ORIGINAL ARTICLE

Designing and using professional development resourcesfor inquiry-based learning

Malcolm Swan • Daniel Pead • Michiel Doorman •

Ad Mooldijk

Accepted: 11 July 2013

� FIZ Karlsruhe 2013

Abstract This paper describes an attempt to design,

analyse and refine professional development (PD) resour-

ces that encourage the implementation of inquiry-based

learning (IBL). We describe the iterative development of

the resources in England with over 100 mathematics

teachers from secondary, tertiary and adult education and

then analyse the impact these resources had on teachers’

beliefs and practices and the issues arising. This evaluation

revealed that teachers had moved away from transmission-

based orientations, encouraged by the use of less structured

tasks and sample lesson plans, but some found it difficult to

adopt IBL pedagogies. The most significant issues for

teachers may be summarised as: confusing IBL with ‘dis-

covery’ learning; developing and managing collaborative

cultures within the classroom; and planning lessons that

adapt to the emerging needs of learners.

1 Introduction

We are currently witnessing an explosion of international

interest in mathematics teacher education, fuelled partly by

the desire to make mathematics accessible to all (Adler,

et al. 2005) and also by a growing concern to reverse the

decline in students’ enthusiasm for mathematics and the

consequent drop in the numbers of students pursuing sci-

entific careers (Mullis, et al. 2008; Rocard 2007). In

response, many initiatives have appeared in order to support

teachers in moving away from traditional transmission-based

approaches towards ‘inquiry-based’, ‘student-centred’,

collaborative pedagogies, such as those currently com-

missioned by the EU. One such project is the project

PRIMAS: Promoting Inquiry-based learning in Mathe-

matics and Science (PRIMAS 2012).

In the context of this reform agenda, the design of rep-

licable professional development models is an increasingly

important focus for research. Most teacher education

research is conducted with pre-service or practicing teach-

ers with whom researchers normally work and is not ‘scaled

up’ (Adler, et al. 2005). In our work, we are engaged in a

design-research process that seeks to progressively design,

analyse and refine professional development processes and

products that may be used to help stimulate PD led by

others. In this paper we describe the design of a professional

development (PD) intervention program for mathematics

teachers, describe its impact on the beliefs and practices of

the participants and the pedagogical issues that arise.

The PD program was intended to support the investi-

gation by teachers of effective inquiry based-learning

(IBL) pedagogies. In the ideal IBL classroom, students are

active participants. They observe and formulate questions;

if problems are too complex, they simplify or model; they

make reasoned assumptions, collect and analyse data, make

representations, and make connections with what they

already know. They interpret findings, check that they are

sensible and share them with others. The teachers’ role is

not to stand back and expect students to discover every-

thing for themselves; it is rather to scaffold the processes of

inquiry through the use of carefully designed tasks and

structured lessons (Artigue & Blomhoej 2013).

M. Swan (&) � D. Pead

Centre for Research in Mathematics Education, School of

Education, University of Nottingham, Jubilee Campus,

Wollaton Road, Nottingham NG8 1BB, UK

e-mail: malcolm.swan@nottingham.ac.uk

M. Doorman � A. Mooldijk

Freudenthal Institute for Science and Mathematics Education,

Universiteit Utrecht, PO Box 85170, 3508 AD Utrecht,

The Netherlands

123

ZDM Mathematics Education

DOI 10.1007/s11858-013-0520-8

Inquiry-based learning is a complex, multifaceted con-

struct. The PD program was designed to help teachers

study and synthesise effective component teaching strate-

gies. These include: choosing substantive tasks that are

extendable, encourage decision-making, creativity and

higher order questioning (Ahmed 1987; Watson & Mason

1998); working collaboratively and developing dialogic

talk (Alexander 2008; Mercer 1995); building on students’

prior knowledge (Black & Wiliam 1998); encouraging

students to critically examine alternative approaches and

achieve consensus (Inoue 2011). It is not the purpose of

this paper, however, to evaluate IBL or attempt to answer

its critics. Suffice it to say that following an international

report (Rocard 2007), the promotion of IBL is currently a

priority area for science and mathematics education in the

European Union.

In this paper, we seek insights into two research

questions:

1. How might PD resources be designed to foster

pedagogies supporting IBL?

2. What impact might such resources have on the

practices and beliefs of teachers, and what pedagogical

issues arise as they use them?

The first question will be addressed through a discussion

of the theoretical basis for the PD (Sects. 2 and 3) and its

iterative development (Sects. 4 and 5) using a design

research paradigm. The second question will be addressed

by considering empirical evidence for the effects of the PD

during its design and subsequent take-up as part of the

PRIMAS project (Sects. 6 and 7).

2 A theoretical approach to designing PD

Our work may be characterised as one of ‘design research’.

Design-based research is a formative approach to research,

in which a product or process (or ‘tool’) is envisaged,

designed, developed and refined through cycles of enact-

ment, observation, analysis and redesign, with systematic

feedback from end-users (Barab & Squire 2004; Bereiter

2002; Cobb, et al. 2003a; DBRC 2003, p. 5; Kelly 2003;

Swan 2006a, 2011b; van den Akker, et al. 2006). Its goals

are to create innovative tools for others to use, to describe

and explain how these tools function, account for the range

of implementations that occur and develop principles and

theories that may guide future designs. In this case, the

tools are a PD process and associated resources.

The design of the PD drew on social constructivist and

socio-cultural theories of learning. From this perspective,

teachers learn just as students do; they participate in social

practices and through active engagement develop and

internalise new ways of speaking and thinking. Goal-

directed and tool-mediated activity are fundamental to this

process and these are influenced by the rules (e.g. the

curriculum goals); the communities within which teachers

play a part (within the PD events and within schools) and

the roles of participants within these communities (Enge-

strom 1999; Jaworski 2008; Vygotsky 1978; Wertsch

1991).

