Supporting Mathematical Thinking through the Eight Cultural Forces

At the heart of mathematics are a set of connected thinking dispositions. The mathematician uses these dispositions as the cognitive tools of their trade. While the traditional imagining of mathematics might be all about the accurate application of well-rehearsed algorithms and processes, in the real world of mathematics, it is all about the thinking. As we consider what our students need from their mathematical education, we should not overlook the importance of these dispositions.

The five components of Working Mathematically describe how content is explored or developed − that is, the thinking and doing of mathematics. (NESA, Mathematics K-10 Syllabus)

In Mathematics, the key ideas are the proficiency strands of understanding, fluency, problem-solving and reasoning. The proficiency strands describe the actions in which students can engage when learning and using the content. (ACARA, Australian Mathematics Curriculum)

The references above capture the importance of these thinking dispositions within the Australian and New South Wales Curriculum. In New South Wales students are expected to engage with mathematical content and concepts through reasoning, communicating and problem-solving. In doing this, they develop fluency as they choose procedures and think flexibly, accurately and efficiently with the content they learn. The result of this is an understanding of the content that allows them to apply what they know to unique situations and to make connections between concepts. Peter Sullivan, the curriculum writer for ACARA, describes understanding, fluency, problem-solving and reasoning as the verbs of mathematics. “This approach has been adopted to ensure students’ proficiency in mathematical skills develops throughout the curriculum and becomes increasingly sophisticated over the years of schooling”.

For teachers, the challenge is how do we ensure that our students develop these dispositions. The research of Ron Ritchhart from Project Zero and the author of Creating Cultures of Thinking is a useful starting point. Ron shares that “We can’t teach dispositions, we must enculturate them”. To support this process, he identifies eight cultural forces which shape our classrooms and that we may manipulate to create an environment that serves our purposes. Being aware of the shape of the cultural forces requires close attention to the nature of our learning environments. Armed with an understanding of the forces which are shaping the culture of our classrooms allows us to notice how they are currently enacted. From that point, we begin to make adjustments to achieve the desired effect. In this sense, the cultural forces can be both a lens with which to see and lever with which to shift our practice. For more on understanding the Cultural Forces as a Lens & a Lever - Read this. And then read Creating a Culture of Thinking for the full picture.

Image courtesy Ron Ritchhart

The following questions are offered as a starting point for the teacher of mathematics looking to audit how the cultural forces are impacting learning in their classroom. Good questions should allow you to see things from a new perspective. Great questions should lead you to more questions. Hopefully, this list does both.

Opportunities

• What opportunities are you currently creating for problem-solving?

• When are students required to explain their reasoning? How might you add other opportunities for this?

• What opportunities do students have to communicate their thinking? To whom and how?

• What opportunities are there for students to select strategies? Are they required to evaluate and justify their choice?

• How many ways are there to solve the problems students engage with? Can you alter the problem to make it open-ended?

• Do the problems your students solve have one answer or many?

Time

• What time is currently allocated for problem-solving, reasoning, communicating?

• When is time allocated for problem-solving, reasoning, communicating? Do students begin their learning journey with a problem, or is his the bonus question at the end?

• What changes if you alter the order of a unit? How might things change if you begin with a problem?

Modelling

• What modelling of mathematical thinking do students benefit from? Do students see you having to think? Do they see what mathematicians do when they are stuck?

• How might modelling of mathematical thinking be increased?

• How might we use modelling of mathematical thinking to enhance learning?

Language

• What language do you currently use when teaching mathematical thinking?

• Are answers right or wrong? What might change if answers were interesting, surprising, challenging, beautiful?

• Do we praise answers or the struggle of the progress? Does our language choice suggest that struggle is a bad thing? How might we change that?

• How might language be used to scaffold, name and notice mathematical thinking?

• Are our students labelled as mathematicians? How might this change their mindset?

Environment

• How does the environment of our classroom promote mathematical thinking?

• Do we show works in progress or solutions? Do we celebrate answers more than questions?

• Do our displays emphasise rote memorisation of facts or encourage flexible thinking?

• How might the environment of our classroom promote mathematical thinking?

Expectations & Interactions

• What expectations do we have for mathematical thinking in our classrooms?

• How are these expectations communicated and lived?

• What do our interactions with students communicate about our expectations? How might we change that?

• What expectations for mathematical thinking might we wish for?

• Do we believe all students can be mathematicians? How do we communicate this belief?

• What would change if all of our students believed they could be mathematicians?

• Do our interactions encourage persistence or do we rescue our students?

Routines

• What routines are currently used to support mathematical thinking?

• What routines might we use to support mathematical thinking?

• How might we make mathematical thinking routine?

• What does our student’s mathematical thinking look like?

• How might we use visible thinking routines to make their thinking visible? What will we do with the information we gain from this?

• Might our students select and use visible thinking routines to enhance their mathematical thinking?

By Nigel Coutts

The Conditions Required for 'Learner Flow'

How do we design learning experiences that our students will want to participate in? How do we maximise engagement and participation in the courses we design?

A presentation on attendance rates for university students sparked these questions. It shared the results of an ongoing research project that is seeking to understand factors influencing attendance rates. One particular finding stood out. During one of the two semesters each year, attendance rates were seen to decline earlier in one semester than the other. This was attributed to the timing of examinations and the reality that students would miss lectures and tutorials to do independent exam prep. It revealed that the clear intent for the students was to achieve high examination marks rather than engage fully with the learning opportunities that the course offered. The students were there to achieve marks rather than learning. Further was the realisation that success in the examination was achieved best by means other than attending the course, but perhaps that raises questions about the nature of the assessment.

What might it take to change this scenario? What might it take to ensure students choose to be in our courses because the value of the learning achieved through mindful attendance is such that they would not want to be anywhere else?

Making the learning fun was offered as one solution. We have probably found ourselves in a class or professional development programme where the presenter has made an effort to make the learning fun. In the moment, sometimes reluctantly, sometimes against our natures, we are swept up in the momentum of the singing, the dancing, the humour. The presenter may be highly engaging and does a great job of lifting the energy levels in the room. As the course continues, participation in the activities increases and the learning concludes with a rousing round of applause before everyone spills out of the room buzzing with excitement.

It is not until the next day or week that you begin to question the outcomes achieved. Despite all of the fun, it seems there wasn’t a great deal of learning. You are left wondering what was the purpose of it all, what was the core message, what were you supposed to do with it all? Was it that you missed something? Did others leave with great ideas that they are now putting into practice or did the fun hide a lack of substance?

Making the learning fun seems to miss the point. If we want genuinely engaging learning, we need to take a closer look at why we might want to engage in learning in the first place.

Mihaly Csikszentmihalyi describes the peak experience of engagement in a project as ‘flow’. Might we begin with this idea and consider what it would require to design learning experiences which create the necessary conditions for our learners to enter a state of flow. Such thinking should uncover the conditions required for truly engaging experiences, which by design result in learning.

“Flow is being completely involved in an activity for its own sake. The ego falls away. Time flies. Every action, movement, and thought follows inevitably from the previous one, like playing jazz.” (Csikszentmihalyi. 1997)

I propose that we aim for something a little more specific to our goal as educators than is achieved by a direct reading of Csikszentmihalyi’s description of flow. I suggest we seek to understand ‘learner flow’, the state where learning achieves a level of meaning, purpose and relevance that the learner becomes completely immersed in the experience. As educators, we want to ensure that the result of all this engagement and thinking is learning. We want to be focused on creating an environment in which the participant’s actions, thoughts and movements lead them towards a meaningful learning goal. This requires becoming comfortable with the discomfort experienced when we step beyond the limits of our existing capabilities, it demands an openness to exploring new ideas from fresh perspectives, and it results in us achieving new capacities.

There are conditions which overlap between those broadly required for flow and those that might be referenced in ‘learner flow’ while some are unique. The following are my initial and early thoughts on this.

Agency is perhaps most important. Learning is best achieved when it is driven by the learner. When the learner owns the process and when their success in the learning endeavour results from the strategic actions that they take ‘learner flow’ becomes possible. When the key decisions are made for the learner, when the learning requires that they merely follow directions, when learning happens to you rather than because of you, engagement declines. The worse scenario is where the learning environment merely requires the learner’s attendance. In this instance, learning is assumed to occur because you were present for the allocated number of hours even if during this time, your mind was elsewhere.

Purpose and relevance seem to come next. The learner must be aware of why the learning matters to them either because of the direct benefits that result from the learning or because of the significant place of the learning in a more extensive learning arc. In response to calls for learning with relevance and purpose, it is frequently argued that there are some things which we just have to learn. The implication is that these things are requisites for later learning that is full of relevance and purpose. When you speak to advocates of this ‘foundational model’ it becomes clear that they are able to see and understand the relevance or purpose of what they are describing. Our role as educators is to reveal this to our learners. When we are clear on why the learning matters, we are more likely to present this content in ways that make this clear. Creating experiences for our learners that require their use of these foundational skills is one way to enhance the visibility of its relevance and purpose. Spending large blocks of time drilling students on foundational knowledge in isolation from experiences which reveal its purpose has the opposite effect. A simple analogy helps us to understand such an approach. In sport, it is our desire to play the game well that encourages us to engage in training drills. Flip this pattern and introduce a potential player only to drills they are unlikely to ever discover a passion for the game.

Extrinsic motivators serve to limit the possibility of learner flow. When grades or certificates motivate us, we will do what is required to achieve a satisfactory grade, and nothing more. Learner flow requires that the learning opportunity engages the learner. Feedback must be delivered in ways that reveal to the learner where they are with their learning and how they might proceed. When positive feedback becomes the goal, learning is limited. If the goal is a high mark the likelihood of the learner choosing to engage with learning that is challenging is bound to decline; why risk a poor score when you can choose an easier path and be assured of success.

An achievable level of challenge is a must. If the learning is easily achieved, we will master it quickly and move on; learner flow is never achieved as it is not required. If the challenge is far beyond our capability, we are just as likely to give up or find alternate strategies to protect our ego. When there is sufficient challenge, when we are challenged in ways that encourage us to try and persist and when we can see ourselves succeeding, we are much more likely to achieve a state of learner flow. In this process, the teacher, mentor or coach has a vital role to play and how we approach this will shift the learner either towards flow or away from it. When we see our learners struggling with learning, our first response can be to rescue them. Rescuing blocks flow and removes learner agency. Sometimes doing nothing is a better response. The skilled teacher or mentor will know when their learner needs a great listener who provides them with time and space to reflect on where they are, why they are stuck and what they might do to resolve the issue. Asking the right question, providing the required nudge, suggesting an alternate perspective on the problem are all ways that the learner can be supported without having to surrender agency. Knowing when to offer advice, guidance or scaffolding is an art.

Each element described above seems to play an equally important role in establishing the conditions for learner flow. Each can and should become a part of our learning environment and considering these as we design learning experiences should support our goal of increasing learner flow.

By Nigel Coutts

AAMT Why Maths? - Inspiration beyond the classroom

This week I spent three days in Brisbane attending the Australian Association of Mathematics Teachers’ national conference. The theme of the conference was “Why Maths?” and along with 500 other mathematicians, we looked to find inspiring answers to this provocative question beyond the classroom. Here are my key takeaways from this event.

“How to think brilliantly and creatively in maths” was the title of the opening keynote from James Tanton. James is a research mathematician deeply interested in bridging the gap between the mathematics experienced by school students and the creative mathematics practised and explored by mathematicians. Having been a college professor for a decade, James realised that high school mathematics was where he could have the most significant impact. Today he is the Mathematician in Residence at the Mathematical Association of America in Washington D.C. James offered the following strategies for being brilliant at mathematics.

1. Do Something - replace the fear of not knowing how to start by taking some action, evaluate the results and adjust your strategy accordingly.

2. Use Visualisations - rather than relegating visualisations to our early years of mathematical learning, we should embrace the approach as a life-long path to brilliantly thinking. Visualisations not only help us to solve problems, they also help us to build deep-understandings and see patterns and connections which otherwise may remain invisible

3. Work Hard to avoid working hard - one of the best ways to be truly brilliant at mathematics is to look for an easier way to solve a problem. Simple, elegant and beautiful solutions should be our goal. If our thinking relies upon complex methods, maybe we don’t truly understand what we are doing.

