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Guozhong Ding
Primary mathematics Department, People s Education Press
Abstract
As the Standards of Mathematics Curriculum for Compulsory Education (of China) advocated, pupils should experience and understand mathematics knowledge in real situations. Current status shows this principle has been accepted by most of teachers and has influenced mathematics classrooms deeply. Pupils study mathematics in an interesting and lively way. So teaching mathematics in real situations is becoming more and more popular in todays classrooms.
But what happening in mathematics classes makes us worried. Teachers pay less attention to the nature of mathematics at the same time of creating real situations, so mathematics classes are becoming less and less mathematical and the efficiency of class is becoming lower and lower.
Based on the experiences of textbook writing, I summary three different models to explain when and how to design situations and emphasize the importance of using situations in a more mathematical way. Sometimes, we should concern the mathematical structure hiding behind the real situations, sometimes we can teach mathematics knowledge in mathematical situations, and some mathematics concepts mustnt be taught in real situations, or else we will make mistakes.
Key words
situation, mathematical
As the Standards of Mathematics Curriculum for Compulsory Education (of China) advocated, pupils should experience and understand mathematics knowledge in real situations. The standards suggested that teacher should create situations based on pupils reallives and knowledge background when they were designing mathematics lessons. The authors of mathematics textbooks were requested to provide materials which pupils were familiar with and interested in into textbooks. The purpose of using situations, especially real situations in mathematics textbooks and mathematics classrooms was to change the condition that mathematics knowledge was often represented in an abstract way. In a long time, pupils thought mathematics was composed of a series of numbers, symbols, definitions, theorems and mathematics knowledge had no relationships with their lives. Many pupils were afraid to study mathematics or had difficulties in understanding abstract knowledge.
Now this condition is being changed. The principles that we should teach mathematics in real situations have been accepted by most of teachers and have influenced mathematics classrooms deeply. Pupils can obtain abstract mathematics knowledge through a reallife situation which is provided by textbook or created by teachers. In their eyes, mathematics is no more boring and meaningless, but interesting and lively. So teaching mathematics in real situations is becoming more and more popular in todays classrooms. You can find many papers describing the benefits of representing mathematics concepts in real situations. I do agree with these inspiring results but I dont want to repeat it any more.
Now what I am worrying about is that teachers always create real situations when they teach, despite that there are many mathematics concepts unsuitable to be taught in real situations. Teachers are paying less attention to the nature of mathematics. On the other hand they focus on the situationdesign which takes them much time. Mathematics classes are becoming less and less mathematical. Too much reallife factors are diluting the mathematical taste and reduce the fascination of mathematics itself. We cant help asking ourselves: what is the main factor to inspire pupils interest to study mathematics? Is it an interesting story or fairy tale? Or a mathematical fact around us? Or the intrinsic fascination of mathematics itself such as the abstract, terseness, precise? Which one is the most important factor? Maybe its hard to answer in a simple way. An advisable choice is that we should distinguish what kinds of mathematics concepts are necessary to be taught in real situations and then determine what situations are suitable.
What happening in our classrooms is worth to be noticed.
The first thing we should pay attention to is that teachers spend a plenty of time in creating situation when they prepare for lessons. They have no time and energy to investigate the structure of mathematics concepts, and they dont spend enough time to organize all the mathematical materials in a reasonable way. They dont have interests to know about the developmental level of different pupils. If teachers havent the capability to grasp the key points of mathematics, the lesson will still be a failure even if the real situations they created are very fascinating. So I think its more important for a teacher to put much time into exploring the process of mathematics knowledge forms and develops.
Another phenomenon is that too much information irrespective to the theme in real situations disturb the teaching process and reduce the efficiency of class. Much mathematics knowledge can be taught in real situations, but these real situations often include much information which has no relationship with the topic of mathematics being discussed. Its very important for teachers to form a mathematical model from the real situations as quickly as possible in the limited classroom time. For example, when we teach the concept of dividing averagely, we can create a real situation of spring travel, in which pupils need to prepare and distribute food and fruits. We should put emphases on following mathematical problems:
How many pieces of bread (or other food and fruits) are distributed?
How to divide?
How many portions to be divided?
How many pieces one portion has?
Are there any pieces of bread left?
Is there common characteristic in different methods of division?
In this real situation, we shouldnt spend much time in arguing which fruit we should choice among apple, orange and peach, or discussing what we should pay attention to during the travel. These problems are so far from the nature of mathematics.
In fact not each mathematics concept is suitable or necessary to be taught in real situations. In a broader view, situations include not only real situations but also mathematical situations. Sometimes, using real situations even will make pupils confused.
When we deal with the mathematics materials in textbook writing and in classroom, there are three models of using situations. I will illustrate the viewpoints using some examples in our textbooks.
