Professional Learning and Development

Regular (or even semi-regular..or even the occasional) readers will know that professional learning is one of my soap-box issues.

It was therefore very interesting to read the just-published New Zealand’s Education REview Office report on Professional Learning and Development from Derek Wenmoth’s blog. Well worth a look. Well worth considering in the Australian context. Well worth the reflection. Well worth the thinking. Worth it for teachers’ learning and for better learning outcomes for our students.

An eclectic mix

“…sharing the same fears. Time it was, and what a time it was, it was. A time of innocence, a time of confidences..”  (from Simon & Garfunkel’s Bookends)

Yes, it’s reporting time in Victoria. And what a time it is. Although I’m not sure if we all share the same fears! A time of innocence? Maybe. A time of confidences? Sometimes…depends on how honest the comment, I suppose! Certainly a time of angst. A time when some teachers resent the time that must be taken to write them out of their already crowded and stressed existences, a time when old and barely concealed umbrage and bitterness comes to the fore. A time for martyrdom and indignation (‘How could they ask me to do that as well as write these reports?’) Oh well…

However, because I also have reports to write, this will be a brief post.

Over the last few days, I have discovered that Art Garfunkel has degrees in mathematics and art history (I knew there was an affinity there…that explains why I felt the need to perm my hair in the 80s!!) and that John Brack, the iconic Melbourne artist whose extensive exhibition I viewed yesterday, was also good at mathematics and was always looking for patterns in human behaviour and then creating a visualisation of this in ways that could not be expressed with words. I have written before about mathematics being more like a philosophy than a science and more creative than many give it credit for. Mathematics should, perhaps, be more about looking, musing and wondering.

The second item of this eclectic mix is Elizabeth Wynhausen’s essay On Resilience (part of the Melbourne University’s Press series of little books on big themes). In this wonderful reflection on, mainly her mother’s story, she examines the difference between stoicism and resilience and mentions a study that has identified three traits resilient people share: a resolute acceptance of reality, a sense that life is meaningful and an exceptional ability to improvise. Elsewhere she also writes about a playfulness that is evident in resilient people. I have also written about resilience in students of mathematics elsewhere and how important it is for teachers to try and engender such resilience. Too many of our students endure the subject with something akin to stoicism rather than engaging with its challenges and rolling with the punches, playing with ideas and learning something about the discipline, and themselves as learners, as they do so.

Finally, I have been fortunate to have spoken with, and heard Ron Ritchhart, of Project Zero out of Harvard University, present over the past month or so. He is always an interesting speaker and says things about education that make me think and wonder. He is also an ex teacher of mathematics. He has recently started up his own website here and there you will find some of his recent presentations from his Melbourne visit. It is well worth a look.

The Shaping of Mathematics in the National Curriculum

A very short post today:

The paper on the shape of mathematics for the national curriculum has just been released. I highly advise reading what is contained in it. You can obtain a digital version here. You can also register to receive email alerts when anything is added to the National Curriculum Board website relevant to mathematics.

Of interest to me are the following aspects:

*at last, the official ‘study design’ features content (what is to be taught) AND methodology (how this content is explored and developed ie the thinking and doing) – what they are calling content strands and proficiency strands (of understanding, fluency, problem solving and reasoning). This is a great step forward from the content-focused curriculum documentation we have seen for our subject from government bodies of the present and the past.

*a consideration of how to engage more students in their learning of mathematics

*a stated aim of reducing the amount of ’stuff’ to be ‘covered’ to encourage the development of big, key ideas in the subject

*a consideration of how assessment and pedagogy can create a coherent mathematics curriculum ie assessment linked to key ideas and reporting of students’ performance in both content and proficiency strands

There is a focus on teaching for understanding, curriculum documentation emphasising big ideas and essential questions and assessment for learning. I am feeling very hopeful about the proposed curriculum.

Brief feedback

Well…I’ve done the lesson on simultaneous equations as I outlined in the previous post and it all went very well.

We started with two simple equations as part of their daily quiz – to solve for a and b in a + b = 3 and b = a + 1 (ie something they could do relatively easily by trial and error). I then challenged them with 2a – 3b = -1 and a = b + 1. We briefly talked about needing a strategy for more complicated ones.

I then handed out the grids and blocks. The little activity itself generated interest, engagement and a challenge. In a class of 22, I had three students who found one solution in 10 minutes and I asked them to try and find another whilst walking around the room and checking on what the rest of the class was doing. I probably wouldn’t keep it going for any longer than 10 minutes. We then all went to the desk of a student who hadn’t found a solution as yet and I asked her to fill in the rows with an even number of blocks and not worry about the columns as yet. She was a trifle daunted by the prospect of doing this in front of everyone but, although I don’t think she enjoyed the ‘exposure’, I think it pushed her thinking and learning behaviours in positive ways. She needed some assistance to then finish the problem but the majority of students ‘got it’ and they went back to their own problems to get at least one solution happening.

