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Instructional Science 32: 33–58, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands. 33
Designing Instructional Examples to Reduce Intrinsic Cognitive
Load: Molar versus Modular Presentation of Solution Procedures
PETER GERJETS1,∗, KATHARINA SCHEITER2AND RICHARD
CATRAMBONE3
1Knowledge Media Research Center, Konrad-Adenauer-Str. 40, 72072 Tuebingen, Germany;
2University of Tuebingen, Tuebingen, Germany; 3Georgia Institute of Technology, Atlanta,
GA 30332, USA (∗author for correspondence: e-mail: p.gerjets@iwm-kmrc.de)
Abstract. It is usually assumed that successful problem solving in knowledge-rich domains
depends on the availability of abstract problem-type schemas whose acquisition can be
supported by presenting students with worked examples. Conventionally designed worked
examples often focus on information that is related to the main components of problem-
type schemas, namely on information related to problem-category membership, structural
task features, and category-specific solution procedures. However, studying these examples
might be cognitively demanding because it requires learners to simultaneously hold active
a substantial amount of information in working memory. In our research, we try to reduce
intrinsic cognitive load in example-based learning by shifting the level of presenting and
explaining solution procedures from a ‘molar’ view – that focuses on problem categories
and their associated overall solution procedures – to a more ‘modular’ view where complex
solutions are broken down into smaller meaningful solution elements that can be conveyed
separately. We review findings from five of our own studies that yield evidence for the fact
that processing modular examples is associated with a lower degree of intrinsic cognitive load
and thus, improves learning.
Keywords: cognitive load, cognitive skill acquisition, example design, schema acquisition,
worked examples
Designing instructional examples from a cognitive load perspective
It has often been argued that probably the most important prerequisite for
successful problem solving consists in the availability of abstract problem-
type schemas (Gick and Holyoak 1983; Reed 1993), that is, representations
of problem categories together with category-specific solution procedures.
“Schemas are defined as mental constructs that allow patterns or configura-
tions to be recognized as belonging to a previously learned category and
which specify what moves are appropriate for that category” (Sweller and
Cooper 1985: 60). Once a problem has been identified as belonging to a
known problem category, the relevant schema is retrieved from memory,
is instantiated with the information that is specific to the to-be-solved
34 PETER GERJETS ET AL.
problem, and finally the category-specific solution procedure attached to the
schema is executed in order to produce a solution to the problem (cf. Derry
1989; VanLehn 1989). Mayer (1981: 153) reports that “several groups of
researchers have shown that students try to find out what ‘type’ of problem is
presented and then to use a solution strategy appropriate for that type”.
Schema-based problem solving is considered to be very efficient and
therefore often seen as a marking feature of experts’ problem solving
(VanLehn 1996). Accordingly, a substantial amount of research on skill
acquisition has focused on the question of how such schemas can be acquired.
A ubiquitous answer to this question is that studying concrete instances of
problem categories (i.e., examples) is necessary for schema construction.
In particular, worked examples (i.e., example problems together with step-
by-step solutions) seem to play an important role in schema acquisition
(cf. Atkinson, Derry, Renkl and Wortham 2000; Sweller, Van Merriënboer,
and Paas 1998). Studying worked examples is superior to directly teaching
abstract principles as well as to actively solving training problems – at least
with regard to initial skill acquisition. This ‘worked-example effect’ is usually
explained by pointing to the fact that studying worked examples imposes
lower levels of cognitive load on the learner than solving training problems
– mainly because no extensive search processes with regard to the correct
solution steps are involved. As a result, more cognitive resources might be
left for the learner to engage in processes of schema construction. Moreover,
it is assumed that studying worked examples (in contrast to attempting to
solve training problems) focuses the learner’s attention on information that
is relevant to schema construction. For instance, studying worked examples
might draw the learner’s attention to structural task features that define to
which problem category a particular problem belongs or it might draw the
learner’s attention to the solution rationale behind a category-specific solution
procedure.
Conventional example design: Molar presentation of solution procedures
Analyzing cognitive skill acquisition based on the notion of problem-type
schemas quite naturally has specific implications for the design of instruc-
tional examples. Sweller et al. (1998) propose that “learners’ attention must
be withdrawn from processes not relevant to learning and directed toward
processes that are relevant to learning and, in particular, toward the construc-
tion and mindful abstraction of schemas” (p. 264). For instance, if worked
examples are intended to foster schema construction their design should
focus on the information that is related to the main components of problem-
type schemas, namely on information related to problem-category member-
ship, structural task features, and category-specific solution procedures.
REDUCING INTRINSIC COGNITIVE LOAD 35
These considerations fit rather well to the way instructional examples are
conventionally designed – at least in textbooks on knowledge-rich and well-
structured domains like physics, mathematics, or programming. Examples
that can be found in these textbooks often tend to present solution proce-
dures in a molar way as ‘recipes’ that are appropriate for particular problem
categories. In order to enable learners to apply these ‘recipes’ the examples
demonstrate how to categorize problems by considering multiple (and often
abstract) structural task features. For instance, with regard to algebra word
problems Reed (1999) notes that “the solution of algebra word problems
typically begins with the categorization of the problem based on the situ-
ation described in the problems” (p. 92). Mayer (1981) extensively elaborated
the categorical structure of word problems by collecting 1100 story prob-
lems from 10 major algebra textbooks used in California secondary schools
and developing a taxonomy consisting of families, categories, and templates
that describe the category membership of problems at different levels of
abstraction. Atkinson, Catrambone and Merrill (2003) note that mathema-
tical problem solving is often characterized by ‘computationally-friendly’
molar solution approaches in which multiple solution steps are collapsed
into a single formula that represents the solution procedure. These ‘recipe-
like’ formulas allow one to easily calculate solutions by simply inserting the
correct variable values.
