Inquiry and online learning

posted 27 Jun 2020, 07:18 by Andrew Blair   [ updated 1 Jul 2020, 03:21 ]

Inquiry and online learning
During the lockdown, I was adapting the Inquiry Maths workshop I run for teachers to make it suitable for an online session with trainees at a university. In the workshop, the participants and I go through an inquiry - from posing questions about the prompt and noticing its properties to final presentations of the lines of inquiry. The whole process is collapsed into between 30 and 60 minutes depending on the time available, with me 'coming out' of the inquiry on occasions to interject theoretical and practical observations. I wondered if the approach could work online. This set me to thinking about a broader question: What kind of online environment would make inquiry learning in the virtual classroom as effective as it is in the real one?

What do we miss online?
In the real classroom, the inquiry teacher learns about how her students think and feel during the inquiry process. She learns about their habits of mind and their strengths and weaknesses as learners. She learns about how they interact with others, how confident they are feeling, how much guidance they might need. In short, she knows her students as individuals and as a collective. At all times in the classroom she is making assessments about students' needs as individual and collective learners: What level of cognitive challenge is appropriate? What amount of emotional support is required? The power of inquiry lies in its child-centred philosophy. With such a wealth of knowledge, the teacher can facilitate deeper learning and engage with more motivated learners than is possible in the traditional instruction-led classroom.
However, it is true that a diet of instruction and drill is easier to deliver online than the nuanced practice of the inquiry teacher. In the inquiry classroom, the teacher responds to students by, for example, gently coaxing from a hesitant child the thought that is on the tip of her tongue or skilfully sequencing contributions to build ideas into a coherent concept. This is very difficult to accomplish on an online platform. How do we spot the breakthrough made by a child who is normally content to follow others' directions? How do we reproduce the buzz of excitement as students spontaneously interact to share questions or co-construct conjectures? If we know the class, we can fall back on preconceived notions and invite generally confident individuals to lead online inquiries. However, this seems the antithesis of inquiry teaching.
The same is true for adults. I want to replicate the teacher's inquiring disposition during a workshop.Furthermore, I want the participants to be aware of the way I direct or guide the inquiry without appearing to do so. How is 
the inquiry teacher able, in the words of Dewey in the The Child and the Curriculum, “by indirection to direct"? When I ran a workshop in Bratislava, the participants (who were, admittedly, vastly experienced inquiry teachers) commented on how I had achieved my aims for the workshop without seeming to impose them. We reflected on how I had laid emphasis on one contribution or question, which I judged to have greater potential, than on others. I might, for example, have discussed it in greater depth or linked it to another observation and, thereby, influenced a particular line of inquiry. I doubt that an online environment is sensitive enough for an observer to notice the nuanced practices of the inquiry teacher, even if the teacher were able to orchestrate the inquiry in the normal way.
Inquiry as research
Of course, there are different conceptions of inquiry learning. My own conception of mathematical inquiry focuses on interweaving individual and collective exploration with inductive and deductive reasoning. The teacher offers graduated support and guidance depending on the nature of the inquiry and the experience of the individual or class. This is an inquiry process responsive to 30 individuals in which each student can learn from another, join a collective search for examples and counter-examples or set off on their own line of inquiry.
Another conception of inquiry is based on research. This form of inquiry is more relevant and authentic to the social sciences than mathematics. Online access for individual students would seem to be a prerequisite for this type of inquiry. Of course the two conceptions are not mutually exclusive. In mathematics, the number line inquiry, which bridges between number and algebra, would be enhanced by research into the origins of algebra in the Arabic world. However, students would not be learning the specific form of mathematical inquiry in this way; rather, they would be learning about the history of mathematics through carrying out an historical inquiry. Using algebra as a mathematical tool (and, thereby, being able to understand the advances made in ninth century Baghdad) requires participation in mathematical inquiry.
How could we replicate classroom inquiry online?
The sophisticated interaction, flexible collaboration and immediate feedback (from teacher to student and from student to teacher) of the inquiry classroom are difficult to replicate in the blunt online environment. When lessons went online at the start of lockdown, many teachers had to adapt platforms designed for business meetings in which the highest level of interactivity amounts to sharing a screen or typing a message in a chat box. Even though it is possible to use additional tools alongside the platforms (such as tools that give students the opportunity to interact with a slideshow, type ideas onto a bulletin board and reply to those of their peers or see others' inquiries in progress on a shared document), online environments are generally more conducive to telling about rather than inquiring into
That is certainly the case in higher education. The market leading platform promotes the transmission of knowledge and severely limits the possibility of student agency. For example, switching from one breakout group to another, which is one of the very few ways a student could take an independent decision, is contingent on the lecturer setting up breakout groups in the first place and giving permission for students to move from one to another.
If the virtual inquiry classroom were to approach the sophistication of the real one, it would give students the opportunity to direct their own learning. They would participate in the design of the online environment by being involved in setting permissions for various functions (although, of course, the teacher would only sanction the settings in light of safeguarding concerns). And one last point: the platform and all its functions would be free. Currently hard-pressed state schools cannot afford to pay for the extra functionality provided by online tools. The social inequality and related educational disadvantages that have been exacerbated in lockdown should not be a reason to deny students remote access to learning through inquiry.

Andrew Blair
June 2020

Curriculum, knowledge and inquiry

posted 28 Mar 2020, 04:36 by Andrew Blair   [ updated 15 Aug 2020, 08:04 ]

Curriculum, knowledge and inquiry
The curriculum has become the focus of developmental work in schools throughout England since September 2019. That is because the curriculum looms large in Ofsted's new inspection framework. Schools must provide a broad and balanced curriculum so that students have the opportunity to learn about the "best that has been thought and said." Ofsted elaborate on curriculum models in their research document, which has been published to provide evidence in support of the new framework. Here we come across the old dichotomy between knowledge and skills. In the debate about curriculum, inquiry teachers are often perceived to be on the back foot. Opponents characterise inquiry as skills-based, anti-knowledge and even hostile to the very notion of a set curriculum. None of these are true.

