This document analyzes two mathematics lessons about quadratic inequalities taught by a pre-service teacher (Z) and an in-service teacher (Z1) with 12 years experience. The lessons were coded using categories to analyze teacher and student utterances. For teacher Z's lesson, students were able to correctly identify the parabola graph of y=x2-x-6, its two roots of -2 and 3, and its upward direction. The teacher drew the graph based on student responses.
This document discusses problem-based learning and its implementation in a classroom action research study. It provides background information on problem-based learning and discusses its benefits. The study aimed to enhance learning outcomes and group cooperation through problem-based learning. It was conducted in an 8th grade class over multiple cycles. Data was collected through observation and tests to analyze student achievement and cooperation. The criteria for success was achieving a good rating on observations of student activity and an 85% pass rate with average scores above the minimum requirements.
Enhancing students’ mathematical creative problem solvingAlexander Decker
This document discusses a study that examined the effect of situation-based learning (SBL) on students' mathematical creative problem solving (CPS) ability. The study used an experiment group that received SBL instruction and a control group that received conventional instruction. Results showed that the experiment group had greater gains in mathematical CPS ability compared to the control group. Specifically, the experiment group improved their average score from 16.32 to 32.91, while the control group improved from 16.21 to 24.66. Additionally, the strongest aspect of mathematical CPS ability developed was fact finding, while the weakest was acceptance finding. Therefore, the study concluded that SBL is more effective than conventional learning at enhancing students' mathematical
The document provides a mathematics teaching plan focusing on critical teaching skills for presenting information. It includes examples of using visual supports and demonstrations to explain concepts in geometry like circles, pyramids and volumes. The examples emphasize using real objects, clear language, and checking for student understanding to effectively present new mathematical concepts, especially for English language learners.
Analysis of Students in Difficulty Solve Problems TwoDimentional Figure Quadr...IOSRJM
This research is a descriptive study that aims to determine the difficulty of concepts, principles difficulty and skill difficulties experienced by students of SMPN 8 Makassar to solve problems, specifically about waking flat rectangle. The subject of this research was the seventh grade students of SMPN 8 Makassar in the academic year 2015/2016 consisting of 5 classes of 200 students, while research subjects were students of class VII SMPN 8 Makassar as many as 35 students. The data collection is done by providing an instrument in the form of a test which consists of 5 items essay in the classroom in order to obtain a score of each kind of level of difficulty with descriptive analysis. The results were obtained percentage score of the degree of difficulty concept was 71.43% categorized as moderate difficulty level, the difficulty level of the principles is 25.71% categorized as very low level of difficulty, the difficulty level of skill 20% categorized the degree of difficulty is very low.
This document discusses applying mathematical problem solving to improve student results in an 8th grade classroom in Bengkulu, Indonesia. It begins by outlining the background and importance of problem solving in mathematics learning. The author then states the problem formulations, purpose, benefits, and scope of the study. Next, the document reviews literature on learning mathematics, study approaches, problem solving approaches, and types of math problems and questions. It concludes by discussing learning outcomes and aspects of the problem solving approach. The overall purpose is to determine how applying problem solving can improve student activity and learning outcomes in mathematics.
while practicing for ctet I came across some word which i didn't find in syllabus. so in this PPT I am discussing all those key words. wish it will help you in your studies. if you find any other words which I this PPt doesn't contain then plz let me know I will definitely try to find out.
This document discusses problem-based learning and its implementation in a classroom action research study. It provides background information on problem-based learning and discusses its benefits. The study aimed to enhance learning outcomes and group cooperation through problem-based learning. It was conducted in an 8th grade class over multiple cycles. Data was collected through observation and tests to analyze student achievement and cooperation. The criteria for success was achieving a good rating on observations of student activity and an 85% pass rate with average scores above the minimum requirements.
Enhancing students’ mathematical creative problem solvingAlexander Decker
This document discusses a study that examined the effect of situation-based learning (SBL) on students' mathematical creative problem solving (CPS) ability. The study used an experiment group that received SBL instruction and a control group that received conventional instruction. Results showed that the experiment group had greater gains in mathematical CPS ability compared to the control group. Specifically, the experiment group improved their average score from 16.32 to 32.91, while the control group improved from 16.21 to 24.66. Additionally, the strongest aspect of mathematical CPS ability developed was fact finding, while the weakest was acceptance finding. Therefore, the study concluded that SBL is more effective than conventional learning at enhancing students' mathematical
The document provides a mathematics teaching plan focusing on critical teaching skills for presenting information. It includes examples of using visual supports and demonstrations to explain concepts in geometry like circles, pyramids and volumes. The examples emphasize using real objects, clear language, and checking for student understanding to effectively present new mathematical concepts, especially for English language learners.
Analysis of Students in Difficulty Solve Problems TwoDimentional Figure Quadr...IOSRJM
This research is a descriptive study that aims to determine the difficulty of concepts, principles difficulty and skill difficulties experienced by students of SMPN 8 Makassar to solve problems, specifically about waking flat rectangle. The subject of this research was the seventh grade students of SMPN 8 Makassar in the academic year 2015/2016 consisting of 5 classes of 200 students, while research subjects were students of class VII SMPN 8 Makassar as many as 35 students. The data collection is done by providing an instrument in the form of a test which consists of 5 items essay in the classroom in order to obtain a score of each kind of level of difficulty with descriptive analysis. The results were obtained percentage score of the degree of difficulty concept was 71.43% categorized as moderate difficulty level, the difficulty level of the principles is 25.71% categorized as very low level of difficulty, the difficulty level of skill 20% categorized the degree of difficulty is very low.
This document discusses applying mathematical problem solving to improve student results in an 8th grade classroom in Bengkulu, Indonesia. It begins by outlining the background and importance of problem solving in mathematics learning. The author then states the problem formulations, purpose, benefits, and scope of the study. Next, the document reviews literature on learning mathematics, study approaches, problem solving approaches, and types of math problems and questions. It concludes by discussing learning outcomes and aspects of the problem solving approach. The overall purpose is to determine how applying problem solving can improve student activity and learning outcomes in mathematics.
while practicing for ctet I came across some word which i didn't find in syllabus. so in this PPT I am discussing all those key words. wish it will help you in your studies. if you find any other words which I this PPt doesn't contain then plz let me know I will definitely try to find out.
Childrení»s mathematics performance and drawing activities a constructive cor...Alexander Decker
This document summarizes a study that examined how drawing activities can improve children's performance in mathematics. The study used observation, interviews, achievement tests, and a pictorial Likert scale to analyze relationships between discovery learning methods and mathematics performance for 62 pupils at a primary school in Ghana where mathematics performance was low. The results revealed that incorporating drawing activities into mathematics lessons significantly improved pupils' mathematics performance and engagement. The study concluded that using arts-integrated approaches can help establish relationships between concepts and enhance understanding in mathematics.
The document discusses key topics in mathematics pedagogy for CTET exams, including:
- Defining pedagogy and mathematics.
- The nature of mathematics as both a science of discovery and logical processes.
- Guiding principles and vision for mathematics in the NCF-2005 curriculum.
- Strategies for teaching mathematics like written work, oral work, group work and homework.
- Reasons for keeping mathematics in school curriculums like its basis in other sciences and role in developing logical thinking.
- The language of mathematics including concepts, terminology, symbols and algorithms.
- Approaches like community mathematics and mathematical communication to engage students.
EFFECTIVENESS OF SINGAPORE MATH STRATEGIES IN LEARNING MATHEMATICS AMONG FOUR...Thiyagu K
The Singapore math method is child-focused, and seeks to make sure that the student gains a full and complete understanding of the fundamental mathematical concepts, rather than merely memorizes a rote collection of facts. This approach not merely enhances mathematical learning; it also offers a firm foundation from which broader mathematical principles can be extrapolated. The present study tries to find out the effectiveness of Singapore math strategies in learning mathematics among fourth standard students. Two equivalent group experimental-designs are employed for this study. The investigator has chosen 64 Fourth standard students for the study. According to the scoring of pre-test, 32 students were chosen as control group and 32 students were chosen as experimental group. Finally the investigator concludes; (a) the experimental group student is better than control group students in their gain scores. (b) There is no significant difference between control group and experimental group students in their pre test scores and post test. (c)There is significant difference between control group and experimental group students in the scores of posttest attainment of knowledge, understanding and application objectives.
REALITY – BASED INSTRUCTION AND SOLVING WORD PROBLEMS INVOLVING SUBTRACTIONWayneRavi
This study was conducted to determine the effect of reality based on the solving word problems involving subtraction. Descriptive-Comparative research design using paired sample T-test was used to utilized in the study. The study was carried out in Tibungol Elementary School to Fifty student of Grade Three section 1. Results revealed that there was a significant difference on the pretest and post test scores of pupils in reality based approach. Further, the reality based approach is effective in improving the performance of student.
