# Paideia High School/Fingerprints

"Number: The Language of Science" (pub. 1930).
These guidelines address teachers for the purpose of guiding instruction. See Paideia Learning Plan for the student's point of view.

## Column One

Column One teaching and learning should make up about 10% to 15% of the total scheduled instructional time. It is didactic in nature and uses teacher lectures, text books or other didactic instructional materials, and questioning appropriate to this mode of education. Teaching in this mode encompasses of three facets: Exoridium, Interpretation, and Erudition.

### Exordium

The Exordium is the teacher's didactic introduction to the work that is the subject of a Paideia Unit Plan. This introduction consists of both an oral and physical (or electronic) presentation of the work. For longer works, the teacher may limit the oral presentation to key parts of the work. The teacher should read texts live distinctly, accurately, and intelligently. Other works should be orally presented similarly as appropriate to the type of work. In addition, the teacher should provide high quality audio and video recordings of works if possible. Students should have a consumable print copy of the work both electronically and in hard-copy if possible.

#### Oral Presentation

* Because no oral performances were available, I recorded myself reading (set to background music comprised of several modern and jazz compositions, each of which is entitled "Fingerprints") this first chapter of Tobias Dantzig's book, "Number: The Language of Science." This recording was first played in its entirety for the students in a setting where their only task was to listen. The recording will be played a second time for the same students, at which point they will have a complete copy of the text before them and will be asked to follow along, underlining any words they are unfamiliar with and any passages they do not understand. As well, they will be encouraged to take margin notes to indicate any sections that are of particular interest, and to jot down any questions that arise throughout the reading.

#### Written Presentation

* Fingerprints is available online as a viewable PDF at Google Books.
* Available in Gateway to the Great Books
* Also available in Danzig's book Number: The Language of Science.
Notable quotes within or pertaining to the text:
* "This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands." - Albert Einstein
* "The harmony of the universe knows only one musical form - the legato; while the symphony of number knows only its opposite - the staccato. All attempts to reconcile this discrepancy are based on the hope that an accelerated staccato may appear to our senses as a legato." - Tobias Dantzig
* "Man is the measure of all things." - Protagoras

### Interpretation

Textual Interpretation refers to the teacher's didactic analysis of a written work in terms of the four major questions a demanding reader should ask of a text. Using the term work instead of book, these questions are: (1) What is the text about as a whole? (2) What is being said in detail, and how? (3) Is the text true, in whole or part? and (4) What of it? The Exordium begins to answer the first question because it introduces the whole text both orally and in writing (although the oral presentation may be limited in the case of longer works). It is in the Interpretation stage of Column One instruction that the teacher didactically begins to thoroughly unlock the second question. The third and fourth questions are relevant to Interpretation, but question two receives most of the teacher's attention. It is only when the student begins Column Two activities that a fuller grasp of questions three and four begins to mature in the student's mind. Consistent with the purpose of Column One instruction, the teacher is simply introducing elements of proper interpretation for the student to build on during Column Two and Column Three learning.
These questions can be appropriately modified for analysis of a work in science, mathematics, or the fine arts. Analysis of these types of works proceeds in a manner analagous to that of a text. Inasmuch as imaginative literature can be considered a work of fine art, the analogies drawn in Adler's and Van Doren's How to Read a Book can serve as a guide to other types of works too.

#### Interpretation of <Words and Terms> or <Elements>

In textual interpretation, the teacher didactically presents the author's key terms in the stage of Interpretation of Words and Terms. In addition, the teacher anticipates unfamiliar general vocabulary. These words and terms are given didactically, either orally or in writing, or both, but the task of interpreting is a Column Two skill of learning that the teacher must coach. Consequently, the purpose of this Column One stage is merely to point out key terms and potentially unfamiliar terms. The student must learn how to interpret. Again, this planning can be adapted to other kinds of works.

