SUMMARY OF MECHANISMS IN LEARNING
Constructivism (Piaget)
At the foundation of radical constructivism are the two principles: (1) “Knowledge is not passively
received but built up by the cognizing subject” and (2) “The function of cognition is adaptive and
serves the organization of the experiential world, not the discovery of ontological reality” (von
Glasersfled, 1995, p. 18). In other words, constructivist theory is based on the premise that “all
knowledge is necessarily a product of our own cognitive acts. We can have no direct or unmediated
knowledge of any external or objective reality. We construct our understanding through our
experiences, and the character of our experience is influenced profoundly by our cognitive
lenses.”(Confrey, 1990, p. 108). This does not mean a rejection of the external reality. Rather, from
an epistemological perspective it is to say that “we can rationally know only what we ourselves have
made” in Vico’s terms (von Glasersfeld, 1995, p.6).
Piaget makes a distinction between physical knowledge and logical mathematical knowledge. While
the former refers to the knowledge about objects based on experiences through physical acting
(figurative activities), the latter applies to structures that are abstract and constructed through
mental operating (operative activities). These lead to two types of abstraction described by Piaget:
1) simple (empirical) abstraction which is derived from the object that we act upon, 2) reflective
abstraction which is drawn from the action itself (von Glasersfeld, 1994). The second one is
reflective in a sense that “at the level of thought a reorganization takes place” (Piaget, 1970, p. 18).
Examples from Piaget (1970): A child holding objects in his hand notices the difference in their
weights and realizes that usually the big ones are heavier than the small ones. However, small things
sometimes can weigh more than the big ones. In this experiential realization, the child’s knowledge
is abstracted from the object, i.e. empirical or simple abstraction. Piaget shares a story about his
mathematician friend’s experience to illustrate the latter: “When he was a small child, he was
counting pebbles one day; he lined them up in a row, counted them from left to right, and got ten.
Then, just for fun, he counted them from right to left see what number he would get, and was
astonished that he got ten again. He put the pebbles in a circle and counted them, and once again
there were ten. He went around the circle in the other way and got ten again. And no matter how he
put the pebbles down, when he counted them, the number comes to ten.” (pp. 16-17). In this
example, the child discovers the commutative property (changing the order of the quantities doesn’t
change the sum) using physical objects, such as pebbles, but it was the child who changed their
order from line to circle and counted up all to get the sum. So the knowledge was derived from the
actions that he carried out on the pebbles (i.e., reflective abstraction), rather than their physical
attributes.
Reflective abstraction always involves coordinated actions in mental processes (Piaget, 1970). So,
“constructivism not only emphasizes the essential role of the constructive process, it also allows one
to emphasize that we are at least partially able to be aware of those constructions and then to
modify them through our conscious reflection on that constructive process.”(Confrey, 1990, p. 109).
According to Piaget (1970), there are two types of actions: individual actions, such as throwing,
pushing, touching, and coordinated actions from these individual ones, (additive, sequential,
correspondence, intersections) Piaget considered coordinated actions as the basis of logical
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mathematical thoughts. For him, coordinated actions become mental operations and a system of
these operations constitutes a structure. His notion of scheme refers to “whatever is repeatable and
generalizable in an action” (ibid, p. 42). Hence, scheme theory functions to explain the stability and
predictability of actions (Confrey, 1994). Like the actions, the schemes can be coordinated with each
other and the logical mathematical structures emerge from these coordinations (Piaget, 1970). For
instance, when a child wants to reach an object on the blanket and pulls the blanket toward himself
to get it, that scheme includes some subschemes, involving the relationship between the child’s
hand and the blanket, the relationship between the blanket and the object, and another relationship
between the object and its position. According to Piaget the way that the subschemes are included
in the whole scheme lays the basis for the inclusion relation, like the relationship of class inclusion in
the logical mathematical structure (e.g., there is a class of objects called apples; and there is also a
class called fruits. If all apples are fruits, then the class of fruits includes the class of apples).
