Teaching math in the primary grades by ctu_sanfran

A Course Module for

Teaching Math in the Primary Grades Riza C. Gusano Mark Zoel 3 . Masangkay Lady Angela M. Rocena Melanie C. Unida Authors

Greg Tabios Pawilen Coordinator

TEACH Series

OLTTCOMESBASED

UNIT I: THE MATHEMATICS CURRICULUM IN THE PRIMARY GRADES Lesson 1: Mathematics in the Primary Grades..............................................................2 Lesson 2: Mathematics Curriculum in the Primary Grades........................................... 7 Lesson 3: Constructivist Theory in Teaching Mathematics in the Primary Grades.........13

UNIT II: INSTRUCTIONAL PLANNING Lesson 4: The Teaching Cycle...................................................................................... 18 Lesson 5: Things to Consider in Planning Instruction in Mathematics in the Primary Grades..................................................................................23 Lesson 6: Instructional Planning Models..................................................................... 29

UNIT III: INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES Lesson 7: Problem Solving .......................................................................................38 Lesson 8: Inductive Learning ................................................................................... 49 Lesson 9: Concept Attainment................................................................................. 56 Lesson 10: Mathematical Investigation.....................................................................64 Lesson 11: Design Thinking........... ........................................................................... 71 Lesson 12: Game-based Learning............................................................................ 80 Lesson 13: Use of Manipulatives ..............................................................................85 Lesson 14: Values Integration...................................................................................90 Lesson 15: Collaboration..........................................................................................99 Lesson 16: Teaching by Asking............................................................................... 106

UNIT IV: ASSESSM ENT STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES Lesson 17: Assessing Learning................................................................................. 114 Lesson 18: Traditional Assessment...........................................................................123 Lesson 19: Authentic Assessment........................................................................... 130 Lesson 20: Designing Learning Portfolios................................................................. 141 References...................................................................................................................... 149 Index............................................................................................................................... 151

iV | TEACHING MATHEMATICS IN THE PRIMARY GRADES

In line with the latest K-12 mathematics curriculum, this book illustrates how to creatively incorporate this new curriculum into teaching. Prepared by a combination of Department of Education (DepEd) specialists and University of the Philippines Integrated School classroom teachers, this book is a good blend of theory and practice. It aims to inspire future teachers to lead pupils toward meaningful mathematics. Finally, the book is written in the hopes of sharing the excitement found in the teaching of mathematics. To achieve this, each chapter was designed following the TEACH format - Think, Experience, Assess, Challenge, Harness. Below is a brief description of each component. □ Think Provides notes that will get the readers to think about the learning theories within an educational setting □ Experience Includes illustrative examples on how the theories are applied in the actual classroom experience □ Assess Poses questions that assess the readers' understanding of the Think and Experience contents □ Challenge Challenges the readers to answer questions and do activities that allow self reflection. □ Harness Requires the readers to engage in an activity that harnesses creativity.

This unit will give you an overview of what it means to teach and learn mathematics in the primary grades. It will provide you with basic information about the curriculum, the learners, and the learning theory that governs mathematics.

Objectives Describe the importance of mathematics in the primary grades

Introduction Mathematics in the primary grades lays the foundations not only of mathematics content but also of life skills essential in the 21st century. This lesson briefly discusses the characteristics of children in the primary grades and the importance of teaching mathematics at this level.

Think Contrary to the common assumption that math is difficult, especially for children, young learners actually tend to enjoy and appreciate math. Children naturally find wonder in patterns and take delight in challenging puzzles. Therefore, it is important to harness this innate love for mathematical procedures to develop a positive disposition toward math. This can be done by preparing activities that bridge the abstract concepts of mathematics to the concrete world that the children know about. As a teacher, your role is to create an engaging and encouraging environment for young learners to thrive in. Even before children enterformal schooling, they already have intuitive and informal understandings about numbers and shapes. As they enter school, they begin to formally learn concepts of numbers and the operations that govern them. In the Philippines, primary grades refer to Kinder until Grade 3. Generally, primary grades completers are expected to have a good sense of numbers, perform simple operations like addition and subtraction, have the capacity to progress from using physical examples and written calculations to carrying out operations mentally, and have the ability to identify proper strategies and techniques needed to come to conclusions and carry out calculations when solving mathematical problems. In the succeeding lessons, you will learn more about the content of the mathematics curriculum in the Philippines. Some experts argue on whether it is a healthy practice to start formally teaching math at an early age. They point out that children may not yet be developmentally prepared to learn abstract concepts. However, Jean Piaget (1963), a Swiss psychologist 2 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

known for his theory on children's cognitive development, believed that children's developmental stages in cognition were done through stages of thought process— sensorimotor, preoperational, concrete operational, and formal operational. These stages suggest that the idea of numbers can be already learned by children at an early age. Those who pursue teaching mathematics in the primary grades have the following arguments: 1. Learners find mathematics relevant in everyday life—telling time, handling money, measuring objects, etc.; 2. Teaching math in the primary grades provides opportunities for children to develop their thinking skills as they solve problems. These skills which include reasoning and creative thinking are important in upper-grade math and in life; and 3. Numerous researches have proven that the mathematical ability of children in the early grades predicts their performance not only in higher-level mathematics but also in reading.

Experience Teachers can develop and implement effective lessons only when they know their learners. Following are the general characteristics of learners that you can expect when you enter a primary grade classroom. Physical

extremely active and enjoys physical activities

Social

beginning to have a choice of friends but frequently quarrels with them

Emotional

. becoming sensitive to criticism and other's feelings; eager to please the teacher

Cognitive

sees patterns, engages in problem-solving activities, learns a lot through self-talk

Taking into account these characteristics will aid you in designing efficient, effective, and motivational learning activities.

UNIT I * THE MATHEMATICS CURRICULUM IN THE PRIMARY GRADES | 3

Assess Answer the following questions to verbalize your understanding of teaching mathematics in the primary grades. 1. In your own words, describe the importance of teaching mathematics in the primary grades.

2. Why is it important for children to develop a positive disposition in learning mathematics?

Challenge The following activity will practice your research skills and reasoning ability. It will also strengthen your opinion on the importance of learning math in the primary grades. Research about some arguments of educators who believe that mathematics should not be taught in the primary grades. Write each argument in the left column. Then, write a rebuttal for each claim. 4 | TEACHING MATHEMATICS !N THE PRIMARY GRADES

Arguments on Why Mathematics Should Not Be Taught in the Primary

j Rebuttals

Grades

Argument 1:

Argument 2:

Argument 3:

Harness The next activity will expose you to an actual mathematics class. You will do ■. serous classroom observations throughout this module. In this particular activity, you m focus your observation on the characteristics of primary grade students. This activity ■ be part of the learning portfolio which you will compile at the end of this module. Observe a Grade 1 math class. Focus your observation on the characteristics of —e students. On the right column of the table below, write specific examples of how —e characteristics on the left column are exhibited by the children you observed.

UNIT I * THE MATHEMATICS CURRICULUM IN THE PRIMARY GRADES |

Characteristics of Primary Learners

Physical

Examples Based on Classroom Observation

extremely active and enjoys physical activities

Social

beginning to have a choice of friends but frequently quarrels with them

Emotional

becoming sensitive to criticism and other's feelings; eager to

■r

please the teacher Cognitive

sees patterns, engages in problem-solving activities, learns a lot through self-talk

What other characteristics have you observed?

S u m m a ry Children's physical, social, emotional, and cognitive developmental characteristics give evidence of their innate interest and ability to learn mathematics and these must all come together in designing an engaging and encouraging mathematics learning environment.

6 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

Objectives Demonstrate understanding and appreciation of the mathematics curriculum in the primary grade

Introduction "Mathematics is a skills subject. It is all about quantities, shapes and figures, functions, logic, and reasoning. Mathematics is also a tool of science and a language complete with its own notations and symbols and grammar rules with which concepts and ideas are effectively expressed" (DepEd Mathematics Curriculum Guide, August 2016). This lesson will guide you in understanding the Philippines' mathematics curriculum in the primary grades.

T h in k The K-12 mathematics curriculum categorizes content into five content areas: Numbers and Number Sense, Measurement, Geometry, Patterns and Algebra, and Statistics and Probability. 1.

Numbers and Number Sense - concepts of numbers, properties, operations, estimation, and their applications

Measurement-the use of numbers and measures to describe, understand, and compare mathematical and concrete objects; attributes such as length, mass and weight, capacity, time, money, and temperature, as well as applications involving perimeter, area, surface area, volume, and angle measure

Geometry - properties of two- and three-dimensional figures and their relationships, spatial visualization, reasoning, and geometric modeling and proofs

Patterns and Algebra as a strand studies patterns, relationships, and changes amongshapesand quantities; use ofalgebraicnotationsand symbols, equations, and most importantly, functions, to represent and analyze relationships

UNIT i * THE MATHEMATICS CURRICULUM IN THE PRIMARY GRADES

Statistics and Probability as a strand is all about developing skills in collecting and organizing data using charts, tables, and graphs; understanding, analyzing, and interpreting data; dealing with uncertainty; and making predictions about outcomes

The primary grades include Grades 1 to 3. At the end of Grade 3, students are expected to demonstrate understanding and appreciation of key concepts and skills on the following: Numbers and number sense • whole numbers up to 10000 • the four fundamental operations (including applications on money) • ordinal numbers up to 100th, basic concepts of fractions Measurement • time, length, mass, capacity • area of square and rectangle Geometry • two-dimensional and three-dimensional objects • lines • symmetry • tessellation Patterns and algebra • continuous and repeating patterns • number sentences Statistics and probability • data collection and representation in tables • pictographs • bar graphs • outcomes

8 I TEACHING MATHEMATICS !N THE PRIMARY GRADES

The mathematics curriculum is not simply a list of competencies. It is logically arranged and organized. For the teachers' reference, content standards, performance standards, and the learning competencies are explicitly stated. See an example below.

Content Standards Contents The learner...

Performance

LEARNING

Standards

COMPETENCY

The learner...

Grade 1 FIRST QUARTER

Numbers and Number Sense

demonstrates understanding of whole numbers up to 100, ordinal numbers up to 10th, money up to phplOO and fractions Vi and

1. is able to recognize, represent, and order whole

1. visualizes and represents numbers from 0 to 100 using a variety of materials;

numbers up to 100 and money up to php 100 in various forms and contexts; and 2. counts the number of objects in a given 2. is able to recognize,

set by ones and tens; and represent and ordinal numbers up 3. identifies the to 10th, in various number that is one forms and contexts. more or one less from a given number. The content standards are broad descriptions of what the students should learn. The performance standards outline what the students should be able to do once the concepts and skills are taught. The learning competencies are logically arranged objectives that must be aimed in classroom instruction for the students to achieve the required content and performance standards. The Philippines' mathematicscurriculumframeworkputcriticalthinkingand problem solving skills as the goals in learning and teaching mathematics. This is the goal across levels in each topic of mathematics contents. The important principles in teaching and learning mathematics such as reflective learning; active and student- centered teaching/learning; communications allowing the learners to articulate their understanding or express their thoughts; making connections is so important that prior learning/prerequisite UNIT I * THE MATHEMATICS CURRICULUM IN THE PRIMARY GRADES j 9

skills are always considered; and mathematics in the context of real-life situations should be the main considerations in designing mathematics activities.

Experience In Piaget's four stages of cognitive development, students in the primary grades fall under the second stage—the preoperational stage. Children in this stage begin to think symbolically as they use words and pictures to represent real objects. However, they still tend to think about concepts in very concrete manner. Now study the curriculum of Grades 1 to 3. Do you think the mentioned cognitive characteristics of children in the preoperational stage were considered in the content standards of Grades 1 to 3 mathematics? Explain your thoughts.

Assess A lot of teachers in the field are confused about the difference between content standards, performance standards, and learning competencies. It is important that you understand them and their importance as they serve as the skeleton of the mathematics curriculum. In your own understanding, explain the differences between content standards, performance standards, and learning competencies. What is the importance of each?

10 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

Challenge We can truly comprehend our own curriculum when we get a clear picture of :thers' curricula. This is the context of comparative studies in education. The following activity will challenge you to study other countries' curricula in order to better understand the Philippines' mathematics curriculum. Read about the mathematics curriculum of the Philippines, Singapore, and United States. Compare and contrast the curricula of the three countries in terms of the five content areas. Content Area

Philippines

Singapore

United States

Numbers and Number Sense

Measurement

Geometry

Patterns and Algebra

Statistics and Probability

Harness In every math lesson, the teacher must keep three things in mind—(1) what is to be learned, (2) where the students are coming from, and (3) where the students are

UNIT I • THE MATHEMATICS CURRICULUM IN THE PRIMARY GRADES | 1 1

going with what they will learn. The following activity will help you develop the skill of mapping every competency you teach. This will be part of the learning portfolio which you will compile at the end of this module. Choose three learning competencies in Grade 2. In each competency, find the prerequisite competencies in Grade 1 and the competencies in Grade 3 wherein your chosen Grade 2 competency is a prerequisite of. Prerequisite Grade 1 Competency

Grade 2 Competency

Future Grade 3 Competency

S u m m a ry The Philippines' mathematics curriculum under the K-12 program promotes critical thinking and creativity. Moreover, content standards, performance standards, and learning competencies are explicitly stated to guide teachers in developing their lessons.

