THE STUDY OF UNDERSTANDING IN ARITHMETIC EDUCATION
UNDERSTANDING PHASE AND LEARNING PROCESS MODEL
-IN CASE OF AREA-LEARNING FOURTH GRADERS-
Hideharu Fujita: Hirano Primary School Attached to Osaka Kyouiku University
Shigeki Yamamoto: Sakai Municipal Eishou Primary School
This study focuses on analyzing the pupils' understanding process of arithmetic. To
analyze this is far-reaching and beyond our ability. Then we made the first step in
viewing the example we get through practice. One is to present the fourth graders'
understanding process of area-learning. The other is to present instruction learning
process to improve a deeper understanding. The former is modality model concerning area learning and the latter is process model. The purose of this study is tomake this, arrangement will be made of the several grounds concerning understanding process. Then through comparing those grounds with models, general logical hypothesis will be given as to model structuring and the result will be considered.
1. Introduction;
We are interested in a fact when pupils have solved F-1 problem after finishing area-learning. Most pupils tried to solve the problem using (length × width) formura
carefully. Then they can solve area problem-A but in case of problem-B in spite of
the time sent dividing into pieces, we got a lot of incorrect answers because of using
the formura many times. However, there were some pupils who solve the problem by formulating adequately voluntary unit (1/8 of the figure of problem-A) after watching
the figure sometime without trying to solve at once. To those pupils, this problem was like a puzzle and the given answer was correct. This was quite a shock to us and made us think what it is to understand deeply the concepts of area.
It is so happend that a phrase has changed in the course of study and guidelines for arithmetic. "Have pupils to understanding "has changed into "pupils understand". What seems to be a drastic concerning understanding process has not been seriously considered. Most people don't pay a serious attention to the change. We have focused our study upon "understanding" because we were doubtful of the conventional teaching and we wanted know the understanding process of arithmetic.
What is it for pupils to "understand" arithmetic. This is exactly what we try to find out in our study. But to solve this is far reching and beyond our ability. Then we resent our first step through one example we got in the classroom. It may be difficult to investigate pupils underastanding process of arithmetic but it may not be so difficult to investigate what it is like for pupils to underatand the area-learning(fourth grade). We will elaborate upon it in the following chapter.
2. Proceding Study:
(1) The interpretation of R.R.Skemp
The core of the famous work of Skemp is the distinction between relational understanding and instrumental understanding. He stated in the report, "He distingushes these two understanding: relational understanding and instrumental understanding. The former means that both how and why are understood. Instrumental understanding was not considered as understanding until recently. Formerly it was called regulation without reasoning".
Without knowing the reason one can obtain the circle-area only with the knowlege that radius × radius × 3.14. This is instrumental understanding. On the other hand, relational understanding is knowing the reasoning of the circle-area.
(2) The interpretation of S.I.Brown
The core part of Brown is the distinction between internal understanding and external understanding. Brown defined the two understandings as follows. "To understand X means is to know the relationship inside the X itself and to undestand X externally is to know how it is related with others while considering X as a whole." As a matter of fact , to obtain the circle-area by adoptong "radius × radius × 3.14 formura, and also to know the formura is in the category of internal understanding. On the other hand, "What is the use of inducing area formura in measuring area? " and "Why do people try to measure the area?" seem in the category of external understanding.
(3) The interpretation of Yutaka Saeki
Yutaka Saeki mentions in his book "Understanding" as follows: " To understand deeply things or phenomenn will coincide with the following accumulated factors. This is :@ solving the practical problems (problem-solving), A showing the reasoning of things(grounds for it ), B correlations between actual society and cuture (practicing socially ), C widening of correlated society (developmental enlargement)". That is , understanding occurs when the practical problems are solved, the reasoning is shown, and accordingly combination of actual society and culture, then widening of correlated society.
3. Modlity model and Process model in area-learning:
We made the study of understanding, focusing upon "understanding the area" in the children's behaviour. As a result, based upon the F-1 reseach problem, the other problem, which you will find later in the paper, suggested that there are phases in the understanding of area learning. This is obviouly endorsed by the result from the problem 1 solving the area of 3 cm × 5 cm rectngle. When we had our pupils explain "as a result of the formura" and the other could explain how many times of area unit perspective. We thought that the framework of the two groups correspond s with that of Skemp's instrumental understanding and relational understanding.
