Domain of Validity framework: A new instructional theory for addressing students' preconceptions in science and engineering

Abstract

We propose a new instructional theory, the Domain of Validity (DoV) Framework, which offers a new way forward for designing teaching for conceptual change, while also resolving conflicts between existing theories related to common, difficult-to-change conceptions students have about particular scientific topics. We propose that knowledge consists of two connected elements: a model and a domain of validity (or DoV). Foregrounding the notion of DoV for given models allows us to reconceptualise and diagnose many problematic preconceptions as examples of an oversized DoV. Mapping the different elements of knowledge – both the model and its domain of validity – allows teachers to pinpoint precisely the cognitive conflict that students need to confront in a conceptual change approach to teaching. We highlight the instructional implications related to these scientific learning difficulties and conclude by proposing particular teaching strategies based on this new framework, emphasising the domains of validity of particular scientific models.

Keywords: misconception; conceptual change; science education; model-based learning; threshold concept

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Domain of Validity framework:
A new instructional theory for addressing students’ preconceptions in
science and engineering
Raoul Sommeilliera, Kathleen M. Quinlanb and Frédéric Roberta
aBio Electro and Mechanical Systems (BEAMS), Université libre de Bruxelles, Brussels, BELGIUM;
bCentre for the Study of Higher Education, University of Kent, Canterbury, UK
We propose a new instructional theory, the Domain of Validity (DoV)
Framework, which offers a new way forward for designing teaching for
conceptual change, while also resolving conflicts between existing theories
related to common, difficult-to-change conceptions students have about particular
scientific topics. We propose that knowledge consists of two connected elements:
a model and a domain of validity (or DoV). Foregrounding the notion of DoV for
given models allows us to reconceptualise and diagnose many problematic
preconceptions as examples of an oversized DoV. Mapping the different
elements of knowledge – both the model and its domain of validity – allows
teachers to pinpoint precisely the cognitive conflict that students need to confront
in a conceptual change approach to teaching. We highlight the instructional
implications related to these scientific learning difficulties and conclude by
proposing particular teaching strategies based on this new framework,
emphasising the domains of validity of particular scientific models.
Keywords: misconception; conceptual change; science education; model-based
learning; threshold concept
Introduction
Since the early 20th century, science education research has emphasised the importance
of understanding students’ prior knowledge (National Research Council, 1999; Rittle-
Johnson, Star, & Durkin, 2009). Researchers from various backgrounds, periods and
nationalities have studied how difficult-to-change prior knowledge impacts the way
students learn, yielding a variety of terms and concepts that are similar, but have subtle
distinctions. In this paper, we clarify existing constructs before proposing an integrative
framework from which we derive instructional strategies that make explicit, rather than

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leave implicit, the domains of validity of given concepts.
First, we briefly summarise seminal ideas that address similar topics, have
similar purposes, and contribute to an understanding of students’ learning in sciences.
The six constructs we review are: 1) misconceptions and preconceptions, 2) alternative
and anchoring conceptions, 3) phenomenological primitives (p-prims), 4) threshold
concepts, 5) cognitive obstacles and 6) conceptual change.
Second, we propose a new framework that builds on those key constructs, but
explicitly attends to the context of application, or “domain of validity” (DoV) of a given
model. This DoV framework makes explicit an aspect of existing theories that has
remained largely implicit or overlooked. In doing so, it reconciles some of the tensions
found between existing constructs and allows us to propose associated teaching
strategies, which we emphasise in the last section.
Third, we reinterpret the constructs from existing literature in the new terms of
our formalised DoV framework. This last step shows the coherence, limitations and
contributions of the proposed framework in relation to the extant literature. Finally, we
elaborate specific teaching strategies with examples to highlight the practical elements
of the framework for enhancing teaching for conceptual change.
1. Literature Review
Research on “prior knowledge” is organised here into six constructs. Rather than an
exhaustive review of research in the field, we examine the foundations of each of these
constructs to point out their common ground and points of divergence. For each
construct, we succinctly summarise its key features, the contributions it makes beyond
previous literature, and major criticisms it has faced.

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1.1. Misconceptions and Preconceptions
Since the 1960's, the science education literature has referred to students' conceptions
under several names, including: misconceptions (Helm, 1980), preconceptions (Novak,
1977), alternative conceptions (Driver & Easley, 1978), children’s science (Gilbert,
Osborne, & Fensham, 1982), and synthetic models (Vosniadou, 2012). Some authors
use other terms (e.g. intuitive conceptions, naïve beliefs, pre-instructional conceptions,
spontaneous knowledge, folk knowledge, personal models of reality) to mean the same
idea (Özmen, 2004; Peşman & Eryilmaz, 2010). The variety of terms in use today
remains quite wide. Some authors justify their choice by explicitly defining the terms
they use, but others do not.
The choice of terminology sometimes depends on the researcher’s view of the
status of knowledge (Gilbert and Watts, 1983) and on the semantics of the terms. For
many authors (Demirci & Çirkinoglu, 2004) preconception and misconception refer to
the same concept: a student’s idea that is in conflict with scientifically accepted ideas
(Gilbert & Watts, 1983; Viennot, 1985). For others, not all preconceptions are
misconceptions (Brown & Clement, 1987; Clement, 1991, 1993; Clement, Brown, &
Zietsman, 1989).
Some researchers do not characterise the relationship between preconceptions or
misconceptions and scientifically accepted ideas in terms of conflict (Hammer, 1996;
Hamza & Wickman, 2008; National Research Council, 2012b; Smith, diSessa, &
Roschelle, 1994). For Driver and Easley (1978), the main property of a preconception is
that it is naïve, immature, or under-developed compared to a scientific concept. A
review of this literature (National Research Council, 2012b) uses the term
misconceptions to mean understandings or explanations that differ from what is known
to be scientifically correct, while recognising that some researchers refer to these

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explanations as alternate conceptions, prior understandings, or preconceptions, and that
the different terms can reflect different perspectives.
Other authors emphasise temporal aspect of these concepts. For Fetherstonhaugh
and Treagust (1992), a preconception is simply a conception in a certain area that is
present in the student’s mind prior to relevant instruction (see also Subsection 1.2).
Vosniadou (2012) reserves the term misconception for erroneous ideas that students still
hold after instruction. And for Carey (1986), misconceptions may exist either before or
after students have had relevant instruction.
The previous references demonstrate the variable meaning of these two first
terms. The meaning of the most commonly used term, misconception, has evolved over
time. According to the most common use of the term today, a misconception refers to
ideas that differ from what is known to be scientifically correct (Hamza & Wickman,
2008; National Research Council, 2012b). Furthermore, misconceptions can be said
(Hammer, 1996; Smith et al., 1994) to have four properties: a misconception (1) is
knowledge that affects how students understand natural phenomena and scientific
explanations, (2) involves stable (strongly held, difficult to change) cognitive structures,
(3) differs from expert conceptions, and (4) must be eliminated or dismantled for a
student to progress.
However, the third and fourth properties of misconceptions– an intrinsic
incorrectness of the prior knowledge and consequent need for removal – have been
questioned (Chi, 2008, 2013; Clement et al., 1989; Michelet, Adam, & Luengo, 2007).
We incorporate these last two criticisms into our use of the term preconception to
depict a more general and neutral concept than misconception (Brown & Clement,
1987; Clement, 1991, 1993; Clement et al., 1989). Thus, we define preconception as
knowledge that has the two first properties mentioned above, but is mainly defined by

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its presence before formal instruction (Fetherstonhaugh & Treagust, 1992) and by its
role as an obstacle to further learning, without necessarily requiring dismantling (Chi,
2008, 2013; Clement et al., 1989; Gilbert & Watts, 1983; Michelet et al., 2007). Our
definition then is: preconceptions (1) are knowledge affecting how students understand
natural phenomena and scientific explanations, (2) are stable (strongly held, difficult to
change) cognitive structures, (3) have been constructed by students (via experiences,
formal learning, etc.) prior to the considered instruction and (4) can act as barriers to
new learning. The review of constructs provides further elaboration for our broadened
definition of preconceptions.
1.2. Alternative conceptions and anchoring conceptions
According to Gilbert & Watts (1983), researchers of misconceptions assumed that
students’ scientifically incorrect ideas were flaws in the knowledge base that needed to
be eliminated or repaired for new knowledge acquisition to be successful. Yet,
dismantling prior knowledge is inconsistent with constructivist views of learning (see
discussion of Piaget in Subsection 1.5) that emphasise building on existing knowledge,
which is seen as the product of an individual’s intentional, active, and ongoing
construction of meaning. Gilbert & Watts (1983) thought that students’ scientifically
“incorrect” ideas should be respected as “personally viable constructive alternatives” to
scientific knowledge, thus preferring the terms alternative frameworks and alternative
conceptions, after the work of Driver & Easley (1978).
Therefore, compared to misconception, which suggests via the prefix “mis-” an
intrinsic incorrectness of the knowledge, alternative conception highlights the idea that
students’ prior knowledge (preconceptions) may be meaningful and useful in the
learning journey. By the end of the 20th century, it was generally accepted that “prior
knowledge can either interfere with or facilitate new learning” (National Research