12 j TEACHING MATHEMATICS IN THE PRIMARY GRADES

O b je ctiv e s • Demonstrate understanding and appreciation of the constructivist learning theory • Determine how the constructivist learning theory is applied in teaching mathematics in the early grades

Introduction The constructivist learning theory states that learning is an active process of creating meaning from different experiences. In other words, students learn best by trying to make sense of something on their own with the teacher as a guide. DepEd (2016) specifically noted constructivist theory as the backbone of the curriculum. According to DepEd, knowledge is constructed when the learner is able to draw ideas from his/her own experiences and connect them to new ideas. In this lesson, you will learn about the constructivist learning theory and how it is applied in teaching mathematics in the primary grades.

Think Constructivism was conceptualized by educational theorist Jean Piaget. Do you remember him from your psychology classes? Piaget believed that young children learn by doing, constructing knowledge from experiences rather than from adults telling them about their world. According to Piaget, and others who practice what is known as constructivist education, the method most likely to truly educate students is the one in which they experience their world. Constructivism is appropriately so applied in teaching mathematics since math is a cumulative, vertically structured discipline. One learns new math by building on the math that has been previously learned. Brooks & Brooks (1993) listed the following characteristics of constructivist teaching. 1.

Constructivist teachers invite student questions and ideas.

2.Constructivist teachers accept and encourage students' invented ideas.

UNIT I • THE MATHEMATICS CURRICULUM IN THE PRIMARY GRADES | 13

3. Constructivist teachers encourage student's leadership, cooperation, seeking information, and the presentation of the ideas. i

4. Constructivist teachers modify their instructional strategies in the process of teaching based upon students; thought, experience and or interests. 5. Constructivist teachers use printed materials as well as experts to get more information. 6. Constructivist teachers encourage free discussions by way of new ideas inviting student questions and answers. 7. Constructivist teachers encourage or invite students' predictions of the causes and effects in relation to particular cases and events. 8. Constructivist teachers help students to test their own ideas. 9. Constructivist teachers invite students' ideas before the student is presented with the ideas and instructional materials. 10. Constructivist teachers encourage students to challenge the concepts and ideas of others. 11. Constructivist teachers use cooperative teaching strategies through student interactions and respect, sharing ideas, and learning tasks. 12. Constructivist teachers encourage students to respect and use other people's ideas.

Experience So how is a constructivist classroom different from a traditional classroom? In the constructivist classroom, the focus shifts from the teacher to the students. The classroom is no longer a place where the students are seen as empty vessels to be filled by the teacher. In a constructivist classroom, the students are actively involved in their own learning. The teacher functions as a facilitator who guides, prompts, and helps students to develop and assess their own understanding.

1 4 I TEACHING MATHEMATICS IN THE PRIMARY GRADES

The table below compares the traditional classroom to the constructivist one. Notice differences in the foci of the curricula and the roles of teachers and students. Traditional Classroom

Constructivist Classroom

Curriculum

Curriculum begins with the parts of the whole, emphasizing basic skills.

concepts, beginning with the whole and expanding to include the parts.

Teacher's role

Teachers disseminate information to students; students are recipients of knowledge.

Teachers have a dialogue with students, helping students construct their own knowledge.

Student's role

Students work primarily alone.

Students work primarily in groups.

Curriculum emphasizes big

Assess Answer the following questions to verbalize your understanding of the constructivist learning theory. 1. What is the constructivist theory? Explain it in your own words.

2. Expound why the constructivist theory is applicable in teaching mathematics.

Challenge How well do you understand the constructivist learning theory? Consider the following scenarios and answer the questions that follow. UNIT I • THE MATHEMATICS CURRICULUM IN THE PRIMARY GRADES I 15

Scenario 1

\ \ A teacher told the students, "Four glasses of water will fill this pitcher."

Scenario 2 A teacher provides a glass and lets the children pour water into the pitcher. They are learning how much water it takes to fill the pitcher. In which scenario do you think will the students learn better? Why do you think so?

Harness The following activity will direct your observation skills to the teaching style of the teacher. Note that this is not an activity to criticize the teacher. The purpose is for you to develop keen observation skills on teaching styles implemented in the classroom, and later on, suggest ways to improve the learning activities. This activity will be part of the learning portfolio which you will compile at the end of this module. Observe a Grade 3 mathematics class. Did the lesson develop in a constructivist way? If yes, describe the part of the lesson that followed constructivism. Otherwise, explain how you would revise the lesson in order to facilitate a constructivist lesson.

Summary The constructivist learning theory states that learning takes place when we build on what students already know. Moreover, it is student-centered, allowing the students to take ownership of their own learning. 16 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

In this unit, you will learn about how to plan, develop, and execute lessons in -lathematics for the primary grades. You will go over the learning cycle, the things to consider in lesson planning, and the different instructional planning models.

Objectives Demonstrate an understanding and appreciation of the instructional planning cycle

Introduction The work of a teacher does not start and end in teaching per se. The teaching process is not a linear activity that starts with planning and ends with testing. Instead, it is a cycle of repeating stages until the students acquire an understanding of the targeted concepts and skills. You may think of the teaching cycle as a spring—you go through the same process over and over again, but each time with a more informed objective and a better understanding of what it means to learn and teach mathematics.

Think There are many models of the teaching cycle that various educators have developed over the years. However, all models boil down to six common stages: (1) identify objectives, (2) plan instruction, (3) implement plan, (4) check for understanding, (5) reflect on teaching, and (6) assess learning and reflect on results. The cycle that involves these stages is illustrated below. Identify objective

Assess learning and reflect on the results

instruction

Reflect on teaching

Check for understanding

Implement plan

The Teaching Cycle

18 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

Study the figure. What do you observe? Do you now get the idea of the teaching rrocess as a cycle? The following describes each stage of the learning cycle. 1.

Identify objectives What knowledge and/or skills do the students need to learn? You must be guided by the content standards, performance standards, and the learning competencies that are found in the curriculum guide.

Plan instruction What strategies must be implemented for the students to achieve the objectives targeted in the previous stage? In planning instruction, it is important that you have mastered the content of the lesson that you are about to teach. It is also beneficial to be familiar with your students—what they know, how they learn, etc. You will learn more about instructional planning in the next chapter.

Implement plan This is the stage where you conduct the learning activities that you have prepared during the planning stage. A word of advice—even though you have carefully and delicately planned for the lesson, you must be flexible with the possible changes that you need to accommodate. How will you know whether change is needed? Read on to the next stage.

Check for understanding Teaching is about helping students learn. During the implementation of the lesson plan, you must every now and then check whether the students have understood what you have covered so far. Facial reactions and verbal cues help in assessing whether or not the students can move on to another concept or skill. If not, you might need to give a more elaborate explanation, more examples, or whatever you think is needed based on the students' reactions. This stage also makes use of formative assessments which you will learn more about in Chapter 17.

Reflect on teaching You must evaluate every teaching period that you finished. Were the objectives achieved? Were the implemented strategies effective? How can instruction be improved? Your answers to the last two questions will give you an insight on how to improve instruction the next time you teach the same lesson. However, if your answer in the first question is no, i.e., the UNIT it • INSTRUCTIONAL PLANNING

I 19

objectives were not met, then you need to plan again. What do you need to do differently in order to achieve the objectives? 6.

Assess learning and reflect on the results This stage gives you a concrete measure of what the students have learned. In math, this is usually through a paper-and-pen examination. However, some authentic assessments may also be implemented as you will learn in the later chapters of this book. Take note that this stage does not end in assessing learning. You need to reflect on the results. What can you learn about student learning and teaching practice based on the results? After assessment and reflection, you will once again identify the next learning goals and so the cycle continues.

Experience The following is a narrative of how a teacher might experience the teaching cycle. 1.

Identify objectives Teacher Gina identified "multiplication of whole numbers up to two digits" as the goal of her next lesson.

Plan instruction Teacher Gina thought it is best to apply a constructivist approach to help her students learn techniques in multiplying whole numbers. She planned a lesson which incorporates the problem-solving strategy.

Implement plan The class went on smoothly. The activities that Teacher Gina prepared were successfully done by her students.

Check for understanding To make sure that her students understood the lesson, Teacher Gina gave a three-item exercise as an exit pass.

Reflect on teaching Based on the exit pass, Teacher Gina found out that many of the students have difficulty multiplying numbers that involve the digit 8. So, she decided to do a find-your-error activity the next day for the students to realize their mistakes. She also planned to give a short drill on skip counting by 8.

|TEACHING MATHEMATICS IN THE PRIMARY GRADES

Assess learning and reflect on the results Teacher Gina, later on, gave a multiplication quiz. Ninety percent of the students passed. She planned to give remedial exercises to those who failed. This teaching cycle taught Teacher Gina that students can discover concepts on their own. However, they must still be guided by a teacher because misconceptions may arise.

Assess Answer the following questions to verbalize your understanding of the teaching cycle. 1. In which stage/s of the teaching cycle are the students involved? Explain.

2. Which stage/s of the teaching cycle requires the teacher to reflect on teaching and learning? Explain.

Challenge The next question will challenge your reasoning skill. What do you think is the most important stage of the learning cycle? Why do you think so?

UNIT II • INSTRUCTIONAL PLANNING

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Harness Aside from classroom observations, a lot can be learned from conversations with teachers in the field. The following activity will require you to interview math teachers and summarize what you learn from them in a diagram. This activity will be part of the learning portfolio which you will compile at the end of this module. 1. Interview two mathematics teachers. Ask them about the stages of the teaching cycle that they follow. Then, create a diagram illustrating their common answers.

2. How is the diagram you created in #1 similar or different from the cycle that was presented in this lesson?

Summary Teaching involves a repetitive cycle of defining objectives, planning and implementing instruction, assessing learning, and reflecting on teaching and learning. Each part of the cycle provides a better understanding of what it means to teach and learn mathematics and should result in better instruction in the next repetition of the cycle. 22 I TEACHING MATHEMATICS IN THE PRIMARY GRADES

Lesson 5

o Consider in Planning uction in Mathematics in the Primary Grades Objectives Demonstrate understanding and appreciation of the things to consider in planning instruction in mathematics in the primary grades

Introduction In education, planning refers to the designing and preparation of learning activities for students. In lesson planning, the teachers thoughtfully contemplate about the lesson objectives, the activities that will meet these objectives, the sequence of those activities, the materials needed, how long each activity might take, how the class would be managed during those activities, and the evaluation method to assess how far the objectives were met. This lesson enumerates the things to consider in planning instruction in mathematics in the primary grades.

Think There are three important elements in lesson planning that you need to consider— content, objectives, students, learning environment, and availability of resources. 1.

Content Research the subject matter that you will be teaching. You should consult the curriculum and teaching guides published by DepEd. Aside from books, you can also visit websites which will give you information relevant to your subject area. You should master the contents of your lesson before you teach it. Remember, you cannot give what you do not have. Moreover, you would not want to teach wrong contents to the students. It is easier to learn than to unlearn; it is difficult to take back wrong contents that have already been taught. You have a big responsibility as a teacher—master your content!

Objectives Before you begin planning, you need to know what specific knowledge and skills you want your students to develop during the lesson or unit. UNIT II • INSTRUCTIONAL PLANNING I 2 3

Teachers often focus too much on knowledge, forgetting about developing skills which in the long term are more important than knowing mere facts. So, in planning your instruction, always consider both knowledge and skills. 3.

Students Get to know your students—where they came from, what their interests are, what they already know, their learning style, attention span, special needs, etc. These will all help you determine your students' needs. Remember that you need to prepare your lessons with all your students in mind and that your main goal should be to meet their needs and offer them enabling environments to learn their preferred way. Knowing your students will also help you build rapport with them which is important if you want your students to be freely sharing their ideas with you and their classmates. Another important consideration that needs serious attention in teaching, especially mathematics, is the students' mindset. You may have all things considered—lesson mastery, focused objectives, and comprehensive understanding of students—and yet still find that the lesson is not coming through the students. This may be because the students have closed their doors toward math. Many school children have come to believe that math is difficult and they can never be good at it. This is called a fixed mindset. Students with fixed mindsets believe that their math skills cannot be improved, which results in underperformance in the subject. Reasons for fixed mindset include influence from adults who dislike math, previous unpleasant experience in math class, and others. Your goal as a teacher is to develop students with growth mindset. Students with growth mindsets believe that they can be better at math. They know that their efforts are not wasted and that they can learn even in their failures. Many studies have proven that students who have a growth mindset perform better in school than those who have a fixed mindset. So, in planning your lesson, you must consider how to encourage growth mindset in class.

Learning environment Aside from the physical environment where the learning takes place, it is also important to consider the social and emotional learning environment of the class. You need to make sure that you promote a positive environment where students are motivated and are supportive of each other's growth. The students must feel safe to express their thinking, without fear of being

TEACHING MATHEMATICS IN THE PRIMARY GRADES

embarrassed because of mistakes or different views. Most importantly, you must create an atmosphere where students are open to learning through the activities you prepared and interactions with their classmates. 5.

Availability of resources Take into consideration the instructional materials that you will be needing before you write your lesson plan. Is a blackboard available? If not, can you improvise? Are there specific manipulatives that you need? Where can you get them? Can you make them instead? Do you need technology resources? Have you checked whether your devices are compatible with what are available in school? These are some of the questions that you can reflect on.

Experience The next activity will delve into the experiences of math teachers and will give you insights on effective lesson planning. Interview three experienced primary grades (Grades 1, 2, or 3) mathematics teachers. Ask him/her the following question: "If you were to give a piece of advice about lesson planning to your rookie teacher self, what would it be?" What are common about their responses? Write them down below.

Assess Answer the following questions to verbalize your understanding of the things to consider when planning instruction in mathematics. 1.

In addition to what has been discussed, explain why content, objectives, students, learning environment, and availability of resources are the essential considerations in planning a lesson.

