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SOME ISSUES in the study of affect
and mathematics Learning
DRAFT VERSION FOR DISCUSSION IN TSG30, ICME-11
July 8, 2008, Monterrey, Mexico
Gerald A. Goldin
Rutgers University
geraldgoldin@dimacs.rutgers.edu
The domain of affect and motivation in the psychology of mathematics education urgently requires greater research attention. This paper discusses some challenging problems in how we can approach the study of complex affect in the learning of mathematics. We consider both the individual and classroom levels. Among the issues raised are those of construct validity, reliability, and the reproducibility and generalizability of the research. These are especially difficult to achieve in the context of theoretical perspectives on affect that take account of its complexity and its subtlety.
The Nature of the Problem
The prevalence of math anxiety in the general population, at least in the United States, is well known. Many prospective elementary school teachers feel considerable unease about mathematics, while even people who have been successful in mathematics and like the subject can recall painful or discouraging experiences associated with mathematical learning in school. Slow learners and gifted children alike continue to share in profound frustrations or dissatisfactions.
But this is far from the whole story. In some classrooms, and in some homes, the learning of mathematics occurs in an overall emotional climate of enjoyment, excitement, and satisfaction. Here children seem to develop beliefs in their own capabilities for insight and reasoning, a sense of ownership of the power that mathematics confers, and positive attitudes or orientations all contributing to what is sometimes called intrinsic motivation.
Despite these generally-accepted observations, the affective domain is not highlighted in the 2000 Principles and Standards for School Mathematics of the National Council of Teachers of Mathematics.
The federal No Child Left Behind (NCLB) Act, which mandates sanctions for schools which fail to make adequate yearly progress (AYP) in achieving threshold percentages of students demonstrating mathematical proficiency (as evidenced by standardized state tests), is influencing the direction of mathematics teaching. The orientation is toward the procedural learning of computational and other routine performance skills through repetition, posing additional challenges to the creation of classroom environments that foster intrinsically motivational affect. The broad political support for NCLB reflects a highly-prevalent set of beliefs about what effective mathematical learning is, which tend to discount the value of measures other than test performance.
In addition, the U.S. Department of Education has sought (with some justification) to encourage research that studies the (causal) effects of various teaching methods and curricula, using quantitative experimental designs based on inferential statistics. While such studies are important and of practical utility, they tend to require the definition and measurement of dependent (outcome) variables based on instrumentation suitable for large numbers of students or classes. This in turn can discourage the study of more complex variables or structures requiring more labor-intensive methods. In the cognitive domain we again have a tilt toward test scores, as opposed to interview data or alternative assessments. In the affective domain, survey data (e.g., to study attitudes and beliefs) are easily obtained from large numbers of participants. But the study of emotions and affective structures in individuals, or the affective climate and social interactions among children at the classroom level, is extremely difficult to scale up.
In addition, the consideration of emotions and affective structures related to mathematics in individual learners to be inferred from observations of their mathematical behaviors, from the analysis of individual interviews, or both raises unsolved and daunting problems of construct validity and reliability. Facing these difficulties, how can we anticipate achieving research findings with some claim to reproducibility or to generalizability across wider populations of learners?
Theoretical Constructs
Some of the theoretical ideas that I brought to the study of affect at the individual level (over the dozen years or so beginning around 1988) were arrived at through the study of mathematical problem solving in one-on-one, task-based interviews. One focus of analysis was the interplay of problem solvers emotional feelings with their strategic and heuristic decisions.
The key ideas, many of which come from joint research with Valerie DeBellis, include the following:
(1) The idea of affect as an internal system of representations, in which emotional feelings have meanings (significations) and carry essential information.
(2) A broad, tetrahedral model for describing the affective domain in connection with mathematics, which includes (a) states of emotional feeling, (b) attitudes, (c) beliefs and structures of belief, and (d) values, ethics and morals.
