Children's Mathematical Frameworks: The Volume of a Cuboid Is V = L x B x H. Part OneAuthor(s): Kathleen HartSource: Mathematics in School, Vol. 16, No. 3 (May, 1987), pp. 16-18Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214214 .

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CHILDREN'S MIATHEMVIATICAL FRAMIEWORIS

The Volume

of a Cuboid

is

V= Lx B x H

by Kathleen Hart

P,-XRT ON-E 16

The research project "Children's Mathematical Frame- works" (CMF) was financed by the ESRC at Chelsea College during the years 1983-85. The investigation was designed to monitor a child's transition from a practical or material based approach to mathematics to the formal and symbolic mathematical language used in the secondary school. The volume of a cuboid was one of the seven topics studied; all taught by volunteer teachers as part of their normal class programme.

Methodology The methodology employed during the research was as follows:

(i) the recruitment of volunteer teachers; (ii) the writing of a scheme of work or teaching plan, in

some detail, by the teacher. These were discussed with the researchers who did not seek to influence the way the teachers taught;

(iii) the interviewing of six children in the class to be taught, before the specific teaching of a topic took place;

(iv) the reporting by the teacher of any changes to the lesson plan and the provision to the researcher of work cards and assignment sheets, etc. used in class;

(v) the interviewing of the six children immediately before the formalisation experience was planned to take place;

(vi) the observation and tape recording of the formali- sation lesson(s);

(vii) the interviewing of the six children immediately after the lesson(s);

(viii) the interviewing of the same children three months later.

(ix) the transcribing of all the tape recordings; (x) the discussion and analysis of the transcripts.

The observation and taping of the lessons in which the formula was verbalised, enabled the researchers to see how the transition was presented to the children. The four interviews gave insights into misconceptions held by the children as well as providing evidence of the progress made by them during the teaching.

Sample The work of three groups of children who were being taught the formula for finding the volume of a cuboid is discussed here. They were aged 10-11 years and from three different primary schools. Class A was the first set of children to take part in the CMF research and all their lessons on volume of a cuboid were observed by the researchers. There were 31 pupils in the class and they were taught altogether, working in groups using worksheets. Class B was composed of six children who were withdrawn from their class and taught either singly or in pairs but not as a group. Class C had over 20 children who were taught together.

Content The intention of each teacher was that the children should acquire and use the formula V = I x b x h when appropriate. Each class initially worked with cubes, building cuboids as well as filling boxes with them. Classes A and B worked towards the finding of a "layer" and then progressed to the formula by calculating the number of layers. In Class C very early in the formalisation lesson a child suggested that the volume could be found by multiplying three numbers and this was adopted and used by the class.

Most of the children were new to the formula and the teachers designed their lessons to best introduce the sym- bolic statement. The interviews and observation of lessons have indicated conclusions made and routes taken by

Mathematics in School, May 1987

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children which were unlikely to result in correct under- standing. Sometimes these could be traced to prerequisite skills not taught or conventions not known. The ideas in this article are intended to illustrate the assumptions made by adults which are not part of the knowledge of all children.

Volume Piaget identified two meanings of volume; "interior vol- ume" where the space is confined within boundaries and "occupied volume" where the volume is viewed in relation to other objects around it. The children in our sample were presented with examples of each type by their teachers.

When shown a box the question most often asked a child was "How much does it holda" Unvoiced was the additional question "How many cubes 1 cm by 1 cm by 1 cm would it hold if they were packed togethera" The answers "40 tea bags" or "200 tissues" were equally valid when the boxes shown the children were so labelled. One child when asked if a pencil had volume said "No. It's not hollow is it." Another, in Class B, when asked to compare the volume of two boxes, filled the smaller with 1 cm cubes and the larger with 1 inch cubes since it would be quicker. These children have answered the questions they were asked but not necessarily the questions that were meant.

Measurement To find the dimensions of a cuboid built from unit cubes most of the children counted one cube at a time along an edge. Given a box they found the height by standing a column of cubes along an edge. It was noticeable how seldom a child used a ruler to measure a length. In Class A only one child of the seven interviewed used a ruler when asked during an interview to find the dimensions of a box. In Class C the children referred to the length of boxes as a number of cubes, e.g. "9 centimetre cubed", "9 cubes high". Little wonder that the wrong units appear in answers when the child is encouraged to measure length with cubes but is expected to abstract only the linear attribute.

Diagrams All the children interviewed were asked the question shown in Figure 1. It appeared on the CSMS Measurement test paper and is adapted from one used by Piaget.

