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We analyze the interrelations between prospective and practicing teachers' learning of the mathematics of change and the development
of their emerging understanding of effective mathematics teaching. The participants in our study, who were all interested
in teaching secondary mathematics, were mathematics majors who had significant formal knowledge...
Contexts in source publication
Context 1
... student reasoned that the clown would be at 18 meters after three seconds. He then plotted the points (3,18) and (0,0) and then drew a line segment connecting these two points as shown in Figure 5a. Ellen then asked him to draw a velocity vs. time graph showing the clown's velocity at each second. ...
Context 2
... then asked him to draw a velocity vs. time graph showing the clown's velocity at each second. In response, the student drew the graph shown in Figure 5b. Because this graph did not match Ellen's expectations of what the student would draw, namely a velocity graph showing a constant speed of 6 m/s for 3 seconds, she dismissed it as incorrect. ...
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Citations
... Some calculus education researchers have started to pay more attention to how students understand, apply, and use the derivative concept in contexts outside of mathematics because of the derivative's importance in science and engineering fields of study and the challenges students face when using it in those fields (Berry & Nyman, 2003;Roorda, Vos, & Goedhart, 2007). However, the vast majority of the mathematics education research dealing with applications of the rate of change and the derivative is centered on the contexts of position, velocity, and acceleration (Berry & Nyman, 2003;Bezuidenhout, 1998;Bowers & Doerr, 2001;Hale, 2000;Marrongelle, 2004;Roschelle, 2000;Schwalbach & Dosemagen, 2010;Zandieh, 2001). ...
The difficulties experienced by students in studying derivative material are difficulties understanding the definition of derivatives and representations of these derivatives. Derivatives are a material that is quite difficult to develop mathematical creative abilities because they have many functions and symbols. This difficulty indicates a barrier to thinking in students. This research is descriptive qualitative research with five research subjects of Tadris Mathematics at the Curup State Islamic Institute. Each student can think creatively mathematically on the indicators of flexibility, fluency, and originality, but the levels are different. The level of mathematical creative thinking ability can depend on the questions or problems and the material being studied. However, one of the indicators of the ability to think creatively mathematically, which is very weak for students, is originality because students seem rigid with what they have obtained from lecturers and books. Students find it difficult to come up with ideas in solving problems. Barriers to thinking creatively mathematically can occur due to several factors, including 1) due to a lack of prior knowledge of students in understanding problems and determining ideas in planning solutions; 2) a lack of strong concepts possessed by students. One way for lecturers to overcome these obstacles is to provide scaffolding. The provision of scaffolding is carried out using Treffinger learning with several stages, namely basic tools, practice with process, and working with real problems. In practice with process stage, students solve problems by using analogical reasoning to develop their mathematical creative thinking abilities. The stages of analogical reasoning used to consist of the stages of recognition, representation, structuring, mapping, applying, and verifying. This study's findings are that two new stages are added to the early stages of analogical reasoning, namely the stages of recognition and representation.
... D'autres auteurs voient par contre l'utilisation d'un contexte d'application pour la dérivée comme un support à la compréhension. Il faut noter cependant que ces études ont lieu dans le cadre d'expériences d'enseignement interdisciplinaires (Marrongelle et al., 2003;Schwalbach et Dosemagen, 2000), d'ingénieries didactiques (Gantois et Schneider, 2012;Schneider, 1992), ou d'activités développées spécifiquement (Berry et Nyman, 2003;Bowers et Doerr, 2001;Nemirovsky et Rubin, 1992). Nous en savons donc peu sur ce qui se passe dans un cours régulier et aucune étude en didactique des mathématiques ne s'est intéressée à la dérivée dans le cours de physique lui-même. ...
Research report (PARÉA).
