Teacher Turnover (Research - Quantitative)

Level: Primary   Learning Theory: Social Cognitive Theory  

Imitative Aggression in Children (Research - Quantitative)

Important early study of observation learning-- children exhibited aggresive behaviors after observing aggressive cartoons as well as after observing peer models. The term “imitation” is used to refer to the learned behavior.

Level: Secondary   Learning Theory: Social Cognitive Theory  

Is Self-Efficacy More General than Previously Thought? (Research - Quantitative)

Abstract: The generality of academic self-efficacy judgments was examined among 588 high school students. Students rated their confidence for solving 42 problems in English, Spanish, U.S. history, algebra, geometry, and chemistry. Confirmatory factor analyses showed that students’ efficacy perceptions prevailed beyond the boundaries of specific problems. The 1st-order model with a separate self-efficacy factor for each school subject displayed the best fit. Verbal and Quantitative Academic Self-Efficacy illustrated the relations among the 1st-order factors better than General Academic Self-Efficacy. The generality of academic self-efficacy partly depended on the degree of perceived similarity among tasks. When asked to rate their efficacy toward 8 pairs of isomorphic algebra and physics problems, students reported more comparable strengths of self-efficacy as they perceived greater similarity between the problems.

Learning Theory: Social Cognitive Theory  

Self-Concept Vs. Self-Efficacy (Research - Quantitative)

Bong and Clark (1999) compare and contrast self-concept and self-efficacy along theoretical and empirical dimensions. They cite Shavelson et al’s (1976) definition of self-concept as the “perceptions of one’s self” that develop through an interplay of experiences, cognition, and affect. Self-concept is characterized as organized, multifaceted, hierarchical, stable, developmental, evaluative, and differentiable. The cognitive aspect of self-concept includes descriptive and evaluative components, and that the evaluative components often relate to comparisons to others (inferiority and superiority). Self-esteem refers to one’s general feelings of self-worth in which one is treated as a global entity. Self-efficacy, by focusing on one’s perception of capability to complete a given task in a specific context, is conceptually different, but often confused with self-concept. Some research suggests self-efficacy predicts self-concept. Self-concept effect sizes on achievement are often not definitive (average correlation of .21), while self-efficacy is a better predictor of both performance (.38) and persistence (.34). Self-concept has stronger relationships to anxiety, apprehension, intrinsic motivation, and value than self-efficacy. There is strong research evidence that self-concept and self-efficacy influences achievement more than the reverse effect, especially for older students (after 4th grade). The authors suggest that self-concept research could benefit from assessing cognitive components (self-concept of ability) and affective components separately, thus improving predictive power of the construct.

Elementary vs. Secondary Teaching Majors (Research - Quantitative)

Education vs. Non-Education Majors (Research - Quantitative)

Level: College   Learning Theory: Social Cognitive Theory  

Problem-Solving Calibration of Canadian College Students (Research - Quantitative)

Looked at 64 Canadian college students’ self-efficacy judgments on cognitive performance, problem-solving strategies, and the accuracy of self-evaluation of responses. Author concludes “self-efficacy is a viable construct for comprehending performance, particularly on academic tasks required sustained self-monitoring” (p. 353).

Math Anxiety, Prior Experience, Confidence in Preservice Teachers (Research - Quantitative)

Statistical Characteristics of Future Teachers (Research - Quantitative)

Level: Secondary  

Gender Gap and Socialization (Research - Quantitative)

interesting theoretical view of the gender gap... pertains to high-achieving students (like math majors)

Level: College   Learning Theory: Social Cognitive Theory  

Manipulating Mastery Experiences and Self-Efficacy (Research - Quantitative)

How does performance influence self-efficacy, task interest and self-evaluations of performance?

Level: Middle   Learning Theory: Social Cognitive Theory  

Self-Efficacy Calibration (Research - Quantitative)

Abstract: This study investigated seventh graders' math self-efficacy calibration and its effect on students' math performance, individual differences, such as gender, as well as academic variables, such as previous math achievement, post-performance effort judgment, and post-performance self-evaluation. According to Bandura (1986), students' self-efficacy beliefs about their capability to perform affects how they make choices of activities, courses of action, amount of effort to exert, and length of time engaged on a task. To date, the issue of the accuracy judgment of self-efficacy beliefs, termed calibration , has received little investigation. In the present study, it was measured in two ways: accuracy, which measures the magnitude of judgment errors; and bias, which measures the direction of judgment errors. In addition, the design of the study enabled the researcher to assess the relationship between students' personal processes (e.g., self-efficacy judgments of math capability, calibration, effort judgment, and performance evaluation) and variations in context (e.g., specific math problems and their difficulty level).

The results revealed that students' calibration accuracy significantly increased the predictiveness of their self-efficacy beliefs. Path analysis showed that calibration accuracy had both direct and indirect effects on math performance, with the indirect effects mediated through the students' self-efficacy beliefs. Self-efficacy played a direct role in predicting students' math performance, post-performance self-evaluation, and post-performance judgments of effort. The effects of prior math achievement on math performance were mediated largely through the students' self-efficacy beliefs. Unexpectedly, the effect of self-efficacy on post-performance judgments of effort was negative, indicating that high efficacy students needed to spend less effort in solving the math problems than low efficacy students. As for the individual differences in gender, the study found no statistical differences on any of the dependent measures, although boys had numerically higher self-efficacy, post-performance self-evaluation, and lower effort judgment than girls. In conclusion, the results revealed that students' self-efficacy beliefs play an important role in their acquisition of mathematical competence. Such information can be vital in assisting educators to tailor interventions that will enhance students' beliefs in their capability to learn math and as well as their actual success

Level: Middle   Learning Theory: Social Cognitive Theory  

Self-Efficacy Calibration (Research - Quantitative)

