Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 1
Measurement Concepts and Measuring Attributes with
Instruments
Students’ Understanding of Measurement
A report on primary school students’ understanding of science and technology in
Australia (Pattie, 1995) revealed that the effect of teaching was evident for the sample
of 1161 children from 34 government and non-government schools. The study indicated
that students performed at the higher levels when measuring temperature, length, and
mass, reading a classification chart, and using space relationships—mathematical skills
used in science and technology. However, such skills need to be further related to the
scientific context. There was no overall difference in performance between males and
females, or between children in urban, suburban and rural schools but there was a high
correlation between socio-economic status and children’s understanding of science
concepts and their performance of science skills.
Measurement and Conservation
Piaget originally thought that since pre-operational thinkers cannot conserve, then
certain learning experiences for measurement should be delayed. It was previously
considered that students could not explore the attribute, measure directly or indirectly
until able to conserve but experience in classrooms clearly indicates that the experiences
assist students to conserve.
Children’s Development
Piaget’s early research suggested that conservation of length, area, volume were
issues in learning. He felt volume was much later than length and area.
Later research suggested that informal play with capacity (filling containers with
sand or water) showed that conservation of area is more difficult than expected and that
volume (cm3 ) is more difficult than litre capacity.
Mass is also quite difficult to sense without hefting in your hands and is visually
confused with volume or capacity. A large, light object has a low density, and it has a
low mass. However, a small, heavy object has a high density, and it has a large mass.
Experiences with different types of objects is important.
Later research suggests that the issues of conservation were clouded by perceptual
dominance and experience and that measuring as a concept needs to link to everyday,
situated learning e.g. ruler.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 2
Learning Tasks for the Reader
Self-check on Conservation
experiencing
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 3
Generic Development of Measurement Concepts
No matter what attribute of measurement, certain critical aspects of measurement need
to be developed. These are:
Recognition of the attribute
The different representations of the attribute,
e.g. for length, we have length, height, width, distance, curved line, perimeter
e.g. for area, we have flat, horizontal surface, curved surface, combined surfaces
The idea of measuring
comparing size directly
comparing size indirectly
needing to be more versatile, precise, non-visual
The idea of a unit
The idea of a composite unit made of joined units
The idea of different base units
Count Me In Measurement provides six levels of development. The project provides
lessons for each level for each attribute: length, area, mass, capacity and volume.
Level 1: Identification of the attribute includes directly comparing and ordering
quantities
Level 2: Informal measurement includes finding the number of units to cover, pack or
fill a given quantity without overlapping or leaving gaps; knowing that the number of
units used gives a measurement of quantity; using these measurements to compare
quantities and realising that a quantity is unchanged if it is rearranged (the principle of
conservation)
Level 3: Unit structure includes replicating a single unit to cover, pack or fill a given
quantity, either by drawing or visualising the unit structure; and realising that the larger
the unit, the fewer units will be needed
Level 4: Recognise, measure and record in conventional units
Level 5: Use relationships between units and from the geometry to measure and
calculate in smaller and larger units
Level 6: Knowing and representing large units, consolidating and converting units, use
scale
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 4
Learning Tasks for the Reader
Measurement Activities 1
experiencing

Find an interesting object in the room, measure different aspects
of it and in different ways. What were critical aspects of your
measurement?

How many people can stand in the room? Compare the different
ways people do this and also think about short-cuts, especially
consider the natural units in the room.

Make circular, triangular, square, other prisms with the lateral
faces made by folding an A4 sheet of paper.

Describe which prism and why has the largest volume from A4
paper.

Show the length of string which is equal to the average height of
your group of three

Draw different shapes with an area of 12 square units. What did
you consider?

Make stacks of 24 cubes. Try making some stacks which are not
rectangular prisms. Make them with different surface areas.
Select a surface area and make stacks of different numbers of
cubes.

What did you have to know about to measure an object? Was it
the only thing that you could have measured about the object?
What does that tell you about attributes of objects and
measurement?

