Content uploaded by Ference Marton
Author content
All content in this area was uploaded by Ference Marton on Apr 30, 2015
Content may be subject to copyright.
Content uploaded by Ference Marton
Author content
All content in this area was uploaded by Ference Marton on Apr 30, 2015
Content may be subject to copyright.
Meanings are acquired from experiencing differences against a background of sameness,
rather than from experiencing sameness against a background of difference: Putting a
conjecture to the test by embedding it in a pedagogical tool
Ference Marton
University of Gothenburg, Sweden
Ming Fai Pang
The University of Hong Kong, Hong Kong SAR, China
1
ABSTRACT
In helping learners to make a novel meaning their own, such as when helping children to
understand what a word means or teaching students a new concept in school, we
frequently point to examples that share the aimed-at meaning but differ otherwise. This
type of approach rests on the assumption that novel meanings can be acquired through the
experience of sameness against a background of difference. This paper argues that this
assumption is unfounded and that the opposite is the case: we make novel meanings our
own through the experience of differences against a background of sameness. We put this
conjecture to the test in an experimental study by embedding it in a computer game. The
results supported the conjecture.
2
THE CONJECTURE
The origin of meaning
It is commonly believed that a child—or an adult for that matter—can learn the
meaning of a word by observing a number of examples of what the word refers to that
share this meaning but differ in other ways. For example, we point to a dog and say dog,
point to another dog and say dog, point to a third dog and say dog, and then expect the
child to understand what the word “dog” means (refers to), i.e., it refers to a certain kind
of animal whose representatives have “dogness” in common. It is possible that the child
has already noticed differences between dogs and cats, for instance, and that by so doing,
he or she has discovered one meaning of “dog” (in the sense that it differs from “cat”),
but has not learned that dogs are called dogs. In this case, he or she can easily learn the
word “dog”—and link it to the meaning that he or she discovered earlier. If, however, the
child had not noticed any such differences between cats and dogs previously, then he or
she will not be able to grasp the meaning of “dog” simply by looking at different dogs
that have “dogness” in common and seeing how they are alike. If a child is unable to see
the “dogness” of a dog in one case, then he or she will be unable to see how two—or
indeed 22—dogs are the same (see Fodor, 1980).
Providing different examples of the same thing is not only the most common
method of helping young children to build a vocabulary, but probably also the most
common method of teaching concepts, principles, and problem-solving methods in
school. Stigler and Hiebert (1999) describe such an approach as the typical way of
teaching mathematics in U.S. schools, and the highly authoritative volume How People
3
Learn urges teachers to provide “…many examples in which the same concept is at
work.” (Bransford, Brown & Cocking, 2000, p 20)
Looking at cases that are the same in one respect but differ in others to determine
what they have in common is called induction. According to Fodor (1980), this is the
only idea that exists to explain how conceptual learning takes place, and it simply does
not work at all for the reasons already cited. It follows then that no explanation of how
we learn, find, create, or appropriate new meanings is possible. Hence, by default, Fodor
concludes that meanings (concepts) are innate. In our view, however, even if the concept
(meaning) of “dog” were innate, you would never be able to separate that meaning from
the meaning of “animal” if you had never encountered any animals other than dogs.
Regardless of whether dogs were then called dogs or animals, the meaning of “dog”
would be exactly the same as the meaning of “animal.” Hence, you still would not have
acquired the meaning of “dog.” Nor of “animal” for that matter.
Similarly, if we lived in an entirely green world, then we would be unable to notice
the greenness of everything. Hence, whether or not concepts (meanings) are innate, we
must encounter alternatives to them if we are to be able to notice and grasp these
concepts. Awareness of a particular number presupposes awareness of other numbers (or
at least one other number), and awareness of a particular color presupposes awareness of
other colors (or at least one other color). You cannot possibly understand what Chinese is
simply by listening to different people speak Chinese if you have never heard another
language, and you cannot possibly understand what virtue is by inspecting different
examples of the same degree of virtue. Nor can you understand what a “linear equation”
is by looking only at linear equations.
4
The conjecture that was put to the test in the study discussed in this paper is this:
meaning (or a way of seeing something) does not originate in sameness, but rather in
difference, or, more accurately, from the experience of difference (cf Marton, 2006).
You cannot arrive at a novel meaning through induction, but you can through
contrast. In induction, the focused meaning, i.e., the one that you are trying to help
another to make his or her own (e.g., “dog,” “virtue,” “Chinese,” “linear equation”) is
kept invariant, while the other features of the same entity (e.g., smallness or largeness,
male or female, formal speech or casual conversation, x and y, or a and b) vary. In
contrast, it is just the other way around. The focused meaning is a component part of
variation, whereas the other features of the same entity are kept invariant. As an example,
let us consider two different patterns of variation and invariance used to help someone to
grasp the meaning of “Chinese,” as shown in Figure 1.
Insert Figure 1. Induction (on the left) and contrast.
According to the present line of reasoning, learning to make a novel meaning one’s
own amounts to discerning an aspect or part (or several aspects or parts) of something,
through variation (difference) in the focal aspect (that to be discovered) and invariance
in other aspects. We call this conjecture—and the system of corollaries (also referred to
as conjectures in the following) that it implies—somewhat immodestly the Variation
Theory (of Learning) (Marton, 2012; Marton & Tsui, 2004).
One way of interpreting our conjecture is this: for a particular feature to be
discerned, it must be separated from that which it is a feature of. If everything is green,
then the color green cannot be separated out. Hence, it cannot be seen. To be able to
5
separate out greenness, you must see something that is green (a ball, for instance) and the
same thing in another color (say blue). If you hear only one language all of the time, then
you cannot possibly separate “Chinese” from “Language,” or “Language” from
“Chinese” for that matter. The two are glued together. To be able to “think apart” the two
and come up with two concepts (“Language” and “Chinese”) rather than one, you must
encounter another language. By experiencing the difference between two languages—
Chinese and Korean, for instance—you not only appropriate a rudimentary version of the
concepts of “Language” and “Chinese” but, in this case, even the concept of “Korean.”
Now, “Language” is superordinate to “Chinese” (and to “Korean”). According to
the present line of reasoning, you cannot grasp a particular concept (meaning) without
grasping the superordinate concept. In our case, the latter is seen not in terms of what is
common among the subordinate concepts (languages in our example), but in terms of
how they differ. Hence, the subordinate concept cannot be grasped without grasping the
superordinate concept. Nor, however, can the superordinate concept be grasped without
grasping at least two of the subordinate concepts. The two must be grasped
simultaneously. In our framework, the superordinate concept (i.e., Language) is a
“dimension of variation” (with “aspect” as a synonym), and the different subordinate
concepts (i.e., languages) are “values” in that dimension (with “features” as a synonym).