EFFECTIVENESS OF INTEGRATING RIDDLES IN TEACHING MATHEMATICS AMONG VIII STAND...Thiyagu K
Mathematics is considered as dry subject and students do not find anything interesting in it. This impression about Mathematics can be reversed with the help of recreational activities in Mathematics. The present study tries to find out the effectiveness of integrating riddles in teaching mathematics among eighth standard students. Two equivalent group experimental-designs are employed for this study. The investigator has chosen 40 eighth standard students for the study. According to the scoring of pre-test, 20 students were chosen as control group and 20 students were chosen as experimental group. Finally the investigator concludes; (a) There is a significant difference between the means of students thought through conventional method and puzzles and riddles way of learning group. (b) There is a significant difference between the means of the Post-Test scores of control group and experimental group students with respect to the knowledge, understanding and application objectives.
The document discusses issues students with disabilities face in math including perceptual, language, reasoning, and memory challenges. It then describes considerations for instruction including differentiated instruction, metacognitive strategies, progress monitoring, and the use of instructional technology and Universal Design for Learning to address diverse needs. Specific strategies are provided such as concrete-representational-abstract instruction, mnemonic devices, graphic organizers, and technology tools to enhance math curriculum.
This study aims to produce a learning trajectory using the mathematical modeling in helping students to understand the concept of algebraic operations. Therefore, the design research was chosen to meet the research aims and to give in formulating and developing local instructional theory in learning algebraic operations.Learning trajectory designed in the early phases and tested on 34 seven-grade students in SMP N 10 Palembang. Data collection was conducted through observation by recording the learning process that occured in the classroom and students’ group work was evidenced by video and photos. Data was analyzed qualitatively by describing actual learning which happened in pilot experiment and teaching experiment. There are 3 learning activities in the design of this study. These 3 activities are designed based on the steps of the Mathematical Modeling, activity 1 meaning of algebraic expressions, activity 2 addition of algebraic and activity 3 subtraction of algebraic. Based on the result, it can be concluded that activity which has been designed can help the students in learning algebraic operations using mathematical modeling. Used mathematical modeling can help student solve the problems and understand concept are structured using the assumptions and model start they design so gradually developed into formal mathematics.
READ AND ACT APPROACH AND RETENTION SKILLS IN MTBWayneRavi
This study was conducted to determine the effect of read and act approach on the Retention Skills in MTB. Specifically, it sought to find out if read and act approach significantly contribute in the retention skills of student in MTB. Descriptive-Comparative research design using paired sample T-test was used to utilized in the study. The study was carried out in Beam Extension School of Guinobatan annex of Paradise Embac Elementary School to Thirty student of Grade Two section 3. Descriptive statistics (mean & SD), Paired-Sample T-test and Eta2were used as tools in the analysis of data. Results revealed that the pre test score was high; the post test score was very high. There was a significant difference on the pretest and post test scores of pupils in read and act approach. Further, the read and act approach has large effect on the retention skills of student in MTB.
KEYWORD APPROACH AND SOLVING WORD PROBLEMS INVOLVING ADDITION OF WHOLE NUMBER WayneRavi
This study was conducted to determine the effect of keywords approach on solving word problems involving addition of whole numbers with sums up to 99,999 including money following the steps in problem solving. Results revealed that the pre test score was low; the post test score was high. There was a significant difference on the pre test and post test scores of keywords approach. Further, the keywords approach has large effect on the solving word problems involving addition of whole number.
Algebra difficulties among second year bachelor of secondaryJunarie Ramirez
This study examined the difficulties that second year Bachelor of Secondary Education students experienced with algebra. Questionnaires were used to understand what causes difficulties and specific algebraic topics that posed problems. The results showed that ineffective teaching strategies were the primary cause of difficulties. Special products/factoring and rationalizing denominators were the most challenging topics. The researchers concluded that teachers must find better ways to explain concepts and cater their lessons to students' needs.
The Effect of the Concrete-Representational-Abstract Mathematical Sequence o...Janet Van Heck
This document describes a study that examined the effects of using the concrete-representational-abstract (CRA) teaching sequence with explicit instruction to teach addition skills from 0 to 9 to kindergarten students struggling in math. The CRA sequence begins by using manipulatives, then representations like pictures, before moving to abstract problems. Three kindergarten students identified as needing math support through RTI screening were given scripted CRA lessons and their test scores were measured. The study found the CRA sequence improved students' conceptual understanding and performance on addition facts from 0 to 9 when delivered with explicit instruction.
The purpose of this study is to determine the mathematics teacher performance category in building high order thinking skills (HOTS) of students. The study was conducted on 560 students taken randomly from ten junior high schools and eight high schools from eight districts in North Sumatra Province. Data collection techniques and instruments are carried out by giving questionnaires to students which contain a number of questions about students' assessment of the mathematics teacher's performance in constructing the HOTS indicator. Based on descriptive analysis, it was found that the performance of mathematics teachers built HOTS indicators, namely (1) understanding of concepts, (2) mathematical communication, (3) creativity, (4) problem solving, and (5) reasoning is enough category. The results of analysis of variance show that teacher performance builds (1) understanding of concepts, (2) mathematical communication, (3) creativity, (4) problem solving, and (5) reasoning significantly influences students' abilities.
The lesson plan is for a junior high school mathematics class on similarity and congruence. It involves having students work in groups on a project to identify similar planes by calculating proportional sides and congruent angles. Over two class periods, students will design and present their projects, then conclude what defines similarity. The teacher will use questions, discussion, and individual assessments to guide students in understanding similarity through observing and calculating proportions of similar shapes.
The document summarizes key aspects of teaching mathematics, including:
1) The goals of mathematics are critical thinking and problem solving.
2) Mathematics should be taught using a spiral progression approach, revisiting basics at each grade level with increasing depth and breadth.
3) Effective mathematics teaching employs methods like problem-solving, concept attainment, concept formation, and direct instruction.
MATHEMATICS and How to Develop Interest in Maths?Shahaziya Ummer
Meaning of Mathematics, Definition of Mathematics, Nature of mathematics, Need and significance of learning Mathematics, How to develop and maintain interest in mathematics?,
This document provides information about a precalculus and trigonometry workbook created by The Great Courses. It includes a biography of the workbook's author, Professor Bruce H. Edwards of the University of Florida. The workbook is designed to accompany Professor Edwards' Great Courses lecture series on precalculus and contains 30 lesson guides on topics ranging from functions and complex numbers to trigonometric identities, vectors, and conic sections. It is published by The Great Courses, an educational media company located in Chantilly, Virginia.
This document discusses Bloom's taxonomy, which is a classification of learning objectives into levels of complexity and specificity. It describes the three domains of cognitive, affective, and psychomotor. Within the cognitive domain are the levels of knowledge, comprehension, application, analysis, synthesis, and evaluation, ordered from simple to complex. Examples are given for how objectives could be written for each level within topics like arithmetic, algebra, geometry, and trigonometry.
Mathematics anxiety is feelings of tension and anxiety that interfere with manipulating numbers and solving math problems. It can be caused by poor instruction, testing pressures, and negative past experiences with math. Teachers, parents, students, and even other teachers can experience mathematics anxiety. It often occurs when doing math tasks but can also happen just by thinking about math. Reducing mathematics anxiety requires improving teacher-student relationships, using positive reinforcement, and teaching math concepts understanding rather than just memorization.
Mathematics anxiety is feelings of tension and anxiety that interfere with manipulating numbers and solving math problems. It can be caused by poor instruction, testing pressures, and negative past experiences with math. Students, parents, and teachers can all experience mathematics anxiety, especially when participating in math-related activities or tests. Reducing math anxiety requires improving teacher-student relationships, making math more relevant and comprehensible, and creating a supportive learning environment where mistakes are accepted.
An action film genre involves heroes facing physical challenges and threats through fights, chases and explosions. The genre originated in silent Western films and evolved through war, crime and spy films. Iconic films like Bullitt, Enter the Dragon and Raiders of the Lost Ark incorporated car chases, martial arts and adventure themes that influenced the action blockbusters of the 1980s starring actors like Schwarzenegger and Stallone. By the 1990s, action films parodied their conventions while utilizing new CGI effects.
The document summarizes Outburst 3D Node Knockout 2011, a real-time multiplayer game created by four developers in 48 hours. It discusses using WebGL and Three.js to render 3D graphics in the browser without plugins. The game shares code between a Node.js server and client using WebSockets for multiplayer. It transmits input and world state packets 50 times per second using delta compression to reduce bandwidth.
