A reply to Mr Barton

posted 4 Oct 2018, 13:50 by Andrew Blair   [ updated 7 Nov 2018, 13:15 ]

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 me and my colleagues, 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

The best maths teaching

posted 14 Feb 2018, 07:06 by Andrew Blair   [ updated 14 Feb 2018, 07:09 ]

The best maths teaching
In January 2018, the National Director (Education) of Ofsted, Sean Harford, tweeted about his idea of what the 'best maths teaching' looks like:
The best maths teaching I have seen starts with the basics for a short period for all pupils and then ramps up to the next and more difficult stages (for everybody to try) and then looks to scaffold for those who need it. And involves lots of teacher explanation,demonstration etc
Well, you might say, this sounds like a pretty standard teacher-led maths lesson, which is the staple diet of hundreds of thousands of students in UK secondary school classrooms. When challenged on twitter for evidence that it is, indeed, the best, Harford claimed he did not have time for a conversation. He went on, however, to defend his description by saying that it is "the predominant way of teaching maths in China and the outcomes are accepted as being very good in maths."
There are two issues with Harford's intervention in the debate about maths teaching. Harford is the national director of an inspection body that wields great power over schools. Many leaders of schools in the 'requires improvement' or 'inadequate' Ofsted categories feel forced to look for any indication that will improve their chances of a favourable report on re-inspection. Declarations from Harford about 'best teaching' are bound to attract their attention and will inevitably lead to pressure on innovative maths departments to change their practice.
What is more troubling, however, is Harford's comment seems to directly contradict Ofsted's official position laid out in guidance about inspection 'myths':
Ofsted don’t prescribe any particular teaching style. We know that different things work for different teachers and trainers. Inspectors are only interested in how much progress students make.
Indeed, 'best' maths teaching in the official inspection handbook bears only a passing resemblance to Harford's description. According to the handbook, a maths lesson is to be judged on how well it "fosters mathematical understanding of new concepts and methods, including teachers’ explanations and the way they require pupils to think and reason mathematically for themselves." The focus on the students' depth of understanding is completely lost in Harford's prescription for 'lots' of teacher explanation, demonstration and scaffolding.
Of course, Harford is simply a mouthpiece for government ideology. Official funding now only goes to those who support the Department for Education's 'mastery' programme, which ministers claim to have adapted from practices observed in Shanghai. Just as teachers, consultants and academics are being sucked into the programme through the hubs network and the NCETM, Harford seems to have abandoned his own organisation's neutrality towards teaching styles in favour of the government's teacher-centred approach.

Andrew Blair
February 2018

For a reply to Harford's claim about the advantages of the Shanghai model, see this previous blog post. On the weaknesses of the NCETM's variant of mastery, see here.

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 multi-step 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 out-of-date 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 well-practised 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 ill-conceived. Frequent practice potentially blocks conceptual understanding.
The information processing model of the brain in which ‘facts’ are banked in long-term 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 pre-condition 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). Rittle-Johnson 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 hand-in-hand 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 co-construct 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
August 2017

Is inquiry compatible with instruction?

posted 24 Aug 2017, 03:29 by Andrew Blair   [ updated 31 Aug 2017, 05:51 ]

Is inquiry compatible with instruction?
In schools, students have to acquire mathematical knowledge. How they acquire knowledge – indeed, what constitutes knowledge – and how they use that knowledge are contested issues. In the discussions around knowledge, inquiry and instruction are often presented as opposite (and, even, contradictory) forms of teaching. If they are used in the same classroom, then they appear in a strict sequence: students receive instruction on a particular topic before applying the new knowledge through inquiry. However, teachers using prompts from this website have suggested that the Inquiry Maths model combines both forms of teaching. And, recently, the Executive Director of the reSolve (Mathematics by Inquiry) project in Australia has argued that inquiry is a form of explicit instruction.

