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When Teaching Kills Learning: Types of Mathemathantic Effects

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Instructional research is reviewed where teaching failures have produced students who seem to be less able to use learning skills or had less access to knowledge in some domain than before they were taught. Three general types of "mathemathantic" (i.e. where instruction "kills" learning) effects are hypothesized, theoretical explanations for each effect are examined and representative studies in each area are described. The three types of effects described are where instruction serves to: (1) substitute learning procedures (e.g. novel learning strategies are hypothesized to interfere with the learning of higher general ability learners and inadequate learning strategies are provided to those with lower general ability); (2) impose less desirable motivational goals on learners (e.g. when teaching methods lead constructively motivated learners to believe that failure avoidance has replaced achievement directed goals and, conversely, when defensively motivated students believe that achievement directed goals have replaced the opportunity to avoid failure); and (3) substitute student control for system control over instructional method (e.g. by allowing lower cognitive load instructional methods to be chosen by high general ability, constructive students and/or by allowing higher cognitive load methods to be chosen by defensive students who have low general ability). A four-page bibliography and an outline of situations when mathemathantic effects are more probable are attached. (Author/JAZ)
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Published as, Clark, R. E. (1989) When Teaching Kills Learning: Research on Mathemathantics. In H. Mandl, E. De
Corte, N. Bennett, & H. F. Friedrich (Eds.)
Learning and instruction. European Research in an International Context.
Volume II. Oxford: Pergamon.
WHEN TEACHING KILLS LEARNING: RESEARCH ON MATHEMATHANTICS
Richard E. Clark
University of Southern California
Abstract
Instructional research is reviewed where teaching failures have produced students who are
less able to use learning skills or had less access to knowledge than before they were taught.
Three general types of "mathemathantic" (i.e. where instruction "kills" learning) effects are
hypothesized, theoretical explanations for each effect are examined and representative studies in
each area are described. The three types of effects described are where instruction serves to: 1)
Substitute learning procedures (e.g. Novel learning strategies are hypothesized to interfere with
the learning of higher general ability learners and inadequate learning strategies are provided to
those with lower general ability); 2) Impose less desirable motivational goals on learners (e.g.
when teaching methods lead constructively motivated learners to believe that failure avoidance
has replaced achievement directed goals and, conversely, when defensively motivated students
believe that achievement directed goals have replaced the opportunity to avoid failure); and 3)
Substitute student control for system control over instructional method (e.g. by allowing lower
cognitive load instructional methods to be chosen by high general ability, constructive students
and/or by allowing higher cognitive load methods to be chosen by defensive students who have
low general ability).
ORIENTATION TO THE PROBLEM
A central objective of educational research is to identify "mathemagenic" instructional
methods (i.e. those that "give birth" to learning, Rothkopf, 1970). Researchers have generally
come to expect that when instructional experiments fail, measured achievement or skill levels
will remain at pre-experiment levels. Yet there has been increasing evidence for
"mathemathantic" effects in instructional research (i.e. where instruction "kills" learning) since
the construct was first suggested by Snow (1972). The thread that connects these diverse studies
is their report that instructional treatments unintentionally produced a condition where students
were
less able to use learning skills or have less access to knowledge in some domain than before
entering the experiment.
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For example, in a math task involving the specification, intersection and separation of sets,
Gagne & Bessler (1963) found it difficult to explain why an untreated control group significantly
outperformed an experimental group on a nine week delayed retention test. The treated group
had received training in applying intersection rules to a number of example problems. Sidel &
Rothberg (1964) described a mathemathantic effect in one of the early computer programming
studies. Experimental subjects, taught to state and write the rules for components of computer
programs, performed significantly worse than control subjects who were only taught
programming components. Similarly, in a study where various types of questions were inserted
in an instructional text on Educational Psychology for undergraduates, Shavelson et al (1974)
found that an untreated control group received significantly better quiz scores.
Operational Definition of Mathemathantic Studies
The Mathemathantic effect is operationally defined by one of four different instructional
research outcomes: 1) a control group significantly outperforms a treated experimental group;
2) pre-test scores for an experimental group are significantly higher than post-test scores; 3) there
are significant disordinal interactions between student's aptitudes and instructional treatments;
and 4) where increasing amounts of some instructional treatment produces significantly negative
correlations with learning outcomes. Except for the aptitude-treatment interaction (ATI)
research characterized by condition three above, mathemathantic findings are seldom published
and rarely discussed (Shavelson, Berliner, Loeding, Porteus & Stanton, 1974 and Lohman,
1986).