Childrení»s mathematics performance and drawing activities a constructive cor...Alexander Decker
This document summarizes a study that examined how drawing activities can improve children's performance in mathematics. The study used observation, interviews, achievement tests, and a pictorial Likert scale to analyze relationships between discovery learning methods and mathematics performance for 62 pupils at a primary school in Ghana where mathematics performance was low. The results revealed that incorporating drawing activities into mathematics lessons significantly improved pupils' mathematics performance and engagement. The study concluded that using arts-integrated approaches can help establish relationships between concepts and enhance understanding in mathematics.
The document discusses key topics in mathematics pedagogy for CTET exams, including:
- Defining pedagogy and mathematics.
- The nature of mathematics as both a science of discovery and logical processes.
- Guiding principles and vision for mathematics in the NCF-2005 curriculum.
- Strategies for teaching mathematics like written work, oral work, group work and homework.
- Reasons for keeping mathematics in school curriculums like its basis in other sciences and role in developing logical thinking.
- The language of mathematics including concepts, terminology, symbols and algorithms.
- Approaches like community mathematics and mathematical communication to engage students.
EFFECTIVENESS OF SINGAPORE MATH STRATEGIES IN LEARNING MATHEMATICS AMONG FOUR...Thiyagu K
The Singapore math method is child-focused, and seeks to make sure that the student gains a full and complete understanding of the fundamental mathematical concepts, rather than merely memorizes a rote collection of facts. This approach not merely enhances mathematical learning; it also offers a firm foundation from which broader mathematical principles can be extrapolated. The present study tries to find out the effectiveness of Singapore math strategies in learning mathematics among fourth standard students. Two equivalent group experimental-designs are employed for this study. The investigator has chosen 64 Fourth standard students for the study. According to the scoring of pre-test, 32 students were chosen as control group and 32 students were chosen as experimental group. Finally the investigator concludes; (a) the experimental group student is better than control group students in their gain scores. (b) There is no significant difference between control group and experimental group students in their pre test scores and post test. (c)There is significant difference between control group and experimental group students in the scores of posttest attainment of knowledge, understanding and application objectives.
REALITY – BASED INSTRUCTION AND SOLVING WORD PROBLEMS INVOLVING SUBTRACTIONWayneRavi
This study was conducted to determine the effect of reality based on the solving word problems involving subtraction. Descriptive-Comparative research design using paired sample T-test was used to utilized in the study. The study was carried out in Tibungol Elementary School to Fifty student of Grade Three section 1. Results revealed that there was a significant difference on the pretest and post test scores of pupils in reality based approach. Further, the reality based approach is effective in improving the performance of student.
EFFECTIVENESS OF INTEGRATING RIDDLES IN TEACHING MATHEMATICS AMONG VIII STAND...Thiyagu K
Mathematics is considered as dry subject and students do not find anything interesting in it. This impression about Mathematics can be reversed with the help of recreational activities in Mathematics. The present study tries to find out the effectiveness of integrating riddles in teaching mathematics among eighth standard students. Two equivalent group experimental-designs are employed for this study. The investigator has chosen 40 eighth standard students for the study. According to the scoring of pre-test, 20 students were chosen as control group and 20 students were chosen as experimental group. Finally the investigator concludes; (a) There is a significant difference between the means of students thought through conventional method and puzzles and riddles way of learning group. (b) There is a significant difference between the means of the Post-Test scores of control group and experimental group students with respect to the knowledge, understanding and application objectives.
The document discusses issues students with disabilities face in math including perceptual, language, reasoning, and memory challenges. It then describes considerations for instruction including differentiated instruction, metacognitive strategies, progress monitoring, and the use of instructional technology and Universal Design for Learning to address diverse needs. Specific strategies are provided such as concrete-representational-abstract instruction, mnemonic devices, graphic organizers, and technology tools to enhance math curriculum.
This study aims to produce a learning trajectory using the mathematical modeling in helping students to understand the concept of algebraic operations. Therefore, the design research was chosen to meet the research aims and to give in formulating and developing local instructional theory in learning algebraic operations.Learning trajectory designed in the early phases and tested on 34 seven-grade students in SMP N 10 Palembang. Data collection was conducted through observation by recording the learning process that occured in the classroom and students’ group work was evidenced by video and photos. Data was analyzed qualitatively by describing actual learning which happened in pilot experiment and teaching experiment. There are 3 learning activities in the design of this study. These 3 activities are designed based on the steps of the Mathematical Modeling, activity 1 meaning of algebraic expressions, activity 2 addition of algebraic and activity 3 subtraction of algebraic. Based on the result, it can be concluded that activity which has been designed can help the students in learning algebraic operations using mathematical modeling. Used mathematical modeling can help student solve the problems and understand concept are structured using the assumptions and model start they design so gradually developed into formal mathematics.
READ AND ACT APPROACH AND RETENTION SKILLS IN MTBWayneRavi
This study was conducted to determine the effect of read and act approach on the Retention Skills in MTB. Specifically, it sought to find out if read and act approach significantly contribute in the retention skills of student in MTB. Descriptive-Comparative research design using paired sample T-test was used to utilized in the study. The study was carried out in Beam Extension School of Guinobatan annex of Paradise Embac Elementary School to Thirty student of Grade Two section 3. Descriptive statistics (mean & SD), Paired-Sample T-test and Eta2were used as tools in the analysis of data. Results revealed that the pre test score was high; the post test score was very high. There was a significant difference on the pretest and post test scores of pupils in read and act approach. Further, the read and act approach has large effect on the retention skills of student in MTB.
KEYWORD APPROACH AND SOLVING WORD PROBLEMS INVOLVING ADDITION OF WHOLE NUMBER WayneRavi
This study was conducted to determine the effect of keywords approach on solving word problems involving addition of whole numbers with sums up to 99,999 including money following the steps in problem solving. Results revealed that the pre test score was low; the post test score was high. There was a significant difference on the pre test and post test scores of keywords approach. Further, the keywords approach has large effect on the solving word problems involving addition of whole number.
Algebra difficulties among second year bachelor of secondaryJunarie Ramirez
This study examined the difficulties that second year Bachelor of Secondary Education students experienced with algebra. Questionnaires were used to understand what causes difficulties and specific algebraic topics that posed problems. The results showed that ineffective teaching strategies were the primary cause of difficulties. Special products/factoring and rationalizing denominators were the most challenging topics. The researchers concluded that teachers must find better ways to explain concepts and cater their lessons to students' needs.
The Effect of the Concrete-Representational-Abstract Mathematical Sequence o...Janet Van Heck
This document describes a study that examined the effects of using the concrete-representational-abstract (CRA) teaching sequence with explicit instruction to teach addition skills from 0 to 9 to kindergarten students struggling in math. The CRA sequence begins by using manipulatives, then representations like pictures, before moving to abstract problems. Three kindergarten students identified as needing math support through RTI screening were given scripted CRA lessons and their test scores were measured. The study found the CRA sequence improved students' conceptual understanding and performance on addition facts from 0 to 9 when delivered with explicit instruction.
The purpose of this study is to determine the mathematics teacher performance category in building high order thinking skills (HOTS) of students. The study was conducted on 560 students taken randomly from ten junior high schools and eight high schools from eight districts in North Sumatra Province. Data collection techniques and instruments are carried out by giving questionnaires to students which contain a number of questions about students' assessment of the mathematics teacher's performance in constructing the HOTS indicator. Based on descriptive analysis, it was found that the performance of mathematics teachers built HOTS indicators, namely (1) understanding of concepts, (2) mathematical communication, (3) creativity, (4) problem solving, and (5) reasoning is enough category. The results of analysis of variance show that teacher performance builds (1) understanding of concepts, (2) mathematical communication, (3) creativity, (4) problem solving, and (5) reasoning significantly influences students' abilities.
The lesson plan is for a junior high school mathematics class on similarity and congruence. It involves having students work in groups on a project to identify similar planes by calculating proportional sides and congruent angles. Over two class periods, students will design and present their projects, then conclude what defines similarity. The teacher will use questions, discussion, and individual assessments to guide students in understanding similarity through observing and calculating proportions of similar shapes.
The document summarizes key aspects of teaching mathematics, including:
1) The goals of mathematics are critical thinking and problem solving.
2) Mathematics should be taught using a spiral progression approach, revisiting basics at each grade level with increasing depth and breadth.
3) Effective mathematics teaching employs methods like problem-solving, concept attainment, concept formation, and direct instruction.
MATHEMATICS and How to Develop Interest in Maths?Shahaziya Ummer
Meaning of Mathematics, Definition of Mathematics, Nature of mathematics, Need and significance of learning Mathematics, How to develop and maintain interest in mathematics?,
This document provides information about a precalculus and trigonometry workbook created by The Great Courses. It includes a biography of the workbook's author, Professor Bruce H. Edwards of the University of Florida. The workbook is designed to accompany Professor Edwards' Great Courses lecture series on precalculus and contains 30 lesson guides on topics ranging from functions and complex numbers to trigonometric identities, vectors, and conic sections. It is published by The Great Courses, an educational media company located in Chantilly, Virginia.