Moreover, according to "Adding up and Consideration of Achivement Analysis" at
Tokyo Arithmetic Education Study Group, 1990, there is as low as 47 percent of correct answer to the problem asking the area of classroom among 34143 fourth graders. The problem is a multiple chice: to choose 60 cu, 600 cu, 6000cu, 60 u
, 6 ku. The pupils came into two groups: one group have the ability of calculating i n the situational setting and the othergroup do not have such ability. This shows that there are more phases of understanding besides Skemp's instrumental understanding and relational undrestanding which we discussed before. And there neeed tobae another set phases to explain our result.
Then we call the Skemp's set of phases of instrumental understanding and relational understanding, "shallowing understanding phase". We thought that the set of phases, with which we cannot explain our result, correponds with Brown's external understanding or Saeki's practicing socially and developmental enlargement. And we set up a hypothesis and call it, "deep understanding phase".
Then the fact suggests us that the following four phases(mosality model).
Phase A: Pupils cannot solve the problem by using the formura. Phase B: Pupils can use the formura and cn exlain. Phase C: Pupils can understand the birth of universal unit in addition to phase A and B. Phase D: Pupils can understand the necessity of using larger units or smaller units in addition of all the phases above mentioned.
These models are bases upon the classroom teaching at elementary school and the phase D pupils canbe considered to have reached the "deep understanding" which we believe to be ideal. The consideration of phase D pupils can develop the creative activities themselves after entering a wider societly, such as play-ground and parks, out from the classroom situation. Phase D puppils can also distingush themselves at changing perspectives. Also, they are the children who can name their own unit because of the attachment of the unit based upon their own classroom activities. Using the unit, they further can create the new units one after another according to the new situations. As a result, they will have grown into the children who can undrestand the background of the birth of universal unit.
We considered the phase D pupils to have a good understanding of the area and we made a format of learning process model and practised at some elementary schools.
"LEARNING PROCESS MODEL OF UNDERSTANDING"
FIRST: Understanding why universal unit is produced
(1) Creating the area of my own. (territory-gaining game)
(2) This is the unit of my own. (naming the voluntary unit of one's own finding)
"1 Othello", " 1 Salata" , " 1 Mario", "1 Domino".
(3) This is the unit of our owdn. (producing the common unit usable in classroom sociaty)
"1F" F is the initisl of homeroom teacher.
(4) My size is "14F" and my size is "18F".
SECOND: Understanding how people learn to measure the larger size and smaller size.
(1) The unit of our own usable anywhere: "1F"
(2) Let's create the larger unit ...1F
Let's create the smaller unit ...1F
(3) The size of our classroo;m is 100F.
The size of a telephone card is 8F.
THIRD: Understanding the grounds of area-formura being produced.
Let's produce the formura. It's troublesome to lay units all the while.
(AREA)=(WIDTH)×(LENGTH)
FOURTH: Understanding the reasons of how the universal unit is produced.
We want to know the convenient unit commonly used the world.
cu, u, ku
4. Comparison Between Modlity Model and Understanding Theory:
In the process of our study we were amazed at the fact that Skemp took up as an examle of area-slving when he discusses relational understanding and instrumental understanding. In this case he defines the instrumental understanding the solution getting by the use of the formura width ×Length. He calls the knowledge of why to multiple, relational understanding.
This is to say that our notion of phases arising from solving the area of 3 p×5 p rectangle correspond with distinguishing instrumental understanding and relational understanding.
Let us compare phase-concept we presented with "relational understanding" and "instrumental understanding".
Our phase A of being able to solve the problem by using the formura is in the category of instrumental understanding and our phase B of being able to use the formura and also to explain it, is in the category of relational understanding. But the difficulty lies in the discussion of phase C and phase D pupils. Those pupils who understanding the birth of cu and u in the society, because they have experienced the birth of valid unit. We find it rahter difficult to put those pupils in the category of relational understanding. Another category would be needed.
The same is also true of phase D pupils, who are able to change perspective on a case-by-case basis. Those pupils are also difficult to discuss within the framework of relational understanding. To approach changing perspective mathematically, we think that in addition to relational understanding, we would also need intuitional understanding and logical understanding proposed by two Canadians, Byers and Herscovics.
Here we must distinguish between internal understanding and external understanding before comparing internal understanding and external understanding, as Brown put it, and our understanding model.
