CHAPTER THREE Classroom Complexity and Flow
Introduced briefly near the end of Chapter 1, classroom complexity was hypothesized
to be a contributing factor to intrinsic enjoyment of mathematics. In the pages that
follow, the relationship between complexity and flow is examined, culminating in a
model of optimal classroom complexity used to investigate the effects of mathematical
tasks on students’ motivation and achievement. Classroom complexity refers to the
structures and interactions inherent in the learning environment through which students
realize greater personal integration and differentiation.
A fundamental premise of flow--and why people find it rewarding--is its connection
to psychological complexity. Not only does flow result in greater integration of
differentiated ideas, traits and skills, the process by which this happens is found to be
innately pleasurable (Csikszentmihalyi, 1993; Csikszentmihalyi & Larson, 1984). An
association between task complexity and intrinsic enjoyment is found repeatedly in
studies of flow. Teenagers talented in math (Csikszentmihalyi, et al., 1993), artists who
become completely immersed in their painting (Getzels & Csikszentmihalyi, 1976), many
of the most creative people of this century (Csikszentmihalyi, 1996), whether they know
it or not, enjoy what they do partly because they keep aiming for higher levels of
complexity. Tasks that facilitate flow share some important features. Clear goals,
immediate feedback and a balance between a person’s skills and the challenges of a task
were three mentioned in the last chapter. In the context of classroom activities, these can
be elaborated upon in terms of cognitive complexity, novelty, locus of control,
In addition to a person’s skills, the gradient of challenges inherent in an activity helps
to determine flow (Csikszentmihalyi, 1988b). For example, a person who has begun to
master chess will find little challenge in tic-tac-toe, a much simpler activity.
Like games, learning activities may be relatively simple or complex. One of the ways
they may place different demands on learners is in terms of requisite cognitive skills. If
an activity is not challenging enough, one way it can be made more challenging is by
increasing the difficulty of the material.
Another way would be to make the task more cognitively challenging in terms of the
thought processes needed to perform the work. Bloom’s taxonomy of cognitive
operations (Bloom, et al., 1956) provides such an approach. If asking students to recall
factual information is not engaging enough, they may be more challenged by an activity
that requires them to paraphrase the material, a more complex operation. The real
essence of complexity in the taxonomy is that more challenging operations involve an
integration of differentiated skills. These begin with knowledge and are followed by
comprehension, application, analysis, synthesis and evaluation. Thus, comprehension
requires more skills than knowledge, application integrates both knowledge and
comprehension skills, and so on. Some have found this a useful tool for analyzing the
cognitive demands of math instruction (e.g., Stodolsky, 1988). Others have found a four-
stage hierarchy of cognitive skills just as useful (Burns, 1984): new learning, practice,
One of the ways to enhance the difficulty of subject matter and require higher
cognitive operations at the same time is through the use of discovery methods. As
opposed to presented or known problem solving, discovered problem solving involves
novelty,1 situations that are unfamiliar, in which a correct solution and the process are
both unknown (Getzels, 1964). The National Council of Teachers of Mathematics (1996)
refers to these as genuine problems. Genuine or discovered problem solving may depend
on inventing new rules and finding the right ways to apply them.
In known problem solving, the solution may be apparent to all but the problem solver.
But there is little novelty since the solution can be known by retrieving the rules from
memory and applying them. According to Poincarè (1952), more than sheer memory is
1 Surprising, novel, fantastic, incongruent or discrepant information in an activity may move students to
seek additional information, to explain the unexpected, to resolve inconsistencies, to be imaginative (Bruner, 1977; Malone, 1981; Piaget, 1970). As is true for challenges, optimal discrepancies are believed to be relative to a learner’s knowledge. Ideas too unfamiliar will probably not arouse a subject or be rejected; differences too minor will be assimilated or ignored (Berlyne, 1960; Cofer & Appley, 1964; Lepper & Hodell, 1989).
needed to solve difficult problems. An intuitive grasp of how the steps fit together in a
meaningful way is equally if not more important. The inventive skills needed to discover
solutions to novel problems are among the most complex of all cognitive behaviors
When material is novel, students will probably be more attentive and interested than
when material is redundant (Stodolsky, 1988). Although this assumption was not
thoroughly tested in Stodolsky’s research, it was found that students were more engaged
in work that provided them with greater novelty and complexity. Novelty used for its
own sake, however, may impede intrinsic motivation if students are distracted from the
intended task (Blumenfeld, 1992). In this light, the more appropriate use of novelty is in
conjunction with the unknown outcome of the problem rather than its unique delivery.