Previous Discussion of the Mathemathantic Effect
While a number of mathemathantic studies have appeared in the instructional research literature
over the past 50 years, it was Snow (1972) who first suggested the construct to account for wide
swings in performance by similar students who were exposed to different treatments in ATI
studies. Interpreting these negative research results requires an analysis of the role that
individual differences play in learning from instruction. What is mathemathantic for one
aptitude level may be mathemagenic for another. In ATI research, the same instruction is
presumed to support learning for students at one aptitude level while either producing neutral or
negative effects for the other extreme of the same aptitude. In disordinal ATI's, many student's
at one end of a relevant aptitude distribution have their learning depressed by at least one
treatment. Higher aptitude scorers have their learning depressed by a treatment that benefits
those with low aptitudes, and vice versa. Given the current development of instructional theory,
it is easier to account for increased than for depressed performance.
One interesting and often replicated example of a disordinal ATI was provided by Dowaliby
& Schumer (1973). They contrasted "student centered" (i.e. unstructured, discovery oriented)
and teacher centered (i.e. more directed and externally monitored) instruction for undergraduate
psychology students who differed in manifest anxiety. On two examinations covering different
course content over six weeks of instruction, ATI appeared (Figure 1).
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. . . Insert Figure 1 about here . . .
In both sets of data, higher anxiety students profited from the greater structure provided by the
teacher-centered approach. Tobias (1985) reviews a number of similar studies and concludes that
a more highly structured treatment reduces the demands on working memory imposed by the
"worry" about failure that accompanies high levels of anxiety. Yet, few researchers have
attempted to fully explicate the reasons for the depressed performance of low anxious students
assigned to the more structured treatments in these studies.
In another disordinal ATI example, Dansereau et al (1979) taught the coding of text material
into conceptual networks using either a mapping strategy (i.e., a procedure for locating and
connecting similar conceptual categories) or a traditionally instructed control (expository
description of the task requirements) for students with different grade point averages (GPA). The
results (Figure 2) suggest a significant, disordinal ATI where lower GPA students profited from
the strategy contained in the experimental treatment and were able to perform nearly at the level
of higher GPA students. Presumably, some features of the networking treatment compensated
for the skill deficits represented by lower GPA scores. However, here also it was difficult to
explain the very depressed performance of the high GPA scorers except to suspect that the
networking treatment had somehow "interfered" with their learning.
. . . Insert Figure 2 about here . . .
Clark (1982) speculates that a pattern of significant negative outcomes in instructional
research have been ignored because of the practical goals of educational research. The search for
robust instructional methods have led us away from attempts to explain treatments which depress
learning when evidence of a successful alternative exists. Yet, it is likely that an exploration of
the cause of mathemathantic events will increase our eventual knowledge about learning from
instruction.
PURPOSE
The purpose of this paper is to analyze and classify mathemathantic studies. The analysis will
be conducted in order to provide tentative descriptions of the cognitive processes underlying the
different types (and related subtypes) of effects. A taxonomy will be based on a coding of the
cognitive functions served by the diverse instructional methods used in instructional studies. The
approach taken in this analysis makes the assumption that a critical function of all instructional
methods is to externally support cognitive processing (Clark & Voogel, 1985). It will emerge
from the analysis that all mathemathantic effects appear to have one factor in common -- when
instruction
substitutes inadequate or disliked cognitive processes, goals or learner control for
those that are demonstrably more adequate or better liked, learning will suffer. The evidence
reviewed for this discussion indicates that mathemathantic substitution tends to operate in three
areas: 1)
Learning Procedure Substitution where interfering learning strategies are imposed on
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high aptitude subjects or inadequate learning strategies are forced on low aptitude students; 2)
Motivational Substitution where subjects are forced into learning goals that they dislike; and 3)
Student Control Substitution where instructional systems permit student control over the choice
between alternative instructional methods for learning the same subject matter and those students
at the extreme ends of ability distributions seem often to make inappropriate choices for
themselves.
I. LEARNING STRATEGY SUBSTITUTION
Whenever an instructional treatment encourages students to replace an existing, effective
learning strategy with a dissimilar alternative, learning is depressed. Strategy substitution tends
to take two forms: 1)
Novel Strategy Substitution seems to primarily affect higher aptitude
students by replacing more automatic and effective strategies with less familiar strategies.
Different types of novel learning strategy substitutions seem to interact with different
components of general ability; and, 2)
Inadequate Strategies affect lower aptitude students by
providing them with unsuccessful and/or incomplete learning procedures.
