Table Of ContentRReflect
A Time to
before we launch forward
Kathryn Palmer
Every Child Counts, Numeracy Consultant
<[email protected]>
Reflection, or consciously thinking about our the way in which I communicate mathematics:
experiences, is the key to powerful learning. the Teaching and Learning Coaching Initiative,
Reflection allows us to analyse our experiences, implementing structured problem-solving
make informed changes based on our mistakes, mathematics lessons through Japanese Lesson
maintain successful practices, and build upon Study and specific programs including YuMi
or modify our past understandings based on new Deadly Maths and Back to Front Maths (Problem-
and emerging knowledge. This paper describes based Maths).
my journey through education in Victoria and
provides insights into the elements I have iden-
Teaching and Learning Coaching
tified as integral to successful mathematics
Initiative
education as we launch into the future.
In 2007, the Department of Education and Early
Introduction
Childhood Development (DEECD) worked exten-
sively with Professor Richard Elmore on evaluat-
As teachers we are more than just educators, we ing the school improvement practices in Victorian
also become the best ‘thieves’, ‘samplers’ and Government schools. Elmore (2007) noted that
‘borrowers’. Much of our growth stems from look- human investment was the key strength of the
ing at an idea, resource and activity then using education system in Victoria and suggested that
our professional judgement and knowledge to teachers should view their teaching practice as
adapt it to suit the students in our class, the one in which new and powerful ideas are public
school we are working in or for the professional goods, rather than private practice. Teachers
learning of teachers. We grow from this culture should be exposed to coaching and mentoring
of sharing: the sharing of ideas, research and others as early as possible in their careers.
enthusiasm for our endeavour to provide our In 2009 I was one of 200 teaching and learn-
students and teachers with the best of math- ing coaches seconded to the Teaching and
ematics education. Learning Coaches Initiative and I commenced
I have been extremely fortunate over my working in the Western Metropolitan Region as
teaching career to have participated in some part of the school improvement team. The initia-
valuable numeracy projects and had the privi- tive was underpinned by the key understand-
lege of working and learning from a community ing that student achievement is determined to a
of ‘dedicated sharers’. significant extent by the knowledge and skills of
In this paper I will discuss the projects and teachers in individual classrooms.
programs that have most strongly influenced The aims of the initiative were to improve:
4 amt 69(3) 2013
• student learning outcomes, especially in teacher who is being coached. This shift leads
the areas of literacy, mathematics and/or to the embedding and sustaining of change by
science, for students in identified schools; the classroom teacher.
• teacher knowledge and skills related to effec- Teachers in primary schools require a deep
tive literacy, mathematics and science teach- understanding of mathematics for teach-
ing; and ing and this is a key component in improving
• teacher capacity in the use of ICT, particularly student learning outcomes (Hill, Rowan & Ball,
for online curriculum planning, assessment 2005). My coaching has helped me to identify
and delivery in preparation for the ultranet a common element that is fundamental for
school capacity to support improved student successful teaching in mathematics; teacher
learning outcomes (DEECD, 2013). content knowledge, and the associated under-
The purpose and intention was to provide standing of the continuum of learning that
intensive assistance to identified schools to bring this provides. Where teacher content knowl-
about changes in classroom practices necessary edge is poor, it stands to reason that so too is
to improve student outcomes and build teacher student learning. Developing both the practi-
capacity. Six key elements emerged as the ongo- cal and theoretical aspects through coaching
ing focus for the coaches. The elements were: leads to powerful gains for all stakeholders. It
• building professional relationships within is widely accepted that teachers of mathemat-
the schools; ics require appropriate strength in both content
• building teacher capacity to establish priori- and pedagogical content knowledge. In my role
ties, analyse student results, measure I am trying to address this issue by offering
student progress and use collected data timely professional learning to teachers around
purposefully; a specific key mathematical idea. The purpose
• improving the quality of learning and teach- of these sessions is to give teachers some valu-
ing through purposeful instruction by model- able kinaesthetic activities while building their
ling, observing and providing feedback; own content knowledge and highlighting links
• providing substantive conversations with to class management and organisation ideas.
teachers to elicit goals, prompt inquiry and Whilst initially many teachers viewed coach-
support reflective practice; es suspiciously and as an inconvenience, this
• developing school improvement by working soon changed as they recognised the value to
with the school leadership team and profes- themselves and the children, and most schools
sional learning teams; and across the region employed their own school-
• continuing one’s own self development. based coaches in both literacy and numeracy.
