Empirical data from neuropsychology seem to support that we are born with number sense which facilitates our intuitive grasp of the meaning of natural numbers and their basic arithmetic. But the bad news is: this intuitive understanding of numbers does not include zero, negative integers, fractions, irrational numbers and complex number. In other words, these mathematical entities do not correspond to any innate mental category in our brain, and therefore easily defy our intuition (Dehaene 1997; Geary 1995; Lakoff & Nuñez 2000). This findings gives a very good explanation for the hardship of teaching and learning directed numbers and their basic arithmetic, encountered by most of the pupils and teachers in the middle school years. Recent research suggests that the use of metaphors is both theoretically and pedagogically sound to resolve such problem (Dehaene 1997; English 1997; Lakoff & Nuñez 2000). Popular metaphors include: surplus/deficit in financial management, left/right movement on a number line, positive/negative temperature of a certain object. This paper will first examine the theoretical foundation of quantitative intuitions and the importance of metaphors in mathematical mind. Pedagogical use of metaphors in teaching and learning of directed numbers will also be investigated. Subsequently, the main argument of this paper will be explicated, suggesting that the pedagogical use of metaphors can be further improved by adopting Egan’s (1988a, 1988b, 1990, 1992, 1997) sophisticated framework about the development of educated mind and different kinds of Understanding. Two Chinese examples will be given to support this argument. The first example uses the mythical Yin/Yang idea, which permeates nearly every aspect of the life of ancient Chinese. The second example adopts the story-telling approach. The story is an adaptation of the Fiery Mountains episode of the famous Chinese classical myth – The Journey to the West. The hot/cold metaphor will be incorporated into the teaching and learning activities, when the Monkey King battles with the Bull Demon King in order to lower the temperature of the Fiery Mountains so that his Great Mentor – the Tang Priest can have his way to the West unblocked.
The importance of Mathematics as a school subject is broadly recognized by most of the mathematics educators, school administrators, teachers, parents and students throughout the world. One of the main reasons is that it is important for the development of logical and rational thinking in children’s mind. Unfortunately, due to their narrow definition of good performance, quite many teachers and students are actually supporting teaching and learning without understanding. For instance, what is taught in primary school as arithmetic is, for the most part, not ideas about numbers but automatic procedures for performing operations on numerals to give consistent and stable results. But being able to carry out such operations accurately does not mean that teachers have taught meaningful content and students have learned imaginative ideas about numbers.
At first glance, the mission of developing of logical and rational thinking in children’s mind is sacred. Therefore, the problem can be resolved by removing those implementation obstacles for the development of logical and rational thinking, by shifting emphasis on foundations, formal reasons, problem solving and other big ideas of mathematics development. But the author argues that even if these implementation obstacles are removed for the sacred mission, there is still a possible danger of missing the imaginations. Due to their inadequate view of mathematics and mathematics education, many teachers and students may embrace a narrow, disembodied theory of rationality and logical thinking which supported a hard, calculative, unimaginative and dehumanized form of thought
Different philosophers may propose different antidotes, that is, their own philosophical framework for the design and implementation of the teaching and learning activities. For instance, after rigorous and in-depth philosophical analysis and critical reflection, Ernest (1991) proposed a soft deterministic neo-Marxist view on the reproduction of social hierarchy through education. With reference to mathematics education, he asserted that
"The form of mathematics education plays a central part in the reproduction (or challenging) of the social hierarchy, but it is only one of several such elements, which included the philosophies of mathematics and the theories of school mathematics knowledge. Thus epistemology and the content of education plays a crucial role in recreating or changing the social hierarchy" (Ernest 1991, p.258).
The author does not accept this social constructivist view of mathematics education (Tang, 1999). Instead, an embodied realistic view of mathematics and humanistic view of mathematics education has been adopted. This paper first gives a brief introduction of the embodied realistic view of mathematics, by referring to two recently published books: Philosophy in the Flesh: The Embodied Mind and Its Challenge to Western Though, written by George Lakoff and Mark Johnson in 1999; and Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being, written by George Lakoff and Rafael Nuñez in 2000. Arithmetic and directed numbers, which is the mathematical and educational focus of this paper, will be used for the illustration of this realist theory.
