In my last blog post on listening effectively as teachers, I tacitly assumed that we had already identified what to listen to. In reality, this is not an easy task. A typical classroom is a flurry of activity and commotion, and a teacher can’t possibly be aware of everything that is happening, let alone respond. Our own backgrounds, experiences and beliefs also influence what we deem worthy of our attention.
Like listening, noticing is an essential skill in a teacher’s repertoire. John Mason has written extensively about the discipline of noticing. Early in his book ‘Researching Your Own Practice: The Discipline of Noticing’, Mason writes that ‘the mark of an expert is that they are sensitised to notice things which novices overlook’ (p. 1). With practice and reflection, our ability to notice can be improved.
In this (rather long) blog post, drawn from the literature, I describe the components of teacher noticing and with a focus on student mathematical thinking. I also look at the differences between novice and expert noticers, what we might do to improve our noticing, and what my takeaways are.
Components of teacher noticing
Teacher noticing in mathematics education has been well studied over several decades. Two of the most prolific contributors to the field, Elizabeth van Es and Miriam Sherin, recently summarised the different lines of research (2021). Somewhat surprisingly, one debate is exactly what comprises teacher noticing, although there seems to now be a growing consensus that it involves three interrelated dimensions:
- Deciding what to notice
- Deciding what it means
- Deciding what to do with it
Deciding what to notice requires us to attend, that is, to look for particular types of information, events and interactions, and to identify which are noteworthy and which can be disregarded (van Es and Sherin (2021) and references therein). There is a vast number of phenomena that can be noticed in a learning environment, including student behaviour, classroom climate, teacher pedagogy, modes of dialogue and discussion, and mathematical thinking.
Deciding what it means requires us to interpret by ‘drawing on our knowledge and experiences to make sense of what is observed’ (van Es and Sherin, 2021). For teachers, that includes knowledge and understanding in many domains including: content, teaching, pedagogy for teaching specific content, and individual learners and their contexts.
There are many parallels between interpreting and listening. As described in my last blog post, Brent Davis (1994, 1997) identified three different forms of listening: evaluative which seeks a particular response; interpretive which seeks to make sense of an individual’s thinking; and generative which seeks to transform both individual and shared understandings. In a similar way, van Es and Sherin identified three general stances taken when interpreting: descriptive in which the teacher describes what they noticed; evaluative in which the teacher appraises what they noticed (e.g. as correct or incorrect); and interpretive in which the teacher makes inferences about what they noticed. Like listening, interpreting may require us to probe further to make sense of the student’s thinking.
Deciding what to do with it requires us to choose how to respond to what we have noticed. This phase is more ambiguous in the literature. In their original framework, van Es and Sherin restricted teacher noticing to attending and interpreting. In their revised framework, they added a third dimension called shaping—‘constructing interactions, in the midst of noticing, to gain access to additional information that further supports their noticing’—which begins to blur the boundaries between noticing and teaching. Jacobs et al (2010) focus on the intended response and do not include the execution in their conceptualisation of noticing.
In deciding how to respond, teachers have a range of next actions to choose from. One valid action is also to do nothing. It may be, that having attended to and interpreted a phenomenon, we decide that it is not productive to pursue.
Noticing student mathematical thinking
Noticing and making sense of student thinking, and then deciding how those ideas might be used to shape the learning experience, is both complex and challenging. Stockero et al. (2017) wrote an informative paper focused on in-the-moment teacher noticing of student mathematical thinking that is worth discussing in more detail.
The authors distinguish between noticing within and noticing among instances of student thinking. Noticing within may occur in dialogue with a student or when analysing their written work. The teacher does not need to identify the event or interaction to analyse. Instead, they are attending to what is happening within the instance of student thinking. In contrast, noticing among occurs when a teacher selects from a range of instances by identifying those that they think are significant or intriguing in some way.
Both types of noticing are important in classrooms, and are especially pertinent to orchestrating mathematical discussions using the five practices described by Smith and Stein (2011):
- Anticipating possible student thinking
- Monitoring actual student thinking — noticing within
- Selecting a subset of student thinking to share — noticing among
- Sequencing instances of student thinking to frame the discussion
- Connecting different instances of student thinking to highlight key mathematical ideas
Stockero et al. (2017) highlight that noticing among also includes the work of discriminating between instances [my emphasis] of student thinking ‘since it neither possible nor desirable to build on every student contribution during a lesson’. The complexities of noticing within, among and between student mathematical thinking, and then building on them to shape the learning experience, requires skilful action from us as teachers. To help, the authors have devised a framework with tools for identifying Mathematically Significant Pedagogical Opportunities to Build on Student Thinking (MOSTs).
MOSTs have three characteristics that are encapsulated in the name. A MOST must be mathematically significant, provide a pedagogical opportunity, and be an instance of student mathematical thinking. To determine whether an instance of student thinking is a MOST, the authors use two criteria per characteristic:
- To be characterized as student mathematical thinking, the instance must have student mathematics and a mathematical point. That is, there must be sufficient evidence within the observable student action (statement, gesture, written work) to be able to infer the mathematics that the student is expressing, and we must be able to associate it with a mathematical idea.
- To be considered mathematically significant, the instance must be appropriate and central. To be appropriate, the mathematical point should be one that is accessible but not yet understood by most students in the class. Centrality is determined by whether the mathematical point is related to a learning goal.
- To provide pedagogical opportunity, the instance must have the potential to create an opening or an intellectual need for students to understand the mathematical point, and with the right pedagogical timing during the learning experience.
An illustration of using the MOST framework in a classroom setting to support noticing within and among is given in Stockero et al. (2017), and helps us to understand how this might be enacted.