The difference between explicit and direct instruction
instruction aims to achieve specific behavioural and cognitive outcomes, which are communicated to students. Knowledge and skills are clearly ‘framed’ and teacher-directed interaction occurs within the boundary set by the lesson’s aims.
Direct instruction (of which Engelmann and Bereiter’s ‘Direct Instruction’ is the best known model) is tightly programmed with teachers following a step-by-step, lesson-by-lesson script. The approach is based on stimulus-response and conditioning psychological models. Lessons are tightly paced, follow a prescribed and pre-determined sequence of skill acquisition, aim to maximise time-on-task and involve positive reinforcement of student behaviours.
Explicit instruction gives the teacher more opportunity to respond to students’ prior knowledge and misconceptions than direct instruction, but that response occurs within a restricted boundary or ‘frame’.
Professor Steve Thornton (Executive Director of reSolve) makes the case for inquiry as a form of explicit instruction:
The word explicit comes from the Latin words ex (out) and plicare (to fold). To make something explicit therefore literally means ‘to unfold’. This idea of explicitness is completely in line with our view of inquiry, which focuses on unfolding important mathematical ideas by encouraging students to ask questions and seek meaning.
In the reSolve model, the teacher guides students to ‘unfold’ the mathematical ideas behind a classroom task. This involves modelling, the use of enabling prompts to provide access, attending to misconceptions, and the unpacking of alternative strategies. These teacher ‘interventions’ are conceived of at the task design stage and the timing of some, such as modelling a general form, are pre-determined. The structured reSolve tasks might be said to resemble explicit teaching in that the teacher establishes a boundary and aims to achieve a specific outcome. As we argue here, the tasks are better described as one-off ‘enquiries’, rather than as part of a fully-fledged inquiry model of teaching.
Similarly, teachers who argue that instruction should precede inquiry also conceive of inquiry in a limited way. If a task is used to apply knowledge or a skill that has been recently learnt, then, by its very nature, the task is restricted. It lies within the boundary set by the instructional phase and the outcome is pre-determined. Teacher-directed interactions help to facilitate and structure the students’ application of the knowledge or skill to a particular context. As the potential for open inquiry is precluded in this sequence, the task might also be called an ‘enquiry’. However, even that label is inappropriate if students do not have any creative input at all. Tasks designed for mechanical application cannot be considered to be a form of inquiry.
In Inquiry Maths lessons, teachers have characterised phases of teacher explanation as explicit instruction. In this view, the inquiry itself, rather than the teacher’s intention, acts to ‘frame’ new knowledge. The initial phase of questioning and noticing entices students into the topic area, making them receptive to new knowledge. The teacher then gives the class explicit instruction before students go on to use the knowledge in answering their own questions in the remainder of the inquiry. The benefit of this approach is that students realise why the teacher is explaining; they see the content of the explanation as both meaningful and relevant. In this way, the teacher connects with the students’ intent to answer their own questions. Therefore, I would not characterise this period as ‘explicit instruction’, even if the teacher had pre-planned the explanation and would have given it regardless of the questions. The overall approach is an inquiry because students have autonomy to set and plan their own outcomes (rather than have them communicated by the teacher) within the mathematical field implied by the prompt.
Inquiry and explicit instruction are pedagogical approaches that originate in different epistemologies. Explicit instruction sees knowledge as transmitted from teacher to student and teaching as effective transmission; inquiry sees knowledge as constructed by students and teaching as facilitating that construction. Vygotsky was right when he said in Thinking and Speech that explicit instruction is “pedagogically fruitless”, achieving “nothing but a mindless learning of words, an empty verbalism.” He went on, “the formation of a [mathematical] concept only begins at the moment a child learns a verbal definition”, and the full generalisation arises through and is formed by “an extraordinary effort of his own thought.”
The difference between explicit or direct instruction and inquiry is neatly summarised by Professor Peter Sullivan in the reSolve newsletter. During explicit and direct instruction, the teacher explains first and then students practice using the new knowledge. The questions are normally graded to go from easier to harder, so by the end of the lesson almost every student encounters a problem they cannot do; they “transition from a state of knowing to not knowing”. In inquiry, students begin with a context or prompt that they do not immediately understand, but one that promotes a desire to know more. As the inquiry develops students come to understand; they “transition from a state of not knowing to knowing.”