Of the 
three types of curriculum defined in Ofsted's research document, inquiry is most commonly associated with 'skills-led', occasionally with 'knowledge-engaged' and not at all with 'knowledge rich'. According to Ofsted, the 'skills-led' curriculum uses 'generic knowledge' to promote the development of skills, such as resilience, perseverance and a growth mindset. Inquiry seems to fit this definition because of its supposed emphasis on process over content. Teachers might also see inquiry as compatible with a 'knowledge-engaged' curriculum in which skills are developed once teachers have passed on the requisite knowledge. In this view, the only place for inquiry is after instruction. In Ofsted's third and final type of curriculum, there is no place for inquiry at all. Indeed, the traditional 'knowledge rich' curriculum, which involves the transmission and mastery of a body of subject-specific knowledge, is considered the opposite of inquiry learning. However, and perhaps surprisingly, inquiry turns out to be more knowledge rich than the traditional classroom.
In his classic text The Child and the Curriculum (1902), Dewey argues that the child and the curriculum are two limits on a continuous process:
Just as two points define a straight line, so the present standpoint of the child and the facts and truths of studies define instruction. It is continuous reconstruction, moving from the child's present experience out into that represented by the organised bodies of truth that we call studies…. The facts and truths that enter into the child's present experience, and those contained in the subject-matter of studies, are the initial and final terms of one reality. 
(pp. 11 and 12)
For Dewey, the curriculum could only have meaning when it becomes part of personal experience. In his Laboratory School at the University of Chicago, academic subjects were introduced as “organic outgrowths of the child’s daily experience” (Mayhew and Edwards, p. 345). For example, cooking provided an opportunity to learn about measures and constructing simple machines provided a context to learn about ratio and proportion. A curriculum map, therefore, could not replace a student's actual learning journey: “The logically formulated material of a science or branch of learning, of a study, is no substitute for the having of individual experiences” (The Child and the Curriculum, p. 20).
However, Dewey regarded the curriculum 
as having a pivotal role in the design of stimuli for new experiences. As he wrote in Experience and Education (1938), teachers should select situations with the potential to develop experiences into a fuller, richer and more organised forms that progressively approximate to the way adults learn subject-specific knowledge. Previously, in Democracy and Education (1916), Dewey had insisted that subject-specific knowledge underpins an environment in which the teacher develops desirable intellectual and emotional dispositions. He upbraided his followers at the 1928 convention of the Progressive Education Association for conducting their classrooms as though they regarded the "orderly organisation of subject-matter as hostile to the need of students" (Bowers, 1969, p. 11). The curriculum map, Dewey argued, allows the teacher “by indirection to direct” (The Child and the Curriculum, p. 31), serving as a guide to prevent useless wandering and point towards the paths that lead most quickly and securely to the desired forms of thinking.
Dewey’s notion is different to Ofsted’s definition of a skills-led curriculum because experiences are based not on generic knowledge, but on the "facts and truths ... contained in the subject-matter of studies" 
(The Child and the Curriculum, p. 11). Educative experiences, at least in theory, do not only aim to develop transferable skills; rather, they promote thinking and dispositions associated with the separate subjects. 
However, in practice, Dewey proposed a universal hypothesis-testing model of inquiry that transcended subjects. In How We Think (1910), he described his method in terms of a double movement. Firstly, inductive discovery involves bringing together partial and confused empirical observations in a plausible, yet speculative, hypothesis. Secondly, deductive proof, characterised by further observation, leads to a larger pool of data that can be used to confirm or refute the hypothesis. Thus, for Dewey, deduction incorporates the experimental method of induction to check the plausibility of conjectures and inferences arrived at inductively. This weak definition of deduction is a long way from the rigorous logical approach to reasoning and proof required by mathematicians.
While Dewey’s general method was appropriate for his primary (elementary) school, it is difficult to sustain in secondary school. As subject knowledge becomes more specialised and concepts become more abstract - that is, more distant from everyday experiences -, so the method of inquiry takes on domain-specific characteristics. Secondary school inquiry integrates two types of knowledge, which I have adapted from the OECD’s definitions:
  • Disciplinary knowledge: subject-specific concepts, detailed content and procedural steps taken to accomplish a goal.
  • Epistemic knowledge: the understanding of how expert practitioners within disciplines think and inquire and how they create, validate and revise disciplinary knowledge.
This dual nature of knowledge is reflected in the inquiry-based curriculum. The ‘final reality’ at the end of Dewey’s student-curriculum continuum should contain both disciplinary and epistemic forms of knowledge.
In the inquiry classroom at secondary school, students learn disciplinary knowledge in an enriched process rooted in epistemic knowledge. This is the unique strength of inquiry-based learning. In each subject, students learn the concepts, content and procedures of the subject through the subject-specific method of inquiry endorsed by the domain’s community of experts. In the mathematics classroom, for example, inquiry is based on the community’s notions of induction (exploring, conjecturing and generalising) and deduction (explaining, structural reasoning, proving) along with specialised symbols and signs used in mathematical communication. In acquiring disciplinary and epistemic knowledge, students are not simply complying with specific rules and content; rather, they are "joining ‘communities of specialists’ each with their different histories, traditions and ways of working” (Young and Lambert, 2014, p. 101). Academic rigour is combined with intellectual curiosity as students learn when and how it is appropriate to question knowledge and the authority on which that knowledge is based.
An inquiry curriculum is very different to Ofsted’s three types of curriculum. It is not ‘skills-led’ because, rather than comprising 
a list of hybrid skills, it is built on a deep appreciation of subject-specific epistemic knowledge; it is not ‘knowledge rich’ because, rather than force students into a passive acceptance of content, it engages critically with disciplinary knowledge; and it is not ‘knowledge-engaged’ because, rather than artificially separate knowledge (first) from skills (second), it combines disciplinary and epistemic knowledge in a unified process. 
Each discipline has its own form of inquiry through which disciplinary knowledge is created and validated. Indeed, mathematical disciplinary knowledge could not exist other than through a mathematical epistemic way of knowing that is built on the distinctive social and historical inquiry practices of the domain. This is just as true for the secondary school classroom as it is for the university faculty. You cannot have a mathematics curriculum built exclusively on disciplinary knowledge. It ceases to be mathematics.

Andrew Blair
April 2020

Bowers, M. (1969). The Progressive Educator and the Depression: The Radical Years. New York: Random House.
Young, M. and Lambert, D. (2014). Knowledge and the Future School: Curriculum and Social Justice. London: Bloomsbury.