4. Seek the story behind the topic at hand - Thinking in stories is a powerful strategy to understand what is truly going on. When we start to tell the story behind the mathematics, instead of merely looking at the numbers and symbols, we allow ourselves to build a more complete understanding

5. Got haze - Walk into hazy thinking - There will be times when a confusing haze confronts even the brilliant mathematical thinker. The path forward is unclear, and you will need to confront the unknown. Brilliant mathematicians are comfortable with admitting what they don’t know, but they don’t retreat. Instead they extend their thinking into the haze by building on what they do know.

James is the mathematician behind Exploding Dots, a strategy for visualising and understanding many mathematical concepts, especially in the area of number. Where other methods for explaining place value, the four operations and algebra obfuscate the essential concepts at play, the Exploding Dots method makes this transparent and allows students to develop a true understanding of the mathematics. In a fast-paced fifty minute workshop, James explained the fundamentals of Exploding Dots. For those wanting to explore the possibilities of using Exploding Dots with your class, these websites are the best place to start. G’Day Maths - Exploding Dots

In the image below, we see how Exploding Dots are used to visualise counting with Base 2. The aim is to demonstrate how as the number of dots increase, we show the larger quantity by using the next spot in our place value table. James refers to this as a two into one machine as it takes two dots in any square, explodes them and replaces them with a square in the next one to the left. This sequence shows a single dot in the one column. Then two dots in the one column which in the next image explode and are replaced by a single dot in the twos column. The final image shows what happens when we add a fourth dot. The two dots in the ones column explode, and one dot is added to the twos column. As there are now two dots in the twos column they explode and one dot appears in the fours column. This principle can be extended to any base value and can be used to demonstrate the four operations and algebra.

The value of mathematical reasoning quickly emerged as a common theme for the conference. By asking students to explain their thinking, validate their solutions, test their proofs, solve problems in multiple ways and reframe questions, teachers are including opportunities for students to develop mathematical reasoning. By valuing mathematical reasoning, teachers require their students to think like mathematicians.

Tingalpa State School is supporting mathematical reasoning in many ways. They have created a culture that is tolerant of mistakes and understand that much can be learned by reflecting on the mistakes we make. Their students engage in mathematical reflections through the use of Maths Journals supported by a metalanguage for mathematical understandings. Teachers and students utilise a common set of prompts for mathematical reasoning such as “Convince me...?”, “What stays the same and what changes?” and “Is it just sometimes true, or always true?”. The use of these prompts ensures that the students are thinking throughout their maths lessons and that passive absorption of mathematical methods is banished. These prompts are used alongside visible thinking routines within classrooms which value thinking. This use of a common language for learning, coupled with routines for thinking maximises learning opportunities over time as students engage with new concepts and strategies in a familiar learning environment.

Dr Toh Tin Lam of Singapore’s National Institute of Education shared strategies for developing mathematical investigations. The emphasis once again was on the use of strategies which would require thinking and in particular mathematical reasoning. A mathematical investigation is described by Dr Lam as a task that is open-ended and where the goals are ill-defined. A problem-solving task, by contrast, most likely has a solution and while there may be multiple ways to solve it, the set of possible strategies one might deploy and the mathematical concepts involved are likely to be limited.

One particularly useful strategy for designing a mathematical investigation described by Dr Lam is to explore the common mistakes made by students. He used the example of a common mistake seen when students are investigating fractions. A student may think that the strategy used to simplify the fraction as shown in the example below is valid, particularly as in this example, it results in a correct answer. When students are invited to investigate this further and to test this solution in multiple instances, they begin to understand where they went wrong. Encouraging students to investigate their errors and find an explanation for where their thinking goes wrong seems like a much more effective strategy than returning work full of red crosses and hoping the students correct their strategy before the next assessment.

Libby Foley shared her experience of working in remote regions of Far North Queensland. She reminded us of the importance of building positive, supportive relationships with our students and especially those of Aboriginal or Torres Strait Islander descent. The strong message here is that our pedagogy must always be aligned with the context in which it is practised. Foley’s deep respect for the culture of her students and the community in which her teaching is situated is impressive. There is a range that our faith in Western epistemology as the one path to truth and understand can blind us to the cultural bias of such a view. The wisdom and knowledge of our Indigenous Australians, founded on over 65,000 years of living in harmony with the land cannot be ignored and should not be diminished by cultural elitism.

Cathy Foley, the Chief Scientist for Commonwealth Scientific and Industrial Research Organisation (CSIRO), offered a strong case for rethinking the capacities we focus on in education. The CSIRO has an impressive record in science, and its contribution to our collective understanding should not be overlooked. Foley outlined the work of the CSIRO in response to Australia’s greatest challenges, which could also serve as a starting point for exciting investigations in schools:

• Resilient and Valuable Environments: Enhancing the resilience, sustainable use and value of our environments, including by mitigating and adapting the impacts of climate and global change.

• Food security and quality: Achieve sustainable regional food security and grow Australia’s share of premium AgriFood markets.

• Health and Wellbeing: Help enhance health for all through preventative, personalised, biomedical and digital health services.

• Future Industries: Help create Australia’s future industries and jobs by collaborating to boost innovation performance and STEM skills.

• Sustainable Energy and Resources: Build regional energy and resource security and our competitiveness while lowering emissions.

• A secure Australia and region: Help safeguard Australia from risks (war, terrorism, regional instability, pandemics, biosecurity, disasters and cyber attacks).

In response to these the CSIRO is evolving eight future science platforms each as exciting as the next and all demanding mathematics:

1. Active Integrated Matter - Creating Industry 5.0

2. Deep Earth Imaging - Unlocking our resource potential

3. Digiscape - Digital solutions for the land

4. Environomics - Environmental genomics to care for biodiversity

5. Hydrogen Energy Systems - Next generation energy industry

6. Precision Health - Integrated and tailored health solutions

7. Probing Biosystems - Innovative medical devices and diagnostic technologies

8. Synthetic Biology - Artificial engineering of biological systems

This is but a small taste of al the ideas shared over the three days. As is so often the case much of the best learning came from conversations with other educators along the way. What was clear is that mathematics education in Australia is in good hands. We are fortunate to have many teachers with a genuine passion for mathematics who believe that all learners can be successful in their learning. It is also clear that mathematics has a vital role to play in our collective futures. And, Why Maths?, there were many answers shared but perhaps Dr Cathy Foley offered the definitive answer, because it will help us solve the challenges of today and the future.

By Nigel Coutts

Reflections from The Future of Education Conference

The Future of Education is a topic often discussed, and at the recent gathering of educators in Florence, it was the title and theme for the conference. Now in its ninth year, The Future of Education is an international conference that attracts educators from around the world and across all domains touched by education. The conference is an inspiring two days of discussion and sharing, with the city of Florence, the centre of the Renaissance, providing a constant reminder of what might be possible when creativity and critical thinking combine. Here are my key takeaways from this event.

The thinking of Jon Dewey seems as vital and relevant to education’s future, as when he first shared his vision for a student centred model of learning. In the opening keynote of the conference, we were challenged by Austėja Landsbergienė, Lithuania’s Sir Ken Robinson, ponder the place that John Dewey’s work plays in our imagining of educations future. Originally published between 1900 & 1938 Dewey’s conceptualisation of schooling as a foundation for life-long learning continues as promising yet unrealised vision for education. Rather than trumpeting the far-sightedness of Dewey, she asked us to consider how it is that we have resisted the vision described in his work for 100 years? In this way, the conversation began, ‘do we have a vision for education that we intend to enact or does it exist only in documents’?

Education is inherently Political. Martin Laba of Vancouver invited his audience to ponder the role that political activism by students plays in their education. In a presentation titled “Learning in the Streets”, Martin discussed the part activism plays within the development of ethical understandings, citizenship, stewardship and student agency. Participation in political protests, discussion and debate in classrooms about the ethics and sustainability of the decisions backed by those in power have a part to play in our curriculum. Martin shares that we need a more expansive model of education than that which is defined by an economic imperative or market rationalism and that responding to the needs of industry is not sufficient.

Guido Giuntini of Boise State University shared a powerful project used with students to encourage a deep understanding of economic principles while also engaging them in the debate of significant ethical issues. “What is the value of a life?” As evidenced by a study which revealed that less than 25% of students were able to answer a simple economic question, even after being taught it, Guido wanted to find a better way. Through a student centred exploration of this highly engaging question, students became active participants in discussion and debate while utilising the economic theories and methods that were the course goals. It was inspiring to see this sort of rich inquiry task being used at a University level. As a Primary Educator, it is interesting to see pedagogies and approaches to curriculum which are common in this context used effectively in higher-education.

The importance of a curriculum that engages students in a substantiative social, ethical and political debate was a strong theme across the conference.

Bruce Underwood of South Australia shone a light on the need for greater care in the provision of education to Australia’s indigenous people. For Bruce the challenge is finding teachers who will thrive, not just survive when they adopt teaching placements in what is seen by all but the locals as remote communities. These communities are not well served by missionaries, mercenaries or misfits but require teachers who want to rise to the task of educating in ways genuinely respectful of the communities in which it is situated. While Bruce’s message relates directly to the remote indigenous communities of South Australia, all learning occurs within a community, and all educators need to be respectful of the community in which their practice is situated.

The research on the role of leadership in shaping change in schools by Maria-Antònia Guardiola was shared by Alan Bruce (Maria-Antònia was unable to attend). The importance of leadership was well illustrated, and the need for this leadership to be dynamic, collaborative, facilitative, distributed and open to a constant rethinking of education was made clear. The five concluding points from this research are worth pondering:

• Educational projects based on distributed leadership identify professionals with talent and intention to improve through coordinated engagement and commitment

• Trust is critical

• Leadership can be shared

• Change is ubiquitous

• Organisational learning is the foundation for understanding educational leadership

Another strong theme throughout the conference, although it was less immediately addressed than others was the connection between student engagement and grades. Terry Lansdowne and Yiannis Argyropoulos of Harriot-Watt University in the UK shared their research on attendance rates of students enrolled in courses at the university. One interesting insight they shared is worth mentioning as it says something about the link between engagement and grades. It was noted that the scheduling of exams in one semester was likely to impact attendance as students chose to do exam prep in place of attending face-to-face sessions. What does it say about learning, if students stop attending the course so that they can improve their preparation for exams? Shouldn’t we be concerned that students believe they will get better results by not attending class and doing exam prep instead? One strategy suggested is to look at ways to make the learning more fun, but are we missing the point if we aim to increase engagement by making learning fun? Shouldn’t we make it purposeful and relevant - if we just “fun” up the irrelevant, what really changes? Might we not aim to have courses which are so relevant and powerful in the lives they will lead that the students want to take advantage of every learning opportunity available?

The power of grades and marks was reinforced in a session exploring the challenges of encouraging creative thinking. In this research, students involved in a university course struggled to approach tasks which required creative and collaborative thinking. It was not that they lacked the ability but that they demonstrated a fear of being wrong. Our learners are so used to playing the game of learning, of being the student who quickly and easily responds to the teacher’s question with the right answer. When they are asked to step into the role of the questioner, when there is no right answer, when they need to take risks with their thinking, learning is difficult, and they are unsure of how to proceed. In schools, we sometimes imagine that our young students are not yet ready to be problem-finders and creative risk takers. This study involving university students shows that it is not a matter of being ready for this type of learning but of having experience with it such that this becomes the norm. When we show our students that we truly value the creative process more than the memorisation of correct answers, and when we reveal this in every aspect of our teaching, especially how we grade work, we allow our students to become risk-takers.

This is but a snapshot of all that was shared, and I was only able to see a small piece of all that was on offer. The full list of accepted abstracts and the conference proceedings can be viewed online here - PIXEL FOE Conference

By Nigel Coutts

A New Renaissance - The Future of Education

This week I am in Florence having spent two days at “The Future of Education” conference. Visiting this city, which has played such a significant role in western history, is inspiring. It encourages one to not only look back at what was, but also to look ahead at what might be, especially when the topic of conversation has been as illustrious as the future of education.

You cannot ignore the impact that the Renaissance had on Florence and nor can you overlook the role that this period of western history played in shaping the mindset of the modern world. So much of our artistic and cultural heritage has its roots in this time and in this particular place. The artists, inventors, scientists and philosophers of Firenze left a legacy far greater than that which is on display to the eager tourist exploring the art and architecture of this beautiful city.