Andrew Blair
August 2017

The difference between ‘inquiry’ and ‘enquiry’ in mathematics classrooms

posted 24 Aug 2017, 01:23 by Andrew Blair   [ updated 17 Jun 2018, 04:07 ]

The difference between ‘inquiry’ and ‘enquiry’ in mathematics classrooms
In an Inquiry Maths workshop a few years ago, I was asked what the difference is between enquiry with an ‘e’ and inquiry with an ‘i’. While some people use the terms interchangeably, the dictionary makes a distinction. An enquiry is an informal one-off query; an inquiry is a formal judicial examination of evidence to uncover the truth. I think this distinction is helpful in mathematics education. Enquiry suggests a short, structured and time-limited one-off task; inquiry is more a philosophy of teaching that promotes student agency and aims for open classrooms.
I was reminded of the workshop question when I received the latest newsletter from the government-sponsored reSolve (Mathematics by Inquiry) project in Australia. It includes two classroom tasks that exemplify the project’s approach. The tasks focus on how algebra can develop as generalised arithmetic. They encourage children to reason by exploring and expressing mathematical structure, pattern and relationships.
The year 4 task is called Number Maze. The teacher sets pupils the task of moving through a number grid in a specified way so that the sum of the numbers in the cells they pass through is odd. The aim is summarised in this way: “Through the course of this task, students are encouraged to look at how many odd and even numbers are in each pathway. They see that an odd number of odds is always required to give an odd total.”
The year 9 task Addition Chain follows the same course. The teacher requires a student to choose two numbers from which to start a chain where each term is the sum of the two previous terms. Once the chain has ten numbers, the teacher asks the class to find their total. Using the ‘trick’ that the total is 11 times the seventh number, the teacher announces the answer to the surprise of the class before students have the chance to begin the calculation. The teacher has used a property of the Fibonacci sequence. Starting with two numbers (a, b), the seventh term of the sequence is 5a + 8b and the sum of the first 10 terms is 55a + 88b.
From these two examples, we can identify the reSolve model. It combines the two key mathematical processes of inductive exploration and deductive reasoning. Students choose a particular case to explore before being introduced to an explanation of the general structure. In both tasks, the source of inquiry and the teacher’s role are also the same. The teacher starts by giving instructions that allow students little flexibility in how to carry out the task. In year 4 the children choose their own routes through the maze, but the teacher provides the maze used in the task and pupils can only move in prescribed ways. In year 9 students choose the two starting numbers, but again the process they follow is laid out in the teacher’s instructions. Once the class has reached the realisation about odd numbers in year 4 or has understood the trick in year 9, the teacher’s role is to generalise from particular cases. In year 4, the teacher introduces a visualisation through which pupils can ‘see’ why an odd number of odds is required. Similarly, the teacher introduces the algebraic form of the Fibonacci sequence to year 9.
Student questioning and regulation
Students’ questions, which we at Inquiry Maths hold to be fundamental as a source of inquiry and as a precursor to teacher explanations, seem to have a limited role in reSolve classrooms. The description of the year 9 task states that with the introduction of the algebraic form “the door is opened here to many more mathematical investigations.” There follows a number of questions about how the task could proceed. It is not clear where the questions have come from. Are they examples of questions that students have posed in classrooms or are they suggested extensions from the task designers? In his introduction to the newsletter, Steve Thornton (reSolve Executive Director), says that “at each step of the lesson students learn through the teacher’s active intervention.” This suggests that the teacher poses the questions and students have the choice of which ones to follow.
The restricted potential for students’ questions has a serious consequence when students do not or cannot follow the path laid out by the task designers. In the year 4 task, for example, it is not clear how pupils can influence the course of the inquiry if they do not notice what they are required to notice. Steve Thornton says that pupils are not expected to discover results in reSolve classrooms, but in the year 4 task they are encouraged to ‘see’ a specific mathematical property. While the distinction between discovering and ‘seeing’ might seem to rest on semantics, the more important point relates to how students can contribute to resolving the impasse caused by not noticing. The reSolve model lacks a student-driven mechanism (be it questions to the teacher or, as in the Inquiry Maths model, regulatory cards) for overcoming an obstacle to inquiry. Ultimately, the teacher has to tell the class what to ‘see’ in line with the design of the lesson.
There seems little scope for students’ agency in the reSolve tasks. The teacher provides the source of the inquiry and its direction and the task designer determines the timing of the explanation. In the tasks we have reviewed the students have the opportunity to decide their own path through the maze or to select a pair of numbers to use, but these are limited responsibilities within closely defined parameters. In contradistinction, Inquiry Maths prompts establish a wider ‘landscape’ or ‘zone’ for exploration in which students have the space to ask questions and participate in directing the inquiry. 
From an Inquiry Maths perspective, we might call the reSolve tasks ‘enquiries’. They are restricted to a pre-determined outcome, structured by the designer and directed by the teacher and fit neatly into a predictable time frame. Of course, each task could open up into a wider inquiry by encouraging students’ agency in developing their own pathways. Year 4 pupils could suggest changes to the maze or to the rules for moving between the cells or to the property of the result. Year 9 students could suggest changes to the rule for summing the terms of the sequence. While the task designers say they welcome “alternate representations”, the reSolve model does not make students’ questions and suggestions an integral or essential part of inquiry.