We can also consider “Language” as a value in a higher-order dimension of variation,
“semiotic systems,” for instance, in which case “Language” gains another meaning
component, originating in what language is not, for example, through comparison with
other semiotic systems.
6
We thus acquire the meaning of something by learning what it is not. Therefore, by
listening to a language other than Chinese and noticing the difference between the two,
we become able to separate “Chinese language” from “Language.” If, however, the
Chinese that we have heard our entire life is a particular dialect of Chinese, then Chinese
and this dialect are for us identical, and we have a rather narrow understanding of what
Chinese is. Through contrast, we have separated “Chinese” from “Language,” and
learned that there are different languages of which Chinese is one. Now, we have to
separate “Chinese” from our own particular dialect, and learn that there are different
dialects of Chinese of which that spoken in our village is just one. We can do so by
encountering another dialect—or several other dialects—of Chinese. The pattern of
variation and invariance is illustrated in Figure 2. Now, “Chinese,” which was a focused
feature (value) previously, becomes a focused aspect (dimension of variation) that is
generalized across different dialects, all of which are Chinese. Here, it is not the meaning
of “dialect” that is highlighted; rather, we are enriching the meaning of “Chinese” by
separating it from any specific dialect and allowing for variation in dialects.
Insert Figure 2. Chinese language and its dialects.
We call this pattern generalization, and the focused aspect (Chinese) is invariant. We are
not attempting to locate the meaning of the focused feature, as in induction. “Dialect” is
an optional attribute of “Chinese”; regardless of the dialect, the language is Chinese.
According to this line of reasoning, learning amounts to the increased
differentiation of experienced entities through the discernment and separation of features
from other features and from that which they are features of. This view of learning is akin
7
to going from an undivided whole to a differentiated and integrated whole, which
resembles Werner’s (1957) orthogenetic law and Gibson and Gibson’s (1991) way of
describing (perceptual) learning. The latter authors distinguish between two ways of
seeing learning: as enrichment or as differentiation. The former way implies that what is
perceived is interpreted using memory resources, and thus develops into a full-blown
novel meaning: the learner must add to his or her originally meager perception. Our way
of seeing something is always richer than that which we see. The latter way of seeing
learning implies that the originally vague impression is differentiated—“the world gets
more and more properties as the objects in it get more distinctive . . .” (p. 34). What we
see is always richer than our way of seeing it.
This second alternative means that in the course of learning, we learn to distinguish
one thing from another one feature from another, and one feature from one thing. Doing
so involves experiencing both sameness and difference, which involves the simultaneous
experience of those entities that are the same and those that are different. Once we have
learned to tell them apart (through separation), we learn how to put them together
(through fusion), that is, how to see them at the same time (simultaneously). The different
patterns of variation through which this process takes place are shown in Figure 3.
Insert Figure 3. Patterns of variation and invariance.
What does it take to discern a particular aspect or feature of something or to
separate one particular feature from other features and from what it itself is a feature of?
A necessary condition for the former is that the learner experiences variation (or
8
differences) in relevant respects against a background of invariance in other respects.
Accordingly, learning is more likely to occur when there is variation to be experienced.
Origins of the conjecture
Where does our conjecture come from? To a large extent, it originates in
phenomenography, a research approach which posits that a limited number of
qualitatively different ways in which a phenomenon is seen, takes on meaning, and is
understood, can be identified. These different ways of seeing, acquiring meaning, and
being understood, can be captured in a set of categories of description that are logically
related to one another. Phenomenographic research has shown that this set of
qualitatively different ways of seeing can be rewritten in terms of features, some of which
are discerned by certain learners but not by others. In the present theoretical framework,
the feature set of the specific entity discerned and focused upon defines how that entity is
seen (Marton & Booth, 1997), which in turn defines its meaning (or rather one of its
meanings). The next question is “What does it take to discern a particular feature?” From
the early 1970s to the early 1990s, our research focused on the relationship between
differences in what students learn, on the one hand, and differences in how they go about
learning it (how they “approach it”), on the other (Marton, Hounsell, & Entwistle, 1984).
We noticed that students who acquired a better grasp of the meaning of what they were
studying displayed certain patterns of variation and invariance. To reveal the deeper
meaning of what they were studying, they frequently tried out different meanings, as it
were. They appeared keen to compare different views of the same object of study, often
reading the same text from different perspectives, which seemed to be a more powerful
strategy than repeatedly reading it in the same way (Bowden & Marton, 2004; Marton,
9
Carlsson, & Halasz, 1992). Further studies of Chinese learners indicated that what
appeared to be repetition was frequently variation (or invariance in what was read, but
variation in how it was read) (Marton, Wen, & Wong, 2005).
In addition to the already noted Heinz Werner and Eleanor and James Gibson, a
number of other researchers have pointed to the important role played by differences.
Ferdinand de Saussure (1986), in particular, argued more than a century ago that the
meaning of a word is a function of how it differs from other words. All words in a
language form a gigantic system, and the meaning of every word therein is determined by
its position in the system. In addition, the Swedish-German philosopher Ernst Cassirer
challenged the idea of induction as the origin of meanings. Cassirer (1953) remarked that
concept formation in the fields of mathematics and science is not at all like induction.
Instead of disregarding what is particular, he wrote, a scientific concept offers “… a
universal rule [italics in original] for the connection of particulars” (p. 20). The formula
(x-x0)2 + (y-y0)2 = r2 (where x and y are the co-ordinates of any point on any circle in the
system of Cartesean co-ordinates, x0 and y0 are the co-ordinates of the middle-point, and r
stands for the radius) applies, for instance, to all circles. The relation between the terms is
valid for all cases, and by inserting specific values into the formula, we can identify any
specific case. In the case of scientific concepts, what is general and what is specific are
interconnected. The meaning of every specific instance does not derive from any “real
essence” (substance) therein (corresponding to the concept that it is an instance of), but
rather from its position in a system of instances and from the position of the concept that
it is an instance of in a system of concepts. In other words, the meaning of a number does
10
not derive from the number itself, but rather from its position in the system of numbers to
which it belongs.
Not very frequently, but frequently enough, we find a similar focus on difference
(variation) rather than sameness in a variety of fields and theoretical orientations (see, for
instance, Bateson, 1980; Luhmann, 2002; Spencer-Brown, 1972).