This is the closest article to my proposed dissertation design. It really should be memorized word-for-word. Using path analysis techniques, Chen found significant and independent effects of calibration, self-efficacy, and prior math achievement (as measured by the ITBS) on a mathematics test based on TIMMS items. Chen found different results when students rated their confidence on the same task versus different tasks. The generalizability and power of the study is limited by a relatively small sample size and sample of seventh graders at a catholic school in Tennessee. Chen’s findings of the significance of variables in her model are particularly helpful for my design, including her finding that gender was not a significant predictor of any other variable in the model. Chen also incorporated task difficulty in the model as a “level” variable. Good quote: “As a group, seventh-grade students overestimated their math capabilities, but their inaccuracies did not relate to the strength of their self-efficacy beliefs. Both high and low self-efficacy students were overly optimistic about their performance.” (p. 91)

Level: Middle   Learning Theory: Social Cognitive Theory  

International Calibration Comparison (Research - Quantitative)

This article builds on Chen’s dissertation by applying the same protocol for assessing Taiwanese students’ self-efficacy, performance, and calibration. In the review of literature, the authors point to the work of Bol and Hacker (2001), Ewers and Wood (1993), and Pajares and Graham (1999) in suggesting that “accurate estimations of capability may be important to the academic success of gifted or highly achieving students” (p. 223). This may have implications for my study, because many mathematics majors would qualify as “gifted or highly achieving.” A major focus of this study is on the role of task-difficulty in self-efficacy judgments. Because Taiwanese content is more difficult than American, the authors compared seventh-grade U.S. students to sixth-grade Taiwanese students. The researchers found “more similarities than differences” in the self-efficacy ratings and calibration scores for students in the two countries, although there were differences favoring Taiwanese students in performance and effort.

Level: Secondary  

Gender and Ethnicity Differences in Course Taking (Research - Quantitative)

Very small sex differences in mathematics course taking found in the 1990 NAEP data.

Level: College  

Mathematics Confidence Scale (Research - Quantitative)

Dissertation addresses the reliability and validity of a mathematics confidence scale for college women.

Level: Primary   Learning Theory: Social Cognitive Theory  

Mathematics Self-Efficacy by Gifted and Sex Variables (Research - Quantitative)

Gifted and average-ability fifth graders show no differences in math self-efficacy by gender. Gifted students have higher self-efficacy than average students. Also addressed Prediction Calibration. Small study.

Level: College   Learning Theory: Social Cognitive Theory  

Statistics Self-Efficacy-- Current and To Learn (Research - Quantitative)

Abstract: We developed measures of current statistics self-efficacy (CSSE) and self-efficacy to learn statistics (SELS) to address whether statistics self-efficacy is related to statistics performance, and whether self-efficacy for statistics increases during an introductory statistics course. Both instruments yielded reliable, one-factor solutions that were related positively to each other and to two measures of statistics performance (i.e., specific statistics problems and overall course performance). The CSSE and SELS also were related positively to math self-efficacy and attitudes towards statistics, but related negatively to anxiety. Changes between two different testing occasions using the CSSE indicated that statistics self-efficacy increased almost two standard deviations over a 12-week instructional period

Level: College   Learning Theory: Social Cognitive Theory  

Effects of Major and Gender on Calibration in Mathematics (Research - Quantitative)

This important first study on calibration includes path analysis and regression approaches to assessing self-efficacy and performance in college men and women. In arguing for self-efficacy as a predictor of career decision making, Hackett and Betz cite Bandura’s contention that mathematics anxiety is a consequence of low self-efficacy, and thus self-efficacy is a more important predictive variable. One finding includes “Hackett (1985) reported the results of a path analysis indicating that mathematics self-efficacy contributed more significantly than sex, years of high school mathematics, ACT mathematics score, or mathematics anxiety to predicting the choice of a mathematics-related college major.” The authors found no gender differences in calibration or performance. Contrary to subsequent studies, self-efficacy outweighed prior performance in influencing achievement on the mathematics performance measure.

Level: College   Learning Theory: Social Cognitive Theory  

Manipulating Mastery Experiences and Self-Efficacy (Research - Quantitative)

experimental manipulation of self-efficacy by passing or failing math problems

Level: K-12  

Inconsistent Relationship between Self-Esteem and Performance (Research - Quantitative)

Abstract: This meta-analysis examines the relationship between the various self-measures and measures of performance and achievement. The statistical results of 128 studies are transformed to a common measure, namely, correlation coefficients. These studies represent a total sample of 202,823 persons and produce a data base of 1,136 correlations between self-ratings and performance measures. A range in the relationship of -.77 to .96 was reported with an “average” correlation of.21. It was found that this average relationship was modified by a number of variables. The more significant modifiers of the average relation- ship were the grade-level of subjects, socioeconomic status, ethnicity, ability of subjects, self-term used in the study, name of self-test used, type and name of performance/?achievement measures, and the reliability of both the self-ratings and performance/?achievement measures.

Level: College  

Mathematics Attitudes and Beliefs of Prospective Elementary Teachers (Research - Quantitative)

Abstract: This research establishes the Prospective Elementary Teachers’ Mathematics Attitudes and Beliefs Survey with the following four dimensions or subscales: (1) the Prospective Teachers’ Personal Confidence About Mathematics (2) Usefulness of Mathematics Content (3) Perception of Former Teachers’ Attitudes and Beliefs About Mathematics Ability and (4) the Prospective Teachers’ Attitudes and Beliefs on Teaching Mathematics to Elementary Students. The Prospective Elementary Teachers’ Mathematics Attitudes and Beliefs Survey is administered three times: once at the beginning of the Mathematics Teacher Education Course, once at the end of the Mathematics Teacher Education Course, and once during the fall of the prospective teachers’ first year teaching elementary students.