(3) The essential role of meta-affect in transforming affect, where meta-affect includes affect about affect, affect about cognition about affect, the affective context of affect, and affective monitoring.
(4) Affective pathways, or recurring sequences of emotional states, which interact with mathematical strategies and heuristics, and which (as they repeat) lead to the development of global affect in the individual.
(5) Affective structures, which (in analogy with cognitive structures) incorporate affective pathways and meta-affect with cognition.
(6) Mathematical intimacy (a way of being with mathematics that entails engagement, importance, depth, and vulnerability) mathematical integrity (open awareness and acknowledgment of mathematical understanding or the absence and limitations of understanding), and mathematical (self)-identity (the persons sense of who I am in relation to mathematics, including self-concept and self-efficacy).
Currently (since 2005) I have been working under the auspices of the MetroMath Center at Rutgers University with a substantial group of faculty and graduate students to study the affect of students in three urban, inner-city 8th-grade classrooms engaging in conceptually challenging mathematics. Based on the preliminary, qualitative analysis of classroom videotapes, together with retrospective, visually-stimulated recall interviews with selected children, Yakov Epstein, Roberta Schorr, and I have proposed the construct of an archetypal affective structure (or an engagement structure). This is an idealized pattern, situated in the individual, that includes: characteristic behavior, affective pathways, information or meanings encoded by the emotional states, self-talk and associated affective responses, meta-affect, interactions with (including evocations of) problem-solving strategies and heuristics, interactions with the individuals beliefs and values, and interactions with the individuals structures of self-identity, intimacy, and integrity, and socioculturally-dependent as well as idiosyncratic expressions of affect.
Examples of such structures include:
(1) Dont Disrespect Me. This involves the persons experience of a perceived threat to his or her status, dignity, well-being or safety, that can occur when ideas are challenged during a mathematical discussion. Maintaining face comes to supersede the mathematical content under discussion.
(2) Check This Out. This involves the individuals realization that successful engagement with the mathematics can have a payoff, leading to increased (intrinsic) interest in the task itself, or increased (extrinsic) interest in an external benefit.
(3) Stay Out Of Trouble. This involves the persons avoidance of interactions that may lead to conflict or emotional distress, so that aversion to risk comes to supersede the tasks mathematical aspects.
(4) Its Not Fair. Here the experience of a sense of unfairness within a group problem-solving effort, e.g. with the level of participation by others in the group, leads to a disinvestment in the mathematical ideas in the task, and a desire just to get it done.
Evidently some of these patterns, when they occur, contribute to intimate mathematical engagement, while others may contribute to a different sort of engagement or impede further engagement. To this point, we have found the inference of such constructs to be extremely helpful in our understanding of classroom interactions around conceptually challenging mathematics, and in our ability to characterize the apparent emotional and motivational consequences of particular teacher interventions during mathematical activity.
Scientific and Methodological Issues
It is apparent that attitude surveys, or even structured interviews with individuals, are likely to be insufficient or inapplicable on a large scale to measure and characterize such constructs as meta-affect, mathematical intimacy and integrity, or the highly complex affective interactions such as those we seek to describe via archetypal affective structures. Here I would like to raise as questions some of the scientific issues involved.
Let me note that one approach to qualitative research, sometimes characterized as a postmodernist paradigm, fundamentally rejects the possibility or value of objectivity. Adopting this view may be an attractive way for some to escape the challenges discussed here. It renders the description of classrooms and of students affect as an inherently subjective, interpretive story, about how an observer/participant interacted emotionally with a group of learners of mathematics. The study of affect then takes on many of characteristics of literary fiction, and we obtain, at best, a powerful, inspirational story (or, possibly, an unpersuasive or discouraging account). But we relinquish from the start the hope of learning something true or valid about mathematical affect and its role in students learning, on which we can build systematically to improve mathematics education.
To be able to proceed scientifically, we need to address some fundamentals.
