Bringing concepts to early learning in Mathematics

Our beliefs about mathematics play a significant role in how we approach learning within the discipline. These beliefs are established by the nature of our early engagement with mathematics and are difficult to change once established. For many people mathematics is viewed as a subject that is not for them. Indeed the situation is so bad that many people will say that they are not a maths person and approach mathematics with fear and anxiety.

The beliefs one holds regarding mathematics have their roots in the manner by which we were introduced to the discipline. Sadly, many learners, are taught mathematics through methods which emphasise rote learning of methods that result in singular correct responses. The emphasis is upon speed and accuracy. Talented mathematicians are those who can rapidly recall the required method and then perform each step without error. Learning mathematics is about memorising prescribed knowledge, learning the methods and acquiring a set of strategies which make the process easier. Understanding, creativity, critical thinking, questions and reasoning have little or no place in this version of mathematics.

Our earliest experiences with mathematics shape our beliefs about mathematics. The challenge is to alter the nature of this early experience. In place of engaging with mathematics as a set of processes to be mastered, our long term learning is best served when we are encouraged to understand the concepts upon which mathematics is founded. Too often, the idea of building a conceptual understanding is considered too complicated for young minds. In this model, conceptual understanding is achieved once mastery of the basic processes is achieved. It is imagined that after many years of disciplined study, the individual will have a zen moment and achieve a conceptual understanding. But, we should be engaging with the concepts first. When teachers consider not the steps in the process but the nature of the concepts, they approach the learning from a very different perspective. By asking what the concept here is and how might I build an understanding of this, we begin to plan learning that achieves a conceptual understanding. This approach also allows the learner to see mathematics as a beautiful and ever-expanding discipline that invites creativity and critical thinking.

A conceptual approach to mathematics has a further advantage to teachers who feel they are struggling with a cramped curriculum. When one considers the essential concepts of early mathematics, you find that there are as few as five or six fundamental concepts and perhaps twenty essential ideas which evolve from each concept. If a child enters Kindergarten with an understanding of these concepts and a belief that mathematics requires creative and critical thinking, they have a solid foundation for success.

Several researchers have contemplated what the essential concepts of early mathematics are. One model derived from the research of Diezmann & Yelland (2000), and Fromboluti & Rinck (1999) describes the essential concepts or dispositions as Number Sense, Representation, Spatial Sense, Measurement, Estimation, Patterns and Problem Solving. This model has some overlap with the necessary skills within mathematics. An alternate approach would be to consider both concepts and mathematical skills or dispositions. This model aligns well with those adopted by the Australian Curriculum and Reporting Authority (ACARA), New South Wales Educational Standards Authority (NESA) and in the United States of America the Common Core State Standards. In these models, students engage with concepts through learning that builds and incorporates fluency, understanding, problem-solving, communicating and reasoning.

Reid and Andrews describe an alternate, and preferred model, of the essential concepts for early mathematical understanding in “Fostering Understanding of Early Numeracy Development”. (2016) In this they outline six significant areas of early numeracy:

• numbers and counting

• sharing, number comparison and counting

• calculations

• patterns

• shapes

• measurement

This model is unpacked further in the diagram below. Each of the concepts described by Reid and Andrews has a significant role to play in mathematical learning both in the early years and later as life-long learners of mathematics. Engagement with these concepts should be through experiences which are not aimed at mere memorising. Take the essential skill of counting with numbers. Listening to many children count from one to ten reveals that they have learned to recite this as they may learn to recite a poem. Interrupt them partway through, and they need to begin again from the start. When students learn to count through rote learning methods, they miss vital components of their learning. An understanding that seven comes after six and that five comes between four and six is not developed by lyrically parroting the numbers. Students need to experience the counting numbers and the concept of cardinality in connection with objects and collection which they count.

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What is required from the teacher is a deep questioning of what each concept involves. When we take the time to ask “What does it mean to understand this?”, “What does understanding this look like?” and “How might it demonstrate my understanding?” we also begin to see what it takes to build this understanding. While a teacher might like to be shown processes for teaching each concept, such an approach has the danger of not permitting the teacher to truly understand why the pedagogical moves they have learned work. This is the equivalent of learning mathematics as a process to be applied in the manner of a robot. Teachers who are allowed the time and who collaborate towards a true understanding of the concepts they teach are much more likely to engage their students.

Teaching mathematical concepts in the early years might require additional time and thought, but it is a much better fit with the philosophy of early years learning. Most importantly, it sets a strong foundation for all future learning in mathematics and shows the young learner that mathematics is for them and is a subject of beauty and creativity.

By Nigel Coutts

Reid, K. & Andrews, N. (2016) Fostering Understanding of Early Numeracy Development. ACER; Camberwell Victoria

What it takes for deep learning in primary education?

Our goal might be to support Deep versus Surface Learning, but what does this mean in practical terms. What are the beliefs and dispositions which support teaching for deep learning, and what are the implications of this in terms of the pedagogy we adopt?

Ron Ritchhart describes encouraging Deep vs Surface Learning as one of the five expectations that help shape a culture of thinking. The five expectations describe a continuum along which our teaching practices fall. It is not that we spend all our time at one end of this continuum; the intent is not to create a dichotomy of good versus bad practices. It is more a case that we aim to spend more of our time leaning towards one end of the continuum than the other and that we look for strategies which take our learners with us in this direction.

Surface strategies focus on memory and knowledge gathering, whereas deep strategies are those that help students develop understanding. In designing any episode of learning, effective instructors tend to prompt their students to employ certain modes of processing. This prompting can be done either explicitly as part of the assignment itself, as with the use of thinking routines, or implicitly by signalling the use of what have become commonly expected modes of processing within that learning group for completing such tasks. - Cultures of Thinking

In their recently published research, Jal Mehta and Sarah Fine seek to explore the characteristics of teachers who encourage deep learning. From their search for Deeper Learning in American high schools, they found that there was a remarkably common set of beliefs amongst teachers who were successful in teaching for deep understanding. Such teachers had a high level of pedagogical knowledge and a preferred approach to teaching. They were deliberate and consistent in their approach and believed in the methods they used. They had a stance towards teaching as an act of igniting a spark, encouraging curiosity and interest more so than filling a bucket with knowledge. They could describe seminal experiences which had shaped their approach as teachers. Defining moments of understanding from which they came to see the role they might play if they adopted a particular stance. They had these characteristics, and they perceived the discipline they taught not as a body of knowledge to be learned but as a way of making sense of the world.

To a person, they saw their disciplines as open-ended rather than close-ended fields, meaning that they saw their fields as places where people had constructed provisional knowledge, rather than as places where there was a finished set of answers that needed to be passed on or “professed” to others. . . If teachers saw their fields as fixed or inherited bodies of knowledge, teaching as transmission seemed like a logical and efficient approach. . . . Conversely, if the fields were understood as places where different people would develop different interpretations, experiments, and approaches to problems, it seemed natural to invite students into this process of inquiry, connecting them to the generations of scholars and seekers of knowledge who had come before. (Mehta & Fine 2019 p352)

These teachers understood the true nature of their discipline. They saw themselves as members of a profession that was alive and to which they might contribute new knowledge. Their most valuable knowledge is an understanding of the epistemological foundation of the discipline. They may also possess sound discipline-specific knowledge, but they know that possessing this alone is not sufficient. A scientist is not defined by their recall of the periodic table but by the manner in which they approach puzzles and ambiguity. An author may require a sound knowledge of grammar, but they are defined by their approach to communication as a creative act between their language choices and their audience. Each discipline has its unique epistemological foundation, and deep learning is achieved when teachers invite their students to become participants in this.

This presents a particular challenge for primary and early years teachers. The role requires expertise across multiple disciplines, and while we have some exposure to each, it is unlikely that this is from working as a professional in the field. Undergraduate teaching courses will attempt to provide an overview of the content to be taught, and the curriculum documents unpack this further. Once on the job, professional development is available, which in some cases is specific to teaching within a particular discipline. Courses provide insight into the pedagogical moves which support the learning of specific content. Teachers share project ideas which work, mathematical puzzles which challenge students in new ways and strategies for writing which enhance the quality and creativity of students responses. What is missing from all this are opportunities to engage in the true work of the discipline. There are few opportunities for teachers at any level to be a scientist, a geographer, a historian or an engineer. The result is that most have a limited view of what it is that these people do and a bias towards the historical knowledge base of the field.

The effect of this can be seen in the general level of confidence with which teachers engage with learning across the disciplines. The life of an author or writer is perhaps not too far removed from the experience of the typical teacher. Both are very much about communication. Authors and teachers are in the business of telling stories. Both manipulate language to achieve a desired effect. When it comes to teaching English, most teachers exhibit a reasonable degree of confidence and have an understanding not just of the knowledge an author requires but of what an author does. Maybe the average author disagrees with this perception of what they do, but the teacher at least feels confident. Step into science or mathematics or geography, and the typical teacher relies upon the knowledge they possess, the facts and figures they can recall. They know the scientific method, they know what the symbols on a map mean, they know the process for long division. They know more than their students and are able to use their general teaching skills to transfer this knowledge into the minds of their students.

