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.

Introducing Inquiry Maths into a department

posted 2 Jul 2016, 02:09 by Andrew Blair   [ updated 8 Jan 2017, 07:33 ]

Introducing Inquiry Maths into a department

At an Inquiry Maths conference workshop recently, Rob Smith (@Mrrismithmaths) raised the issue of introducing inquiry into a secondary school department. Rob, the leader of the Maths Department at Northampton Academy (UK), wanted advice on the best way to promote Inquiry Maths with his team. Andrew Blair, who has led maths departments in three schools, replied:

The approach I would take depends on the culture and practice of the department. If we assume that the department is like the majority described in Made to Measure, then we face a situation in which teachers do not expect students to solve multi-step problems or reason mathematically on a regular, or even infrequent, basis. In this context, the introduction of Inquiry Maths is unlikely to succeed without support and training. It is not sufficient, for example, to write prompts into the schemes of learning as suggested activities. This approach underestimates the obstacles teachers face. 
Firstly, Inquiry Maths is not simply another resource that can be assimilated into existing practice; the full model might involve fundamental changes to a teacher's practice. Secondly, teachers in the UK state sector rarely have the time to engage with new ideas on their own. Hence, teachers will reject the suggested inquiry in favour of tasks that fit with their existing practice. 
I will now consider three cases based on the level of support for or opposition to Inquiry Maths within the department:
(1) The majority of the department is interested in inquiry learning. In this case, I would require all members of the department to try out the same inquiry, with each colleague choosing one of their, for example, year 7 or 8 classes. This initiative would be the main professional learning to occur in the department over a half term, taking up the majority of time in departmental meetings. The departmental leader would organise sessions on trying out the inquiry, considering the level of inquiry appropriate to each class, discussing potential pathways that might arise, preparing resources for those pathways and, afterwards, evaluating the inquiry. In the next half term, the team could go through the same process or teachers might select their own inquiry to work on in smaller teams. Whatever approach is taken, however, departmental time must be given over to planning, implementing and reflecting. By the end of the first year, the leader might feel confident to include inquiries as required elements of the schemes of learning.

(2) A minority of the department is interested in inquiry learning. In my experience, this is a more common situation. Perhaps two other teachers are prepared to embrace new ideas and are excited by the prospect of negotiating aspects of learning with their students. In this case, I would work with the interested teachers using a lesson study model. (Read Helen Hindle's report on an inquiry lesson study here.) We would observe each other running the same inquiry, focusing on the learning of identified students. Once again, time is a key issue. I would endeavour to arrange time off timetable to prepare the lesson study. Even without that, however, I have found that teachers interested in Inquiry Maths are those most committed to collaborative development and they are likely to give up their own time to improve practice. The aim would be for the two teachers to become advocates for inquiry within the department, spreading their enthusiasm to others. This phase might last for a year or less depending on how quickly others are drawn in. In the second year, the two advocates
would lead their own lesson studies with four other colleagues and in the following year - that is, the third year - inquiries become required.
(3) The members of the department are opposed to inquiry learning. It is difficult to envisage a situation in which you would find yourself leading a department that is opposed to inquiry if you wanted to introduce Inquiry Maths. If you are applying for departmental leadership posts, my advice is to consider other schools because this situation inevitably leads to frustration and acrimony. There are, however, two ways you might end up in this position. Firstly, you have been promoted internally and want to stay at the school. Hopefully, the relationships you have built up during your time in the department will lead others to trust you and you are able to identify one or two colleagues who are at least prepared to try out inquiry. Secondly, you are promised by the headteacher at interview that you will have the full support of the senior leaders to change practice in the maths department. This is a cynical ploy to use you as the battering ram against a recalcitrant department. Think carefully before accepting the post because, in my experience, the success of departmental leadership rests more on the relationships you have with classroom teachers than those with senior leaders. If the department thinks you are a stooge of the headteacher, then they will resist the introduction of Inquiry Maths (and any other initiative for that matter) even more.
It is difficult to introduce Inquiry Maths into a department without the possibility of developing at least one advocate. Even if the department is supportive, the process involves cycles of planning, implementation and evaluation until, after a year, inquiry prompts become part of the schemes of learning. Of course, the special nature of inquiry learning means the process is never complete. Each inquiry starts with a unique set of students' questions and observations, has the potential to develop into new pathways and requires the teacher to decide each time on the nature of structure or guidance offered to the students. Eventually, discussion between colleagues about inquiry will become an everyday feature of the department's culture, but the leader must always be prepared to reinforce that process by giving over time in formal meetings to analyse the team's ever-deeper understanding of inquiry processes.