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Council, 1999). This early debate in the literature, though, helps accounts for the mix of
terms for students’ scientifically "incorrect" ideas in the education literature today.
Many researchers continue to use the term misconception, but actually mean alternative
conceptions. Insofar as they do not assume students hold an "inadequate ‘picture’" of
the world: “Students' alternative frameworks (frequently regarded as misconceptions)
play a crucial role in science concept learning” (Nussbaum & Novick, 1982, p. 183).
Teaching to overcome inappropriate alternative conceptions “could mean modify the
domain of, displace, modify and improve, replace, or suppress a conception, depending
on what is most appropriate” (Clement, 1993, p. 1242).
Clement and colleagues (Brown & Clement, 1987; Clement, 1991; Clement et
al., 1989; Hammer, 2000) insisted further on the positive facet of some preconceptions.
Their research led to the notion of anchoring conceptions: “Although many
preconceptions are detrimental to learning, some are largely in agreement with accepted
physical theory. These will be referred to here as "anchoring conceptions" (or more
briefly, as anchors).” (Clement et al., 1989). Based on their studies, Clement et al.
(1989) observed that students’ correct answers to a particular example can trigger an
anchoring conception that is a starting point for building an expert-like conception.
Clement (1993) reported success in using a teaching approach in mechanics that
extended students’ physical intuitions from an anchoring example (e.g. a spring) to the
misunderstood example (e.g., the “springiness” of molecules in a seemingly rigid table)
through a series of carefully sequenced, intermediate examples. The key relationship or
structure (e.g. springiness) is the major relationship in the situation that the student
needs to transfer to other situations.
1.3. Phenomenological primitives (or p-prims)
DiSessa distinguished two views in the literature. The first, what he called the

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conceptual framework view, includes the constructs of misconceptions, alternative
frameworks and naïve conceptions because they all: (1) regard students’ ideas to be
fully formed, stable, and connected (diSessa, 2014; Hammer, 1996; Taber, 2008); (2)
emphasise the differences between novice and expert knowledge (Smith et al., 1994);
and (3) must be challenged (Taber, 2008), eliminated or overcome (Hammer, 1996) in
order for students to develop scientific understanding.
In contrast, diSessa proposed a new perspective he called the knowledge in
pieces view. Firstly, he argued that students’ thoughts may not be stable, theory-like,
content-dependent, stored knowledge structures. Instead, they may be spontaneous,
transient, context-dependent constructions that arise from the activation of small,
intuitive, and more fundamental knowledge elements to deal with an immediate
situation. They arise in the moment, without students having thought about it
consciously before. He called these abstract fragmentary knowledge structures
phenomenological primitives or p-prims (diSessa, 1986, 1993; Hammer, 1996; Smith et
al., 1994). Thus, diSessa questioned the first and second properties of
misconceptions/preconceptions in the definition above (Subsection 1.1).
For example, when students are asked why it is hotter in the summer than the
winter, they will often explain that it is because the Earth is closer to the sun (Sadler,
Schneps, & Woll, 1989). From a conceptual framework view, this explanation is part of
a coherent structure of knowledge about the solar system. From a knowledge in pieces
perspective, the students constructed that explanation in the moment based on the
fragment of knowledge that moving closer to the sun would make the Earth hotter.
Thus, students activated the p-prim connecting proximity and intensity: “closer means
stronger”. Other p-prims have been identified by diSessa (1993) including "maintaining
agency", "actuating agency", and "dying away". Like "closer means stronger", they are

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not incorrect in and of themselves, but their activation in certain contexts is
inappropriate.
Secondly, this new knowledge in pieces view challenged another aspect of the
conceptual framework view: its perception of the discontinuity between naïve and
expert knowledge (Smith et al., 1994). By studying physics experts’ thinking as well as
novices’ thinking, they showed that intuitive ideas were still present in expert
knowledge (diSessa, 1993; Smith et al., 1994). Instead of naïve thinking and expert
thinking being completely different, the knowledge in pieces view held that learning at
all levels was a process of reorganising intuitive ideas into better knowledge (diSessa,
1993; Hammer, 1996; Smith et al., 1994).
1.4. Threshold concept
More recently, Meyer and Land coined the term threshold concept which has been
widely applied in higher education research, but not at other educational levels (Cousin,
2006a; Land & Meyer, 2006; Meyer & Land, 2003; Meyer, Land, & Baillie, 2010).
Their thinking was influenced by David Perkins’ troublesome knowledge (Perkins,
1999, 2006), which shifts the focus from students’ cognitive constructions to the nature
of the knowledge itself. Thus, instead of locating the problem of understanding in the
student, it situates the problem as a property of the knowledge itself.
They formulated key features of threshold concepts (Cousin, 2006a; Cullen,
2010; Land, Cousin, Meyer, & Davies, 2005), though these have not been well-debated
or differentiated in the literature (Quinlan, Male, Baillie, & Stamboulis, 2013). We
focus here on transformativeness, integrativeness, boundedness and troublesomeness as
key features, and on what they called the “state of liminality”.
A threshold concept is transformative in that, once understood, students
experience a significant shift in the perception of a subject and often of themselves. For

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example, biological variation has been proposed as a threshold to understanding
evolution (Batzli, Knight, Hartley, Maskiewicz, & Desy, 2016). Variation is needed for
natural selection to take place; appreciating its role allows students to understand the
randomness of evolutionary change, shifting their view of biology and their own place
in nature. Thus, Meyer and Land (2003) add assumptions about the transformative
nature of certain key concepts. Misconceptions, alternative conceptions and p-prims
may be difficult to alter and common, but no claims are made about the importance of
those prior conceptions or the centrality of them to progressing in the discipline. By
emphasising the transformative nature of thresholds, Meyer and Land give them special
curricular significance.
A threshold concept is also integrative insofar as it exposes the previously
hidden interrelatedness and connections with other concepts or parts of the discipline.
Thus, their integrative nature is another notable feature not claimed by other traditions
reviewed earlier. This property of “integrativeness” is now emphasised more in recent
research on conceptual change, notably in the framework theory approach developed
by Vosniadou and her colleagues (2008) (see Subsection 1.5).
A threshold concept is often, though not always, bounded, in that any conceptual
space will have terminal frontiers, bordering with thresholds into new conceptual areas.
It might be that such boundedness serves to demarcate different disciplinary areas
(Meyer & Land, 2003).
Finally, a threshold concept is also likely to be troublesome, involving
“troublesome knowledge”: “different kinds of knowledge – inert, ritual, conceptually
difficult, and foreign – are likely to prove troublesome for learners in different ways”
(Perkins, 1999, p. 8), they can also be tacit or using troublesome language (Meyer &

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Land, 2003) or troublesome due to fear of uncertainty (Baillie & Johnson, 2008, pp.
137–138)
These authors also suggest that the process of learning a threshold concept
involves an in-between unstable state in which the students oscillate between earlier and
emergent understandings (Land & Meyer, 2006). They called this the state of liminality,
referring to work by Brousseau (1989a) on epistemological obstacles (see Subsection
1.5).
Threshold concepts are described differently than the constructs reviewed above
insofar as the threshold concept is the targeted concept to be understood, rather than the
prior knowledge relative to it. Moreover, this theory discusses both the nature of
knowledge and the process of acquiring new knowledge without making a clear
distinction between these two foci.
In other words, this area of research is more focused on identifying the steps and
the result of a process necessary to grasp the threshold than on entering the minds of the
students’ as learners to understand their prior knowledge of this threshold. It is,
therefore, more focused on the structure of the discipline, curriculum and course design
than the cognitive structures or reasoning processes of individual learners (Cousin,
2006a). This shift in focus is a key step in building our own theory of DoVs because it
prompts us to consider the structure of knowledge itself.
Their discussions of liminality also (Land et al., 2005) add insights about the
learning process and implications for teaching. They argue that teachers need to support
students through liminal states, trusting that their students can tolerate confusion. The
learning process is recursive and excursive: there is no simplistic, linear, learning
approach from easy to difficult. The mastery of a threshold often involves many “takes”
and looping back on the material (Cousin, 2006a).

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1.5. Cognitive obstacles
The francophone educational literature from earlier in the 20th century also addressed
many of the same issues as the later – mostly Anglo-American – literature discussed
above. Constructivism is generally associated with Piaget research during the 1920-
30’s. In Piaget’s view, learning is based on the interactions between the prior cognitive
structures (or schemes) of learners and their environment (Piaget, 1967).
In parallel, Bachelard (1938) introduced the idea that students’ mistakes are
evidence of knowledge, arguing that the process of awareness of a contradiction and
then overcoming it is the driving force of learning. Didactics has later articulated these
two views:
“The mistake is not merely the effect of ignorance, uncertainty, or happenstance, as
is assumed in empirical or behaviourist theories of learning, but the effect of prior
knowledge that had its role, but now is shown to be false or simply inappropriate.”
(Brousseau, 1983)
From these fundamentals, some authors attempted to define obstacle. For
instance, Fabre (1995) and Astolfi (2006) elaborated six key features: inwardness, ease,
positivity, ambiguity, polymorphy, and recursiveness (translated from French by the
authors
1
). While the first three features are a part of the previous constructs reviewed
here, we focus on the last three because they bring different contributions that we build
on in our proposed theory.
According to Brousseau (1989b, 1989a), an obstacle has ambiguity because it
can be either functional or blocking, depending upon its context. Knowledge in some
contexts, provides suitable solutions to problems, but outside of those contexts, leads to
1
Terms translated by the authors from the French corresponding terms: “intériorité”, “facilité”,
“positivité”, “ambiguïté”, “polymorphie” and “récursivité”.