##### Interpretation of Key Terms

The key terms are those few words or phrases the author uses in unique, special, or important ways.[3]. In the Column Two stage, students are coached both to find these key terms and to "unlock" them on their own. This skill is essential to analytical reading. In this Column One stage, however, the teacher points out a list of such terms. Students should understand that this listing is not necessarily exhaustive. In addition, the teacher provides students with a handful of these terms with definitions worked out in detail for consideration; it should consist of about three to five terms.
* base of numeration
* cardinal number
* number sense
* one-to-one-correspondence
* ordered succession
* ordinal number
* tactile number sense
* visual number sense

##### General Vocabulary

The teacher may anticipate general unfamiliar vocabulary and point out the more difficult words either orally or in writing or both. However, defining words using context, a dictionary, or a knowlegable friend or adult (including the teacher) is a Column Two skill that must be coached. Teachers must help students build good habits of knowing the meanings of words. Dictionaries, in both electronic and book form must be available, and students must be taught how to use them. They must also be taught how to ask for definitions--a perfectly acceptible life skill commonly used by demanding readers!
* abstract
* advent
* ambiguity
* anthropological
* anthropomorphic
* artifice
* attribute
* attrition
* concrete
* conjecture
* devise
* discriminate
* economy
* elegant
* eminent
* essence
* etymology
* exhausted
* explicit
* extraction
* faculty
* heterogeneous
* homogeneous
* impenetrable
* inarticulate
* kinship
* latent
* latter
* metamorphosis
* metaphysical
* motley
* mystic
* numeration
* obliterate
* obsolete
* parity
* permeate
* philology
* physiological
* plausible
* precedence
* preliminary
* Providence
* quaint
* reckon
* relic
* rosary
* rudimentary
* ruse
* scope
* simultaneously
* speculation
* superfluous
* supersede
* supplanted
* symmetric
* tactile
* Thimshian
* typify
* vintage
* wrought
Note: If the teacher points out words in anticipation of their potential difficulty, this should be done in context by giving citations or electronically highlighting the words.
After students have read and marked passages and vocabulary, have them create a personal glossary, looking up marked words and notating the correct definitions for this usage. I choose to give an open note vocabulary quiz at this point. The rules were simple--student could use only their copy of the text (to look at context) and their glossary notes to complete the following quiz:
“Fingerprints” Vocabulary Quiz
1. Ruse
2. Faculty
3. Corroborate
4. Plausible
5. Impenetrable
6. Render
7. Consolidated
8. Heterogeneous
9. Plurality
10. Derive
11. Exhausted
12. Adept
13. Supersede
14. Tacit
15. Permeate
16. Surmise
17. Philology
18. Imperceptibly
19. Attrition
20. Obliterate (-ed)
21. Rudimentary
22. Conjecture
23. Metamorphosis
24. Anthropomorphic
25. Suffices
26. Providence
27. Physiological
28. Irreducible
29. Ambiguity
30. Proposition
A. A maneuver designed to trick or deceive
B. To have merged two or more elements together into one
C. To remove or destroy completely, so that no trace is left
D. The quality of state of being uncertain
E. Highly skilled and/or expert in performing a task
F. Figure out through the use of logic
G. All used up
H. A process of change in which one thing becomes something that appears entirely different
I. To be completely spread or diffused throughout a medium—as water is to a soaked sponge
J. Is enough
K. The study of languages and their development to better understand human culture
L. A likely or believable on the surface
M. To speculate or suppose based on little evidence
N. A statement to be argued as true or false
O. Change so subtle it almost can’t be noticed
P. Cannot be moved or seen through
Q. To make
R. Having to do with the physical body
S. A grouping consisting of items that are significantly different from each other
T. Divine guidance—the hand of God
U. To back up evidence with additional evidence or testimony
V. The ability to act or do
W. Unspoken, but understood
X. To take the place of something
Y. Having human characteristics
Z. Speculation or guessing
AA. The rate of loss
BB. Having a varied and large number or quantity
CC. Basic
DD. Cannot be transformed to a simpler condition
The goal here is not vocabulary building per se, but development of a sense of what words are both important enough and unfamiliar enough to warrant looking them up--which is a highly individualized situational decision. A secondary developmental goal is effective use of the dictionary in determining which meaning is relevant based on context.

#### Interpretation of Sentences

One point of didactic interpretation at the level of interpreting sentences is to use grammar to get at the author's meaning. As with interpreting words and terms, this level of interpretation is also a Column Two skill that must be coached. Consequently, at the Column One didactic level, a teacher should choose a handful of the most difficult sentences in the text for demonstration. The teacher will always unlock the grammar of a few important and more difficult sentences for students independent of whether these sentences are key premises to an argument. As appropriate to the text, a teacher should also consider demonstrating the grammar of sentences that work together as propositions in the author's most important arguments.
Another important point is to demonstrate the meter and prosody in both poetry and prose texts. This aim is in great danger of being completely overlooked or forgotten in a world where oral reading is not nearly as common as it once was. Nevertheless, great speechs often succeed in part because the author understands prosody. Lincoln's Gettysburg Address and King's I Have a Dream serve as striking examples. The teacher should select sentences or verses to demonstrate both meter and rhythm.
* <First Sentence>: <explanation of the grammar and how it helps to unlock the author's meaning>
* <Second Sentence>: <etc.>
* <Third Sentence>: <explanation of meter and rhythm, consideration of overall prosody>
* <Fourth Sentence>: <etc.>

