# Paideia High School/Beyond the Googol

Beyond the Googol by Edward Kasner and James Newman is a Paideia Unit Plan. 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

* <Enter instructions or information about live presentation of the work.>
* <Enter information and links to quality audio or video recordings of the work.>

#### Written Presentation

* <Enter a link to Wikisource or an external link to the work if possible. Do not enter the actual work markups or links. Versions with such elements can be entered under the Interpretation section (with vocabulary markups, for example) or the Erudition section (with links to background information, for example).>
* <Enter references to available books, prints, scores, or recordings and include a reference as follows A reference template may be added to update this template later because Majrjejm is sure there's a better way but doesn't know it yet.>
* <Enter any notes about the work, whether it's in the public domain, under copyright, out of print, available through used booksellers, other vendors etc.>

### 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.. 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 worked out in detail for consideration; it should consist of about three to five terms.
* <First Term>
* <Second Term>
* <Etc.>

##### 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!
austere
monogamous
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.

#### 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 premeses 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 versus 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>

### 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

* <information relevant to anthropological issues and discoveries--include citations--delete if not used>

#### Antiquarian References

* <information about references to things no longer commonly understood because of their situation in ages past--include citations--delete if not used>

#### Archaeological References

Papyri

#### Biographical References

Edward Kasner was a professor of mathematics at Columbia University.
J.R. Newman was a mathematician and mathematical historian.

#### Cultural References

* <information and links to cultural references--include citations--delete if not used>

#### Ethical References

* <information and links to ethical references--include citations--delete if not used>

#### Geographical References

* <links to relevant maps--delete if not used>
Nile
Pisa
Woolsthorpe

#### Historical References

Chaldea
Heraclitus
Pythagoras

#### Mathematical References

Quanta
Relativity

#### Political References

* <information and links to political references--include citations--delete if not used>

#### Religious References

Exegesis
Pythagoreanism
Theologian

#### Scientific References

Nebulae

## 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. It's 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 exercised 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

* <links to additional quality recordings--include citations--delete if not used>
* <information relevant to listening for grammatical, logical, rhetorical, poetical, and stylistic elements of the work--include citations--delete if not used>

##### 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

* <read hard-copy with a pencil in hand--always!>
* <marking up a text both with a pencil and electronically>
* <inspectional reading notes as taught in Adler's and Van Doren's How to Read a Book>
* <analytical reading notes as taught in the same work>

##### 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>