After the long dark years of Medieval times, the Renaissance signified a new awakening of interest in our capacities for thought. Creativity and critical thinking exploded in these times of fervent collaboration and competition between thinkers. Philanthropists and patrons understood that the greatness of their growing cities would be measured by the quality of the creative works produced by the minds of their citizenry. This combination of inventive potentialities and belief in the economic value of creative works produced the perfect circumstances for a revolution whose impact continues to be felt today.

In the times prior to the realisation of the Renaissance as something more than a few isolated works but as a revolutionary time, there must have been a period where the norms of Medieval life seems to no longer work. These in-between times are intriguing periods in human history although we often want to skip ahead to the excitingly new times which emerge out of them. In some ways, we are living in such times now. Sardar refers to these as post-normal times. “. . . we have entered postnormal times, the in-between period where old orthodoxies are dying, new ones have not yet emerged, and nothing really makes sense.” Surely the period between times clearly marked by the mindset of Medieval man and those of Da Vinci, Michelangelo or Brunelleschi as the quintessential Renaissance man were like this.

What then comes next for humanity as we head into the second decade of the twenty-first century?

It is increasingly difficult to turn away from the challenges collectively confronting us. This week has seen large parts of Europe reach record-breaking temperatures, a pattern that is global and so much the norm that it barely makes headlines; we expect each summer to be hotter than the one before. In India, cities wait desperately for rain as their water supplies run drastically low. Some 600 million people in India are living with high to extreme water stress.

Despite the clear evidence and broad consensus that these weather patterns are driven by climate change resulting from our dependence upon carbon, politicians and large parts of the media enthusiastically spread misinformation. Truth and scientific reason are dismissed easily and readily by those with alternate agendas and reputation trumps all. This is the reality of a post-truth era defined by Matthew d’Ancona as “circumstances in which objective facts are less influential in shaping public opinion than appeals to emotion and personal belief’. Our debasing of truth, reason and science in this way appears as a slap in the face to the great thinkers of the Renaissance who searched for these things in all that they did.

While in some areas of human endeavour, any form of thinking seems to be on hiatus, in others we rush headlong towards the unknown inventing new technologies and opportunities even while paying little heed to the consequences. Artificial intelligence is either our saviour or damnation. It has the potential to provide us with new opportunities to understand our world. It might bring with it new ways of working and collaborations between humans and machines which give birth to a new era of prosperity. It may also remove many of the employment opportunities which we relied on and create a new underclass reliant on charity. In other realms of science we meddle with the genetic structures which define life, and yet we have not fully considered the implications. While a technology like CRISPR may allow us to engineer life and by design eradicate many diseases and disabilities, it is a technology which is unlikely to be made equitably available. Will we see a future world where humanity is bifurcated into those who have the capacity to edit their genes and those who are left with the inferior code provided by nature?

We need a new Renaissance.

To move forward and out of these post-normal times requires a rebirth of our passion for thinking and creativity. A new desire to seek out true understanding and to dig deep into the marrow of life. We need to value creativity, critical thinking, inventiveness and reason. We need philosophers and scientists and artists and inventors who engage in fruitful dialogue about what the world might be.

We need to love learning.

The Renaissance we need today must begin with education. It requires that we value education as a life-long endeavour in which learning becomes the constant thread weaving its way through our lives. Only by becoming self-navigating life-long learners will we overcome the epistemological threat posed by the post-truth brigade. Our continuous engagement in learning will allow us to rise up to whatever new challenges confront us and transform each into new opportunities.

But we need a new model of education. Traditional schooling served purposes now extinct. Transfer of the essential knowledge required by citizens of the industrial era is a goal that education must no longer serve. Education for the new Renaissance must enrich our capacities for creativity, critical thinking, analysis, experimentation, exploration and discovery. In an era where information is ubiquitous, the last thing a learner needs from their teacher is more information. Rather we need to learn to manage this flood of data and understand how it may be transformed into wisdom. We need to become problem finders who are imbued with empathy such that we see the wrongs in the world and through our awareness of our agentic capacities take action to make amends.

A new Renaissance built upon a revitalised education system is full of hope for humanity. The Future of Education, should we choose a path towards a new Renaissance, is bright and allows learning to take its rightful place as the most empowering of all our capacities.

By Nigel Coutts

How might we prepare our students for an unknown future?

How might we prepare our students for an unknown future?

If we accept that we are living in times of rapid change and that the world our children will inhabit is likely to be very different from the world of today, or perhaps more importantly, different from the work our current education system was designed to serve, what should we do to ensure our children are able to thrive?

One approach to this conundrum is to contemplate what that world of the future might be like. We can expect that technology will continue to accelerate and that it is likely to expand its sphere of influence. The first industrial revolution resulted in the replacement of human labour by machines, the next is expected to remove humans from much of the cognitive labour we currently perform. Artificial intelligence is set to expand and while we can guess at some of the ways this might impact our lives, the full impact is yet to be imagined. We might readily imagine a world where the transport industry is revolutionised by driverless vehicles and can perhaps fathom many routine information processing tasks being taken over by computers, but what happens when AI allows computers to move into domains that we believed required a human touch. Will we accept that the telephone counsellor who listens tirelessly to our woes and offers sage advice might be a robot?

When we look back to 2007 we begin to see somewhat of the challenge that we confront when we try to predict the future and make plans based upon today's circumstances. 2007 was a busy year. Apple introduced the iPhone and with it the notion of a device that fits in your pocket, performs many of the tasks which previously required a computer and thanks to an always-on internet connection gave you access to all of the world's information, anywhere you are. The iPhone was just the shiny tip of the iceberg for 2007. Big data became a big thing thanks to a still little known company called Hadoop. GitHub became the go to repository for computer code and by making it easy to share code accelerated the pace of software development. Twitter expanded, Google bought YouTube and we all became video stars. The Kindle eBook reader launched. Airbnb launched us into a new world of sharing and with it a new economic model was born. 3G communication meant that mobile data was beamed into our devices at a pace that made it practical.

Thinking about the consequences of the technologies launched in 2007 is made more interesting when you consider that most children currently in Primary School were not alive then. They have grown up in a world where all of this technology was neither new nor shiny, but normal. Trying to predict what the world will be like in 2030, when our current Kindergarten students leave high schools seems like pure folly and yet the challenge confronting school systems is to prepare our students for this world.

What thinking might guide us? Is there a way we can approach this question that does not rely on potentially flawed predictions of what the future might be like? Can we prepare our young people in ways that will allow them to rise to whatever challenge the future brings?

Such thinking leads us to identifying a set of dispositions and capabilities which are flexible, adaptable and when sought in combination perhaps most importantly uniquely human. A machine may possess some of these capabilities but it is at least for now inconceivable that a machine would possess them all.

At the heart of this is a perceived demand for graduates capable of creative and critical thinking, collaboration, problem finding/solving, self-organisation, empathy, innovation and agency. Students who have this skill set should be able to quickly develop the technical expertise required of specific tasks and it is this skill set that prepares students for the unknown. As writer and futurist Alvin Toffler (1970) puts it ‘the illiterate of the twenty first century will not be those who cannot read and write, but those who cannot learn, unlearn, and relearn.’

My belief is that our focus should be on developing our students' ability to think and to do so in ways that allow them to confront unfamiliar situations with confidence backed by relevant educational experiences. David Perkins says we have impoverished models for thinking. This results from excessive emphasis in schools on the transmission of knowledge. 'Education is a process of self transformation that enables a person to negotiate changes that are as-yet indeterminate, as well as the changes that must surely come.’ (Kalantzis & Cope 2012 p92) For this to become the reality our students experience we need to empower them to become the driving force behind their learning now and beyond school. Thus the output of our education systems shifts from being educated people to being people well prepared for a life of education; true life-long learners.

Beyond accepting that education shall be a life-long endeavour, our children will need to embrace their agentic potential and understand that they have the capacity to shape their world. By combining a sense of Agency with empathy we prepare our young people to make sense of the challenges faced by themselves and others and to then take action which makes a difference. This combination of empathy and empowered agency if anything is the distinction between the human and machine world that matters most. It is said that a machine charged with making paper clips will do so until all matter is transformed into a paperclip. A human will understand that there are richer goals to be achieved and change course.

The challenge of the future is real but now is not the time for despair. Education surely has a central role to play and learning and the dispositions of the learner have greater value now than perhaps ever before. Now is the time for new Renaissance for education not as preparation for an unknown future but as the one constant which flows through our lives and allows us to flourish amidst unceasing change.

By Nigel Coutts

Kalantzis, M. & Cope, B. (2012). New learning: a charter for change in education. Critical Studies in Education, 53, 1, 83-94.

Toffler, A. (1970). Future shock. New York: Random House.

Becoming a reflective practitioner

There are particular behaviours and a mindset that accompanies effective reflective practice. Understanding and applying these allows us to become reflective practitioners. The process for most people will begin with a recognition that reflective practice has value and a part to play in our professional learning. It is also true that it can be easier to act our way into a new mindset than it might be to change our beliefs. To that end, by adopting the behaviours of reflective practice, we may find ourselves valuing the change, and that might result in a new mindset. 

Reflective practice takes time and making time for deliberate reflection is an excellent place to start. Setting aside time for thoughtful reflection may not be easy and requires a degree of planning for most people. Some are fortunate or are perhaps forced to spend time alone with their thoughts. Maybe this results from a long commute or time spent waiting for children to finish sports training or the like. For those who do not have time for reflection built into their day by default it is worth scheduling time for it in the same way that you might schedule a meeting; set up a recurring event with yourself as the sole attendee. 

Now that you have time for your reflective practice consider the environment in which this will occur. If you are hoping to turn your mind inwards, it is essential to be in a place that is free of distractions. A quiet mountain top might be the gold standard but is perhaps not accessible on a daily basis. What works for the individual will vary, but you should avoid places or proximity to devices which will draw your attention away from your reflective mind. Research shows that a phone, even when set to silent, is likely to be a distraction as you wonder if a notification has arrived. 

To be effective, our reflective practice should be grounded in our close noticing of the events of the day. This requires us to be attentive in the moment. If we are not looking closely at the impact we are having, at the reactions our actions cause, at the ripples which are formed by our presence or the results of our effort, we are likely to fall into ponderous navel-gazing. Our reflection on our practice as a teacher begins by making space in our teaching to step back and observe our learners. Our goal should be to notice them in the act of learning. We can do this through our strategic use of routines which allow us to become an observer of their learning. 

Now is also the ideal time to consider how we may document or capture our students thinking. One of the strengths of employing a selection of Visible Thinking Routines is that they include opportunities to capture the thinking which our teaching is facilitating. By deploying strategies which make our students’ thinking visible, we gain access to insights which can inform how we subsequently guide or nudge our students in the desired direction. When we include this documentation as evidence within our reflective practice, we enhance the impact that both the capturing of the thinking and the reflection upon our teaching can have. 

This leads us to the question of what we might consider to be evidence of learning. The evidence gathered from our considered use of thinking routines, student responses to rich tasks and our observations of students engaged in the act of learning provides us with invaluable information. Tasks which offer us numerical data alone provide a different perspective. When we look at a data set, it should lead us towards questions and wonderings. If a data set makes us wonder why a student achieved a particular result and what this might reveal about their learning, then it has a valuable role to play. If shifting the data becomes more important than the learners, which it purports to measure, we may have forgotten our purpose. 

Having observed our learners practising their craft, we can begin to consider what other information we might include in our reflections. What questions will we ask that might reveal the thinking that our learners are engaging with? Carefully designed questions, framed through language that encourages our students to describe where they are with their thinking, and where they are going are most effective. Questions which test knowledge or lead the learner to believe their task is to guess what the teacher is thinking, should be avoided. A question which uncovers the thought processes of our students is one where the answer is only revealed by the student’s response. If the answer to the question can be found in a text or online, then it is not probing our student’s thinking. 