Andrew Blair
August 2017

Inquiry is not discovery learning

posted 25 Jun 2017, 09:23 by Andrew Blair   [ updated 2 Jul 2017, 08:49 ]

Inquiry is NOT discovery learning
If we were to believe the critics, classroom inquiry is just another variation of discovery learning. Citing their favourite article, they conflate teaching models under the umbrella term ‘minimal guidance’. Rather than analyse the specific nature of each model, the critics lazily dismiss one by association with the perceived weaknesses of another. In the learning of mathematics, discovery and inquiry are very different processes.
In discovery learning, students are expected to derive a procedure or concept from an activity devised by the teacher. For example, a class might be required to work out the areas of squares on the sides of right-angled triangles and then notice that the sum of the areas of the squares on the two short sides equals the area of the square on the hypotenuse. This ‘discovery’ of Pythagoras’ Theorem can be a memorable and exciting experience. The theorem can seem novel, even when students find out later that it is well known.
However, the discovery classroom is often an uncomfortable place for the teacher, especially in a subject like mathematics that is built on axioms and proof. The first problem occurs when students do not make the required discovery and ask for direction or clues. Teachers are forced into subterfuges such as pretending not to hear the student or replying that they are “not at liberty to say” or they “don’t know”. Another approach sees teachers assert that it is not in the interest of the students to be told and that finding the concept independently will “help them learn more”.
A second problem occurs when the student makes the wrong discovery. In an attempt to tackle the misconception, while simultaneously preserving the potential for a correct discovery, the teacher gives hints such as “it’s not quite right” or asks whether the student has considered an alternative approach. A third problem arises when one student experiences the ‘aha’ moment and wants to share the discovery. The teacher is forced into attempting to quieten that student to avoid ruining the experience for the rest of the class.
procedure or concept that appears at the end of the discovery process is incorporated into the course of an inquiry. It is used to answer students’ questions and develop their observations. In the right-angled triangles inquiry (see prompt right), for example, Pythagoras’ Theorem is deliberately introduced to pursue an inquiry pathway. The key issue for the teacher becomes how and when to introduce the theorem.
in the initial phase of the inquiry, a student will ask if the length of the hypotenuse forms a linear sequence in the same way as the lengths of the short sides. (The word ‘hypotenuse’ could be introduced by the teacher when she reformulates a question about the ‘longest’ side.) Alternatively, if the question does not arise and the teacher aims to 'cover' the theorem through the inquiry, she might pose the question herself. 
In whatever way the question arises, the teacher has a number of options over how to proceed. She could decide to explain Pythagoras’ Theorem immediately; she could use the selection of the regulatory card 'Ask the teacher to explain' to justify an explanation; or, alternatively, she could ask students to research the theorem and report their findings to the class. The decision would depend on her evaluation of the appropriate level of inquiry for the class. An immediate explanation is characteristic of a structured inquiry, the use of the cards would form part of a guided approach and student research might indicate a more open inquiry.
In discovery learning, the teacher attempts to preserve the pretence of discovery, even to the extent of withholding knowledge; in inquiry, the teacher, as a participant in the classroom activity, aims to introduce subject-specific knowledge when it is most relevant and meaningful to her students. 