Present status of the conjecture
Most work on variation theory has been carried out in the form of Learning Studies,
the inspiration for which is the Japanese Lesson Study, which came to wider attention
through the publication of Stigler and Hiebert’s (1999) best-selling book The Teaching
Gap. In this type of study, a group of teachers teaching a particular subject at a particular
level together choose an object of learning (something to be learned) that is vitally
important for students’ continued learning and that has earlier been found to present
difficulties for them. The teachers plan a lesson together, and one of them carries it out—
usually in his or her own class—while the others observe. Afterwards, the group analyzes
and discusses what happened in the classroom. The Learning Study is a hybrid form of
Lesson Study and Design Experiment. It is a theory-based research undertaking whose
important components include exploration of students’ ways of making sense of the
object of learning before and after the lesson(s). A Learning Study usually comprises
three cycles, each building on the conclusions of the previous. Finally, a Learning Study
is documented, frequently in publishable form. The model was originally developed right
after the turn of the millennium in Hong Kong, where about 350 such studies have been
11
carried out in the past 10 years. A similar number have been carried out in other countries
in the same period, most of them in Sweden (Lo, 2009).
The main (quantitative) results of the studies published to date can be summarized
as follows.
•In nearly all of the studies, students’ results were better after the lesson(s)
than before (Lo, Pong, & Chik, 2005). (Although this may appear self-evident, it
is not. Unfortunately, there are many school lessons in which students learn
nothing, or at least not what the teacher had hoped they would.)
•Students with weaker learning prerequisites usually learn the most. Hence,
not only does the average rise, but the spread diminishes (Lo et al., 2005).
•In cases in which what the students had learned was observed not only
immediately after the lesson but also on a later occasion, the results were often
found to be better at the later time (thus indicating a content-specific “learning to
learn” effect) (Holmqvist, Gustavsson, & Wernberg, 2008).
•Results on national achievement tests increased for classes that had
participated in several Learning Studies, an effect that in all likelihood was
mediated by changes in teachers’ regular ways of teaching (Maanula, 2011).
•When the same object of learning is dealt with in a Learning Study and in
a Lesson Study by groups of equally well-qualified teachers, the quality of
learning turns out to be strikingly higher in the former (Marton & Pang, 2006,
2008; Pang, 2010; Pang & Marton, 2003).
12
•When the three cycles of a Learning Study are compared, the results from
the third are usually better than those from the second, and those from the second
are usually better than those from the first (Lo, 2009).
John Elliot, one of the founders of the “action research” movement in education,
has evaluated two large-scale Learning Study projects carried out in Hong Kong. He
concluded:
“The evaluation gathered convincing evidence of the positive impact of the process
on teachers’ and students’ learning …. Learning Study is focused on realizing new kinds
of pedagogical roles. From the evidence gathered in this evaluation it has enormous
potential in this respect”. (Elliott, 2004)
It seems, in other words, that the Learning Study approach has been something of a
success story. What about our conjecture? Has it been supported in Learning Study
research?
In our Learning Studies, every lesson was initially planned to be consistent with
variation theory, and hence consistent with our conjecture. Differences between cycles
were related to differences between different interpretations of the same ideas. Although
this approach may be a good way to improve lessons, it is not really suitable for testing a
theoretical conjecture. Accordingly, we carried out a few studies using comparison
groups, controlling for the assumed generally positive effects of the co-operative Lesson
Study model. Two groups of teachers, randomly selected for the two conditions (i.e., a
Learning Study and Lesson Study condition), agreed on a particular object of learning.
Together, they explored their students’ understanding of that object, and planned a lesson
on the basis of what they found and on their previous experience of teaching the same
13
object of learning. One of the teachers then carried out the lesson, while the others
observed. After the lesson, the group again explored students’ understanding of the object
of learning, and the lesson was analyzed in light of the results.
A researcher was present as a resource person during both the discussions and
lessons. The only difference between the two conditions was that in the Learning Study
group, the researcher introduced variation theory, which he did not do in the Lesson
Study group. Although he participated in the discussions in both groups, he tried to act in
a reactive rather than active (initiating) manner. The focus of the studies was a
comparison of students’ results under the two conditions in relation to a comparison of
the patterns of variation and invariance brought about in those conditions (Marton &
Pang, 2006; Pang & Marton, 2003, 2005). As these patterns were controlled by the
teachers—and by the students, of course—these comparisons had to be post hoc. To
sharpen the comparison of patterns of variation and invariance, the researcher must be
able to ascertain exactly what patterns are being compared. In quasi-experimental
comparisons, such as that described here, there are usually no consecutive cycles. Even if
a researcher tries to be as blind to the two conditions as possible, we can hardly claim that
he or she has succeeded completely. In our case, the “theory group” may have had an
advantage beyond that originating in the theory itself. Furthermore, the comparisons were
made between the conditions in terms of the patterns of variation and invariance observed
by the researcher, which means that they were post hoc, as noted, and hence the matter of
empirical support for variation theory is not entirely straightforward.
There are no teaching experiments
14
A fair number of studies have been published in recent years in which the outcomes
of learning have been found to be systematically related to the patterns of variation and
invariance inherent in the conditions of learning. The lived object of learning (learning
outcome) in these studies has generally been found to be related to the enacted object of
learning (teaching and classroom interaction) in ways entirely consistent with our
conjecture. The outcomes of learning, and differences therein, can be made sense of in
terms of the patterns of variation and invariance or the differences in these patterns that
are inherent in the conditions of learning (see, for instance, Fraser, Allison, Coombes,
Case, & Linder, 2006; Fraser & Linder, 2009; Linder, Fraser, & Pang, 2006; Marton &
Pang, 2006; Pang, Linder, & Fraser, 2006; Pang & Marton, 2005).
If lessons are to provide stronger evidence, then they must be defined in advance,
and their effects on learning must also be predicted in advance. Kullberg (2010) carried
out an interesting study in which she instructed teachers to teach particular objects of
learning in terms of the critical features identified and patterns of variation and invariance
employed in previous successful studies. The teachers were familiar with variation
theory, according to which critical features and patterns of variation and invariance are
powerful tools for communicating ways of handling a certain object of learning. Even
when Kullberg’s (2010) results supported her expectations, however, there were several
cases in which the enacted pattern of variation and invariance differed from that
expected. Although in some cases, the teacher had failed to open up dimensions of
variation to make it possible for the students to discern certain critical features, in others,
the students opened up dimensions of variation that they were not supposed to under their
specific condition, but that were critical for learning. In such cases, the class was meant
15
to serve as a control, and the unpredicted changes may have strengthened or weakened
the results.
