Readinessbased differentiation in
primary school mathematics: Expert recommendations and
teacher selfassessment
Emilie
J. Prast^{a}, Eva Van de WeijerBergsma^{a},
Evelyn H. Kroesbergen^{a},
&
Johannes E.H. Van Luit^{a}
^{a }Utrecht University, The Netherlands
Article received 8 April 2015 /
revised 18 June 2015 / accepted 3 August 2015 / available
online 21 August 2015
Abstract
The
diversity
of students’ achievement levels within classrooms has made
it essential for teachers to adapt their lessons to the
varying educational needs of their students
(‘differentiation’). However, the term differentiation has
been interpreted in diverse ways and there is a need to
specify what effective differentiation entails. Previous
reports of low to moderate application of differentiation
underscore the importance of practical guidelines for
implementing differentiation. In two studies, we
investigated how teachers should differentiate according to
experts, as well as the degree to which teachers already
apply the recommended strategies. Study 1 employed the
Delphi technique and focus group discussions to achieve
consensus among eleven mathematics experts regarding a
feasible model for differentiation in primary mathematics.
The experts agreed on a fivestep cycle of differentiation:
(1) identification of educational needs, (2) differentiated
goals, (3) differentiated instruction, (4) differentiated
practice, and (5) evaluation of progress and process. For
each step, strategies were specified. In Study 2, the
Differentiation SelfAssessment Questionnaire (DSAQ) was
developed to investigate how teachers selfassess
their use of the strategies recommended by the experts.
While teachers (N = 268) were moderately positive about
their application of the strategies overall, we also
identified areas of relatively low usage (including
differentiation for highachieving students) which require
attention in teacher professional development. Together,
these two studies provide a model and strategies for
differentiation in primary mathematics based on expert
consensus, the DSAQ which can be employed in future studies,
and insights into teachers’ selfassessed application of
specific aspects of differentiation.
Keywords:
differentiation; mathematics; primary school; teacher
selfassessment; Delphi method
Corresponding
author: Emilie Prast, Faculty of Social and
Behavioural Sciences, Department of
Pedagogical and Educational Sciences, Heidelberglaan 1, 3584
CS Utrecht, The
Netherlands, e.prast@uu.nl
DOI: http://dx.doi.org/10.14786/flr.v3i2.163
1. Introduction
Every
day, primary school teachers are faced with the task of teaching
students of diverse academic ability and achievement levels.
Therefore, teachers should adapt their lessons to the diverse
educational needs of their students (Corno, 2008). Such
adaptations are often promoted using the term differentiation or differentiated instruction,
defined by Tomlinson et al. (2003, p.120) as “an approach
to teaching in which teachers proactively modify curricula,
teaching methods, resources, learning activities, and student
products to address the diverse needs of individual students and
small groups of students to maximise the learning opportunity
for each student in a classroom”.
The
international trend towards inclusive education makes the need
for differentiation especially urgent. Within response to
intervention models, general education teachers are required to
provide both universal support  i.e., a good general education
for all students (Tier 1)  and targeted support (Tier 2) such
as smallgroup instruction for struggling students (Fuchs &
Fuchs, 2007; McLeskey & Waldron, 2011). Smallgroup or
individual interventions carried out by an educational
specialist (Tier 3) are only available for a limited number of
students whose problems persist despite the supports provided by
the general education teacher. Thus, general education teachers
have the primary responsibility for providing a good education
to all students, regardless of their achievement level.
Attending
to the educational needs of students with a broad range of
ability and achievement levels is a challenge for teachers.
Successful differentiation requires advanced
subject matter knowledge, pedagogical skills and classroom
management skills (VanTasselBaska & Stambaugh, 2005).
Consequently, a need for professional development in the area of
differentiation has been identified repeatedly (Johnsen,
Haensly, Ryser, & Ford, 2002; Van den Broekd’Obrenan et
al., 2012; VanTasselBaska et al., 2008).
To
design effective professional development programmes, it is
important to know what teachers should do in their daytoday
teaching to differentiate their lessons for students of diverse
achievement levels. What constitutes best practice? In two
studies, we investigated how teachers should differentiate
according to experts as well as the degree to which teachers
already apply the recommended strategies. The focus was
exclusively on mathematics since strategies for differentiation
may vary across subject areas. Moreover, domainspecific
guidelines or strategies tend to be more concrete and may
therefore provide stronger guidance to teachers.
Differentiation
is
an umbrella term that may be used to refer to one or several of
a variety of instructional modifications. It may involve
modifications of the content (what students learn), the process
(how they learn it), or the product of learning (how students
demonstrate their learning) (Tomlinson, 2005). Various student
characteristics may serve as a ground for differentiation. For
example, Tomlinson et al. (2003) distinguish between
differentiation by student readiness (representing the current
level of knowledge and skills in the subject area), learning
profile (a student’s preferred ways of learning, such as a
preference for visual input) and interest (topics about which
the student wants to learn more).
In the current
study, the focus is on differentiation by student readiness.
Readiness is influenced by a child’s natural ability as well as
learning experiences and is reflected in the child’s current
knowledge and skill level. The importance of differentiation by
student readiness is supported by the theoretical constructs of
the zone of proximal development (Vygotsky, 1978),
challengeskill balance (Csikszentmihalyi, 1990),
aptitudetreatment interaction (Cronbach & Snow, 1977), and
adaptive teaching (Corno, 2008). Vygotsky (1978) stated that
learning occurs when a child engages in activities that fall
within its zone of proximal development (ZPD), i.e. that are
slightly more difficult than what the child already masters
independently. When children within one classroom have widely
varying readiness levels, their zones of proximal development
also differ. A task that is just within reach for
averageachieving students (i.e. in their ZPD) may be too
difficult for children with lower readiness levels when the gap
between existing knowledge and skills and the task is too big.
Conversely, children with higher readiness levels may already
master the task and in this case they are not challenged to
reach beyond what they can already do. This implies that
children within the same classroom may need different
instructional treatments to work in their ZPD. To work in the
ZPD, the skill level of the students should be in balance with
the difficulty level of the tasks. Such a challengeskill
balance may result in effective and engaged learning, while
tasks that are much too difficult or too easy may lead to
frustration, boredom, and withdrawal from learning
(Csikszentmihalyi, 1990). Additionally,
certain characteristics of the learning environment may be
useful for some learners but not for others, depending on the
aptitude of the student (Cronbach & Snow, 1977). Because of
the variation in student aptitudes and the resulting diversity
of educational needs, teachers should adapt education to the
needs of their students (Corno, 2008). What these theories have
in common is the idea that students with different readiness
levels have different educational needs and that instruction
should be matched to these needs, which is exactly what
differentiation aims to do.
Roy, Guay, and
Valois (2013) took a step towards clarification of the term
differentiation by identifying two main components of
readinessbased differentiation: academic progress monitoring
and instructional adaptations. Ideally, the developments in
students’ achievement or understanding are closely followed, for
example using frequent formal or informal tests, and adaptations
are then made to ensure a good fit between the readiness of the
student and the instruction.
Most
approaches to differentiation include these two components in
some way. Nevertheless, the way in which progress is monitored
and the nature of instructional adaptations strongly vary across
intervention studies (e.g. Brown & Morris, 2005; McDonald
Connor et al., 2009; Reis, McCoach, Little, Muller, &
Kaniskan, 2011; Tieso, 2005; Ysseldyke & Tardrew, 2007).
Students’ achievement may be measured with standardised,
curriculumbased, or informal assessments. In some cases, the
results of these assessments are used to determine the
instructional treatment for an extended period of time (weeks or
months) whereas other interventions continuously monitor
progress and adapt the instructional treatment accordingly.
Adaptations may be at the level of individual students or
subgroups of students. When grouping is used, such groups may be
betweenclass or withinclass, fixed or flexible (Tieso, 2003).
Adaptations may entail modification of the amount of
instruction, the content or type of instruction, the content or
type of independent practice tasks, or combinations of these
elements. Given the diverse interpretations of
the term differentiation, there is a need to specify what
effective differentiation entails.
One line of
research has examined the effects of various types of ability
grouping. The best results are obtained when
students can switch between groups based on changes in their
educational needs (the progress monitoring component of
differentiation) and when instruction is tailored to the needs
of the students in the groups (the instructional adaptations
component) (Kulik & Kulik, 1992; Lou et al., 1996; Slavin,
1987; Tieso, 2003). When
these conditions are met, homogeneous withinclass ability
grouping has demonstrated positive effects on student
achievement across multiple studies (Kulik & Kulik, 1992;
Lou et al., 1996; Slavin, 1987; Tieso, 2005). In contrast, slight negative
effects of withinclass ability grouping in primary school were
found across three studies in which variations in instructional
treatment were not explicitly described (Deunk, Doolaard,
SmaleJacobse, & Bosker, 2015). So, it seems to be important
to use the grouping arrangement as a means to provide the
different subgroups with the instruction that they specifically
need, i.e. to differentiate instruction. Another issue in the
literature on ability grouping is the potential existence of
differential effects, i.e. different effects for students of
different ability levels. While Slavin (1987) reported a higher
median effect size for lowability students than for
averageability and highability students, other reviews have
found different patterns with smaller or even negative effects
for lowachieving students (Deunk et al., 2015; Kulik &
Kulik, 1992; Lou et al., 1996). More research is necessary to
determine in which situations such differential effects may
arise.
A recent
review (Deunk et al., 2015) examined the effects of various
readinessbased differentiation practices on student
achievement. Although the authors aimed to include all
highquality studies published about this topic since 1995, only
sixteen studies about differentiation in primary school could be
included. Most of these sixteen studies were either too narrow
(ability grouping without explicit instructional
differentiation) or too broad (interventions in which
differentiation was one of several components of a comprehensive
school reform initiative) to be informative about the effects of
differentiation on student achievement. However, promising
results were obtained with two computerised interventions for
differentiation: Individualizing Student Instruction (McDonald
Connor, Morrison, Fishman, Schatschneider, & Underwood,
2007; McDonald Connor et al., 2011a; McDonald Connor et al.,
2011b) and Accelerated Math (Ysseldyke et al., 2003; Ysseldyke
& Bolt, 2007). The Individualizing Student Instruction
programme provides the teacher with recommendations about the
amount and type of literacy instruction needed by individual
students based on their scores on a computerised test.
Accelerated Math is a technological application which
continuously monitors students’ progress, adapts practice tasks
to students’ individual skill level, and informs the teacher
when students struggle with certain types of problems. Both of
these interventions, which clearly contain both components of
differentiation (progress monitoring and instructional
adaptations), have demonstrated significant positive effects
across multiple studies.