Structured inquiry: a contradiction in terms?

posted 23 Jul 2019, 10:14 by Andrew Blair   [ updated 20 Feb 2020, 02:39 ]

Structured inquiry: a contradiction in terms?
At a recent Inquiry Maths workshop, a teacher suggested that structured inquiry is a contradiction in terms. She defined inquiry as an open process in which students follow their own ideas. Providing an external structure, her comment implied, was incompatible with inquiry. This view, which is not uncommon, has deep roots in educational discourse. It is not only false, but also dangerous.
The idea that classroom inquiry is, by its very nature, open and free has its origins in US progressive schools during the inter-war period. The First World War was a turning point in progressive circles as the doctrine of creative self-expression took hold:
"... [J]ust as prewar Progressivism had given rise to a new educational outlook, one that cast the school as a lever of social reform, so this postwar protest developed its own characteristic pedagogical argument: the notion that each individual has uniquely creative potentialities and that a school in which children are encouraged freely to develop these potentialities is the best guarantee of a larger society truly devoted to human worth and excellence." (Cremin, 1964, pp. 201-202)
The 'new education', as Dewey called it, defined itself in opposition to traditional classrooms and aimed to be everything they were not. Dewey characterised this as Either-Or thinking. The progressives counterposed the imposition of authority from above to the cultivation of individuality, external discipline to free activity and learning from textbooks to learning through experience. As Cremin observes, "In too many classrooms license began to pass for liberty, planlessness for spontaneity, recalcitrance for individuality, obfuscation for art, and chaos for education - all justified in the rhetoric of expressionism" (p. 207).  
Dewey argued that just because traditional teaching emphasised external control of students' experiences and paid little attention to internal child-centred factors did not mean progressives were justified in their focus on the internal to the exclusion of the external. The traditionalists' violation of the principle of interaction from one side, he claimed, did not mean progressives should violate it from the other:
When external authority is rejected, it does not follow that all authority should be rejected, but rather that there is need to search for a more effective source of authority. Because the older [traditional] education imposed the knowledge, methods, and the rules of conduct of the mature person upon the young, it does not follow, except upon the basis of the extreme Either-Or philosophy, that the knowledge and skill of the mature person has no directive value for the experience of the immature. On the contrary, basing education upon personal experience may mean more multiplied and more intimate contacts between the mature and the immature than ever existed in the traditional school, and consequently more, rather than less, guidance by others. (Dewey, 2015, p. 21, my italics)
Dewey continues that children who escape the control of the traditional teacher have only the illusion of freedom if their actions are then controlled by whim and caprice. He concluded that, "Since freedom resides in the operations of intelligent observation and judgement by which a purpose is developed, guidance given by the teacher to the exercise of the pupils' intelligence is an aid to freedom, not a restriction upon it" (p. 71, my italics). 
For the contemporary teacher, Dewey poses the question of how we can guide students towards open inquiry in a way that does not lead to confusion and disillusionment. In my experience most inquiry teachers would approach this task with a taxonomy of inquiry types, building through the levels towards the final aim. Trevor MacKenzie, for example, lists four types of inquiry in his book Dive into Inquiry (see illustration).
Trevor relates how agency shifts from the teacher to students as the year progresses. At the start his class learns through a structured inquiry model before transitioning to controlled inquiry and then pushing on to a guided approach. If the students are ready, Trevor finishes the year with a free inquiry. He explains why viewing inquiry as an open process from the start is problematic: 
"In my experience, without flipping control in the classroom, empowering student learning, and scaffolding with the Types of Student Inquiry, students will not feel as confident, supported, or empowered through their inquiry journey.... By the time students encounter Free Inquiry, they have a variety of tools to help them successfully curb the perceived risks of inquiry." (pp. 27 and 30-31).
The types of student inquiry in Dive into Inquiry are similar to the levels published on this website. Both involve a gradual relinquishing of teacher control as students take a greater role in directing the inquiry. However, in my secondary school classroom it takes much longer to reach open inquiry. Students might be experiencing inquiry learning only in their mathematics lessons, which means it can take up to three or four years before students have the habits of mind and level of self-regulation to inquire independently.
Even then, I am aware that a class is never a homogeneous group. It is inconceivable to me that all students in a class will be engaged in exactly the same type or level of inquiry at any one time. As Kath Murdoch says in The Power of Inquiry, when the teacher begins to offer more choice through, let's say, a guided approach it is common for a few students to struggle to use their time effectively. One of the main reasons for this, Kath contends, is because the teacher has not sufficiently developed routines, systems and structures to support the student who finds self-direction challenging. In such circumstances, with control being yielded to the students, the teacher has to remain "a vital force in 'holding the space' and keeping students connected to their learning, accountable to expectations and, most of all, inspired" (p. 124). Indeed, Kath recently tweeted that inquiry is always guided in some way.

Structure, then, is one means by which teachers reach their aim of open inquiry. Structured inquiry is far from a contradiction in terms. As Dewey says, inquiry might necessitate more structure and guidance than occurs in the traditional classroom. If we see classroom inquiry as a form of free self-expression, we face the danger of educational chaos with its concomitant student disengagement and low levels of learning.

Andrew Blair
July 2019

Cremin, L. A. (1964). The Transformation of the School: Progressivism in American Education, 1876-1957. New York: Vintage Books.
Dewey, J. (2015). Experience and Education. New York: Free Press (first published 1938).
Mackenzie, T. (2016). Dive into Inquiry: Amplify Learning and Empower Student Voice. Irvine, California: EdTechTeam Press.
Murdoch, K. (2015). The Power of Inquiry. Northcote: Seastar Education.

Taking the inquiry out of Inquiry Maths: A reply to Naveen Rizvi

posted 4 May 2019, 12:57 by Andrew Blair   [ updated 3 Dec 2019, 11:41 ]