This represented a significant change in My role as a regional coach became that of the
focus for school improvement policy to one that main resource for the school based numeracy
more directly supported teachers within the coaches. At the end of 2011 any lapsing govern-
classroom and shifted the onus of accountabil- ment projects were not re-funded and unfortu-
ity to the individual. For schools the rationale nately the coaching initiative was one of these.
and incentive for teacher coaching was based on The network that I was working in had pooled
research that increasing teacher capacity had their National Partnership Funding to support
the most direct impact on improving student school improvement and to distribute resources.
achievement (Hattie, 2003). In 2012 the network funded me to continue
Coaching offers differentiation of profes- the work as Network Numeracy Coach for the
sional learning for teachers within schools and 22 schools within the network. As a result my
recognises teachers are at different stages of role within the network allows teachers to work
their careers, and possessing varied levels of collaboratively and engage in reflective practic-
knowledge and skill. It successfully overcomes es together. There has been a cultural change
the difficulty of external professional learning where professional learning is embedded into all
transferring into their classroom. With many schools I work with. The model I have developed
coaching models to choose from, the region I within my network has been to divide school-
was working in adopted the Gradual Release based coaches into small working groups, who
of Responsibility framework. In this framework, meet regularly to achieve a specific goal. As the
the responsibility for the new learning in this network is extremely diverse (with P–9, primary,
case gradually shifts from the coach to the secondary and small rural schools) this model
amt 69(3) 2013 5
allows me to meet the needs of all schools I am the lesson fits within the broader school
working with and supports a collective approach. curriculum, linking the lesson topic and
I am extremely passionate about my profes- skills to previously learned content, and to
sion and everyday enjoy the challenges my job. content that will be learned in future grades.
Coaching has allowed me to focus on my great- This lesson plan template also focuses on
est passion: the teaching of mathematics. I ways to assess student thinking during the
would not say that I am a great a mathematician lesson.
(I am not) but I am a good coach. In my position • The lesson is taught by a member of the group
I have the rare opportunity of going into a school and observed by the other members, as well
and working closely with teachers and leader- as other teachers in the school and usually
ship teams. I am a critical observer who is not some outsiders. The focus of the observation
involved in the politics of the everyday running of is on student thinking not on the teacher’s
the school and therefore can make suggestions. abilities.
Building relationships is a crucial component of • The group then gets together to discuss the
my work. Teaching is a very social occupation, lesson and their observations.
interacting with students, parents, teachers This is usually done on the same day. They
and the wider community. Coaching can be an evaluate the components of the lesson.
isolated position and for me the rewards come – Who was the lesson just right for?
when I am working with a teacher who makes – Who was too challenged?
the ideas that I had shared with them previ- – Who was under challenged?
ously their own. When they are so excited to see – What would we do differently next time?
me the next time I am in their school to share – What are our next teaching steps for these
something that worked well, that enthusiasm students?
becomes infectious. Revisions are made to the lesson, based on
these observations and analysis, and another
member of the group may possibly be select-
Implementing structured problem-
ed to teach the lesson again. This experience
solving mathematics lessons
can lead to valuable insight into student
through Japanese lesson study
thinking, strengths and weaknesses.
