Can students learn fluency through inquiry?

posted 24 Aug 2017, 11:36 by Andrew Blair

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from a science that is under continual revision would suggest that the rudimentary techniques of drill and recall are out-of-date before the teacher arrives in the classroom.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 practice 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 24 Aug 2017, 03:39 ]

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.
The reSolve tasks follow the same model. They combine 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

Maths Inquiry Template

posted 29 Jan 2017, 12:27 by Andrew Blair   [ updated 29 Jan 2017, 12:28 ]

Maths Inquiry Template
Amelia O'Brien, a grade 6 PYP teacher at the Luanda International School (Angola), has shared her Maths Inquiry Template with Inquiry Maths. The template helps students think about concepts relevant to the prompt and plan the inquiry. In their most recent inquiry, Amelia's pupils posed generative questions that opened up new pathways for inquiry (see a report here under the title 'Question-driven inquiry'). You can follow Amelia on twitter @_AmeliaOBrien.

The teacher’s role in inquiry

posted 8 Jan 2017, 12:46 by Andrew Blair   [ updated 9 Jan 2017, 13:07 ]

The teacher’s role in inquiry 
It is a common misconception that the inquiry teacher tries to do as little as possible in the classroom. For those who caricature inquiry as a discovery model, the teacher is obliged to let students develop concepts by themselves. For those who define inquiry as exclusively open learning, the teacher must refrain from intervening in order to allow students the freedom to find their own pathways. 
The first approach can lead to awkward interactions when the teacher refuses to give knowledge for fear of denying students the satisfaction of discovering it for themselves. The second approach often leads novice inquirers to complain that they “do not know what to do”. 
Even Dewey, who advocated experiential inquiry based on children’s lives in and outside school, did not encourage a passive role for the teacher. In Democracy and Education, he explains that the opposite of the teacher’s role in traditional teaching is not inaction, but rather participative activity
This does not mean that the teacher is to stand off and look on; the alternative to furnishing ready-made subject matter and listening to the accuracy with which it is reproduced is not quiescence, but participation, sharing, in an activity. In such shared activity, the teacher is a learner, and the learner is, without knowing it, a teacher and upon the whole, the less consciousness there is, on either side, of either giving or receiving instruction, the better. (p. 188)

Participation in the inquiry process requires more skills than in traditional classrooms. The teacher is a learner in the sense that she is continually assessing students’ understanding and taking on-the-spot decisions about whether to structure or guide the inquiry or encourage students to set off on their own. 
Although the Inquiry Maths model aims for participative activity in Dewey’s sense, it is also the case that participants adopt roles consciously and make them an object of reflection. The regulatory cards allow students to make suggestions about how the inquiry should proceed. This includes the activity they will embark upon at a particular stage of the inquiry and whether new knowledge is required to make progress. 
Importantly, the cards provide a mechanism for students to ask for an explanation from the teacher. At such a time, the inquiry classroom might take on the appearance of the traditional transfer of knowledge. However, as the 'transfer' is both meaningful in and relevant to an inquiry process partly directed by students, it is consistent with the teacher's democratic intent to give students control over their own learning.
The role of the teacher in conveying knowledge is the most misunderstood point of all when in comes to views of inquiry. The teacher, as the representative of the discipline of mathematics, introduces a new concept or procedure when it overcomes an obstacle to inquiry. 
Even the leaders of Dewey’s school, Mayhew and Edwards report, changed their model in recognition of the advantage of subject specialists over general teachers:
One of the reasons for this modification of the original plan was the difficulty of getting scientific facts presented that were facts and truths. It has been assumed that any phenomenon that interested a child was good enough, and that if he were aroused and made alert, that was all that could be expected. It is, however, just as necessary that what he gets should be truth and should not be subordinated to anything else. (pp. 35-36)
In this description, the teacher is the arbiter of what constitutes facts and truths. In the Inquiry Maths model, the teacher might also instruct students (although attempting to co-construct knowledge as much as possible) when participants in the inquiry, including the teacher, identify the need.

Andrew Blair
January 2017

PISA 2016 and 'enquiry-based teaching'

posted 11 Dec 2016, 07:39 by Andrew Blair   [ updated 11 Dec 2016, 22:17 ]

PISA 2016 and 'enquiry-based teaching' in science
There is good reason why this blog has never before discussed inquiry in science education. The inquiry processes in science and maths are completely different: science develops and adapts hypotheses based on experimental results; mathematical inquiry involves generalisation (based on pattern spotting or structural analysis) and proof by deductive reasoning.
However, the PISA 
2016 volume dedicated to science teaching, published this week, has been received as confirmation of the superiority of ‘teacher-directed’ over ‘enquiry-based’ lessons. Traditional teaching practices, it is claimed, produce better test performance. The evidence seems compelling. As the report says, “In all but three education systems ... using teacher-directed instruction more frequently is associated with higher science achievement” (p. 65).
However, the report also says that teacher-directed instruction is used much more frequently than enquiry. Might it be the higher frequency, rather than the superiority of the practice itself, which accounts for the association with test performance?
Let’s look at teacher-directed practices first. PISA identified four characteristics of traditional teaching and asked students to report how often they featured in their lessons. I have grouped the four possible responses into two, combining ‘many lessons’ with ‘every lesson or almost every lesson’ and ‘some lessons’ with ‘never or almost never’.

