Teachers assume that by the completion of primary school, student should understand the concepts associated with volume. For example, the NSW Mathematics K-6 syllabus indicates that by year six, students should be able to estimate and measure volume and capacity using formal units. However, lack of authentic measurement questions in textbooks (Dept. of Education and Training, 2004b, p.1) as well as teachers’ approaches to mathematics, which is governed by rules and formulae (Szydlik, Szydlik & Benson, 2003), has lead to many students misunderstanding questions about volume and capacity as poorly disguised multiplication and addition questions. Hence there is a need to revise teaching approaches to incorporate practical learning experiences, which are essential to the development of students’ conservation of volume (Dept. of Education and Training, 2004a, p.82).
The aim of this paper is to look at the teaching and learning of volume in the primary classroom, by providing an overview of its attributes and making suggestions for teaching strategies and resources that will address some common misconceptions amongst students.
Understanding Volume and Capacity
With the greater recognition of numeracy/mathematics in the primary sector on both national and international levels (Zevenbergen, 2005, p. 3), there is a need for teachers to strengthen their mathematic discipline knowledge with particular attention to knowledge of volume.
Volume is a complex concept consisting of five aspects that are interrelated; capacity, volume of model blocks, interior volume, exterior volume and displacement (Dept. of Education and Training, 2004a, p.82) which teachers need not confuse students with. It is appropriate to teach volume and capacity as separate concepts before drawing the relationships between them. Volume and capacity can be defined as the following:
Volume is “the amount of space occupied by an object or substance” (Board of Studies NSW, 2002, p.201). Hence volume refers to the measurement of three dimensional space and not surface area, to which students may confuse.
Capacity on the other hand is “the amount that a container can hold” (Board of Studies NSW, 2002, p.193). Students need to eventually understand that capacity relates to the “same three dimensional space [of volume] but only refers to the [maximum] measure of substances that can be poured [into a container]” (Bookers, Briggs, Davey & Nishet, 1992, p.272).
There are two units of measure for volume and capacity, militres and cubic centimetres. The identification of when to use which unit may be overwhelming for students and confusing for teachers. Below is a table which identifies the unit of measurement to the type of measurement used.
Measurement | Units |
Volume of substances | Mililitres |
Volume of solids | Cubic centimeters |
Capacity | Mililitres |
Table 1: Which type of measurement with which unit?
The capacity of a container is the maximum volume of liquid that it can hold. Therefore, capacity is measured in the same units of liquid volume.
Resources and teaching strategies
According to the measurement framework (Dept. of Education and Training, 2004a, p.12-15) students begin exploring volume by identifying the attributes of volume. Excellent ways to achieving this is to have students participate in activities that involve filling and packing containers (Zevenbergen, Dole & Wright, 2004, p.268). For example, filling a container with cubes and then taking them out and rearranging them in a different shape (Beaumont, Curtis & Smart, 1986, p.25-26). This will aid develop students’ sense of volume because the number of cubic cubes do not change even if the shape does. This can also be performed with liquids by pouring from one container to another with the same volume (Figure 1). This will help to dispel the misconception that tall containers hold more than short containers.
Figure 1: Pouring fluid from one container to another of the same volume but different sizes.
It is important that teachers limit the use of workbook and worksheet based activities, although they reduce planning time and are convenient to print off. Textbooks tend to lack authentic measurement questions (Dept. of Education and Training, 2004b, p.1) and not to mention fail to engage students cognitively. The use of two-dimensional representations of three-dimensional objects on paper (Figure 2) often causes confusion for students between the concepts of surface area and volume (Zevenbergen, 2005, pp.13-14).
Figure 2: Two-dimensional representation of three-dimensional block
Hence it is necessary for students to explore volume and capacity using three dimensional units, such as wooden/plastic cubic centimeter cubes (Wilson & Osborne, 1988, pp.98-99) or unifix cubes (Figure 3).
