An Aesthetic Representation of Alfred North Whitehead's Philosophy of Mathematics Education

FOOTNOTES

[^1] A. N. Whitehead believes education should achieve automaticity so that the mind is free to study higher-level problems (Ocken, 2007). He disagreed with the idea that people must always think about what they are doing. He believed thought is useful only at decisive moments (Whitehead, 1911), and it should come with a decisive consciousness, which results from concentration and elimination of the irrelevant (Whitehead, 1938d).

[^2] The book, with no numbers and no practical problems, is a reference to Alfred Whitehead’s unsuccessful effort with Bernard Russell to reduce mathematics to logic in Principia Mathematica (1910). Despite all their effort to put certainty back into the foundations of mathematics, the book was a failure in creating a closed system of truth. Nevertheless, it turned out to be an open-ended starting point for Alfred N. Whitehead (Ernest, 2000), as well as for Ali. However, Ali was the one who discovered that the symbols were changing themselves, as they were causing a change in his brother’s mind. The brother was unaware of that change, and for him the book was only a source to understand the static matter-of-fact world around him.

[^3] Both real numbers and natural numbers are infinitely large, as well as the daisies and the real numbers in the number plane. Cantor showed that the real numbers’ infinity is larger than the infinity of natural numbers. Interested readers may want to read Dauben (1979) who opposed that actual infinity is an expression of any sort of reality.

[^4] A. N. Whitehead’s process philosophy considers life as an organic entity. The reality (and learning) is beauty-centered and holistic (Ernest, 2000). At this point, Ali sees the world as composed of disconnected bodies that are in constant competition with each other.

[^5] Rabbit’s inability to talk explains why A.N. Whitehead thinks failure of language is the great problem of philosophy in the finite world (1938c). This relates to our inability to express concepts of the infinite world (where important things come from) by using the tools of the finite world (language). Secondly, the rabbit’s inability to talk metaphorically represents how A.N. Whitehead separates humans and animals from vegetation; yet he still emphasizes that the main characteristic of the animals is their lack of having the same level of expression as humans.

The Whitehead in the story went through a Kafka-style metamorphosis in Ali’s world and became a rabbit. Therefore, he doesn't possess the ability to express himself as humans do. The only communication tool that remained was what he used to have in his original world - speaking without uttering words.

[^6] Understandably, Ali doubts the reality of the rabbit as he has not met a talking rabbit before. Utilizing such a knowledge that is not connected to past experiences is a difficult task for Ali. A. N. Whitehead formulizes this by saying, “each actual occasion is a distillation of the totality of the past (McMahon, 1999, chap. 3)”.

[^7] A. N. Whitehead (1938a) explains the notion of importance as the “interest, involving [the] intensity of individual feeling which leads to publicity of expression” (p. 11). Ali’s interest in the story is to build a daisy chain, which is a definitely the way he chose to express himself. However, this interest becomes important for him to such a degree that he ignores all the other things around him, including a talking rabbit.

According to A. N. Whitehead, importance should be based on the unity of the universe, whereas the interests stem from the individuality of the details. As interest always modifies expression, Ali is in need of guidance that would emphasize higher order thinking skills — connected to his individual interests— that will lead him to the unity.

Importance based on the unity of the universe comes from infinity and learning becomes important as it leads to generalizable facts. Expression is displayed in the temporal and finite world which is shaped by individual interest. A. N. Whitehead’s education connects the finite with the infinite, and aims to bring importance and interest together in the classroom.

[^8] (Whitehead, 1938b, p.28)

[^9] (Whitehead, 1938b, p.28)

[^10] This reflects the organic connectedness of everything in the changing temporal world. According to A. N. Whitehead (1938a, p 30) “human body is that region of the world which is the primary field of human expression”, thus the correct answer to who or what we are should be in whatever way we express ourselves. Because our body and mind are in constant change, each of the atoms within our body has been, and will be, a part of something else.

[^11] Evangelista Torricelli is the inventor of Gabriel's Horn figure which has infinite surface area, but finite volume. The name refers to the Archangel Gabriel who blows the horn to announce Judgment Day. In the story, it is the metaphorical connection between the finite and the infinite worlds.