• At the end of this process, the group may
Lesson study is an ongoing, collaborative, profes- produce a report that outlines what they
sional development process that was developed learned in regards to their research theme
in Japan. Teachers systematically examine and goal.
their practice in order to become more effec- One of the fundamental elements of lesson
tive instructors. The Trends in International study is that it is an ongoing process. The
Mathematics and Science Study (TIMSS) iden- process focuses on the key actions of collabo-
tified Lesson Study as a powerful, ongoing rating, planning, teaching, observing, reflecting
process for improving the quality of teaching in and revising.
mathematics. (Stigler & Hiebert, 1999). All lessons are informed by research and pre-
Lesson Study involves a number of steps. reading. The team of teachers share the plan-
• Teachers select a research theme. ning and responsibility of the lesson. The team
This theme focuses on a broad research ques- matches the lesson to their students and agrees
tion regarding their students that involves on roles for the lesson.
skills or attitudes they would like to foster. I was fortunate to be able to attend the
• The research team selects a goal and a unit of IMPULS Lesson Study Immersion Program in
study on which to focus. Tokyo, Japan this year as a result of my involve-
They research their students’ abilities and ment in the Deakin University Implementing
needs within this unit of study. The team Structured Problem-solving Mathematics Lessons
researches and shares ‘best practice’ ways to through Lesson Study project over the past
teach this. year. The professional learning of teachers is
• The team creates the lesson. an ongoing process of knowledge building and
Teachers select a lesson within the unit to skill development in effective teaching practice
develop, and follow an established lesson (NPEAT, 2003). This project became the vehicle
plan template. This template focuses on how to build teacher pedagogical content knowledge.
6 amt 69(3) 2013
The collaboration between the schools and Reality
teachers involved, as well as the support from • Ensure existing knowledge prerequisite to
both our university colleagues at Deakin and the idea is known.
the network, all resulted in teachers involved • Construct kinaesthetic activities that intro-
being more reflective of their practice. A high- duce the idea (and are relevant in terms of
light of the project was watching the confidence local experience).
of the teachers in their teaching of mathemat-
ics build over the 12 months and being seen in Abstraction
their own schools as leaders of numeracy. • Develop a sequence of representational activ-
ities (physical-virtual-pictorial-language-
symbols) that develop meaning for the
Influential programs
mathematical idea.
• Develop two-way connections between real-
Numerous mathematics programs, texts and ity, representational activities, and mental
support materials emerge, seemingly constantly. models through body → hand → mind activities.
One of the challenges faced by teachers is to • Allow opportunities to create own representa-
select programs that will work successfully for tions, including language and symbols.
their students and for themselves. I found two
programs, YuMi Deadly Maths and Back to Front Mathematics
Maths, particularly useful for creating connec- • Enable students to appropriate and under-
tions and encouraging genuine student dialogue. stand the formal language and symbols for
the mathematical idea.
YuMi Deadly Maths • Facilitate students’ practice to become famil-
iar with all aspects of the idea.
In 2010 I was invited by the Sunshine/Deer • Construct activities to connect the idea to
Park Network to attend the YuMi Summit at other mathematical ideas.
Queensland University of Technology as a criti-
cal friend, to offer insights and help evaluate Reflection
whether this project would be beneficial to the • Set problems that apply the idea back to
students in our schools. The summit was an reality.
opportunity for schools involved in the project to • Lead discussion of the idea in terms of real-
professionally engage, share and showcase their ity to enable students to validate and justify
YuMi Deadly Maths experiences and journeys their own knowledge.
with colleagues. • Organise activities so that students can
‘YuMi’ is a Torres Strait Islander Creole word extend the idea (use reflective strategies—
meaning “you and me” but is used with permis- being flexible, generalising, reversing, and
sion in this project to mean working together as changing parameters).