To sum up, math textbooks often present mathematical problem solutions
in a molar way and use ‘category-focusing’ instructional examples that are
designed to illustrate how these ‘recipes’ are applied. However, formulas
are usually restricted to solving a narrow range of problems that fall into
predefined problem categories corresponding to the solution formula. Addi-
tionally, the strong focus on problem categories might cause learners to
“memorize stereotypic solutions to problems based on their categorization”
(Reed 1999: 95). Thus, although schemas and problem categories are obvi-
ously very useful as a means for organizing knowledge representations and
for guiding problem solving, they are not unequivocally advocated by math
educators. For instance, Sowder (1985) argues that students should also be
enabled to solve problems without relying on problem categories.
To illustrate the notion of problem-type schemas in greater detail and to
elaborate on how conventional molar (or category-focusing) examples are
usually designed, we will refer to the domain of probability word problems
that is illustrated in Table 1. The left column of Table 1 contains a convention-
ally designed molar solution for an example problem related to calculating
complex event probabilities. The right column of Table 1 displays an alterna-
tive modular solution approach that we developed to improve example-based
learning and that will be explained later. Because we used problems like the
36 PETER GERJETS ET AL.
one in Table 1 for experimentation in all studies reviewed in the second part
of this paper, we will introduce the domain of calculating complex event
probabilities in this section in some detail. Problems of calculating complex
event probabilities are related to situations where the probability of selecting
a particular configuration of elements randomly out of a set of elements
has to be determined. A typical example includes calculating the probability
of winning a lottery, for instance, calculating the probability of correctly
guessing the six winning numbers out of a set of 49 numbers.
Four different problem categories are commonly distinguished in this area
of probability theory (permutations and combinations, each with and without
replacement) that differ with regard to two structural features. The first is
whether the order in which elements are selected is important; the second
is whether selected elements are replaced after being chosen. Depending
on these two structural features different problems will require different
formulas for their solution. Accordingly – like in many other mathematical
areas – the calculation of complex event probabilities can be taught by means
of category-specific solution formulas. The rationale of this approach is to
divide the number of acceptable complex events by the number of possible
complex events; category-specific solution formulas are used to calculate
the number of possible complex events. The solution procedure based on
this approach comprises four steps that are illustrated in the conventional
molar example format in the left column of Table 1, namely, (1) identify task
features, (2) apply formula, (3) insert values, and (4) calculate probability.
This solution approach is a convenient and fast way of calculating complex
event probabilities. The conventional molar example format illustrated in
Table 1 might be well suited for conveying this approach by explaining how
to categorize problems and how to apply category-specific solution formulas.
Profitable example processing and patterns of cognitive load
When providing learners with conventional worked examples like the one
presented in the left column of Table 1, instructors usually intend to trigger
the construction of an appropriate problem-type schema that will allow
learners to solve all problems that belong to the same problem category as the
example problem. Research in the domain of learning from worked examples
over the last 15 years has however demonstrated that the mere availability
of conventional instructional examples does not seem to be sufficient to
promote an adequate representation of problem categories, an understanding
of category-specific solution procedures, and problem-solving transfer.
Rather, learners are often described as having difficulties identifying
relevant information in worked examples (i.e., their structural task features)
and as being distracted by the surface features of the examples (Ross 1989).
REDUCING INTRINSIC COGNITIVE LOAD 37
Table 1. Molar and modular example formats used for experimentation
100 m-sprint example
At the Olympics 7 sprinters participate in the 100 m-sprint. What is the probability of correctly
guessing the winner of the gold, the silver, and the bronze medals?
Molar example format Modular example format
IDENTIFY TASK FEATURES FIND 1ST EVENT PROBABILITY
This problem is a permutation-without- In order to find the first event probability you
replacement problem. Problems of this type have to consider the number of acceptable
have two important features: First, the order choices and the pool of possible choices. The
of selection is important, and second, there is number of acceptable choices is 1 because only
no replacement of selected elements. We are 1 sprinter can win the gold medal. The pool of
not interested only in finding out just which 3 possible choices is 7 because 7 sprinters partici-
of the 7 sprinters win medals, rather we want pate in the 100 m-sprint. Thus, the probability
to know specifically which sprinter wins which of correctly guessing the winner of the gold
medal. Therefore, the order of selection matters. medal is 1/7.
A sprinter can win at most only one medal.
Thus, this problem is a problem without FIND 2ND EVENT PROBABILITY
replacement. That is, after a sprinter wins a In order to find the second event probability you
medal he is not eligible for being selected again. again have to consider the number of acceptable
choices. The number of acceptable choices is still
APPLYFORMULA 1 because only 1 sprinter can win the silver medal.
For this type of problem the following formula The pool of possible choices is reduced to 6
should be applied: A=n!/(n−k)! with nbeing because only the remaining 6 sprinters participat-
the number of all sprinters and kbeing the ing in the 100 m-sprint are eligible to receive
number of sprinters that have to be correctly the silver medal. Thus, the probability of correctly
guessed. guessing the winner of the silver medal is 1/6.
INSERT VALUES FIND 3RD EVENT PROBABILITY
In the given example there are 7 sprinters to In order to find the third event probability
choose from. This is the set of elements for you again have to consider the number of
selection (n = 7). As we want to find out the acceptable choices. The number of acceptable
probability of correctly guessing the winner of choices is still 1 because only 1 sprinter
the gold, the silver, and the bronze medals, can win the bronze medal. The pool of possible
3 sprinters out of these 7 sprinters have to choices is reduced to 5 because only the
be selected. Therefore, the number of selected remaining 5 sprinters participating in the
sprinters equals k = 3. Inserting these values into 100 m-sprint are eligible to receive the bronze
the formula for permutation without replacement medal. Thus, the probability of correctly
yields 7!/(7−3)! = 210 possible permutations. guessing the winner of the bronze medal is 1/5.
CALCULATE PROBABILITY CALCULATE THE OVERALL PROBABILITY
In order to calculate the probability of correctly The overall probability is calculated by multi-
guessing the winner of each of the three medals, plying all individual event probabilities. Thus,
divide 1 (the particular permutation we are inter- the overall probability of correctly guessing the
ested in) by the number of possible permutations. winner of each of the three medals is
Thus, the probability of getting this permutation 1/7∗1/6∗1/5 = 1/210.