Prior
research has shown that there is room for improvement in
teachers’ implementation of differentiation. The Dutch
Inspectorate of Education recently found that adequate
adaptations to diverse educational needs are only made at about
half of the schools (Van den Broekd’Obrenan et al., 2012). In US middle schools, both
teachers and students reported low usage of differentiation
strategies (Moon, Callahan, Tomlinson, & Miller, 2002). In a
recent study on Canadian elementary schools, teachers
selfreported moderate use of differentiation strategies, but
strategies requiring more time to implement were used relatively
infrequently (Roy et al., 2013). Similarly, studies about
adaptations for students with learning disabilities found that
teachers tend to implement typical adaptations which can be easily
implemented for all students rather than specialised
adaptations, i.e. adaptations targeted at the unique educational
needs of individual students (McLeskey & Waldron, 2002,
2011; Scott, Vitale, & Masten, 1998). However, a recent
study carried out in Finland found that teachers do provide more
individual support to struggling students (Nurmi et al., 2013).
For highachieving or gifted students, low levels of
differentiation have generally been found (Reis et al., 2004;
Westberg, Archambault, Dobyns, & Salvin, 1993; Westberg
& Daoust, 2003). In sum, prior studies have generally found
low to moderate use of differentiation strategies, although the
degree of implementation of differentiation seems to vary
depending on the specific strategies for differentiation
examined, the targeted population of students, and perhaps also
the country in which data are collected. Specialised adaptations
as well as adaptations targeted specifically at highachieving
students seem to be used relatively infrequently.
In
conclusion, there is a clear need to apply differentiation based
on differences in students’ readiness and teachers could use
some help in doing this. The literature shows that
differentiation should include progress monitoring and
instructional adaptations. However, the ways in which this can
be done effectively are less clear. Promising results have been
obtained with two computerised interventions. However,
highquality research about the achievement effects of
interventions in which differentiation is mainly implemented by
the teacher himself is scarce. There is a
need for general guidelines for differentiation that can be
applied in a wide array of schools, independently from
curricular methods or technological applications. Therefore,
Study 1 sought to achieve consensus among a consortium of
mathematics experts about a feasible model and associated
strategies for differentiation. Study
2 linked the results of Study 1 to teachers’ daily practice by
examining how teachers selfassess their use of the strategies
for differentiation recommended by the experts.
2.
Study 1
2.1
Aims
Study 1
The aim of
Study 1 was to operationalise the concept of
differentiation by achieving consensus among a consortium of
mathematics experts about a coherent set of strategies for
differentiating primary school mathematics education. The result
of the consensus procedure needed to be feasible for use by
general education teachers in daily mathematics teaching.
Additionally, it needed to be applicable in diverse schools,
independent from curricular method.
Expert
consensus procedures can be valuable when scientific literature
provides insufficient information to make complex decisions
(Landeta, 2006) and have been applied before to achieve
consensus about effective teaching (Teddlie, Creemers,
Kyriakides, Muijs, & Yu, 2006). While several individual
experts have made recommendations for differentiation in primary
mathematics in books and journals for practitioners, consensus
among various experts could provide a more solid foundation. For
differentiation in mathematics, teacher educators with expertise
in the didactics of primary mathematics are the relevant group
of experts. Teacher educators may have gained practical
knowledge regarding the effectivity and feasibility of diverse
strategies for differentiation. Making use of this experiential
knowledge has the potential to complement the scientific
literature and strengthen the link between theory and practice.
2.2
Method
Study 1
2.2.1
Participants
The
consortium of experts was designed to include distinguished
Dutch preservice and inservice teacher educators with a
professional focus on mathematics education. To be eligible for
participation, potential members had to be experts in their
field, as demonstrated by their (1) experience in providing
preservice or inservice teacher training about teaching
mathematics (2) regular presence as invited speaker at
educational conferences and (3) role as a consultant to the
Ministry of Education, Culture and Science to discuss new
educational policy. The senior authors approached potential
candidates with these criteria in mind. All experts
who were invited to participate agreed to join the consortium.
This resulted
in a consortium of eleven experts (seven men, four women)
representing eight large national and regional institutes for
pre and inservice teacher training spread across the
Netherlands. The members had experience in at least two of the
following areas: inservice teacher training for mathematics (M = 8.6 years, SD = 8.5 years),
preservice teacher training for mathematics (M = 5.4 years, SD = 6.3 years),
carrying out educational evaluation studies (M = 25.0 years, SD = 21.2 years) and
teaching (M = 5.7
years, SD = 5.4
years). The current daily work of the
consortium members included educating preservice teachers,
providing professional development for inservice teachers, and
guiding schools in the implementation of new educational
approaches including differentiation.
2.2.2
Consensus
procedure
Focus
group discussions (Liamputtong, 2011) and the Delphi method
(Hasson, Keeney, & McKenna, 2000) were used to investigate
the experiential knowledge of the experts on differentiated
mathematics education systematically. Focus group discussions
are structured discussions with a group of persons involved in
the topic in which certain roles (e.g. a discussion leader, a
timekeeper and a secretary) and rules (e.g. only ontopic
contributions) are specified and adhered to. The Delphi
technique entails the repeated administration of a questionnaire
in order to achieve consensus among experts. After the first
round of administration, the initial responses are presented
anonymously to the participants, who are then asked to fill out
the questionnaire again. This procedure is repeated until
consensus (specified with a consensus criterion) is reached. The
order of focus group discussions and Delphi rounds in the
current study is presented in Figure 1. The whole procedure took
place between November 2011 and January 2012.
In
the first threehour focus group discussion, the experts were
invited to share their knowledge, prompted by eight core
questions about differentiation (see Figure 1). These questions
were deliberately left open to elicit broad input. No particular
theoretical perspective was chosen a priori apart from the
assumption that student readiness would be an important ground
for differentiation (see questions 6 and 7). Rather, the
questions were asked from a practical point of view (what does
and does not work in practice and how can this be improved). In
principle, the questions were discussed one by one in the listed
order, but in practice, the discussion sometimes moved back and
forth between the various questions because of their high
interrelatedness. After the first focus group discussion, the
first author restructured the meeting minutes in terms of
(initial) answers to the eight core questions.
Second,
based on this input, the researchers constructed an online
Delphi questionnaire (Round 1). During the first focus group
discussion, one of the experts had listed five general themes
that are central to differentiation: organisation, goals,
instruction, practice, and learning styles. These themes were
used to structure the Delphi questionnaire. The theme
‘differentiation in kindergarten’ was added to account for
aspects of differentiation specific to kindergarten
(kindergarten is integrated in the Dutch primary school system).
For each theme, the first author summarised the main ideas of
the focus group discussion and proposed this to the other
authors. Apart from some minor changes, the other authors agreed
that these summaries accurately reflected what had been said in
the discussion. These summaries were included in the Delphi
questionnaire as oneparagraph concepts for differentiation (see
Appendix 1). For each theme, statements about specific elements
of the concept were also included (for example: ‘the
lowachieving subgroup profits from extended instruction’). The
experts rated their agreement with the concepts and with the
specific statements on a Likert scale ranging from 1 (do not agree at all)
to 5 (fully agree).
Additionally, open questions prompted participants to provide
any comments they had.
Third,
a Round 2 Delphi questionnaire was developed which included only
those questions on which no overwhelming consensus had been
reached in Round 1. The consensus criterion for Round 1 was that
all responses should be at one end of the scale (i.e. either 4
and 5 or 1 and 2), with a maximum of one neutral response (3).
The questions on which consensus had not yet been reached were
presented to the participants again accompanied by a bar chart
of the responses in Round 1. Additionally, comments provided by
the participants in Round 1 were included as new questions in
Round 2. Using a more lenient consensus criterion of maximally
three neutral responses and the rest at one end of the scale,
the researchers determined for which items consensus was
achieved in Round 2.
Fourth,
the researchers presented the results of the Delphi
questionnaire to the experts during the second focus group
discussion which lasted two hours. The items on which consensus
had not been achieved were discussed to clarify
misunderstandings (especially about the open comments provided
by the participants in Round 1) and resolve conflicting
opinions.
Fifth,
the first author reviewed the meeting minutes of the focus group
discussions and the responses to the Delphi questionnaire to
synthesise the input, resulting in a proposed model for
implementing differentiation. The other authors agreed with this
model.
Sixth,
the proposed model for differentiation was sent to all
consortium members and discussed during a third onehour meeting
of the focus group.
2.2.3
Attendance
rates of consortium members
Of
the eleven consultants, six (54.5%) attended the first two focus
group discussions and completed the two Delphi questionnaires
and four (36.4%) completed three out of four components (i.e.
either both discussions and one questionnaire or both
questionnaires and one discussion). One participant only
completed the Delphi questionnaire, after being informed about
the content of the first focus group discussion in a separate
meeting with the researchers. All members received the proposed
model for differentiation by email and were given the
opportunity to send any comments or questions, and six
participants (54.5%) attended the third meeting in which the
cycle was discussed.
2.3
Results
Study 1
In
Round 1 of the Delphi questionnaire, the experts agreed with the
concepts for differentiation in instruction,
differentiated goals, differentiated practice, differentiation
based on learning styles, and differentiation in kindergarten
which had been formulated based on the first focus group
discussion. Positive consensus on the remaining
concept (organisation of differentiation) was reached in Round
2. Table 1 provides an overview
of the degree of consensus achieved on the specific statements
in the two rounds of the Delphi questionnaire. Consensus
was
reached on 35 items in the first round and on an additional 25
original items in the second round, amounting to consensus on
74.1% of original items after two rounds. Regarding the new
statements that were derived from the open comments in Round 1,
consensus was reached on 46.0% of these statements in Round 2.
The items on which consensus had not been reached after the
second Delphi questionnaire were discussed in the second focus
group discussion. Differences in interpretation of certain items
were resolved and consensus was reached about the main issues.
Items on which no consensus had been reached in the Delphi
questionnaires often concerned issues about which the experts
were unsure or had no pronounced opinion, including the
importance of specific elements (e.g. videotaped instruction,
mind maps, games, student choice) and preference for certain
grouping formats (e.g. pairs or small groups). In the second
focus group discussion, the overall conclusion about these
elements and formats was that they all have their merits and
that the choice is dependent on the situation, but that they are
not crucial for differentiation.
Table 1
Overview of consensus on statements in the
Delphi questionnaire