To interpret Piaget’s scheme theory, one needs to understand two fundamental processes:
assimilation and accommodation. According to Piaget, assimilation involves an incorporation of new
experiences and perceptions of the world to the existing schema whereas accommodation refers to
adaptation of the existing schema to a new structure. Moreover, these two processes work in a
dialectical relationship (See Figure 1). When a scheme leads to a perturbation, the problematic will
be called to action. Then, accommodation takes place in order to maintain or re-establish the
equilibrium. Note that equilibrium takes place both in the forms of assimilation and accommodation.
New Situation/ Perturbation
ASSIMILATION
Disequilibrium
ACCOMMODATION
Equilibrium
Assimilation
Existing Schema
Equilibrium
New Situation
Figure 1. Dialectic relationship between assimilation and accommodation in Piaget’s theory
For example, once a child constructs the idea of commutative property of addition in natural
numbers, he can assimilate it to the addition of fractions as well (i.e. 1/4+1/2=1/2+1/4). This means
that in von Glasersfeld’s term, the child fits an experience into existing conceptual structure during
the assimilation process. However, when he tries to assimilate the addition in natural number to
fractions, such as 1/4+1/2= 2/6, it may raise some problems and calls for an accommodation.
Representational Re-description (RR)
Karmiloff-Smith, who is a student of Piaget, proposes a mechanism of cognitive development that
brings Piaget’s constructivism and Fodor’s nativist approach together (Karmiloff-Smith, 1992). For
knowledge acquisition she claims that “a specifically human way to gain knowledge is for the mind
to exploit internally the information that it has already stored (both innate and acquired), by
redescribing its representations or, more preciously, by iteratively re-representing in different
representational formats what its internal representations represent.” (Karmiloff-Smith, 1992, p.
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15). This representational re-description refers to a knowledge change from implicit to explicit
allowing for flexibility and creativity. In other words, the acquired knowledge initially stored
implicitly and the development occurs through the redescription of that knowledge into accessible,
explicit forms at various levels.
Karmiloff-Smith describes the RR model as it “attempts to account for the way in which children’s
representations become progressively more manipulable and flexible, for the emergence of
conscious access to knowledge, and for children’s theory building.” (p. 17). Her view of how children
develop understanding of concepts in math, science, or language indicates that “redescription of
knowledge into increasingly exploit formats which ultimately enable the child to provide verbal
explanation is at the heart of the RR model’s account of how children subsequently develop their
intuitive theories about different domains. Theories are built on explicitly defined representations.”
(p. 110). In other words, a shift from initially procedural knowledge to knowledge with a theoretical
status (i.e., abstract concepts) requires an explicit representation of that knowledge.
In the RR model the development of representations (from implicit to explicit) takes place in three
recurring phases:
Phase 1. Representational adjunctions are created based on data from external
environment. Representations are independently stored rather than being linked in a coherent
system. Consistently successful performance through these representations leads to ‘behavioural
mastery’ at this phase (behavioural change).
Phase 2. The focus is on internal representations and not in external data anymore. The
neglect of features of the external environment can lead to a decline in performance at the
behavioural level, rather than in representational level.
Phase 3. There is a reconciliation of internal representations and external data. A balance is
achieved to restore new knowledge with a complete representation when a child is able to succeed
in a task and reports on the representation verbally.
Example: Like maybe many other people solving puzzles, Karmiloff-Smith explains how she had to
ignore her conciseness when solving Rubik’s Cube. She had to try to analyze her steps until she could
solve it. During the solution process, she initially developed a proprioceptive solution that she could
carry out fast, but it was difficult for her to repeat it at a slower pace. She claims that her current
“knowledge” was embedded in the procedural representations through which she could do it
rapidly. This did not stop her from reiterating the solution several times. When she began to
recognize certain states of the cube, she was able to see whether she was in the right direction
towards her solution. Her knowledge was still not flexible enough to interrupt the solution and to
keep solving the puzzle from any state. Then she realized that she could predict the next step before
really executing them. Eventually she could explain her solution to her daughter. She observed that
when her daughter tried to solve the Rubik’s Cube, she moved from procedural to explicit
knowledge in the same way, rather than using her mother’s explicit solution steps. Regarding
development she hypothesizes that “children are not satisfied with success in learning to talk or to
solve problems; they want to understand how they do these things.” (p. 17).