I then asked about a possible connection between the activity they just did and the two equations. Quite a few of the students realised the connection and got excited. We talked about how we could obtain the value of one variable then find the other. I showed them the substitution method. I kept repeating the main idea – that to solve two equations with two unknowns by hand, we had to keep using the strategy of finding one variable first then thinking about calculating the value of the other.

In subsequent lessons, we talked about pairs of equations that didn’t have a ‘letter on its own’ and what we could do but I made sure to always come back to the strategy – how can we solve for one pronumeral first? I was very pleased with the way the activity added to their understanding that the two algebraic methods of solution were just different forms of the same strategy and the way in which it engaged and made all feel as if they were in control of what they were learning – not just copying down a procedure that someone else had thought about.

Successful Students Think

I attended a one-day conference on the Gold Coast April 17 with this title. It was subtitled Engaging Students in Mathematics Learning.

The keynote – The Psychology of Adolescent thinking and connections to Mathematics Learning – was given by Andrew Fuller (see his website at www.andrewfuller.com.au) then there were presentations from Judy Hartnett (Lecturer – Queensland Uinversity of Technology), Judith Selby (Head Teacher at Cowra High in NSW), Julie Wright (Canobas High), Steve Flavel (Education Consultant, WA), Matt Skoss (NT) and Charles Lovitt.

Andrew Fuller challenged my thinking. He started with a list of what adolescent brains do well and what they do poorly.

Do Well                                             Do poorly

Vigilance                                              Transfer

Self-focus                                             Self-Appraisal

Challenge                                            Coping with anxiety

Taking risks                          Motivation in the face of ‘failure’

Forming new patterns                   Recognising patterns

Playing games (competitive element)   Auditory processing

                                                               Sequencing

I had a few problems with some of these items – particularly how they relate to girls. In my experience of teaching girls for the last twenty-odd years, I have found that girls are pretty adept at self-appraisal and recognising patterns. I have also found that girls are not big risk-takers and sometimes don’t enjoy the competitive challenge of games. They also tend to have more of a social justice focus rather than self-focus. I did, however, agree with his inference as it applies to teaching: that it is useless asking “Does everyone understand that?” as some students may think they understand and not respond in the negative and some may say they don’t ‘get’ it as they always err on the side of under-achieving to avoid possible ‘failure’ if they say they understand then get ‘found out’. It is important that the teacher asks the right questions to elicit the right information about whether their students really understand the underlying concepts or not.

He then went onto say that it was a myth that students can multi-task and learn. They can do many things at the same time but not well enough to do so in a learning environment. Focus is needed to learn effectively. He said that learning:

• involved imitation and experiencing difference
• was designed to create meaning
• was motivated through mastery
• was best achieved in a social setting that allowed for interaction
• involved skills and habits, not just content

He followed this with a summary of how brain behaviour could positively affect the learning environment for adolescents and advocated immersing students in high-quality experiences that will train the brain to ’skill up’. This, of course, is the same idea as what Martin Westwell from Flinders University has been saying for a while…and, to be perfectly honest, says much better than Andrew did. My notes from Matrtin’s address at last year’s ACEL conference follows.

The way our brains are ‘wired up’ depends on the number and type of connections made. These connections are determined by the experiences we have. These lines of communication between brain cells consequently determine the learning formed. It is the interconnectivity of ideas between cells that transforms information into learning. Repetition of these experiences re-inforces the connections made. (Me: important, therefore, to ensure the connections made are those that produce quality thinking rather than regurgitating a learnt script). It isn’t important as to how the information gets into the brain but what the brain does with it when it receives it.

Anxiety (especially long term), just like how we are educated, changes the way we think…there is an emotional component. Anxiety can physiologically prevent us from achieving our potential, it inhibits learning. (Me: So intervening to improve learning means intervening when affective learning behaviours are not going to produce optimal learning as well as intervening when cognitive behaviours aren’t conducive)

An experiment done with a group of young black boys in the US produced the following. These boys were all given an IQ test. Half of them were just given the questions. The other half were first asked to tick a box to describe their ethnicity. Even though the groups’ ability make-up were very similar, the second group produced significantly less IQ points as the other. (Hattie’s “the best predictors of a child’s achievement are the child’s predictions”…if the child believes that they are going to perform badly then they will.) Students who think of their intelligence as fixed usually have achievements that decrease over the course of their schooling. Those who believe intelligence is malleable are more resilient, can come back from failure, don’t give up as easily and show a positive trajectory in terms of their achievements.

The executive functions of the brain that we should be encouraging and promoting in the way we teach are:

·        Concentration

·        Resisting temptation

·        Delayed gratification

·        Self-directed/interdependent learning (note to self: use these terms instead of independent/group work)

·        Problem solving

·        Creativity/Innovation

The environment we create in classes and schools can affect how students develop their intelligence.

