Putting students in the driver’s seat of their learning

The true purpose of getting an education is to apprentice students into becoming their own teachers. We want them to be self-directed, lifelong learners , and to have curiosity about the world. We want them to have the tools they need to formulate their own questions, pursue meaningful answers, and through metacognition, be aware of their own learning in the process.                   (Hattie, Fisher and Frey, 2017, p176)

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.

Why do I need this?

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.

Venn diagram

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:

  • should be selected to encourage connections
  • should be of higher complexity with higher difficulty
  • may not have clear entry points
  • may have multiple steps
  • may not have one correct solution – students have to make judgments about the best solution and justify their thinking with evidence
  • incorporate maths ideas that can be applied in other learning areas
  • may take more than one lesson to solve
  • should have relevancy
  • should be developmentally appropriate.

Metacognition, the ability to think about our thinking, promotes transfer learning. Palinscar (2013) described metacognitive awareness as having three parts:

  • knowledge about our learning selves
  • an understanding of the task demands and necessary strategies to complete it
  • the means to monitor learning and self-regulate

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.

Age appropriate pedagogies conceptual framework

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.

Problem-solving teaching in mathematics

It is similar to the design thinking necessary for the new Australian Curriculum learning area of Technologies.

Design thinking in AC Technologies – thanks to Grant Smith (one of my regional colleagues)

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.

How to design a school garden?

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. :

  1. predicting – students use knowledge of mathematics and the information provided to predict what is happening and to decide what mathematics may be needed to solve the problem.
  2. clarifying – students make lists of information that they might need to solve the problem – unfamiliar vocabulary, known facts and necessary information to solve the problem.
  3. solving – students use their problem-solving strategies to find one or more solutions to the problem and are encouraged to use multiple representations.
  4. summarising – this is a time for individual self-reflection  that includes justifying the solution, reflecting on the effectiveness of the selected strategies and evaluating their participation.

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.

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.

Going beyond surface learning in mathematics

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

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.