Our observations and our questioning reveal a picture of learning in our classroom. We should be able to discern what our students are understanding from our instruction and also what is causing confusion. We will also create the conditions necessary for us to notice and name what we value from and for our students. An honest observation of our practice in the classroom should reveal who does most of the talking, which students are engaged and which could have spent the lesson elsewhere. There will be learner behaviours which are puzzling ad these can be the most interesting fodder for our reflections. The student who seems to struggle to verbalise a particular letter sound might lead us to a deeper understanding of the decoding skills being deployed. A group of learners with a shared misconception of a mathematical concept may reveal an opportunity for us to represent the idea in a new way. It is only through the close-noticing of what is occurring and the practice of reflection that we are likely to capitalise on these moments.

Finally, reflective practice is made more effective when it is shared. By sharing our reflections with others, we engage in a mix of practices which support a deeper understanding. As discussed previously, explaining our thinking to others allows us to better understand our own thinking. Engaging in co-construction of reflections opens our mind to fresh perspectives as our collaborators notice things we may not or offer a perspective which challenges our stance. And, the process of engaging with others as a participant in their reflective practice allows us to achieve new insights into our thinking. Further, we may wish to open our reflections to ongoing scrutiny by ourselves and possibly others by capturing it in some way. By recording our reflections, we build an archive which we can review and which can itself prompt further reflection when we take the time to look back. 

By Nigel Coutts

Realising the benefits of reflective practice

It is generally accepted that learning is enhanced by the inclusion of deliberate, reflective practice. Indeed the act of reflecting on the impact that our actions have towards the achievement of any goal (learning oriented or other) is shown to have a positive impact. Reflective practice is defined as the praxis (interdependent and integrated theory, practice, research, thought and action) of individuals or groups to move from ‘better thinking to better action' as a result of reflection for, in and on learning (Harvey et al. 2010 p140). With this in mind, it is worth considering what reflective practice might look like and to consider it in a range of contemporary contexts.

Design thinking is a highly iterative process in which a problem is attacked through a cyclical process. A typical design thinking cycle includes:

• developing an understanding of the context of the problem with empathy for the user being a common element

• a deliberate process of ideation to allow many ideas to emerge

• a convergent thinking process where ideas are evaluated and set aside

• a prototyping phase where ideas are brought to life and tested

• an ongoing process of refinement and consolidation

As Looijenga et al. identify ‘Iteration during the design process is an essential element’ (Looijenga et al. 2015 p1), and this process is facilitated in part by reflection. 'Reflection, critical thinking and developing metacognitive skills concern: Planning, evaluating, and justifying inquiries, designs, explorations, investigations, actions, performances, etc.’ (Looijenga et al. 2015 p20). Further, a key to successful design iteration identified by Looijenga et al. (2015) is a process of collaboration. In design thinking, the iterative process is driven by ongoing reflective practice inside a collaborative environment. Ideas are proposed and shared within the design team and with stakeholders outside of the team. The process moves forward because of the collective embrace of a reflective process.

It should not be assumed that this collective embrace of the reflective process will occur. A team is unlikely to benefit from a design thinking process if its members are unwilling to have their ideas criticised, modified, blended with others and potentially rejected or set aside. If the team does not value a gradual move from divergent thinking where many ideas are shared towards convergent thinking where one or two ideas emerge as dominant, the process is likely to be flawed. A team culture that values correct answers, where there is a fear of being wrong, where questions are not valued is unlikely to promote divergent thinking. Divergent thinking requires a culture where risk-taking and open sharing of ideas is the norm. Individuals must know that they are respected for what they offer the group even when their ideas do not survive to the end of the project. Building such a culture requires deliberate attention to the establishment of norms which support the free flow of ideas into and out of the design thinking process.

While we often think of reflective practice as something that is done by the individual for their benefit, the potential of reflecting on the thinking and learning of others offers many advantages. Peer assessment is shown to encourage deeper thinking and reflection (Cheng & Warren, 1999) and encourages individuals to think deeply about what they have observed (Stefani 1994). Li, Liu and Steckelberg (2010) showed in their study that the quality of feedback provided by individuals in the role of assessor improved the quality of work they subsequently produced and the authors recommend that efforts to enhance the quality of feedback should be pursued. However, Li, Liu & Steckleberg (2010) found in their research that the quality of feedback provided did not have the same impact for those it was provided to and cite research suggesting feedback could be more constructive and detailed. The implication here is somewhat contrary to the perceived wisdom of who is best served by the provision of feedback. This research reveals that the person providing the feedback has the most to gain and that the act of reflecting carefully on the thinking and learning of others enhances our learning more than it improves the learning of those we have observed. The implication here is that if we wish to gain the most from reflective practice, we should include opportunities to reflect upon the work of our peers.

Sandi et al. (2011) found that metacognitive skills can be developed through collaboration and that skills developed in this way are transferable to the individual solution of a problem. Sandi et al. provided students with a problem and allowed time for reflection and collaboration enabled by prompts enacting meaningful social interaction, which they found enhanced metacognition. Hausmann et al. (2004) have gathered evidence that supports three mechanisms to describe why collaboration is effective in enhancing understanding and task performance:

• Other-directed explaining where the individual takes the stance of a teacher or instructor.

• Co-construction of reflections with elaboration or critical evaluation of a peer’s contributions to the reflective practice

• Participation in self-directed explaining by listening to others’ self-explaining.

The implication is that reflective practices which occur in isolation are less effective than collaborative reflection. Collaborative reflection allows us to benefit from explaining our thinking to others and in doing so to clarify our thinking as we seek to be understood. In explaining ourselves to others, the social cues inherent in conversation with an active listening partner provide us with instant feedback on the clarity of our self-explaining, and we are able to modify our reflections in response to this. The collaborative, reflective dialogue allows opportunities for us to also modify our reflection as we co-construct meaning with our collaborators. When we share our reflection and expose our thinking to questioning and alternate perspectives we create space in which fresh ideas might emerge; a process enhanced when we recruit collaborators with diverse perspectives who might offer alternative interpretations. Lastly, by listening to the reflective practices of others, we allow new ideas to enter into our stream of consciousness and in doing so, generate a fresh perspective on our reflections.

Becoming a reflective practitioner has much to offer. Reflective practices are an essential element of learning, and it might be argued that true learning cannot occur without reflection on learning. The quality of our reflective practice is worth considering, and we can enhance our process by adopting an open mindset and through engaging in collaborative reflection. Amidst the business of our daily lives, it can be easy to fall into patterns where our thinking shifts rapidly from one task to the next without time to pause and reflect on what we have been doing or thinking. If we understand the value that quality reflective practices play in our learning, problem-solving, creativity, social interaction and personal growth, we are more likely to make time for it.

By Nigel Coutts

Cheng, W. & Warren, M. (1999) Peer and teacher assessment of the oral and written tasks of a group project, Assessment and Evaluation in Higher Education, 24(3), pp. 301–314.

Harvey, M; Coulson, D; Mackaway, J. & Winchester-Seeto, T. (2010). Aligning reflection in the cooperative education curriculum. Australia Pacific Journal of Co-operative education, 11 (3), 137-152.

Hausmann, R. G. M., Chi, M. T. H., & Roy, M. (2004, August). Learning from collaborative problem solving: An analysis of three hypothesized mechanism. Paper presented at the proceedings of the 26th annual conference of the Cognitive Science Society, Chicago, IL, pp. 547–552.

Li, L., Liu, X., & Steckelberg, A. (2010). Assessor or assessee: How student learning improves by giving and receiving peer feedback. British Journal Of Educational Technology, 41(3), 525-536.

Looijenga, A., Klapwijk, R., & de Vries, M. (2014). The effect of iteration on the design performance of primary school children. International Journal of Technology & Design Education, 25(1), 1-23. doi:10.1007/s10798-014-9271-2

Sandi‐Urena, S., Cooper, M., & Stevens, R. (2011). Enhancement of metacognition use and awareness by means of a collaborative intervention. International Journal Of Science Education, 33(3), 323-340.

Stefani, L. (1994). Peer, self and tutor assessment: Relative reliabilities. Studies In Higher Education, 19(1), 69-75.

Four perspectives on truth, normality and education in times of rapid change

We are living in interesting, frightening and rapidly changing times. Where rapid changes and transformations through technology, politics, globalisation and the climate, conspire against normality. These times demand a fresh approach to education, one that provides learners with the thinking dispositions they need to turn challenges into opportunities.  "All that was 'normal' has now evaporated; we have entered postnormal times, the in-between period where old orthodoxies are dying, new ones have not yet emerged, and nothing really makes sense.” But what thinking might guide us through this time of volatility, uncertainty, complexity and ambiguity?

The four reads shared below are a good starting point. Each presents a particular perspective on the modern era and suggests a way forward. Each has its own bias and particular flaws. The purpose of this list is not to point the reader towards a solution but to start a conversation.

21 Lessons for the 21st Century by Yuval Noah Harari

In this essay, the author takes us on a journey through our history to argue that the situation in which we find ourselves today is perhaps not as unique as we might imagine. The author shows how a superficial reading of our history and the stories we deploy as we make sense of the world blocks us from achieving a more nuanced understanding of how we arrived where we are. It invites the reader to engage in a close inspection of our histories and an even closer investigation of our stories. Yuval is blunt in his attacks on social conventions and clear in his belief that humanity has more in common than we might imagine. His detailed description of the challenges one might face in organising an event like the Olympic games in 1016 compared to today clarifies this point well. For educators, there are many points worth noting but the standout is perhaps, "In such a world, the last thing a teacher needs to give her pupils is more information. They already have far too much of it. Instead, people need the ability to make sense of information, to tell the difference between what is important and what is unimportant, and above all to combine many bits of information into a broad picture of the world.”

Post-Truth: The New War on Truth and How to Fight Back by Matthew d’Ancona

"The notion of science as a conspiracy rather than a world-changing field of inquiry used to be confined to cranks. No longer. It seems to me intolerable that this should be so.”

In 2016 the Oxford dictionary declared Post-truth the word of the year indicating the move of the phrase into mainstream use. If as is often stated, we are living in the post-truth era, what are the implications for education? If knowledge, wisdom and expertise are no longer trusted and easily dismissed, how do we armour the intellect of our learners against such rebuttals? How do we produce young people who will seek the truth and question those who spread falsehood? These are the questions that d’Ancona tackles in “The New War on Truth”. Who and what lies behind this war on truth, what part do individuals play, and how might we tip the balance back towards a society that seeks truth and values wisdom over emotion and base reputation?

“Welcome to PostNormal Times” by Ziauddin Sardar

In this article, Sardar proposes that we have moved into post-normal times, where all that we once relied upon has dissolved, and we are left with no sense of normal. Change is rapid and unpredictable. The structures and stories which helped us to navigate our world and predict the near future have fallen and we are yet to replace them. In this in-between time, we struggle to make meaning, find certainty and know who or what we may trust. "We will have to imagine ourselves out of postnormal times and into a new age of normalcy—with an ethical compass and a broad spectrum of imaginations from the rich diversity of human cultures.” So how might educators prepare learners for post-normality, and how will we ensure that they leave our schools with the ethical compassion they require? What is the ideal preparation for thriving amidst complexity, chaos, contradictions and uncertainty?

In Search of Deeper Learning: The Quest to Remake the American High School by Jal Mehta & Sarah Fine

Having dived deep into the “why” of a rethink of what and how we teach, this book describes the authors' search for the antidote in the schools of America. What started as a research project to examine the practices of America’s more innovative schools quickly transformed into an analysis of what was missing and a search for answers. Even in schools with a reputation for being cutting edge, the authors found large numbers of students best described as disengaged from the learning process. From this starting point, the authors embark on a search for examples of teaching practice that support ‘deep-learning’. For anyone interested in developing an insight into what school might be like as we approach the challenges of this post-normal, post-truth era, this book at least asks the right questions.

By Nigel Coutts

Learning about change from a home cooked meal.

Last week I decided that a good home cooked meal was in order. Lacking inspiration I turned to a recipe book I had been gifted the previous Christmas and found what appeared to be a tasty and nutritious option. I read on with enthusiasm and was soon imagining myself dining on this wholesome meal. If the end result looked half as good as the glossy picture that accompanied the recipe, I would be in luck.

I began the process of preparing this culinary delight by reviewing the list of ingredients. Keeping in mind that the recipe had been prepared by an expert chef with years of experience, I had no qualms at all planning a few adjustments to this list based on my personal taste and what I knew I already had in the cupboard. Other items would need to be sourced from the supermarket and despite many claims that they are the fresh food people, some of the ingredients had to be sourced from the tinned foods aisle while others came snap frozen. At least one of the ingredients came in a box, and even though the recipe made no mention of a ‘light batter’ I felt confident that this substitution would work well.