The current form of embodied realism may not be adequate for the design of the imaginative teaching and learning activities for students. Therefore, Egan’s (1988a, 1988b, 1990, 1992, 1997) sophisticated framework about the development of educated mind and different kinds of understanding has been adopted. With reference to his humanistic framework, two Chinese examples for imagination in teaching and learning of directed numbers are proposed. Since this cannot be a lengthy paper. Therefore, only instructional suggestions and teaching and learning activities will be given whereas implementation problems will not be investigated. Finally, the author argues that the use of game and story-telling in mathematics education can be a serious endeavor, and with fun!
Embodied Realism, Metaphors, Arithmetic and Directed Numbers
Embodied Realism and Metaphors
Embodied realism is founded on three major findings of cognitive science: the mind is inherently embodied, thought is mostly unconscious and abstract concepts are largely metaphorical (Lakoff & Johnson 1999). And these three major findings require not only a new way of understanding reason and the nature of a person, they also require a new way of asking philosophical questions. Lakoff and Johnson argue that they have only taken a small first step for the emerging philosophy: “A philosophical perspective based on our empirical understanding of the embodiment of mind is a philosophy in the flesh, a philosophy that takes account of what we most basically are and can be” (Lakoff & Johnson 1999, p.8).
George Lakoff and Mark Johnson have their supporters in the field of philosophy of mathematics. George Lakoff and Rafael Nuñez have worked with together to publish a book in 2000: Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being. Goodbye Descartes: The End of Logic and the Search for a New Cosmology of the Mind, written by a mathematician Keith Delvin in 1997, is another important book for the elaboration of an embodied philosophy of mathematics.
In their 2000 book, George Lakoff and Rafael Nuñez have explored how the general cognitive mechanisms used in everyday nonmathematical thought can create mathematical understanding and structure mathematical ideas. They argue that
“a great many cognitive mechanisms that are not specifically mathematical are used to characterize mathematical ideas. these include such ordinary cognitive mechanisms as those used for the following ordinary ideas: basic spatial relations, groupings, small quantities, motion, distributions of things in space, changes, bodily orientations, basic manipulations of objects (e.g. rotating and stretching), iterated actions, and so on” (Lakoff and Nuñez, 2000, p.28)
And they suggest that, with reference to general cognitive mechanisms, conceptual metaphors play a major role in characterizing mathematical ideas. There are two basic kinds of metaphorical mathematical ideas: grounding and linking metaphors. Grounding metaphors yield basic, directly grounded ideas. Examples: addition as adding objects to a collection, subtraction as taking objects away from a collection, sets as containers, members of a set as objects in a container. These usually require little instruction. Linking metaphors yield sophisticated ideas, sometimes called abstract ideas. Examples: numbers as points on a line, geometrical figures as algebraic equations, operations on classes as algebraic operations. These require a significant amount of explicit instruction.
Metaphors, Arithmetic and Directed Numbers
Empirical data from neuropsychology seem to support that we are born with number sense which facilitates our intuitive grasp of the meaning of natural numbers and their basic arithmetic. Intuition about numbers is thus anchored deep in our brain.
“Number appears as one of the fundamental dimensions according to which our nervous system parses the external world. Just as we cannot avoid seeing objects in colour ….., in the same way numerical quantities are imposed on us effortlessly through the specialized circuits of our inferior parietal lobe. The structure of our brain defines the categories according to which we apprehend the world through mathematics” (Dehaene 1997, p.245)
Together with the inventiveness of our ancestors who developed an extremely powerful numeration system which reduce computation with numbers to the routine manipulation of symbols on paper. Lakoff and Nuñez (2000) have succinctly given this piece of cultural evolution history an embodied explanation:
“Our mathematics of calculation and the notation we do it in is chosen for bodily reasons – for ease of cognitive processing and because we have ten fingers and learn to count on them. But our bodies enter into the very ideas of a linearly ordered symbolic notation for mathematics. Our writing systems are linear partly because of the linear sweep of our arms and partly because of the linear sweep of our gaze. The very ideas of a linear symbol system are at the heart of mathematics. Our linear, positional, polynomial-based notational system is an optimal solution to the constraints placed on us by our bodies (our arms and our gaze), our cognitive limitations (visual perception and attention, memory, parsing ability), and possibilities given by conceptual metaphor” (Lakoff and Nuñez, 2000, p.86)
There is also bad news: this intuitive understanding of numbers does not include zero, negative integers, fractions, irrational numbers and complex number. In other words, these mathematical entities do not correspond to any innate mental category in our brain, and therefore easily defy our intuition (Dehaene 1997; Geary 1995; Lakoff & Nuñez 2000). This findings gives a very good explanation for the hardship of teaching and learning of fractions, zero, negative integers and their basic arithmetic, encountered by most of the pupils and teachers in the middle school years. In other words, our teaching and learning are constrained by our biology and psychology.