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In the RR model, Karmiloff-Smith proposes that there are at least four levels at which knowledge is
represented and re-represented throughout the development. These different levels of
representational formats are part of a reiterative cycle both within different microdomains1 and
during the developmental period. The representations formed at the initial level develop towards a
higher level through this reiterative process of representational redescription.
Implicit Level (I): The nature of representations is procedural and implicit when used in
response to an external stimulus. A procedure is a closed entity as a whole and therefore it
components are not available to other operators. For instance, “One-to-one correspondence is an
implicit feature of successful counting procedures. The principle embedded in the procedure must
then be abstracted, redescribed, and represented in a different format independent of the
procedural encoding. This level-E1 representation, once it is lifted from its embedding in the level-I
counting procedure, can then be used for unspecified quantities.” (p.109)
Explicit Level 1 (E1): Representations at this level are the result of a process of redescription
of level-I representations encoded in procedural formats. The resulting compressed formats are
reduced descriptions, that is to say many details of the procedurally encoded information are
discarded but some essential features are exploited. [For example, a child groups all different kinds
of triangles together and all squares and rectangles together by distinguishing them with respect to
the number of sides, but fails to differentiate rectangles from squares or equilateral triangles from
isosceles triangles. That’s my understanding at least. -SK] Representations are redescribed into E1
formats explicitly. So cognitive system has access to them, but this is not conscious yet.
Explicit Level 2 (E2): At this level E1 representations are recoded and the resulting E2
representations become accessible to consciousness but are not verbalized yet (possible only at level
E3). For example, when we draw diagrams of problems that we cannot verbalize, E1 spatial
representations are redescribed into E2 spatial representations which are consciously accessible
(Kamiloff-Smith, 1992). So, the end product is actually a visual representation of similar knowledge
existing at level E1 with less detail and explicitness. (She notes, “No research has thus far been
directly focused on the E2 level (conscious access without verbal report).” (p. 23)
Explicit Level 3 (E3): Knowledge is recoded into a format in which it can be stable and
communicated verbally. At this higher level of redescription, representations become manipulable
and available to both consciousness and verbal report. She argues that some knowledge is learned
through verbal interactions with others. These representations are directly stored at level E3
(Kamiloff-Smith, 1992). She points out that
However, knowledge maybe stored in linguistic code but not yet be linked to similar
knowledge stored in other codes. Often linguistic knowledge (e.g., a mathematical
principle governing subtraction) does not constrain non-linguistic knowledge (e.g.,
an algorithm used for actually doing subtraction) until both have been described
into a similar format so that inter-representational constraints can operate.” (p. 23)
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Karmiloff-Smith refers to a microdomain as a subset within a particular domain, such as language,
mathematics, etc.
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This claim implies that a child can perform a mathematical procedure, such as computing the mean
of a data set, without really understanding the reasoning behind the procedure. [sort of similar to
Vygotsky -SK] But some children can have both understandings at the level E3.
In the RR model, the abstraction emerges during these redescriptions. [Question: Does it mean that
an abstraction is possible without understanding how the algorithm works? –SK]
According to Karmiloff-Smith, a child uses the existing Level I representations in certain tasks
involving speed and automaticity and the representational redescriptions in situations where explicit
knowledge is required.
In the RR model, failure, incompletion, or conflict can lead to behavioural mastery, but in the
transition between phases she proposes a success-based view of cognitive change. In contrast to
Piaget’s process of equilibration in the presence of a disequilibrium, Karmiloff-Smith argues that “for
the RR model certain types of change take place after the child is successful (i.e., already producing
the correct linguistic output, or already having consistently reached a problem-solving goal).
