12th International Congress on Mathematical Education
Program Name XX-YY-zz (pp. abcde-fghij)
8 July – 15 July, 2012, COEX, Seoul, Korea (This part is for LOC use only. Please do not change this part.)
STRIVING TO MAXIMISE CHILDREN’S LEARNING OF MASS
MEASUREMENT
Andrea McDonough
Australian Catholic University
[email protected]
Jill Cheeseman
Sarah Ferguson
Monash University
Australian Catholic University
[email protected]
[email protected]
This paper reports on one key element of a design experiment involving the teaching and learning of
the measurement of mass with 119 children aged between 6 and 8 years. A sequence of lessons was
developed to offer the children opportunities to experience and grapple with rich and challenging
tasks. We designed hands-on lessons focused on key measurement understandings of comparison and
unit and provided opportunities for children to work with both non-standard and standard units. The
children’s engagement with the tasks and the mathematical ideas, evident in the rich discussions that
ensued and results from assessment data collected before and after the teaching suggest that we had
some success in working towards our goal of striving to maximise learning of mass measurement. We
acknowledge that it is likely that along with the tasks, other factors including teacher actions, are
likely to have contributed to children’s growth in understandings.
Mass measurement; Foundational ideas; Rich tasks; Early years
INTRODUCTION
Teachers’ major concern is students’ learning and they strive to provide effective learning
opportunities in their mathematics classrooms. We know that teachers play a key role in
improving learning (e.g., Sullivan & McDonough, 2002). Our investigations into the learning
of mass measurement were informed primarily by research on the learning of measurement
particularly taking account of the development of foundational ideas of measure. Our use of
tasks was influenced also by research on effective teaching of mathematics. There is
widespread concern for effective teaching of mathematics (e.g., Australian Association of
Mathematics Teachers, 2006; National Council of Teachers of Mathematics, 2000). Research
publications report practices of effective teachers of mathematics (e.g., Brown, Askew, Baker,
Denvir & Millett, 1998; McDonough & Clarke 2003; Muir, 2007) that include having a clear
mathematical focus; requiring thought rather than practice; emphasising and fostering
meanings and connections; using open-ended tasks; and allowing autonomy for students to
develop and discuss their own methods and ideas. Knowledge of such practices can inform
teaching of measurement, including the design of tasks and the ways that those tasks are used
in the classroom.
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McDonough, Cheeseman, Ferguson
In the teaching and learning of measurement, students learn about several attributes of
measure including length, mass or weight1, area, angle, time, temperature, and volume. They
also develop understandings related to foundational or key ideas such as comparison,
identical unit, unit iteration, number assignment, and proportionality (Lehrer, Jaslow, &
Curtis, 2003; Wilson & Osborne, 1992). However, while “the basic idea of direct
measurement is quite simple” (Wilson & Osborne, 1992, p. 91), there are complex mental
accomplishments within measuring (Battista, 2006; Stephan & Clements, 2003) which are
often downplayed in typical measurement teaching (Stephan & Clements, 2003). Kamii
(2006) believed that typical instruction focuses more on measurement as an empirical
procedure, such as placing paper clips along a pencil and counting them, rather than a
procedure requiring reasoning. The incorporation of opportunities for children to reason, with
the purpose of coming to understand foundational or key ideas of measurement, can be
enhanced by task design and teacher actions in the utilization of those tasks. In this paper we
examine a selection of tasks and the ways they were utilized in lessons as the teachers strove
to maximize children’s learning of mass measurement.
In an earlier paper (Cheeseman, McDonough & Clarke, 2011) we reported data from 479
Australian Year 1 students which showed that little progress in understandings of mass
measurement had been made over the period of a year and thus we argued that rich
experiences involving measuring mass were needed, particularly at the Year 1 level (6 years
of age). We also emphasized the importance of teachers assessing children’s understandings
of mass measurement and structuring learning opportunities to build on and extend those
understandings. The research reported in the present paper was developed in response to that
advice.
FOUNDATIONAL IDEAS OF MEASURE
Young children’s learning of measurement involves not only the use and understanding of
procedures but also the development of conceptual understandings. In the literature these are
commonly discussed in relation to length measurement (e.g., Battista, 2006; Lehrer et al.,
2003), but they can be transferred to some other measurement systems (Wilson & Osborne,
1992) or attributes. Some of those foundational or key ideas, as relevant to the focus of the
tasks discussed in this paper, are considered here.