Arithmetic: Children Are not Machines
Some may think that drill-and-practice with hardwork is one of the best learning strategy to overcome our constraints, both biological and psychological. Recent research suggests that the use of metaphors is a much better alternative, which is both theoretically and pedagogically sound, to resolve such problem.
Lakoff and Rafael (2000) argue that we are not machines. We have four grounding metaphors for arithmetic: Arithmetic As Object Collection, Arithmetic As Object Construction, The Measuring Stick Metaphor, and Arithmetic As Motion Along a Path. They are the basic cognitive mechanisms that create arithmetic understanding and structure arithmetic ideas. With regard to classroom teaching and learning strategy, Dehaene (1997) bemoans that children are not machines. We don’t want them to “be reduced to purely formal manipulations of digital symbols, in exactly the same way that a computer follows an algorithm without ever understanding its meaning” (p.87). He argues strongly for an early use of electronic calculator. By releasing children from the tedious and mechanical constraints of calculation, he is in full conviction that they can concentrate more on the fascinating regularities, magics and meaning about arithmetic. They can be taught how to make friends with numbers instead of despising them.
Directed Number: Formalism versus Intuitionism
Negative numbers and their basic arithmetic is also another difficult topic for many teachers and students in middle school years. During the Modern Mathematics period several decades ago, some mathematics educators proposed the use of the language of set, logic, axioms and formal rules to replace drill-and-practice for the teaching and learning of mathematics, including this topic. Again, recent research suggests that the use of metaphors is the third alternative which can be much better.
Lakoff and Rafael (2000) the four grounding metaphors for arithmetic can extend innate arithmetic considerably. For instance, the Arithmetic Is Motion Along a Path metaphor allows the path to be extended indefinitely from both sides of the origin. Zero and negative numbers can then be conceptualized as point-locations on the path. Fractions can then be conceptualized as point-locations between the integers, at distances determined by division. In order to achieve closure in basic arithmetic for zero, negative numbers and fractions, extended or stretched metaphors are needed.
Dehaene (1997) argues that since nature has only endowed us an intuitive picture for positive integers. Therefore, in order to help students understand zero, fractions and negative numbers, teachers have to help student piece together novel model for new intuitive understanding. And he argues that it is both theoretically and pedagogically sound for teachers to use metaphors, such as surplus/deficit in financial management, left/right movement on a number line, positive/negative temperature of a certain object.
The author, on the one hand, strongly agrees with this line of thought, both from theoretical and practical point of view. On the other hand, the author finds that this framework seems to be inadequate when designing classroom teaching and learning activities. The author wants to argue that, in the remaining two sections of this paper, the pedagogical use of metaphors can be further improved by adopting Egan’s (1988a, 1988b,1990, 1992, 1997) sophisticated curriculum model about the development of educated mind and different kinds of understanding. Two Chinese examples will be given for a more effective, imaginative and meaningful learning of directed numbers.
Five Kinds of Understanding and Two Chinese Examples: Directed Numbers in Focus
Educated Mind and Five Kinds of Understanding
Egan’s (1988a, 1988b, 1990, 1992, 1997) sophisticated framework about the development of educated mind and different kinds of understanding is original and innovative. His basic idea is that there are five distinctive strands or layers of our understanding: Somatic, Mythic, Romantic, Philosophic, and Ironic. They are generated by different mediating tools – such as language or literacy – which shape our perception of the world.