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errors. This ambiguity explains why some knowledge may resist instruction and
continues to be mobilised by learners. In other words, the knowledge has proven to be
correct and useful in some (but not all) situations, making it useful to retain. Although
the concept of ambiguity is consistent with arguments for the value of some
preconceptions, as discussed under Subsection 1.2 above, it highlights the context of use
of a given concept, an element we foreground in our DoV framework.
Polymorphy means that an explanation system rooted in a deep and coherent
network of ideas may manifest in different forms in seemingly unrelated situations.
Thus, polymorphy resembles the integrative property of the threshold concept.
Polymorphy also means that the cognitive obstacles may be in relationship with
multiple other dimensions: motivational, affective, emotive, etc. Even if regularly
mentioned in the literature, those aspects have received less attention than cognitive
ones (see, as exceptions, Pintrich, Marx and Boyle (1993); Tyson, Venville, Harrison
and Treagust (1997); Zembylas (2005)) We pick up on these emotional aspects in our
own theory, insofar as teachers need to be mindful of them.
Recursiveness means that an obstacle is perceived as such only by people who
can link the prior conceptions to refuting knowledge. So, an error can be identified as an
obstacle only after having overcome this obstacle, thus highlighting the relationship
between the obstacles and a metacognitive loop back on learning. Given this definition
and as the term “recursive” is used with changes in meaning in different constructs
(including our DoV framework), we refer to this feature as retrospection rather than
recursiveness.
This cognitive obstacle perspective opened the doors to much science education
research. For instance, research about students’ reasoning highlighted cross-cutting
regularities of obstacles such as spontaneous reasoning in elementary dynamics

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(Viennot, 1979, 1985), sequential reasoning in electrokinetics (Closset, 1983), and
causal linear reasoning in thermodynamics (Rozier, 1988), as well as the theory of
conceptual fields of Vergnaud (1994, 2009), or the notion of goal-obstacle of
(Martinand, 1995). This construct also laid the foundations for more recent, elaborated
and learning process-oriented theories related to the next construct: the theory of
conceptual change (Subsection 1.6).
Early constructivism (Piaget, 1967) viewed learning as the result of a dynamic
process tending toward equilibrium between the learner and his environment. It rested
upon two complementary processes: assimilation and accommodation. Assimilation is
the integration of new information into the cognitive structure of the learner without
modifying the learner’s cognitive structure. An accommodation occurs when the
external situation drives a change in the learner’s cognitive structure to accommodate
the new information.
According to Piaget, an accommodation mechanism can take place if,
previously, an attempt at assimilation has failed and if the resistance creates disturbance
in the learner’s mind. This unstable state is called cognitive conflict. These kinds of
conflicts may lead to a cognitive rupture; in the face of disturbances, the learner may
switch from one set of representations to another. Other authors have refined how this
process takes place (Posner, Strike, Hewson, & Gertzog, 1982; Strike & Posner, 1985).
Bachelard (1938) – and later, in mathematics, Brousseau (1983) – went beyond
giving these learning obstacles the status of knowledge. For him, some of this
knowledge is essential to learning students have to temporarily learn “false” knowledge
because the awareness of this erroneous aspect would be constitutive to the construction
of the target knowledge. He called these unavoidable obstacles that act as
steppingstones to more sophisticated knowledge epistemological obstacles.

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According to the cognitive obstacle perspective, a problem in classical learning
processes is that the teacher focuses on the target knowledge and sees students’
representations (the obstacles) as a barrier to acquiring the target knowledge. For
students, though, these pre-existing representations are valuable intellectual tools to be
used as long as they work.
1.6. Theory of conceptual change
The constructs in Subsections 1.1, 1.2 and 1.3 mainly focused on the nature of the
student's knowledge. With the threshold concept and the cognitive obstacle
(Subsections 1.4 and 1.5) perspectives, the focus shifted to the process by which
learners evolve from one set of concepts to another. This process was formalised by
Posner, Strike and colleagues under the name conceptual change (Posner et al., 1982;
Strike & Posner, 1985).
Starting from a Piagetian position that learning is a rational activity, Posner et al.
(1982) were concerned with “how students’ conceptions change under the impact of
new ideas and new evidence.” They drew a parallel between, respectively, assimilation
and accommodation processes and the “normal science” and “scientific revolution”
phases developed by well-known philosophers of science (Kuhn, 1970; Lakatos, 1978).
Posner and colleagues (1982) focused their research on accommodation.
Posner’s theory of accommodation (Posner et al., 1982) proposed four
conditions of accommodation: (1) the existing conception (prior knowledge) must be
dissatisfying, (2) the new one must be intelligible, (3) plausible, and (4) fruitful (i.e. it
should have the potential to be extended). They define a student’s conceptual ecology as
the existing concepts that will influence the selection of a new central concept. Two
kinds of concepts are particularly important determinants of the direction of an
accommodation: anomalies and fundamental assumptions about science and knowledge.

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Posner’s approach to replacing prior knowledge with scientifically acceptable
knowledge – now called the classical conceptual change approach – became the
leading paradigm in science education. But it was progressively subject to criticism,
including that it offered a simplistic view of misconceptions with no relation to other
concepts, the context, or motivational, emotional and affective dimensions (Caravita &
Halldén, 1994; Zembylas, 2005) and ignored students’ productive ideas (Smith et al.,
1994).
Taking these criticisms into account, Vosniadou and colleagues (Vosniadou,
2006; 2007) have developed a re-framed conceptual change approach to learning: the
framework theory approach. This theory, as well as the Synthetic Models Approach
(Vosniadou, 2012), is constructivist and sees misconceptions as part of a knowledge
system consisting of many different elements organised in complex ways built through
successive (conflicting) assimilations, that is, gradual change rather than sudden
restructuring. While cognitive in focus, it is not incompatible with affective,
motivational or socio-cultural factors (Vosniadou, 2006).
2. Why a new theory is needed
From the preceding review, we see that there is a number of unresolved issues between
science education concepts that explain students’ difficult-to-change conceptions. In
sum, the constructs vary in terms of whether they see prior knowledge as intrinsically
incorrect and in need of removal or as positive and useful; whether they see
transformation as continuous or discontinuous; the degree of recursiveness; and even
the stability of prior knowledge.
In particular, we note many constructs seem to “circle around” the notion of the
“context of application” of students’ conceptions rather than placing it at the centre of
their theories. This notion has been mentioned, but not elaborated, in anchoring

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conceptions, p-prims and cognitive obstacle constructs. There is also a link between this
idea and the “boundedness” feature of the threshold concept construct.
Hence, one motivation for an additional theory is to reconcile points of
divergence between the abovementioned constructs. As we’ll detail in Section 3, the
first hypothesis of our theoretical proposal is that this “context of application” – or
what we call the “domain of validity” (DoV) – is an essential part of the individual’s
knowledge. This proposition resolves some of the conflicts cited above (see Section 4).
However, our main motivation and contribution lies in the operational power of
our framework to “map” the nature of obstacles to learning and to directly infer from
this map a practical teaching strategy. This power relies on the second hypothesis of our
framework, that explicitly defines preconceptions as an overgeneralisation of the DoV.
To explain why this is a significant shift in perspective, we discuss briefly the
instructional power of previous constructs.
The existing theories mainly have an explanatory value and are not intended to
be theories of instruction. On one level, all of these perspectives have similar
instructional implications: they see students’ incorrect statements as reflecting (shared)
cognitive structures rather than as individual, nonsensical mistakes (Bachelard, 1938).
Thus, they suggest it is important for an instructor to explore students'
misunderstandings, uncertainties and prior knowledge and to look for the sense behind
students’ incorrect statements.
However, there is less consensus on teachers’ actions after this “diagnostic”
phase. Moreover, the links between the teaching implications and the key features of the
learning difficulties described in each construct are rarely straightforward. Viewing
student’s conceptions as inherently incorrect prompts attempts to remove, dismantle or
replace in contrast to reorganising students’ existing understanding or intuitions

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(Clement et al., 1989; diSessa, 1993; Hammer, 1996; Smith et al., 1994). Exploiting
prior knowledge as resources for new learning can take different shapes, though. On the
one hand, the teacher could choose to recognise students’ preconceptions as a creative
and productive act of meaning-making and then use these acts of reasoning as steps
toward more expert-like understanding. On the other hand, the teacher may create a set
of conditions conducive to supporting students through cognitive conflicts (or liminal
states), by making a new conception more intelligible and plausible (Posner et al., 1982;
Strike & Posner, 1985), by tolerating learner confusion (Cousin, 2006a), by using
teaching techniques such as metaphors and analogies (Clement, 1993), by changing the
semiotic representations (Duval, 1993), by creating socio-cognitive conflicts in class
(Brousseau, 1989b; Closset, 1992), or sometimes by acting as an adversary confronting
the students when they attempt problematic assimilations (McDermott, 2005; White &
Gunstone, 2014).
In this paper, we privilege the practical challenges of teaching and argue that
moving the domain of application to the foreground (and the model, to some extent, to
the background) is the critical shift in perspective proposed in our framework. Two of
us are engineers, which may explain the different focus we promote here. The job of
scientists is to create models and theories that describe the world and then demonstrate
the generalisability of those theories (that is, ever-wider domains of applicability). Thus
the model is central. The job of professional engineers, or applied scientists, is to solve
(sometimes very complex) problems. To do so, they focus first on understanding the
context of application, before selecting among various available models, a process
called “modelling the problem” (Quinlan et al., 2012). Engineers are constantly
assessing the trade-offs between complexity (accuracy) and efficiency as they apply to
the demands of a given situation. For example, to design a power supply, why would an