The area of the room could be measured using rectangular units.
What does that tell you about units? What does it tell you about
the development of the formula for the area of a rectangle? What
did you learn about composite units if you tried short-cuts?

What did you learn about the number of units needed when you
measure with smaller units?

Look at the three attached diagrams of squares on grids. Discuss
how these require an understanding of a grid and the idea of
composite units.

Was the comparing of the volume of different prisms, a task
about area or volume? Square centimetre paper could be used
for greater accuracy. How accurate would the measurement of
the base of the prisms be?
connecting ideas
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Ch. 7, page 5
Measurement Concepts and Measuring Atributes with Instruments
summarise
and record

The average height activity helps in developing the concept of
addition of lengths and the meaning of average. Explain.

Can a space measure a square unit but not be a square? What do
these activities tell you about shapes, the distinction between
shape and area, and the effect of perception on estimating area?
Do all the shapes for 12 square units have to be polygons? Why
and why not? How might the variance in shape for 12 square
units be used in advertising or packaging?

What do you learn about formulae for area and volume from the
tasks?

What previous
approaches?

Did you change your procedure/thinking during the
problems? Why?

What do these experiences tell you about the value of
problem solving?

What concepts did you use in finding the solution? How did
you refine your understanding of these concepts?

What do you think an area unit is?

