Introducing Volume and Capacity to Students: Laying the Foundation for Further Learning
For many teachers, introducing the concepts of volume and capacity to their students can be a daunting experience; they may doubt their own understandings of the concepts or may struggle to find strategies for early investigation amongst existing resources that emphasise the use of formal standard units. This paper aims to address some of these issues, first by clarifying the concepts of volume and capacity. Subsequently it discusses research assessments of the effectiveness of introducing the measurement of volume and capacity in the early years of primary school. Finally, by adopting established frameworks of conceptual development, this paper suggests some simple learning experiences that can introduce volume and capacity in a supportive and effective manner.
Volume and Capacity: Is there a difference?
Although the terms ‘volume’ and ‘capacity’ are regularly used interchangeably, there is an important distinction to be made between the two terms, so that teachers can model appropriate language use for their students. Both the NSW K-6 Mathematics Syllabus (BOS NSW, 2006, p. 102) and the definitions provided by the NSW DET, in Teaching Measurement (2004, p. 82) concur that capacity refers to the “[maximum] amount a container can hold”. Conversely, volume refers to the amount of space taken up by an object or substance (ibid.). Usually capacity is measured in millilitres or litres (mL/L) and volume in centimetres or metres cubed (cm3/m3), however this is not always the case, for example, when measuring the capacity of a lunchbox in terms of how many unifix cubes it can hold. The distinction between these two concepts can be difficult for students to grasp, especially if students are only introduced to situations where the capacity of a container (for example, 10 scoops of sand) is also the volume of sand measured.
When should teachers introduce the concepts of volume and capacity in their classrooms?
Research into student comprehension of volume and capacity can appear to suggest that the teaching of such concepts to younger students is ineffective. Much of this research assesses student mastery of concepts in terms of being able to conserve volumes. This appears logical as conservation (that a quantity is unchanged if it is rearranged) is identified by the NSW DET (2004, p. 8) as being “fundamental to understanding measurement”.
Piaget, as part of his research into children’s cognitive development, argued that students who are in the pre-operational stage of development (up to around seven years of age) are unable to grasp key attributes of volume and capacity (cited in Wadsworth, 1996, p. 74). Specifically, Piaget focused on the aspect of conservation and demonstrated that many younger children were unable to attend to the transformations that occurred when pouring a liquid from one container to another container of equal capacity but different proportions.
Similarly, Twidle (2006) conducted research on primary-aged students in the UK which also used the ability to conserve volume as an indicator for comprehending concepts of volume in both liquid and solids. This research indicated that “a significant proportion of children entering secondary schools had not mastered some of the concepts we might have expected”, including conservation of volume (Twidle, 2006, p. 94). Twidle’s research did however reveal that conservation is easier for students to understand in terms of conservation of liquid volumes (Twidle, 2006, p. 93). This article therefore suggests it may be most effective to introduce volume and capacity with investigations that focus upon measuring liquids, or solids that are easily poured with no visible gaps, such as sand. This is supported by Outhred, Mitchelmore, McPhail & Gould (2003, p. 85) who observe that volume which involves filling, rather than packing, does not require a prerequisite knowledge of area.
It is important to emphasise that the above research has investigated mastery of the concept of volume conservation. However, both Wadsworth (1996, p. 75) and the NSW DET (2004, p. 8) recognise that conservation is an aspect of measurement that is developed over time, as the result of experimentation and manipulation of different substances and volumes. Teachers promote development by providing opportunities for students to continually test and re-evaluate their understandings.
Early Stages of Development in Understanding Volume and Capacity
The Measurement Framework (NSW DET, 2004, p. 11) and the framework for the Count Me into Measurement program (Outhred et al, 2003, p. 85) suggest very similar patterns of conceptual development in the early stages of studying volume and capacity, which also apply to other areas of measurement:Level/Stage One: Identifying the attribute being measured by making direct comparisons, and ordering containers by their capacity and ordering different volumes.Level/Stage Two: Making informal measurements of volume and capacity and comparing capacities and volumes using appropriate units.These two stages align with the NSW K-6 Mathematics Syllabus which outlines “direct comparison” as a means of addressing volume and capacity in Early Stage 1 (2006, p. 102), and promotes the introduction of “informal units” in Stage 1 (2006, p. 103). Furthermore, Outhred et al. (2003, p. 82) cite both Australian and international research that recognises that inadequate understanding of these two stages is a common cause for error in more sophisticated volume and capacity tasks.