The goals of the PD design were to support the collab-

orative investigation by teachers of effective IBL pedago-

gies. Conceptual and physical resources (the tools) were

found necessary to stimulate activity toward these goals.

These included novel ‘genres’ of classroom activity,

illustrated with classroom videos, and sample lesson plans.

The design was planned to work at two levels; activities for

the PD events were designed to mirror the intended

classroom experiences of students. Rules and communities

have a profound influence on the impact of PD. Teachers

must integrate IBL into the culture of the school with its

own curriculum goals, and they are expected to interact

with and justify their actions to colleagues. As for roles, the

power relations within PD are important. We do not want

to appear authoritarian, telling teachers how to teach and

limiting their creativity; neither do we want to deprive

teachers of the wisdom of well-researched models. We see

our role as PD designers, as selecting and suggesting

potentially powerful stimuli and tools (e.g. PD structures,

lesson plans) leaving it for teacher educators and teachers

to adapt, apply and evaluate in local contexts. These tools

should enable teacher educators and teachers to be chal-

lenged and to act reflectively in new ways.

Our PD is intended to allow cycles of discussions of

pedagogical issues, classroom challenges, and reflective

feedback (Even 2005; Muller, et al. 2011; Swan 2011a).

First, teachers work on collaborative classroom activities

that illustrate the pedagogical challenges of IBL. Second,

they watch other teachers using these same activities on

video. These are expected to provide a ‘challenge’ to

existing practices. Third, teachers are encouraged to adapt

and try the activities in their own classrooms, supported by

sample lesson plans. Fourth, teachers meet together to

share their classroom experiences, discuss the pedagogical

implications and reflect on the growth of new practices and

beliefs. This cyclical process is repeated at successive PD

events, as new pedagogical issues are addressed. This

process has resonances with Japanese Lesson Study (with

its pattern of investigation, planning, research lesson and

reflection (Lewis, et al. 2009; Stigler & Hiebert 1999))

though there are also major differences. Teachers are given

sample lesson plans to adapt and it is not assumed that

teachers are able to observe each other’s lessons nor that an

outside expert is available to support the extensive analysis

of all lessons. These differences we hope increase support

and scalability in a European context.

M. Swan et al.

123

3 Characterising teachers’ beliefs and practices

Ultimately, PD attempts to influence the beliefs and prac-

tices of teachers. Research has shown that teachers hold on

to their beliefs and practices tenaciously, and attempts to

dislodge or replace them through rational argument and

persuasion are usually unsuccessful (Kagan 1992; Nespor

1987). In our PD, rather than seeking to persuade teachers

to change beliefs so that they will behave differently, we

invite teachers to take risks and adopt new practices so that

they have cause to reflect on and perhaps modify their

beliefs (Fullan 1991). As Guskey (1986) noted, profes-

sional development programmes are mostly unsuccessful in

modifying beliefs, but when teachers are encouraged to

adopt a procedure and find that it improves student moti-

vation and achievement, significant changes in attitude

may be attained. The design of our PD thus attempts to

offer teachers opportunities ‘to doubt, reflect and recon-

struct’ IBL pedagogies in an unhurried, ‘safe’ environment

(Wilson & Cooney 2002). In evaluating our trials of the

materials we therefore sought to develop a framework for

describing the evolution of beliefs and practices.

Our description of the beliefs of teachers are based on

characterizations described by Ernest (Ernest 1991a,

1991b) and Askew (1997), elaborated in Table 1.

Ernest suggests that a teacher’s belief system has three

components; the teacher’s conception of the nature of

mathematics as a subject, of the nature of mathematics

teaching and of the process of learning mathematics.

Askew characterized the orientations of teachers towards

each of these components as transmission, discovery or

connectionist. These categories are ‘ideal types’ and an

individual teacher’s beliefs may combine elements of each

of them, even where these appear to conflict. These

categories were originally derived from studies of primary

teachers and we recognize that they take on different

interpretations in other settings. For example, we have

found secondary and further education teachers attended

particularly to the individual learning and collaborative

learning aspects of the ‘discovery’ and ‘connectionist’

orientations respectively. It may further be noted that both

discovery and connectionist orientations may foster IBL,

however both empirical and theoretical analyses would

suggest that connectionist orientations are more effective

(Askew, et al. 1997; Kirschner, et al. 2006).

Many researchers report inconsistencies between what

teachers say they believe about teaching and learning and

what they do in practice (Fang 1996). Beliefs may be

compromised in practice by day-to-day realities of class-

room management, student expectations, available resour-

ces and so on. One must also be suspicious of self-reporting

by teachers as they commonly claim changes that appear

exaggerated to an outsider. Often cited is the case study of

Mrs Oblier (Cohen 1990), who reported revolutionary

shifts to her practice yet, when observed closely, these

were revealed to be cosmetic. In evaluative trials of the PD

material, where opportunities for direct classroom obser-

vation was often limited, we therefore sought students’

views of teachers’ practices as well as teachers’ self-

reported practices. The evolution of the questionnaires used

is described more fully in Swan (2006b).

4 Structure and purpose of the PD units

The PD units were designed to support the investigation by

teachers of effective IBL pedagogies. The units are activ-

ity-based; and are built around the development and

Table 1 Beliefs about mathematics, teaching and learning

Mathematics is…Transmission: A given body of knowledge and standard procedures. A set of universal truths and rules that need to be conveyed to learners

Discovery: A creative subject in which the teacher should take a facilitating role, allowing learners to create their own concepts and

methods

Connectionist: An interconnected body of ideas which the teacher and the learner create together through discussion

Learning is…Transmission: An individual activity based on watching, listening and imitating until fluency is attained

Discovery: An individual activity based on practical exploration and reflection

Connectionist: An interpersonal activity in which learners are challenged and arrive at understanding through discussion

Teaching is…Transmission: Structuring a linear curriculum for the learners; giving verbal explanations and checking that these have been understood

through practice questions; correcting misunderstandings when learners fail to ‘grasp’ what is taught

Discovery: Assessing when a learner is ready to learn; providing a stimulating environment to facilitate exploration; and avoiding

misunderstandings by the careful sequencing of experiences

Connectionist: A non-linear dialogue between teacher and learners in which meanings and connections are explored verbally.

