ERIC EJ898703: Teaching Early Mathematics "Smarter Not Harder": Using Open-Ended Tasks to Build Models and Construct Patterns

2010·0.49 MB·English

by  ERIC

Teaching Early Mathematics “Smarter not Harder”:
Using Open-ended
Tasks to Build Models and
Construct Patterns
O
pen-ended tasks focused on content
AMy MCKNIGHT & specific features are regarded as an
effective way to promote particular
JOANNE MULLIGAN concept development and to elicit higher-
order thinking (Sullivan, Griffioen, Gray, &
discuss the Powers, 2009). Such tasks may vary widely in
their focus and approach and they can be
construction and formulated without unnecessary complexity.
When mathematics tasks are designed in an
pattern work produced open-ended fashion they can also provide
flexible opportunities to cater for a range of
by young children in
differing student abilities. In this article it is
shown that they can be appropriately utilised
response to open-
in the development of measurement, space
and geometry, number, and pattern concepts
ended tasks.
to achieve a range of learning outcomes.
Current research suggests students’
exploration of space and geometry is not
sufficiently encouraged in early mathematics
classrooms (Casey, Andrews, Schindler, Kersh,
Samper, & Copley, 2008). It is recommended
that the development of spatial awareness
be fostered using block building activities
that reflect students “intuitive and informal
capabilities” (Clements & Sarama, 2007,
p. 139). In a study of preschoolers and
kindergarteners, Papic (2007) found that
children can represent, symbolise, abstract
and generalise by exploring patterns in a
variety of ways, including repeating, growing,
and spatial forms. A strong awareness of
pattern and structure is also thought to
provide a basis for understanding space and
4 APMC 15 (3) 2010
Teaching Early Mathematics “Smarter not Harder”: Using Open-ended Tasks to Build Models and Construct Patterns
geometry concepts (Mulligan, Mitchelmore, in previous lessons. The task presented a
Kemp, Marston, & Highfield, 2008). Each realistic problem with the aim of fostering
of these studies employed the careful design a high level of engagement and allowing
and implementation of open-ended tasks that full participation for students of all abilities.
explicitly linked space and geometry, and The task was as follows:
pattern and structure. Remember the tower you built last term; it
was next to a house. This house showed a
pattern too. The house was knocked down
The year 1 classroom study
and you have been asked to rebuild it. It
must show a pattern or several patterns.
A classroom-based action research study Imagine what your house looks like. Use the
investigated the development and cubes to build it. Draw it. Write about your
implementation of a series of open-ended house. Explain how you made a pattern.
tasks focused on two-dimensional and The teacher discussed the students’
three-dimensional shape, patterning, and earlier explorations of patterning
spatial awareness through the working completed in previous lessons. This
mathematically processes central to the NSW encouraged them to build on their prior
Mathematics K–6 syllabus. learning by creating and applying simple
The participants comprised one mixed and complex repetitions in new ways. In
ability class of 19 Year 1 students, aged this study, students constructed and
6 to 7 years, from a Sydney metropolitan re-constructed their models and were then
public school, drawn from high socio- required to draw them with the model in
economic, white Anglo-Saxon backgrounds. view. They were encouraged to think about
The majority was identified by the classroom the pattern structure and spatial features
teacher as capable of achieving at or beyond within their models and to explain or justify
Stage 1 mathematics outcomes (Board of their observations in writing while the teacher
Studies NSW, 2002). The classroom teacher took digital photographs. They described
acted as participant researcher and is the features of their houses including any walls,
first author of this paper. windows, doors, or any shapes or patterns that
were evident and wrote a sentence justifying
the pattern: “My house shows a pattern
Task description, implementation and
because …”. Responses were not shared until
feedback the final session in order to allow students
ample time to reflect on their work and
Several tasks were implemented during make adjustments to their thinking in
regular class mathematics time comprising consultation with the teacher. The teacher
six sessions of one-hour duration spaced assisted some students who had difficulty
throughout a school term. Students were scribing their responses.
