Numeracy in health and Physical Education.
Peters, Colleen ; Geiger, Vince ; Goos, Merrilyn 等
Context
My school is in Cowell, a small coastal town in South Australia about 100 km from the nearest regional centre. The Area School has 180
students from Reception to Year 12. Often in such schools there are
composite classes for Years 6/7 and 8/9; however, at Cowell these year
levels are in separate classes, some with quite small numbers of
students. (Middle school is Years 7, 8, 9.) The school grounds are
attractively maintained, with well kept gardens and shaded areas for the
students to play or sit. An outdoor learning area beside the Year 8
classroom was built with the support of federal funding obtained by
another teacher and myself. It comprises a large concrete slab the size
of half a basketball court, together with a separate shelter shed that
has a concrete floor and a whiteboard on one of the walls. I use these
areas for outdoor learning activities; the students and I enjoy getting
out of the classroom.
The classrooms are in timber buildings grouped in Primary, Middle,
and Senior clusters. The rooms are brightly decorated with posters and
student work samples. All of these buildings are due to be replaced with
new, prefabricated buildings to be brought up from Adelaide. This is a
major refurbishment project, federally funded, that has captured the
interest of the whole school community. The school itself is a community
hub and enjoys strong support from parents. Many of the staff are
long-serving members of the school community and there is generally an
air of commitment to and connection with the school and its community.
Background
I have taught at Cowell Area School for around 20 years. My class
for the DECS Numeracy in the Learning Areas Project is Year 8 (11
students), and I teach them mathematics, Health and Physical Education
(HPE) and English. I was on leave in Term 1 of 2009 and thus unable to
attend the Professional Development (PD) day at which the numeracy
project was launched. Vicki, my teaching partner within the project
attended and brought back information from the PD day for us to share.
School visit: Round 1
Planning and preparation
My initial thoughts were about planning a unit based on the novel A
Fortunate Life, but I abandoned this idea because I thought a numeracy
focus would take too much away from the story. Instead I developed a
unit around Barbie and body image, triggered by an activity presented at
the PD day. I incorporated this activity within the HPE learning area. I
borrowed a Barbie doll from a colleague and noted that students'
first reaction was to say that Barbie could not be
real--"That's ridiculous; look how small her waist is!"
One of the activities I introduced to students involved them scaling up
a Barbie to a version of the same height as an average 18 year-old
Australian girl (164 cm). This information was obtained from the
Australian Bureau of Statistics website and, in itself, was a point of
interest with students who commented on how short this seemed.
Students spent a lesson drawing a grid (5 cm squares) on large
sheets of Butchers' paper in preparation for transferring an
outline of a scaled-up Barbie to the grid. I was surprised how long it
took students to draw these grids and mused that I could have used a
grid on an OHT instead to save time. The Barbie activity was completed
mostly in HPE but also referred to in Maths where students in a
subsequent lesson on ratio commented that "This is what we were
doing with the Barbie thing!". The students worked in pairs and
produced a poster/life-size drawing of Barbie together with annotations
on: her proportions; whether her proportions were realistic; and what
messages this conveyed about women. The models were displayed around the
classroom and resulted in many interesting discussions.
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My participation in the project, at that stage, increased my
awareness of the numeracy demands of the various learning areas and how
much time students need to deal with these demands. Previously, I tended
to focus on the concepts of the learning area rather than its numeracy
demands, and this meant I was not attending to students' numeracy
learning issues. Doing hands-on activities revealed some of the problems
students had with numeracy. These included drawing a grid with 5 cm
squares, which showed that some did not know how to measure using a
meter rule, that is, they did not know which side to use, metric or
inches or did not start measuring from the zero point (which could be at
the very end of the ruler or a small distance from the end) or did not
know how to mark units on the grid (putting the scale points in the
middle of the side of the grid squares, instead of the ends). These were
all things that I expected students to be able to do, much like using a
tape measure in HPE.
