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Yuwen Li
Mathematics Education Research Institute, Dezhou College, Dezhou, Shandong 253023, P.R.China
Abstract: How to promote creativity for all students in mathematics education is always a hot topic for mathematics educators. Based on the theory study and practice in the project "Open Mind Questions in Mathematics" granted by Ministry of Basic Education Curriculum Study Center in China, the paper reported the effect of "Open Mind Questions in Mathematics" on the way to change the development of thinking ability, to inspire students to develop Thinking flexibility, to expand their imagination, to stimulate their interest in learning, and to foster students creativity.
Keywords: Open Mind Questions in Mathematics, Quality Education, Stimulate Interest, Creativity
Introduction
I attended the International Congress on Mathematical Education held at the Shanghai Normal University during Aug. 15 21, 1994. I was deeply impressed by Dr. Pingtiangensans presentation and discussed with him in detail after his presentation. We exchanged our latest books after the discussion. There are many interesting topics in his book. I introduced some topics in his book For Primary School Teacher into Chinas Primary School Mathematics Education Community and caused a lot of attention.
The Basic Education Curriculum Center in China provided me a grant for studying Open Mind Questions in Mathematics for Primary School Students. As the leader of Mathematics Education Research Institute in Dezhou College, I focused on the open mind mathematics questions study for more than two years. As a reward, we published two books titled Mathematics Questions for Inspiring Interests in Mathematics. One book is for primary school students from 1st grade 1 to 3rd grade, published in 2004. The other book is for primary school students from 4th grade to 6th grade, published in 2005.
Requirement for Open Mind Question (OMQ) Study
My advisor Dr. Dianzhou Zhang presented me one of the latest edition of his book Mathematics Quality Education Design when I visited him in summer 1996. There is one paragraph impressed me a lot, which talks about how to use open mind questions to promote creativity in mathematics and gives some examples of open mind questions used in middle school mathematics education in foreign countries. For example, If the distance between A and the school is 5,000 meters, the distance between B and the school is 10, 000 meters, what is the distance between A and B?
This question appears as simple as an arithmetic question for primary school students. However, it actually include many contents such as adding nature numbers, subtracting rational numbers, geometry orbit of a circle, point distance, circle parameters, complex number subtraction and etc. The question is open mind and also can be calculated. The conditions for this open mind question can be freely added according to your mathematics knowledge. The open mind questions usually leave a lot of imagine space for readers. At the same time, they can also actively involve the readers and inspire them thinking from different points of view.
On the contrary, the mathematics questions in China are restricted to exam questions like mathematics questions in College Entry Exam and all kinds of Mathematics Competition exams. The questions in those exams often have common characteristics such as sufficient conditions, specific process and unique conclusions. This type of questions is necessary for mathematics education, but it is not enough to promote students creativity and open mind thinking style.
As Dr. Zhang said, in the 1980s, the basic science education attracted a lot of attention with the development and application of science and technology in Japan. They paid a lot of attention to its own creativity instead of just imitate the West in science and technology. In education, they highly advocate creative thinking training and promote creativity. The emergence of Open Mind Questions is the natural fruit of this kind of development.
The economy in China developed greatly after the reform and open policies took action. The talent competition is very fierce in the market, which gives new requirements for educators. In 1990, China proposed Quality Education. The Quality Education marked the new era of Education in China. With the development of teaching reform in education, the reform of education material is natural based on the requirement of talents who can express their opinion and have creativity ability. In this situation, the study of open mind questions is very necessary.
I mailed our newly published book Stimulate Students Interest and Enthusiasm in Mathematics Class to Zaiping Dai, who replied to me expressed his satisfaction and encouragement. He thought the study for Open Mind Questions is valuable for Quality Education especially for promoting creativity.
Studies in Open Mind Question
One characteristic of Open Mind Question is multiple answers for the same question. To have perfect answers for Open Mind Questions, one need to both master the knowledge from textbook and thinking actively and imagine boldly. Therefore, open mind questions are helpful to promote students creativity and make them smarter.
An example of Open Mind Question: There are many square cards with two different strips like the following. Take 12 cards from them and see what kind of closed graphics can be made by the strips on those cards?
