Raise of Primary Students’ Mathematic Ability
- Observing the objects of experiment: According to the teaching conditions of teachers, for the convenience of controlling experimental operation and experimental variables, the school chose (A) class of grade one as experimental controlling class, which is grade six at present; (B) class of grade one as the experimental class of “The Combination of Three Types of Arithmetic”, which is grade six at present. These two classes are observing the objects of experiment.
- Control of variable: The students are distributed to experimental class and corresponding class in random when entering school, so the numbers of students in these two classes are the same. School prescribed that students can not been transferred out or into other classes during the experimental period. Literature and mathematic teachers are arranged according to their education?teaching period? teaching achievement?age close and the arrangement of other subjects are similar. Corresponding class applies conventional teaching method and measures without any control while experimental class applies the teaching material of “The Combination of Three Types of Arithmetic” edited by China National Institution for Educational Research.
- Research materials: (1)
*Examination and Evaluation of Primary Students’ Mathematics Ability*(5) for grade six, edited by Mr. Zhao Yuchun in chief; (2)*Test of Our Children*(6), edited by Mr. Zhang Shiquan; (3)*Mansa The IQ Test Book*(7), written by Philip Carter 0Ken Russell (England), translated by Wang Fan and Men Rui. - Research procedures: Test the above students separately with single desk for a person to collect data. The Educational Research Department trained the examined teachers professionally to help them strictly implementing the examining tool’s operating requirements of instruction and time management etc.
Through six years’ experimental contrast, the mathematic ability of students in experimental class is raised rapidly and obviously. It is well-known that knowledge does not equal to ability but they have close relationship. First: ability is the premise of mastering knowledge; Second: ability embodies in the process of mastering knowledge; Third: ability is developed on the base of knowledge and it gradually come into exist with the generalization and widespread the transfer of knowledge. Therefore, students’ mathematic ability can be examined through knowledge. Mathematic ability is one of the necessary basic abilities in students’ studying activity. Mr. Cruqski, the psychologist of former Soviet Union considered that mathematic ability shall include nine components of the ability of formalizing mathematic materials and operating with relation and connection?the ability of summarizing mathematic materials?the ability of operating with mathematic and other symbols?logical reasoning ability?simple thinking ability?the ability of inverting psychological process?free thinking ability?numbers memory and spatial concept etc., among which, the summarizing ability is the core of mathematic ability. Mr. Lin CongDe, the psychologist of our country, considered that mathematic ability is a multi-side?multi-level?multi-pattern and multi-connection system, which consists of three abilities of operating ability?logical thinking ability and spatial imaginary ability and five thinking types of agility?flexibility?profundity?originality and criticalness. From the examining condition of mathematic ability in the two classes, experimental class is superior to corresponding class in number concept as probability and projection. The total score in experimental class is 146, which is 62 higher than that in corresponding class of 84 (The contrast is very obvious). The standard difference in experimental class is less than that in corresponding class, which indicates that the scores in experimental class are even and outstanding. The number concept as the relation between fraction and percentage?the multiple of fraction?the operation of multiply-divide of fraction in experimental class is superior to that in corresponding class. The total score in experimental class is 160, more than twice of that in corresponding class of 75, which indicates that the students in experimental class have stronger ability on number concept. In the terms of number concept as reasoning operation and application problem of quantitative relations, the total score in experimental class is 308, and is 100 score higher than that in corresponding class of 209. Students in experimental class have stronger ability of analyzing?solving and reasoning problems, in which aspect, there is obvious difference between two classes. In the terms of basic abilities as series reasoning? estimating?analyzing summarizing and operating mathematic rules, the score in experimental class is about twice of that in corresponding class. In the terms of spatial relationship as finding concealed graph?rotating dimensional graph and counting the surfaces of cube, students in experimental class have great advantage and outstanding achievement, which indicates that the teaching strategy of experimental material of “The Combination of Three Types of Arithmetic” has obvious effect. With it, children’s right brain is exploited and their imaginary thinking is developed. (As shown in Table 1) Research result reflects that there is obvious difference between experimental class and corresponding class, among which, the most obvious difference is basic ability and spatial relationship for the average score in experimental class is about twice of that in corresponding class, which indicates that the experiment of “The Combination of Three Types of Arithmetic” is a good method to cultivate students’ mathematic ability. Long-term reckoning abacus with two hands four fingers? mental abacus calculation and oral calculation is favor for the development of their right brain and good cultivation of their imaginary thinking.