What they don’t have is a true understanding of the discipline as an open-ended field to which individuals contribute new knowledge and interpretations through a discipline-specific epistemology. This results in lessons which focus on finding engaging methods to teach the content rather than genuinely engaging the learner in the art of the discipline. Mehta and Fine share the metaphor developed by David Perkins of a pedagogy which invites young learners to play the ‘whole game’. When learning to play a sport, the novice player comes to develop a passion for the game not from drills but from the opportunities to play the game at a junior level. Aspects of the game are adjusted to suit their lack of experience, maybe a softer ball or a smaller field is used, but the fundamentals are the same. When applied to learning within a discipline, deep learning is achieved by teachers with sufficient confidence and understanding of the discipline they teach that they can deliver a suitable ‘junior’ version to their learners.

The challenge then is to provide our excellent primary and early years teachers with experiences which reveal to them what it is to be a scientist, geographer, historian etc. Rather than expanding their knowledge of the field, they require experiences which allow them to engage in the practices and the epistemology of the discipline. Armed with this understanding and with their deep understanding of pedagogy, they will become empowered to invite their learners to engage fully with the disciplines they teach.

By Nigel Coutts

Mehta, J. & Fine, S. (2019) In Search of Deeper Learning: The Quest to Remake the American High School Harvard University Press.

Perkins, D. (2009) Making Learning Whole: How seven principles of teaching can transform education. Josey Bass, San Francisco

Shifting towards student centred learning

Particular patterns of pedagogy have been of most interest to me across the years, particularly those that shift the focus from what the teacher does to what the student does. With this shift comes an emphasis on understanding how students learn and with this knowledge in mind developing learning experiences that will allow them to develop their skills for learning.

This pattern comes out of the emergence of a number of elements impacting education. One is the rise of ICT and the shift that this brings to the importance of content knowledge. When the teaching of content knowledge was an important role for teachers the emphasis was on the transfer of this from the teacher to the student. Teaching was about how effectively this transfer could take place and how this transfer may be measured. The student’s role in this process was relatively passive and receptive.

Now that access to content is ubiquitous the value of pools of knowledge stored in long term memory has declined. But the challenge of ubiquitous knowledge is compounded by trends towards the deliberate perversion and falsification of knowledge. "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.” (Matthew d’Ancona) Our students require the cognitive tools which allow them to seek the truth and falsehood in the information they confront and while a base load of knowledge may assist with this, skills and dispositions for truth seeking are vital and must be nourished.

Overall the emphasis is on what students are able to do with the knowledge they possess and we must provide opportunities for this to occur. (Wagner & Dintmarsh 2015) Further we must consider that the opportunities that students have to have to be publishers of knowledge, ideas and products has risen alongside other technological changes. Our students can readily become creators of content and will enter a workplace where this is an expected skill.

For pedagogy the consequence of this is that we shift towards a student centred learning model in which the students are empowered to be learners. Seeing students as creators of works, finders of problems, metacognitive learners and global connected collaborators brings a shift in the role of the teacher to one of guide and mentor. (Lough ran 2013) Much is made of measuring who does most of the talking in classrooms and the shift is towards a classroom dominated by the students' voices. (November. 2012). We can set up scenarios in our classes that allow students to fail and in doing so explore iterative learning cycles of trial and error through which students learn ‘grit’ and expand their ability to grapple with complex ideas and solve ‘wicked problems’. The assessment in these classrooms is more interested in evaluating the processes of problem or inquiry based learning utilised by the students rather than the recall of content. Students learn to identify a meaningful problem, structure it in a way that facilitates inquiry, gather and evaluate information and share the results with an interested audience.

The difficulty that all of this shift in pedagogy is that neither the curriculum or the ‘High Stakes Testing’ of NAPLAN and HSC have kept pace with the change. While teachers struggle to adapt their pedagogy to better fit this new model they do so with a narrow, content heavy curriculum and in a climate of testing that focuses on base skills in limited curriculum areas. That compliance with the curriculum, curriculum knowledge and performance on standardised tests are measures of school and teacher success makes the task of delivering a student centred pedagogy more difficult. For the students they are confronted by a conflict in the three message systems that play a most significant role in prioritising education; curriculum (the what might be taught), pedagogy (how teaching and learning is delivered) and assessment (what is valued by its measurement).

The result is we have students who are engaged by learning that focuses on their long-life skill development and challenges them with meaningful learning experiences linked to their interests and real world problems and yet they are measured against a curriculum that overemphasises specific content knowledge and tested in ways that do not allow them to use their skills for creativity, collaboration and connectedness. The result is a grammar of schooling which while excellent preparation for a narrow assessment regime is less than ideal preparation for life.

By Nigel Coutts

d’Ancona, M. (2017) "Post-Truth: The New War on Truth and How to Fight Back”. London, Elbury Press.

Loughran, J. (2013). Pedagogy: Making Sense of the Complex Relationship between Teaching and Learning. Curriculum Inquiry. 43, 1, 118-141.

November, A. (2012) Who owns the learning?: Preparing students for success in the digital age. Solution Tree Press; Bloomington IN

Wagner, T.& Dintmarsh, T. (2015) Most likely to succeed: Preparing our kids for the innovation era. Simon & Schuster; New York

Reflecting on report writing time - How might we maximise the value?

For schools in Australia and many parts of the world, we are heading towards the end of another school term and year. That means report writing season. For the next few weeks, teachers across the country will be huddled in front of computer screens, writing reflections on the progress their learners have made. Mark books will be opened, assessments consulted, work samples will be reviewed. All so that in the first week of the long Summer vacation students can sit and read their report and make plans for how they will enhance their learning in the coming year.

Or at least that is what we hope will happen, but surely few of us actually believe it will.

Reporting is in most systems a significant component of how we provide feedback on the learning that has occurred. It occupies a great deal of time, energy and mental space in the annual schedule of most teachers. Each school has their way of doing it. There are some consistent elements which have persisted over time. Most use some form of grading system. Most include a teacher comment either on the child holistically or broken out by disciplines or both. There are some progressive elements appearing such as comments or tick boxes which share perceptions of learner dispositions and mindset. We report on academics and social & emotional learning. There is an inevitable time crunch as deadlines loom, and in the final weeks of term, we transform from educators to editors.

Fortunately, or surprisingly, we don’t ask ‘why’ we do this all that often.

Surely the goal with reporting is to provide the learner, their parents and their future teachers with information about where they got to with their learning. To an extent, reporting achieves at least parts of this. Once they decode the teacher speak, parents gain some idea of what their child learned and perhaps what they are yet to master. They may have an idea of where their child sits in comparison to their peers. They may be able to see growth from one year to the next. Dig beneath the surface, and you might find that some of the details are a little vague. This is inevitable given the complexity of learning, contemporary curriculums and the subtle nature of an individual’s personal growth over time. To capture all that occurred throughout a semester of learning would require a much longer document or maybe a short film (think short like Ben Hur). If the following year’s teacher reads the reports, they too will have some idea of what the child learned. Again the limitations of the reporting process hinder the utility of this information and a good conversation with the previous teacher is generally considered of greater utility.

This leaves the learner, the person at the centre of all this effort. What purposes for the learner does the report serve?

Sadly, the truth is not much. After all they were there for the whole journey. They experienced the successes and the failures. They sat wondering what the teacher was on about. They sought to understand the new concepts, answered questions, completed tasks, collaborated with their classmates. They possibly read and maybe even wrote down countless learning objectives. They received marks, grades and feedback. They sat with their teacher and listened to feedback, asked questions and sought help. The report to them is like a postcard from a vacation. Nice to share with someone else, but, “you should’ve been there”.

As a form of feedback, which surely they are, reports are pretty ordinary.

In a study which considered evidence from well-designed studies conducted over a ninety-year period, Kluger and DeNisi (1996) found that feedback actually made performance worse in 38% of the studies. What this study and others like it reveal is that the manner in which most feedback is provided is at best of little value to most learners and at it’s worst is damaging. This points to a need to alter how we provide feedback. Lipnevich & Smith (2008) report that “Detailed feedback specific to individual work was found to be strongly related to student improvement in essay scores, with the influence of grades and praise more complex. Overall, detailed, descriptive feedback was found to be most effective when given alone, unaccompanied by grades or praise.” Further, research by Pulfrey, Buchs, and Butera (2011) show that “grades and grades accompanied by comments incited equally lower levels of intrinsic motivation”.

In this video, Dylan Wiliam describes what makes feedback effective. It is noteworthy that the largest impact on learning is feedback provided to the student in the moment. While the learning is taking place, while the learner has opportunity to integrate advice and take immediate action is the ideal time to provide feedback. It seems vital that we understand this. Wiliam makes this point well using the analogy of driving by looking through the windscreen or the rearview mirror. Traditional feedback is like looking through the rearview mirror at where you have been. Effective feedback shows you the view ahead, where you are going, and how you will get there. It is about action on reflection.