Original statements 

New statements in Round 2 

Theme 
Subtheme 
No. of statements 
Consensus after Round 1 
Consensus after Round 2^{a} 

No. of statements 
Consensus after Round 2 
Organisation of differentiation 
11 
4 
7 

14 
2 

Differentiation
in instruction 








General 
5 
3 
4 

14 
4 

Wholeclass
instruction 
9 
2 
6 

10 
6 

Subgroup
instruction for lowachieving students 
10 
3 
8 

7 
3 

Subgroup
instruction for highachieving students 
12 
6 
8 

4 
3 
Differentiated
goals 
10 
4 
8 

23 
10 

Differentiated practice 
14 
5 
9 

6 
3 

Differentiation based on learning styles 
3 
1 
3 

25 
15 

Differentiation
in kindergarten 
7 
7 
7 

10 
6 

Total 

81 
35 
60 

113 
52 
^{a} total
amount of items on which consensus was reached, including items
on which consensus had already been reached in Round 1
The
experts approved the model for differentiation which was created
based on their input. The model, dubbed the cycle of differentiation,
consists of the following five steps: identification of
educational needs, differentiated goals, differentiated
instruction, differentiated practice, and evaluation of progress
and process (see Figure 2). A distinction is made between
instruction and practice. Instruction refers to moments during
which the teacher provides instruction to the whole class,
subgroups of students, or individual students, whereas practice
refers to moments during which students work on tasks,
individually or in groups. These two can happen simultaneously,
for example when the teacher provides instruction to a subgroup
while other students are working on practice tasks. In the
following paragraphs, we describe the key recommendations for
each step in the cycle of differentiation provided by the
experts in the consensus procedure.
Figure 2.
Cycle of differentiation. (see pdf)
Organisation
is placed centrally in the cycle, because successful
implementation of differentiation depends on a facilitative
organisational structure
and good classroom management. A key
organisational
characteristic of the model for differentiation agreed upon by
the experts is the assignment of students to subgroups based
on achievement level, allowing teachers to make instructional
adaptations for subgroups of students with similar educational
needs. Only the remaining individual educational needs that
are not met within the subgroup call for individual
accommodations.
The
first step in the cycle of differentiation is the
identification of educational needs. Initially, the teacher
should assign students to subgroups (typically a
lowachieving, an averageachieving and a highachieving
subgroup) based on their results on standardised tests and
curriculumbased tests. In the course of the schoolyear,
teachers continuously gather new and more detailed information
about students’ educational needs, for example with the
analysis of daily work, informal observations and diagnostic
conversations. The
subgroups should be flexible, i.e. students should be able to
switch groups based on changes in their educational needs.
Based on the
educational needs of the students, differentiated goals should
be set. Overarching objectives (the material that students
should master at the end of primary school) and lesson goals
(goals for a specific lesson) are distinguished. Overarching
objectives should not only be formulated for averageachieving
students but also specifically for lowachieving and
highachieving students (see also Appendix 1, differentiation
in goals). The overarching objectives should be translated
into concrete lesson goals, which are provided mostly by the
curriculum. However, only some of
the mathematics curricula available in the Netherlands
differentiate lesson goals for three achievement levels.
When the curriculum does not differentiate goals
sufficiently, the teacher should formulate challenging but
realistic lesson goals for all subgroups.
Based
on the educational needs and the goals that have been set, the
teacher differentiates instruction through broad wholeclass
instruction, subgroup instruction tailored to the needs of
that subgroup, and individual adaptations. During wholeclass
instruction, the teacher should serve a broad range of
educational needs by varying the difficulty level of
questions, stimulating all children to think about the answer
to a question by giving thinking time, teaching at various
levels of abstraction, and using several input modalities
(e.g. visual, verbal, tactile). In subgroup instruction, the
teacher should adapt instruction to the educational needs of
lowachieving and highachieving students. It is assumed that,
in general, lowachieving students need more guidance (e.g.
explicit instruction) and instruction at lower levels of
abstraction (e.g. using blocks to represent and calculate a
sum) while highachieving students need more exploratory
instruction about advanced content with a focus on conceptual
understanding (e.g. the relation between multiplication and
division). To the extent possible, the teacher should also
take into account individual differences during subgroup
instruction and while giving individual feedback.
In
the practice phase of the lesson, the subgroups need
quantitatively and qualitatively different tasks. For the
lowachieving subgroup, completing all regular tasks is often
not realistic, so the tasks that are crucial for mastery of
the objectives for lowachieving students should be selected
(the remaining tasks can still be completed when students have
time left). For the highachieving subgroup, the regular
material should be compacted. For the most popular Dutch
mathematics curricula, guidelines exist to inform the teacher
which tasks can be skipped by highachieving students. In
other cases, the teacher should remove most of the repetitive
tasks and select the tasks that are crucial steps towards
mastery of the objectives. The time freed up by compacting
should be spent on enrichment. Some curricula provide
enrichment tasks and these tasks can be used in some cases,
but they are often not sufficiently challenging for very
highachieving students. Therefore, supplemental enrichment curricula
should also be used. Technological applications
such as mathematics websites and instructional computer
programmes can also be valuable tools for individual
differentiation, provided that they are used deliberately for
additional practice in areas the student does not master yet
or for enrichment at an appropriate challenge level.
The
final step in the cycle of differentiation is the evaluation
of progress and learning process. Based on daily work and
achievement tests, the teacher should evaluate whether the
students have met the lesson goals. Regarding process, the
teacher should evaluate whether the applied adaptations of
instruction and practice had the desired effect. For example,
when a teacher has intentionally taught the lowachieving
subgroup at a lower level of abstraction, the teacher should
evaluate whether this was helpful for these students. To gauge
the effectiveness of the accommodations made, the teacher may
supplement achievement results with informal measures such as
observations or diagnostic conversations. The evaluation phase
informs the teacher about students’ current achievement level
and about instructional approaches that work for these
students, completing the cycle and serving as new input for
the identification of educational needs.
2.4 Concluding
summary
Study 1
The
aim of Study 1 was to operationalise the concept of
differentiation by achieving consensus among a consortium of
mathematics experts about a coherent set of strategies for
differentiating primary school mathematics education. A
combination of focus group discussions and the Delphi
technique was used to investigate the experiential knowledge
of eleven experts in mathematics education systematically.
Consensus was reached on all summarising concepts and on the
majority of specific statements from the Delphi questionnaire.
The input from the experts was synthesised into a cycle of
differentiation consisting of the following five steps:
identification of educational needs, differentiated goals,
differentiated instruction, differentiated practice, and
evaluation of learning progress and process. For each step,
strategies were specified, providing teachers with concrete
guidelines for implementing differentiation in primary school
mathematics.
3. Study
2
3.1
Aims Study 2
Study 2
linked the results of Study 1 to teachers’ daily practice by
investigating teachers’ selfassessed use of the strategies
for differentiation recommended by the experts. Therefore, we
developed the Differentiation SelfAssessment Questionnaire
(DSAQ) which covers the recommended strategies in five
subscales corresponding with the five steps of the cycle of
differentiation (see also section 3.2.2). Since the DSAQ was
newly developed, we also aimed to investigate its statistical
properties, including its factor structure and relation with
other scales.
The
development of a new instrument was necessary to ensure
coverage of the broad set of strategies recommended by the
experts in Study 1. Another recently developed instrument to
measure teachers’ selfreported use of differentiation is the
Differentiated Instruction Scale (DIS; Roy et al., 2013). This
instrument was based on a similar theoretical framework and
there is overlap between the content of the items of the DIS
and the DSAQ. However, the DIS has only twelve items and is
not sufficiently specific to measure all strategies
recommended by the experts. Regarding progress monitoring, for example, the DIS
includes the rather general item ‘analyse data about students’
academic progress’ while the DSAQ distinguishes between the