This document discusses Bloom's taxonomy, which is a classification of learning objectives into levels of complexity and specificity. It describes the three domains of cognitive, affective, and psychomotor. Within the cognitive domain are the levels of knowledge, comprehension, application, analysis, synthesis, and evaluation, ordered from simple to complex. Examples are given for how objectives could be written for each level within topics like arithmetic, algebra, geometry, and trigonometry.
Mathematics anxiety is feelings of tension and anxiety that interfere with manipulating numbers and solving math problems. It can be caused by poor instruction, testing pressures, and negative past experiences with math. Teachers, parents, students, and even other teachers can experience mathematics anxiety. It often occurs when doing math tasks but can also happen just by thinking about math. Reducing mathematics anxiety requires improving teacher-student relationships, using positive reinforcement, and teaching math concepts understanding rather than just memorization.
Mathematics anxiety is feelings of tension and anxiety that interfere with manipulating numbers and solving math problems. It can be caused by poor instruction, testing pressures, and negative past experiences with math. Students, parents, and teachers can all experience mathematics anxiety, especially when participating in math-related activities or tests. Reducing math anxiety requires improving teacher-student relationships, making math more relevant and comprehensible, and creating a supportive learning environment where mistakes are accepted.
An action film genre involves heroes facing physical challenges and threats through fights, chases and explosions. The genre originated in silent Western films and evolved through war, crime and spy films. Iconic films like Bullitt, Enter the Dragon and Raiders of the Lost Ark incorporated car chases, martial arts and adventure themes that influenced the action blockbusters of the 1980s starring actors like Schwarzenegger and Stallone. By the 1990s, action films parodied their conventions while utilizing new CGI effects.
The document summarizes Outburst 3D Node Knockout 2011, a real-time multiplayer game created by four developers in 48 hours. It discusses using WebGL and Three.js to render 3D graphics in the browser without plugins. The game shares code between a Node.js server and client using WebSockets for multiplayer. It transmits input and world state packets 50 times per second using delta compression to reduce bandwidth.
This document discusses different types of film distributors that could distribute an independent film. It identifies major studios like 20th Century Fox and Warner Bros. that distribute big budget films. It also discusses smaller independent distributors that are better suited to independent films with less financial backing. The document recommends two potential distributors - Pathé, a French distributor known for films in their genre, and Fractured Films, a UK independent distributor specializing in similar films.
This document provides information about HIV/AIDS in 3 paragraphs. It defines HIV/AIDS, describes how it is transmitted through bodily fluids and discusses prevention methods. It also outlines government programs that provide voluntary testing and treatment clinics, and distribute clean needles and condoms to prevent transmission. The overall message is that individuals should protect themselves and others from HIV/AIDS.
Knockout.js is an open source JavaScript library for building rich client-side web applications. It uses the Model-View-ViewModel (MVVM) pattern to bind HTML markup and JavaScript code together. Knockout.js provides automatic dependency tracking and integrated templating capabilities. It supports declarative bindings that allow for two-way communication between the view and view model components. This enables rich interactivity and removes the need for overlapping event handlers in the code.
Digipack Deconstruction 2: AC/DC - High VoltageTomSmellsGood
The document summarizes the album artwork and packaging for AC/DC's album "High Voltage". Key elements are prominently featured, including the band's logo, lead guitarist Angus Young in his schoolboy outfit, and lightning bolts representing the album title. These visual motifs are used consistently throughout to establish the band's brand and reinforce the high-energy themes of their live performances and music that are represented by the album title "High Voltage".
Deconstruction 1: Queen - I Want To Break FreeTomSmellsGood
This video summary analyzes the Queen music video for "I Want to Break Free". It notes that the video tells the story of Freddie Mercury trying to break out of domestic expectations through performance and narrative elements. Unlike typical rock videos, it does not feature on-stage performances. However, it does include close-ups of Freddie and the band as demanded by the record label to promote their image. The video also has strong visual connections to the song's lyrics about desiring freedom from social constraints.
Enter Shikari is a post-hardcore band formed in 2003 in St Albans, England. The band combines post-hardcore with other genres like electronica, drum and bass, and occasionally dubstep. They have released four studio albums between 2007-2012 to critical acclaim. Enter Shikari is known for incorporating political and social commentary into their lyrics. They have voiced opinions on issues like government overreach and environmental destruction. The band creates in a style that blends rock instrumentation with electronic elements to produce an energetic live show.
The document summarizes ways that music videos, album artwork, and advertisements establish genre and represent artists. It discusses how videos use shots that link to the music's genre. Album covers highlight band names and use colors associated with genres. Advertisements prominently display band names and use colors linked to genres. Photographs of bands playing rock instruments also help establish genre.
Presentasi ini membahas materi-materi matematika yang terdiri atas empat bab yaitu persamaan kuadrat, membuat grafik, memasukkan gambar, dan membuat art. Presentasi ini disusun oleh Surianto Sitorus untuk mengajar matematika.
Presentasi ini membahas materi-materi matematika yang terdiri atas empat bab yaitu persamaan kuadrat, membuat grafik, memasukkan gambar, dan membuat art serta menjelaskan rumus persamaan kuadrat.
Konsep diri adalah ide, pikiran, kepercayaan dan pendirian yang diketahui individu tentang dirinya sendiri. Konsep diri terbentuk secara bertahap oleh pengaruh lingkungan, orang-orang terdekat, dan pengalaman pribadi. Konsep diri mempengaruhi interaksi sosial dan psikologi individu.
The document discusses the origins and evolution of rock music from the 1940s-1950s in the United States. It then covers several iconic rock bands that helped shape the genre including The Beatles, The Rolling Stones, Nirvana, and Guns N' Roses. The document also explores some rock subgenres like alternative rock, punk rock, and nu metal. It provides examples of songs from each band and genre discussed.
Alfred Adler developed Individual Psychology, also known as Adlerian Therapy. Some of the key concepts in Adlerian Therapy include feelings of inferiority, compensation, lifestyle, and social interest. The therapeutic process in Adlerian Therapy involves building a relationship, assessing the client's lifestyle, providing insight and interpretation, and reeducating and reorienting the client to adopt a new lifestyle.
This document discusses several key concepts in social perception:
1. Nonverbal communication plays an important role in social perception. Facial expressions, eye contact, body language, posture, and touching can all reveal emotional and mental states. Basic emotions are often expressed through specific facial movements.
2. Attribution refers to how people seek to understand the behaviors of others by inferring underlying traits or motives. Correspondent inference theory holds that behaviors perceived as freely chosen and distinctive are more likely to be attributed to internal traits. Kelley's theory examines how attribution is influenced by consensus, consistency, and distinctiveness.
3. Impression formation is the process by which people combine diverse information to form unified impressions of others. Initial
Teachers’ Knowledge about Students’ Errors in Word Problems at Elementary Ma...Aquarius31
The document summarizes the research of a Bangladeshi student on elementary teachers' knowledge of students' errors in mathematical word problems. The research included preliminary and final surveys of students and teachers. The preliminary survey found that students made various errors in word problems and teachers were aware of common causes and solutions. The final survey analyzed test responses from 124 students at 4 schools and interviewed teachers. It identified specific errors students made and teachers' knowledge of errors, causes, and teaching strategies. Overall, the research examined elementary students' errors in Bangladesh and teachers' understanding of those errors to improve mathematics instruction.
INTERPRETING PHYSICS TEACHERS’ FEEDBACK COMMENTS ON STUDENTS’ SOLUTIOijejournal
This paper investigates teachers’ intentions, when providing their feedback comments to hypothetical
students’ written solutions to linear motion tasks. To obtain an in-depth understanding of the teachers’
thinking when responding to student written solutions, a qualitative case study approach was employed
using two different data sources: a Problem Centred Questionnaire (PCQ) and a Problem Centred
Interview (PCI). Data processing was conducted in two main phases: Initial and Comparative. In both
phases we explored patterns about teachers’ foci across student strategies and motion tasks. A main finding
of this research is to categorising teachers’ interpretations and feedback on student solutions, based on the
extent of teachers’ attentions to Student Thinking and Disciplinary Thinking. This analysis approach
refines the previously held view that a high level of teacher content knowledge, and a concurrent focus to
both ‘student thinking’ and ‘disciplinary thinking’ are required to provide meaningful feedback on student
solutions. The findings indicated that their level of teachers’ propositional
Model of Mathematics Teaching: A Fuzzy Set ApproachIOSR Journals
This document presents a fuzzy set approach to modeling mathematics teaching. It defines four dominant views of mathematics teaching and represents them as fuzzy sets based on the membership of teachers. A classroom experiment evaluates 12 teachers and their performance across the three states of the model - learner focused, content focused, and classroom focused. The teachers' profiles are represented as elements of the Cartesian product of the fuzzy sets. Membership degrees, possibilities, and probabilities are calculated for the teachers' profiles based on their performance with two different groups of students. The combined results provide a quantitative analysis of teachers' skills during the mathematics teaching process.