#### Interpretation of Passages

At the level of passages, the full trivium (grammar, logic, and rhetoric) come into play. First, a teacher should choose one or two of the author's most important arguments for a demonstration of how to use logic as a key to interpreting a text. Next, the teacher should select several passages to demonstrate how they conform (or not) to rhetorical, poetical, and sylistic rules. These rules, of course, must be didactically taught as prerequisites to interpretation of texts.
* <Argument>: <demonstration of both the grammar of the sentences comprising premeses and of the concluding proposition along with the logic of the argument followed by an explanation of how the arguments help to unravel the author's meaning>
* <Passage Illustrating Rhetorical Rules>: <demonstration of the rhetorical rules and their value in the success of the author's purpose>
* <Passage Illustrating Poetical Rules>: <demonstration of the poetical rules and their value in the success of the author's purpose>
* <Passage Illustration Stylistic Rules>: <demonstration of the stylistic rules and their value in the success of the author's purpose>

#### Outline of Fingerprints
##### 1) Man possesses a rudimentary number sense.

       a. Number sense precedes counting.
          i. Only humans can count.

##### 2) Only a few other species possess a crude number sense akin to man.

       a. Solitary wasps possess number sense (5, 12, 24 grubs).
          i. Genus Eumenus (5 grubs/males – 10 grubs/females).
       b. Many birds possess number sense.
          i. The squire and the crow example.
       c. Prospective arguments against points a and b above.
          i. Species possessing number sense are exceedingly few.
             1. This argument stands.
          ii. Number sense in animals is limited in scope.
             1. But number sense is also extremely limited in man!

##### 3) Scope of rudimentary number sense in man quite limited.

       a. Human number sense is difficult to disentangle from counting.
       b. Direct visual number sense rarely extends beyond four.
       c. Tactile number sense is even more limited.
       d. Anthropological evidence supports this argument.
          i. Australian Aborigines.
          ii. African Bushmen.
          iii. English language studies corroborate.

##### 4) Rudimentary number sense is the precursor to the more sophisticated human concept of number.

       a. Sophisticated number concept is a prerequisite of higher level thinking.
       b. Counting truly sets us apart from all other species.

##### 5) The origins of counting.

       a. Concrete preceded the abstract.
       b. Heterogeneous (concrete) numbers → homogenous (abstract) number concept.
       c. One-to-one correspondence → no counting required to compare two collections.
          i. Matching/tallying are examples of one-to-one correspondence.
          ii. Model collections afford comparisons with concrete collections.
          iii. Cardinal number develops.
             1. Number words originally derived from concrete collections → abstract.
             2. Number words are based on one-to-one correspondence → no counting required.
          iv. Ordinal number concept develops.
             1. Cardinal number words.
             2. Ordered succession.
             3. Phonetic scheme for repeating.
          v. From systems of counting evolved basic arithmetic, multiplication, and all higher math.

##### 6) Finger counting provides pathway ←→ between cardinal and ordinal number.

       a. Fingers raised simultaneously → cardinal number.
       b. Fingers raised in succession → ordinal number.
       c. Philological evidence highly supports this claim.
          i. Five is frequently equated to “hand.”

##### 7) Man’s articulate fingers a prerequisite for various realms of higher thinking.

       a. Precursor to calculation.
          i. Finger counting.
          ii. Finger arithmetic.
          iii. Finger multiplication.
          iv. Finger calculators.
       b. Math the basis of all exact sciences.
       c. Exact sciences → man’s material and intellectual progress.

##### 8) Number language predates written history by thousands of years.

       a. Number words extraordinarily linguistically stable.
          i. Yet with exception of “five” – no trace of original meaning discernible.
             1. This due to change over time of original concrete object words.