Misunderstandings are made explicit and worked on

PD resources for IBL

123

analysis of real lessons. Each unit includes a PD session

guide for a facilitator (who may be a teacher) and handouts

for participants, as well as sample classroom activities and

lesson plans. The video sequences show teachers trying

these materials with their own classes for the first time.

Their status is not to exemplify ‘best practice’ but rather to

illustrate the issues that arise as teachers attempt to modify

their own practice. They invite participants to share in a

learning process.

Moving towards an inquiry-based learning (IBL)

approach raises many pedagogical issues for teachers: How

can I encourage my students to ask and pursue their own

questions? How can I help students to follow up these

questions in profitable ways? How can I teach students to

work cooperatively and to learn from each other? The PD

is structured in a way that systematically tackles such

questions in seven units (Table 2). These units are freely

available online (Swan & Pead 2008, 2011) and are

intended to provide a cumulative experience, and the

gradual integration of the pedagogical challenges are

illustrated by lesson plans.

We illustrate the activity sequence with one unit, Asking

Questions that Promote Reasoning.

1. Make existing beliefs and practices explicit. Teach-

ers are asked to identify the different types of questions

they ask, the different functions served, the frequency

they use them, and the most common mistakes they

make when asking questions.

2. Consider ‘contrasting practices’. Teachers are asked

to reflect on the types of questions that are most likely

to encourage student inquiry, and are offered princi-

ples, drawn from research, for effective questioning

(Watson & Mason 1998). Participants are shown a

video of a teacher attempting to put such principles

into practice, are invited to identify occasions where

the teacher employs each principle and then consider

the effects of such questioning.

3. Develop a lesson plan. Teachers are offered a situation

and are asked to prepare an introduction, ground rules

for interaction (e.g. ‘think, pair, share’), intervention

strategies, and specific questions that may be asked at

different points in the lesson. To support this, they are

offered a lesson plan showing a possible structure for a

1-h lesson. An outline of this is shown in Table 3. The

full lesson plan is more detailed, offering sample

questions teachers might ask at various stages of the

lesson and typical student responses.

4. Teach a lesson. Teachers use their collaborative lesson

plan incorporating the questioning strategies they have

considered. They are encouraged to audio record some

of the questions they use to share in the post-lesson

discussion.

5. Analyse the lesson. After teaching the lesson, partic-

ipants meet again to share experiences of questioning

and reflect further at how students may be made more

aware of the value being placed on reasoning. Further

strategies for making reasoning more ‘visible’ are

discussed.

The important role of the teachers in the development of

these units must be mentioned. During early drafts of the

units, groups of teachers were invited to use them. Each

group met for 1 h to reflect on existing practice and discuss

the pedagogical focus of the unit. Teachers then taught

their own adaptations of the sample lesson plan, paying

particular attention to its pedagogical focus. Finally they

met up again to discuss what they had learned. This process

was videotaped and analysed. The issues raised by teach-

ers, their enacted lessons and their reflections were then

incorporated into the revised, now video-enhanced PD

resources. Thus the teachers’ own interpretations of the

pedagogical ideas were systematically built into the

resources.

5 The historical development of the PD units

The PD units evolved slowly over several years and have

now been adapted for different contexts. When opportunity

and funding has allowed, empirical data have been col-

lected to evaluate the impact of the PD and lesson plans on

teachers and students. In this section we describe some data

gathered from three early cycles of the design. These

involved a study with 44 teachers of low-attaining

16–19 year old students, a study with 24 teachers of adult

numeracy classes and a study with 30 secondary school

teachers (Table 4). Later in the paper, we report on more

qualitative data from later trials of the material as they have

been used more widely. At each stage of development, the

content of the PD was adapted to incorporate issues raised

by teachers.

The PD programs during the trials in Table 4 each

consisted of an initial two-day residential workshop, fol-

lowed by a number of 1-day follow-up meetings. In

between, teachers were invited to use the lesson plans in

their own classrooms.

The first version of the PD resources was constructed in

a limited domain: the use of collaborative learning activi-

ties in the teaching of algebra to low-attaining students

aged 16–19. A series of algebra lesson plans were devel-

oped with eight teachers over 1 year. These teachers were

filmed using and discussing the lessons and this was

incorporated into the PD. The lessons included specific

‘genres’ of classroom activities that were designed to

engage students’ concept development: classifying

M. Swan et al.

123

mathematical objects, interpreting multiple representa-

tions, evaluating mathematical statements, creating and

solving students’ own problems, and generalizing existing

problem situations. In the second year the resource was

used within a four-day course for 44 teachers spread over

6 months. In between meetings, teachers tried a variety of

classroom tasks within each genre. The results of this

empirical study are summarized below (Swan 2006a,

2007). The resulting resources were published and copies

were sent to all Further Education Colleges in England by

the Learning and Skills Development Agency, a govern-

ment body (Swan & Green 2002).