UNIT II • INSTRUCTIONAL PLANNING I 2 5

2. Sketch an infographic about the difference of growth mindset and fixed mindset.

- -

•

Challenge The following questions will challenge your reasoning and critical thinking skills. It will also initiate reflection on the kind of mathematical mindset you had as a student. 1. Why is it important to be in consultation with the curriculum guide when planning instruction?

2. Why do you think is having a fixed mindset a setback in learning? Can you think of specific examples when you were a student and had a tendency of having a fixed mind pattern?

2 6 I TEACHING MATHEMATICS IN THE PRIMARY GRADES

Harness You will come face-to-face with an actual lesson plan in the following activity. This aims to give you an initial exposure to the components of a lesson (which will be discussed in the next chapter) while focusing on how content, objectives, and students were given attention to in the plan. This activity will be part of the learning portfolio which you will compile at the end of this module. Borrow a lesson plan from a primary grade mathematics teacher. Give specific examples in his/her lesson plan wherein you saw the conscious consideration for content, objectives, students, learning environment, and availability of resources.

Content

Objectives

Students

Learning environment

Availability of resources

UNIT II • INSTRUCTIONAL PLANNING I

Summary Before writing a lesson, teachers are expected to thoughtfully contemplate on the objectives, review the content, and get to know the learners. Doing these will help them plan a relevant and effective lesson for the learners.

2 8 i TEACHING MATHEMATICS IN THE PRIMARY GRADES

O b je ctive s Demonstrate understanding and appreciation of the most commonly used instructional planning models in the Philippines

Introduction Now that you have learned the things to consider when planning instruction, you are ready to create one yourself. Teachers usually plan lessons following a specific model. In this lesson, you will learn about the two most commonly used instructional planning model in the Philippines and their common features.

T h in k There are many instructional planning models that mathematics educators have constructed, but the two most widely used in the Philippines are the ADIDAS and the Five Es models. ADIDAS stands for Activity, Discussion, Input, Deepening, Activity, and Summary. Activity. The lesson begins with an activity that will later facilitate a meaningful discussion about the topic of the session. In other words, the activity introduces the topic to the students. This activity must be motivating and engaging to catch the attention of the students. Discussion. The lesson proceeds with the processing of the activity. In this part, the students, as facilitated by the teacher, talks about, their experiences during the activity. Here, the questioning skills of the teacher is important because he/she must be able to direct the discussion toward the targeted lesson. Input. In a traditional classroom, the Input is where the teacher lectures. However, in a constructivist classroom, this is the part where the students would share the concepts that they learned based on the activity and the discussion. Nevertheless, no matter which learning theory is applied in the lesson, this is the part where the concepts are clearly established. Deepening. Here the teacher asks questions that will engage the students to critical and creative thinking. Nonroutine mathematical problems or real-life word UNIT II » INSTRUCTIONAL PLANNING | 2 9

problems may be given. The purpose is to give the students the opportunity to deepen I their understanding of the concepts that they have just learned. Activity. In mathematics, this is the part where the students verify what they have just learned by solving mathematical problems. Depending on the need, the students may be engaged in guided practice and/or individual practice. Sometimes, the teacher facilitates games in this part of the lesson. Synthesis. The last part of the ADIDAS model is the Synthesis. Here the students are given the opportunity to express what they have learned by verbally giving a summary of what transpired in class and what they have learned. The students may also be given a short assessment to give the teacher feedback on what they have learned. Another commonly used instructional planning model in our country is the Five Es. The Five Es are Engage, Explore, Explain, Elaborate, and Evaluate. Engage. This part activates the students' prior knowledge and engages them into new concepts by doing short activities. The aim of this part is to arouse the students' curiosity. Explore. In this part, the students are exposed to different experiences that will facilitate the discovery of new concepts. Explore may involve observation exercises, simulations, or manipulations of instructional materials. The goal here is for the students to discover something new. Explain. Here the students explain what they have experienced in Explore. The role of the teacher is to facilitate the discussion that should lead to students seeing patterns that will help them to describe the new concept in their own words. Elaborate. The Elaborate part of the lesson allows students to expand their understanding of the concept by applying the concept that they have learned in solving mathematical problems. Evaluate. The last part of the Five Es model, Evaluate, lets the teacher and the students evaluate their learning. Though giving short exercises are usually the mode of evaluation, the teacher can be creative by implementing other evaluation activities.

Experience Aside from the components of whatever instructional planning model, an instructional plan also reflects basic information about the lesson like prerequisite

[ TEACHING MATHEMATICS IN THE PRIMARY GRADES

knowledge and skills, time allotment, materials needed, etc. Below is a sample template of a lesson plan.

Subject:

-a s:

Grade Level: Duration: Objectives: At the end of the session, the student will be able to: • • • Prerequisite Concepts/Skills:

New Concepts/Skills:

Materials:

References:

UNIT if * INSTRUCTIONAL PLANNING |

Lesson Proper: Activity

A sse ss

Duration (Number of Minutes)

Teacher's Role

Students' Role

Answer the following questions to verbalize your understanding of instructional planning models commonly used in math. 1. Did you notice any similarity between the ADIDAS and the Five Es model? Match the components of the two models to summarize the similarities that you saw. Activity

• Engage

Discussion

• Explore

Input

• Explain

Deepening

• Elaborate

Activity

• Evaluate

Synthesis

2. Explain the matching you did in #1.

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Challenge Even though ADIDAS and Five Es are commonly used, they also receive criticisms such as not being applicable to some topics in math. The following questions will challenge your reasoning skills regarding this issue of applicability of instructional planning models. 1. Do you think the ADIDAS or the Five Es model is applicable to planning any lesson in mathematics? Explain your thought.

2. What if, in the school where you will be employed, a different instructional planning model is used. Do you thinkyou will have a hard time adjusting? Explain.

Harness In this activity, you will be asked to refer to the lesson plan you previously studied in Chapter 5. This time focus your analysis on the different components of the lesson plan in relation to the ADIDAS and Five Es models. This activity will be part of the learning portfolio which you will compile at the end of this module.

UNIT II • INSTRUCTIONAL PLANNING I 3 3

Refer to the lesson plan you collected in the previous chapter and do the following. 1.

Extract parts of her lesson plan that exhibits the components of: a. ADIDAS Activity

Discussion

Input

Deepening

Activity

Synthesis

b. Five Es

Engage

Explore

Explain

Elaborate

Evaluate

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2. Are there components of the ADIDAS/Five Es model that were not reflected in the lesson plan? If you are to fill in these missing parts, what would you write?

Summary ADIDAS and Five Es are just two of the many instructional planning models applied in math. All models usually boil down to common components such as activities, discussions, and evaluations.

UNIT II • INSTRUCTIONAL PLANNING I

This unit will equip you with various learning strategies anchored in the constructivist learning theory. These strategies are research-based and have been proven to be effective in developing mathematical thinking in young children. The activities in this unit will engage you in planning out your lessons using these strategies.

O b je ctive Plan a lesson that uses problem-solving strategy

Introduction Not all word problems promote problem-solving skills. In this lesson, you will learn the characteristics of a good word problem, when it is best to give a word problem, and how to process students' varied solutions.

Think The problem-solving strategy involves students being challenged to collaboratively solve real-world math problems which they have not yet previously encountered. It is student-centered and promotes critical and creative thinking skills, problem-solving abilities, and communication skills. The integral part of this strategy is the time given to the students to struggle with the problem and its beauty is in the varied solutions that the students would produce. There are three main elements of problem-solving that you should take note of: (1) the word problem, (2) the time given for the students to struggle with the problem, and (3) the mathematical discourse that happens during the struggle and during the processing of the student-generated solutions.

The word problem In many Filipino classrooms, word problems are given at the end of the lesson and students are expected to answer them by applying the concept or skills that had just been taught to them. In most cases, the teacher first demonstrates how to solve a problem and then the students would independently answer a similarly-structured problem. In this practice, the students are not doing problem-solving—they already know how to solve the problem! They know that the just-taught lesson is the key to solve the problem and they pattern their solutions to what the teacher demonstrated. In using the problem-solving strategy, the problem serves as the starting point of the learning experience. Therefore, it is given at the beginning of the lesson. The challenge for you, the teacher, is to choose or create a problem, which can be solved using 3 8 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

~~e target concept of the lesson at hand but can also be answered using previously earned knowledge and skills. How you present the problem also matters especially for the primary grades. It is not always helpful to introduce the problem by posting it on the board; doing this may -rimidate some students and reading and comprehension skills may intervene. Instead, * is suggested to narrate the problem in a story-telling manner to engage the learners. Encourage the students to imagine the scenario and allow them to clarify information if :ie y find some details confusing. Showing drawings or real objects might help.

The time given to struggle with the problem The goal is for the students to collaborate—share their ideas with each other— to come up with a solution. Encourage the students to use their previously-learned knowledge and skills to solve the problem, and to communicate their ideas with their classmates through words, equations, and/or illustrations. It is natural for the students to find this phase burdensome especially when it is their first time to engage in such an activity; critical thinking and communicating ideas are not easy tasks after all. So, it is the task of the teacher to encourage the students to think out of the box. Tell the students that there is more than one way to solve the problem, so they do not need to worry about their solution being wrong as long as every step they did is meaningful in solving the problem.

The mathematical discourse This is the most exciting element of the problem-solving strategy. While the students are working in small groups to solve the problem, you get to move around and enjoy the mathematical talk that the students are engaging in. Of course, you may intervene in the students' discussions when corrections and clarifications are needed but be careful not to give hints. It may be tempting to do so especially when the students are struggling but do not. As you encourage your students to think, believe that they actually can. Allow yourself to be amazed at how the students would defend their thinking, correct each other's ideas, and figure things out on their own. Remember that all the student-generated solutions, as long as correct, can be directed to the concept or skill that is the objective of the lesson. The challenge is how you would process those various solutions, make sense of each of them, and use them to generalize or come up with a solution that makes use of the knowledge/skill that is the objective of the lesson. In this phase comes the importance of the teacher's fluency of the subject matter. UNIT III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES j 3 9

Experience Study the following lesson plan. Take note that the plan only shows the development of the lesson, which involves the problem-solving strategy. Other parts are not included.

Topic: Apply properties of multiplication to mentally multiply whole numbers up to two digits Grade level: 3 Target learning competency: By the end of the lesson, the learners will be able to mentally multiply whole numbers up to two digits. Prerequisite knowledge and skills: Multiplication of whole numbers up to two digits Presentation of the problem: Mentally multiply 18 and 5. Present the problem above in a narrative approach which will engage the students. See an example below. "Hannah is next in line to pay at the counter. She will buy 5 pieces of bread which cost 18 pesos each. She would like to know how much she needs to pay for all the bread. Her hands are full so she couldn't write her solution nor use her phone calculator. She needs to solve mentally! If you were in Hannah's shoes, how would you solve it?" Generation of solutions: Students will work in pairs or triads. Encourage the students to think about the problem and share their thoughts with their classmates. Assure them that there is no one right solution. They may do calculations or draw; any solution is welcome as long as they can explain why they did such. The problem calls for mental calculations but for the sake of discussion and to facilitate mathematical communication through writing, instruct the students to write down their thoughts as they explain to their groupmates/partner.

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Watch out: The students should realize that the given is a multiplication problem. Gi/e guide questions to pairs/groups who may have not realized this. In giving guide questions, determine first what the students know and then build on that. Processing of solutions: Have some pair/group write their solutions on the board and explain. Possible solutions: (1)

18x5

9x2x5

9x10

90 (2)

18x5

(10 + 8) x 5 10 x 5 + 8 x 5

V 50

(3)

V 40 =

18 x 5

= (20x5)-(2x5) =100 - 10 = (4)

18 x5 90

Use solution (1) to introduce the commutative and associative properties. Solutions (2) and (3) will aid in the discussion of the distributive property. Solution (4) is the usual algorithm in multiplying but done in the imagination of the student.

UNIT 111 • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 41

Guide the students to realize that the different solutions all lead to the same answer, which is 90. Ask the students who among them did the same as the ones presented in class. Tip: Name a solution after the students who shared it. For example, the illustration is "Paolo and Jose's technique." Doing this would (1) deepen the students' sense of ownership of their learning and (2) motivate them to think of unique solutions when given the same task in the future.

Assess Answer the following questions to solidify your understanding of the problem solving strategy. 1. How are the three possible solutions shown in EXPERIENCE different from each other? What goes in the minds of the students who would possibly give those solutions?

2. As a teacher, how would you ensure that the word problems you will give genuinely promotes problem-solving?

I TEACHING MATHEMATICS IN THE PRIMARY GRADES

Challenge As discussed, the task of the teacher is to present the problem in an engaging way and to be able to anticipate possible solutions from students. Do the following to practice these. 1. Browse the DepEd mathematics curriculum guide. Choose a topic from K to Grade 3. Write the topic and grade level below.

2. Browse the DepEd mathematics teaching manual for the grade level you chose. Find a problem from your chosen topic. Write the problem below as how it is written in the teaching manual.

UNIT iff * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES |

3. How would you present your chosen problem in an engaging way? Pu situation where the students can relate to. You may use your own expe too (many times children are interested in what's happening to their te; lives ©)! Imagine you are talking to your students. Write your script b( you plan to use materials, write a note about it.

4. The possible solutions in the sample lesson plan are generated by real Gra students. So be convinced that Filipino students are actually capable of thin! As would-be teachers, your task is to anticipate such possible solutions. H Consult with experienced teachers! Show your problem to some teac and ask them how they think the students would answer if they are given problem for the first time. You may also ask the children themselves. Tal your nephew/niece, godchild, neighbor, etc. Share with them your prob and have them explain to you how they think they can solve it. Write be three of the possible solutions that you have gathered.