In addition to these internal aspects of the work, other ways to define complexity are
suggested by Rathunde’s (1988, 1989) analysis of families and flow. Complex families
differ from simple ones in terms of locus of control, the quality of family members’
interactions, and their activity selections. Inasmuch as the dynamics that make families
simple or complex are also part of the classroom environment, these concepts may have
corporate motivational significance there as well.
Members of complex families are integrated by a sense of mutual encouragement and
support which allows them freedom to develop their individuality. Children are provided
with choices and feelings of control, yet they are not left completely on their own.
Complex families support the autonomy of their members. Locus of control is collective
rather than coercive or centralized in one individual. Findings indicate that autonomy-
supporting experiences in the context of one’s family account for more productive and
enjoyable work (Csikszentmihalyi, Rathunde, & Whalen, 1993; see also Deci, 1995).
This kind of autonomy depends on each family member’s ability to self-regulate plus
a strong commitment to interaction. Otherwise the family just becomes more
differentiated, possibly fragmented. Family integration occurs through common activities
as well as discussion of its goals so that there is clarity, understanding and ownership.
Members of complex families listen to each other, finding ways to affirm their
differences and still maintain family harmony. Parents serve as models, encouraging
interaction and by being thoughtful listeners. Consequently, family members feel that
others are responsive to them. By comparison, members of simple families tend to
perceive that their opinions do not matter much.
Family complexity is also evidenced by the way time is allocated to activities.
Complex families invest proportionately more time in two areas: interaction
(communicating, socializing, etc.) and productive work (homework and study).
Alternatively, they devote less time to housekeeping routines (eating, dressing, chores,
etc.).2 In this connection, it is not uncommon for gifted children to be excused from
doing chores so they can devote more time to their area of talent (Bloom, 1985).
The types of activities which complex families choose to do provide them with
greater challenges and generally require more differentiated skills than activities that
seem to be preferred by simple families. (These activity selections, especially in the
latter’s case, may be more by default than by deliberate decision.) For example, one of
the more prevalent leisure time activities is television viewing, a simple activity
infrequently associated with flow (Massimini & Carli, 1988). Feelings of boredom and
apathy are the most prevalent affects for time spent in front of the TV. One reason for
this is that viewing encourages passivity. Its requirements are not demanding: sitting
still, watching and listening. Besides requiring no special skills, watching television
involves practically no challenges--at least as those who watch it report
(Csikszentmihalyi, 1993). Not surprisingly, TV provides persons with about as much
flow as cleaning the house or trying to sleep.3 Although talented teens spend a good deal
of time watching TV and other leisure pastimes no matter their family context, complex
families devote proportionately more time to activities with greater inherent challenges
such as communicating and schoolwork than do simple families (Rathunde, 1988).
The dynamic tensions which distinguish complex families from simple ones may also
be observed in the classroom in terms of locus of control, interactions, and task
preferences, in which cognitive challenge and novelty figure prominently. Of all these,
locus of control is the most studied motivational dimension. For over twenty years
2 The source of this information is Csikszentmihalyi, Rathunde, & Whalen (1993).
3 Nonetheless, television exerts a strong attraction. Talented teens tend to watch more TV than average
teens, but it remains an activity that provides practically no flow (Csikszentmihalyi, Rathunde, Whalen, 1993).
researchers have reached a similar conclusion: intrinsic interest declines when students
are made to feel like pawns rather than origins (deCharms, 1968), when teachers exert
excessive control over their students (Valås & Søvik, 1993). In the context of
instruction, goals and rules, the types of choices students are free to make, and whether
feedback is controlling or informative are aspects that may differentiate classrooms in
The extent to which interactions take place in a classroom may be another telling sign
of complexity. Are students encouraged to share their ideas? Is there a climate of give-
and-take among students and between teacher and students? Interactions may be
observed in discussions, students working together, times the teacher spends listening to
As this implies, the kinds of activities that comprise a typical lesson may serve as a
guide to classroom complexity. The types of tasks found and the amount of time devoted
to them may reveal preferences that have motivational importance. If a great deal of time
is spent in housekeeping duties (e.g., collecting and distributing papers) to the detriment
of more challenging work (e.g., tasks that require application-level thinking), then it may
be expected that less flow and less learning will take place.