a community for the betterment of education for
all. “Deadly” is an Aboriginal word used to mean The project has involved teachers in 12
smart, in terms of being the best one can be in schools in Victoria being trained using a ‘train
learning and life. (QUT 2010) the trainer’ and research-based models. My role
The YuMi Deadly Maths Program is based has been to support trained teachers and facili-
on two imperatives: first, that mathematics can tate a professional learning team with all trained
empower all people’s lives if understood as a teachers to continue the learning and ongoing
conceptual structure, life-describing language implementation. In 2011, I accompanied teach-
and problem-solving tool; and second, that ers from six Victorian schools to the sharing
all people can excel in mathematics if taught summit in Brisbane. I was able to highlight the
kinaesthetically, contextually, with respect network approach to implementing the program
and with high expectations. It centres around into Victoria. In May this year I coordinated all
a framework focussing on Reality, Abstraction, schools in the project to showcase their learn-
Mathematics, and Reflection (RAMR). ing and highlight best practice with delegates
from schools not involved in the project at the
Victorian sharing summit.
amt 69(3) 2013 7
The YuMi Deadly Maths project identified problem that they do not yet know how to solve.
key elements that I believe to be important to This requires the students to think mathemati-
mathematics education. The train the trainer cally, and experiment to try and work out a solu-
model and research-based professional learning tion. Next, they look for student misconceptions
supported the need to improve teacher pedagog- (where the student has a fundamental misun-
ical content knowledge. The ideas shared were derstanding of a concept), and help the students
kinaesthetic and engaging to both the teach- to analyse their ideas to see if they really work.
ers and students, but to me the ‘aha’ moment When students self-correct their misconcep-
came when I saw children in Queensland doing tions, their mathematical understanding deep-
the abstraction component of the RAMR frame- ens and they learn concepts far more quickly
work. If an idea can be represented in a diagram, (Kennedy, 2010).
table or graph then it could be modelled kinaes- Finally, and possibly most importantly, the
thetically and then students make their own teacher’s primary role is one of asking really
representation of where they were in the reality searching questions that encourage students to
with symbols and language. In my experience think deeply about a problem, access their prior
of observing many schools across Victoria, the knowledge about it, experiment with different
abstraction component was the missing link, ideas and then analyse how well these work
what we were not doing. And yet it was so simple. when applied. Teachers help students to focus
For example, in a class we get students to use on the fundamental principles and patterns in
their bodies to get into ‘groups of three or four’ mathematics, therefore enabling deep under-
and discuss how many groups have been made. standing and developing less need for repetition
If we then do not get them to draw or represent and memorisation.
what that looked like, we jump straight from the As educators, “we understand something if we
reality to the maths. We expect somehow as a see how it is related or connected to other things
child sits back at their table that they have made we know.” (Hiebert et al., 1997) Students formu-
the link from the engaging ‘groups of’ activity to late and solve problems when they use math-
the multiplication worksheet in front of them. ematics to represent unfamiliar or meaningful
situations, when they design investigations and
Back to Front Maths plan their approaches, when they apply their
existing strategies to seek solutions, and when
Last year I received a phone call from a numer- they verify that their answers are indeed reason-
acy coach I had met in Queensland telling me able. When students are told rather than being
she had just attended an extremely valuable able to explore they tend to develop a fragment-
professional learning. With my passion for great ed set of rules and procedures that do not repre-
maths teaching I was inspired to find out more sent what the teacher intended (Williams, 2010).
and discovered Back to Front Maths (Kennedy, At a facilitators training with Tierney Kennedy,
2010). It is a problem-based teaching resource the mastermind behind Back to Front Maths, I
and contains a series of novel or unfamiliar observed teachers having those ‘aha’ moments—
problems that are used to introduce new topics, calling in unison “we tell our students too much”.
uncover student misconceptions, stimulate The training is unique. Within the first hour of
interest and experimentation and ultimately a two day training, teachers observe a Back to
lead to building new mathematical understand- Front lesson in a classroom. By the completion
ing in students. of the lesson the observers have many insights
I am extremely fortunate that I get to observe into what these students know and do not
numerous problem solving lessons in various know; the purpose of the problem-based task is
settings. A constant issue during the lesson to identify student misconceptions. The second
introduction is that the teacher tells the students day of training involves participants trialling a
how to get the answer and they lose the oppor- problem based lesson with students they have
tunity for students to develop their own capac- no prior knowledge of and in a school which is
ity for logical reasoning and analytical thought. not their own. This form of professional learn-
I try to highlight to teachers that I work with ing gives participants a rare opportunity to trial
‘never tell a student something they can find out their learnings and reflect on their own practice
for themselves’. Back to Front Maths supports with the assistance of the facilitator and other
teachers to start asking students to solve a participants.