##### 9) Ten fingers the foundation of our base 10 number system → a physiological accident!

       a. Base 10 by far most widespread.
          i. Highly linguistically documented.
       b. Remaining bases predominantly 5 and 20 → fingers and toes.
          i. Romans.
          ii. Axtecs.
          iii. Mayans.
       c. Binary the only other evidenced alternative.
          i. Economy of symbols.
          ii. Most fundamental system.
             1. Logic is binary based.
             2. Computers are binary based.
             3. Can everything be reduced to yes-no/off-on?
       d. Other bases more practical.
          i. Base 12 provides great number of divisors (2,3,4,6).
          ii. Base 11 dramatically reduces redundant expression of rational numbers.
          iii. Base 10 neither prime nor highly divisible (2,5)

##### 10) Man is the measure of all things (Protagorus).
### Erudition

Erudition refers to all manner of background information assumed by or necessary to understanding or fully appreciating the text.
<Note that these categories are presented alphabetically for ease of organization>

#### Anthropological References

Cultural Anthropology
* Auvergne was a historic province in south central France. It was originally the feudal domain of the Counts of Auvergne. It is now the geographical and cultural area that corresponds to the former province.
* Ancient Aztecs: The Aztec people were certain ethnic groups of central Mexico, particularly those groups who spoke the Nahuatl language and who dominated large parts of Mesoamerica in the 14th, 15th and 16th centuries, a period referred to as the late post-classic period in Mesoamerican chronology.
* Emperor of China refers to any sovereign of Imperial China reigning between the founding of China, united by the King of Qin in 221 BCE, and the fall of Yuan Shikai's Empire of China in 1916. When referred to as the Son of Heaven (Chinese: 天子 tiānzǐ), a title that predates the Qin unification, the Emperor was recognized as the ruler of "All under heaven" (i.e., the world). In practice not every Emperor held supreme power, though this was most often the case.
* Maya Indians: The Maya people constitute a diverse range of the Native American people of southern Mexico and northern Central America. The overarching term "Maya" is a convenient collective designation to include the peoples of the region who share some degree of cultural and linguistic heritage; however, the term embraces many distinct populations, societies, and ethnic groups, who each have their own particular traditions, cultures, and historical identity.
* The Roman Empire (Latin: imperium romanum) was the post-Republican period of the ancient Roman civilization, characterised by an autocratic form of government and large territorial holdings in Europe and around the Mediterranean. The term is used to describe the Roman state during and after the time of the first emperor, Augustus.
Linguistic Anthropology
* Philology is the study of language in written historical sources; as such it is a combination of literary studies, history and linguistics. Classical philology is the philology of the Greek, Latin and Sanskrit languages. Classical philology is historically primary, originating in European Renaissance Humanism, but was soon joined by philologies of other languages both European (Germanic, Celtic, Slavistics, etc.) and non-European (Sanskrit, Persian, Arabic, Chinese, etc.). Indo-European studies involves the philology of all Indo-European languages as comparative studies.
* Australian Aborigines are those people regarded as indigenous to the Australian continent. In the High Court of Australia, Australian Aborigines have been specifically identified as a group of people who share, in common, biological ancestry back to the original occupants of the continent. Edward Micklethwaite Curr (1886). The Australian race: its origin, languages, customs, place of landing in Australia and the routes by which it spread itself over the continent. J. Ferres. http://books.google.ca/books?id=qLw0AAAAIAAJ.
* Bushmen of South Africa: The indigenous people of southern Africa, whose territory spans most areas of South Africa, Zimbabwe, Lesotho, Mozambique, Swaziland, Botswana, Namibia, and Angola, are variously referred to as Bushmen, San, Sho, Barwa, Kung, or Khwe. These people were traditionally hunter-gatherers, part of the Khoisan group and are related to the traditionally pastoral Khoikhoi.
* The Greek language, an independent branch of the Indo-European family of languages, is the language of the Greeks. Native to the southern Balkans, it has the longest documented history of any Indo-European language, spanning 34 centuries of written records. Its writing system has been the Greek alphabet for the majority of its history; other systems, such as Linear B and the Cypriot syllabary, were previously used. The alphabet arose from the Phoenician script, and was in turn the basis of the Latin, Cyrillic, Coptic, and many other writing systems.
* The Mongolian language is the official language of Mongolia and the best-known member of the Mongolic language family. The number of speakers across all its dialects may be 5.2 million, including the vast majority of the residents of Mongolia and many of the Mongolian residents of the Inner Mongolia autonomous region of China.
* Old German (Old High German) refers to the earliest stage of the German language and it conventionally covers the period from around 500 to 1050. Coherent written texts do not appear until the second half of the 8th century, and some treat the period before 750 as 'prehistoric' and date the start of Old High German proper to 750 for this reason. There are, however, a number of Elder Futhark inscriptions dating to the 6th century (notably the Pforzen buckle), as well as single words and many names found in Latin texts predating the 8th century.
* Semitic: In linguistics and ethnology, Semitic (from the Biblical "Shem", Hebrew: שם‎, translated as "name", Arabic: ساميّ‎) was first used to refer to a language family of largely Middle Eastern origin, now called the Semitic languages. This family includes the ancient and modern forms of Akkadian, Amharic, Arabic, Aramaic, Ge'ez, Hebrew, Maltese, Phoenician, Tigre and Tigrinya among others. As language studies are interwoven with cultural studies, the term also came to describe the extended cultures and ethnicities, as well as the history of these varied peoples as associated by close geographic and linguistic distribution.
* Thimshian language: The Tsimshian are an indigenous people of the Pacific Northwest Coast. Tsimshian translates to Inside the Skeena River. Their communities are in British Columbia and Alaska, around Terrace and Prince Rupert and the southernmost corner of Alaska on Annette Island. There are approximately 10,000 Tsimshian. Their culture is matrilineal with a societal structure based on a clan system. Early anthropologists and linguistics grouped Gitxsan and Nisga'a as Tsimshian because of linguistic affinities. Under this terminology they were referred to as Coast Tsimshian, even though some communities were not coastal. The three peoples identify as separate nations. There are many other ways to spell the name, such as Tsimpshean, Tsimshean, Tsimpshian, and others.