Take, for example, the experiment done with mice who were deliberately injected with Huntington’s Disease…a disease that withers the brain. Huntington’s is a genetically inherited disease. If you have the gene, you develop it…or do you?

Only 20% of the infected mice who were placed in a rich environment full of wheels, crawl tunnels etc actually developed the disease. 100% of the infected mice, who were placed in an environment in which no stimuli were provided, developed the disease.

So..what is an enriched environment for schools? One that is multi-sensory, relevant, that has emotional content, interpersonal interactions, exercise, good nutrition and hydration and one that has sufficient blue light (eg sunlight)

Another automatic reaction of the brain (leftover from animalistic days when we needed to protect ourselves from harm) is its reaction to risk. This has huge implications for both teaching/learning and change agents of systems, such as education. We have impulsive preferences for certainty. This limits the potential for innovation. Our brains want us to ‘go back to what we know’…don’t risk the uncertainty. We see this whenever anything new is suggested or introduced. For example: technology. In the UK, when the internet meant that students were plagiarising their coursework component, the system reacted by making more assessment external and assessed by examination. As with anything new, however, the challenge is not to dismiss its existence in our reaction, but to be judicious and deliberate in our use of it to support, promote and encourage what is the essence of education: learning. The other mistake is to go overboard in its use. Not everything new is ‘good’ for learning – a lot of the educational technology games may lead to greater short term engagement but not to long term learning. Keep in mind the purpose. It’s not the technology per se that changes what and the way students think, it’s about you and what you do with it.

 

Based on his knowledge of brain behaviour in adolescents, Andrew recommended the folowing strategies:

• High level of feedback
• Repetition
• Ample time for wondering, being intrigued
• Opportunities for challenge, experiencing difference
• Making meaning

Implications for teachers of mathematics? Mini maths review quizzes at the start of every lesson, more physicality in activities, more opportunities for student discourse and interaction, more problem solving. To expand their memory capacity, increase repetition of key ideas and processes, use new information deliberately to link into already-held knowledge and describing things in different ways. He also recommended teaching the ‘concept’ – but I don’t think he knew that a concept was different to a fact – before relational questions were asked, giving bullet-proof definitions then a series of examples and non-examples.

 

It was this last set of recommendations that made me realise the message was getting lost due to the fog created by his lack of understanding about the nature of mathematics and the learning of mathematics. He, like many others, including a substantial number of mathematics educators, believe that mathematics is a set of definitions, rules and processes to be memorised and practised rather than a set of ideas and understandings and his examples demonstrated this.

 

In fact, I was a little disappointed, truth be known, in the conference as a whole. I got the distinct impression that this was the underlying mindset of most of the presenters – that mathematics is a set of skills that define the content and that the best we can do as teachers is prepare engaging activities (like the Maths 300 ones that were used extensively as examples) that use these skills in interesting ways or provide insight into what ‘working like a mathematician’ means….activities that our students enjoy and thus engender a positive relationship to the discipline.

 

I believe, we, of course, can do so much better than this.

 

I really found it difficult to justify separating classroom experiences into ‘problem solving’ and ’skills’. As readers would know, I believe it is possible, desirable, no; imperative, to teach mathematics for understanding. Start with purpose and carefully consider what it is we want our students to be able to do (skills), understand and know for every lesson in every topic, then deliberately align all learning activities to those learning objectives and assess for our students’ learning at regular intervals. The Maths 300 lessons are great but sometimes go off on tangents, in my opinion, that take students (and teachers!) away from the concept one is trying to instil. Here I agree with Andrew Fuller – adolescents need repetition of the ‘big idea’, they need to be focused on that big idea to develop understanding and create their own meaning. That’s the job of the teacher: to decide what the big idea is and design learning activities that deliberately address and continually re-inforce this big idea.

 

As an example, take the Radioactivity Maths 300 activity. I use this as my introduction to exponential functions in Year 10. We start with 1/6 as the rate of decay, we do the die activity, we plot the class results as a graph. My learning objective is to develop the understanding that multiplying by the same constant for each successive value of the indepedent variable, always results in a similar curve that we call exponential decay (or growth) and that the magnitude of this constant determines the rate of decay, or growth, of these curves. Hence, my next step in my lesson plan is to stop and discuss the shape of the graph obtained. Why isn’t it linear? (Opportunity here to repeat the conditions that result in a linear graph) Why is it coming down? We talk about half life. We do it again with a different rate of decay. We stop and compare. What’s the same as before? What’s different? Why? What if we ‘grew’ rather than decayed? What if the number of atoms doubled each year? etc. I have a clear purpose in mind for the lesson. I keep bringing back my students’ focus to that learning objective. I would then transfer that understanding to some skills-based questions on exponential rules and graphing.

 

How is this different to what was recommended at the conference? Skills seemed to be done in isolation and prior to ‘using’ them. With the lesson plan above, I started with ‘the whole game’ – as David Perkins would say – then drew out the understanding I wanted and then used this as a justification for learning about the related skills. This is how I think mathematics is best taught. With understanding comes engagement. Students can ‘work like mathematicians’ every single lesson!