Since the activities found in school are not necessarily the same as those found at
home, the kinds of activities that may be considered complex may differ. Therefore, the
types of school activities and their potential for complexity are examined in more detail.
4 Differentiated classrooms support student autonomy and choices--without a doubt the most
consentaneous recommendation made by motivational researchers interested in educational improvement
Classroom activities may be described in terms of structural dimensions. As
employed in educational research, a number of these dimensions have been useful in
characterizing classroom experience: instructional formats, pacing, task goals, cognitive
requirements, types and nature of feedback, teacher and student behaviors, interactions,
emotional climate, the physical environment and the materials used (e.g. Berliner, 1983;
Burns, 1984; Doyle, 1978; Grannis, 1978; Stodolsky, 1988). As the preceding discussion
ought to have made clear, several of these dimensions may directly affect students’
The meta-structure that holds the various dimensions in place is the activity segment.
Activity segments reveal how classroom experience is organized, how a lesson is divided
into distinguishable parts, without fragmenting the instructional experience into
meaningless particles (Stodolsky, 1988).
An activity segment is usually distinguishable by an instructional format. The main
formats that generally occur in school are seatwork, recitations, lecture, demonstrations,
reports, group work, discussion, contests and games, testing, checking work, giving
instructions, transitions and so on (Berliner, 1983; Stodolsky, 1988). Listing the formats
(e.g., Flink, et al., 1990; Green & Foster, 1986; Ryan, Connell, & Deci, 1985; Valås and Søvik, 1993). Undifferentiated classes minimize personal choices and self-determination.
used during a class period is a basic way to segment a lesson for in-depth analysis of
Instructional formats are the building blocks of a lesson. Each format is a unique
combination of elements that all formats have in common. Seatwork, for example, may
involve students with topics and problems through an organization that includes pacing,
goals, cognitive requirements, feedback, teacher and student behaviors, emotional tone, a
physical setting and instructional technology. Generally speaking, lecture is comprised of
the same elements but configured differently. Whether students find these activities
intrinsically rewarding or not depends on the emphasis given to each element and how
The way time is allocated in the classroom is measured by pacing (Gump, 1967;
Grannis, 1978). Pacing helps to establish who controls the rate of work in a segment;
pacing may be determined by a teacher, student, a machine (such as a VCR or computer),
or by a group of students working cooperatively (Stodolsky, 1988). Pacing intersects
with all other segment elements, making it possible to compare the relative use of one
format against another, to calculate the amount of time devoted to specific task functions.
Lesson pacing also provides incisive information about locus of control. Recalling
Deci’s seminal studies (1971, 1972), and many thereafter, individuals who freely chose
activities and further exercised their autonomy by determining how long or how fast they
would work were more intrinsically motivated. In most students’ experience, lesson
pacing is determined by the teacher. However, when a class becomes involved in a group
discussion or a student makes a presentation, it becomes less clear who controls the clock.
Formats may also be described in terms of their goals, which have significance for
flow in terms of clarity. Goals can lack clarity if they are poorly worded or if the work
with which they are associated is too challenging. Both conditions are bound to be
greeted with confusion on the part of students.
Goals can also convey intrinsic or extrinsic orientations. If in justifying the purpose
of an activity it is stated or implied that students are expected to conform to a given
standard, self-regulation may be directly confronted. Or if the utility value of the activity
is emphasized, for example, doing x number of problems in order to earn extra credit
points, once again a potential extrinsic orientation is introduced whether students need it
or not to get them moving. On the other hand, if the goal is for students to see if they can
discover other ways to solve the problem, that may allow them greater freedom and place
fewer obvious extrinsic incentives in their path.
Also relevant to goals and motivation is the implicit task challenge. As indicated
earlier in this chapter, one way to gauge this is by ascertaining the complexity of the
cognitive operations required. In classroom studies such as Stodolsky’s (1988)
comparison of fifth grade social studies and math lessons, she was able to identify
cognitive challenges across activity segments according to their complexity: those “with
no cognitive goal, those emphasizing facts, those oriented to comprehension or concepts,
those in which students learned research skills, and those containing application or other
higher mental process activities” (Stodolsky, 1988, pages 78-79).
