Numeracy in secondary school mathematics.
Gibbs, Melissa ; Goos, Merrilyn ; Geiger, Vince 等
Context
I teach at Mount Gambier High School, located in a large regional
centre (population 24 000). The main industries of the region are
forestry, fishing, dairying and agriculture, but there is a growing
emphasis on clean energy production (geothermal, wind energy etc). The
high school was founded in 1908, and currently has over 1000 students in
Years 8 to 12. The size of the school is seen as an advantage in giving
it the ability to offer a wide range of subjects and extra-curricular
activities and the school has a strong reputation in the local community
for providing students with an excellent education. Staff turnover is
low, around 10-13% (mainly due to retirements in recent years).
Students mainly come from blue collar families and there is little
cultural diversity in the school. Because of the fairly stable regional
industry base, many are enrolled in VET courses through the school.
(Only about half of Year 12 students are eligible for a Tertiary
Entrance Rank.) The University of South Australia has a campus in the
city, offering Nursing, Social Work and other programs, but most school
leavers who aspire to tertiary education are thought to move to Adelaide
and Melbourne for their studies.
The school promotes the "Student Voice" via the Student
Representative Council (SRC) and student input is actively sought in
decisions that affect the school. For example, students designed and
helped build a courtyard and created sculptures and murals that decorate
the grounds. They have a strong sense of ownership of the school and
there is little if any vandalism evident. Parents are also very much
involved in the school, as many were students here themselves.
Background
I came to the DECS Numeracy in the Learning Areas Project as I
started my second year of teaching. I am qualified to teach Mathematics,
Chemistry and Junior Science, although I was teaching only Mathematics
at the start of the project and took on a Chemistry class later in the
year. My project class was a Year 8 Mathematics class, due mainly to the
fact that my only junior school classes were Mathematics, and choosing
the Year 8s enabled me to work with the same group of students all year.
Due to the complexity of the school timetable and curriculum structures,
this would not have been possible with a Year 9 class.
School visit: Round 1
Planning and preparation
The initial phase of the project was quite difficult, as it seemed
that I was 'mutating' the project a little as I was looking at
'numeracy across the learning areas'--but within Mathematics.
After the initial project meeting, and still a little unsure of my
objectives, I began looking at trying to bring practical and real-life
applications more into focus in my Mathematics classes. My initial foray
into this was the Clipsal500 Project, which required students to
complete a range of tasks using skills with decimals and information
from tables of data. The context was the Clipsal500, which is
Australia's largest motor sport event and is held annually in
Adelaide in March. Most tasks involved the students using decimals in
the context of time, to order lap times and determine starting
positions, or estimate the time taken to complete the race, given the
fastest lap times. Some of these manipulations proved quite difficult
for the students; for some students, just locating the relevant data
from the large tables of data proved to be a skill that needed practice,
while most students found it quite difficult to multiply lap times, as
they needed to convert between minutes and seconds as a part of their
calculation.
The Clipsal500 Project, which was written in conjunction with
another Mathematics teacher, was an idea developed before my involvement
in the Numeracy Project. As I reflected on the positive and negative
aspects of this first project, I decided that my focus needed to change
as part of my preparation for writing the next one. I was still too
focussed on the fine, quite technical mechanics of manipulating
decimals, rather than the 'big picture' skills. For example,
the next time I taught this project, I decided the focus could shift to
making reasonable estimates, rather than necessarily carrying out the
precise calculations. For some students, the difficulty of the
calculations involved decreased their participation in the task.
After discussions with other teachers, led by university
researchers, about the importance of numeracy and the numeracy model, I
entered into discussions with the Mathematics coordinator about what the
next project should be. Discussions began around ensuring that there was
a clear context, and that students had the chance to learn some
practical skills that they might need later in life. In initial planning
sessions this next project took shape as planning an adventure holiday,
but as the planning continued, it became The Amazing Race, based, of
course, on the television programme in which contestants compete to
reach a common destination from a starting point by choosing the best
route and means of transport available. Over 4-5 weeks we assembled a
booklet for the students which included their itinerary for the two week
race, as well as challenges which were to be completed along the way.
Students worked in pairs to book flights and accommodation. They started
with a $10 000 budget, as well as having the potential to earn up to $2
000 for successfully completing the three challenges, which included
Diving with Sharks in Cairns, Skiing in Switzerland, and visiting The
Roman Colosseum. These challenges had a focus on using Directed Number
in context, a topic which students had studied in the previous weeks. In
the Roman Colosseum challenge students were also required to use
formulae in the context of comparing areas of the Colosseum and the
Melbourne Cricket Ground, as well as looking at exchange rates and
converting between currencies.