Ce projet de recherche a pour but d’étudier et renforcer la collaboration entre les cours de calcul différentiel et de mécanique dans le cadre d’un jumelage implémenté au niveau collégial. En particulier, les cours de calcul ont des taux d’échec et d’abandon importants, tant à niveau international (p. ex. Rasmussen et Ellis, 2013) qu’au Québec (p. ex. Tcheuffa Nziatcheu, 2015) Dans ces cours, la notion de dérivée occupe une place centrale et est nécessaire dans les différents parcours en sciences, technologie, ingénierie et mathématiques ; cependant, la recherche en didactique des mathématiques a identifié des difficultés importantes dans l’apprentissage de cette notion (p. ex. Ferrini-Mundy et Graham, 1994; Jones et Watson, 2017; Orton, 1983; Park, 2015), mais aussi des difficultés à l’appliquer dans d’autres contextes non mathématiques (p. ex. Başkan et al., 2010; Christensen et Thompson, 2012) et, en particulier, en mécanique (p. ex. Basson, 2002)
Pour aider les étudiants à mieux apprivoiser la notion de dérivée, un jumelage avec le cours de mécanique, a été mis en place de sorte que les étudiants puissent mieux apprivoiser cette notion en calcul et d’approfondir certaines notions de mécanique (vitesse et accélération). Un premier cycle de notre recherche-action collaborative a été mené lors de l’implémentation du jumelage au trimestre d’automne 2019, ce qui a mené à la deuxième itération de l’automne 2020. La pandémie a fortement ralenti notre recherche et nous n’avons pu étudier que l’implémentation de l’automne 2021. Le suivi du jumelage a été effectué par des observations des classes jumelées, des rencontres régulières avec les enseignants et trois questionnaires étudiants distribués à différent moment dans le trimestre.
Les résultats obtenus montrent que les étudiants ayant participé au cours jumelé établissent plus des liens entre les contenus liés à la dérivée dans les deux cours, ainsi que leur capacité à utiliser des approches d’un cours pour résoudre des tâches dans l’autre cours. Les étudiants ont également développé de meilleures compétences pour résoudre certaines tâches présentées dans un contexte graphique. Finalement, nous soulignons aussi des retombées importantes pour la communauté enseignante au collège Dawson et des possibilités réelles de pérenniser cette innovation.
... Dreyfus (1999) mentions the related difference between providing chronological accounts of actions carried out and pointing out connections and implications. Cobb et al. (2003) and Bowers and Doerr (2001) also make a similar distinction between an explanation of a process without reason and an explanation of the reasons related to a concept. This also relates to the work of Popper (1934Popper ( /2002, who argues for a sharp distinction between the context of discovery (how did you find this out?) and the context of justification (how can we decide if this is true?). ...
Exploring student explanations: What types can be observed, and how do teachers initiate and respond to them? Nordic Studies in Mathematics Education, 26 (1), 53-72. Exploring student explanations: What types can be observed, and how do teachers initiate and respond to them? ove gunnar drageset This article presents different types of student explanations that can were observed, and how these were initiated and responded to. The research is based on the practice of five teachers, with all interactions having been analysed and categorized to develop the concepts. First, three distinct types of student explanation were found: explaining actions, explaining reasons, and explaining concepts. Secondly, the teach-ers' initiations were inspected, by studying the turn before each student explanation. Strong connections were found between the initiation and each type of student explanation. Thirdly, teachers' responses to the students' explanations were inspected, with three main types of response being found to all three types of student explanation: pointing out what to notice, requesting further detail, and confirming and moving on. The main contribution of this article is the conceptualization of students' explanations and the explanation of how these are initiated and responded to. During classroom conversation, students contribute with different types of interaction. Drageset (2014) suggests that these interactions can be separated into five types: explanations, initiatives, teacher-led responses, unexplained answers, and partial answers. Of these, students' explanations might be of the greatest interest for further exploration. It is, of course, possible to focus on who explains and how frequently they do so, but instead, this article presents the development of concepts describing student explanations, as part of the classroom conversation, related to the following research question: What types of student explanation can be observed, and how are these initiated and responded to?
... At least four studies have investigated secondary teachers' meanings for rate of change (Bowers & Doerr, 2001;Coe, 2007;Person, Berenson, & Greenspon, 2004;Thompson, 1994b). Data from each study supported the claim that many secondary teachers have meanings for rate that are chunky or indexical. ...
... Confusing rate with amount or rate with a change is consistent with schemes that do not entail considering the multiplicative comparison of two quantities. Bowers and Doerr (2001) reported over half of the fifteen secondary mathematics teachers in a university course inappropriately applied the formula d=rt in situations with non-constant rates of change (Bowers & Doerr, 2001, p. 124). Despite the mathematical relationship between ∆d/∆t = r and ∆d = r∆t, we suspect that many people apply these formulas calculationally in different circumstances without connecting their quantitative meanings. ...