different types of progress monitoring recommended by the
experts, ranging from standardised tests to diagnostic
interviews. Thus, the added value of the DSAQ is that it is a
detailed measure of the specific strategies recommended by the
experts in Study 1.
The first
aim of the current study was to examine the factor structure
of the DSAQ. The literature reviewed in the introduction
indicates that effective differentiation entails two
components: progress monitoring and instructional adaptations.
The steps in the cycle of differentiation reflect these
components: identification of educational needs and evaluation
of progress and process involve progress monitoring, while
differentiated goals, instruction, and practice involve
instructional adaptations. For the DIS, Roy et al. (2013)
found that a model with these two factors provided a better
fit to the data than a model in which all items loaded on one
general differentiation factor. Therefore, we investigated
whether the DSAQ has a similar factor structure by comparing
the fit of a twofactor model with one factor for progress
monitoring and one factor for instructional adaptations to the
fit of a onefactor model.
The second
aim was to examine the convergent and divergent validity of
the DSAQ by investigating its relationship with other teacher
selfreport scales. Teacher selfefficacy is a
multidimensional construct which comprises teachers’ perceived
ability to perform various aspects of teaching (Skaalvik &
Skaalvik, 2007; TschannenMoran & Woolfolk Hoy, 2001).
Theoretically, selfassessed usage of differentiation as
measured by the DSAQ should be more closely related to aspects
of selfefficacy related to differentiation than to other
aspects of teacher selfefficacy. Specifically, we expected
stronger correlations with scales that measure teachers’
selfefficacy for instruction to students of diverse
achievement levels, for adapting education to individual
students’ needs, and for selfassessed prerequisite knowledge
for differentiation (which would support convergent validity)
than with scales that measure teachers’ selfefficacy for
motivating students, for coping with changes and challenges,
and for classroom management (which would support divergent
validity).
The third
and main aim was to investigate teachers’ selfassessed use of
the strategies for differentiation recommended by the experts.
Besides examining teachers’ overall usage, we
also aimed to identify strategies which were relatively
infrequently used. Such information may provide starting
points for teacher professional development by indicating in
which areas teachers perceive most room for improvement. Based
on the literature reviewed in the theoretical background as well as
on the input from the experts in Study 1,
we hypothesised that average scores would be low to moderate
and that specialised
strategies aimed at individual students’ unique educational
needs as well as strategies targeted specifically at
highachieving students would be used relatively infrequently.
3.2
Method Study 2
3.2.1
Participants
and procedure
The sample
consisted of 268 primary school teachers working at 31 schools
participating in a largescale project about differentiation.
Schools were informed about the project through flyers and
advertisements and could register themselves for participation
on a project website. The schools were located in rural and
urban areas spread across the Netherlands and were diverse in
terms of school size, student population, religious
background, and mathematics curriculum used. All 325 teachers
of Grade 1 through 6 of the participating schools were invited
by email to fill out an online questionnaire containing the
DSAQ and related scales. The
questionnaire
was administered at the beginning of the 2012  2013 school
year. A total of 268
teachers (83%) completed the questionnaire and gave informed
consent. The remaining teachers did not give informed consent
(n = 3), completed
the questionnaire only partly (n = 7) or did not
respond at all (n =
47). On average, participants had 15.6 years of teaching
experience (range 0 – 40 years). Seventyone teachers (26%)
taught a
multigrade class. Fiftyfour teachers (20%) worked
fulltime, whereas most teachers worked two, three or four
days a week (61, 81 and 55 teachers respectively).
3.2.2
Instruments
The DSAQ was
developed to examine how teachers assess their use of the
strategies recommended by the experts in Study 1. Each
subscale represents one step of the cycle of differentiation
and covers core strategies for differentiation
belonging to that step. Subscales
and sample items are provided in Table 2. Organisational
aspects of the model for differentiation were not captured in
a separate scale but were partly covered in the subscales
corresponding to each step of the cycle. In an earlier pilot
study, a pilot version of the DSAQ had been completed by 27
teachers recruited at four schools. Based on the analysis of
the internal consistency of the pilot version, which was
acceptable to good, some adaptations were made in the final
version of the DSAQ. The internal consistency of the final
version, obtained in the current sample (N = 268), is reported
in section 3.3.1.
To
assess the convergent and divergent validity of the DSAQ,
subscales from two wellestablished multidimensional teacher
selfefficacy scales were selected. The Norwegian Teacher
SelfEfficacy Scale (Skaalvik & Skaalvik, 2007) was
developed with special attention for adapting education to
individual educational needs. It consists of six subscales
with acceptable to good internal consistency which load on six
primary factors, which in turn load on a secondorder factor
for general teacher selfefficacy (Skaalvik & Skaalvik,
2007). For the current study, the subscales for Instruction 
which emphasises instruction to students of diverse
achievement levels  and
Adapting
Education to Individual Students’ Needs[1]
were selected to assess the convergent validity of the DSAQ
while the subscales for Motivating Students and for Coping
with Changes and Challenges were selected to assess the
divergent validity. The subscales were translated into Dutch
and the wording was adapted to make the items domainspecific
for mathematics instruction.
To
further examine the divergent validity, the subscale for
classroom management from the wellestablished Ohio State
Teaching Efficacy Scale (OSTES; TschannenMoran & Woolfolk
Hoy, 2001) was administered in Dutch translation (Goei,
Bekebrede, & Bosma, 2011). The OSTES consists of three
subscales which load on a firstorder factor and on a general
teaching efficacy factor. The subscales have demonstrated high
internal consistency and can be used independently with
inservice teachers (TschannenMoran & Woolfolk Hoy,
2001).
As
a third potential support for convergent validity, a
selfassessment scale about Prerequisite Knowledge for
Differentiation was adapted from an informal scale that had
already been used to assess the level of prior knowledge in
professional development programmes (Nationaal
Expertisecentrum Leerplanontwikkeling, 2010). Teachers
selfassess the extent to which they already possess the
knowledge necessary for implementing differentiated
instruction.
3.2.3
Analyses
Because
we wanted to compare the fit of two specific models based on
theory and previous findings, we used confirmatory factor
analysis (CFA) to investigate
the factor structure of the DSAQ. We first tested a onefactor
model in which all DSAQ subscales loaded on one general
differentiation factor. Second, we tested a twofactor model
with one factor for progress monitoring (subscales
Identification of Educational Needs and Evaluation of Progress
and Process) and one factor for instructional adaptations
(subscales Differentiated Goals, Differentiated Instruction
and Differentiated Practice). Version 7.3 of the Mplus
statistical package (Muthén & Muthén, 19982012) was used.
Model fit was evaluated with the chisquare statistic, the
comparative fit index (CFI), the TuckerLewis Index (TLI), the
root mean squared error of approximation (RMSEA), and the
standardised root mean square residual (SRMR). Values above
.95 for the CFI and TLI and values below .06 and .08 for the
RMSEA and SRMR, respectively, indicate good model fit (Hu
& Bentler, 1999). The maximum likelihood estimator was
used. In the standardised solution, the variance of the
factors was fixed to 1 so all factor loadings could be
estimated freely.
Correlational
analyses were performed to assess the convergent and divergent
validity of the DSAQ. We expected moderate to strong positive
correlations with Prerequisite Knowledge for Differentiation,
Adapting Education to Individual Students’ Needs, and
Instruction (for students of all achievement levels). For the
latter two scales, we expected that correlations would be
lower for the factor Progress Monitoring than for the factor
Instructional Adaptations, because Progress Monitoring does
not focus on the instructional phase. Regarding divergent
validity, we hypothesised that DSAQ scores would be less
strongly related to selfefficacy for Motivating Students,
Coping with Changes and Challenges, and Classroom Management,
although we still expected positive correlations because these
dimensions of teacher selfefficacy can be helpful when
implementing differentiation.
To identify
areas of relatively low use, we compared the means of all
single items to the mean of their factor. If a mean was more
than one standard deviation below the mean of the factor, it
was classified as relatively low.
Table 2
Sample items and descriptive statistics (N =
268) of the administered scales
Scale 
Sample item 
Response options 
No. of items 
α 
M 
SD 
DSAQ 