Andrew Blair
July 2016

Inquiry and problem solving

posted 2 Jan 2016, 10:34 by Unknown user   [ updated 16 Feb 2017, 13:05 by Andrew Blair ]

Inquiry and problem solving
The last post about problem solving featured a discussion with Dan Meyer (@ddmeyer). It centred on the notion of openness in maths classrooms with Dan arguing that openness is "a spectrum, not a switch." I was not convinced by this argument in relation to problem solving.
Inquiry can be more or less open (see levels of inquiry), but problem solving only has the 'open middle'. While the the teacher poses the problem and knows the answer, the solving process can be carried out in different ways. Skills required by students relate to extracting relevant information from the problem, identifying similar problems that they have solved before, selecting methods, checking progress and verifying the solution. Inquiry has the potential for an open beginning, middle and end. The teacher will have an idea of different pathways that could arise, but they will not be exhaustive. Moreover, students' initial questions and findings might be novel. Key processes include questioning, noticing, conjecturing and proving.
The distinction is reinforced in the National Curriculum in England. Reasoning through inquiry and problem solving make up two of the three separate aims of the curriculum:
Unsurprisingly, the GCSE assessment objectives (AO), which are based on the National Curriculum, feature the same separation. AO2 covers reasoning, interpreting and communicating (which might be broadly classed as elements of inquiry) and AO3 lists problem solving steps (below).
The distinction came up again in a discussion I was having with a new head of a maths department (
@PythagnPi). She wrote to Inquiry Maths about how to develop mathematical reasoning: "I was thinking an inquiry approach would lead to encouraging the students to question what they do and why they do it, then lead to helping them with prompts for solving a problem. Do students not need to be on the path of fluency of inquiry before they can embark on the problem solving approach?" The question implies that mathematical inquiry is a precursor to problem solving.
During the discussion, one teacher said “it all depends on how you define a problem.” If you define a problem in terms of the questions that appear in public examinations, then, as Mike Ollerton (@MichaelOllerton) contends, problems are “pseudo-problems which undermine mathematical thinking and all that is creative in maths." They have one closed answer that students are required to find. Yet, as another participant in the discussion remarked, a problem set in a classroom does not necessarily imply the answer is known. In problem-based learning, for example, problems are open-ended, even if the starting point (the problem) is closed from the students' perspective. Geoff Wake (@geoffwake1), Associate Professor in Mathematics Education at the University of Nottingham, posted the following comment: "It's useful to think about the difference between solving a problem and problem solving." I took this to mean that ‘solving a problem’ is a restricted process with a closed beginning and end, but ‘problem solving’ is a creative process of applying generic skills, such as Polya’s heuristics, to an open-ended problem.
The discussion echoes a distinction between two different types of problems that Polya himself identifies in How to Solve It. On the one hand, a problem to find aims to "to find a certain object” and, in order to achieve the aim, the solver must know the problem’s principal parts, the unknown, the data and the condition. On the other hand, a problem to prove aims to show conclusively that a certain clearly stated assertion is true or false. Its principal parts are the hypothesis and the conclusion of the theorem to be proved or disproved. We note that, while the problem to find could be “theoretical or practical, abstract or concrete, serious problems or mere puzzles,” the problem to prove lies exclusively in the domain of mathematics.
It is a short step to linking a problem to find with the problem solving strand of the National Curriculum (including 'non-mathematical' contexts) and the problem to prove to the mathematical reasoning strand. However, the problem to prove is not synonymous with inquiry. In promoting students’ questions, exploration and conjectures, inquiry involves far more than a deductive proof of a theorem. While in Inquiry Maths lessons the teacher might help students formalise their ideas into a problem to prove, the starting point of inquiry (the prompt) cannot be likened to either of Polya’s problem types.
Paradoxically, inquiry could involve what Polya describes as routine problems. These focus on the mechanical performance of operations and "can be solved either by substituting special data into a formerly solved general problem, or by following step by step, without any trace of originality, some well-worn conspicuous example." When students select a regulatory card to practise a procedure, the teacher might suggest answering routine problems, although restricting students exclusively to this type of problem is, according to Polya, “inexcusable" and, we might add, antithetical to the principles of inquiry.
After this discussion, we must amend our characterisation of the relationship between inquiry and problem solving. We maintain the distinction – and even the separation – between classrooms in which problems to find (with their ‘open middles’) predominate and classrooms that emphasise inquiry processes linking exploration and deduction. However, problem solving can develop into inquiry when, for example, students change the conditions in the problem and study the relationship between the new solution and the old one. Nevertheless, I cannot conceive of a situation when or a reason why an open inquiry would be restricted to a problem-solving process. While inquiry skills enable students to attempt to solve problems, the converse is not true. Heuristics employed in problems to find would not, on their own, enable students to generate mathematical inquiry.