Besides cognitive complexity, another important aspect of challenge is mathematical
aptitude. Are the goals set too high or too low for one’s ability? Without the potential to
perform an activity, the experience is bound to be frustrating, anxiety-producing or even
meaningless. With sufficient skills but lacking a challenge, a person will probably
become bored. Therefore, students’ affective experiences may be relied upon to measure
the dynamics between mathematical tasks and aptitude. These may be measured by
asking the students about their feelings; they may also be deduced from behaviors.
The importance of feedback has already been mentioned. Without immediate or
unambiguous feedback it may be very difficult to tell whether one’s efforts are on track or
not. In this light, a teacher, another student, or a mechanical device can provide the
necessary information to determine the accuracy or appropriateness of one’s efforts.
Often overlooked in classroom research is the possibility that a task may be a source of
informational feedback on its own. The involvement of other people may be
unnecessary; in fact, the feedback they provide may be controlling in nature and
encourage an extrinsic motivational focus. It is also conceivable that no feedback may
occur, that no information or external control is forthcoming to guide one’s efforts in
completing a task. Since access to a student’s perception is limited, it is no easy task to
tell whether the feedback is perceived as controlling or not. But it may be possible to
describe the source of the feedback and its intended purpose.
Activity segments also provide information about teacher and student behaviors,
which are a potential source of information about locus of control, perceived competence
and challenges. In most cases, teachers respond to students and vice versa. Relative to
skills and challenges, if students are not “getting it,” a responsive teacher may be
observed adjusting the difficulty of the task. If the students are doing fine, the teacher
may provide little more than passive assistance, co-participate in the activity, or not be
involved at all. When things start to go awry, the teacher is likely to attempt to regain
Student behaviors are also important in making this determination because they can
disclose whether an activity holds their attention or not. If something is not interesting
because it is too easy, too hard, or not rewarding enough, attention to the prescribed task
will dissipate. Instead of reading, for example, students may start to talk or gaze around
the room. Some other task that is more interesting will probably divert their attention. If
students are interested in a task to start with, however, and then the teacher employs
controlling strategies, the more likely it becomes that students’ intrinsic motivation will
be undermined (Deci, Nezlec, & Sheinman, 1981; Valås & Søvik, 1993). In either case,
the actions of teacher and students may be used to interpret the motivational climate of
Little more needs to be said at this point about interaction except to that it may be
observed in certain formats more often, for example, group work and discussion.
Interaction may involve as few as two individuals or as many students as there are in
Clearly, behaviors do not always divulge what classroom participants are
experiencing, whether they are really thinking about what they appear to be listening to
(Jackson, 1984). But behaviors can act as a barometer by which to tell if tasks are being
attended to or not. Tied to observable behaviors is the energy level with which students
conduct themselves quite candidly. These encompass high energy (when subjects are
animated and excited about a task, or in a negative sense, agitated), average energy (the
more common passive state of classroom experience), and low energy (a condition in
which the approach to work is lethargic and unmotivated). Generally, students’ energy
levels provide clues to their task interest and the emotional climate of the room.
Finally, every segment has a physical contextin which segment elements occur. The
classroom environment, instructional media, temperature, time of day and distractions are
all part of the milieu that may affect what occurs in a segment. As clinical studies have
shown, distractions and discomforts can effectively negate intrinsic motivation (Deci &
Ryan, 1985). Moreover, the environment of instruction can say a lot about who is in
control and should not go unheeded (Goodlad, 1984). From wall displays to the
arrangement and the condition of the furniture, the agenda of control is present in three
dimensions. How the room is used is potentially important as well. For example,
students who are assigned to seats or who are allowed to sit wherever they want may
come to different conclusions about their freedom. Before external locus of control can
bring about a loss of intrinsic motivation, however, an individual must perceive a threat
to self-determination. Therefore, it cannot be observed what the effect of the physical
setting is without also listening to students.