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engineer use a thousand-equation model if a ten-equation model is much faster to
compute and provides the same mathematical result within 1% accuracy? The ten-
equation model may be applicable to fewer situations (less generalizable, less complex),
but still useful, convenient and efficient for this specific situation. In this example, it is
important to note that the choice of models is not the choice between models used in the
everyday lifeworld versus the symbolic domain (Solomon, 1983), but between two or
more models used in the symbolic domain.
We bring this way of “thinking like an engineer” (Quinlan et al., 2012) to
problems of teaching and to the analysis of students’ errors. First, in analysing students’
errors, we saw how they were related to students’ choice of symbolic models (rather
than Solomon’s (1983) everyday vs symbolic models). We saw that their errors were a
problem of applicability to a given situation, rather than a problem with the model itself.
In reviewing the literature on students’ preconceptions, we saw a tendency to focus
primarily on the model (preconception) itself, rather than its domain of validity.
Second, we sought not only to describe students’ understandings, but to solve a
practical, instructional problem. That is, our theory is focused on deriving teaching
strategies. Our proposal is not intrinsically “better” than the ones proposed before, but it
is more oriented towards building teaching strategies, as we’ll illustrate in Section 5.
Our theory directs teachers’ attention to the “paradoxical experience”, as well as
offering a unique graphical representation of the cognitive shift required, which has not
previously been documented.
In summary, we claim that our proposal, by foregrounding the “domain of
validity” as an essential part of individual’s knowledge itself (first hypothesis), explains
the nature of preconceptions more explicitly and operationally than previous constructs
(second hypothesis). The increased operationality comes from making the domain of

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validity central, rather than secondary, when elaborating teaching strategies intended to
overcome preconceptions. And we believe this shift in the gaze for teachers, students
and even researchers, offers a novel contribution to the literature that builds on, rather
than contradicts, earlier references to applicability of models (e.g. context sensitivity in
the knowledge-in-piece perspective (Hammer, 1996), lifeworld vs symbolic models
(Solomon, 1983), learning impediments (Taber, 2001, 2008), framework theories
(Vosniadou, 2012), constructive alternativism and range corollary of the Kelly’s
personal construct theory (Fransella, 2003; Kelly, 2005) or contextual effects on
students’ reasoning (Palmer, 1997)).
While science teachers may agree with the principle of teaching the range of
applications of models, this may not be happening in practice. For example, in
chemistry,
Drechsler and Schmidt
(2005) s
howed that textbooks often illustrate
models of acids and bases in successive order, but fail to draw connections
between them or explain how, under differing circumstances, one model may be
better than another. Furthermore, they found that teachers rarely explicitly taught
models and frequently did not understand the differences between two of the
most common models.
When models are taught, they may be taught from a science, rather than an
engineering perspective. That is, the focus may be on the generalisability of models –
that is, expanding the domain of validity – rather than on shrinking it or testing it for its
utility (alongside other candidate symbolic models) in a given situation. This emphasis
on expanding the domain of validity is consistent with the much more commonly
understood challenge of transfer (Salomon & Perkins, 1989). Thus, we argue that
neither teachers nor researchers are routinely describing preconceptions as a problem of
an overgeneralised domain of validity. Nor are teachers routinely using an instructional

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sequence designed to trigger the process of downsizing a domain of validity. Our theory,
while not the first to acknowledge domains of validity, nevertheless makes a significant
contribution to science education.
Finally, in addition to its operational power, our proposal also resolves
discrepancies between existing constructs. These are the elements we will now develop
in more detail. In Section 3, we elaborate the concept of “domain of validity” at the
heart of our DoV framework. In Section 4, we show how our theory of DoV reconciles
discrepancies across the constructs reviewed above. Based on the DoV framework, we
present a teaching strategy that we have developed and trialled which involves the
creation of paradoxical situations. This strategy is explained in Section 5, along with
examples highlighting DoV-centric teaching strategies.
3. Domain of Validity framework
We present in this section a proposal for a conceptual framework whose core element is
the notion of domain of validity (DoV): the DoV framework. The origin of this
framework is our study of the processes occurring during courses in an engineering
school at university level when teachers confront students’ learning difficulties
apparently caused by prior knowledge. This approach prompted us to model the
observed phenomenon and derive an explicit teaching strategy to address these
difficulties.
Given the aims of this formalised tool (mainly helping teachers to identify
students’ preconceptions and to develop effective, theory-based teaching strategies), this
DoV framework is compatible with most of the concepts cited in Section 1 and
reconciles some of the oppositions noted in Section 2.

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As any model
2
, it does not pretend to address all aspects of learning. After
presenting and illustrating this DoV framework (based on two main hypotheses), we
discuss its links with existing literature.
3.1. A first hypothesis about knowledge: K=M+DoV
As seen in the review in Section 1, in existing constructs, the knowledge, conception or
model is typically considered to be the central (if not sole) element to be learned. That
is, the knowledge and the model are treated as the same thing. Discussion across the
various constructs then tends to centre on how compatible that knowledge (conception
or model) is in relation to expert views. Instead, we hypothesise that a person’s
knowledge consists of two connected elements: a model and a domain of validity (or
DoV).
A model is a tool that allows its user to economically understand, describe and
predict real world behaviour using abstraction. The American Framework for K-12
Science Education (2012a) defines conceptual models as “explicit representations that
are in some ways analogous to the phenomena they represent” which “allow scientists
and engineers to better visualize and understand a phenomenon under investigation or
develop a possible solution to a design problem” and “include diagrams, physical
replicas, mathematical representations, analogies, and computer simulations” (National
Research Council, 2012a, p. 56).
Whatever the type, all models contain approximations and assumptions
(Halloun, 2011; National Research Council, 2012a). Confronted with their environment
(“experiences”), humans need a way to reduce its complexity or abstract it. Thus,
knowledge involves generalisation: creating a set of concepts and links between those
2
The term “model” is used here in accordance with the definition given in Subsection 3.1.

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that extract and formalise some regularity across individual situations, so we can
communicate and reason about the world around us. In science and engineering, these
generalisations take the form of concepts, principles (links between concepts, that may
typically appear as equations), procedures or (meta)cognitive strategies (Anderson (Ed.)
et al., 2001; Krathwohl, 2002). A few examples will be given in the coming sections.
In addition to a model, we argue that “knowledge” contains a second key
element: the domain of validity (or DoV). The DoV is the bounded area within which
the model properly describes real-life experiences. It guides the selection of a specific
model when facing a specific situation.
One advantage of this definition is its ease of visual formalisation, statically as
well as dynamically. 1 illustrates this view: a piece of knowledge (whole figure) is the
association of a model M1 and a domain of validity DoV1 (represented by the rounded-
corner box). The dots represent various experiences: these include situations students
may face in everyday life (observations, experiments, phenomena, etc.) as well as
situations created by the teacher (e.g. exercises, labs, problems, etc.). Some of the dots
are inside DoV1 (white dots), while others are outside DoV1 (black dot): M1 properly
describes the three “white-dot” experiences, but not the “black-dot” experience.
Figure 1. In the DoV framework, knowledge is the combination of a model and a domain of validity (DoV)

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We assume the DoV is also built by generalisation. However, we hypothesise
that the DoV is more implicit and unconscious than the model itself. The implicitness
of the DoV is illustrated in the review of constructs in Section 1, in which the model is
usually considered equivalent to knowledge. As pointed out in the review, Clement
(1993) is an exception when he notes that modifying the domain of a conception is an
option. Brousseau’s (1983, 1989a, 1989b) “ambiguity” takes us much farther in
explicitly recognising that concepts may be appropriate in some contexts and not others.
Hammer (1996) stated that “context sensitivity is easier to understand from a p-prims
perspective than it is to understand from a misconceptions perspective”. And the
“boundedness” of threshold concepts (Cousin, 2006a; Cullen, 2010; Land et al., 2005)
also points us toward considering the boundaries of a concept, but research in that
tradition has not emphasised that feature.
Our first hypothesis has two important consequences
3
. Firstly, a model –
assuming it reproduces or explains at least one experience from the perspective of its
owner – cannot be intrinsically false because it successfully describes something in the
surrounding world. Thus, once a model allows us to predict what will happen in some
situations (DoVs), it can be considered “valid”. Focusing on the DoV, then, allows us to
depart from characterising models as “right or wrong” or even “naïve or expert”, as
much of the literature discussed in Section 1 does (Clement et al., 1989; Gilbert &
Watts, 1983; Smith et al., 1994).
For example, in engineering, Maxwell equations (model) represent very well an
extremely large set of experiences of electromagnetic phenomena (“white dots”), but
they do not represent the electrochemical reaction that occurs in a battery to produce the
3
For these consequences, we do not need to assume that the DoV exists inside individuals’
minds, so the following considerations may be thought of as outside of individuals.