Andrew Blair 
February 2017

Shanghai Maths

posted 29 Nov 2015, 04:03 by Unknown user   [ updated 13 Jul 2016, 13:15 by Andrew Blair ]

Shanghai Maths: teacher led and student centred?
In November 2015 the Shanghai Exchange reached the secondary school stage with maths teachers from China teaching year 7 and 8 classes in English schools. The maths hubs organised observations of the lessons, which were accompanied by an introduction to Shanghai Maths from teachers who had visited China in September and an analytical discussion after the lessons. The two teachers from the Sussex Maths Hub who ran the event I attended did an excellent job of reflecting on their experiences in Shanghai in an insightful way. Unlike the Schools Minister who writes that the exchange is about showing teachers Shanghai's “perfect formula for learning," they acknowledged the good practice that already exists in the UK. The two teachers from Shanghai also demonstrated a highly professional and critical approach when they analysed their lessons with the observers.
It is indicative of the consistency of practice in Shanghai that the event did not teach me much more about the system than I had learned from attending a hub event during the primary exchange. In the post I wrote after that event, I characterised Shanghai Maths as teacher-led and focused exclusively on mathematical concepts and procedures (as opposed to inquiry which can be student-led and involves reflection and regulation). However, since then, the organisers of the exchange have described Shanghai Maths as “teacher led but student centred.” While this phrase is intriguing, it (or, rather, half of it) proves to be misplaced.
If we take ‘child-centred’ to mean that teachers adjust their teaching by taking account of students’ levels of understanding, then at no stage did the lesson I observed appear to be child-centred. There was neither assessment for learning, nor 
The lesson
The 50-minute lesson I observed was about multiplying fractions and involved a year 8 class (set 2). The teacher based the four phases of the lesson on twelve expertly-crafted questions, with each one increasing the level of complexity or introducing a new concept or procedure. The first phase was typical in combining a short episode of whole-class interaction followed by practice. It started with 4/5 x 2/3which, after a very brief discussion, one student answered correctly. The teacher then showed a pictorial representation of the calculation on a 3 by 5 square grid, confirming the answer as 8/15. (In discussion with me afterwards, the teacher said he might have extended this phase in Shanghai by asking students to create their own representations and showed me the blank squares of paper he had prepared for this.) Students then practised by finding the answers to four more questions. This phase ended with the teacher presenting the general equation:
There was no discussion of why q ≠ 0 and n ≠ 0. 
The next stage of the lesson was designed to teach students to cancel down before completing the calculation. The second of two questions used to show the value of this procedure (5/48 x 24/15) led to one of the two extended whole-class episodes (‘extended’, that is, in the context of this fast-paced lesson) because students did not identify 24 as a factor of 24 and 48. One student eventually arrived at 24 after the teacher had refused to accept other common factors. The third phase involved just one question: 3/4 x 7. The students invariably gave the answer as 3/4 because 3/4 x 7 = 21/28 = 3/4. The teacher explained how this could not be true by using repeated addition. The second ‘extended’ whole-class episode then started when students struggled to recognise that 7 could be written as 7/1, which would have allowed them to see multiplying a fraction by a whole number as a special case of the general equation. Each of the students’ suggestions was written on the board, but was not discussed, and was rubbed off when superseded by the next suggestion. With time running out, the teacher tried to initiate the fourth stage by introducing 4/9 x 131/2, but the bell went signalling the end of the lesson. In discussion, he told me that he had designed one more stage involving the multiplication of a proper fraction, a mixed number and a whole number.