seated individually, located at a distance
from others so as to encourage independent
thinking and recording of their own Collecting and analysing responses
responses. This arrangement differed
markedly from that used for regular small- The teacher made notes on her observation
group collaborative problem solving and of 10 of the 19 students during the lesson and
routine mathematics activities. immediately following the lesson. Students’
One of the open-ended problems related drawings and their written explanations
to a patterning “tower” task, adapted from the were collected and analysed for elements
work of Papic (2007), had been completed of mathematical thinking (representing,
APMC 15 (3) 2010 5
McKnight & Mulligan
verifying/applying and justifying); and for She created the model by aligning a total
characteristics of pattern structure, two- of 18 towers in a border style rectangular
and three-dimensional properties, and pattern recognising the ABCD repetition by
transformation skills. stating: “It goes orange, white, green, blue
each time”. She incorrectly counted the
number of towers because she did not realise
General findings that the corner towers need not be counted
twice. She said, “It has 22 blues… 22 greens…
All students successfully made a three- 22 whites… 22 oranges…”. Grace used
dimensional model depicting the structure multiplicative thinking to describe numerical
of a house rather than a simple tower in features of her pattern. For example, she said,
vertical formation that they had been limited “There are four sides, there are four in each
to in earlier tasks. Sixteen of the 19 students row, you count by fours… it has 20 rows.”
modelled and drew their houses depicting Her use of the term “row” was meant to be
the structure of one or more complex “column”. Grace’s drawing (Figure 2) shows
repeating patterns, a border or cyclic pattern. a complex three-dimensional configuration
The remaining three students were unable in two dimensions using the unit of repeat
to construct patterns successfully. The 16 (even though she misaligns the number of
students were able to explain and justify units on each side of the model). She also
the patterns they created and some of the makes visual transformations by drawing a
relationships between parts of their model in “flattened” perspective.
drawn and/or written form. Several students Evan’s model , shown in Figure 3, depicts
were able to rotate their structure mentally a border as a second layer showing an ABBA
to view it from other orientations before repetition. He explained: “It shows a pattern
doing so physically. The following examples because red, green, green, red and this is
represent a range of responses. (Pseudonyms another way, blue, red, red, green and I know
have been used to preserve the students’ what comes next… blue at the corners”.
anonymity). Thus he identified another unit of repeat
vertically as “blue, red, red, green.” Evan
depicted both horizontal and vertical
Individual student responses patterns in the structure and made it clear
that he understood the different units of
Grace built on her prior experience of simple repeat and the row and column structure.
AB repetitions in single towers to construct a Evan constructed four identical
rectangular pattern showing an ABCD unit faces by making the model in segments
of repeat as a complex three-dimensional (“walls”). The model also shows a
structure (Figure 1) .
Figure 1. Grace’s Figure 2. Grace’s
model using single drawing of her
towers from top model.
to bottom (orange,
green, white and
blue).
6 APMC 15 (3) 2010
Teaching Early Mathematics “Smarter not Harder”: Using Open-ended Tasks to Build Models and Construct Patterns
Figure 3. Evan’s
model of a house
with corners built top Figure 4. Evan’s
to bottom as blue, drawing of the front
red, red green. view of the model.
symmetrical pattern. He said that his Figure 6 shows the ABB unit of repeat
house “is shaped like a cube”, recognising represented more times than the model,
the approximately equal-sized length, and he drew the second layer as a concentric
width and height, if the “chimney/turret” pattern. He attempted to represent the
line is visualised. His drawing showed how model accurately using the centre and two
he visualised the faces of the house by layers, and may have focused on ensuring his
replicating the same pattern on each side cubes were repeating in a circular motion
(Figure 4). Evan’s representation showed rather than being accurately joined without
that he could reflect more deeply about gaps. When questioned about the spatial
common features of the structure, such as features, he stated that he saw a variety of
noticing that the walls must be congruent. two-dimensional shapes including triangles
Andrew created an unusual model of and a “diamond” [rhombus] and the idea
a house, shown in Figure 5. It comprised of a centre: “My house has a triangle hole in
two identical layers, each one approximately the middle.”
circular and using an ABB pattern with Dylan’s model, shown in Figure 7, was
a triangular centre (A as green, and B). a rectangular prism that was hierarchically
His model also showed an integrated and complex. He made the model by constructing
complex unit of repeat in another way: an AB four individual towers, and then connecting
pattern using colour; green (two blocks) and the towers to create a series of bridges. He
blue (one block). showed an awareness of pattern and structure
Andrew showed consistency in the by aligning the vertical and horizontal frames
application of the pattern structure by on each side of the model. This demonstrated
replicating it and aligning it as a second flexibility in his thinking and an ability to see
“layer”. relationships between the parts.
Figure 5. Figure 6. Andrew’s
Andrew’s model drawing of his
of a house. model depicting a
circular pattern.
APMC 15 (3) 2010 7
McKnight & Mulligan
Figure 7. Dylan’s model Figure 8. Dylan’s
of a house (top to drawing of his
bottom as orange, yellow model.
alternating cubes and
yellow in the middle
column).