A critical experiment
The only possible way to ensure that what is being compared is what we want it to
be seems to be to build a pattern of variation and invariance into pedagogical tools: texts,
tasks, examples, illustrations, problems, and the like. Variation and invariance—as far as
the conditions of learning are concerned—can then be defined in terms of the
relationships between the constituent parts of the pedagogical tools that are used. A study
of this kind was carried out by Ki and Marton (2003). They investigated how non-native
speakers of Cantonese could be helped to learn to attend to both the tonal and segmental
(the sound but not the tone) aspects of Cantonese words simultaneously to identify their
meanings. Cantonese is a tonal language in which the distinctions between six tones are
of vital importance. The difficulty that speakers of non-tonal languages have when they
try to learn it is not so much their inability to distinguish between two juxtaposed tones
(Stagray & Downs, 1993) as their inability to link variation in pitch at the word level to
variation in word meanings. Variation in pitch exists in all languages, but its significance
in non-tonal languages is at the sentence- rather than word-level. Learning to pay
attention to differences in pitch at the word level as a cue to differences in word meanings
requires reorganization of the attentional field. Ki and Marton (2003) employed a set of
nine words grouped in two ways. In the first, they were grouped to constitute three
triplets, each characterized by one tone (the same within each triplet, but differing from
16
the other two). In the second, three segments were grouped to constitute three triplets,
each characterized by one segment and three different tones (see Figure 4).
The participants’ task was to learn to identify the meaning of the word they heard
by selecting its English equivalent. If we consider each triplet as a sub-task, then to be
able to come up with the meaning of each word, a participant must be able to differentiate
between the three words. If the three words in the sub-task have a tone in common, then
the participant must learn to distinguish between the three different segments and link
them to the three different meanings. If, instead, the three words in the sub-task have a
segmental in common, then the participant must learn to distinguish between the three
different tones and link them to the three different meanings. Hence, when the segments
vary, you learn segmentals, and when the tones vary, you learn tones.
Insert Figure 4. Three triplets characterized by one segment and three different
tones (if read by column) and by one tone and three different segments (if read by
rows).
The two ways of grouping the words can be seen as a comparison between two
patterns of variation and invariance, that is, as induction and contrast from the point of
view of tones. If we believe that language learners learn tones (i.e., differentiate between
them) best if we offer them different examples of the same tone, then we group words
into triplets, within which each has the same tone but a different segment, and ask
learners to compare them. If we believe instead—as our conjecture suggests—that
meaning (in this case, “the meaning of tones”) derives from variation, then we group the
words into triplets, within which the tones differ but the segment is the same, and ask
17
learners to compare them. In Ki and Marton’s (2003) study, the participants clearly
learned to distinguish words more effectively by means of tones in the condition in which
the tones were varied during the lesson and the segment remained the same, than in the
condition in which the tone was invariant and the segments varied. The study thus
demonstrated that learning is more effective under the contrast condition than under the
induction condition, as predicted by our main conjecture (see also Guo & Pang, 2011).
Another way of putting the conjecture to the test
Above, we have argued—in agreement with Fodor (1978, 1980)—that induction is
the most common means of trying to help others to acquire novel meanings, but it is
certainly not the only one. In our own studies of the teaching and learning of Economics
(Pang & Marton, 2003, 2005; Marton & Pang, 2006, 2008), we found that teachers
spontaneously used neither induction, nor contrast. They differed from the teachers using
Variation Theory by not only varying the focused aspect but also varying the unfocused
aspect. The teachers not using Variation Theory actually used more variation than the
teachers using Variation Theory.
The comparison between induction and contrast depicted in Figure 1 above can be
re-written in the following form:
induction contrast
focused aspect unfocused aspect focused aspect unfocused aspect
i v v i
However, the comparisons between ways of teaching Economics consistent with the
conjecture and ways not consistent with it are more like this:
18
not aligned to theory aligned to theory
focused aspect unfocused aspect focused aspect unfocused aspect
v v v i
The conjecture also predicts that learning will be more effective when the focused aspect
varies and the unfocused aspect is invariant than when both aspects vary.
In a further study, Ki, Ahlberg, and Marton (2006) tested this corollary
experimentally , in the above discussed context of learning Cantonese tones, when the
aim was for students to learn both tones and segments (both were considered focal
aspects). According to the main conjecture, to discern a focused aspect (tonal or
segmental), we must vary it while keeping the unfocused aspect invariant and then do the
opposite: vary the unfocused aspect and keep the focused aspect invariant. Then, we need
to allow all aspects to vary at the same time (in this case, the third and unfocused aspect,
meaning, also varies). To test this corollary to the main conjecture, we have to compare
the foregoing sequence with one in which all aspects vary from the beginning. The
comparison can thus be depicted as shown in Figure 5.
Insert Figure 5 Comparing patterns of variation and invariance consistent with
(left) and not consistent with (right) the conjecture.
This is exactly the comparison that Ki at al. (2006) carried out, demonstrating that
the participants in the condition that was consistent with the main conjecture learned
better than those in the condition that was not. Moreover, the conjecture was built into the
pedagogical tools they used in the study, a computer-administered program that afforded
19
variation, invariance, and feedback to the participants. In the first experiment, one of the
aspects, tone, was considered focused (what is to be learned) and the other, segment, was
considered unfocused. In the second experiment, both aspects were considered focused
(they had to be learned).
Discerning an aspect amounts to separating it from other aspects. Two aspects can
be distinguished from each other if one varies and the other is invariant. Furthermore, if
there are two focused aspects that learners are expected to learn to discern, then they
should be varied one at a time, rather than simultaneously. If we want these learners to
relate the two aspects, then we should vary them simultaneously, but only after they have
been discerned. In the second experiment carried out by Ki et al (2006), there was a third
aspect, meaning, that was assumed to be recognized by the learners (they were expected
to make sense of the pictures representing the meanings). This aspect is a function of the
other two aspects and cannot be kept invariant when any of the other aspects vary; nor
does it interfere with the experience of variation in the other aspects, of which it is a
function. A third kind of comparison was carried out in a study reported in the next
section. In this case , in addition to the two focused aspects (demand and supply) and the
unfocused aspect being a function of the two (meaning), there was an additional
unfocused aspect involved (good), which according to the theory was supposed to remain
invariant. Again, two patterns of variation and invariance - one consistent with, and one
not consistent with our main conjecture - thus subjecting it to a third kind of experimental
test.
THE STUDY
Understanding pricing
20
The point of departure for this study was an earlier study in which 10-year-old
children were taught to discern price as a function of demand and supply (Lo, Lo-fu,
Chik, & Pang, 2005). That study, in turn, built on an earlier study of qualitatively
different ways of understanding price and pricing (Dahlgren, 1978). In both of these
studies, with minor differences, it was found that most children—and many adults—see
price as a function of the attributes of goods. For instance, if something is expensive, then
it is because it is big, beautiful, tastes good, etc. Price is thus seen as an attribute of the
good in question, and linked to its other attributes , not as a function of market conditions
(notably demand and supply), as economics tells us that it is. Some see price as a function
of demand only, and others as a function only of supply. For others still, price is a
function of both demand and supply, or rather of the relationship between the two, which
is roughly in accordance with the canonical conceptualization of price in classical, liberal
economics. We use the expression “learning to see something in a certain way” as
synonymous with “making a novel meaning your own” or “appropriating a meaning.” All
three refer to the capability to discern certain aspects of a phenomenon and focus on them
simultaneously.