... chunk) of time as opposed to the relative size of the measure of distance travelled and the measure of elapsed time to travel that distance. In interviews and prior research some teachers used the formula d = rt without considering quantitative relationships that this formula entails (Bowers & Doerr, 2001). ...
This article reports an investigation of 251 high school mathematics teachers’ meanings for slope, measurement, and rate of change. The data was collected with a validated written instrument designed to diagnose teachers' mathematical meanings. Most teachers conveyed primarily ad- ditive and formulaic meanings for slope and rate of change on written items. Few teachers conveyed that a rate of change compares the relative sizes of changes in two quantities. Teachers’ weak measurement schemes were associated with limited meanings for rate of change. Overall, the data suggests that rate of change should be a topic of targeted professional development.
... Hence, to gain a deeper understanding of the ways learners approach the study of kinematics, it is recommended to involve student teachers in researching this topic for the following reason. Some researchers explain that by acting as students, student teachers have the opportunity to gain pedagogical insights, and by acting as teachers, they have the opportunity to gain content insights (Bowers & Doerr, 2001). Therefore, by putting student teachers in the role of students, we may gain additional pedagogical insights. ...
The modern way to teach physics, as prescribed by high school physics curriculum, requires teachers to take on new roles and modify their concepts about the nature of science and the acquisition of scientific knowledge (Anderson, 2002). Some researchers (Aiello-Nicosia & Sperandeo-Mineo, 2000) propose that student teachers experiment for themselves the scientific approach in an environment similar to that in which they will teach. Having student teachers reflect on their own learning process in this type of environment and providing them with educational tools for teaching physical models, aims to favor within the student teachers a better understanding of physical concepts of motion, of ways to plan an experiment to link the results to an initial hypothesis, and of the usefulness of models in planning and executing experiments. Hence, to gain a deeper understanding of the ways learners approach the study of kinematics, it is recommended to involve student teachers in researching this topic for the following reason. Some researchers explain that by acting as students, student teachers have the opportunity to gain pedagogical insights, and by acting as teachers, they have the opportunity to gain content insights (Bowers & Doerr, 2001). Therefore, by putting student teachers in the role of students, we may gain additional pedagogical insights. This additional benefit helps us achieve our research goals: as learners, the student teachers enable us to ‘model learning of a group of students given specific conditions ... in a video based laboratory’ and as future teachers, they help us identify ways of improving student learning. As a consequence, this paper tries to answer the following research questions: 1) How do preservice teachers as learners interact with specific kinematics software within the context of the teaching-learning sequence proposed on relative motion? 2) From an analysis of their learning paths, and their ongoing comments and reflections about it, which teaching strategies help preservice teachers develop conceptual understanding of relative speed?
... Hence, to gain a deeper understanding of the ways learners approach the study of kinematics, it is recommended to involve student teachers in researching this topic for the following reason. Some researchers explain that by acting as students, student teachers have the opportunity to gain pedagogical insights, and by acting as teachers, they have the opportunity to gain content insights [9]. Therefore, by putting student teachers in the role of students, we may gain additional pedagogical insights. ...
This exploratory study intends to model kinematics learning of a pair of student teachers when exposed to prescribed teaching strategies in a video-based laboratory. Two student teachers were chosen from the Francophone B.Ed. program of the Faculty of Education of a Canadian university. The study method consisted of having the participants interact with a video-based laboratory to complete two activities for learning properties of acceleration in rectilinear motion. Time limits were placed on the learning activities during which the researcher collected detailed multimodal information from the student teachers’ answers to questions, the graphs they produced from experimental data, and the videos taken during the learning sessions. As a result, we describe the learning approach each one followed, the evidence of conceptual change and the difficulties they face in tackling various aspects of the accelerated motion. We then specify advantages and limits of our research and propose recommendations for further study.
... While all of these studies have added significantly to an understanding of teachers' practices and interactions with students and technological tools, we do not know much about how novice and pre-service teachers learn to interact with students while using a technology tool to solve a task. Bowers and Doerr (2001) and Lee (2005) investigated pre-service teachers' interactions with students while using a technology tool. Both found that pre-service teachers were able to use representations available within the technology environments to focus students' attention on important ideas in a task or to pose additional questions for them to consider. ...
... The researchers indicate that this "didactical cycle" was carefully orchestrated by the teacher in response to her analysis of students' uses of the tools and mathematical meanings they were developing. Unlike the studies conducted by Bowers and Doerr (2001) and Lee (2005), teachers in this study had the opportunity to formulate questions, tasks, and new activities in response to their interpretations of students' geometric and technological work. ...