Representational re-description is a process of “appropriating” stable states to extract the
information they contain, which can be used more flexibly for other purposes.” (p. 25). She puts
more emphasis on the “role of internal system stability as the basis for generating representational
redescription.” (p. 26). Her idea of representational re-description is similar to Piaget’s assimilation.
However, von Glasersfeld (1995) notes that “if an unexpected result happens to be a desirable one,
the added condition may serve to separate a new scheme from the old. In this case, the new
condition will be central in the recognition pattern of the new scheme.” (p. 66). This seems to imply
that cognitive change can occur when there is unexpected success.
Cognitive Conflict
In the Piagetian tradition, the role of “cognitive conflict” or “socio-cognitive conflict” to probe
students’ learning has been studied as a mechanism of social interaction in peer collaboration
through a series of experiments conducted by a group of psychologists (e.g., Mugny & Doise, 1978).
In this approach students with moderately different perspectives are paired up so that when they
work on a mutual task, the incompatibilities in their ideas or strategies creates disequilibrium. This
socially created cognitive conflict then leads them to cognitive reorganization as they try to reach a
consensus. According to Mugny and Doise (1978), “social coordination of actions facilitates and
precedes the individual coordination of actions.” (p. 191).
The socio-cognitive conflict mechanism is usually studied through peer interactions and also viewed
as a way of implementing a co-construction of knowledge (Cesar, 1998). According to Cesar, this
process is “a powerful way of confronting pupils with one another’s solving strategies and that made
them decentralize from their own position and discuss the one another's conjectures and
arguments.” (1998). For instance, among several studies conducted by Mugny and his colleagues the
researchers examined the effectiveness of the socio-cognitive conflict in the context of conservation
(see Mugny, 1982). First, they administered a pre-test individually to assess whether a child is
conserver or nonconserver. Then, the children are paired, such as a conserver child with a
nonconserver, to ensure a potential difference in perspectives as they work on the same task during
the training in conservation. For instance, in a task where children were to determine whether there
is an equal number of candies in two given rows after the candies were spread out further in one of
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them, children were asked to come up with a joint decision (cited in Tudge & Rogoff, 1989). Based
on Murray (1972) and Silverman and Stone (1972), Murray (1982) reported that these studies
showed children’s success in various conservation problems through having the nonconservers
argue with their conserving peers until they all agree. The results revealed that in the individual
post-test, 80-94% of the nonconservering children made significant gains in conservation.
Social Constructivism
Building upon the two principles of radical constructivism mentioned earlier, Ernest (1999) refers to
social constructivism as a philosophy of mathematics and defines additional assumptions of the
existence of social and physical reality as a basis of social constructivism:
1) The personal theories which result from the organization of the experiential world must 'fit'
the constraints imposed by physical and social reality;
2) They achieve this by a cycle of theory-prediction-test-failure-accommodation-new theory;
3) This gives rise to socially agreed theories of the world and social patterns and rules of
language use;
4) Mathematics is the theory of form and structure that arises within language.
More specifically, social constructivism focuses on “the development of “knowledge communities”
as a larger unit of analysis provided it is connected to its effects on independent reasoning patterns
for individual students, as also a target unit of analysis.” (Confrey & Kazak, 2006, p. 319). In other
words, the main concern of social constructivists is still the individual aspects of knowledge with the
acknowledgement of the social interaction as a secondary focus (Ernest, 1994). For example, the
approach taken by Yackel and Cobb (1996) “reflects the view that mathematical learning is both a
process of active individual construction (von Glasersfeld, 1984) and a process of acculturation in to
the mathematical practices of wider society (Bauersfeld, 1993)” (p. 460).
In the social constructivist tradition, social processes are important mechanisms through which
participants negotiate meaning and co-construct knowledge in collaborative learning environments.
Co-construction of knowledge refers to the internalization of knowledge by individuals from semiotic
and tool-based mediation, i.e. shared language and tools (e.g., Hull & Saxon, 2009).