Novel Strategy Substitution
When instruction imposes an unfamiliar learning strategy on higher aptitude students
their learning is depressed as they attempt to employ the new approach. These novel learning
strategies interfere with the skill repertories of higher ability students. Nevertheless, the same
strategies that are mathemathantic for higher aptitude learners, often benefit low aptitude
students who simply lack an effective approach. Snow (1980) has argued that aptitude for
learning is best characterized by the efficient use of effective cognitive skill assembly and
control processes. A number of specific cognitive correlates of novel strategy substitution have
been described recently by Lohman (1986). He notes that various component's of higher aptitude
processes may be interfered with by different types of strategy substitution. It is important to
note that there may be two main types of replacement strategies that interfere with the learning of
high aptitude scorers -- either
procedural or declarative learning strategies. This distinction is
compatible with similar categories used to identify the procedural and declarative differences
between learning tasks commonly employed by cognitive psychologists (e.g. Rumelhart &
Norman, 1981). Procedures are specific sequences of behavior that have a goal (e.g. steps in a
procedure for adding three digit numbers). Declarative strategies are thought to be involved in
the connecting of knowledge between domains (Corno & Mandinach, 1983). Here, conceptual
strategies such as analogy and metaphor are used to create new knowledge to solve novel
problems (e.g. the math teachers use of the "slice of pie" analogy to introduce addition and
subtraction of fractions).
The substitution of either declarative or procedural learning strategies may produce dissimilar
mathemathantic effects for students who possess different levels of crystallized or fluid ability
(Snow, 1980). Fluid ability depicts flexible adaptation to
novel learning tasks and is represented
by scores on the Progressive Matrices (Raven, 1962) scale. Crystallized ability represents a
learner's skill at more familiar, school-based tasks. One way to illustrate the possible
interactions between learning strategies embedded in instructional methods and learner aptitude
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would be in a 2 x 2 table similar to the one depicted in Table 1.
. . . Insert Table 1 about here . . .
Declarative Substitutions for High Fluid Aptitudes
Research on novel substitution of declarative learning strategies for high fluid learners is
meager. There is some evidence that when declarative instructional methods such as analogies
are imposed on high fluid scorers, it depresses their performance (Clark & Voogel, 1986; Royer,
1979). Holyoak (1984) noted that only high fluid ability learners seem to spontaneously employ
analogies during learning and problem solving. Sternberg (1985) has argued that the ability to
use analogies successfully characterizes fluid skill. Clark & Voogel (1986) maintain that
analogies serve to facilitate "far transfer" (Royer, 1979) which is the critical skill represented by
fluid ability. Presumably, when instructional methods impose analogies they conflict with the
more successful elaborations that are spontaneously generated by high fluid learners. Students
with high crystallized but low fluid scores on the other hand, might be expected to benefit from
provided analogies. This may be part of the reason why analogical treatments sometimes aid
learning for high ability students and sometimes have mathemathantic effects (e.g. Gick &
Holyoak, 1983; DiVesta & Peverly, 1984; Lohman, 1986).
Procedural Substitutions for High Fluid aptitudes
The substitution of a novel procedure for fluid learners often involves the imposition of
productions that are thought to underlie transfer of learning. Sternberg (1985) describes typical
procedures employed by successful analogy users and solvers. Using an analogy requires the
mastery of procedural steps such as mapping the similarities in both the source and target domain
and excluding negative transfer elements. When these procedures are taught to high fluid
learners, they compete with their existing idiosyncratic procedures that have been serving to
facilitate the use of analogies. The result is presumably that the new procedure is not learned
effectively and the "old" procedure is weakened (Anderson, 1983). For example, Skanes et al
(1974) sought to teach children to solve complex letter series problems at a "high transfer" level
by using either direct practice (a procedure for generalizing rules across cases) or an indirect
practice condition that simply provided rules and offered opportunities to solve problems. The
high transfer, direct practice treatment interfered with students who's Progressive Matrices
(Raven, 1962) scores were above 133. Since the Raven's measure is accepted as one of the
general indicators of fluid skill, one might assume that the higher fluid scoring subjects
experienced interference when they attempted to use the transfer procedure employed in his
direct practice treatment.
Declarative Substitutions for High Crystallized Aptitudes
Novel strategy substitution also produces mathemathantic effects in studies where treatments
provide declarative substitution for high crystallized subjects. Crystallized aptitude tends to
represent expertise in subject matter domains. Here, experience in identifying and employing
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more declarative elaborations of facts, concepts and principles permit domain specific learning
and problem solving to occur. This is probably the explanation for the lack of a relationship
between crystallized aptitude and learning when help and debugging screens were used in a
computer-based training study reported by Snow (1980). In a small scale (n=29) experiment
whose results are only suggestive, Snow explored the relationship between fluid and crystallized
aptitude and various features of computer-based training for teaching college students to solve a
large number of specific computer programming problems. The instructional program included
conceptual models for "debugging" programs and elaborative "help" for other learning problems.
The results of the study might suggest that higher crystallized students did not profit from the
more declarative help and debugging strategies.