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electromotive force (“black dot”), which is, however, a very common element in
electrical circuits. Hence saying that Maxwell equations are either “right” or “wrong” is
a poor description of their validity. They are “right” in situations within their DoV, but
not outside it.
The second consequence of our hypothesis that knowledge contains two
interrelated components (model and domain of validity) is that different models are not
mutually exclusive. There is no single “right” model surrounded by “false” models;
models just coexist, having different DoVs. Moreover, depending on the context,
different models offering different levels of precision can be used to describe the same
experience. For example, the full Maxwell equations can coexist without contradiction
with a simplified set of quasi-static equations in which propagation effects do not exist,
and even with a third model where individual electric charges simply attract or repel
themselves depending on their distance and their electric charges. “Experts” may use
any of them given the level of specificity they seek.
For an example from everyday life, the model of the Earth as flat is extremely
useful and highly accurate when building a house, but disastrously inaccurate when
launching a satellite. Another well-known example is the coexistence of classical
mechanics (Newton) and the theory of relativity (Einstein).
Introducing the DoV concept allows us to capture the fact that a model is
sufficient in many but not all circumstances. This point is depicted at the top of 2
(representing the teacher’s cognitive structure) where two models M1 and M2 are both
valid but in different domains of validity.
This two-component perspective invalidates the idea of “wrong” knowledge (in
the sense where the model is wrong) in many situations. Coming back to the modelling
of Earth, “the Earth is flat” contradicts scientific understanding. However, as we pointed

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out, this model is sufficiently “true” to build a house. The contradiction doesn’t come
from the model itself, but from the failure to recognise and discuss models together with
their associated domains of validity. So, our shift consists in replacing the “true/false”
character of a model with a much more nuanced concept: the domain of validity. This
brings us back to the idea of alternative conceptions (Subsection 1.2), with the nuance
that we clarify that the “correctness” of a model depends upon being inside its DoV.
These ideas are aligned with other constructs (Hammer, 1996; Palmer, 1997;
Smith et al., 1994; Taber, 2008) but we claim that this formulation, emphasing the
context-sensitivity of knowledge, offers a powerful tool for teachers. Ultimately, this
formulation should help teachers understand how to teach models and modelling,
important over-arching concepts in science (Halloun, 2006; National Research Council,
2012a). As we’ll see, emphasising the domain of validity as a part of the knowledge
itself, on equal footing with its associated model, opens new paths in terms of teaching.
In effect, we are suggesting taking an engineering perspective, rather than a scientific
perspective, on the teaching of models and modelling.
In summary, the DoV is the (often implicit) part of the cognitive structure that describes
the area within which a model succeeds in describing and predicting real-world
behaviour. DoVs lead us to abandon the idea that a conception is “right” or “wrong”
and opens the door to multiple valid conceptions coexisting. Thus, even if researchers
and teachers agree that “all models contain approximations and assumptions that limit
the range of validity of their application and the precision of their predictive power”
(National Research Council, 2012a, p. 56), explicit formalisation and recognition of the
importance of domains of validity is lacking in existing literature and teaching practices.

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Figure 2. Graphical formalisation of the DoV framework applied to a simple teaching situation
3.2. A second hypothesis about preconceptions: P=M+ODoV
The second proposition of the DoV framework is that the very nature of some
preconceptions is an overgeneralised domain of validity (ODoV): a domain of validity
too wide relative to what the associated model can really represent (Figure 3). Believing
that “the Earth is flat in all circumstances” is an overgeneralisation from everyday
experiences of the domain of validity of this model: it serves us in the situation of
housebuilding, but not when putting a satellite into orbit. This simple hypothesis
explains many phenomena related to prior knowledge. It also suggests that the typical
blocking situation experienced in learning may arise when either the teacher or the
student focuses only on the model (without its associated domain). Although this basic
example focuses on the difference between lifeworld (everyday) and symbolic
(scientific) models (Solomon, 1983) for ease of reading, problematic preconceptions
also arise when one scientific/symbolic model is applied in place of another

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scientific/symbolic model.
Figure 3. In the DoV framework, a preconception is a knowledge consisting of the combination of a model and
an overgeneralised domain of validity (ODoV)
For instance, the different atomic models taught in middle and high schools
(generally Ancient Greek model, Dalton’s, Thomson’s, Rutherford’s and Bohr’s
models) are numerous (Cokelez & Dumon, 2005; Justi & Gilbert, 2000) and each can
be considered as valid to some extent. The Ancient Greek model (or ball model) is
relevant to understand and teach that matter is not “continuous and infinitely divisible”
but “composed of very small indivisible corpuscles” (Justi & Gilbert, 2000). But this
model offers no basis for distinguishing between types of atom (a contribution from
Dalton) and it obviously cannot explain the internal structure of atoms (electrons
(Thomson) and nucleus consisting of protons and neutrons (Rutherford), the former
orbiting the latter (Bohr)). Hence, the domain of validity of the Ancient Greek atomic
model doesn’t include isotopes or chemical bonds (Özmen, 2004; Taber, 2005).
Believing that molecules are groups of ball model atoms is, in the DoV framework, due
to a preconception consisting of an overgeneralised domain for the Ancient Greek
atomic model.

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Other examples of symbolic models with varying DoVs include the Maxwell
equations mentioned earlier, Newtonian physics versus relativity or the different
definitions of acids and bases (
Drechsler & Schmidt, 2005; Taber, 2008).
Returning to the cognitive structure depicted on the top half of Figure 2, this
teacher has two models in mind, with different DoVs, both coexisting without
contradiction. One experience (black dot) is properly described by M2 but not by M1.
For instance, electromagnetic propagation effects are properly described by full
Maxwell equations (M2) but not by the quasi-static ones (M1), the launch into orbit of a
satellite can be properly described on a spherical Earth (M2) but not on a flat Earth
(M1). We will follow this last basic example throughout the next explanation to make it
more explicit.
The bottom part of Figure 2 depicts the cognitive structure of students who are
likely to possess a preconception related to M1: the students possess the same model
M1 as the teacher but associated with an ODoV (here including the “black dot”
experience covered by M2)
4
. According to our theory, the difference in DoVs
associated to M1 between the teacher and the student may explain many phenomena
described as problematic preconceptions.
4
The existence of this cognitive structure is consistent with the ideas that (1) humans tends to
generalise their experiences in order to find some regularity and (2) the DoV is more
implicit than the model itself. Indeed, such a cognitive structure could, for example, result
from the following situation: (1) The teacher, having presented “white dot” experiences to
the student (including exercises in class, for example), has always told students they were
“right” when giving correct answers using M1; (2) the student has not been exposed to the
“black-dot” experience yet; (3) the student, by generalising these white dot experiences,
implicitly builds an overgeneralised DoV of M1 since neither the student nor the teacher
consider knowledge as having any component other than the model itself.

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Starting from the situation depicted in Figure 2, when the teacher presents the
students with a “black dot” experience (satellite launching) for the first time (for which
M2 is a better fit), the students will use M1 (flat Earth) according to their own cognitive
structure, especially if the student is not conscious of a structural difference between
this black-dot experience and the white-dot experiences (basically, a difference in
scales). Thus, the student is confident in M1 because using it in the past has resulted in
positive feedback from the teacher or everyday life. In a classical teaching situation
focused only on models, the student’s answer will be considered “wrong” because it
does not include M2 (round Earth) as the teacher expects (“Earth is not flat, it’s
round!”)
5
.
For students focused only on the model without awareness of DoVs, the first
encounter with the black-dot experience (satellite launching) will result in three
simultaneous discordances:
1. At a cognitive level it is difficult to understand, in absence of additional
elements, why M1 (flat Earth) suddenly “does not work” when it has always
worked before. This can be linked to the ambiguity of the cognitive obstacle
(Astolfi, 2006; Brousseau, 1989b, 1989a; Fabre, 1995) and to the troublesome
aspect of the threshold concept (Meyer & Land, 2003; Perkins, 1999).
2. At an affective/emotional level, it may be upsetting when the model M1 in
which a student has confidence built on past experiences (e.g. housebuilding)
suddenly doesn’t work anymore. Students may feel a sense of betrayal,
5
It is worth noting that, scientifically speaking, the planet Earth is neither flat nor round. Using
the round Earth model (M2) without taking on board its domain of application (for
instance by considering this model as appropriate for any context) may consist in a new
preconception. Therein lies the recursive aspect of learning in the DoV framework.

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especially when a pattern of positive teacher feedback for use of M1 is broken.
Insofar, considering such a model as an individual’s representation of the world,
if the distinction between the model and the world is unclear in the student’s
mind, modifying the model would mean, disturbingly, modifying the world
itself! This affective aspect can be linked to the emotional discordance
highlighted in the thresholds literature (Meyer et al., 2010; Schwartzman, 2010)
and to the affective aspects in the conceptual change literature (Duit, Treagust,
& Widodo, 2013; Tyson et al., 1997; Zembylas, 2005).
3. At an epistemological level, it is difficult to understand why one should throw
away a model that works in practice. This epistemic aspect can be linked to the
works of Piaget (1967), Bachelard (1938) and Posner et al. (1982).
We call the black-dot situation a paradoxical experience for two reasons, which both
include a paradox:
1. A paradox common to all situations: in the absence of additional explanation (as,
for example, that M2 could be a superset of M1), “abandoning” M1 is
paradoxical because it worked well up to that moment.
2. A paradox specific to the situation: the black-dot experience is the very element
that will create an incongruity in student’s mind, in relation to the content of the
targeted preconception. Hence the black-dot experience is the practical tool to
use to trigger the overcoming of the preconception, as we’ll see in Subsection
3.3.
When left unexplained, the three discordances above may impede students’ access to
M2 (round Earth). In theories that consider only the model without its DoV, students
might be advised to discard M1 (“erase it from memory”), although M1 has been

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coherent and successful with their experiences thus far (white dot experiences:
housebuilding).
But the discordances cited above are a consequence of paying attention only to
the model, as if it were all of the knowledge involved. The DoV framework offers an
alternative view of knowledge. There is no need to “throw away” M1 (third
discordance), and the first two discordances are mitigated if the teacher explicitly
teaches the concept of DoV when explaining that M1 “does not work here” (in this
context, at this level of analysis, or for this purpose). Moreover, the overgeneralisation
process that leads the student to build an ODoV for M1 is more likely if teachers and
students only discuss knowledge as models, rather than models associated to DoVs.
Our hypothesis also explains why, even when students understand M2, they may
continue to apply M1 (the oscillation mentioned in Subsection 1.4). It is different to
remember, understand, explain or even apply a model (which involves only the model
itself) than it is to analyse a situation or experience and then select an appropriate model
(which involves both the model and its DoV). Students could have learned and
remembered a model M2 without having modified the DoV of a model M1. This failure
to modify the DoV is more likely if teachers focus primarily on models without
explicitly attending to DoVs. Since the DoV is related to the interaction with the
environment and is often unconscious, it takes time and repetition to modify the
associated model selection process.
In sum, the idea that knowledge is made of coexisting models each with their own DoV
explains many of the learning phenomena addressed by the misconceptions and
conceptual change literature in science education.