[Side notes: To maintain clarity, I’ve made two stylistic choices in this section: (1) To use the acronym, even though I’m not a fan of acronyms in general or this one in particular, as it’s not immediately obvious how to avoid the cumbersome phrase (2) Not to continually cite the authors, even though the description above draws heavily on Stockero et al. (2017) which also cites their earlier work in Leatham et al. (2015).]
Improving our noticing
Teachers notice all the time. In-the-moment noticing rarely affords us the time to fully analyse an instance of student thinking, and we often make instinctive decisions on the fly when selecting and building on contributions. While noticing skills can improve with experience, they also develop with sustained practice and reflection.
In their research over more than twenty years, van Es and Sherin have used video clubs, in which teachers watch and discuss videos from their own classroom, to improve participants’ noticing. They’ve shown that teachers can learn to shift their attention to the substance of mathematical thinking, and to move from primarily describing and evaluating to consistently interpreting observations.
A study of 131 prospective and practicing teachers by Jacobs et al. (2010) examined whether expertise in noticing student mathematical thinking increased in relation to two factors: experience and sustained professional development. The prospective teachers were undergraduates; the practicing teachers had a range of teaching experience (between 4 and 33 years) and were further grouped by time (0, 2 or 4 years) in the professional development program. Participants examined two artifacts of K-3 classroom practice (a classroom video clip and a set of written work), and then wrote responses to prompts about attending, interpreting, and deciding how to respond to student mathematical thinking.
Prospective teachers struggled with all three elements of noticing. Of this cohort:
- 42% provided evidence of attention to the details of children’s strategies
- 47% provided some, albeit limited, evidence of interpretation of children’s understandings
- 14% provided some, albeit limited, evidence of planning their next move on what they had learned about the children’s understandings
None of the prospective teachers demonstrated robust evidence of interpreting or deciding how to respond. Practicing teachers who had not yet started the professional development program but had more teaching experience improved in attending (65%) and interpreting (16% robust; 61% limited) but still struggled with deciding how to respond (3% robust; 23% limited). The noticing of teachers in the professional development program improved markedly, although a significant proportion still showed a lack of evidence in responding to student thinking. The details are more nuanced, of course, and I encourage those seeking more explanation to read the paper.
The study serves to reinforce that teacher noticing of student mathematical thinking is both complex and challenging. As an educator of prospective teachers, the results highlight that my students need multiple opportunities to practice and reflect, and that I need patience as they develop and grow. Jacobs et al. (2010) make three points that I want to keep in mind:
- ‘Attending to children’s strategies requires not only the ability to focus on important features in a complex environment but also knowledge of what is mathematically significant and skill in finding those mathematically significant indicators in children’s messy, and often incomplete, strategy explanations.’ (pg. 194)
- ‘To interpret children’s understandings, one must not only attend to children’s strategies but also have sufficient understanding of the mathematical landscape to connect how those strategies reflect understanding of mathematical concepts.’ (pg. 195)
- Deciding how to respond is particularly complex, as there is a range of teacher moves to choose from. For example, we may decide to: ask a clarifying question, probe a student’s thinking, introduce a tool, suggest an alternative strategy, encourage a different representation, pair students up to help each over, select the next problem, to name just a few avenues available to us.
The authors also list six growth indicators that I’ve included word-for-word below:
- A shift from general strategy descriptions to descriptions that include the mathematically important details
- A shift from general comments about teaching and learning to comments specifically addressing the children’s understandings
- A shift from overgeneralizing children’s understandings to carefully linking interpretations to specific details of the situation
- A shift from considering children only as a group to considering individual children, both in terms of their understandings and what follow-up problems will extend those understandings
- A shift from reasoning about next steps in the abstract (e.g., considering what might come next in the curriculum) to reasoning that includes consideration of children’s existing understandings and anticipation of their future strategies
- A shift from providing suggestions for next problems that are general (e.g., practice problems or harder problems) to specific problems with careful attention to number selection.
Many elements need to come together to improve our skills in attending, interpreting, and deciding how to respond. Jacobs et al. (2010) caution against addressing these skills independently and in sequence, and suggest they should be developed mindfully in an integrated way.
I don’t have a snappy conclusion to this blog post, except to say that the literature gives me lots to think about further. I already know that I will be approaching the task of helping pre-service teachers with a lot more understanding of both the structure of noticing and the time that it takes to develop the skill. Perhaps down the track I will summarise some of my observations and reflections of this endeavour.
Jacobs, Victoria R., Lisa L. C. Lamb, and Randolph A. Philipp. 2010. “Professional Noticing of Children’s Mathematical Thinking.” Journal for Research in Mathematics Education 41 (2): 169–202. https://doi.org/10.5951/jresematheduc.41.2.0169.
Mason, John. 2001. Researching Your Own Practice: The Discipline of Noticing. London: Routledge. https://doi.org/10.4324/9780203471876.
Smith, Margaret Schwan, and Mary Kay Stein. 2011. 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA : Thousand Oaks, CA: National Council of Teachers of Mathematics ; Corwin.
Stockero, Shari L., Rachel L. Rupnow, and Anna E. Pascoe. 2017. “Learning to Notice Important Student Mathematical Thinking in Complex Classroom Interactions.” Teaching and Teacher Education 63 (April): 384–95. https://doi.org/10.1016/j.tate.2017.01.006.
van Es, Elizabeth A., and Miriam G. Sherin. 2021. “Expanding on Prior Conceptualizations of Teacher Noticing.” ZDM – Mathematics Education 53 (1): 17–27. https://doi.org/10.1007/s11858-020-01211-4.