What then are “ways of seeing something”? They are categories of description used
to depict the various appearances of something or the different ways in which it is
experienced (or its different meanings). The aforementioned research specialization of
phenomenography (Marton, 1981; Marton & Booth, 1997; Marton & Pang, 2008) is the
study of categories of description depicting appearances, experiences, and meanings. It
posits that if a learner exhibits a certain way of seeing something, then this does not
imply that he or she has that way of seeing (as a mental representation, for instance).
21
What then does it imply? It implies that he or she is seeing—or has seen—a particular
phenomenon in a particular way under particular circumstances. Further, the fact that he
or she is so seeing implies that he or she is able to see that particular phenomenon in that
particular way under the given particular circumstances. Accordingly, what we might
wish to explore is the extent to which the same person can see the same phenomenon in
the same way under different circumstances. If he or she can, then this could be
interpreted as demonstrating that he or she has separated the particular way of seeing this
particular phenomenon from the particular circumstances. Becoming an “expert”
frequently amounts to being able to see particular phenomena in particular ways under
widely varying circumstances (cf. Chi, Feltovich, & Glaser, 1981; Goodwin, 1994;
Marton & Booth, 1997, p. x; Sandberg, 1994).
Hence, phenomenography does not tell you what individuals’ ways of seeing
something are. It tells you how their ways of seeing something vary (between people
under the same circumstances and/or within people under different circumstances). The
different categories of description together constitute the outcome space (of how the
particular phenomenon might be experienced). As previously mentioned, studies have
established four categories of description that together constitute the outcome space of
the experience of price.
Making it possible to learn to see price in a more powerful way
Are the different ways of seeing price equally powerful? We do not believe that we
can always—or even most of the time—find a universal ordering of how valid and
powerful different ways of seeing the same thing are. In a planned economy, and
according to Marxist economics, for example, price is not a function of demand and
22
supply. However, we can delimit a set of contexts and settle for ordering the different
options within that set. We could thus argue that it is better to enable learners to see
something in an additional way that we believe to be powerful—in certain contexts, that
is—than not doing so.
Accordingly, we may try to help learners to see something in a new way, that is, in
a way that they have previously been unable to. Although we can certainly try, we can
never be certain of success. At best, we can ascertain that this new way of seeing might
have been instilled, that is, that under the conditions given, it is possible that the learners
learned to discern certain critical features, which is exactly what Lo et al. (2005) did in
five primary school classes (Grade 4) in Hong Kong in the context of a Learning Study.
The aim of each lesson in this study was the same: to enable the students to see price as a
function of demand and supply in novel situations. After the lesson, a novel question was
used to probe their way of seeing price.
The enacted object of learning
A double-lesson was used to help the students to learn to discern demand and
supply, and the relationship between the two, as determinants of price. During the lesson,
the students formed groups and participated in an auction of four items (a mechanical
dinosaur, a doll, a dinosaur card, and a stationery set). The auction was repeated several
times, with variations. To encourage the students to focus on and discern the critical
aspects of supply and demand separately, changes were made in supply (by varying the
number of items available) while demand was kept invariant, and then changes were
made in demand (by varying the purchasing power through changes in the auction money
23
afforded the groups) while supply was kept invariant. After each auction, they were asked
what would be a reasonable price for a new, limited-edition mechanical dinosaur if
people had more money to spend. After the groups had written their answers on a
worksheet, the teacher engaged the class in a discussion of the case of supply going down
and demand going up.
Did the teachers who took part in this study achieve their goal? If so, to what extent
did they do so? As can be seen from Table 1, their attempts were not especially
successful, with the possible exception of class 4B (see the frequencies for category D,
considered the canonical conception here).
Insert Table 1. Distribution of conceptions in pre- and post-tests in Learning Study
carried out by Lo et al. (2005).
Rather than ask whether (and why or why not) seeing price in terms of demand and
supply is too difficult for 10-year-old children, we are more eager to understand the
striking difference in results between class 4B and the other classes. Did something
happen in this class that did not in the others? Or did something happen in all of the
classes except 4B? Prompted by the same curiosity, Lo et al. (2005) did indeed come up
with an interpretation for the discrepancy in their results: the necessary conditions for
discerning a simultaneous variation in demand and supply were present only in class 4B,
which was the only class in which the unfocused aspect (the item for auction) was
invariant throughout the entire sequence of variation and invariance in the focused
aspects (demand and supply). Differences of this kind (the focused aspect varying and the
unfocused aspect remaining invariant versus both aspects varying) have also been found
24
in two other studies, and in both cases were linked to rather dramatic differences in what
the participants had learned (i.e., to the outcome measures) (Marton & Pang, 2006; Pang
& Marton, 2003). The conjecture that we want to put to the test here has two component
parts: what is expected to vary in sequence (the focused aspects) and what is expected to
remain invariant (the unfocused aspect) throughout. In the study reported here, we
wanted to compare two conditions: one consistent with the second component part (the
unfocused aspect remaining invariant throughout) and one not consistent with it. Could
we replicate the findings of the aforementioned study, which served as our point of
departure, with the same difference built into pedagogical tools? Figure 6 shows the
comparison carried out.
Insert Figure 6 Comparing patterns of variation and invariance, consistent (left)
and not consistent (right) with the conjecture.
Design of the study
To reduce the number of factors that could affect the outcome, we tried to build the
pattern of variation and invariance (which we assumed to be necessary) into the task
structure of the learning resources in such a way that the entire experiment would be an
interaction between students and the auction game tool: the computer. Students were
invited to attempt to achieve the object of learning by using two different computerized
learning resources during an independent learning session that lasted approximately one
and a half hours and was held in the multi-media learning center of the participating
school. In line with the Lo et al. (2005) study, in both learning resources, the economic
principle to be dealt with was the determination of the market price through the
25
interaction of supply and demand. An auction game was used to embody the variation in
the dimensions of supply and demand. To test whether it is crucial to keep the auction
item in question invariant, so as to enable students to focus on and discern the critical
aspect of the interaction between supply and demand more readily and effectively, the
two learning resources were identical in all respects but one: one resource made use of
the same product (i.e., boxes of candy) throughout the auction game, whereas the other
featured different products within and across each round.
Seventy-eight Grade 4 students from four classes of one school in Hong Kong
participated in the study. Within each class, students were randomly divided into two
groups, with each given one of the two learning resources. To minimize the teacher
effect, learning took place in an autonomous manner, with the students involved playing
the computerized auction game on their own, although the researcher gave a five-minute
summary at the end of the session to remind the students of the key learning points. (Note
that it was impossible for the researcher to know under which condition each student was
working. The only difference between the two conditions was that students in the same
multi-media learning center received one or the other of the two versions of the learning
resource, with the distribution of the two completely randomized.)