Students were introduced to this project about midway through the
term and their initial reactions were mixed. Some students seemed quite
excited and enthusiastic about the break from 'normal' maths,
while others found the magnitude of the project quite intimidating. The
project was quite large, but was designed so that there was a lot of
scaffolding once students had time to read through the tasks. Students
were provided with a list of websites which might assist them to
complete the tasks, examples of how they could record their information,
and the challenges which they needed to complete. Presenting such a
large project to the students was nerve-wracking, as they were not a
class that always coped well with activities that were different from
'normal,' and a lot of hours (more than 20) had been invested
into developing the tasks.
Lesson observation: The Amazing Race
Having a lesson observed by so many people (the research team) was
daunting, so it was good that this task did not require a lot of direct
instruction from me, out the front of the class. Instead, the students
were mostly able to be self-directed, and begin work immediately upon
entering the computer room. Students who were working together would
generally log onto two computers, but often they would be found working
collaboratively on one machine anyway. Most students made a positive
beginning to the task, learning how to search for and book flights and
accommodation online. For some, who had never been on a plane before,
this was very challenging, and some students needed reassurance that as
long as they were not putting their names anywhere, or entering a credit
card number, that they were not actually going to book a flight!
Some students became very involved in the race. For example, when I
checked where one student was up to he responded "I'm on my
way to Paris!" In the initial weeks of the project most students
were quite self-motivated and did not require a lot of reminders to stay
on task; however, as the project continued, and students encountered a
few technical issues with using the computers and Internet, it did
become more difficult to keep them focused. While students were working
to complete tasks I encouraged them to try to work out how to do things
themselves, but I also scaffolded as necessary. Conversations ranged
from making sure connecting flights were not departing before the
passenger arrived, ways of dealing with different currencies (some
students found currency converters online, others converted between
Euros and dollars using a conversion rate), using directed numbers in
context, how to calculate means, percentages and areas.
Student feedback on The Amazing Race varied, depending on their
engagement levels. While some students found the project fun, saying
they would rather learn through tasks like this than 'normal'
maths, others said they would prefer the normal lesson structure. Due to
some network and Internet issues, on some days it took students a very
long time to search for their flights and accommodation, and this
created 'down' time for students to be off task. The waiting
time allowed students to become bored, and this was reflected in their
feedback. Other feedback was more encouraging in identifying the things
they had learned, such as how to book flights and find accommodation,
make calculations using exchange rates, use percentages, and produce
line graphs on the computer.
Upon reflecting on the Amazing Race project once it was finished, I
decided that its length needed to be shortened, as the original version
took around six weeks in the computer room (originally it was scheduled
to take 2-3 weeks) and a number of students lost interest or became
discouraged. A few students seemed almost relieved to return to routine
bookwork for a while after this project! Also, students became very
focussed on booking the flights and accommodation, and did not
necessarily make a good attempt at the (mathematical) challenges which
they were required to complete along the way. This meant that they were
repeatedly using the same skills, and not broadening their skill base
through the challenges. Another tactic that was considered was to break
the project up and allocate some time each week towards the project and
other lessons to more conventional Mathematical learning.
The next time
While the task booklet for the Amazing Race project did not look a
lot different (facts and figures were updated but the majority of the
task was very unchanged) the focus was very different this time. Because
students had spent so much time booking flights and accommodation in my
first attempt at the project, this time I provided them with details of
some flights and accommodation at the beginning. This meant that they
only had to book a few of these online. This approach did decrease the
amount of time that students spent on making bookings, although some
groups became a little competitive and looked up their own costs to see
if they could find cheaper alternatives.
One big change was the time that students were given to complete
the race. Instead of spending a large block of time on the project,
students were introduced to it about four weeks into the term, after
completing the directed number topic, and given a week to make a start.
This allowed students to familiarise themselves with the project and
develop some confidence in their ability to complete all the tasks it
contained. After this first week, students used a weekly double lesson
to continue with the project, while for the remaining 3 lessons of the
week the Maths curriculum was learnt in a more traditional manner. This
approach proved to have a few positive effects. Firstly, most students
looked forward to the double lesson and the opportunity to continue the
race and do something a bit different. They did not have as much
opportunity to become bored with the project, although there were still
a couple students who would have preferred to do 'normal' work
in this lesson as well. Mostly the students were able to work
independently through the tasks, although sometimes I had to provide a
small amount of explicit instruction when it became apparent that most
students did not understand or know about a specific concept or skill.