TEACHING MATHEMATICS IN THE PRIMARY GRADES

Topic:________________________________ Grade level:___________ Target learning competency: By the end of the lesson, the learners will be able to

Prerequisite knowledge and skills:

—

Presentation of the problem:

4 6 I TEACHING MATHEMATICS IN THE PRIMARY GRADES

Seneration of solutions:

—

I '1 --------------- ,

■

Processing of solutions:

UNIT li! * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 4 7

Summary Not all word problems develop problem-solving skills. A good use of the problem solving strategy involves a realistic problem, ample time for students to think about the problem, and a venue to engage students in mathematical discourse.

4 8 I TEACHING MATHEMATICS IN THE PRIMARY GRADES

bjective Plan a lesson that allows students to inductively learn a concept

troduction In our contemporary society, teachers are discouraged to spoon-feed information learners. Instead, teachers are to provide opportunities for students to discover icepts on their own. One way of doing this is through the inductive learning strategy.

link The inductive learning strategy, sometimes called discovery learning, is based on principle of induction. Induction means to derive a concept by showing that if it ■ue to some cases, then it is true for all. This is in contrast to deduction where a cept is established by logically proving that it is true based on generally known s. The inductive method in teaching is commonly described as "specific to general," icrete to abstract," or "examples to formula." Whereas the vice versa are used to :ribe the deductive method. In an inductive learning lesson, teachers design and facilitate activities that guide learners in discovering a rule. Activities may involve comparing and contrasting, iping and labeling, or finding patterns. In mathematics classes, learners engage iductive learning when they observe examples and then, later on, generalize a or formula based on the examples. There are four processes that the students hrough when given an inductive learning activity—(1) observe, (2) hypothesize, ollect evidence, and (4) generalize.

serve Children love looking for patterns! When given a lot of examples, it is natural for i to look for similarities and assume rules. So, the key is to give them examples to rve. These examples must be well-thought-of so that the students would eventually 5 at a complete rule. For instance, if you want your students to discover the rule in iplying by powers of 10, it is better to use the examples in set B than those in set A. UNIT III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 49

B 6 x 10 = 60

6 x 10 = 60

18 x 10 = 180

321x10 = 3,210

10x321 = 3,210

457 x 10 = 4,570

40 x 10 = 400

Both sets will lead students to discover that the technique in multiplying by 10 is placing a 0 after the number being multiplied. However, the variety of examples in Set B allows students to establish that the rule works even when exchanging 10 and the other factor (18 x 10 = 180) and if the other factor ends with a zero, that zero is not neglected (40 x 10). Set B allows students to have a more comprehensive understanding of the rule.

Hypothesize The students form rules in their minds as they observe. In this stage, encourage the students to share their thoughts. Assure them that there are no wrong hypotheses. Acknowledge the variety of the students' ideas but also streamline them to, later on, test only the unique hypotheses.

Collect evidence Here the students would test their hypothesis. How? By applying their hypothesis to other examples. If there are more than one hypothesis generated by the class, intentionally give a counterexample for them to test.

Generalize Finally, the students would now formalize their hypothesis to a rule. Support the students so that they would use mathematical terms in stating their rule. Doing this would develop the students' mathematical vocabulary and therefore their overall mathematical communication skills.

Experience Study the lesson plan on the next page. Take note that the plan only shows the development of the lesson, which involves the inductive learning strategy; other parts are not included. In this lesson, inductive learning was not used to discover a rule but rather to discover a relationship. 5 0 |TEACHING MATHEMATICS IN THE PRIMARY GRADES

Topic: Multiplication and Division as Inverse Operations Grade level: 2 Target learning competency: By the end of the lesson, the students will be able to describe multiplication and division as inverse operations. Observe 12 -f 2 = ___

1 5 * 3 = ___

24 * 6= ___

6 x 2 = ___

5 x 3 = ___

4 x 6 = ___

3 6 * 4 = ___ 9x4 =

Ask the students to fill in the blanks by dividing or multiplying. Then lead them to observe each pair of division and multiplication number sentences. Give some time for the students to observe the examples. Fast learners may become too excited to share their hypothesis but do not allow them to spill it. The goal is for all students to have the "Aha!" moment. Hypothesize Struggling students may not see the pattern right away. Help them by focusing their attention to the quotient and the first factor. Call some students to explain their hypotheses. After each explanation, ask who has the same hypothesis. Collect evidence Apply the hypotheses to each example to see if they always work. Generalize Based on the result of the "collect evidence" stage, ask the students which hypothesis is true for all. Then instruct the students to write, using their own words, the rule in their notebook. Have two to three students read aloud what they have written.

UNIT lit • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 51

Assess Answer the following questions to verbalize your understanding of inductive learnin 1. Explain how inductive learning is related to the constructivist theory of learnir discussed in the previous unit.

2. What possible hypotheses would the students come up with given the probler in EXPERIENCE?

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Challenge "he following activity will engage you in identifying mathematical concepts that csr r-e taught using the inductive learning strategy. 1 Browse the DepEd mathematics curriculum for Kinder to Grade 3. Write five mathematical rules that you can teach using the inductive learning strategy.

2. The key to effective inductive learning is well-thought-of examples. Choose one topic from your list in #1 and write examples which you can use in class to allow discovery. What were your considerations in choosing your examples?

UNIT III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES I 5 3

Harness Write a lesson plan which allows the students to discover a rule inductively. If appropriate, use the same topic as in your HARNESS in Lesson 7. This activity will be part of the learning portfolio which you will compile at the end of this module.

Observe

Hypothesize

5 4 I TEACHING MATHEMATICS IN THE PRIMARY GRADES

Collect evidence

Generalize

Summary Inductive learning is about the students discovering the mathematical concepts by themselves with the teacher as a guide. In this strategy, students observe, hypothesize, collect evidence, and generalize.

UNiT 111 • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES

I 55

Objective Plan a lesson that applies concept attainment strategy

Introduction The inductive learning strategy in the previous lesson is most useful in discovering rules. In mathematics, students do not only study rules, but they also need to remember and understand many definitions of terms. For better retention, it is best for students to discover the meaning of the different mathematical terminologies that they encounter. The concept attainment strategy is useful for this, purpose.

Think Concept attainment is another instructional strategy anchored to the constructivist learning theory. In this strategy, the concept is not directly taught to students. Instead, the students understand and learn concepts by identifying common attributes through comparison and contrast of examples and non-examples. Since concept attainment is used in understanding meanings, it is often applied in English vocabulary lessons. However, it is also useful in learning mathematical terminologies. There are five simple steps in the concept attainment strategy: (1) presentation of examples and non-examples, (2) listing of common attributes, (3) adding student-given examples, (4) defining the mathematical term, and (5) checking of understanding.

Presentation of examples and non-examples Alternately give examples and non-examples. The students should be able to guess some common attributes based on the examples alone so non-examples are given to confirm their guesses.

Listing of common attributes List the common attributes given by the students. This may be done as a whole class or by pairs or traits first. Some listed attributes may be later on crossed out as the listing of examples and non-examples go on. 5 6 I TEACHING MATHEMATICS IN THE PRIMARY GRADES

Adding student-given examples Ask students to provide their own examples based on the listed attributes. Then confirm whether their suggestion is indeed an example. Based on the students' answers, some of the attributes may be revised to make them clearer for the students.

Defining the mathematical term Help the students come up with a word or phrase for the concept. The exact term may not come from them, especially when it is too technical (e.g., polyhedron), but the etymology of the word may be derived from them (e.g., many polygonal faces).

Checking of understanding To verify that the students have understood the concept, give them a list and ask them whether each item on it is an example or a non-example.

E x p e rie n ce Study the lesson plan below which applies the concept attainment strategy. The goal of the lesson is for the students to define a prism. Instead of listing examples on the board, real object examples and non-examples will be provided. Topic: Square Grade level: 1 Target learning competency: By the end of the lesson, the learners will be able to define a square, draw examples of a square, and identify whether a given figure is a square or not. Prerequisite knowledge and skills: Definitions of: 1.

Straight and curvy lines

Plane figures

Solid figures

Identifying common attributes based on examples and non-examples: Tell the students that they will be detectives for today. Their goal is to discover the common characteristics of the figures which will be shown to them.

UNIT HI • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES

j 57

Give the following examples and non-examples by batch.

5 8 |TEACHING MATHEMATICS IN THE PRIMARY GRADES

Below are the expected common attributes that the students will provide, refined through the batches. Ask guide questions if the students do not arrive at these. Batch

Common Attributes

Made of straight lines; no curvy lines

Has four sides

The sides have equal length

All the angles are right

It is a plane figure

In between batches, ask the students to look around the room and give examples of what they think are squares. Classify the student-given objects as examples or non-examples of prisms. Defining the mathematical term Lead the students to agree that a square is a four-sided plane figure whose side lengths and angle measures are equal. Checking of understanding Show 10 real objects and let the students identify each as a square or not.

Assess Doing the following activity will strengthen your understanding of the concept attainment strategy. 1. Use the Venn diagram below to compare and contrast inductive learning and concept attainment strategies.

UNIT ill • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 5 9

2. What is the importance of giving the examples and non-examples given by batch? Will there be a difference if all of them are presented at once? Explain.

Challenge The following activity will challenge you to ponder on things to consider when thinking of the order of examples and non-examples that you present when applying the concept attainment strategy. 1. Interview a Kinder, Grade 1, 2, or 3 mathematics teacher. Ask him/her what mathematical term the students have a hard time remembering, or that which they find confusing. List three mathematical terms and explain why each term is difficult to remember. Term 1 :______________________________________________

Term 2:

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2. Choose one math term from your answers in #1. List examples and non-examples of the concept and group them by batch according to how you would present them. What were your considerations in grouping them? Term:______________________________________________ Examples

Non-examples

Considerations:

UNIT III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 61

Harness

Write a lesson plan which allows the students to discover the meaning mathematical term. Use the topic you answered in Challenge. This activity will b of the learning portfolio which you will compile at the end of this module.

Topic:--------------------------------------------------------------------------Grade level:----------------------

Target learning competency: By the end of the lesson, the learners will be able 1

Prerequisite knowledge and skills:

Identifying common attributes based on examples and non-examples:

I TEACHING MATHEMATICS IN THE PRIMARY GRADES

Defining

the mathematical term:

Checking of understanding:

S u m m a ry Mathematics is considered a language with its own set of jargons. Mathematical terms can also be defined through discovery by applying the concept attainment strategy. Concept attainment involves the presentation of examples and non examples, listing of common attributes, adding student-given examples, defining the mathematical term, and checking of understanding.

UNIT III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 6 3

Objective

Generate investigative tasks and anticipate possible problems that may arise from the task

Introduction

Contemporary leaders in mathematics education revolutionized the goal o mathematics teaching and learning from passive learning dictated by the curriculun to an active process where the students are developed to think like mathematicians Mathematical investigation is a strategy that may be implemented to achieve this.

Think

Mathematical investigation is an open-ended mathematical task which involve: not only problem-solving but equally importantly, problem-posing as well. In thi: strategy, the word "investigation" does not refer to the process that may occur wher solving a close-ended problem, but an activity that in itself promotes independen mathematical thinking. To illustrate, consider the two mathematical tasks below.

Task A - Problem solving

There are 50 children at a playground and each child high-fives with each of the other children. Find the total number of high-fives.

Task B - Mathematical investigation

There are 50 children at a playground and each child high-fives with each of the other children. Investigate.

In Task A, there is a specific problem to solve. Some students might attempt tc solve it by drawing diagrams for smaller numbers of children and then investigating the pattern that may arise. This investigation is a process that may occur in problem solving. On the other hand, the problem in Task B is not specified. Students may 0 1 may not choose to find the total number of high-fives. Some students may want tc investigate a more general case where they would want to know how many high-five: there would be given a certain number of children. Some may want to find out hov\ 6 4 j TEACHSNG MATHEMATICS IN THE PRIMARY GRADES

many high-fives there would be if instead of once, the children would high-five each other twice or thrice. Some children may even decide to work on a problem that the teacher has not thought of. This is investigation as an activity itself. As illustrated, what sets mathematical investigation apart from other strategies that have been discussed in this unit by far is that the goal of the investigation is not specified by the teacher; the students have the freedom to choose any goal to pursue. In problem-solving, the students are encouraged to think outside the box; in mathematical investigation, there is no box to start with. The students are placed in a space where they can play around whichever way they want. This makes mathematical investigation a divergent and learner-centered strategy. So, like in the problem-solving strategy, it is crucial that the teacher chooses or creates a situation that is engaging and caters mathematical investigation. Tasks A and B show that a close-ended word problem can easily be converted into an open-ended investigative task by simply replacing the question with an instruction to investigate. There are three main phases of a mathematical investigation lesson: (1) problem-posing, (2) conjecturing, and (3) justifying conjectures. In the problemposing phase, the students explore the given situation and come up with a mathematical problem that they would want to engage in. The conjecturing phase involves collecting and organizing data, looking for patterns, inferencing, and generalizing. In the final phase, the students are to justify and explain their inferences and generalizations. Always remember that although mathematical rules or theorems may arise as results of the mathematical investigation, they are notthe objectives of an investigative lesson—the objective is the investigation itself; the exercise of creative thinking and problem-solving that the students underwent as they investigated. Mathematical investigation is not after the teaching and learning of some competency in the curriculum; it is about developing the mathematical habits of the mind.

