Mathematics Trainees' Module PDF

Federal Democratic Republic of Ethiopia Ministry of Education Special Capacity Building Training Program for Secondary School Teachers Mathematics Trainees’ Module June, 2024 Ministry of Education...

Federal Democratic Republic of Ethiopia Ministry of Education Special Capacity Building Training Program for Secondary School Teachers Mathematics Trainees’ Module June, 2024 Ministry of Education Addis Ababa  Federal Democratic Republic of Ethiopia Ministry of Education Special Capacity Building Training Program for Secondary School Teachers Mathematics Trainees’ Module Part I- Teaching Chemistry Trainees’ Module Developer: 1. Abera Abate (PhD) 2. Berie Getie (MSc) Part II- Educational Technology-EdTech 1. Developer: Inku Fasil 2. Reviewer: Alemu Tesfaye June, 2024 Ministry of Education Addis Ababa i ii | Page Table of Contents Part I- Teaching Mathematics....................................................................................................................... 1 Module Introduction..................................................................................................................................... 1 Unit 1: Problem-solving-based Mathematics Instruction (4 hrs.)............................................................... 10 Session 1.1: Why and how of problem-solving approach (1 hr.)............................................................ 10 Session 1.2: Nature of Problem-solving Tasks (1 hr.)............................................................................ 14 Session 1.3: Lesson Structure of Problem-solving Based Approach (2 hrs.)......................................... 18 Unit 4: Teaching Exponential and Logarithmic Functions (9 hrs.)............................................................ 51 Session 4.1: Teaching Exponential Functions (2 hrs.)........................................................................... 52 Session 4.2: Teaching Graphs of Exponential Functions (2 hrs.)........................................................... 55 Session 4.3: Teaching Logarithmic Functions (2 hrs.)........................................................................... 60 Session 4.4: Teaching Graphs of Logarithmic Functions (3 hrs.)........................................................... 63 Unit Summary......................................................................................................................................... 67 Unit 6: Teaching Coordinate Geometry and Solid Figures (9 hrs.)............................................................ 80 Introduction............................................................................................................................................. 80 Session 6.1: Distance between Two Points (2 hrs.)................................................................................ 82 Session 6.2: Teaching Division of a Line Segment (2 hrs.).................................................................... 85 Session 6.3: Teaching Solid Figures: Prisms and Cubes (2 hrs.)............................................................ 88 Session 6.4: Teaching Solid Figures: Cylinders and Cones (3 hrs.)....................................................... 92 Unit Summary......................................................................................................................................... 96 Unit 9: Teaching Statistics and Probability (11 hrs.)................................................................................ 117 Introduction........................................................................................................................................... 117 Session 9.1: Teaching Measures of Central Tendency of Raw Data (2 hrs.)........................................ 118 Session 9.2: Teaching Measures of Central Tendency of Grouped Data (2 hrs.)................................. 122 Session 9.3: Teaching Measures of Dispersion: Mean Deviation from the Mean (2 hrs.)................... 127 Session 9.4: Teaching Measures of Dispersion: Variance and Standard Deviation (2 hrs.)................ 131 Session 9.5: Teaching Probability of Simple Events (3 hrs.)................................................................ 134 Unit Summary....................................................................................................................................... 139 Unit 11: Teaching Introduction to Calculus (6 hrs.)................................................................................ 153 Introduction........................................................................................................................................... 153 Session 11.1: Rates of Change............................................................................................................. 153 Session 11.2: Teaching the Derivative –The Slope Function.............................................................. 153 Session 11.3: Teaching the Integral-Riemann Sum Investigation........................................................ 153 Session 11.1: Rates of Change (2 hrs.)................................................................................................ 153 Session 11.2: Teaching the Derivative –The Slope Function (2 hrs.).................................................. 159 Session 11.3: Teaching the Integral-Riemann Sum Investigation (2 hrs.)............................................ 162 Unit Summary....................................................................................................................................... 165 iii | Page Part II – Educational Technology............................................................................................................. 168 Introduction............................................................................................................................................... 168 UNIT ONE: UNDERSTANDING EDUCATIONAL TECHNOLOGY.................................................. 172 Learning Objectives.............................................................................................................................. 172 Key Topics............................................................................................................................................ 172 Session One: Locally available digital resources and their function.................................................... 172 Session Two: The Concept of Educational Technology....................................................................... 175 Key Ideas.............................................................................................................................................. 177 Implication to Teaching........................................................................................................................ 178 Self-Assessment (Unit One)................................................................................................................. 178 Reference Materials.............................................................................................................................. 179 UNIT TWO: BASIC DIGITAL SKILLS................................................................................................. 180 Learning Objectives.............................................................................................................................. 180 Key Topics............................................................................................................................................ 180 Session One: Computers and Smartphones.......................................................................................... 180 Session Two: Using Web Browsers...................................................................................................... 181 Key Ideas.............................................................................................................................................. 