Subsequently, government funding (from the then DfES

Standards Unit) permitted us to develop a second version

of the PD resource and make this freely available to all

post-16 providers in England (Swan 2005). This was

Table 2 The PD units for PRIMAS

These are downloadable from: http://www.primas-project.eu

1. Student-led inquiry. In this unit, teachers are presented with phenomena and are then invited to pose and pursue their own questions. They

thus experience what it feels like to think like a mathematician or scientist and are then invited to try a similar activity with their own

students and reflect on the outcomes

2. Tackling unstructured problems. This unit invites teachers to consider the decisions that we make for students when we present them with

structured problems. It invites comparison of structured and unstructured versions of problems and considers the demands and challenges

unstructured problems present in the classroom. Teachers are invited to try out unstructured versions of textbook problems in their own

classrooms and report back on their experiences

3. Learning concepts through IBL. This unit considers how the processes of inquiry-based learning may be integrated into the teaching of

content. Often, these two aspects of learning are kept separate: we teach content as a collection of facts and skills to be imitated and

mastered, and/or we teach process skills through investigations that do not develop important content knowledge. The integration of

content and process raises many pedagogical challenges. The processes under consideration here are: observing and visualizing, classifying

and creating definitions, making representations and translating between them, finding connections and relationships, estimating,

measuring and quantifying, evaluating, experimenting and controlling variables

4. Asking questions that promote reasoning. This unit contains a selection of stimuli designed to help teachers to reflect on: characteristics of

their questioning that encourage students to reflect, think and reason; ways in which teachers might encourage students to provide

extended, thoughtful answers, without being afraid of making mistakes; the value of showing students what reasoning means by ‘thinking

aloud’ as problems are worked on in collaborative classroom environments. As before, teachers systematically try to develop their own

classroom questioning and report back on what happens

5. Students working collaboratively. This unit is designed to offer teachers the opportunity to reflect on the characteristics of student–student

discussion that benefit learning; to recognize and face their own concerns about introducing collaborative discussion; to explore researched

techniques for promoting effective student–student discussion; and to consider their own role in managing student–student discussion.

They plan and carry out discussion-based lessons and report back on what happened

6. Building on what students already know. This unit considers the different ways teachers can use formative assessment techniques to make

effective use of students’ prior knowledge. It focuses on the following questions: How can problems be used to assess performance? How

can this assessment be used to promote learning? What kinds of feedback are most helpful for students and which are unhelpful? How can

students become more engaged in the assessment process?

7. Self and peer assessment. This unit encourages discussion of the following issues: How can we help students to become more aware of IBL

processes, and their importance in problem solving? How we can encourage students to take more responsibility for their own learning of

IBL processes? How can students be encouraged to critically assess and improve each other’s work?

Table 3 A generic lesson plan for a 1-h lesson

Teacher introduces lesson (5 min) Students are given an unfamiliar task, for which they have not been explicitly prepared. The teacher

engages interest in the problem, introducing the context and encouraging students to formulate

questions

Individual thinking time (5 min) Students are encouraged to work independently and develop initial questions/ideas for approaching

the task

Students work collaboratively (20 min) Students work in groups of two or three on the task. Teacher monitors work and seeks ideas that

might be shared with whole class later. Teacher interacts with students that are stuck offering

strategic guidance rather than technical help

Whole class shares approaches being

used (10 min)

Teacher selects students to present their approaches to whole class. Teacher introduces fresh ideas

that have not been considered. (‘‘Here are ideas I observed in another class’’). Students discuss the

possible approaches—strengths and weaknesses

Students continue to work collaboratively

on task (10 min)

Students have second attempt using ideas that have been shared

Whole class reports on their reasoning.

(10 min)

As students report on reasoning used, teacher draws attention to significant ideas, makes connections

and generalisations to other work. Teacher may conclude by modelling reasoning for class

PD resources for IBL

123

developed over 2 years, and expanded the content from

algebra to relate to other content. Over 50 lesson plans

were written and piloted with 90 teachers. Again videos

were made and incorporated into the PD materials.

Unfortunately, however, no systematic study of the effects

of this PD was made. During regular inspections of post-16

education institutions, however, Ofsted1 came across these

resources in use and they noted their use in their report:

These materials encouraged teachers to be more

reflective and offered strategies to encourage students

to think more independently. They encouraged dis-

cussion and active learning lessons. While some

colleges were just dipping into the resources, a few

had used the full package to transform teaching and

learning across an entire mathematics team.

(Ofsted 2006 paras 32, 33).

Then a third UK government agency (the Department

for Education and Skills), embarked on a further program

(Maths4Life) to improve the teaching of adult literacy and

numeracy. We were offered funding to develop PD

resources for this and used this as an opportunity to further

develop the resources. This time, a formal evaluation was

conducted (Swain & Swan 2007, 2009; Swan & Swain

2010) and some of the results from this are shown below.

Up to this point, the emphasis in the PD resources had

been concerned with the pedagogical skills necessary for

teaching mathematical concepts. A further project, funded

by an independent charity, Bowland Maths, allowed us to

further extend this work to consider the development of

mathematical processes with lower secondary classrooms.

These processes included those involved in mathematical

modelling: simplifying and representing, analysing and

solving, interpreting and evaluating, communicating and

reflecting. This was the first version of the resources that

incorporated the IBL units in Table 2 (Swan & Pead 2008).

This was evaluated with funding from one local district in

England (Somerset). The results are summarized below

(previously unpublished). Currently, the PD materials are

being adapted for secondary classrooms around Europe as

part of the PRIMAS project (Swan & Pead 2011). This

adaptation involves removing specific cultural content

(such as references to the English National Curriculum)

and including some emphasis on science learning. We are

currently redeveloping the resources to support the devel-

opment of the mathematical practices (or processes) in the

common core state standards in the US (Swan, et al. 2012).

Evaluation of the use and impact of these is still in its early

stages.

6 The impact of the trials of the PD units on teachers

Teachers were asked to rate their own beliefs by assigning

weightings to each of nine descriptive categories (Table 1).

These were cross-validated with more qualitative descrip-

tions which showed remarkable consistency, when checked

by independent researchers. The results concerning chan-

ges in beliefs of the teachers in England during the pilot

studies (averaging over Mathematics, Learning and

Teaching) are given in Table 5.

The FE teachers and the secondary teachers both began

with predominantly transmission views. Both groups were

focused on getting students through the examinations. The

adult numeracy teachers, on the other hand, had lower

expectations, and worked with much smaller classes that

included students with severe learning difficulties. These

teachers had a different profile, and the discovery orien-

tation was more evident in this group.

