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Situated Learning and Education

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This paper provides a reviezo of the claims of situated learning that are having an increasing influence on education generally and mathematics education particularly. We review the four central claims of situated learning with respect to education: (1) action is grounded in the concrete situation in which it occurs; (2) knowledge does not transfer between tasks; (3) training by abstraction is of little use; and (4) instruction must be done in complex, social environments. In each case, we cite empirical literature to show that the claims are overstated and that some of the educational implications that have been taken from these claims are misguided. ollowing on the so-called "cognitive revolution" in psychology that began in the 1960s, education, and particularly mathematics and science education, has been acquiring new insights from psychology and new approaches and instructional techniques based on these insights. At the same time, cognitive psychologists have being paying increasing attention to education as an area of application of psychological knowledge and as a source of important research problems. As research in cognitive psychology progresses and increasingly addresses itself to educational issues, even closer and more productive links can be formed between psychology and mathematics education. However, some educational opinion, including opinion that is quite contrary to the body of empirical evidence available on these matters, is presented as deriving from cognitive psychology. For instance, Lesh and Lamon (1992) write in the introduction to a recent book they edited:
Situated Learning and Education I
JOHN R. ANDERSON LYNNE M . REDER HERBERT A . SIMON
This paper provides a reviezo of the claims of situated learning
that are having an increasing influence on education generally
and mathematics education particularly. We review the four
central claims of situated learning with respect to education: (1)
action is grounded in the concrete situation in which it occurs;
(2) knowledge does not transfer between tasks; (3) training by
abstraction is of little use; and (4) instruction must be done in
complex, social environments. In each case, we cite empirical
literature to show that the claims are overstated and that some of
the educational implications that have been taken from these
claims are misguided.
Educational Researcher, VoL 25, No. 4, pp. 5-11
F
ollowing on the so-called "cognitive revolution" in
psychology that began in the 1960s, education, and
particularly mathematics and science education, has
been acquiring new insights from psychology and new
approaches and instructional techniques based on these
insights. At the same time, cognitive psychologists have
being paying increasing attention to education as an area
of application of psychological knowledge and as a source
of important research problems. As research in cognitive
psychology progresses and increasingly addresses itself
to educational issues, even closer and more productive
links can be formed between psychology and mathematics
education.
However, some educational opinion, including opinion
that is quite contrary to the body of empirical evidence
available on these matters, is presented as deriving from
cognitive psychology. For instance, Lesh and Lamon (1992)
write in the introduction to a recent book they edited:
Behavioral psychology (based on factual and procedural
rules) has given way to cognitive psychology (based on
models for making sense of reaMife experiences), and
technology-based tools have radically expanded the
kinds of situations in which mathematics is useful, while
simultaneously increasing the kinds of mathematics that
are useful and the kinds of people who use mathematics
on a daily basis. In response to these trends, professional
and governmental organizations have reached an un-
precedented, theoretically sound, and future-oriented
new consensus about the foundations of mathematics in
an age of information. (p. 18-19)
As in many recent publications in mathematics educa-
tion, much of what is described in the Lamon and Lesh
book reflects educational implications that have been
drawn from two movements--"situated learning" and
"constructivism"--which have been gaining influence on
thinking about education and educational research. Much
of what is claimed by these movements is not "theoreti-
cally sound." This paper will focus on situated learning.
Elsewhere (Anderson, Reder, & Simon, 1995) we have writ-
ten on the excesses of constructivism. However, construc-
tivism is primarily a philosophical position, whereas
situated learning has strong empirical consequences that
are not always borne out. In this paper, we want to con-
centrate on empirical evidence and its implications for
mathematics education.
Situated learning (e.g., Lave, 1988; Lave & Wenger, 1991;
Greeno, Smith, & Moore, 1992) emphasizes the idea that
much of what is learned is specific to the situation in which
it is learned. While implications have been drawn from
situated learning for all aspects of education, this paper
will focus primarily on mathematics education. This is
because many of the examples from the situated learning
literature involve mathematics, and it has particularly in-
fluenced researchers in mathematics education (e.g., Cobb,
Yackel, & Wood, 1992; Lesh & Zawojeski, 1992; Resnick,
1994). Particularly important has been situated learning's
emphasis on the mismatch between typical school situa-
tions and "real world" situations such as the workplace,
where one needs to deploy mathematical knowledge.
