Using photographic images to enhance conceptual development in situations of proportion.
Hilton, Annette ; Hilton, Geoff ; Dole, Shelley 等
Introduction
The ability to think proportionally does not always occur naturally
in students and often takes considerable time to develop
(Bangert-Drowns, Hurley & Wilkinson, 2004). For students to
understand situations of proportion, they need to able to use
multiplicative thinking, to have a sense of co-variation and to
recognise multiplicative relationships in situations of comparison
(Behr, Harel, Post & Lesh, 1992; Lesh, Post & Behr, 1988).
Multiplicative thinking refers to the capacity to recognise and solve
problems that involve the use of multiplication or division (Siemon,
Izard, Breed & Virgona, 2006). Having a sense of co-variation
requires students to be able to recognise situations and solve problems
in which two quantities are related in such a way that a change in one
quantity results in either a directly or inversely proportional change
in the other. For example, a child may understand that if they walk the
same distance at twice the speed, they will take half the time.
The main proportional reasoning problem types include rate problems
(involving both commonly used rates, such as speed, and rate situations
in which the relationship between quantities is defined within the
question, such as birthday cake slices per child); part-part-whole
(e.g., ratio problems in which two complementary parts are compared with
each other or the whole, such as comparing numbers of boys to girls in a
class or boys to total students); and stretchers and shrinkers (growth
or scale problems; see Lamon, 2005). Concepts such as relative and
absolute (e.g., such as when comparing Group A: 4 people with 3 pizzas
and Group B: 3 people with 2 pizzas, who has the most pizza relatively?)
are also pivotal in students' understanding of proportionality. As
highlighted by Van Dooren, De Bock, Verschaffel and Janssens (2003), it
is also important to give students practice at differentiating
proportional situations from non-proportional.
While formal instruction around ratio and proportion may not occur
in mathematics curricula until early secondary schooling, research shows
that the concepts and skills that are needed by students if they are to
successfully engage with problems that involve ratio and proportion take
a long time to develop. Exposing children to foundational concepts (such
as fractions and part-part-whole comparisons), which are needed for
proportional thinking, can build on the intuitive understanding that
they often have of such relationships (Irwin, 1996; Parish, 2010).
Teachers often express confidence in teaching procedural skills,
such as multiplicative, fractional, and relational thinking, which are
required for proportional reasoning. However, the development of
underlying conceptual understanding can be challenging for students and
it is difficult for teachers to identify ways to assist students to
recognise and deal with situations involving proportional reasoning
(Sowder, Armstrong, Lamon, Simon, Sowder & Thompson, 1998; Sowder,
2007). Staples and Truxaw (2012) argued that a key aspect of focusing on
conceptual understanding is the use of mathematical language.
They suggested that unless teachers focus on and are aware of
mathematical language demands, they might inadvertently shift a task
from one that engages students in conceptual discourse to a less
demanding procedural task. In this article, we describe a sequenced
activity that used selected images to target the conceptual development
of proportional thinking and focus on the language associated with
comparison and proportional thinking. We also describe the participating
teachers' perceptions of the activity.
Impetus for the development of the activity
During a two-year multi-state Australian research project, which
focused on enhancing proportional reasoning, (see Hilton, Hilton, Dole
& Goos, 2013; Hilton, Hilton, Dole, Goos & O'Brien, 2013;
Hilton, Hilton, Dole, Goos & O'Brien, 2012), some upper primary
school teachers requested assistance in finding ways of engaging their
students in the underlying concepts of proportional reasoning without
necessarily focusing on algorithmic or procedural skills. They felt that
their students often could not recognise the difference between
proportional and non-proportional situations, could not identify the
different types of proportional situations, and regularly did not have
the language to competently engage in or describe the proportional
situations. In response to these requests from teachers, we decided to
trial the use of photographic images to promote students'
understanding and discussion of real life examples of varied
proportional situations. This trial involved 15 upper primary school
teachers. As noted earlier, this article focuses on the activity itself
and the teachers' perceptions of it. Students' feedback and
diagnostic test results were also collected during the study. These are
beyond the scope of this article and some have been reported previously
(see Hilton, Hilton, Dole & Goos, 2013).
The decision to combine images (visual representations) and
discussion (verbal/linguistic representations) was based on research
findings that have identified a range of learning advantages associated
with having students learn through the use of multiple representations.