Returning home, I placed the ingredients out on the bench and began the meticulous process of preparing my meal. Once again, some adjustments needed to be made. Some of the equipment that the recipe called for were lacking from my kitchen, others were in the wash, some just seemed unnecessary. Sadly the quality wine which the recipe suggested was consumed in advance and so that didn’t make it into the pot (which should have been a small cast-iron pan). After an initial process of sautéing a selection of mostly fresh vegetables in a light fish sauce (which had become Soy Sauce, because I don’t like fish sauce), I realised that in my rush I had left some of the vegetables in the fridge. Not to worry, there seemed like there was enough greenery going into the meal.

Twenty-five minutes later, I removed the battered ocean trout from the oven. This was not a highlight of the process as the recipe had strongly suggested that the trout spends not more than fifteen minutes in the oven. My thinking was that the barramundi I had substituted for the trout would need longer, but it turned out (as revealed from the plumes of smoke emanating from the oven) that twelve minutes might have been ideal.

I was nearing completion of this task with just the sauce to complete. I had forgotten to pick up a lime while at the supermarket, but I did have a bottle of what was described as freshly squeezed lemon juice in the fridge, something I think might have come with the refrigerator, I can’t recall ever having purchased it. The sauce came together quite quickly, mostly because I had managed to find a sachet of what sounded like a very similar finishing sauce and which required just ten minutes in the microwave (actually two minutes but who cooks with their glasses on).

The meal was ready. I plated it up with care, added a splash of lemon juice to the sauce, partly for flair but mostly to reconstitute it after its extended stay in the microwave and sat myself down for what was bound be a splendid meal.

A few emails later I tucked in, and it was . . .

Well to be very honest it was terrible. Not only was it cold, (perhaps it was more than a few emails) but it also lacked any real flavour. The fish was like leather. The sauce tasted like lemon room deodoriser. The sautéed vegetables were both soggy, crunchy, burned and in parts still frozen, how any of that is possible I have no idea. The whole sorry affair was a complete letdown. It held absolutely no resemblance to the picture in the book. Clearly, I had been cheated.

The recipe book is now in the recycling bin where it belongs and good riddance to bad ideas.

So what does this have to do with education? Well often, when we are implementing a new idea, we do precisely what I did with my glorious home-cooked meal. We fail to implement our great idea in full. We substitute in ideas which are incompatible with our stated objective. We leave out some of the key components of the plan, either deliberately or accidentally. We let some parts of the plan sit and stew unattended for far too long. Other parts are rushed and never given the degree of thought they deserve. We seek out advice from experts or look to comprehensive research papers to guide our thinking but then promptly ignore the advice and implement something entirely different.

We do all of these things and then genuinely stand back and wonder why the plan failed. Clearly, the whole idea was flawed and indeed when we are asked why we have abandoned it, we will explain at length how we knew all along it wasn’t going to work.

Perhaps there is something wrong with our approach to change.

By Nigel Coutts

Learning And Teaching for Understanding - A day of learning with PZ Sydney Network

Today I had the pleasure of joining over three-hundred educators for a day of learning and sharing. That this was a Sunday and that the event was organised as a free event for educators by educators speaks volumes of the quality and care that educators bring to their role.

This event was organised by Project Zero Sydney Network, which is a collective of educators who share an interest in the ideas which emanate from Harvard’s Project Zero. The networks aim is to share professional learning opportunities with educators at no cost and in doing so, facilitate the expansion of pedagogical philosophies that are supportive of deep-understanding, critical thinking and lifeworthy learning. Drawing on Project Zero pedagogy and practice, PZ Sydney Network believes that:

• learning is a consequence of thinking

• understanding is not only something you have, but something you do

• intelligence is not one thing, but many, and is something that can be learned

• thinking skills alone are not sufficient; we must also have the disposition to use them

• thinking and learning are processes that are deepened when we make them visible

• collaboration is the stuff of learning.

Today’s event is the third annual conference offered by the network and interest continues to grow. Its founding members are committed to learning, and there is extensive experience within their midst. What unites them is their desire to collaborate with others and to facilitate learning through dialogue and facilitative coaching rather than banner waving or dictatorial instruction. Attending a PZ Sydney event is more like participating in a conversation with a trusted friend than listening to a lecture. There is a clear understanding that learning is best when it is mutual and that questions drive learning in new directions while answers suggest an ending.

On this particular Sunday, we were joined by the inspiring Tina Blythe. Tina has been a researcher at Project Zero for nearly 30 years. She is part of Project Zero’s online learning leadership team and is the education chair of the Project Zero Classroom summer institute. She is also a Lecturer on Education at the Harvard Graduate School of Education. Central to her research and teaching are how to create and sustain learning environments that support understanding. Collaborative inquiry and the collaborative assessment of student and teacher work are key focuses of her work

Among many gems, Tina shared a story of her personal learning. It was a story I have shared through this blog previously and involves Tina’s reflections on a professional learning event which did not live up to expectation. Tina utilised the metaphor of searching for driftwood on a beach to describe the process of connecting learning opportunities with our particular needs. The beach wanderer passes by many items which do not meet their needs. Only a few of the pieces of detritus will fit the wanderer's needs, maybe fitting into a particular project or perhaps offering enough possibility to be collected for one yet to be imagined. For the professional learner, the journey is the same. They engage with many ideas, strategies and solutions, but only a few are implemented or saved for future use.

“Sometimes, I go to the beach and I know exactly what I’m looking for: sea glass of a certain color, shells of a particular shape or size. Sometimes, I see something that catches my eye, and I’ll pick it up and bring it home, even though I’m not sure at that moment what I’m going to do with it. - you never bring home the whole beach.”

I found a few pieces of driftwood today that I will keep and make use of, but perhaps my favourite was one of those delightful wonderings that force you to reconsider what we might be best focusing on with our teaching. “What questions will our students inherit?” We know that the students we teach today will become adults in a world shaped mostly by the questions or challenges that we create for them today. The questions of climate change, displaced communities, a global population on the move in a world dominated by globalisation and rapidly evolving technologies are unlikely to be adequately answered in the short term and so will be passed on to the children of today. With this in mind, how will we educate our children so that they may find answers to the questions we have created?

Checkout #pzsyd on Twitter to see what others found.

By Nigel Coutts

Focusing on What Matters - From Identifying to Enacting our Big Rocks

There is a delightful story frequently told of a philosophy professor who shares a valuable life message with students through the story of the “Big Rocks”. There are many versions of the story, and the protagonist varies from philosopher to business guru to teacher, it goes something like this.

The professor places a large glass jar on the bench at the front of the hall. It is empty, and this fact is verified when the glass is turned upside down. From a small bag, the professor places one rock after the other into the jar until it can take no more. He holds the jar aloft and asks the students who are wondering what the point of this demonstration might be, “Is the jar full?”. In unison, the class responds “yes”. The professor returns the jar to the table and takes a second bag from under the table. Delicately and with some shaking of the jar to facilitate the process the professor now empties the contents into the jar. Out of the bag tumbles a stream of gravel which fills the spaces between the larger rocks already in the jar. Again the jar is held aloft, and the question asked, “Is the jar full?” and again the answer comes, louder this time, “Yes!”.

The professor’s face reveals little emotion as a third bag is pulled from beneath the table. This time fine sand is poured into the jar and with some gentle shaking encouraged to fill the small spaces between the rocks and the gravel. A faint smile is evident on the professors face as the jar is once more held up for review, and the students are asked, “Is the jar full?”. The students respond with a resounding and confident “Yes!” suspecting that the professor must surely be close to beginning the serious work of the lecture.

The now heavy jar is placed delicately back on the bench. The professor pauses for a moment, takes a small sip from the coffee cup which has sat as always on the bench beside a large pile of papers. He pauses as though distracted by some thought of times long past, before pouring the nearly full contents of the cup into the jar before repeating the process with a second cup from beneath the bench. A grin moves across a wise face as the jar is once more held aloft, and the professor proclaims triumphantly “Now the jar is full!”.

The message is now unpacked for the class. The jar represents our lives, and the challenge is to decide what we will fill our lives with. The large rocks represent those things which matter most in our lives. The gravel and sand the small things which occupy our time and keep us from what matters most. If we place the gravel and sand into our jar first, if we attend to the little things first, we have no space for the large rocks. The cup of coffee shows that no matter how full our lives might be, there is always time for a cup of coffee with a friend.

It is an essential story for educators to ponder as we decide not only what fills our lives but what fills the lives of our learners. It is a story that requires its audience to stop and ponder what are their “Big Rocks”, what are the things that truly matter. When we consider what we emphasise in our teaching the question needs to be asked, are we focusing on what we believe will truly matter in the lives that our learners are likely to live? Too often we spend time and energy on tasks which are not well aligned with what matters most to us and to our learners. The daily minutia of being a teacher pulls us away from the actions which we value most. Our students are required to engage in tasks which result in them developing a false view of what school and learning are about. We allocate our time and their time to activities which we know are low impact even when we know this is not what matters most. We remain in what Stephen Covey describes as the realms of the not important and fire fighting.

Engaging with the process of identifying our big rocks is well worth the time but that it should only be the first part of the process. Once we determine our big rocks, how will we make sure that our actions and the messages we send to our students, our team members and ourselves align with these ideals and move us closer to achieving our goals? It is always hard to move out of the zone of the urgent, low order tasks and move our thinking into the more strategic realm. In a recent article, Jeff Bezos of Amazon reflected that his role required him not to do lots of work but to make one or two significant decisions a day. The hard part is making the right decision. Maybe if we consider the magnitude of the impact that each action we take might have on our learners and on our school, we will gain a better understanding of how we allocate our time and make better decisions. 

By Nigel Coutts

Local Wisdom versus Global Assessments

A significant shift continues to occur within global education markets. It is signified by the manner in which it makes sense to speak of a global education market. It is driven by neo-liberalism and the expansion of markets into all aspects of our lives and it is made possible by manipulation of the third messaging system within the educational triad of curriculum, pedagogy and assessment. It is a drive towards accountable, comparable and productive education systems fine-tuned to maximise the return on investment and provide industry with the workforce it desires. What must be asked is how does this trend impact students and are these the forces that should be driving change in our education systems?

The concept of an International Education market has its origins in the emergence of global testing initiatives. Without measures for meaningful comparisons between national systems, the very idea of a market remained abstract. With the emergence of international tests such as PISA (Programme for International Student Assessment), it has become possible for direct comparisons of national education systems to be made using a score and with that ranking tables. Assertions such as 'We are falling down the international rankings and our students are performing at a lower level in some subjects than they were a decade ago, according to the OECD’ (Jensen, Hunter, Sonnemann & Cooper 2014 p3) become possible, and nations compete to out-educate each other.

PISA brings with it an interest in particular aspects of educational systems through its manipulation of the messaging system of assessment. What PISA measures is given value and focus but the true breadth of PISA is rarely discussed while the traditional domains of reading, writing, mathematics and science take the limelight. This narrow focus on select disciplines results in a narrowing of the curriculum and the logic behind this focus needs to be analysed.

While reading, writing and mathematics have always been considered of importance the emerging STEM (Science, Technology, Engineering and Mathematics) discipline is easily seen as a response to emerging trends in industry, a trend that predates PISA. 'The role of schools has become increasingly regarded as being to supply appropriate human capital to serve the needs of business and industry.’ (Angus 2009 p38) Imaginings of national prosperity are increasingly linked to the quality of graduates in the STEM fields as seen in reports from Australia’s Office of the Chief Scientist which state ‘Australia is now the only country in the OECD not to have a current national strategy that bears on science and/or technology and/or innovation.,’ (Chubb. 2014 p10) and PwC suggests a focus on our capacity for innovation linked to OECD reporting 'But we need to lift our game; the OECD recently rated Australia as only ‘average’ in its competency and capacity to innovate.' (PwC 2015 p13)

In 2018 PISA added an evaluation of students’ global competence and an optional assessment of financial literacy. For 2021 an assessment of creative thinking is planned and work on developing appropriate measures for this are underway. It will be interesting to see how the results of these assessments are reported by the mainstream media and the influence that they might have on educational policy. Will we see a renewed emphasis on global competence and creativity fuelled by reports that nations are slipping on these dimensions compared to a new global education elite?

