Graphing Calculator Art in the Teaching of Functions(3rd International RME Conference in Boulder)

Fukui National College of TechnologyTakehiro TSUBOKAWAtubokawa@fukui-nct.ac.jp

1

What is Graphing Calculator Art like?

1. How and when to create it:

(1) draw pictures on the display by creatively utilizing functions

(2) use graphing calculators or PCs

(3) during the summer / winter vacation as an assignment

2. Aims of the Graphing Calculator Art Assignment:

(1) to motivate students to become familiar with functions

(2) to gain firsthand experience in the practical application of the knowledge of functions acquired in the classroom

2

Let’s take a look at some pieces of graphing calculator art.

Cheetah, the first prize entry in 2004 Saturn (2004)

Clock (2005)

Mt.FUJI,the first prize entry in 2005 3

Graphing Calculator Art Contest History (1)

1. 2001  Start to use graphing calculators in the classroom

(1) Math with the TI-89 graphing calculator becomes compulsory for first-year students at the Fukui National College of Technology.

(2) There are always some students who stop listening to their teachers and become absorbed in creating art with their graphing calculators. We decided to give them the graphing calculator art assignment.

2. In fact,  we got the idea of using graphing calculators in the classroom when we took part in conferences held in the US and Japan. Bert K. Waits and Franklin D. Demana’s way of thinking about Support & Confirm impressed us deeply.  We also learned that there was a college of technology in Japan which had introduced TI-83s into the classroom and the students had been given a graphing calculator art assignment.  We held a “Graphing Calculator Conference” at our school in 2001, and since then we have continued to hold it at our school every year.

4

Graphing Calculator Art Contest History (2)3. 2001  Start to hold a school-wide contest

(1) We gave our 1st year students the graphing calculator art summer vacation assignment and held class-wide contests in September. Students presented their entries in each class and they exchanged opinions and impressions.

(2) All entries were exhibited at the school festival in October and visitors were requested to vote for the best entry. Entries that got many votes were awarded.

(3) We have continued to have the school-wide contest since then.

4. 2004  Start to hold a nationwide contest

(1) We gave a presentation on our school-wide contest at the “Nationwide Graphing Calculator Conference in Tokyo.” The presentation was favorably accepted and a positive suggestion that the contest should be nationwide was made.

(2) We started to call for entries from all over the country in July for the 1st nationwide graphing calculator art contest. 5

Graphing Calculator Art Contest History (3)

5. The 1st “Nationwide Graphing Calculator Art Contest” is held

(1) There were about 800 entries from all over the country.

(2) The list was closed in December. The finalists list were made public after the first and the second screenings in January and February respectively.

(3) The first screening is done by math teachers at the Fukui National College of Technology.

(4) The second screening is done by a committee consisting of 6 members: 2 university professors, 3 national college of technology math teachers, and 1 CEO.

(5) The Fukui National College of Technology is responsible for maintaining records and other secretarial duties.

(6) Top entrants receive awards in form of certificates of merit and small prizes.

6

Graphing Calculator Art History (4)

6. The contest was renamed “the Nationwide Function Graphing Calculator Art Contest” in 2005. The word “function” was added to the original.

7. A total of 7 nationwide contests were held as of 2010.

(1) There are 15-25 schools that send entries. Some new schools join, and some old schools leave. Probably due to the transfer of math teachers who are interested in the contest.

(2) There are about 1,000 entries each contest. About 100 entries pass the first screening and about 25 of them pass the second screening.

8. March 2008  The 1st Graphing Calculator Art Conference is held

(6) Our intent is to provide an opportunity for top entrants to present their entries and describe their creative process

(7) The conference is a good chance for contest judges to ask entrants questions in person.

(8) The originality and mathematical elegance of the entries are acknowledged through the presentations.

7

Posters

8

Graphing Calculator Art Contest Application Requirements

1. Three Categories:

(1)　Category with prerequisites (only use the “y=f(x)”type of equation)

(2)　Category with no prerequisites (use functions of any kind including

parametric equations and polar co-ordinates)

(3)　Category with a theme (display the next year’s animal from the Chinese

zodiac)

2. Graphing calculator art entries are screened separate from PC art entries. Therefore the total number of categories is six.

3. The current categories were created in 2009. The PC category and the category with a theme were newly introduced in 2009. There were only two categories before then.