To obtain students’ existing understanding of the object of learning before they
engaged with the learning resources and to form a baseline for comparison of the learning
outcomes of the two groups, a pre-test was administered to all students. Then,
immediately after the independent learning session, they were required to complete a
post-test to allow evaluation of their mastery of the object of learning.
26
In both tests, the students were asked to consider a problem relating to a real-life
scenario embodying the principle in question, i.e., the interaction of supply and demand
in determining the market price of a good. They were also asked to elaborate upon the
factors they had considered in setting that price. The questions in the pre- and post-tests
were essentially identical, except that the product in question varied. Mirroring the Lo et
al. (2005) study, a hot dog and a box of biscuits were used. Students who were asked a
question about the hot dog in the pre-test were asked about a box of biscuits in the post-
test, and vice versa. The pre- and post-test questions were as follows: Have you ever tried
the hot dogs (biscuits) sold in the school shop? Do you know how much they cost?
Maybe you know or you don’t know. Anyway, just for your information, hot dogs are (a
box of biscuits is) now sold at HK$5. Suppose that you are the new owner of the shop.
What price would you set for a hot dog (box of biscuits)? Would you set the current
price, or a different price? What would you consider when you set the price?
The students’ answers were analyzed and described in terms of the aforementioned set of
four categories of understanding.
The learning resources
To build a relevance structure (Marton & Booth, 1997, p. 143) that would enable
students to appropriate the object of learning, they were given the task of bidding on
goods for an upcoming New Year’s celebration through the computerized auction game.
In the first round, students were introduced to the basic rules and operation of the game.
Each student was given HK$400 in auction money and asked to bid for and thus try to
obtain as many items as possible from the nine being auctioned, which were displayed on
27
screen with their base prices shown. Each round of the auction came to an end after three
minutes or once the student had used up all of his or her money, whichever came first.
The average prices of the goods auctioned were then calculated and shown to the student
so that he or she could associate possible changes in those prices with changes in the
conditions of each round of the auction, such as the amount of auction money provided,
the number of goods to be auctioned, or both.
As previously noted, the only difference between the two learning resources was
that for the “different goods” group the nine items, which included different kinds of
snacks such as potato chips, chocolate bars, biscuits, and so on, differed both within each
round and between rounds, whereas for the “same goods” group the nine items were all
the same, i.e., every item was a box of candy.
In the second round (see Figures 7and 8), to bring students’ focal awareness to bear
upon the dimension of demand, demand was deliberately varied (by varying students’
purchasing power by changing the amount of auction money they were given) while the
supply of goods was kept invariant. Each student’s auction money was cut by HK$200,
thus diminishing their purchasing power and demand for goods. The supply of goods for
auction, however, remained invariant, with the number of items kept at nine. Everything
was identical for both learning resources except that the nine items for auction remained
invariant in the “same goods” design (the same nine boxes of candy as in the first round,
whereas the type of goods varied in the “different goods” design, changing from the nine
kinds of snacks in the first round to nine kinds of soft drink in the second.)
Insert Figure 7. Same goods design (round 2).
Insert Figure 8. Different goods design (round 2).
28
In the third round, to help students to shift their focal awareness to the dimension of
supply, supply was deliberately varied while demand was kept invariant. The number of
items for auction was reduced from nine to seven, whereas the amount of auction money
remained the same (HK$200). However, the only—but critical—difference between the
two learning resources was that all seven items in the “same goods” design remained
boxes of candy, whereas the seven items used in the “different goods” design now
differed from those in the two previous rounds, with participants being asked to consider
different kinds of balls in this round.
Unlike the classroom study carried out by Lo et al. (2005), we introduced a fourth
auction round in which variation was introduced in both the demand and supply of goods
in a simultaneous manner. Our purpose was to help students to focus on the dimensions
of both in determining the market price of a good. To this end, the auction money given
to students was increased from HK$200 to HK$400, and the number of items to be
auctioned was reduced from seven to six. This round thus involved a simultaneous
variation in the supply of goods and variation in purchasing power (demand for the
goods), the aim of which was to enable students to discern the critical aspects of
experiencing price and pricing. As before, the only difference between the two learning
resources was that the six items in the “same goods” design remained the same, whereas
a new set of items (six different kinds of decorations) was introduced in the “different
goods” design.
Lastly, similar to the procedure in the earlier study (Lo et al., 2005), students were
asked a question (for instructional purposes) about what would happen to the price if the
supply were increased and purchasing power decreased. In the current study, they were
29
invited to predict the direction of change in the market price, that is, whether the price
would go up or down, if the amount of auction money was decreased from HK$400 to
HK$100 while the number of items to be auctioned increased from six to 11.
As noted, the learning session concluded with a five-minute summary delivered by
the researcher to remind students of the key learning points in the computerized learning
resources. He simply read the following PowerPoint slides to the two groups of students
at the same time.
1. (Slide 1) “Compare the auction game on Days One and Two. As the
auction money given to you on Day Two was less than that on Day One, your
income decreased. When your income decreased, your purchasing power also
decreased. This made your demand for goods decrease. As the supply of goods on
Day Two was the same as that on Day One, the average price of goods was
lower.”
2. (Slide 2) “Compare the auction game on Days Two and Three. As the
auction money given to you on Day Three was the same as that on Day Two, your
income and purchasing power remained unchanged. Your demand for goods also
remained unchanged. As the cost of production, such as the prices of raw
materials, electricity, and labor increased, the supply of goods decreased. As a
result, the average price of goods on Day Three was higher than that on Day
Two.”
3. (Slide 3) “Compare the auction game on Days Three and Four. As the
auction money given to you was more than that that on Day Three, your income
30
and purchasing power increased, and your demand for goods also increased. At
the same time, the increase in the cost of production made the supply of goods
decrease. As demand increased and supply decreased, the average price of goods
on Day Four was higher than that on Day Three.”
4. (Slide 4) “The price of a good is determined by its supply and demand.
The supply of a good is affected by its cost of production, such as the prices of
raw materials, electricity, and labor, whereas the demand for a good is affected by
people’s income and purchasing power. When businesses set the price of a good,
they need to consider the factors affecting supply and demand at the same time.”
Results and findings
The results presented in Table 2 show that students who belonged to the group using the
learning resource with the “same goods design” outperformed their counterparts using the
learning resource with the “different goods design” in the post-test, in which statistically
significant difference was observed between the two groups (
2
χ
= 10.36, p = 0.03 (<
0.05); effect size = 0.32). (Note, in particular, the relative frequencies for the target
understanding— Category D — in the post-tests for the two conditions.)