The influence that PISA has on education is profound, and many of the problems it creates are outlined in an open letter written to Dr Andreas Schleicher, director of the OECD's Programme for International Student Assessment by a group of concerned academics. Chief amongst their concerns is that PISA brings a focus on the use of quantitative data and with that an emphasis on aspects of education which are easily measured. 'By emphasising a narrow range of measurable aspects of education, Pisa takes attention away from the less measurable or immeasurable educational objectives like physical, moral, civic and artistic development, thereby dangerously narrowing our collective imagination regarding what education is and ought to be about.’ (Letter to OECD)

That PISA is a product of an organisation whose focus is on economic development is also seen as problematic. 'But preparing young men and women for gainful employment is not the only, and not even the main goal of public education’ (Letter to OECD). The writers go on to state that 'OECD's narrow focus on standardised testing risks turning learning into drudgery and killing the joy of learning’ (Letter to OECD). The globalisation of education that PISA allows for sets an agenda that sits divorced from local issues and stands separate from cultural traditions and values. Driven by economic forces and linked to growth targets as it is, PISA shifts our thinking of what is important for education in one specific direction.

In this trend towards globalisation, the place that schools and education play within local communities is lost, replaced by national and global imperatives, 'the policy framework within which schools now work is characterised by new public management, top-down leadership, performativity and remote ‘standards’ (Angus. 2009 p37). In this market economy, 'Students and teachers are expected to turn themselves into the kinds of people demanded by ostensibly ‘high performing’ and ‘effective’ schools that succeed in market competition.’ (Angus 2009 p38)

The realities of how this transformation may occur are less clear and it is apparent from the dialogue that it is the responsibility of the individual to transform. PISA and NAPLAN create a setting in which ‘highly effective’ schools, teachers and students are believed to thrive and those who do not achieve success are the deviant minority. 'This minority has to be cajoled or coerced into being achievement-oriented, industrious, self-helping and properly trained for the workforce, and this is the strategy to overcome social exclusion.’ (Connell. 2013 p5) This blaming of individuals ignores the clear research that assesses the effect size of various influences on educational outcomes and reveals '85% of the effect being due to family circumstances’ (Angus. 2009 p38) What truly works or does not work for any individual child, or even any particular community is not easily measured and surely cannot be measured by one assessment.

The complexity of international comparisons is given little attention in the mainstream press. Where this complexity has been analysed and studied a more complete picture of what is working for some nations but also why this might not transfer to another system emerges. In ‘The Smartest Kids in the World’ Amanda Ripley (2013) looks at the education systems of some of the reportedly high-achieving nations including Finland, Germany, South Korea and Poland. Through this analysis, it becomes clear that each education system is a reflection of the culture in which it functions. While there may be some lessons to be learned and applied from one system to another, it would not be possible or effective for Australia to wholesale adopt another nations systems and process for education. It is deeply flawed to believe that such a move would be positively transformative and so one needs to ask what purposes are achieved through simplistic comparisons of national systems.

If one approach from international systems must be widely adopted, perhaps it should be the one that seems to deliver positive results with greatest consistency across multiple systems. Identified by the OECD and discussed in depth by Jensen et al, (2014) it is a commitment to professional learning programmes for teachers. The consistent features of such programmes are that they give time for teacher development, collaboration, mentoring, feedback and teacher driven research. Such programmes have the power to transform schools for the betterment of all. However, if such programmes are not allowed to develop in response to locally identified needs and are not driven by teachers and researchers working within local communities, they are bound to produce limited results.

Globalisation and market forces will not improve the educational success of all students. 'Broadly, we know how to make schools work even in environments of poverty: build up local experience, develop relevant curricula, create social solidarity and mutual help, put in serious resources’. (Connell 2013 p6). For education to thrive, we need a more honest discussion of what our schools need, one driven by richly complex and humanistic data and understandings, not global test scores.

by Nigel Coutts

Angus, L. (2009) Problematizing neighborhood renewal: community, school effectiveness and disadvantage Critical Studies in Education Vol. 50, No. 1, February 2009, 37–50

Chubb, I. (2014) Science, Technology, Engineering and Mathematics: Australia’s Future. Office of the Chief Scientist; Australian Government, Canberra

Connell, R. (2013) Why do market ‘reforms’ persistently increase inequality? Discourse: Studies in the Cultural Politics of Education

Jensen, B., Hunter, J., Sonnemann, J. and Cooper, S. (2014), Making time for great teaching, Grattan Institute

Jensen, B & Sonnemann, J. (2014) Turning around schools: it can be done Grattan Institute

PwC (2015) A smart move: PwC STEM Report April 2015 (Accessed online April 2015) -https://pwc.docalytics.com/v/a-smart-move-pwc-stem-report-april-2015

Ripley, A. (2013) The smartest kids in the world: and how they got that way. New York; Simon & Schuster Paperbacks

Why we don't cook frogs slowly and other thoughts on change

In the film An Inconvenient Truth, Al Gore relays the story of a frog that jumps into a pot of lukewarm water. We are told that if a frog jumps into a pot of boiling water, it will hop straight back out again, but the frog in the lukewarm water will stay there, even as the temperature rises. The frog's survival depends upon the presence of a sympathetic rescue party. 

It is a cinematic moment that has the desired effect. It is one of the moments from the film that the audience remembers long after the credits roll. I have often thought about how this metaphor applies to change and particularly the way that change operates in schools.

For this purpose, the story requires a small change. The frog has not happened to land in the pot by accident, it has been very deliberately placed in the pot, the pot is full of stock, and our goal is to transform the hapless amphibian into a tasty dish of cuisse de grenouilles (frogs legs). The frog goes in, the burner is lit, now we wait and consider which wine will make the perfect accompaniment. As the water warms the delightful aroma of gently cooking frog fills the room and the small party of diners assembled for the feast contemplate their choice of condiments.

This scenario is more like what happens when we are introducing a change. The process is initiated by deliberate and planned action inspired by a sensitivity to the need for a change. We have in mind a desired change, and we may even have a rough idea of how we might get there. At the beginning of our endeavour, there is almost always a degree of apprehension. Experience tells us that while everyone might want change, very few people want to change. Change is frightening and uncomfortable. We know that some people are going to embrace the change. They are either of the type that embraces, at least superficially, every change. Or the change is precisely what they have always wanted and so supporting the change is in their best interests. We know that some people will be ambivalent. Maybe they just don’t mind, or perhaps they have encountered so many changes in their time that they approach any new change with the belief that ‘this too shall pass’.

There will be those who find the change very uncomfortable. They will not agree with it, imagine it as the beginning of the end and a sure sign that management has lost all sense of direction. As the change progresses, they will shift their approach from sullen disquiet resistance to outright discontent.

Back to our culinary metaphor. As did our changemakers our cooks set out to achieve a particular purpose; a tasty and exotic meal (unless you are in France in which case it is a domestic classic or 'common as’). We knew that at least one of the dinner guests (froggy) would find the process more than a little uncomfortable. As the water warms, froggy moves from enjoying the warmth after a long spell in a bucket of ice water, to finding the temperature change quite disturbing. At this point, the polite thing to do would be to turn the burner all the way up and drown any feelings of guilt with a nice glass of something with too many bubbles.

Our change makers are facing a similar conundrum. They have initiated the change process and are beginning to notice that the anticipated discomfort is taking shape. What should they do?

Unfortunately, in too many cases they decide to slow down the process. The consensus is that a gradual change will achieve the same ends and give everyone the time they need to adjust. In the kitchen, this is equivalent of leaving the frog in the pot but turning down the burner. We are still cooking the frog but now as the temperature slowly climbs the poor little fellow develops a heightened sense of anxiety; the water is after all still getting warmer, or the change is still happening. In time our cooks and change makers notice this anxiety, and so the pace of the change and the cooking is again slowed. At this point, we are confronting a range of other problems. The change is not having the desired effect. The dinner guests are hungry and running out of champagne. The need for the change has only grown. If we don’t get something on the table soon, our guests will leave. The frog begins to smile.

After an extended soak in warm water, the frog is plucked from the pot and released back into the wild smelling oddly of onions and parsley. Our exotic dinner is replaced by home-delivered Pizza with a side of chips (not even french-fries). Our change effort is abandoned, and any future change efforts are put on hold for at least the foreseeable future. The status quo reigns supreme.

If we want to cook a frog, we need to accept that it will not end well for the frog. As Steve Jobs said of the role of leaders “If you want to make everyone happy, don’t be a leader - sell ice cream”. If we recognise that change brings with it a degree of discomfort, then when we are confronted by it, we need to find a way forward rather than beat a retreat. This does not imply that we push forward with no regard for those who are finding the change difficult. We do not turn away from empathy and understanding or move forward blindly paying no heed to those experiencing the change most immediately, but we also do not persist on the same route but at a continually slowing pace. We listen, we adapt, we empathise, we explore alternatives and build a shared understanding of how we can achieve our goals together. In the end, we might develop a vegan equivalent of cuisse de grenouilles and enjoy a lovely guilt-free meal.

By Nigel Coutts

Mathematical thinking presents teachers and students with new challenges

Mathematics is a discipline that is central to all national and state curriculum models. It is one of the core subjects studied by students across the world, and an understanding of mathematics is considered an essential skill for success beyond school. Mathematics is broadly acknowledged as playing an essential role in our lives and its importance in modern times is amplified rather than diminished by our ever-increasing reliance on technology. As is the norm across education, questions are being debated about the nature of mathematical learning and how we might ensure our students leave school with the skills, knowledge and dispositions they need. What does it mean to be a mathematical thinker? What mathematical skills and knowledge will our students require in the lives they are likely to lead? How do we increase mathematical confidence? Which pedagogies are most appropriate if our goal is mathematical understanding? What place do calculation and basic skills occupy? How do we increase participation rates of all students in advanced mathematics courses, particularly groups which are currently underrepresented? 

The response to this is a growing body of research that describes an emerging approach to mathematics. This contemporary research is reflected in the designs of the modern mathematical curriculum. In Australia both the Australian Curriculum and the New South Wales curriculum place emphasis on the ‘doing’ of mathematics.

In Mathematics, the key ideas are the proficiency strands of understanding, fluency, problem-solving and reasoning. The proficiency strands describe the actions in which students can engage when learning and using the content. - Australian Curriculum

The proficiencies reinforce the significance of working mathematically within the content and describe how the content is explored or developed. - Australian Curriculum

Students develop understanding and fluency in mathematics through inquiry, exploring and connecting mathematical concepts, choosing and applying problem-solving skills and mathematical techniques, communication and reasoning. As an essential part of the learning process, Working Mathematically provides students with the opportunity to engage in genuine mathematical activity and develop the skills to become flexible and creative users of mathematics. - New South Wales Educational Standards Authority (NESA)

The five components (communicating, problem solving, reasoning, understanding & fluency) of Working Mathematically describe how content is explored or developed − that is, the thinking and doing of mathematics. - (NESA)

This growing focus on mathematical thinking, reasoning, problem solving and understanding is plainly evident and in response to this teachers are exploring new pedagogical models. The shift away from teaching for the rote memorisation of prescribed methods requires teachers to rethink their approach to the discipline. With this new pedagogy comes a need to understand the processes of mathematical thinking in ways not previously required. When we require our students to be able to reason and problem-solve through unique challenges we also require our teachers to have an understanding of the mathematical moves that their learners are likely to call upon. This is challenging if the teachers only mathematical learning has occurred in classrooms with a strong focus on accurate replication of set methods. A teacher may know that the method for solving a question that requires division with fractions involves inverting one fraction and multiplying, but may not know why this method works.

According to the Australian Curriculum "Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations”, and as the curriculum’s writer, Peter Sullivan shares 'Students are not solving problems if they have been told what to do’. Peter Sullivan adds that 'students are not reasoning if they are merely repeating an argument developed by someone else – the reasoning needs to be their own’. This presents a real challenge to the average primary teacher and requires a shift not only pedagogy but access to professional development that addresses potential gaps in their mathematical understanding. Even to those who may be described as experts might have achieved this status through their mastery of ‘mathematical tricks’ or ‘shortcuts’ that allow a solution to be quickly determined but obfuscate why and how the method works.