Comparison
Critical in the learning of measurement is an understanding of comparison, that is, that “like
properties can be compared to see which is greater” (Wilson & Osborne, 1992, p. 92). In
considering masses we make comparisons by hand (hefting), and by use of units and scales.
Outcomes from non-unit comparison of mass are described with mathematical language such
as “heavier” and “lighter”. While comparison without units or numbers can be seen as
non-measurement reasoning (Battista, 2006), the understanding of comparison underpins
more sophisticated measuring that utilizes numbers and units. Indeed, Kamii (2006) argued
that it is essentially for the purpose of making comparisons that one measures.
Two concepts related to comparison that are generally considered by researchers (Stephan &
Clements, 2003) including Piaget, Inhelder and Szeminska (1960) as necessary for complete
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understanding of measurement are conservation and transitivity. Conservation is the
understanding that an object retains its size when moved or subdivided; transitive reasoning
involves understanding the relationship between the measures of three items.
Unit
Another key measurement idea is unit (Lehrer et al., 2003; Wilson & Osborne, 1992). For
mass, examples of non-standard units are counters or metal washers; standard units include
the gram and kilogram. Balance or spring scales are used as appropriate.
For length measurement, the unit is placed end to end (iterating) alongside the object, that is,
the object is subdivided (mentally and physically) by the unit. In this process the size of the
unit remains the same, the units or subdivisions must be identical, the unit is seen as part of
the whole, and the unit is translated successively. For mass measurement, while the size
(mass) of the unit must remain the same, the unit is not laid directly in line with the object. For
example, when using non-standard units a balance scale is required to make the measure.
Measuring in standard units of mass may involve the use of balance scales with physical
weights, or some type of spring or digital scale.
Foundational ideas of measure, of which the above are just some identified by researchers,
provide the conceptual context in which to consider the tasks discussed in this paper.
OUR VIEW OF LEARNING
A social constructivist perspective on learning informed our development of the teaching
experiment discussed in this paper and the tasks used within it. That is, we recognized that
knowing is active, “individual and personal, … that it is based on previously constructed
knowledge” (Ernest, 1994, p. 2), and that the knowledge is not fixed, but rather is socially
negotiated and is sought and expressed through language. Agreeing with Stephan and
Clements (2003), we believe that while measurement instruction may typically focus on the
procedures of measuring, that is, on teaching children how to measure, these physical
activities are underpinned by conceptual activity related to the development of foundational
ideas of measure. With an awareness of the importance of children constructing
understandings of foundational ideas of measure, we sought to develop rich tasks that had the
potential to lead to conceptual understanding in mathematics, that challenged children to
think and that fostered the development of mathematical reasoning.
METHODOLOGY
The study for which the tasks discussed in this paper were designed might be considered
design research (Cobb, Confrey, DiSessa, Lehrer & Schauble, 2003) as it involved a
classroom experiment in which the research team collaborated with a teacher who took
responsibility for instruction and who was also a member of the research team. Of the three
phases in the study, the first was the administration of individual one-to-one, task-based
assessment interviews (see Cheeseman et al., 2011) to collect baseline data on children’s
understandings of mass measurement; the second was the teaching phase of five lessons
(three weeks after the initial interviews); the third was another set of one-to-one interviews
for further assessment of children’s learning (three weeks after the teaching). A detailed
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discussion of the data and their collection can be found in Cheeseman, McDonough and
Ferguson (in press).
In addition to these quantitative data, observational data were collected to gain further
insights into young children’s thinking about mass. Notes, photographs and audiotapes were
used to document elements of the lessons.
A sequence of lessons on the topic of mass measurement was developed by the authors for use
with the 119 Year 1 and 2 children in five mixed ability classes at one school. Each class was
taught five one-hour lessons over one week, with three of these taught by the third researcher
who was also a practising primary school teacher, and two by the classroom teacher (who had
observed the third researcher teach the lesson earlier that day).