185 Implication to Teaching........................................................................................................................ 187 Self-Assessment.................................................................................................................................... 188 Reference Materials.............................................................................................................................. 189 UNIT THREE: THE USE OF SOCIAL MEDIA FOR EDUCATIONAL PURPOSES.......................... 190 Learning Objectives.............................................................................................................................. 190 Session One: Social Media for Educational Purpose............................................................................ 190 Key Ideas.............................................................................................................................................. 193 Implication to Teaching........................................................................................................................ 195 Self-Assessment.................................................................................................................................... 195 Reference Materials.............................................................................................................................. 195 UNIT FOUR: ONLINE EDUCATIONAL TOOLS................................................................................. 196 Learning Objectives.............................................................................................................................. 196 Key Topics............................................................................................................................................ 196 Session One: Virtual Communication and Collaboration Tools........................................................... 197 Session Two: Assessment Tools........................................................................................................... 201 Session Three: Creative Tools (Artificial Intelligence)........................................................................ 208 Key Ideas.............................................................................................................................................. 210 Implication for Teaching....................................................................................................................... 212 Self-Assessment.................................................................................................................................... 212 Reference Materials.............................................................................................................................. 214 UNIT FIVE: CREATE AND ACCESS TO DIGITAL CONTENTS...................................................... 215 Learning Objectives.............................................................................................................................. 215 iv | Page  Key Topics............................................................................................................................................ 215 Session One: Digital Contents.............................................................................................................. 215 Session Two: Creating Digital Contents............................................................................................... 217 Session Three: Open Educational Resources........................................................................................ 221 Key Ideas.............................................................................................................................................. 228 Implication for teaching........................................................................................................................ 230 Self-Assessment.................................................................................................................................... 230 Reference Materials.............................................................................................................................. 231 UNIT SIX: DIGITAL CITIZENSHIP...................................................................................................... 233 Learning Objectives.............................................................................................................................. 233 Key Topics............................................................................................................................................ 233 Session One: Internet Risks and e-Safety............................................................................................. 233 Session Two: Computer and Smartphone safety.................................................................................. 236 Key Ideas.............................................................................................................................................. 236 Implication for Teaching....................................................................................................................... 238 Self Assessment.................................................................................................................................... 239 Reference Materials.............................................................................................................................. 239 v | Page  Part I- Teaching Mathematics Module Introduction The basis for the change of the secondary school (Grade 9-12) Math curricula was based on MoE national assessment of the study done by the Cambridge international (2019). The study figured out major drawbacks in Math Education: lack of relevance of contents; Prevalence of difficult and overloaded contents in the text book; Absence of interactive learner-centered methodologies; Absence of proper implementation of continuous assessment and lack of elements of 21st century skills like collaboration have been affecting students’ performance, interest towards the subject, and the quality of education in general. After discussing the strengths & weaknesses of the existing Math curriculum. the new Ethiopian general education Math curriculum is developed. Secondary education learning themes were designed for all grades (grades 9-12); 9 units for G-9, 7 Units for G-10, 8 Units for G-11 and 5 Units for G-12 was outlined; Math for social science as well as natural science was merged together for G-11 & 12; Periods for each unit is given; The content flow chart (CFC) formulated; minimum learning competence (MLC) for each title is developed; Learning strategies and evaluation techniques suggested. Finally, after these preliminary activities, both the ST and TG are developed. Thus, the following is the main difference between new and old mathematics content flow chart: New Math CFC Old Math CFC Grade 9 Unit 1’: Further on Sets Unit 1: The number System Unit 2: The number System Unit 2: Solving of Equations Unit 3: Solving of Equations Unit 3: Further on Sets Unit 4: Solving inequalities Unit 4: Relations and Functions Unit 5: Introduction to Trigonometry Unit 5: Geometry and Measurement Unit 6: Regular Polygons Unit 6 Statistics and Probability Unit 7: Congruency and Similarity Unit 7: Vectors in Two Dimensions Unit 8: Vectors in two dimensions Unit 9: Statistics and Probability Grade 10 Unit 1: Relations and Functions Unit 1: Polynomial Functions Unit 2: Polynomial Function Unit 2: Exponential and Logarithmic Functions Unit 3: Exponential and Logarithmic Functions Unit 3: Solving Inequalities Unit 4: Trigonometric functions Unit 4: Coordinate Geometry Unit 5: Circles Unit 5: Trigonometric Functions Unit 6: Solid figures Unit 6: Plane Geometry 1 | Page  New Math CFC Old Math CFC Unit 7: Coordinate Geometry Unit 7: Measurement Grade 11 Unit 1: Relations and Functions Unit 1: Further on Relation and Functions Unit 2: Rational Expressions & Rational Unit 2: Rational Expressions and Rational Functions