(the winner of each of the three medals) equals
1/210.
Note: In experimental conditions with instructional explanations the example solutions
contained all information stated in the relevant table column. Conditions without instructional
explanations contained only the information printed in bold.
38 PETER GERJETS ET AL.
Furthermore, Renkl (1999) suggests that students often suffer from illusions
of understanding when learning from worked examples. That is, they might
have the false impression of having grasped the solution rationale of an
example problem. Similarly, Catrambone (1998: 355) notes that learners
“tend to form solution procedures that consist of a long series of steps –
which are frequently tied to incidental features of the problems – rather
than more meaningful representations that would enable them to success-
fully tackle new problems”. That is, learners have difficulties generalizing
solutions from examples to novel problems (Catrambone and Holyoak 1989;
Reed, Dempster and Ettinger 1985).
To sum up, numerous findings indicate that students experience serious
difficulties in example-based learning resulting in the acquisition of rather
shallow representations of problem categories and solution procedures.
Research on the role of example elaborations in learning has demonstrated
that learners need to draw inferences concerning the structure of example
solutions, the rationale behind solution procedures, and the goals that are
accomplished by individual solution steps (e.g. by relating example-specific
information to more abstract information; Chi, Bassok, Lewis, Reimann
and Glaser 1989; Pirolli and Recker 1994; Renkl 1997) in order to over-
come these difficulties. Example elaborations may in particular help to foster
learners’ skills in solving novel problems that do not fall into known problem
categories and that require an adaptation of procedures illustrated by worked
examples.
Beyond example elaborations, learners have to engage in profitable
processes of example comparison in order to notice structural features that
differ among problem categories and that are shared by all problems within a
category. If learners compare examples within and among problem categories
with regard to their differences and commonalities they might be more likely
to identify the relevant features of worked examples and to avoid confusion
due to the surface features of the examples (Cummins 1992; Quilici and
Mayer 1996).
Unfortunately, it has often been observed that learners do not spontane-
ously engage in these profitable processes of example elaboration and
example comparison when studying worked examples (e.g., Chi et al. 1989;
Gerjets and Scheiter 2003; Gerjets, Scheiter and Tack 2000; Schuh, Gerjets
and Scheiter 2003). Rather, learners seem to need additional instructional
support and carefully designed learning materials in order to make the
most of instructional worked examples. To address the issue of develop-
ing improved instructional settings that ensure profitable processing of
worked examples, we refer to the instructional-design framework provided
by cognitive load theory (Sweller et al. 1998). According to this theory,
REDUCING INTRINSIC COGNITIVE LOAD 39
constructing a problem-type schema might impose cognitive load on learners,
that is, it demands working-memory resources because it requires learners to
simultaneously process all information units that are to be integrated into that
schema. These working-memory demands are particularly high for learners
with low domain-specific prior knowledge. These learners lack complex
knowledge structures that would otherwise help to increase the amount of
information that can be held in working memory simultaneously by chunking
individual knowledge elements into a single element. Within cognitive load
theory, three types of cognitive load are distinguished:
•Intrinsic cognitive load: The number of elements that are to be inte-
grated into a to-be-learned schema and therefore have to be processed
in working memory simultaneously is referred to as intrinsic cognitive
load. Intrinsic cognitive load depends on the relational complexity of the
to-be-learned content (so-called element interactivity) and the learner’s
degree of prior knowledge (i.e., schema availability). It is usually
assumed in cognitive load theory that intrinsic cognitive load cannot be
altered by instructional design.
Beyond intrinsic cognitive load there might be additional cognitive load
due to the instructional presentation of the material and the activities learners
are engaged in. This load can be influenced by instructional design and can
be categorized according to whether it is beneficial for schema construction
(i.e., germane cognitive load) or not (i.e., extraneous cognitive load).
•Germane cognitive load: When intrinsic task demands (resulting in
intrinsic cognitive load) leave sufficient cognitive resources available,
learners might “invest extra effort in processes that are directly relevant
to learning, such as schema construction. These processes also increase
cognitive load, but it is germane cognitive load that contributes to,
rather than interferes with, learning” (Sweller et al. 1998: 264). Germane
cognitive load is imposed by adding higher-level cognitive processes to
the mere simultaneous activation of elements in working memory; these
processes integrate the elements into a schema. In the case of learning
from worked examples, germane cognitive load might be imposed on the
learner by cognitively demanding activities like example comparisons
and example elaborations.
•Extraneous cognitive load: Extraneous cognitive load is the result of
implementing “instructional techniques that require students to engage
in activities that are not directed at schema acquisition” (Sweller 1994:
299). For instance, these activities might comprise processes of finding,
relating, or integrating particular pieces of information within instruc-
tional materials (whereas with a redesign such processing might not
be required to the same degree). Extraneous cognitive load might
40 PETER GERJETS ET AL.
impede learning, as it requires cognitive resources that can no longer be
devoted to mindful cognitive processes that are associated with germane
cognitive load. Furthermore, cognitive resources required by extraneous
cognitive load might result in an overall cognitive load that exceeds the
limits of working-memory capacity.
From a cognitive load perspective, an important objective of instruc-
tional design in example-based learning is to reduce extraneous cognitive
load and to encourage learners to invest unused resources in higher-level
cognitive processes such as example comparisons and example elaborations
that are associated with germane cognitive load. Accordingly, several sugges-
tions have been made on how to design instructional materials in order to
foster a profitable utilization of worked examples and thus to improve the
resulting pattern of cognitive load during learning. With regard to example
comparisons it could be shown that the provision of multiple examples can
support schema induction (Cummins 1992). Additionally, providing multiple
examples with different surface features might further improve example
comparisons (Quilici and Mayer 1996). With regard to example elaborations
it has been shown that training can be used to encourage learners to engage
in self-explanations (Chi, de Leeuw, Chiu and LaVancher 1994). Addition-
ally, it has been demonstrated that grouping solution steps according to their
subgoals (Catrambone 1998) is effective because it provides affordances for
learners to self-explain the meaning of individual solution steps (Chi et al.