 

I did come away with a couple of lovely ideas I will put into practice this coming term. I hadn’t seen the Maths 300 Soft Drink Cans activity. It was recommended as a ’strategy lesson’ at the conference – to teach students about breaking a problem down into manageable chunks, considering one aspect at a time. For those who don’t know it, students are given a 6×4 rectangular grid and 18 cubes. The problem is to place the cubes onto the grid in such a way that every column and every row has an even number of cubes in it. This proves to be quite tricky and frustration tolerance is tested. After about 5 to 10 minutes, the class is stopped and the teacher brings everyone around one student’s table. He/she asks the student to arrange the blocks on the grid so that every row has an even number, forgetting about the columns for the time being. When this is done, the teacher says…”Now, slide cubes along the rows – but keeping the rows intact in total - to make the columns all even” The problem becomes incredibly easy with this simple instruction. The Maths 300 lesson goes onto extend the problem in various ways BUT this won’t help students transfer knowledge of this strategy into other situations. I immediately thought of its application to solving simultaneous equations. The whole idea of solving simultaneous equations is to deal with one variable first then come back and find the other one. I’m going to try doing this little acitivity first then, in the same lesson, start simultaneous equations and try and embed the strategy in some skills work straight away so that their brains can make the connection and thus strengthen the procedural link.

 

I’ll let you know how it goes. Good luck for Term 2.

 

All Systems Go

Well…I’m very glad that first term is over. It has been a pretty rough time for everyone, I think. The tremendous heat affected us all, then the emotional and psychological impact of the Victorian bushfires and a seemingly never ending series of little workplace ’battles’  – those working in any institution would be familiar with these - added up to an intensely tiring term.

On this Easter Sunday, however, my thoughts turn to a more hopeful future. It’s not ‘good’ for the soul to dwell on the past. Easter Sunday is symbolic of new beginnings, new ways of thinking about things and the emergence (or re-emergence) of bright ideas and an engaging aesthetic….plus…delight in the world cannot be repressed. As Leonard Cohen said when he toured here earlier this year: “I’ve been studying the philosophies of the religions…but cheerfulness kept breaking through”.

Earlier this year, I attended a brief presentation by Ron Ritchhart, of Project Zero at Harvard University, that was about embedding leading and learning in schools. During this talk he asked the questions:  “How can we know if classrooms are changing?” “How do we know if schools are becoming cultures of thinking?”

He said that behaviours were ‘indicators’ of cultural forces, that behaviours were easily seen but that behaviours could also be misleading, in terms of what they showed about the authenticity of the enculturation of a thinking, learning community in schools. Behaviours can sometimes only indicate superficial changes. All teachers, may, for example, all be required to write up their lesson plans in a particular way or draft ‘essential questions’ for each topic taught or engage their students in self-assessment. These things may be occurring but is a culture of thinking really embedded in teachers’ practice, in their thinking and in their classrooms?

In classes, do students ask questions that focus on procedural things or questions that relate to the learning? Is there real interplay in the discussions in classrooms? That is, not ‘just’ student question, teacher response; teacher question, student response but student-student discourse that is spontaneously generated and isn’t filtered through the teacher. One of my greatest joys in the classroom comes when a student is doing a problem on the board and others start commenting on what this student is doing or has done….commenting, not to me, but directly to the student. I just sit at the back of the room and lap it up. “How did you get that answer?” “I got it by doing x, y and z”  ”Why did you do y first and then x?”  “Can anyone help me out here? I don’t know what to do now” Sure, there are still moments when these are directed my way but I am doing my utmost to encourage real dialogue between my students, real ‘argument’ (in a Socratic, academic way), real application of mathematical thinking. This discussion helps me too. It gives me information about their thinking that informs my future instruction. When students have ‘completed’ their problems, I give them the opportunity to think about it and change their responses if they want to. I emphasise the importance of being able to make mistakes and learn from them. As Ron went onto say, teachers have the dominant voice in classrooms and it’s important we are continually open to think of ways in which we can increase the student voice…to increase the level of ‘tentative talk’ as students think and wonder and engage with the ideas in order to develop their own conceptual frameworks.

So…behaviours are what Ron calls ‘first order changes’. The deeper, more authentic, sustainable changes are second order changes. How do we get to these? What do they look like? What things stand in the way of them occurring? What are the structures and processes that support the creation of a critical mass of teachers in a school who then help a school become a real culture of thinking for learners and leaders of learning?