Functions Unit 3: Matrices Unit 3: Coordinate Geometry Unit 4: Determinants and their properties Unit 4: Mathematical Reasoning Unit 5: Vectors Unit 5: Statistics and Probability Unit 6: Transformation on the Plane Unit 6: Matrices and Determinants Unit 7: Statistics Unit 7: The Set of Complex Numbers Unit 8: Probability Unit 8: Vectors and Transformation of the Plane (For NS Students) Unit 9 Further on Trigonometric Functions (For NS Students) Unit 10 Introduction to Linear Programming (For SS Students) Unit11 Mathematical Applications in Business (For SS Students) Grade 12 Unit 1: Sequences and Series Unit 1: Sequences and Series Unit 2: Introduction to Calculus Unit 2: Introduction to Limits and Continuity Unit 3: Statistics Unit 3: Introduction to Differential Calculus Unit 4: Introduction to Linear Programming Unit 4: Applications of Differential Calculus Unit 5: Mathematical Applications in Business Unit 5: Introduction to Integral Calculus Unit 6 Three-Dimensional Geometry &Vectors in Space (For NS Students) Unit 7 Mathematical Proofs (For NS Students) Unit 8 Further on Statistics (For SS Students) Unit 9 Mathematical Applications for Business & consumers (For SS Students) The students’ textbook is also “unitized,” that is, divided into respective lessons with the principle of “one lesson, one topic” while each one lesson is basically composed of four components: Activity, Definition/Theorem/Note, Examples and Exercises (ADEE, or DEE or EE) The trainees are advised to follow the four components of each lesson and provide the required assistance to the students regularly. For detailed discerption of the four components of each lesson refer mathematics students’ text book “Introduction about the Students’ Textbook”. Thus, this training module can be considered as part of the TPD and developed based on national survey conducted by MoE collected from trainees that they have difficulty to teach mathematics content at each grade level from grade 9-12. Purpose of the module This training Module on Mathematics Education for trainees is aimed to: 2 | Page  ❖ Provide content and pedagogical knowledge to trainees based on new mathematics curriculum. ❖ Provide trainees with opportunities to explore their own values, attitudes and knowledge regarding the new mathematics curricula; ❖ Provide trainees with opportunities to use the knowledge and skills acquired during training to promote appropriate conceptual understanding of mathematics Pedagogical Approaches Active and participatory teaching/learning approaches are employed throughout the module. Activities are mainly experiential in nature, and trainees are required to actively engage as participants in all of the lessons. In particular, the following pedagogical approaches are employed. Problem-based Learning (PBL): Problem-based learning develops trainees’ critical thinking and problem-solving skills. It is a highly structured, cooperative learning mode to enhance both individual and collective knowledge by engaging teacher trainees’ in critical and deep enquiry of real-life problems. Real-life applications: Real-life applications are essential visualization techniques in mathematics teaching and learning. A typical goal of school mathematics courses is the ability to relate mathematics to real life (Gravemeijer & Terwel, 2000; NCTM, 2000). Trainees have to strive to make mathematics more alive, realistic, and accessible. They have to encourage more individuals to learn mathematics by making it more alive, even though it is a difficult subject. They to demonstrate why mathematics is needed in everyday life by making it more realistic. They have to make mathematical problem-solving skills accessible to as many students as possible by making it more accessible, even if everyone has varied potential and prospects in this area of mathematics. They frequently employ real world problems in our schooling to put our beliefs into reality. It's crucial to remember that these problems are typically complex, requiring a wide variety of knowledge and experience to handle (Agata, 2000; Blum, 2002). Real-life mathematical applications are frequently recommended as a strategy to motivate trainees. It can also help students understand mathematical concepts better, motivate them, and improve their attitudes toward mathematics (Gainsburg, 2008). Examples and non-examples: Trainees' visual abilities in mathematics are developed through examples and non-examples. Examples are an important aspect of mathematical thinking, learning, and teaching, especially when it comes to conceptualization, generalization, 3 | Page abstraction, argumentation, and analogical reasoning. Non-examples that are related with conceptualization and definition and help to emphasize essential elements of a concept are also included in our study of examples. Non-examples can help to clarify distinctions and provide a deeper knowledge of mathematical concepts. Non-examples and examples can appear anywhere in a unit or lesson, but they're most commonly utilized to introduce new concepts. Using examples allows students to generalize concepts and identify connections between related concepts. Non-examples, on the other hand, educate students how to distinguish between ideas and concepts so that they do not repeat them. Guided Discovery Method: Under this approach, the trainee assists the students either through the art of questioning or by explaining to them what they are to do or in the recall and/or the application of relevant principles; the trainee then allows them to work on their own, carrying out the activities designed for the period. With the appropriate guidance provided by the trainee, the students usually discover the concept to be learnt; or the generalization to arrive at. We should be clear here that the trainee guides the students and does not simply provide the solution to them. Questions and Answering Method of Teaching: The question-answer teaching method focuses on asking students to answer sets of questions in replacement of lecturing. The designed questions should encourage and challenge students to correct their misconceptions and understand the concept by themselves. Students are required to express and discuss their own ideas with other students and instructor. The instructor needs to provide opportunities and prompt students to answer the questions by conducting discussions while controlling the scope of discussion to not go off the topic Manipulative: Visualization techniques in mathematics include manipulatives like real objects or models, as well as paper folding. Manipulatives are actual things that trainees can see and manipulate to teach or represent abstract mathematical concepts. Tangrams, cubes, and base- ten blocks are among the colorful shapes and objects featured. According to learning theory, mathematical understanding in young children is tightly linked to sensory perception and real experience. After experiencing the ideas on a physical level, students learn to understand symbols and abstract concepts. Manipulatives are useful tools in mathematics because they assist trainees’ progress from a concrete to an abstract understanding. Trainees who see, touch, sort, disassemble, and manipulate physical objects create clearer mental images and can better express abstract ideas than those who have fewer 4 | Page concrete experiences. Trainees who learn mathematics through manipulative experiences are more likely to bridge the gap between their everyday lives and the abstract world of mathematics. GeoGebra Software: GeoGebra is a frequently used software program in mathematics classrooms (Tatar & Zengin, 2016). GeoGebra is notable for combining the properties of both Computer Algebra Systems (CAS) and Dynamic Geometry Environments (DGE) into a single software package. This software allows users to visualize any mathematical object in its algebraic, graphical, and spreadsheet forms at the same time. As a result, using GeoGebra helps students have a deeper understanding of intended mathematics topics. Furthermore, math education studies found that successful