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3.3. Overcoming the preconception
Our propositions above have important consequences in terms of instructional design.
Overcoming ODoV-based preconceptions does not involve throwing away or even
modifying the initial model itself (M1), but simply reducing its associated DoV (and
accepting a multiplicity of models that are applicable in different situations). What
needs to be “abandoned” is only its use in relation to the paradoxical experience.
According to our theory, the downsizing of the ODoV opens the door for the student to
consider a second model with a DoV appropriately covering the black-dot experience.
Reducing the applicability of a model may seem counterintuitive when a key
goal of teaching is to enable students to transfer what they have learned, i.e. apply a set
of principles or model to a broader range of situations than originally taught. Transfer,
a key pedagogical challenge (Salomon & Perkins, 1989), can also be described as
dependent upon awareness of the appropriate DoV associated with a model (i.e. finding
new "white dots": situations when the model is appropriate). Transfer, though,
emphasises the expansion of a model’s domain of validity. Overcoming a preconception
involves reducing an overgeneralised DoV (i.e. excluding the "black dots": situations
for which the model is inappropriate). Therefore, the DoV framework presents a unique
extension of research on transfer, not a contradiction. Making explicit the concept of a
domain of validity and our hypothesis of a two-component model of knowledge links
these two pedagogical challenges (misconceptions and transfer) theoretically. To our
knowledge, this linkage has not been made before, though a full investigation of transfer
is beyond the scope of this paper.
In conclusion, our DoV framework suggests shifting focus from the model itself to the
domain of validity of a given model and directs the attention of the teacher towards
reducing overgeneralised DoVs instead of replacing “wrong” or “naïve” models. In

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Section 5 we build an explicit teaching strategy that attends to both components of
knowledge. First, we discuss further how the proposed framework aligns with the
constructs reviewed earlier.
4. The DoV framework as an integrative step
In this section, we reinterpret the constructs from existing literature described in Section
1 in the new terms of the formalised elements constituting the DoV framework as
explained in Section 3. We show the coherence and integrative power of the DoV
framework in relation to the extant literature. Obviously, as with any model, the DoV
framework does not have an infinite range of application (domain of validity). We note
this limitation and call for empirical testing in a range of situations beyond university-
level engineering and physics courses.
4.1. Ideas and issues
In Section 2 we highlighted several discrepancies found in the existing literature
reviewed in Section 1. Each of these, in the six key constructs, is reconciled by our DoV
framework.
One discrepancy is whether prior conceptions must be removed to reach expert
conceptions (misconception) (Hammer, 1996; Smith et al., 1994) or whether they can be
positive and useful on the way to accessing a new model (Brown & Clement, 1987;
Clement, 1991, 1993; Clement et al., 1989; Hammer, 2000). Introducing a two-
component view of knowledge (model plus domain of validity) allows both perspectives
to be valid: the model is useful and needs to be retained, while the domain of validity
needs to be altered.
Another discrepancy is between a discontinuous versus continuous view of
learning, discussed in Subsection 1.3 and Section 2. According to the DoV framework,

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the process may be seen as both continuous and discontinuous. Continuity is found (1)
insofar as initial models are kept and ideally linked to new models at the end of the
process and (2) through recursiveness: when a learning cycle is complete, the
association of a new model M2 with another overgeneralised DoV may be the next
preconception. This process can be seen as an extension of the relationship between
cognitive obstacles and the retrospective aspects of learning (Astolfi, 2006; Bachelard,
1938; Fabre, 1995). So, the “naïve/expert” divide is replaced by a potentially infinite
sequence of conceptions subject to continual refinement of both models and their
domains of validity. This gradual process is similar to the anchoring example teaching
approach (Clement, 1993), the reorganisation of intuitive ideas in the p-prims construct
(diSessa, 1993; Hammer, 1996; Smith et al., 1994), the excursive journey through
liminal states in the threshold construct (Cousin, 2006b; Land et al., 2005), the cognitive
conflict process (Brousseau, 1989a; Piaget, 1967) and the successive (conflicting)
assimilations in the framework theory approach (Vosniadou, 2012; Vosniadou et al.,
2007). This recursive and potentially infinite evolution of students’ knowledge
explained by the DoV framework echoes the way philosophers of science describe the
never-ending progress of scientific knowledge (Kuhn, 1970; Lakatos, 1978).
Discontinuity is found (1) in restricting the DoV of the initial model (hence
abandoning that model to represent a specific experience) as with the context-sensitivity
of p-prims (diSessa, 1993; Hammer, 1996; Smith et al., 1994), boundedness of the
threshold concept (Cousin, 2006a; Cullen, 2010; Land et al., 2005)), but more
importantly (2) in accessing a new model (M2) as with cognitive rupture (Brousseau,
1989a; Piaget, 1967), conceptual change by accommodation (Posner et al., 1982) that
allows a new interpretation of the world (first misconception’s property (Hammer,
1996; Smith et al., 1994), transformativeness of the threshold concept), and (3) making

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links previously unknown between this new model and the initial one (integrativeness
of the threshold concept).
In Subsection 3.2, we described three different types of discordances (cognitive,
affective and epistemological) that may occur when facing the paradoxical experience
and argued that the DoV approach could ease or eliminate those discordances. Defining
knowledge incorporating both a model and its domain of validity accommodates the
various key ideas contained in existing literature and resolves discrepancies between
different constructs. As such, adopting a two-component view of knowledge mitigates
all three learner-felt discordances.
4.2. Reinterpreting existing key constructs
The DoV framework uses a small number of concepts: knowledge, model, domain of
validity (as a part of an individual’s cognitive structure) and experiences (including
paradoxical experiences). To build further definitions, we use the term conception as a
synonym of knowledge. As defined in Subsection 3.1, knowledge consists of an
association of a model and a domain of validity.
We call the initial conception the level N knowledge (M1 in Figure 2), and the
target conception the level N+1 knowledge (M2). The initial conception (N) is the prior
knowledge of the student. The target conception (N+1) is the knowledge the teacher
wants the student to learn. The framework we propose is recursive (unfolding over a
series of iterative learning cycles), so that there is a succession of learning cycles
moving from N to N+1.
In this section, we explicitly reinterpret key constructs from existing literature
using the terminology of the DoV framework.

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Preconception
In Subsection 1.1, we used existing literature to define a preconception as: (1)
knowledge which affects how students understand natural phenomena and scientific
explanations; (2) stable (strongly held, difficult to change) cognitive structures; (3)
constructed by students (via experiences, formal learning, etc.) prior to the considered
instruction; (4) forming barriers to new learning.
Building on the DoV framework, we propose two additional defining features of
preconception: (5) knowledge is the association of a model and a domain of validity; (6)
a preconception is knowledge containing an overgeneralised domain of validity that
includes experiences that are not properly addressed by the associated model.
Thus, a preconception including properties (5) and (6) could be called a DoV-
based preconception. We do not claim that all preconceptions are DoV-based
preconceptions.
As explained in Subsection 3.2, a preconception is stable (item (2)) because it
has been useful to understanding the world until encountering the paradoxical
experience.
Misconception, alternative conception and anchoring conception
We put the term misconception (as defined in Subsection 1.1) aside because it suggests
knowledge may be intrinsically false and should be removed, which is not aligned with
our first hypothesis (Subsection 3.1). Alternative conceptions (Clement, 1993; Gilbert
& Watts, 1983; Nussbaum & Novick, 1982) and anchoring conceptions (Clement,
1993; Clement et al., 1989; Hammer, 2000; Hestenes, 1987) are initial conceptions in
which "valid" knowledge would be included and even used to build target knowledge.
This is exactly what our model describes in more detail, suggesting that (nearly) all

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initial conceptions are alternative conceptions and possibly anchoring conceptions (as
defined in Subsection 1.2).
However, embedded in all those constructs is an understanding of "conceptions"
as focusing on models only, with insufficient attention to the domains of validity of
those models. Our term DoV-based preconception makes the source of the problematic
knowledge element clear.
P-prims
In p-prims, there are two key ideas: (1) the students’ answer is an on-the-spot
construction, and (2) lower level (abstract) fundamental primitives are used to build
higher level models (diSessa, 1993; Hammer, 1996; Smith et al., 1994).
Both of these elements are compatible with the DoV framework. Though, like
alternative conceptions and anchoring conceptions, they also focus primarily on models,
rather than being explicit about domains of validity. By reducing models to their
primitive parts, p-prims come closer to recognising the problem of how and when
different “pieces of knowledge” are invoked in a given situation. That construct also
highlights the importance of the situation (“context-sensitivity”) (Hammer, 1996),
though it does not explicitly explain the association between models and domains of
validity.
Troublesome knowledge and threshold concepts
According to the DoV framework, troublesome knowledge could be defined as a target
conception (rather than the initial conceptions described by the other constructs) for
which assimilation is troublesome for the students. The origin of this troublesome
character has been discussed in Subsection 3.2, as occurring when knowledge is seen as
focusing only on the model without attending to its domain.