An Investigation Of Secondary Teachers Understanding And Belief On Mathemati...Tye Rausch
1) The document discusses secondary teachers' understanding and beliefs regarding mathematical problem solving in Indonesia. It investigates how teachers understand problem solving concepts like problems, strategies, and instructional practices, as well as their self-reported difficulties.
2) The study found that teachers have a good understanding of pedagogical problem solving knowledge but a weaker understanding of problem solving content knowledge such as strategies. Teachers reported that their main difficulties are determining precise mathematical models and choosing suitable real-world contexts for problems.
3) The study also examined teachers' beliefs about mathematics and learning, finding they tend to view mathematics as static but believe problem solving should be taught dynamically to engage students.
The document summarizes Roslinda Rosli's dissertation defense on integrating problem posing in teaching and learning mathematics. It provides an overview of the dissertation and introduces the research questions and methodology used in four studies on the effects of problem posing on student learning, preservice teachers' problem solving and posing abilities, and preservice teachers' knowledge and attitudes towards fractions. The theoretical framework draws on constructivist learning theory. Key findings include that problem posing can benefit student mathematics learning, and preservice teachers demonstrated understanding in solving mathematical problems but posed many low-quality problems initially.
This study aimed at analyzing and describing Various Methods used by mathematics teacher in solving equations. Type of this study is descriptive by subject of this study comprised 65 mathematics teachers in senior, junior, and primary schools respectively 15, 33, and 17 in numbers. The data were collected from the answer to containing four problems of equation. Data Coding was conducted by two coding personnel to obtain credible data. The data were then analyzed descriptively. It has been found that the teachers have implemented a method for solving equation problems by means of operation on one side of equation and procedural operation. This method has been dominantly used by the teachers to solve to the equation problems. The other method was doing operation on both sides of the equation simultaneously by focusing on similar elements on both sides of the equation.
The document summarizes a study that examined the effects of integrating mathematical modeling on students' problem solving performance and math anxiety. The study found that:
1) Both the experimental group taught using mathematical modeling and the control group taught using guided practice showed significant improvements in their problem solving scores and reductions in math anxiety from pre-to-post testing.
2) The experimental group showed significantly higher post-test problem solving scores and lower math anxiety than the control group, indicating that mathematical modeling was more effective at improving problem solving and reducing anxiety than guided practice alone.
3) The study concluded that both mathematical modeling and guided practice can increase students' problem solving skills, but mathematical modeling more significantly improves problem solving performance and
The document summarizes a research study on self-evaluation of practice teaching by pupil teachers. The study aimed to evaluate the effectiveness of pupil teaching through self-evaluation. It assessed 70 pupil teachers (30 boys and 40 girls) from two colleges on 21 components of teaching effectiveness using a checklist. Statistical analysis found no significant differences in teaching efficiency based on gender, faculty, or type of college. The study concluded that self-evaluation helped pupil teachers identify areas for improvement, but they generally lacked satisfaction in their teaching abilities.
This 3-day lesson plan teaches high school honors geometry students how to determine the distance formula, midpoint formula, and slope formula through problem-solving activities. On day 1, students work in groups to discover the distance formula using the Pythagorean theorem. On day 2, groups present their work and the teacher explains how this relates to the distance formula. Day 3 has students find the midpoint and slope formulas through similar exploration and explanation. The goal is for students to understand these geometric concepts and formulas through hands-on learning experiences.
Problem solving is a complex cognitive process that is important for developing critical thinking skills. This study aims to assess the attitudes of freshmen students towards problem solving by determining their socio-demographic profile and attitudes, and identifying negative attitudes. The participants were freshmen students at Southern Christian College who completed a questionnaire on problem solving attitudes. The findings will help educators develop strategies to improve students' problem-solving skills.
A Structural Model Related To The Understanding Of The Concept Of Function D...Cheryl Brown
1) The document discusses a study that examines secondary students' understanding of the concept of function across three dimensions: definition, representation (recognition and interpretation), and problem solving.
2) Confirmatory factor analysis confirmed these four dimensions comprise the conceptual understanding of functions. The ability to define the concept was found to influence performance on the other dimensions.
3) The study aims to verify the proposed theoretical structure of conceptual understanding of functions and examine how the dimensions are interrelated, as well as identify differences in student performance at different grade levels.
This study aimed to understand how secondary mathematics teachers engage
with learners during the teaching and learning process. A sample of six
participants was purposively selected from a population of ordinary level
mathematics teachers in one urban setting in Zimbabwe. Field notes from
lesson observations and audio-taped teachers’ narrations from interviews
constituted data for the study to which thematic analysis technique was then
applied to determine levels of mathematical intimacy and integrity displayed
by the teachers as they interacted with the students. The study revealed some
inadequacies in the manner in which the teachers handled students’
responses as they strive to promote justification skills during problem
solving, in particular teachers did not ask students to explain wrong answers.
The teachers indicated that they did not have sufficient time to engage
learners in authentic problem-solving activities since they would be rushing
to complete syllabus for examination purposes. On the basis of these
findings, we suggest teachers to appreciate the need to pay special attention
to the kinds of responses given by learners during problem-solving in order
to promote justification skills among learners.
A Case Study of Teaching the Concept of Differential in Mathematics Teacher T...theijes
In high schools of Viet Nam, teaching calculus includes the knowledge of the real function with a real variable. A mathematics educator in France, Artigue (1996) has shown that the methods and approximate techniques are the centers of the major problems (including number approximation and function approximation...) in calculus. However, in teaching mathematics in Vietnam, the problems of approximation almost do not appear. With the task of training mathematics teachers in high schools under the new orientations, we present a part of our research with the goal of improving the contents and methods of teacher training
A study of students perception of class room behavior of mathematics teachers.Alexander Decker
This study examined students' perceptions of mathematics teachers' classroom behavior in secondary schools in Jammu City, India. 171 students in grades 8-10 were surveyed about their teachers' teaching behaviors and personality dimensions using the Lamsal scale. Results showed that boys and girls had similar perceptions of their teachers' classroom behaviors and did not differ significantly in their ratings of teaching behaviors or personality dimensions. Overall, students were found to have a favorable perception of their mathematics teachers' classroom behaviors.
The effects of collaborative learning on problem solving abilities among seni...Alexander Decker
This study investigated the effects of collaborative learning on problem solving abilities among senior secondary physics students learning about simple harmonic motion (SHM). The study found that:
1) Most physics teachers in the schools studied lacked proper teaching qualifications.
2) Students faced difficulties with SHM problems due to lack of understanding terms, basic math skills, and confidence. They benefited from collaborative learning and computer simulations.
3) Students taught with collaborative learning scored significantly higher on problem solving tests than those taught with traditional methods, showing collaborative learning improved problem solving abilities in SHM.
Effects of Task-based language teaching on grade five students reading perfor...Mesfin Eyob
This document summarizes a study on the effects of task-based language teaching (TBLT) on the reading performance of grade 5 students in Mekelle City, Ethiopia. The study used a quasi-experimental pre-test post-test design with an experimental and control group. Results showed that students in the experimental group, who received TBLT, performed significantly better on the post-test than the control group. Additionally, high-achieving students benefited more from TBLT than low-achieving students. Challenges encountered included limited English proficiency and time constraints in completing tasks. The pilot study informed revisions for a full study on the impact of TBLT on reading skills.
A Pair-Wise Analysis Of The Cognitive Demand Levels Of Mathematical Tasks Use...Joe Andelija
This study analyzed the cognitive demand levels of 66 paired mathematical tasks used for classroom instruction and homework assignments in an 8th grade classroom over one school year. The study found that approximately two-thirds of the time, the cognitive demand levels of the mathematical tasks assigned for homework differed from those used during classroom instruction. The implications for student learning, teaching practices, homework, and further research are discussed.
Teacher educator perspectives on pedagogical modelling and explaining in desi...DTGeek
Abstract:This paper builds on a previous study on the demonstration as a signature pedagogy in design and technology, this paper explores teacher educators’ values on teacher modelling and explanation. In a previous study the participating teachers identified “competent management of the learning experience” as a significant factor in effective demonstrations, and in particular teacher competency, clarity and subject knowledge. The demonstration is a fundamental pedagogical tool for practical subjects where procedural knowledge is developed over time from observation and imitation to independence and adaption of technique. As such, it tends to align itself at the restrictive end of an expansive-restrictive continuum. This study builds on the developing exploration of the nature of the demonstration, exploring the subjective values of teacher educators. Q Methodology is used to compare and analyse the responses of the participating teacher educators. A Q-Set of statements, developed and refined with D&T teacher educators, relating to modelling and explaining, represents the concourse of opinions and perspectives. The sample is purposive, comprised of teacher educators. The findings represent a snapshot of subjective values, informing the wider discourse on signature pedagogies in design and technology education.
