805 research outputs found
Pre-Service Teachers\u27 Understanding the Measurement of the Area of Rectangles
No full textExisting research has identified pre-service teachers\u27 (PTs\u27) difficulties with measuring the area of rectangles and explaining their approaches to area measurement. Yet, little is known about elementary PTs\u27 understandings of area measurement. Existing research has focused on children\u27s understanding of area measurement in the context of rectangles and has identified mental processes they use to construct area measurement. The study described in this report contributes to and extends this literature by exploring elementary PTs\u27 understandings of area measurement in the context of rectangles. Clinical interview methodology was used to investigate four PTs\u27 mental processes used in area measurement tasks. Study participants engaged in a 45-minute explanatory interview during which they solved area measurement tasks, including finding areas of rectangles with whole and non-whole number side lengths. Retrospective analysis of interview transcripts was conducted using Battista\u27s framework to identify mental processes used by PTs. While PTs correctly measured the area of rectangles with whole number side lengths, their approaches included identifying side lengths by counting units of area rather than units of length. This approach caused significant difficulties for PTs as they tried to find the area of rectangles with non-whole number side lengths (e.g. length of 0.1 cm). Findings include descriptions of the mental processes used by PTs to find the area of rectangles and are used to describe understanding needed to address measurement of area of rectangles with whole and non-whole number side lengths
Pre-Service Teachers\u27 Understanding of Geometric Reflections in Terms of Motion and Mapping View
No full textIn this manuscript, I describe a study of pre-service secondary mathematics teachers’ (PTs) understanding of geometric reflection in terms of a motion and a mapping views. PTs often have a motion view of geometric reflection based on their understanding of reflection line, domain, and plane. A motion view is a preliminary perspective developed prior to the construction of a mapping view. PTs need a mapping view of geometric reflection, and to be conscious of sub-concepts of a mapping view involved reflection line, domain, and plane. However, there is no clear evidence documenting how a learner’s motion view evolves to produce a mapping view. A clinical interview methodology was used to describe how mental structures occur in the movement between PTs’ motion view and the mapping view. Also, factors critical to the transition from a motion view to a mapping view were explored. Four case studies were constructed from transcript audio records, videos, and written works. Ongoing and retrospective analyses using Dubinsky’s action, process, object and schema (APOS) framework were used to examine PTs’ mental structures. The results indicated that the motion view transforms into the mapping view through the development of mental structures associated with three important sub-concepts of geometric reflection. These three sub-concepts are reflection line, domain, and plane. The results further indicated that there are series of factors that impact the development from the motion view to the mapping view. These factors are perpendicularity and equidistance properties, the role of reflection line, type of figures (circle, semicircle, interior and exterior points of the figures), the operational definition of the plane, and relations between figure and plane
Algebraic reasoning in and through quantitative conceptions
No full textIn a 4-month constructivist teaching experiment, I investigated how one 7th grade child and two 8th grade children constructed algebraic reasoning as a transformation in their multiplicative schemes. Algebraic reasoning is the purposeful functioning of a child’s schemes and operations in contexts I defined as algebraic. The algebraic contexts explored in my study required a child to create equality between two multiplicative situations (compilations) by operating on/with the individual units (1s) and composite units that constitute the compilations. Algebraic reasoning emerges in two ways from these contexts. First, when a child operates on and with the structure of his or her schemes, the child is reasoning algebraically. Second, if a child’s schemes form the underpinnings for formal algebraic conceptions, the child is reasoning algebraically. I identified three specific schemes of equality, a unidirectional scheme of equality, a relational scheme of equality, and a quantitative relational scheme (QRE) of equality, that were constructed by the children in my study as they produced equality between two compilations. A QRE scheme of equality is comprised of operations on composite units across two compilations. The QRE scheme constitutes algebraic reasoning because a child operates on the structure of their multiplicative schemes with the structure of their additive schemes. A child also operates on the structure of their equality scheme with the structure of their multiplicative schemes. Moreover, because the QRE scheme is comprised of operations on composite units across two compilations, it forms the cognitive roots for the algebraic concepts of the distributive property, quantitative conservation, and solving linear equations. The construction of the QRE scheme by a child Joe was crucial to my study because it provided evidence of the existence of the hypothesized QRE scheme. The second scheme of equality, a relational scheme, consists of balancing the total 1s from each compilation via operations on 1s. This scheme also constitutes algebraic reasoning because a child increases the smaller total via addition and decreases the larger total via subtraction until they produce equality. They operate with the structure of their additive scheme on the structure of their equality scheme. A child with the third scheme of equality, a unidirectional scheme, creates equality by transforming the total from one of the given compilations to produce the second total. A child who operates with this scheme is not reasoning algebraically. I suggest a trajectory for the development of these three equality schemes. A child begins with a unidirectional scheme of equality. Next, they construct a relational scheme of equality as their scheme progresses to include operating on both totals with 1s to create balance. A second progression occurs to the most advanced equality scheme, a QRE scheme, when a scheme is constructed that moves past operations on 1s to include operations on composite units. My research goes beyond a description of the three schemes as it provides evidence as to how a child transitions from not having a QRE scheme to having one. Moreover, my research informs the teaching of mathematics as the UDS and QRE tasks offer elementary school teachers a way to give children opportunities to form the foundations for the formal algebraic concepts of the distributive property, solving linear equations, and quantitative conservation