Level: College  

Pedagogical Content Knowledge (Research - Quantitative)

Level: College   Learning Theory: Social Cognitive Theory  

Testing the Motivational Efficiency Hypothesis (Research - Quantitative)

Abstract: A regression design was used to test the unique and interactive effects of self-efficacy beliefs and metacognitive prompting on solving mental multiplication problems while controlling for mathematical background knowledge and problem complexity. Problem-solving accuracy, response time, and efficiency (i.e. the ratio of problems solved correctly to time) were measured. Students completed a mathematical background inventory and then assessed their self-efficacy for mental multiplication accuracy. Before solving a series of multiplication problems, participants were randomly assigned to either a prompting or control group. We tested the motivational efficiency hypothesis, which predicted that motivational beliefs, such as self-efficacy and attributions to metacognitive strategy use are related to more efficient problem solving. Findings suggested that self-efficacy and metacognitive prompting increased problem-solving performance and efficiency separately through activation of reflection and strategy knowledge. Educational implications and future research are suggested.

Level: Primary  

Relationship between Formative Assessments and Goal Orientations (Research - Quantitative)

Abstract: This study examined the role of formative assessment in the development of student goal orientations. Students in one elementary school were examined over the course of a school year as they participated in a reading program using continuous formative assessment. Mastery orientation remained consistently high and performance orientation decreased. Students’ personal goal orientations were significantly related to their perceptions of their teachers’ goals for reading.

Teacher shortages (Research - Quantitative)

Level: College  

Performance/elf-Efficacy for Preservice Math Teachers (Research - Quantitative)

Within a cross-sectional design of pre-service middle school mathematics teachers in Turkey, Isiksal (2005) used explored students’ mathematics self-efficacy and performance in relation to gender and year-in-program effects. Isiksal found women consistently modestly outperformed men in the program, but that men and women held similar self-efficacy views. The author also found that self-efficacy ratings increased with year in the program, which he interpreted as evidence supporting Bandura’s four sources of self-efficacy. In other words, Isiksal interpreted the apparent increase in mathematics confidence to the cumulative effect of mastery experiences in college mathematics (and education) courses.

Level: College  

Content Preparation of Preservice Math Teachers (Research - Quantitative)

LISREL (Research - Quantitative)

Common Structural Equation Modeling Software

Level: College  

Math Educators and Technology Use (Research - Quantitative)

Mathematics educators talk about how little they use technology in their methods, etc. classes.

Level: K-12   Learning Theory: Social Cognitive Theory  

Overconfidence of Students with Learning Disabilities (Research - Qualitative)

Level: College   Learning Theory: Social Cognitive Theory  

Efficacy and Career Interests in Math Performance (Research - Quantitative)

Lapan et al cite research that suggest high school mathematics preparation (ACT scores, mathematics courses taken) and mathematics self-efficacy ratings explain significant and independent portions in observed sex differences in men and women’s choice of mathematics-related careers (math, science, engineering). Good quote: “Results from the present study strongly support the key role of math self-efficacy (Betz & Hackett, 1983; Hackett, 1985) as a critical filter (Sells, 1980) in the developmental process through which women either embrace or reject math/cience college majors. In this study, choice of a math/cience major was largely a function of adapting to self-efficacy (Bandura, 1977) and vocational interest patterns (Hansen & Campbell, 1985) that predated student entry into college.” (p. 289)

Level: College   Learning Theory: Social Cognitive Theory  

Sources of Self-efficacy and the relationship to Career Choices (Research - Quantitative)

sources of self-efficacy helped explain gender differences in math self-efficacy

Learning Theory: Social Cognitive Theory  

Sources of Self-Efficacy (Research - Quantitative)

Learning Theory: Cognitive Information Processing  

Calibration in Postdictions (Research - Quantitative)

Abstract: Two experiments attempted to improve the quality of people's probability assessments through intensive training. The first involved 11 sessions of 200 assessments each followed by comprehensive feedback. It produced considerable learning, almost all of which was accomplished after receipt of the first feedback. There was modest generalization to several related probability assessment tasks, but no generalization at all to two others. The second experiment reduced the training to three sessions. It revealed the same pattern of learning and limited generalization. About one-third of all subjects appeared to use probabilities quite appropriately on some tasks before training began. Further research is needed to understand why the training worked as well as it did, why that training did not always generalize, and why some individuals seemed to need no training at all. [Relates to calibration research in other arenas; specifically, on probability assessments in which people estimate the probability that a given statement is true. With training, this article says there is improvement.]

Level: K-12   Learning Theory: Social Cognitive Theory  

Reading Calibration (Research - Mixed Methods)

Reviews literature surrounding “calibration of comprehension” for students engaging in reading tasks. The authors place calibration in the context of metacognition (specifically evaluating knowledge instead of regulating cognition) and stress the importance of multiple measures of calibration (not just a single task) “Comprehension is a continuous variable and should be measured by multiple questions.” (p. 367) The review looks at 34 studies of young adults (college students) . Results include (1) students tend to use both self-beliefs of ability and information from tasks when rating their confidence of comprehension, (2) there is little research relating pretest and posttest calibration, (3) interest in a domain may be used to assess confidence on tasks, (4) there is an “illusion of knowing” effect related to overconfidence expressed by students on moderate and difficult tasks, (5) students tend to rate their likelihood of correctly answering an item at around 70 to 75%, (6) there is little research into the effect of item difficulty on pretest ratings. Good quote: “There is a tendency for adult students to generate unrealistic feelings of knowing when it comes to evaluating outcomes of learning. As can be seen in the present review, overconfidence is a common phenomenon among young adult students that may result in inadequate learning due to premature termination of cognitive processing.” (p. 384)

Level: Secondary   Learning Theory: Social Cognitive Theory  

Sources of self-efficacy in High School (Research - Quantitative)

Level: College  

Estimates of Mathematics Majors (Research - Quantitative)

Level: College  

U.S. Math Departments (Research - Quantitative)

Page 127 or 137 gives percents of depts using graphing calculators, specifically for calculus.