Experience The only planning that the teacher needs to do is to create or choose an appropriate task and anticipate possible problems that the students would pose. Below is an example of a close-ended word problem transformed into a mathematical investigative task and the problems that the students would possibly come up to.

UNIT ill • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 6 5

Close-ended problem: Find the perimeter of the triangle whose side lengths are 3 units, 4 units, and 5 units. Investigative task: Distribute: 12 popsicle sticks per pair/group Instruction: Investigate the following.

/X / /

Possible student-generated problems: 1. What is the perimeter of the given triangle? 2.

How many triangles can be formed using 12 popsicle sticks?

3. What types of triangles can be formed using 12 popsicle sticks?

Assess The following activity will broaden your understanding of the mathematical investigation strategy. 1. Use the Venn diagram below to compare and contrast problem-solving and mathematical investigation.

66 |TEACHING MATHEMATICS IN THE PRIMARY GRADES

2. In what ways does mathematical investigation develop students who think like mathematicians?

Challenge Even though the students are the ones who would identify the problem given a situation, the teacher must be able to anticipate some of the problems that may come up. To develop this skill, the teacher must him/herself undergo mathematical investigation. The following activity will engage you in mathematical investigation and allow you to reflect upon your experience. 1. Pose a problem, make a conjecture, and justify your conjecture given the following situation. This task is adapted from Orton and Frobisher's Insights into Teaching Mathematics (1996). Investigate the following number trick. 854 -458 396 + 693 1089

UNIT ill • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES I

2. Write about your experience. How did you feel before, during, and after the task?

Harness Choose a close-ended problem from the DepEd mathematics teaching materials for Kinder to Grade 3. Transform it to an investigative task then list the possible problems that the students would pose given the task. This activity will be part of the learning portfolio which you will compile at the end of this module.

6 8 |TEACHING MATHEMATICS IN THE PRIMARY GRADES

Close-ended problem:

----------------------------

UNIT III * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES

Summary Mathematical investigation is an open-ended teaching strategy that capitalizes on the students' ability to identify a problem. Any word problem can be transformed into a mathematical investigation by limiting the given information and omitting the specific question that it is asking.

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Objective Execute the empathize, define, ideate, prototype, and test stages of the design thinking process

Introduction Students find learning mathematics most engaging when they are involved in a thinking process that results in an output that can be applied to a relevant context. The design thinking process engages the students in such a thought-provoking and purposeful activity.

Think Design thinking is a progressive teaching strategy that allows students to look for real-world problems and finding creative solutions. Students do this by focusing on the needs of others, collaborating for possible solutions, and prototyping and testing their creations. This can be summarized in five stages: empathize, define, ideate, prototype, and test. These stages are adapted from the Institute of Design at Stanford University.

Design Thinking Framework (Institute of Design at Stanford, 2016)

UNIT III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES I 71

Empathize The goal of design thinking is for students to respond to a particular need (a real-world problem), so it is fitting that the first stage is empathy. In this stage, the teacher needs to be explicit in guiding the students to put themselves in the shoes of others through activities like immersed observation and interviews. According to the developmental stages, it is not natural for children in the primary grades to be empathetic toward others. It is a common observation by teachers that students at these levels often do not realize that their actions affect others. So, applying design thinking in the classroom gives the children the opportunity to cultivate empathy, and at the same time, develop their problem-solving skills.

Define The next stage is for the students to define the specific problem or issue that they want to address. First, the students will identify an audience—the future users of the product they will develop. Their audience can be students, teachers, family members, or just anyone in their community. Then the students will use the information they gathered from the Empathize stage and focus on one aspect of the problem. It is important that the students be able to identify a true problem because it is impossible to successfully complete the design thinking process without a meaningful problem to solve.

Ideate The third stage of design thinking is the generation of ideas to solve the identified problem. This involves brainstorming and research. The students are to be encouraged to think out of the box and produce radical ideas. What sets this stage apart from the usual brainstorming is that all ideas must be written or illustrated. Ideas are usually written or drawn on sticky notes and students, later on, organize them into a mind map. It is at this stage that the students will be able to apply their mathematical knowledge and skills. Aside from being able to operate their problem-solving skills, they will also be able to apply specific content knowledge like measurement, proportion, geometry, and statistics.

Prototype and test Finally, the students go through a repetitive cycle of prototyping and testing. A prototype is anything that a user can interact with in order to, later on, provide feedback about it. It can be made of easily accessible materials like paper, cardboard, 7 2 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

s ~: «. :apes, recycled plastics, and so on. Once a prototype is created, they test it or ale* a user to test it and then make improvements, or possibly overhaul the design, sec-ending on their observations and the feedback of the user. In these stages, it is - : : 'tant to emphasize that it is totally fine to fail at the first attempt of prototyping. trial- and-error aspect of the design thinking process is glorified because it is x e.ed that the students learn a lot through their failures. Even though a physical : :r^ct is the expected output of design thinking, it should be emphasized that ;: 'g through the process is what is more important because it is where the learning tr <es place.

Experience Below is a template of a sample worksheet that will guide the students : "trough the design thinking process. This is a simplified version of Stanford's sample *eTiplate. In this example, the students are to create a project about their playtime experience.

■our challenge is to redesign your school’s playtime experience. Em pathize

Observe your classmates and teachers during playtime. Take time to casually interview some of them about their usual playtime experience.

r Notes from your observation:

k __

Notes from your interview:

Discuss your observation and interview notes with your groupmates. Do you have similar notes?

UNIT III « INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES j 7 3

Define Identify a user and define a problem that you want to address. Your group must come up with one user and problem to address.

user

needs user's needs

because ' insight

Ideate Write/Sketch at least four innovative ways to address your user's needs. Be specific with your measurements and/or proportions, if needed.

Idea 2:

Idea 1:

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| TEACHING MATHEMATICS IN THE PRIMARY GRADES

Share your ideas with your groupmates. Get feedback. As a group, generate a unified solution that incorporates the ideas and feedback from the members. It is not necessary that all ideas will be seen in the unified solution. What is important is that all members agree that the necessary ideas are integrated in the solution. Sketch your group's unified solution below.

Prototype Create a prototype of your unified solution. Use readily available materials like papers, cardboards, coloring materials, sticky tapes, popsicle sticks, etc. Be accurate with your measurements by using a ruler, compass, and/or measuring tape. Test Share your prototype with a user. Write your observations below. What worked:

What can be improved:

More ideas:

-A*.

UNIT II! • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 7 5

Assess This lesson introduced design thinking as a thought-provoking and purposeful strategy. Elaborate why you think design thinking was described as such.

C hallenge Many teachers are not informed about design thinking because it is a new, if not the newest, strategy in teaching mathematics. The following activity will challenge you to convince a teacher about the benefits of using this strategy. 1. Make an infographic about design thinking for the teachers who have not yet heard about it. Sketch a draft of your infographic below.

I TEACHING MATHEMATICS IN THE PRIMARY GRADES

2. Share your infographic to a Grade 4, 5, or 6 mathematics teacher. What are the teacher's questions or comments about design thinking? Were you able to answer his/her questions? Do you share the same sentiments with him/her about design thinking?

Harness The design thinking process is best learned when done. Go over the steps yourself with a partner. Empathize, define, ideate, prototype, and test to redesign your school's lunch experience. This activity will be part of the learning portfolio which you will compile at the end of this module. Your challenge is to redesign your school's lunch experience. Empathize Observe your classmates, teachers, and canteen managers during lunchtime. Take time to casually interview some of them about their usual lunch experience. Notes from your interview:

Notesfromyourobservation:

-A , Discuss your observation and interview notes with your partner. Do you have similar notes? BIBgwBfiB

18 Kssas

UNIT 111 • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 7 7

Define Identify a user and define a problem that you want to address. You and your partner must come up with one user and problem to address.

user needs user's needs because

insight

ideate Write/Sketch at least four innovative ways to address your user's needs. Be specific with your measurements and/or proportions, if needed.

| TEACHING MATHEMATICS IN THE PRIMARY GRADES

s Share your ideas with your partner. Get feedback. Then generate a unified. Sketch your unified solution below. ................................................................................................. ........... ..........

Unified Solution

-J: Prototype Create a prototype of your unified solution. Test Share your prototype with a user. Write your observations below.

"'V' What worked:

~\ More ideas:

What can be improved:

Summary Design thinking is a contemporary teaching strategy that fosters creativity by allowing students to come up with concrete and tangible solutions to authentic problems that the students themselves identified. UNIT Hi * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES |

Objective Develop a game to motivate students, cater mathematical investigation, or practice a mathematical skill

Introduction Play is children's work and they love it! Well-designed lessons using game-based learning strategy takes advantage of children's natural love for play to lead them toward complex problem-solving.

T h in k Children find games both motivating and enjoyable, so it is not a surprise that teachers harness games to cater to learning. There are many ways in which games are used in the classroom. Games are sometimes used as lesson starters to get the students engaged. In some lessons, games are used to explore mathematical concepts and processes or cater mathematical investigation. But most of the time, games are used to practice mathematical skills. Not only do games make the lesson engaging for young learners but they also create a relaxed environment in a mathematics class. Games associate mathematics with positive feelings like excitement, victory, and fun competition. So, students who might have developed mathematics anxiety, or those who simply "hate" math, might begin to open up and be more receptive. The students' love for play may translate to love for math. Moreover, games give a venue for students to communicate and defend their ideas while at the same time learning from each other in a fear-free environment. And because in every game a goal has to be achieved, students naturally develop strategic and creative thinking and problem-solving skills. Games that require students to work in groups advance their social skills as well. However, not all games that involve mathematical processes are considered to have instructional value. For example, the game of Monopoly involves computations and strategies to maximize scores, but it is considered to have little instructional

8 0 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

value. According to Koay Phong Lee (1996) in his article "The Use of Mathematical Games in Teaching Primary Mathematics" a game that has instructional value has the following characteristics: 1.

The game has two or more opposing teams

The game has a goal and the players have to make a finite number of moves to reach the goal stated. Each move is the result of a decision made.

There is a set of rules that govern decision-making.

The rules are based on mathematical ideas.

The game ends when the goal is reached.

The 4th characteristic is what separates mathematical games from other games. This suggests that a good mathematical game is not only about "having fun" but also about "doing math" in itself. A teacher has three important tasks in a lesson that implements game-based learning strategy: (1) lay down rules clearly, (2) observe, assess, and process students' understanding, and (3) work with students who need additional help.

Experience Following a guessing game to develop the students' sense of weight. Weight is probably the most abstract measurement for children because they do not see it (unlike length and area). This game will help them makes sense of weight.

Topic: Estimating weight Materials (for each pair of players): 1. Weighing scale 2. Various objects (may be brought by students) Mechanics: 1. Before the game starts, let the students carry a 1 kg weight and a 500 g weight. Providing this experience will give the students a basis for their estimates. 2. Divide the class into two groups. UNIT lii • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 81

3. Each group will take turns to estimate the weight of a given object. 4. In each turn, the guessing team will announce their estimate then the other group will say whether they think the actual object weighs higher or lower than their opponent's estimate. 5. If the non-guessing team's answer is correct, they get the point. Otherwise, the guessing team earns the point. 6. For the students to have a basis of their estimate, pass the object around before the guessing starts in each turn.

Some students might want to go back to carrying 1 kg or 500 g and then compare it to the given. This may be allowed for the first 2 items. The game may be modified to accommodate other measurements like length, area, or volume.

Assess Answer the following questions to verbalize your understanding of game-based learning. 1. What are the benefits of using games in mathematics lessons?

2. What do you think are some disadvantages of game-based learning strategy?

82 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

Challenge Developing a game that has instructional value is challenging so there may be instances when instead of creating a new one, you would adapt an existing game but modify it a bit to meet your learning goal. The following activity will challenge your creativity as you think of ways to modify the game presented in Experience. 1. Modify the game presented in Experience. How will you change the mechanics to target subtraction of fractions?

2. A game like the one presented in Experience requires materials. If resources are limited, how would you modify the game to make it a group activity so that fewer materials need to be prepared?

UNIT 111 • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES

I 83

Harness Develop a mathematical game that has instructional value. This activity will b part of the learning portfolio which you will compile at the end of this module.

Topic:____ Materials:

Mechanics:

Summary Game-based learning is a strategy that takes advantage of children's love fc games. Applying this strategy is good in reducing math anxiety.

| TEACHING MATHEMATICS IN THE PRIMARY GRADES

Objective Develop a manipulative to aid mathematical instruction

Introduction Mathematics is an abstract subject which is the reason why many students find it difficult. One way to concretize mathematics for young learners is through the use of manipulatives.