In designing the lessons, which were to replace the existing school planning for mass
measurement, we were aware that traditionally the teaching of measurement concepts is
sequenced from nonstandard to standard units (Outhred & McPhail, 2000) and that
commonly children explore each attribute informally long before encountering standard units
or standard tools for measuring (Clements & Battista, 1986). However, we were cognisant
also that more recent studies have questioned whether this sequence is the best approach
(Clements, 1999; Clements & Bright, 2003). We were aware also of the Australian
Curriculum (Australian Curriculum Assessment and Reporting Authority, 2012) that
indicates that for the Foundation level (first year of school) the expectations relate to use of
comparisons to determine which is heavier, that for Year 1 does not include a specific
statement regarding mass, that for Year 2 identifies the comparison of mass through use of
balance scales as the expectation and that for Year 3 includes use of metric units to measure,
compare and order mass. We took account of these various messages, but also of our own
advice, as identified above, that came from previous research (Cheeseman et al., 2011) and
that advocated the use of rich and challenging tasks.
The lessons in the present study were designed to follow typical growth of children’s
concepts of measurement identified in the literature (Outhred & McPhail, 2000) and
accordingly they began with comparing mass and moved toward quantifying mass using
standard units of measure. However, the difference was that we included a more rapid move
towards standard units, with both Year 1 and 2 children being given the opportunity to work
with these.
THE TASKS
The five lessons were designed to engage the children in rich mathematics experiences
including real world applications, problem solving and play-based approaches (Baroody,
2009; Copley, 2006) and children were involved in hands on measuring during each lesson,
following the advice of Wilson and Osborne (1992, p. 119) that “children should measure
often, preferably on real problem tasks … [that] encourage discussion to stimulate the
refinement and testing of ideas and concepts through oral interactions”. Each task is described
in turn, starting with the title and mathematical focus, followed by a list of materials, a brief
description and some discussion in light of our efforts to provide rich tasks that maximised
learning opportunities.
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Task 1: Party Bag Surprises
Mathematical focus: Comparing and ordering by hefting and use of balance scales
Materials: Objects in the room; opaque party bags; balance scales
The lesson began with a discussion of heavy and light, with children identifying items in the
classroom to demonstrate what they meant by these terms. They then worked in pairs,
choosing and placing objects from the classroom in small opaque party bags to create three
sealed bags of different masses to be ordered by hefting, that is, by holding a bag in each hand
and comparing masses. The children were challenged to make their bags “tricky”, so they
deliberately made them similar in mass which led to careful comparisons by other children.
Once each pair had filled and sealed their three bags, another pair of children was challenged
to order by hefting (without being able to see what was inside each bag, that is, without visual
cues). Where agreement could not be reached, balance or digital scales were available if
children chose to use them for further comparison of mass.
Discussion: Of the foundational ideas of measure discussed above, this lesson focused on
comparison. With a focus on comparison and without use of units, the lesson may appear
simple. However, it presented challenges requiring the children to reason about the
mathematics. Firstly, because of mass being the amount of matter in an object and being
invisible, the lesson posed challenges beyond those of the appearance based comparison that
occurs, for example, when measuring lengths (Battista, 2006). When asked to heft and order
three bags, children also needed to reason about the relationships between the masses of the
bags in a transitive manner (e.g., if A<B, and B<C, then A<C). The teachers commented that
some children appeared to have difficulty comparing the three bags methodically. To
illustrate, one child held two bags in one hand and one in the other and unsuccessfully tried to
order the three bags. Another child suggested he should “find out which one [of two bags] is
lighter then hold the other one and see what that is like with the first one”. Letting the children
grapple with this task allowed the difficulty with transitive reasoning to become apparent and
showed that children of 6-8 years will work together to solve such problems. A further
challenge became apparent in one class when at the end of the lesson a child, who the teacher
described as confident and quite capable, shared with the class that he believed that the first
bag put in the scale would always be the heavier bag. Another child offered an explanation:
“no, it was just because it went in first, it has to go down”. The teacher intentionally held back
from telling and the class discussed and tested the two children’s theories. The teacher
responses in these two instances might be described as allowing autonomy for students to
develop and discuss their own methods and ideas.
The Party Bag Surprises task allowed a focus on comparison of mass in a context that was not
only engaging for the children but that also challenged them to estimate, compare, reason and
explain. The success of the task was due partly to the actions of the teachers who held back
from telling, elicited children’s thinking, and encouraged testing of children’s hypotheses.