Insert Table 2. Distribution of conceptions, pre- and post-test.
CONCLUSION
Only in a restricted sense was this study a replication of Lo et al’s (2005) investigation.
We wanted to find out if invariance or variation in an unfocused aspect can really have
such a strong impact on the learning of the focused aspects as was interpreted to be the
31
case in the previous study. The question could be answered in the affirmative and the
conjecture was thus supported.
It should be noted that in both the original and follow-up studies, variation was
restricted. When the supply was invariant, demand went down (instead of going up in one
case and down in another), and when demand was invariant, the supply went down
(instead of going up in one case and down in another). In the last round, only one of the
four combinations of demand (up/down) and supply (up/down) was realized. We decided
not to include all four combinations as it would have made the task too difficult for such
young participants. In all of the circumstances considered, it is of course possible that the
results would have differed had the students been exposed to more of the possible
differences among the patterns of variation and invariance.
Quite a few of the students in the comparison group managed to learn to discern the
critical features of pricing, even though the good was not invariant. In general, even if
there is variation in several dimensions, learners may be able to block out all dimensions
but one, that on which they happen to focus. There is an interesting twist concerning how
this question appeared in the experiment, however. As noted, in the target group, a
number of items of the same type of good were offered in each round at the same base
price. In the comparison group, the same number of items as in the target group were
offered at the same base price. Further, whereas the target group considered the same
type of good both within and between rounds, the comparison group considered different
goods in each round. The difference between the conditions was illusory, however: all the
relevant factors (the number of items available, the amount of money participants had,
and the base price of goods) were exactly the same. The only element that differed was
32
the irrelevant labels placed on the goods. The only thing those in the comparison group
had to do was to separate what was relevant for their decisions from what was not, and
bracket the latter. If all of them had done so, then the conditions for the two groups would
have been the same. As we can see from the differences in outcome, however, this was
not the case. Our finding that the comparison group was affected by the irrelevant
differences in the item labels implies that quite a few learners in that group failed to
separate relevant information from irrelevant information, and therefore failed to see the
former.
The main contribution of this study is the support it provides for our conjecture: if
both the focused and unfocused aspects of the object of learning vary, then it is more
difficult to discern the focused aspects and relate them to one another than if the
unfocused aspect remains invariant while the focused aspects vary. We found this to be
true in the current study, even though the unfocused aspect was completely redundant.
However, the conjecture also addresses the question of how we can acquire new
meanings (or how we can learn to see certain things in certain ways). As mentioned
earlier, Fodor (1978, 1980), and others, claim that there is no answer to this question and,
in fact, there cannot be any. Meanings are innate.
However, we argue that regardless of whether meanings (concepts) are innate, or of
the sense in which they are (or are not) innate, we have to learn to discern them as aspects
of the world around us, and for this to happen, there are necessary conditions. These
necessary conditions are specific to particular meanings and to learners’ particular
experiential history. They can be formulated in terms of patterns of variation and
invariance among instances that do and do not have that particular meaning. Our
33
conjecture is thus very straightforward, as is the way in which it can be put to the test. We
simply have to create the necessary conditions in one case and ensure that they are absent
in another, as Ki and Marton (2003) did in their aforementioned study. Then, we can
compare the two cases and determine whether, as expected, all participants in the first
case learn the target meaning, whereas none of those in the second do. If these are indeed
the results, then the conjecture is strengthened.
Obviously, this is not what happened. Even if we can demonstrate that contrast is
more powerful than induction as far as the learning of new meanings is concerned, we
cannot demonstrate that new meanings cannot be learned through induction. After all,
some learners seem to learn in that condition too, and certainly not all learners will learn
even if all possible steps are taken to make it possible for them to do so. The relationship
between what is learned, on the one hand, and the conditions of learning, on the other, is
stochastic rather than deterministic. But why is this so?
Returning to the experiment reported in this paper, beyond the fact that the target
principle (price as a function of the relationship between demand and supply) was made
explicit to both groups, there is a more general answer to the foregoing question. Our
conjecture concerns the pattern of variation and invariance as experienced by the learner,
whereas a pattern of variation and invariance that can be controlled by the researcher
refers to the patterns seen by the researcher. What might the relationship between the two
look like? One condition of experiencing variation is that there is variation to be
experienced. Making sure that this condition is met is the first step toward making
learning possible (which in our view is what teaching is all about). However, variation
can also be experienced because of previous experiences. Experienced variation is thus
34
not necessarily the experience of what is present in the learning situation as seen by the
observer. On the other hand, even if there is variation, it is not necessarily experienced by
all learners. In conclusion, when comparing two randomly selected groups, we would
expect more learners to experience variation if it is present than if it is not.
However, experiencing variation not only concerns the variation to be experienced
in a relevant dimension; it also presupposes invariance in other dimensions. In other
words, variation can be experienced only against a background of invariance. In this
sense, experienced variation is a function of invariance, and, as previously stated,
experienced variation is also a function of variation. Our desire in the present study was
to illustrate that learning (in the sense of the discernment of the necessary features of a
phenomenon) is a function of experienced variation (by the learner), which is a function
of both variation and invariance (as seen by the observer). This we did, with a focus on
the latter (invariance).
We have thus obtained support for what we refer to as “the differentiation view of
learning” (as opposed to “the enrichment view”). We have shown that introducing
redundant information (different goods) that is correlated with a variation in critical
aspects (a change in demand and supply) significantly reduces the likelihood of learners
being able to discern the critical features of the object of learning. A seemingly subtle
difference between two conditions, both representing 90 minutes of pedagogical effort, is
proved to play a key role in learning outcomes.
Acknowledgments
The research reported here was financially supported by the Swedish Research Council.
35
References
Bateson, G. (1980). Mind and nature: A necessary unity. Toronto: Bantam.
Bowden, J. A., & Marton, F. (2004). The university of learning. London: Routledge.
Cassirer, E. (1953). Substance and function. New York: Dover.
Chi, M. T. H., Feltovich, P. J., & Glaser, R. (1981). Categorization and representation of
physics problems by experts and novices. Cognitive Science, 5(2), 121-152. doi:
10.1207/s15516709cog0502_2
Dahlgren, L. O. (1978). Effects of university education on the conception of reality.
Reports from the Institute of Education, University of Goteborg. Gothenburg:
Institute of Education, University of Gothenburg.
de Saussure, F. (1986). Course in general linguistics (R. Harris, Trans.). New York:
Open Court.
Elliott, J. (2004). The independent evaluation of the PIPS project. Hong Kong: Hong
Kong Institute of Education.
Fodor, J. A. (1978). The language of thought. Hassocks, Sussex: Harvester.