#### Biographical References

* About the author: Tobias Dantzig
* List of Citations
** Comte de Buffon, Georges-Louis Leclerc (7 September 1707 – 16 April 1788) was a French naturalist, mathematician, cosmologist, and encyclopedic author. His works influenced the next two generations of naturalists, including Jean-Baptiste Lamarck and Georges Cuvier. Buffon published thirty-six quarto volumes of his Histoire naturelle during his lifetime; with additional volumes based on his notes and further research being published in the two decades following his death. It has been said that "Truly, Buffon was the father of all thought in natural history in the second half of the 18th century."
** Curr, Edward Micklethwaite published many reports and several books throughout his career, including Pure Saddle Horses in 1863, an account of his travels through Europe and the Middle East in the early 1850s, and The Australian Race: Its Origins, Languages, Customs in 1886. However, his most widely known work is Recollections of Squatting in Victoria, which was first published in 1883 but republished as an abridged version in 1965, recounting Curr's experiences managing his father's properties in northern Victoria forty years earlier, including his interactions with the local Aboriginal Australians.
** Grimaldi, Francesco Maria born April 2, 1618 in Bologna (Italy) and died on December 28, 1663 in Bologna, was an Italian Jesuit priest, mathematician and physicist who taught at the Jesuit college in Bologna. See Catholic Encyclopedia article for Francesco Maria Grimaldi.
** Laplace, Pierre-Simon was a French mathematician and astronomer whose work was pivotal to the development of mathematical astronomy and statistics. He summarized and extended the work of his predecessors in his five volume Mécanique Céleste (Celestial Mechanics) (1799–1825). This work translated the geometric study of classical mechanics to one based on calculus, opening up a broader range of problems. In statistics, the so-called Bayesian interpretation of probability was mainly developed by Laplace.
** Leibniz, Gottfried occupies a prominent place in the history of mathematics and the history of philosophy. He developed the infinitesimal calculus independently of Isaac Newton, and Leibniz's mathematical notation has been widely used ever since it was published. He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685[4] and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is at the foundation of virtually all digital computers.
** Protagoras (ca. 490 BC – 420 BC) was a pre-Socratic Greek philosopher and is numbered as one of the sophists by Plato. In his dialogue Protagoras, Plato credits him with having invented the role of the professional sophist or teacher of virtue. He is also believed to have created a major controversy during ancient times through his statement that "man is the measure of all things." This idea was very revolutionary for the time and contrasting to other philosophical doctrines that claimed the universe was based on something objective, outside the human influence.
** Russell, Bertrand is generally credited with being one of the founders of analytic philosophy. He was deeply impressed by Gottfried Leibniz (1646–1716) and wrote on every major area of philosophy except aesthetics. He was particularly prolific in the field of metaphysics, the logic and the philosophy of mathematics, the philosophy of language, ethics and epistemology. When Brand Blanshard asked Russell why he didn't write on aesthetics, Russell replied that he didn't know anything about it, "but that is not a very good excuse, for my friends tell me it has not deterred me from writing on other subjects."