What might be meant by composite units of area?
experiences
helped
you
with
your
Length
Boulton-Lewis, Wilss, & Mutch (1994) asked students to measure the length of two
lines made from joined matchsticks; each configuration had a recognisable pattern.
Younger students were likely to choose the familiar ruler rather than the unfamiliar
measurement units (sticks) in order to attempt to compare the lengths of the two lines.
Boulton-Lewis et al. suggested that the idea of introducing measurement with arbitrary,
informal units may not be appropriate for students if they are to grasp the concept of
measuring length because they do not have a mental model based on familiarity and
past experience of the arbitrary units. By contrast, many syllabus documents have
suggested that there be experiences with informal units before formal units. Willis
(2005) suggests that there is much more about using units than what type. To begin with
a unit is an abstract idea. The stick is only representative of that idea. Similarly, when
talking about gaps and overlaps in tiling for area, the tile (even if it is 1 square
centimetre) is a thing not the idea of an area unit that could take any shape. Early
research also showed that young children could recognise that they would need fewer of
a larger unit than a smaller unit to measure a fixed line.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 6
Area
There have been several studies on various issues related to this concept. Doig,
Cheeseman, and Lindsey (1995) investigated the effects of different material—paper
squares, Dienes’ blocks and wooden tiles—on children's success in measuring the area
of a rectangle. They found that children were least successful when they used the paper
squares and most successful when they used the wooden tiles. Children who used
Dienes’ blocks were most likely to confuse measurement of area with that of perimeter
or length. However, the use of paper squares revealed inadequate understandings of area
because students were more likely to overlap or leave gaps between the paper squares.
Consequently, practice in tiling with rigid materials may not help students’
understandings of area (Outhred, 1993). If mathematical concepts are to emerge, it
seems important that concrete experiences of covering areas should engage students’
visual imagery and analysis (Owens, 1994b) and should also involve student-student
and student-teacher interaction about the ideas needing development (Hart & Sinkinson,
1988; Owens, 1994a).
Clements and Ellerton (1995) interviewed a large sample of students on several test
items used in basic skills tests in Australia. They showed that there were notable
proportions of mismatches between correct/incorrect answers and nonunderstanding/understanding as probed during interviews. One of the items was to find
the area of a trapezium consisting of a square and a triangle (half the size of the square)
with only some lengths given. The study showed a large number of students did not
understand the concept of area, how lengths relate to area, why different shapes have
different formulae for calculating, or the value of visual/spatial knowledge. In another
study Clements (1995) illustrated a lack of conceptual understanding for a student able
to calculate the area of a triangle.
Young children often hear the word area referring to place, and may think of area as
somewhere to go—for example, the assembly area or the reading area—without
considering it as a region. They do not seem to realise that such regions are twodimensional (2D) spaces enclosed by boundaries and that they can be covered with units
(e.g., sheets of newspaper).
Students may have covered small regions such as desks, books, and chairs with
informal units and perhaps compared the two by counting the number of units needed to
cover them. Such activities, intended to be introductory to the concept of area
measurement, may in fact confuse students. The use of irregular shapes and informal
units (e.g., potato prints) may result in the activity being dominated by counting, while
ideas crucial to the concept of area measurement (e.g., overlaps, gaps, and congruent
units) are ignored (Outhred, 1993; Willis, 2005). Willis emphasised that students who
had counted to decide on the measure of an object’s length or mass were then unable to
use this information to answer a question about whether the object was heavier than
another or longer than another. A greater appreciation of the concept of covering would
seem to be necessary if older students are to calculate areas meaningfully (see
Mitchelmore, 1983, and Clements, 1995, for examples of typical student difficulties
with area calculations).
In observations of pre-school children covering squares, rectangles, and triangles
with smaller cut-out rectangles, squares, and triangles, Mansfield and Scott (1990) have
shown that students vary in their ability to choose appropriate unit shapes, in their
persistence, and in their turning and flipping tactics. The most difficult shape for the
children to cover was an equilateral triangle with a point facing down. Familiarity with
the shape to be covered seemed to increase success on the task.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 7
In a study by Wheatley and Cobb (1990), students were asked to cover a large
square by selecting shapes from a collection comprising a square, several triangles, and
a parallelogram. Some students chose only the parallelogram and tried to cover the
square with it, an approach which suggested to Wheatley and Cobb that the students
were matching lengths. Alternatively, students may have chosen the shape that appeared
to be largest. Other errors included leaving gaps, especially on the sides, and
overlapping pieces or the sides of the square.
Drawing may be one way of linking experiences with concrete materials to students’