Identification
of educational needs^{a} 
I
analyse the answers on curriculumbased tests to
assess a student’s educational needs 
1 = does
not apply to me at all, 5 = fully applies to me 
5 
.69 
3.64 
.55 
Differentiated
goals^{a} 
I set
extra challenging goals for highachieving students 
1 – 5 as
above 
6 
.79 
3.78 
.55 
Differentiated
instruction^{a} 
I adapt
the level of abstraction of my instruction to the
educational needs of the students 
1 – 5 as
above 
7 
.72 
3.81 
.42 
Differentiated
practice^{a} 
I select
the most important elaboration activities for very
lowachieving students 
1 – 5 as
above 
8 
.72 
3.46 
.55 
Evaluation
of progress and process^{a} 
I use
diagnostic conversations to evaluate whether specific
students have met the lesson goals 
1 – 5 as
above 
7 
.86 
3.56 
.57 
Additional scales 






Instruction^{b}

How
certain are you that you can explain central themes in
mathematics so that even the lowachieving students
understand? 
1 = not
certain at all, 4 = absolutely certain 
4 
.74 
3.13 
.37 
Adapting
education to individual students’ needs^{b} 
How
certain are you that you can adapt instruction to the
needs of lowachieving students while you also attend
to the needs of other students in class? 
1  4 as
above 
4 
.78 
2.91 
.44 
Coping
with changes and challenges^{b} 
How
certain are you that you can manage instruction
regardless of how it is organised (working with
subgroups, multigrade classes with 3 grades, etc.)? 
1  4 as
above 
4 
.76 
2.99 
.43 
Motivating
students^{b} 
How
certain are you that you can get students to do their
best even when working with difficult problems? 
1  4 as
above 
4 
.76 
2.95 
.42 
Classroom
management^{c} 
How much
can you do to control disruptive behavior in the
classroom? 
1 =
nothing, 9 = very much 
7 
.92 
7.17 
.77 
Prerequisite
knowledge for differentiation^{d} 
I know
the different solution strategies that are used by
children 
1 = does
not apply to me at all, 5 = fully applies to me 
10 
.84 
3.79 
.41 
^{a }Newly deloped^{ }DSAQscales ^{b} Adapted
from Skaalvik and Skaalvik (2007) ^{c} Taken from Goei, Bekebrede, and
Bosma (2011) ^{d} Adapted from Nationaal
Expertisecentrum Leerplanontwikkeling (2010)
3.3
Results Study 2
3.3.1
Properties
of the DSAQ: internal consistency and factor structure
As
reported in Table 2, the internal consistencies of the DSAQ
subscales were acceptable to good (Streiner, 2003).
The
results of the confirmatory factor analysis indicate that the
onefactor model in which all five subscales loaded on a
general differentiation factor did not fit the data well: χ^{2}(5)
= 55.126, p <
.001; RMSEA = .193 (90% CI .149  .241); CFI = .912; TLI =
.824; SRMR = .050. The twofactor model had a good fit: χ^{2}(4)
= 5.637, p = .228;
RMSEA = .039 (90% CI .000  .107); CFI = .997; TLI = .993;
SRMR = .017. For the factor Progress Monitoring, standardised
factor loadings were .84 (SE = 0.03, R^{2} = .71)
for Identification of Educational Needs and .85 (SE = 0.03, R^{2} = .73)
for Evaluation of Progress and Process. For the factor
Instructional Adaptations, standardised factor loadings were
.77 (SE = .04, R^{2} = .59)
for Differentiated Goals, .75 (SE = .04, R^{2} = .56)
for Differentiated Instruction, and .74 (SE = .04, R^{2} = .54)
for Differentiated Practice (p < .001 for all
factor loadings). The correlation between the factors was .78
(p < .001). Since
the twofactor model provided a better fit, the two factor
scores (average of the subscale scores comprising that factor)
were used in subsequent analyses.
3.3.2
Convergent
and divergent validity: correlations with other scales
The
correlations between the two DSAQ factors and related scales
are reported in Table 3. In support of convergent validity,
the correlation with Prerequisite Knowledge for
Differentiation was strong for both factors. As hypothesised,
correlations with selfefficacy for Instruction and
selfefficacy for Adapting Education to Individual Students’
Needs were moderate to strong for the factor Instructional
Adaptations and somewhat lower for Progress Monitoring.
Regarding divergent validity, the correlations with Motivating
Students and Classroom Management were less strong, although
still in the moderate range. Contrary to expectations, Coping
with Changes and Challenges correlated strongly with the
factor Instructional Adaptations.
Table 3
Correlations (p < .001) between DSAQ factor
scores and related scales
Scale 
Progress Monitoring 
Instructional Adaptations 


r 
95% CI 
r 
95% CI 
Selected
for convergent validity 




Prerequisite knowledge for
differentiation 
.62 
.54  .68 
.70 
.64  .76 
Instruction (to students of all
achievement levels) 
.40 
.30  .49 
.47 
.37  .56 
Adapting education to individual
students’ needs 
.38 
.28  .48 
.56 
.48  .64 
Selected
for divergent validity 




Motivating students 
.30 
.19  .40 
.37 
.27  .47 
Classroom management 
.34 
.23  .44 
.40 
.30  .50 
Coping with changes and challenges 
.42 
.31  .52 
.58 
.49  .65 
3.3.3
Distribution
of DSAQ scores: mean scores and infrequently reported
strategies
The mean
factor scores were 3.60 (SD
= 0.52) for Progress Monitoring and 3.68 (SD = 0.43) for
Instructional Adaptations. With a range from 1.83 to 4.86 for
Progress Monitoring and from 2.45 to 4.94 for Instructional
Adaptations, the factor scores were normally distributed at
the high end of the scale. Table 2 provides the means and
standard deviations of all subscales. Taken together, the
mean factor and subscale scores reflect moderate to high
selfassessed use of differentiation strategies.
Table 4
provides the means and standard deviations for each item of
the DSAQ. Five items  numbers 3.7, 4.2, 4.4, 4.8, and 5.5 
had a mean score at least one standard deviation below the
mean of their factor. Two of these  the use of diagnostic
conversations to evaluate whether the learning goals have been
met and the adaptation of type of practice to students’ needs
 reflect specialised strategies
because they involve the refined diagnosis of and adaptation
to individual students’ needs. Other specialised strategies
(items 1.5 and 5.7) also had somewhat lower means, although
these means were within one standard deviation of the factor
mean. The three remaining infrequently reported items
concerned adaptations for highachieving students, namely additional onlevel instruction or
guidance, curriculum compacting, and the use of computer
programmes for additional
challenge. Nevertheless, two other strategies targeted at
highachieving students (items 2.5 and 4.5) were frequently
reported.
Table 4
Means and standard deviations of DSAQ items
(scale range 1  5)
DSAQ item 
M 
SD 
Subscale
1: Identification of educational needs 


