ERIC EJ1093244: Measurement: Five Considerations to Add Even More Impact to Your Program

2015·0.15 MB·English

by  ERIC

Measurement
Five considerations to
add even more impact
Derek Hurrell
to your program
University of Notre Dame, Australia
<[email protected]>
The potential of using Measurement as a way of “tuning students into mathematics” is
demonstrated. Five ideas that can form the basis of focusing on measurement to access
other strands of the mathematics curriculum are examined.
In this article I would like to look at some key understanding of measurement in order to be
considerations which have proven to be very properly understood. Further, as Reys et al.
handy in the teaching of measurement in the (2012) wrote:
primary classroom. Developing an understand-
ing of the following considerations and keeping Another reason measurement is an impor-
them up-front in my teaching has given my tant part of the mathematics curriculum
measurement lessons much more focus than they is not so much the mathematical but the
once had, and has allowed me to deliver a more pedagogical. Measurement is an effective
comprehensive and hopefully engaging program way to engage many types of student, some
of work in this important strand. of whom would be less motivated to learn
One of the refrains you can hear coming from other topics. They often see the usefulness
the mathematics classroom is: “When will I ever of the tasks as it relates to them personally
use this?” This is not an accusation that should be (p. 404).
legitimately used about the measurement strand.
The ubiquitous nature of measurement in dealing This engagement stems from the fact that if
with our societal, professional and personal lives taught properly, measurement requires action,
means that anyone who does not have the capac- socialisation and reflection. A measurement lesson
ity to be efficient and effective with measurement should not be a ‘passive’ lesson.
could not really call themselves numerate (Serow, If you teach in a school in which mathematics
Callingham & Muir, 2014). On any given day seems to cause anxiety and is not well received,
we require measurement to operate in the world. then it may just be worth considering centring
Measurement is a rich place to think about the mathematics program on measurement for
mathematics and can be used as a way of giving some of the school year. Rather than making
students a contextual need for learning and using the emphasis of the mathematics program come
other mathematical strands, particularly number from a focus on the number strand, some very
and place value, but also geometry and probability deliberate and purposeful teaching and learning
(Van de Walle, Karp & Bay-Williams, 2013). of number authentically rising from the need to
Measurement is also a strand of mathematics complete measurement tasks may be a way in
which can be shown to naturally link with many which to proceed.
other parts of the school curriculum. Examples of I would like to share five ideas that I have
measurement can be readily found in areas such found useful to be aware of regarding the teaching
as art, science, history, geography and music. of the measurement strand.
There is also a plethora of literature which is
predicated on measurement or requires an
14 APMC 20 (4) 2015
Measurement: Five considerations to add even more impact to your program
There is a teaching sequence
• Once the unit has been chosen it must stay the
same. It is not appropriate to start a measure-
There is a well-researched and well-established ment in hand-spans and then change to hand-
sequence to teaching measurement (Booker, lengths. This is an understanding which many
Bond, Sparrow & Swan, 2010; Serow, Callingham young students find challenging (Clements &
& Muir, 2014; Van de Walle, Karp & Bay- Sarama, 2007).
Williams, 2013). This sequence works across • The largest number of the same unit represents
all attributes of measurement. the greatest amount of the attribute being
measured, that is, (under the constriction that
Identify the attribute to be measured the units are the same ‘size’) four cubes is ‘more’
The first step in the measurement process is to ask than three cubes.
the question about what is being measured. For • Estimation should always precede measurement.
example, if a box is placed in the middle of the • Understand the degree of accuracy appropriate
room and the question of how big it is is asked, in a given situation and work within an
the first response from the students should be acceptable level of tolerance of error.
about what needs to be measured: is it the weight • There can be no gaps or over-laps when lining
of the box, the height of the box, the width of the up the units.
box, the volume of it, the capacity of it, or even • Measurement relies on an inverse relationship,
the surface area? that is, the larger the unit, the less of that unit
is required to measure an object. For example,
Compare and order five heavy washers may be the same mass as 12
A key and arguably, the key to any measurement light washers.
is the idea of comparison (Booker et al., 2010; • A smaller unit gives a more exact measurement
Drake, 2014). Comparison is part of human (Booker, et al., 2012; Reys, et al., 2012; Serow,
nature. On an individual level we seem to be Callingham & Muir, 2014).