use of GeoGebra helped and improved students' conceptual understanding and performance in a range of mathematics courses, including geometry and algebra. The below figure show snapshots taken by using GeoGebra software. Multiple representations: Duval (2006) claims that representations are necessary to express meaning extracted from or evident from mathematical objects and concepts, regardless of their form. Trainees can visualize (for instance, through diagrams and graphs) mathematical concepts, create connections, and develop new ideas, for instance, by using different representations. Through exercises with multiple representations, including different forms of expression, trainees develop their mathematical illustration skills and get a deeper understanding of mathematics (e.g., speaking, writing). Trainees and students may have a clear understanding of how to solve a mathematical problem with multiple representations. Graphs, tables, pictures, and written explanations are examples of multiple representations that can help students visualize the bigger picture of a mathematical task. However, if trainees and students are unfamiliar with solving problems using multiple representations, understanding mathematical concepts may be difficult. 5 | Page Case Study: Case studies are often real-life stories that describe in detail what happened to an individual, family, school or community. Case studies are powerful catalysts for thought and discussion. The case study method involves an in-depth examination of a single instance or event. This method helps student teachers make use of real-life incidents to see how theoretical knowledge might be applied to real cases. Case study offers a systematic way of looking at the events, collecting data, analyzing information, and reporting the results, which in return supports enquiry learning among student teachers. Student teachers are able to gain a deeper and more thorough understanding of why the events or instances happened as they did Brainstorming: In brainstorming, student teachers actively generate a broad variety of ideas about a particular topic or question in a given, often brief period of time. Quantity of ideas is the main objective of brain-storming. Evaluating or debating the ideas occurs later. Brainstorming allows student teachers to generate ideas quickly and spontaneously. It helps them use their imagination and break loose from fixed patterns of response Class discussion (in small or large groups): In class discussion, the class examines a problem or topic of interest with the goal of better understanding an issue or skill, reaching the best solution, or developing new ideas and directions for the group. Class discussion provides opportunities for trainees to learn from one another and practice turning to one another in solving problems. It also enables trainees to deepen their understanding of the topic and personalize their connection to it. Class discussion helps trainees develop skill in listening, assertiveness, and empathy. Assessment Recommendations To ensure that the trainees have acquired the relevant knowledge, skills and attitudes, both formative and summative assessment techniques will be employed. Process evaluation will be carried out to appraise the implementation of the curriculum. This will help to identify challenges and successes on the delivery of the module, trainees’ receptivity and administrative support. Findings will identify areas for improving the curriculum and its delivery. End of course assessment will also be administered to gain a summative appraisal of the module. To have information about the impact, a follow-up evaluation will be conducted on trainees after they have completed the training and after a reasonable time has elapsed. The follow-up may be conducted using a questionnaire. 6 | Page This module is prepared based on the national Technical Teams framework for Summer-Based Special Teacher professional development for school Teachers. The module consist eleven units. The objectives, activities, key ideas and some implications for teaching are identified as follows. Each topic is linked to specific learning objectives which state the intended outcomes of working on particular topics. Learning objectives are defined and stated in the way the section can be implemented. Activities are provided to allow participants to be engaged in learning about the topics raised The topics are followed by key notes, which provide description or explanation for each major issue to be addressed. It provides answer to questions that may be raised by trainees teaching mathematics. Implications of each topic are incorporated to indicate the conclusions that can be drawn from the sessions. This could be put either in the form of questions and/or in the form of specific recommendations. Structure of the Module Unit 1: Problem-solving-based Mathematics Instruction (4 hrs.) Session1. 1: Why and how of teaching through problem-solving (1 hr.) Session 1.2: Nature of problem-solving tasks (1 hr.) Session 1.3: Lesson Structure of problem-solving based approach (2 hrs.) Unit 2: Teaching Relations and Functions (G10, G11) (12 hrs.) Session 2.1: Teaching the concept of relation (4 hrs.) Session 2.2: Teaching the concept of Function (2 hrs.) Session 2.3: Teaching Power Functions (4 hrs.) Session 2.4: Teaching Composition of Functions (2 hrs.) Unit Summary Unit 3: Teaching Polynomial Functions and Rational functions (G11) (5 hrs.) Section 3.1: Teaching polynomial functions (2 hrs.) Section 3.2: Teaching rational functions (3 hrs.) 7 | Page Unit Summary Unit 4: Teaching Exponential and Logarithmic Functions (G10) (9 hrs.) Session 4.1: Teaching Exponents Functions (2 hrs.) Session 4.2: Teaching Graphs of Exponential Functions (2 hrs.) Session 4.3: Teaching Logarithmic Functions (2 hrs.) Session 4.4: Teaching Graphs of Logarithmic Functions (3 hrs.) Unit Summary Unit 5: Trigonometric Functions (G9, G10) (9 hrs.) Session 5.1: Teaching trigonometric Ratios based on right angled triangle (3 hrs.) Session 5.2: Teaching trigonometric Values of different angles (3 hrs.) Session 5.3: Teaching graphs of sine, cosine and tangent functions (3 hrs.) Unit Summary Unit 6: Teaching Coordinate Geometry and Solid Figures (G10) (9 hrs.) Session 6.1: Teaching Distance between Two Points (2 hrs.) Session 6.2: Teaching Division of a Line Segment (2 hrs.) Session 6.3: Teaching Solid Figures: Prisms and Cubes (2 hrs.) Session 6.4: Teaching Solid Figures: Cylinders and Cones (3 hrs.) Unit Summary Unit 7: Teaching Vectors (G11) (4 hrs.) Session 7.1: Teaching Vectors and their representations (2 hrs.) Session 7.2: Teaching Vector cross product and its applications (2 hrs.) Unit Summary Unit 8: Teaching Transformations on the Plane (G11) (7 hrs.) Session 8.1: Teaching line Translation (2 hrs.) Session 8.2: Teaching Reflection of a circle in the line (2 hrs.) Session 8.3: Teaching Rotation when the center of rotation is about the point (3 hrs.) Unit Summary Unit 9: Teaching Statistics and Probability (G9, 11 and 12) (11 hrs.) Session 9.1: Teaching Measures of Central Tendency of Raw Data (2 hrs.) Session 9.2: Teaching Measures of Central Tendency of Grouped Data (2 hrs.) Session 9.3: Teaching Measures of Dispersion: Mean Deviation from the Mean (2 hrs.) 8 | Page Session 9.4: Teaching Measures of Dispersion: Variance and Standard Deviation (2 hrs.) Session 9.5: Teaching Probability of Simple Events (3 hrs.) Unit Summary Unit 10: Teaching Introduction to Linear Programming (G12) (8 hrs.) Session 10.1: Teaching graphical Solutions to the System of Linear Inequalities (2 hrs.) Session 10.2: Teaching Graphical method of solving linear programming problems (3 hrs.) Session 10.3: Teaching solving Linear Programming Problems Using Microsoft Excel (3 hrs.) Unit Summary Unit 11: Teaching Introduction to Calculus (G12) (6 hrs.) Session 11.1: Rates of Change (2 hrs.) Session 11.2: Teaching the Derivatives –The Slope Function (2 hrs.) Session 11.3: Teaching the Integral-Riemann Sum Investigation (2 hrs.) Unit summery 9 | Page Unit 1: Problem-solving-based Mathematics