We should be honest then with the way that we perceive reporting and consider what actions we might take alongside what our systems and parent bodies require from us. What dialogue might we have with learners as we conclude our year of learning with them that will allow them to build on what we have achieved? How do we encourage them to reflect on the learning journey they have had and then cast their mind forward to the actions they are yet to take? How do we communicate this to their parents and their future teachers? While we as teachers are engaged in this extensive process of reflecting on our learners' learning seems like the perfect opportunity to invite them into the dialogue. Doing so might make the whole reporting process somewhat more valuable for all, but especially to whom it should matter the most.

By Nigel Coutts

Anastasiya A. Lipnevich. & Jeffrey K. Smith. (2008) Response to Assessment Feedback: The Effects of Grades, Praise, and Source of Information. Educational Testing Service

Caroline Pulfrey, Ce ́line Buchs, and Fabrizio Butera (2011) Why grades engender performance-avoidancegoals: The mediating role of autonomous motivation. Journal of Educational Psycholog; Vol. 103, No. 3, 683–700

Kluger, A. N., & DeNisi, A. (1996). The effects of feedback interventions on performance: A historical review, a meta-analysis, and a preliminary feedback intervention theory. Psychological Bulletin, 119(2), 254–284

All learning is a consequence of thinking

All learning is a consequence of thinking.

I have these words printed and posted on the wall above my desk. It is a reminder of what I believe is a vital understanding. The consequences of this one statement are quite profound. They fundamentally shape what I do as an educator and the experiences I hope to create for my learners.

They are words adopted from the research of Project Zero. They point towards a dispositional model of learning where the learner is actively involved in the process. The implication of this is that even the most effective teaching can only create the possibility of learning. Unless the learner is actively engaged in the act of learning and deploys strategies which achieve this purpose, learning is unlikely to occur. Our role as teachers then is to create the conditions most likely to engage the learner in this process. We must make our learners think. We must allow time for them to think and to reflect upon their thinking. We must notice, name and value the thinking they do and guide them towards thinking strategies which maximise the impact their thinking has.

Looked at from this perspective, learning is an event that occurs within the mind of the learner. It is the result of the mind’s interaction with a stimulus such that a change occurs in long term memory. The individual forms and reforms schemas that bring new capabilities and knowledge into their repertoire. This learning may be stimulated by the directed actions of an educator. It may occur while we are in groups or on our own. It can be the result of a deliberate effort to learn or as a consequence of our daily interactions with the world. Regardless of where or how it originates, learning requires the individual mind to process the experience and from this form new schemas. This is what learning is.

We do hope to make this process more efficient and leave less of it to chance. There are certain actions and behaviours which are more likely to result in learning. As Cohen (2008) points out, autonomous learning requires learners to rehearse and apply learning strategies, structure his/her own learning, and critically reflect upon her/his own learning processes. The role of the educator is at least in part to introduce our learners to the strategies they might deploy to enhance their learning.

Critical to becoming a master of learning is a collection of metacognitive strategies. In simple terms, metacognition is an awareness or noticing of one’s thinking. As described by Seifodin Rajabi, metacognition is the ability to reflect on what you know and do and what you do not know and do not do. The metacognitive learner actively deploys strategies which guide them towards a goal, shine a light on how their understanding is evolving and pose questions for further inquiry. O’Malley and Chamot (1990) share that “students without metacognitive approaches are essentially learners without direction or opportunity to plan their learning, monitor their progress, or review their accomplishments and future learning directions”.

For this model to apply to the self-regulation of learning the teacher must re-imagine themselves as an expert learner, problem finder and problem solver who models the methods and dispositions of learning. Our students need to see that we are knowledgeable, as a result of our learning experiences and skilled in the processes of acquiring, analysing and applying the fruits of our learning. In short, we need to model our capacity to learn. This requires that our students see us engage with ideas which we have not yet mastered, where we need to draw on our capacity to learn.

This transformative process, where the teacher comes to understand that they require more than content knowledge, is described by Mehta & Fine. They share their observations of and interview with a Philosophy teacher who shares his altered awareness of his role in the learning process. “I was no longer the authority on the text... ‘Well, I have my own thoughts about it but they’re not actually what matter,’ or, ‘I don’t claim to know the bottom line about this book. I can tell you my own personal feelings about it but I don’t have the answer. I think that’s really scary for a lot of teachers ’cause basically what you’re doing then is you’re walking into a big question mark. In fact, you are the question mark.” The story of this teacher’s transformation from the source of knowledge to question mark is compelling because of the level of engagement and learning that this approach produces. By focusing on transmitting powerful messages about what learners do, by modelling the process of asking and exploring questions, he establishes conditions within which his learners can think and therefore learn.

For some educators realising that all learning is a consequence of thinking can come with a degree of fear. Some will cling to the belief that if we teach it and teach it well, it will be learned. There is a sense that the act of teaching is diluted when we believe that even our very best efforts may produce no learning if our students are not engaged in thinking. When we embrace the importance of thinking, teaching is elevated above a service for the transmittal of information. We become provocateurs and catalysts for thinking, sparkers of imagination and guides to wonderment and exploration. This is why I am happy to agree that “All learning is a consequence of thinking”.

By Nigel Coutts

Cohen, A. D. (2008). Speaking strategies for independent learning: A focus on pragmatic performance. In S. Hurd, & T. Lewis (Eds.), Language learning strategies in independent settings (pp. 119-140). Bristol: Multilingual Matters.

O’Malley, J.M., & Chamot, A. U. (1990) Learning strategies in second language acquisition. New York: CUP.

Mehta, J. & Fine, S. (2019) In Search of Deeper Learning: The Quest to Remake the American High School Harvard University Press.

Rajabi, S. (2012) Towards self-regulated learning in school curriculum. Procedia: Social and Behavioral Sciences 47, 344–350

What if questions are the way to the solution

Children have amazing imaginations and love to ask questions. Small children are well known for asking “Why?”. Children of all ages, in any form of transport, ask “Are we there yet?” with a regularity bound to send any driver around the twist. Somewhere along the way, they discover “What if...?” questions and the adults around them experience a new version of creative torture made only worse when the question is transformed into “But, what if...?”. No level of logic or science can defeat a small child who imagines they are on a winning streak with a set of increasingly impossible to answer “What if...?” questions.

Adults love to ask “What if...?” questions too. It is a model that can be a catalyst for change when used in the right way and when combined with some other questions.

When you begin to ask “What if...?” questions you open the door to a fresh perspective. It is a particular framing of a question that invites creativity and hints at a shift in the status quo. It can be readily applied to questions in education. Indeed the power of it as a question is why participants in Project Zero’s Creating Cultures of Thinking begin their course by sharing their “What ifs?”. As a coach in this online course, I get to contemplate some inspiring questions. Matthew of Manhasset Secondary School asked, “What if students were provided with an environment where they could find true purpose in tasks and become more informed thinkers and citizens?”. Rebecca also of Manhasset proffers “What if we provided opportunities to be “smarter together,” rather than competing separately?” Silvana of Dufferin-Peel Catholic District School Board wonders, “What if schools where a place of global learning, where teachers could take a genuine interest in their students and how their students make sense of their learning?”

Each “What if” question reveals a deep puzzle that the individual is struggling with. At their best, they emerge from a realisation that something could be made better. They are a response to noticing that the way we do things now is less than ideal and they direct our thinking to what we might change. They fit well within the design thinking process, and for that reason, I have previously discussed the benefits of asking “What if...?” in the context of student inquiries. In January 2017, I wrote:

Another take is borrowed from the writing of Warren Berger and ‘A More Beautiful Question’. The idea here is that students generate big ‘Why...’ questions which identify a problem they have encountered. From here they move to ‘What if...?’ questions thinking individually or in collaboration and pose possible solutions. With a list of interesting’ what ifs’ they move to ‘how might’ questions where they focus their thinking on a gradual move towards implementing a possible solution. A nice way to introduce this is with examples from the world of start-up companies which have exploded on to the market thanks to thinking differently about common problems. Starting with a ‘Why’ question like ‘why can I not get a cab when I need one’ led to the founders of Uber asking ‘What if I could pay for one of the many empty seats in the cars driving past me’ and then on to the ‘How might we turn empty seats in cars we don’t own into a global business’. Similar examples can be found in the story of Air BnB among others and a list of such ‘Beautiful Questions’ can be found on Warren’s site: A More Beautiful Question

As educators we can engage with the same process as we move from noticing something that is not serving our purposes, to a “What if . . ?” that hints at a way forward and then on to a possible strategy. Unfortunately, the cycle too often pauses at the “What if...?” assuming it even makes it that far. When we pause at the why, we are merely complaining. Why is our curriculum so crowded? Why do we have to grade every piece of student work? Why can’t we focus on deep thinking? Each “why” identifies a problem, but if we don’t plan to transform our noticing of a tension into an action, we are just having a whinge. If we go no further than asking “What if...?” we are merely dreaming.