Drawing upon the studies by Goodwin (1995), Roschelle (1992) and Säljö (1999), Rojas-Drummond,
Albarr’an and Littleton suggested that the process of children’s co-construction of meaning and
knowledge to achieve their goals on a collaborative task entailed “joint planning; taking turns; asking
for and providing opinions; sharing, chaining and integrating of ideas; arguing their points of view;
negotiating and coordinating perspectives; adding, revising, reformulating and elaborating on the
information under discussion and seeking of agreements.” (p. 186). [When examining coconstruction of knowledge in collaborative learning environments, looking for these actions can be
useful, I guess. –SK]
The work of Cobb and his colleagues (e.g., Yackel & Cobb, 1996; Cobb, 1999) focuses on the analysis
of sociomathematical norms of mathematical difference (what counts as a different solution) and
mathematical sophistication (what counts as a sophisticated or an efficient solution) to foster the
intellectual development in the classroom community through social interactions between teacher
and students. Sociomathematical norms refer to “what counts as an acceptable mathematical
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explanation and justification” when teacher and students interactively participate in a classroom
discussion (Yackel & Cobb, 1996, p.461). Yackel and Cobb argue that in addition to the participation
in mathematical discussion, the sociomathematical norms enable students to engage in higher level
cognitive activity. When the sociomathematical norms are negotiated in the classroom, students are
encouraged to make an effort to understand others’ perspectives. As a result, taken-as-shared
understanding of the mathematical value of explanations, reasoning, and solutions evolves. In this
process, there needs to be a shift from students’ making sense of an explanation for themselves to
anticipating how others might make sense of it (Yackel & Cobb, 1996).
For instance, Yackel and Cobb (1996) provides an example of the negotiation of the meaning of
mathematical difference in the following classroom dialogues where students respond to a mental
computation problem “7 8 - 53 = ?“ (p. 463)
Dennis: I said, um, 7 and take away 50, that equals 20.
Teacher: All right.
Dennis: And then, then I took, I took 3 from that 8 and then that left 5.
Teacher: Okay. And how much did you get?
Dennis: 25....
...
Teacher: Ella?
Ella: I said the 7, the 70, I said the 70 minus the 50 ... I said the 20 and 8 plus 3,...Oh,I added, I said
8 minus the 3, that'd be 5.
Teacher: All right. It'd be what?
Ella: And that's 75 ... I mean 25.
Dennis: (Protesting) Mr.K ., that's the same thing I said.
In this exchange, Yackel and Cobb point out that Dennis takes the initiative to use the norm of
mathematical difference when he protests Ella’s way of solving the problem. Here he compares
Ella’s solution and his. Then he arrives to a conclusion that it is not different from the one he
described earlier. Furthermore, Yackel and Cobb suggest that “the sociomathematical norm of what
constitutes mathematical difference supports higher-level cognitive activity.” (p. 464). They argue
that in order to compare differences and similarities in two solutions Dennis reflected on his own
activity. According to Yackel and Cobb, “Such reflective activity has the potential to contribute
significantly to children's mathematical learning.” (p. 464).
In another classroom-based design study, Cobb and his colleagues aimed at developing 12-year-old
students’ conceptual understanding of key statistical ideas and topics as they organize and structure
data using computer-based minitools and describe the characteristics of distribution through
statistical reasoning (Cobb, 1999). The researchers designed data sets for analyses in which students
could make decisions in the context of a real-life problem situation. Moreover, students were
encouraged to justify their reasoning in the whole-class discussions to develop their own statistical
understandings through these tasks.
As students initially tended to focus on data as individual points rather than as distributions (an
aggregate view of data), Cobb (1999) describes the emerging mathematical task when students used
the first minitool (like value bars showing two different brands of battery’s life) as “exploring
qualitative characteristics of collections of data points.” (p. 17). When students started to use the
second minitool (like split dot plots showing the speeds of 60 cars before and after the speed trap), a
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student began to describe a data set with qualitative terms by focusing on its global features, i.e., its
shape, such as “like hills” and “bunched up close.” (p. 19). According to Cobb, when one of teachers
showed the “hills” by marking them on the projected data, student’s interpretation was legitimized.