Level: College  

Statistical Summary of Mathematics Programs (Research - Quantitative)

Level: College  

Metacomprehension accuracy = Calibration (Research - Quantitative)

Abstract: The authors investigated absolute and relative metacomprehension accuracy as a function of verbal ability in college students. Students read hard texts, revised texts, or a mixed set of texts. They then predicted their performance, took a multiple-choice test on the texts, and made posttest judgments about their performance. With hard texts, students with lower verbal abilities were overconfident in predictions of future performance, and students with higher verbal abilities were underconfident in judging past performance. Revised texts produced overconfidence for predictions. Thus, absolute accuracy of predictions and confidence judgments depended on students’ abilities and text difficulty. In contrast, relative metacomprehension accuracy as measured by gamma correlations did not depend on verbal ability or on text difficulty. Absolute metacomprehension accuracy was much more dependent on types of materials and verbal skills than was relative accuracy, suggesting that they may tap different aspects of metacomprehension.

Level: Middle  

Performance Avoid=Bad, Mastery=Good, Performance Approach=? (Research - Quantitative)

Abstract: This study extends previous research on the relations among students’ personal achievement goals, perceptions of the classroom goal structure, and reports of the use of self-handicapping strategies. Surveys, specific to the math domain, were given to 484 7th-grade students in nine middle schools. Personal performance-avoid goals positively predicted handicapping, whereas personal performance-approach goals did not. Personal task goals negatively predicted handicapping. Perceptions of a performance goal structure positively predicted handicapping, and perceptions of a task goal structure negatively predicted handicapping, independent of personal goals. Median splits used to examine multiple goal profiles revealed that students high in performance-avoid goals used handicapping more than did those low in performance-avoid goals regardless of the level of task goals. Students low in performance- avoid goals and high in task goals handicapped less than those low in both goals. Level of performance-approach goals had little effect on the relation between task goals and handicapping.

Level: College  

Statistics of Mathematics Teachers (Research - Quantitative)

Level: College  

Content Preparation of Preservice Math Teachers (Research - Quantitative)

Level: College  

Preparation/ualifications of Math Teachers (Research - Quantitative)

Level: College  

Sex-differences in Expectations for Success in Math (Research - Quantitative)

most overconfident, men more so than women

Level: College   Learning Theory: Social Cognitive Theory  

Self-Efficacy, Gender, and Career Interests (Research - Quantitative)

O’Brien, Kopola, and Martinez-Pons (1999) describe an investigation into a literature-based model of 400 secondary students’ interest in mathematics-related careers. The authors tested a model that incorporated a general mathematics self-efficacy construct (Hackett & Betz, 1985), gender, ethnic identity, and SES. Self-efficacy was the strongest correlate of career interest, prior mathematics score (PSAT), and ethnic identity, although there was no correlation between gender and self-efficacy. In fact, the only variable that correlated to gender was interest in mathematics-related careers (males were more interested than females).

Assumptions of Multiple Linear Regressions (Research - Quantitative)

Lists assumptions and gives preactical reasons for testing for them.

Level: Middle   Learning Theory: Social Cognitive Theory  

Motivational Factors, Self-Efficacy, and Performance in Middle School Math (Research - Quantitative)

Pajares and Graham (1999) set-out to assess the effects of a variety of motivational factors in predicting mathematics performance among (N=273) middle school students. The authors also sought to assess whether these effects change during students’ first year in middle school. In a review of self-efficacy research, the authors say, “Across ability levels, students whose self-efficacy is higher are more accurate in their mathematics computation and show greater persistence on difficult items than do students whose self-efficacy is low.” (p. 125) In terms of gender differences, Pajares and Graham describe six studies that found no differences in performance between boys and girls, but that boys held higher confidence in mathematics than girls starting in middle school and persisting through high school. Other motivational variables that predict academic performance include math anxiety, self-concept, self-efficacy for self-regulation, perceived value, and academic engagement (persistence and effort). The article contains detailed explanation of self-efficacy and calibration instrumentation, with justification for the choices of measures in the literature. The authors also highlight the implications of administering self-efficacy surveys on high stakes assessments, which is atypical for studies of self-efficacy and academic achievement. The authors found no gender differences, but did find that gifted students performed better, held higher self-efficacy ratings, and better calibrated than non-gifted students. After controlling for all other motivational variables, self-efficacy was the largest predictor of performance, and the only significant predictor on both administrations of exams.

Level: Middle   Learning Theory: Social Cognitive Theory  

Path Analysis of Self-Efficacy and Self-Concept on Achievement (Research - Quantitative)

Level: Secondary   Learning Theory: Social Cognitive Theory  

Measuring Self-Efficacy and Performance (Research - Quantitative)

Tested three methods of measuring calibration when students were asked to rate their confidence in doing tasks and then completed open-ended and multiple choice tests. Calibration was lower for the open-ended test format, but did not differ significantly based on method of measurement or pre-alg vs. algebra students. The authors found no gender differences on any of the self-efficacy, calibration, or performance measures. The authors caution that "using identical self-efficacy and performance indexes in an effort to closely match belief and criterion may lead to positively biased estimates of effects from self-efficacy to performance outcomes. Thus, researchers are encouraged to use similar rather than identical items or tasks to assess self-efficacy beliefs and performance criteria" (p. 220).

Level: College   Learning Theory: Social Cognitive Theory  

Teacher Prep and Attitudes toward Mathematics (Research - Quantitative)

Teacher prep can improve teachers attitudes toward mathematics, including self-efficacy

Level: Adult  

Calibration in Postdiction Consumer Knowledge with General Self-Efficacy (Research - Quantitative)

Learning Theory: Cognitive Information Processing  

Calibration of Postdictions (Research - Quantitative)

Path Analysis Guidelines (Research - Quantitative)

contains criteria for path analysis and reporting... VERY GOOD

Path Analysis Step-by-Step (Research - Quantitative)

Some procedural steps for path analysis.