As Figure II.2.14 (below) shows, the frequencies with which the four practices occur are mirrored exactly by their position in the ranking of 'score-point difference'. For example, 'the teacher explains scientific ideas' occurs most frequently and is associated with the highest positive score-point difference; 'a whole class discussion' occurs least frequently and is associated with the only negative score-point difference.
We turn now to PISA’s curious characterisation of ‘enquiry-based instruction’. While the most frequent feature ('students explain ideas') is perhaps more applicable to enquiry than teacher-directed lessons, the next two ('teacher explains') might just as easily occur in teacher-directed lessons. Confusingly, PISA offered students four different responses this time. Again I have grouped them into two, combining ‘all’ with ‘most’ lessons and ‘some’ lessons with ‘never or hardly ever’.
The features that might be described exclusively as enquiry (that is, those linked to experimentation and investigation) occur, in the main, far less frequently than the other categories. Once again, however, there is a very close correspondence between frequency and score-point difference (see Figure II.2.20 below).
What conclusion should we draw from this? The dominant narrative this week is that teacher-directed practices are superior to enquiry because they lead to higher test performance. However, we could just as easily say that the most frequently used teaching practices (regardless of the specific type) lead to higher test scores. Two questions follow from this: Why do science teachers employ traditional techniques more frequently? And why do they use enquiry-based techniques much less frequently?

The PISA report gives answers to the first question: teacher-directed techniques are less time-consuming and easier to implement. In answer to the second question, a survey of European science and maths teachers showed a negative correlation between ‘systems restrictions’ and ‘routine use’ of inquiry-based learning (IBL) – that is, the more restrictions, the lower the use. The restrictions included:
  • The curriculum does not encourage IBL
  • There is not enough time in the curriculum
  • My students have to take assessments that don’t reward IBL.
Thus, on the one hand, teachers are under pressure to get through a curriculum that discourages inquiry processes and, on the other hand, students face assessments (such as PISA tests) that do not reward IBL. That teachers use inquiry less frequently means they, as a professional body, are less experienced in its use. Similarly, students are less skilled in inquiry processes to take full advantage of the potential for learning they offer. We could surmise that traditional practices get results (measured by test performance) because both teachers and students are more accustomed to them.
The message to be taken from PISA 2016 is that the teaching practices used most frequently in classes are associated with higher test results and those methods are used because of restrictions imposed by curricula and assessments. The PISA review of science teaching says very little about the relative merits of teaching practices and far more about how authorities define and measure learning.

Andrew Blair
December 11, 2016

Inquiry and mixed attainment classes

posted 6 Nov 2016, 11:19 by Andrew Blair   [ updated 7 May 2017, 04:56 ]

Inquiry and mixed attainment classes
Mathematics is the most heavily setted subject in the secondary school curriculum. The most recent reliable figures published for England show over 80% of classes for students aged between 11 and 14 were set and, no doubt, the percentage was higher for older secondary students. Mike Ollerton characterises setting as "educational apartheid" in which the powerful exercise control over the powerless. Bottom set students are taught "repetitive, procedural, fragmented, disjointed, simplified mathematics" (Watson et al.); top set students are accelerated through the curriculum often to their detriment (see here for a selection of research papers).
It is, therefore, welcome that more maths departments today are considering mixed attainment classes. However, an examination of the reason for the growth of interest suggests there remains cause for concern.
The main reason is the mastery movement’s promotion of mixed attainment teaching. Supporters of mastery argue that students should move through the curriculum together, studying the same topics from the same materials. Yet, students are not treated equitably in the mastery classroom. Only when a topic has been 'mastered' do students get the opportunity to solve problems and reason deeply. Inevitably this two-stage model of learning leads to a two-tier classroom. Students who do not master a topic as quickly as their peers are denied access to the creative aspects of mathematics. As NRICH says here, mastery "may be insufficient for developing the potential of young mathematicians."
The problem with the mastery approach is its insistence that solving and reasoning provide an opportunity to apply new knowledge; it rejects the notion that learning can occur in the process of solving or reasoning. Yet, it is when students are involved in a mathematical process that learning new knowledge becomes relevant and meaningful. When mastering a procedure is part of a wider aim to solve a problem or put forward a convincing argument, students are less likely to question the need to practise and more likely to become fluent in that procedure.

Inquiry Maths was devised and developed in mixed attainment classrooms. Its design is ideally suited to promote learning at multiple levels:
  • Students’ questions and observations about the prompt unite the class in a mathematical process that ranges from relatively basic definitions and procedures to more sophisticated conjectures;
  • The regulatory cards allow students to determine their own access points to the inquiry; 
  • The teacher introduces new knowledge for an individual, a group or the class when required by the development of the inquiry; 
  • The inquiry pathways involve students working on a common aim from different directions and at different levels of mathematical reasoning. 
The unity of purpose guarantees equity as all contributions add to the findings of the inquiry. Each student's selection of an approach and mathematical level (guided by the teacher when necessary) ensures challenge and progress for all.
Mixed attainment classes have their roots in social justice. Justice is not served by restricting one set of students to knowledge acquisition, while their peers move on to creative tasks. As Jerome Bruner says here, students should learn by both 'leaping' and 'plodding':
Let him go by small steps. Then let him take great leaps, huge guesses. Without guessing, he is deprived of his rights as a mind. (p. 531)
The current mastery classroom consigns some students to plodding. The rights of learners are being denied. The philosophy of inquiry, in contrast, promotes inclusiveness, cohesion and equity.

Andrew Blair 
November 2016

New on the website

posted 27 Jul 2016, 14:40 by Andrew Blair   [ updated 27 Jul 2016, 14:41 ]

New on the website: Inquiry and curriculum
As Inquiry Maths becomes more widely known, teachers are asking how they can incorporate the prompts on the website into their schemes of learning. More broadly, they are asking whether inquiry classrooms that promote curiosity and student agency are compatible with covering the content of a mathematics curriculum. In response, we have created a new page called Inquiry and Curriculum that can be reached through the link on the menu bar or by clicking here.

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