Andrew Blair
June 2017

In response to the post, Mike Ollerton (@MichaelOllerton) wrote: I see discovery learning as a complementary subset of enquiry-based learning. I do not see them in terms of a binary divide. At issue is when I choose to tell students something and when I choose not to; the intervention or interference continuum.
Andrew Blair replies: There is a continuum in inquiry, but it relates to the level of control students have in directing the learning process. The aim is to develop their ability to regulate a mathematical inquiry. Rather than the teacher deciding when or when not to intervene, students learn how to overcome an impasse by requesting new knowledge.
"I've discovered ..."
In a recent Inquiry Maths workshop at a conference in Birmingham (UK) when participants were feeding back on progress in an inquiry, one teacher said "I've discovered ..." before correcting himself. He reminded the participants of a slide from a presentation at the start of the workshop that said inquiry is not discovery learning. However, inquiry does not preclude the discovery of novel pathways or applications of mathematics to the prompt. The point is that discovery of a concept or procedure is not the aim of inquiry.

Leigh Taylor (@leigh_taylor13) inspired this post by asking on twitter for clarification about Inquiry Maths and discovery learning.

Is inquiry age-related?

posted 9 Apr 2017, 01:18 by Andrew Blair   [ updated 9 Apr 2017, 10:06 ]

Is inquiry age-related?
Recently, on social media, Alycia Corey (@corey_alycia) asked if the levels of Inquiry Maths (structured, guided and open) are affected by the age of learners? This is an excellent question.
Inquiry Maths was devised for secondary school classrooms. Unless children have been through an inquiry-based curriculum (such as the PYP programme), there is little opportunity for them to learn how to inquire into academic domains. In consequence, structure is often necessary for secondary students to inquire constructively. Yet, at the other end of schooling, young children's inquiry might be inhibited by structure. They inquire naturally through play. Paradoxically, we might characterise early years as a time of open inquiry and secondary school as one of structured inquiry.
The development from structured to open inquiry established in the hierarchical levels of Inquiry Maths appears to be reversed. This is the situation in most school systems. As children are institutionalised into the culture of traditional classrooms, they either learn to conform and comply, as is the case with the majority, or become the subject of ‘behaviour interventions’. Either way, inquiry processes disappear from formal schooling. A teacher wishing to introduce inquiry at secondary level faces obstacles created by conventional classroom practices and power relations.
In most schools, then, the levels of inquiry are linked to the students’ prior experience of inquiry and the extent to which they demonstrate initiative and independence. These considerations are not related to age.
That is not to say, however, that inquiry is not age-related. Four years ago when advising a new 4-19 school about inquiry learning at different stages of schooling, I drew up a diagram of how the nature of mathematical inquiry changes. The diagram (right) assumed children are involved in open inquiry processes across the age ranges.

The changes that occur in the three phases do not relate to inquiry processes per se, but rather to the consciousness children have of those processes in relation to the object of inquiry. While curiosity, noticing and questioning underlie all phases of inquiry, their content and form develop as children learn to direct inquiry at higher levels of subject knowledge. Firstly, children become more able to regulate their activity in a manner consistent with the domain-specific method of inquiry. Secondly, the object of inquiry changes: immediate perceptions in early years, experience of surroundings in primary and de-contextualised stimuli in secondary. In mathematics, for example, students learn increasingly complex (and abstract) concepts, while simultaneously developing a more sophisticated understanding of the mathematical form of inquiry.
Reflecting now on the diagram, it implies a rigidity between the age groups that is not warranted. The idea of play, for example, endures in the exploratory phases of later inquiries. Similarly, applications of abstract mathematics can be studied in practical projects at secondary level; just as prompts that focus on a mathematical object can be used at primary level when supported by concrete apparatus.
Even if the phases of inquiry do not fit into neat categories, it is the case that open inquiry is age-related; self-consciousness develops and the object of inquiry changes as children grow older. However, the levels of Inquiry Maths are not related to age because they are designed for classrooms in which students do not normally have prior experience of inquiry processes.

Andrew Blair
April 2017

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