Greater emphasis should be given to the relati6nship
between what is learned in the classroom and what is
needed outside of the classroom, and this has been a valu-
able contribution of the situated learning movement. How-
ever, while it is important to have our consciousness raised
about this issue, the claims from the situated learning
camp are often inaccurate. Moreover, the educational
implications taken from these claims (not always endorsed
by the original situated authors) are often mistaken.
Two of us have been involved in past reviews relevant to
situated learning--Simon in support of the mutual com-
patibility of modern information processing theory and
situated cognition (Vera & Simon, 1993) and Reder in an
assessment of the effectiveness for training of techniques
located at various points along the scale of "situatedness"
(Reder & Klatzky in a report of the National Research
JOHN
R. ANDERSON
is a professor of psychology and computer
science at Carnegie Mellon University, Department of Psychol-
ogy, Pittsburgh, PA 15213 (ja@cmu.edu). He specializes in
learning and computer-based instruction.
LYNNE M. REDER
is a professor in the Department of Psychology
at Carnegie Mellon University (reder@cmu.edu). She specializes
in learning and memory.
HERBERT a . SIMON
is a professor of psychology and computer
science at Carnegie Mellon University (has@cs.cmu.edu). He
specializes in learning and problem solving.
MAY 7996 5
Council, 1994). We will focus on the four claims of situated
learning identified in the NRC report.
Claim 1: Action Is Grounded in the Concrete Situation
in Which It Occurs
That action is situationally grounded is surely the central
claim of situated cognition. It means that the potentialities
for action cannot be fully described independently of the
specific situation, a statement with which we fully concur.
However, the claim is sometimes exaggerated to assert that
all knowledge is specific to the situation in which the task
is performed and that more general knowledge cannot and
will not transfer to real-world situations. One supposed
example is Lave's (1988) description of Orange County
homemakers who did very well at making supermarket
best-buy calculations but who did m uc h worse on arith-
metically equivalent school-like paper-and-pencil mathe-
matics problems. Another frequently cited example is
Carraher, Carraher, and Schliemann's (1985) account of
Brazilian street children w h o could perform mathematics
when making sales in the street but were unable to answer
similar problems presented in a school context. As Lave
(1988) asserts in summarizing this research:
In sum, arithmetic practices are made to fit the activity
at hand, and there are discontinuities between the tech-
niques used to solve arithmetic problems in school-like
situations and in the situations of shopping, selling pro-
duce, cooking, making and selling clothes, and assem-
bling truckloads of dairy products. Place-holding algo-
rithms do not transfer from school to everyday situations,
on the whole. On the other hand, extraordinarily success-
ful arithmetic activity takes place in these chore and job
settings. (p. 149)
Even if these claims are valid and generalizable beyo nd
these specific cases, they demonstrate at most that particu-
lar skills practiced in real-life situations do not generalize
to school situations. They assuredly do not demonstrate
the converse. That is, it does not follow from these exam-
ples that arithmetic procedures taught in the classroom
cannot be used by a shopper to make price comparisons or
a street v endor to make change. Such observations call for
closer analyses of the task d emands and the use of the
analyses to devise teachable procedures that will achieve
a balance between the advantages of generality and the
advantages of incorporating enough situational context to
make transfer likely. They also call for research on the fea-
sibility of increasing the application and transfer of knowl-
edge by including ability to transfer as a specific goal in
instruction, a skill that is given little attention in most cur-
rent instruction.
At one level, there is nothing new in this claim about the
contextualization of learning. There have been numerous
demonstrations in experimental psychology that learning
can be contextualized (e.g., Godden & Baddeley, 1975;
Smith, Glenberg, & Bjork, 1978). For instance, Godden and
Baddeley found that divers had difficulty remembering
un der water what they learned on land or vice versa.
However, it is not the case that learning is wholly tied to a
specific context. For instance, G odden and Baddeley's
divers could remember some of what they learned in the
other context. In fact, there are many demonstrations of
learning that transfer across contexts and of failures to find
any context specificity in the learning (e.g., Fernandez &
Glenberg, 1985; Saufley, Otaka, & Bavaresco, 1985)--a fact
that has often frustrated researchers who were looking for
context sensitivity.
Ho w tightly learning will be bound to context depends
on the kind of knowledge being acquired. Sometimes
knowledge is necessarily bound to a specific context by the
nature of instruction. Thus, to give a mathematics example,
one would not be surprised to learn that, given typical in-
struction, carrying is bound to the context of doing base-10
addition and would not generalize to another base system.