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Thus, the DoV framework offers a different explanation for why some target
conceptions are “troublesome” for students. Namely, troublesomeness isn’t inherent in
the model but in a lack of awareness of the association between a model and its domain
of validity.
There is ambiguity in the definition of the threshold concept insofar as some key
features clearly refer to a target conception (e.g. the bounded, integrative and
troublesome features of the threshold concept), while others refer to the process of
shifting from an initial conception to a target conception (e.g. the transformative,
irreversible and troublesome features of the threshold concept). The DoV framework,
by defining the transition (ODoV downsizing process) between an initial conception
(M1) and a target one (M2), offers disambiguation.
Finally, the DoV framework is consistent with the five key characteristics
associated with threshold concepts (see Subsection 1.4). The property of “boundedness”
(Cousin, 2006a; Cullen, 2010; Land et al., 2005) comes closest to capturing the
boundedness of a DoV, although the literature on thresholds does not elaborate
“boundedness” and does not distinguish between models and DoVs. In our terminology,
confronting a paradoxical experience is likely to be troublesome and once a new model
is introduced (following the shrinking of the ODoV of the initial model), the result may
be transformative (and potentially irreversible). Likewise, students may experience a
liminal state when the domain of validity of the initial conception is being challenged
and re-sized (during cognitive conflict). However, the DoV framework does not require
that all those properties are present, though their presence is not incompatible with our
propositions.
Cognitive obstacle and conceptual change
According to the classical frameworks (Astolfi, 2006; Brousseau, 1989b, 1989a; Fabre,

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1995), a cognitive obstacle is an initial model present in the student’s mind and acting
as an obstacle to new knowledge. In the DoV framework, the element preventing access
to a new model is not so much the model, but the overgeneralisation of the domain of
validity of the initial model.
Referring to Piaget’s definition (Subsection 1.5), cognitive conflict is the
discomfort experienced by a student originating from various discordances as described
in Subsection 3.2. In the DoV framework, Piagetian cognitive conflict is stimulated by a
paradoxical experience.
Finally, conceptual change is the transition from an initial conception to a target
conception involving a cognitive rupture process (Piaget, 1967; Posner et al., 1982;
Strike & Posner, 1985). It corresponds in the DoV framework to a series of ruptures
leading to the target conception: (1) understanding that the initial domain of validity
was overgeneralised (for instance by considering differences between “black dot” and
“white dot” experiences) and subsequently reducing it, (2) the discovery of the target
model and its better fit than the initial model with the new situation, and (3) the
discovery of the links between the initial and target models that makes overcoming the
preconception acceptable.
In sum, the DoV framework accounts for and resolves discrepancies between a number
of existing constructs in the literature. Nevertheless, while it explains some of the
divergences observed in previous research, it is not intended to explain all these types of
learning difficulties. The reviewed constructs are often seen as complementary. For
example, the authors of the knowledge in pieces perspective did not claim that all
knowledge is structured as p-prims, but that this new perspective offered a previously
undescribed level of knowledge structure (diSessa, 2014; Hammer, 1996). They
suggested that an inclusive view admitting both conceptual framework and knowledge

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in pieces perspectives is likely necessary to fully explain student knowledge. This
complementary approach is consistent with other attempts at syntheses that explain
disparate individual observations of difficult-to-change preconceptions (Brown, 2014;
Chi, 2013; Hammer, 1996; Hammer, Elby, Scherr, & Redish, 2005; Taber, 2008).
Likewise, we offer a new perspective that complements and extends existing major
explanations.
5. An effective teaching strategy derived from the DoV framework
Coming back to the DoV framework itself, we now discuss its operational power by
illustrating how it underpins a general teaching strategy to overcome preconceptions,
that may be used in a various of science education situations.
Since a preconception is hypothesised to be due to an overgeneralisation of a
DoV, when teachers face an “error” (and more specifically a recurrent and difficult-to-
overcome “error”) from one or several students, the DoV framework approach suggests
the following general strategy to facilitate the ODoV preconception overcoming
process:
1. Map student’s knowledge and teacher’s (target) knowledge using models and
DoVs, as in Figure 2. If the observed situation reveals an oversized DoV by the
student(s), proceed with the following steps.
2. Imagine a situation that is located between the DoV and the ODoV of M1
(black-dot experiment): inside student’s ODoV but outside teacher’s DoV.
Present this situation to the student.
3. Ask the student to reason about it, guiding him/her if necessary, to the
contradiction that using M1 for that situation inevitably provokes. Help the
student to concurrently (a) realise that the contradiction comes from the

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unsuitable application of M1 to that specific situation (hence the need to reduce
students’ M1 DoV) and (b) imagine a different model (M2) that would be
applicable to the situation.
4. Help the student integrate all these elements (M1 with its reduced DoV, M2 with
its own DoV, relationships – if any – between M1, M2 and their DoVs) in a new
and coherent structure.
The same four-step strategy can then be applied if the student associates an ODoV to
the new model M2, or if the teaching sequence or curriculum involves a series of
intermediary target knowledge successively building on each other. The DoV
framework is therefore a simple and powerful tool able to describe static situations
(knowledge’s mapping and preconceptions’ diagnosis) as well as dynamic situations
(teaching strategy and learning process, including the recursive and potentially infinite
evolution of students’ knowledge).
As an example, let us suppose an 8-year-old European girl asks “why don’t Australians
fall off the Earth?” The reaction of a teacher would be:
1. To map the situation as:
o M1 is the model “gravity attracts matter downwards” (initially the only
model owned by the student).
o M2 is the model “gravity attracts matter towards the centre of the Earth”
(or an alternative, wider one: “matter attracts matter by gravity and Earth
is a lot of matter”), also held by the teacher.
o Student’s DoV for M1 (oversized): M1 is applicable to the situation of
Earth (and people on it) in Space, supposing a “downward” direction that
would be in the direction of the ground for European people.

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o Teacher’s DoV: M1 is applicable locally on Earth; M2 is applicable to a
wider set of situations (explain the presence of atmosphere, discuss
whether people would fall or not “in Space” depending on their latitude
on Earth, etc).
2. In this example, the question initially raised by the girl is already a paradoxical
experience (Australians obviously do not fall off and thinking that they would is
the result of applying M1 outside of a local situation). So, one option would be
to simply confirm to the girl that her question is relevant and interesting, and
reinforce the troublesomeness of it: “Yes, it’s true that the Australians do not
fall. It’s strange. How it is possible?”. Another option would be to question the
question itself: “What makes you think they would fall off?” This option has the
advantage of triggering metacognition by the student. A third option would be to
identify another paradoxical experience in the same area of the DoV: “What is
our latitude?”, or more simply: “Do you think we are on top of the Earth?”. But
in any case, what is sent to the student is a question, encouraging her to reason.
3. Mainly by asking questions
6
, the teacher encourages the girl to reason herself
and to be confident in her reasoning by confirming the relevant elements she
provides. For example: “Yes, we are attracted downwards; yes, the Australians
do not live at the same position on Earth, and yes they do not fall. (So again,)
what makes you think they would fall off?”. In this case, this should make the
girl express, maybe using a drawing, that she places (unconsciously) the Earth in
a vertical referential in Space. Which allows the teacher in turn to ask “then why
6
We follow here a constructivist approach since the teacher, instead of immediately giving the
final explanation, guides the student via questions and reasoning to make him/her infer it
him/herself.

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do the Moon and other planets not crash into the Earth?”, reinforcing the
contradiction again. Step by step, she can be guided to express that “vertical
direction” is relative to the place on Earth, not to a general direction in Space
(asking for example: “when Australians drop an object, where does it fall?”). So,
the double conclusion follows: (a) M1 is not applicable to discuss whether
people fall or not in Space, but only to a more limited set of situations, and (b) a
suitable model for that question is “people are attracted towards the centre of the
Earth” (M2).
4. Help the girl express that these conclusions are not contradictory and most of her
reasoning was relevant: M1 is applicable locally and is a simplification of M2,
which is more suitable at a larger scale. Her only “error” was applying M1 to a
non-local situation, where it is not supposed to be applied.
It may be that many science teachers do have that specific conversation with their
students. However our point here is to illustrate that what guides the teacher is (a) a
goal: making the student downsize the ODoV of M1 (to allow him/her to imagine the
possibility of another model M2) and (b) a tool, using a paradoxical experience to push
the student to reason in the appropriate direction. This method helps even a young
science student learn that knowledge consists of both a model (gravity pulls
“downward”) and its domain of validity (local to one’s own spot on Earth).
In this procedure, a key point is to make students become confident in their
reasoning, which is a key resource for them. Asking the student questions will also
allow the teacher to confirm, disconfirm, or refine the initial mapping as needed. From
our experience, the main resource needed by the student is confirmation on specific
points by the teacher, so that they have the courage to reason further on alternative
models (M2) that may initially seem strange.