Instruction (4 hrs.) Introduction In this unit, the shift from the rule-based, teaching-by-telling approach to a problem-solving- based approach to mathematics teaching is explained and illustrated with mathematics examples. Problem-solving approach is a method of teaching that can allow learners to ‘do mathematics’ and through this build up their understanding of mathematics. This unit contains three main sessions which are: why and how of problem-solving approach, nature of problem solving tasks, and a three-phase lesson structure. Unit Outcomes After completing this unit, you will be able to: Appreciate the need for a shift in thinking about mathematics instruction. Understand the key principles of problem-solving-based approach to teaching mathematics. Apply the principles to design and teach effective problem-solving-based mathematics lessons from the textbook. Key contents Session 1.1: Why and how of teaching through problem-solving (1 hr.) Session 1.2: Nature of problem-solving tasks (1 hr.) Session 1.3: Lesson Structure of problem-solving based approach (2 hrs.) Unit Summary Session 1.1: Why and how of problem-solving approach (1 hr.) In this session, we discuss about the need to shift from the rule-based approach to a problem- solving-based approach to mathematics teaching. We begin the discussion with an activity so that we can build upon your own current experience and understanding. This should help you better to engage with the discussion that follows. Session Objectives: Upon completion of this session, you will be able to: 10 | Page  Critically reflect on the value of teaching with problems. Activity-1: Teacher Reflection (30 minutes) Reflect on the following questions from your prior knowledge and experiences. a) What mathematics teaching methods/approaches do you know? b) What are the main characteristics of each method/approach? Are they different from the methods used in other subjects? c) What mathematics teaching methods/approaches are often used in Ethiopia? How effective are they? Why or why not? Activity-2: Solving Linear Inequalities as a Case Study (30 minutes) Consider the following two instructions as case studies that Teacher A and Teacher B taught the same concept to grade 9 students. Read through the two case studies and then answer the questions that follow. Share your answer to your nearby colleague, and then to the whole class. a) Which of these two approaches is most like the way that you teach? b) Which of these two approaches do you prefer and why? c) Which of these two approaches allows for meaningful construction of ideas? Explain your answer. Case study-1: Teacher A Instructional activities After checking students’ homework, the teacher asks: Who can remember what you have learned last period? One student gives response: We learn about linear equation. Teacher A then asks: What is linear equation? Another student says: Which involve equal sign. The teacher then says: A linear equation is an equation which can be written as ax + b = 0. Teacher A gives two sample linear equations emphasizing: a, and b are constants, x is a variable, and the value of x is a solution. The teacher announces the title of the current lesson: Solving linear inequalities. The teacher writes the following definition: Definition: An inequality is said to be linear inequality if it is written in one of the following forms: ax + b < 0, ax + b ≤ 0, ax + b > 0, or ax + b ≥ 0, for a ≠ 0. Teacher A then provides examples to support the definition. x + 3 > 0, x + 4 ≤ 0, 2x + 1 < 0, and 3x + 7 ≥ 0 are linear inequalities. Teacher A writes the addition, subtraction, multiplication and division properties of inequalities on the blackboard. She then demonstrates the properties as follows. 2 5 2 1 5 1 < and − < − 4>2 and 4×3>2×3 3 3 3 3 3 3 10 < 15 and 10 ÷ 5 < 15 ÷ 5 The teacher then provides the following example: Find the solution set for the inequality x + 4 < 7 Teacher A then demonstrates the solution interacting with students. x+4 565 at w > 33 week. b) Hence, 582 – 8w < 422 Week Amount in Tolcha’s account 582 – 8w +8w < 422 + 8w 582 < 422 + 8w 1st 582 - 8 582 – 1 × 8 582 – 422 < 422 – 422 + 8w 2nd 582 - 16 582 – 2 × 8 160 < 8w 3rd 582 - 24 582 – 3 × 8 160/8 < 8w/8 4th 582 - 32 582 – 4 × 8 20 < w 5th 582 - 40 582 – 5 × 8 That is, w > 20 week. 6th Hence, his money in his account is less than 422 after the 20th week. 582 – 8w c) Based on the answers in a) and b), we have: 400 + 5w > 582 – 8w 5w + 8w > 582 – 400 13w > 182 12 | Page  w > 182/13 w > 14 week Helen’s money is equal to Tolcha’s money at week 14 and greater at week 15 and more. The teacher provides the following exercises, and ordered them to work individually. Solve each of the following inequalities. a) x + 5 < 8 b) y - 3 > 2 c) 2z < 10 d) 2x - 3 > 7 e) 4y - 1 < 3y + 8 The teacher walks around the classroom to check every student’s work. The teacher provides summary orally through questioning. The teacher gives emphasis for comparing solving equations and inequalities, based on the number of solutions and solution strategies. Finally, she provides home work exercises from the textbook. Key Ideas The first approach in Case Study 1 involves going straight into formal mathematics and using 'rules'. One advantage of this approach is that it is quick and easy for the teacher, and some students may be able to answer similar questions correctly. A disadvantage of the first approach is that it offers little opportunity for learners to do mathematics. Since learners are to follow the rules demonstrated by the teacher, they may not construct a deep understanding of the content. It does not encourage learners to create and invent their own constructions of the concept. On the other hand, the second approach described in Case Study 2 requires more time initially, both in terms of planning outside of class and during in-class activities. Since the teacher does not provide specific rules to follow, she/he must be prepared to handle various types of thinking, including divergent and sometimes incorrect ideas. Despite this challenge, one benefit of this approach is that students are able to discuss and explore concepts in a meaningful way. Through collaboration with peers and guidance from the teacher, students can work through fundamental principles and tackle more complex examples. This approach allows students to construct their own understanding of new content before formal definitions and formulas are introduced by the teacher. Overall, the second approach emphasizes problem-solving as a means of enhancing students' comprehension of the subject matter. It also aids in the development of students' problem-solving skills. Implications to Teaching What did you learn from this session and how will you apply the notion of this session to your real classroom? How would you help your students benefit from this? Write a short paragraph summarizing the key things you learned and things that you want to apply. 13 | Page ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- --------------. Takeaway Resources 1) https://lessonresearch.net/teaching-problem-solving/overview/ 2) https://fhsu.pressbooks.pub/ecumath/chapter/chapter-4-teaching-mathematics-through-problem-solving/ Session 1.2: Nature of Problem-solving Tasks (1 hr.) In this session, we look at a variety of different problem-solving tasks in order to identify the characteristic features that will help you to design your own tasks. We will start with an activity that explores integer subtraction to illustrate some general issues. Session Objectives: Upon completion of this session, you will be able to: Identify key characteristics of problem-solving tasks in mathematics. Select and analyze appropriate tasks and problems for learning mathematics. Activity-1: Problem-solving Tasks? (30 minutes) Discuss on the following questions in small groups of three to five members. 1) What is problem-solving? What is the difference between problem solving and the doing of routine exercises? 