Taking the inquiry out of Inquiry Maths 
A reply to Naveen Rizvi

It is always welcome when a teacher engages with the Inquiry Maths website. Naveen Rizvi has done just that in her post Teaching an Inquiry Maths Problem and, indeed, gone further by recommending the site to her readers. She acknowledges the depth of the mathematical ideas in the prompts, focusing her post on the percentages prompt. From my point of view her post is interesting because it demonstrates how two completely different models of teaching might develop out of the same idea. Although starting with the prompt from the website, Naveen's intention is to take the inquiry out of Inquiry Maths.
The difference between our approaches is apparent when Naveen writes, “There is so much mathematics that needs to be communicated.” It quickly becomes clear that the teacher is doing the communicating. Naveen proceeds to break the teaching of the prompt into very small steps. The teaching sequence, which maps out all the calculations students might need to manipulate and extend the prompt, is impressive for its thoroughness and detail. Based on Engelman's idea of Direct Instruction, Naveen uses a carefully constructed script and encourages students to generalise the calculations by listing the procedures they should follow to solve problems with the same structure. The presentation of one concept at a time, Naveen claims, excludes the possibility of students forming a misconception. Furthermore, she contends, this will “increase the probability that all pupils can be successful in learning the subject content.” Such an approach, according to Engelman, constitutes logically faultless communication.
The “so much mathematics that needs to be communicated” in Naveen's approach becomes the “so much mathematics that needs to be explored” in my own. The mathematical concepts and procedures might be the same, but the way we want students to interact with them are very different. My view of mathematics follows Polya's in How to Solve It: “Mathematics presented with rigour is a systematic deductive science but mathematics in the making is an experimental inductive science.” The experimental side of mathematics involves questioning, exploring and, as Polya says in Mathematics and Plausible Reasoning, guessing. Yes, mathematics involves small steps, but it also involves great leaps of imagination. 
The inquiry classroom aims to incorporate these two sides of the subject by presenting the prompt, which appears midway through Naveen’s teaching sequence, at the start of the inquiry. The prompt functions as a proto-typical mathematical structure that requires students to specialise and generalise if they are to understand and extend that structure.
Specialising and generalising from 
Mason et al., Thinking Mathematically
The form of generalisation in an Inquiry Maths classroom is different to Naveen's. Rather than the transfer of a set of procedures to a new problem, inquiry involves identifying an underlying structure, creating more examples with the same structure (and consciously transforming that structure), making and testing conjectures and using mathematical language (including algebra) to express generalisations. Mason et al. call this process "the essence of mathematical thinking."
Perhaps the biggest difference between our approaches lies in the relationship between the teacher and student. In Naveen's approach, it is one-way - teacher communicating to student. In the inquiry classroom, students are encouraged to participate (under the guidance of the teacher) in the direction of the inquiry. Rather than follow the pre-programmed steps of Direct Instruction, students and teacher explore together what is known and what is not known and, moreover, what needs to be known to specialise and generalise with purpose. In this way, students begin to understand that mathematics is a tool with which to understand different contexts and over which they can develop control.
Two definitions of 'mathematics'
When I read Naveen's post, I was reminded of Skemp's seminal paper on relational and instrumental understanding – not, I hasten to add, because I want to characterise Naveen's approach as instrumental and inquiry as relational. Rather, I was reminded of Skemp's main point: “I now believe that there are two effectively different subjects being taught under the same name, ‘mathematics’." As in relational and instrumental classrooms, the subject matter of Direct Instruction and inquiry is the same, but, paraphrasing Skemp, the approaches are so different that we must regard them as being based on two different definitions of 'mathematics'. On the one hand, we have a definition of mathematics, at least insofar as the student is concerned, as a logical chain of small steps that limits generalisation to applying a procedure to a new problem; on the other hand, we have a definition of mathematics that interweaves induction and deduction, involves small steps and leaps, and combines specialising and generalising from proto-typical structures.
Now, at this point, teachers who advocate Direct Instruction might argue that secondary school students are 'novices' who are incapable of studying the mathematics that I describe. Only once students have understood the atomised content can they be expected to reason mathematically without cognitive overload. The problem I see in this line of argument (aside from the imponderable of how much content is sufficient to reason like an 'expert') is that when the time comes students will not know how mathematicians reason or, more importantly, why they reason in the way they do. Once the teacher's small steps are withdrawn, the danger is that mathematics will cease to have any independent meaning over which students feel they can exercise control.
Inquiry aims to replace the over-reliance on the teacher that occurs in the Direct Instruction model by giving students agency to act on their questions and observations about the prompt. Inquiry puts students in the position to act like mathematicians (not, nota bene, act as mathematicians). In a paper about fostering communities of inquiry, eminent professor Alan Schoenfeld writes that parallels can be drawn between the mathematics community and the inquiry classroom because "some of the felt experience [of the mathematician] is the same from the students' point of view." Indeed, he continues, "there are remarkably strong parallels between the two environments - despite the fact that one is 'real' and one (by some measures) 'artificial'. What matters, I think, is that both are communities dedicated to exploration and sense-making."

Andrew Blair
May 2019

Positives and negatives of the first inquiry

posted 3 Feb 2019, 04:37 by Andrew Blair   [ updated 2 Apr 2019, 10:34 ]

Positives and negatives of the first inquiry
Ania Maxwell, 2i/c in a secondary school mathematics department, used the area prompt with her year 8 class. The students had middle to low prior attainment. Ania attended an Inquiry Maths conference workshop and decided to try out her first inquiry.
Ania reports that the prompt "brought out so many misconceptions" (see the students’ questions and observations above) that she would not have known about if she "just launched into instruction". However, the emergence of the misconceptions left Ania in a quandary, "How can I expect the students to understand more complex ideas without dispelling their misconceptions?" 
Ania had prepared to use the resource on the website (the rectangle, triangle and circle on squared paper), but "discovered how unconfident my students were even with that task." On reflection, she feels she was not prepared enough and needed more resources to address the different directions the inquiry took. Ania was very glad when a student asked how to work out the area of a circle using a formula so that she could direct the class to follow that line of inquiry and "resort to [her] usual style of teaching".
Overall, Ania says that the class "felt like mayhem" and she felt "stripped of confidence". She was not sure what the class had learned and concludes, "It's going to take some perseverance, more time and a lot more reading to get good at inquiry!" 

Andrew Blair replies:
Thank you very much, Ania. I think this feedback is very important and invaluable to other teachers thinking of using inquiry. As I said in the workshop, closing down the inquiry is advisable if the teacher and class are new to the process. 'Closing down' can mean reverting to your normal teaching style. For example, if you had prepared to focus on the area of the circle, I would have gone with that whatever questions came up. Then I could have thought about how to address the questions in the subsequent lessons. Overall, I think inquiry is very different to normal classrooms and it takes time to create a negotiated culture of learning based on following a line of inquiry. Once the culture is different, however, learning is accelerated with students developing into self-directed learners. The final point I would make comes from Kath Murdoch's The Power of Inquiry: "
Inquiry can be messy. Not in the literal sense (although that too sometimes!) but 'messy' in that, by being more responsive to what students say, do and reveal, there is less the teacher can tightly control from the outset.... Inquiry moves the act of teaching out of the realm of control and authority to one of complexity, nuance and some shared responsibility." I hope you try out an inquiry prompt again.

How to get started with Inquiry Maths

posted 30 Dec 2018, 07:50 by Andrew Blair   [ updated 15 Aug 2020, 01:42 ]