This study investigated the experiences of two special educators who experienced math phobia. Through interviews and journal entries, the educators discussed their struggles with mathematics as students and teachers. As students, they avoided math courses and received tutoring. As teachers, they felt unprepared to teach higher-level math and sought help from other teachers. They believed math was important but blamed their students' poor performance on tests and lack of critical thinking rather than their own math difficulties. Both avoided math in their personal lives and one refused to participate in a math evaluation for the study. The study provided insights into how math phobia impacts special educators and their ability to teach mathematics effectively.
The application of reciprocal teaching on the subject of straight line equati...Alexander Decker
This document summarizes a study on applying reciprocal teaching to teach straight line equations in secondary school mathematics. The study found that:
1) Students who learned using reciprocal teaching had better learning outcomes than those taught through conventional methods.
2) Reciprocal teaching was an effective model for teaching straight line equations. Both students and teachers responded positively to this approach.
3) Reciprocal teaching helped students communicate knowledge through discussion, ask questions, and summarize what they learned. It trained them to be more independent learners.
How to Manage Your Lost Opportunities in Odoo 17 CRMCeline George
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Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
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International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
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This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
বাংলাদেশের অর্থনৈতিক সমীক্ষা ২০২৪ [Bangladesh Economic Review 2024 Bangla.pdf] কম্পিউটার , ট্যাব ও স্মার্ট ফোন ভার্সন সহ সম্পূর্ণ বাংলা ই-বুক বা pdf বই " সুচিপত্র ...বুকমার্ক মেনু 🔖 ও হাইপার লিংক মেনু 📝👆 যুক্ত ..
আমাদের সবার জন্য খুব খুব গুরুত্বপূর্ণ একটি বই ..বিসিএস, ব্যাংক, ইউনিভার্সিটি ভর্তি ও যে কোন প্রতিযোগিতা মূলক পরীক্ষার জন্য এর খুব ইম্পরট্যান্ট একটি বিষয় ...তাছাড়া বাংলাদেশের সাম্প্রতিক যে কোন ডাটা বা তথ্য এই বইতে পাবেন ...
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2. 56 Study on Effect of Mathematics Teachers’ Pedagogical Content Knowledge
knowledge. They set forward different views. For example, Shulman proposed
a framework for analyzing teachers’ knowledge that distinguished different
categories of knowledge: knowledge of content, general pedagogical
knowledge, curriculum knowledge, pedagogical content knowledge (PCK),
knowledge of students, knowledge of educational contexts and knowledge of
educational ends, purposes and values. He emphasized PCK as a key aspect to
address in the study of teaching.
In later studies, the notion of PCK was often combined with other
theoretical ideas. For example, Klein and Tirosh (1997) evaluated pre-service
and in-service elementary teachers’ knowledge of common difficulties that
children experience with division and multiplication word problems involving
rational numbers and their possible sources. They summarized the findings
saying that “most prospective teachers exhibited dull knowledge” of the
difficulties that children’s experience with word problems involving rational
numbers and their possible sources, whereas “most in-service teachers were
aware of students incorrect responses, but not of their possible sources.”
Throughout these studies on the effect of mathematics teachers’ PCK
on mathematics teaching, we thought there were two problems. One was that
teaching objects, teaching structure, and notion explaining were often noticed
in these studies, whereas a teacher’s education views, teaching emotion,
teaching design, teaching language, students’ mathematics thinking, students’
learning attitude were not noticed fully. The other was that most teachers in
these studies were primary mathematics teachers. In other words, how middle
mathematics teachers’ PCK or high mathematics teachers’ PCK affected their
teaching was not studied adequately.
Based on the above, we would analyze two mathematics lessons about
one-variable quadratic inequality in this article. One was taught by pre-service
high mathematics teacher Z, the other was taught by in-service high
mathematics teacher Z1 who had 12 years of teaching and was responsible for
Grade Ten’s mathematics teaching in a high school in Xiaogan.
Method
Kawanaka and Stigler (1999) thought, in classrooms, speakers were
mainly teachers or students. They asked questions, answered questions,
provided feedback to answers, provided information, provided directions, and
so forth. they developed coding categories to capture these behaviors:
3. Miao Li & Ping Yu 57
soliciting, responding, reacting and utterances. The categories and definitions
were listed as follows:
(1)Elicitation (E): A teacher utterance intended to elicit an immediate
communicative response from the student(s), including both verbal and
nonverbal responses.
(2)Information (I): A teacher utterance intended to provide information to
the student(s) that did not require communicative or physical response
from the student(s).
(3)Direction (D): A teacher utterance intended to cause students to perform
immediately some physical or mental activity. When the utterance was
intended for future activities, it was coded as information even if the
linguistic form of the utterance was a directive.
(4)Uptake (U): A teacher utterance made in response to a student’s verbal
or physical responses. It was intended only for the respondent. When the
utterance was clearly intended for the entire class, it was coded as
information instead of uptake.
(5)Teacher Response (TR): A teacher utterance made in response to a
student elicitation.
(6)Provide answer (PA): A teacher utterance intended to provide the
answer to the teacher’s own elicitation.
(7)Response (R): A student utterance made in response to teacher
elicitation or direction.
(8)Student elicitation (SE): A student utterance intended to elicit an
immediate communicative response from the teacher or from other
students.
(9)Student information (SI): A student utterance intended by a student to
provide information to the teacher or to other students that did not
require an immediate response.
(10)Student direction (SD): A student utterance intended to cause the
teacher or other students to perform immediately some physical or
mental activity.
(11)Student uptake (SU): A student utterance intended to acknowledge or
evaluate another student’s response.
(12)Other (O): An utterance that did not fit into any of the previous
categories or that was not intelligible.
Further, they classified elicitation requesting subject-related
information into three categories to capture the cognitive demands of teachers’
questions, and the definitions of the three categories were as follows:
4. 58 Study on Effect of Mathematics Teachers’ Pedagogical Content Knowledge
(1)Yes/no (YN): Any content elicitation that requested a simple yes or no as a
response.
(2)Name/state (NS): Any content elicitation that (a) requested a relatively
short response, such as vocabulary, numbers, formulas, single rules,
prescribed solution methods, or an answer for computation; (b) requested
that a student read the response from a notebook or textbook; and (c)
requested that a student chose among alternatives.
(3)Describe/explain (DE): Any content elicitation that requested a description
or explanation of a mathematical object, nonprescribed solution methods, or
a reason why something was true or not true.
Example 1. Yes/no
[E][YN]Teacher: Wait a second. Do you need this inside too?
[R]Student: This isn’t needed.
Example 2. Name/state:
[E][NS] Teacher: How do you solve this?
[R]Student: You cross-multiply them.
Example 3. Describe/explain:
[E][DE] Teacher: So how do you think you should explain?
[SE]Student: Oh, my explanation?
[TR]Teacher: Yes. That’s what we want to hear.
[R]Student: Um…one…one…un…so a figure in which this and this
are connected …and…[explanation continues].
Next, we would use these categories to analyze how teacher Z’s and
teacher Z1’s PCK affect her/his mathematics teaching.
Teaching Episode
Teaching Episode 1. Pre-service teacher Z
1[I] T (teacher Z): We will learn how to solve one-variable quadratic
inequality in this lesson. (Teacher Z writes the topic on the blackboard.) First,
let us to look at some questions:
a. Give the graph of y=x2-x-6
b. Find the roots of x2-x-6=0 through the graph of y=x2-x-6.
c. Find the solution set of y=x2-x-6>0 through the graph of y=x2-x-6.
2[E][NS] T: The first question: give the graph of y=x2-x-6. What is the
graph of y=x2-x-6?
3[R] S (Student): Parabola.
5. Miao Li & Ping Yu 59
4[E][NS] T: Does the graph of y=x2-x-6 intersect the x axis? If does,
how many intersection points are there?
5[R] S: yes, there are two intersection points.
6[E][YN] T: Could you find the two intersection points?
7[R] S: yes.
8[E][NS] T: What are they?
9[R] S: -2, 3.
10[E][NS] T: (Teacher Z draws the coordinate axis, then draws (-2,0)
and (3,0))What is the direction of y=x2-x-6?
11[R] S: Up.
12[U]T:(Teacher Z draws the graph of y=x2-x-6 approximately
( Figure 1)) Good!
y
x
-2 O 3
Figure 1. The graph of y=x2-x-6.