[^12] A. N. Whitehead claims “language is thought, and that thought is language,” (1938b) and it is the first step to achieve in the precision stage of a child’s cyclic growth after the initial romance period. Ali’s interest to build an infinitely large daisy chain was the first step in his romance period, where his interests were shaped. At this stage, Ali is about to enter the first precision stage, and he needs to know the perfect method to communicate with the elements of the World of Heavens in order to reach the stage of generalization. Such a language doesn’t exist in the temporal world, but the ripples of his words with this perfect language will help him reach the final stage.

[^13] (Whitehead, 1985, p. 346).

[^14] This “…is a rhyme which fits onto the tradition respecting Dr. Whewell, who was Master of Trinity College, Cambridge, about eighty years ago. The rhyme is well-known, and runs thus:

I am Master of this College;

And what I know not, is not knowledge.

This attitude is always prevalent in the learned world. It sterilizes imaginative thought, and thereby blocks progress” (Whitehead, 1938c, p.59).

By providing this example, Whitehead opposes the God-role played by the teachers, who see themselves as the owner of eternal knowledge. According to A. N. Whitehead, it is not possible for a teacher to know everything that is necessary for pupils. Students need to learn not only specialized knowledge of the teacher, but also how it is connected to life and all other subjects. A. N. Whitehead’s (1929, p.4) definition of education is, “…the acquisition of the art of the utilization of knowledge”. As his definition suggests, the role of the teacher is to teach how to learn this art.

[^15] As this is not the real world, no such teacher exists in the temporal world. I am in no way implying that the teachers should play the role of God as the source of knowledge. I assume the sarcasm in these sentences is clear, as it targets those teachers who are very much fond of such a role.

[^16] (Whitehead, 1938c, p. 60)

[^17] A. N. Whitehead (1929) claims, “[the] present contains all that there is. It is the holy ground; for it is the past, and it is the future”. According to him, this is the type of knowledge we should be providing to our children, instead of teaching depreciation of the present or the knowledge of the past, which doesn’t equip them for the present. For example, when we teach about history of mathematics we are interested in bringing attention to the experiences of the past mathematicians and showing their life so that students can relate those experiences to their own lives, and learn from the process. A. N. Whitehead believes any other type of knowledge of the past is a deadly harm to the minds of our students.

[^18] Squaring the circle is the historical challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and ruler. Informally, one can also claim that Ali’s job of making a daisy chain with an infinite radius is squaring the circle and logically not possible. On the other hand, Anaxagoras was an ancient Anatolian philosopher who is known for his work on squaring the circle. It was much later proven that this task is not possible because the number pi was proved to not be an algebraic irrational number.

[^19] Ernest (2000) describes A. N. Whitehead’s notion of organic connectedness of all things as an inspiration to ecological philosophers which “posits human knowing and human culture as an open-ended phenomenon” (p. 226). According to A. N. Whitehead, various forms of knowledge are linked, and the pupil should be thrown into every possible combination. The student doesn’t need to be taught too many subjects; but what is taught should be taught thoroughly. The most important subject matter of knowledge should be life itself (Whitehead, 1929).

[^20] A. N. Whitehead believes that uniform external examination is a deadly practice because there will always be some issues that are not covered by the individual teacher of the pupil, which will mean tricking the student. On the contrary, the educational system proposed by A N. Whitehead depends on the character of the students and the genius of the teacher to teach them how to learn about life (Whitehead, 1929)..

He also opposes a uniform curriculum filled with inert knowledge because educators deal with the human mind, not with dead matter. The human mind is not an instrument that needs to be sharpened by learning inert ideas and skills. Humans are social beings, and each of them is a part of a culture; you cannot postpone life until you have sharpened their minds (Whitehead, 1929)..

[^21] As Ali gained the automaticity of basic knowledge and gained an understanding of the well-connectedness of life around him, it is now time for him to go back to his own world. At this time, he completed the precision stage, and is ready to apply his knowledge to further generalizations in his own world.

Author Notes: M. Sencer Corlu is a graduate student of mathematics education at Texas A&M University with research interests in social constructivist theories, teacher education, and integration of mathematics and science.