One of the reasons, I have enjoyed this text so much is the parallels of thought with my own eclectic beliefs around teaching and learning. Hattie, Fisher and Frey’s final chapter in their book, Visible learning for mathematics explores the importance of assessment, feedback and meeting the needs of all learners.
Rather than using the terminology of formative and summative assessment, the authors use formative and summative evaluation. Their argument for this choice is that as the evaluator, it is the teacher’s role to interpret where a student is during and after a lesson and how to improve his or her teaching in response to this evaluation.
Hattie, Fisher and Frey state that it is daily formative evaluation that allows to teachers to make instructional decisions about what will occur next. Carol-Anne Tomilson, the guru of differentiation states that formative assessment is the heart of differentiation as it provides the evidence as to what students know, don’t know and when done correctly, formative assessment provides both the teacher and student with information as to what to do next. The authors suggest a process for formative evaluation with the following key elements:
Hattie has several internal questions for learners to self-regulate and for teachers to ask about their instruction:
Some examples of strategies to check for student understanding include: asking questions; numbered heads together; response cards; purposeful sampling; exit tickets; and lots of opportunities to self-assess.
Hattie, Fisher and Frey discuss feedback for the teacher (adjusting teaching) and feedback for the student (adjusting learning). Hattie considers the four levels of feedback (task, process, self-regulatory and self) and considers each type’s impact. Not surprisingly the most common types of feedback are on task and process. However, for students to work towards self-regulation, self-regulatory feedback is crucial. It plays a prominent role during deep and transfer learning. The final type of feedback, self (Excellent job! You are so smart!) appears to have a zero to negative impact on learning. The effect size of feedback that fosters learning and perseverance has an effect size of 0.75.
The authors discuss the importance of differentiation in order to meet the needs of all the learners. They quote Tomilson and write that differentiated instruction is the use of a variety of instructional approaches to modify content, process, and/or products in response to the learning readiness and interest of academically diverse students (1995, p.80).
They conclude the chapter with some evidence-based provocations.
This year, the Metro region of Qld State Schools has done a lot of work around what makes a standards based curriculum (such as the Australian Curriculum) work well. One of those aspects is teacher expertise that embeds assessment into pedagogical practice. When I begin planning a unit of work or the teaching of a concept, I backward map from the summative assessment task. It is important to have the end goal in mind. The best way I can really understand the demands of the task is to actually do the assessment myself and to unpack the marking guide, so I can provide opportunities for students to have unlimited growth and personal learning.
I then consider what I already know about the students in relation to the demands of the assessment and the important concepts of the unit and design a quick pre-assessment for things I don’t know. Finding out what students know and don’t know allows precision of teaching.
As the unit progresses, I then have quick formative assessments in mind, those I will do daily and those that I use as more formal check ins with my fellow year level teachers. These allow me to give students ‘just-in-time, just-for-me’ information when and where it can do the most good.
One quick formative assessment that I use consistently is the exit ticket. Not only am I able to give ‘just-in-time’ feedback to individual students, but I am able to see trends and patterns and adjust my teaching and groupings to cater for these. This particular image of one I used in a year 4/5 class gave me so much information. I could see what mental strategies students were applying, what stage of multiplicative thinking they were at, whether they could justify their thinking and a self-reflection on where they felt they were in their application of mental strategies.
This embedding of assessment (and feedback) within pedagogical practice means no surprises when the summative assessment is given. The students and I know how they are going and what their next steps for learning is. I am also able to adjust my teaching in response to real-time data and make good decisions about instruction. The brilliant thing is that the data gathered in the summative assessment can then be used to inform the next unit.
Where are you when it comes to assessment, feedback and meeting the needs of your learners particularly in the learning area of mathematics? This final chapter explores these ideas in detail. Hattie, Fisher and Frey have produced a fabulous book that is easy to read, easy to apply, makes learning visible and puts the student at the centre of teaching and learning. Well worth a read.
Hattie, Fisher and Frey mention three phases of learning – surface, deep and transfer. My previous two blogs explored the first two phases of learning: surface and deep. The final phase is that of transfer. In mathematics, this phase occurs when students are able to make connections among mathematical understandings and then use those understandings to solve problems in unfamiliar situations, while being very aware of what they are doing. Because learning is cyclical, transfer occurs all the time.
The authors use the example of Linda who understood the mechanics of mathematics (surface learning mixed with a little bit of deep learning) but had no idea how to apply that knowledge to real world contexts.