2) Based on the temperature change task given below; reflect on the following questions. a) Would you use tasks like this in your teaching? Why/Why not? b) What are the key characteristic of the problem-solving task given below? c) Does it provide for learners to explore the concept in different ways? d) Does it provide opportunity for learners to solve non-routine problems? 3) Can you give or design similar types of tasks to teach integer subtraction? Temperature Change Task Based on the information given below, answer the following problems. Temperature Change 14 | Page  1. When we travel from Addis Ababa (+12 0 C) to City-1 (-7 0 C), does the temperature rise or fall? How much is the change in temperature? Describe how you reached to this answer. 2. What is the temperature change when we travel from City-2 (-5 0 C) to City-3 (+22 0 C)? Describe how you reached to this answer. 3. If the temperature of a city in Ethiopia is greater than that of City-1 by 15 0 C, then what is the temperature of this city? 4. Determine the temperature of City-5? Describe how you reached to this answer. 5. Determine the temperature of City-6? Describe how you reached to this answer. Key Ideas A mathematics problem-solving task is a task/problem that poses a question to be answered but students do not have a readily available memorized rules, or solution methods for answering it (Lester & Cai, 2016; NCTM, 2000). A problem-solving task is one that engages the learners in thinking about and developing the important mathematics they need to learn. This can be contrasted with the traditional approach to teaching in which teachers explain a rule, provide an example and then drill the learners on similar examples. It must be stressed that whether something is a problem or not is dependent on the level of sophistication of the problem solver. A learner in grade 8 may be required to solve a problem in which the method and solution are not obvious, and yet the same problem given to older students, for example, in 10th grade, may be quite routine. The temperature change task given above illustrates a key characteristic of a problem-solving approach: it provides for learners to explore the concept in practical and different ways. It also provides a chance to apply and connect previously constructed mathematical understanding. Another key characteristic of the problem is the focus on equipping learners to tackle non-routine problems. 15 | Page Problem solving should be a primary goal of all mathematics instruction and an integral part of all mathematical activity. Learners should use problem-solving approaches to investigate and understand mathematical content. What is critical, though, is that if the mathematics is to be taught through problem solving, then the tasks or activities are the vehicle by which the desired curriculum is developed. Teachers don’t teach the concepts first, and then require learners to do exercises – the problem-solving activity is the vehicle through which the concepts are taught. Activity-2: Routine or Non-routine? (30 minutes) Given below is a list of six problems suitable for primary school learners to learn area and perimeter of rectangles. a) Select the problems that you would consider as routine for the learners. b) Select the problems that you would consider as non-routine. c) What are your criteria to classify the problems as routine or non-routine? 16 | Page  Key Ideas The nature of instructional tasks or problems presented to students is a key aspect of teaching and learning mathematics. It highly influences students’ level of engagement, and, thus, the learning outcomes achieved. However, not all mathematical tasks afford the same opportunities for the same type of student learning. Tasks which are connected to real-life contexts; tasks which require students to apply multiple solution approaches and multiple representations provide maximal learning opportunities. These tasks are called non-routine. Non-routine tasks require students to explore and understand mathematical concepts, processes, or relationships by using complex, non- algorithmic thinking. While, tasks which require students only to recall and apply previously learned facts, rules, procedures, formulae, or definitions are routine and cannot help them to do mathematics. The focus of these types of tasks is on correct answers rather than understanding. They do not need real comprehension of concepts involved, and often focus on recall of basic facts and proficiency in computational techniques, and hence they do not require much cognitive demand on the students. When you are planning a lesson, part of your work is to select (or design) tasks that will facilitate the learning of the mathematical content to be covered in that lesson. Most teachers use their textbooks as the everyday guide to the curriculum. Many of the new textbooks are written with the learner in mind and they contain challenging and stimulating activities in which learners can be engaged. However, there is always an opportunity for teachers to adapt textbook activities so that they are more suitable for the particular situation of their learners. Teachers can also design their own activities with the specific needs of their learners in mind. Implications to Teaching What did you learn from this session and how will you apply the notion of this session to your real classroom? How would you help your students benefit from this? Write a short paragraph summarizing the key things you learned and things that you want to apply. 17 | Page ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- --------------. Takeaway Resources 1) https://lessonresearch.net/teaching-problem-solving/overview/ 2) https://fhsu.pressbooks.pub/ecumath/chapter/chapter-4-teaching-mathematics-through-problem-solving/ Session 1.3: Lesson Structure of Problem-solving Based Approach (2 hrs.) Each of the main three phases of a lesson – before, during and after – are considered critical for successful problem-solving lessons. To learn more about lesson phases in problem- solving based approach, let you participate actively in the following activities. Session Objectives: Upon completion of this session, you will be able to: Describe, with insight, the three-phase lesson format for problem solving referred to as before, during and after. Critically describe the teacher’s actions in the before, during and after phases of a problem-solving lesson. Design effective problem-based lesson from the textbook and other resources. Activity-1: Lesson Phases– before, during and after (30 minutes) Discuss the following questions in small groups of three to five members. Write a report of your discussion. a) What must be the teacher’s action BEFORE students start solving the problems? b) What is the teacher’s and students’ role DURING problem-solving (that is, during students solve the problems in small groups)? c) What is the teacher’s and students’ role AFTER problem-solving (that is, after they solve the problems in small groups)? d) Compare the structure of lessons in the two case studies in session 1.1 in terms of the before, during and after lesson phases. What are the implications of this comparison to your teaching? Key Ideas Teaching through problem-solving does not mean simply providing a problem or task, sitting back and waiting for something to happen. The teacher is responsible for making the atmosphere and the lesson work. To this end, Van de Walle (2004) sees a lesson as consisting 18 | Page GETTING READY Activate student’s prior knowledge which is of three main phases: before, during and after. He proposes the following simple three-phase structure for lessons when teaching through problem solving: Before LEARNERS' WORK Let students start doing the new problem in During small groups. Listen carefully and provide hints. Observe and assess. WHOLE-CLASS DISCUSSION After Accept learner solutions without evaluation. Conduct discussions as learners justify and Activity-2: More on Three-phase evaluate resultsLesson Format (30 minutes) and methods. For this activity, you are required to use the temperature change task given in session 1.2. Or you can use your own ones from the textbook. Discuss the following