1989). Finally, completion problems (Van Merriënboer 1990) and fading
procedures (Renkl and Atkinson 2003) have been introduced to prevent
learners from being overly passive in studying instructional examples.
Nearly all instructional interventions conceived to improve learning
from (conventionally designed) worked examples are intended to either
increase germane cognitive load (i.e., profitable example processing) and/or
to decrease extraneous cognitive load (i.e., activities not directed at schema
acquisition). In our own research we try to go beyond this general rationale
and to find a way to reduce intrinsic cognitive load in example-based
learning by abandoning the molar structure of conventionally designed
worked examples.
Reducing intrinsic cognitive load in example-based learning
As mentioned before, cognitive load theory usually assumes that intrinsic
cognitive load cannot be manipulated by instructional design because it
depends directly on the number of elements that are to be integrated into a
to-be-learned schema and therefore have to be processed in working memory
simultaneously. According to the theory, this element interactivity depends
only on the relational complexity of the to-be-learned content and on the
REDUCING INTRINSIC COGNITIVE LOAD 41
learner’s degree of prior knowledge (i.e., on schemas already available).
However, the assumption that intrinsic cognitive load cannot be manipulated
is not uncontroversial (cf. Van Merriënboer, Kirschner and Kester 2003).
Different instructional approaches have been proposed that aim at reducing
intrinsic cognitive load associated with learning materials.
•Part-whole sequencing: When the content of learning pertains to solving
complex tasks, it is a well-known instructional approach to break down
the complex task into simpler subtasks that can be conveyed separately.
When learners have mastered the subtasks they may be instructed on
how to solve the total complex task (cf. Gagné 1962). This part-whole
sequencing strategy is suitable for reducing intrinsic cognitive load.
First the load associated with acquiring the component tasks is lower
than the one that would be imposed by starting with acquiring the total
complex task right from the beginning. Second, when learners are finally
instructed on how to solve the total complex task later in the instructional
sequence they will already possess a certain amount of domain-specific
prior knowledge (in terms of schema availability) from their exposure
to the component tasks. This prior knowledge will reduce the intrinsic
cognitive load imposed by the need to finally acquire the skill of accom-
plishing the total complex task. However, it has to be noted that the
part-whole sequencing strategy reduces the amount of intrinsic cognitive
load during learning by changing the learning task in the early phases
of the instructional sequence. So it might on the one hand be argued that
this strategy does not predominantly change the cognitive load associ-
ated with a task but that it simply changes the task itself. On the other
hand, the total complex task that is acquired later in the instructional
sequence is the same task that would have been used without the part-
whole sequencing strategy. From the latter perspective this instructional
strategy does not change the learning task but instead improves learners’
knowledge and skills before being confronted with the learning task
and thereby reduces intrinsic cognitive load during learning due to an
increased availability of related schemas.
•Simplified whole tasks: Van Merriënboer et al. (2003) argue that the
fragmented approach of part-whole sequencing might make it difficult
to integrate and coordinate subtasks into total complex tasks. There-
fore, they advocate a different approach to lowering intrinsic cognitive
load during learning, namely, to start learning with a simplified whole
task with lower element interactivity. That is, the number of subtasks
out of which the to-be-taught complex task consists is kept constant
during training. However, each of these subtasks is first taught in a
simplified version and then the difficulty of the subtasks and thereby
42 PETER GERJETS ET AL.
of the whole task is gradually increased. As with part-whole sequenc-
ing, this approach might be considered as changing the learning task at
the beginning of the instructional sequence by confronting the learners
with simpler problems initially. It is only at the end of the instructional
sequence – when learners already possess sufficient prior knowledge –
that they are presented with the task in its full complexity. Again, the
increased prior knowledge at that point in time results in a lower level of
intrinsic cognitive load when learning how to accomplish the complex
task.
•Modular presentation of solution procedures: Inspired by the afore-
mentioned approaches to reduce the intrinsic cognitive load associated
with learning tasks, we tried to design instructional examples that allow
learners to start with a total complex task right from the beginning
but nevertheless reduce intrinsic cognitive load during learning. Our
approach was to develop a more modular solution procedure in examples
that required a learner to keep only a limited number of elements
active simultaneously in working memory. The basic idea behind this
modular example format is to present solution procedures in a way
that completely avoids references to conventional molar concepts like
problem categories, clusters of structural task features, and category-
specific solution procedures. It can be argued that these molar concepts
– that refer to complex entities – usually impose high levels of cognitive
load onto the learner. In contrast, modular examples focus on smaller
meaningful solution elements and their relation to individual struc-
tural task features. This approach substantially decreases the number
of elements that have to be considered at the same time. The modular
examples that we constructed differ very much from a part-whole
sequencing strategy because we convey right from the beginning how
to solve total complex tasks rather than teaching how to solve smaller
subtasks in isolation first. Furthermore, our approach is also different
from using simplified whole tasks, because we do not alter the diffi-
culty of learning tasks in the course of learning in order to reduce
intrinsic cognitive load. Instead, we use exactly the same example prob-
lems for designing modular examples as we used for designing molar
examples. The next section illustrates in greater detail how examples
with a modular presentation of solution procedures might differ from
examples with a conventional molar presentation of solutions.
Designing modular worked examples
In the construction of modular worked examples we started with a careful
analysis of the pattern of cognitive load imposed by conventional molar
REDUCING INTRINSIC COGNITIVE LOAD 43
examples. A main feature of conventional examples is that they are designed
to explain how to categorize problems according to multiple structural task
features and how to apply category-specific solution formulas. Thus, molar
examples usually demonstrate a convenient and fast way of solving prob-
lems that is quite similar to the approach a domain expert would choose
for problem solving. However, a strong focus on problem categories might
be rather overwhelming for novice learners because they might not possess
sufficient prior knowledge (in terms of schema availability) to cope with the
cognitive load imposed by this approach. A category-based approach requires
learners to keep in mind all category-defining structural features of a problem
before they are able to accurately decide on its problem category and the
appropriate formula needed for its solution. Accordingly, studying molar
examples requires that learners consider multiple structural task features at
the same time in order to understand the problem’s category membership.