The next step towards a solution requires that we ask “How might...?”. The language is very deliberate. It is an open invitation to ponder possible actions. Matthew could consider “How might we begin to build an environment where the true purpose is thinking and active citizenry?”. Rebecca could propose “How might we value our collective smarts?” and Silvana could wonder “How might we understand how our learners make sense of their world?”. Each “How might...?” question moves us from noticing a challenge towards implementing a solution.

But questions only get us so far. Once we get to the point of a well constructed “What if...?” and a well-matched “How might...?”, we need to clarify what our first step will be and then we need to take it. All this thinking in questions must, at some point, transform into action. All significant transformations begin with step one and are followed by step two and three etc. As Millard Fuller says, it is easier to get people to act their way into a new way of thinking than to get people to think their way into a new way of acting. Hopefully, then we set out to find the tensions that stand in the way of the impact we wish to have as educators. Having found them, we ask questions that move us towards a solution. And then we identify our first step towards the change we have imagined, and we confidently take it.

By Nigel Coutts

Reflections from Mathematics: The Greatest Show

I have just had the opportunity to spend the weekend learning alongside a large group of mathematics teachers. After a day and a half of talking and thinking about mathematics teaching, I am excited to get back to school and try out some new ideas. There were also some key takeaways for me that I share below. These are the questions or wonderings that my mind wandered to while listening to the numerous talented speakers over the past two days. A provocative presentation creates space for the audience to reflect on the ideas shared and how the new ideas which emerge from the experience challenge our existing practice. The presenters at the MANSW Annual conference certainly gave their audience plenty to think about.

It was nice to spend time surrounded by people who believe that mathematics is a creative endeavour. Being around people who understand this aspect of mathematics, people who see the beauty of its patterns and operations makes you question why we feel the need to add Art to STEM. Not that Art and Mathematics do not make excellent companions but that the mix is too often advocated for by those hoping to add a creative element where they imagine it is missing. There is a need to be aware of an individual’s history with mathematical thinking before we launch into lessons requiring a creative approach. If a student has not been exposed to creative approaches to mathematics early in their learning journey, such methods are likely to cause confusion. Students trained to believe mathematics is about learning a set of procedures to be applied as taught, will find requests for creative solutions are like asking a non-swimmer to ‘just dive in and see what happens’. The result is likely to be an increased level of maths anxiety and an even greater desire to be shown what to do and when to do it. The answer is to ensure that creativity is a part of mathematical explorations from the earliest days of a child’s learning. As this learning is formalised, opportunities to play, explore, create, and wonder need to be maintained. A playful exploration of mathematical ideas should be enhanced when an experienced mathematical thinker joins the child on their learning journey. It should not be replaced by a forced march to the expert’s drum.

Students require mathematical freedom. Kirsty Thorpe beautifully shared this idea. Teachers like to be organised. We like to have our resources ready to go. Our lesson planning includes notes on how the lesson should evolve and what the students are going to need. With this in mind, the well-organised teacher prepares the resources that students are going to need and has them ready for easy distribution. Except when we do this, sometimes we are limiting how our students engage with their learning. As Kirsty noted, if we supply them with counters, don’t be surprised if all of their solutions involve manipulating sets of counters. Allowing students to select the materials they will use to visualise their mathematical thinking gives them mathematical freedom. It also allows more opportunities for students to demonstrate their reasoning skills. When students select the materials they use to visualise their thinking; they are making choices which they should then have to defend. Working with my colleague Stellina Sim, we have created Number Talk Bags. Each bag contains an assortment of commonly used mathematical manipulative and students make choices about which ones allow them to share their thinking with their classmates. Having the Number Talk Bags ready to go maintains a smooth flow to the lesson while enabling the students to make choices.

There are, of course, times when a limited set of resources can produce better thinking. Having to work with limitations can be a part of a challenge and is typical of problems we encounter in the real world. Limiting students to a particular set of manipulative can also guide them towards the thinking that is required for the learning objective. When we started exploring patterns with kindergarten, we limited the resources they had access to. Early on, it was almost impossible for the students not to arrange the objects we offered them into a pattern. Yes, we were limiting their freedom, but we ensured that they were forming, seeing and describing patterns. Once the foundation was laid, we offered other possibilities from an expanded palette.

Mathematical confidence requires a culture that supports mathematical exploration and risk-taking. Students who believe mathematics is about right answers are going to struggle in open classrooms. For teachers who are discovering new approaches to mathematics and exploring pedagogies where learning from mistakes is celebrated a word of caution. Some students will find this very confronting. If your entire mathematical career has led you to believe that good mathematicians don’t make mistakes, and now your mistake is shared on the whiteboard, you are going to want to dig a hole and climb in. As we change our mindset as teachers, as we explore the use of challenging tasks and mathematical inquiries, we need to take our students with us on the journey.

And it’s not just changing how we perceive mistakes. Some of our students love mathematics because they believe that there is just one right answer and one best way to find it. They imagine themselves to be confident and competent mathematicians because they can find that one correct answer and do so using the right method and quickly too. Now we are asking them to find another way and telling them that there is not one right answer. In essence, we are taking away the rock upon which their mathematical confidence rests. Some students will love this and see mathematics in a whole new light. Others will feel that their teacher has turned off the light and hidden the switch.

Less is more. When we teach less, we make time for tasks to go as long as they need to. We need to understand that a challenging task taught in ways that require students to genuinely engage and explore in multiple ways does not fit in an hour block. The advantage of teaching in this way is that the learning should stick. If we get it right, we shouldn’t have to start each year with an introduction to fractions. We also streamline our teaching when we allow students to make use of their mathematical knowledge in contexts which span multiple strands of the syllabus. Instead of teaching two weeks of multiplication and then two weeks of area, rich tasks allow us to to do a better job of teaching both concepts together and in less time. Maybe three weeks full of challenging inquiry into area fuelled by meaningful opportunities to build an understanding of multiplication.

The best learning can occur when students have to share their thinking, and this is more important than solving the problem. In many classes, the focus of problem-solving is on finding a solution, once the problem is solved, we move on and the learning stops. If you just solve the problem, you are not creating opportunities for communicating and reasoning and students only see one solution. In this way, opportunities for flexibility are missed. When you include ample time for students to share their solutions, explain the choices they made and compare options, you create opportunities for mathematical reasoning and communication. Problem-solving is great, but it is only one part of working mathematically.

And, Learning Outcomes written on the board for all to see might not be the panacea we are all looking for. Yes, the teacher needs to have an understanding of what they hope their children will learn, but it should not be assumed that publishing this enhances learning. Such a reading of the research is easily shown to be flawed. We should also consider the sequence of learning rather than just the endpoint. We need to ask more questions about the path our learners might be taking towards an objective. What are the waypoints on our journey towards the learning objective? How do we know where we are and where we are going next? What paths are learners likely to take, what mistakes might that make, what do their attempts reveal about their emerging understanding?

Fiona Foley shared with us the Five Practices for Productive Mathematics Discussion. This is a process which allows teachers to understand better the impact that their planned teaching might have and to shape their plan to maximise this. It evolved from the research of Margaret Smith and May Stein working for the National Council for Teachers of Mathematics (NCTM). It includes anticipating how students might respond, monitoring how they respond, selecting which students share solutions, sequencing the solutions shared to support learning and helping students to make connections. The Sequencing component is a particularly useful tool for teachers hoping to clarify the learning journeys students are likely to take. A group of teachers begin by anticipating the solutions and methods students are likely to use, both successful ones and those which are flawed. The teachers then consider which of these they will unpack with the class first and why. The intent is to understand how the mathematical conversation that results from a challenging task can lead the learners towards the desired learning. The result is a discussion where students are best able to learn from the methods attempted and the mistakes which were made. When undertaken as a collaborative process, it extends the mathematical understanding of the teachers significantly and is well worth the time it takes.

So the conference is over, but the learning will continue. It is reassuring to know that there are many teachers with a great depth of knowledge to share and a genuine passion for teaching mathematics. Thanks to organisations like Mathematics Association of New South Wales (MANSW) for providing ways to connect teachers and to share resources. A rising tide lifts all ship as they say and for mathematics, the tide is coming in.

By Nigel Coutts

Five reads for September

For teachers in Australia, the long Term Three is drawing rapidly to a close. Indeed as I write this just ten days remain before a two-week break. This is the perfect time to consider a holiday reading list. Just enough time to raid the school library or place an order with your favourite book store. Here is what’s currently occupying space on my nightstand.

1. Limitless Mind: Learn, lead and live without barriers by Jo Boaler

Boaler’s new book is high on my reading list. Having gained so much from previous books including “Mathematical Mindset”, “The Elephant in the Classroom” and the hugely practical “Mindset Mathematics” series, this book is highly anticipated. The question to be answered is does this book build a compelling case that anyone can learn anything. Do our beliefs about intelligence and ability hinder our capacity to learn and might we be limiting the learning our children are capable of?