- Introduce abacus computation from grade one and explore the inner relationship among oral arithmetic?written calculation and abacus computation as much as possible to make them promoting each other with their own particular advantages and operating alternatively in order to promote the raise of students’ calculation ability. The abacus computation is introduced from grade one, which makes primary students learning oral arithmetic?written calculation and abacus computation with a “up one and down four beads” small abacus while reading?recognizing? writing numbers and reckoning. Abacus is not only the important teaching tool for teachers but also the main learning tool for students. With abacus as direct teaching tool, depending on the iconicity of abacus computation, teachers can help students forming number concept?understanding number’s composition and partition and getting familiar with finger method combining with the reckoning. When learning the reading and writing of multi-digit numbers, the abacus characteristic of clear files helps students understanding the relationship between digits and number, which is easy to learn and has low error rate, and moreover, it can helps students of grade one to master the recognition of numbers within ten thousand (the recognition of numbers within ten thousand is the teaching content in grade two prescribed by general teaching material.) When students learning decimal addition and subtraction, it is difficult for students of grade one and it has high error rate. But with the help of abacus, students can see and touch the operating process, so they can master them quickly and correctly. Its principle is to make the theory and operation of abacus computation in accordance with that of oral arithmetic and written calculation without applying the pithy formula; “reckoning in” means addition and “reckoning out” means subtraction, with which the result can be gotten easily. Promote the oral arithmetic with abacus computation and also promote the abacus computation with oral arithmetic to make the students easier to master the oral arithmetic of sum as 5 (1 and 4, 2 and 3) and five couples of complementary numbers of sum as 10 (1 and 9, 2 and 8, 3 and 7, 4 and 6, 5 and 5), which is good for learning abacus computation. When learning addition and subtraction of “numbers within 100”, with the common rule of adding and subtracting after aligning the same digits?learn the integer of ten-digit addition and subtraction?integer of ten-digit addition and subtracting with one-digit number?two-digit number adding and subtracting with one-digit number or ten-digit integer?two-digit number adding and subtracting with two-digit number. Students can reason the addition and subtraction of two-digit numbers easily according to the abacus-based operation of addition and subtraction of one-digit numbers, with which, abacus computation can promote and examine the written calculation and “three arithmetic” inter-promote each other; at the same time, it embodies students’ studying initiative.
When learning multiplication, accumulative adding of the same numbers with abacus to make students understand the origin and significance of multiplication. Through editing multiplication pithy formula, make students mastering and operating 81 sentences of “Great 99” multiplication time table, which largely raises the ability of oral arithmetic. When learning division, through visual demonstration and abacus computation of one number accumulatively subtracting the same numbers, make students to understand the origin and significance of division easily. First teaching the inclusive division (double division) and later teaching the equal division; students can easily embody the significance of these two types of division with their contrast. The operation of quotient division is in accordance with that of oral arithmetic and written calculation. Students can first learn written calculation and then learn abacus computation. These two calculations can easily inter-examine and inter-promote with each other. - The basic skills of calculation can gradually transfer to perfect skills in the training opportunities. In the teaching of “The Combination of Three Types of Arithmetic”, for abacus computation raises the calculation speed and increases the quantity of applying problems, the training time is increased. Abacus, as a learning tool, reduces students’ mental burden. Moreover, training types are various, such as abacus-beads calculation through viewing, abacus-beads calculation through listening, mental arithmetic through viewing, mental arithmetic through listening, abacus-beads translating to numbers and numbers converting to abacus-beads etc., which greatly stimulates students’ studying interest and makes the classroom teaching atmosphere active. Moreover, it makes students’ cerebral cortex often keep exciting status and high studying mood and raises the quality of classroom teaching perfectly.