#### Geographical References

* <links to relevant maps--delete if not used>
* <information relevant to geographical references--include citations--delete if not used>

#### Mathematical References

* Cardinal number: In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number – the number of elements in the set.
* Finger counting, or dactylonomy, is the art of counting along one's fingers. Though marginalized in modern societies by Arabic numerals, formerly different systems flourished in many cultures, including educated methods far more sophisticated than the one-by-one finger count taught today in preschool education. Finger counting can also serve as a form of manual communication, particularly in marketplace trading and also in games such as morra. Finger counting is studied by ethnomathematics.
* Number Sense: In mathematics education, number sense can refer to "an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations." Many other definitions exist, but are similar to the one given. Some definitions emphasize an ability to work outside of the traditionally taught algorithms, e.g., "a well organised conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms". The term "number sense" involves several concepts of magnitude, ranking, comparison, measurement, rounding, percents, and estimation
* One-to-one correspondence: The notion of one-to-one correspondence is so fundamental to counting that we don't even think about it. When we count out a deck of cards, we say, 1, 2, 3, ... , 52, and as we say each number we lay down a card. Each number corresponds to a card. Technically, we can say that we have put the cards in the deck and the numbers from 1 to 52 in a one-to-one correspondence with each other. If we count out a second deck and in the same way and count to exactly 52 when we lay out the cards, we know that both decks have the same number of cards. This means that we could pair up the cards, one from each deck, and not have any cards left over when all the pairs had been made. By putting the cards in each deck in a one-to-one correspondence with each other, we are sure that both decks have the same number of cards. These things are so obvious they seem silly. However, if we want to know the size of an unknown quantity, but the counting task is tricky, we can try to put the unknown quantity in one-to-one correspondence with some known quantity. This is the strategy that Georg Cantor used to compare different sizes of infinity.
* Ordered succession
* Ordinal number: In linguistics, ordinal numbers are the words representing the rank of a number with respect to some order, in particular order or position (i.e. first, second, third, etc.). Its use may refer to size, importance, chronology, etc. They are adjectives. They are different from the cardinal numbers (one, two, three, etc.) referring to the quantity.
* Reckoning

#### Philosophical References

* Metaphysics is a branch of philosophy dealing with theories of existence and knowledge. A person who studies metaphysics is referred to as a metaphysician. Metaphysicians are concerned with explaining the fundamental nature of being and the world. The metaphysician attempts to clarify the fundamental notions by which people understand the world, including existence, the definition of object, property, space, time, causality, and possibility. Traditionally, metaphysics attempts to answer two basic questions in the broadest possible terms:
** "What is there?"
** "What is it like?"

#### Religious References

* Agnosticism is the view that the truth value of certain claims—especially claims about the existence or non-existence of any deity, but also other religious and metaphysical claims—is unknown or unknowable. Agnosticism can be defined in various ways, and is sometimes used to indicate doubt or a skeptical approach to questions. In some senses, agnosticism is a stance about the similarities or differences between belief and knowledge, rather than about any specific claim or belief.
* Catholicism is a broad term for the body of the Catholic faith, its theologies and doctrines, its liturgical, ethical, spiritual, and behavioral characteristics, as well as a religious people as a whole. Jesuits are men who are members of a religious order called The Society of Jesus (Latin: Societas Iesu, S.J., SJ, or SI), who follow the teachings of the Catholic Church.
* Providence In theology, divine providence, or simply providence, is God's activity in the world. By implication, it is also a title of God. A distinction is usually made between "general providence" which refers to God's continuous upholding the existence and natural order of the universe, and "special providence" which refers to God's extraordinary intervention in the life of people

#### Scientific References

* Solitary Wasp: The general life style of a solitary wasp consists of a lone female mating and then preparing and provisioning one or more nests each containing one or more cells with food for her young. The egg hatches and the larvae consumes the supplied food without ever leaving the cell. After pupation the new adults emerge and seek a mate after which the males (being shorter lived in most species) die and the females go on to restart the cycle.
** Genus Eumenus: Many insects also have a well-developed number instinct, especially among the solitary wasps. The mother wasp lays her eggs in individual cells and provides each egg with a number of live caterpillars on which the young feed when hatched. Some species of wasp always provide five, others 12, and others as high as 24 caterpillars per cell. The number of caterpillars is different among species, but it is always the same for each sex of the eggs. The male solitary wasp in the genus Eumenus is smaller than the female, so the mother supplies him with only five caterpillars; the larger female receives ten caterpillars in her cell. In other words, she can distinguish between both the numbers five and ten in the caterpillars she is providing and which cell contains a male and which contains a female.