These include allowing students to communicate their understanding to
others, to make connections between familiar and unfamiliar or novel
situations and concepts, and to clarify and refine their ideas
(Ainsworth, 2008; Chittleborough & Treagust, 2008; Lesgold, 1998;
Prain, Tytler & Peterson, 2009). According to Lemke (2002), it is
important for teachers to use a combination of natural language and
mathematical and visual representations when engaging students in
mathematics learning.
The activity
A series of photographic images portraying differing proportional
contexts was selected and accompanying discussion prompts were designed.
The purpose of the activity was to provide students with opportunities
to consider the underlying concepts and develop language associated with
the inherent proportional situation in the images. The images
represented the main proportional reasoning problem types identified by
Lamon (2005) and each problem type was presented more than once to allow
students and teachers to re-engage with them on multiple occasions. It
is important that students are explicitly taught mathematical language
so that they can clarify and communicate their thinking (Queensland
Studies Authority, 2013). The integration of mathematical language in
the discussion prompts for the images allowed the teachers to address
this need. Some examples of the images and accompanying prompts follow.
These examples represent some of the different proportional reasoning
problem types.
Example 1: Relative and absolute
The image in Figure 1 shows hippopotamuses (very large powerful
animals) standing next to an elephant (an even larger powerful animal).
The image was used to illustrate that while a hippopotamus is a very
large animal (absolute), it appears to be relatively small in the photo
when compared to the elephant.
In this example, the accompanying prompts attempted to elicit
notions of relative size with suggestions for questions that included:
1. Hippopotamuses grow to approximately 3.5 m in length and can
weigh up to 1800 kg which compared to humans is quite large. Why do the
hippos not look so large in this photograph?
2. Could we say that hippopotamuses are relatively large compared
to humans but relatively small compared to elephants?
3. Give other examples where something can be thought of as large
or small depending on the situation? Use the word
'relatively'.
[FIGURE 1 OMITTED]
Example 2: Fractional thinking, part-part-whole relationships
A photograph of a lizard, which is missing its tail, shown in
Figure 2, was used to prompt discussion about part-part-whole fractional
concepts and to allow students to use fractions and percentages in their
explanations.
[FIGURE 2 OMITTED]
The script included the following:
1. What fraction of the lizard's length do you think is
missing, and what fraction of its original length do you think is
present (part of a whole expressed as a fraction or percentage)?
2. Do these two responses add to a whole?
3. How many times longer do you think the existing part is than the
missing part and vice versa (part-part relationship)?
Example 3: Ratio
The image in Figure 3 shows herds of zebra and wildebeest massing
at the crocodile infested Mara River in Kenya. They are waiting to cross
the river. They often wait for many hours until thousands of animals
arrive and then they cross at the same time. The purpose of this image
was to engage the students in ideas related to ratio.
It also provided teachers with the opportunity to engage students
in considering ratio in a context involving chance--the odds of being
attacked are decreased when the number of animals increases.
[FIGURE 3 OMITTED]
The script elicited notions of ratio through the following prompts:
1. Why do you think the animals do not cross as soon as they arrive
at the riverbank?
2. By waiting for other animals to arrive, what happens to their
chance of survival when they cross the river? (Changing ratios.)
3. What is meant by safety in numbers? (Draw out the concept of
ratio.)
4. Have you (students) ever been in a situation where you felt
safer in a crowd? (When and why?)
The implementation process
The teachers used 15 photographs and discussion prompts to engage
the students in discussion. The concepts were repeated in subsequent
weeks to reinforce the ideas and language. A description of each image
with its inherent proportional reasoning concept is shown in Table 1.
One new image was projected on the whiteboard each day for three
weeks. It was intended that the activity only take around 10 minutes per
day. In Table 1, a sample discussion prompt has been included for each
image with relevant strands of the Australian Curriculum; however, the
nature of the discussion prompts and the students' responses will
influence which strand or strands are relevant to each image. The
discussion prompts were provided as a guide for the teachers; however,
they were optional for the teachers to use. If the teachers felt
confident with the language and the underlying concepts portrayed in the
photographs, they could choose to use their own ideas. Similarly, if the
students' ideas went beyond the suggested prompts, the teachers
were free to extend or expand the discussion or to extend the time
allocated to the activity.