Level: College  

Possible to Improve Calibration (Research - Quantitative)

Abstract: In two experiments, it was examined whether the accuracy of comprehension monitoring (metacomprehension accuracy) was improved by summarizing texts. College students read texts and then some wrote a summary of each text (either immediately after reading or after a delay—the delay between reading and summarizing was filled by the reading of the remaining texts), whereas others did not (the control group). All the students then rated their comprehension of each text. Finally, they completed a test of the material covered in each text. In both experiments, metacomprehension accuracy, operationalized as the correlation between ratings of comprehension and subsequent test performance, was dramatically greater for the group of students that wrote summaries after a delay than for the control group or the group of students that wrote summaries immediately after reading a text. These findings are described in the context of a discrepancy-reduction model of self-regulated study.

Learning Theory: Social Cognitive Theory  

Meta-analysis of Sources of Self Efficacy Research (Research - Mixed Methods)

Complete synthesis of the literature on sources of self-efficacy in education. Includes a nice summary of the very limited qualitative research into the sources, along with a recommendation for more of it. Thorough discussion of the quantitative and construct validity issues surrounding measurement of the sources.

Level: Primary   Learning Theory: Social Cognitive Theory  

Self-Efficacy and Goals in Academic Motivation (Research - Quantitative)

Abstract: The causal role of students’ self-efficacy beliefs and academic goals in self-motivated academic attainment was studied using path analysis procedures. Parental goal setting and students’ self-efficacy and personal goals at the beginning of the semester served as predictors of students’ final course grades in social studies. In addition, their grades in a prior course in social studies were included in the analyses. A path model of four self-motivation variables and prior grades predicted students ‘final grades in social studies, R = .56. Students’ beliefs in their efficacy for self-regulated learning affected their perceived self-efficacy for academic achievement, which in turn influenced the academic goals they set for themselves and their final academic achievement. Students’ prior grades were predictive of their parents’ grade goals for them, which in turn were linked to the grade goals students set for themselves. These findings were interpreted in terms of the social cognitive theory of academic self- motivation.

Level: Middle   Learning Theory: Radical Constructivism  

Cognitively Guided Instruction (Research - Quantitative)

Level: Middle   Learning Theory: Situated Cognition  

Street Mathematics (Research - Mixed Methods)

Level: Primary   Learning Theory: Behaviorism   Methodology: Case Study  

Individual Programmed Instruction (Research - Qualitative)

Benny is a student who develops incomplete understanding of mathematics by working for several years in a individualized programmed instruction curriculum. Benny "learns" that mathematics is sometimes like magic and that there are multiple answers for a given mathematical problem, but that equivalent answers may be incorrect because they do not follow the form on the answer sheet. This early example of a qualitative study was influential in mathematics education because it provided a counterexample to the benefits that behavioral researchers attributed to programmed instruction that was founded on Skinner's principles of conditioned responses. Though Benny was excelling in his program, Erlwanger was able to gain insight into Benny's many misconceptions through tasked-based interviews with qualitative follow-up questions. Benny had invented many "rules" to fit the feedback he received from the answer keys, but understood very little mathematics. Poor Benny.

Level: Primary  

Evaluating a Reform Initiative (Research - Quantitative)

Level: Primary   Learning Theory: Social Constructivism   Methodology: Case Study  

Mathematical Discourse in Teaching Exponents (Research - Qualitative)

Level: Middle   Methodology: Comparative Case Study  

Teachers' Views of Mathematics (Research - Qualitative)

Level: K-12  

Informal Probability Understanding (Research - Quantitative)

Fischbein and Schnarch (1997) administered a probability questionnaire to 20 students in each of grades 5, 7, 9, 11 as well as 18 preservice teachers in college. The instrument (p. 98-100) asks questions directly related to seven intuitive (as opposed to logical) misconceptions identified in probability research. While the authors claim their instrument was designed to study the evolution of students' misconceptions over time, they did not collect time series data, choosing instead to conduct independent random samples. Their results suggest students at all ages frequently hold the misconception that a 5-6 and a 6-6 dice throw are equally likely. Five of the main misconceptions consistently increased with age, except college students held fewer misconceptions regarding the effect of sample size on probabilistic events. The only misconception that appeared to reduce with age was the "conjunction fallacy", defined as the mistaken belief that the probability of an event appears to be higher than the probability of the intersection of the event with another event (e.g., If Dan wants to be a doctor and enrolls in school, it might seem more likely Dan is enrolled in medical school than in school). The seven misconceptions, in order of the their frequency in the populations, were 1) Compound and Simple Events, 2) Effect of Sample Size, 3) Heuristic of Availability, 4) The Conjunction Fallacy, 5) Representativeness, 6) Negative Recency, and 7) The Time Axis Effect. Each effect, along with its roots in literature, is described on p. 100. Many fifth graders did not respond to some of the items, suggesting their understanding of probability was too low for intuitive misconceptions to have formed.