Generally speaking, in the post-questionnaire in all three

studies, teachers reported a shift away from transmission

orientations (see Table 4). Initially the percentages of

Table 4 Three formative trials of the PD development

Further education (FE) teacher study Adult numeracy teacher study. Secondary teacher study

Teachers

involved

44 teachers from 44 post-16 colleges (35

provided data)

24 teachers from 12 organizations

(17 provided data)

30 teachers from 17 schools (21 provided

data)

PD structures 4 days over 6 months (2 ? 2 9 1 days) 5 days over 9 months

(2 ? 3 9 1 days)

6 days over 10 months.

(2 ? 4 9 1 days)

Mathematical

focus

Algebraic concepts Mathematical concepts Mathematical concepts and processes

PD content

addressed

(Table 2)

3, 4, 5. 2, 3, 4, 5, 6. 1, 2, 3, 4, 5, 6, 7.

Data collection Teacher and student questionnaires.

Limited lesson observation. Interviews

Teacher questionnaires. Extensive

lesson observation. Interviews

Teacher and student questionnaires.

Limited lesson observation. Interviews

1 Ofsted, the Office for Standards in Education, is an independent

national inspection service that reports directly to Parliament in the

UK.

M. Swan et al.

123

teachers expressing overall preferences for Transmission,

Discovery, Connectionist were respectively 44:22:23. After

the PD these ratios were 16: 21:53.

In Table 5 we have aggregated each of the 71 teachers’

reported orientations and allocated them to the most pre-

dominant one.

The trajectories of these changes are particularly inter-

esting. Some of the transmission teachers claimed little

change in their beliefs. These beliefs had been long-held

and were associated with a need to be ‘‘in control’’ and a

belief that students could not cope with IBL pedagogies:

Students are more confident with an imitation

approach. Only stronger students can cope with non-

linear dialogue.

(FE teacher).

Others blamed examination pressures for an unwilling-

ness to change:

I feel that these (lessons) are very good for learning

breakthroughs, but I don’t think they are going to get

the bulk of my students through their exams. I think

you need a ‘crammed’ approach. This is the big issue

for me. I’d be quite happy to use these materials

every lesson, the time went ‘like that’ and it’s great to

see people not yawning and actually enjoying them-

selves. You don’t have discipline problems. But, I

feel that for an exam, I’ve got to feel that I am giving

them the knowledge that they need to pass that exam

and I feel that I can do that through the traditional

approaches and a bit of bullying.

(FE teacher)

Some teachers moved from ‘transmission’ towards

‘discovery learning’ orientations. These teachers clearly

recognized the shortcomings of transmission methods and

that ‘telling’ was not an effective way of helping learners to

develop concepts and processes. In reaction to this, it

appeared that they moved to a position in which they

seemed uncertain of their own role in the classroom,

beyond that of ‘facilitator’:

It is allowing them to make discoveries for them-

selves rather than you writing it up on the board […]

It is their discovering, not mine; it is nothing to do

with me really. I have just to keep an eye on it.

(Adult numeracy teacher).

Three-quarters of the teachers that began with a dis-

covery orientation moved towards a connectionist orien-

tation. These teachers appear to have adopted a more

interventionist role, and were prepared to challenge and

discuss concepts and approaches with groups of students in

a collaborative manner.

These and other qualitative data suggested to us that

many teachers may follow a transmission to discovery to

connectionist trajectory as they at first withdraw support

from students and then recognise the need to redefine their

own role in the classroom. If, as some researchers assert

(Askew, et al. 1997; Kirschner, et al. 2006), discovery

approaches are less effective than transmission-based ones,

this may have the consequence that teachers at first become

less effective, before they become more effective as they

develop along this trajectory.

6.1 The impact on practices

We further examined both teachers’ and students’ percep-

tions about the changes that had occurred in teachers’

practices. Both open and closed response questionnaires

were used. These were given by the researchers to teachers

during the PD sessions and by the teachers to students

in the classrooms. The rationale and design of these

questionnaires is given in Swan (2006b). On pre- and post-

questionnaires, teachers were asked to describe how

frequently they now displayed 25 classroom behaviours on

a 5-point scale: 1 = almost never to 5 = almost always.

The results (Table 6) are ranked in order of the overall

changes in means. Teachers described their own practices

as mostly ‘teacher-centred’ at the outset of the PD and

reported that these had changed towards more ‘student-

centred’ practices by the end. This is reflected in the table

Table 5 Trajectories of changes in predominant beliefs (%)

From To

Transmission (%) Discovery (%) Connectionist (%) No overall orientation (%) Total (%)

Transmission 15.1 11.0 11.0 6.8 43.8

Discovery 0.0 2.7 16.4 2.7 21.9

Connectionist 0.0 5.5 17.8 0.0 23.3

No overall orientation 1.4 1.4 8.2 0.0 11.0

Totals 16.4 20.5 53.4 9.6 100.0

PD resources for IBL

123

by the S statements increasing in frequency, while the T

statements decreased in frequency (these labels did not

appear on the questionnaire). Teachers reported more col-

laboration and discussion (including the discussion of

mistakes) giving more choice about the questions to tackle,

fewer graded exercises, students were encouraged to be

more creative, lessons were more responsive to student

needs and less predictable.

As we have said, self-reporting is unreliable, so to

confirm teachers’ accounts, we asked students to describe

their teachers’ behaviours. In the FE project, this was only

done on one occasion as the course was short and at the

Table 6 Changes in teachers’ practices (self-reported)

Further

education

n = 35

Adult numeracy

n = 17

School teachers

n = 21

All teachers

n = 73

Pre-

mean

Post-

mean

Pre-

mean

Post-

mean

Pre-

mean

Post-

mean

Pre-

mean

Post-

mean

Change

Post-

pre

16 S Learners work collaboratively in pairs or small

groups

2.5 3.6 3.4 4.3 2.9 3.5 2.8 3.7 0.9

15 S Learners learn through discussing their ideas 2.6 3.5 3.7 4.3 3.1 3.7 3.0 3.7 0.7

5 S Learners choose which questions they tackle. 1.9 2.6 2.4 2.9 2.1 2.9 2.1 2.8 0.6

12 S I am surprised by the ideas that come up in a

lesson

2.0 2.6 3.0 3.5 2.3 2.9 2.3 2.9 0.5

28 S I jump between topics as the need arises 2.5 2.8 3.0 3.8 2.4 3.0 2.6 3.1 0.5

27 S I encourage learners to make & discuss mistakes. 2.7 3.2 3.7 4.2 3.1 3.6 3.0 3.5 0.5

17 S Learners invent their own methods 1.9 2.4 2.6 3.2 1.8 2.4 2.0 2.6 0.5

7 S Learners compare different methods for doing

questions.