Investigating Latina/o mathematics students\u27 perceptions of interpersonal relationships in college mathematics
No full textIt is known that interpersonal relationships are significant to the success of the Latina/o mathematics learner. Unfortunately, interpersonal relationships in mathematics are understudied. For this reason, this study seeks to answer the following question: How do Latina/o mathematics students perceive their interpersonal relationships with their mathematics professors, peers in mathematics, and families with respect to mathematics? To explore the role of such relationships in Latinas/os experiences in mathematics, I define a meaningful relationship in mathematics education (MRIME) as interactions between two parties in which one party acknowledges the other party as a sociocultural being and fosters their mathematics identity. This definition draws from research on care, mathematics identity and mathematics students as sociocultural beings. This study was done using case studies. Four Latina/o students who were mathematics majors or had a bachelor\u27s degree in mathematics were recruited using a Facebook page. The data collection took place over a period of one year. Three hour-long telephone interviews were conducted with each participant. Analysis of the interviews was based on identifying evidence of components of MRIME including the fostering of the participants mathematics identity and acknowledgement of the participant as social and cultural beings. The results showed that the participants felt that their relationships with their mathematics professors and peers in mathematics with the most meaningful while their relationships with their families were the least meaningful when the focus was on mathematics. The results also showed that the interpersonal relationships that the participants had were the weakest when it came to the sociocultural element of MRIME. The results of this study imply that there exist factors, which are possibly connected to mathematics students\u27 perceptions of interpersonal relationships, which may contribute to persistence in mathematics. The data also suggest that further research should be conducted on interpersonal relationships in mathematics educational settings
Two Mathematics Teachers\u27 Personal Practical Knowledge: Experiences Making Curriculum Within the 3D Inquiry Space
No full textThis study is an effort to expand conceptualizations of mathematics teachers\u27 knowledge where teachers\u27 experiences are considered an important source of their knowledge. Using Schwab\u27s (1969, 1983) concept of curriculum and Dewey\u27s (1938) theory of experience as theoretical lenses, the researcher, a former secondary mathematics teacher, explored an eighth-grade mathematics teacher\u27s personal practical knowledge ([PPK], Elbaz, 1981, 1983). The mathematics aspect of this study was the emerging concept of area as students worked on a lesson involving real-world context. Field and research texts were constructed through narrative inquiry (Clandinin & Connelly, 2000). This study illustrates how under a four-year collaboration: 1) the researcher learned about the mathematics teacher\u27s PPK and her personal PPK in relation to the teacher; 2) mathematics teachers make curriculum (Clandinin & Connelly, 1992) by paying attention to their experiences, readings their students, and following Dewey\u27s (1938) principle of continuity; and 3) the teachers\u27 previous experiences with the area concept informed their in-the-moment understandings of the concept and their interactions with students, and between themselves, allowed them to develop new understandings of area and its teaching
Characteristics of Critical Friendship that Transform Professional Identity
No full textWe met at CASTLE 2018, two trained mathematics teacher educators (MTEs), interested in mathematics, and teaching elementary mathematics methods to preservice teachers (PTs). Melva’s self-study research, focused on improving her online methods course, was approaching its second year and her second critical friend had lost interest in continuing. Melva invited Signe to be her critical friend (Schuck & Russell, 2005) and Signe agreed. Explicit expectations of our critical friendship included weekly meetings. Our critical friendship seemed to follow an expected trajectory for, “supporting/coaching the transformation of another’s teaching” (Stolle, et al., 2019, p. 20). However, there were implicit ways our critical friendship evolved, drawing from connected, entangled threads of our individual expectations and our MTE identities
Learning of trigonometry: An examination of pre-service secondary mathematics teachers\u27 trigonometric ratios schema
No full textMathematics education research has emphasized the importance of trigonometry in advanced mathematical learning and highlighted students\u27 difficulties with trigonometry, which stem from underdeveloped foundational concepts of angle and angle measurement. The purpose of this dissertation was to investigate pre-service secondary mathematics teachers\u27 (PSMTs) constructions of relationships between angles and side lengths in a right triangle (RASR) and their relationship to constructions of trigonometric ratios. Dubinsky and his colleagues\u27 (Arnon et al., 2014; Asiala et al., 1996; Dubinsky, 1991, Dubinsky & McDonald, 2001) APOS theory was used to define knowledge as a collection of physical and mental constructions and operations with them (action, process, object, and schema). To explore constructions of RASR and relationships to constructions of trigonometric ratios, evidence of types of processing was sought. Using Clements\u27 (2000) clinical interview methodology, this study utilized a series of controlled interviews to gather evidence of four PSMTs\u27 constructions of angles, angle measurement, RASR, trigonometric expressions and trigonometric ratios. The analysis consisted of case study and cross-case study analysis. The case study analysis focused on constructing a model for each individual PSMT from observations during task-based interviews. Inferences about constructions of the APOS levels associated with each concept were developed. After each model was created for the PSMT, similarities and differences in actions, processes, objects, and