13[E][NS] T: The second question: find the roots of x2-x-6=0 through
the graph of y=x2-x-6. That is to say, in the expression of y=x2-x-6, order
y=0, then, ……
14[R] S: -2, 3.
15[E][YN] T: The third question: find the solution set of y=x2-x-6
through the graph of y>x2-x-6. That is to say, in the expression of y=x2-x-6,
order y>0, all right?
16[R] S: yes.
17[I] T: y>0 means the graph locates the upside of the x axis (Teacher
Z marks the corresponding graph with a red chalk.) and the corresponding x is
this section or that section (Teacher Z points to the graph of y=x2-x-6) That is
to say, x<-2 or x>3.
18[E][YN] T: Yes or no? Is the result true or not true?
6. 60 Study on Effect of Mathematics Teachers’ Pedagogical Content Knowledge
19[R] S: No.
20[I] T: The result should be the solution set{x|x<-2 or x>3}. Let us
summarize the findings: First, we draw the graph of y=x2-x-6. Second, find
the corresponding x. Third, find the solution set of y=x2-x-6. That is to say,
the solution set of y>x2-x-6 is found through the graph of y>x2-x-6. So,
one-variable quadratic equality and one-variable quadratic inequality can be
studied through corresponding graph, and it embodied the mathematics idea
that a function is combined with its figure closely. This is a specific example.
Next, we will study its general instance ax2+bx+c>0 (a>0) or ax2+bx+c<0
(a>0)……
(The all time is 6:38)
Teaching Episode 2. In-service teacher Z1
1[I] T (teacher Z1): We have learned the quadratic function and
learned its graph, its properties and its solution. Next, let us to solve three
questions:
a. Solve the equation of x2-x-6=0.
b. Give the graph of y=x2-x-6.
c. Find the solution set of x2-x-6>0 through the graph of y=x2-x-6.
2[E][NS] T: First, solve the equation of x2-x-6=0. What about?
3[R] S: x=3 or x=-2.
4[U] T: x=3 or x=-2.
5[E][NS] T: Second, give the graph of y=x2-x-6. How do we draw the
graph?
6[R] S: list table, draw points, link points.
7[I] T: We omit the process of draw points and
y
only draw some specific points.
8[E][NS] T: corresponding coordinate, this is …
9[R] S: (-2,0).
10[E][NS] T: The coordinate of this point is… x
-2 O 3
11[R] S: (3,0).
12[E][NS] T: The coordinate of this point is…
13[R] S: (0,6).
14[E][NS] T: The coordinate of this point is…
1 25
15[R] S: ( 2 , - 4 ). (0,-5)
Figure 2. The graph of y=x2-x-6.
7. Miao Li & Ping Yu 61
16[E][NS] T: (Teacher Z1 draws the graph of y=x2-x-6) Please
observe this graph. The graph is given by listing table, drawing points and
linking points. So, if there is a point and its horizontal coordinate is 4, what is
its vertical coordinate?
17[R] S: 6.
18[U] T: 6.
19[E][NS] T: If a point of this graph is given and its vertical coordinate
is found(Teacher Z1 points to the corresponding graph), we can see its vertical
coordinate ……
20[R] S: Its vertical coordinate is bigger than 0.
21[U] T: Its vertical coordinate is bigger than 0, all right?
22[E][NS] T: If a point of this graph is given and its vertical coordinate
is found (Teacher Z1 points to the corresponding graph), we can see its
vertical coordinate ……
23[R] S: Its vertical coordinate is smaller than 0.
24[U] T: Its vertical coordinate is smaller than 0. Good!
25[E][NS] T: What about these two points? What are their vertical
coordinate?
26[R] S: 0.
27[U] T: 0. Good!
28[I] T: The vertical coordinates of two points is 0 respectively. That is
to say, (Teacher Z1 points to the corresponding graph and says) when x=-2 or
3, its corresponding y is 0. And if a point is given in this place, its
corresponding y is bigger than 0, if a point is given in that place, its
corresponding y is smaller than 0
29[E][NS] T: Can we find the solution set of x2-x-6>0 through the
graph of y=x2-x-6?
30[R] S: Yes.
31[E][NS] T: x2-x-6>0….What is the relationship between it and
y=x2-x-6?
32[R] S: In y=x2-x-6, y>0.
33[R] T: In y=x2-x-6, y>0
34[E][NS] T: Through the graph, y>0 means the corresponding graph
locates in…
35[R] S: Up.
36[I] T: Yes, y>0 means the graph locates the upside of the x axis.
37[E][NS] T: So, the value of y corresponding to this graph is…
38[R] S: y>0.
8. 62 Study on Effect of Mathematics Teachers’ Pedagogical Content Knowledge
39[E][NS] T: This?(Teacher Z1 points to the corresponding graph)
40[R] S: y>0.
41[I] T: That is to say, x2-x-6>0. If a point in this graph is selected,
its vertical coordinate is bigger than 0, vice versa.
42[E][YN] T: Does this point have the property (Teacher Z1 points to
the graph )
43[R] S: No.
44[E][YN] T: Does that point have the property?( Teacher Z1 points to
the graph )
45[R] S: No.
46[E][YN] T: This point?
47[R] S: No.
48 [E][YN] T: These points? Are these points’ vertical coordinate
bigger than 0?
49[R] S: Yes.
50[E][NS] T: If these points are put in the x axis, what can we get?
51 [R] S: x>3 or x<-2.
52[I] T: We can describe x>3 or x<-2 in the x axis with red chalk.
53[E][NS] T: So, what is the solution set of x2-x-6>0?
54[R] S: { x|x>3 or x<-2}.
55[E][NS] T: And, what is the solution set of x2-x-6>0?
56[R] S:{x|-2<x<3}.
57[I] T: The solution sets of x2-x-6>0 and x2-x-6<0is found through
the graph.
This is a specific example. Then, how about the general instance? This
is what we will learn today, ax2+bx+c>0 (a>0) or ax2+bx+c<0 (a>0).
( Teacher Z1 writes the topic on the blackboard)
Analysis
Through previous teaching episode 1 and teaching episode 2, we could
see pre-service teacher Z had finished her teaching from the special example to
the general instance within six minutes. However, in-service teacher Z1 used
ten minutes when he taught the same content. Based on Kawanaka and Stigler
(1999) coding categories, the differences about the questions posed by teacher
Z and teacher Z1 were as follows:
(1) The sequence for posing three questions was different, especially
the former two questions.
9. Miao Li & Ping Yu 63
First, teacher Z let students draw the graph of y=x2-x-6; second, let
students find the roots of x2-x-6=0; third, he let students find the solution set
of x2-x-6 > 0. On the other hand, teacher Z1 first let students solve the
equation of x2-x-6=0; second, he let students draw the graph of y=x2-x-6;
third, he let students find the solution set of x2-x-6<0 through the graph of
y=x2-x-6.
From the sequence for posing three questions by teacher Z and teacher
Z1, we could see teacher Z thought the latter two questions were solved based
on the first question. That was to say, the latter two questions could be solved
through the graph of y=x2-x-6. But, teacher Z1 thought the roots of x2-x-6=0
should be found before the graph of y=x2-x-6 was drawn, namely, the roots of
x2-x-6=0 were special points of the graph of y=x2-x-6 (This view could also
be seen from 7th row to 11th row in the teaching episode about teacher Z1).
Thus, the different teaching design between teacher Z and teacher Z1 was
reflected.
(2) As to the question “finding the roots of x2-x-6=0” or “solving the
equation of x2-x-6=0,” this question was easy for students in high school. So,
two teachers let students answer this question directly. (This view could be
seen from 13th row to 14th row in the teaching episode about teacher Z and
from 2th row to 4th row in the teaching episode about teacher Z1). The only
difference was that teacher Z1 often repeated students’ answers.
(3) As to the question “drawing the graph of y=x2-x-6,” this question
was also not difficult for students in high school.
Teacher Z asked 5 questions (Please saw 2th row to 12th row in the
teaching episode about teacher Z): What was the graph of y=x2-x-6? ──Did
the graph of y=x2-x-6 intersect the x axis? If it did, how many intersection
points were there? ──Could you find the two intersection points? ──What
were they? ──What was the direction of y=x2-x-6?
Teacher Z1 also asked 5 questions (Please see 5th row to 15th row in the
teaching episode about teacher Z1): How did we draw the graph? ──drew
some specific points──The coordinate of this point was……──The
coordinate of this point was……──The coordinate of this point was……──
From the first question, we could see teacher Z’s and teacher Z1’s
different emphasis. Teacher Z particularly emphasized visual thinking and let
students recall the form of the graph. On the other hand, teacher Z1
particularly emphasized logical thinking and let students recall the step about
drawing the graph of function.