- Teaching of “The Combination of Three Types of Arithmetic” can fully embody the subjectivity of students. “They who know the truth are not equal to those who love it, and they who love it are not equal to those who delight in it.” Primary students are curious and sportive by nature, especially their hands, and they can not concentrate on some aspect for a long time and often need changing content and mode of activity, which is the performance of their age characteristics. Teaching of “The Combination of Three Types of Arithmetic” satisfies primary students’ natural disposition, especially the abacus, not only as a good teaching toll, but also as a learning tool in accordance with children’s age characteristics, even as their favorite toy. In the classroom teaching, through eye watching?ear listening?mouth reading?hands dialing?brain thinking?pencil writing etc. primary students’ multiple sense organs make their movement at the same time and are used alternatively, which makes them under game status. In the studying practice, students can recognize and discover some things in regularity through analysis and compare, for example, carry one when excess-ten?decompose one when insufficient ten?empty-bead means “0” etc. As long as mastering the “carry addition” and “decomposition subtraction” of one-digit numbers, students can learn “carry addition” and “decomposition subtraction” of the numbers within hundred, even they can summarize the rule of written calculation and abacus computation of addition and subtraction of long numbers. Through self practical activity, students can discover new things they do not know before and master new knowledge easily, which maximum promote students’ activity and stimulate them to play their subjectivity during the studying process. Therefore, teaching of “The Combination of Three Types of Arithmetic” changes the situation of students learning in passiveness and helps them from “those who know it” into “those who delight in it”.(9)
- Teaching of “The Combination of Three Types of Arithmetic” complies with students’ law of cognition, from concrete to abstract and sensibility to rationality and promotes the knowledge structure transferring to the cognition structure. It increases practical operative opportunities and changes the former classroom teaching structure of teachers’ lecturing and students’ listening?teachers’ demonstrating and students’ watching into the new teaching structure of teachers and students acting together.
The direct-viewing function of abacus instructs students processing practical reckoning operation and helps them forming number concept. For example, to recognize “3”, first to know three books, three ducks and three bars represent “3”, and then reckon 3 lower beads to the central beam of abacus; they can form visual image of “3”, as the physical pillar. Through the teaching process of object (it can be a picture) →abacus beads → numbers (reading, writing), abacus computation is as the bridge from concrete numbers into abstract numbers. For long numbers, through the learning process of reckoning?writing and reading, with the clear shift of abacus and decimal system of digits, it is easy for students to read and write long numbers. Mr. Hua Luogeng, a famous national mathematician and educator, has a saying: “numbers originate from numbers”. The best way to count numbers is to move them. Mr. Jean Piaget, the Swiss psychologist, represents through research that moving operation is the key to form number concept. Students are reckoning beads while counting, carry one to tens digit when the unit exceeds ten, carry one to hundreds digit when the tens digit exceeds ten…until to great digits as ten thousands, hundred millions, which can avoids the error of writing 21 as 201 by mistake. Teaching process fully makes use of abacus’s visual direct-viewing operation to reduce the duration time of forming number concept, which is good for consolidation and proficient of four-basic arithmetic operations of integrals and decimals. (10) - Teaching of “The Combination of Three Types of Arithmetic” makes full play of abacus’s educational function, enlightens children’s thinking and promotes children’s intellectual development as well. According to the research of archaeologists, ancient ape men obtained food with their both forelimbs and there was not forming a great difference between right and left hemispheres. Later, for the gradual division of right and left hands, right activity increased, which led to left brain more developed. Learners put forward: brain awareness and hands movement are subjected to brain and on the other hand, frequent movement of both hands stimulates the brain and promotes its development. Working with both hands can stimulates the left and right hemispheres simultaneously and to promote the development of whole brain.
Fingers are the “outpost” of intelligence. Human’s intellectual activity is mainly completed with the movement of fingers. Writing?drawing? playing?embroidering and engraving can not be done without the movement of fingers, especially the thumb and forefinger, which are indispensable to fine skills. Mr. B· A· Cyxomunicin, the Soviet famous educator, pointed out: “There is extensive connection between hand and brain playing part in two aspects: hands make brain developing and more intellectual; brain makes hands developing and as a tool of creating intelligent and a measure and mirror of thinking”. “Clever and practical” does represent the relationship between hands and brain. Clever is the reason and practical is the result; on the contrary, practice inevitably promotes cleverness. From table 2, the intelligence of students in experimental class is obviously higher than that in corresponding class. Especially in graphs, the average score of students in experimental class is 8.02, near 3 scores higher than that in corresponding class. The difference is due to the teaching of “The Combination of Three Types of Arithmetic”, which greatly stimulates the development of right and left hemispheres of human-brain through training right and left hands reckoning beads at the same time. - Teaching of “The Combination of Three Types of Arithmetic” is favorable for cultivating students’ memory capacity. Mr. Carl Gustav Jung, the famous psychologist, said: “human’s all thought is just the accumulative memory, so is the history.”