constantly comparing ourselves to those around
us. Early on, the comparison is often quite general A second consideration which may prove
in nature; questions such as “Who is taller? What challenging is that standard units are actually no
is heavier? Which is longer?” [If you have any more accurate than non-standard units. That is,
doubts about how inquisitive we are about these if something is 12 paper clips long, this is just as
sorts things, just consider the sales for Guinness accurate as saying something is 24 centimetres
World Records (formerly The Guinness Book of long. The issue with non-standard units is not
Records) which between 1955 and 2013 had sold with accuracy but more about when we try to
132 002 542) copies. These sorts of questions transport or communicate these measurements
then give way to a desire to be more specific and to others. Can they replicate the length of my
this is when the need for quantification starts to 24 paper clips exactly, or will the length of one
become apparent. of their paper clips be different from the ones
I am employing and therefore the over-all length
Use non-standard units be different?
It may be surprising to some people that the
majority of understandings that will be eventually Use standard units
developed to work with standard units are the A quick scan of the understandings which can be
same understandings which are developed with developed through the use of non-standard
non-standard (or informal) units. Understandings units reveals that these are the same understandings
such as: that are required to work efficiently and effectively
with standard units. In Australia the standard units
• The unit of measurement chosen is chosen are derived from the metric system, a very uniform
through convenience and appropriateness. system which is predicated on a base-10 under-
It is more appropriate to measure the length standing and therefore has a strong link between
of a netball court with strides than hand-spans measurement and number. Students first need
or with metres than centimetres. to develop a familiarity with the standard unit,
APMC 20 (4) 2015 15
Hurrell
the basic size of the units, which units are 2014; Van de Walle, Karp & Bay-Williams, 2013)
appropriate to which situations for a required as they regard money as a measure of value.
level of precision, and what the relationships There are also three less well explored attributes of
are between commonly used standard units measurement in the primary school setting: speed,
(Siemon et al., 2011; Van de Walle, Karp & density and probability (Reys et al., 2012).
Bay-Williams, 2013). When working with all attributes of measure-
Standard units are important when measure- ment there seems to be a natural hierarchy
ments need to be recorded for later use, or need to of questions.
be communicated over time or distance, or are to
be used in a calculation. As standard units are an • Which is biggest (longest, widest, tallest,
agreed measure, a person can be sure that if they heaviest, etc.)?
order a piece of piping to be shipped from Europe • How big (long, wide, tall, heavy, etc.) is it?
to Australia that is to be three metres long and has • How much bigger (longer, wider, taller,
a diameter of 1.5 centimetres, then that is what heavier ,etc.) is it than the other one?
will arrive. Asking someone in Europe for a piece
of pipe which is two and a bit arm spans long and Each of these questions shows an increasing
that my index finger will fit snuggly into may not level of mathematical sophistication. The first
provide you with exactly what is required! question asks for a perceptual comparison; quite
simply, if the items start at a common baseline
Application of measurement
(for example the table top), which is taller? This
(formulas)
then can be developed from a direct comparison
of objects to an indirect comparison of objects.
Once students are proficient with using standard If I cannot move two objects together to make
units they can be provided with opportunities to a comparison, I can (for example) cut a piece of
use formulas and apply their measurement skills string the length of the first object, and place it
and knowledge. The formulas for area, perimeter, next to the second object make the comparison.
volume and surface area are usually developed in Question two asks students to quantify the
the later years of primary school. Whilst formulas attribute. Whether the quantification is using
are very necessary, they should be a product of standard or non-standard units, the notion is to
carefully developed processes and understandings, determine how many units ‘big’ something is, for
not a replacement of them. example, 24 paper clips long, eight marbles heavy,
13 centimetres tall. It should be noted that there
There are several different attributes
is quite a pronounced difference between using
of measurement that need attention
non-standard units and additively determining
the size of an object, and reading a scale from an
In discussing the teaching sequence, the attributes instrument to determine its size.
of measurement have often been mentioned. As in question two, question three is about
There are seven common attributes of measure- quantifying, but in this case, establishing the
ment taught in primary schools and these are: difference between two or more objects or shapes.