Dylan’s model showed a simple AB a pattern) and their three-dimensional
repetition as “yellow, orange, yellow, spatial/geometric concepts were made more
orange…” vertically. He also identified a transparent than if they had completed
letter-shape sequence as “A, n, o, A, n, o" simple linear AB repetitions using blocks or
and described the A, n and o letter shapes symbols. The teacher was able to challenge
on the face of the model. In doing so, the students further based on their level of
Dylan showed that he recognised embedded integrated conceptual knowledge.
shapes (shapes within shapes) and embedded In terms of curriculum requirements,
patterns (pattern within a pattern). Dylan the students were engaged in a range of
identified these shapes in his model and mathematics learning outcomes related to
was then able to identify another pattern measurement, space and geometry, number
as “space, yellow, space, yellow”, or “orange and pattern, as well as reasoning about, and
yellow, orange yellow" by looking at his justification of, their responses. This open-
model from a variety of perspectives. This ended but simple task allowed the teacher
showed his use of transformation skills to to glean much information about students’
find the pattern repetition. Figure 8 depicts mathematical knowledge in a short time.
his attempt to show three-dimensionality in It proved to be an effective way to cater for
the drawing by showing that the structure the varying needs of students in the
is not flat. This is unusual and impressive classroom, extending individuals at their own
at this age because he uses oblique lines to pace and level. It was also possible to capitalise
show depth. on the rich opportunities for engagement
in visual and practical mathematical
processes to develop students’ expressive
Implications for pedagogy, curriculum
language.
and assessment The proposed Australian Curriculum
(2009) recommends the use of open-ended,
The wide variety and complexity of responses rich tasks to stimulate the development
to this task allowed us to see the potential of mathematical proficiencies. Open-ended
of the students and the possibilities that tasks are also recognised within the NSW
an open-ended approach can offer. Quality Teaching Model as providing high
All of the students were engaged in intellectual quality as students engage
sustained investigation and independent in higher-order thinking and opportunities
mathematical thinking. Their understanding for explanation of concepts explored (NSW
and application of a unit of repeat (as DET, 2003).
8 APMC 15 (3) 2010
Teaching Early Mathematics “Smarter not Harder”: Using Open-ended Tasks to Build Models and Construct Patterns
Casey, B., Andrews, N., Schindler, H., Kersh, J., Samper,
Concluding comments
A., & Copley, J. (2008). The development of
spatial skills through interventions involving block-
building activities. Cognition and Instruction, 26(3),
To use a colloquialism, teachers and learners
269–309.
can “work smarter not harder” by posing well Mulligan, J. T., Mitchelmore, M. C., Kemp, C.,
Marston, J., & Highfield, K. (2008). Encouraging
formulated open-ended tasks that integrate
mathematical thinking through pattern and
a range of mathematical concepts in an structure. Australian Primary Mathematics Classroom,
efficient way. The formulation of the task 13(3), 10–15.
National Curriculum Board (2009). The shape of the
presented in this paper had an explicit
Australian curriculum: Mathematics. Barton, ACT,
purpose but was implemented in such a way Australia: Commonwealth of Australia. From
as to promote individuals’ conceptual ideas. Australian Curriculum, Assessment and Reporting
Authority (ACARA) at http://www.acara.edu.au/
It was built on practical ideas from research
publications.html
with young children. New South Wales Department of Education. (2003).
Quality teaching framework. Sydney: NSW Department
The development of patterning and
of Education and Training.
space/geometry concepts and multiplicative
Papic, M. (2007). Promoting repeating patterns with
skills through this open-ended task is only young children: More than just alternating colours!
Australian Primary Mathematics Classroom, 12(3),
one way to promote children’s mathematical
8–13.
creativity. Increasingly, new research may Sullivan, P., Griffioen, M., Gray, H., Powers, C. (2009).
require us to raise expectations of young Exploring open-ended tasks as teacher learning.
Australian Primary Mathematics Classroom, 14(2), 4–9.
children’s abilities and at the very least, urge
us to extend children beyond traditional,
single focus programs.
Amy McKnight (MEd student)
Joanne Mulligan
References Macquarie University
<[email protected]>
Board of Studies NSW, (2002). Mathematics K–6 syllabus
<[email protected]>
2002. Sydney: Board of Studies NSW.
Clements, D. & Sarama, J. (2007). Effects of a preschool
mathematics curriculum: Summative research on APMC
the Building Blocks project. Journal for Research in
Mathematics Education, 38, 136–263.
APMC 15 (3) 2010 9