Linda reminds me very much of myself when I attended secondary school and continually battled with the subject of maths. I loved it, particularly the challenge of the discipline but was extremely frustrated in that I did not know how to apply what I had learned to different situations or real world contexts. I also constantly questioned why I had to learn certain things, such as sine, cos and tan. I was never shown how trigonometry is used in contexts outside of school. Relevancy is a major condition of transfer which is basically answering the questions, “Why do I need to learn this and when will I ever use it?”
Learning becomes more meaningful when learners see what they’re learning as being more meaningful in their own lives.
Hattie, Fisher and Frey state that it is also extremely important that teacher knows the learner developmentally and experientially in order to promote transfer of learning. The authors recommend two stages of transfer: near and far. Near transfer occurs when a new situation is paired to a context that a student has already experienced. A great way for students to develop transfer learning is to consider similarities and differences between the new and a recently completed idea.
Using graphic organisers such as the Venn diagram can help students consider the similarities and differences.Hattie writes that comparing and contrasting new with old problems has an effect size of 1.23.
The leap is bigger for far transfer as the student has to make connections to more removed situations.
Hattie, Fisher and Frey mention a few things to consider when planning tasks for transfer. Tasks:
Metacognition, the ability to think about our thinking, promotes transfer learning. Palinscar (2013) described metacognitive awareness as having three parts:
To develop metacognitive skills, students need to learn the art of self-questioning. Self-verbalisation and self-questioning have an effect size of 0.64. Self-questioning enables students to track their understanding and re-align when off target. Hattie recommends using pre-lesson questions (What are today’s goals?) and post-lesson questions (What was today’s goals?) to encourage metacognition.
Self-reflection is a follow-up technique after the lesson that encourages students to understand where they were and where they are now.
Rich class discussion promotes transfer learning. In mathematical discussions, students should be able to represent their thinking to others, pose questions, and engage in disagreements respectfully. The caveat is that it is not unrelated storytelling, random opinion sharing or teachers doing most of the talking. The teacher’s role is to use questioning to lift student thinking, press for evidence and have students make links amongst concepts.
For students to take the driver’s seat of their own learning, they need a sense of how the mathematics that they know is organised. Creating an organisational structure for mathematical knowledge is a powerful tool.
Peer tutoring requires the tutor to connect what the tutee needs to learn to what the tutee already knows and understands. Peer tutoring has an effect size of 0.55 and works best when programs are structured, the tutor receive training and the tutor and tutee are of different ages. It allows the tutor to solidify his/her understanding and the tutee to improve their learning.
Giving students opportunities to explore their own ways of using what they have learned is critical for transfer learning and should be woven into classroom life. Students should be continually challenged to develop projects and investigations across the school day.
This makes me think of a recent initiative being used in Queensland: Age Appropriate Pedagogies which encourages the use of a range and balance of pedagogical approaches in the classroom.
The conceptual framework includes six approaches to teaching and learning: event-based, project, explicit instruction, inquiry learning, play-based learning, direct teaching/instruction and a blended approach (Department of education and training, Qld Government). Only ever using one approach such as direct instruction will limit a student’s ability to transfer learning.
Problem-solving teaching also encourages the transfer of mathematics learning and is different to a problem-solving process.
It is similar to the design thinking necessary for the new Australian Curriculum learning area of Technologies.
Problem-solving teaching helps students engage in the process of determining the cause of a problem; using multiple perspectives to uncover issues; identifying, prioritising, and selecting alternatives for a solution; designing an intervention plan; and evaluating the outcomes. (Hattie, Fisher and Frey, 2017, p193)
Design challenges for any age group can be identified by listening to students’ wonderings about the world around them. An example may be designing a school garden to grow vegetables for cooking.
Reciprocal teaching also assists students to transform their mathematical thinking and is a literacy strategy that develops comprehension of the mathematical problem. Students work collaboratively to build their understanding of texts. There are four stages and students are encouraged to keep a record of their thinking and work. :
Students need the opportunity to apply what they have learned in mathematics to unfamiliar situations but it would be nearly impossible to transfer without surface and deep learning. Transfer learning should not be left to chance and requires caring teachers to deliberately design opportunities by knowing their students and using effective strategies.
Stay tuned. My blogs have slowed down as I am back at work but I have one more blog for this text. It will look at the last chapter – Assessment, feedback and meeting the needs of all learners and will be posted within the next couple of weeks.