Model 1: Looking at real situations in a more mathematical viewpoint
Some contents are suitable to be represented in real situations. But we should focus on the nature of mathematics hiding behind the situations. Especially in the lower grades, when pupils havent the ability to understand mathematics in an abstract way, learning mathematics in real situations will help them understand mathematics more easily. But for teachers, its necessary to know of the mathematical background and look at the situations in a mathematical way.
Example1: When we teach the knowledge of location and position, we usually provide real situations in which pupils describe the relative directions of two objects or places in reallife with some daily words.
We can find the mathematical models of coordinates in these real situations. There are some cases as below.
Case1: The book is on the desk and the desk is under the book, Mike is in front of John and John is behind Mike, Mike stands left to John and John stand right to Mike. Figure 1 shows xaxis, yaxis and zaxis in a threedimensional coordinates represent respectively three directions of leftright, frontbehind and updown in real life.
Figure 1
Case2: Mike sits at the first line and the second row. How to find Mikes seat easily? Figure 2 tells us its just a point in coordinates.
Figure 2
Case3: How to locate a ship in ocean? What s the direction? How long it runs away from the dock? If we use the concept of polar coordinates, it s easy to get answers. Two parameters (r,) of figure 3 can help us find the ship quickly.
Figure 3
All these pictures show the relationships between school mathematics and life mathematics.
Model 2: Using mathematical situations
Some knowledge can be taught in real situations, but if be taught in more mathematical method, still can arise pupils willing to explore the world of mathematics. Sometimes mathematical situations show more significant effects in helping pupils to understand mathematics.
Example 2: We designed an activity about the numbers within one hundred in grade one. Its a manipulative activity but involves multiple mathematics concepts, skills, abilities and approaches. We let pupils put some beads respectively into tens digit and ones digit of a digital table. If there is only one bead, we can get two numbers: 1(put the bead into ones digit) and 10(put the bead into tens digit). If there are two beads, we can get three numbers: 2(put both of the beads into ones digit), 11(put one bead into ones digit and the other into tens digit) and 20(put both of the beads into tens digit). Then we can add the beads one by one.
Pupils are requested to think about the following problems when they manipulate:
With the increasing of the amount of beads, what numbers can you get? Are there any rules among these numbers?
With the increasing of the amount of beads, does the amount of the numbers you get always increase along with the increasing of the amount of the beads?
If there is no bead in hand, can you write out these numbers directly?
Through Pupils manipulation and the instruction of teacher, they can get the perfect results shown in the following graphs as figure 4.
amount of beadsnumber created with beads111022112033122130441322314055142332415066152433425160771625344352617088172635445362718099182736455463728190101928374655647382911129384756657483921239485766758493
If pupil wants to get more results in this approach, they can try more beads. When they draw another picture to show the result like figure5. Who can be suspicious of the beauty of mathematics? Its so interesting and funny.
0123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899
Figure 5
In this activity, pupils master mathematics concepts such as place value and decimal system. They develop the abilities of concluding, reasoning and thinking systematically, and experience the process of forming guess, overthrowing guess, finding out the reason and exploring the rules. Pupils can enjoy the power of the mathematics. It is the inherent fascination of mathematics that attracts pupils to explore the mathematics world unknown to them.
Model 3: Not using real situations
A noticeable fact is that there are some contents which mustnt be taught in real situations and have to be represented in an abstract way. The power of mathematics development is not only from the need of the society development, but also from the development of mathematics itself. If taught in real situations, some contents will become hard to understand.
Example3: Multiplesteps operations involved in primary school have two rules, that is: if there are parentheses, work out the operations in the parentheses firstly, then finish the other operations; if there is no parenthesis, finish the operations of multiplication or division firstly and then do the operations of addition or subtraction. The purpose People established these two rules of operation were to ensure the unique result when different people operated the same formula, not because the results of multiplication or division always need to be worked out firstly in real life.
Here are two situations. One is A pen costs $2, Peter have $7, if he wants to buy three pens, how much will he left? The other is A pen costs $7, and the shop provides $2 off for each one. Peter want to buy three pens, how much will he pay? In the first situation, we need to finish multiplication operation firstly, but in the second situation, we need to finish subtraction operation firstly. Only when we have mastered the rules of multiplesteps operation(which should have been taught in a mathematical way), can we write respectively out the correct foumulas:7
23 and7
2 3. But if teachers try to teach the rules of multiplesteps operation in these two real situations, it will be proved to be a failure.
As a conclusion, providing real situations in mathematics classes or mathematics textbooks is very important to strengthen the relationships between the life experiences of pupils and the world of mathematics. Through real situations, pupils can find mathematics is not so boring any more and know that much mathematics knowledge comes from real life and can be applied in their lives. At the same time, we should realize that situations should serve for the nature of mathematics as a carrier of mathematics knowledge and we should use situations in a more mathematical way.
Reference
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Primary mathematics Department (2001). Mathematics Textbooks in Compulsory Education. China: Peoples Education Press.
Qihua Zhang, Youlin Pen(2006). Do situations you create make sense? Journal of peoples Education, 8.
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