In other cases, how contextualized the learning is depends
on the way the material is studied. If the learner elaborates
the knowledge with material from a specific context, it be-
comes easier to retrieve the knowledge in that same con-
text (Eich, 1985), but perhaps harder in other contexts. One
general result is that knowledge is more context-bound
when it is just taught in a single context (Bjork & Richard-
son-Klavehn, 1989).
Clearly, some skills such as reading transfer from one
context to another. The very fact that we can engage in a
discussion of the context-dependence of knowledge is it-
self evidence for the context-independence of reading and
writing competence. Many of the demonstrations of con-
textual-binding from the situated camp involve mathemat-
ics, but clearly mathematical competence is not always
contextually bound either. Although the issue has seldom
been addressed directly, the psychological research liter-
ature is full of cases where mathematical competence has
transferred from the classroom to all sorts of laboratory
situations (sometimes biz a r r e - - t h e intention was never to
show transfer of mathematical skills--e.g., Bassok &
Holyoak, 1989; Elio, 1986; Reder & Ritter, 1992). It is not
easy to locate the ma ny published demonstrations of math-
ematical competence generalizing to novel contexts; these
results are not indexed u nder "context-independence of
mathematical knowledge" because until recently this did
not seem to be an issue.
The literature on situation-specificity of learning often
comes with a value judgment about the merits of knowl-
edge tied to a nonschool context relative to school-taught
knowledge and an implied or expressed claim that school
knowledge is not legitimate. Lave (1986, 1988, p. 195) goes
so far as to suggest that school-taught mathematics serves
only to justify an arbitrary and unfair class structure. The
implication is that school-taught competences do not con-
tribute to on-the-job performance. However, numerous
studies show modest to large correlations between school
achievement and work performance (e.g., Hunter &
Hunter, 1984; Bossiere, Knight, & Sabot, 1985) even after
partialing out the effects of general ability measures (which
are sometimes larger).
Claim 2: K n o w ledge Do e s Not Transfer Between Tasks
This second claim of situated co gnition--of the failure of
knowledge to transfer--can be seen as a corollary of the
first. If knowledge is wholly tied to the context of its
acquisition, it will not transfer to other contexts. However,
even without assuming extreme contextual dependence,
one could still claim that there is relatively little transfer
beyond nearly identical tasks to different physical con-
texts. Indeed, Lave (1988) argues that there is no empirical
evidence for such general transfer and asserts:
6 ED UCATIONAL RESEARCHER
It is puzzling that learning transfer has lasted for so long
as a key conceptual bridge without critical challenge. The
lack of stable, robust results in learning transfer experi-
ments as well as accumulating evidence from cross-situa-
tional research on everyday practice, raises a number of
questions about the assumptions on which transfer
theory is based--the nature of cognitive "skills," the "con-
texts" of problem-solving and "out of context" learning,
the normative sources of models of good thinking and
less than perfect "performances." (p. 19)
Contrary to Lave's opinion, a large body of empirical
research on transfer in psychology, going back at least to
Weber in 1844 and Fechner in 1858 (Woodworth, 1938,
chap. 8), demonstrates that there can be either large
amounts of transfer, a modest a m ount of transfer, no trans-
fer at all, or even negative transfer. H ow much there is and
whether transfer is positive depends in reliable ways on
the experimental situation and the relation of the material
originally learned to the transfer material.
The more recent psychological literature (for two rela-
tively recent reviews, see Perkins & Salomon, 1989; Singley
& Anderson, 1989) contains many failures to achieve trans-
fer (e.g., Gick & Holyoak, 1980; Hayes & Simon, 1977;
Reed, Ernst, & Banerji, 1974; Weisberg, DiCamillo, &
Phillips, 1985), but also contains many successful demon-
strations of transfer (e.g., Brown, 1994; Brown & Campi-
one, 1994; Kotovsky & Fallside, 1989; Lehman, Lempert, &
Nisbett, 1988; Pennington, Nicolich, & Rahm, 1995;
Schoenfeld, 1985; Singley & Anderson, 1989; Smith, 1986).
Indeed, in the same domain (Tower of Hanoi isomorphs),
quite different amounts of transfer occur depending on t h e
amount of practice with the target task and on the repre-
sentation of the transfer task (Kotovsky & Fallside, 1989).
Representation and degree of practice are critical for deter-
mining the transfer from one task to another.