There are also significant questions to be addressed here about the philosophical beliefs which underlie our approach to mathematics. If we believe that mastery of fundamental basic knowledge is required before students have opportunities to engage in problem-solving we will prioritise this learning above all else. If we require students to demonstrate accurate and rapid recall of number facts and in particular times tables we will dedicate time and energy to achieving this capacity and in doing so send strong messages to our students about what mathematical knowledge is and what the doing of mathematics involves. When we do introduce our students to mathematical thinking, understanding, reasoning and problem-solving, they are likely to approach it with a mindset that limits their capacity to learn. In place of a willingness to embrace challenges and engage with uncertainty, they are likely to come to mathematics with a belief that speed and the accurate application of a prescribed method is what the subject is all about. When we only experience problem-solving as an extension task at the end of a unit of learning it should not be surprising that student and teacher view it as a black-art that exists slightly beyond what they know or even what they might want to know.

Learning mathematics creates opportunities for and enriches the lives of all Australians. . . The curriculum focuses on developing increasingly sophisticated and refined mathematical understanding, fluency, reasoning, and problem-solving skills. - Australian Curriculum

The study of mathematics provides opportunities for students to appreciate the elegance and power of mathematical reasoning and to apply mathematical understanding creatively and efficiently. The study of the subject enables students to develop a positive self-concept as learners of mathematics, obtain enjoyment from mathematics, and become self-motivated learners through inquiry and active participation in challenging and engaging experiences. - NESA

It is enlightening to read the rationale and aims of these two curriculum documents. They describe mathematics as a beautiful and empowering discipline full of opportunities for creative and original thinking that leads to generalisations and abstractions. It is a discipline founded on essential dispositions such as problem-solving, communicating, understanding, representing and reasoning. It is perhaps a shame that the online presentation of both curriculums make it easy for the aim and rationale to be skipped over by teachers looking for the content they are to teach. The thinking behind these curriculums is not unique to Australia. A quick look at the Common Core Standards from the United States of America reveals a similar emphasis  in the identification of essential practices.

"These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections.” Common Core State Standards Initiative

As is the case in Australia, students in America explore content areas such as geometry, number, measurement and data through their engagement with and application of these proficiencies. Mathematical thinking in all cases is built on rather than extending by approaches to mathematical thinking, reasoning and problem solving. 

What is now required is a conversation around how we make this thinking accessible to all who are charged with teaching mathematics, particularly those charged with introducing students to the discipline and those whose experience with mathematical thinking might be limited.

By Nigel Coutts

Contemplating questions of work life balance

Visit the Apple website and have read of their “Jobs at Apple” page and you will be enticed by an organisation is all about creating the conditions where people can engage in work with impact.

Do more than you ever thought possible. Have more impact than you ever imagined. - This is where some of the world’s most passionate people create the world’s most innovative products and experiences. Join us and you’ll do the best work of your life — and make a difference in other people’s lives.

It sounds like the perfect description for a teaching role. After all, is this not how we see our roles as educators and is this not the sort of person we hope to have leading the learning of our children. As educators we are in the very special position of being that person with the opportunity by actions to shape a life, to lay a foundation upon which a lifelong love of learning is built and to empower the next generation. Surely this is a position that even an Apple engineer would be envious of.

Oddly lately I have been pondering how schools responds to the question of a work life balance. Let me try to explain my thinking. I am still trying to clarify my thinking here, so please bear with me.

I think it is a positive step that the question of work life balance is regularly an agenda item. The hero narrative of the teacher giving there all for the good of their students has probably served no one well. Claims that teaching is a calling and not a profession seem to undervalue the importance of professionalism and give permission for the deeply thoughtful and reflective practices of educators to be devalued. We should be proud of our profession and value what our professionalism brings to the care and nurture we provide to young people.

And we need to have conversations about how we take care of ourselves and how we meet our needs. This is where the question of work-life balance is of importance. For ourselves, for our families, for our students, our employers and all who rely upon us, our health and well-being matters. I wonder though if the current work-life balance agenda is based on a somewhat limited interpretation of what it means to achieve a work-life balance. I wonder how the work-life balance mix that is typically advocated for in this current agenda might be misaligned with the life-goals of some.

Don’t get me wrong, for many people the dialogue around striking a balance between life and work is important. Any move away from workplace cultures that place unceasing demands on time and availability, an always on lifestyle where sleep deprivation and constant business is a badge of honour, is a positive thing. Messages that it is ok to turn the phone off, disconnect from email and take a lunch break are all positive and necessary. I just wonder if we have perhaps have adopted a restrictive one-size fits all imagining of what it means to strike a work-life balance?

In our search for a happy work life balance, maybe we approach this goal from the wrong direction. At the core the concept is that individuals have their work which occupies a large amount of their time and energy and almost diametrically opposed to this their life which includes family and friends. Striking a work life balance becomes a balancing act between time at work, time with family and friends, between doing what we have to do for work and doing what we enjoy. And this is important for many people, but has the pendulum swung to far and are we now creating cultures where individuals confront pressure if their lives do not align with the traditional imagining of work and life.

Maybe the traditional approach to work and life does not serve the needs of all and ignores that for some of us the questions around work and life require a different perspective. Phil Libin the former CEO of Evernote, the popular cloud based solution for note-taking. When interviewed for Traingulation by Leo Laporte, Phil provided the following insight to this question of balance in life and work:

"If you think of it as life as being separate from work, then you’re going to have this conflict, if you think of them together and you try to have a job that’s sufficiently epic that you want to spend more time, that no matter how much time you spend at it you want to spend more then its great."

Astronaut Chris Hadfield echoes this sentiment in these words, “Decide in your heart of hearts what really excites and challenges you… Don't let life randomly kick you into the adult you don't want to become.” And these thoughts seem to fit with the sort of people who read the Apple jobs page and think, “yes, that’s where I want to be”, putting a ding in the universe. The challenge is how might we create cultures in our workplaces that give those who need it permission to switch off while also providing those who want to ‘do more than they ever thought possible’ the conditions they need to do just that.

It seems it is about empowerment and agency. If we are in control of our work and our life then we can make choices that provide us with the balance we require, whether that be through switching off or powering on. When our choices are restricted through policy or culture we will find balance a struggle but where we have control and can act strategically for our own well-being true balance becomes achievable.

By Nigel Coutts

Number Talks for Number Sense

"Number Talks" is an approach to the teaching and learning of Number Sense. Rather than relying on the rote-memorisation of isolated number facts achieved through drills of "table-facts", Number Talks aim to build confident, number fluency, where learners recognise patterns within and between numbers and understand the properties of numbers and operations. Number Talks are a "mind on" learning task that engages students in an active learning process as they search for patterns, decompose and recompose numbers and develop a flexible understanding. It is achieved through direct instruction methods and facilitative dialogue with the teacher or between groups of peers who have had experience with the number talks methodology. It becomes one of the routines of a classroom focused on mathematical reasoning.

Number talks are a valuable classroom routine for developing efficient computational strategies, making sense of math, and communicating mathematical reasoning. A number talk is structured to help students conceptually understand math without memorizing a set of rules and procedures. (Nancy Hughes)

Number talks are:

a brief daily practice where students mentally solve computation problems and talk about their strategies, as a way to dramatically transform teaching and learning in the mathematics classroom. Something wonderful happens when students learn they can make sense of mathematics in their own ways, make mathematically convincing arguments, and critique and build on the ideas of their peers. (Humphreys & Parker)

Number Talks should be a regular routine within the Mathematics Programme; a tool for building the Mathematical Fluency that underpins an understanding of the "Big Ideas"

Number Sense or "Making Friends with Numbers"

Number Sense is fundamental to success in mathematics and involves developing an understanding of number, patterns inside numbers, patterns throughout sets of numbers and the effect that operations have on numbers. It is much more than memorisation of table facts and unlike learning by memorisation develops a deep and flexible understanding that promotes mathematical confidence and is a solid foundation for reasoning and problem solving.

Number sense is important because it encourages students to think flexibly and promotes confidence with numbers. . . . The fact is, students who lack a strong number sense have trouble developing the foundation needed for even simple arithmetic, let alone more complex mathematics. A large body of research has shown that number sense develops gradually, over time, as a result of exploration of numbers, visualizing numbers in a variety of contexts, and relating to numbers in different ways. (Keith Devlin)

Research shows that students who are taught to rely on memorisation of number facts and mathematical processes do not perform as well as students who learn in an environment that emphasises number sense. Memorisation may help with less challenging questions, but is of little use as the questions become more challenging.

Students who avoid making an effort to understand mathematics concepts may succeed in some school environments; but a lack of deep, critical and creative thinking may seriously penalise these students later in life when confronted with real, non-routine problems. PISA results show that, across OECD countries, perseverant students, students with positive attitudes towards problem solving and mathematics, including high instrumental motivation to learn mathematics, interest in mathematics, high self-efficacy and self-concept, and low mathematics anxiety are less likely to use memorisation strategies. - OECD PISA Analysis - Is Memorisation a good strategy for learning mathematics?

This commonly shared example shows a student who has rote learned the vertical algorithm for subtraction but with little understanding. Rote practice is fraught with danger. Left unchecked, it can reduce rich mathematical concepts to a slew of rules and procedures that feel arbitrary and confusing to students. (Junaid Mubeen)

10 Guiding Principles for Number Talks

• All students have mathematical ideas worth listening to, and our job as teachers is to help students learn to develop and express these ideas clearly.

• Through our questions, we seek to understand students' thinking.

• We encourage students to explain their thinking conceptually rather than procedurally.

• Mistakes provide opportunities to look at ideas that might not otherwise be considered.

• While efficiency is a goal, we recognise that whether or not a strategy is efficient lies in the thinking and understanding of each individual learner.

• We seek to create a learning environment where all students feel safe sharing their mathematical ideas.

• One of our most important goals is to help students develop social and mathematical agency.

• Mathematical understandings develop over time.

• Confusion and struggle are natural, necessary, and even desirable parts of learning mathematics.

• We value and encourage a diversity of ideas.

• (Source - Humphreys & Parker)

Questions to Ask when facilitating Number Talks

Carefully selected questions are used to focus the students on sense making at the beginning of the problem:

• What's going on here?

• What are you noticing?

• What do you wonder?

• Tell me something about the problem.

• Forget about the question for a second. What's going on in this situation?

• What do you estimate the answer might be?

• What do you predict the solution might look like?

• Let’s try to visualise that?

• What makes you say that?

• Is there a pattern here?

• Can you build on that idea?

• Can we find a new way of looking at this?

• What if . . .? & How might . . .?

The language of a number talk is very deliberate and designed to invite participation and alternate perspectives. We ask ‘Who has an idea for a solution?’ rather than ‘Who knows the answer?”. All responses are valued and no response is dismissed. Each time a student is told their thinking is wrong, they learn that in mathematics there is a right answer and the goal is to guess what that answer is. Number talks work best within a culture that values mathematical thinking more than calculation. Students should be encouraged to share their ideas, debate responses, look for other solutions and demonstrate their thinking in multiple ways.

Making Number Talks Routine

Number Talks are most effective when they are a routine part of the students mathematical thinking and learning. A Number Talk is an ideal warm-up activity before other mathematical learning. A daily number talk can take between ten and fifteen minutes and this routine engagement with mathematical thinking builds number sense and fluency.

A ten to fifteen minute Number Talk should be a routine part of every learner's day.

By Nigel Coutts

Adapted from http://globalcognition.net/index.html

Resources that Support Number Talks

YouCubed - Number Talks

Fluency without fear: Research evidence on the best ways to learn Math Facts - Read Online

Sherry Parrish: Number Talks - Building numerical reasoning - 1hour 15 minutes - YouTube Video

Making Number Talks Matter - by Cathy Humphreys & Ruth Parker - Amazon Australia Link

In the Moment: Conferring in the Elementary Classroom - by Jen Munson - Amazon Australia Link

Number Sense Routines: Building mathematical understanding every day in grades 3-5 by Jessica Shumway - Amazon Australia Link

Classroom-Ready Number Talks for Third, Fourth and Fifth Grade Teachers - by Nancy Hughes. Ulysses Press. - Amazon Australia Link

Growth = Mindset + Action

"Sometimes people overcomplicate things”

It was this line in a Growth Mindset video that made me stop and take notice. It has a truth to it that we can easily miss. It directs our thinking in a new way, towards what lies at the heart of the matter. In this case it shines a light on what we do if we have a growth mindset and what we should be doing when we have a fixed mindset. 

