Inquiry Maths posts
Structured inquiry: a contradiction in terms?
posted 23 Jul 2019, 10:14 by Andrew Blair [ updated 23 Jul 2019, 10:34 ]
Structured inquiry: a contradiction in terms? The idea that classroom inquiry is, by its very nature, open and free has its origins in US progressive schools during the interwar period. The First World War was a turning point in progressive circles as the doctrine of creative selfexpression took hold:
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 EitherOr 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 childcentred 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:
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). 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 3031). 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 selfregulation 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 selfdirection 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. Andrew Blair July 2019 Cremin, L. A. (1964). The Transformation of the School: Progressivism in American Education, 18761957. 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 prototypical mathematical structure that requires students to specialise and generalise if they are to understand and extend that structure. 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 oneway  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 preprogrammed 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 prototypical 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 overreliance 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 sensemaking." 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 Andrew Blair replies: 
How to get started with Inquiry Maths
posted 30 Dec 2018, 07:50 by Andrew Blair [ updated 9 Nov 2019, 06:36 ]
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 email 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 coconstructed 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 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 had decided on running a structured inquiry, then I would prepare resources for a single main line of inquiry. In subsequent lessons, I might 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 postit 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 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 losing vitality. Students can find it difficult to reengage with their questions and 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 to 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 classwide inquiry; sometimes, the inquiry is diverse and multifaceted. 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 5 Dec 2019, 11:19 ]
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 volteface as the one Craig delivers. However, in the style of a zealous new convert, Craig has a onesided view of history. He relates in the book how he used the surds prompt (pp. 9798). 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. 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 inquiryoriented instruction is positively
related to student outcomes, which "lends support to decades worth of reform to refocus mathematics
instruction toward inquiry and conceptbased 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 metaanalysis (2016) starts by saying that "research has consistently shown that inquirybased 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 teacherled 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. 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. 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 oneoff 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 nonformal 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. Planning inquiry lessons.pptx 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 reorientate 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. 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 2r^{2} < A < 4r^{2}. The class seemed to have settled on 3r^{2} 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 = πr^{2}. 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):
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 onetoone support before the next wholeclass session. The discussion develops in a sequence from basic contributions to sophisticated reasoning:
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 counterexamples 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 recasts 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 exUS 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 8yearold pupils who were trying to guess his identity:
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 "inquiryfilled 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 highpotential 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:
During the slowdown, the teacher and students coconstruct the foundations that will lead to a selfperpetuating inquiry based on a classwide 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! 
The best maths teaching
posted 14 Feb 2018, 07:06 by Andrew Blair [ updated 14 Feb 2018, 07:09 ]
The best maths teaching


Can students learn fluency through inquiry?
posted 24 Aug 2017, 11:36 by Andrew Blair [ updated 31 Mar 2018, 08:22 ]
Can students learn fluency through inquiry? or How drill obstructs mathematical learning Currently in mathematics teaching, there’s an idea that the subject cannot be taught through inquiry. More even, it is a dereliction of teachers’ duty not to drill students to become fluent. This claim is normally accompanied by reference to a contentious theory about cognitive load and to research on memory that is in its infancy. Of course, the meaning of ‘fluency’ itself is contentious. To some, fluency is developed through repetitive practice and demonstrated by the immediate recall of basic number facts and the accurate application of procedures. To others, fluency means something different (and more). The NCTM, for example, expect students who are mathematically fluent to demonstrate flexibility by transferring procedures to different contexts, building or modifying procedures from other procedures and recognising when one strategy or procedure is more appropriate than another. In this post, I will argue the following: firstly, using drill and recall to promote fluency in classrooms rests on flimsy