As problems are grouped together in a problem category because they
share a common solution procedure at some level of abstraction, the solution
procedure associated with a problem category (i.e., applying an appropriate
solution formula) can usually be characterized as a molar entity. For instance,
in mathematical problem categories, multiple solution steps are frequently
collapsed into a single complex formula that represents the solution and can
be used as a recipe that allows one to calculate solutions in a fast and compu-
tationally convenient way. However, one has to consider that during schema
acquisition all information units that are to be integrated into that schema
have to be simultaneously activated in working memory (Sweller et al. 1998).
Therefore, molar examples will result in a high level of intrinsic cognitive
load – in particular for novice learners – depending on the number of struc-
tural task features that have to be kept in mind concurrently and depending
on the complexity of the solution formula needed. As already mentioned
before, a substantial amount of intrinsic cognitive load might prevent learners
from engaging in profitable processes of elaborating or comparing examples
that are necessary to overcome shallow representations of problems and their
solutions.
The modular example format that we developed in response to this
analysis was intended to impose less intrinsic cognitive load on learners by
avoiding the need for learners to consider multiple structural task features
or multiple solution steps simultaneously. The rationale in the construc-
tion of this example format was to isolate task features and meaningful
solution elements that can be conveyed and understood separately, thereby
reducing intrinsic cognitive load. Besides reducing cognitive load, a second
advantage of this approach is that it might allow learners to understand
relations below the category level, that is, relations holding irrespective of
44 PETER GERJETS ET AL.
category membership such as relations between individual structural task
features and individual solution steps. As a result, learners might acquire
meaningful knowledge on modular solution elements that enables them to
directly translate individual structural task features into characteristics of the
problem solution. This knowledge might be much more helpful than conven-
tional knowledge on problem categories and solution recipes for adapting
solution procedures to novel problems beyond the known problem categories
(cf. Catrambone 1998).
When we considered how to ‘rethink’ the domain of calculating complex
event probabilities in order to develop instructional examples that do not refer
to problem categories, their defining structural features, and category-specific
solution formulas, we relied on the fact that problems in probability theory
can be solved by breaking down complex events into sets of individual events.
Accordingly, the calculation of a complex event probability by means of a
formula can be decomposed into a sequence of simpler calculations, that
is, calculations of individual event probabilities. In line with our approach,
the calculation of individual event probabilities allows one to directly relate
individual structural task features and individual characteristics of solution
steps. This is not true for the molar approach of using a solution formula! The
solution procedure based on the modular approach we developed comprises
four steps that are illustrated in the worked example in the right column of
Table 1 (the left column displays the conventional molar solution approach for
solving the same task). In this example the probability of a complex event is
calculated by determining the probabilities of all individual events that make
up the complex event (steps 1 to 3) and then multiplying these individual
event probabilities to calculate the overall probability (step 4).
When calculating a particular individual event probability one has to take
into account how the number of possible and acceptable choices changes
from the preceding to the current trial. These changes depend on whether
previously selected objects are replaced or not after having been selected
(problem with or without replacement), and on whether there is more than
one acceptable choice in a given trial (order of selection important or not).
For problems without replacement the number of possible choices decreases
from trial to trial (otherwise it remains the same). For problems in which the
order of selection is important there is only one acceptable choice on each
trial (otherwise there might be more than one acceptable choice).
The fact that the two structural features used to categorize problems in this
domain correspond to particular characteristics of individual solution steps
makes it easier to adapt the modular approach to novel problems. The solution
procedure illustrated by the modular example format does not require one to
categorize problems before solving them. Rather, decisions with regard to
REDUCING INTRINSIC COGNITIVE LOAD 45
individual structural task features can be directly translated into modifica-
tions of individual solution steps (i.e., changes in the number of possible
and acceptable choices from trial to trial). The reasoning exemplified in the
modular example format thus should help learners to understand relations
below the category level that hold irrespective of category membership. What
is even more important – in contrast to the category-based approach – is that
this reasoning can be understood by holding only a rather limited amount
of information in working memory simultaneously. Thus, this format should
impose less intrinsic cognitive load than a molar example format and accord-
ingly free cognitive resources that can then be used by learners to engage in
profitable processes of elaborating and comparing example problems.
To sum up, compared to traditional molar examples, the alternative
example format we constructed is modular because solution procedures are
broken down into smaller meaningful solution elements that can be under-
stood in isolation without holding large amounts of information active in
working memory. These elements can be separately transferred when solving
novel problems. In this respect, modular examples conform to the subgoal
learning model that proposes to group sets of solutions steps according to
the subgoals they aim to achieve in the solution procedure (Catrambone
1998).
In the second part of this paper we will review the available evidence that
an example format characterized by a modular presentation of solution proce-
dures can reduce cognitive load, improve learning from worked examples,
and foster transfer to novel problems.
Review of experimental evidence
In order to evaluate the evidence for the claimed superiority of a modular
example format we can refer to five studies that are reported in Catrambone
(1994), Gerjets, Scheiter, and Kleinbeck (in press), and Gerjets, Scheiter, and
Catrambone (in press). An overview of these studies will be given in the
following sections.
