japanese culture, paper- folding and logarithmic … culture.pdfusing origami and paper folding to...

Japanese Culture, Paper-Folding and Logarithmic Spirals

A Trifecta for Learning!

Nick Weldy, Ph.D. and Richelle Zbinden, NBCT, Ohio Master Teacher Presentation for Ohio ACTE Conference 2013

The Miami Valley Career Technology Center is located in Clayton, Ohio just North of Dayton. We have approx. 1700 high school students from 27 different associate schools and serve a five county region. These students have the opportunity to chose from more than 40 career-technical programs. In addition, we serve approx. 5500 adult students through, long and short term classes, ABLE/GED courses, workforce development testing, and other opportunities.

Quick Facts About Japan

Government: Parliamentary with Constitutional Monarchy

Population: 127,078,679

Education System: National (prefectural)

Literacy Rate: 99%

Unemployment Rate: 4%

The Japanese Educational System

The school structure in the Japanese education system is segmented. 6 years of primary or elementary school (including 2 years of Kindergarten); 3 years of middle or junior high school; 3 years of high school; and 4 years of university.


A school year has three terms: summer, winter and spring, which are each followed by a vacation period. The school year begins in April and ends in March of the following year.

Nine years of schooling are considered compulsory. (elementary school 6 years and junior high school 3 years)

School attendance rate for the nine years of compulsory education is 99.98%.


The school system was based upon the American system of education and was instituted in 1947 during the U.S. Occupation.

The average size of a Japanese classroom is 35-40 students across all grade levels.

Children learn early on (beginning in preschool) to maintain cooperative relationships with their peers; to follow the set school routines; and to value punctuality (from their first year in elementary school).

Classroom management emphasizes student responsibility and stewardship through emphasis on daily chores such as cleaning of desks and scrubbing of classroom floors. (Constructing their own desk)

Students are encouraged to develop strong loyalties to their social groups, e.g. to their class, their sports teams, their after-school circles, e.g. baseball and soccer teams. Leadership, as well as subordinate roles, are learned through assigned roles for lunchtime, class monitor or class chairperson and other such duties.


The elementary school curriculum covers Japanese, social studies, mathematics, science, music, arts and crafts, homemaking and physical education. At this stage, much time and emphasis is given to music, fine arts and physical education.


Noisy and lively classrooms, the absence of teacher supervision, and the effective use of peer supervision are noted in elementary school classrooms. Homework workload is not overly heavy at this stage, daily portions typically comprise kanji (Chinese characters) or kokugo (Japanese language) worksheets and one or two pages of arithmetic worksheets. Various after-school hamako, or club activities, or remedial classes may be held by individual home-room teachers (or schools) as they see fit.

High schools may be classed into one of the following types:

Elite academic high schools collect the top of the student population and send the majority of its graduates to top national universities.

Non-elite academic high schools prepare students for less prestigious universities or junior colleges, but in reality send a large number of their students to private specialist schools, which teach subjects such as book-keeping, languages and computer programming. These schools constitute mainstream high schooling.

Vocational High Schools that offer courses in commerce, technical subjects, agriculture, home economics, nursing and fishery (to name a few). Approximately 60% of their graduates enter full-time employment.

Correspondence High Schools offers a flexible form of schooling for 1.6% of high school students usually those who missed out on high schooling for various reasons.

Evening High Schools which are used to offer classes to poor but ambitious students who worked while trying to remedy their educational deficiencies. But in recent times, such schools tend to be attended by little-motivated members of the lowest two percentiles in terms of academic achievement.



Juku and Exams

High school environment shifts the student to a lecture-centered and systematic learning mode which is alternatively lauded for its high levels of achievement in math and science and criticized for its monotony and lack of creativity during a time geared towards competitive examinations when an intensive selection process occurs. From middle-school to high school years, students are affected more by the after-school activities and juku culture. 59.55% of middle-school students attend juku usually the large-scale cram school chains

Japan has about three million students enrolled in 1,200 universities and junior colleges and consequently the second largest higher educational system in the developed world. Japan also has one of the largest systems of private higher education in the world. The 710 odd universities in Japan can be separated into 3 categories: highly competitive, mildly competitive and non-competitive (the schools that are first-tier being the infamously difficult to enter ones). Public universities are generally more prestigious than their private ones with only 25 percent of all university-bound students being admitted to public universities.

More than 65 percent of high school graduates continue their studies; of these, over 70 percent are enrolled in private colleges and universities. Only about 10 percent of private institutions receive their financial resources from public funding, with most public funds on higher education being spent on the national and local public universities. Despite the impressive statistics, Japanese universities are considered to be the weakest link in the country's educational system.


Facilities Overall, Japanese educational

facilities are older but very well kept.

They are in the process of remodeling some older schools.

Student life Students are extremely busy. They start at 7:30-

8a and stay at school until 4-6p.

At the end of school, ALL students clean the school.

Immediately after cleaning, students participate in club activities until 7-9p every night.

Homework levels steadily increase through grade-levels and peak in high school

Teacher’s Life

Teachers must take a national exam before being allowed to have students. The exam, depending on grade level, consists of a swimming, piano, and dancing test, in addition to rigorous academic assessments.

The test takes a minimum of three days to administer.

Only about 40% of teachers pass the test on the first attempt.

Many teachers spend all day Saturday working individually with students or making home visits.

Teachers eat lunch with students in their classroom everyday.