How to get started with Inquiry Maths
This week I received an enquiry from a teacher new to an international school: "I am just about to start using inquiry in my maths classroom. Your website is just what I need. However, it all seems a little overwhelming at the moment. Where is the best place to start in terms of resources?" This brought to mind a similar e-mail from a maths teacher who had looked over the Inquiry Maths website. She liked the ideas and wanted to carry out an inquiry: "The prompts look interesting and I can see how they could develop into inquiries, but how do I start?"
The questions of how and where to start also arise frequently in workshops. As inquiry is so different from conventional maths lessons, it can be difficult to visualise an Inquiry Maths classroom. On occasions, I have been advised to produce videos to show how it's done. However, to me, this is contrary to the spirit of inquiry that I would hope to engender not only in students, but in teachers as well. An inquiry disposition is essential if the teacher is to respond to students' ideas with empathy, curiosity and genuine interest. Inquiry is not a model to be copied; it is a way of being in the classroom - for student and teacher. I would encourage all teachers to find their own way without thinking there is one right way.
Having said that, I know it can seem daunting to conduct your first inquiry. One 
teacher in an Inquiry Maths session described using investigations, but felt that inquiry seems like "another step again." If students are participating in setting the agenda through their questions and responses to the prompt, then teachers new to inquiry can become concerned about meeting specific learning objectives. If students participate in directing the lesson, then they can be fearful of losing control of the lesson. Inquiry requires a complete cultural shift in which learning is co-constructed through promoting student initiative and agency.
Navigating the first few steps towards an inquiry classroom can, therefore, seem daunting. With that in mind, I think the following seven points are worth considering when getting started with Inquiry Maths. 
(1) Choose a prompt linked to your scheme of learning. 
The prompts on the website relate to standard mathematics curricula and can lead into pathways that cover content descriptors. If you are concerned that inquiry learning involves too much ambiguity around content coverage, then you can select process objectives as the aims of the lesson (see this lesson plan for objectives such as devising questions, making conjectures and justifying and reasoning). In my response to the enquiry five years ago, I recommended ignoring content objectives as a 'hindrance'. After all, the more the teacher attempts to address curriculum content, the less space there is for students to inquire. However, this is a luxury most teachers don't have. Now I would counsel teachers to balance the requirements from school authorities with the desire to promote student exploration. As long as students follow a pathway linked to the curriculum (however loosely tied to the specific content in the scheme of learning), then that, for me, is legitimate mathematical activity.
(2) Choose a prompt that you think will intrigue the class. 
The prompt should be set just above the prior attainment of the class in order to generate curiosity and encourage questioning and noticing with which to launch the inquiry. In one school, for example, the percentages prompt was adapted (see the box) for classes with different prior attainment in order to provide the right level of intrigue.
10% of 50 = 50% of 10
40% of 70 = 70% of 40
47% of 74 = 74% of 47
20% of 30% of 40 = 40% of 30% of 20
The percentages prompt was adapted to provide an appropriate level of intrigue to classes that were set on prior attainment (going from lower to higher in descending order).
(3) Inquire into the prompt yourself.
The best lesson preparation is to inquire into the prompt yourself. In that way, you are aware how different inquiry pathways could develop. It also allows you to model an inquiry disposition in the classroom more easily. You might be genuinely surprised when a student notices something you had not considered or curious to see how a student expands on an observation you had discarded.
(4) Decide if you are going to run a structured, guided or open inquiry.
Over the last five years, I have given the difference between levels of inquiry greater importance in Inquiry Maths workshops. Many teachers associate inquiry with free discovery and have a preconceived notion of classrooms in which students do what they want. This is not the case. Students must learn how to inquire alongside learning subject knowledge. For a class new to inquiry, I would advise teachers to plan for structured inquiries, even if some of their students show signs of independence and preparedness to take risks.
(5) Prepare resources related to the prompt.
Inquiry requires careful planning. Although Inquiry Maths prompts draw on a specific area of mathematics, they have the potential to develop along different pathways. If I decide to run a structured inquiry, then I might prepare resources for a single main line of inquiry. In subsequent lessons, I would offer resources to support different pathways that develop out of students' questions and observations. When I talk of resources, I include stems to promote questioning and noticing (see here for examples), diagrams to support explanations, models of different types of cases (that students could use when, for example, practicing a procedure) and further prompts that structure students' activity.
(6) Plan how students will communicate their questions and observations to the class
I take feedback about the prompt in the same way at the start of each inquiry. As individuals think about the prompt, I circulate around the classroom to see what they have come up with. Students pair up (in order to emphasise the collaborative nature of inquiry), agree on a question or observation and then one of them reads it out as we go round the class. I do not necessarily follow a set order, although I might aim to take lower level questions, such as requests for definitions, first and build up to more sophisticated ideas, such as conjectures or even generalisations. I write out all contributions on the board for public inspection and as a record of our initial thoughts. Other teachers have used post-it notes that are stuck up in the classroom for discussion or a Padlet wall that allows students to contribute to ideas on a virtual board. 
One way to give yourself time to prepare after this phase is to take students' questions and observations about the prompt at the end of the lesson before the one in which you intend to run the inquiry. However, this approach runs the risk of dissipating the energy and excitement created through the process. Students can find it difficult to re-engage with their questions and might not find them as immediate and intriguing as they did first time round. One way to encourage the students to take ownership again is to ask them to group the questions and observations into those with similar themes or decide on the ones they think they could answer immediately, in that lesson or over a longer period.
(7) Review the questions and observations
Next I review the students' questions and observations by talking aloud. I might classify them into groups based on similar types of questions (such as what, how and why) or on similar subject matter. I might feel that the class can address questions immediately by giving a definition or demonstrating a procedure. If so, I would orchestrate a discussion with the aim of unearthing such knowledge.
(8) Use the regulatory cards t
o decide what to do next
You might decide not to use the cards in your first inquiry and direct the inquiry yourself instead. However, the aim of Inquiry Maths is to involve students in setting aims and deciding upon the direction of the inquiry. As the cards provide a mechanism for doing that, I would encourage you to try using a set in subsequent inquiries. I note down the cards chosen (by ticking them on the board) and draw lines between them or number cards when students start to select more than one card. If a pair of students propose a different card than others have done, I will ask them to justify their choice. Sometimes, the choice of cards allows you to run a unified class-wide inquiry; sometimes, the inquiry is diverse and multi-faceted. You decide based on the mathematical validity of the cards chosen.

Andrew Blair
December 2018

A reply to Mr Barton

posted 4 Oct 2018, 13:50 by Andrew Blair   [ updated 23 May 2020, 02:56 ]

A reply to Mr Barton
Earlier this year Craig Barton's book How I Wish I'd Taught Maths appeared. Many colleagues told me of their disappointment that Craig (the TES adviser) had decided to listen to a limited field of researchers. A few teachers have even expressed dismay that they had booked to hear Craig talk at one of his frequent conference appearances only to feel deceived by his turn towards cognitive science. I began to hear that Inquiry Maths figures in the pages of the book. With a sense of foreboding I recently got hold of a copy to see what Craig had written. Unfortunately, there is no index in the book, which seems a curious omission for a book with pretensions to synthesise academic research. The references to Inquiry Maths - three in total as far as I can see - come at the beginning of chapter 3 on explicit instruction. It is this part of the book and, in particular, section 3.3 ('When and why less guidance does not work') that I will address in this post. 
It seems that the two Mr Barton podcasts I have been invited to do (the first in 2013 and the second in 2017) bookend a transformation in Craig's thinking. Of course, everyone has the right to change their mind, although I cannot think of such a  dramatic volte-face as the one Craig delivers. However, in the style of a zealous new convertCraig has a one-sided view of history. 
He relates in the book how he used the surds prompt (pp. 97-98). The example is curious in respect to Craig's conversion to explicit instruction. He showed his students the prompt at the end of a unit on surds that presumably involved some of the explicit instruction that Craig now extols. The fact that the students' insights and conclusions were "rarely complete and often erroneous" might say more about the effectiveness of the instruction than it does about inquiry. I would have used the prompt at the beginning of the unit to develop curiosity and questioning that often makes instruction more meaningful by giving it a context that students have helped to create. 
However, the main point I wish to make about the re-writing of history concerns how Craig evaluates the use of the prompt. He claims that the students' discussion and debate and the teacher's corrections and re-explanations wasted time, concluding, rather dramatically, that "it broke my heart." Now, in June 2013 (at the time of the first podcast) Craig's evaluation of using the sequences prompt was completely the opposite. Even allowing for the hyperbole that goes with publicising a podcast, Craig's tweet (below) expresses his enthusiasm for Inquiry Maths.