Construct validity. It should be an early priority to determine whether or not we are speaking of something meaningful, identifiable, and observable when we entertain notions such as affective pathways of emotional feeling, meta-affect, and so forth. Let us take a meta-affective example. Is it valid to say that while different individuals may experience frustration during mathematical problem solving, some have overall negative feelings about their feelings of frustration (e.g., evocative of possible failure), while others have more positive feelings about their feelings of frustration (e.g., evocative of heightened interest in the problem)? One approach to this question is to develop several distinct ways to make observations from which such inferences can be drawn, and to look for consistency. In this example, such methods might include (a) individual interviews to determine whether people recognize such feelings about feelings in themselves (spontaneously or when prompted), (b) interviews to determine whether people describe others as having such feelings about feelings, (c) inferences drawn from the observation of uninterrupted problem-solving behavior, such as differences in subsequent behaviors when problem solvers show (verbal or non-verbal) evidence of frustration, (d) the consistency of inferences drawn from uninterrupted problem-solving behavior with inferences drawn from retrospective, stimulated-recall interviews, (e) inferences drawn from a problem solvers response to questions at the moment of occurrence of the feeling, i.e., interrupting the problem-solving to gather affective information, (f) problem-solvers written responses to survey questions, or written accounts of their affective experiences. While each of these methods has drawbacks and limitations, a degree of consistency among them can not only strengthen our evidence for the validity of the construct, but can sharpen our definition of it.
Likewise, we need to develop distinct lenses through which we can approach the validity of our still more complex constructs in classroom-level studies. In our current work, we find some confirming evidence in the retrospective interviews for particular archetypal affective structures that are inferred from classroom observations. But we must develop ways to investigate their validity systematically, including study of the recurrence of the patterns described in a variety of different classrooms and mathematical task contexts.
Reliability of inferences. In most of our work up to now, one or more researchers draw inferences, which are shared in small-group discussions. Differences of opinion are resolved through further review of videotapes and transcripts, and either a consensus opinion is reached (regarding, for example, a childs emotional state, or the meaning of a particular social interaction), or (often) we conclude that the information is insufficient to allow us to distinguish among several possibilities. We need to move to a stage of investigating inter-observer reliability of inferences regarding affective events. This requires careful the definition of classification alternatives in a coding scheme based on what we want to observe. For example, we have identified the notion of a classroom environment that is emotionally safe for childrens interactions around conceptually challenging mathematics. Supposing this to be valid construct, a sufficiently detailed description is needed to allow different observers to independently arrive at estimates of the degree to which such emotional safety is present so that the extent of agreement can be evaluated, the points of agreement identified, and possible points of disagreement resolved.
Generalizability and reproducibility. We are certainly quite far from being able to claim to have identified definitively the teaching techniques or interventions that lead children to develop powerful affective structures for mathematics. Rather we are at a much earlier stage of formulating some key constructs and offering tentative and preliminary hypotheses. Nevertheless, it is important to move from the level of just a few individual classroom studies to the point where we can perform comparable studies in a set of classrooms, and can look for cognitive and affective outcomes across different groups of students.
To achieve this, we must develop in-depth instrumentation for the systematic study of more complex affect during mathematical activity, with evidence of validity and reliability. An important challenge is to do this in a way that permits scale-up, without sacrificing our ability to observe the most important constructs of interest. Then we can look for correspondences (or lack of correspondences) between information gathered through survey data, and the more fine-grained information about affect obtained from the use of in-depth instrumentation.
This paper is intended mainly to raise questions and offer possibilities suggested by current research directions, rather than to propose definite recommendations. I look forward to a discussion of these and related issues in Topic Study Group 30 at ICME in 2008.
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This research is supported by the U.S. National Science Foundation (NSF), grant no. ESI-0333753 (MetroMath: The Center for Mathematics in Americas Cities). Any opinions, findings, and conclusions or recommendations are those of the author and do not necessarily reflect the views of the NSF.
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