Beliefs, attitude and emotions: new views of affect in
mathematics education
Douglas B. McLeod
Reference section 17 on “Affect and Mathematical
Problem solving: A new Perspective” by Douglas B.
McLeod and Verna M. Adams, springer-Verlag New,
1989.
“On a more positive note, Brown and Walter(1983) discuss how making conjectures can
be a source of great joy to mathematics students. Similarly, von Glasersfeld (1987) notes
the powerful positive emotions that often go along with the construction of new ideas or
the cognitive reorganization of old ideas. Lawler (1981) also documents the positive
emotional responses that accompany that moment of insight when a child first sees the
connections between two important ideas.” P. 250
“This approach could be particularly important for gender-related differences in
mathematics education (Tittle, 1986), where the impact of the social context and affective
factors is likely to be particularly strong.” P.254
“Affect and Mathematical Problem solving: A new
Perspective” by Douglas B. McLeod and Verna M.
Adams, springer-Verlag New, 1989.
“Seeing the pattern not only gives students the answer, but it gives them confidence as
well.” P 26
“ Problem solvers who find a solution have a sense of achievement that fuels their “
looking back” and taking stock in a “review” phase. “ p27
“ students need help in experiencing the joys of setting up conjectures (Mason, burton,
And stacy, 1982) the aid of problem posing (brown and walter, 1983), and the social
rewards that come from group work on solving problems.” P29
“ Instruction in problem solving should help students deal with negative emotions, such
as the inevitable frustrations of trying to solve nonroutine problems. It should also
provide those positive experiences that help students enjoy problem solving.” P 29
“The relatively short duration of emotional states in problem solving can be contrasted
with the typical conception of attitude toward mathematics (Halodyna, Shaughnessy and
shaughnessy, 1983). Attitude is usually defined as a general, longterm emotional
disposition toward the subject.” P 30
“If we can help problem solvers become aware of their emotional reactions, they should
improve their ability to control automatic responses or unconscious responses to
problems. Because the automatic responses are frequently inadequate, greater awareness
should improve the problem solver’s chances for success.” P 30
References
Beck and Energ, 1985 anxiety disorders and phobias: a cognitive perspective. New york:
Basic Books
Meichenbaum, 1977. Cognitive behavior modificatiob: An interactive approach. New
York: Plenum Press.
“ Metacognitive and managerial processes seem particularly susceptible to the influence
of emotions; for example, decisions about whether to persevere along a possible solution
path may be heavily influenced by student confidence or anxiety.” P 31
Reference:
Mandler G. (1975) Mind and emotion. New york: Wiley
“ Another perspective on attitudes toward mathematics is that the best way to foster
positive attitudes in students is to increase the level of understanding of mathematics.” P
38
“…some educators posit a causal connection between attitudes toward mathematics and
achievement in mathematics with a change in attitude bringing about a corresponding
change in achievement. “ p 38
“Retrieval of information from memory depends on upon where the information resides
within the network. Information that has many links to other nodes usually has a higher
probability of retrieval than knowledge that is unlinked because there are more paths
through the linked network.” Page 51
“…this activation may also inhibit another set of nodes through negative connections.”
P51
reference:
Marchall, Pribe and smith, 1987: Schema knowledge structures for representing and
understanding arithmetic story problems. Arlington, VA: Office of Naval Research.
*Meaningful Learning
 Conditional Learning: assimilating based on what will be asked
“ While solving the first story problem, the child encodes in memory certain aspects of
situation.” P 53
“ As the child encounters the second problem and makes an error, the link between the
positive affect node will be weakened, a competing link will be formed with a negative
affect node. Repeated failures will strengthen this link. Repeated successes weaken it
and strengthen the positive one.” P 53
“ One can surmise that it may be easier to change affective responses that are coded
simultaneously than to alter posterior encoding…Because the simultaneous encodings
result from specific instances, they have links to identiable parts of the schema. If
positive experiences can be created that link to these same parts, a tension can be
generated between the positive and negative responses to the same features of the
problem.” P 54
“ There were fairly obvious individual differences among subjects in the tendency to
express aesthetic judgements. Two factors appeared to affect the likelihood that aestheti
evaluations would appear in a subject’s protocol: problem difficulty and think-aloud
interference.” P 69
“…When a subject was having difficulty with a problem, he or she was less likely to
express any aesthetic feeling. We assert that this phenomenon is due to the taxing
cognitive load carried by an individual who is having difficulty solving a complex
problem. Conversely, we assert that when an expert knew that he or she could solve a
given problem, the expert had the cognitive luxury of aesthetic considerations. “ p 69
“…that aesthetic monitoring is not strictly cognitive but appears to have a strong
affective component; that is, decisions or evaluations based on aesthetic considerations
are often made because the problem solver “feels” he or she should do so because he or
she is satisfied or dissatisfied with a method or result.” P70
“ Instead of showing our students only the finish products, we must involve them more
actively in the processes of shaping, refining, and polishing.” P 72
From chapter 6 (self confidence, interest, beliefs, metacognition; lester, garofaldo and
kroll)
“ …an individual’s failure to solve problem successfully when the individual possesses
the necessary knowledge stems from the presence of noncognitive and metacognitive
factors that inhibit the appropriate utilization of this knowledge. These factors are of at
least four types: affects and attitudes, beliefs, control, and contextual factors.” P 75
“ Kerr(1977) have suggested that tolerance of ambiguity and resistance to premature
closure are also important correlates of problem-solving performance.” P 76
Interest: a liking for or willingness to engage in solving problems
Confidence: Believing his or her ability to succeed in solving even hard problems
Self-confidence: a consequence of a belief about self.