His theory suggests that our initial understanding is Somatic. Each other kind of understanding results from the development of particular intellectual tools that we acquire from the societies we grow up in. Working with ‘tool’ of oral language leads to the Mythic understanding with new perspective on the world and experience, and new style of sense making. The Romantic layer is a little more complicated. It has been identified not simply with the ‘tool’ of alphabetic literacy, but with a cluster of further, related social and cultural developments in ancient Greece. The Philosophic understanding is shaped by an even more diffuse ‘tool’. It requires not only a sophisticated language and literacy, but also a particular kind of communication to support and sustain it. Finally, Ironic understanding is an implication of self-conscious reflection about the language one uses.
These kinds of understanding are not neat and discrete categories. They do not represent irreconcilable features in the mind of their users. They are more like different perspectives than different mentalities, by means of which particular features of the world and experience are brought into focus and prominence. Furthermore, each kind of understanding does not fade away and would not be replaced by the next, but rather each properly coalesces in significant degree with its predecessor.
Since they have developed in evolution and cultural history in a particular sequence and coalesced to a large extent as each successive kind has emerged. Therefore, Egan (1997) argues that “education can best be conceived as the individual’s acquiring each of these kinds of understanding as fully as possible in the sequence in which each developed historically” (p.4). Thus, his theory of educational development is based on a new recapitulation theory. The following introduction of Somatic and Mythic Understanding draws heavily from his recently published book in 1997, The Educated Mind: How Cognitive Tools Shape Our Understanding, which is recommended by Howard Gardner as the best introduction of his important body of work.
Egan espoused an embodied philosophy by suggesting that our body is the most fundamental mediating tool that shapes our understanding. Somatic understanding “refers to the understanding of the world that is possible for human beings given the kind of body we have” (p.5). Sequentially, it precedes the Mythic, Romantic, Philosophic and Ironic understanding. But it does not fade away or to be replaced by language development and other kinds of understanding. Rather, it remains with us throughout our lives and continues to develop within other kinds of understanding, maybe with some modification. Therefore, his thought is, in certain sense, in line the embodied and intuitive view of mathematics.
The two great epic poems of ancient Greece, Iliad and Odyssey, traditionally attributed to Homer, are good examples for showing the features of Mythic understanding. Binary structuring is one of the characteristics of Mythic understanding. Male/female, culture/nature, rational/emotional, self/other, figure/ground and Chinese Yin/Yang are some of the examples. Another feature is fantasy. “Young children, apparently universally, delight in fantasy stories full of talking clothed rabbits, bears, or other animals, also dislocated from anything familiar in their everyday waking experience” (p.44). Other features include abstract thinking, metaphor, rhythm and narrative and images. According to Egan, these features are inevitable consequences of oral-language development, whether in oral societies throughout the world and throughout history or by children throughout the world as they grow into language-using environments.
Journey to the East
As mentioned before, binary structuring is one of the characteristics of Mythic understanding. Chinese Yin/Yang is one of the examples used by Egan (1997). Adopting this mythic idea and its impressive diagram for teaching and learning of directed number is quite common in Chinese society. Appendix 1B is extracted from a local Hong Kong textbook. Similar approach has been observed in a Taiwan book for mathematics educator also. Teachers are usually advised to ask students to produce several +1 and –1 figures, so that they can play with the figures to explore the principle of addition and subtraction, then multiplication and division (for the purpose of demonstrating division, some teachers may prefer the use of +4 and –4 instead). This design of teaching and learning activities for directed number is not only effective. It has its implications on the use of metaphor and students’ development of Mythic Understanding. In order to understand such implications, a brief introduction of Chinese Yin/Yan may be helpful.
The following introduction of Chinese Yin/Yan draws heavily from Fritjof Capra’s 1975 book: The Tao of Physics. More than two thousand years ago, the Chinese sages believed that there is an ultimate reality which underlies and unifies the multiple things and events we observe. They called this reality the Tao. The Tao is the cosmic process in which all things are involved, and the world is seen as a continuous flow and change. The idea of cyclic patterns in the motion of the Tao was given a definite structure by the introduction of the polar opposites Yin and Yang (see Appendix 1A). The diagram is a symmetric arrangement of the dark yin and the bring yang, but the symmetry is not static. From the very early times, the two archetypal poles of nature were represented not only by bright and dark, but also by male and female, firm and yielding, above and below, Heaven and Earth. The diagram is a rotational symmetry suggesting, very forcefully, a continuous cyclic movement.