Study 1 (Catrambone 1994)
In Experiment 1 of Catrambone (1994), 66 learners studied a pair of worked
examples in which both examples belonged to the same single problem
category (permutation without replacement). Each worked example demon-
strated how to calculate a complex event probability that was related to
humans picking a particular configuration of objects by chance. Learners
were asked to study carefully the booklet with the two examples at their
46 PETER GERJETS ET AL.
own pace and they were told that they would be asked later to solve some
problems without looking at the examples. After studying, learners wrote
a description of how to solve problems in the domain. Finally, they solved
four test problems that either belonged to the same problem category as the
learning examples (isomorphic problems: permutations without replacement)
or to a different problem category (novel problems: combinations without
replacement). The experimental manipulation was whether the two worked
examples for learning were designed to be molar or modular. The molar
examples explained the appropriate solution formula for solving the problems
and demonstrated how to insert the correct variable values into the formula.
The modular examples explained how to consider the complex event in ques-
tion as a sequence of individual events and how to integrate individual event
probabilities into an overall complex event probability. The modular-example
group outperformed the molar-example group on transfer performance. These
results provided initial evidence that modular examples might help to foster
learning from worked examples.
Study 2 (Catrambone 1994)
In Experiment 2 of Catrambone (1994) the findings of Study 1 where repli-
cated under slightly different conditions. To further improve learning the
participants (N = 78) received three worked examples of a single problem
category (permutation without replacement) in the learning phase. Partici-
pants were not required to describe the solution procedure before solving the
test problems as they were in Study 1. Finally, the wording of the test prob-
lems was slightly modified. The results of Study 2 were similar to those from
Study 1 in that learners who received modular examples clearly outperformed
the molar group in transfer performance.
However, both of the initial studies of Catrambone (1994) demonstrating
the superiority of a modular example format were characterized by two
limitations with regard to the claims elaborated in the current paper. First,
no measurement of learning time and no measurement of cognitive load
was administered; thus we do not know whether the molar and modular
example formats differ in their processing demands as can be expected from
the theoretical considerations outlined in this paper. Second, the experiments
did not address the ecologically more natural situation of learners studying
multiple problem categories in a domain. The latter concern may easily result
in an artificial bias in favor of the modular example format: When students
are confronted with multiple problem categories during learning and problem
solving they might strongly profit from being able to distinguish among these
categories with regard to their structural task features, an ability that might be
better supported by molar examples. Thus, the initial results of Catrambone
REDUCING INTRINSIC COGNITIVE LOAD 47
(1994) might be valid only in the rather restricted situation in which learning
only one problem category is required.
Study 3 (Gerjets, Scheiter and Kleinbeck, in press)
The first experiment reported in Gerjets, Scheiter, and Kleinbeck (in press)
changed the design of Study 1 and 2 used in Catrambone (1994) in that
learners had to acquire multiple problem categories by using a nonlinear
hypertext learning environment that allowed the experimenters to measure the
time for learning and later problem solving by means of logfiles. These time
parameters were used to measure the processing demands of the two different
example formats. In this experiment 52 learners could study two example
problems for each of six different problem categories related to calculating
complex event probabilities. These examples contained instructional explana-
tions as illustrated in Table 1. The examples could be retrieved by means
of hyperlinks in the learning environment used for self-paced study. In the
introductory instructions of the experiments, participants were informed that
they would have to solve six probability test problems on their own after
having studied the worked examples. To avoid memory artifacts – in partic-
ular in the molar group that had to remember six different solution formulas
– a re-examination of the instructional examples during the test phase was
possible. In the test phase every participant had to solve three isomorphic and
three novel test problems. Isomorphic test problems differed from the instruc-
tional examples only with regard to their surface features, whereas novel test
problems were constructed in a way that two complex event probabilities
had to be found and then multiplied in order to calculate the required
probability.
Logfiles were used to measure example-study time, time for retrieving
instructional examples during problem solving (re-examination time), and
time for solving the test problems. As a performance measure the percentage
of correctly solved test problems was registered. A declarative pretest was
used to distinguish between high and low prior-knowledge learners within
the two groups learning with molar or modular examples. Prior knowledge
was used as an additional independent variable because it is strongly related
to intrinsic cognitive load according to the cognitive load theory (cf. Sweller
et al. 1998).
The results of Study 3 showed that learning with modular examples led to
better problem-solving performance for isomorphic as well as novel problems
irrespective of learners’ level of prior knowledge (cf. Figure 1). Additionally,
learning with modular examples required significantly less example-study
time as well as re-examination time during problem solving. No differences
with regard to problem-solving time were obtained.
48 PETER GERJETS ET AL.
Figure 1. Problem-solving performance (% correct) for isomorphic and novel problems,
example-study time, re-examination time, and problem-solving time (in seconds) for isomor-
phic and novel problems as a function of example format and prior knowledge (Study 3).
In sum, Study 3 demonstrated that a modular example format is not only
superior when a single problem category has to be acquired but also in case
of multiple problem categories. This finding rules out the concern from Study
1 and 2 (Catrambone 1994) that the superiority of a modular example format
might be an artifact due to the restricted learning situation used. It seems
that students’ ability to distinguish among different problem categories with
regard to their structural task features will not be better supported by molar
examples than by modular examples – although molar examples are explicitly
designed to convey knowledge on problem categories and their structural task
features. To explore this issue in greater detail we conducted a replication
REDUCING INTRINSIC COGNITIVE LOAD 49
of Study 3 that used slightly different instructions and that contained two
additional dependent measures.
Study 4 (Gerjets, Scheiter and Kleinbeck, in press)
The second experiment reported in Gerjets, Scheiter, and Kleinbeck (in
press) used exactly the same procedure as in Study 3 (including the same
set of worked examples and test problems). However, a different task was
announced to learners at the beginning of the experiment. Instead of telling
participants that they would have to solve test problems later on, they were
instructed to study the instructional examples with the goal of acquiring
structural features of problem categories. Participants were informed that
theywouldhavetoworkonaclassification task and on a comparison task
after having studied the instructional materials. They were made aware of
the importance of knowing about structural features of different problem
categories by familiarizing them with these two tasks they later would have
to accomplish in the test phase.
Classification task. To accomplish this task, participants had to identify
one out of six word problems that was most similar to a given test problem
with regard to its structural features. Four of the instructional examples that
were already known from the learning phase were used as test problems.