The teachers' perceptions of the activity
During the activity, the teachers recorded their ideas as to the
success or usefulness of each image in eliciting student understanding
of the concepts of proportional reasoning. At the conclusion of the
activity, we asked the teachers about its benefits for student learning.
They emphasised a number of different aspects.
Student engagement and interest
The unanimous response from teachers was that using the photographs
as discussion starters was very effective for engaging their students
with proportional reasoning concepts. The teachers noted that students
participated in the discussions and found them enjoyable. This positive
disposition towards mathematical discussions was a powerful outcome of
the activity. Teachers noted students' "willingness to discuss
the images" and their "interest in the activity" and that
the "images certainly engaged the children and generated a lot of
discussion".
The teachers also noted that "students had an open-ended
opportunity to engage in discussion about the ideas that the images
provoked" and "(the images) led to discussion and debate... it
allowed the discussion to be open and illustrated different ways to
answer the same question".
Language and conceptual development
The benefit to students' language skills and conceptual
understanding of proportional reasoning was regularly noted by the
teachers. For example, one teacher stated, "By the end of the
images, they (students) were used to thinking proportionally and were
using language such as relative, proportion, and ratio", while
another reported, "They (students) began to use the language, terms
and phrases of proportional reasoning". Throughout the activity,
the students were using visual representations of proportional concepts
(the images) and engaging in verbal representations through which
varying notions about the concepts were shared and discussed. The
teachers' observations that discussion of concepts led to powerful
learning, reflect research findings regarding the benefits of learning
through multiple representations mentioned earlier.
The teachers felt that the sustained (daily for three weeks)
engagement with proportional reasoning concepts in the classroom allowed
significant conceptual and language development to occur over time. For
example, "concept development was a slow progression--a little bit
but often" and "it got better as it went on because now they
could look at a picture and identify what type of (proportional)
situation was involved".
Others stated, "the benefit was the regular use of the
language of proportion and the regular familiarity with the general
concepts of proportion" and "by the end, they (students) were
able to express themselves mathematically using proportional reasoning
words". This is an example of the social construction of knowledge
emphasised by Kozma and Russell (2005) and Lemke (2001). This approach
to engaging students in the use of mathematical language also helped to
address the problem noted by McKendree, Small, Stenning and Conlon
(2002) that teachers often assume students' understanding of the
language associated with mathematical concepts.
Making connections
The teachers noted several ways in which the activity helped
students make connections. Firstly, they noted that engaging the
students with different images but similar proportional reasoning
concepts over the three-week period allowed the students to make
connections between similar concepts in different images, which they
felt helped the students to begin to recognise the different concepts
involved. For example, one teacher said that when an image involving
relative thinking was shown in Week 2, his students noted that the
thinking and language were similar to an image discussed in Week 1.
Another student learning benefit noted by teachers was that the
photographs allowed the students to use and make connections to their
background knowledge and previous experiences. One teacher summarised
her perception of this learning benefit by saying, "I really
enjoyed that they (students) would look at a picture and think
mathematically".
Some images provided a simple and powerful opportunity to engage
students in concepts and language through unfamiliar contexts. For
example, the plaque in Figure 4 shows the total number of soldiers
mobilised from each country during World War 1 with the total killed,
followed by the percentage losses. The discussion prompts focused on the
absolute and relative losses. For instance, Germany (Deutsches Reich)
had the greatest number of soldiers mobilised and killed but they did
not have the greatest losses relative to soldiers mobilised; Bulgaria
(Balgarija) had the highest relative losses at 22%. While some teachers
noted that students initially had difficulty understanding the
difference between relative and absolute, many teachers reported that
this image elicited great interest and provided an excellent opportunity
to engage the students in these concepts.
Conclusion
The photographs used in this activity represented authentic
situations. The teachers noted that they interested the students and
encouraged a logical link between mathematics and the real world. By
engaging students in the proportional reasoning concepts in this new
way, the teachers had broadened the students' exposure to the
concepts through visual and oral modes. This powerful strategy requires
students to make connections between visual images, mathematical
concepts and mathematical language and helps them to construct their
understanding of the mathematical relationships involved without the
need to engage in algorithmic or procedural aspects. Indeed, through
this activity, the teachers reported becoming more aware of the need for
a general broadening of focus from procedural teaching and learning to
include a greater emphasis on concept and language development.