Level: Middle  

Students Conceptions of Algebraic Proof (Research - Mixed Methods)

Healy and Hoyles aimed to investigate the impact of the new proof component in England’s National Curriculum using mixed methods. The authors used hierarchical linear modeling (students at Level 1, class, school, teacher, and curriculum at Level 2). Students and teachers were asked separately to choose proofs of number theoretic statements based on the one that most resembled a proof they might give as well as the proof they thought would receive the best marks. The most popular choice for “own method” was the least popular choice for “get the best mark”; however, this could be due to the survey instrument because the task of “List the one proof that most closely resembles the method you prefer” was IMMEDIATELY followed by “List one that might receive the best mark”. The authors point to evidence that students “transported” (or transferred) the investigations approach in their curriculum when trying to construct proofs of number conjectures. Students used techniques like the ones they were taught. Students tended to use empirical arguments, but were aware that they were not considered valid for their teachers (“they knew more was expected of them”). The students' viewed the purpose of proof to be verification, explanation, examples, or ritual (p. 417) Moreover, the study found gender differences: “Higher general mathematics competence is associated with better constructed proofs, with girls performing better than boys” (p. 421) The study originally aimed to compare schools and teachers, but found no differences in students’ choices associated with teacher effects-- found more unexplained variation within schools than between schools.

Level: K-12  

Informal Probability Understanding (Research - Quantitative)

This article tests a theoretical model for students' informal understanding of probability called the outcome approach. The outcome approach posits that students view probabilistic statements, e.g., "their is a 70% chance it will rain", as aids in predicting single outcomes rather than frequency based statements (i.e., it's probably going to rain vs. it will rain 7 out of 10 times). Students with an outcome orientation tend to think of probabilistic events as causal relationships with weighted influences (the object will land face up because it is heavier on the bottom and has a small side on the top), which may lead to misconceptions. To test his model, Konold conducted and analyzed task-based interviews of 16 undergraduates for instances of incorrect informal reasoning. These initial problems (the Weather Problem, the Misfortune Problem, and the Bone Problem) were chosen to vary along dimensions relating to the sample space, chance factors, and cultural tendency to view the problem statistically. The students responses were coded according to the outcome approach model in order to predict students' performance during follow-up task-based interviews. The predictive strength of the model was strong, suggesting that students who used outcome approaches in the initial tasks used them again in the follow-up tasks. Konold connects an outcome orientation to a "personalist interpretation" of probability, whereby students use prior experience and "feelings of certainty" to assist their probability judgments. These strong prior conceptions, based on experience, may be viable in everyday probability settings but might also interfere with the learning of formal probability. The article also includes appendices of problems that tend to illicit outcome approaches from students.

Level: Secondary  

Algebraic Reasoning (Research - Quantitative)

Suggesting that "teachers' beliefs about students' ability and learning greatly influence their instructional practices" (p. 168), Nathan and Koedinger (2000) set out to investigate the degree of correspondence between students' performance on algebraic problem solving tasks and rankings made by (n=67) secondary teachers and (n=35) mathematics education researchers regarding the relative difficulty of the algebra problems. Informed by a review of literature, the authors distinguish between start-unknown (SU) and result-unknown (RU) tasks in Algebra I. Nathan and Koedinger further categorized the tasks into story problems (prose and context), word equation problems (prose but no context), and symbolic equations (of the "solve for x" type). Combining the two categorizations, the authors designed and administered an algebra exam that contained each of the six types of elementary algebra problems to (n=245) students who completed Algebra I in middle school or high school (Koedinger & Nathan, 1998). In descending order of percentage-correct, students were most likely to successfully complete RU Story, RU Word, SU Story, RU Equation, SU Word, and SU Equation. The most successful student solution strategies were guess-and-check and "unwinding" (working backwards). Though the sample of teachers and researchers was a convenience sample, the beliefs implicit in the rankings of the teachers did not accurately match the students' performance. When asked to rank-order the six types of algebra problems from "easiest" to "hardest" for students in Algebra I, the mean teacher and researcher list was RU Equation, RU Word, RU Story, SU Equation, SU Story, and SU Word. Thus, the teachers and researchers appeared to stress the result-unknown/tart-unknown order above the roles played by word, story, and equation problems. Based on the results of the surveys, the authors propose an alternative developmental model for understanding algebra that they call the Verbal Precedence Model (VPM). The VPM more successfully categorized students' solution patterns than the Symbol Precedence Model implicit in the teacher and researcher rankings and most Algebra I textbooks.

Level: Primary   Learning Theory: Radical Constructivism   Methodology: Case Study  

Informal Probability Understanding (Research - Qualitative)

Nicolson (2005) reports on a series of three videotaped interviews with a fifth grader named Paul. The task-based interviews introduced Paul to common probability activities surrounding flipping a single coin (once and then ten times), drawing from candy bags with different distributions of raspberry and blueberry candies, rolling a single die, and spinning a spinner with colored regions. Though well-articulated, Paul's subjectively based interpretations of chance-- level 1 according to Jones' (1997) framework-- were based largely on prior experiences and incorrect generalizations from small trials. Paul believed that occurrences involving chance were entirely unpredictable unless a physical (deterministic) explanation could be found, and thus did not fully understand representativeness. For example, Paul reported that when he flips coins they usually come up heads, but when he flipped a coin ten times it came up tails six times. Paul reasoned that the result was different because the "tail" of a penny was lighter than the head and that since he usually flipped a coin by reversing the coin onto the back of his hand at the end the results actually supported his theory that the heavier side lands face up. Paul used similar logic to describe dice and spinner outcomes. Nicolson found Paul's "misconceptions" remained after six hands-on classroom lessons on probability, suggesting that Paul's beliefs were not changed by classroom experience. Nicolson concludes that probability understanding may not improve from empirical activities (e.g., with coins or spinners) because variations in distributions of results may be explained by "luck", "loaded dice", "extra effort", etc., and thus may not give students persuasive reasons to abandon subjectively based probability beliefs. Nicolson instead advocates for more "real-world" experiences that are not based on repeated trials; for example, What is the probability of me picking a student's name out of a hat with a summer birthday?