## Column Two

Column Two teaching consists of coaching. This mode of instruction aims at helping students to form habitual skills in the language, scientific, and fine arts. Thus, a teacher must correct students as they practice listening, speaking, reading, writing, observing with the senses unaided, observing with the aid of scientific apparatus, measuring, estimating, calculating, and exercising dexterity in the musical and visual arts. Each of these arts in turn rely upon the aquisition of fine and gross motor, imagination, and memory skills. These rules for developing Paideia Unit Plans address teachers. For their counterpart written for students, see Paideia Learning Plan.
Column Two learning comprises 65% to 75% of scheduled learning time. Its chief charactaristic is student activity. Students must be practicing some skill or skills while the teacher corrects him or her. While athletic coaching is an obvious example of this type of instruction, debate coaching, directing a drama, art instruction, and piano lessons also represent coaching. In order to coach well, a teacher must have a repertoire of activities carefully designed to exercise desired skills. Additionally, the teacher must have a clear idea of how to correct the skills as students practice them to ensure their habitual formation.

### Activities
#### Prerequisite Activities
##### Fine & Gross Motor Skills

<delete if not used>

##### Sensory Imagination Skills

<delete if not used>

##### Mnuemonic Skills

<consider this category for every unit>

#### Language Arts
##### Listening Skills

After listening to the audio of the text for vocabulary and meaning, students are offered the opportunity to listen to it one more time for accuracy. They are able to earn points for the number of reading errors against the text they can successfully locate on a third listening.

##### Speaking Skills

* <imitation of quality oral readings--should be a part of every unit>
* <recitations from memory--should be some part of every unit>
* <oral presentations of written work--should be a part of most units>
* <class discussions--should be a part of every unit>
* <recitation of the rules of reading from Adler's and Van Doren's How to Read a Book or similar rules>

##### Reading Skills
###### Outline of Fingerprints
####### 1) Man possesses a rudimentary number sense.

       a. Number sense precedes counting.
          i. Only humans can count.

####### 2) Only a few other species possess a crude number sense akin to man.

       a. Solitary wasps possess number sense (5, 12, 24 grubs).
          i. Genus Eumenus (5 grubs/males – 10 grubs/females).
       b. Many birds possess number sense.
          i. The squire and the crow example.
       c. Prospective arguments against points a and b above.
          i. Species possessing number sense are exceedingly few.
             1. This argument stands.
          ii. Number sense in animals is limited in scope.
             1. But number sense is also extremely limited in man!

####### 3) Scope of rudimentary number sense in man quite limited.

       a. Human number sense is difficult to disentangle from counting.
       b. Direct visual number sense rarely extends beyond four.
       c. Tactile number sense is even more limited.
       d. Anthropological evidence supports this argument.
          i. Australian Aborigines.
          ii. African Bushmen.
          iii. English language studies corroborate.

####### 4) Rudimentary number sense is the precursor to the more sophisticated human concept of number.

       a. Sophisticated number concept is a prerequisite of higher level thinking.
       b. Counting truly sets us apart from all other species.

####### 5) The origins of counting.

       a. Concrete preceded the abstract.
       b. Heterogeneous (concrete) numbers → homogenous (abstract) number concept.
       c. One-to-one correspondence → no counting required to compare two collections.
          i. Matching/tallying are examples of one-to-one correspondence.
          ii. Model collections afford comparisons with concrete collections.
          iii. Cardinal number develops.
             1. Number words originally derived from concrete collections → abstract.
             2. Number words are based on one-to-one correspondence → no counting required.
          iv. Ordinal number concept develops.
             1. Cardinal number words.
             2. Ordered succession.
             3. Phonetic scheme for repeating.
          v. From systems of counting evolved basic arithmetic, multiplication, and all higher math.

####### 6) Finger counting provides pathway ←→ between cardinal and ordinal number.

       a. Fingers raised simultaneously → cardinal number.
       b. Fingers raised in succession → ordinal number.
       c. Philological evidence highly supports this claim.
          i. Five is frequently equated to “hand.”