mental models of tessellations. Several researchers (e.g., Mitchelmore, 1983; Outhred,
1993; Outhred & Mitchelmore, 1992) have suggested that drawing is an important tool
in developing students’ knowledge of rectangular arrays and in making links to
multiplication. Outhred (1993) found that many students had difficulties visualising or
drawing tilings of square units to cover rectangles when the squares were only shown on
adjacent sides of the rectangle or indicated by side marks, particularly for rectangles
with large dimensions. Some students’ drawings suggested that they did not understand
what features of arrays were important in constructing tessellations of squares. Owens
(1992, 1993) found that students in Years 2 and 4 had difficulties imagining tilings of
squares, rectangles, and triangles to cover larger shapes. For example, in the activities
illustrated in Figures 1a and 1b, they had difficulty in predicting the number of smaller
triangles that would be needed to cover the larger ones. Very few students commented
on the amount of space covered when asked what was the same about different
arrangements of five squares (pentominoes); nearly all focused on the number of tiles
(Owens, 1993).
(a) Tangram triangles
(b) Pattern-block triangles
Figure1. Shapes made during spatial activities (Owens, 1993).
The studies mentioned above, especially those by Outhred (1993) and Owens
(1993), emphasise the importance of spatial thinking and visualising when students
cover and compare shapes. To learn about tiling, students need to identify suitable units,
to transform shapes to other orientations, to recognise and partition shapes, and to
identify key features of shapes (e.g., matching parts such as right angles or equal
lengths).
Owens (1993, Owens & Outhred, 1998) examined the drawings of students in
Years 2 and 4 who were asked if specific units (squares, rectangles, right-angled
triangles, and equilateral triangles) could be used to tile figures and how many units
would be needed. Three factors that seemed to influence children's responses were
summarised by Owens and Outhred (1998):
1. Size of tiles. While children seemed to know that there was a pattern for filling
the space, some seemed to retain the shape but not the size of the tiling unit. Children
who drew tessellated tiles without regard for size usually felt that the space was being
adequately filled.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 8
2. Recognition of tile features. Students frequently used the sides and corners to
begin filling in spaces with tiles. The type of corner seemed important to some students
in deciding if a particular tile would be likely to fit. Students had difficulty recognising
a trapezium as a composite of tessellated right-angled triangles, despite prior experience
with concrete materials.
3. Judgments about drawings. Some students decided that overlaps or size or gaps
did not matter if the result was "close" enough, that is, they based their judgments on
their spatial sense and ignored slight discrepancies in their drawings. Children who
thought the tiles should fit often filled gaps with additional tiles, disregarding shape or
overlap. Others were influenced by inaccuracies in their drawings and said that the
shapes could not be made by fitting tiles together.
While Owens was carrying out her study, Outhred (1993) was independently
exploring children's difficulties in representing arrays and how such difficulties are
related to performance on area measurement tasks. Her research suggests that
knowledge of array structure provides a link between measurement and multiplication
concepts in the context of rectangular area measurement. She found that knowledge of
array structure was essential for children to relate the lengths of adjacent sides of a
rectangle to the number of squares that would cover it. These results indicate why
activities with concrete materials may not be sufficient to help children understand the
formula for the area of a rectangle. When measuring the area of a rectangle using
concrete materials children do not require awareness of row and column structure
because the structure is determined by the materials, rather than by the child's thinking.
The effects of specific types of instruction on children's use of lines to represent
rows and columns was investigated with children in Years 1 and 2 (Outhred, 1993). The
findings suggested that teaching children that squares in an array were all the same size
was not the most effective method to help children to perceive array structure. Teaching
children that the squares are aligned or that there are the same number of squares in
each row (column) seemed to be more effective methods for moving children from
drawing squares individually to representing rows (columns) of squares using lines.
McPhail (1997) has continued to explore how young students develop knowledge
about tessellations and area. With a series of four lessons she has shown that young
students in Years 1 and 2 can learn about area. Her lessons allowed students to first
make their own area enclosed by a length of braid. The children also painted or rubbed
large triangles and squares which were later used to make large visual displays of
tessellations. In addition, the students had many small squares and triangles. They were
asked to make large squares and triangles as well as covering given ones. The cardboard
tiles had the edges in black so that arrays were easily seen. This seemed to facilitate
children drawing tilings using arrays rather than individual tiles. Interestingly, the
children applied many number facts in telling the teacher how many tiles they had used.
Some used the ideas of repeated addition of rows of tiles, others counted a number and
then added on the subitised remaining number of tiles. Students could explain the
patterns of the triangular covering.
Willis (2005) provides another challenge in saying that a tile is a measuring