Fodor, J. A. (1980). Fixation of belief and concept acquisition. In M. Piattelli-Palmarini
(Ed.), Language and learning: The debate between Jean Piaget and Noam
Chomsky (pp. 142-162). London: Routledge & Kegan Paul.
Fraser, D., Allison, S., Coombes, H., Case, J., & Linder, C. (2006). Using variation to
enhance learning in engineering. The International Journal of Engineering
Education, 22(1), 102-108.
36
Fraser, D., & Linder, C. (2009). Teaching in higher education through the use of
variation: Examples from distillation, physics and process dynamics. European
Journal of Engineering Education, 34(4), 369-381.
Gibson, J. J., & Gibson, E. J. (1991). Perceptual learning: Differentiation or enrichment.
In E. J. Gibson (Ed.), An odyssey in learning and perception (pp. 291-304).
Cambridge, MA: MIT Press. (Reprinted from: Psychological Review, pp. 32-41,
1955).
Goodwin, C. (1994). Professional vision. American Anthropologist, 96(3), 606-633. doi:
10.1525/aa.1994.96.3.02a00100
Guo, J.-P., & Pang, M. F. (2011). Learning a mathematical concept from comparing
examples: The importance of variation and prior knowledge. European Journal of
Psychology of Education, 26(4), 495-525. doi: 10.1007/s10212-011-0060-y
Holmqvist, M., Gustavsson, L., & Wernberg, A. (2008). Variation theory: An organizing
principle to guide design research in education. In A. E. Kelly, R. A. Lesh & J. Y.
Baek (Eds.), Handbook of design research methods in education: Innovations in
science, technology, engineering, and mathematics learning and teaching (pp. 111-
130). New York: Routledge.
Ki, W. W., Ahlberg, K., & Marton, F. (2006). Computer-assisted perceptual learning of
Cantonese tones. Paper presented at the 14th International Conference on
Computers in Education, Beijing, China: Asia-Pacific Society for Computers in
Education (APSCE).
Ki, W. W., & Marton, F. (2003). Learning Cantonese tones. Paper presented at the
EARLI Biennial Conference 2003, Padova, Italy.
37
Kullberg, A. (2010). What is taught and what is learned: Professional insights gained
and shared by teachers of mathematics. Doctoral dissertation, University of
Gothenburg, Acta Universitatis Gothoburgensis, Göteborg.
Linder, C., Fraser, D., & Pang, M. F. (2006). Using a variation approach to enhance
physics learning in a college classroom. The Physics Teacher, 44(9).
Lo, M. L. (2009). Building a teacher learning network for developing the ability to teach
for learning. Paper presented at the 13th Biennal Conference of EARLI,
Amsterdam, The Netherlands.
Lo, M. L., Lo-fu, Y. W. P., Chik, P. M. P., & Pang, M. F. (2005). Two learning studies.
In M. L. Lo, W. Y. Pong & P. M. P. Chik (Eds.), For each and everyone: Catering
for individual differences through learning studies (pp. 75-116). Hong Kong: Hong
Kong University Press.
Lo, M. L., Pong, W. Y., & Chik, P. M. P. (Eds.). (2005). For each and everyone:
Catering for individual differences through learning studies. Hong Kong: Hong
Kong University Press.
Luhmann, N. (2002). Theories of distinction: Redescribing the descriptions of modernity.
California: Stanford University Press.
Maanula, T. (2011). Resultat från nationella prov i matematik m m. Available from
tuula.maunula@telia.Com (unpublished manuscript).
Marton, F. (1981). Phenomenography—Describing conceptions of the world around us.
Instructional Science, 10(2), 177-200. doi: 10.1007/bf00132516
Marton, F. (2006). Sameness and difference in transfer. Journal of the Learning
Sciences, 15(4), 499-535. doi: 10.1207/s15327809jls1504_3
38
Marton, F. (2012). Necessary conditions of learning (manuscript).
Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, N.J.: L. Erlbaum
Associates.
Marton, F., Carlsson, M. A., & Halasz, L. (1992). Differences in understanding and the
use of reflective variation in reading. British Journal of Educational Psychology,
62(1), 1-16. doi: 10.1111/j.2044-8279.1992.tb00995.x
Marton, F., Hounsell, D., & Entwistle, N. J. (Eds.). (1984). The experience of learning:
Implications for teaching and studying in higher education. Edinburgh: Scottish
Academic Press.
Marton, F., & Pang, M. F. (2006). On some necessary conditions of learning. Journal of
the Learning Sciences, 15(2), 193-220. doi: 10.1207/s15327809jls1502_2
Marton, F., & Pang, M. F. (2008). The idea of phenomenography and the pedagogy for
conceptual change. In S. Vosniadou (Ed.), International handbook of research on
conceptual change (pp. 533-559). London: Routledge.
Marton, F., & Tsui, A. B. M. (2004). Classroom discourse and the space of learning.
Mahwah, NJ: Lawrence Erlbaum Associates.
Marton, F., Wen, Q., & Wong, K. (2005). Read a hundred times and the meaning will
appear ... Changes in Chinese university students’ views of the temporal structure
of learning. Higher Education, 49(3), 291-318.
Pang, M. F. (2010). Boosting financial literacy: Benefits from learning study.
Instructional Science, 38(6), 659-677. doi: 10.1007/s11251-009-9094-9
39
Pang, M. F., Linder, C., & Fraser, D. (2006). Beyond lesson studies and design
experiments: Using theoretical tools in practice and finding out how they work.
International Review of Economics Education, 5(1), 28-45.
Pang, M. F., & Marton, F. (2003). Beyond "lesson study'': Comparing two ways of
facilitating the grasp of some economic concepts. Instructional Science, 31(3), 175-
194. doi: 10.1023/a:1023280619632
Pang, M. F., & Marton, F. (2005). Learning theory as teaching resource: Enhancing
students’ understanding of economic concepts. Instructional Science, 33(2), 159-
191. doi: 10.1007/s11251-005-2811-0
Sandberg, J. (1994). Human competence at work: An interpretative approach. Göteborg,
Sweden: BAS.
Spencer-Brown, G. (1972). Laws of form. New York: Julian Press.
Stagray, J. R., & Downs, D. (1993). Differential sensitivity for frequency among speakers
of a tone and a non-tone language. Journal of Chinese Linguistics, 21(1), 143-163.
Stigler, J. W., & Hiebert, J. (1999). The teaching gap: Best ideas from the world's
teachers for improving education in the classroom. New York: Free Press.
Werner, H. (1957). The concept of development from a comparative and organismic
point of view. In D. B. Harris (Ed.), The concept of development: An issue in the
study of human behavior (pp. 125-148). Minneapolis: University of Minnesota
Press.
40