First lesson observation: Directed numbers
After meeting the project researchers for the first time (Merrilyn,
Shelley and Vince), I launched into a traditional Maths lesson. The
students were learning about Directed Numbers, something they often
struggle with and a topic where I find it hard to explain using real
life situations when using addition and subtraction.
I began the lesson by asking students to say anything they
remembered about directed numbers. Students volunteered information such
as "moving to the right is positive and to the left is negative;
the opposite of positive 1 is negative 1; temperatures can be above or
below zero; East is positive and West is negative; and depositing money
in the bank is positive while withdrawing is negative." A pretty
good start! I then explained that the class would go to the outdoor area
to work with a number line that they would draw on the floor slab with
chalk. The aim was to work out how to add and subtract directed numbers.
I demonstrated the method via a number line drawn on the blackboard,
which was also illustrated on a handout I distributed to students.
Students were to stand on the first number listed and face in the
positive direction if the operation was to be addition or the negative
direction if the operation was to be subtraction. They were then to walk
the number of steps indicated by the second number, walking forward if
this number was positive and backwards if it was negative. The number at
which they arrived via this process was the answer to the problem. The
handout provided a systematically developed list of problems involving
adding and subtracting positive and negative numbers, including a
"long walk" with seven operations in succession. Two questions
then required students to describe any patterns they observed in their
walks and to explain some of the rules they discovered while adding and
subtracting.
After an initial ten minutes of teacher instruction, students moved
outside to complete the activity. One drew a chalked number line and I
gave instructions to another as this student "walked" a couple
of problems. At this stage I felt like I had 'out instructed'
a sergeant major and I could see some of the students reeling from too
much information!
After 30 minutes of outdoor activity the class moved inside and I
asked students what they had been thinking and feeling during the
activity. How did they know if they were on the right track? Students
seemed willing to say they were confused, others simply said that the
activity was fun. I explained to them that they had been using a model
that would help them understand the thinking they would be doing in the
next few lessons on adding and subtracting directed numbers. I drew
their attention to the questions about patterns and rules, and asked
them to try these for homework. I then modelled the number line
representations of:
-3 + +2 = -1
-3 + -2 = -5
-3 - +2 = -5
-3 - -2 = -1
and asked if anyone could tell me something about what they saw,
reiterating that this was to be done for homework. I find that sometimes
students are able to clarify their learning by moving (literally)
through the steps. The problem was there may have been too many steps
for some to follow.
The mathematical knowledge dealt with in this lesson was addition
and subtraction of directed numbers. At the start of the lesson I
elicited some of the real life contexts in which directed numbers
appeared. I used a representational tool--number lines drawn on the
blackboard and on the ground--to help students create patterns and
explain rules concerning these operations on directed numbers, although
I did not have time to elicit and evaluate students' ideas during
the lesson. At one point I addressed students' dispositions towards
mathematics and the learning activity by asking them how they felt about
the lesson. There was no scope in the lesson for developing a critical
orientation to this subject matter, something I have tossed around in my
head many times but found difficult to implement at this stage. Overall
this seemed a bit like another pointless maths lesson, taught just in
case and not just in time. I knew I had missed a vital point here.
School visit: Round 2
Planning and preparation
After much reflection I decided to do some things differently. The
Year 8 class was unique in that it had a small number of students with
few behavioural issues and it was an ideal time for me to change. One of
the goals I set myself was to take a more exploratory and investigative
approach, particularly in dealing with teaching aspects of numeracy
across all learning areas. In the past I had taken an approach that left
students with little freedom to explore a topic and to ask their own
questions (too busy getting on with content). I had made the realisation
that the nature of numeracy means that students needed to learn how to
explore an issue or problem and then to make choices or decisions.
Within any real world context these choices and decisions need to be
justified. I always knew this but could never justify the time it took
to teach. As part of this less direct approach, I intended to encourage
students to support each other in their exploration of a problem or
topical issue. I believe there is great value in fostering student
interdependence that can be utilised in other aspects of everyday life.