For each of the four test problems, six word problems were presented as
multiple-choice items that differed with regard to the problem category they
belonged to. For each test problem participants had to identify the structurally
most similar word problem, that is, the word problem that belonged to the
same problem category as the test problem.
Comparison task. In this task, participants had to compare pairs of word
problems with regard to their structural similarities and differences. Six prob-
lems from different problem categories were used as test problems. These
problems were already known from the learning phase. From the 15 possible
problem pairs that could have been made up from these six problems, we
selected six problem pairs that were presented to participants for the compar-
ison task. For each pair participants had to fill in a form that asked for
structural similarities and differences of the two word problems.
Following the learning phase participants had to accomplish three
different tasks. First they had to solve the same test problems as in Study
3 without having been informed about this task in advance. Subsequently,
they had to work on the two tasks that had been announced and explained to
them at the beginning of the experiment. The same time data as in Study 3
were obtained by means of logfiles.
50 PETER GERJETS ET AL.
Figure 2. Problem-solving performance (% correct) for isomorphic and novel problems,
example-study time, re-examination time, and problem-solving time (in seconds) for isomor-
phic and novel problems as a function of example format and prior knowledge (Study 4).
The results of Study 4 were similar to those of Study 3 and again supported
the claim that learning with modular examples led to better problem-
solving performance for isomorphic as well as novel problems irrespective
of learners’ level of prior knowledge (cf. Figure 2). Again, learning with
modular examples required significantly less example-study time as well as
less re-examination time during problem solving. No differences with regard
to problem-solving time were obtained.
With regard to the comparison task and the classification task that were
used to investigate the influence of example formats and prior knowledge
on the acquisition of structural features of problem categories, we found
that performance in the classification task was affected only by participants’
REDUCING INTRINSIC COGNITIVE LOAD 51
prior knowledge. Both example formats were equally effective in conveying
knowledge on structural features of problem categories necessary to solve
the classification task. The same pattern of results was obtained for the
comparison task.
These findings can be seen as indicating that learners might experi-
ence substantial difficulties in extracting and understanding information on
abstract structural task features and problem categories from molar worked
examples although these examples are explicitly designed to convey that
information. In line with our initial cognitive load analysis, learners studying
molar worked examples might have few cognitive resources left to engage
in profitable processes of example elaboration and example comparison
which are, however, often shown to be necessary to construct more abstract
knowledge from specific example problems.
An explanation might be that the understanding of molar example solu-
tions requires learners to hold a substantial amount of information simultane-
ously in working memory. In contrast, understanding modular example
solutions might be less demanding for students so that they are able to develop
an understanding of structural problem features and to infer the categorical
structure of the domain by themselves without being explicitly provided
with this information. Accordingly, they accomplished the same levels of
performance in the classification task and the comparison task that were
used to measure this particular aspect of students’ knowledge. In our view,
this can be seen as evidence that students learning from modular worked
examples may possess unused cognitive resources that might be invested in
(germane) processes that are directly relevant to learning and understanding
(e.g., self-explanations).
Study 5 (Gerjets, Scheiter and Catrambone, in press)
To provide more direct evidence for the line of reasoning outlined above,
we conducted another experiment (described as Experiment 2 in Gerjets,
Scheiter and Catrambone, in press) that involved a cognitive load measure-
ment in order to test whether the cognitive load is indeed lower for studying
modular example solutions than for studying molar example solutions.
Beyond including measures of cognitive load we manipulated the availability
of instructional explanations in Study 5. Two competing hypotheses can
be formulated with regard to the impact of instructional explanations: On
the one hand, learners using the modular example format might have suffi-
cient cognitive resources available to engage in self-explanations, whereas
learners using the molar example formats might suffer from cognitive over-
load when trying to understand molar solution procedures. Accordingly,
students learning with modular examples might not need highly elaborated
52 PETER GERJETS ET AL.
examples compared to students learning with molar examples. On the other
hand, the substantial amount of instructional explanations provided in Studies
3 and 4 (cf. Table 1) might have imposed cognitive load on learners that is
especially harmful for those studying molar worked examples. According to
this line of reasoning, learners with molar examples might benefit from a
more condensed and therefore less overwhelming presentation of solution
procedures, whereas learning from modular examples might be less vulner-
able to cognitive overload due to the provision of instructional explanations.
This ambiguity with regard to the expected effectiveness of instructional
explanations is consistent with the existing literature on this instructional
manipulation (cf. Renkl 2002).
In this experiment 68 students from the Georgia Institute of Technology
learned with a shortened and linearized version of the learning environment
used in Studies 3 and 4. Learners were provided with two worked examples
for each of four problem categories taught. After studying the examples parti-
cipants solved three isomorphic and six transfer problems. Novel problems
were constructed similarly to the ones used in Studies 3 and 4 (Gerjets,
Scheiter and Kleinbeck, in press). In contrast to these experiments, however,
participants were given no opportunity to re-examine instructional examples
in the test phase. Instead, solution formulas were provided during problem
solving for those participants learning with molar examples. The formula list
was provided during problem solving in order to allow direct access to the
solution formulas and to rule out the possibility that a potential inferiority of
the molar example format may be traced back to participants merely forget-
ting formula details (or having difficulties in finding the appropriate formula
when re-examining instructional examples).
As in the previous studies, the worked examples were either presented
in the molar or the modular example format. Additionally, we varied the
degree of instructional explanations between subjects. Half of the participants
learned from the highly elaborated examples used in Studies 3 and 4 while the
other half studied a rather condensed version of the examples. Whereas the
highly elaborated examples provided detailed justifications for solution steps,
the condensed examples focused on the mathematical structure of example
solutions without providing instructional explanations (cf. Table 1).