instrument. Rigid tiles for area may not cover a thinner object without discussing cutting
up the tile. Unlike the fluid units used in capacity that flow around and fill the container,
this does not automatically happen with area. They concluded that understandings
should include:
(a) the instrument we choose to represent our unit should relate well to the attribute
to be measured and be easy to repeat to match the thing to be measured;
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 9
(b) to measure consistently we need to use our instrument in a way that ensures a
good match of the unit with the object to be measured;
(c) units are quantities and so we can use different representations of the same unit
so long as we do not change the quantity (Willis, 2005).
Volume
Toh (1993) compared the effectiveness of computer simulated experiments with
that of parallel instruction involving hands-on laboratory experiments for teaching
volume-by-displacement concepts. The purpose of the simulation was to have students
test their misconceptions rather than simply being told about erroneous misconceptions.
The study consisted of 389 students from 6 Malaysian schools. The results indicated
that the computer-assisted group was significantly better in terms of learning gains in
the cognitive categories of knowledge and application. The computer simulated
different conditions such as same shape, different masses; or same mass, different
shapes. While it is not easy to do this with concrete materials, nevertheless the
experience assists in moving students on from limited conception about the volume by
displacement.
In summary, teachers
 should be aware of the confusion between area and perimeter because students
do not develop the concept of area, they do not develop area formulae, or they
are just told to use the formula (with the meaningless idea of lengths becoming
area rather than the formula involving numbers only without the unit attachment
and that it is a formula associated with a specific shape, e.g. rectangle);
 can establish the area concept through painting an area, tiling, discussing no
gaps, and the nature of shapes;
 should not use just solid tiles that structure exercise and prevent abstraction, or
lead to counting exercise rather than an area experience;
 should know the value of tiles that are not square;
 should know the value of recognising patterns and drawing grids (and discuss
drawing difficulties);
 should recognise the importance of visualising and estimating;
 should build on children’s intuitive dissecting of areas to assist in calculating.
Units
Students need to develop concepts like area that are measured. Spatial experiences
and knowledge about shapes will help when comparing informally or directly or when
selecting a unit for measuring. Later knowledge of the properties of shapes will assist
students to calculate areas.
We also use standard units so that there is no confusion. Students need to be
familiar with these, to know about how big they are, and to be able to estimate in these
units. In particular, students need experiences in (a) selecting objects that represent the
unit, (c) estimating, and (d) measuring in order to develop a sense of these units.
Composite Units
An important concept in measurement is that of a composite or iterable unit. For
example, when the young students put tiles in rows and count by rows, for example, 3,
6, 9, they are using the row as a composite unit made up of 3 units. This idea is also a
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 10
basic concept in both multiplication and our place value system with a ten being a
composite unit of 10 ones.
While students first develop a sense of units which are within their grasp, they
expand these into larger and smaller units using the notion of composite unit. Students
eventually need to be able to change from one unit into another. A good understanding
of the composite unit will assist this procedure.
Metric Units and the Place Value System
Measurement can assist students to develop their understanding of the place value
system. For example, students can read off the length of an object from a ruler in (a)
metres, (b) metres and centimetres, and (c) centimetres. The ruler can establish the idea
that a metre is a composite of 100 centimetres but also that a hundred is a composite of
100 ones. More importantly, the idea of one being a composite of 100 hundredths is
also established. So, for example, a string might be 1.23 m or it can be written as 123
centimetres or 1 m 23 centimetres. Activities allowing for these various descriptions
will assist students to make the links. They will take time.
Recognising Structure
Students need to recognise structure in order to develop their measurement
concepts. Mulligan expresses this in the following diagram. See your earlier activities
on area of a room, making a ruler and area sheets for units.
Learning Tasks for the Reader
Self-check on Measurement Sense
experiencing
1. At what temperature does a person suffer from hypothermia?
a. 25o
b. 30o
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 11
c. 35o
d. 40o
e. 45o
2. The height of a child in Year 2 is about
a. 1 cm
b. 50 cm
c. 100 cm
d. 150 cm
e. 1 000 cm
3. A small fish tank would hold about
a. 1 L
b. 20 L
c. 100 L
d. 1 mL
4. The area of floor in front of the desks is about
a. 1 000 m2
b. 100 m2
c. 10 m2
d. 0.5 km2
5. A house block has a house with gardens covering the same area
as the house. The area of the block is about
a. 0.5 hectares
b. 1 hectare
c. 1.5 hectares
d. 2 acres
6. An A4 sheet is about
a. 10 cm2
b. 100 cm2
c. 1 000 cm2
d. 10 000 cm2
7. The space taken up by an engine of a small car is about
a. 1.4 L
b. 1.4 m3
c. 1.4 m
d. 1.4 kg
8. A litre of water has the same mass as:
a. a commonly available bag of rice
b. a house brick
c. a can of condensed soup
d. a dozen eggs
e. 2 apples