2. Transformational Professional Learning: Making a difference in schools by Deborah Netolicky

We have all sat through professional learning that leaves us wishing we could wind back time and retrieve the hours we just lost. Why does so much professional learning fail to transform our practice? Why is it that within a profession that is all about teaching and learning, we get it so wrong when it comes to professional learning. As a practising academic who comfortably straddles the boundary between the practical world of the classroom and the informative space of academia, Deborah is well placed to offer insights with impact.

3. In Search of Deeper Learning: The quest to remake the American High School by Jal Mehta and Sarah Fine

Join with the authors on this tour of American High Schools as they unpack why some schools are delivering pockets of deep learning, but few if any, have found a model that truly works. This is not the usual banner-waving with stories of transformative practice. This book is deep dive into the pedagogy and curriculum of schools which claim to be delivering deep learning and look at the truth behind the claims. If education is going to transform itself, we need to be able to honestly assess what is working and what is not. Bold claims about school transformation are easily made; this book uncovers a more complex reality.

4. Nuance: Why some leaders succeed and others fail by Michael Fullan

“The more complex the problem, the more that people with the problem must be part and parcel of the solution.” Schools are particularly complex places; an observation that would not surprise anyone who has spent much time in one. Large organisations which exist to serve a diverse population in varying stages of development and with hugely differing needs and wants is bound to result in near chaos levels of complexity. Leadership in schools, therefore, needs to be adept at managing this complexity. In “Nuance”, Michael Fullan unpacks the leadership style that schools and education require. This is a book that every teacher should read. Don’t leave it on the shelf for the Principal to read. Every teacher has a part to play in school leadership, and an understanding of nuanced leadership will enhance the impact we can all have.

5. The Power of Making Thinking Visible: Using routines to engage and empower learners by Ron Ritchhart and Mark Church

If you are making use of visible thinking routines, if you have read “Making Thinking Visible” or “Creating Cultures of Thinking”, you will be waiting for this new release by Project Zero’s Ron Ritchhart and Mark Church. The big enticement is the promise of 21 new thinking routines which will bring new ways of making thinking routine in our classrooms. Beyond this, the book promises to explore the ongoing research behind why visible thinking is a powerful tool for learning. “Exclusive to this book will be a careful examination of what it means to engage and empower students as learners working with big ideas, reaching into the world to take action, and collaborating with others.”

By Nigel Coutts

Do We Truly Understand Place Value?

Number Talks are a wonderful way to see where our students are with their mathematical thinking. As a part of a daily routine a Number Talk promotes number sense and mathematical reasoning. In this post I take a closer look at what a Number Talk can reveal about our students’ understanding of mathematics and how they might be used to promote a fresh perspective.

James Tanton shattered my understanding of the vertical algorithm. More than that, he helped me to see how poorly I understood place value and that many of my students function with the same misunderstanding. What made the experience more humbling was that it took him less than two minutes to do this. Imagine a simple addition scenario involving two three digit numbers, something like 236 + 543 = How do you solve this? The mathematically inclined will know that there are many ways to achieve an answer. Undoubtedly the mathematics teachers reading this will be well armed with strategies involving rounding, or partitioning that make the addition more manageable. Most people with years of experience in the traditional mathematics classroom deploy the vertical algorithm. It probably looks something like this:

The vertical algorithm worked left to right.

The average person knows that to solve the equation you work right to left. If you ask a student to verbalise the process you hear something like, “first you add the 6 and the 3 to get 9, then you add the 3 and the 4 to get 7 and the 2 and the 5 to get 7, the answer is seven hundred and seventy three”. The fun begins when you demonstrate how to solve this but reverse the order. Instead of working right to left, work left to right, just like you do when you are reading. “First you add the 2 and the 5 to get 7, then you add the 3 and the 4 to get 7 and then you finish by adding the 6 and the 3 to get 9, the answer is seven hundred and seventy nine”.

Do this with a class of students and by this point they will be howling. “You did it wrong!”, “That’s not how you do it” or my favourite “You have to start with the 6”. Claiming that the answer you got is the same as the answer they got doesn’t help. Some will point out that it only works because you picked small numbers. Some throw words at you like “trading”. Many will resort to the highest form of classroom reasoning and argue “But that’s not how you do it”.

Another example of the vertical algorithm worked left to right. The answer might leave some unhappy.

So you offer to change the numbers. Make them larger, be sure that when the digits in each place value are added they surpass the magic number of ten. Try something like this:

Again explain to the students how you solve this beginning with the seven in the top left corner. If you want to really mess with their heads, start with the four but be prepared for claims that you always have to start with the top row. “First you add 7 and 4 to get 11. Then you add 6 and 9 to get 15. Then you add 8 and 5 to get 13. The answer is 11 hundreds and 15 tens and 13 ones or what might be playfully expressed as eleven hundred, fifteenty and thirteen”. In the interests of conventional counting it can and should be seen that we can unpack this number into a simpler form. Our fifteen ones allow us to add one to our collection of tens. We now have 16 of those and we can easily move ten of these into our collection of hundreds. We end up with 12 in our hundreds column, 6 in our tens column and 3 in our ones column and can call our answer one thousand, two hundred and sixty three.

What does this reveal? Our students have learned to follow the vertical algorithm but they may not truly understand how or why it works. The fact that we can work it backwards, or middle out, or upside down should not come as a surprise. We should see that in our numbers we have collections of ones, and tens and hundreds etc. and that we can combine these and have totals of any value. I proved this thinking to a student by offering them $10 notes. They didn’t mind the idea of having eleven such notes, or twelve or more, even though the idea of putting 15 in the tens column just minutes earlier seemed like the work of the devil.

What does this have to do with Number Talks? I have taught many classes who can perform page after page of vertical algorithms without error. There are any number of text books which provide just this sort of practice. I can dress up the question by wrapping it in a seemingly real world problem, something like “John has 768 watermelons, he buys 495 watermelons at the market. How many watermelons does John have?” (The only sensible answer here is too many) Regardless of whether the students get the answers right or wrong, a page full of vertical algorithms tells me very little about their understanding of the fundamental aspects of place value that it exploits. But, a short number talk will.

In a number talk I am inviting and requiring students to explain their thinking. Mathematical reasoning becomes more important than correct answers. Ask students to solve an addition like 68 + 95 in a number talk and you will know which students understand place value. While participating in the Number Talk students share numerous approaches to each question. They share and hear a range of strategies. Provide students with a whiteboard so they might make their thinking visible and you open new possibilities. Include the option of an extended Number Talk using concrete materials and you allow for diverse representations of mathematical thinking. In each instance the students are revealing how they understand number and each response offers new insights to the teacher for future learning. Number Talks by design close the gap between student performance and teacher action to address and remediate misunderstandings.

The particular misunderstandings revealed in our reversal of the vertical algorithm are beautifully addressed by Tanton’s use of “Exploding Dots”. The basic premise is simple. You can add dots into a place value box until it reaches a set value. In Base Ten that value is 10. Once you have more than ten dots in a box they explode and one dot appears, as if by magic, in the box one place to the left. If you model the above addition problem with dots the process becomes very visual and it is much easier to understand why you can start with any column. The process is not done justice when explained in words, it is one of those things you have to try for yourself. The website Exploding Dots is a great place to start. The diagrams show the three stages in the process.

Above the question 768 + 495 is modelled in dots. There are orange dots to represent seven hundred and sixty eight and green dots to represent four hundred and ninety five. Clearly some of our boxes have more than ten dots, so we get some explosions as below.

Finally we get to an arrangement that is mathematically stable and we can easily read off an answer that everyone is likely to be happy with.

Modelling the addition question we posed above with dots might not be the norm. It might take longer and require more space, but it does ensure that students understand what is going on. Exploding Dots can be used for so much more than addition. As an introduction to place value, perhaps beginning with binary counting, Exploding Dots provides a strong foundation from which mathematical understanding can be built. If you are keen to correct some misunderstandings amidst your students, definitely explore the world of exploding dots. It can be a great addition to you Number Talk routine.

By Nigel Coutts

Thinking and learning in the postnormal era

These are times of chaos, complexity and contradiction [1] where education is challenged to reimagine how it prepares young people of today for their worlds of tomorrow. Confronted by rapid change from a conflation of transformative forces society appears to be in a state of flux. The grand unifying socio-political stories and underlying structures that we have relied upon in the past seem to have dissolved under our feet leaving us bewildered [8]. The beliefs, values and philosophies which we once relied upon for guidance, trust in reason and science, the valuing of human intellect and our understanding of fundamental political systems have been replaced by volatility, uncertainty, complexity and ambiguity [9]. The exponential acceleration of the capabilities of our digital systems carries with it a transformative potential with far-reaching consequences and opportunities [2]. Similarly, our reliance on technological innovations that emerged during the first industrial revolution is today driving climate change and represents what David Attenborough describes as our greatest threat [10]. "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." [1 p. 435]

Five forces have combined to deliver this state of flux between stories which adequately informed our responses to societal challenges and our imagining of new stories capable of propelling us beyond these postnormal times towards a new age of enlightenment. Technology, climate change, globalisation, the internet and the dawn of the post-truth era. Each of these forces carries implications for the purposes and practices of education, and none is likely to cease driving change in the near future nor is the pace of change likely to slow. This paper explores technology and the dawn of the post-truth era.