In this last term, I have become convinced that it is imperative for the totality of a school’s systems and processes to be integrated into a coherent whole that recognises and works towards aligning their goals, outputs - however you want to put it – for optimal learning. We have heard it often enough: “Learning is the core business of schooling”…but how often do we actually ensure that ALL processes and systems in place have better learning as their focus? There are too many mixed messages in schools. On one hand we are encouraged to create professional learning teams, get more involved in assessment for learning, use technology to a greater extent to broaden students’ experiences BUT the systems in place sometimes end up acting against these. Not deliberately, not overtly, not much. But every process or system sends out its own message about what is important, what is valued and what is not. Ron calls this ’symbolic conduct’.

For example, if professional learning teams are important then their meeting times need to be protected and supported. Other meetings should not be scheduled for the same time as this dilutes the attention and sends out a symbolic message.

Resources, especially time, within a school are precious and limited. The available resources and expertise need to be accessed and utilised in the most efficacious way in order to maximise learning. If it doesn’t positively affect learning then it needs to go. Resources need to be intelligently used. Systems need to be held to the same levels of high expectation and efficacy as to which teachers are held. There needs to be a focus on what matters most, sustaining improvement over time and building on expertise. Systems and teaching need to be structured to ensure success and success is judged by that which matters most. Initiatives in both systems and teaching should be tailored to the overall attainment of quality learning, learning for students, their teachers and administrators.

I believe that a culture of leadership is a pre-condition to a culture of thinking. Not a culture of management. Strong, visionary leaders are needed in schools who can drive improvement within a school and continually reflect on the systems in place and change these if necessary to create those that are carefully and deliberately aligned with a whole-school focus on developing a culture of thinking and learning for all within.

Let’s go back to school with open minds and hearts. But also with a critical eye. A vibrant, professional thinking culture can be achieved for us, as leaders of learning. Simultaneously, a thinking culture can be achieved for our students. It is very clear to me, however, that these things cannot be achieved well if school resources and systems are not deliberately aligned to the same purpose – that of achieving optimal learning. That is, everyone and everything all working together to focus attention to that which directly impacts on student learning and to ensure any performance tail in any system or process is identified and reduced.

Marty’s Lament

Marty Ross wrote an opinion piece in yesterday’s Education Age : How Maths Became the Sum of Many Failings.

In this article, he rails against:

• Text books
• The training of teachers of mathematics
• The mathematics curriculum
• Calculators

I would like to respond to a number of points made in this article.

I have stated elsewhere that textbooks should not define the mathematics curriculum of a school. They are just one resource available to teachers and students. If the text IS the syllabus then this is, indeed, a problem in my opinion. Too many mathematics syllabi are still comprised of a list of content and the exercise(s) from the text that address this content. It is my belief that teachers determine what concepts, ideas and knowledge are important for students to learn (albeit within the constraints of either statewide or, to come, national standards) then formulate the assessment that will determine if students have learnt these things, then determine a set of learning activities that specifically and deliberately targets these understandings. A text is a good source of questions that can be used to assess students’ formative understandings…but only if the questions chosen are those that address the desired understandings. This is the job of the teacher; a teacher who knows what they want their students to learn and understand, who knows what misunderstandings were evident in the classroom and who wishes to address those misconceptions in order to improve learning.

Of course, in order for a teacher of mathematics to do the above, they need to know their subject well and the associated pedagogy of teaching mathematics. This is where teacher training comes in – or ongoing professional learning that addresses pedagogical issues. Too many student teachers will repeat the behaviours of their own teachers and these behaviours may be less than the ideal. It is difficult. It is becoming increasingly harder to find school placements for student teachers. Student teachers rely on their supervising teachers to assess them so are unwilling to go against any advice offered. Consequently, student teachers may replicate their supervising teachers’ methodologies rather than try out something else associated with what they might have learnt in college/uni. If these methodologies are ones that aren’t conducive to a greater understanding of mathematics, the cycle goes on.

Mathematics is all about reason. It’s about ideas. I have blogged previously about what follows but it’s worth re-stating it here, I think.

There is a poem by Peter Hooper which is partly given below:

Poetry isn’t in my words

It’s in the direction I’m pointing

If you can’t understand that

And if you’re appalled at the journey

Stick to the guided tours

Perhaps teaching and learning are much like this – the journey is the real education; the content merely the vehicle by which we explore the landscape. In what direction do we point our students with the what, the why and the how of teaching? From W.S. Anglin: ”Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness where the explorers often get lost”

Do we sometimes forget about the journey in our desire to get to the destination?

The teaching of ideas should be the journey in the teaching of mathematics. These ideas are taught through the vehicle of content topics that develop mathematical thinking and behaviours (acting like a mathematician) as well as a set of mathematical skills.

Mathematical thinking is about looking for patterns, testing these for the circumstances under which they prevail (proof) and abstracting them in order to generalise and predict. In order to inculcate such thinking in our students, teachers should strive to build on connections, coherence, creation, collaboration and conversations. Understanding mathematics is the destination. Assessment then addresses these understandings deliberately.

Teaching for understanding involves thinking – deep, sustained thinking that leads to students being able to construct an abstraction in their minds that actually makes sense of many distinct pieces of knowledge.