2.2 2.9 3.1 3.7 2.7 2.7 2.5 3.0 0.5

6 S I encourage learners to work more slowly 1.9 2.3 2.4 3.2 1.7 2.2 2.0 2.5 0.5

11 S I draw links between topics and move back and

forth between topics

3.0 3.4 4.1 3.9 2.5 3.1 3.1 3.4 0.3

23 S I teach each learner differently according to

individual needs.

2.3 2.8 3.5 3.6 2.6 2.6 2.7 2.9 0.3

22 S I find out which parts learners already understand

and don’t teach those

2.2 2.6 2.4 2.4 2.8 3.0 2.4 2.7 0.2

3 T Learners use only the methods 1 teach them 3.0 3.0 2.3 1.9 3.2 2.9 2.9 2.7 –0.2

2 T Learners work on their own, consulting a

neighbour from time to time.

3.5 3.1 2.7 2.9 3.0 2.7 3.2 2.9 –0.3

21 T I only go through one method for doing each

question

3.3 2.8 1.7 1.5 2.2 2.2 2.6 2.4 –0.3

25 T I tend to teach each topic separately. 3.2 2.9 2.7 2.2 3.2 3.0 3.1 2.8 –0.3

8 T I teach each topic from the beginning, assuming

they know nothing.

3.2 2.8 2.2 1.5 2.4 2.2 2.7 2.3 –0.4

10 T I try to cover everything in a topic 3.8 3.2 2.3 1.6 3.1 3.2 3.2 2.8 –0.4

14 T I tend to follow the textbook or worksheets

closely

3.3 2.7 1.9 1.4 2.4 2.2 2.7 2.3 –0.4

9 T I teach the whole class at once. 3.9 3.4 3.1 2.8 3.8 3.3 3.7 3.2 –0.5

1 T Learners learn through doing exercises 3.7 3.2 2.9 2.6 3.4 2.8 3.4 2.9 –0.5

13 T I avoid learners making mistakes by explaining

things carefully first.

3.2 2.7 2.6 2.1 2.7 2.2 2.9 2.4 –0.5

26 T I know exactly what maths the lesson will contain. 3.9 3.5 3.2 2.5 3.6 3.0 3.7 3.1 –0.5

19 T I tell learners which questions to tackle 4.1 3.5 2.6 2.4 3.6 2.9 3.6 3.0 –0.6

4 T Learners start with easy questions and work up to

harder questions

4.0 3.3 3.8 3.1 3.7 3.2 3.9 3.2 –0.6

Means of frequencies of self-reported practices before and after the PD, ordered by changes. 1 = almost never; 5 = almost always. First column

indicates order of items on questionnaire (standard deviations are omitted from the table for clarity, for all statements 0.7 \ SD \ 1.2)

M. Swan et al.

123

beginning students were unable to comment on teachers’

practices. We also found it difficult for the adult education

teachers to administer questionnaires to students of very

limited attainment. The results from the secondary schools

are reported below in Table 7. The student statements were

derived from the teacher statements by excluding those that

refer to a teacher’s motivation for behaving in particular

ways and strategic issues such as curriculum design and

coverage. It was felt that students would be unable to

respond to these. Noyes (2012) also used these items in a

large survey with over 2,900 11-year-old students in state-

funded comprehensive schools in the Midlands of England

and the means from this data are presented for comparison.

The rank order of teaching behaviours reported in the

pre-questionnaires for the Noyes data and the secondary

schools is almost identical. The most predominant style,

according to students, appears to be worksheet or textbook-

led; the teacher selects the questions, demonstrates the

method, expects students to practice this intensively,

working mostly on their own.

Comparing the pre- and post-questionnaire data, there is

evidence that the incidence of pair and group work, dis-

cussion and making connections between topics have all

increased, and teachers are beginning to discuss multiple

ways of doing questions. Comparing with the Noyes data,

in these schools, there is perhaps less emphasis on fol-

lowing textbooks and imitating procedures than in other

schools. A factor analysis was conducted on these state-

ments and this showed that they do not form a uni-

dimensional scale. The S statements taken alone, however,

are acceptably consistent (Cronbach alpha = 0.722) and

the pre-post changes using this measure are statistically

significant (Table 8).

7 Issues arising for teachers

In the above projects, and as the PD resources have been

disseminated more widely across the EU as part of the

PRIMAS project, we have had many opportunities to

observe and interview teachers. In this section we draw

together and exemplify some of the major findings that

have emerged in the United Kingdom and in the Nether-

lands. In order to preserve the relationship between the

findings and the contexts involved (adult education, school

education; UK, Netherlands), we present these issues study

by study rather than in a thematic manner. We then syn-

thesise the common issues that have emerged.

The principles underlying IBL are complex and teachers’

appropriation of them is often gradual, and at first, partial. In

the UK-study involving adult numeracy teachers (Swain &

Swan 2009), for example, 49 semi-structured teacher inter-

views and 110 classroom observations were carried out, and

each teacher was observed between 3 and 6 times. These

findings showed that some teachers had initially adopted

superficial understandings of IBL pedagogies, seeing IBL as

merely adding ‘fun’ or ‘enrichment’ to an otherwise dull

curriculum. Others, as noted above, confused IBL with

‘discovery learning’ and did not support students sufficiently

in their learning. This study also attempted to identify

aspects of IBL pedagogy that appear to be more accessible

for teachers and those that appear more problematic. By the

end of the PD, a majority of the adult education teachers

were observed effectively and consistently using unstruc-

tured, collaborative tasks, organising co-operative group

work and using more probing questions to assess and pro-

mote reasoning. Teachers, however, found it much more

difficult to develop connections and build on students’ prior

knowledge. These aspects require sound content knowledge

and an ability to flexibly adapt teaching to students’ needs

‘in the moment’. They also found it particularly difficult to

focus on teaching process skills for problem solving.

