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.

Accountable Talk

I have just finished reading through a paper entitled Accountable Talk Sourcebook: for classroom conversation that works (Michaels, O’Connor & Williams Hall, 2013).  Again, I was introduced to accountable talk in the Metropolitan sessions led by Lyn Sharratt in February this year.

accountable talk 2

Anchor chart from https://tackk.com/fzmr3b

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

Shared as a part of Lyn Sharratt’s presentations to Metro schools

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.

accountable talk for littlies

from Susan Jones (thank God it’s First Grade blog)

Anchor chart

Accountability to the learning community

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.

Accountability to accurate knowledge

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.

Accountability to rigorous thinking

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?

How to set up accountable talk in the classroom

  1. Begin with a focus on academic purposes –  this is critical. As a teacher,  what are the academic goals for the lesson? What are the key concepts students need to learn?  What are the big ideas they need to grapple with? How do the ideas linked to work already done?

    accountable talk poster

    Anchor chart from https://tackk.com/fzmr3b

  2. Ask what kind of instructional talk will support the accomplishment of those purposes – it needs to provided points of entry and opportunities for engagement for all student
  3. Plan – there needs to be a clear introduction of the task, time for student activity and a clear recap of the point, the big ideas that have been discussed and the new understandings that have been arrived at
  4.  Consider the best class format to facilitate these academic goals – whole group, small group or partner work? Should the topic be set up and then stopped at a certain point for a partner discussion?

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.

My thoughts

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.

Poster from Cheryl

Thanks to Cheryl for sharing this with me.