####### 7) Man’s articulate fingers a prerequisite for various realms of higher thinking.

       a. Precursor to calculation.
          i. Finger counting.
          ii. Finger arithmetic.
          iii. Finger multiplication.
          iv. Finger calculators.
       b. Math the basis of all exact sciences.
       c. Exact sciences → man’s material and intellectual progress.

####### 8) Number language predates written history by thousands of years.

       a. Number words extraordinarily linguistically stable.
          i. Yet with exception of “five” – no trace of original meaning discernible.
             1. This due to change over time of original concrete object words.

####### 9) Ten fingers the foundation of our base 10 number system → a physiological accident!

       a. Base 10 by far most widespread.
          i. Highly linguistically documented.
       b. Remaining bases predominantly 5 and 20 → fingers and toes.
          i. Romans.
          ii. Axtecs.
          iii. Mayans.
       c. Binary the only other evidenced alternative.
          i. Economy of symbols.
          ii. Most fundamental system.
             1. Logic is binary based.
             2. Computers are binary based.
             3. Can everything be reduced to yes-no/off-on?
       d. Other bases more practical.
          i. Base 12 provides great number of divisors (2,3,4,6).
          ii. Base 11 dramatically reduces redundant expression of rational numbers.
          iii. Base 10 neither prime nor highly divisible (2,5)

####### 10) Man is the measure of all things (Protagorus).

The following questions are then a good point to begin class discussion of the actual content of the text. These questions are intended to lead students to an understanding of the question, "What does this piece specifically say?" and thus prepare them to better answer the companion questions, "What is this piece, as a whole, about?" and "What of it?":
“Fingerprints” by Dantzig
1. What is “number sense” as Dantzig uses the term?
2. Give an example from the text of an animal demonstrating number sense, and explain in your own words HOW that example demonstrates the animal having number sense:
3. In paragraph nine, who is the author referring to when he speaks of “. . .those savages. . .”?
4. What skill is it that the author contends has allowed us to move beyond our number sense to be able to really use numbers to understand the universe?
5. What is the difference between cardinal and ordinal numbers?
6. What are the “two principles which permeate all. . .realms of thought. . .”?
7. In your own words, explain what each of these two principles (from question number six) mean:
8. What specific type of counting always precedes or accompanies any counting system?
9. What word is the most probable origin for the word for the number “five” in most languages?
10. What does the “base” number of a counting system mean?
11. What is the base number for the vast majority of counting systems around the world and throughout history?
12. Why is that (the base identified in question eleven) the base most human cultures use?
13. What evidence does Buffon give in support of the practical usefulness of a base 12 system?
14. In what way does our counting system support the idea that: “Man is the measure of all things”?
Have students answer the questions individually, and turn them in for individual scoring. Hand the papers back, and go over them in class, calling on students who have given particularly appropo, insightful, or interesting answers to share theirs in class, and as an instructor, share what distinguished those answers. Students may correct their papers and turn them in as "redos" after this activity.

##### Writing Skills

* <Choose phrases from text and express them in different ways>
* <Reconstruct a previously disarranged passage from the text>
* <Compose verses or lines in imitation of the author>
* <Change a passage or poem of one kind into another kind>
* <Imitate a passage>
* <Write a composition imitating the author>
* <Translate sentences or passages into Latin or another language>

#### Scientific Experimentation
##### Observation Skills - Estimation Skills - Calculation Skills
###### Visual Number Sense

Students will be asked to observe screen shots of sets of dots flashed momentarily before their eyes via the overhead projector. Some of the sets will be random groupings of white dots against a black background, while others will present random groups of different colored dots, and yet others will present dots in more recognizable patterns, for example on the face of one or more game die. Students will be asked to estimate/count/calculate how many of the given objects exist in specific screen shots. The purpose is for students to get a sense of how symmetric pattern reading, counting, and/or calculating augment their visual number sense to a great degree.
Students will record and turn in their own data throughout the experiment, and analyze the results in a write-up portion of the lab.

###### Aural number sense

Students will also listen to prerecorded clips of combinations of tones produced by a tone generator, and will be asked to identify the number of tones blended in the individual clips. They will then listen to the same combinations of tones, but this time with each tone added to the mix in rapid succession such that their counting skills will be enabled in determining the number of tones in a given clip.
Students will record and turn in their own data throughout the experiment, and analyze the results in a write-up portion of the lab.