Belief: Individual’s mathematical world view. Subjective knowledge
“ Beliefs often interact with and, at times, shape attitudes and emotions, and beliefs
influence the decisions made during problem solving.” P 77?
“ the sorts of interactions students have among themselves and with their teachers, as
well as the beliefs, values, and expectations that are nurtured in school contexts, shape
not only what mathematics is learned, but also how it is learned.” 78
“ Desire for a good grade was stronger than desire to avoid solving boring problem.” P84
“ …the beliefs a person holds about his or her ability to do mathematics, about the nature
of mathematics, and about problem solving are dominant forces in shaping that person’s
behavior while engaged in work on a mathematics task.” P 85
From Information Technologies and Affect in Mathematical Experience by Kaput.
“ The overall strategy in applying multiple, linked representations is to take abstract or
feature-base representations that are hard to learn and use, and link them with concrete
and feature rich representations; thus, a student working in the abstract representation can
ask for feedback in a more concrete, feature-rich, and “readable” representation
whenever desired.” P 90
“By checking on the boundary to activate the window, the student can view the
consequences of his or her input in any window, including the icon window, which most
important for our purposes.” P 94
“Ownership of the decision process as well as the repair process (If needed) is in the
hands of the student, although a teacher, or even the computer, can be called upon for
help if needed.” P95
“Affective consequences of this ownership dimension in such learning environments
need to be researched.” P 95
“Given that the cybernetically linked representations can be cognitively linked, then the
cognitive structures associated with any one representation are enriched by connections
to those cognitive structures associative with the other representations. “ p 95
“Mandler’s approach is based on the notion that affective experience follows from the
interruption of schema-based plan.” P 96
“…two steps are essential for solving a traditional algebra word problem: a) extract from
the text enough information about the quantitative relationships expressed there in to
build a cognitive representation of the situation described in the text, and b) translate
those quantitative relationships into the representation system of algebra to solve one or
more equations whose solution helps answer the question posed by the original problem.
Traditional pedagogical approaches have emphasized the translation process (stepb) but
have ignored the importance of building a cognitive representation that is sufficiently
elaborated to support that translation process; thus, the translation of the problem into
algebra is separately a very difficult step.” P 97
“The schema and plans that result from the reading of the original problem, which
necessitate writing an equation and then solving that equation in order to solve the
original problem, are immediately blocked, and the student has no choice but to feel
frustrated or to engage in trivial translation processes (e.g., key-word matching).” P 97
“ …school mathematics as expressed by most students is compartmentalized into
meaningless pieces that are isolated from one another and from the students’ wider world.
Symbols are manipulated without regard to the meanings that might be carried, either by
referents of the symbols or by actions on them. “ p 99
The sources of mathematical reasoning:
a. Transformations within and operations on a particular representation system.
b. Translations across mathematical representation systems
c. Translations between non-mathematicallly described situations and mathematical
representations
d. D. The consolidation and reification of actions, procedures, or webs of related
concepts into phenomenoligcal objects that can often serve as the bases of new
actions, procedures, and concepts at a higher level of organization. “ p 100
“The first source involves the recognition or elaboration of patterns in one’s symbol
manipulations-a matching or elaboration of previously learned cognitive structures with
new ones generated by the surface forms of the symbols that one is transforming. “
p100
“…the cognitive structures associated with each representation system become linked,
leading to what Lawler has called an elevation of control (Lawler, 1981, 1985). The
multiple-representation software environments are intended to help support these kinds of
meaning building.” P 100