length, area, volume, capacity, mass, time and This usually means measuring all of the objects
temperature. An eighth attribute is angle but which are being compared and then performing a
mention of this is not made in the Australian mathematical operation (usually, but not always, a
Curriculum in the measurement strand but rather subtraction) to calculate the difference. It is at this
the geometry strand (ACARA, 2014). However point when a ‘problem’ can become apparent, and
most mathematics educators consider it to be a this will be discussed in the following ‘idea’.
measurement attribute (e.g. Reys et al., 2012;
Many measurement lessons are not
Serow, Callingham & Muir, 2014; Van de Walle,
measurement lessons
Karp & Bay-Williams, 2013).
It may be surprising, but many mathematics
educators also include money as a ninth attribute The ‘problem’ that can become apparent is that
(Booker, et al., 2010; Serow, Callingham & Muir, once calculations start to become necessary in a
16 APMC 20 (4) 2015
Measurement: Five considerations to add even more impact to your program
measurement lesson, it can quite easily become the estimation of the teacher (Muir, 2005),
the point where students stop engaging in some rather than an expected and necessary part
of the necessary skills and understandings of of the measuring process.
measurement, and the lesson essentially becomes
Understandings developed well in
a number lesson. It can happen that entire
one attribute can be transferred to
measurement lessons can be conducted without
other attributes
the students necessarily engaging in any actual
measurement. They are completing calculations
using measurement as the context, but not actually As is illustrated in the previous four ideas, if
engaging in measurement. That is not to suggest students deeply understand measurement ideas
that these calculation lessons are not important about one kind of attribute, they can transfer
and worthwhile, but they should not be categorised their knowledge to other attributes (Booker et
as measurement lessons. It is always worth remem- al., 2010). This is good news for busy teachers.
bering, that in every measurement lesson, an act What it means is that the teaching and learning
of measurement should occur. experiences for the attribute of length (length is
the attribute that most students encounter first
Estimation is important
(Reys, et al., 2012)) should be systematic, well-
considered and aiming for depth of understand-
I would like to propose that in every measurement ing. By doing so, when the next attribute is being
lesson (and in fact in every lesson) the students are taught and learned, the teaching sequence, how
required to use and develop their powers of estima- students think about the attributes and the need
tion, and therefore that as teachers we develop the for estimation, along with sound pedagogical
notion that that there is not measurement without practices have already been established. Investing
estimation. Estimation is an important and, I time and energy into teaching the attribute of
believe, a much under-utilised skill. Estimation is length well, will have many positive benefits.
important because: The measurement strand is mathematically
powerful and has rich pedagogical possibilities.
• without it, the focus of the mathematics lesson It is a part of the mathematics curriculum which
can be on number rather than measurement is replete with possibilities to engage students in
(Booker, Bond, Sparrow & Swan, 2010); meaningful, rigorous and rewarding mathematics.
• it reinforces the size of units and the relation- Many of the understandings needed to be
ship among units (Reys, et al., 2012; Van de efficient and effective with measurement are
Walle, Karp & Bay-Williams, 2013); relatively straightforward and readily applicable,
• it has practical implications (Reys, et al. 2012) something which allows access into the world
as it is an essential skill for anyone involved of mathematics success for some students. The
in trade to determine the amount of materials potential of the measurement strand for tuning
that may be required to complete a job; students into mathematics is great and should
• it helps focus on the attribute being measured not be underestimated.
and the measuring process (Van de Walle, Karp
References
& Bay-Williams, 2013);
• it provides intrinsic motivation for measure-
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Education Australia.
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Williams, 2013.)
mathematics learning. In F.K. Lester, Jr. (Ed.), Second
handbook on mathematics teaching and learning (pp.
461–555). Charlotte, NC: Information Age.
The significance of estimation as an everyday
Drake, M. (2014). Learning to measure length: The
and natural aspect of measurement needs to be
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view estimation as difficult, where success is Guinness World Records. (2015). Retrieved from Best selling
annual publication, http://www.guinnessworldrecords.com/
judged by how close the estimation is to that of world-records/best-selling-copyright-book. April, 2015.
APMC 20 (4) 2015 17
Hurrell
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18 APMC 20 (4) 2015