For example, I found it necessary to teach skills concerning formulae,
timelines (using BC and AD), percentages, and using Excel to produce
line graphs. Nevertheless, students were also often able to help each
other and pool their knowledge to achieve tasks.
What really improved the effectiveness of the project was the
change of focus from booking flights and accommodation, to ensuring that
students attempted the challenges. More emphasis was put onto completing
these and students knew that they were expected to complete them by
certain deadlines. This resulted in many more successfully completed
challenges, a more satisfying experience for high achieving students as
well as those who were struggling a little, and more opportunities for
all students to demonstrate their learning.
While the Amazing Race is still a large project, interspersing it
with lessons where students learned the usual curriculum allowed them to
increase their skills in a range of areas through routine classroom
activities, while stretching themselves through more challenging tasks.
This allowed for a much more balanced range of learning outcomes to be
achieved.
School visit: Round 2
Planning and preparation
After the lessons learned in the Amazing Race project I gave a lot
of consideration to the types of tasks I wanted to design for the next
research cycle. The major problem with the previous project was its
length, which required a certain amount of determination (learning
stamina!) from the students in order to try and complete it. While its
sheer length was discouraging for some students, absences tended to
increase the problem because students would use this as an excuse not to
complete it. The next planning phase began with this in mind. Discussion
centred around the fact that I wanted the next project to be comprised
of a series of smaller tasks which would be less daunting to students.
Also, if students were absent for an extended period it would be easier
to negotiate catch up work or exclude a task if necessary.
I decided I wanted to plan a unit of work based on the topic of
Data. I wanted the tasks to have relevance but also an overarching theme
to tie them together. I also wanted a clear purpose to the tasks that we
were doing--to what I was asking the students to 'find out.'
This came from the emphasis in a previous research project meeting on
having a 'critical orientation' to the tasks that we are
asking students to do. In the ensuing brainstorm it was eventually
decided that we would use Approaches to a Healthy Lifestyle as the loose
theme for the unit of work. This would require students to collect and
use data about themselves and their classmates, as well as finding,
interpreting and manipulating data from other sources such as the
Australian Bureau of Statistics.
Within the series of eight tasks, students would record the heights
of people in their class, calculate the mean and median, summarise the
data in frequency tables, construct histograms, and look at the modal class. Students then collected data on their walking speeds by measuring
40m on the running track and timing how long it took them to walk this
distance. Using these data, students learned how to calculate their
average walking speeds in metres per second, metres per minute and
kilometres per hour. They next had to apply this knowledge of their
walking speed to the problem of how long it would take them to walk to
school. Another task required students to analyse the class height and
walking speed data sets to decide whether height was related to walking
speed, then use their data to make predictions if possible. This task
required students to use Microsoft Excel to represent their data as
scatterplots. They had further opportunity to interpret data by looking
at statistics on the aging population and childhood obesity.
The teaching style for this unit was completely different from that
of the Amazing Race project. I put a lot more focus on explicitly
teaching the skills and concepts that students would need for each task,
either before starting the task, or during the task at the time students
required the knowledge (a 'just in time' approach). This meant
I taught students about concepts such as the mean, median and mode,
frequency tables, and various types of graphical representations,
produced both by hand and in Excel. Many students had limited knowledge
of Excel so it was necessary to spend some time explicitly teaching the
skills of graphing in Excel, including such things as adding in titles,
labels and changing axes. Some students acquired these skills quickly
and easily, while many others needed a lot of support and time to ask
questions to learn the basics of graphing in Excel. It was very
interesting to watch those students who were positive about learning new
skills in using the software, and also those who were wary of the
program or even negative about their ability to learn and successfully
use the software. When teaching these skills, I felt encouraged when
students were comfortable and willing to help those around them and this
helped them to sort out any issues more quickly.
The research team observed the task that required students to
investigate whether there was any relationship between height and
walking speed. Having already done the background work and explicitly
taught graphing in Excel, I was pleased to see that students were able
to work mostly autonomously in producing their graphs. As the teacher,
my role became one of assisting students who needed some support with
some of the skills, and, just as importantly, questioning students'
conclusions. The other interesting aspect of this task was that it used
real data, so that the graphs that were obtained were not as idealised and simple as those often found in textbooks. Having graphed the
student-collected data before the lesson, I could see that the
correlation between height and walking speed was quite low. I was not
sure how students would approach this, especially as the data did not
really support the predictions of most students.