As mentioned in my previous blog, surface learning is the beginning of conceptual understanding. The challenge for teachers is not to over rely on it and to know when and how to move students from surface to deep learning. Deep learning (the focus of Hattie, Fisher and Frey’s fifth chapter in their text Visible learning form mathematics) provides students with opportunities to consolidate mathematical understandings and to make deeper connections among ideas.
One of the reasons I write this blog is to take my own reading to a deeper level and to make connections with other research and focuses in my work.
With appropriate instruction, surface learning transforms into deep learning. (Hattie, Fisher and Frey, 2017, p35)
In deep learning, mathematical tasks will be cognitively more demanding, more open-ended and have multiple ways of solving them or multiple solutions. As teachers, we have to make judgments about how much surface learning is required in preparation for this deep learning.
Who is doing the talking?
To promote deep learning, students need opportunities for mathematical talk. Accountable talk is promoted by Lyn Sharratt in her work in the Metropolitan region. Hattie, Fisher and Frey state that students should be drenched in accountable talk and this chapter provides a number of language frames that scaffold the use of language to support a mathematic topic. A number of other supports for accountable talk are also mentioned.
Who is doing the thinking?
When it comes to mathematical thinking in whole class or group discussion, the authors mention three sociomathematical norms, norms that promote true mathematical discourse. Reading the text gives an insight into how a skilled mathematics teacher can promote the following:
Are students being encouraged to collaborate?
Collaboration is another important dimension of deep learning. Humans learn better when they interact with other humans and the authors recommend that 50% of class time over a week be devoted to student discourse and interaction with their peers. Effective collaboration and cooperative learning tasks must:
How are students being grouped?
Students should be grouped strategically. The authors point out that fixed ability grouping does not help students to understand the math they are learning. It can affect motivation and make students dislike maths. The most effective grouping strategy is one that is flexible and balanced and allows for a moderate range of skills.
Are you using manipulatives and encouraging multiple representations?
Just as in the surface learning chapter, this one wraps up with a look at the importance of multiple representations and the strategic use of manipulatives to promote deep learning. Students demonstrate stronger problem-solving abilities and deeper mathematical understanding when are encouraged to represent mathematics in a variety of ways. Representations can be physical, visual, symbolic, verbal and contextual.
In the Metropolitan region of Queensland, my colleagues and I have always promoted the think board (an idea we found in First Steps in Mathematics), It can be used in a variety of ways as a graphic organiser for different representations and a way of showing the connections.
Manipulatives shouldn’t be limited to surface learning and can be used to make concepts concrete and visible. They assist students to see patterns, make connections and form generalisations. Again the Concrete Semi abstract Abstract (CSA) model (also promoted by my Metropolitan colleagues) is a useful frame to assist teachers with choices around the instructional sequence from physical to visual to symbolic representations.
All of the above have high effect sizes and assist in making deep learning visible. My next blog will look at transfer learning, the application of surface and deep learning.
Surface learning is the focus of Chapter 4 of Hattie, Fisher and Frey’s text Visible learning for mathematics. When students are at the the surface phase of learning, high-impact approaches that foster initial conceptual understanding are followed by linked procedural skills. The tasks are efficient, often more engaging and designed to raise student motivation. When considering the fluency quadrant, many tasks will be of low difficulty and low to moderate complexity. There should be discussion and opportunities to connect the tasks to students’ learning and understanding.
…observing someone doing something uses many of the same neural pathways as when you perform the action yourself. (Hattie, Fisher and Frey, 2017, p107).
This is why number talks, guiding questions, worked examples and direct instruction are so powerful. I am not going to go in-depth into these as the text does it beautifully. What I will say it that I was first introduced to number talks a couple of years ago when I was doing some work around the development of mental calculation. I became an immediate advocate and have bought a number of texts that illustrate how number talks are conducted.
I will also mention that direct instruction is especially useful in developing students’ surface-level learning but it has the following caveats. It should not:
All four activities enable teachers to demonstrate the kinds of questioning and listening strategies that exemplify excellent mathematical practice as well as give their students opportunities to begin building these habits of mind (Hattie, Fisher and Frey, 2017, p119). Self-verbalisation and self-questioning, skills that help students think critically, have an effect size of 0.64. The text also provides some simple sentence frames that can be used to facilitate student metacognitive thinking (which has an effect size of 0.69).