Think about this problem. If you hold a man’s handkerchief
diagonally, how many are needed for the length of a horse?

Before wheelie bins, how high was the garbage bin?
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
connecting ideas
Ch. 7, page 12

What role does a sense of size play in measurement?

How did (a) visualising; (b) experience with the unit; and (c)
prior experience impact your decision-making?

What influences the necessary degree of accuracy?

What are some ways that you can encourage students to develop
a good sense of volume and area?

Why should students have many activities to encourage them to
construct both a sense of area and formulae for the area of a
rectangle and a triangle, and the volume of a prism.

Why are square units so useful?

Look in the Syllabus at the experiences that students need for
developing their own area formula for a rectangle, a rectangular
prism and a triangle

Include in your summary for area comments on:
summarise
and record
(a) the importance of number of rows and number of square
units per row
(b) how to deal with sides that are not whole numbers (e.g.
folding rectangular areas)

Include in your summary for volume comments on:
(c) Number of layers and number of cubic units in each layer
(d) The link between 1 cm3 and 1 mL.
Outcomes for NSW Mathematics K-6 Syllabus
Table 1 gives the NSW outcome statements for measurement.
Table 1
Outcome Statements for Measurement
Early Stage 1
Stage 1
Stage 2
Length
Area
MES1.1 Describes
length and distance
using everyday
language and
compares lengths
using direct
comparison
MES1.2 Describes
MS1.1 Estimates,
measures, compares
and records lengths and
distances using
informal units, metres
and centimetres
MS1.2 Estimates,
MS2.1 Estimates,
measures, compares
and records lengths,
distances and
perimeters in metres,
centimeters and
millimeters
MS2.2 Estimates,
Stage 3
MS3.1 Selects and uses
the appropriate unit and
device to measure
lengths, distances and
perimeters
MS3.2 Selects and uses
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Volume
and
Capacity
Mass
Time
Passage of
time, its
measurem
ent and
representat
ions
area using everyday
language and
compares areas
using direct
comparison
MES1.3 Compares
the capacities of
containers and the
volumes of objects
or substances using
direct comparison
MES1.4 Compares
the masses of two
objects and
describes mass
using everyday
language
MES1.5 Sequences
events and uses
everyday language
to describe the
duration of
activities
measures, compares
and records areas using
informal units
Ch. 7, page 13
MS1.4 Estimates,
measures, compares
and records the masses
of two or more objects
using informal units
measures, compares
and records the areas of
surfaces in square
centimeters and square
metres
MS2.3 Estimates,
measures, compares
and records volumes
and capacities using
litres, milliltres and
cubic centimeters
MS2.4 Estimates,
measures, compares
and recordsmasses
using kilograms and
grams
the appropriate unit to
calculate area,
including the area of
squares, rectangles and
triangles
MS3.3 Selects and uses
the appropriate unit to
estimate and measure
volume and capacity,
including the volume
of rectangular prisms
MS3.5 Selects and uses
the appropriate unit and
measuring device to
find the mass of objects
MS1.5 Compares the
duration of events
using informal methods
and reads clocks on the
half-hour
MS2.5 Reads and
records time in oneminute intervals and
makes comparisons
between time units
MS3.5 Uses twentyfour hour time and am
and pm notation in
real-life situations and
constructs timelines
MS1.3 Estimates,
measures, compares
and records volumes
and capacities using
informal units
Measuring Instruments
Students need to understand measuring instruments. For example, the scale on a jug
looks like a ruler for measuring length but it is indicating volume. How can we get
students to understand that? Most rulers have gradations that are equally spaced, i.e. 1
and 2 are spaced at the same distance as 2 and 3. This is not always the case depending
on the purpose of the instrument. Students also have to understand how to read the
gradations that are not marked and they need to know that the number is not the point
but the measure from the start. For example, it is the amount of water in a cup; at zero
the cup is empty.
What Instruments can we Make for Measuring
If students are going to appreciate how a ruler works and that the numbers on the
ruler are representing the length from the start of the ruler, then they need to make a
ruler. It can be done by lining up some base 10 long blocks and marking off and
numbering a strip of paper, cloth or plastic. Alternatively they can make lengths of 1
cm, 2 cm etc, put each length on their long strip of paper with the starts together and
marking the length and writing down the length on their long strip.
0
5
Figure 1. Lengths of paper being used to make a ruler.
A volume measure can be made by using a jar and lids. As the student puts in
another lidful, they mark where the water comes to and the number of lids now in the
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 14
jar. This helps students to see the linear marks on the measuring jar as representing
volume.
Students can bring to school all sorts of measuring instruments that can be found in
the home and garage. All sorts of gauges are used. These can be discussed even if the
students do not fully understand what is being measured. They can see the needle
moving through the numbers on bathroom scales; they can look at the different widths
of the sparkplug gauges; they can watch an amp metre needle swing.
Students can make various time clocks.
Learning Tasks for Readers
Making Measuring Instruments
experiencing

Make a pair of calipers to pinch the fat on your back to measure
fat.