Singley and Anderson (1989) showed that transfer
between tasks is a function of the degree to which the tasks
share cognitive elements. This hypothesis had also been
put forth very early in the development of research on
transfer (Thorndike & Woodworth, 1901; Woodworth,
1938), but was hard to test experimentally until we ac-
quired our modern capability for identifying task compo-
nents. Singley and Anderson taught subjects several text
editors one after another and sought to predict transfer
(savings in learning a new editor when it was not taught
first). They found that subjects learned subsequent text
editors more rapidly and that the number of procedural
elements shared by two text editors predicted the amount
of this transfer. In fact, they obtained large transfer across
editors that were very different in surface structure but that
had common abstract structures. Singley and Anderson
also found that similar principles govern transfer of math-
ematical
c o m n e t e n c p a c r n ~ rn 1 1 1 t i n l p c l n m a i n c a l t h n l ~ ¢ h
to other domains. As they note, there have been wildly
optimistic claims about such transfer and disappointing
results. Klahr and Carver show that one can get transfer if
one performs a componential analysis of the structure of
LOGO debugging and the structure of the transfer task and
provides instruction in LOGO designed to teach the com-
mon components.
What about the situations in which subjects have sl~own
relatively little transfer? In one famous series of studies
(Gick & Holyoak, 1980, 1983), subjects were presented with
Duncker's (1945) classic radiation problem: "Suppose you
are a doctor faced with a patient who has an inoperable
stomach tumor. You have at your disposal rays that can
destroy human tissue when directed with sufficient inten-
sity. H ow can you use these rays to destroy the tumor with-
out destroying the surrounding healthy tissue?" (adapted
from Gick & Holyoak, 1983). Prior to their exposure to the
target problem, subjects read a story about an analogous
military problem and its solution. In the story, a general
wishes to capture an enemy fortress. Radiating outward
from the fortress are many roads, each mined in such a
way that the passing of any large force will cause an
explosion. This precludes a full-scale direct attack. The
general's plan is to divide his army, send a small group
down each road, and converge on the fortress. The com-
mon strategy in both problems is to divide the force, attack
from different sides, and converge on the target. After
reading this story, however, only about 30% of the subjects
could solve the radiation problem, which is only a "lim-
ited" improvement (although an improvement by a factor
of three) over the 10% baseline solution rate (Gick &
Holyoak, 1980).
A striking characteristic of such partial failures of trans-
fer is how relatively transient they are. Gicl< and Holyoak
increased transfer greatly just by suggesting to subjects
that they try to make use of the problem about the general.
Exposing subjects to two such analogs also greatly in-
creased transfer. The amount of transfer appeared to de-
pend in large part on where the attention of subjects was
directed during the experiment, which suggests the desir-
ability of instruction and training on the cues that signal
the relevance of an available skill. A number of studies
converge_on the conclusion that transfer is enhanced when
training involves multiple examples and encourages learn-
ers to reflect on the potential for transfer (e.g., Bransford,
Franks, Vye, & Sherwood, 1989; Brown & Kane, 1988;
Ghatala, Levin, Pressley, & Lodico, 1985; Pressley,
Borkowski, & Schneider, 1987).
In research on transfer, there has been a tendency to look
for it where one is least likely to find it. That is, research
tends to look for transfer from little practice in one d o m a i n
to initial performance in another domain. Superficial dif-
f ~ r p n r ~
h ~ l ' w ~ n
t h ~ t ~ r n A n m ~ i ne w i l l h ~ z a I - h a i r l ~ r r ~ c ~
material, there can be either large amounts of transfer, a
modest amount, no transfer at all, or even negative transfer.
(2) Representation and degree of practice are major
determinants of the transfer from one task to another, and
transfer varies from one domain to another directly with
the numb er of symbolic components that are shared.
(3) The amount of transfer depends on where attention
is directed during learning or at transfer. Training on the
cues that signal the relevance of an available skill should
probably receive more emphasis in instruction than it now
typically receives.
Claim 3: Training By Abstraction Is of Little Use
The claim that training by abstraction is of little use is also
a corollary of the claims just discussed. Nonetheless, one
might argue for it even if one dismisses the others. Claim 3
has been extended into an advocacy for apprenticeship
training (Brown, Collins, & Duguid, 1989; Collins, Brown,
& Newman, 1989). As Collins, Brown, and Newma n assert:
The differences between formal schooling and appren-
ticeship methods are many, but for our purposes, one is
most important. Perhaps as a by-product of the relegation
of learning to schools, skills and knowledge taught in
schools have become abstracted from their uses in the
world. In apprenticeship learning, on the other hand, tar-
get skills are not only continually in use by skilled practi-
tioners, but are instrumental to the accomplishment of
meaningful tasks. (p. 453-454)
What is meant by advocacy of apprenticeship training can
vary from advocacy of certain rather traditional pedagogi-
cal strategies such as modeling in traditional classrooms to
the claim that the most effective training is real apprentice-
ship to workers in their real-world environments. The
stronger ;~zersions of this claim clearly challenge the legiti-
macy of.school~based instruction.