When technology is used in classrooms new interactions among students, the teacher, and technology are enabled. The purpose of this study was to examine the ways pre-service mathematics teachers implemented technology-based tasks with individual advanced middle-school students. Pre-service teachers posed questions that focused students on features of technology and geometry in different classifiable ways. In particular, there were instances when teachers focused only on mathematics or technology. There were also instances when the teacher suggested students use the technology for the purpose of noticing mathematics and other times when the teacher would pose a mathematics question or statement with the assumption that students would use technology in response. Analysis of six pre-service teachers’ is provided along with a classification system.
... However, Peggy was inclined to think that speed is an index of "fastness", so all of the changes in speed throughout the trip might seem important to take into consideration. Additional U.S. studies of calculus students and secondary teachers are related to teachers' understandings of rate of change (Bowers & Doerr, 2001;Stump, 1999;Weber & Dorko, 2014). These studies suggest that teachers' meanings for rate of change might be inadequate for making sense of average rate of change. ...
... These studies suggest that teachers' meanings for rate of change might be inadequate for making sense of average rate of change. For example, Bowers and Doerr (2001) investigated 26 secondary teachers' thinking about the "mathematics of change" in two university technology based mathematics classes. They designed the first two instructional sequences to help the participants understand the Fundamental Theorem of Calculus by exploring relationships between linked velocity and position graphs (Bowers & Doerr, 2001, p. 120). ...
This study explores teachers' meanings for average rate of change in U.S.A. and Korea. We believe that teachers convey their meanings to students and teachers who have productive mathematical meanings help students build coherent meanings. We administered a diagnostic instrument to 96 U.S. teachers and 66 Korean teachers. Some of teachers' responses revealed particular problematic meanings for average rate of change that should be addressed in professional development. Our analyses suggest that Korean teachers' meanings for average rate of change are substantially stronger than U.S. teachers' meanings.
... Much has been written on student and teacher understandings of the curricular topics connected to conceptions of relative size such as fractions, rates of change and derivatives (Armstrong & Bezuk, 1995;Bowers & Doerr, 2001;Harel & Behr, 1995; Byerley!Thompson! ! ...
... We suspected that teachers with chunky meanings for speed might choose j-s, an answer that is only true for the first one-second interval. There is some evidence in the written work and interview data to support this hypothesis, an example of which is provided in Figure 2. In teacher responses to other items, interviews, and in the literature, we also noticed teachers using the formula d = rt inappropriately and thought that some teachers may expect to see a product as part of the answer (Bowers & Doerr, 2001). For example, some teachers used the formula d = rt to find the total distance traveled on a trip with a non-constant rate of change by simply selecting the rate of change at the end of the trip. ...
This paper explores the usefulness of understanding quotients as measures of relative size in mathematics. The paper characterizes the types of thinking displayed by high school mathematics teachers on two novel tasks designed to reveal teachers' meanings in contexts where making comparisons of relative size is productive.
... Many of the ways in which Lewis and Susan represented and approximated the figured world of reform mathematics in their course are not uncommon in teacher education courses. Although others have noted that positioning teachers as both students and teachers in mathematical activity can enable novices to build and practice a repertoire of pedagogical tools and moves (e.g., Ball & Forzani, 2009;Bowers & Doerr, 2001), the concern for models of identity also turns our attention to the ways in which instructors use these approximations of practice to help novices recognize and reason about their multiple and often conflicting expectations for learners and teachers in any given context. As illustrated above, these expectations shape and constrain how subject matter comes to be defined and enacted and how pedagogical resources are taken up. ...
Starting from the assertion that traditional and reform mathematics pedagogy constitute two distinct figured worlds of teaching and learning, the authors explore the initiation of prospective teachers into the figured world of reform mathematics pedagogy. To become successful teachers in reform-oriented classrooms, prospective teachers must learn more than pedagogical tools and moves: They must understand what it is to participate in the figured world of reform pedagogy, develop models of identities for participants in this world, and negotiate new constructions of mathematics. In this article the authors present three episodes from an elementary mathematics teacher education class where positions of “teacher” and “child” were offered by instructors in activities designed to approximate practice in the reform figured world. Students negotiated new models of identity and conceptions of mathematics as they took up these positions in varying ways.