Task 2: The Three Bears
Mathematical focus: Choosing and using non-standard units to quantify mass
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Materials: Plastic “Three Bear” teddies of graduating sizes, balance scales, objects to weigh
This lesson was the first within the five-lesson sequence in which there was a focus on
measuring and quantifying mass. The children were asked to choose objects from within the
classroom, and measure and record the mass of the objects. Choosing from three possible size
bears as the unit, for each measure the children noted the type of bear and the number of these.
A discussion followed regarding the mass of items they had weighed using different types of
bears. They found, for example, that less Papa bears were needed than Baby bears to weigh
the same object. Using a mix of bears to weigh the same object was also discussed.
Discussion: An understanding of unit is a foundational idea of measure. Following the
traditional approach, our instruction sequence included non-standard units before standard
units. In this lesson, which included use of non-standard units, children had opportunities to
think deeply as they were required to select the size of the teddy unit as well as to choose the
objects to be weighed. Thus there was the possibility that their findings would not be “neat”
and therefore might challenge them. Perhaps it was because of a difficulty in having partial
units of mass with the plastic teddies that some children chose to record with mixed units, for
example 3 Papa bears and 1 Baby bear. The task also allowed the children to consider how
lighter masses, the Baby bears, allowed for a more accurate measure. Not only did the lesson
give the children the opportunity to investigate and discover in relation to non-standard units,
but also to explore and experiment to find how the balance scales worked. Some children
became intrigued by one characteristic or feature of the scales. Those who became fixed on
the pointer or on having the beam across the two pans sit flat appeared to lose sight of what
they were weighing, thus putting a combination of teddies and objects in both sides of the
scales to achieve the balance. The challenges and openness of the lesson gave opportunities
for the children to learn about units of mass measurement and the tools used to measure, and
gave the teachers opportunities to learn observe, listen to and build on children’s responses.
Task 3: Post Office Parcels
Mathematical focus: Using standard units to measure mass in grams
Materials: Balance scales, parcels, cubic centimetre blocks (“centicubes”) in singles and
sticks of ten, Post-it notes
This lesson involved children measuring the mass of various envelopes and packages in a
“Post Office”. The teacher began by asking the children if anyone had heard the word gram
and what they thought it meant. The teacher took the opportunity to listen to the children’s
ideas without judgement. Following the discussion, the teacher held a centicube and said,
“This is a centicube. It weighs one gram”. This centicube was passed around so that each
child could hold it. Then the teacher showed the class a stick of 10 centicubes and established
that it weighed 10 grams. The children were invited to measure the mass of various post office
parcels using the balance scales and sticks of ten centicubes and the loose centicubes. The
recording of the mass in grams was modelled. The children worked in pairs to measure the
mass of at least two parcels and record the result in grams.
Discussion: This lesson was a good way to bring in a real world application of weighing
items. Once again the focus of this task was the foundational idea of unit, but with the
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standard unit of the gram being introduced. At the end of the lesson the teacher asked if
anyone had found any “surprises” or made any “discoveries” during the task. Many children
spoke of being surprised at how light one gram was. A boy who during the initial discussion
within the lesson had said that he weighed 30 grams now stated “I realised that I couldn’t”.
This activity may have helped the children develop a feel for or a benchmark for one gram.
The relationship between the physical appearance of the parcels and their mass also provided
“surprises” for some children who noticed that the taller cylinder was lighter than a shorter
cylinder. This led to a discovery that how an object looks does not necessarily determine its
mass.
Task 4: Weighing stations
Mathematical focus: Quantifying mass in non-standard and standard units
Materials: Various materials including fruit and vegetables, packaged foods, balance scales,
rice, pasta, non-standard units such as teddy bears and metal washers, standard mass weights
(e.g., 5g, 10g, 20g, 1kg), analogue kitchen scales, and digital scales
In this lesson children were offered a selection of experiences with a range of objects to be
weighed and a range of measuring equipment. The tasks from which the children could
choose included: estimate and then measure to fill a plastic zip-lock bag with 100g of rice (or
another material as selected); choose five potatoes, order from heaviest to lightest and check
using units (balance scales and non-standard units and standard mass pieces were available);
investigate the masses of kitchen items noting the label on the packing and the mass as
weighed with kitchen scales; investigate the masses of various fruits and vegetables through
estimating by hefting, ordering and checking with standard mass pieces and balance scales.