schema were noted. This cross-case analysis allowed the construction of a cognitive model which describes how the PSMTs\u27 mental constructions were related to their constructions trigonometric ratios schema. PSMTs participating in the study all had a schema related to 2-line angles and angle measurement. However, constructions of 1-line and 0-line angles and angle measurement were at lower levels. Yet, constructions of 1-line and 0-line angles and angle measurement were not required to operate with right triangles. The schema level for 2-line angles and angle measurement was sufficient for constructions of schema for RASR and trigonometric ratios in a right triangle context. The findings also support the claim that a schema of right triangles and RASR is necessary to construct trigonometric ratios schema. Particularly, having a schema of RASR provided PSMTs opportunities to act on dynamic right triangles presented in dynamic geometric software (DGS), specifically in GeoGebra, and reason about RASR. If students reached the schema level for RASR, they could then flexibly act on the trigonometric ratios considering the RASR. A cognitive model which explains relationships in PSMTs\u27 constructions that are related to their constructions of trigonometric ratios schema is shared. Findings also suggest that reaching the schema level for 2-line angles, right triangle, RASR, and trigonometric expressions as well as reaching the object level for ratios was sufficient to construct a trigonometric ratios schema in a right triangle context. In addition, it was found that using the mnemonic, SOH-CAH-TOA, and the special right triangles [45°-45° -90° and 30°-60°-90°] supported PSMTs\u27 construction of trigonometric ratios schema
Factors influencing teacher instructional practice in mathematics when participating in professional development
No full textIn this research, I investigated teachers’ interpretations of the goals of professional development and factors that contributed to enacted instructional practices. A multiple-case study design was used to examine the interpretations of four high school teachers participating in a year-long professional development program with a standards-based framework for mathematics education. Data collection included information about the professional development program, the intended and enacted curriculum (Stein et al., 2007), the teachers’ interpretations of the professional development goals, and context factors that influenced instructional planning and implementation. The data were used to create a description of the professional development, a case study of each teacher that included a description of the enacted curriculum and a description of context factors that influenced the instructional practices. Additional examination included a cross-case analysis to identify common themes between the teachers. Each teacher provided an interpretation of the goals of the professional development that was consistent with the professional development, but often focused on a narrower objective for each of the goals. The teachers’ interpretations of the goals influenced their use of ideas from the professional development in their classrooms. Four additional context factors were identified as influences on enacted instruction: perception of classroom control, attitude towards standards-based instruction, usefulness of professional development activities in relationship to grade levels or courses taught, and concerns about student success due to a lack of experience with standards-based instruction. The findings of this research have implications for providers of professional development for K-12 teachers of mathematics. First, professional development providers need to spend time learning about teachers’ interpretations of the goals of the professional development. Second, professional development providers should use a framework of content to be learned that is aligned with the goals of a professional development program. Finally, learning activities and sample lessons during the professional development should be grade level or course appropriate
The development of mathematics-for-teaching: The case of fraction multiplication
No full textThe parallel research traditions of explicit-objective and tacit-emergent vary greatly in how they define, assess, and enable development of teacher mathematical knowledge. Despite these diversities, widespread agreement exists in mathematics education research that a teacher\u27s mathematical knowledge is a key competency of an effective teacher. This research report investigates the nature and development of teacher mathematical knowledge of fraction multiplication defined from a tacit-emergent perspective. Questions about the nature and development of teacher mathematical knowledge for fraction multiplication were investigated in this report at the individual and collective levels. In addition, this research report also investigated the developmental links between these levels. The concept study design and the framework for teacher knowledge used in this report derived from the work of Davis and colleagues (Davis & Simmt, 2006; Davis & Renert, 2014). The results from this report were multifaceted for both the individual and collective levels of mathematical knowledge. Teachers\u27 individual mathematics-for-teaching (M4T) knowledge of fraction multiplication developed throughout their participation in the mathematical environments of the concept study. Furthermore, two types of collective action emerged as proposed links between the collective and individual development of teachers\u27 M4T knowledge of fraction multiplication. These proposed links, titled synergistic realizations and recursive elaborations emerged in this report as patterns of mathematical action existent in moments of coaction. Recursive elaboration defines the decision-making mechanism where the collective expands the realm of what is possible for a single mathematical realization. Synergistic realization defines the collective decision action in which all previous realizations are abandoned for one innovation in the mathematical realization of a mathematical concept. A discussion of the implications for defining teachers\u27 mathematical knowledge of fraction multiplication as nested systems of individual and collective knowledge is included in the conclusion of this report
Snow cover fraction on land fromt MODIS at the three GEM sites
No full textSnow cover fraction on ground from MODIS.
Gaps due to cloud cover are filled using the last available observation. A description paper is in preparation.Please contact the author directly with any questions
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