scientific arguments and does not work; secondly, we have to view fluency as encompassing both procedural and conceptual understanding (although, as I will go on to say, this distinction is not helpful); and, thirdly, fluency can be developed through inquiry. The arguments for drilling rest on shaky foundations. Even if the Cognitive Load Theory is not at “an impasse, and dissatisfaction with it is growing” as this post claims, the idea that a ‘limited working memory’ should dictate how we teach completely ignores the social side of classrooms. Teachers have been supporting learning for years by using proxies for working memory, such as ‘holding’ provisional results during a multistep calculation in their own memory. Furthermore, to devise teaching methods from a science that is under continual revision would suggest that the rudimentary techniques of drill and recall are outofdate before the teacher arrives in the classroom. If the latest finding on how memory works “may force some revision of the dominant models of how memory consolidation occurs”, then will it also force a revision in teaching methods? However, the main problem with drill comes later. Once students have memorized and practiced procedures, they have less motivation to understand their meaning or the reasoning behind them (NCTM). Drilling in facts and procedures interferes with conceptual development in three ways (Pesek and Kirshner): (a) cognitive interference results from the development of such strong routines that students block subsequent learning; (b) attitudinal interference occurs when they see no point in attempting to connect wellpractised and successful rules with other representations that might give them a deeper meaning; and (c) metacognitive interference arises when conceptual learning threatens to draw away mental resources required to maintain a procedural competence. In light of these conclusions, the fluency aim of the National Curriculum (England), which requires frequent practice “so that pupils develop conceptual understanding” (my italics), is illconceived. Frequent practice potentially blocks conceptual understanding. The information processing model of the brain in which ‘facts’ are banked in longterm memory is only one way of understanding how we think and learn. An alternative model focuses on concept formation and emphasises the growth of concepts and their relationship to other concepts in a connected network or ‘schema’ (Skemp, The Psychology of Learning Mathematics). In the classroom, the model leads teachers to prioritise opportunities to make links between facts, propositions and principles. The degree to which a student understands mathematical ideas or procedures is determined by the number, strength and richness of the connections in the network. For example, students drilled about types of triangles and other polygons bank them as disconnected facts. In a conceptual approach, students broaden the concept of a triangle through categorising different types and deepen the concept by, for example, linking the triangle to the construction of other polygons. Conceptual learning is important because, by developing relationships and links, students have a wider repertoire of approaches to solve a problem. Drilling might work if the structure of problems does not change; the student simply applies the same procedure each time. However, faced with a novel situation, the student needs to identify properties of the problem and their links to other mathematical ideas. By linking concepts, the student can generate a new procedure. Conceptual knowledge becomes a precondition for “adaptive” or flexible procedural expertise (Baroody et al.). Having promoted conceptual learning over the drilling of facts or procedures, it is nevertheless the case that researchers have become increasingly uneasy with the separation of different forms of mathematical knowledge. Making a distinction between conceptual and procedural knowledge has been described as limiting and an impediment to the study of mathematical learning (Star). RittleJohnson et al. argue that “conceptual and procedural knowledge develop iteratively, with increases in one type of knowledge leading to increases in the other type of knowledge, which trigger new increases in the first” (p. 346). Conceptual knowledge allows for a deeper structural analysis of a mathematical situation, which leads to more flexible procedural approaches; and correct procedural knowledge helps students represent key aspects of situations, which underlies advances in conceptual understanding. While we might take issue with the idea that there is only one (iterative) relationship between the two types of knowledge, the idea of a relationship is important in the development of mathematical fluency. Different types of knowledge develop handinhand in inquiry classrooms. In the inquiry on fractions and decimals that O’Connor observed, students discussed mathematical ideas directly, but conceptual understanding also developed from computational activity. The same occurs in Inquiry Maths lessons. The prompt 24 x 21 = 42 x 12 motivates students to practise multiplication facts, requires students to multiply accurately and also encourages them to reason about the structure of the equation. The time spent on each aspect and the teacher’s actions to support each one vary from class to class. The teacher makes the decision based on the students’ questions and observations in the initial phase of the inquiry and on their selection of regulatory cards. As the inquiry develops, students devise (or coconstruct with the teacher) new pathways to which they transfer their learning. They evaluate the relevance of the facts, procedures and concepts from the original pathway to the new situation and, if necessary, modify (or seek the teacher’s help to modify) them. In this way inquiry combines all forms of mathematical thinking and relates them to each other. Drilling, on the one hand, obstructs conceptual learning. It leaves facts and procedures isolated and unconnected and, furthermore, discourages students from developing a deeper understanding. Inquiry, on the other hand, links different forms of mathematical thinking in a unified process. It promotes the NCTM’s idea of an enhanced fluency. Andrew Blair 