Procedural Substitutions for High Crystallized Aptitudes
Anderson (1985) has carefully described the consequences of attempting to replace steps or
entire sequences of automatized learning procedures. This type of procedural substitution
operates like the one described above for high fluid aptitude scorers. The difference is that the
procedures embedded in these treatments seek to replace the school and/or experience-based,
content-rich procedures that characterize crystallized ability (Snow, 1980). Here, instructional
methods force students into an unfamiliar procedure for solving problems in familiar domains.
This was probably the phenomenon that accounts for the data reported by Dansereau and his
colleagues (Dansereau, McDonald, Collins, Garland & Holley, 1979). A "networking" treatment
presented a procedural method to college subjects for a hierarchical mapping technique intended
to facilitate the transformation of text into "node networks" (concept-link relationships) to
facilitate prose processing. A control group was allowed to use their existing strategies for prose
processing. Note in Figure 2 that the imposed networking procedure actually resulted in
significantly lower achievement scores by high GPA students. Here, grade point is presumed to
be representative of crystallized aptitude. The concern with this type of treatment stems from
Anderson's (1983) claim that procedural interference may lead to long-term negative effects.
The mathemathantic effects described thus far affect higher aptitude scorers. Since lower
aptitude learners do not have composed procedures for attacking learning tasks, when other
things are equal, they tend to be helped by the strategies that inhibit the learning of those with
higher aptitude. Yet, there are a class of mathemathantic effects that seem to be specific to those
with low levels of crystallized and fluid skill.
Inadequate Procedural Strategy Substitution
Rohwer (1979), in a discussion of "how the smart get smarter" complained that teachers often
attempt to provide inadequate learning strategies for slow or novice learners. These inadequate
procedures are sometimes derived from analyses of the learning strategies apparently used by
skilled performers at a task. He noted, for example, that mathematics teachers discourage the
concrete "finger counting" that slower math learners use while performing arithmetic operations.
Mathematics teachers tend to encourage their own more "abstract" learning strategies that seem
not to help the lower aptitude or younger subjects. When expert task analyses are used to
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construct procedures to be taught to novices, they are often "incomplete" because they contain
automatized "chunks" of procedures as single steps (Anderson, 1983). Lower aptitude learners
are characterized by a lack of effective learning strategies. In many cases these learners have
never acquired a workable approach to mastering certain learning or problem solving tasks. So,
mathemathantic instructional methods "substitute" an unworkable or irrelevant strategy for no
strategy or for one that is incomplete. Studies by Case (1978) that describe the procedural
defects experienced by children in their performance of developmental tasks such as
conservation of liquids, support this description of mathemathantic effects.
Inadequate Procedures in Computer Based Instruction Studies
Evidence from many short-term instructional studies suggest that "...situations of optimum
learning require a great deal of preparation. If we do experiments in learning with only
superficial preparation -- instructions, "training", etc., of short duration -- then the rare things get
swamped by statistical noise." (p. 6, Hawkins, 1966, in Cronbach & Snow, 1977 p. 130). The
reason for this noise seems to be that students are also swamped. Research subjects and
classroom students seem to require "tuning", i.e. an opportunity to develop procedures that
permit them to utilize new teaching methods and media. When novel and complex instructional
media are introduced to students without sufficient tuning, learning seems often to be depressed.
For example, Clark (1985) describes a number of computer-based instruction studies where
lower general ability subjects were apparently not given sufficient procedural support in the use
of the computer medium. As a consequence, they were unable to take advantage of the lower
cognitive load instructional programs that the computer offered. Clark (1985) cautioned that this
weak preparation element in computer-based instruction studies has confounded the results of
those studies. Because inadequate procedural preparation was offered in a number of key
studies, reviewers in that area have sometimes concluded that computers provide more learning
support for higher ability students (e.g. Kulik, Kulik & Cohen, 1980).
Inadequate Declarative Strategy Substitution
There are also weaker declarative strategies that interfere with learning. Samuels (1967) taught
word recognition using pictures and a no-picture control where the experimenter merely repeated
the word that was presented. For low word-recognition pretest scorers, the no-picture controls
outperformed the picture conditions. The pictures may have been distracting and irrelevant for
low aptitude subjects. In a study by Sullivan, Okada and Nidermeyer (1971) slow readers with
an inadequate letter combination (whole word) treatment performed significantly below their
pretest level on word elements and new word recognition. And, an investigation of note taking
on memory for lecture information for undergraduate students (Peters, 1972) reported that
instructions to take notes significantly reduced the test performance of poor listeners. In
addition, poor listeners in the note-taking condition who were observed
not to be actively taking
notes performed at about the same rate as their untreated control cohorts. One might speculate
that note-taking was irrelevant to the measured performance.