In the last part of the chapter, the authors consider how to use vocabulary, manipulatives, spaced practice with feedback, and mnemonics strategically.
Vocabulary instruction is important in all the phases of learning but it is often in the surface phase that students begin to build academic language. Researchers such as Beck that as teachers, we need to support students in the appropriate terminology but not give away the mathematical challenge. A focus on vocabulary has always been a focus in the work my colleagues and I have done in mathematics and numeracy. There are many vocabulary games that can be used in warm-up or transition times and we have always promoted the use of word walls and graphic organisers, the latter two being addressed in this chapter.
Using manipulatives in all the learning phases helps build the links between the object, the symbol and the mathematical idea. It has an effect size of 0.5. Once again, my colleagues and I have promoted the use of manipulates and have commonly referred to the Concrete Semi-abstract Abstract (CSA) model.
Using the dual approach of spaced practice and feedback teaches students about the task and about their thinking. Spaced practice, giving multiple exposures to an idea over several days to attain learning and then spacing the practice of skills over a long period of time, has an effect size of 0.71. Feedback about the process (and not just the task) moves students into deeper learning and has the added advantage of giving a student agency so that they rely lesson on outside judgments and have a more positive mindset.
The use of mnemonics has an effect size of 0.45 and assist students to recall a substantial amount of information.
The authors give readers an insight into research based tasks that have a high effect size in the surface phase of learning. This strong start sets the stage for meaningful learning. Too often, however, the learning ends at the surface level. Teaching for deep and transfer learning must occur. My next blog will look at making learning visible in the deep learning phase.
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In primary school, I can remember the blackboard (yes, I am that old) being covered in numerous columns of mathematical operations, all very similar but with increasing difficulty. I don’t remember receiving too many problems to solve and I certainly wasn’t allowed to discuss my solutions or ways of working. In fact, there really was only one way to do it – the traditional algorithm as modelled by the teacher – on my own and in silence.
In the beginning of my teaching career, I emulated some of what I experienced in my own education – lots of operations for students to solve. I made it harder by setting them out horizontally or increasing the size of the numbers. Problem solving was on a Friday. If students finished quickly, I gave them more. Thankfully, my understanding of higher order thinking, collaboration, discussion, and the importance of problem solving and reasoning increased as my experience as a classroom teacher increased.
Imagine my concern last year, when during another heated discussion over homework, my son in year 10 argued, ‘Why do I have to do twenty of these, Mum, when I have already got the first three right? They are all the same?” I had no answer as he was absolutely right. Completing extensive, repeated, context-free exercises is not going to develop the transferable and flexible understanding of mathematical processes (Hattie, Fisher and Frey, 2017).
Procedural fluency cannot be developed without true and meaningful comprehension and ‘drill-and-kill’ exercises without understanding can harm students’ mathematical understandings, their motivation level, and the way they view mathematics. (Hattie, Fisher and Frey, 2017, p75)
Mathematical tasks and talk that guide learning is the title for Chapter 3 in the text, Visible learning for mathematics.
When it comes to planning and implementing maths lessons, Hattie, Fisher and Frey (2017) mention three important ideas to take into consideration.
1. Spaced practice is more effective than mass practice with an effect size of 0.71.
2. Math is not a speed race and speed is not a part of fluency. Speed can damage students’ self-perception of their maths ability.
3. Tasks should not focus solely on procedures as conceptual understanding can be sacrificed.
Good mathematical instruction involves reasoning, exploration, flexible thinking and making connections. Students should know that learning isn’t easy and that there is an enjoyment in meeting the challenges of learning demands.
There are typically two types of tasks – exercises and problems. Exercises make up most of traditional text book practice, are without context and are provided to allow students to practise a particular skill. Problems are generally written in words and apply or provide a context for a mathematical concept. There are those that focus on using a particular concept and those that are non-routine or open ended.
Problems should be used to introduce an idea so that students can model a situation and begin to develop conceptual understanding. Non-routine or open ended problems involve more that applying a mathematical procedure for solutions. Students need a toolkit of strategies and some ‘out-of-the-box’ thinking to solve these.