T h in k Manipulatives are concrete objects like blocks, tiles, and geometric figures, that students can interact with (touch and move) in order to develop conceptual understanding of mathematics concepts. Use of manipulatives is not at all new; manipulatives have helped people learn mathematics since ancient times. Forexample, the early Chinese had the abacus and the Incas used knotted strings called quipo to aid in counting. In modern times, educators Friedrich Froebel and Maria Montessori were the ones who advanced the use of manipulatives in classroom instruction. At present in the Philippines, the DepEd mathematics curriculum calls for manipulatives to be used in teaching a variety of competencies. Aside from helping the students acquire a deeper understanding of mathematics, the use of manipulatives also gives you, the teacher, the chance to genuinely assess their students' mathematical thinking. You can move around, observe, and take note of students' discussions and ways of manipulating. Moving around will let you give immediate feedback and taking notes of observations will help you improve your future lesson. One drawback in using manipulatives is that it may cause confusion, especially to struggling students, if they are not presented with proper guidance and instruction from the teacher. Moreover, careless use of manipulatives might result in students believing that there are two different worlds of mathematics—the manipulative and the symbolic. It is, therefore, importantthattheteachercarefully plans howto integrate

UNIT III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 85

manipulatives in classroom discussion in such a way that there is a smooth transition from concrete to abstract. Following are some guidelines in using manipulatives in the classroom. 1. Orient the students on how to use the manipulative. Give some time for the students to play with the manipulative. Allow them to explore the object and what they can do with it. 2. Give clear and specific instructions. State the goal of the activity and how the manipulative can help them achieve the goal. 3. While the students are at work, pay attention to their mathematical talk. Use their ideas to enhance the discussion that follows after the activity. 4. If some students are struggling, ask them "why" and "how" questions to scaffold their way through the activity. Many manipulatives are commercially available; the common ones are base 10 blocks (for learning value, place value, decimals, etc.), geoboards (for learning properties of plane figures), play money, and paper clock. However, you may also create manipulatives using readily available materials like popsicle sticks, buttons, boards, fasteners, etc. Making your own manipulatives is much cheaper and it gives you the benefit of customizing them according to your need.

Base 10 blocks

Geoboards

E X P E R IE N C E Below is an example of a do-it-yourself manipulative. It is a protractor customized for young children and useful for measuring angles of inclined surfaces. Using this manipulative will help primary students develop a sense of angle measure without the intricacies of using an actual protractor which is more suitable for older children. 8 6 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

How to make: 1. Print out the following on a heavy-duty paper or a piece of cardboard.

2. Use a circular fastener to attach the arrow to the protractor. Make sure that the arrow is loose; it must be always facing downwards however the orientation of the protractor. .

How to use: 1. Align the straight part of the protractor to the inclined surface. 2. Allow the arrow to fall. 3. The measure to which the arrow points to is the measure of the angle of inclination.

In the example above, the angle of inclination is 609. Now the students will have an idea of how wide the opening of a 60-degree angle is.

Assess Answer the following questions to verbalize your understanding of the use of manipulatives in mathematical instruction. UNIT ill • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES

| 87

1. Give an analogy that involves the use of manipulatives and the use of games in classroom instruction. manipulatives : _____________________ games : _____________________ 2. The first guideline in using manipulatives is to let the students play with the material. Why do you think is this so?

Challenge The following activity will challenge your creativity as you try to modify the manipulative presented in Experience. 1. Make the customized protractor in Experience. 2. Try to use your D-l-Y protractor. Describe your experience.

3. What improvements can be made on the manipulative based on your experience?

Have you noticed? This Challenge is a design thinking task! © 8 8 j TEACHING MATHEMATICS IN THE PRIMARY GRADES

Harness Invent your own manipulative for a topic of your choice. Draw your design and ac-e mportant parts of it. Then explain how to use your invention. This activity will be z--~of the learning portfolio which you will compile at the end of this module.

Topic.---------------------------------------- _________________ Name of manipulative:_________________ _______________ .....

Design:

Instructions:

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S u m m a ry Manipulatives have long been used to facilitate learning mathematics. Although nany manipulatives are available in the market, nothing beats the manipulatives :reated by the teacher who knows exactly what his/her students need. UNiT III * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS SN THE PRIMARY GRADES | 89

Objective To plan a lesson in which values education can be incorporated into existin mathematics curricula

Introduction You do not just teach math. You teach students—with young and impressionabl brains. Primary teachers play an important role in developing young learners' heart and minds. What students learn in primary years can mold the persons they wi become. Instilling good values in children early will help them grow into successfi and responsible citizens of the nation. Mathematics can be used as a toot for value integration. Values such as honesty, patience, and resilience in facing failures are somi of the many values that can be developed through mathematics. In this lesson, yoi will learn how to deliberately integrate positive values into your mathematics topics.

Think

Integrating Math into Other Subject Areas Integrating mathematics into the curriculum can be quite challenging anc rigorous. However, math is connected to a lot of disciplines and should not be isolatec from other subjects. Our complex brain looks for patterns and interconnections as its way of making sense of things. Our learners develop an appreciation for mathematics and a deeper understanding of concepts when they make connections with prior experiences or with different areas of learning.

Tapping into the Affective Domain Dr. Benjamin Bloom classified three domains of educational learning: cognitive affective, and psychomotor. In the formal classroom set-up, the bulk of the teache's lesson planning focuses on the cognitive and psychomotor aspects of the teachinglearning process. The third domain, which is the affective domain, is often overlooks± The affective domain includes the manner in which we deal with things emotiona * such as feelings, values, appreciation, motivations, and attitude (Kratwohl, 1964). Th 5 90 |TEACHING MATHEMATICS

IN THE PRIMARY GRADES

particular domain, when tapped during the learning process, can really make students reflect on the connection between mathematical concepts and values or standards of behavior that will help them deal with the pressures and difficulties in life. As future teachers, you want to form not only competent students but students with moral courage, clear values, and excellent character.

Values Integration and Retention of Information Associating values or standards of behavior with mathematical concepts can serve as a source of motivation for students. Values integration will help students get life lessons through math. If students find a learning material engaging and meaningful, then they will ask for more (since curiosity will start to kick in). Curiosity is the force behind lifelong learning!

Experience The valuing part can be done before closing the lesson. Listed below are the mathematical concepts vis-a-vis the sample questions and/or moral lessons that you might want your learners to reflect on.

You may ask your students to reflect on and write about mathematical concepts in relation to values or standards of behavior related to their lesson. Math Topic 1. Zero as a placeholder

Values Integration Point Zero is considered as a number that does not worth anything on its own. But in reality, zero makes a lot of difference. For example, the number 5,034. Without the number zero, how you express the substantial difference between 5,034 and 534? Sometimes, we develop a feeling of worthlessness but remember that we are irreplaceable and valuable to other people. We make a difference in their lives.

2. Equivalent fractions

Notice that despite the differences in form, color, ethnicity, gender, faith, socioeconomic status, mental health condition, etc. our life is worth the same and worth saving..We are born equal, but not the same.

UNIT HI • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES I 91

3. Shapes

Let the students reflect on this. What is the SHAPE of your life right now? Spiritual gifts (What are the blessings that you received?) Hindrances (What are the problems that you encountered?) Abilities (What are the lessons that you understand? do not understand?) Persons (Who inspires you? Who makes you feel sad?) Experiences (How have your past experiences helped you grow as a student?)

4. Whole Numbers

Connect the lesson to the concept of 'wholeness7—comprising the full quantity, the start of forming a complete and harmonious whole, the state of being unbroken and undamaged. 1. What were your experiences in the past that make you feel 'whole' or 'complete'? 0

2. Bullying can make a person's heart break/not whole. Cite a specific happening in the past wherein you or a classmate of yours have experienced bullying (in any form). How did you respond to the situation? What can you do to stand up against persons who break or damage people's heart? 5. Order of Operations

Relate this lesson to the importance of obeying rules/order for self-management and doing things one step at a time. 1. Why are rules important? 2. What aspect of obeying rules did you find quite challenging in the past? 3. What step-by-step process do you follow in solving your problems?

6. Factors and Multiples

Associate this with the idea of organizing things. You group all items that have a common factor together. 1. What are some benefits of being organized? 2. How do you deal with people who are having a hard time organizing things?

9 2 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

7. Addition and Subtraction of Fractions

Relate this to the idea that most of us tend to be attracted to people who are similar to ourselves. 1. What are the qualities would you like your friends to have? Do you also possess these qualities? 2. Reflect on the saying: "Opposite attract." Do you believe in this saying? 3. Should you listen to the opinion of a person that is not similar to you? Why or why not? Fractions that are dissimilar can still be combined. You just have to do some modifications to the denominators to make them similar. Just like in real life, you live in a very diverse world. Even if two people are different (in faith, gender, faith, socioeconomic status, etc.), they can still work harmoniously. A key value that you need to develop is modifying your attitude and genuinely respecting other people.

8. Geometry (Triangles)

9. Patterns

Relate this lesson to the rigidity of triangles. Other polygons can be easily deformed. If you make a rectangle or a square from metal wires with hinges at the corners, you will find that it does not stay in that orientation. It can be transformed into an ordinary parallelogram. In a triangle, each edge is supported by the other two edges. This characteristic makes a triangle stable. You have to act like triangles, you have to make sure that you have a strong support group. A person develops a pattern of behavior if he/she repeats an activity over and over again. 1. What are the personal behavior patterns that you wish to break? Why? 2. What are the personal behavior patterns that you wish to form? Why?

UNIT III * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES j 9 3

10. Data Presentation

Give examples where the use bar graph or a pictograph can be abused. You should not let your students be easily manipulated by misleading statistics. What's wrong with this? 1. Both graphs are presenting the same data set

What can you say about the 1st graph? 2nd graph? Which graph presents a more accurate reflection on the increase in house prices? 2. Which graph is misleading? Why? o f singles :

Number of singles sold 200,000

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Year

1997 1998

9 4 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

3. Which graph is misleading? Why? n

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11. Polygons

Increasing the number of sides of a polygon approximates a circle. In ancient times, a circle is considered as a perfect shape. In life, if you want to improve yourself, you shall undergo a lot of trials. Increasing the number of trials means gaining new insights/ perspectives. These new insights will make you a better person.

12. Units of

Why are units important?

Measurement

Units are like names. Their names help identify who they are. Objects also have names. They can be identified and described further using their units.

In our society, it is vital to educate people on the traditional values of our country. There is a growing demand for teachers to deliberately teach values by setting a good example and discussing/processing moral issues to learners. It is therefore crucial to the formation of students that you deliberately use an eclectic mix of methods to convey the important values that the students have to uphold. UNIT in • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES I 95

Assess Do the following to help you think of creative ways to integrate values in your lesson. 1. Browse the DepEd mathematics curriculum guide. Choose a topic from Grades 1 to 3. Write the topic and grade level below.

How would you inject values to this particular topic? Put it in a situation where the students can relate to. Imagine you are talking to your students. Write your script below. If you plan to use materials, write a note about it.

2. What is the most memorable life lesson you have learned from your former teacher? Explain why.

| TEACHING MATHEMATICS IN THE PRIMARY GRADES

Challenge Reflect on the following questions. 1. Do you foresee problems or difficulties in integrating values into your curriculum?

2. In what mathematics topics in Grades K to 3 do you think is this strategy most appropriate? Why do you think so?

Harness This activity will test your skill in spontaneously integrating values in a math class setting. This activity will be part of the learning portfolio which you will compile at the end of this module. Consider this situation. A student consulted with you and raise the following points. "Hi, Teacher! Our lesson now on addition is not that hard. But why do I need to study addition? My gadget can actually add for me? Why do I have to do it on my own?"

UNIT III * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES

| 97

How will you tap into your student's affective domain for him/her to understand the relevance of your lesson in his/her life?

Topic: Addition of Whole Numbers

Possible Values integration Point ----------------------------------------------------------------------------------------------------------------------------------------- -—

................................

Summary There is a growing demand for teachers to deliberately teach values and this is possible even in the mathematics classroom. All teachers need do is to be intentional about it and reflect on ways to inject values in their lessons.

| TEACHING MATHEMATICS IN THE PRIMARY GRADES

Objective Design collaborative activities that will encourage involvement, interdependence, and a fair division of labor among students

Introduction When transitioning from preschool to primary grades, children develop a really strong bond with one friend. Some child psychologists point out that it easier for some kids to relate to just one co-learner rather than socializing with a big group at the same time. Teachers, however, can provide primary graders with many opportunities for interaction. Within collaborating groups, children learn to try things out, conjecture, explore, justify, evaluate, and convince others of their findings. Collaborative tasks provide enriching opportunities for learners to explore other students' perspectives that may differ from their own. Thus, these can develop a stronger sense of empathy among students. Group activities, if facilitated carelessly, could waste classroom time. Because of this, it is important for teachers like you to ensure that group activities are carefully designed and successfully implemented. This lesson aims to help you prepare, monitor, and process collaborative tasks in your classroom that will maximize your student's capacity to socialize with each other.

Think Vygotsky’s Social Learning Theory Collaborative learning branches out from the Zone of proximal development theory of Vygotsky. Vygotsky defined the zone of proximal development as follows: "The zone of proximal development is the distance between the actual developmental level as determined by independent problem-solving and the level of potential development as determined through problem-solving under adult guidance or in collaboration with more capable peers." unitin•Instructional'sfkA TEG iESformathematicsintheprimarygrades j

In the zone of proximal development, the learner is close to developing the new skill, but they need supervision and assistance. For instance, if a student has already mastered basic addition of fractions, then basic subtraction may enter their zone of proximal development, that is, they have the capacity to gain mastery of subtraction of fractions with assistance. The assistance may not be directly provided by the subject teacher. A child seeks to understand the actions or instructions provided by any skillful peer and internalizes the information, using it to guide or regulate their own performance. It is, therefore, necessary that learners should be given the opportunities to work with their peers in broadening their learning experience, allowing small groups of students to work together to share knowledge, exchange ideas, and to solve problems together. As learners collaborate with their classmates and teachers, they adopt some of the learning heuristics and develop more skills in problem-solving.