As this chapter and the one before it have sought to illustrate, there are several ways
tasks may predictably affect intrinsic motivation. Basic to the relationship between tasks
and motivation is the premise that activities which satisfy the needs for autonomy,
competence and self organization are inherently enjoyable, attracting and holding an
From this discussion of task elements and motivational effects it is possible to distill a
model activity, one that optimizes opportunities for flow. Assuming that the prevalence
of certain task dimensions (and the absence of others) results in greater intrinsic
enjoyment, what was hypothesized at the end of chapter 1 regarding complexity in the
classroom can now be stated more precisely.
Simple and complex classes will differ in their use of formats. There will be more
opportunities for students to interact and make choices in complex classes. In terms of
activities there will be more group work, projects, discussion and student presentations in
complex classes. In contrast, formats that limit social interaction and student choices
(e.g., teacher presentations, seatwork and recitations) will be more prevalent in simple
classes. Just as an emphasis on chores characterizes simple families, grading papers, time
spent getting organized and other classroom housekeeping will characterize simple
Because students are afforded more autonomy in the complex classroom, they may
behave in ways that are more characteristically reserved for the teacher: initiating
discussions, posing new problems, and helping to evaluate solutions. Teachers will
recognize and encourage student participation at this level.
There will be greater novelty in the types of problems assigned to students in complex
classes. Students will not be given a steady diet of “skill and drill” problems in which an
algorithmic procedure is repeated until it is mastered. Novel problems, that is, ones in
which the algorithm to use is not predetermined, will be used to integrate mathematical
skills. Incentives in the classroom will be based on discovered problem solving.
Extrinsic rewards such as extra credit or an emphasis on the long-term, utility value of
mathematics (i.e., what good will it be someday) will be subordinate to the intrinsic,
short-term satisfaction of discovering solutions and finding new ways to solve problems.
Even though discovery-type problems may be used, student questions and behaviors
will not reflect significant confusion about the goals of the activity or what is expected.
Some of the potential confusion should be eliminated when students become accustomed
to the use of discovery problems. Before this can happen, the goals of the activity will
need to be stated clearly, and if there are questions, the teacher will take care to explain
what is expected in a way that results in understanding and acceptance. Ultimately, the
goal may be for students to participate in finding new discovery problems to solve.
Feedback in the classroom will be informative rather than controlling, providing
insights to the learner on the adequacy of his or her actions. In a more complex
environment, feedback will come from sources other than the teacher, for example, other
students and the task itself. (If the activity is balanced with the learners’ skills, the only
feedback necessary may come from the task itself.) Feedback will be immediate to the
task, not delayed, except in cases where students are guided to discover solutions on their
own. Formats in which students actively engage in focused problem-solving such as
seatwork, group work and discussion may be expected to increase perceptions of flow.
More time will be spent in tasks that are appropriately challenging; consequently,
boredom and frustration will be minimal. Students will not appear to be anxious or
lethargic when tasks are just right for their abilities. Teachers will not have to intervene
to correct student behaviors when this balance is achieved. When this balance is optimal,
less time will be spent experiencing discomfort or distraction in the classroom, even
though those elements may be present. Plus, students will create fewer distractions
themselves when their attention is devoted to the task.
Complexity may not depend on variety, which is listed sometimes as a possible
motivational effect.5 Flow provides an interesting footnote concerning variety: students
who engage in a narrow range of behaviors are not necessarily less motivated. Persons
who experience flow are known to engage in the same activity for long periods of time
(Csikszentmihalyi & Csikszentmihalyi, 1988). Intense enjoyment does not seem to
depend to a great extent on variety alone (differentiation) but on complexity, the
integration of differentiated task element and skills in a specific task. Therefore, flow in
math may not be related to task variety as much as the judicious selection of tasks.
The purpose of a model task comprised of these dimensions is to test whether or not it
helps to explain optimal experience in math. By observing classroom activities, the
behaviors of teachers and students, the choices offered and the related events of
instruction, it may be possible to determine if students who experience more of these
optimal elements of tasks manifest greater levels of flow. It may turn out that certain
elements aid in the formation of intrinsic interest more than others. As it is, the model is
not expected to be perfect but primarily a tool by which to understand classroom
experiences in light of flow and achievement. The findings are reported and discussed in
5 For example, Ames (1992) suggested five dimensions of classroom experience that may have
motivational importance: variety, novelty, control, challenge and meaningfulness. (Meaningfulness is assumed to be implicit in flow.)
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