Level: K-12  

Probability and Coins (Research - Quantitative)

As part of her dissertation, Rubel's (2007) investigation of student's reasoning with probability related to coin tasks is thorough and remarkably well-set in related literature. The article is a great example of integrating clinical or "teaching interviews" into a well-designed quantitative study of students' reasoning. The diagnostic task asked grade 6-12 students at a private New York high school to complete tasks (adapted from existing instruments) related to coin-toss scenarios and common misconceptions regarding sample space, randomness, compound events, and independence. Using Chi-squared analysis and interviews of 33 students, Rubel found much of the errors in students' responses could be explained using a theoretical framework combining representativeness heuristics, the negative recency effect (p. 533), core beliefs about coins (p. 534), and Jones' classification of student reasoning (subjective, transitional, informal quantitative, numerical). A unique strength of Rubel's Probability Inventory was that it asked for justification of all solutions, meaning that students' reasoning could be classified into general categories. The clinical interviews added great validity to Rubel's claims, with transcripts of students' explanations providing some of the most compelling evidence for how students think about independence and distributions of outcomes (see especially "Bob" on p. 545).

Level: Middle   Methodology: Basic  

Trial Size and Empirical Probability (Research - Qualitative)

Level: Middle   Methodology: Basic  

Probability Simulation (Research - Qualitative)

Stohl and Tarr's qualitative case study of students' learning of statistical inference through probability simulations summarizes a twelve-session instructional intervention using a technology-based learning environment. The researchers developed a series of activities using the software program Probability Explorer, which were then implemented in a sixth grade classroom of average ability students. The activities were designed to engage students in thinking about common conceptual difficulties regarding probability, including connections between theoretical probability and expected distributions at differing sample sizes. The researchers videotaped three pairs of students during all class sessions and analyzed the recordings using Powells analytic method. The final report focuses on two students, Manuel and Brandon, and their developing understanding through three critical instruction sessions referred to as Mystery Marble Bag, the Spinner Simulation task, and Schoolopoly. Transcripts of conversations between and among the students and the teacher-researcher support claims of developing understanding. The authors conclude that carefully designed problem solving tasks can foster sound understanding of probability and statistical inference in a technology-enhanced learning environment.

Level: K-12  

Assessing Best Practices (Research - Quantitative)

Level: K-12  

Fairness and Probability (Research - Quantitative)

The authors suggest students and teachers should consider the fairness of dice (and other probability tools) as an important and questionable property. They report on a longitudinal study of 78 grade 3, 5, 6, 7, and 9 students in Australia and Tasmania. Initial survey data of students' conceptions of fairness in dice was followed by clinical interviews of a subsample of 44 students 4 years later. The students responses to the survey and subsequent interviews supported previously reported observations that students may hold a variety of conceptions about the fairness of dice, from idiosyncratic beliefs that dice are unfair (based on experience), inconsistent beliefs that dice are fair but extremes (1 and 6) are less likely, unistructural beliefs that dice are fair for theoretical reasons (their shape, number of sides, etc.), multistructural beliefs that dice are fair subject to how they are manufactured and rolled, and relational beliefs that short term outcomes may vary but long term behavior of dice represents fairness. Students' strategies for determining whether a particular set of dice is fair ranged from Ikonic (intuition and luck-based), Untestable (they are always fair), Observational (unsystematic trials or inspection of physical dimensions), and Empirical (large or small sampling based on recording trials). The results of the study indicate that many students progressed from an idiosyncratic or inconsistent belief of dice fairness to a unistructural belief system. Most students' strategies for evaluating dice did not change over time; students consistently used the Observational approach. The authors did not employ an educational intervention to improve students' strategies for assessing dice fairness, so suggest the students' normal education over the four years did not help them to develop empirical strategies or begin to see dice as fair only under certain conditions. They suggest experimenting with loaded dice or unfair coins could help students learn about theoretical and empirical distributions as well as variation and inference.

Level: Secondary   Methodology: Case Study  

A Computer-Probability Paradox (Research - Qualitative)

Wilensky is a mathematician associated with the Connected Mathematics project who conduction 17 in-depth interviews with participants ranging from 14 to 64 years old (averaging 7 hours each). The interview questions related to statistical and probability concepts, and the case study he presents is of Ellie, a computer programmer, who is asked to give a solution to Bertrand's Paradox: "From a given circle, choose a random chord. What's the probability that the chord is longer than a radius?" Since "choose a random chord" is not a well-defined term, there are multiple "right" answers to the question, and Ellie used a solid argument to show the result is 2. Wilensky, seeking to challenge Ellie's reasoning, gave a logical argument for the probability equaling 3. Ellie struggled with the concept, wrote a computer program to simulate her reasoning, and was not able to resolve the cognitive conflict until she considered her code in the context of real life (i.e., "we could drop pins on a circle, and see which way they pointed too", "it depends on what you mean by random"). Wilensky believes the interaction with the problem, together with Ellie's attempts to simulate her reasoning on a computer (using thousands of "turtles" in StarLogo), helped her to better understand the epistemological foundations of "randomness" and reflect on the possibility of multiple correct interpretations of the problem. Wilensky also warns of "black box simulations" that do not allow students access to the underlying code of a simulation. He argues for giving students access to controlled programming environments based on the perceived conceptual thinking of Ellie, but since the "student" in his interview was a professional computer programmer, his claims may not transfer to typical secondary student populations.