In 1965, Gordon Moore, co-founder of Intel, predicted that the number of transistors and resistors on a chip doubles every twenty-four months and so introduced Moore's Law [2]. Subsequently embraced by the microprocessor industry as the benchmark to be achieved, Moore's Law has become a self-fulfilling prophecy. The human mind does not easily handle the concept of exponential growth behind Moore's Law [2]. Since 1965, based on Moore's biennial cycle of doubling, the capabilities of our microprocessors have doubled twenty-seven times. If we assign a measure of 1 to the capabilities of a 1965 processor, a chip of similar size and cost today would achieve a measure of 134,217,728. By 2031, the year in which our current Kindergarten concludes their formal education, this number climbs to over 8.5 thousand-million. While once we waited for each new processor to come to market in the hope that it might better meet our needs, today and into the future, we will seek to exploit the opportunities of our extensive computational powers.

In the Industrial Revolution of the 1780s, human labour was replaced by machine power and the workplace was transformed. Workers retrained into the newly created jobs as humans shifted from providing physical labour to low-order cognitive labour. In today's industrial revolution, artificial intelligence (AI), born from the exponential growth in computational power and algorithms that replicate or exceed human cognition is forcing us to re-evaluate our position in the economy [8]. Two examples and a glimpse into the near-future demonstrate some of the implications that AI has for education. In 2011, IBM demonstrated that the human capacity to associate memories with natural-language questions could be replicated by a machine when Watson defeated its human competitors in the game 'Jeopardy' [2]. If education is merely about transferring knowledge from one generation to the next, then Watson reveals that we are developing a capacity that is easily supplanted.

The 1997 defeat of Gary Kasparov, the then world chess champion, by Watson's forerunner Deep Blue, is often heralded as a milestone for computers; however the events of December 2017 trump this in significant ways. The creatively named 'Stockfish 8' was the standing world computer chess champion until it encountered Google's 'AlphaZero'. While 'Deep Blue' required humans to programme it with the moves necessary to defeat Kasparov, AlphaZero learned to play chess the old-fashion way; by playing chess [8]. According to the' 10,000-Hour Rule', made popular by Malcolm Gladwell in "Outliers" [11] it should have taken AlphaZero some 416 days to achieve sufficient mastery, but it went from novice to world champion in four hours and used moves considered unconventional demonstrating a level of creativity that might have been considered the realm of only a human mind [8]. If the purpose of education is to teach students to code or solve anything besides the most complex multi-dimensional problems requiring creativity and responsive dexterity, we are wasting our time.

If we believe that matters of the heart or emotional intelligence will separate humans from machines, we might be surprised when the next wave of AI handles this too. Merging machine learning with sensors allowing algorithms to read faces, monitor changes in pulse and body-temperature, listen for changes in our voice and learn our preferences, we unlock machines capable of responding to emotion, with emotion [8].

The post-truth era brings a different set of challenges. Post-truth is defined as "relating to or denoting circumstances in which objective facts are less influential in shaping public opinion than appeals to emotion and personal belief" [12]. "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." [13 p. 180] and "Experts are vilified as an ill-intentioned cartel" [17 p.8]. A combination of factors, including the rise of social media, twenty-four-hour news and widespread disenfranchisement from our knowledge elites, has led us to a time where reputation [15] and opinion trump wisdom and science [14]. In a world aflood with information, fake news and disinformation have risen to the fore[16]. Now more than ever, our students need to be skilled in the art of discerning truth from falsehood, to be ever questioning sceptics.

Oxford Dictionary’s word of the year in 2016 ‘Post-Truth’

‘circumstances in which objective facts are less influential in shaping public opinion than appeals to emotion and personal belief’.

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

For education the challenge of postnormal times is immense and yet now is not the time to advocate despair. In imagining what education might offer our students as preparation for the lives they are likely to live, by seeking an understanding of lifeworthy learning [17], we see immense opportunity. Our perception of what matters in education must change. Mere factual knowledge, mimicry of methods, solving already solved problems, learning in isolation and a belief that education is a phase of our lives that terminates with graduation are ideas we must move beyond. We must look closely at what we emphasise in our curriculum and whether what we focus on is likely to feature in their future lives. “A vision of education that’s more “future wise,” reflecting our best guesses about what’s most likely to happen and foregrounding flexible knowledge likely to inform whatever does happen.”[17]

Our children will need a sense of agency empowered by capacities required to activate or perform their intentions [6]. "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" [4 p. 77]. They must become creative problem finders through learning opportunities that allow them to "sense that there is a puzzle somewhere or a task to be accomplished" [18 p. 95] and respond strategically, creatively and collaboratively towards solutions devised with empathy and a long-term view of impacts and real-costs [19]. Our students must be shown the value of acquiring deep-understandings through weaving ideas together, going beyond information and figuring things out [4].

This demands that our students are routinely engaged with learning that requires them to do thought-provoking things with what they know, such as considering different viewpoints, reasoning with evidence, uncovering complexity and building explanations [6] [7]. Such complex thinking does not occur automatically, and our students will need to master structures which support this. Visible thinking strategies assist teachers to make deep thinking a routine part of their classrooms and allow them to 'see' the way their learners are engaging with ideas [6]. All learning is a consequence of thinking and schools must transform themselves into "cultures of thinking" [4]. This demands a continual evaluation of the culture that is experienced by students and teachers. We must come to value thinking in all its forms and appreciate that our collective futures depend upon the quality of our thinking.

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"The fullest representations of humanity show people to be curious, vital, and self-motivated. At their best, they are agentic and inspired, striving to learn; extend themselves; master new skills; and apply their talents responsibly" [3]. When educators consider how every move they make, every message they send aligns with achieving the fullest representation of humanity, we ensure our children are well prepared for their tomorrows.

By Nigel Coutts

Originally published as:

Thinking and learning in the postnormal era: How might we respond to a curriculum that embraced diverse perspectives and contested issues? - Conference Proceedings of The Future of Education International Conference - ISBN code (978-88-85813-45-8) - Publisher - Filodiritto Editore (The Future of Education).

References

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

2. Friedman, T. "Thank you for being late”, Great Britain: Picador, 2016, 461 p.

3. Ryan, R. & Deci, E. "Self-determination theory and the facilitation of intrinsic motivation, social development, and well-being”, American Psychologist, 55(1), 2000, p. 68-78.

4. Ritchhart, R. "Creating cultures of thinking: The eight forces we must truly master to transform our schools”, San Francisco, Josey-Bass, 2015, 368 p.

5. Ritchhart, R., Church, M. & Morrison, K. "Making thinking visible: How to promote engagement, understanding and independence for all learners”, San Francisco, Josey-Bass, 2011, 294 p.

6. Clapp, E., Ross, J., Oxman Ryan, J. & Tishman, S. "Maker-centered learning: empowering young people to shape their worlds”, San Francisco, Josey Bass, 2017, 225 p.

7. Blythe, T. "The teaching for understanding guide”, San Francisco, Jossey-Bass Publishers, 1998, 144 p.

8. Harari, Y. "21 Lessons for the 21st Century”, London, Random House, 2018, 416 p.

9. USAWC. "Origins of VUCA” Accessed online 5.5.19 - http://usawc.libanswers.com/faq/84869

10. Attenborough, D. "Sir David Attenborough: Climate change our greatest threat” accessed 5.5.19 - https://www.bbc.com/news/science-environment-46398057

11. Gladwell, M. “Outliers: The story of success”. London, Allen Lane, 2008, 299 p.

12. Oxford Dictionary, Accessed online 5.5.19 - https://en.oxforddictionaries.com/word-of-the-year/word-of-the-year-2016

13. Brotherton, R. "Suspicious Minds: Why We Believe Conspiracy Theories" (2016) London, Bloomsbury Publishing, 304 p.

14. d’Ancona, M. "Post-Truth: The New War on Truth and How to Fight Back”, 2017, London, Elbury Press, 167 p.

15. Origgi, G. “Reputation: What it is and why it matters”, Oxfordshire, Princeton University Press, 2018, 273 p.

16. [16] Kakutani, M. “The death of truth” London, William Collins, 2018, 208 p.

17. Perkins, D. “Future Wise: Educating our children for a changing world”, San Francisco, Josey-Bass, 2014, 274 p.

18. Csikszentmihalyi, M. "Creativity: Flow and the psychology of discovery and invention”, New York, Harper Perennial, 2013, 456 p.

19. Kelley, D. "Creative Confidence: Unleashing the Creative Potential Within Us All”, London, Harper Collins, 2013, 281 p.

20. M. Jefferson & M. Anderson. “Transforming Schools: Creativity, Critical Reflection, Communication, Collaboration”. Bloomsbury Publishing, 2017, 202 p.