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While the example above is one of dialogue, there are other ways to guide
students through the key steps of the general strategy. For example, a written exercise
for an entire class may focus on common errors (from past examinations, for example),
and creation of appropriate paradoxical experiences.
This principle of confronting students with a paradox may be used via many
techniques, including the predict-observe-explain (POE) strategy (White & Gunstone,
2014), demonstrate-observe-explain (DOE) strategy (Champagne, Klopfer, &
Anderson, 2005), elicit-confront-resolve (ECR) strategy (McDermott, 2005), and the
elicit-confront-identify-resolve-reinforce (ECIRR) strategy (Wenning, 2009). Although
using similar processes, our teaching strategy makes explicit what is being confronted:
the failure of a model in the context of a new experience. It focuses on identifying the
student’s assumed domain of validity and its appropriateness, in order to use these
conceptual change teaching techniques most effectively.
We tested such a strategy in circuit theory sessions in an engineering course at
university level (Sommeillier & Robert, 2016, 2017). After having identified the nine
most common preconceptions about electricity among second-year engineering
students, we developed and implemented a teaching strategy derived from the DoV
framework by modifying the content of two exercise sessions. The reference and
modified sessions shared the same intended learning outcome: being able to solve
electrical circuits (i.e. finding the voltages across and the currents through every
component). The reference exercise sessions consisted mainly in drilling the students in
solving circuits from A to Z, through many computations without intuitive, qualitative
or interpretative questions, and without any DoV-based teaching strategy. The modified
sessions followed a teaching strategy similar to the one described above with various
questioning techniques (traps, incorrect demonstrations, qualitative questions, open-

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ended problems, etc). The teaching assistants were briefed on the new approach. An
experimental pre-test/post-test design – the intervention group following the modified
sessions and the control group following the previous versions of the sessions – found
that the intervention group of students achieved significantly better results (75%
success) on the post-tests in comparison with the control group (50% success). Full
details are reported elsewhere (Sommeillier & Robert, 2020).
Our second example, still focused in solving electrical circuits (network
analysis) at university, level, uses the DoV framework to analyse the curriculum itself
and suggest a revision to its structure and sequence. As reported in (Theunissen,
Sommeillier, & Robert, 2020), applying the framework to this problem made us
formalise that choosing the appropriate method to solve an electrical circuit may be
described by 4 models (representing different circuit solving methods) and their 4
associated DoVs. These last ones may be defined using 3 binary criteria (introducing a
distinction between 23=8 types of circuits). Figure 4 represents the models, the DoVs
and the criteria. To give a couple of examples:
• M3 which is the use of phasors (circuit solving in the frequency domain)
is suitable to solve resistive (R) or reactive (C/L), DC or AC circuits in
steady state (so without transient, as explained by the fact the switch is
outside the DoV associated to M3).
• M4, the use of long-term behaviours of reactive elements, is suitable
only to solve reactive DC circuits in steady state.

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Figure 4. Application of the DoV framework to electrical circuit solving (network analysis)
Using this representation suggested three different (and complementary) ideas to revise
our own electricity course and related teaching practices.
The first one is the possibility of mapping in more detail recurring students’
errors according to the way they select solving methods (models), and the possibility to
help them overcome their underlying learning difficulties – or what we call
“methodological preconceptions” using the strategies detailed above.
The second one is to write an online application to investigate the possibility of
automating (to some extent) the overcoming of preconceptions for this specific problem
(electrical circuit solving). A first small-scale pre-test/post-test design with 37 students
showed that after just 15 min of using the application 64% of the participants improved
their scores.
Last but not least, this way of modelling the choice of the method when solving
electrical circuits made us rethink the teaching sequence of methods, according to the

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size of their DoVs. The classical way of teaching circuits consists of exploring resistive
circuits and reactive circuits (including switches) with continuous sources, then reactive
circuits with sinusoidal sources. However, the idea of following an increasing size of
the DoVs of the different models suggests teaching reactive circuits with sinusoidal
sources before reactive circuits (with switches) with continuous sources. The classical
way of teaching implies first expanding and then shrinking the DoV, rather than
progressively expanding or shrinking it. We hypothesise that it is the comings and
goings in terms of DoV size that create learning obstacles unnecessarily for students, in
a subject that is notoriously difficult for them to understand.
While other theories have touched upon domains of validity, the instructional
implications have received little attention. For example, Givry and Tiberghien (2012)
extensively analysed (via written worksheets and hours of video data collected in the
classroom and in the lab) the interaction between two 15-year old students in a physics
course related to gases. Having identified three processes of learning: “(1) establishing
links between ideas, (2) increasing the domain of applicability of ideas, or (3)
decreasing the domain of applicability of ideas” (Givry & Tiberghien, 2012, p. 223),
they observed than the third process occurs much less often than the second one. They
concluded their paper with implications for teaching strategy: one is about the first
process (which is not of interest here), and the second “concerns the process of
increasing the domain of applicability […] Consequently, we suggest designing
teaching sequences in which the same scientific concepts will be involved
systematically by students in several situations. Furthermore, we consider that this
learning process should be explicitly assessed by teachers to help students develop a
better understanding of these aspects of their learning.” (Givry & Tiberghien, 2012, p.
245)

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Hence while Givry and Tiberghien (2012) explicitly observed and identified
downsizing of the “domain of applicability” they do not discuss it in relation to teaching
strategies. Furthermore, they suggest the opposite teaching strategy that, arguing that
students already often do it spontaneously. In fact, one could argue that students need
more help with downsizing domains of validity because they rarely do it on their own.
Using very similar elements (models and domains of validity), but linking them
explicitly to preconceptions, our framework comes to opposite conclusions. We have
built strategies specifically oriented towards Givry and Tiberghien (2012) third process,
suggesting its power in overcoming recurrent students’ errors.
This example (Givry & Tiberghien, 2012), of reinforcing a generalisation
strategy, brings us back to our earlier points in relation to transfer (Subsection 3.3). The
authors do not seem to consider that two different situations exist between a model and
a new experiment: the “white dot” situation (for which generalisation is the way to go,
but does not involve a problematic preconception) and the “black-dot” situation (for
which downsizing the ODoV is required).
These various examples illustrate that the DoV framework suggests a general method
that can be applied to many situations. It offers practical strategies to help overcome
preconceptions in very local situations (discussion with a student, exercise session) as
well as more general course design (sequence to teach electrical circuit solving, choice
of teaching strategy in relation to expected learning process). In many cases, it suggests
new ways of teaching because of its foregrounding of the domain of validity as the key
element when trying to address preconceptions.

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6. Conclusion
By embracing a two-component view of knowledge and explicating the domain of
validity of various models, we propose a new theoretical framework in science
education useful to both understanding and addressing difficult-to-change prior
knowledge. Our model explains the obstacle to learning as an overgeneralised DoV.
The significance of the DoV framework lies not only in its explanatory value,
but also in its action-oriented approach. We propose an instructional technique in which
students confront a paradoxical situation so that the student realises the limits of the
original DoV and subsequently both reduces the domain of validity of the original
model and searches for an alternative model. This instructional approach also
emphasises the importance of teaching not just models, but their domains of validity.
This instructional approach, then, also means being explicit about the two components
of knowledge.
We have demonstrated the integrative power of the DoV framework in relation
to six scientific constructs related to prior knowledge, firstly by resolving apparent
discrepancies between these constructs, and secondly by redefining (or at least linking
with our model) known concepts in the literature using a small set of simple and
precisely defined terms. Doing so clarifies the relations between initial conception,
target conception and the process of going from the former to the latter via cognitive
conflict and cognitive rupture.
Although we claim that our framework has high integrative power, it has its own
domain of validity like any other model. It does not address all the issues related to prior
knowledge and conceptual change. While our experiments in the field of electrical
engineering show promising results (Sommeillier & Robert, 2016, 2017, 2020), further

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research is needed to demonstrate its broad applicability across fields of science and
levels of education.
Notes on contributors
Raoul Sommeillier is a Ph.D. candidate in science education specialising in didactics in
applied sciences, a teaching assistant in electricity and electronics at the Bio-, Electro-
And Mechanical Systems (BEAMS) department, and an engineer in electromechanics.
His research focuses on higher education students’ preconceptions in scientific fields
and the development and testing of teaching strategies to help students overcome these
learning obstacles. He holds a double Master of Engineering in mechatronics and
constructions from Université libre de Bruxelles (ULB) and Vrije Universiteit Brussel
(VUB), an Advanced Master in technological & industrial management from Solvay
Business School and an upper secondary teaching certificate in engineering sciences.
https://orcid.org/0000-0002-4903-3386
http://beams.ulb.ac.be/users/raoul-sommeillier
https://be.linkedin.com/in/raoul-sommeillier
Kathleen M. Quinlan is Professor of Higher Education and Director of the Centre for
the Study of Higher Education at the University of Kent. Before joining the University
of Kent in 2016, she held academic and leadership positions in educational development
at the University of Oxford, The Australian National University and Cornell
University’s College of Veterinary Medicine. She holds a Ph.D. in education from
Stanford University and a bachelor’s degree in psychology from the University of
Maine. Her research focuses on teaching and learning in higher education, with special
attention to discipline-specific pedagogical concerns and the development of students’
interest in their subjects.
https://orcid.org/0000-0003-3606-4148
https://www.kent.ac.uk/cshe/people/staff/quinlan2.html
https://www.linkedin.com/in/kathleen-m-quinlan-12a0111b/
Frédéric Robert is Dean and Professor in electricity and electronics at the Brussels
Faculty of Engineering. He has held various positions within the Université libre de
Bruxelles (ULB): Advisor to the Rector for teaching and learning in higher education,

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on 5 Oct 2020 available online: http://www.tandfonline.com/doi/full/10.1080/03057267.2020.1824472
Vice-Dean of the École polytechnique de Bruxelles and Head of the Embedded
Electronics research unit of the Bio-, Electro- And Mechanical Systems (BEAMS)
department. From 2005, he supervised the engineering education department of the
École polytechnique de Bruxelles: the Bureau d’Appui Pédagogique en Polytechnique
(BAPP). His research interests include science education (project-based learning,
conceptual change, cognitive obstacles) and both advanced engineering in electricity
and electronics (high frequency transformers, high energy physics instrumentation,
industrial electronics). He holds a Ph.D. in Applied Sciences and a Master of
Engineering in electronics and telecommunications.
https://orcid.org/0000-0001-6520-5873
http://beams.ulb.ac.be/users/frédéric-robert
https://www.linkedin.com/in/frédéric-robert-3881253/
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