As dependent variables, problem-solving performance for isomorphic and
novel problems as well as example-study times and problem-solving times
were recorded. In order to test the assumption that using modular examples
would lead to a reduction of intrinsic cognitive load during learning, we addi-
tionally assessed different aspects of cognitive load after the learning phase
by administering a modified version of the NASA-TLX questionnaire (Hart
and Staveland 1988). We preferred the NASA-TLX to the usual cognitive
REDUCING INTRINSIC COGNITIVE LOAD 53
Figure 3. Cognitive load (scale values) as a function of example format and degree of instruc-
tional elaborations (Study 5).
load questionnaire introduced by Paas and Van Merriënboer (1994) because it
allows for a more detailed analysis. Each of the three cognitive load items was
rated on a scale ranging from 0 (low cognitive load) to 100 (high cognitive
load). The following subscales were used: ‘Effort’ (How hard did you have
to work in your attempt to understand the contents of the learning environ-
ment?), ‘Stress’ (How insecure, discouraged, irritated, stressed, and annoyed
did you feel during the learning task?), and ‘Task demands’(Howmuch
mental and physical activity was required, e.g., thinking, deciding, calcu-
lating, remembering, looking, searching etc.? Was the learning task easy or
demanding, simple or complex, exacting or forgiving?).
Analyzing the cognitive load data revealed that the example formats influ-
enced learners’ subjective experience of cognitive load whereas the degree
of instructional explanations had no reliable impact (Figure 3). First, with
regard to the effort participants believed they had to invest in the task, they
indicated that they had to work less hard in order to understand the instruc-
tional contents when learning with modular examples. Second, participants
experienced far less stress during learning with modular examples. However,
there was no effect of example format on the task demands associated with the
learning task, but there was a significant interaction between example format
and instructional explanations. The interaction indicated that participants
judged the learning task as being less demanding with modular examples
than with molar examples – but only when instructional explanations were
provided – whereas there was no difference between the two example formats
when no explanations were given.
As expected, participants who had learned with modular examples clearly
outperformed participants learning with molar examples with regard to
54 PETER GERJETS ET AL.
Figure 4. Problem-solving performance (% correct) for isomorphic and novel problems,
example-study time, and problem-solving time (in seconds) as a function of example format
and degree of instructional elaborations (Study 5).
problem-solving performance for isomorphic as well as for novel problems
(Figure 4). There was, however, no effect of instructional explanations nor
did any of the factors interact. Finally, analyzing the time data revealed
that not only were participants learning with a modular example format
more successful with regard to problem-solving performance, but they also
needed less time studying the examples than participants learning with molar
examples. Rather naturally, the examples that included explanations took
longer to process. There was no interaction between the two factors. As in the
previous studies there were no effects with regard to problem-solving time.
REDUCING INTRINSIC COGNITIVE LOAD 55
Discussion
Much of the existing research on learning from worked examples has been
based on the notion of problem-type schemas as a central prerequisite for
proficient problem solving. In this paper, we have argued that this conven-
tional approach of designing examples in a way that focuses on teaching
schemas might result in molar examples that impose high levels of cognitive
load on learners. The cause for this problem might be that all the informa-
tion that has to be integrated into the schema (e.g., multiple task features,
formulas) has to be kept active in mind simultaneously. This high degree
of cognitive load associated with molar, schema-based examples might
prevent learners from implementing higher-level processes like elaborations
and comparisons. Instructional designers have addressed this problem by
inventing a variety of techniques to either foster profitable processing of
molar examples (and thereby increasing germane cognitive load) and/or to
reduce unnecessary demands imposed on the learner (and thereby decreasing
extraneous cognitive load).
In our work we took another approach by trying to design examples in
a way that reduces the task-related, intrinsic load. Similar approaches of
reducing intrinsic load have already yielded promising results recently (Van
Merriënboer et al. 2003). The basic idea of our modular approach was to
break down solution procedures into smaller, more meaningful, pieces that
can be conveyed and understood separately. In five studies we provided
evidence that indeed these modular examples are superior to molar examples
with regard to problem-solving performance for isomorphic and novel prob-
lems, different measures of learning time, and cognitive load. The positive
effects of modular examples were found regardless of the number of problem
categories taught, the learning task announced, and the amount of instruc-
tional explanations given. Furthermore, modular examples proved to be
superior for learners with low as well as with high prior knowledge. There-
fore, the advantages of modular examples for teaching problem-solving skills
seem to be very stable over a variety of instructional conditions.
One possible critique that we are aware of is that the design of modular
versus molar examples might be restricted to this specific area of probability
theory, that is, calculating complex event probabilities. We do not believe that
this is the case; rather we are convinced that this approach might be extended
to other well-structured domains where problems fall into categories. In
fact, Catrambone (1994) has already successfully attempted to extend this
approach to the domain of algebra word problems – a line of research that we
would like to follow in our future work.
Another possible critique is that modular examples might be helpful only
for calculating complex event probabilities when rather small numbers are
56 PETER GERJETS ET AL.
involved, whereas the strength of using formulas comes into play for prob-
lems dealing with larger numbers. However, we are convinced that in order
to be able to apply a formula to novel cases, a learner must have already
achieved an understanding of the domain. Therefore, we recommend using
modular examples for initial skill acquisition in order to foster this kind
of understanding and to switch to a molar approach in later stages of skill
development. Learners would have at that point sufficient prior knowledge
available to cope with the complexity of the formulas, that is, for them even
a molar approach might be characterized by a low level of element inter-
activity. That is, we do not propose that the solution procedures conveyed by
means of modular and molar examples are mutually exclusive; rather it is the
instructional designer’s important and difficult task to decide when to quit a
modular approach in favor of using formulas and to rely on learners’ ability
to categorize problems according to their structural features. This decision
should be based on empirical evidence with regard to finding the most suit-
able transition point between these two approaches. However, this empirical
evidence is not yet available and has to be obtained by future research.
Acknowledgements
The work reported in this paper was supported by the German Research
Foundation DFG (Collaborative Research Center 378: Resource-adaptive
Cognitive Processes) and by the Alexander von Humboldt-Foundation
(TransCoop-Program). We thank Julia Schuh for helpful comments on an
earlier version of this paper.
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