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Between research team visits I worked with the students on a
project in HPE where we investigated media coverage of sports. We
collected sports reports from an Adelaide newspaper each day for a week
and then measured and calculated the area of the space devoted to both
female and male sports. The students acknowledged that there was not an
equal representation given to female and male sports in the media and we
had an interesting discussion around this topic. At last I had begun to
address numeracy in HPE and it seemed to create a deeper student
understanding of the concepts and processes covered.
I also began work on a project in HPE where students were asked to
investigate their physical activity during a typical week. This activity
was measured through the use of a pedometer that the students were
required to wear for one week. The collected data, that is, the number
of paces walked or run, were entered into a shared Excel spreadsheet
every day. It was my intention that students analysed their own data by
using facilities available through Excel, for example, the graphing
tool, and then for students to compare each others' results. In the
initial work on this activity I drew on the skills of students who were
already competent in the use of Excel to assist other students with data
entry and initial data analysis, for example, trying different types of
graphs to represent their data. I was a bit apprehensive of this project
as my experience with Excel was limited. I asked a colleague to give me
some support and then relied heavily on students helping each other (and
me!).
In an upcoming lesson, I planned for students to convert their
total daily and total weekly paces into kilometres. The aim of this
activity was to give students a sense of how far they typically walked
in the course of a day or a week. The task was also designed to help
students realize that the distance they walked was not determined by the
number of paces alone as an individual's pace length was also a
factor. When planning the work I was struck by the range of different
mathematical concepts and skills that were embedded in the task, for
example, the use of calculators, an understanding of decimal fractions
and how to round results appropriately. I viewed the need to recall how
to make use of these skills as a powerful way for students to reinforce
previous learning and also to understand the integrated way in which
mathematical skills are usually used in real world contexts.
Lesson observation: Measurement--Converting paces to kilometres
I began the lesson by outlining what students were expected to
achieve. The introduction also included revisiting previous work on how
to convert an individual's number of walking paces to kilometres,
as well as directions for the activity that was to follow. When I asked
the students to recall the conversion work they had completed in the
previous lesson I did so via prompts and probing questions rather than
direct instruction.
We then went outside to the footpath in front of the school where
students were asked to estimate 100 metres from a common starting point and then walk to this position. After the students started walking to
their estimated positions, the researchers and I followed the students
while laying out a measuring tape. This approach had the effect of
surprising students who had underestimated the distance as they could
see we were catching up and then passing them. Students who
overestimated the distance were also somewhat bemused to see us stop,
well short of their positions, and just wait.
After calling all the students to the 100 metre mark, I asked them
to estimate how many "normal" paces it would take for each of
them to walk back to the starting point. Students were then asked to
walk back to the start and to record their number of paces.
Once back in the classroom, I introduced a template designed to
assist students to convert paces to kilometres. The template was
displayed via the electronic whiteboard that was positioned at the front
of the classroom. I began by filling in my own result: 119 paces. I then
asked the students how to complete the conversion.
With a little support, via prompts and clarifications from me, the
students developed an example which made use of a 'ratio'
approach.
119 paces = 100 metres
119 x 10 paces = 100 x 10 metres
1190 paces = 1000 metres
1190 paces = 1 kilometre
After this I displayed an Excel spreadsheet summary of
students' individual pedometer results that were recorded each day
over a week. I asked students how they would convert their results to
kilometres. Students found it difficult to begin this task so I again
displayed the conversion template, entered my data and encouraged
students to help me "discover" a method. After some discussion
we produced the following result:
117 581 paces in a week.
From the previous calculation I take 1190 paces per kilometre.
So 117581 must be divided by 1190 to find the number of kilometres.
Students were amazed by the distance I had travelled in a week with
one student exclaiming, "Wow! That's nearly all the way to
Whyalla!"