The uncomplicated version of the story goes like this: Belief leads to Action, and Action leads to Growth. If I believe I can do something I am more likely to take action to achieve my goal. I am more likely to practise, more likely to persist with a challenge, more likely to listen to feedback and more likely to incorporate feedback into my actions. If I am taking actions that are likely to result in growth, then I am more likely to see growth than if I don’t take action. My mindset or the beliefs which underpin it, when backed by action make growth possible. The opposite is also true, if I believe I can’t grow or learn, I am less likely to take action, less likely to persist and less likely to listen to feedback. I will fail to take action if I believe that my innate talent means I don’t have to try and without action I will fail to grow. 

Henry Ford was right, "Whether you think you can, or you think you can’t, either way you are right.” 

Having the right mindset is alone not sufficient for growth. I might talk the talk about a growth mindset and believe I can learn a new skill but unless I back that belief with action, it is just talk. Indeed, I probably need to look at why I am not taking action to achieve new learning, maybe I have a false growth-mindset. If I am not willing to take the risk involved in trying something new, if I am unwilling to begin a learning journey, if I talk about potential but avoid demonstrating it, I am using the language of a growth mindset to protect or hide my fixed mindset. 

Taking action alone is also not enough to ensure growth. There are many ways I can practise at achieving the perfect golf swing but there is a very good chance that my unguided efforts will go to waste. Practising a flawed method ten thousand times can only result in failure and build habits which will be hard to unlearn. If action is to result in growth and success it needs to be the right type of action and my chances of achieving this increase if I seek support in my learning from an expert. But hours spent in the company of an expert is also never going to result in growth. At some point I need to do the work, take the action, practise and practise some more. If I do this and take on board feedback that will refine my actions than I have a chance of achieving my goal. 

If I am a school interested in student growth through the adoption of a growth mindset programme, but I only target a part of this process I am likely to achieve little success. Many schools have adopted strong beliefs about the value of a growth mindset. Their beliefs are evident in the posters and slogans adorning their walls. “F.A.I.L = First Attempt in Learning”, “The Power of YET”, “Everything is hard, before it is easy”

All of this can be good but advocating for a growth mindset alone is not going to result in a change. Unless the belief that we can all grow is coupled with action that is designed to achieve that growth we will remain where we are.

What can be remarkably powerful is to apply the behaviours we adopt in our growth mindset contexts to those that trigger a fixed mindset. If I can recognise a context where my thinking is fixed but then apply the behaviours I bring to my growth mindset contexts I open the door to growth. Maybe I believe I can learn a new piece of software, but I can’t learn to play the piano. Instead of giving up on the piano, I apply the actions I would take to learn a new piece of software. I seek expertise, I give time to the task, I persist when it gets hard, I adjust what I am doing, I practise and practise some more. I know these actions work for learning software so I shouldn’t be surprised when they work with the piano. I also remind myself that one of the biggest obstacles is my mindset. Instead of trying to change that and then take action, I begin by taking the action I would take if my mindset was different. Actions are much easier to change than mindsets. 

So, by keeping things simple and focusing on beliefs, actions and growth I can achieve my goals and tackle new challenges and I have an approach to apply where my mindset lets me down by believing in the power of taking action towards a goal. 

By Nigel Coutts

Watch the video that inspired this.

The Curse of False Expertise

A growing body of research reveals details of the “Curse of Expertise” in which it is shown that as an individual’s level of expertise increases, their ability to communicate their knowledge to a novice declines. The extent to which an expert assumes information to be common knowledge can be so large that they fail to see the gaps which exist in the understanding of a novice. The take-away is that the expert needs to consciously think like a novice and be deliberate in seeking an understanding of where the novice’s knowledge of a subject strikes its limit.

Physicist and author, Richard Feynman could be considered a master at overcoming the curse of expertise. He understood that the best indication of a truly deep understanding of a concept was revealed in one’s capacity to describe it to a child. His advice: "When we speak without jargon, it frees us from hiding behind knowledge we don’t have. Big words and fluffy “business speak” cripples us from getting to the point and passing knowledge to others.” Feynman understood that his expertise would prove to be a barrier to his students learning and that as such he would need to take actions to ensure his knowledge was accessible; something all educators should do.

But what if our expertise is imagined or false. What if what we think is so, just ain't so. This might be more common than we care to admit, and it is worth considering the source of this difficulty and its implications.

Not just what we were taught but also the way that we were taught things in school shapes our beliefs about what matters, and these messages can be hard to undo. Consider the average mathematics class that the typical teacher experienced when they were forming an understanding of what mathematics is all about. The emphasis was almost certainly on accurate calculations and application of prescribed methods which would result in the correct solution. Today that teacher is likely to believe that mathematics demands this sort of knowledge and that an expert mathematician is one who can quickly and accurately perform calculations. The trouble is this is false expertise as revealed by comparing these beliefs with the way that maths is described by a modern syllabus; "Mathematics is a reasoning and creative activity employing abstraction and generalisation to identify, describe and apply patterns and relationships….The study of mathematics provides opportunities for students to appreciate the elegance and power of mathematical reasoning and to apply mathematical understanding creatively and efficiently.” (NESA, 2017) In place of speedy and accurate calculation the syllabus speaks of “enjoyment”, active participation” and “challenging and engaging experiences”.

The same is undoubtedly true of other disciplines. If you spent time in a typical Science classroom you would likely believe that the work of a scientist revolves around correctly filling in a science lab report. The emphasis of the learning is on accurately filling in the template and a knowledgeable scientist would know the template by heart. In History the lessons seemed to revolve around remembering a list of dates, names and places and expertise could be measured by the number of facts which one might recall on demand. Somehow this does not fit with the rationale for the study of History according to the syllabus: "History is a disciplined process of inquiry into the past that helps to explain how people, events and forces from the past have shaped our world.” And if Geography is a “rich and complex discipline" that “build(s) a holistic understanding” and "is the study of places and the relationships between people and their environments”, how does it get reduced to knowing where you find Muscat on a map. False Expertise.

The implication of this unquestioned false expertise is that it becomes self-repeating. We believe that our knowledge base and underlying beliefs about the disciplines we teach are sound. Our teaching methods are founded upon this knowledge and these beliefs and so we present our students with a view of learning within these disciplines which is aligned with them. We perpetuate false expertise.

We must unlearn and relearn what we know and in doing so question the beliefs upon which our expertise is constructed. We need to examine closely the rationales for what we teach and understand deeply the concepts our students are needing to learn. We need to ask always, What will students actually do with the skills and knowledge they are acquiring and what underpins my belief that this learning will matter in the lives they are likely to live.

By Nigel Coutts

Learning From the Feynman Technique

The Curse of Expertise

NSW Educational Standards - Curriculum and Syllabuses

In Postnormal Times our Students need to be Brave

In contemporary times, we understand the importance of creativity, imagination and ideation. We are living in times shaped by chaos, complexity and contradiction where we confront ‘wicked problems’ such as climate change, globalisation, rapid technological transformations. These are times of rapid accelerations and ‘post normality’. The patterns of action and the organisational structures which served our needs in the past no longer apply and we are constantly confronted by challenges for which we do not have ready-made solutions. Only through the deployment of our most creative and innovative dispositions will we think our ways forward. 

“In an uncertain and often volatile world, successful organisations and their leadership must embrace ambiguity, disruptive change, risk, and the exponential quickness of the digital 4th industrial revolution.” (Jefferson & Anderson, 2019)

In “Normal” times, we had action stories which served our purposes well. In schools, these action stories revolved around the transfer of knowledge and skills from one generation to the next. Normality was normal because we had a clear understanding of what was required for young people to participate in the world of adults. There was an obvious body of knowledge and set of skills which would serve the school leaver well. Reading, writing and arithmetic, combined with a knowledge of some history, world place names, an understanding of some basic science and solid interpersonal skills mostly centred on following social norms was sufficient for a citizenry in normal times. The structures of our schools served these needs sufficiently well and our exit slips provided future employers with the evidence they required to determine which of our students would be the most suitable candidates. 

All that was ‘normal’ has now evaporated; we have entered postnormal times, the in-between period where old orthodoxies are dying, new ones have not yet emerged, and nothing really makes sense. (Sardar, 2010)

But now we are in Postnormal times and what once worked is at best no longer sufficient, at worst it is outright dangerous. The skills our students need to develop are centred around dispositions not readily learned in a teacher directed classroom. Creativity, critical thinking, collaboration, communication, problem-finding, ideation, hypothesising, and innovation are now critical skills where but a short time ago they may have been viewed as the innate properties of certain individuals. Each requires that students are learning within environments which provide frequent opportunities that demand these skills. Each is learned not through direct instruction but through immersion in a culture where thinking in all its forms is the norm. 

“We need to transform our schooling system from a predominantly test-driven transmission model of learning to a place where creativity, collaboration, communication and critical reflection are central to learning.” (Jefferson & Anderson. 2018)

Coupled to this drive towards creativity and innovation is an understanding of the importance of individual and collective agency. In these times of volatility and chaos it is easy to see the world as something that happens to us and turn towards despair. The postnormal era is also the Post-truth era and it is easy to lose faith in the institutions in which we once placed great trust. Our politicians seem to play games with the truth, science is openly questioned and painted with the same brush as opinion. 

‘Our beliefs come first; we make up reasons for them as we go along. Being smarter or having access to more information doesn’t necessarily make us less susceptible to faulty beliefs.’ (Brotherton, 2019)

Empowering us against despair is our sense of agency; the degree to which we are able to imagine our capacity to influence and shape events towards our purposes. Where the individual full of knowledge and possessing a pre-defined set of skills achieved through a traditional education may find themselves lacking in times of rapid and volatile change, the creative and agentic innovator will see opportunity and new possibilities. 

“agency,” is the ability to make choices and direct activity based on one's own resourcefulness and enterprise. This entails thinking about the world not as something that unfolds separate and apart from us but as a field of action that we can potentially direct and influence. (Ritchhart. 2015)

If we are to cultivate the dispositions required in these times of postnormality and post-truth we need to establish cultures in our classrooms which will allow them to thrive. As we strive to do so it is likely that we will recognise the need for a classroom culture that is safe and where risk-taking and mistake making are not only tolerated but embraced. As we move towards a valuing of creativity we move away from notions that there is a right or wrong answer. There is not a known method and it is unlikely that success will be achieved on the first attempt. In building this culture we seek to eliminate messages that are likely to restrain or limit thinking and we value the creative process at least as much as the creative product. 

In this drive towards the provision of environments which are safe and risk-tolerant there is a danger that we might go too far. We have done this in the past when in an effort to create safe playgrounds, we removed every possibility of risk. The result is a generation of children whose safety is ensured by the structures in which they spend their days but who have not developed the ability to evaluate risks. 

Creative, innovative, agentic thinking demands bravery. If, in our efforts to provide environments which support risk-taking we remove all risk or deny the bravery demonstrated by individuals who take risks with their thinking, we do our students a disservice. Rather than removing risk, we need support all learners to take risks and to be brave. We need to acknowledge brave thinking as a step towards learning. We need to ask our students to be brave in their imagining, their questioning, their appraisals of truth, their confronting of power and their agentic actions. We need to show that we are brave and that bravery is a part of our lives. As Nelson Mandela shares "I learned that courage was not the absence of fear, but the triumph over it. The brave man is not he who does not feel afraid, but he who conquers that fear."

By Nigel Coutts

Rob Brotherton (2016) Suspicious Minds: Why We Believe Conspiracy Theories. Bloomsbury Publishing.

M. Jefferson & M. Anderson. (2019) Transforming Organizations: Engaging the 4Cs for powerful organisational learning and change. Bloomsbury Publishing.

M. Jefferson & M. Anderson. (2017) Transforming Schools: Creativity, Critical Reflection, Communication, Collaboration. Bloomsbury Publishing

Ritchhart, R. (2015) Creating cultures of thinking: The eight forces we must truly master to transform our schools. SanFrancisco: Josey-Bass  

Sardar, Z. (2010). Welcome to postnormal times.Futures, 42(5), 435-444.