Suggested Learning Experiences which Address Aspects of Each Developmental Stage
Direct Comparisons of Capacity
· Sand (equal to the exact capacity of one container provided), with a scoop and spill tray (e.g. an icecream container)
· Funnels (can be cones of paper)
· Containers of varying capacity, some able to hold more, less and equal the amount of sand provided. Containers should be labelled for ease of reference (e.g. Container A, B etc.)
1. Introduce and clarify, using visual examples, with students some terms which indicate the level of the sand in a container, e.g. ‘full’, ‘nearly full’, ‘overflowing’ etc. (Adapted from NSW DET, 2003, p. 88). Ask students how they know a container is full, that is, no gaps. It may be necessary to attempt this activity prior and establish terms which are appropriate for the resources provided.
2. Have students predict one container they think will hold the exact amount of sand provided.
3. In pairs or small groups, ask students to experiment with pouring the sand into the different containers provided. Emphasise the need to use the same amount of sand in each measurement. They should record the level of the sand using the terms established. Students should also revisit their earlier predictions: were they accurate?
4. Engage students in a dialogue about which containers can hold the most. Suggest that the containers that can hold the most don’t get full or overflow when others do, and that containers that hold the least overflow the most.
5. Ask students to use their observations to record which container can hold the most sand, which container can hold the least, and which containers hold about the same amount. These results should also be communicated in simple sentences, which can be completed in a cloze-like fashion, for example, Container ____ can hold the most sand.
Using Informal Units to Test Capacity
· Water (preferably coloured)
· Funnels (plastic)
· Containers (Same as used in previous activity)
· Various cups that can be used as ‘units’ for measuring water (e.g. small Tupperware, plastic cups, Chinese food containers)
1. Working in small groups, have students order the containers from what they think will be the smallest to largest capacity, that is, from the container they think will hold the least water to the one they think will hold the most.
2. Provide each group with one type of unit for measuring water. Have students record the number of times they need to fill and pour the smaller container in order to fill each larger container (Adapted from NSW DET, 2003, p. 96).
Container A B C Prediction of capacity from smallest to largest smallest largest middle No. of vegemite lids needed to fill the container 10 12 9 No. of Chinese food containers needed to fill the container 2 2 (plus a bit more) 2
3. Rotate ‘unit’ measures. Have students repeat Step 2.
4. Discuss with students which ‘units’ they preferred using, was one more appropriate than the other? One may have been faster but was it as accurate? Also discuss whether their initial predictions about the capacity ordering were correct. If not, have they noticed anything about each container now that might help them make a more accurate prediction next time? That is, were any compensation strategies utilized such as “this container is a similar height but is wider so will probably hold more water”.
5. The teacher should model and assist students in writing a reflection about their predictions and whether or not they were accurate and why this might be.
These learning experiences are intentionally similar, both addressing capacity, in order to highlight the difference between each level of development. Other experiences which focus upon volume would also be necessary in a balanced teaching program.
Teachers should be encouraged to introduce lessons of volume and capacity in early primary classrooms. By focusing on frameworks of early developmental stages, and teaching strategies such as direct comparison and informal units, strong foundations are created for the formalised learning of volume and capacity that occurs in the later years.
Board of Studies NSW. (2006). Mathematics K-6 Syllabus. Sydney: Author.
NSW Department of Education and Training (DET). (2003). Teaching measurement: Early stage 1 and stage 1. Sydney: Author.
NSW Department of Education and Training (DET). (2004). Teaching measurement: Stage 2 and stage 3. Sydney: Author.
Outhred, L., Mitchelmore, M. C., McPhail, D., & Gould, P. (2003). Count me into measurement: A program for the early elementary school. In D. H. Clements & G. Bright (Eds.), Learning and teaching measurement (pp. 81-99). Reston, VA: National Council of Teachers of Mathematics.
Twidle, J. (2006). Is the concept of conservation of volume in solids really more difficult than for liquids, or is the way we test giving us an unfair comparison? Educational Research, 48(1), 93-109.
Wadsworth, B.J. (1996). Piaget’s theory of cognitive and affective development: Foundations of constructivism. White Plains: Longman.
Despite extensive searching for an appropriate multimedia or ICT resource to support early investigations of volume and capacity I encountered two problems. One, many resources focused upon sophisticated formal units of measurements. Two, an emphasis on was made on displacement or predicting the height of water levels when transferring from different containers, rather than emphasising different capacities.