It is known that memory plays an important role in human’s survival and development. Without it, human’s life has no significance at all. It can be viewed from table 3 that the memory power of students in experimental class of “The Combination of Three Types of Arithmetic” is obviously raised higher than that in corresponding class, which is due to teaching strategy of “The Combination of Three Types of Arithmetic” for students in experimental class are cultivated on their memory with specific target and task in each class, for example, stimulation of listening and watching mental arithmetic?watching and listening abacus computation, promotion of studying motion? development of studying interest and definition of studying purpose, which help strengthening students’ conscious memory obviously. - Teaching of “The Combination of Three Types of Arithmetic” is favorable for reducing students’ schoolwork burden and raising teaching quality. It is a reform of teaching thought. It attaches importance on students’ subjectivity and emphasizes on self-discovery and self-exploration under the instruction of teachers. The teaching transforms from “spoon-feed” to “enlightenment”?from “single statement” to “multiple statements”. Students’ multiple sense organs as hands?eye?mouth and brain are running in coordination; with the cooperation of oral arithmetic?written calculation and abacus computation, students can learn actively with great interest and remember firmly. Moreover, this teaching has many advantages, such as fast calculating speed?more opportunities of classroom training and immediate feedback of assignments etc. Most of the training can be completed in class without left homework after class, which greatly reduces students’ schoolwork burden. During six years, the average score of subjects in experimental class is about 81, that in corresponding class is about 70 and that around the whole county is about 40. The effect of teaching of “The Combination of Three Types of Arithmetic” also influences other related subjects and therefore raises teaching quality in a larger scope.
- Teaching of “The Combination of Three Types of Arithmetic” trains students’ calculating ability and thinking capacity. To train primary students’ calculating ability is an important task in primary mathematic teaching. The teaching program requires: students can correctly do four-basic arithmetic operations of integrals?decimals and fractions and reach a certain familiar extent for some basic calculation with rational and flexible calculating method. The test of oral arithmetic?written calculation of four-basic arithmetic operations of integrals?decimals and fractions and fractional application problems indicate that students in experimental class have faster calculating speed?higher accuracy rate and stronger thinking ability. See the result in table 4.
It can be seen from table 4 that students in experimental class have received the teaching of “The Combination of Three Types of Arithmetic” and their scores in oral arithmetic?written calculation?abacus computation and application problems are higher than that in corresponding class and the difference is very obvious. The three tests are limited by time but not by quantity with the time of oral arithmetic of 10 minutes. For 120 problems with each of 1 score, students in experimental class make out 66 problems on an average and students in corresponding class make out 40 problems on an average in the fixed time of 10 minutes; for 40 written problems, students in experimental class make out 30 problems on an average and students in corresponding class make out 15 problems on an average in the fixed time of 40 minutes; for 38 application problems, students in experimental class make out 24 problems on an average and students in corresponding class make out 12 problems on an average in the fixed time of 40 minutes. Students in experimental class can finish the examination with half of the time in the final examination. It is proved that students in experimental class have faster speed and less time in solving out problems, and the accuracy rate is higher. Students in experimental class have formed good studying habit; they have stronger ability to analyze and solve problems?active thinking?rapid response?unique methods to solve out problems?novel creativity and rich imagination.