As mentioned in the first section, Brown defines the distinction between the two understandings. Hirabayashi mentions that we hold an "understanding rope" and that new object is to be hung upon the rope to identify its location. This is what he thinks external understanding and pupils begin examining the object "hung on the rope". This is like a spider approaches the hunt on the web and this process can be concsidered to be internal understanding. By *(3),
this idea of external and internal understandings can be thought to be set in a definite framework. Accordingly, the problem is what kind of framework is to be set. Hirabayashi says, "Math education in Japan can be charactarized as perhaps something underpinning internal understanding of materials and lacking external understanding. This is exactly like "Being unable to see the wood for the trees" which is often quoted to imply today's educational system. Therefore, the difference between the framework of internal understanding and external understanding is that the concept of internal understanding is analyzing and external understanding is how to locate the concept in the whole area. In connection of our model, phase A in which pupils cananalyze its formura canbe thought to be internal understanding. And to be able to understand the birth of universal unit is to be able to understand its background process, so this could be framed into external understanding. Phase D in which pupils can understand in addition to the phases above mentioned, the necessity of using larger units or smaller units could be in the framework of external understanding.
This brings us to believe that "understanding phases" we presented, that is, modality model, is the model which includes both internal and external understandings. The development of class through "learning and studying process model" which created phase D pupils can the thought to be the learning-process model bases upon external understanding, the type of model never developed in Japan before.
Saeki mentions in his book *(4) that there are four conditions in the process of understanding @ problem-solving, A grounds for it,
B practising socially, C developmental enlargement of it. This is to say, the phase in whivh concrete problems can be solved and the reasoning can be presented correeponds with modelA and B. If praciticing it socially is to be something including universal unit of social background, it corresonds with phase C. And if fits exactly phase D and the model we presented.
Accordingly, what Skemp, Brown, Saeki say about understanding corresponds wonderfully with our model.
5 Results and Consideration:
In order to verify the effectiveness of "understanding learning models" we propose which helps pupils reach deep understanding (phase D) , we made evaluating problems 1 to 5 and considered the results. Here we evaluated phase A as problem 1, phase B as problem 2, phase C as problem 3, and phase D as problem 4 and 5.
"RESULTS OF UNDERSTANDING MEASUREMENT"
PUPILS IN EACH PHASE RENERAL GROUP HYPOTHITICAL GROUP
staying in phase A 41% 0% staying in phase B 34% 11
staying in phase C 16 15
having reached phase D 9 74
Note: Phase A pupils can only solve the problems by using formura.
Phase B pupils can understand the meaning of formura.
Phase C pupils can understand the birth of universal unit.
Phase D pupils can understand the larger and smaller units.
The results indicated that among pupils in general grtoup those who are not able to understand the meaning of formura and try to obtain the answer formally consist 41%.
On the other hand, those who stayed in phase A consist 0%. This means that pupils in general group stayed in Skemp's instrumental understanding , whereas most pupils who had experienced this learnig model attained relational understanding. The excessively low ratios (74%) among experimental group indicate that pupils who experienced this learning model have reached not only inner understanding but also external understanding , as Brown put it. We should like to omit the other researches and problems for lack fo space. But in discussing the pupils in the process of experimenting this learning model, the phase in which pupils solve the practical area-problem and are able to explain the grounds for it corresponds with Saeki's solving the practical problems and showing the reasoning of things. And those who reached phase C have grown into pupils who find voluntary unit and create new units one after the other because of a deep attachment to units. They go on to understand the background of the birth of universal unit now being used. In other words, they recognize the necessity of creating common unit when a sociaty meets another society. This can be regard as practising socially. And phase D , the deepest form of understanding we proposed involves, added to the three other phases, the ability to conceive units corresponding to each scene. Pupils in phase D, we just
discussed are those who develop activities which correspond to wider society out of classroo;m situation, such as play-grounds or parks. And in these activities they can create their own units and see things in a different perspective. This can be looked upon as widening of correlated society.
For reason mentioned above, "understanding phade model" we proposed is a new concept on understanding (framework of understanding) which is not only based on Skemp's concept but also combined with the understanding concepts of Brown and Saeki. Our "understanding model" surely helps pupils attain what we call"deep understanding". We firmly believe the validity of our learnig model.
*1 「Mathematics teaching」 R.R.Skemp 1976
*2 「Mathematics In The Primary School」 R.R.Skemp
*3 「算数教育における活動主義的展開」 平林一榮監修
*4 「わかるということの意味」 佐伯 ゆたか
*5 「学力実態調査の集計と考察」 東京都算数教育研究会
「算数教育学のパ−スペクティブ」 平林一榮
「考えることの教育」 佐伯 ゆたか
「Young Children Reivent Arithmetic(IMPLICATIONS OF IAGET'S THEORY)」
Constance Kazuko Kamii 平林一榮監修
(The End)
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