10. 64 Study on Effect of Mathematics Teachers’ Pedagogical Content Knowledge
Next, teacher Z let students find the two intersection points between
y=x -x-6 and the x axis and the direction of y=x2-x-6. Then, teacher Z drew
2
the graph y=x2-x-6 approximately under right-angle coordinate system
without unit length. On the other hand, teacher Z1 let students find some
specific points, including two intersection points between y=x2-x-6 and the x
axis, one intersection point between y=x2-x-6 and the y axis and the vertex of
the parabola. These specific points could ascertain the approximate form of the
graph. Then, teacher Z1 drew the graph y=x2-x-6 under right-angle coordinate
system with unit length.
It was a good question set forth by teacher Z when he asked students to
recall the form of the graph. Actually, I once met such a thing in one
mathematics teaching lesson: when I asked students to draw the graph of a
function, they knew how to list table and how to draw points, but didn’t know
how to link points, namely, they didn’t know these points were linked by
linear form or by curvilinear form and didn’t know these points were linked by
continuous form or by discontinuous form. Thus, some mistakes were made
when students were linking points. For example, some students didn’t notice
the function domain and the linking exceeded the function domain. So, if
possible, mathematics teachers should let students ascertain the approximate
form of the graph before they draw the graph since it is helpful for students
learning the process of drawing the graph. Nevertheless, teacher Z didn’t pay
attention to cultivating students’ precise mathematics thinking. For example,
teacher Z didn’t notice the importance of the vertex of the parabola in
ascertaining the graph of y=x2-x-6. Furthermore, teacher Z drew the graph of
y=x2-x-6 under right-angle coordinate system without unit length. All these
teaching behaviors would lead students to form bad learning habits when they
are drawing the graph without necessary precise demand. Just as one teacher
said, “Although it is not crucial for students learning in this lesson, drawing
the graph is an important mathematical skill. For example, drawing a graph
accurately is helpful for solving questions in solid geometry. Furthermore,
drawing graphs accurately also shows mathematics beauty. ”
Teacher Z1 emphasized the step of drawing graphs from the beginning
and paid attention to cultivating students’ logical thinking. However, he didn’t
let students recall the form of graph. Through our talking with teacher Z1, we
knew that teacher Z1 thought this question was easy for his students and
needn’t spend much time on it, but this question should be noticed according
to students’ learning content in other occasions. As to the latter two questions,
teacher Z1 didn’t list the table and draw points casually, whereas he let
11. Miao Li & Ping Yu 65
students find some key points that ascertain the form of the graph. From these
teaching behaviors, teacher Z1’s mathematics understanding about function
graph could be seen. Furthermore, teacher Z1 noticed necessary criterion about
drawing graphs, such as three elements of right-angle coordinate system,
including origin, unit of length and positive direction. He was also aware of
students’ enjoying mathematics beauty from the graph.
(4) As to the question “finding the solution set of x2-x-6>0”, how to
solve this question was crucial for students’ mathematics learning in this
lesson, and was helpful for students understanding the solution set of ax2+bx+c
>0 (a>0) or ax2+bx+c<0 (a>0).
First, teacher Z used a “Y/N” question and gave the solution set by
herself. Then, she used a “Y/N” question again and reminded students to
notice how to write the solution set. After the two questions were answered,
teacher Z summarized the findings and gave the general instance (This view
could be seen from 15th row to 20th row in the teaching episode about teacher
Z).
However, teacher Z1 spent much time on the process of finding the
solution set of x2-x-6>0 and posed many questions to accelerate students’
thinking. These questions were as follows (This view could be seen from 16th
row to 56th row in the teaching episode about teacher Z1): choosing a special
point of the graph and letting students to think about its horizontal coordinate
and vertical coordinate── letting students to think about some points in some
scope, including these points’ vertical coordinate were bigger than 0, equal to
0 and smaller than 0── summarizing the findings── asking students to find
the solution set of x2-x-6>0── leading students to analyze the relationship
between x2-x-6 > 0 and y=x2-x-6── leading students to observe the
relationship between the value of x and the value of y ── giving three
reverse examples and one positive example to strengthen the relationship ──
leading students to get the relationship between the function value and the
value of x ── getting the result. After all this learning, teacher Z1 didn’t
summarize these solving processes immediately but posed another question:
“what is the solution set of x2-x-6<0?” If students could get the solution set
of x2-x-6>0, they should get the solution set of x2-x-6>0. That meant the
solution set of x2-x-6>0 could promote students to understand the solution
set of x2-x-6>0. Only when all these were answered, teacher Z1 summarized
the findings and extended the findings to the general instance.
Based on the above, we thought teacher Z1 holds better PCK than
teacher Z about this question.
12. 66 Study on Effect of Mathematics Teachers’ Pedagogical Content Knowledge
First, transition. Teacher Z didn’t give any transition from the former
question to this question; namely, she posed this question directly after the
former question. But teacher Z1 gave some transition from the former question
to this question in order to lead students to discover the findings.
Second, relationship: Although teacher Z asked students to find the
solution set through the graph, teacher Z didn’t lead students to think and
many students didn’t know how to do this. Maybe some students who were
good at mathematics would know how to do, but most of the students didn’t
know how to find the solution set through the graph. Specially, the process of
finding the solution set through the graph was just the key for students’
mathematics learning in this lesson. Obviously, teacher Z omitted this
difficulty in student’s mathematics learning. On the other hand, teacher Z1,
with more than 10 years of teaching, especially paid attention to this difficulty
in student’s mathematics learning. He led students to discover the relationship
among one-variable quadratic equality, one-variable quadratic inequality and
quadratic function. Furthermore, he gave three reverse examples to explain
this relationship. Based on these learning, students could understand the
relationship and get the solution set of one-variable quadratic inequality
through the graph of quadratic function.
Third, the manner of question: Two questions posed by teacher Z were
“Y/N” question and only needed students to answer yes or no. Thus, the scope
of students’ thinking was limited invisibly. On the other hand, all questions
posed by teacher Z1 were “N/S” question, which demanded students to search
corresponding knowledge from their cognitive structure. So, the scope of
students’ thinking was widened greatly and the extensity and flexibility of
students’ thinking were developed too.
Fourth, uptake: There were more than 4 times uptakes in teacher Z1’s
teaching (This view could be seen from 18th 21th24th27th row in the teaching
episode about teacher Z1). These “uptakes” were helpful for forming better
classroom atmosphere where teacher Z1 and students had harmonious emotion
communicative and students thought mathematics questions actively. But
teacher Z gave more attention to her own instruction and students’ answers
seemed to respond to her instruction. Thus, the principal status of students in
mathematics learning was weakened and the atmosphere in the classroom was
inactive and tedious. So, teacher Z had to say: “Please follow me, otherwise, I
will …” Based on the above, the two teachers’ education views could be seen,
namely, teacher Z1 was aware of the importance of students’ initiative, but
teacher Z wasn’t.
13. Miao Li & Ping Yu 67
Fifth, mathematics language: When teacher Z was narrating
mathematics language, she didn’t notice the correlative criterion about
mathematics language. For example, from the 8th and 9th row in the teaching
episode about teacher Z, we could see that when students said the intersection
point was -2, 3, she didn’t correct the mistakes. Obviously, she didn’t notice
that a point should have its horizontal coordinate and vertical coordinate.
However, teacher Z1 noticed the correlative criterion about mathematics
language and no mistakes appeared in his teaching. Thus, two teachers’
different mathematics understandings were reflected.
Certainly, we could also see, any “D/E” question didn’t appeared in the
teaching of teacher Z and teacher Z1. There were many reasons for it. One
reason dealt with the learning content. Because students had learned one-
variable quadratic equality and quadratic function in middle school, the
difficulty in this lesson was: (1) understanding the relationship among one-
variable quadratic inequality, one-variable quadratic equality and quadratic
function; (2) understanding the solution set; (3) permeating function thought.
So, two teachers emphasized the relationship and the mathematics thought and
didn’t use “D/E” question. Furthermore, they didn’t also use “SE” “SI” “SD”
“SU” questions.
Conclusion
(1) How a mathematics teacher’s PCK affected his/her mathematics
teaching could be seen not only from his/her teaching objects, teaching
structure, notion explaining but also from his/her education views, teaching
emotion, teaching design, teaching language, students’ mathematics thinking,
students’ learning attitude and so on.
(2) Mathematics teacher’s PCK was combined with his/her other
knowledge, such as mathematics knowledge, general pedagogical knowledge,
curriculum knowledge, knowledge of students and so on.
(3) Just as “Knowledge is infinite,” a teacher’s teaching had his /her
excellence and a teacher’s teaching should be improved in teaching practice.
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Authors:
Miao Li
Hubei Xiaogan University, China
Email: limiao403@yahoo.com.cn
Ping Yu
Nanjing Normal University, China
Email: yuping1@njnu.edu.cn