The graphs that students produced did not all look the same
depending on the scale that they used and whether they thought to adjust
the axes. Most students were able to describe a general trend in the
data, even if the trend was weak, and use this, or other reasoning, to
make a prediction about their teacher's and the Principal's
walking speeds from their heights. Some of the reasons students used
were perhaps not mathematically valid, but it was good to see students
trying to give reasons for their predictions. Interestingly, students
were focused on their own data point within the scattergram with
comments such as "This is me (pointing at the appropriate data
point) and "This is how tall I am and how fast I walk". Using
personal data seemed to be effective for engaging students with the
task. From a student's perspective, the activity was about them and
how they compared to the rest of the class.
Students expressed surprise that the scattergram was not linear, so
that taller people did not necessarily walk faster. My role was then to
challenge them to explain why this should be the case. Some groups
suggested that alternative variables--with associated alternative
hypotheses--should be explored, including, for example, the relationship
between walking speed and leg length or between walking speed and stride
rate. One group suggested there might be a stronger relationship between
a person's height and their maximum walking pace rather than their
natural walking pace.
I was able to use a context that made use of students'
personal details to introduce the mathematical knowledge that was used
in this lesson. The use of personal data encouraged positive
dispositions towards involvement in and completion of the task. This
task required knowledge of how to produce a scatterplot from a data set
using Excel and the capacity to make predictions from trends in the
data. Students were asked to use representational tools such as
scatterplots and digital tools in the form of computers and Excel. By
challenging students to explain the variance in their data from the
anticipated linear relationship, it was possible to introduce a critical
orientation to the task.
Final reflections
During the course of the project I had the chance to develop and
trial new tasks and styles of teaching and learning numeracy. I became
increasingly comfortable with using more extended tasks for learning
mathematics. However, I believe that it is beneficial if larger projects
are made up of self-contained sub-tasks that allow students to move
towards smaller, achievable goals. Surveys of students indicated that
many of the class appreciated the variety of tasks that they had the
opportunity to tackle, and many were quite happy to complete exercises
from the textbook interspersed with these other tasks to practise their
skills.
While I originally aimed to incorporate two extended units per
semester in the future, I have since found that time constraints have
made this difficult to do. However, the improved level of engagement
that I observe from many students when they are working on thematic
activities has seen me try to add to the relevance and
'real-life' applications of all of the mathematics units that
I teach. More recently I have worked, in conjunction with a numeracy
coach, to find ways to teach introductory algebra and equations so that
students can see where they can apply algebra in everyday situations,
and hopefully to make it more accessible to all students. Throughout
these units there was an increased emphasis in students learning
together and being able to verbalise the equations they were working
with, and use symbolic representations to write rules for things such as
working out the AFL scores for the weekend.
When the project was presented to other staff and governing
council, it was positively received, and started discussions among a
number of staff around how they are already incorporating numeracy into
their learning areas. Ideally it would be good if there could be some
coordination between units that students are doing in mathematics and in
other subject areas. There are a lot of possibilities to coordinate the
themes that students are doing across subjects, but the difficulty is in
being able to coordinate this with colleagues, and have the time and
opportunity to formally plan these connections across the curriculum.
This process began with a desire to improve my teaching by
increasing my focus on embedding student learning in engaging contexts.
I believe this is vital in helping students to understand why they need
to gain mathematical knowledge. Throughout the course of the project I
also had an increased focus on developing activities that provided a
critical orientation towards the use of mathematics. It also became
obvious how vital the role of positive dispositions is in encouraging
students to try approaches to solving a problem for themselves rather
than relying on the teacher to provide all the answers. Students need to
have the confidence to attempt a problem, even if they are not
successful in getting an answer straight away. As part of this, it was
important to increase students' opportunity to use tools to aid
them in the problem-solving process. There was a particular emphasis on
digital tools throughout the project as these could assist in analysing
data in authentic contexts which could be brought into my classroom.
Melissa Gibbs
Mount Gambier High School, SA
<melissa.gibbs@mghs.sa.edu.au>
Merrilyn Goos
The University of Queensland
Vince Geiger
Australian Catholic University
Shelley Dole
The University of Queensland