9

The Previous Contest Results

Year best prizespecial recognition award/

2nd,3rd best prizehonorable mention

1st/2004

2nd/2005

3rd/2006

4th/2007

5th/2008

6th/2009

7th/2010

3 entrants 6 entrants /4 entrants 15 entrants

2 entrants 3 entrants /11 entrants 11 entrants

1 entrant 6 entrants 19 entrants

2 entrants 7 entrants 21 entrants

2 entrants 1 entrants /4 entrants 14 entrants

3 entrants 11 entrants 24 entrants

2 entrants 3 entrants 15 entrants

10

Graphing Calculator Art Conference

1. The 1st Graphing Calculator Art Conference was held in March 2008 at the end of the 2007 academic year

(1) The 1st conference ( held in Fukui in March 2008)      16 presentations

(2) The 2nd (held in Tokyo in March 2009)      11 presentations

(3) The 3rd          (held in Tokyo in March 2010)       14 presentations

(4) The 4th          (held in Nagoya on March 11, 2011)   20 presentations

2. Conference Agenda:

(1) Award winners (10-20) who passed the 2nd screening present their entries.

(2) They are supposed to describe their creative process including how functions were employed, where they had difficulty, and what they learned through this activity.

(3) All the participants are involved in evaluation of the entries and selection of the best one.

11

The 4th Conference held on March 11, 2011 (videotaped)

1. As you know, March 11, 2011 was the day on which the Great East Japan Earthquake occurred. We felt an earthquake during the conference, but we continued it to the end.  Some students from Tohoku District could not go home that day. It was a week later when they were finally able to go home.

2. Activity Sample “A Road Sign of Kangaroo” created by Junichi Yamaguchi, a 2nd year student at the Fukui National College of Technology.

12

13

Graphing Calculator Art and Mathematics (1)

1. Functions that students often use are as follows:

(1) Polynomial Functions

(2) Power Functions

(3) Fractional Functions

(4) Irrational Functions

(5) Trigonometric Functions

(6) Exponential Functions / Logarithmic Functions

(7) A combination of some functions mentioned above

14

Graphing Calculator Art and Mathematics (2)

2. Mathematical techniques that students often employ are as follows:

(1) Setting a Domain

(2) Adjusting the Range (or Co-Domain)

(3) Translation of Axes (or Parallel Displacement)

(4) Symmetric Transformation

(5) Dilatation

(6) A combination of some of the techniques mentioned above.

15

Graphing Calculator Art and Mathematics (3)

3. Students do various kinds of calculations:

(1) to connect a line or curve to another one

(2) to find the most efficient translation of axes

(3) to create perspective or draw subtle shadows

(4) to get the gradient and intercepts of a line that is fit for the intended purpose

4. The calculations mentioned above show that students independently do calculations for problem-solving through the activity. These calculations are different from tedious algebraic-manipulation practice. Some skeptics suggest that calculator use prevents students from developing calculation skills. Such is not the case in our graphing calculator art contest.

16

Connection with Math Classes

1. Graphing calculator art is not in the regular curriculum.

2. The graphing calculator art assignment requires functions 1st and 2nd year students have learned before the assignment is given. The assignment helps deepen their understanding of these functions.

3. The motivation behind the assignment is not to provide an answer to a stale question posed by a teacher such as “What kind of graph do you think this equation represent?” but is rather the students’ urge to draw a picture.

4. Students advance beyond their classroom learning on occasion and employ function they have not yet learned.

5. A few students have independently attained advanced mathematical skills such as interpolation.

17

The Future of the Graphing Calculator Art Contest1. There is a growing number of entrants who create entries for the contest

using PCs or tablet PCs. As a result, it is becoming difficult to screen those entries based on the current criteria for evaluating graphing calculator art entries.In Japan in-class computer usage has been spreading quickly across the country, while in-class graphing calculator usage is hardly spreading.

2. Graphing calculator art creation develops creativity and originality. Nurturing these traits is one of the main purposes of mathematical education. This activity could contribute to complementing passive curriculums most schools are now using.

3. Our ambition is to evolve the nation-wide contest into a world-wide one. The English website is crucial in this attempt. Your understanding and help would be greatly appreciated. A selection of entries can be seen on the website below:http://www.ge.fukui-nct.ac.jp/~math/graph_art/

18

In conclusion I would like to present recent awardwinning entries.

Marionette (2009)

Snowman (2009)

Euler’s dream (2009)

Kendou (2009)

Hinamaturi (2009)

19

Crane (2010) Hayabusa falcon (2010)

Eiffel Tower (2010)

There are rabbits on the Moon (2010) Crayfish (2010)

20

Enjoying the moon (2010)

A morning glory (2010)

The javelin throw (2010)

Daruma (2010)

The cosmos (2010)21