Instead of asking the question “What topics do we need to cover?”,we need to aim to develop processes and approaches which result from asking “What are the powerful ideas and processes that we believe are important for young people to learn in mathematics ?”

Our goal is then to develop syllabi around desired understandings – the big ideas – then ask “What information and what experiences do we engage the students in, in order for them to develop these understandings?”

The focus of assessment is in terms of the further question “If students understand this idea, how can they demonstrate it?”

The task of the teacher is not to put knowledge where it does not exist, but rather lead the mind’s eye so that it might see for itself. Our goal should be to provide learning opportunities within the curriculum that develop the students’ capacity to construct their own understandings and deliberately plan lessons that offer, support and develop rich and authentic thinking.

Research has shown that students who solve difficult problems on their own — without the help of other students or teachers — often gain a better understanding of mathematics concepts.

The learning comes with the struggle.

Let’s endeavour to give students opportunities to struggle through a problem, and refrain from directly telling them how to solve it. We can support and encourage risk-taking and the making of mistakes as a natural part of learning. We can emphasise understanding as the goal rather than just looking at results. We can encourage students to reflect on their learning and consider how they learn best in order to improve future learning. We can encourage students to be creative and open-minded in their thinking and consider multiple perspectives and develop alternative pathways to solution.

It’s a question of being more mindful about our purpose in the teaching of school mathematics.

This idea of purpose is an important one that should guide all teaching. Proof need not be the dry and largely esoteric horror people may recall from their own schooling. The capacity for absolute proof is unique to mathematics. It is important. But it can be as simple as asking “how do you know?”. In the same vein, the use of the calculator, including the CAS models, can assist mathematical understanding and free up teachers and students to focus on big ideas and deep thinking, instead of spending large amounts of time doing calculations. It all comes back to purpose. What is the purpose of the learning activity? If it is to do arithmetic and develop quick thinking in numeracy, then a calculator isn’t the appropriate tool. If, on the other hand, the purpose is to investigate continuity and differentiability of functions, then a calculator can enable these concepts to be ’seen’ and understood a lot better without the ‘distraction’ of numerous calculations that have the effect of providing intellectual  ’white noise’ and distract the learner from the concepts.

I certainly agree with Marty’s final paragraph: if mathematics curricula are not written in the right ’spirit’ that reflect the heart and soul of the learning enterprise we call mathematics, then students will not enjoy mathematics nor will they understand it.

AAMT gets political on assessment

The AAMT (Australian Association of Mathematics Teachers) released their position paper on The Practice of Assessing Mathematics Learning late in 2008. The paper describes practice teachers should endeavour to work towards, in mathematics assessment. It is based around three themes. Students’ learning of mathematics should be assessed in ways that
• are appropriate
• are fair and inclusive
• inform learning and action
Under each of these themes, the paper outlines expectations for teachers and, more interestingly, for assessment programs of education authorities. The latter, I believe, is the first public inclusion, of which I am aware, of advice to government bodies from a subject association. This advice includes statements such as:

Large scale assessment programs are expensive. Funds applied to these should be proportional to the benefits from the program to students, teachers, parents, schools and the education authorities themselves. Excessive public expenditure on assessment programs cannot be justified in the context of limited overall funding for education.

It is well worth a careful reading…more than this, it deserves a careful reading, an open-minded reflection on current assessment practices and then a thorough audit of these practices, leading to change if appropriate. It is also of value to other disciplines and is available here.

Whilst I’m on the topic of assessment, I’m planning to make assessment for learning my theme for this coming academic year. I would like to explore ways in which assessment can be made both more formative and more informative. Two references I have found invaluable so far for ideas are:
Thinkers – A Collection of Activities to Provoke Mathematical Thinking by John Mason et al and
Securing Their Future – Subject based assessment materials for the School Certificate by Doug Clarke et al
The former is available through AAMT and the latter was produced for the NSW Government and I’ve hotlinked the title to the pdf of the materials for you.

Both contain excellent ideas for assessing students’ understanding in ways that uncover misconceptions and lead students and their teachers to a greater understanding of both the mathematics and the underlying cognition.

What IS mathematics?

I’ve been clearing out my offices at home and at school recently. I came across this gem. It’s Professor Julius Sumner Miller’s response to the question posed in this post’s title, as quoted in the AMT (Australian Mathematics Teacher) Volume 4 2006.

Mathematics, when properly viewed, properly taught, properly learned, arouses the spirit, cultivates imagination, stirs curiosity, invites further learning. It is indeed one of the noblest creations of the human mind. It is, moreover, the product of the greatest minds of all time. It interprets nature, it unveils the harmonies of the universe. It is the priestess of clarity.

From the structure of mathematics emerges a sense of truth. It has no room for opinion or conjecture. It gives to our understanding what music is to the ear, what beauty gives to the eye. It is to the head what poetry is to the heart.