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Inadequate Substitutions for Students With Different Aptitudes
It is likely that a weak strategy substitution influences subjects who are low on crystallized
aptitude differently than those low on fluid aptitude. The study by Skanes et al (1974) strongly
suggests that the aptitude and task differentiation is as necessary to fully understand weak
substitution as it is to explain novel strategy substitution. Skanes and colleagues found that low
crystallized middle school students who were pretested on Thurstone type letter and number
series problems performed significantly worse than non-pretested students. However, higher
crystallized scorers and all levels of fluid aptitude students performed better with pretests. The
experimenters concluded that pretests served as an inadequate and procedural "practice session"
for low crystallized students and imposed on a "muddled background" of skill and served to
"consolidate confusion". Rumelhart & Norman's (1981) discussion of the use of inadequate or
limited analogies in learning are also strongly suggestive of mathemathantic effects for low fluid
scorers when they perform more declarative tasks. They explore evidence that analogies
commonly used by teachers may facilitate short term learning goals at the expense of longer term
goals. For example, the "slice of pie" analogy helps low fluid learners "decontextualize"
declarative understandings of fractions. The analogy apparently helps low fluid ability students
transfer operations from slicing pies or cakes to the addition and subtraction of fractions.
However, the same declarative learning strategy seems to inhibit the learning of the
multiplication and division of fractions through negative transfer (Rumelhart & Norman, 1981;
Voogel, 1987).
II. MOTIVATIONAL GOAL SUBSTITUTION
Learning strategy substitution does not explain another large group of studies where
mathemathantic effects are often found. In the learning strategy substitution studies, ability
differences influenced mathemathantic outcomes. In the next series of studies to be discussed,
general ability differences were controlled and learning differences remained. These learning
differences appear to be due to motivational variables. For example, Domino (1968, 1971,
1974) varied the amount of external direction and control imposed on students who differed in
their scores on the California Psychological Inventory subscales for achievement through
independence (Ai) or Achievement through Conformity (Ac). In his 1968 study, a 2 x 2 design
varied conformity or independence conditions for those with high Ai or Ac scores. When, for
example, conforming behavior was required of high Ai students, their learning suffered
compared to high Ai's with an independent treatment. In his 1971 study, college psychology
students with extreme Ai or Ac scores were assigned to instructors who either required high
levels of conformity or independence. Scores on a variety of learning measures indicated that
while main effects for achievement were generally nonsignificant, disordinal interactions
between aptitude and treatments were highly significant. Learning and enjoyment were both
depressed when students were assigned to teachers whose style was antagonistic to their
preferences. In his 1974 study Domino used Guilford's tests for divergent and convergent
thinking and compared a less directive with a more guided tutorial treatment in a college poetry
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course. There were no main effects but a predicted, significant disordinal interaction was found.
Divergent thinkers achieved more with the traditional open course and convergent thinkers
benefitted most from the structure imposed by the tutorial treatment.
Constructive and Defensive Motivation
The instructional research literature provides many similar examples of studies where varying
the amount of structure and direction imposed on students produces disordinal interactions with
personality variables in a complex that Cronbach & Snow (1977) called "constructive and
defensive motivation". This complex includes achievement through conformity (Ac) and
independence (Ai), measures of trait anxiety (Ax) and need for achievement (nAch) and to avoid
failure. Extreme scores on this complex have been interpreted as expressing strong goal
preferences either to avoid failure or to achieve success. The more "defensive" (i.e. anxious,
conforming and failure avoiding) student seems to work with the belief that failure is more
controllable with higher levels of structure and direction during instruction. Conforming
students seem not to invest sufficient effort in unstructured, more independent settings and
therefore achieve less. The more "constructive" (relaxed, independent and achievement oriented)
student places high value in success and seems to believe that greater freedom during learning
will produce more success. When faced with instructional methods that impose relatively high
amounts of external learning control (e.g. methods that contain more structure, external
monitoring, specific directions and frequent progress testing) the constructive student reduces the
amount of effort they invest in learning. For example, when constructive students were assigned
to organized, sequenced, directive instruction they reduced their effort and achieve less in a study
by Clodfelter (1969). When he gave frequent quizzes to high Ai physics students, they achieved
less than an equivalent group who received no quiz and no feedback. Defensive students (high
Ac) in his study, achieved more with frequent quizzes than in the no quiz condition. There were
no significant differences in GPA between Ai and Ac students.
Replacing Failure Avoiding or Success Directed Motivation Goals
One plausible interpretation of this literature may be found in cognitive motivation theory
(Mook, 1986). Students seem to believe that different types of instructional methods support
different learning goals. The more structure, direction and external monitoring imposed by the
method, the more the instruction is believed to support the avoiding of failure. More
independent, discovery oriented and unstructured methods are presumed to support the
achievement of success. The evidence from existing studies gives some support to the notion
that beliefs seem to underlie motivational mathemathantic effects.