1.1 I analyse the
answers on curriculumbased tests to assess a
student’s educational needs 
4.02 
0.77 
1.2 I analyse the
answers on standardised tests to assess a student’s
educational needs 
3.49 
0.91 
1.3 I assess
specific students’ educational needs based on daily
maths work 
3.75 
0.72 
1.4 I assess specific students’
educational needs based on (informal) observations
during the maths lesson 
3.76 
0.77 
1.5 If necessary, I conduct diagnostic
conversations to analyse the educational needs of
specific students 
3.20 
0.90 
Subscale
2: Differentiated goals 


2.1 I set different goals for the
children, dependent on their achievement level 
3.62 
0.79 
2.2 I set extra challenging goals for
highachieving students 
3.57 
0.83 
2.3 I set wellconsidered minimum
goals for very lowachieving students 
3.75 
0.76 
2.4 I know the opportunities for
differentiation offered by the curriculum 
4.03 
0.68 
2.5 I use the opportunities the
curriculum offers for differentiation for
highachieving students 
3.88 
0.84 
2.6 I use the opportunities the
curriculum offers for differentiation for
lowachieving students 
3.83 
0.82 
Subscale
3: Differentiated instruction 


3.1 I adapt the level of abstraction
of instruction to the needs of the students 
3.95 
0.55 
3.2 I adapt the modality of
instruction (visual, verbal, manipulative) to the
needs of the students 
3.82 
0.62 
3.3 I adapt the pace of instruction to
the needs of the students 
3.95 
0.56 
3.4 I deliberately ask openended
questions during wholeclass instruction 
3.82 
0.67 
3.5 I deliberately ask questions at
various difficulty levels during wholeclass
instruction 
3.69 
0.73 
3.6 I regularly
provide lowachieving children with additional
instruction (extended instruction, preteaching) 
4.25 
0.64 
3.7 I regularly provide highachieving
students with additional instruction or guidance at
their level, in a group or individually 
3.20 
0.92 






Table 4 (continued) 


DSAQ item 
M 
SD 
Subscale
4: Differentiated practice 


4.1 I vary different
types of practice during the maths lesson (e.g.
individual or group work, solution spoken, written or
drawn) 
3.53 
0.78 
4.2 I adjust
different types of practice to the needs of the
students in the classroom (e.g. having a specific
child complete exercises on the computer because this
child learns more in this way) 
3.04 
0.83 
4.3 I select the
most important tasks for very lowachieving students 
3.73 
0.73 
4.4 I use curriculum compacting
for highachieving students 
3.20 
1.25 
4.5 I provide highachieving
students with enrichment tasks 
4.00 
0.87 
4.6 I also use
computer programmes or maths websites in my maths
lessons 
3.68 
0.97 
4.7 I use computer programmes
and/or maths websites to offer children focused
practice in a skill that they do not sufficiently
master 
3.32 
0.96 
4.8 I use computer
programmes and/or maths websites to offer specific
children additional challenge in the maths lesson 
3.15 
1.05 
Subscale
5: Evaluation of progress and process 


