# How to Solve It

“How to Solve It” (first published in 1945) is a small volume by mathematician George Pólya describing methods of problem solving.  The book suggests the following steps when solving a mathematical problem:

1. Understand the problem.
2. Devise a plan to solve it.
3. Carry out the plan.
4. Revise and extend: look back on your work.

https://en.wikipedia.org/wiki/How_to_Solve_It

https://en.wikipedia.org/wiki/Heuristic

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# Yonkoma

From Wikipedia:

Yonkoma manga (4コマ漫画, “four cell manga” or 4-koma for short), a comic strip format, generally consists of gag comic strips within four panels of equal size ordered from top to bottom.

And also:

Traditionally, yonkoma follow a structure known as kishōtenketsu. This word is a compound formed from the following Japanese kanji characters:

• Ki (起): The first panel forms the basis of the story; it sets the scene.
• Shō (承): The second panel develops upon the foundation of the story laid down in the first panel.
• Ten (転): The third panel is the climax, in which an unforeseen development occurs.
• Ketsu (結): The fourth panel is the conclusion, in which the effects of the third panel are seen.

https://en.wikipedia.org/wiki/Yonkoma

https://en.wikipedia.org/wiki/Kish%C5%8Dtenketsu

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# The Golden Ratio

Here are shown various ways to write the Golden Ratio, a favorite theme of mathematical recreation and investigation:

• The equality of two ratios of simple algebraic quantities in two unknowns
• The infinite limit of the ratio of successive Fibonacci numbers
• In an infinite continued fraction representation
• As a specific irrational number involving the square root of five (but cross multiply and 4 = 4 is explicit)

These instances remind me a bit of the Four Causes in their representations:

• Relative and relational, but indefinite (Efficient)
• Idealized as an infinite sequence and infinite completion (Formal)
• Infinitely nested and recursively reductive (Material)
• Definite but with embedded irrationality (Final)

https://en.wikipedia.org/wiki/Golden_ratio

https://en.wikipedia.org/wiki/Fibonacci_number

https://en.wikipedia.org/wiki/Irrational_number

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# Four Dimensions of Knowledge

What is the best way to educate, to teach and learn? Ideally, students shouldn’t merely memorize facts and recall them on demand, although retaining well accepted knowledge is important. Certainly, students need mental structures to organize these facts, so that they form associated groups of categories and classification. Additionally, methods are needed for accepting and rejecting facts, and procedures for organizing facts, although facts may often be revisited for truth, or to reorganize them, and so on.

Benjamin Bloom et al. developed a taxonomy for educators of six mental aspects for the acquisition of knowledge, which was revised later into six cognitive actions or processes by L. Anderson, B. S. Krathwohl and others. These six actions form a sort of food pyramid for knowledge, so that lower actions form a broader base for higher ones. Both taxonomies sort a dimensional hierarchy of knowledge operated by these aspects or actions from concrete to abstract: Factual, Conceptual, Procedural, and Metacognitive (the last added when revised).

In the revised taxonomy, “Remember” is the lowest cognitive action, described by “remember facts and basic concepts,” which deals with the Factual and Conceptual. Above it is “Understand,” described by “explain ideas or concepts,” and I imagine an idea can be a fact. Next is a procedural action “Apply: use information in new situations,” and in fact all six actions are procedural by being actions. At the top of the pyramid is “Analyze,” “Evaluate,” and “Create”. The Metacognitive dimension (“thinking about thinking”) is for thinking about these six actions and these four dimensions, as to how they are related and differ.

These taxonomies are well considered and there are many resources to investigate.

http://edtheory.blogspot.com/2016/03/knowledge-dimensions-factual-conceptual.html

Bloom’s Taxonomy

Revised Bloom’s Taxonomy

Bloom’s Taxonomy Revised

https://www.goconqr.com/en/mindmap/12973371/four-types-of-knowledge

Pedagogy of book and chapter organization

Click to access krathwohl.pdf

Anderson, L., Bloom, B. S., Krathwohl, D., & Airasian, P. (2000). Taxonomy for learning, teaching and assessing: A revision of Bloom’s Taxonomy of Educational Objectives (2nd ed.). New York: Allyn & Bacon, Inc.

Anderson, L. W., Krathwohl, D. R., Airasian, P. W., Cruikshank, K. A., Mayer, R. E., Pin- trich, P. R., … & Wittrock, M. C. (2001). A taxonomy for learning, teaching, and assessing: A revision of Bloom’s taxonomy of educational objectives, abridged edition. White Plains, NY: Longman.

https://en.wikipedia.org/wiki/Metacognition

https://en.wikipedia.org/wiki/Concept

https://plato.stanford.edu/entries/concepts/

Concepts, Theory-Theory of

https://en.wikipedia.org/wiki/Cognition

https://plato.stanford.edu/entries/innateness-cognition/

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# Yojijukugo

A Yojijukugo (四字熟語) is a unit of four kanji characters that usually represents an idiomatic saying in Japanese. It is itself a yojijukugo, even though it isn’t idiomatic, since the term can also broadly refer to a non-idiomatic phrase of four characters.

Shunkashūtō (春夏秋冬) is a nice one that means the four seasons of the year, and so is a fourfold written in four kanji characters.

Eshajōri (会者定離) is hopefully appropriate, meaning “every meeting must involve a parting”.

https://en.wikipedia.org/wiki/Yojijukugo

https://en.wikipedia.org/wiki/Shunkash%C5%ABt%C5%8D

https://www.tofugu.com/japanese/yojijukugo-idioms/

https://www.nippon.com/en/nipponblog/m00111/

https://en.wikipedia.org/wiki/Japanese_proverbs

And these are just interesting:

https://en.wikipedia.org/wiki/List_of_kanji_by_stroke_count

https://gogonihon.com/en/blog/japanese-characters

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