A number of separate studies that provide supporting evidence for this motivational effect
(See Table 2) were described by Cronbach & Snow (1977). More recent discussions of cognitive
theories of motivation (e.g. Schunk, 1984; McClelland, 1985; Morgan, 1984; and Salomon,
1984, Mook, 1986) portray motivation as the consequence of a rational belief system.
. . . Insert Table 2 about here . . .
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Most cognitive theories rest on the assumption that motivation stems from beliefs about one's
self and the learning task. Bandura (1982) and McClelland (1985), tend to agree on the variables
that influence decisions to invest effort in learning. Bandura's theory implies that all human
beings implicitly ask themselves two questions before investing effort, "What is required of me
in this task?" and "Do I have what it takes to achieve my goals?". McClelland (1985) adds
"value" as a different yet significant consideration. He suggests that students are generally
concerned about whether any given segment of instruction has value or incentive for them.
Salomon (1984) characterizes Bandura's task question as the "perceived demand characteristics"
(PDC) in learning and the skill question as "perceived self efficacy" (PSE). He offers evidence
that the relationship between both PDC, PSE and effort is an inverted U (See Figure 3). With
either exceptionally low or high amounts of either PDC or PSE, very little effort will be
expended by learners. McClelland argues that the value one identifies in a task determines
whether one will choose to become engaged.
. . . Insert Figure 3 About Here . . .
It may be that constructive learners value the independent pursuit of success. When
instructional methods allow them freedom to employ their own skills in their own way they
become more engaged in a learning task. In general, their perceived self-efficacy may be
moderately high and moderate self efficacy may often be combined with task demands that are
moderately difficult. Unstructured learning tasks are considered to produce a "higher cognitive
load" (Snow, 1980) for these students. The combination of moderate task load and moderate
self-efficacy is the point where Salomon (1984) predicts the greatest effort expenditure. Yet,
when the increased structure and direction typical of low load treatments cues them to a disliked
"failure avoiding" goal a mathemathantic spiral may begin where: 1)the structured treatment
lowers their perception of the demands of the task
and changes their view of its goal; 2) their self
efficacy judgments grow to very high levels; and 3) their motivation then may sink below the
level necessary to produce the effort necessary to learn. A similar scenario could be imagined
for defensive students. They may have lower self-efficacy (which leads them to value the
avoiding of failure). They also perceive learning tasks as more difficult and demanding than do
more constructive learners. When more unstructured, discovery oriented methods are employed,
defensive may sense the increased probability of failure. This begins their mathemathantic spiral
where: 1) the unstructured treatment increases their perception of the difficulty of the learning
task
and changes their view of its goal; 2) their self efficacy judgments sink to even lower
levels; and 3) their motivation then falls to the point where they are unwilling to invest the effort
needed to learn.
III. SUBSTITUTING STUDENT CONTROL FOR SYSTEM CONTROL OF METHOD
The final mathemathantic effect to be discussed can be understood as a special case of a higher
order interaction of types I (learning strategy substitution) and II (learning goal substitution)
described above. It appears to occur when students with either higher or lower general ability
are allowed to choose between instructional methods differing in amount of structure and
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direction but presenting the same curriculum content. So, for example, college students may be
offered the option of enrolling in a course presented either by a highly directive, structured,
computer-based instructional program that provides constant feedback on progress or they might
select a version that is more traditional and less structured (i.e. lecture with reading and a
midterm and final examinations). Clark (1982) reviews ATI studies where students either
choose or were assigned to treatments that differed in structure and where both learning and
"enjoyment of instructional method" measures were collected (Table 3). He found that there was
a pattern of "antagonistic" outcomes in these studies that fit the definition of mathemathantic
effects. In most of the studies selected for review, students with either high or low general
ability scores reported liking the instructional methods from which they learned the least. Lower
cognitive load methods were preferred more by higher general ability students who liked them
more but learned less from them than from higher load methods. Conversely, the less structured
and more demanding higher load methods were liked better by lower general ability students
who learned much less from them than from the lower load methods.
. . . Insert Table 3 About Here . . .
Clark (1982) and Salomon (1984) have explained this mathemathantic effect using motivation
theory. It appears that when both high and low ability students are allowed to choose between
instructional methods, they make estimates of the "demands" of the alternatives. Clark's (1982)
review it was the low ability students who most often chose higher cognitive load methods but,
as would be expected, learned more from lower cognitive load methods. Why do lower ability
subjects choose higher load methods? Clark suggested that lower ability students tend to choose
the more difficult alternative because these instructional methods permit more freedom and
anonymity during learning. The methods also allow a great deal of freedom for students to
schedule and direct their own time and effort on learning tasks. The openness of the higher load
options enables students to invest
less effort and to fail in private. Clark's explanation for the
tendency of higher ability students to choose lower load methods was also motivational. He
thought that high aptitude learners expected the more structured and directive methods to provide
a more direct and less difficult route to higher test scores. With the benefit of recent
developments in motivation theory, we can expand on Clark's explanation. It is likely that the
antagonism effects depicted in Figure 3 are the result of an interaction of ability and motivational
style. From the studies reviewed in the previous section we might expect that the higher general
ability students in Clark's (1982) review who chose lower load methods were more conforming.