Answers:
For example
First, we should let our students understand that the length of strip in each of those cards is fixed. Thus, it is an activity of making a closed graphics using square cards with same length strips. As figure1 shows, the strip in card A is a straight line. Assume the length of strip on A is one unit, then the acute angle of strip on card B is composed of half unit + half unit = one unit. According to above reasoning, we will know that the strip length on card A is the same as the strip length on card B. It also means the strip length is always one unit no matter which card you are using. The circumference of the closed graphics is fixed value 12 when you use 12 such cards to make a Closed Graphics. However, the areas are different when the closed graphics are different. Alternatively, the closed graphics could be different even the areas are the same. The purpose of this activity is making students realize the difference.
Figure1. Two square cards with same length strips
In addition, it is necessary to let students have the ability to distinguish closed graphics and unclosed graphics for 1st grade students at the beginning of teaching. Figure2 shows examples of closed graphics and unclosed graphics.
closed graphics unclosed graphics
Figure2. Two examples for closed graphics and unclosed graphics
These handson activities not only increased students interest in learning mathematics but also made them realize many mathematics principles. These activities will greatly benefit the development of students mathematics thinking skills.
Another open mind question for 2nd grade student:
Please fill in appropriate numbers in % and do the calculation.
%%
+%%
%%%
It is a question about adding two doubledigit numbers with the result of a tripledigit number. There are 8,100 formulas if you list all the sum of two doubledigit numbers from 10 + 10 to 99 + 99 . Among those formulas, there are 4,860 formulas can get a tripledigit number results from 10 + 90 = 100 to 99 + 99 = 198.
If the students can write down all of those formulas, they actually practiced the sum calculation many times without awareness.
When listing those formulas, the students can list all of them without losing one if they order those formulas by certain criteria.
The activity itself is a mathematics activity. At the beginning, students might list those formulas at random. However, they will find the law gradually.
Students generally find many formulas such as 10 + 90, 11+89, , 90+10 etc. They all have the same answer 100 in the formula. Some students can find the law after listing a lot of formulas: Fix the first number in the additive formula, from 10 + 90, 10 + 91, 10 + 92, , 10 + 98, 10 + 99 to 89 + 11, 89 + 12, , 89 + 98, 89 + 99, the number of all types formulas are 10, 11, , 89. From " 90 + 10, 90 + 11, , 90 + 98, and 90 + 99 " to " 99 + 10, 99 + 11,, 99 + 98, and 99 + 99 ", the number of formulas are all 90 for those formulas belong to these 10 categories. Therefore, it is easy to get the total number of all formulas:10+89 8029010=4860.
To answer this type of questions, students usually can find out some formulas that satisfy the condition. At this time, if teacher can ask them to find out the total number of formulas that satisfy the condition and list all of them, students can think harder from different point of view and apply the classification theory and the knowledge about serial adding unwittingly.
Another example for 3rd grade students: Team Marathon
Table 1 is the result of team marathon with four persons in each team.
Table1. The result of team marathon
Rank of team membersTeam A 471013Team B 261214Team C18916Team D 351115
Please give the rule for determining the rank of each team.
Different rules will give different ranks for each team. The purpose of this activity is applying the arithmetic knowledge to figure out the determining rule.
For example, if we use the best rank in each team as the rule for ranking the teams, then team C ranks first, team B ranks second, team D ranks third and team A ranks fourth. If using the sum of all players ranks in each team as the rule for ranking the team, then these four teams have the same rank, and so on.
Finding out the answers from different point of view, one can think out many ways to determine the ranks.
Conclusions and Future Work
For a long time, the exercises in mathematics textbooks are designed for the purpose that students can remember the mathematics theorems and conclusions, and familiar with the arithmetic process. Therefore, students only focus on remembering things instead of involving in the learning process<=# P Q Y Z  ~
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dhWD` and thinking critically. From the examples of those Open Mind Questions, it can be seen that designing the Open Mind Questions according to the age and cognitive level of primary school students can greatly inspire their wisdom, improve their critical thinking and reasoning skills, challenge their imagination, stimulate their interest in learning mathematics and foster their creativity. In the future, we will continue our efforts in the study of Open Mind Questions in mathematics and continue make our contribution to the development of mathematics education in China and around the world.
References
Dianzhou Zhang, Mathematics Quality Education Design (1996), Jiansu Education Publisher.
Dianzhou Zhang, Mathematics Quality Education Fine Plans (2000), China Youth Publisher.
Yuwen Li, Mathematics Questions for Inspiring Interests in Mathematics (2004), Beijing Normal University Publisher.
About the author
Yuwen Li
Dezhou College
Dezhou, Shandong 253023
P. R. China
Email: lyw_534@163.com
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