Monthly Archives: April 2017

Going beyond surface learning in mathematics

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Too often, learning ends at the surface level. (Hattie, Fisher and Frey, 2017, p131) How do we go beyond surface learning and into deeper learning?

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:

  1. Explanations that are mathematical arguments and include justifications.
  2. Errors as opportunities to reconsider problems from a different point of view.
  3. Intellectual autonomy that recognises participation.

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:

  • be complex enough that students need to work together.
  • allow for argumentation.
  • include sufficient language support.
  • provide individual and group accountability.

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.

The thinkboard can be used as pre-assessment, formative assessment or collaborative learning

An example of an annotated think board

Using the CSA model to develop use of the Multiplication and Division Triangle (a strategy for problem solving)

 

 

 

 

 

 

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.

 

NB The main reason I blog is to give teachers (who are time poor) access to current research texts. The blog is constructed from my notes which are only snippets of the chapters and are my impressions. I recommend purchasing and reading the book for the in-depth information and resource support.
Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics What works best to optimize student learning Grades K-12. Thousand Oaks, CA: Corwin.

Surface learning – setting the foundation

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The instructional goals of any lesson need to be a combination or balance of the three phases of learning – surface, deep and transfer. An expert teacher knows when and how to help students move from surface to deep and transfer learning. Despite whatever phase of learning the two key reminders are:

  • Learning intentions and success criteria need to be clearly signalled.
  • The instructional strategy with the highest importance is student 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).

One of my favourite references. Number talks helping children build mental math and computation strategies by Sherry Parrish

Making number talks matter by Cathy Humphreys and Ruth Parker

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:

  • be used as the sole means of teaching mathematics.
  • consume a significant part of the instructional time available for learning.
  • always have to be used at the beginning of a lesson.

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.

The CSA model used for problem solving strategies

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.

Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics What works best to optimize student learning Grades K-12. Thousand Oaks, CA: Corwin.

 

 

Mathematical tasks and talks

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What were your mathematics lessons like as a student? What would students say about your current mathematics lessons?

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, 2017, p77

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

  • When students need to build automaticity
  • After mastering conceptual understanding and learned thinking strategies and procedures
  • If limited to this quadrant – learning isn’t going to be robust

High Difficulty Low Complexity

  • Builds perseverance
  • Problems or exercises that extend current knowledge to a more difficult situation
  • Students work independently before consulting with a peer, then returning to problem individually a second time

Low Difficulty High Complexity

  • Extends understanding to more complex situations
  • Supported by students working collaboratively and justifying their thinking

High Difficulty High Complexity

  • Pushes students to stretch and extend their learning

 

Hattie, Fisher and Frey (2017) mention a number of characteristics that assist students to build understanding and confidence in mathematics. These include:

  1. Teachers using questioning and prompts to build understanding.
  2. Mistakes being valued and seen as opportunities.
  3. Students considering different approaches and noting their similarities and differences.
  4. Lessons having an element of productive struggle which is accompanied by perseverance.
  5. Rich discourse.

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.

Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics What works best to optimize student learning Grades K-12. Thousand Oaks, CA: Corwin.

Teacher clarity

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Visible learning for Mathematics

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:

  • are critical for students of mathematics.
  • should be shared with students.
  • should be referred to throughout the lesson.
  • do not always have to be shared at the outset of a lesson as they can be withheld until a learning experience has occurred.
  • can be used to compare with what they have learned from the lesson.

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.

Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics What works best to optimize student learning Grades K-12. Thousand Oaks, CA: Corwin.

Making learning visible in mathematics

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Graduation – Masters of Education 2005

 

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:

  • helps teachers identify the attributes and influences that work.
  • helps teachers better understand their impact on the learning of their students.
  • helps students become their own teachers.

    http://www.jjfbbennett.com/2016/04/visible-learning-quotes.html

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:

  • designing mathematical instruction around rigorous mathematical tasks.
  • monitoring student reasoning.
  • providing many opportunities for skill- and application-based experiences.

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…

Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics What works best to optimize student learning Grades K-12. Thousand Oaks, CA: Corwin.

Visible Learning for Mathematics

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It has been a year since I have visited my site. In that time, I have read and used a number of texts but just haven’t had time to share. My goal this Easter was to re-visit my blog and share a current text. A copy of  Visible Learning for Literacy by Hattie, Fisher and Frey was given to all Queensland State School principals at a recent conference. As pre-reading for a professional learning session that I was conducting, I was given a couple of chapters to read and make reference. This pre-reading whetted my appetite and when I found that these authors had also produced a text called Visible Learning for Mathematics, I immediately bought both.

Visible learning in Mathematics

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.

 

Hattie, J., Fisher, D., & Frey, N. (2017). Visible learning for mathematics What works best to optimize student learning Grades K-12. Thousand Oaks, CA: Corwin.