find the midpoint of the block, not the midpoint of the back and forth actions! This theory explained success with several symmetrically weighted blocks, but it eventually met with contradictory data, too, as learners went on to explore asymmetrically weighted blocks (blocks with more mass on one side, e.g. a ramp shaped block with lead in the tip.) At first these were even deemed emphatically by children as “impossible” blocks to balance, given their theories!

Throughout the sessions exploring the blocks, even these very young learners continued to generate possibilities and develop progressively new models to explain balance. Each new perspective resulted in a temporary structural shift in thinking resulting from problem posing and solving, reflection, sense-making, and interpretation, evidence of the active nature of thinking and learning.

As Exemplified in the Space Design Competition

By engaging students in the formation of companies focused on offering solutions and designs for the development of living environments in space, the Space Design Competition (SDC) involves students in an experience designed to engender powerful learning. It takes seriously what it means to do science and treats scientific thinking and learning as the active creative processes they are. Rather than just applying previously learned concepts and procedures, participants are asked to seek novel solutions to complex problems. They are asked to generate unique designs and construct and defend concepts as they work.

In the words of one participant,

The designs we are challenged with often feed on what students have already touched upon within their schoolwork, however they stretch their understanding much more than any learning environment at school. I had learnt about the concept of solar panels in GCSE Physics; however the SDC pushed me to research which types of solar panels are most effective, what their power ratings are, how their energy could be stored, and whether there were better alternatives such as nuclear fission. Similarly, I learnt about power, aeroponics, aquaponics, and motors through the SDC long before they were introduced to me at school. For example, one settlement had to travel along the edge of Mercury in order to follow the sunset for optimum temperatures; I had to calculate how much power would be required to drive the settlement forwards. I became certain that these concepts were useful because of the SDC.

The Role of Language, Representation and Argumentation

When mathematicians and scientists solve problems they must provide evidence in convincing ways for their ideas to be accepted in their communities of discourse. Mathematicians write proofs and scientists collect confirming empirical data as evidence. These are then put forth as viable arguments of their ideas to their peers. Only when the arguments are sound do the ideas gain acceptance, and even then the acceptance is always temporary as a new idea can always take its place given new evidence.

Representing, building models and convincing others are processes that engender even further learning. The mathematician, Hans Freudenthal (1960, p. ix), explains this point well:

No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty.

The very act of representing ideas and solutions within a medium such as language, paint and canvas, mathematical symbols and equations, scientific model, or architectural design, in an attempt to convince others, can create a dialectical tension beneficial to thought. That is because each medium has its own attributes and limits and thus produces new constructions, new variations on the contextually embedded meaning (McLuhan, 1964; Olson, 1975; Eisner, 1993). In fact, it was this very point, the beneficial effect of the act of representing on thought, that led the sculptor, Henry Moore , to write, “I always draw something [first] to learn more about it,” the writer, Donald Murray, to comment, “I write to surprise myself,” the mathematician, Gauss, to say, “I have had my results for a long time: but I do not yet know how I am to arrive at them,” and the scientist, Kelvin, to say, “When you can measure what you are talking about and express it in numbers, you know [learn] something about it.”

The philosopher, Nelson Goodman (1978, 1984), pushes this idea even further arguing that there is no unique “real world” that pre-exists independently of human mental activity within a medium. Instead, what we call the world is a product of minds whose symbolic procedures construct the world by interpreting, organizing, and transforming prior worldviews, thereby constructing new symbols. For Goodman, the difference between the arts and sciences, for example, is not subjectivity vs. objectivity, but the difference in constructional activities and the symbolic systems that result.

Howard Gardner (working with Nelson Goodman at Harvard Project Zero) researched the development of early symbolization to characterize the different modes of operation by which intelligence expresses itself. In Frames of Mind (1985) he presents evidence for multiple, different “intelligences” that are the result of “minds which become specialized to deal in verbal or mathematical or spatial forms of world making, supported by symbolic means provided by cultures which themselves specialize in their preference for different kinds of worlds (Bruner, 1986, p. 103).” Thus the world a musician builds using a symbolic system that employs rhythm, cadence, and tone is indeed a different world than the one constructed by a visual artist employing space, line, repleteness, and color. Language, too, becomes its own context, for it involves the uses of signs to organize and plan sign-using activity itself. This building process is developmental because constructions within a medium serve as building blocks to new constructions.

Domains of discourse that we generate become part of our existence and constitute the environment with which we couple. Just as single cell organisms couple with their surround symbiotically via activity and self-organization, humans do so in communities of discourse for identity and adaptation, a natural process of evolution. Maturana and Varela (1998, p. 234) describe this process well:

We humans, as humans, exist in the network of structural couplings that we continually weave through the permanent linguistic trophallaxis of our behavior. Language was never invented by anyone only to take in an outside world. Rather it is by languaging that the act of knowing, in the behavioral coordination that is language, brings forth a world.

To that point, even science ideas such as time and space can be seen as the use of languaging to bring forth a world. As pointed out by Poincare (1905, p.13):

It is not nature that imposes time and space upon us; it is we who impose them upon nature because we find them convenient.

As Exemplified in the Space Design Competition

By requiring students to present their designs to others (the jury as well as other student ‘companies’) with supporting evidence that their design will work, the SDC is providing rich opportunities for students to experience the role of representation and argumentation in science. During the process of developing a viable defence of their designs, contradictions and/or insufficiencies in thinking arise causing students to re-examine aspects of their product and the criteria used in its formation. One participant, Rachel von Maydell, describes how collaborative discourse helped her team achieve further clarity as they prepared to present to the jury,

The process of preparing for the presentation changed our designs largely because we realized where there were inconsistencies. We found that we often changed our outlook based on what our goal was; for example, one person stated that we tried to use as little material and space as possible in order to save cost, but another said that no expenses were spared on the living quarters so that inhabitants would be healthier psychologically. The insight into what the priorities of others were, whether it was cost, human factors or efficiency, widened our outlook on the settlement as a whole. We were no longer working in a niche department such as water recycling, but were encouraged [and even required] to consider how everything worked together and intertwined.

