Level: Secondary   Methodology: Basic  

Clickers (Research - Qualitative)

Clickers.

Level: College   Methodology: Basic  

White Future Teachers and Racism (Research - Qualitative)

Case and Hemmings (2005) chronicled the communication strategies of preservice white teachers enrolled in a course focused on the effects of White racism in education and society. The authors' qualitative analysis of interview data and in-class conversations suggested the preservice teachers used a number of distancing strategies to avoid talking about racism, including silence in the classroom and among friends and family, dissociation from racist labels, claiming to be color-blind, and blaming reverse racism. They concluded, however, "metadialogic" strategies can be used to engage preservice teachers in reflecting upon the "white talk" of others and thus depersonalizing a topic many preservice teachers associate with feelings of guilt.

Level: Secondary   Methodology: Basic  

TI-Navigator (Research - Qualitative)

Level: Secondary   Learning Theory: Social Constructivism   Methodology: Basic  

Qualitative Study of Graphing Calculators (Research - Qualitative)

Qualitative study on issues surrounding the graphing calc., including its negative effects on cooperative learning.

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   Learning Theory: Situated Cognition   Methodology: Basic  

Why Doctoral Math Students Leave and Stay (Research - Qualitative)

Herzig (2002) summarizes her dissertation investigation into graduate students in mathematics and faculty members at a large doctoral mathematics program. Using a situated learning perspective, and focusing on participating in mathematics communities and authentic activities, Herzig found that some students left the program due in part to a lack of positive experiences with faculty members. Herzig also relates the importance of how graduate students and faculty viewed qualifying exams and mentorship opportunities.

Level: K-12   Methodology: Basic  

Examples of "Good" CRT (Research - Qualitative)

Level: K-12   Methodology: Basic  

Teachers' Views of Technology (Research - Qualitative)

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: College   Methodology: Basic  

Concept Image and Definition (Research - Qualitative)

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   Learning Theory: Social Constructivism   Methodology: Case Study  

Mathematical Discourse in Teaching Exponents (Research - Qualitative)

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: 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: Secondary   Learning Theory: Situated Cognition   Methodology: Comparative Case Study  

Reform Curriculum and Transfer (Research - Mixed Methods)

Using a mixed methods approach, Boaler (1998) investigated the nature of learning of students (ranging in age from 13 to 16) at two British schools, one process-based school where students focused on projects and applications problems and one content-based school where students focused on algorithms and memorization of concepts. Data collection included student and teacher interviews, student questionnaires, open ended tests, short answer tests, student demographic information, lesson observations, and standardized exam grades. The results showed that the students who learned mathematical processes (process-based) scored higher on open ended questions and performed as well as students who learned mathematical procedures (content-based) on procedural questions. In addition, the content-based students have worse attitudes toward mathematics than the other students do. Implications for teaching consist of the idea that students who learn through activity based instruction (process-based) perform better on applied problems and as well as students taught using algorithms on short answer, content-based problems. [by Ann Wheeler]

Level: College   Methodology: Comparative Case Study  

The TI-Navigator and Feedback (Research - Qualitative)

Level: College   Methodology: Comparative Case Study  

Actor-Oriented Transfer and Understanding Slope (Research - Qualitative)

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

U.S. and Chinese Elementary Teachers' Pedagogical Content Knowledge (Research - Qualitative)

Ma reports on the results of her dissertation investigation into "above average" U.S. elementary mathematics teachers' and a variety of Chinese elementary teachers' understanding of the mathematics needed to teach elementary school. Ma found that U.S. teachers focused largely on procedural aspects of mathematical tasks and held fragmented views of arithmetic operations. Chinese teachers, in contrast, focused on the need to know both the how and the why of algorithms and relayed a multiple ways of representing arithmetic operations and varied models for calculating with numbers. Ma's tasks included (1) subtraction with regrouping, (2) multiplication of three digit numbers, (3) division of fractions, and (4) relating perimeter and area of a rectangle. U.S. teachers were competent at performing calculations, but lacked "profound understanding of fundamental mathematics".