Both teachers continue to capitalize on the student’s contribution for the rest of the classroom
discussion. As another student attempted to describe these qualitative differences in data sets in
terms of quantitative ways (e.g., “most people are from 50 to 60” in one data set and “most people
were between 50 and 55” in the other one), the teacher presented her analysis as a way of the
global shift proposed by the other student earlier. Through this revoicing of the student’s
contribution, Cobb argued, “it gradually became taken-as-shared that the intent of an analysis was
to identify global trends or patterns in data that were significant with respect to the issue under
investigation.” (p. 20). As the tasks evolved through use of different minitools and the classroom
mathematical practices emerged as students developed statistical competencies, Cobb et al.
observed shifts in students’ statistical perspectives through the taken-as-shared ways of reasoning
and arguing about data. Cobb further argued that the shifts in mathematical practices and meanings
in a collective mathematical activity were to attributed to the classroom community (the unit of
analysis in the study) rather than the individuals. He also acknowledged the tool-use as a mediator in
a Vygotskian perspective as well as the role of argumentation in relation to developing the
sociomathematical norms in the classroom.
How do we know that a mathematical practice (i.e., interpretation of data sets as distributions) as
taken-as-shared? When the student used the notion of “hills” for the first time, she provided a
warrant for her interpretation explaining her interpretation (Cobb, 1999). However in the following
task, the legitimacy of the hill interpretation was not questioned when students discussed the data
sets. This is Cobb’s way of interpreting that seeing data sets as distributions was taken-as-shared!??
Socio-Cultural Theory (Vygotsky)
In contrast to Piaget, Vygotsky assumes that higher mental functioning in the individual emerges
from the social context. Thus, the direction of intellectual development is from social to individual.
According to Vygotsky’s general genetic law of cultural development, any mental function occurs
first in the social plane and then in the psychological plane (Wertsch, 1985).
Mediation is a key concept in Vygotsky’s theory to describe the shift from social plane to
psychological plane. Vygotsky argued that higher mental processes are mediated by tools (i.e.,
technical tools, such as a calculator, a graphic paper, etc.) and signs (i.e., psychological tools, such as
language, numerical system, algebraic symbols, and so on). Development takes place when these
different forms of mediation create a transformation in mental functioning. Individuals have access
to the mediational means as part of a socio-cultural context, from which individuals “appropriate”
them (Wertsch, 1985).
Example of mediation by a technological tool. In a teaching experiment conducted with ten 7th grade
students participated in an after-school program in Turkey, students engaged in a series of tasks
involved data analysis and probability by using TinkerPlots (Konold & Miller, 2004) as a computer
tool for data analysis, data modeling and probability simulations (Kazak, 2012). In one of these tasks,
students were first asked to determine whether it is fair to choose a winner among three players by
flipping a coin twice in a game where player 1 wins if both outcomes are tails (TT); player 2 wins if
both comes up heads (HH); and player 3 wins if one tails and one heads turn up (HT/TH). Some
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students initially predicted that the game was not fair because after the first flip either player 1 or
player 2 would be eliminated but player 3 would always be in the game. This was a reasonable
explanation. To test this idea, students were asked to build a Sampler to play this game in
TinkerPlots (TP) and to run it 12 times individually. The results from each student were recorded and
discussed whether the game was fair as a class. Looking at the combined results (24 TT, 26 HH, and
46 HT/TH) all students thought that the game was not fair since one tails-one heads came up about
twice as many as the other outcomes. When asked to explain why one tails-one heads is twice as
likely as the other outcomes, S1 said “since TH, HT are the same, it occurs more.” But it was not clear
what he meant by “TH, HT are the same.” During the class discussion, another student (S2) made a
comment that “there [pointing to H;T/T;H on the TP plot] we combined the two chances.” In S2’s
explanation, the action involved in showing the results on the plot (i.e., combining two joined
outcomes, HT and TH, by dragging one into the other) enabled her to recognize that there were two
different ways to get one T one H (an understanding that leads to the idea of sample space).