Designing Group Activities Collaborative activities encourage active participation from learners. Instead of passively accepting information from the teachers, learners discover new insights by cooperatively working with other learners. As mentioned earlier, teachers should be keen in selecting appropriate learning activities for students. Listed below are some tips about preparing, monitoring, and processing collaborative tasks in your classroom that will maximize your student's capacity to socialize and learn from and with each other. • Identify the instructional objectives. When deciding whether or not to use group work for a specific task, reflect on the following questions: What does the activity aim to achieve? How will that objective be furthered by asking students to work in groups? Is the activity complex enough that it requires group work? Will the project require true collaboration? Is there any reason why the assignment should not be collaborative? Are the objectives attainable within a given time frame? • Determine the group size. How many students will be assigned to each group? The size you choose will depend on the total number of students in your classroom, the size of the venue where the activity will be held, the variety of students needed in a group, and the task assigned. If you want to have a diverse, productive, active,

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and cohesive groups, then try dividing your class into groups with four to five members. Decide how you will divide the class. Will you group them based on proximity? Will you group them according to their own preference? The fastest way to group students is to divide the class based on proximity. You might also want to randomly assign students to groups by counting off and grouping them according to number. Another idea is to let students get a piece of chocolate from a basket of different chocolates and group students according to the flavor they chose. You may also strategically assign them to groups instead of randomly assigning them. Prepare a list with names vis-a-vis his/her prevailing attitude toward the subject. Divide the students accordingly based on this list. Make sure that each group has a good mixture of personalities. Other possible factors that you should consider include gender, race, ethnicity, and behavior. Give a teambuilding task before assigning the actual task. Give a preliminary task that will help each student establish a good rapport with his/her group. These primer activities should be designed in such a way that positive relationships will be built and mutual respect between and among members will be established. You may prepare a simple activity like asking each member to answer questions about his/her favorite foods, books, places, or hobbies. Students will be given the opportunity to find connections—things they have in common with one another. (Note: Feel free to remove this part if the class is already bonded and cohesive.) Delegate a specific task to each member of the group. How do you get students to participate in the task? Come up with a task wherein different roles are assigned to group members so that they are all involved in the process. Each member should feel responsible for the success of their groupmates and realize that their individual success depends on the group's success. If a student feels that other people are relying on them, then he/she will be motivated to accomplish his/her part excellently. Have a contract signed by your participants. Establish how group members should interact with one another. Make them sign an agreement that explicitly states their expectations of one another. UNIT III * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES I 101

The contract should also include the behaviors that you want them to avoid and the values that you want them to observe and uphold. • Share your reason/s for doing collaborative activities. The reason for doing collaboration has to be clearly articulated to your students. Students must understand the benefits of collaborative learning. Explicitly connect these activities to larger class themes and learning outcomes whenever possible. • Give your instructions clearly. Giving instructions is not something that you take for granted. Giving a clear set of instructions contributes to the good performance of students in an activity. Failing to do so can lead to a huge waste of time. If the students do not understand the given task, then this will result to a lot of interruptions. As a facilitator of the activity, you should tell exactly what your students have to perform and describe what the final output of their group task will look like. • Go around and keep your ears open. As students accomplish their group task, go around and answer questions about the task. Make sure to keep your ears open. Listen to their collaborative dialogue.. Pay attention to the interesting points that will surface from the discussion. Talk about these interesting points during the subsequent closing/ processing of activity. Try not to interfere too much with the group's way of proceeding; give your participants the time to think about their own problems before getting involved. Consider leaving the venue for a few minutes. Your absence can increase students' willingness to share uncertainties and disagreements. If you find a group that is experiencing some sort of uncertainty or disagreement, refrain from giving the answers or resolving the disagreement. Allow your participants to feel some stretch/to experience struggle—within reason—to accomplish the task. • Provide closure to the group activities. Conclude the activity by having a session wherein students give a report. You can ask each group to give an oral report or submit a written report. The reporting should revolve around their insights. You may also ask them to reflect on how they performed in the group. This will also give you an idea of their

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perceptions about group work. Relate the points raised to your current lesson and the objectives of the activity.

Experience The following are sample collaborative activities that you might want your iearners to engage in.

Activity: Jigsaw As in Jigsaw puzzles, each student holds a vital piece of the puzzle. They need to assemble in order to complete the whole puzzle. In this collaborative activity, each member of the group will be given a task to perform. The goal of the group is to gather every member's input to come up with a final output. Topic: Graphical Presentation of Data Objective: To work cooperatively with other students to ensure the attainment of your group goals Time frame: 55 minutes Task: Divide the class into groups. Ask each group to think of a question they would like to survey their fellow classmates on. For instance, they might want to ask their classmates about their favorite pizza flavor. Allot enough time for students to walk around the room buzzing with each other to gather data. Instruct them to write down the responses of their classmates. Once they are done collecting data, ask them to represent their results graphically. Remind them to label their graphs. Take a photo of each student's graph, which you can later print out to create a class collage to display.

Assess Answer the following questions to verbalize your understanding of collaboration as a teaching strategy. 1. What are the possible drawbacks to collaborative activities? What can you do to address these issues?

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Drawback

Response

2. Consider the activity given in the Experience unit. How do you plan to divide the task equally among the members if this activity is to be accomplished by a group with five members?

Challenge The Snowball Technique is a way for students to teach each other important concepts and information. Students begin by working individually. Next, they collaborate with a partner. After that, partners form groups of four. Groups of four join together to form groups of eight. This snowballing effect continues until the entire class is working together as one large group. Identify a topic in primary mathematics in which this collaborative activity can be used. Write down a sequence of increasingly complex tasks that can be given in this activity.

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Harness Design your own collaborative activity. Explain the mechanics of this activity succinctly. This activity will be part of the learning portfolio which you will compile at the end of this module.

Summary Group activities can foster collaborated when thoughtfully designed and carefully facilitated. Group size and composition are some of the considerations in designing group activities. It also helps to explain to the students why doing the activity as a group is essential in learning the lesson where it is applied.

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O b je ctive Formulate purposeful questions that encourage students to participate in classroom discussions

Introduction In a mathematics class, effective questioning is essential. Students will get bored if his/her teacher merely states facts. An effective teacher does not just tell the definitions and theorems but rather asks meaningful questions that lead the learners to the correct ideas. Also, the teacher gets to identify students who are having a hard time with the lesson and those with more advanced skills through questioning. It is through questioning that a teacher gets to know the misunderstandings of the learners. If a teacher is knowledgeable about the misunderstandings of the learners, then the teacher will have the greatest understanding of his/her learners. It is, therefore, necessary that teachers deliberately frame questions that will keep the class discussion moving. The goal of this strategy is to keep the learners' voices at the forefront of every classroom session. The challenge for you now is to think of questions that you could ask that would get your students engaged.

Think Discussion vs. Lecture In the discussion-based strategy, the teacher's role is to engage the learners in a question-oriented dialogue. The teacher spends a significant amount of time to ask scaffolding questions to help students understand an idea deeply. The interaction in this method leans on both the teachers' and students' equal participation. This type of strategy is different from that of a lecture. In a lecture, the teacher is the chief source of information.

Art of Questioning Not all questions are created equal. Some questions can be answered by a simple yes or no. Some questions would require students to think more meaningfully. Asking 1 0 6 I TEACHING MATHEMATICS IN THE PRIMARY GRADES

the right questions will help you understand what your learners know, do not know, and need to know. Asking questions is an art. As with most arts, no specific formula will work in all situations all the time. This lesson will enumerate general ideas for your careful consideration when framing essential questions. • Avoid 'one-word-response' questions Refrain from asking questions which only require a yes or no answer. In general, questions that would require one-word answers do not provide much information to check your learners' thought processes. This type of questioning may not stretch the mental muscles of your learners. Questions are posed to help students articulate themselves, clarify concepts, challenge known assumptions, examine reasons, and make significant connections to mathematical concepts. • Foster a climate conducive to learning and questioning Make sure that your learners feel comfortable to express his/her ideas and/or ask questions at any time. Some students are reluctant to speak up because they are afraid of what the teacher or classmates might think if they give an incorrect response. Listen attentively to what your learners have to say. If your learners feel that you are listening to their ideas, then a good working relationship with them will develop. Do not focus on hearing "correct responses" but rather focus on listening to the message that the learners are trying to send across. Avoid directing a challenging question to a student if your goal is just to discipline him/her for not behaving well in class. Challenging questions are posed to stimulate critical thinking. Create a classroom environment where learners feel heard and recognized. • My Question, My Answer is a no-no! Do not answer your own questions. If you are not able to elicit responses from your students, try rephrasing your question. Do not rush learners to give responses instantly. Give students some time to ponder and hypothesize deeply about ideas. You might also give some leading questions to help them level up their conceptual understanding. If your student does not answer correctly, you should continue to listen and ask clarificatory questions. Thinking should be respected and valued even if the response contains many misconceptions.

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• Frame questions that are accessible to all learners Remind your students that the question is for all members of the class. Try not to label the degree of difficulty of a question. Avoid saying: "I expect my fast learners/challenged ones to answer this question." Give open-ended questions from time to time. The answers to open-ended questions vary from person to person. This type of questioning encourages students to communicate their thoughts since there are multiple answers to open-ended questions. Moreover, this allows all types of learners to contribute their ideas to the discussion. Get ready to hear surprising answers from your learners! • Learners should be active questioners, too! Demand your students to ask questions. Learners should practice directing questions not only to you but also to their co-learners. You should give other students the time to develop an answer to the question that their co-learners have posed. Keep in mind that in a discussion, you do not always provide a ready answer. You want you the voices of your students to be at the center of every classroom session! After hearing a response from one student, follow up by channeling it to another learner for feedback. This prompts students for further participation.

Experience The table provides examples of classroom scenarios along with possible questioning techniques. Situation

1. You want to help a student who got stuck on a problem

Questioning Technique

• What part/s of the problem is/are difficult to understand? • What are the given pieces of information? • Can you identify some strategies to help you understand the problem? • Would drawing a diagram help? • How would you describe the problem in your own words? • What happens if you try it with smaller numbers?

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2. You want your students to listen and respond to the answers of his/ her classmates

• What do you think about your classmate's answer? • How would you restate your classmate's reasoning? • Did anyone get the same result but with a different solution method? • Why is your classmate's assumption correct/incorrect? • Would you like to comment on any of the previous answers?

3. You want to lead your students to the right conjecture, definition, or generalization

• Does this formula always work? Why? • How do perimeter and area differ? • How does the radius of a circle relate to the diameter of the circle? • Will this solution method work if some conditions about the problem are changed? • Do you notice any patterns? What can conjectures can you give about this?

• What mathematical law/s support/s this statement? • How are fractions related to decimals? • Can you give examples and non-examples of integers?

Assess The following activity will practice your questioning skills. Supply the table with an appropriate questioning technique/s to address the indicated classroom scenario. .

...........

Scenario -■

Questioning Techniques

1. The teacher gave examples and non examples of polygons. When students were asked to give a definition, they were unresponsive. 2. Only the bright students are answering the questions. The challenged ones do not raise their hands.

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3. When a student was asked answer a question, he/she responded with an angry stare. 4. A student answered the question in a manner that is not comprehensible to other students.

Challenge The following questions will challenge your understanding of the teaching by asking strategy. 1. If the class is too big, it is difficult for everyone in the class to participate. What accommodations are you willing to do to encourage all your students to participate in classroom discussions?

2. What strategies would you try if you suspect that students who do not understand the lesson are hesitant to ask questions?

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3. What questions could be asked to your students that would encourage them to ask questions?

Harness Imagine that you are going to teach multiplication of whole numbers. Write a script containing all the scaffolding questions that you will ask to lead your students to the correct multiplication rules. This activity will be part of the learning portfolio which you will compile at the end of this module.

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Summary Teachers can effectively facilitate a meaningful discussion by asking the righ questions. Questioning is a beautiful art which scaffolds student learning. In thi strategy, it is important that you have already created a learning environment that i open to questioning.

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f UNIT IV MENT STRATEGIES FOR CS IN THE PRIMARY GRADES

This unit discusses one of the most important aspects of teaching - assessment. - :- is unit you will learn how to assess student's learning for formative and grading z-'ooses alike. You will also learn about the contemporary types of authentic assessments.

O bjectives • Demonstrate understanding and appreciation of assessments • Differentiate formative and summative assessments

Introduction A very important concern that demands urgent attention is the assessment, specifically the classroom assessment which should be within the K to 12 Basic Education framework and aligned with the learning standards of the enhanced curriculum. Due to the need, the Department of Education issued the DepEd Order No. 8, s. 2015, which is the Policy Guidelines on Classroom Assessment for the K to 12 Basic Education Curriculum. This lesson will help you understand assessment and how it is used in the classroom.

Think Assessment is defined as a process that is used to keep track of learners' progress in relation to learning standards including that of the development of 21st century skills which is part of the new K to 12 education framework. Thus, assessment should be aligned with curriculum standards and on the 21st century skills assessment framework. Every assessment you give must be aligned with the objectives of the lessons to which the assessment was made for. This way, you are sure that you are testing what you intended for the students to learn. The process of assessment is anchored to the framework of Zone of Proximal Development (ZPD) of Vygotsky. In the center of the process is the nature of the learner. Assessment shall recognize the diversity of learners inside the classroom which requires multiple ways of assessment measures of their varying abilities, skills, and potentials. The ZPD assessment framework put premium consideration on the recognition of learner's zone of proximal development at the heart of the assessment. A learner-centered assessment supports the learner's success in moving from guided to independent display of knowledge, understanding, and skills, as well as assimilation of these in future situations. The ZPD adheres the learning and teaching within a degree that is not difficult yet challenging for the learner. It facilitates the 1 1 4 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

ultimate objectives of the K to 12 program that each learner is to develop higher-order thinking and 21st century skills. From this view, there is unity between instruction and assessment. Instruction is assessment, and vice-versa. And assessment is not delimited to written examinations; it is part of the day-to-day lessons and classroom activities and transcends to real-life setting. The enhanced curriculum of the K to 12 basic education is standards-based. The assessment measures shall be anchored on the attainment of these standards and competencies. Assessment is aimed at helping learners perform well in relation to these learning standards. There is a recommended type, component, period, and approach of assessment for each learning standard lifted in the policy guidelines of the Department of Education. Standards-based Curriculum Curriculum Standards

Learning Competencies

Content Standards

Performance Standards

Type

Diagnostic

Summative

Formative

Approach

Assessment "for" Learning

Assessment "of" Learning

When

Before the Lesson

During the Lesson

After the Lesson

Written Work

Performance Task

Quarterly Assessment

Components

■>

Principles of Assessm ent 1. Assessment should be consistent with the curriculum standards. The teacher should make sure that the assessment measures the attainment of the learning objectives set at the beginning of the lesson or unit. 2. Formative assessment needs to scaffold the students in the summative assessment. The results of formative assessment are not graded but it is important to keep documents of these to study the patterns of learning demonstrated by learners to prepare them in taking the summative assessment.