[GRAPHIC OMITTED]
After modelling the procedure, I asked students to complete
conversions of their own pace totals to kilometres. Some students were
able to proceed with this task independently while others required
assistance. I supported students through questions and probes rather
than simply telling them how to perform the procedure directly. I also
suggested that students compare their kilometric distances with each
other and to discuss why they were different and to check if their
answers seemed realistic.
I finished the lesson by indicating the next session could include
an investigation of the number of paces Usain Bolt takes during a 100
metre sprint.
Measurement (estimation and converting units), number (ratio) and
chance and data (collection, organizing and representing data)
constituted the mathematical knowledge used in this lesson.
Students' learning was situated in the real life context of an
outdoor activity that required them to convert personal
information--paces walked in a week--into standard measures
(kilometres), which in turn were used to compare each student's
level of activity with that of others. I used a range of tools through
the lesson, physical tools such as tape measures to make out 100 metres,
representational tools like the template I designed in order to scaffold the conversion of paces into kilometres, and digital tools in the form
of electronic calculators and the Excel spreadsheet used for recording
students' data. By embedding learning in an outdoor activity that
made use of students' personal information, I was attempting to
encourage positive dispositions towards learning mathematics. In
addition, by taking an approach where I supported students'
learning via prompts and clarifications rather than directly responding
to enquiries, I was attempting to foster students' confidence in
their own capacities to think flexibly and to make use of mathematical
knowledge to solve problems in unfamiliar circumstances. This lesson
also incorporated aspects of critical orientation as students made
judgments about the reasonableness of results and were also asked to
consider why the distances they had travelled differed from each other
in relation to both pace length and to students' different levels
of activity over a week.
Final reflections
After the lesson I was asked what I had learned through the
project. At a personal level I had learned about the need to address
numeracy as the opportunity arose, which requires me to be more flexible
and to avoid being driven by my teaching plan alone. In my role as a
Coordinator I believe I am now better equipped to challenge colleagues
to address numeracy within their own teaching areas because of my new
understanding of what it means to become numerate and what experiences
are necessary to develop students' numeracy. Further, I want to
emphasise with other teachers that numeracy is not just about number and
that there is great potential to address other strands of mathematics
within all learning areas.
I also described how I was attempting to take a more exploratory
and investigative approach to teaching as I believe that this is vital
in developing positive dispositions in students towards taking risks and
becoming flexible thinkers and problem solvers. I believe that
encouraging students to discuss problems and share possible solutions is
an important part of an investigative approach to learning. In previous
years I had completed a similar pedometer project by strictly
controlling students' procedures for data collection, analysis and
representation of data as the focus was on the acquisition of skills
with little consideration given to how and when students would make use
of these skills in any context outside of the classroom. My feeling now,
is that students are more positive about the use of mathematical
knowledge and skills within a physical education lesson and that I was
also learning a great deal more about my students through the use of
different types of tasks and from conversations with students, which
flowed naturally from an inquiry based approach to learning.
During this session I was asked to reflect on my changing
understanding of numeracy in terms of the model presented at the start
of the project. My desire to improve students' dispositions marked
my entry point to the model, and I attempted to do this by exploring the
numeracy demands of different learning areas and real world contexts.
This necessitated a change in teaching practice towards a less directive
and more inquiry-oriented approach, a "letting go" process
that I found difficult but more effective for enriching students'
mathematical knowledge and promoting a critical orientation to
evaluating information and answers. Once I began to give students more
responsibility for their learning I became more willing to experiment
with unfamiliar tools, such as spreadsheets, for problem solving. While
my entry point into enhancing students' numeracy was through
attempting to improve students' dispositions through the course of
the project, I addressed all aspects of the numeracy model and through
this process changed my approach to teaching in a fundamental way.
Colleen Peters
Cowell Area School
<colleen.peters@cowellas.sa.edu.au>
Vince Geiger
Australian Catholic University
Merrilyn Goos, The University of Queensland
Shelley Dole
The University of Queensland