Figure 1. Model of the Coin game in TinkerPlots and the outcomes in 100 trials. A single mixer device
is set to draw twice with 100 repetitions. The table next to it shows the results of each repetition as
they are drawn. The graph on the left-hand side displays the percent of outcomes for each event. In
the graph on the right-hand side, the two outcomes (HT and TH) are combined into a single bin by
dragging one into the other.
In studying the development of thinking as a dialectic process between thought and language,
Vygotsky focuses on the word meaning as a unit of verbal thought (Vygotsky, 1978). From
Vygotskian perspective, words have certain meanings for adults/experts and children/less
competent students begin to develop them through social interactions. It is also noted that “word
meaning continues to develop long after the point when new words (that is, sign vehicles or sign
forms) first appear in children’s speech. He argued that although it is tempting to attribute a
complete understanding of word meaning to children when they begin to use word forms in what
seem to be appropriate ways, the appearance of new words marks the beginning rather than the
end point in the developments of meaning.” (Wertsch, 1985, p. 99). Furthermore, using the
Vygotsky-Shpet Perspective on implications for instruction, Wertsch and Kazak (2011) argue that
“the act of speaking often (perhaps always) involves employing a sign system that forces us to say
more (as well as perhaps less) than what we understand or intend, more in the sense that
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interlocutors may understand us to be conveying a higher level message than our mastery of the
sign system really justifies.” (p. 156)
Example. Students can use the word “standard deviation” after they learn it in the class. Their initial
understanding could be only the formula (
) so it means only an algorithm that
they employ to find a number, not the same meaning that an expert would have. Eventually we
expect that the meaning will evolve and they begin to use the word as a measure if variation from
the mean (or expected value).
The notion of intersubjectivity is a relevant idea in collaborative learning environments. From the
Vygotskian perspective, it refers to a process in which a less competent child/student can become
involved with a more experienced adult/teacher and through which he can extend his understanding
to a higher level (Wertsch & Kazak, 2011).
Vygotsky, unlike Piaget, emphasized more on adult-child interaction. He introduced the notion of the
zone of proximal development (ZPD) (Vygotsky, 1978):
[The ZPD] is the difference between [a child’s] actual development level as determined by
independent problem solving and the level of potential development as determined through
problem solving under adult guidance or in collaboration with more capable peers. (p. 86)
He argued that intellectual development occurs within the ZPD of the child. Moreover, in his
formulation of the ZPD, Vygotsky pointed to the role of imitation in learning. He argued that
“children can imitate a variety of actions that go well beyond the limits of their own capacities. Using
imitation, children are capable of doing much more in collective activity or under the guidance of
adults” (ibid, p. 88). For him, with assistance of an adult or a more capable peer every child can do
more than he/she can by him/herself.
To elaborate on Vygotsky’s ZPD, Bruner and his colleagues used the notion of scaffolding as a way of
structuring the learning task by an adult to help the learner “to internalize knowledge and convert it
into a tool for conscious control” (Bruner, 1985, p. 25).
Example. How can we observe scaffolding during the instruction? Based on their research with
teachers Bliss, Aksew, and Macrae (1996) identify scaffolding strategies in primary school design and
technology, mathematics, and science classrooms:
1) Attempted, but unsuccessful, scaffold with unexpected consequences. When students
interpret teacher’s actions in a different way than the teacher expected, their responses
might lead to unintended consequences. This would create a distraction from the task and
its goal.
2) Unintended scaffolding. Teacher’s actions or words may help students respond in an
unexpected way, but teacher may not be conscious about it.
3) Actual scaffolds. Mostly they involve “approval, encouragement, structuring work, or
organizing people scaffolds.” (p. 47)
4) Props scaffolds. Teacher provides a suggestion that will help students during the task.
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5) Localized scaffolds. When the idea or concept is too big and complex, it might be difficult to
provide students help with the overall idea. So, teacher scaffolds one part of it which could
help student move towards the more general concept.
6) Step-by-step scaffolds. When arguments get a bit difficult in teaching, leading students step
by step in a series of questions can help instruction moving.