Hattie, Fisher and Frey advise teachers to consider the level and type of challenge a given task provides and have developed a fluency quadrant to assist with this consideration:
Low Difficulty Low Complexity
High Difficulty Low Complexity
Low Difficulty High Complexity
High Difficulty High Complexity
Hattie, Fisher and Frey (2017) mention a number of characteristics that assist students to build understanding and confidence in mathematics. These include:
The chapter finishes with a focus on posing purposeful questions and the use of prompts and cues. Purposeful questions help teachers check for understanding and the use of prompts and cues encourages students to do the cognitive work, moving them on in their learning.
Identifying the right approach at the right time is critical. Misalignment between tasks and types of learning (surface, deep and transfer) puts students at risk.
Giving students appropriate tasks at the right time in their learning cycle is crucial to move students from surface to deep and transfer learning (Hattie, Fisher and Frey, 2017, p73).
My next blog will explore what tasks the authors have found to work at each stage of the learning.
Think about when you set yourself a goal. It might be a health goal, a work goal, a personal goal… How did you go about achieving it?
My goal this Easter holidays was to revisit my blog and publish at least one post about a current research text. To achieve this goal, I had to begin by clarifying exactly what I wanted to achieve by defining the end point.
I then had to evaluate my starting point in relation to that end point. I recognised that I hadn’t made a post since Easter last year and that I needed to select a text and to re-familarise myself with my blog.
I then had to consider all the things I needed to do to be successful to reach that end point. Creating an interesting and appealing post; taking notes from the text; considering how to frame the post; writing a personalised summary; considering what images would enhance the post; imposing a time frame; considering copyright. For each of these I had to seek the resources to help achieve my end point.
Prior to publishing and as I constructed my draft post, I had to check in on what was needed to be successful and to seek or give myself feedback.
What I was doing is something that ultimately we want students to be able to do. Student goal setting or having students determine their own criteria has been shown to boost achievement (Hattie, Fisher and Frey, 2017). As teachers, having a clear path does not leave learning to chance.
Learning intentions, success criteria, preassessments and checking for understanding all contribute to teacher clarity.
Teacher clarity, which according to Hattie has an effect size of 0.75, is the focus of the second chapter in Hattie, Fisher and Frey’s text, Visible learning for mathematics. It begins with careful consideration when planning a unit or lesson and the composing of learning intentions that make the unit or lesson very clear to both the teacher and students. This is extended to consistent evaluation of where students are in the learning process and describing success criteria so that students can assess their own progress and teachers can monitor how students are progressing with a mathematical idea or concept.
Learning intentions describe what teachers want students to learn – when students know the target, it is more likely that the target will be achieved. The following are some of my take-aways about learning intentions. They:
Hattie, Fisher and Frey also mention the importance of establishing language and social learning intentions. Language learning intentions focus on the vocabulary and academic language needed to master the content. We all know the language demands of mathematics are immense, particularly for English as an Additional Language or Dialect (EALD) students. In addition, a focus on language goals will enhance student reasoning, something that many students fall down on in the marking guide. High quality maths lessons also require collaboration and establishing social learning intentions will foster collaboration and communication.
Success criteria describe what success looks like when the learning goal is achieved. They are specific, concrete and measurable statements. Lyn Sharratt promotes co-construction of success criteria and Hattie, Fisher and Frey’s research supports this. Student goal setting or having students determine their own criteria boosts achievement and has an effect size of 0.56. Allowing students to draft success criteria in relation to the marking guide promotes maximum acceptance and effectiveness. An important note: the more complex the summative assessment, the more time that is needed to understand and deconstruct success criteria.
Hattie, Fisher and Frey describe learning intentions as the bookend for learning and success criteria as the bookend used for measuring success. Strategic use of learning intentions and success criteria promote student self-reflection and metacognition which has an effect size of 0.69. One of the most important things a student can learn is internal motivation and Hattie, Fisher and Frey state that learning intentions and success criteria increase student motivation.
Preassessment helps determine the gap between a student’s current level of understanding and the expected level of achievement. It allows teacher to be precise in determining learning intentions and establishing success criteria. Teachers are also able to effectively differentiate and meet the instructional needs of their students.
Formative assessment allows that checking for understanding. What do my students need to learn today and how will I know that they have learned it? What feedback can be given to students to help their learning progress. A great example is the use of exit tickets which allow teachers to gauge progress.
As far as my goal of revisiting my blog and publishing at least one post about a current research text over the Easter break goes, I achieved it (in fact, exceeded it as I am up to blog #3). It was through making my goal very clear (learning intention), establishing what was necessary to be successful (success criteria), evaluating my starting point (preassessment) and evaluating how I was going with what I had deemed to be successful (checking for understanding).