questions in pairs and share the agreed upon answer to the whole-class. a) If you were using the temperature change task in your classroom, what specifically would you do during each of the ‘before, during, and after’ phases of the lesson? Describe each phase clearly. b) What do you expect the learners to do during each of the phases of the lesson? Activity-3: Lesson Design (as a homework activity) (1 hr.) Now let us put into practice what we have explored in this unit. This may take more time to finalize. If it is not finalized within the allotted time, it can be a homework activity. a) Look through your current teaching plans. Choose a concept/topic which you had planned to teach in a more traditional way based on past experience but which you realize you could now use problem-solving for. b) Redesign and teach the lesson using problem-solving as your main teaching strategy. c) Write a comparison between the new lesson and your previous one and answer the following questions. (i) What can you and the learners do differently? (ii) Do learners learn any better or worse when you use this new approach? Key Ideas The above activity focuses on planning in the problem-based classroom – in an attempt to give you the confidence and competence to pursue this approach with your learners. You know that people involved in a teaching-learning situation would find it difficult to teach 19 | Page effectively without using a good lesson plan. The three-phase lesson format described above provides a basic structure for problem-based lessons. That basic structure resulted from the need for learners to be engaged in problems followed by discussion and reflection. Implications to Teaching What did you learn from this session and how will you apply the notion of this session to your real classroom? How would you help your students benefit from this? Write a short paragraph summarizing the key things you learned and things that you want to apply. ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------- --------------. Takeaway Resources 1) Website: https://fhsu.pressbooks.pub/ecumath/chapter/chapter-4-teaching-mathematics- through-problem-solving/ 2) Book: Van de Walle, JA (2004). Elementary and middle school mathematics – teaching developmentally. New York: Pearson. Unit Summary Problem-solving is an essential strategy in mathematics education that empowers students to develop critical thinking, logical reasoning, and mathematical proficiency. It is an integral part of all mathematics learning. Learning takes place as a result of problem solving and the mathematical ideas are the outcomes of the problem-solving experience. When using problem- solving approach in mathematics, learners become actively involved in doing mathematics. They are engaged in selecting strategies, justifying solutions, extending and generalizing problems. There is cooperation and questioning as learners acquire, relate and apply new mathematical ideas through sharing, inquiring and discussing. Learners are also investigating relationships, and the problems act as a catalyst for connecting mathematical concepts and skills. Teaching mathematics through problem solving is a transformative approach that fosters mathematical thinking and prepares students for both academic and practical success. By 20 | Page implementing effective strategies, providing a supportive environment, and engaging students in meaningful problem-solving tasks, educators can empower students to become confident and capable problem solvers. For effective implementation of problem-solving approach to teach mathematics, teachers need to consider the following key strategies. a. Select appropriate problems: Problems should be challenging, yet accessible, and encourage multiple solution paths. b. Provide a problem-solving environment: Create a classroom culture where students feel comfortable taking risks and collaborating. c. Guide students through problem-solving process: Use scaffolding techniques, such as probing questions and providing hints, to support students without giving away the answer. d. Encourage multiple solution strategies: Emphasize that there may be different ways to solve a problem and celebrate diverse approaches. e. Reflect on solutions: Encourage students to analyze and discuss their solutions, and identify areas for improvement. Unit 2: Teaching Relations and Functions (12 hrs.) Introduction: This unit aims at helping trainees establish an understanding about the concepts of relations functions. In this unit, you will discuss on methods of teaching relations and functions. That means, you will discuss how to teach relations in a variety of ways, including patterns, words, tables, graphs, equations, diagrams, visual representation and technology. Relations describe connections between elements in sets, while functions represent specific types of relations that assign each element of one set to exactly one element of another set. Introducing students to relations and functions involves exploring concepts such as domain, range, mapping diagrams, and function notation. By understanding relations and functions, students develop critical thinking skills, logical reasoning, and problem-solving abilities that are crucial in various mathematical contexts. Through interactive examples, visual representations, and practical applications, educators can inspire students to appreciate the significance of relations and functions in modeling real-world scenarios, analyzing data, and solving mathematical problems. This foundation in relations and functions sets the stage for students to delve deeper into advanced mathematical topics and applications. By investigating into these concepts, 21 | Page Trainees will develop critical thinking skills, problem-solving abilities, and a deeper comprehension of how mathematical structures can describe various phenomena in the world around them. Through engaging lessons, interactive activities, and real-world examples, you can inspire your students to explore the intricacies of Relations and Functions, fostering a strong foundation in algebra that will benefit them in their academic pursuits and beyond. In general, this unit is subdivided in to four sections. In all the sections, you will see main points that will help you teach relations and functions. Unit Outcomes: After completing this unit, you will be able to: Outline how to teach relations and functions. Apply the methods of teaching real-life application of relations and functions. Use different techniques of teaching relations and functions. Select appropriate teaching aids to teach relations and functions. Plan to connect relations and functions to real life situations in teaching Key contents Session 2.1: Teaching the Concept of relation (4 hrs) Session 2.2: Teaching the Concept of Function (2 hrs) Session 2.3: Teaching the Concept of Power Functions (4 hrs) Session 2.4: Teaching the Concept of Composition of Functions (2 hrs) Unit Summery Session 2.1: Teaching of the concept of Relation (4 hrs.) Session Objectives: After completing this session, the trainees will be able to: ✓ Discuss various ways of teaching the concept of relation. ✓ Understand the concept of Relations. ✓ Identify effective teaching strategies for teaching relation. ✓ Demonstrate how Relations are used in practical situations 22 | Page  ✓ Describe the key conceptual challenges students and teachers face in understanding the concept of relation. ✓ Adapt the discussion to their own situations and then plan to use it to their own classrooms. Activities Activity 1: Teacher Reflection on their Teaching (25 minutes) How did you teach the concept of relation to you students? (you can frame your reflection in terms of the following key components of an instruction) ✓ Activating students’ prior knowledge ✓ Nature of tasks/activities provided (context, multiple representation, routine/non-routine, etc.) ✓ Students classroom discussion and teacher’s role in the discussion ✓ Summary/consolidation ✓ Challenges you face in teaching. Activity-2: Real-life Application of Relation(25 minuties) In small group discuss real-life application of relations and reflect to the whole class? Key Ideas In academic institutions, relations are applied to track the enrollment status of students in different courses. By defining relations between students and courses they are enrolled in, schools can manage class schedules, academic progress, and course prerequisites efficiently. Businesses use relations to manage and analyze their interactions with customers. By categorizing customers based on their purchase history, preferences, and behavior, businesses can tailor their marketing efforts and maintain strong relationships with their customers. Relations are utilized in airline flight scheduling to determine connections between different flights, airports, and routes. By establishing relations between departure and arrival times, flight connections, and aircraft availability, airlines can optimize their flight schedules and enhance operational efficiency. 