The advantage of using Yin and Yang is that the mythic primordial pair of opposites permeates Chinese culture and determines all features of the traditional Chinese way of life. As a nation of farmers, the Chinese had always been familiar with the movements of the sun (Yang) and the moon (Yin). A healthy diet consists in balancing the Yin and Yang elements. Traditional Chinese medicine, too, is based on the balanced Yin and Yang in the human body, and any illness is seen as a disruption of this balance. Furthermore, the story does not stop here. By studying various combinations of Yin and Yang, the Chinese also developed a system of cosmic archetypes. The system is elaborated in the I Ching, or Book of Changes.
The book is a work that has grown organically over thousands years and consists of many layers stemming from the most important periods of Chinese thought. The starting point of the book was a collection of sixty-four figures, or ‘hexagrams’, which are based on the Yin-Yang symbolism and were used as oracles (See Appendix 2). More importantly, the purpose of consulting the I Ching was not merely to know the future. But rather to discover the disposition of the present situation so that proper action could be taken. This attitude lifts the book above the level of an ordinary book of soothsaying and made it a book of wisdom.
Journey to the West
With regard to Mythic Understanding, Egan (1997) argues that another feature of this type of Understanding is fantasy. “Young children, apparently universally, delight in fantasy stories full of talking clothed rabbits, bears, or other animals, also dislocated from anything familiar in their everyday waking experience” (p.44). Journey to the West is a well-known Chinese mythological novel of such kind, with many talking animals, sea creatures and even insects.
Journey to the West was an extremely imaginative and creative work basing on folk tales, and was probably put into its present form in the 1570s by Wu Cheng’en. Many different editions of the novel have appeared over the past 400 years. In recent decades, great amount of comic books, movies, TV episodes and cartoons have been created by adopting or adapting features of the whole or part of this novel. Of all the Chinese fantasy novels published in the sixteenth and seventeenth centuries, it is the only one to have become so central to Chinese culture and remains so popular. The following brief introduction of the story draws heavily from W.J.F. Jenner’s translated version published in 1994.
The story starts with the earlier life of two of the chief characters, Monkey and the Tang Priest Sanzang, and of the circumstances in which Emperor Taizong sends Sanzang off on his journey to the Western Heaven to visit the Tathagata and fetch the Mahayana scriptuers. The second part of the story tells about how the lone pilgrim Sanzang acquires his three disciples, Monkey, Pig and Friar Sand, as well as his white dragon horse. The next part is about the adventures of the four travellers and their horse on their fourteen-year journey, are not one but dozens of stories. In each episode, the travellers are presented with a problem that they have to deal with before they can continue on their way. The last part is about the travellers’ arrival at the Thunder Monastery on Vulture Peak in India, their rather shabby treatment there, their return to China, and their final reward.
The author suggests the adoption of story-telling approach in this second example. The story is an adaptation of the Fiery Mountains episode of this famous novel. In order to lower the temperature of the Fiery Mountains so that his Great Mentor – the Tang Priest can have his way to the West unblocked, the Monkey King has to battle with the Bull Demon King. Students will be divided into groups of two to play the battle game: one as Monkey King and one as Bull Demon King. Each student will be given two dices, one red and one blue. In the beginning, the temperature is 37oC. Each player has their alternative turn to throw the blue (cold) or red (hot) dice, depending on their own decision. Since the Monkey King want to lower the temperature, whereas the Bull Demon King want to increase it. Therefore, Monkey King may use red dice to take away heats (+1s) or blue dice to add colds (-1s) whereas the Bull Demon King may use red dice to add heats (+1s) or blue dice to remove colds (-1s). After six to ten rounds, each group may come up with a problem of simple addition and subtraction of directed numbers. The final task of the players of each group is to do the calculation and find out who is the winner. For less capable students, their final calculation may be supported with addition and removal of real red and blue cards after each round of their fight.