The teachers suggested that this was a useful and valuable
approach, not only for engaging students in concepts associated with
proportional reasoning but in other mathematical topics or even beyond
mathematics. Many of the teachers reported plans to extend the activity
by using photographs that they or their students would generate as well
as using the idea to enhance learning of broader mathematical concepts.
[FIGURE 4 OMITTED]
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Annette Hilton
Aarhus University, Copenhagen, and University of Queensland
<annette.hilton@edu.au.dk>
Geoff Hilton
University of Queensland
<g.hilton@uq.edu.au>
Shelley Dole
University of the Sunshine Coast
<sdole@usc.edu.au >
Merrilyn Goos
University of Queensland
<m.goos@uq.edu.au>
Table 1. List of image types and concepts used during
the intervention
Image Inherent proportional reasoning concept
Statue with person Relative size of statue to person;
standing estimation of height of statue using
beside it. person as benchmark.
Incorrectly drawn Non-proportional (seemingly
graph. proportional but not).
Lizard missing Part--part--whole (fractional thinking).
its tail (see
Figure 2).
Elevator sign: Multiplicative thinking (the average
total load mass of persons used in calculating
and person elevator loads varies).
capacity.
Large herds of Ratio (safety in numbers).
animals
crossing a
crocodile
infested river
(see Figure 3).
Elephants and Relative thinking.
hippos
(see Figure 1.)
Nails needed for Non proportional situation (Seemingly
attaching fence proportional situation).
palings.
Medieval bread Fractional thinking (part-part-whole
(1.5 m long, relating modern-day bread size to
0.5 m wide very large medieval loaf).
and 0.3 m
high).
Christmas lights Multiplicative thinking (using
hung across multiplicative strategy to estimate
a street in extremely large numbers).
multiple rows.
Town Map. Scale (using scale to determine times
and distances).
Museum Plaque. (WW1 casualties) (see Figure 4)
Small car with Relative thinking, scale/rate
large
man beside it.
Empty African Multiplicative thinking (considering
classroom the seating capacity of a classroom
showing a set out in rows).
portion of
furniture.
Animal size. Disproportional situations (images
from nature illustrating that
sometimes animals must instinctively
consider proportional situations).
Photo of a Using an object of known size to
fish with benchmark the size of another
a matchbox object--relative thinking/scale.
beside it.
Image Sample question/task Australian
Curriculum:
Mathematics
strand(s)
Statue with person How many times taller is Measurement and
standing the statue than the Geometry
beside it. man? (Linear scale.)
Incorrectly drawn Represent this Statistics and
graph. information Probability
Lizard missing proportionally. Number and Algebra
its tail (see
Figure 2).
Elevator sign: Compare the different Number and Algebra
total load lift notices. Are Statistics and
and person they all based on the Probability
capacity. same assumptions
(e.g., average mass)?
Large herds of Number and Algebra
animals Statistics and
crossing a Probability
crocodile
infested river
(see Figure 3).
Elephants and Number and Algebra
hippos
(see Figure 1.)
Nails needed for Why would using more Number and Algebra
attaching fence and more nails to
palings. attach fence
palings not
multiply their
holding strength?
Medieval bread If you wanted to buy Number and Algebra
(1.5 m long, a piece of medieval Measurement and
0.5 m wide bread of a size Geometry
and 0.3 m similar to a modern
high). day loaf, what
fraction of the
medieval loaf might
you buy? (Answers
can vary but
reasoning should be
provided.)
Christmas lights Can you suggest a Number and Algebra
hung across multiplicative
a street in method that
multiple rows. would allow
you to estimate the
number of Christmas
lights hung
across the street?
Town Map. Discuss why is it Measurement and
important that a Geometry
walking map for
tourists has a scale
on it.
Museum Plaque. Absolute/relative (using Number and Algebra
casualty numbers in Statistics and
absolute and relative Probability
terms).
Small car with Why or how would the Measurement and
large relative size of a Geometry
man beside it. vehicle influence its
uses (e.g., family
car, pizza delivery
car, builder's
vehicle)?
Empty African How would the seating Number and Algebra
classroom capacity of the room
showing a vary according to the
portion of age and size of the
furniture. students?
Animal size. How do animals sometimes Measurement and
have to instinctively Geometry
think proportionally
(e.g., in places or
spaces in which they
live)?
Photo of a Why is an object with Measurement and
fish with a known size sometimes Geometry
a matchbox included in
beside it. photographs?