23 | Page  By defining relations between patients, healthcare providers, medications, and diagnoses, healthcare facilities can improve patient care and ensure accurate record-keeping. In the study of social networks, relations are used to analyze connections between individuals, groups, and communities. By defining relations based on interactions, friendships, and collaborations, influence patterns, and information flow in social networks. ❖ By incorporating real-life applications of relations in teaching, trainees can make mathematical concepts more relevant and engaging for students. ❖ Real-world examples help students see the practical significance of relations, fostering a deeper understanding of abstract mathematical concepts. Activity-3: Lesson analysis (1hrs) Consider the following two instructions as case studies that mathematics teacher A used pattern recognition as a teaching strategy to help students understand relation concepts. The teacher started the lesson by introducing the idea of patterns and how they can be used to identify relationships between elements. The teacher presented students with a series of patterns, such as number sequences, geometric shapes, or color patterns, and asked them to identify the underlying relations. Students were encouraged to look for commonalities and differences in the patterns to determine the rules governing the relationships. By analyzing patterns, students were able to see how elements are related to each other and apply this understanding to relation concepts in mathematics. The teacher then transitioned to discussing relations as sets of ordered pairs and how patterns can be used to represent these relations graphically. Through pattern recognition activities, students were able to develop a deeper understanding of relation concepts and apply them to solve mathematical problems effectively. Another mathematics teacher B utilized pattern creation and analysis as a teaching method to help students grasp relation concepts. The teacher encouraged students to create their own patterns using numbers, shapes, or symbols and then analyze the relationships between the elements in the patterns. Students were tasked with identifying the rules or formulas that govern the patterns they created and representing them as relations. Through this hands-on approach, students were able to see how elements in a pattern are related to each other and how these relationships can be expressed mathematically as relations. The teacher guided students in translating their patterns into sets of ordered pairs and graphing them to visualize the relations. By engaging in pattern creation and analysis activities, students developed a deeper 24 | Page understanding of relation concepts and gained proficiency in applying them to solve mathematical problems and taught the concept of relation to Grade 9 or 11 students. Let the trainees to read the two case studies and then answer the following questions. 1. Which of these two approaches is most like the way that you teach? 2. Which of these two approaches do you prefer and why? 3. Which of these two approaches allows for meaningful construction of ideas? Explain your answer. Key Ideas Both mathematics teachers followed ADEE approaches which are stated in their text book. Effective teaching practices for the concept of relations in mathematics can help students develop a deep understanding of the topic. Teaching relations using patterns helps develop students' ability to recognize and analyze mathematical patterns that correspond to specific relationships. Trainees can engage students in identifying patterns, making connections between elements, and extrapolating relationships based on observed patterns. Trainees can scaffold learning by starting with simple patterns and gradually introducing more complex patterns to illustrate diverse types of relations. Relating patterns to real-world contexts allows teachers to demonstrate how relations and patterns are used to model relationships in various fields. Trainees can facilitate discussions on how patterns represent relationships in data analysis, sequences, functions, and other applied mathematics contexts. Activity 4. Teaching using Guided Discovery Method (1hrs) Discuss how you teach the concepts of a relation by using guided discovery method in small group? Based on the following leading questions: 1. Give specific examples of relation in your surroundings. Some of them might be: in our daily life, we come across many patterns that characterize relations such as brother and sister, teacher and student, etc. Similarly, in mathematics, you come across different relations such as a number 𝑥 is less than a number 𝑦, 𝑦 is a multiple of 𝑥, and so on. In all these cases, we find that a relation involves pairs of objects in some specific order. Everyday phenomena involve two quantities that are related to each other by some rule of correspondence. The mathematical term for such a rule of correspondence is a relation. 25 | Page 2. Identify the two sets involved in the relation. For example, if they are working with a relation between students and their favorite subjects, one set would represent the students and the other set would represent the subjects. 3. List the elements in each set. For instance, if the students are Alex, Guutaa, and Chaltu, and the subjects are Math, Science, and English, then the sets would be {Alex, Guutaa, Chaltu} and {Math, Science, English}. 4. Specify how elements from the two sets are related to each other. they can represent this as pairs of elements. For example, if Alex's favorite subject is Math, they would represent this relationship as (Alex, Math). 5. Create a visual representation, such as a graph or a table, to show the relation between elements. Key Ideas Relation deals about relationship between two quantities or things. A relation establishes a correspondence between elements from different sets, pairing them based on a specified rule or condition. Relations reflect relationships between elements, indicating how one element is related to another based on a specific criterion or property. 𝑅 is the set of ordered pairs (𝑥, 𝑦) such that 𝑦 is the square of 𝑥 The domain of a relation is the set of all input values (𝑥-values) for which the relation is defined, while the range is the set of all output values (𝑦-values) that result from applying the relation to the elements in the domain. For instance, the domain of the relation in the above relation is the set containing all real numbers such as {1, 2, 3, 4, 5,...} and its range is the set consisting of all square of real such as {1 4, 9, 16, 25,...}. There are different types of relations such as one-to-one, many-to-one, one-to-many, and many-to-many. Each type of relation shows the number of elements paired between the sets. Activity 5: Teaching sketching the graphs of relations involving inequalities using tables or GeoGebra software (1hrs) Be in groups of four and discuss how you can teach sketching the graphs of relations involving inequalities using tables or GeoGebra software with the following steps. 1.Discuss on the basic inequality notation (">", "=","