Make a tapered diameter measure to see how big a ring you
make when you touch your thumb and forefinger.

Make a tiny trundle wheel to measure around your leg or waist
in cm (the wheel can have a circumference of 10 cm.)

Make a balance - spring balance or an equal arm balance.
 What is a measuring instrument?
 Outline some important early and later experiences that
students should have for establishing the concept of mass,
connecting ideas

What important concepts in measurement should students learn?

What are important concerns in teaching about measurement?
summarise
and record
Using Measuring Across the Curriculum
Clearly measuring is an important skill in Science and Technology. Students
should investigate in science and use measuring as a tool. For example, take different
brands of nappies and investigate which is the best.
For older students, two great tasks are from MCTP Activity Banks (Lovitt &
Clarke, 1989) called Danger Distance and Map of Australia. Both encourage
visualisation with measurement.
Finding old measuring instruments around the farm, old mine site or dump,
designing and making measuring instruments can also be fun.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 15
Measuring and reading measurements and interpreting data are all important ideas
in investigating nature and studying, for example, animals.
Human Society and Its Environment provides plenty of opportunities for using
measuring and interpreting information. Map work and built environments are just two
areas.
For Personal Development, Health, and Physical Education, there are many
ideas. What does it mean to have a pulse rate of 60 beats per minute? What does it
mean to have diversity in height at a particular age group? What does it mean to have
10g of fat per 100g, compared to 1g of fat per 100g?
Learning Tasks for the Reader
Mathematics and the Human Body
What is our lung capacity?
experiencing
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What is meant by lung capacity?
How can we find out the volume of air in our lungs?
There are machines which can be discussed.
One approximate way is to blow up a balloon with one
breath.
Discuss how you can get the volume of the balloon. One way
is by putting the balloon in a full bucket of water and
measuring the displacement of water.
Remember that fit people and non-smokers improve their
lung capacity. Lung capacity grows from childhood to
adolescence.
How much skin do we have?
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Estimate how many sq cm for the sole of your foot.
What is the skin mathematically and what is meant by the size
of our skin?
A discussion on this questions should be about covering of the
surface and surface area. Areas can be in different forms
including surface areas and this can be modeled by wrapping
with newspaper.
Discuss how to find the area of different parts of the body.
Arms can be represented by curved cylinders which can be
flatten out to rectangles (or near rectangles). Discuss ways of
measuring the rectangles with informal square units.
If you add up the size of all the parts of the body surface in sq
cm and try to convert to sq m, ask yourself whether the answer
seems sensible. How do you convert sq cm to sq m?
A rule-of-thumb for estimating the total amount of skin is to
multiply the size of the sole of the foot by 100. This is then a
quick way of deciding on the percentage of skin burnt on a burn
victim. Try out this rule-of-thumb and decide what percentage
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
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connecting ideas
Ch. 7, page 16
of skin an arm would be.
Compare the sizes and ratios for a small child and for an adult.
Discuss the following suggestions for activities. Think about:
 the equipment that might be needed,
 the different approaches that might be taken,
 questions that students may ask and how to answer their
questions, and
 how to get them to investigate.
How much carbohydrate food is in a packet of crisps? How
much in a potato? Which is better food?
(Health note: We need quite a bit of carbohydrates with fibre but
not much fat each day.)
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Students undertaking this investigation may:
- investigate the nutrition information
- ask about why grams are used, learn that g is the symbol for
grams
- weigh potatoes, decide what an average one might be
- compare the amount of oil in a packet of crisps by weighing
that in cooking spoons
Do we drink enough water?
(Health note: Children should drink about 8 cups per day.)
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Compare drinking bottles, glasses, and other drink containers.
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Discuss how we can compare - cups, L, mL (depending on
age).
(Teachers need to think about how they can measure bubbler
