Researchers indicated many errors happened in arithmetical task’ pupils, this could be pointed to difficulty with place value concept in decimal number especially. To perceive the full understanding of place value, pupils conceptualize multi-digit numbers as collections of units not as multi-unit structures based on powers of 10. For instance, 37 are looked as a collection of 37 units rather than as three tens and seven units. This present study taken a sample of 262 pupils aged between 7 and 12 years from four primary schools in the Greater Belfast area with variety of pupils’ social background.

This research suggested the question of whether or not pupils can perform routine mathematical tasks, and to examine more closely the development of place value concepts. It was also known little comparatively that the structure of pupils’ multi-digit number concepts in the early stages of development and it might be here that the roots of later weaknesses are to be found. Therefore, the present study set out to examine mental models of number throughout the primary school years.

In addition, in this present study, this property was used to explore multi-digit number concepts at primary school level, through examination and analysis of pupils’ concrete representations of written numerals, together with verbal representations and explanations. Further deeply explanation for pupils’ understanding of number concepts was divided into four particularly attention, such as: understanding of the apparatus, quantity and type of materials used, spatial organization of materials, and explanations of spatial organization.

Understanding of the apparatus meant that the pupils knew how to use the apparatus (media of learning material) and perceived concrete materials as a crutch in understanding of number concepts. For example, those who did not use the materials were proud of this; those who could not use the materials often admitted the fact rather shamefacedly plus explanation that they were not good at mathematics. This data is taken up with pupils’ ages from 7 till 12 years old representing tens and hundreds numbers and also with pupils’ ages from 10 to 12 years only for representing thousands numbers. The result is 75% of pupils know it or guessed it correctly, 20% counts the quantities contained in sticks and flats, and 5% of pupils give an incorrect answer or unable to tell what the materials represented.

Quantity and type of materials used activity examined the pupils’ knowledge of quantity of material and name a numeral correctly. From all ages, 7 till 12 years old, the data shown that one hundred per cent of pupils correctly set out single-digit numbers and two-digit numbers up to 13 and the number 20, 97% for two-digit numbers greater than 20. For three-digit numbers, pupils who can set out the correct quantity only is 85% and who can read the number only is 95%. The last is four-digit numbers, only for 10 until 12 pupils’ ages, 97% can set out quantity correctly.

Spatial organization of materials showed frequencies of responses which ignored place value by aged and number. For instance, in representing 11 or 13, unit cubes are placed directly in line with a stick representing 10. Almost 30% of pupils ignored account of place value, 45% of pupils always positioned the material in the same order as the digits represented, and 25% of pupils said that positioning of material is related to number value. Pupils in the youngest ages were likely to take account of place value than are older subjects.

Explanations of spatial organization measured a case where a numeral was read correctly, pupils were able to set out an equivalent concrete representation. Only 45% of pupils perceive of existence of place value in concrete representations of 10, for instance, by placing a green stick directly below the digit 1 and no material under the digit 0. 30% of pupils are not consistently followed conventional place value order, and 25% place value order was followed less frequently for smaller values than larger values and least frequently in representations of 10. It was not affected by gender or school, but strongly influenced by numerical value represented and pupils’ ages also.

If I compare the explanation of place value from this research “Learning multi-unit number concepts and understanding decimal place value” by Jennifer Hunter and Irene Tuner with the book of “Young mathematicians at work” by Catherine Twomey Fosnot and Maarten Dolk, I can get an overview of the similarities and the differences. The similarities that existed pupils always do the wrong perception in understanding of place value. They are not had a good concern for this thing whereas it is essential thing for development understanding number concept. Another similarity is pupils always make representation for material or problem in order to simplify it.

The differences are looked from the way of research work. In this research, it seems pupils are given numbers problem directly even though the materials are given also. But the problem is pupils have found the function of various materials to the problem, not only one. While from the book, teacher has chosen one material that certainly appropriate with the problem. Another is different problem, from the book, the problem is likely a short story of sentence. While in this study, the problem directly a number form.

By Ekasatya Aldila Afriansyah

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