assessment of learning. Questioning was focussed on getting the required answer and did not probe students’ understanding. Exercise books were treated as note books without any evidence of a teacher-student dialogue. While it is easy to discuss a lesson in terms of what it did not contain, I have seen teachers in the UK threatened with competency proceedings for teaching lessons that did not include regular assessment from which to show progress. One fellow observer commented to me that a UK teacher might be in trouble if observed teaching the lesson. 
Interestingly, then, the Department for Education’s promotion of Shanghai methods might founder on the systemic demands it already makes of teachers through Ofsted and senior leaders. Of course, these considerations in no way count against Shanghai Maths, but they do remind us how difficult it will be to transplant the method into the English education system.

The description of ‘child-centred’, it transpires, relates more to the norms embedded in the culture of Shanghai classrooms. The exchange organisers herald such norms as “students commenting on each other’s work” and a “relentless insistence on pupils giving reasons." These were not evident in the lesson I observed. Indeed, the lesson was designed purely at a mathematical level with a focus on precise questions that aimed to develop understanding in small steps. There was no attempt to develop what Professor Paul Cobb has called social and socio-mathematical 
norms about how to discuss and explain.
It seems to me that if these norms are to become common in English classrooms, then some thought will have to be given to their development. Paradoxically, the Shanghai model might not be the best vehicle to do this because the social and socio-mathematical seem to be taken for granted at the stage of designing the lesson. Rather, an inquiry model, in which students learn how to construct mathematical understanding, is better suited to achieving these norms.

Even if students had been expected to give reasons in the lesson, Shanghai Maths still cannot be considered to be child-centred. There is no acknowledgement of or adjustment for students’ different levels of prior knowledge, no alternative routes to understanding the concept, no encouragement of student questioning and certainly no opportunity for students to participate in the direction of the lesson as would occur in inquiry lessons. Rather, Shanghai Maths is mathematics-centred or, it is more accurate to say, centred on a conception of the subject as a series of tiny increments in a logical progression.
This definition is far from the idea of mathematics as a creative human construction that you would find in inquiry classrooms. The teacher has a script – an expertly designed script, but a script nonetheless. Shanghai Maths is, therefore, a teacher-led, tightly-controlled model of teaching.

Andrew Blair
November 2015

Postscript: It was argued on social media that this post makes general comments about Shanghai Maths based on the observation of only one lesson. Even if this were true, another post about a different Shanghai lesson in south-west London shows that the lesson I observed is not atypical. The lessons were similar and, in some respects, identical. The criticism on social media might have originated in the fact that the analysis above ignores the hubs' preferred analytical framework. The hubs encouraged observers to study lessons in terms of ideas, such as concept/non-concept and intelligent practice, that are said to underpin Shanghai teaching. However, the concepts of child-centred and social and socio-mathematical norms have broader relevance and greater validity when analysing the introduction of a model of teaching into a new environment. 

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