5.1 I use scores on standardised and
curriculumbased tests to evaluate whether the
learning goals have been met 
4.04 
0.73 
5.2 I analyse the answers on
curriculumbased tests to evaluate whether the
learning goals of that unit have been met 
4.06 
0.72 
5.3 I regularly evaluate whether all
students have met the learning goals based on their
daily maths work 
3.75 
0.85 
5.4 I evaluate whether all students have
met the lesson goals based on (informal) observations
during the maths lesson 
3.45 
0.86 
5.5 I conduct diagnostic conversations to
evaluate whether specific students have met the lesson
goals 
2.85 
0.87 
5.6 I evaluate whether the type of
instruction and practice chosen by me were effective
for the majority of the students in the class 
3.44 
0.77 
5.7 I evaluate whether a specific type of
instruction was effective for specific students 
3.32 
0.80 
3.4
Concluding summary Study 2
Study 2
investigated teachers’ selfassessed implementation of
differentiation using the DSAQ. The first goal was to examine
the psychometric properties of the DSAQ. The subscales of the
DSAQ were internally consistent and loaded on two correlated
but distinct factors: Progress Monitoring and
Instructional Adaptations.
Confirmatory factor analysis demonstrated that this twofactor
structure provided a better fit than a onefactor model, which
converges with the findings reported by Roy et al. (2013). The
second goal was to examine the convergent and divergent
validity of the DSAQ. The pattern of correlations between the
DSAQ and other scales supported its convergent and divergent
validity. As expected, strong to moderate correlations with
Prerequisite Knowledge for Differentiation, Adapting Education
to Individual Students’ Needs, and Instruction were found. As
hypothesised, the correlations with the scales selected for
testing the divergent validity were lower, except for the
correlation with Coping with Changes and Challenges which was
unexpectedly strong.
The third
and main goal was to examine teachers’ perceived usage of the
strategies recommended by the experts. With factor means in
the moderate to high range, teachers assessed their use of
differentiation strategies more highly than we had expected.
Five items with relatively low means were identified. In
support of our hypothesis, these items concerned specialised
strategies and strategies targeted at highachieving students.
4. General
discussion
Teachers
are required to implement differentiation for students of
diverse achievement levels. However, the term differentiation
had been used in diverse ways and the literature did not
provide sufficient information regarding the most effective
strategies to provide teachers with general guidelines for
implementing differentiation. To fill this gap, Study 1
operationalised the
concept of differentiation by achieving consensus among a
consortium of experts about a model and strategies for
differentiation in primary school mathematics. Study 2
investigated the degree to which Dutch teachers already
implement the strategies suggested by the experts.
Study
1 resulted in a model for differentiation consisting of five
steps: identification of educational needs, differentiated
goals, differentiated instruction, differentiated practice,
and evaluation of progress and process. These steps reflect
the two core components of differentiated instruction
identified by Roy et al. (2013). Progress monitoring is
captured by the steps of identification of educational needs
and evaluation of progress and process. The component of
instructional adaptations is represented by the steps of
differentiated goals, instruction, and practice. Study 2
demonstrated that a twofactor model in which the subscales of
the DSAQ load on these two factors provides a better fit than
a onefactor model. Our findings converge with the findings
reported by Roy et al. (2013), supporting the idea that
progress monitoring and instructional adaptations are two
distinct but related components of differentiation.
New
in this study is expert consensus on how progress should
be monitored and how
goals, instruction and practice should be adapted to the
learning needs of students with diverse achievement levels.
Regarding progress monitoring, the experts recommended to use
standardised and curriculumbased tests first to divide
students over achievement groups. More refined and informal
measures such as the analysis of daily work should be used
frequently to monitor shortterm progress, to diagnose unique
educational needs, and to determine whether a (temporary)
adjustment of the groups is necessary. Compared to
technological applications which tend to make use of one or
two types of assessment to monitor progress (e.g. McDonald
Connor et al., 2009; Ysseldyke & Tardrew, 2007), the
experts recommended a broader range of strategies and
indicated how they can be used together. The strategies have
complementary purposes: while relatively formal and
standardised tests are useful to get an overview of what a
student can do, more informal and qualitative measures such as
diagostic conversations and the analysis of daily work provide
valuable information about why a student struggles with a
certain problem and what the student needs.
The
use of withinclass homogeneous achievement groups provides
the opportunity to tailor subgroup instruction to similar
educational needs and has demonstrated positive effects (Kulik
& Kulik, 1992; Lou et al., 1996; Slavin, 1987; Tieso,
2005). In line with Slavin (1987), the experts stressed the
importance of flexibility, i.e. allowing students to switch
between groups based on changes in their educational needs.
The literature indicates that the effects of withinclass
ability grouping may depend upon student achievement level,
with smaller or even negative effects for lowachieving
students (Deunk et al., 2015; Lou et al., 1996). Nevertheless,
the experts clearly perceived smallgroup instruction as a
good way to provide lowachieving students with the
instruction they specifically need. Also, students are only
grouped for part of the lesson and participate in the
wholeclass instruction for students of all ability levels as
well. Future research should establish whether these
conditions ensure that lowachieving students also profit from
this type of withinclass flexible ability grouping.
Regarding
instructional adaptations, the experts recommended a coherent
set of strategies to differentiate goals, instruction and
practice. This comprehensive approach is somewhat broader than
technologybased interventions which have tended to focus on
differentiation of either instruction (Individualizing Student
Instruction) or practice (Accelerated Math). Many of the
strategies recommended by the experts are supported by
previous research, including the adaptation of practice tasks
to the skill level of the student (Ysseldyke & Tardrew,
2007), the use of explicit instruction and visual
representations for lowachieving students (Gersten et al.,
2009) and the use of compacting, enrichment, and instruction
at challenge level for advanced students (Rogers, 2007). To
use teachers’ time efficiently, the experts recommended to
teach the whole class when possible, to use subgroups when the
diverse educational needs of subgroups require this, and to
serve remaining unique educational needs individually. Thus,
the experts recommended both universal supports (supports for
all students such as varying the difficulty level of questions
in broad wholeclass instruction) and targeted supports
(supports specifically for lowachieving and highachieving
students including smallgroup instruction and differentiation
in practice tasks). The experts also recommended some
adaptations to individual students’ educational needs (e.g.
the adaptation of type of practice to the preference of
specific students), but they realised that such specialised
adaptations were advanced and primarily suitable for teachers
who already master basic strategies for differentiation.
To link the advice
provided by the experts to teachers’ daily practice, Study 2
investigated teachers’ selfreported usage of the recommended
strategies. Overall, DSAQ scores were moderate to high,
exceeding the expectations we had based on previous studies.
Perhaps, the different context (primary schools in the
Netherlands versus middle schools in the United States) can
explain the discrepancy with the low use of differentiation
strategies reported by Moon et al. (2002). Our findings are
more similar to those of a recent study with Canadian primary
school teachers in which moderate usage was reported (Roy et
al., 2013). Nevertheless, the moderately high selfassessments
in the current study seem discrepant with the finding of the
Dutch Inspectorate of Education that adequate adaptations to
students’ diverse educational needs are only made at about
half of the schools (Van den Broekd’Obrenan et al., 2012).
Also, the experts in Study 1 clearly perceived a need for
professional development about differentiation. Perhaps, the
inspectors of education and the experts from our consortium
have high standards for the quality of
implementation which are not captured by the DSAQ. Teachers
might also overestimate their implementation. Refined
observational studies are necessary to examine whether
teachers’ high selfassessed usage of differentiation
strategies can be confirmed by external observers.
In
line with previous studies (McLeskey & Waldron, 2002,
2011; Reis et al., 2004; Scott et al., 1998; Westberg et al.,
1993; Westberg & Daoust, 2003), specialised studies and
strategies targeted at highachieving students were used
relatively infrequently. Two specialised strategies  the use of diagnostic
conversations to evaluate whether the learning goals have been
met and adaptation of the type of practice to specific
students’ needs  were relatively infrequently reported. This
corresponds with the view expressed by the experts that
individuallevel differentiation is advanced and primarily
suitable for teachers who already implement groupbased
strategies for differentiation successfully (Van Groenestijn,
Borghouts, & Janssen, 2011).
Three
strategies targeted at highachieving students  curriculum compacting, the use of computer programmes for additional challenge, and targeted
instruction for these students  were used infrequently. The
difference between the use of instruction targeted at
highachieving students versus lowachieving students is
especially striking. Perhaps, teachers are not aware that highachieving
students
also need guidance when working on sufficiently challenging
enrichment tasks (VanTasselBaska & Stambaugh, 2005). Many teachers do implement some
differentiation in practice tasks. However,
there is still a lot of room for improvement, since it seems
that only few teachers use a complete approach including
challenging goals, curriculum compacting, enrichment tasks and
onlevel guidance. Low usage of differentiation for
highachieving students has repeatedly been attributed to a
lack of the specific attitudes, knowledge, and skills this
requires (Latz, Speirs Neumeister, Adams, & Pierce, 2009;
MegayNespoli, 2001; VanTasselBaska & Stambaugh, 2005).
Many teacher educators feel that initial teacher training does
not adequately prepare teachers to differentiate instruction
for highachieving students (Schram, Van der Meer, & Van
Os, 2013). Thus, it seems important that this topic receives
sufficient attention in teacher training and professional
development programmes.
The
following limitations should be taken into account. First, the
results of consensus procedures are inherently restricted by
the participating experts. The risk that other experts might
have provided different input cannot be eliminated but was
diminished in this study by recruiting experts working for
several different institutions for both preservice and
inservice teacher training. Second, some experts missed some
components of the procedure (a focus group discussion or a
round of the Delphi questionnaire). This limitation was
compensated for by the repetitive nature of the procedure:
participants who missed one component could still provide
comments and additional input in the subsequent component.
Third, it is possible that teachers provided socially
desirable answers since a selfreport questionnaire was used
in Study 2. The variability between the items provides an
indication that teachers did not simply rate themselves highly
on all items to create a favourable impression. Nevertheless,
we state again that observational studies are necessary to
investigate how selfreported use is related to observed use
of differentiation strategies. Fourth, it is unknown whether
nonresponders differed from teachers who did respond to the
questionnaire, although the response rate of 83% is quite
good.
a
strong combination of methodologies was used. Study 1 employed
an innovative methodology which combined the advantages of two
methods: focus group discussions are suitable for creating
shared understanding and generating ideas, while the Delphi
procedure gives all participants an equal and anonymous say in
the systematic evaluation of those ideas. This combination was
fruitful and efficient and we recommend it for future
research. Moreover, the collaboration with experts who were
familiar with the daily practice of teaching enhanced the
feasibility of the findings. Based on their experience in
various settings, the experts perceived these strategies as
effective and feasible. In addition to this expert
perspective, Study 2 examined the results of Study 1 from a
teacher perspective. The fact that the teachers in Study 2
reported to use most of the strategies recommended by the
experts in Study 1 shows that teachers acknowledge the need to
differentiate and that the recommended strategies are largely
compatible with teachers’ daily practice. At the same time,
the discrepancy between the teachers’ and experts’ perception
of the degree of application of differentiation opens up an
interesting avenue for future research. Thus, the inclusion of
two complementary sources of information provides a richer
perspective on differentiation. Despite these methodological
advantages, future research is necessary to test empirically
whether the implementation of the strategies recommended by
the experts leads to higher student achievement.
The
results of the current studies contribute to scientific
research as well as to educational practice. Although the
experts in Study 1 departed from a practical rather than a
theoretical perspective, the elements of the cycle of