The problem to many children was one of finding squares of four cubes and mentally rearranging them, not one of multiplication and division. In Class A just before the formalisation lesson three of the six children interviewed spoke of moving sets of four cubes to stand on each other and continued to do so three months later. The diagram is cross-hatched to show the cubes. Would the responses have been the same if only dimensions had been givena Frank, in Class A, tried to cover the diagram with a skin of cubes (as he had done when given a box). Two girls tried to find the length of a side by comparison with a drawn line which was labelled "2". The diagram appeared to have much greater reality for certain children, than was intended. It is likely that a diagram drawn on the blackboard and labelled with numbers to show dimensions is seen as a picture of a particular block. In one class the teacher, in order to provide further examples retained the diagram but replaced the numbers (three times as big in some cases). The assumption was that the children would realise that the diagram was a stylised form of all cuboids and simply provided a frame on which to hang numbers. This is very different from seeing it as a particular stack of specific moveable layers.

Mathematics in School, May 1987

This block is made by putting some small cubes together

How many cubes make this block "C" if there are no gaps insidea

All the cubes from block "C" are put into a pile:

I am now going to use all these cubes from block "C" to build a "sky-scraper" so that the bottom floor is four cubes

How many cubes high would this "sky-scraper" be from the grounda

Fig. 1 The Skyscraper Question

Multiplication The CSMS research showed that the operation of multi- plication was not used by many secondary-aged children. It was replaced by addition and counting. In the early stages of the teaching of volume the cubes in a block could be counted and even when "layers" were being considered it was common for the children to count how many in a layer and then repeat that number for more layers. The observed classes did not practice area (as l x b) prior to the final stage of using V = lx b x h but instead found the number of cubes in a layer. So it was not obvious that the volume would be found by multiplication. The formula involves the multiplication of three numbers but only one class was positively taught how to do this. Larry in Class A, tried to multiply three numbers by writing them

12 4 7

This formation does not suggest a method and Larry failed to find the product. Three months after the teaching he added the three dimensions saying "I always forget how to times three numbers, I can do two but ...". The Table shows responses from the sample when just before the formalisation lesson the children were asked to multiply three numbers.

Results Three months after the teaching all the children in the sample were asked to find the volume of a cuboid and shown

17

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Product of three Numbers (3, 2, 7) and (5, 2, 7)

Class A Peter. Gerald. Larry. lan. Fra n k. 527 57 x 2 initly. 5 (5 x 2) x7 Not

2 Incorrect Asked. "7

Sarah and Louise Correct

Class B Elaine. Ken. Tina. Lindsay. (3 x 2) + (2 x 7) (3 x 2) + (2 x 7) 1 can't (3 x 2) x7

do times

Ruth and Sheila Not Asked

Class C Linda. 3 2 x 7

Adam. (2 x 3) + (2 x 7)

Edna, lan and Ben Correct.

diagrams. In 13 of the 20 cases where a correct general method was remembered it was recalled as finding a "layer" and then a number of "layers" rather than as a product of three numbers. The "layer" method can be extended to non-cuboid shapes but it has the disadvantage of being

difficult to visualise when the height measurement is a fraction. What does 21 layers meana Ken of Class B although using the "layer" method when whole numbers were involved balked at a fractional dimension "I done the base but on that one it's got quarters".

A popular belief of teachers is that children better understand and accept a formula if they see how it is found. In Class A the formula V = 1 x b x h followed the tabulation of examples of layers, number of layers and layer number multiplied by height, which the children had compiled for themselves. In Class A three months after the teaching none of the six children interviewed linked the formula V = l x b x h to the tabulation but said it was quicker than counting cubes. Asked how the formula was arrived at Ian said "I've no idea, somebody invented it for us".

In Class B where the children were taught individually, when asked three months later to find the volume of a box, they tended to measure the dimensions and perhaps multi- ply two of them but then fill the box with cubes and count them.

The investigation disclosed examples of children failing to make connections which adults assumed and assuming properties (e.g. of diagrams) of which their teachers were unaware. All teachers have to a large extent experienced successful schooldays and many have had a successful mathematical career in school. May be when children they made the same leaps and connections as their teachers but many children do not. A worthwhile guide is to ask oneself the question "would I understand this explanation if I did not already know ita".

Other interviewers on the Volume investigation were Lesley Booth, Julia Matthews, Marion Dunkerly and Peter Lockley.

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18 Mathematics in School, May 1987

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