Figure 3: Unifix cubes
Some students conceptualize volume as being equal to capacity and vice versa (Wilson & Osborne, 1988, p.98) because the capacity of a container is the maximum volume that it can hold. To undo this misconception, activities that involve measuring objects that have varying volumes will allow students to explore the relationship between volume and capacity.
For example, have students fold a paper box, and then take measurements of the length, breath and height to calculate the volume. Once they have calculated the volume, ask students how much rice will the rectangular prism hold if filled level with the top? Because the sides of the paper box are not rigid it will expand to hold more rice than students had calculated (Dept. of Education and Training, 2004b, pp.1-2). An addition to this lesson could be having students write about what they have learnt. This can be an indication for teachers on how effective the lesson was planned and executed.
The teaching and learning of volume and capacity should not be restricted to classroom activities. Teachers can set home experiments and investigations to relate to students everyday life experiences.
Once competent, students may also engage in computer generated internet programs to draw connections between volume and nets of prisms and discover the formula for volume. Cubes is a website that allows students to fill in the net shape with single cubes, a row of cubes or even a whole layer of cubes. This website will assist students in their understanding of fractional layers of volume. This website can be found here: http://illuminations.nctm.org/ActivityDetail.aspx?ID=6.
Although great resources can be found online, they are not always suitable as resources for students. Many great websites and webquests are based on imperial system units and hence not appropriate for an Australian classroom. However regardless of measurement systems, there is an abundance of teaching resources in varying forms such as video clips, lesson plans and activities that are readily accessible on the internet.
Video clips on what is volume and how to teach it are available on websites like the Virginia Department of Education. This web site can be found at: http://www.doe.virginia.gov/VDOE/middle-math-strategies/#. There are other websites that act as a search engine to filter for specific educational resources. The Primary School website provides not only lesson plans but links to other mathematic websites. This website can be found here: http://www.primaryschool.com.au/mathematics-lessons.php.
Before planning a unit on volume and capacity, teachers should assess their knowledge of the topic before attempting to develop a unit in collaboration with other colleagues.
In conclusion, teachers need to revise teaching approaches to limit textbook based teaching and to incorporate practical learning activities in volume units. It is important that pedagogy shifts to a more practical approach in order for students to achieve the all mathematic outcomes and indicators.
References
Beaumont, V., Curtis, R. & Smart, J. (1986). How to… Teach Perimeter, Area, and Volume. Virginia: The National Council of Teachers of Mathematics, Inc.
Board of Studies NSW. (2002). Mathematics K-6 Syllabus. Sydney: Board of Studies.
Booker, G., Briggs, J., Davey, G. & Nishet, S. (1992). Teaching Primary Mathematics. Melbourne: Longman Cheshire.
Department of Education and Training (2004). Teaching Measurement: Stage 2 and Stage 3. Bankstown: Department of Education and Training.
Department of Education and Training. (2004). Teaching Measurement in the New Syllabus: How Much will it Hold?. Curriculum Support for Primary Teachers. 9(2), 1-2.
Haylock, D. (2001). Mathematics Explained for Primary Teachers. London: Paul Chapman Publishing.
Szydlik, J. E., Szydlik, S. D., & Benson, S. R. (2003). Exploring changes in pre-service elementary teachers’ mathematical beliefs. Journal of Mathematics Teacher Education, 6(3), 253–279.
Wilson, P. S. & Osborne, A. (1988). Foundational Ideas in Teaching About Measure. In T. R. Post (Ed.), Teaching Mathematics in Grades K-8: Research Based Methods (pp. 98-142). Massachusetts: Allyn and Bacon, inc.
Zevenbergen, R. (2005). Primary Preservice Teachers’ Understandings of Volume: The impact of Course and Practicum Experiences. Mathematics Education Research Journal, 17(1), 3-23.
Zevenbergen, R., Dole, S. & Wright, R. J. (2004). Teaching Mathematics in Primary Schools. Crows Nest: Allen & Unwin.