Culture, Community, Identity, and Empowerment

We do not act alone; humans are social beings. Throughout our evolution, from the hunter/gatherer days to the technological present, we have sought to establish communities, societies, forms of communication, and thus cultures as an adaptive mechanism. We attempt to survive collectively, rather than individually; we procreate, communicate and teach our young. Given this fact, it is not surprising that communities of discourse around common interests and ways of “world-making” have evolved into the disciplines we know today, such as science, art, mathematics, and so on.

Is social interaction important to learning, and if so how and why? If learning is a case of self-organization and internal restructuring, then what role do discourse and the social community play in its development? Some evolutionary biologists and neuropsychologists have argued that the encephalization of the brain (and the resulting ability for mental imagery and highly developed language forms) was an adaptation that was viable in that it enabled Homo sapiens to make major social changes (Oatley, 1985). According to Maturana and Varela (1998, p. 234),

Consciousness and mind belong to the realm of ‘social coupling’ and the domains of discourse that we generate become part of our domain of existence and constitute part of the environment in which we conserve identity and adaptation.

The biological and the social are neither separable, nor antithetical, nor alternatives, but complementary. So much of how we define ourselves within our peer group determines who we believe ourselves to be and what we decide to pursue as a future career. Our interests and even our identity are directly linked to the experiences we have had in prior schooling. Sadly, few students during the primary and secondary years have opportunities to form identities as scientists. Even when they have done well in these disciplines, they have rarely had experiences working with real scientists, exploring space frontiers, and engaging in the doing of real science. How could they possibly form identities as young, competent, scientists?

The case is no different in mathematics. To this point, Lambert (2005, p. 2) argues:

As children grow up learning mathematics first at home and later at school, they begin to develop a relationship with the discipline. And that relationship all too often grows into a negative one—exacerbated because claiming incompetence in mathematics remains socially acceptable in American culture. (However, claiming incompetence in reading or writing is socially unacceptable.) Historically, teachers and parents, and then students, have come to understand doing mathematics as memorizing facts and successfully performing predetermined procedures. Such a narrow conception of mathematics has led—and continues to lead—many students to become uninterested or frustrated by rote instruction disconnected from any meaningful context. Not surprisingly, then, some people develop and sustain a negative relationship to the discipline. Even students who are successful in a traditional mathematics curriculum may finally reject mathematics because they believe it has no room for creativity or because they simply do not enjoy it.

Students who engage in a strong mathematical and science community develop important habits and dispositions. In other words, what students learn is less about the content of each lesson than the repeated behaviors they do day after day, and the role they see themselves playing in the development of the content of this community. When their community appreciates the ideas and solutions they construct and defend as viable contributions to a body of knowledge, they develop an identity and a sense of empowerment within the field. They feel ownership of the ideas and a pride in their contribution. Ultimately, they gain confidence and enjoyment in the process. A learner with a positive mathematics and science disposition understands that the ‘doing’ of mathematics and science is about the posing of problems, the asking of poignant questions, the creative invention of solutions, and the developing of viable arguments. They come to view the world through a mathematical and/or scientific lens and persistently enquire and build meaning about the world around them with the tools of these fields. They come to understand the creative, imaginative aspects of the disciplines and to find enjoyment in the fact that mathematics and science learning is not a series of executed steps along a well-cleared path, but a journey into a strange wilderness where the explorers derive enjoyment from getting lost and finding their way out.

As Exemplified in the Space Design Competition

One of the most significant impacts of the SDC is the effect it has on students’ career paths. Experiencing the SDC is stressful and challenging, yet many of the participants become so enthralled, they not only return year after year, they veer from what they previously thought they would pursue and opt instead for space science as a new career choice. Some even stay on as volunteers as they pursue their studies.

Students bond and often form social networks during the experience, which endure long after the experience ends. They stay in touch with a peer group interested in science, and even when their final career choice is not specifically space science, when interviewed they describe the applicability of their experience in the SDC to their ultimate choice. Rachel von Maydell describes the impact it had on her,

After the SDC, our company had bonded. We made a Facebook group and organised reunions. My participation in the SDC was so meaningful and stimulating that I became convinced that engineering, particularly in space, was my calling. The next year, after I had decided to study economics, I returned to the SDC. I found that the SDC is as valuable to a non-science student as it is to an engineer, architect or computer scientist. The skills you practise in marketing and management roles [during the SDC] are equally significant and applicable.

Conclusion

Historically, females and underserved children growing up in poverty only rarely pursued careers in the space sciences. Without working with scientists in those fields on projects where their own creations mattered, they had few opportunities to develop role models in these disciplines or to understand the possible opportunities in the frontier of space exploration and engineering. Ultimately, the identities they formed throughout their schooling often took them in other directions.

The SDC offers new doors. Involving a diverse group of students from schools around the country who ultimately may even compete internationally; the project introduces new worlds to many. It also provides a startlingly rich contrast to the traditional teaching of math and science, which too often are still characterized by transmission, practice, test, and feedback. John Dewey summed up the importance of scientific inquiry with the quote:

When our schools truly become laboratories of knowledge-making, not mills fitted out with information-hoppers, there will no longer be need to discuss the place of science in education.

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[1] See Fosnot and Perry, 2005, for a more in-depth discussion than that provided here of the work of these scientists and the relation of their work to contemporary models of learning.