Unsurprisingly, this does not make it into the book. The reader would have had a more honest picture of Craig's past engagement with Inquiry Maths if it had done.
Before turning to Craig's critique of inquiry, let's get one thing clear that a reader of How I Wish I'd Taught Maths might find hard to believe. Research supports inquiry learning in mathematics. Blazar (2015) concluded that inquiry-oriented instruction is positively related to student outcomes, which "lends support to decades worth of reform to refocus mathematics instruction toward inquiry and concept-based teaching" (p. 27); Bruder and Prescott (2013) report that "the effects of IBL include benefits for motivation, for better understanding of mathematics, and for the development of beliefs about mathematics as well as for the relevance of mathematics for life and society" (p. 819); and Lazonder and Harmsen's meta-analysis (2016) starts by saying that "research has consistently shown that inquiry-based learning can be more effective than other, more expository instructional approaches as long as students are supported adequately" (p. 1).
The final point, which I have placed in italics, is an important one. Inquiry involves different levels of structure and guidance depending on the experience of the class. For Craig, this seems tantamount to cheating. He writes, "Now, of course, we can offer our students support and guidance to help them approach these tasks in a structured and systematic way, but by doing so are we not moving further towards a more teacher-led form of explicit instruction?" (p. 101). My initial response to this question was, "Yes and so what?" However, as I consider it further, the sentence expresses a highly damaging dichotomous view of maths teaching. Either you do inquiry or you use explicit instruct. Apparently, borrowing from the 'other side' negates your approach.
Just this week, a new colleague observed me running an inquiry with a year 7 class. He likened parts of the lesson to explicit instruction. Should I be upset that our inquiry descended into such evils? Certainly not. The context of the instruction had been established by students' questions and observations about the prompt and the situation called for an explanation of indices.
Craig replies that inquiry remains a "less guided approach" that places "greater responsibility in the hands of the student" and leaves students "free to pursue their own paths and lines of inquiry" (p. 100). Yes, but open inquiry in which students are free to choose their own approach is the aim of the inquiry teacher, not the means by which it is achieved. You do not teach students to conduct mathematical inquiry by throwing them into open inquiry. 
The final point I wish to make is a more general one about our aims. Craig acknowledges that "students can and do learn basic skills from these [inquiry] activities, but it will take a lot longer, and with the likelihood of many bumps and bruises along the way" (p. 106). And there's the key. Experiencing bumps and bruises is exactly where our students learn what it means to learn and reason. Even if cognitive scientists try to convince us that teaching can be made 'efficient', the emotional and social aspects mean learning is far more nuanced than the sterile, programmed classrooms envisaged in How I Wish I'd Taught Maths
Furthermore, maths teachers should be in the business of doing more than overseeing the acquisition of basic skills. If we go back to Craig's tweet, we find the emotional and social that he has stripped out of teaching. "Questions and theories were flying," he says. Teachers know how exciting a lesson is when students are engaged as mathematicians in a collaborative and mutually supporting search to answer their own questions or develop their own conjectures. These are the classrooms in which inquiry interweaves the learning of basic skills with the big picture of connecting mathematical concepts.
Craig fears that inquiry is "a bad thing." Well, the most innovative and courageous maths teachers I know use Inquiry Maths prompts regularly. Some teach in international and independent schools; others, like my colleagues and I, teach in comprehensives serving deprived areas in which the vast majority of students are entitled to free school meals. Some teach students who have the highest attainment of their age group; others teach students who have developed a negative attitude towards mathematics. They use their professional judgement to introduce students to the excitement of inquiry. That, Craig, is a wholly good thing.

Andrew Blair
October 2018

Extending an inquiry into the second lesson and beyond

posted 5 Aug 2018, 07:31 by Andrew Blair   [ updated 22 Aug 2018, 02:51 ]

Extending an inquiry into the second lesson and beyond
Teachers have used Inquiry Maths prompts as stimuli for tasks that last one lesson. Through the prompt, they generate one-off deep mathematical discussion or encourage students to connect concepts during a period of exploration. Such approaches go far beyond the majority of maths lessons, which, lamentably, remain limited to learning disconnected 'knowledge' and focus on procedural fluency. However, there is the potential in the Inquiry Maths model for students to extend an inquiry over more than one lesson.
Extending an inquiry has the advantage of developing students' independence and initiative. They have the space to devise and direct their own lines of inquiry. Pursuing inquiry over a longer period also requires perseverance and resilience. These are important attributes, especially when students are required to learn in non-formal settings outside school or with less support in higher education.       