Abstract instruction can be ineffective if what is taught
in the classroom is not what is required on the job. Often
this is an indictment of the design of the classroom in-
struction rather than of the idea of abstract instruction in
itself. However, sometimes it is an indictment, of the job
situation. For instance, Los Angeles police after leaving the
police academy are frequently told by more experienced
officers "now forget everything you learned" (Indepen-
dent Commission on the Los Angeles Police Department,
1991, p. 125). The consequence is that police officers are
produced who, ignoring their classroom training in the
face of contrary influences during apprenticeship, may vi-
olate civil rights and make searches without warrants.
Clearly, one needs to create a better correspondence
between job performance and abstract classroom instruc-
tion, and sometimes this means changing the nature of the
job (including the structure of motivations and rewards)
and fighting unwanted and deleterious effects of appren-
ticeship learning.
Abstract instruction can be very effective. In unpub-
lished research, Singley found that abstract instruction
leads to successful transfer, while concrete instruction can
lead to failure of transfer. He taught subjects to solve alge-
bra word problems involving mixtures. Some subjects were
trained with pictures of the mixtures while other subjects
were trained with abstract tabular representations that
highlighted the underlying mathematical relationships.
The abstract training group was able to transfer better to
other kinds of problems that involved analogous mathe-
matical relations. Perhaps the most striking demonstration
of the benefit of abstract instruction comes from Biederman
and Shiffrar (1987). They looked at the very difficult task o f
sexing day-old chicks--something that people spend years
learning in an apprentice-like role. They found that 20 min-
utes of abstract instruction brought novices up to the levels
of experts who had years of practice.
The issue of choosing between abstract and very specific
instruction can be viewed in the following way. If abstract
training is given, learners must also absorb the money and
time costs of obtaining supplemental training for each dis-
tinct application. But if very specific training is given, they
must completely retrain for each application. Which is to
be preferred, and to what extent, depends on the balance
among (a) the cost of the more general abstract training,
(b) the cost of the specific training, (c) the cost of the sup-
plemental training for application of abstract training, and
(d) the range of jobs over which the learner is likely to have
occasion to apply what was learned. Someone who will
spend years performing a single set of very specific tasks
might be well advised to focus on specific training. But if
the cost of supplemental training is not large (i.e., if there is
substantial transfer over the range of tasks), if technologi-
cal or other changes are likely to alter tasks substantially
over the years, or if the range of tasks the learner is likely
to address over time is substantial, then abstract training
with supplemental applications training is clearly prefer-
able. It is easy to work out an exercise of this kind by as-
signing numbers to the various costs and to the variability
of the tasks encountered and thereby to show that there is
no solution that is optimal for all cases.
Most modern information-processing theories in cogni-
tive psychology are "learning-by-doing," theories which
imply that learning would occur best with a combination
of abstract instruction and concrete illustrations of the
lessons of this instruction. Nume rous experiments show
combining abstract instruction with specific concrete ex-
amples is better than either one alone (e.g., Cheng,
Holyoak, Nisbett, & Oliver, 1986; Fong, Krantz, & Nisbett,
1986; Nesher & Sukenik, 1991; Reed & Actor, 1991). One of
the most famous studies demonstrating this was per-
formed by Scholckow and Judd (described in Judd, 1908;
a conceptual replication by Hendrickson & Schroeder,
1941). They had children practice throwing darts at an
under water target. One group of subjects received an
explanation of refraction of light, which causes the appar-
ent location of the target to be deceptive. The other group
only practiced, receiving no abstract instruction. Both
groups did equally well on the practice task, which in-
volved a target 12 inches u n d e r water, but the group with
abstract instruction did much better when asked to trans-
fer to a situation where the target was now un der only 4
inches of water.
A variation on advocacy of apprenticeship training is
advocacy for using only "authentic" problems (e.g.,
Brown, Collins, & Duguid, 1989; Lesh & Lamon, 1992).