drinks.)
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Compare the different shapes of containers with the same
amount of water.
Students can extend the activity and measure more accurately.
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Discuss how the perceptions of size are used in marketing and
how different shapes help storage and handling. And here we
can link in the eyes and the brain interpreting what we see and
the effects of stored information.
How big is your heart?
The heart is said to be as big as the person’s fist.
Think about and try out how to get its volume (by displacement of
water).
How flexible is our heart and how does it respond to help us?
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
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summarise
and record
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Ch. 7, page 17
Find your pulse and count how often it beats. Do this when you
enter the class after running around at recess. It is usually a little
stronger and it will be easier to find, especially at the neck. Then
discuss how to count it.
Discuss the idea of rate. It is not an easy idea but really
important for some discussion in primary school as it is a major
idea in later studies. One way of doing this is to show that in
quarter of a minute is quarter the number in a full minute. The
ratio is the same.
Take the pulse rate after sitting for awhile and then after some
vigourous activity – the same for the whole class.
Compare different pulse rates and discuss how adaptable the
heart is to meet their needs.
If some studnets are regular swimmers, exercise regularly, or
run around more than others, you might be able to compare their
slower rates after exercise (or how much quicker their rate
returns to normal). Discuss the effect of fitness on heart rate.
Why do students learn more about mathematics through:
 investigations and
 real-life contexts?
How do you make sure that you are covering mathematics and
other Key Learning Area outcomes when giving students
investigations like those listed above?
Can these activities be modified for different age groups? If so,
give some examples.
Planning Events, Times and Calendars
The ancient Babylonians were keen astronomers, astrologers, and travellers. They
made links between the number of days in a year and the time it took for a cycle of
seasons to pass and the earth to rotate around the sun.
We have 365 days as closer to the time taken to complete the cycle but the number
360 is more useful because it can be divided up into many different ways.
Calendars vary from place to place and culture to culture. Yearly and daily periods
of time are described differently in different cultures. A diagram that shows how one
Indigenous Australian tribe describes the time of the year shows the close links between
times and knowing when to fire the grass so that bushfires do not start and when to get
certain food. The overlap of events is easiest represented by segmented concentric
circles. One circle represents the wind seasons, another the plant seasons and so on. Any
period can be determined by the coincidence of these events (Harris, 1989).
Planning Events and Feasts. Many Pacific Islanders organise large feasts. There
is much mathematics involved in deciding on the quantity of food to prepared in the
Marae kitchen, how many bundles (about one for every 50 people) are needed and how
to go about gathering the food together. For people who are growing the food, months
of preparation is involved in planning gardens so that the food is ready for the feast.
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens
Measurement Concepts and Measuring Atributes with Instruments
Ch. 7, page 18
Learning Tasks for the Reader
Planning Events, Times and Calendars
experiencing
connecting ideas
Do you know why we have 360o in a circle?
 What numbers go exactly into 360?
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What events or gatherings do you plan? What mathematics do
you use when you plan for these events?
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Look closely at the lessons on time in the Syllabus. Compare this
with the set of outcomes in Table 1.
Which has greater emphasis in Kindergarten: (a) recognising the
attribute of time by comparing the time taken for events, or (b)
reading the clock.
What are some difficulties that young students will have reading
analogue clocks?
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summarise
and record
Summarise the new ideas that you have learnt about
measurement, especially time, as a result of considering the
differences in cultures.
Summarise the ideas of
 attribute
 composite units
 culturally determined measures
 metric system
 links with decimal place value
Creating Space: Professional Knowledge and Spatial Activities for Teaching Mathematics
Kay Owens