Lower general ability students may have been acting more independently. The only direct
evidence in these studies comes from Peterson (1977, 1979) who collected Ai and Ac scores on
her subjects. As expected, more conforming students liked the lower load methods and more
independent students liked the higher load methods. While there is no direct evidence, it is
plausible to assume that higher general ability students who are also conforming would like
lower load methods but learn more from the higher load. Perhaps the conforming student's
attraction to these methods contributes more variance to the measure of attraction than that of the
independent students. Similarly, Peterson's independent students at all ability levels, expressed
strong preferences for higher load treatments. This preference would be expected to produce a
mathemathantic effect for lower general ability subjects who are allowed to choose methods.
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SUMMARY AND CONCLUSION
In the foregoing discussion, instructional research was reviewed where teaching failures have
produced students who seem to be
less able to use learning skills or had less access to knowledge
in some domain than before they were taught. Three general types of "mathemathantic" (i.e.
where instruction "kills" learning) effects were hypothesized, theoretical explanations for each
effect are examined and representative studies in each area were described. The three types of
effects depicted are those where instruction serves to: 1) Substitute learning procedures (e.g.
Novel learning strategies are hypothesized to interfere with the learning of higher general ability
learners and inadequate learning strategies are provided to those with lower general ability); 2)
Impose less desirable motivational goals on learners (e.g. when teaching methods lead
constructively motivated learners to believe that failure avoidance has replaced achievement
directed goals and, conversely, when defensively motivated students believe that achievement
directed goals have replaced the opportunity to avoid failure); and 3) Substitute student control
for system control over instructional method (e.g. by allowing lower cognitive load instructional
methods to be chosen by high general ability, constructive students and/or by allowing higher
cognitive load methods to be chosen by defensive students who have low general ability). A
summary of these effects can be found in Table 4.
. . . Insert Table 4 About Here . . .
Generally, there appears to be compelling evidence in instructional research for Snow's
(1972) construct of "mathemathantic" effects. Most important is that a psychological
explanation of the types of this negative outcome of instruction requires an analysis of the
individual differences of students. While many different aptitudes may produce harmful
interactions with treatments, the studies surveyed in this report focused on measures of
crystallized and fluid skill (Snow, 1980), and constructive and defensive motivation (Cronbach
& Snow, 1977). Under some treatment conditions it might be expected however, that measures
of prior knowledge, personality and style would also result in mathemathantic effects.
Treatment conditions that appear to combine with aptitude to often produce these effects are
those where 1) learning strategies are embedded in an instructional presentation (they serve as
interfering substitutions for high aptitude learners and sometimes as inadequate learning
procedures for those with low aptitude); 2) the amount of structure and direction provided the
student is either very high or very low (these conditions seem to suggest different motivational
goals to students) and 3) the choice of more or less structured methods is offered (the wrong
choice seems to be made by students who are either very high or low on general ability
measures).
13
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18
30- |
| x Teacher Centered
Quiz | (more structured)
Scores | o
20- |
|
| o Student Centered
| (less structured)
15- | x
=
|
|____________________________
low MANIFEST ANXIETY high
Figure 1. Aptitude Treatment Interaction with anxiety predicting
quiz scores under more and less structured treatments following
Dowaliby & Schumer (1973)
19
45- | x Traditional
| Instruction
|
Detail | o
Mapping 40- |
|
|
|
35- | o
| x Networking
= Training
|____________________________________
low Grade Point Average high
Figure 2. Aptitude Treatment Interaction with school grades predicting
detail mapping on a reading task for treated and untreated groups following
Danserau et al, 1979.
20
TABLE 1.
Studies representing four types of mathemathantic substitutions that result
when either declarative or procedural learning strategies are provided to
students with high levels of crystallized or fluid aptitude.
_____________________________________________________________________________
_______
Examples of Mathemathantic Learning Strategy Treatments
___________________________________________
Declarative Procedural
___________________________________________
High
Gf Holyoak (1984) Skanes (1974)
Ability Analogies Direct practice
APTITUDE
TYPES
High Snow (1980) Danserau et al
Gc (1979)
Ability Debugging Help Networking
______________________________
Gf = Fluid Aptitude
Gc = Crystallized Aptitude
21
High |
|
|
|
|
|
|
Effort
|
|
|
Low |_______________________________
Low PDC or PSE High
Figure 3. Expected relationship between perceived self efficacy (PSE),
perceived demand characteristics (PDC) of tasks and amount of effort expended
during learning (Following Salomon, 1984).