Tune in to my next blog, which will have a brief look at chapter three of this text: Mathematical tasks and talk that guide learning.
How does one become a lifelong learner? How do we develop a love of learning? Hattie, Fisher and Frey (2017) argue that making learning visible assists teachers and students to become lifelong learners with a love of learning. I am proud to say that I am a lifelong learner (as signified in the name of my blog).
After reading the first chapter of Visible Learning for Mathematics, I spent some time considering how my passion for learning came about. I came to the conclusion that I was lucky enough to enter a profession that I love: one that involves constant re-invigoration and one with which I have a natural affinity. I was also lucky to have two brilliant mentors in my education: my English teacher in year 11 and 12 (Mrs Kay Gardiner) and my Children’s Literature lecturer when I completed my Bachelor of Education (Professor Kerry Mallan).
The above-mentioned ladies were instrumental in my love for English and literature and yet I ended up spending four years of my career specialising in numeracy and mathematics. As mentioned in my previous blog, I have always loved the challenge of mathematics but was never very successful in my P-12 grades. My experiences were very much ‘chalk and talk’ and the learning of procedures and formulas. I then struggled when it came to applying those procedures and formulas in a test.
Hattie, Fisher and Frey stress the importance of teachers erasing many of the ways they were taught mathematics and replacing this with intentional instruction, collaborative learning opportunities, rich discussions about mathematical concepts, excitement and persistence when solving complex problems, and applying ideas to situations and problems that matter. I can certainly concur with that – my understandings, application (and marks) would have improved considerably in a classroom such as this.
The first chapter of Hattie, Fisher and Frey’s book looks at making learning visible for both students and teachers; the debate between direct instruction and dialogic approaches; and the balance of surface, deep and transfer learning.
Hattie, Fisher and Frey write that visible learning:
When it comes to the two approaches of direct instruction and dialogic, both focus on students’ conceptual understanding and procedural fluency and neither advocates memorising formulas and procedures. Both models suggest:
Differences include types of tasks, role of classroom discussion, collaborative learning and role of feedback. Hattie, Fisher and Frey (2017) have put together an explicit table that explores the similarities and differences of both approaches and how ‘knowing what strategies to implement when for maximum impact‘ is precision teaching (p26).The authors state that it is not a choice of one approach over the other. Both have a role to play throughout the learning process but in different ways.
Finally, Hattie, Fisher and Frey address the balance of surface, deep and transfer learning. Surface learning provides the toolbox of conceptual understandings, procedural skills and vocabulary and labels of a new topic. It helps students develop metacognitive skills and can be used to address student misconceptions and errors. Students need this toolbox but there is often an over reliance on surface learning and the goal is much more.
Deep learning is where students begin to consolidate understanding of mathematical concepts and procedures and begin to make connections among ideas. This is often accomplished when students work collaboratively, use academic language and interact in deeper ways with ideas and information.
Transfer learning is the ultimate goal. Students have the ability to take the lead in their own learning and apply thinking to new contexts and situations. They are self-directed with the disposition to formulate their own questions and the tools to pursue the answers.
As a classroom teacher, I was eclectic. I believe I used different pedagogical practices and had a balance of direct and dialogic approaches but it was more instinctive rather than intentional and precise. I believe I spent too long in the surface learning. Deep and transfer learning would have happened to some degree but I don’t believe it was always intentional.
As teachers, we have choices. Understanding these phases of learning assists us to make instructional choices that will impact student learning in a positive way. Hattie, Fisher and Frey’s text helps with precision teaching and my next blog will look at the second chapter that focuses on teacher clarity. Until then…
In preparation for another professional learning session, I began reading the mathematics one. It was such an easy read and I read it from front to back. It cleverly combines the effect size work of Hattie and the brilliant pedagogical instruction of Fisher and Frey. Although it is written for mathematics, I can see the ideas transferring to other learning areas.
In the preface to the book, Hattie, Fisher and Frey (2017) write that the denominator of great mathematicians is that they knew how to struggle. Students need to be persistent, enjoy the struggle, and continue to try. As I reflect on my own mathematics education, it is interesting that although I never received brilliant grades, I always enjoyed the challenge of mathematics (and still do to this day). I especially love seeing how other people (including students and my own children) solve problems and have learnt so much by engaging in rich and rigorous discussion about these different techniques. As a teacher, this discussion has transferred to effective mathematics pedagogy and how to enhance mathematics engagement for students.