What is authentic is typically ill-defined but involves
a strong emphasis on problems such as those students
might encounter in everyda y life. A focus on underlying
cognitive process would suggest that this is a superficial
requirement. Rather, we would argue, as have others (e.g.,
8 EDUCATIONAL RESEARCHER
Hiebert et al., 1994), that the real goal should be to get stu-
dents motivated and engaged in cognitive processes that
will transfer. What is important is what cognitive processes
a problem evokes and not what real-world trappings it
might have. Often real-world problems involve a great
deal of busy work and offer little opportunity to learn the
target competences. For instance, we have observed in
high school mathematics classrooms--where we have in-
troduced longer, more real-world-like problems to situate
algebra (Koedinger, Anderson, Hadley, & Mark, 1 99 5 )-
that much of student time is spent on tasks such as tabling
and graphing, which rapidly become clerical in nature. On
the other hand, relatively little time is spent relating alge-
braic expressions to the real-world situations they denote.
To summarize: abstract instruction combined with con-
crete examples can be a powerful method. This method is
especially important when learning must be applied to a
wide variety of (frequently unpredictabl e ) future tasks.
Claim 4: Instruction Need s to be Do n e in Complex,
Social Environments
An elaboration of the previous position is the argument
that learning is inherently a social phenomena. As Lave
and Wenger (1991) argue:
In our view, learning is not merely situated in practice--
as if it were some independently reifiable process that just
happened to be located somewhere; learning is an inte-
gral part of generative social practice in the lived-in
world. (p. 35)
A second argument is that learning should be done on
complex problems (e.g., Lesh & Zawojeski, 1992). These
two ideas are put together in the proposal that learning
should take place in complex, social situations with
varying emphasis on the "complex" and the "social. ''2
Although job training is only one function of education,
this social + complex formula for learning situations is
often justified with respect to preparing students for the
workplace where it is argued they will need to display
their skills in complex, social environments (Resnick,
1987).
While one must learn to deal with the social aspects of
jobs, this is no reason why all skills required for these jobs
should be trained in a social context. Consider the skills
necessary to become a successful tax accountant. While an
accountant must learn how to deal with clients, it is not
necessary to learn the tax code or how to use a calculator
while interacting with a client. It is better to train indepen-
dent parts of a task separately because fewer cognitive re-
sources will then be required for performance, thereby
reserving adequate capacity for learning. Thus, it is better
to learn the tax code without having to interact with the
client simultaneously and better to learn how to deal with
a client when the tax code has been mastered.
In fact, a large body of research in psychology shows
that part training is often more effective when the part
component is independent, or nearly so, of the larger task
(e.g., Knerr et al., 1987; Patrick, 1992). Indeed, in team
training, it is standard to do some part-task training of in-
dividuals outside the team just because it is expensive and
futile to get the whole team together when a single mem-
ber needs training on a new piece of equipment (Salas,
Dickinson, Converse, & Tannenbaum, 1993). In team
sports, where a great deal of attention is given to the effi-
ciency of training, the time available is always divided be-
tween individual skill training and team training.
There are, of course, reasons sometimes to practice skills
in their complex setting. Some of the reasons are motiva-
tional and some reflect the special skills that are unique to
the complex situation. The student who wishes to play vi-
olin in an orchestra would have a hard time making
progress if all practice were attempted in the orchestra con-
text. On the other hand, if the student never practiced as a
member of an orchestra, critical skills unique to the or-
chestra would not be acquired. The same arguments can be
made in the sports context, and motivational arguments
can also be made for complex practice in both contexts. A
child may not see the point of isolated exercises but will
when they are em b e d ded in a real-world task. Children are
motivated to practice sports skills because of the prospect
of playing in full-scale games. However, they often spend
much more time practicing component skills than full-
scale games. It seems important both to motivation and to
learning to practice one's skills from time to time in full
context, but this is not a reason to make this the principal
mechanism of learning.
While there may be motivational merit to embedding
mathematical practice in complex situations, Geary (1995)
notes that there is much reason to doubt how intrinsically
motivating complex mathematics is to most students in
any context. The kind of sustained practice required to de-
velop excellence in an advanced domain is not inherently
motivating to most individuals and requires substantial
family and cultural support (Ericcson, Krampe, & Tesche-
R6mer, 1993). Geary argues, as have others (e.g., Bahrick &
Hall, 1991; Stevenson & Stigler, 1992), that it is this differ-
ence in cultural</