differentiation overlap with elements of more general
didactical models, such as Van Gelder’s didactical analysis
model (Van Gelder, Oudkerk Pool, Peters, & Sixma, 1973)
and De Corte’s didaxology (De Corte, Geerligs, Lagerweij,
Peters, & Vandenberghe, 1976). Apparently, effective
readinessbased differentiation is consistent with the
principles of general good teaching, with the addition that
each stage of teaching needs to be differentiated. A first theoretical
implication is that, rather than studying specific elements
(i.e. differentiated
practice) in isolation, it seems promising to move towards an
integral view of differentiation which involves all stages of
teaching. Second, the experts clearly endorsed the view that
students of different achievement levels have different
educational needs and need different treatments at least part
of the time, echoing the aptitudetreatment interaction
literature (Cronbach & Snow, 1977). This emphasises the
need to consider the potential variation between students in
the design and analysis of educational intervention studies:
what works for highachieving students, may not work for
lowachieving students and vice versa.
At
the practical level, the model and strategies for
differentiation recommended by the experts can be used in
teacher training and professional development, for which
purpose they have also been published in a Dutch journal for
practitioners (Van de WeijerBergsma & Prast, 2013).
Current educational policies require teachers to implement
differentiation and our results provide teachers with concrete
advice on how to do this. The cycle of differentiation can be
used as a framework to structure professional development
about differentiation. It shows teachers that differentiation
requires attention at all stages of teaching in one coherent
approach. The recommended strategies provide teachers with
practical suggestions for each step (these can be found in
section 2.3, Appendix 1, and also in the DSAQitems listed in
Table 4). The focus on mathematics promoted the concreteness
of the results, since domainspecific guidelines can be
applied directly without the need to transfer general
principles. For example, the general guideline that advanced
learners should be adequately challenged was made ready for
use by providing achievement criteria for selecting
highperforming students, suggestions for increasing task
difficulty, guidelines for compacting, and a list of
supplemental enrichment curricula. Nonetheless, the principles
behind these concrete recommendations, including the cycle of
differentiation, seem to be applicable in other domains as
well. Future research could examine whether and how our
results extend to other domains.
Study
2 provides researchers and practitioners with a new tool, the
DSAQ. Researchers can use it, for example, as a pre and
postassessment in intervention studies or to investigate
teachers’ selfassessed implementation in other countries. In
professional development, the DSAQ can inform
trainers about areas in which teachers perceive most room for
improvement. Theoretically, Study 2 builds on the existing
literature by providing support for the twodimensional
structure of differentiation. Moreover, this study is the
first to investigate the selfreported use of a broad range of
strategies for differentiation in mathematics in the
Netherlands. The identified areas of low usage have practical
implications, including the need to pay sufficient attention
to differentiation for highachieving students in teacher
training and professional development.
Keypoints
Effective combination of the Delphi technique and
focus group discussions to achieve consensus among experts
Use of experts’ practical knowledge to enhance
feasibility of the findings
A model and strategies for implementing
differentiation in primary school mathematics, usable in
teacher professional development
A new questionnaire (the DSAQ) to measure
teachers’ selfassessed implementation of differentiation
strategies, usable in future research
Teacher selfassessment indicating moderate to
high usage and identification of relatively infrequently used
strategies
Acknowledgements
This work is part of the research programme ‘Every
child deserves differentiated mathematics education’, which is financed by the Netherlands
Organisation for Scientific Research (NWO), grant number 41110753.
The
NWO was not involved in the collection, analysis,
interpretation, or reporting of the data.
We
thank the consortium members for their valuable input during
the consensus procedure. We are also grateful to the teachers
who completed the questionnaire. Finally, we thank the
anonymous reviewers for their useful comments on a previous
version of this article.
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Appendix 1:
Summarising concepts included in the Delphi questionnaire
What
follows
are the translations of the concepts as they were included in
the Delphi questionnaire. Background information that might be
relevant for nonDutch readers is given in the footnotes.
Organisation
The
starting
point is convergent differentiation[2].
Students are assigned to one of three subgroups based on standardised
tests and / or curriculumbased tests. If curriculumbased tests
are used to assess what students already master, the test score
of the previous unit can be used, but an alternative is to use
the endofunit test of the upcoming unit as a pretest. The
teacher can change the grouping arrangement for a certain unit
or lesson based on test scores. During mathematics classes,
wholeclass instruction, instruction to one of the subgroups (of
lowachieving or highachieving students) and independent
practice are alternated. Average achievers take part in the
wholeclass instruction and receive individual feedback or
guidance during the time for independent practice.
Differentiation in instruction
During
wholeclass
instruction, the teacher serves different levels and educational
needs to the extent that this is possible. The teacher can do
this by teaching at different levels of abstraction and showing
the connection between these different levels. The teacher
should ask questions at varying difficulty levels, implying that
some questions may be too easy or too difficult for some of the
students in the class. During instruction to a subgroup, the
teacher takes into account the educational needs of the students
in that specific subgroup. For example, the teacher spends more
time on lower levels of abstraction when teaching the
lowachieving subgroup, while instruction to the highachieving
subgroup is mainly at a high level of abstraction. Additionally,
it is assumed that the lowachieving subgroup needs more
guidance (more direct instruction) than the highachieving
subgroup (more exploratory instruction). To the extent possible,
the teacher also takes into account individual differences
within a subgroup. For example, the teacher can accommodate to a
student’s need to verbalise a solution strategy himself, or a
student’s need for visualisation. An additional strategy for
differentiation in instruction is the use of instructional
videos.
Differentiation in goals
A
strong awareness of the learning trajectories and accompanying
educational goals is essential for a good lesson. For
differentiated education, this means that different goals are
set for different students. Goals are differentiated primarily
at the subgroup level. Highly competent teachers can also
differentiate goals on an individual basis based on their
professional insight. For the lowachieving subgroup, the
objective is to master the fundamental level (1F)[3]
at the end of primary school. For the averageachieving
subgroup, the objective is to master the target level (1S) at
the end of primary school. For the highachieving subgroup,
mastery of the target level is a minimum requirement, but
additionally, more advanced goals (for example regarding logical
reasoning) are set for these students. The goals for the end of
primary school are converted into specific lesson goals for the
three subgroups based on the curriculum and the professional
insight of the teacher. These lesson goals should be both
ambitious and realistic. The teacher keeps in mind the lesson
goals while preparing and teaching his lesson. After the lesson,
the teacher evaluates whether lesson goals have been met.
Differentiation in the practice phase
The
different subgroups need quantitatively and / or qualitatively
different practice tasks. From the tasks that the curriculum
offers, the tasks at the minimum and fundamental level are most
important for lowachieving students. The highachieving
subgroup can skip a large proportion of the tasks at minimum and
fundamental level. Existing guidelines for compacting[4]
can be used to select the tasks that highachieving students do
need to do. High achieving students spend the time that is freed
up by compacting the regular material on enrichment. The
enrichment tasks provided in the regular curriculum are often
not sufficiently challenging, especially for gifted students.
Additional enrichment should be provided for these students.
Such enrichment may include assignments for which students have
to carry out research or use information from different sources.
Besides the adaptation (selection and supplementation) of tasks,
practice can also be differentiated during instruction to
subgroups. For example, the teacher could use the extended
instruction for lowachieving students to solve the exercises
together stepbystep, while a discussion of the big ideas
behind a certain task may be more useful in the highachieving
subgroup.
Differentiation based on learning styles
The
educational
needs of different students may also vary within subgroups. For
example, students may have a preference for certain formats
(wholeclass instruction, working together, working
individually) and certain input modes (visual or verbal, written
or spoken). It has been mentioned repeatedly that it is
important for some students to express the content themselves or
to have the content explained to them by another student.
Teachers need to be aware of these differences between students
and learn how they can vary their instruction, tasks and formats
to accommodate various educational needs. Especially during the
instruction to subgroups the teacher can accommodate to
individual educational needs, provided that he is able to
identify what kind of instruction or type of task a student
needs to understand the content.
Differentiation in kindergarten
When
the files of students with problematically low mathematics
achievement in primary school are examined, it often turns out
that problems with preparatory mathematics were already detected
in kindergarten but that no or insufficient action has been
taken to tackle those problems in the meantime. Factors that may
play a role in this lack of action are beliefs of the teacher
(‘the child is not ready for preparatory mathematics’ or
‘children of this age should be allowed to play’), inadequate
communication to the teacher of the next grade, and lack of
knowledge of ways to tackle low achievement in (preparatory)
mathematics. In order to respond more quickly to early signals
of problems with acquiring preparatory mathematics skills,
teachers should (a) set more specific and ambitious goals (what
should the child be able to do at the beginning of grade 1?),
(b) be more knowledgeable about levels of abstraction and be
able to demonstrate the connections between various levels of
abstraction, (c) realise that learning can take place in the
process of playing if the activity is well adapted to the
child’s educational needs, and (d) that certain children need
some additional instruction, also in kindergarten. Additionally,
more attention should be given to providing extra challenge to
students with highly developed preparatory mathematics skills.
Table 5
Table of footnotes
No. 
Footnote 
1 
Compared to
the fouritem NTSES subscale for Adapting Instruction to
Individual Students’ Needs, the added value of the DSAQ
is that it provides a more detailed measure of
selfassessed use of a range of differentiation
strategies. 
2 
In
the Netherlands, a distinction is often made between
convergent and divergent differentiation (Gelderblom,
2007). In convergent differentiation, all student in a
classroom work on roughly the same topics at the same
time (even if they might engage in the topic at varying
levels of complexity). In divergent differentiation,
different students work on different learning goals and
topics at the same time. 
3 
In the
Dutch educational system, overarching objectives
(comparable to the common core state standards employed
in the US) have been defined at two levels: the
fundamental level (1F) that should be reached by all
students and the target level (1S) that should be
reached by about 65% of the students (Expertgroep
doorlopende leerlijnen taal en rekenen, 2008). 
4 
An
educational advisory company has published guidelines
for compacting the most popular Dutch mathematics
curricula (Janson & Noteboom, 2004). Children for
whom the material should be compacted receive an
additional booklet with an overview per lesson of the
exercises they should do and the exercises they can
skip. 
[1] Compared to the fouritem NTSES subscale for
Adapting Instruction to Individual Students’ Needs, the
added value of the DSAQ is that it provides a more detailed
measure of selfassessed use of a range of differentiation
strategies.
[2] In the Netherlands, a distinction is often made
between convergent and divergent differentiation
(Gelderblom, 2007). In convergent differentiation, all
student in a classroom work on roughly the same topics at
the same time (even if they might engage in the topic at
varying levels of complexity). In divergent differentiation,
different students work on different learning goals and
topics at the same time.
[3] In the Dutch
educational system, overarching objectives (comparable to
the common core state standards employed in the US) have
been defined at two levels: the fundamental level (1F) that
should be reached by all students and the target level (1S)
that should be reached by about 65% of the students
(Expertgroep doorlopende leerlijnen taal en rekenen, 2008).
[4] An educational advisory company has published
guidelines for compacting the most popular Dutch mathematics
curricula (Janson & Noteboom, 2004). Children for whom
the material should be compacted receive an additional
booklet with an overview per lesson of the exercises they
should do and the exercises they can skip.