The issues arising from the secondary school study were

distilled from both oral and written responses made during

structured debriefing sessions with the 30 teachers. A

variety of issues emerged, which we list below. Teachers

consistently expressed surprise at increased levels of stu-

dent engagement and creativity when engaging in IBL

activity. They reported that the lesson plans were shifting

the emphasis from ‘answer-getting’ to analysing and rea-

soning, particularly when students were encouraged to

assess the work of others (in units 6 and 7). A few claimed

that this revealed misconceptions of which the teachers

themselves had been unaware. Some began to recognise the

need to reduce their expectations of what they might

‘cover’ in a lesson. Rather than completing lists of ques-

tions, they were exploring fewer, richer tasks in multiple

ways. They began to recognise the power of giving for-

mative feedback; rather than ‘correcting’ answers, asking

follow-up questions to cause further reflection.

These teachers also reported many anxieties and strug-

gles. Their main fears concerned the introduction of col-

laborative learning. They commented that discussions

reduced the ‘pace’ of lessons and made it more difficult to

‘cover’ the required curriculum content. Several reported

that their students did not want to discuss or explore

mathematics. Many students had difficulty turn-taking,

listening and elaborating each other’s reasoning, and talked

in ‘disputational’ rather than ‘exploratory’ ways (Mercer

1995). This led to concerns about maintaining classroom

control. Teachers were also concerned with accountability

issues and frequently referred to the ephemeral nature of

discussions and the lack of individual written outputs.

‘‘What hard evidence will I have of learning?’’2 The Cronbach alpha for the T statements was 0.63.

PD resources for IBL

123

In the Netherlands, the PD materials were taken up and

used as part of the PRIMAS project. Twenty teachers from

two secondary schools joined a PD program based on the

resources, from different disciplines. In one school the

teachers chose to join the program (they were allowed to

choose from a school-selected offer of PD-activities). The

teachers from the other school were asked to join by their

school authority as part of their obligatory professional

development. The teachers varied in level of experience.

Teachers were paired, so that they were able to support

each other in planning and observing lessons. They were

asked to report on lessons through structured feedback

forms (including topic, plans, experiences and new

insights). Initially, teachers were sceptical of letting the

students pose their own questions and follow their own

lines of inquiry, but they nevertheless agreed to try. They

met with mixed success. The examples below are drawn

from this work.

A mathematics teacher created a worksheet for

experimenting and explaining parallax in a ninth

grade class. The teacher hoped that the students could

find explanations within small groups. The questions

on the worksheet were open and the students did not

get any guidance during the process of searching

proofs or explanations. The teacher ran from group to

group to solve small questions or misunderstandings.

She reported that: ‘‘these kind of activities might be

suitable for better students, but not for whole class

group work’’.

(Researcher notes)

When asked to report about their experiences with new

pedagogies the teachers also reported that some students

felt uncomfortable with the new approaches. Without

careful guidance and scaffolding of their inquiry process

they became lost and unclear about the purpose of their

activity. This example illustrates again the danger of

equating IBL with ‘discovery’ approaches. One cannot

expect minimal guidance to result in learning ‘unless stu-

dents have sufficiently high prior knowledge to provide

internal guidance’ (Kirschner, et al. 2006). Our lesson

plans, however, do not advocate minimal guidance

throughout the lesson, but rather strategic support at times

of need. Teachers that used these plans met with greater

success:

Table 7 Changes in teachers’ practices (student-reported)

The teacher: Student reports

n = 330

Means

(Noyes)

Pre-

mean

SD Post-

mean

SD Change

Post–

pre

9 S Asks us to work in pairs or small groups 2.6 1.0 3.1 1.0 0.5 2.7

6 S Shows us how topics link together (like algebra and shape) 3.2 1.0 3.4 1.1 0.2 3.2

8 S Expects us to learn through discussing our ideas 3.1 0.9 3.3 1.0 0.2 3.5

4 S Lets us choose which questions we do 1.8 1.0 2.0 1.0 0.2 1.8

14 S Encourages us to make and discuss mistakes 3.5 1.2 3.6 1.1 0.1 3.4

10 S Lets us invent and use our own methods 2.6 1.1 2.7 1.2 0.1 2.6

5 S Asks us to compare different methods for doing questions 2.9 1.0 3.0 1.0 0.1 3.1

1 T Asks us to work through practice exercises 3.6 1.1 3.6 1.0 0.0 3.7

3 T Shows us which method to use, then asks us to use it 3.9 1.0 3.8 1.0 –0.1 4.2

2 T Expects us to work mostly on our own, asking a neighbour from time to 3.6 1.0 3.5 1.0 –0.1 3.8

7 T Expects us to follow the textbook or worksheet closely 3.8 1.0 3.6 1.0 –0.2 4.2

11 T Tells us which questions to do 4.4 0.8 4.2 0.9 –0.2 4.5

12 T Shows us just one way of doing each question 3.0 1.1 2.7 1.1 –0.3 3.2

13 T Teaches each topic separately from other topics 4.0 0.9 3.7 1.1 –0.3 Not used

Means of frequencies of student-reported practices before and after the PD programme, ordered by changes. A five-point scale indicating

perceived frequency of use is used: 1 = almost never; 5 = almost always. The first column indicates the order of items on questionnaire

Table 8 Changes in the ‘Student-centred’ mean rating

Pre-PD Post-PD Change in

meanMean SD Mean SD

‘Student-centred’ mean

rating

2.84 0.65 3.05 0.59 ?0.22**

** p \ 0.01, paired samples t test (n = 329)

M. Swan et al.

123

A physics teacher took an existing, highly structured,

‘‘cookbook’’, practical assignment from his own

curriculum and created a less guided version. The

original task led students step-by-step to deduce a

conclusion about the formula for converging lenses.