Seven Language Moves for Learning

Language unsurprisingly is a powerful force in education. As Ron Ritchhart notes in ‘Creating Cultures of Thinking’, language “is at once ubiquitous, surrounding us constantly, yet we hardly take notice of its subtleties and power.” If we wish to maximise the impact we have, if we hope to achieve particular goals, and if we wish to shape the culture of our classrooms, we must consider the role that language plays. 

Our language choices communicate both intended and unintended messages. In the choices we make, in the subtlety of these choices, lies a truth more powerful than that conveyed by a literal reading of our words. When we look closely and critically at our use of language, we begin to see particular patterns which reveal much about what we genuinely value and expect from our learners. 

There are words which we use with high frequency. Work is one such word. We remind our classes to get back to work, to start their work, to work carefully, to finish all their work. We should not then be surprised that students imagine their role in the classroom is to do the work. This might not be the message we had hoped to convey about learning. Once you notice the number of times that you use the word work, you become open to the idea of using alternatives. In recognising the overuse of this word, we see the possibility that a more deliberate approach to our language choices might have. 

If we are going to leverage language to achieve our goals, we need to become aware of its subtle power. Ron Ritchhart identifies seven language moves that teachers make. Awareness of these seven moves can be the first step towards more facilitative language choices. 

The first is a language of thinking. The teacher who is aware of this language move will utilise questions which engage their students in thinking. Often the most powerful change we can make to our teaching is the addition of one simple question: What makes you say that? By asking this question, I invite my students to move beyond providing what they imagine to be an accurate answer. By asking ‘What makes you say that?’ I require that my students offer a reasoned logic for the responses they provide. By asking this question often, I send the message that I value their thinking. When I then notice and name the thinking moves made by my students, I reinforce this message. If I praise a student for their critical thinking, for their efforts to make connections or to reason with evidence, I send clear signals that learning requires thinking. 

My language choices can build community. Noticing that we are a class of learners and that we have exciting learning to engage with today reflects a subtle choice of language. I might have stated this differently. You are learners, and you have much learning to do is much less inclusive than the first telling of the same set of facts. Use of the words we, and us and our indicate community in ways that you, I and my or mine do not. When the teacher talks about the learning that we are doing the message is clear that the community of learners includes the teacher as a member and that we are all learning together. 

Do we do mathematics or are we mathematicians? Do we study writing, or are we authors? Do we learn about places and spaces or do we think like geographers. By utilising a language of identity, we bring a new mindset to our classrooms. We empower our students to step into the shoes of the expert. When we use a language of identity, we invite our learners to become active participants in a discipline rather than temporary visitors, just passing through. 

When I outline the steps to be taken in a project or activity, I remove opportunities for students to demonstrate initiative. When I ask them to describe their plan, I allow them to take initiative. When a student comes to me because they don’t know what to do, I have a choice to make in how I respond. If I rescue them and show them how to proceed, I promote dependence. If I ask them to describe the steps they have already taken and then prompt them to think of what else they might try, I allow them to retain ownership of the process. Powerful questions such as ‘How might you see this differently?’, ‘What do you think is going on here?’ and ‘What parts of this do you understand?’ are supportive of student initiative. 

In some instances, the power of language is so subtle that it comes down to the choice of one word. If I ask the question “What can we do about this?” my language choice has unconsciously restricted the responses I will receive to those that are imagined to be possible. I have limited the scope of options I will receive back simply because I selected the word ‘can’ instead of more open and inviting ‘might’. When I ask ‘What might we do about this?’ I indicate that I am open to any and all suggestions. The responses offered to questions using a mindful ‘might’ are shown to be more diverse and more creative than those elicited by the use of ‘can’ or ‘could’ or ‘should.’

We might like to be told that we have done a good job. It might be nice to be told we are intelligent and talented. The trouble is that as feedback, this sort of praise is practically useless. More useful is feedback that provides me with specific and actionable detail of what I have done well and what I might want to do less of in the future. In place of praising a student for a great piece of writing, we can notice their choice of discipline-specific vocabulary, their effective paragraph structure and their clear opening sentence which made their argument apparent and understandable. 

The final language move requires us to become skilled listeners. Sometimes this means saying nothing. If we are talking all the time, what space do we leave for other voices to be heard? When we are listening, we make choices about how we respond that indicate how we value the role of listening. Reflective questions which show we desire to clarify our understanding of what the speaker has shared suggest that we appreciate what they have to say. Questions which encourage the speaker to reflect on their understanding or that invite an alternate perspective, allow us to become a valued participant in the speakers thinking.

Becoming aware of the seven language moves might serve to enhance the impact that our language choices have. As with each of the eight cultural forces, language is an inescapable part of our classroom culture. We can leave it to chance and hope for the best or we can practice noticing the choices we make and become more deliberate with the language moves we make.

By Nigel Coutts

A new literacy for understanding and communicating knowledge in the post-truth era.

These are undoubtedly interesting times, post-normal times. This is an era where our norms are reinvented, and our everyday assumptions challenged. This is a time for questions and reflection combined with a great search for understanding.

In this time, our notions of truth and the epistemological foundations of our knowledge are in flux. A battle rages between powerful factions which compete to shape our notions of truth, falsehood and belief. Politicians, corporations, media conglomerates, social networks and special interest groups each bring to the table their version of the truth. Competing constructs for how the world and its events are best understood abound, and evidence in support of any version of the truth is scant and muddied.

Once philosophers and scientists sought to uncover the truth. It was imagined that this one version of the truth existed in the reality of our world waiting to be found. Science was the great tool with which we would make sense of our worlds. As we explored further into the mysteries of the world, we believed that we were building enormous reserves of knowledge. These banks of knowledge, once analysed by those with wit and wisdom, would explain away the mysteries of our age and unlock a new age of enlightenment.

Today, information has become ubiquitous. Each day we generate more and more information, and most of it is never seen. While once those with access to knowledge were made powerful, today recourse to facts and information seems to hinder the power of the stories we tell. Reputation, emotion, status, and likes on social media seem to carry more weight when debating the truth than does scientific research. A compelling fiction wrapped in an appealing narrative appears to carry the day. Facts are most likely to spoil the party and ruin the fun.

As we rush towards 2020, more than ever, it seems our future depends upon our capacity to see true truths. And yet everywhere we look, we find ourselves confronted by multiple re-tellings of a truth. Each truth story has its foundation in some version of reality. Each story is told in compelling rhetoric. Each teller assumes a mantle of confidence in the story that they have to share. Each assumes that all other versions are manifestly false and that this falsehood is tangible to the wise. Each storyteller has their followers who ardently retell the story and add their weight to its validity. Truth becomes not that which is true, but that which is sold to the largest band of the most vociferous followers.

Today’s students enter this world, trusting in their educators to provide them with the skills they require to uncover the truth. It is likely that during their time at school, they will be taught numerous strategies to evaluate claims. They will learn to think like scientists. They will become literary critics. They will interpret historical events and engage in debate over significant ethical and political dilemmas. They will become skilled in the knowledge arts.

But will this be enough? In the post-truth era, does the capacity to utilise knowledge prepare one adequately to do battle against falsehood? When those selling untruths know the falsehood of their stories, does the bright light of truth shine through? Perhaps not.

The young truth-seeker requires a new literacy. The capacity to read each truth story, to understand its origins, its motivations, its place in the world and the purposes it serves. The defence of truth requires a close reading of each truth story rather than a search for the one true version of it. The literate defender of truth must seek to fully understand the beliefs which compel each story’s believers. Armed with this knowledge and their capacity to evaluate one set of claims against another, they can begin to weave a more compelling and truer truth. They will rely on powerful tools and evaluate multiple sources of knowledge as truth-seekers have always done.

What changes is how they share their insights with the world. A full-frontal assault will be avoided. Instead, they will deploy subtlety and nuance. Each audience will receive an alternate telling of a truth story woven from threads of true and tested knowledge. Each story will be personalised to best exploit its audiences weaknesses, appeal to their beliefs and gently guide the listener towards a fresh understanding. This is the deliberate politicisation of the process of spreading truth, wisdom and knowledge. This is empowering those who believe that knowledge founded in reason and logic should be our guide.

“If you know the enemy and know yourself, you need not fear the result of a hundred battles. If you know yourself but not the enemy, for every victory gained you will also suffer a defeat. If you know neither the enemy nor yourself, you will succumb in every battle.”

In The Art of War, Sun Tzu points the way forward. Our students must be taught to know not only truth from falsehood. They must also possess an understanding of the origins of the untruths they seek to extinguish.

By Nigel Coutts

Powerful Provocations for Learning: Sparking curiosity and increasing engagement

Powerful learning begins with the perfect provocation. Creating, refining and skilfully presenting the perfect provocation is an essential capability for teachers hoping to engage their class in rich dialogue. Claims that the percentage of students engaged by their learning declines from 75 percent in fifth grade to 32 percent by eleventh grade suggests a need for a more provocative environment.