Mathematics lays bare the order and the beauty of the great scheme of things – the tides of the seas, the colours of the rainbow, the motion of the planets, the music in the pine trees, the gurgle of the brook, a worm in the good Earth, a bird on the wing. It tells us why raindrops are round; it can give us the geometry of a leaf; it describes the mechanism of light on the eye whereby we see, and sound on the ear whereby we hear.

Mathematics adds vigour to the mind, frees it from prejudice. Borrowing from Francis Bacon: mathematics makes men wise, witty, deep, subtle, able to contend.

The aim of life is to live, and to live means to be aware, joyously, drunkenly, serenely, divinely aware.

Happy New Year to all! May we all venture forth into this new year with the resolution to be more ‘aware’ of things (this post’s title is a Henry Miller quote, one of my favourites). Despite the time pressures of the academic immediate – usually administratively based – and the things that anchor us in the everyday, I believe it is essential that we find the time and the headspace to be aware of knowledge, explore ideas and read and hear voices that influence our teaching and learning.

 

“And thus do we of wisdom and of reach

Find direction…”

(apologies to Polonius in Shakespeare’s Hamlet for modifying his speech to suit my own purposes!!)

 

“For classrooms to be cultures of thinking, schools have to be cultures of thinking for teachers”

Ron Ritchhart – at AISV’s Establishing and Sustaining Professional Learning Communities inservice February 29, 2008

 

I would like to start this year by reflecting on and reviewing a few books that helped increase and deepen my own awareness of things educational in 2008.

 

(1)  Exceptional Outcomes in Mathematics Education by John Pegg, Trevor Lynch and Debra Panizzon

 

This is a small publication I was introduced to at the Summer School for Teachers of Mathematics in early 2008. It is the result of a project conducted jointly by the University of New England, the University of Western Sydney and the New South Wales (NSW) Department of Education and Training. This project identified and explored the factors leading to exceptional outcomes in junior secondary classes in the Australian state of NSW. Primarily, it is of most interest to Heads of Mathematics Faculties but makes interesting reading for anyone involved in mathematics education within a secondary school setting. Chapters include Faculty Staff Characteristics, Faculty Practices, Teacher Practices and Identified Themes Contributing to Exceptional Mathematics Faculties. The stand-out of this book for me was the statement that individuals affect one class at a time. To have school-wide good results, it was necessary to move the whole team of teachers forward. Even ‘islands’ of good teachers weren’t particularly effective. A good school needed

*a mission of high educational outcomes

*a capable and supportive executive (at least one ‘legend’)

*sound organisational and administrational structures in place

*workable student welfare and support programs

It was mentioned that what we sometimes call ‘professional respect’ was actually a way of not scrutinising each other’s teaching and maintaining a protective layer around us that enabled us to get away with not authentically reflecting on our teaching and effectiveness. Interestingly, the research team discovered that the conversations in the science, maths and English classrooms they visited were remarkably similar – all focused on challenge, rigour and teaching for understanding.

Successful faculties were the ones that could align themselves with the school policies better than others.

Aspects of faculties that were high-performing:

*strong sense of team

*high standards expected of colleagues

*faculty seen as ‘family’

*stable staff

*enculturation of new staff

*capable faculty leadership

*high expectations of student performance

*adequate programs and resources

*ongoing mentorship and sharing of resources and ideas by colleagues

*ongoing reflective, informal practice

*professional development occurred regularly

*physical infrastructure promotes teamwork

*disagreements resolved professionally

In many of these faculties, assessment procedures were the catalyst for the above to occur. There was collaborative setting and marking, quick and effective feedback given to students, grading was a focus and assessment used as motivation for students. The most important feature seems to be that teachers are not resting on past laurels and always looking for ways in which to improve.

“Outside show is a poor substitute for inner worth”

 

(2)  Understanding by Design and Schooling by Design by Grant Wiggins and Jay McTighe

 

The Understanding by Design (UbD) book has been around for some time and impressed me greatly. The UbD framework allows us to create a curriculum that enables and supports thinking and understanding. Learning is a product of thinking. Curriculum should also be a product of thinking. Can teachers articulate the reasons for why certain content is to be included in the syllabus? What are the ‘big ideas’ we want our students to understand? What questions could we ask that develop these understandings in our students? How will we know if students have developed these understandings? These are important constructs on which to base a curriculum. This curriculum should focus on challenge, complexity, opportunities for discussion and analysis, no set approach for solution, going beyond the straightforward and stretching students’ thinking to develop understanding.

 

 

Well-documented curriculum is important because it sets the agenda and articulates the vision and provides an enforceable structure, on which improvements to teachers’ practice can be based and referred back to, but we have to simultaneously and continuously work on changing our beliefs about teaching, pedagogy and methodology otherwise the curriculum design will be largely ignored. Teachers ultimately determine how any curriculum will be interpreted and our beliefs and ways of thinking will consequently determine the extent and success of any change. It is the teacher and his/her pedagogical beliefs about how the learning should be shaped that will ultimately determine whether or not learning will occur, how authentic that learning will be and the quality of the learning that happens in our classes.