Parents are the best publicity agents for school. A student in grade one was in the “intelligence-start class” through being tested as feebleminded child; later his parents knew the truth and wanted to make him in normal class rather than in the “intelligence-start class”. They consulted with school leaders for many times and school respected parents’ opinion to transfer this student to experimental class. From then on, this “special student” started his studying process from counting little bars to calculating with abacus. Through three-year’s studying, he ranked in middle of the class and was good at playing drum. His parents were very excited and highly praised experimental class, which once again proved that abacus has a good intelligence start function. Xu Hao, awarded Provincial Excellent Young Pioneer went to school at five and a half. She was younger, her writing speed was slower and it was hard for her to finish the work at class. After school, she was looked after by her 60-year-old grandma for her parents often went on business outside, so she was not in the top of class in grade 3. Through receiving reinforced training of digital writing?listening calculation and watching calculation, she improved quickly and possessed fast abacus speed in grade 4; in grade 5, she was awarded as 1st grade of abacus computation; besides, she had a balanced development among other related subjects and was good at playing electronic organ and dancing; she could deal with life by herself; her parents said: “it was experimental teaching that can cultivate students.” She was the only primary student who won the municipal educational fund. It is worth to be mentioned that Wang Tao, the abacus mental arithmetic excellent player, is approved as 4th grade of abacus computation at 11 years old. He has won various prizes for more than 30 times during six years. He separately participated in provincial?municipal and county contest and won the 3rd prize of individual all-round development in 2nd primary students’ abacus mental arithmetic contest?the 1st prize of individual all-round development in municipal abacus mental arithmetic contest?the 2nd prize of individual event of addition and subtraction in 17th and 18th adult contest?the 1st prize of individual event of addition and subtraction in 19th adult contest etc. Each contest occupies his studying time of one or two weeks on various extents but his score kept ahead. He likes abacus computation very much and focuses on abacus mental arithmetic with great passion. For he lived far away from school, his parents wanted to transfer him to nearer school, but he insisted on studying in experimental class and would go to school by bike rather than leave experimental class. He said that he loved studying in experimental class so he would not leave it. He solves out problems so fast that he can finish various work at class and he never thought study is a burden. Besides fond of football, he is good at playing waist drum and dancing.
- In teaching of “The Combination of Three Types of Arithmetic”, oral arithmetic is the basis, written calculation is the subject and abacus computation is the tool. These three factors should be combined organically with alternate operation and inter-promote each other with right orientation. Neither excessively emphasizes abacus computation’s position and role nor excessively pursues abacus mental arithmetic. Make full use of the intelligence start function of “abacus calculation” and “abacus mental arithmetic” and attach importance on the cultivation of students’ abilities.
- In teaching of “The Combination of Three Types of Arithmetic”, pay attention to deal correctly with the relationship between publicity and improvement to make improvement on the basis of good publicity. Speed should not be too fast, difficulty should not be too big and requirement should not be too high. The teaching should orient all students to help them have all-round development; everyone is able to learn abacus computation and mental arithmetic. The teaching should be in accordance with the direction, target and requirement of education experiment.
- Strengthen the administration and research on experimental class featuring on “The Combination of Three Types of Arithmetic”. Many experimental schools have only an experimental class, which leads to that the teachers in experimental class, can not find objects to communicate, which influences teaching quality. Besides, students have no opponents, which is not favor for their improvement. Do not draw the experimental class under the normal administration, which would attack teachers’ enthusiasm. Writer hopes that the departmental leaders of all levels could support this experiment and strengthen the instruction and research of experimental task.
- Pay attention to the training and cultivation of teachers. Teaching materials of “The Combination of Three Types of Arithmetic” make some reforms on abacus computation, such as getting rid of pithy formula for addition? subtraction?multiplication and division and the division with a divisor of two or more digits. Addition and subtraction are the same with the principle and method of oral arithmetic?written calculation and multiplication and division are similar to the methods of written calculation. In the teaching of abacus mental arithmetic, study the method of transferring multiplication into addition and transferring division into subtraction, which is on the basis of addition and subtraction, so the teaching of addition and subtraction is of importance. Even for the teachers possessing certain basic knowledge and teaching experience, they also require training for two weeks, emphasizing on explaining teaching method and teaching procedures of “The Combination of Three Types of Arithmetic” and which are focal points and difficulties. They should master the new principle of abacus computation. Therefore, teachers on experimental class need training, which enables the teaching reform processing successfully. In the daily teaching, teaching research department and administrative department should give support and proper consideration for teaching business, teachers’ workload?teaching facilities and necessary funds to ensure the successful development of experimental task.
- Teaching experiment of “The Combination of Three Types of Arithmetic” should pay attention to the construction and theoretical research of teaching materials. If the printing quality of experimental materials is not good enough or the materials can not be supplied in time, the experimental teaching schedule will be influenced. Matching theoretical publications and experimental parks are few, so it is advised for educational administrative departments?educational scientific and research departments of all levels to strengthen leading and instruction on this experiment.
References: (8) Huang Jilu: |