Level: K-12   Methodology: Comparative Case Study  

Teacher Change (Research - Qualitative)

Level: Middle   Methodology: Comparative Case Study  

Teachers' Views of Mathematics (Research - Qualitative)

Level: Adult   Learning Theory: Social Cognitive Theory   Methodology: Comparative Case Study  

Self-Efficacy of STEM Men (Research - Qualitative)

Zeldin's (2000) dissertation on career self-efficacy of men with careers in Mathematics, Science and Technology (MST) extends and refines joint research she conducted with her adviser (Zeldin & Pajares, 2000) on the sources of career self-efficacy among 15 women with careers in (MST). Zeldin's report precedes a discussion of semi-structured interviews with 10 Caucasian males by including a thorough review of career self-efficacy constructs and related literature. The qualitative comparative case study-with men and women in MST careers viewed as separate, bounded cases in the sense defined by Merriam (1998)-relies heavily on Bandura's (1997) theoretical framework of four sources of self-efficacy. Though men and women described experiences associated with all four sources (authentic mastery experiences, vicarious experiences, verbal persuasions, and physiological indexes), Zeldin concludes that the women in her study built MST career self-efficacy primarily through verbal persuasions and verbal persuasions while men built career self-efficacy primarily through positive mastery experiences.

Level: Adult   Learning Theory: Social Cognitive Theory   Methodology: Comparative Case Study  

Self-Efficacy of STEM Women (Research - Qualitative)

Level: K-12   Methodology: Ethnography  

CRT and the Standards (Research - Qualitative)

Gutstein, Lipman, Hernandez, and de los Reyes (1997) report on their collaborative reform curriculum project and accompanying research at Diego Rivera Elementary and Middle School in a low-income Mexican American neighborhood in Chicago. The reform initiative included teachers' using the standards-based (NCTM, 1989) program Mathematics in Context (MiC) while engaging in research and professional development efforts aimed at developing a critical orientation to teaching and learning. The authors' present a grounded theory model of intersections between their understandings of culturally relevant pedagogy and the 1989 NCTM standards. The model is particularly grounded in the research teams' observations, interviews, and analysis of Ms. Herrera, Ms. Andula, Mr. Chamarro, and Mr. Simkin. Vignettes describing these teacher's pedagogical orientations, classroom practices, and dispositions toward and involvement with students and parents constitute much of the authors' support for their model. Major aspects of the instructional model include teachers' understanding of critical mathematical thinking, children's informal and cultural knowledge, and orientations to culture.

Level: Adult   Methodology: Ethnography  

Ethnography of Blue Collar Families in Washington, D.C. (Research - Qualitative)

Level: Secondary   Methodology: Ethnography  

Learning Theory in Street Mathematics (Research - Qualitative)

Nasir (2002) summarizes her research into ways African American men (of all ages) learn mathematics while participating in dominoes and basketball play. Pointing to low mathematical achievement among the urban African American students that participated in her research, Nasir contends that better understanding of how these students come to learn mathematics in their out of school practice can inform efforts to improve the students' in-school performance. The author sets much of her narrative in the vocabulary of Wegner's communities of practice learning theory and argues for a theoretical model of learning that focuses on goals, identities, and learning in practice. A unique aspect of Nasir's approach is her choice to use goals and identities instead of, for example, communities or individuals as units for analysis when analyzing young men's participation in dominoes and basketball. Specifically, Nasir describes the mathematical identities and mathematical goals of the dominoes and basketball players using concepts such as imagination and development, suggesting that "development occurs at both an individual level and at the level of the practice itself" (p. 237) as players age and improve. The article concludes with implications for teaching and learning, including an argument for using Nasir's description of goals, identities, and learning in practice as a conceptual framework in mathematics education research.

Level: College   Methodology: Grounded Theory  

Learning Disability through Grounded Theory (Research - Qualitative)

Using many aspects of grounded theory methodology, Troiano (2003) interviewed 9 college students with learning disabilities with the aim of building a conceptual theory of how students come to understand and manage a learning disability. Troiano's narrative is particularly useful as an example of enacted grounded theory strategies and includes a thorough and accessible explanation of the process Troiano went through during his inquiry. While the text explains the broad principles and components of grounded theory well, the authors (usually passive) voice in the article and his implementation of grounded theory methodology limits the value of the author's findings. Troiano also addresses "criteria of soundness" by aligning choices in his research with Lincoln and Guba's (1985) criteria: credibility, transferability, dependability, and confirmability. One limitation in the article can be attributed to the author's apparent lack of attention to the later stages of grounded theory methods, including procedures related to falsifiability and achieving theoretical saturation.

Level: College   Methodology: Phenomenology  

White Teachers and Racism (Research - Qualitative)

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.