22
Table 2
Motivational Studies Demonstrating Mathemathantic Effects
____________________________________________________________
Study Motivational Treatment
Author(s) Aptitude Type
____________________________________________________________
Parent et al, 1975 Locus of Control Teacher Discipline
Karabenick & nAch + Ax Easy vs Difficult
Youssef, 1968 Paired Associates
Snow, Tiffen & Ascendence Filmed vs live Physics
Siebert, 1965 Assertiveness Demonstrations
Clodfelter, 1969 Ai, Ac Physics Quizzes and
feedback on progress
Hiller, 1972 Ax and Self- Inserted questions in
confidence introductory psychology
O'Connor,
Atkinson & nAch + Ax Ability Grouping of Boys
Horner, 1966
Gardner, 1974 nAch Competitive Teachers
Admion & Dependency Direct vs Indirect
Flanders, 1961 Teacher Style
McKeachie, 1963 nAch + Ax Structured Environments
Peterson, Janicki Teacher guided inquiry vs
& Swing, 1980 Ai + Ac Lecture/recitation
___________________________________________________________
nAch = Need for Achievement
Ai = Achievement through Independence
Ac = Achievement through Conformity
Ax = Trait Anxiety
23
Table 3
ATI Studies Where Students Enjoyed Instruction that produced less
achievement (Following Clark, 1982)
_____________________________________________________________
Study Treatment Mathemathantic Effect
_____________________________________________________________
Wispe Permissive(H) Low Ability liked permissive but
(1951) Directive(L) learned best with directive.
Maier & TV(L), PI or Low ability liked the higher load
Jacobs (1964) TV + PI(H) TV but learned best from TV + PI
Maier & Scrambled Low ability liked the scrambled
Jacobs (1966) PI (H) but learned best from the logical.
Logical PI(L)
Morris & Lecture(L) Highs liked Keller but learned more
Kimbrell Keller (H) from Lecture. Lows liked lecture
(1972) but learned more from Keller Plan.
Peterson Low & High Highs liked low structure but
(1977) Structure gained more from high structure.
Lows liked low structure but learned
from high.
Peterson Low & High Highs liked low structure but
(1979) Structure learned more from high. Lows liked low
structure but learned from high
structure.
Peterson & Small Group(H) Lows liked high load small group
Janicki(1979) Large Group(L) but learned more from low load large
group.
____________________________________________________________
PI = Programmed Instruction
L = Low Cognitive Load Treatment
H = High Cognitive Load Treatment
____________________________________________________________
24
Table 4.
Summary of Mathemathantic Effects
______________________________________________________________
Mathemathantics are more likely when instruction serves to:
1) Substitute Learning Procedures, such as when...
A) Novel Learning Strategies Replace Automatic Productions.
Low load methods substitute weak procedures for the more automatic and
successful productions that characterize High G learners. The result is that
the weak procedure interferes with and weakens the successful procedure and
produces lower learning scores than would be achieved with no treatment.
B) Inadequate Strategies are Provided
Low load methods may also substitute faulty procedures for the incomplete
productions that characterize Low G learners. The result is an incorrect
procedure that is becoming automatic through practice. While the procedure
may be learned, the task(s) it is designed to serve will not be learned
because the learning procedure is faulty. e.g analogy in fractions, "learning
to learn" skills.
2) Substitute Motivational Goals, such as when...
A) Failure Avoidance Replaces Achievement Directed Goals.
Low load methods are typically perceived as supporting the need to avoid
failure. High Ai students are described as valuing success and the conflict
elicited by their preferred goals and those implicit in the instructional
method lead them into avoidance behaviors which reduces the amount of effort
they invest and, in turn, their learning.
B) Achievement Directed Replaces Failure Avoiding Goals.
High load methods are typically perceived as supporting the need for
success and achievement. This places high Ac students in conflict with their
need to avoid failure. This conflict leads to avoidance behavior.
3) Substitute Student Control for System Control of Method,
(This mathemathantic effect is the higher order interaction between the
method, ability and motivational types described above). Such as when...
A) Higher Load Methods Chosen by High G Students.
When high G students who are also high Ac are allowed to choose between
higher and lower load methods, they will typically choose on the basis of
their motivational goal (to avoid failure through conformity) and select
lower load methods. While these methods are liked and the student is both
engaged and willing to expend effort, they do not learn as much as they would
from high load methods because of the effect described in #1 above.
B) Lower Load Methods Chosen by Low G Students.
When low G students who are also high Ai choose between methods that place
more and less cognitive load on them, they will typically choose the higher
load method. This choice satisfies their values but leads to less learning
because they require the compensatory procedures and structure of the lower
load method.
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