Conclusion: More Stories, Games, Metaphors and Imagination for More Understanding
The above examples of using stories, games, myths and imagination to teach and to learn have no contradiction with the ideas that arithmetic in particular and mathematics in general are subsystems of the human conceptual system. It only rejects hard, calculative, mechanistic, formalistic, unimaginative and dehumanized form of mathematics and mathematics education. Our mathematics comes from the nature endowment of our intuitive grasp of the meaning of natural numbers and their basic arithmetic, our use of embodied metaphors as cognitive mechanism, and the cultural heritage of our imaginative ancestors. This conceptual subsystems are found to be “precise, consistent, stable across time and communities, understandable across culture, symbolizable, calculable, generalizable, and effective as general tools for description, explanation, and prediction in a vast number of everyday activities, from business to building to sports to science and technology” (Lakoff and Nuñez, 2000, p.50). And the author argues that this is the most important reason for a vision of mathematics for all.
In order to get as close to this ideal as possible, the author also argues that we need more stories, games, metaphors and imagination for teaching and learning with understanding, but not less. This line of thought has many potentials, and some mathematics educators have already started their adventure (e.g. Chiu 2002; Dehaene 1997; Delvin 1997; Lakoff & Nuñez 2000). Starting with adequate support for the development Somatic and Mythic Understanding, students are expected to be developing more Understanding in future: Romantic Understanding, Philosophic Understanding and Ironic Understanding (Egan 1997), not only in mathematics (e.g. Seife 2000), but also in logic (e.g. Kosko 1994), in physics (e.g. Capra 1975), cognitive science (e.g. Varela et al. 1993), so on.
Capra, F. (1975). The Tao of Physics: An Exploration of the Parallel Between Modern Physics and Eastern Mysticism. Boulder, Colorado: Shambhala
Chiu, M.M. (2002). Metaphorical Reasoning: Novices and Experts Solving and Understanding Negative Number Problems. Educational Research Journal, Vol.17, No.1, pp. 19-41.
Dehaene, S. (1997). The Number Sense: How the Mind Creates Mathematics. NY: Oxford University Press.
Delvin, K. (1997). Goodbye Descartes: The End of Logic and the Search for a New Cosmology of the Mind. New York: John Wiley & Sons, Inc.
Egan, K. (1988a). Primary Understanding: Education in Early Childhood. New York and London: Routledge.
Egan, K. (1988b). Teaching as Story Telling : An Alternative Approach to Teaching and the Curriculum. London: Routedge.
Egan, K. (1990). Romantic Understanding: The Development of Rationality and Imagination, Ages 8-15. New York and London: Routledge.
Egan, K. (1992). Imagination in Teaching and Learning: The Middle School Years. Chicago: The University of Chicago Press.
Egan, K. (1997). The Educated Mind: How Cognitive Tools Shape Our Understanding. Chicago: The University of Chicago Press.
English, L.D. (1997). Analogies, metaphors and images: Vehicles for mathematical reasoning. In L.D. English (Ed.), Mathematical reasoning: Analogies, metaphors and images (pp.3–18). Mahwah, NJ: Lawrence Erlbaum Associates.
Ernest, P. (1991). The Philosophy of Mathematics Education. London: The Falmer Press.
Geary, D.C. (1995). ‘Reflections of Evolution and Culture in Children’s Cognition: Implications for Mathematical development and Instruction’. American Psychologist, 50(1), pp. 24-37.
Kosko, B. (1993). The New Science of Fuzzy Logic: Fuzzy Thinking. UK: HarperCollins.
Lakoff, G. and Johnson, M. (1999). Philosophy in the Flesh: The Embodied Mind and Its Challenge to Western Though. New York: Basic Books.
Lakoff, G. and Nuñez, R. (2000). Where Mathematics Come From: How the Embodied Mind Brings Mathematics into Being. New York: Basic Books.
Seife, C. (2000). Zero: The Biography of a Dangerous Idea. NY: Viking Penguin.
Tang, K.C. (1999). Stability and Change in School Mathematics: A Socio-Cultural Case Study of Secondary Mathematics in Macau, Unpublished Ph.D. Thesis, University of Hong Kong.
Verela, F.J., Thompson, E. and Rosch, E. (1993). The Embodied Mind: Cognitive Science and Human Experience. Cambridge, Massachusetts: The MIT Press.
Appendix 1A: Yin/Yan
Appendix 1B: Local Textbook Example
Appendix 2: Two Regular Arrangements of the 64 Hexagrams
(Source: Capra 1975, p.280)