8 amt 69(3) 2013
Conclusion References
Department of Education and Early Childhood
Reflecting on this range of diverse and high Development (DEECD) (2013). Coaching initiatives.
quality projects has allowed me to isolate the Retrieved from http://www.education.vic.gov.au/
about/programs/archive/Pages/coaching.aspx
key elements they have in common, to devel-
Department of Education, Employment and Workplace
op purposeful and genuinely effective maths Relations (DEEWR) (2012). Review of funding for
schooling—Final report. Retrieved from http://foi.deewr.
teaching. The Gonski Review (Department
gov.au/system/files/doc/other/review-of-funding-for-
of Education, Employment and Workplace schooling-final-report-dec-2011.pdf
Elmore, R. (2007). Educational improvement in Victoria.
Relations (DEEWR), 2012) highlights the need
Melbourne: Office for Government School Education.
for extra specialist teachers in the area of litera-
Hattie, J. (October, 2003). Teachers make a difference:
cy and numeracy, and a need to support teach- What is the research evidence? Paper presented at the
Australian Council for Educational Research Annual
ers to sift through the numerous resources and Conference on Building Teacher Quality.
projects available. I believe specialist coaching Hiebert, J. Carpenter, T. Fennema, E. & Fuson, K. (1997).
Making sense: Teaching and learning mathematics with
in mathematics is vital if we are to build teach-
understanding. Portsmouth, NH: Heinemann.
er capacity. Lessons need to be engaging and Hill, H. C., Rowan, B. & Ball, D. L. (2005). Effects of
teachers’ mathematical knowledge for teaching on
challenging whilst making connections to prior
student achievement. American Educational Research
concepts and student interests. I believe these Journal, 42(2), 371–406.
projects have gained momentum because I was Kennedy, T. (2010). Back to front maths. Townsville, QLD:
Kennedy Press. http://www.backtofrontmaths.com.au/
a network resource to coach and mentor teach-
National Partnership for Excellence and Accountability
ers at the point of need to address issues and in Teaching (NPEAT) (2003). Principles of effective
professional development. Research Brief, 1(15).
support trials.
Stigler, J. & Hiebert, J. (1999). The teaching gap: Best
As we launch to the future I can see the ideas from the world’s teachers for improving education
importance of not ‘throwing out the old and in in the classroom. New York: Summit Books.
Student Learning Division Office for Government School
with the new’. As teachers begin to implement
Education Department of Education and Early
an Australian Curriculum, they should take Childhood, Melbourne (2010). Coaching teachers
in effective instruction. Retrieved from http://www.
with them best practice and keep borrowing
education.vic.gov.au/Documents/about/programs/
and trialling those elements that engage our archive/coachteach.pdf
Williams, G. (2010). Australian mathematics curriculum
students, by making links to previous learn-
implementation: Teacher reflections on student
ing, other contexts and to experiences inside thinking informing future directions. In New
curriculum, new opportunities (Proceedings of the 2010
and outside the classroom. It is my belief that
MAV Conference, p. 188). Melbourne: Mathematical
if we are to improve the quality of teaching and Association of Victoria.
Queensland University of Technology (QUT) (2010). YuMi
learning in mathematics, we need to be part of
Deadly Maths. Retrieved from http://ydc.ed.qut.edu.
a community that shares ideas. Therefore we do au/yumi-deadly-maths.html
not need to start from scratch ourselves but can
learn from the teacher next door or the teacher
on the other side of the world or the interna-
tional guru. As Isaac Newton wrote (quoting an
earlier scholar), “we can stand on the shoulders
of giants”.
amt 69(3) 2013 9