7) Hints and slots scaffolds. When usually open ended questions lead to a specific answer, it is
possible to narrow the question down quite further until only one answer can be
appropriate.
Dialogic Theory
Dialogic learning theory as the context for understanding L2L2 (Wegerif)
L2L2 studied in WP2 is not the same kind of learning as the learning of domain specific concepts and
skills studied in WP3. L2L2 is a version of the domain general skill of creative thinking in group.
Whereas learning specific domain concepts could be seen as convergent because it aims at answers,
learning creative thinking could be seen as divergent because it aims at richer and more fruitful
questions, the kind of questions that take more possible perspectives into account. According to
dialogic learning theory (Wegerif, 2013: 2011), creative thinking is learnt in the context of dialogues
and dialogic relations and can be understood as the process of expanding the space of dialogue:
deepening this space by reflecting on framing assumptions and expanding this space through
introducing more voices into the dialogue.
Creative thinking emerges out of the constructive tension of different perspectives held together in
the tension of a dialogue. The dialogic tension that leads to creative thinking can be that between
two specific perspectives but can also be a dialogic tension between a specific focus of attention and
an unspecified horizon of background knowledge. The development of a group capacity for creative
thinking includes learning to see from the perspective of specific others (for example a team-mate),
from the perspective of generalised cultural others (for example the unknown future audience of
scientists who might look at any scientific product) and ultimately from the perspective of the
outside or ‘infinite other’ (being open to original and unanticipated questions).
General dialogic mechanisms behind the development of a capacity for creative thinking that could
be seen in the data are:
• opening a space of dialogue (eg asking open questions, or calling up a brainstorm map)
• deepening the space of dialogue (eg questioning assumptions)
• expanding the space of dialogue (eg introducing new voices and perspectives such as the maps of
other groups or using the attitude cards to re-think plans)
• seeing from the perspective of a specific other (eg listening to and taking on board the comments
of a colleague)
• seeing from the perspective of a generalized cultural other (eg invoking the perspective of the
absent addressee or audience for the product or referring to the generalized other of the
community)
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• reflecting through taking an outside perspective (eg genuinely asking why something is happening
without any presuppositions, looking at it in a new and unexpected way)
A further question to ask is: is evidence of these mechanisms in the data linked to evidence of new
and useful ideas? This does not only mean problem solving ideas but more especially, problem
posing ideas.
An example in the context of statistical thinking: Interpreting results, making conclusions based on
data, generating new ideas from findings and communicating the results are essential to the
conclusions part of the statistical inquiry cycle (Wild & Pfannkuch, 1999). Context is also a crucial
aspect of statistical thinking. Wild and Pfannkuch emphasize that to make meaningful inferences
based on data, one has to make connections between the context that the problem was embedded
in and the results of the analyses.
So in a statistical inquiry task, I anticipate that students will use statistical tools (i.e., mean, median,
range) and technological tools (i.e., TinkerPlots) to make inferences based on data in a real-life
situation, but they need to communicate their conclusions to the others (to group members as well
as the whole class) as well as to connect them to the real life context. This process might initiate a
mechanism that entails “seeing as if from the perspective of others, both real others and virtual
others.” (Wegerif, 2013, p.56). Also connection to the real life context again in the cycle can be
linked to the idea of “dialogue with infinite otherness” (Wegerif, 2013, p.57) because in this
investigative cycle contextual knowledge informs statistical knowledge and vice versa. There is
always an ongoing linking and synthesis of ideas between context and statistics.
Wild and Pfannkuch’s 4-dimensional framework for the statistical thinking in empirical inquiry
includes also some personal qualities, called dispositions, that statisticians use in the statistical
problem solving: scepticism, imagination, curiosity and awareness, openness to ideas that challenge
preconceptions, a propensity to seek deeper meaning, being logical, engagement, and perseverance.
I think emphasis on being open to the new perspectives in the dialogical approach might be useful to
promote these attitudes as well.
References ...to be completed
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