The authors mention four things that make mathematics teaching more powerful: appropriately challenging learning intentions and success criteria; learning that is embedded in discourse and collaboration; students doing more of the thinking and talking than their teacher and; students owning their learning. Hattie adds that greater than an effect size of 0.4 allows students to learn at an appropriate rate. The effect size for classroom discussion is 0.82. Students should be invited to talk in collaborative groups, working to solve complex and rich tasks.
Why am I taking the time to share my insights? People who understand mathematics have a higher quality of life. Our moral imperative is every student succeeding. Therefore, to be successful, students must receive high quality instruction (Boaler, 2015).
There are seven chapters in this book and over the next series of blogs, I will share my insights under the following chapters.
Chapter 1: Making learning visible in mathematics.
Chapter 2: Teacher clarity.
Chapter 3: Mathematical tasks and talk that guides learning.
Chapters 4-6: The three phases of learning, surface, deep and transfer, one chapter for each phase.
Chapter 7: Assessment, feedback and meeting the needs of all learners.
Talking with others about ideas and work is fundamental to learning. We need to be able to organise our thoughts coherently, hear how our thinking sounds out loud, listen to how other respond and hear others expand or add to our thinking.
Michaels, O’Connor and Williams Hall write that to promote learning, classroom talk must be accountable – to the learning community, to accurate and appropriate knowledge, and to rigorous thinking.
Accountable talk takes time and effort to implement. The teacher has to create norms and skills in the classroom by modelling discussion, questioning and probing and leading conversations. Then academically productive talk has to be co-constructed by the teacher and students. It is working towards a thinking curriculum. Setting up predictable, recurring routines that are well-practiced will assist students from different ethnic or cultural backgrounds as everybody will learn and share what is expected in the process of using accountable talk.
Conversations can be at a number of levels – whole class, small group, partner and peer or teacher conferences. All students have the right to engage in accountable talk and these discussions should occur across all year levels and in all learning areas.
Students need to listen to one another and pay attention so they can use and build on another person’s ideas. The idea is to be able to paraphrase and expand on ideas, clarify when meaning is lost and to disagree respectfully. Both students and teachers will need patience, restraint and focused effort. In a classroom, there will be students actively talking together, participation in a variety of talk activities, attentive listening, respect, trust and risk-taking, challenges, and criticism and disagreement will be aimed at the idea, not the person.
When students make a claim or observation, it needs to be as specific and as accurate as possible. In a classroom, students would make specific reference to previous findings to support arguments and assertions and unsupported claims would be questioned for further information, facts or knowledge. Students would be concerned with ensuring what they are saying is true and supportable.
Claims and evidence would be linked together in a logical, coherent and rigorous manner. Evidence is examined critically. Different disciplines vary in the types of evidence. To support how a poem conveys emotions, the speaker may cite multiple pieces of textual evidence to support the interpretation. In history, the speaker may use historical facts to support a position that began as an opinion. In maths, the speaker will use a mathematically relevant basis to support their intuition. Attention will be paid to the quality of the claims – how well support? Is the evidence good? Is it sufficient? Authorative? Relevant? Unbiased?
NB Recurring, familiar events and activities that take place at a certain time, in consistent ways and for consistent purposes will ensure that all students know how to participate in the conversation.
There is a lot of information around accountable talk on the internet. The link below has a video that captures the essence of students using accountable talk. It also has the anchor chart and featured image that I have used for this blog page.
Accountable talk will require lots of modelling and a slow, co-construction of sentence starters and guidelines. It links beautifully with many pedagogies and frameworks already being used in schools, such as Philosophy or yarning circles.
I also love how accountable talk can be used for any year level and in any learning area. Imagine the power of conversation that students can participate in if accountable talk has been fostered from the early years of schooling.
I love the phrase – a thinking curriculum. Accountable talk uses a gradual release model and builds to students doing the thinking and the talking…..about the curriculum with which they are learning.
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So much for my 30 posts in 30 days. Unfortunately, work commitments beat me this week and I am now about five posts behind my goal. My oral language workshop still prevails so tonight’s blog will look at Grand Conversations in Primary Classrooms, an article published as a part of the Capacity Building Series. This series was produced by the Literacy and Numeracy Secretariat to support leadership and instructional effectiveness in Ontario schools and the article I am referring to is 18th in the list.