 

I liked the simple (but not simplistic) three-step plan to designing curriculum and exploring understandings within the discipline. This three-stage approach of the framework referred to as ‘backward design’ involves planning with ‘the end in mind’ by first clarifying the understandings one seeks then thinking about the evidence needed to certify that students have achieved those understandings then planning the means by which those understandings will be attained. It not only provides a well-articulated, coherent and comprehensive structure for curriculum, it also compels those designing the curriculum to explore their own constructs of their knowledge of mathematics and pedagogical practices.

 

The Schooling by Design book explores the wider perspective of the whole school in the quest for teaching for understanding. It is my belief that, for a professional learning team to be involved in productive decision-making that can positively influence the direction taken by a school, its membership needs to include those who have the capacity to do so. Any initiative involving curriculum will need to be actively supported by Heads of Faculty. These people have much influence over the curriculum direction of their areas of responsibility. They are generally very experienced and respected practitioners within their discipline areas. They co-ordinate the content, articulate the vision for the methodology employed and have responsibility for the documentation of the curriculum. They bring knowledge to the group and ‘reach’ to the school. They can ‘seed’ the impetus and drive any reform. Knowledgeable members of staff who are open-minded about their practice and who continually seek to improve what they do in order to provide rich and authentic learning experiences for our students can extend their own reach as well as that of others within their sphere of influence, through their involvement in professional learning teams. In order for the whole school to move forward, it is necessary, in my view, to involve as many members of staff in the process as possible. This book provides leaders of learning with a strong theoretical background and ideas as to how to accomplish this in schools.

 

(3)  Teaching Secondary School Mathematics by Merrilyn Goos, Gloria Stillman and Colleen Vale.

 

Teachers make a difference to the quality of student learning. This book attempts to ‘untangle the complex relationships that exist between teaching practices, teacher characteristics and student achievement’ (taken from p3).

Sections include Mathematics Pedagogy, Curriculum & Assessment, Teaching and Learning Mathematical Content, Equity and Diversity in Mathematics Education and Professional and Community engagement. It is clearly written and easy to read and understand. It contains up-to-date research findings, multiple examples that can be used in the classroom and examples of performance tasks that target specific big ideas. It also has many ‘Review and Reflect’ sections throughout chapters, in which teachers are given tasks that help us to identify our own beliefs, design assessments, think about the whys and hows and extend our own professional knowledge bases. These would make excellent discussion starters in faculty meetings.

Highly recommended.

 

(4)  Improving Student Achievement – A Practical Guide to Assessment for Learning by Toni Glasson

 

From the Introduction: ‘..in an era in which curriculum documents describe specific standards, the teacher’s role is one in which they are asked to make sure that increasingly greater numbers of their students are able to demonstrate the ability to meet those standards. How is this to be done? The answer lies, to a significant extent, in changing the way in which we regard and use assessment in the classroom. In itself, this sounds quite simple, but in reality it requires a major shift in thinking and, indeed, in the very essence of our attitudes to teaching’

 

This book looks at Learning Intentions, Success Criteria, Strategic Questioning, Effective Teacher Feedback, Peer Feedback, Student Self-Assessment & Making Formative use of Summative Assessment. It is based on the seminal research into assessment of Black and Wiliam. It has a firm classroom-based approach and, like the previous book mentioned, it has sections at the end of each chapter headed Professional Learning Focus  that could be used to develop teacher learning during staff meetings or other teacher learning situations.

Also highly recommended.

 

(5)  The Brain that Changes Itself by Norman Doidge

 

I’m only about two-thirds through this book at the time of writing this post, but it has already impressed me greatly.

From the Preface: ‘This book is about the revolutionary discovery that the human brain can change itself…children are not always stuck with the mental abilities they are born with…thinking, learning and acting can turn our genes on or off, thus shaping our brain anatomy and our behaviour…people [can] rewire their brains with their thoughts. The neuroplastic revolution has implications for our understanding of..learning. While the human brain has apparently underestimated itself, neuroplasticity isn’t all good news; it renders our brains not only more resourceful but also more vulnerable to outside influences. Neuroplasticity has the power to produce more flexible but also more rigid behaviours. Once a particular plastic change occurs in the brain and becomes well-established, it can prevent other changes from occurring’

 

As educators, we can help our students (and ourselves!) formulate and strengthen connections that are of benefit to present and future learning and assist them to break free of others that might not be so beneficial for their learning. What teaching and learning strategies and practices can support this? Which ones hinder this? We know that teachers are at the centre of student learning. We have a tremendous responsibility and, although not directly aimed at education, I think that every teacher should read this book as changing the brain is surely what we do for a living!

 

 

 

Best wishes for an informed, productive, joyful year of teaching.