In Japanese society, the teacher rules. They are called Sensei, a very highly regarded title. They tell the parents what to do in many schools.

Unconventional Ways of Learning Math

Using Finger Paints Part 1 I taught this to three classes of students who are enrolled in a course that fits between Algebra 2 and Pre-Calculus. The scope of the course is an in-depth study of functions and trigonometry.

Teaching Inverse Functions Using Finger Paints

Prior Knowledge :

Students have learned how to find:

the inverse of a function by using algebra , paper and pencil.

to use the Horizontal Line Test to predict if a graph’s inverse is a function

to find the domain and range of a function and it’s inverse

It is important to let children play when introducing manipulative to a lesson.

So our lesson began with a simple drawing where students painted on one side of the diagonal of a square. We then folded along the diagonal and squished the paint so that it appeared on both sides. Thereby planting the seed of graphing an inverse.

We also had a discussion on how our fold could represent the line of the equation y = x.


This helped to reinforce the idea of a reflection. It also got the doodling out of their system so they could focus on the purpose of the lesson


Next, I put three equations on the board. Two were quadratic and one was a cubic function.

Each student choose an equation to graph on a Cartesian coordinate system. Then, they determined the inverse, domain and range of each function.

Last, they used finger paints to graph their original function and then folded the paper to graph the inverse.


Examples of each of my classes’ work:

Paper folding, spirals, logarithms, and a touch of Archimedes Part 2

This was used in a Pre-Calculus. The scope of the course is an assortment of abstract concepts that includes: Polynomial Functions, Sequences & Series, Logarithms, Trigonometry, Conics and Parametric Equations.

Sharing Resources

Using Origami and Paper Folding to Explore Spirals

Prior study:

Students have learned:

Sequences & Series

Exponential and Logarithmic Functions

This was a great activity that culminated what we have been studying for the past month and gave a hint to where we are heading with a touch of right triangle trigonometry.

To explain what origami is and why it’s so cool to do as a mathematician, we watched a TED tv video.

Then we created the Wheel of Theodorus by breaking the rules they just learned.

C:/Users/Zbinden/Dropbox/RobertLang_2008-480p.mp4


This introduced the students to mountain and valley folds, as well as how to make crisp creases.

We began with a 1 inch wide strip of paper. The students then needed to use Pythagorean Theorem to determine the length of every side of each right triangle formed in their spiral and determine if there was a pattern.

Then, we did a study on spirals and determined that this is not an example of a Logarithmic Spiral because the ratios of successive radii are not constant even though they appear to be approaching a limit.

The trigonometry came in when they had to determine the interior angle of each right triangle formed in the spiral.


The next day we constructed the three dimensional spiral designed by Tomoko Fuse. Students were required to determine if this is an example of an Archimedean or Logarithmic spiral based on their own research and data collection. They defended their findings in a one page write-up that included these points:

An explanation on how each student created the shape and conducted their research so that someone could recreate the event.

The measurements for each ‘ring’ of the spiral.

How they determine if the origami shape represents an Archimedean spiral, Logarithmic spiral, neither or both.

What would each student do differently if they were to conduct this activity again?

The Students’ Research Found it to be a Logarithmic Spiral

Student statements:

“You may wonder whether this is considered an Archimedean spiral, Logarithmic spiral, both or neither. To determine this, see whether the measurement between each measurement is staying the same or increasing. Since our spiral is increasing in ring measurement, it must be a logarithmic spiral. If it were an Archimedean spiral, the measurement between rings would consistently be the same.” –Julia Anspach, Architectural Engineering, Brookville Junior

“The shape is a Logarithmic spiral because the ratios of the successive radii are equal. The ratio is 1:2, because the length of each radii is 1 cm, 2 cm, 4 cm. If you calculate 1 divided by 2, you get .5, and if you calculate 2 divided by 4, you get .5. Therefore, the ratio is 1:2 and it is Logarithmic.” - Autumn Wright, Allied Health Tech. Prep, Milton Union Junior

“If you look at the ratios of the successive rings, you notice that they are all very close. From this information, it can be concluded that this is a logarithmic spiral because the definition of a logarithmic spiral is that the distance between the ring and the center increases exponentially, thus the ratio of successive rings would be the same.” - Simon Denlinger, Precision Machining, Franklin Monroe, Junior

“Due to this consistent geometric change in value, this spiral is a logarithmic spiral. A logarithmic spiral is a spiral where the distances from the center of the spiral form a geometric sequence. Because the sides of this spiral are not evenly spaced, this is not Archemedean spiral.” - Brad Ratzel, Early Childhood Education, Northmont Junior

Pictures were used as a visual argument by Holden Van

Houten, Digital Design, home schooled

Cultural Significance of the Crane

We took time during class to discuss why the crane is an icon to origami and to the Japanese culture.

Students created cranes in class and were issued a challenge over Thanksgiving Break

Smallest

Largest

Creatively hand decorated

The Results:

1,000 Cranes of Peace (well, not quite 1,000)

Each student decorated one side of a square piece of paper with the theme: “A Day In The Life Of An American Teenager”

We then folded them into cranes and sent them to Japan for the students to see the art work drawn on the cranes.

We’re hoping to get something back!

In Conclusion

Questions?

Contact Information: Nick Weldy: [email protected], 937-854-6272 Richelle Zbinden: [email protected], 937-854-6428

japanese culture, paper- folding and logarithmic … culture.pdfusing origami and paper folding to...

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