I was introduced to the article when Lyn Sharratt (co-author of Putting Face on the Data) visited Metropolitan schools in February this year. Lyn chose an activity whereby each table group member was assigned a section to skim and scan in the article and then jigsawed to share the information gleaned.
Oral language is the foundation for the complex literacy skills that are critical to a child’s success in today’s knowledge society (Grand Conversations in Primary Classrooms, p.1) . The article begins by exploring the difference between gentle inquisitions (one talk pattern where the teacher is in charge and builds on a series of questions and answers) compared to Grand conversations (authentic, lively talk about text that has the potential to foster higher-level comprehension and student’s attitudes to reading). The talk pattern of a grand conversation is conversational where students and teachers exchange ideas, information and perspectives. The teacher is a member of the group, stepping in only when needed to facilitate and scaffold conversation.
This methodology works beautifully when you consider: who is doing the thinking? who is doing the talking?
The critical first step is the choice of text – it needs to be stimulating and to support a grand conversation. It needs to be sufficiently challenging and multi-layered. For non-fiction, choose a text that presents content clearly and at times provides strong visual support. For a fiction text, choose books with interesting plots and characters, detailed descriptions and dialogue. Poetry and wordless picture books are also great choices. The limited text of wordless picture books requires students to infer, make predictions and to express personal thoughts, feelings and opinions. The images must be visible to all participants.
Students need to taught how to consider the ideas presented in the text, how to share and defend their ideas and opinions and how to build on and question ideas of others. Initially, teachers may initiate conversations, ask big questions and model appropriate discussion skills. They will also need to be ready to step in and offer new questions or prompts when redirection is necessary. A teacher’s role will include assisting students to: accept different ideas and opinions; practise turn-taking and discussion techniques; and encourage the quieter students to have-a-go.
Teachers can model and students can practise these skills both in a whole class and small group situation. Anchor charts can be developed with students to record rules and norms for productive conversations.
The teachers’ role in a grand conversation shifts form discussion director to discussion facilitator to participant as students gain greater independence and proficiency (again this fits with the gradual release of responsibility).
• What do you think the author wants us to think?
• How would the story be different if another character was telling it?
• How does the author show his point of view? Do you agree?
• What do you think was the most important thing that happened?
• What was something that confused you or that you wondered about?
• How did you feel about what happened in the story? What made you feel that way?
• Are you like any of the characters?
In what ways?
• Did you agree with what (character’s name) did? Why?
• What do you think will happen next?
What do you think (character’s name) will do? What would you do in the same situation?
• Is there someone in the book you’d like to talk to? What would you say? What makes you want to say that?
Preparing Students for Discussion
The paper lists and describes a number of engaging and innovative strategies to support student thinking about the text prior to classroom discussion. These include:
Grand conversations have many names: literature circles, books clubs, reading response groups and literature discussion groups. Students come together to talk about text they have read or had read to them in order to answer to questions the text as they look at it from different points of view.
My apologies to the authors: the following lists are basically word for word of what is in their article:
Small groups of students (about three) can come together around a common theme or big idea using one or more texts.
Teacher selects books for these small-group discussions based on student needs and interests.
After listening to “book talks” given by the teacher, students may choose the text for their group discussion by holding a vote.
Before beginning the discussion teacher may want to introduce students to various conversational roles – such as discussion director, illustrator, word wizard and connector – as a way of scaffolding student-led conversations.
NB Goal is for students to participate in grand conversation without taking on a specific role.
Heterogeneous small groups that support discussion focused on learning about a concept.
Purpose is to have students build an understanding of a concept through the dialogic exchange of facts and information (Guthrie & McCann, 1996).
Goal is to ensure that each student leaves the group with a clearer, more thorough and more accurate understanding of the target concept.
Multiple concept-related texts, at varying levels of reading difficulty, are provided by the teacher.
Each student reads their selected text, either independently or with a partner, for the purpose of gathering information about the topic under discussion.
Students then bring their information to the circle where the information is shared, clarified, extended and debated in order to co-construct a deeper and more elaborate understanding of the concept.
Being able to think deeply, articulate reasoning and listen purposefully increases student engagement. Grand conversations allow students to be leaders in a collaborative process that promotes the discussion of text in a meaningful way. This supports higher-order thinking skills and increases student learning and achievement. Who is doing the thinking and who is doing the talking in your classroom?
As we move through the gradual release of responsibility, it should move from teacher to student!
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