The children also had the opportunity to explore the Post Office Parcels task again.
Discussion: This lesson had elements of a play-based approach to learning with the children
able to freely explore tasks that interested them, persisting for as long as they chose and
working in fluid groups, pairs or individually. There was sufficient equipment for six children
to be working at each station. For the teacher, the lesson offered the opportunity to observe
and assess the children as they worked and engage them in rich conversations. It was
interesting to note that in this lesson most children completed more than one task but some
focused on just one. For example, one child was observed persisting for some time to find the
mass of one potato using balance scales and a set of mixed weights. After many minutes
absorbed in this task, he was able to proudly say that the potato had a mass of 275 grams.
Other children became engrossed in exploring the packaged foods when they discovered that
the mass given on the packages did not match the mass they found when using the digital
scales. While the teacher listened, a lively discussion ensued. Finally the children agreed that
the factory had weighed only the contents and not the packaging itself, which had led to the
heavier mass they had found for each packaged food. The teacher also observed that a few
children were still mixing units and items to be measured in the balance scales. Others were
using a mix of standard and non-standard units to measure but were essentially considering
the standard units as non-standard “coloured blocks” rather than attending to the grams each
weighed. This lesson was a rich learning experience with the conversations the teachers had
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with the children playing an important part in potential learning. The teachers stated that this
lesson would benefit from being repeated, giving the children other opportunities and more
time to explore the weighing stations, perhaps consolidating their learning.
Task 5: Making a set of weights
Mathematical focus: Creating a set of useful weights to develop benchmarks for common
masses
Materials: playdough, balance scales, standard mass pieces (5g, 10g, 20g), digital scales
This task involved children moving from using pre-made mass pieces to creating their own
objects of a specific mass. The children used playdough and standard masses with balance
scales to create their own set of playdough weights of 5g, 10g, 20g, 50g, and 100g. Before
checking with the balance scales the children needed to estimate how much playdough to
remove or add to create a specific mass. Digital scales were available for a final check.
Discussion: With a focus again on standard units, this task gave the children an opportunity to
experience and build benchmarks for a range of common composite units. Concepts such as
conservation of mass were explored as children realised that the shape of a playdough piece
did not determine its mass. The children were persistent and accurate in creating their weights
and excited to have their own set of weights to take home and share with their families.
CONCLUSION
We developed tasks for the study that we believed would provide an opportunity for children
to further develop their understandings of mass measurement, including of two foundational
ideas, comparison and unit. Our lessons moved more rapidly than normal in moving towards
standard units. Quantitative data collected three weeks before and three weeks after the
teaching experiment showed development in understandings for the majority of children after
just one week of teaching (see Cheeseman et al., in press). The data also showed that the Year
1 students in the present study had a similar profile of thinking as did the Year 2 students in
the original study (Cheeseman et al., 2011). The achievements made by the children in this
study suggest also that the Australian Curriculum underestimates the potential of Year 2
students to learn about standard units of mass.
While we acknowledge that a number of factors may have contributed to the growth in
children’s understandings within the present study, we note that the tasks were developed to
challenge children to think and to reason mathematically, not just to complete empirical
procedures. We wanted these to be rich tasks which, as described by Piggott (2008),
encourage children to think creatively, work logically, communicate ideas, synthesise their
results, analyse different viewpoints, look for commonalities and evaluate findings.
But we also recognise that it is not only the tasks that provide rich learning opportunities for
children. Teachers play a key role in how tasks are enacted in the classroom. Indeed, teacher
actions can contribute to the development of rich classrooms, that is, of “communities of
enquiry and collaboration” (Piggott, 2008). The above discussions of each task give some
insights into our efforts to incorporate characteristics of effective teachers of mathematics in
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our teaching experiment including having a clear mathematical focus, using open-ended tasks,
holding back from telling, and allowing autonomy for students to develop and discuss their
own methods and ideas. Perhaps these contributed to enhancing children’s learning as we
strove to maximise learning of foundational ideas of mass measurement.
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1
There are differences in meaning of Mass and Weight, for example, as defined by De Klerk
(2007) with Mass being “the amount of matter contained in an object” (p. 77), and Weight
“the pull of gravity on an object” (p. 143). In this paper we use the term Mass, as used in the
Australian Curriculum.
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