We were reminded this week that extending the inquiry can be difficult. A teacher contacted Inquiry Maths to tell us about his experiences during an inquiry with his year 10 class. The head teacher who was carrying out a learning walk focused on 'stretch and challenge' observed part of the the first lesson. The feedback was extremely positive. The head teacher remarked on the high levels of independence, continuing: "The students were very engaged and commented that they enjoyed this method of teaching because 'it helped them to discuss their opinions and figure out the answer together'. There was certainly a very high degree of challenge for all learners in evidence that was very pleasing to see." However, the teacher was dissatisfied because he could not sustain his students' engagement and motivation into the second lesson. Students had lost sight of the overall aim of the inquiry and the pathways they had decided to pursue at the end of the first lesson did not seem as intriguing second time round. 
Below, we give advice on how to develop an inquiry over a series of lessons when students have yet to develop the ability to direct their own inquiry from one lesson to the next.
Just above
It is important to set a prompt just above the understanding of the class in order to promote levels of curiosity and questioning that sustain prolonged inquiry. If, on the one hand, the prompt is presented to a class simply as a means of applying recently acquired knowledge, then its potential will be exhausted quickly once the knowledge has been applied. If, on the other hand, the prompt contains properties that are intriguing and unfamiliar, then students will require new concepts to understand it fully - and the new concepts can be built into the course of the inquiry.
'Hanging' lessons on questions
When the prompt is set at the right level, students will often ask a varied set of questions. These normally include questions about how to carry out a procedure or the meaning of a concept. The teacher can use the questions to design a structured inquiry over a series of lessons by 'hanging' lessons on a question or collection of questions. This proves to be far more meaningful for students than simply teaching unconnected lessons. Now each lesson has a context (the inquiry) and a meaningful purpose (answering a student's question). In order to give the teacher time to plan the first lesson of a series, students could be invited to pose questions at the end of the lesson immediately before the one in which the inquiry is due to start.
This presentation shows how the teacher has selected specific
questions (highlighted) on which to 'hang' lessons.
One problem about this approach, it might be argued, arises when the class does not ask the 'right' questions to address the mandatory curriculum content. The teacher can avoid this issue by selecting or designing a prompt that creates a 'conceptual field' linked to a particular area of the curriculum. Then students' questions will be on the terrain established by the teacher. Another problem arises when there are insufficient questions to develop a series of lessons (perhaps because the students are new to inquiry). In this case, the the teacher can take it upon herself to structure lines of inquiry related to the properties of the prompt.
Direction and purpose
Whether the teacher structures the inquiry or opts for a guided or open inquiry (see Levels of inquiry), a student should be clear about the direction and purpose of their inquiry at each stage. When students are exploring or testing cases, they are often concentrating exclusively on mathematical procedures. At the end of such 'search' activity, they will often have to re-orientate themselves towards the original aim in order to appreciate how their results link to the direction of the inquiry. The teacher can achieve this reflection (individual or group) by facilitating regular discussions based on the choice of a regulatory card. When the physical cards, which can stand alone or be sequenced into a series of steps, are laid out on the table, they act as a reminder of the direction and purpose of the inquiry. At the end of each lesson, the teacher can summarise how lines of inquiry have developed, particularly by calling on students to present their work in progress. Referring to these presentations at the beginning of the next lesson is an effective way of bridging between lessons.
Maintaining momentum by learning a new concept
One advantage of setting the prompt 'just above' the level of the class is that students will need new concepts to understand all of its properties. For example, in the inquiry The areas of a rectangle, a triangle and a circle are equal, students will often request instruction on working out the area of a circle. Learning a new concept during inquiry acts to boost the students' momentum and provides the impetus to explore new pathways. (See the box below for a description of a series of lessons with the introduction of a new concept as a central part of inquiry.)
Structured inquiry
Andrew Blair reports on a inquiry he developed with a year 8 mixed attainment class. The nature of the class led him to run a structured inquiry, in which he designed activities that addressed the students' questions and comments (above). Levels of motivation remained high during the inquiry because students could relate their learning to the starting points they themselves had created.

Lesson 1 
While the students had carried out inquiries before, the class had a reputation for being 'challenging' with some students having a poor attitude to learning. Nevertheless, in the initial phase of the inquiry, they all listened attentively as each pair posed a question about the prompt or responded to a peer's comment. Before the lesson, I had decided to restrict the regulatory cards I offered the class to these five.
However, when the time came, I judged the students required an immediate focus and handed out a sheet for them to discuss and work on (see two examples below). Students commented on the connection between the areas using the large squares as a unit of measure and the areas using the small squares. At the end of the lesson, the students had found the areas of the rectangle and triangle and had started to make suggestions for changing their dimensions in order to make the areas equal. We also had estimates for the area of the circle (using the large squares) of between 30.5 and 33.5. Students reasoned that the small squares would give a more accurate estimation because there were more whole squares to count.

Lesson 2
I based the second lesson on the questions about whether it is possible to work out the area of a circle and, if it were, how to do so. We started with a discussion of the diagrams below that link the area of the square on the radius to the area of the circle. 
Students realised that the area of a circle must fall in the range 2r2 < A < 4r2The class seemed to have settled on 3r2 before one girl tried to justify 
“slightly more than 3” because “the circle bends towards the outside.” I then introduced the idea of π as a mathematical constant, which we went on to use accurate to three decimal places. The students practised using the procedure of drawing the square on the radius of a circle and multiplying its area by π
Two students who had independently researched the formula for the area of a circle after the first lesson then presented the formula A = πr2. They modelled how to calculate the area of the circle on the worksheet from lesson one by substituting the length of the radius (3.25) into the formula. The area (33.2 accurate to one decimal place) was towards the top end of the estimates from lesson 1, which led to a short discussion about why that might be. As lesson 2 drew to a close, another student presented her dimensions for a rectangle, a triangle and a circle that have the same area (taking π accurate to three decimal places): 
RectangleTriangle Circle
length 15.71, width 20 base 31.42, height 20 
radius 10

Lesson 3
The final lesson of the inquiry started by addressing the two remaining points from the initial questions and comments. The first related to the question about whether other shapes could have the same area. Students selected one of three tasks:
(1) Draw a rhombus, parallelogram and regular trapezium;
(2) Draw the three shapes with equal areas (by counting squares); or
(3) After completing task 2, make one cut to the three shapes and rearrange the pieces to make rectangles with equal areas.
The second point involved the perimeters of the shapes. After I explained why C = πd, students either practised finding the the lengths of circumferences or tried to establish if the shapes with the same areas (introduced at the end of lesson 2) had the same perimeters. The class decided that if the areas of a rectangle, a triangle and a circle are equal, it would be unlikely they would also have the same perimeter. Many in the class wanted to go further and say it was impossible, but no student could establish a solid reason why this might be so. The contention remained at the level of intuition.
The Inquiry Maths model gives teachers the potential to develop an inquiry over a series of lessons. This can be achieved by setting a prompt just above the level of the class, 'hanging' lessons on students' questions, regularly reviewing and reflecting upon the direction and purpose of the inquiry and maintaining momentum by introducing a new concept that supports new lines of inquiry.

Andrew Blair
August 2018

The zone between knowing and not knowing - Part 2

posted 30 May 2018, 03:32 by Andrew Blair   [ updated 30 May 2018, 03:50 ]

The zone between knowing and not knowing
Part 2: Modelling and orchestrating

In part 1, we talked about slowing down the inquiry as the class enters the zone between knowing and not knowing. The slow down occurs so that students can get in contact with the aims of the inquiry (Alrø and Skovsmose) and the teacher can ensure her intent and the students' intent coincide (Zuckerman). In order to achieve contact and coincidence, the teacher has to take on a specific role during the phase of slowing down that might be different to the ones she adopts later in the inquiry. She aims to encourage students to suspend any doubts they may have about operating in “a twilight of shifting and unclear purposes”  (O’Connor et al., p. 119) by creating an open zone in which questions and contributions are treated respectfully and seriously.