The teacher revised his pedagogical approach, fol-

lowing the lesson plan from the video. At first the

students were asked to think what they could inves-

tigate with the material they had been given. Students

made suggestions concerning the relationship

between the distance between the object and the lens

and the size of the image obtained. After five minutes

the teacher discussed students’ suggestions, priori-

tised possible questions to investigate, invited stu-

dents to choose a research question and then continue

with their investigation. The teacher monitored

groups and restricted feedback to those relating to

process. During the final discussion students’ findings

were presented. While no student had discovered the

formula, they had become more aware of the rela-

tionships with lenses when compared to the tradi-

tional cookbook approach. The teacher was happy

with the increased motivation of the students, their

creativity in trying to find a formula, and wanted to

change practical assignments more often into this

approach.

(Researcher notes)

From an analysis of the feedback from teachers, we

found that when the teacher used the lesson plans provided

to scaffold the students’ inquiry processes, or modelled

their own planning on these examples, the lessons were

consistently reported as more successful. Sixteen of the 18

feedback forms supported this pattern. In summary, the

Dutch teachers removed the structure from textbook

activities for experimenting with IBL-related pedagogies.

The success of their reported experiences was related to the

care with which they prepared lesson plans to structure

student activity. It is important to note here that we are not

talking about simply replacing the structured guidance

embedded within textbook tasks with similar structured

guidance given by the teacher. The structure within the

lesson plans refers not so much to cognitive features, but

rather to structuring the order and type of interaction that

students engage in. Thus a structured plan may include

periods for individual work, pair work, group discussion,

reporting back, evaluation and so on. It also attempts to

anticipate student difficulties and organise planned

responses that lead to students further engaging in reflec-

tion and discussion. The provision of adaptable sample

lesson plans that incorporate such features appear to be

valuable resources for developing new pedagogies for

inquiry-based learning.

We are currently undertaking further design research

into the use of lesson planning to address the issues that

have emerged from this paper. Only one example will be

given. As we have noted, many teachers found it difficult to

anticipate student approaches to open, unstructured tasks.

In consequence, they struggled to respond appropriately

and strategically to student learning needs as they arose in

lessons. In response to this, we are now studying the effect

of sample lesson plans that include the use of assessment

tasks that students complete, individually, before lessons,

together with a list of difficulties that students typically

experience with these tasks and sample strategic questions

that might move students’ thinking forward. In the PD,

teachers are invited to assess their own students’ work on

the assessment tasks, identify issues that are pertinent and

use the sample questions as models for their own ques-

tioning. These questions may then be written into the

teachers’ own lesson plans. These approaches are being

trialled in current PD materials supporting the introduction

of mathematical practices in the US (Swan, et al. 2012).

8 Concluding comments

At the start of this paper we set ourselves two research

questions. We first set out to study how PD resources may

be designed to foster pedagogies that support IBL. Our

discussion illustrates something of the complexity and

difficulty in designing and researching such resources in an

authentic context. The PD exhibited many features that are

reported to be ‘effective’: it was sustained, related to local

contexts (Cobb, et al. 2003b), involved teachers in active

and collective participation (Garet, et al. 1999), focused on

teachers’ knowledge of content, pedagogy and principles

(Hammerness, et al. 2005) and offered support for trans-

lating new ideas into everyday practice (Lee & Wiliam

2005). The ‘tools’ that were devised stimulated reflection

and discussion on key pedagogical issues and the diffi-

culties noted have provided ideas that will be incorporated

into subsequent designs. The design process is continually

evolving. Teachers are involved in the design, particularly

in providing examples of their practice, for other teachers

to scrutinise.

The second question set out to study the potential impact

of such resources and the pedagogical issues that arise for

teachers. The teachers claimed changes to their beliefs and

practices, and that students also confirmed changes to

teachers’ practices, but to a lesser degree. This may be due

to a combination of the teacher unconsciously adopting a

‘melange’ of old and new practices (Cohen 1990), and the

delay between a teacher first adopting and reporting a new

practice and students’ noticing it being used in a sustained,

embedded manner.

PD resources for IBL

123

Teachers frequently reported to us that their under-

standing of the new pedagogies developed slowly, and that

they continued to struggle with embedding and sustaining

them. The most significant broad issues for teachers may be

summarised as: confusing IBL with ‘discovery’ learning,

fostering the development of processes, developing and

managing collaborative cultures within the classroom and

planning lessons that adapt to the emerging needs of

learners.

We recognise many limitations of the work reported.

While we have paid some attention to the culture within the

PD events themselves, we have paid little to the various

school-based cultures in which teachers work. In future

designs we will need to give greater attention to this. In

particular, we are beginning to recognise the importance of

designing resources that teachers can use with colleagues

and senior managers to help inform and evolve these local

cultures.

Finally we note that, with the international effort being

poured into PD that promotes inquiry-based learning there

is an urgent need for design research to provide more

efficient products and processes for PD. We hope that the

products and processes described here and available on the

websites will be a modest, but valuable contribution to this

work.

Acknowledgments This paper is based on work carried out within

the following funded projects: Bowland Maths funded by the Bow-

land Charitable Trust (UK); ‘Learning Mathematics through Discus-

sion and Reflection’ funded by the Learning and Skills Development

Agency (UK); ‘Improving Learning in Mathematics’ funded by the

Standards Unit, Department for Education and Skills (UK); ‘Math-

s4Life’ funded by National Research and Development Centre for

adult literacy and numeracy (UK), and the PRIMAS project funded by

the European Union Seventh Framework Program under grant

agreement n� 244380.

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