Level: College  

Progressivism vs. Transmission and its Effect on Student Perceptions (Research - Quantitative)

Level: College  

Effects of a Quantitative Literacy Course (Research - Quantitative)

Ellington reports on an assessment of the impact of introductory mathematics courses at Virginia Commonwealth University on students' quantitative reasoning skills. Through workshop meetings and collaboration among faculty, the mathematics department developed 16 multiple choice items to test quantitative reasoning skills relative to topics in unit analysis, interpretation of charts and graphs, proportional reasoning, counting, percentages, percent increase or decrease, use of mathematical formulas, average, and exponential growth. Every student taking Contemporary Mathematics, College Algebra, Precalculus, or Calculus I in one of the three semesters between Fall 2002 and Fall 2003 completed a random subset of 4 of the 16 items as part of a required placement test (before taking a math class) and again as part of the final examination in the course. Performance on the placement test items may have been negatively biased by instructions that described a quarter-point penalty for incorrectly answering questions on the placement test. The items listed on final exams were described as bonus items. The summary statistics suggested that a similar number of items were answered in both settings, with a marked increase in the mean percentage of correct responses between the pre- and post-test items. All classes combined, students answered 13 of the 16 items significantly more often after taking one of the introductory mathematics courses. However, students taking the only course that specifically addressed quantitative reasoning (Contemporary Mathematics) both (1) initially scored lower than the combined mean on all of the items and (2) showed less improvement than the combined sample. The final exam mean score was significantly higher than the placement means on only 9 of the 16 items in the Contemporary Mathematics group. The results in the Contemporary Mathematics group might be due to lower initial quantitative reasoning skills, ineffective instruction, or limitations in the validity of the instrument.

Level: College  

Interdisciplinary Algebra (Research - Quantitative)

This article describes the effects of a course developed at Southeastern Oklahoma State University entitled 'Algebra for the Sciences'. The course introduced college algebra topics by first introducing science topics through guest lectures from professors in Biology, Physics, and Chemistry. The research team compared student performance in four sections of the Algebra for the Sciences class to corresponding performance of four sections of College Algebra over the course of a year. Problem-solving skills were assessed by comparing mean scores on common final exam items, critical thinking skills were assessed using the Watson-Glaser Critical Thinking Appraisal, and student attitudes at the end of the course were measured using Likert-type responses to five statements such as "math is important in my life" and "I found this class to be interesting." Students in the two groups performed equally well on the all of the individual common problem-solving items. In terms of critical thinking, the Algebra for the Sciences students outperformed the College Algebra students on only one subscale: "Inference". However, the students taking the interdisciplinary algebra course expressed much more favorable views toward the course than those taking the traditional algebra course (Elliot et al., 2001, Table 3). The results are strikingly more positive in the experimental group, especially on the items "I found this class to be interesting" (84% agreed compared to 42%) and "The materials in this course are related to practical situations" (90% agreed compared to 53%).

Level: College   Methodology: Basic  

Types of Isomorphism during Problem Solving Transfer (Research - Qualitative)

In the context of mathematical cognition, isomorphism refers to when an individual recognizes a surface-level or structural-level correspondence between two mathematical problems or statements. Greer and Harel set isomorphism within the larger context of knowledge transfer, which was extensively researched in the 1970's and 80's (summarized by Lave, 1988). A primary goal of this theoretical article was to propose three models for cognitive isomorphism: surface-level isomorphim, deep isomorphism, and mediated isomorphism. Examples of each proposed model distinguish them from each other. Considerable effort is spent discussing teaching implications of isomorphism research, including the use of analogies, manipulatives, and teacher-imposed isomorphisms as solution aids.

Level: College   Methodology: Spreadsheets in College Algebra  

The researchers' discussion highlighted the likelihood that some mathematical knowledge developed in college can be transferred to the workplace easily. Spreadsheets were identified as being particularly important to this transfer by virtue of their ease of use. Students also appeared to respond positively to the fact that the police inspector used mathematics only in response to a clear need. However, when the role of the spreadsheet was not apparent in necessary calculations, students had difficulty applying their mathematical knowledge to the situation. (Research - Mixed Methods)

This short proceedings article reports on a case study of college mathematics students who are introduced to individuals who use spreadhsheets in the workplace. Participating students with limited college mathematics background met with (1) a police inspector who analyzed job performance of officers in a large city, and (2) a finance specialist at a medium-sized company. Qualitative analysis focused on the role of mathematics in the police inspectors' work as well as the interaction between students and the Inspector as the participants attempted to make sense of the mathematics used in the spreadsheet application.

Level: College   Learning Theory: Social Cognitive Theory  

Guided Exploration and Learning to Search (Research - Quantitative)

Level: Secondary   Learning Theory: Social Cognitive Theory  

Path Analysis of Self-Efficacy (Research - Quantitative)

This is a powerful path analysis attempt at describing the role of self-efficacy in mathematical achievement in problem solving tasks. Using a test of general mental ability (psychometric g), opportunities for students to predict their performance, and problem solving performance tasks, the authors were able to implement aspects of Bandura's theory of self-efficacy to construct a structural equations model for performance that included math anxiety, gender, race, general ability, prior math achievement, self-efficacy, and performance. Students were found to have low calibration (they overestimated their performance ability). "Students' self-efficacy about their math capability had strong direct effects on math anxiety and on mathematical problem-solving performance even when general mental ability was controlled" (p. 17) Race differences in confidence were also found.

Level: Primary   Learning Theory: Behaviorism  

Behaviorism in Arithmetic (Research - Quantitative)

Bonds, or associations, or connections, are reinforced patterns of interacting with the environment. In this early paper on the psychology of learning, Thorndike describes his intuitive theory of how an individual forms and maintains bonds for arithmetic. As the frequency of correct responses to a predetermined type of task increase, we see evidence for strengthened bonds. Thorndike believes that if conditions are set properly, students can develop strong bonds in mathematics through the rewards of successfully and reliably completing basic tasks. Later, students are able to generalize certain abstract properties of numbers from the many examples they complete during practice. The article also argues for inductive approaches to arithmetic instruction, where students naturally discover deeper principles of numbers only after they become comfortable with procedures.

Level: Secondary  

SES and School Size (Replicates O'Callaghan, 1998) (Research - Quantitative)

Level: College  

Cooperative Learning in Remedial Math (Research - Quantitative)

Level: Secondary  

Calculators in Secondary Math (Research - Quantitative)

Level: Primary  

DIF: Testing Items (Research - Quantitative)