Modelling an inquiring mind

The teacher should aim to model the disposition required to be an inquirer and to learn through inquiry:


If we show students what being curious 'sounds like' by regularly and genuinely voicing our own wonderings, we also help teach the art of questioning in a more informal, natural way. The key to fostering an environment where students feel safe to ask questions is to be comfortable with uncertainty ourselves.... Students need to see and hear us in that space, to see and hear our fascinations and uncertainties and finally, to see and hear our willingness to find out when we don't know. (Kath Murdoch, The Power of Inquiry, p. 57)


Modelling an inquiry disposition involves publicly pondering a student’s observation about the prompt or reflecting out loud on the meaning and implication of a question. The inquiry teacher holds back from evaluative statements in favour of seeking clarification and extending ideas. Praise for the depth and mathematical validity of a question can be communicated by expressing interest and musing over a student's contribution during a class discussion. A comment such as “that’s an interesting idea” is preferable to giving overt praise which might interrupt the class discussion and reinforce the impression of the teacher as an authority figure. In this stage of the inquiry, the teacher aims to promote a symmetrical relationship in which she stands as a learner of students’ initial ideas and starting points.

Orchestrating discussion

The inquiry teacher orchestrates productive discussions by giving students time to construct responses to a prompt and then by expecting all students to have something to contribute. While allowing students to ‘pass’ when it comes to their turn, she is aware of the students who regularly opt out and gives them extra one-to-one support before the next whole-class session. The discussion develops in a sequence from basic contributions to sophisticated reasoning:

  • Questions about defining terms;

  • Descriptions of relevant procedures;

  • Explanation of underlying concepts;

  • Conjectures based on empirical features of the prompt (pattern spotting leading to generalisation);

  • Conjectures based on mathematical structure (specialisation in the prompt as representative of a class of objects).

As the students are called on for their turn, the teacher links up ideas and speculates about other connections.

During the class discussion, the teacher oversees the use of two forms of mathematical speech (O’Connor). Through exploratory speech, ideas are generated, conjectures tentatively proposed, and partially developed ideas discussed. In this inductive phase, the teacher might overlook imprecise speculations and might pass over errors in calculations or meet them, not with direct corrections, but with counter-examples that provoke fresh thinking: “When we are in the heavy lifting and framing stages of developing new ideas, stopping to correct every flaw is disruptive to the real work” (O’Connor, p. 177). However, the teacher asserts the deductive side of mathematics when reviewing and summarising the exploratory discussion or when fully formulating an idea that has emerged in the discussion. In summative speech, when the focus of the discussion is clearly defined and stable, the teacher “tightens the criterion levels for precision and correctness” (p. 178). She attends to students’ mistakes and re-casts their ideas using formal mathematical language.


At this point, the class is ready to move on to the next stage of the inquiry. Aims can be negotiated, inquiry pathways can be set out and resources can be gathered. The class is moving further into the zone between knowing and not knowing. Now students and teacher know what they want to find out and how they will go about finding it out.

Andrew Blair
May 2018

The zone between knowing and not knowing - Part 1

posted 17 Mar 2018, 03:25 by Andrew Blair   [ updated 30 May 2018, 04:12 ]


The zone between knowing and not knowing
Part 1: Slowing down
During a recent Inquiry Maths workshop, I was asked how I could expect students to request instruction when they "don't know what they don't know". The questioner found it impossible to conceive of how my claim that students in Inquiry Maths classrooms use regulatory cards to signal a need for new knowledge could work in practice. Yet, for me, this is precisely the distinguishing feature of inquiry. The teacher and students are continually working in the zone between what is known and what is not known. Inquiry is all about unearthing what ex-US Secretary of Defense Donald Rumsfeld called the "unknown unknowns".
I was reminded of the question when I read this article by Andy Hargreaves. He describes his experience with a class of 7- and 8-year-old pupils who were trying to guess his identity:
Much more interesting and engaging [than having the knowledge] for them had been that magical moment before they had the knowledge – the wonderful moment of ignorance. We should cherish this kind of ignorance. It’s not the ignorance that refutes knowledge and expertise. It’s not prejudice or stupidity. It is simply the absence of knowing that invites and anticipates the knowledge that is to come.
The article was brought to my attention in a tweet by Kath Murdoch, the international expert on inquiry learning. Kath commented that the article valued the "inquiry-filled space between not knowing and knowing". She continued that the space "connects to slowing down and tuning in more carefully to students and our own thinking. And nurturing wonder."
For Kath, then, there are five aspects to learning in the zone between knowing and not knowing:
(1) Valuing the space
(2) Slowing down
(3) Tuning in to students
(4) Tuning in, as teachers, to our own thinking
(5) Nurturing wonder.
For me, showing that we value the space, tuning in and nurturing wonder are all predicated on slowing down. If teachers make the time, then they can achieve the other aspects.
However, slowing down is often the hardest to achieve. In inquiry classrooms, students are excited to explore their questions, observations and conjectures and enthusiastic to follow up on their insights and ideas. Even in workshops with teachers and educators, I have found myself asking participants to stop 'doing the maths' and step back to think about how our inquiry will develop and why we plan to develop it in that way. The regulatory cards play the key role in the Inquiry Maths model of slowing down participants by requiring them to consider the direction of the inquiry. 
There are relentless pressures on teachers not to slow down. Curricula are full of content objectives that have to be 'covered' and education departments and boards in many jurisdictions value 'pace' in lessons. Galina Zuckerman, the Russian educationalist and researcher, gives a compelling explanation of why slowing down at the start of inquiry is essential.  After creating "a high-potential field" that energises students’ imagination, arouses their curiosity and evokes questions, the teacher must slow down the "sparks of imagination" so that they are registered by other students. The whole class can then become involved in the process of inquiry by reformulating the original naive questions into aims. What's more, the slowdown is necessary to ensure the ultimate success of the inquiry: 
Like thermal neutrons in a nuclear pile, this slowdown must provide for a self-perpetuating chain reaction of interactions in the class, propagating new questions that lead from the initial chaotic question to hypotheses that can be verified.
During the slowdown, the teacher and students co-construct the foundations that will lead to a self-perpetuating inquiry based on a class-wide understanding of the central questions and aims.
As students enter the zone between knowing and not knowing, the inquiry must proceed slowly. Students have to have time to understand and reflect upon the questions, observations and ideas of their peers. The teacher has to have time to support the process in which the initial aims and direction of the inquiry develop out of the students' contributions. Once the slowdown has served its purpose, inquiry classrooms see an explosion of directed and purposeful activity. Importantly for those jurisdictions that value 'pace', the students' activity, built on high levels of motivation and excitement, moves so rapidly that it easily 'makes up for' the slow down at the start of the inquiry.
We leave the last word to one student who gave her feedback about a series of Inquiry Maths lessons: "Inquiry lessons make us slow down and think about it." Exactly!

Andrew Blair
March 2018

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