A well-crafted provocation should encourage curiosity. Perhaps it has a certain degree of ambiguity such that it demands clarification. Maybe it challenges existing knowledge or beliefs in a way that guides the learner towards a fresh perspective. When we unlock the curiosity of our learners in these ways and provide them with opportunities to engage in inquiry to satisfy their curiosity, we increase learner agency. Matthias Gruber describes the effect that curiosity has on learning; “Curiosity may put the brain in a state that allows it to learn and retain any kind of information, like a vortex that sucks in what you are motivated to learn, and also everything around it,”. A powerful provocation is the seed for such a vortex.

The right provocation sparks questions and invites exploration. Entering a new city for the first time, the traveller experiences varying degrees of cognitive disconnect. There are some vistas which make sense and others which do not fit with any previous experience. There are new sights, and smells and sounds and while at first, the sensation may be overwhelming, the minds innate desire to explore soon takes over. We wander through the city, awash with questions and wonderings. What is around that corner? What are these people doing and why? What is that sound I hear in the distance? We may not be able to take our class to a new city each day, but we can create experiences which provoke a similar reaction. We can seek to provoke a sense of wonder and a desire to explore.

A powerful provocation should challenge our paradigms. We become readily set in our beliefs. We have knowledge that we believe to be true that goes unquestioned until a provocation is encountered that causes us to doubt what we were once certain of. Great art seems to achieve this goal almost effortlessly. Think of the painter who challenges us with a piece that forces us to review our reality from a new perspective. Or the poet, whose subtle choice of phrasing opens our mind to a new interpretation of stories that we once found truth in.

Julie is an exceptional teacher who routinely provokes her students to think, wonder, question and explore. Her class has started reading “Young Dark Emu”, a powerfully provocative book by Bruce Pascoe. By choosing to read this with her class of fifth graders, Julie has set up the possibility for an engaging lesson, but choosing the right book is only the start of the process. Looking for a powerful way to start the learning journey with her class, Julie sought to find the right provocation. Her Year Five teaching team utilises a ‘teaching for understanding’ approach to planning. An important part of this process is the establishment of a Generative Topic that serves to guide the teams teaching towards learning that matters. A generative topic as the name implies should generate opportunities for rich, broad learning around significant topics. This term, Julie’s team is exploring “Perspective” as their generative topic. The hope is that reading “Young Dark Emu” will invite a different way of seeing. Julie describes how she used the book to achieve this goal.

I was introducing the book Young Dark Emu by Bruce Pascoe and we were looking at the first paragraph where he explains that in European astronomy, they look at the constellations to tell the stories, whereas the Aboriginal people look at the darkness between the stars to tell the stories. The outline of the dark emu in the night sky was on the opposite page. The paragraph ended with the sentence – “It’s a different way of seeing”.

Source - Young Dark Emu

This was what Julie needed to spark the curiosity of her class, a powerful provocation to see things differently. An invitation to engage with a reading of the book as a doorway to a new perspective. A chance to see history through a new lens and to challenge preconceived ideas of how the world is to be interpreted.

‘Throughout history, humans have looked to the night sky to explain their existence, but the conclusions peoples draw from the same sky can be remarkably different. European astronomy uses constellations of stars to tell a story, but sometimes Aboriginal Australia uses the darkness between the stars. Dark Emu is a shape in the dark areas between the stars of the Milky Way. It’s a different way of seeing’. Young Dark Emu by Bruce Pascoe

The power of that one sentence sparked so much discussion about perspectives. Students made so many connections with the author’s message and the analogy given made this very clear to them. They spoke about why that paragraph is at the front of the book when considering their predictions of the content. This really highlighted to me the power of a good question or statement in provoking so much rich discussion.

Stellina teaches Kindergarten. She understands that the stimulus materials she chooses and the environment of her classroom acts as the third teacher. With this in mind, Stellina crafts collections of materials which invite her students to explore, wonder, engage and question. Stellina utilises her creativity and understanding of the curriculum to arrange collections that are likely to lead her learners towards the understanding she knows they need. The image below and the accompanying provocations were designed to engage young learners in a mathematical inquiry linked to the concept of number. Enticingly presented, the collection invites the learner to come and explore, to play and make connections. Having unlocked her learners’ curiosity, Stellina now artfully observes how they interact with the collection and decides when and how to interact with them so as to nudge them in the desired direction.

Both of these examples demonstrate the potential of a powerful provocation. They also reveal the added value that a great teacher brings to a provocative stimulus. Julie knew how to manage the conversation. She understood the potential in the text and utilised it as a catalyst for conversation. The culture of her classroom supports debate and her students feel safe contributing ideas. Julie knows when to be the quiet one in the classroom, and when to gently guide the conversation towards an alternate perspective. Stellina creates opportunities for her students to explore and in doing so, makes space for her to become an observer of her students in the act of learning. When she does inject herself into the learning, her actions are well-timed and strategic. Quality learning requires the right provocation and a teacher who can maximise its potential.

How will you provoke your learners? What provocation will you deploy, and how will you maximise its possibility?

By Nigel Coutts

Bringing Mathematical Reasoning into our Classrooms

Reasoning is at the heart of mathematical thinking. It is what mathematicians do. 

The importance of reasoning is acknowledged in most if not all mathematical curriculums. In New South Wales it is a part of working mathematically which is described as the thinking and doing of mathematics. Mathematical reasoning is defined by the writers of the Australian Curriculum as follows:

Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. They are reasoning mathematically when they explain their thinking, deduce and justify strategies used and conclusions reached, adapt the known to the unknown, transfer learning from one context to another, prove that something is true or false, and compare and contrast related ideas and explain their choices. - ACARA

With this in mind and a desire to build the capacity of our students to reason, we do well to ask how this might be achieved. How do we teach our students to bring such high order cognitive dispositions to their mathematics?

We begin with creating opportunities which allow for and which require mathematical reasoning. This is likely to involve a shift in our approach. 

If I begin by teaching my students a procedure and do so by guiding them step by step through the required process, they are unlikely to be called upon to reason. For the students, the reason for step two of the method is that it comes after step one, and the reason that they are doing any of it is that they were told to. If I then proceed to give my students a page of similar questions which closely resemble the ones used in the guided component of the lesson they will continue to apply the learned method, but will never be required to reason.

Perhaps we begin with something different. Maybe we alter the plan, or maybe we work towards a more significant question. 

Let us begin with something different. Let us begin with a prompt that forces us immediately towards reasoning. On the website “Which One Doesn’t Belong” you will find a series of prompts like the one below. The premise is simple; look at the prompt and decide which one doesn’t belong. So far, you have not had to do much reasoning. The next step is the key. Having shared the prompt with a class of students and having received a range of response back from the students, the teacher plays their trump card and asks one of the students to explain and justify their choice. There is no right answer and no wrong answer. What matters is that the student must offer a justification for their response. Repeatedly the teacher calls on a student, asks “Which one doesn’t belong?” and then follows that up with “What makes you say that?”. 

Image Source - Which One Doesn’t Belong - https://wodb.ca/shapes.html

The suggestion is not that we spend our days asking our students to make choices from four possibilities. Nor is that we abandon opportunities to teach our students a mathematical method. What is suggested is that the simple pattern demonstrated by the use of “Which One Doesn’t Belong” style prompts is utilised across the mathematical curriculum. When I choose to demonstrate a mathematical procedure, I include time to ask the students to reflect on what we are doing. Slowing the pace and asking students to explain why the process works, begins to engage them in mathematical reasoning even within a direct instruction model. 

And we should be clear, problem-solving and inquiry-based mathematics is not guaranteed to result in mathematical reasoning. I must also require my students to justify their thinking, explain the choices that they made, prove their working by an alternate method or through a different medium. Unless I do these things, they may get the right answer but could be relying on memory alone. 

And we learn what mathematics is when we are young. If the early years of mathematical life are filled with hours of tasks which only require a good memory, what concept will I have of the discipline? I am likely to pursue a pattern of study where thinking is barely required and creativity is absent? How will I cope when I must transition from learning to memorise methods to reasoning mathematically if I have never been compelled to do this before?

When we are young is also the perfect time to engage with mathematical reasoning. Curiosity is a natural disposition. We learn our way out of it. The young learner wants to know why. It is their favourite question. The clever early years’ teacher will construct environments and learning opportunities which encourage mathematical reasoning. In her book, aptly titled “In the Moment”, Jen Munson describes how this is done. Having placed the child in a situation permissive of mathematical reasoning, the teacher attends to what their learners are doing. They carefully interpret the actions and then decide how they will inject themselves into the moment. Through this process of observing and analysing what is seen, the teacher devises a plan and then provides just the right nudge to allow the child’s thinking to evolve. In classrooms where this pattern is the norm, mathematical reasoning is abundant, and students acquire the foundation they require for a passionate pursuit of the discipline. 

While the prospect of teaching mathematical reasoning may seem daunting, it does not need to be. Careful questioning within a classroom culture where thinking is the norm combined with an expectation that methods will be debated and that answers alone are insufficient provides the right environment for reasoning to thrive.

By Nigel Coutts

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.