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3. Assessment results must be used by teachers to help students learn better. The teacher must seek ways to use assessment to help the students want to learn and feel able to learn. 4. Assessment is not used to threaten or intimidate learners. The main purpose of assessment is to improve learning, not increase anxiety among learners.

Experience There are two fundamental types of assessments—the formative and the summative assessments. Formative Assessment can be viewed in two lenses. It is an "assessment for learning" on the lens of the teacher while an "assessment as learning" on the lens of the learner. Formative assessment can be given at any time, before, during, and after the lesson; it also not confined within the classroom because any interaction with learner is opportunity to assess the learner's abilities. The UNESCO Program on Teaching and Learning for a Sustainable Future defines formative assessment as an ongoing and closely related to the learning process. It is characteristically informal and intended to help students identify his or her strengths and weaknesses in order to learn from the assessment. Formative Assessment as-"assessment for /earn/ng'-provides teachers the evidence about what the learners know and can do. Teachers observe and guide the learners in their task through interaction and dialogue—in the ZPD framework, thus, gaining insights and evidence about the learners' strengths, weaknesses, progress, and needs. The results of these will help the teachers to design instructional activities and to make decisions so that it is suited to the learners' situations and needs. The evidence in the formative assessment shall be documented or recorded in order to track and monitor the learners' progress systematically. But the formative assessment results are not graded, hence not included in the computation for marking or ranking. Formative Assessment as-"assessment as learning"- provides the learners of the immediate information on how they perform on the learning process. The assessment provides information on which areas learners do well or which areas do they need help. This can be through feedback from anyone around them especially from teacher or any individual who is considered more knowledgeable. Formative assessment should also be a learning opportunity which enables the learners to take responsibility for their own learning.

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A formative assessment is effective when instruction is embedded in it to promote learning (McMillan, 2007). The steps begin in giving orientation about the learning goals (black arrow). The detailed steps after orientation of the learning goals are to determine the current status of learners or evidence of prior understanding, next is providing clear, specific and on-time feedback, next is instructional corrections/ adjustments based on the needs of learners, next is move the learners close to the goals/learning standards, next is evaluate the learners' progress, lastly, again to provide feedback of the learners' status. In a case where learner is highly self-regulated, the process begins with the orientation of learning goals, next is determine the status/prior understanding of learners, next it to provide feedback, next is instructional corrections/adjustment, next is evaluation of student progress, lastly, the processes end in the same step which is to provide feedback after evaluation of student progress. These processes omit steps 4 and 5 for learners who are self-regulated. The DepEd guidelines provide the assessment purposes before, during, and after the lesson. Examples are given which teachers may utilize but shall not be limited to: Parts of the Lesson

Purpose For the Learner 1. Know what s/he

Before Lesson

Lesson Proper

For the Teacher 1. Get information

knows about the topic/lesson 2. Understand the purpose of the lesson and how to 2. do well in the lesson 3. Identify ideas or concepts s/he misunderstands 3. 4. Identify barriers to learning 4.

1. Identify one's strengths and weaknesses 2. Identify barriers to learning 3. Identify factors that help him/her learn

Examples of Assessment Methods

about what the learner already knows and can do about the new lesson Share learning intentions and success criteria to the learners Determine misconceptions Identify what hinders learning

1. Provide immediate

1. Agree/disagree 2.

3. 4.

5. 6.

activities Games Interviews Inventories/checklists of skills (relevant to the topic in a learning area) KWL activities (what 1 know, what 1want to know, what 1learned) Open-ended questions Practice exercises

1. Multimedia

presentations feedback to learners 2. Identify what hinders 2. Observations 3. Other formative learning performance tasks 3. Identify what (simple activities that facilitates learning can be drawn from a 4. Identify learning gaps specific topic or lesson)

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After Lesson

4. Know what s/he 5. Track learner 4. knows and does not progress in know comparison to 5. formative assessment 5. Monitor his/her own 6. results prior to the progress lesson proper 6. To make decisions on whether to proceed with the next lesson, re-teach, or provide for corrective measures or reinforcements 1. Tell and recognize 1. Assess whether 1. whether s/he met learning objectives 2. learning objectives have been met for a 3. and success criteria specifies duration 4. 2. Seek support 2. Remediate and/ through or enrich with remediation, appreciate strategies 5. enrichment, or as needed other strategies 6. 3. Evaluate whether 7. learning intentions and success criteria have been met

Quizzes (recorded but not graded) Recitations Simulation activities

Checklists Discussion Games Performance tasks that emanate from the lesson objectives Practice exercises Short quizzes Written work

Summative Assessment is the assessment of learning. This assessment is always given at the end of a particular unit or toward the end of a period because it aims to measure what learners have acquired after the learning process as compared with the learning standards. The results will be used for decisions about future learning or job sustainability. For UNESCO, the judgments derived from this assessment is more beneficial for other than to the learner. The role of summative assessment is to measure if the learners have met the standards set in the curriculum guide. The teacher shall use a method that deliberately designed to measure how well the students learned and able to apply their learning in different contexts. The results of the summative assessment are recorded and reported on the learners' achievement. The results are part of the computed markings and to be reported to parents/guardians, principal/school head, teacher on the next grade level, and guidance teachers. For reiteration, the formative assessment should prepare the learners in taking the summative assessment. And teachers shall provide sufficient and appropriate instructional interventions to ensure that learners are ready to take the summative assessments.

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The summative assessment measures the different ways learners use and apply all are relevant knowledge, understanding, and skills. Learners synthesize the knowledge, jerstanding, and skills during summative assessment and the results will be used as rases for computing the grades. The summative assessment is in the form of unit test a- d quarterly test, it must be spaced properly over the quarter. It has three components, - a-nely, Written Work, Performance Test, and Quarterly Assessment. These components are the bases of computing the grade and different learning areas have unique ways to assess these components and set different percentages for each component.

Components |Written Work

Performance Tasks

Purpose

When to Give

□ Assess learners' understanding of concepts and application of skills in written form

At the end of the topic or unit

□ Prepare learners for quarterly assessment □ This can be individual or At the end of the lesson collaborative over a period of time about a particular topic/skill □ Provide opportunities for learners to demonstrate and integrate their knowledge, understanding, and skills about topic or concept learned to apply in real-life situations through performance

Several times within a quarter

□ Provide opportunities for learners to design and express their learning in diverse ways □ Encourage learners' inquiry, integration of knowledge, understanding, and skills in various contexts beyond the assessment period Quarterly Assessment Synthesize all the learning skills, Once, every end of the concepts, and values learned in a quarter quarter The DepEd guidelines provide a list of assessment tools per learning area. Shown below is for mathematics.

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Components Learning Areas

Math

Written Work (WW)

Performance Tasks (PT)

A. Unit/Chapter Tests A. Products B. Written output 1. Diagrams 1. Data recording and 2. Mathematical Investigatory analyses projects 2. Geometric and 3. Models/making models of statistical analyses geometric figures 3. Graphs, charts, or maps 4. Number representations 4. Problem sets B. Performance-based tasks 5. Surveys 1. Constructing graphs from survey conducted 2. Multimedia presentation 3. Outdoor math 4. Probability experiments 5. Problem-posing 6. Reasoning and proof through recitation 7. Using manipulatives to show math concepts/solve problem 8. Using measuring tools/ devices

Assess Answer the following questions to verbalize your understanding of assessment in mathematics. 1. What does the Zone of Proximal Development say about assessment?

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IL .

F in the Venn diagram below to compare and contrast formative and 5 jmmative assessments.

Challenge The following questions will test your critical thinking skills as it presents an issue n education that has been a cause of debates in recent years. 1. What are your thoughts about graded assessments? Are they necessary?

2. Research about journals and articles related to graded assessments. Do these literatures support your thoughts?

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Harness The following activity will require you to examine an actual assessment given to primary grade students. This activity will be part of the learning portfolio which you will compile at the end of this module. 1. Collect a summative assessment from a Grade 1,2, or 3 math teacher. Describe the assessment in terms of the types of items given.

2. What are the objectives of the lesson to which the assessment material was made for? Do the items in the assessment match the objectives? If not, what had gone wrong and how do you think can this be corrected?

Summary Assessment is an essential aspect of teaching as its results give feedback about students' learning as well as the effectiveness of teaching. Formative and summative assessments are equally important in achieving these.

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Objective Develop a traditional assessment tool

Introduction Assessment has specific purposes, namely, to monitor student's progress, :: gather data for instructional decisions, to evaluate student's achievement and c-erformance, and to evaluate the program. Assessment results can be used differently cy different stakeholders. For the students, it is feedback on their learning; for the :eacher, it is feedback on their teaching; for the curriculum designer, it is a feedback on the curriculum; and for the administrators, it is a feedback on the use of resources.

Think For so long, the most widely used measure to describe learner's achievement and performances is traditional assessment. Traditional assessment is formal and often standardized. In administering the traditional assessment, the learners are given the exact procedures of administering and scoring. It is also described as a single-occasion measure, unidimensional timed exercise which usually in multiple-choice or shortanswer form. The traditional assessment measures are the most widely used measure of student' learning and measure of success in educational goals, it is still considered relevant and acknowledged to be valid assessment measures. There are many critiques on the use of traditional assessment tools. Included is that the tools overemphasis upon narrowly focused skills/abilities and content, the mismatch between the standardized tests and student's experiences in the learning activities, as well as student's motivation to complete such tests. Some issues are relative and apparent vis-a-vis comparison with the authentic assessment. Traditional Assessment

One-shot test Indirect test Absence of feedback to learners Speed exams

Alternative/Authentic Assessment

Continuous, longitudinal assessment Direct test Feedback is part of the processes Untimed exams

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Traditional Assessment Decontextualized task Norm-reference score interpretation Standardized test

Alternative/Authentic Assessment Personalized task Criterion-reference score interpretation Classroom-based test

The comparison puts the traditional assessment in the bad light but according to some authors, though it has many critiques and considered not sufficient assessment measures of learning and success of any program, it is still relevant and necessary in the current situations where the call for alternative assessment is highly recommended. On a positive note, standardized test scores are used to compare a student's performance across different schools since standardized tests control intervening factors such as variety of tasks within a test (Benjamin, 2012). On the other hand, standardized test scores reflect a single measure of student's attribute, performance, or ability but fail to generalize other settings. At present, traditional assessment may have many critiques but still have advantages. To name some, the traditional assessment measures are more objective, valid, and reliable. This is especially true for standardized tests and other types of Multiple Choice tests (Law & Eckes, 1995) while these advantages of traditional assessment measures are the critiques to authentic assessment especially the reliability and subjectivity issues.

Principles of Traditional Assessm ent In deciding which assessment strategy to use, the teacher needs to consider the issues such as content, context, and audience or use of the results (Dikii, 2003). Having clearly defined the objectives, appropriate assessment tools need to be utilized. Depending on the nature of the instruction, a combination of assessment strategies might be useful and to ensure that the assessment tool is meaningful, useful, and honest. There are five main points to consider when designing an assessment tool. 1. The purpose of the assessment and whether the task fulfills that purpose. An essential starting point is to be aware of the reasons why you are assessing the students, and how to design an assessment that will fulfill your needs. To do this it is important to consider the decisions to make, to consider the information you need to gather to make those decisions, and what methods are the most effective for gathering that information. 124

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2. The validity and reliability of the assessment that you are constructing. To ensure that the information to get out of the assessment results is as honest as possible, it is crucial to make sure that the assessment is both valid and reliable. Valid when it tests a relevant skill or ability, while it is reliable when the test gives the same result if taken repeatedly. 3. The referencing of the assessment. To make the assessment meaningful, it is important to compare the candidates' abilities with a common measure. The other common measures for comparison are with other learners, comparison with objective criteria, or comparison with the learner's own performance in other areas. The careful consideration of the purposes of the assessment will help the most appropriate reference frame to become clear. 4. The construction quality of assessment items. For the assessment to become effective, the assessment items must be constructed to an appropriate quality. Judging the quality of items can be complicated but, as a starting point, consider the difficulty level of the items. A good assessment has a difficulty level of the average learners. Consider also how well the assessment differentiates the learners in order to maximize the information that can draw. 5. The grading of the assessment. The grades of the assessment results are very concise summaries of a student's abilities. They are generally designed for the purposes of the institution hence should be clear and easily understood by a layperson. The grading of the assessment is often related to the referencing of the assessment, the grading and the referencing should be considered in tandem.

Experience The following are the most widely used traditional assessment tools that can be used in class. #

•

1. True or False Test True or False items require students to make decisions and find out which of two potential responses is true. Because this measure is easy to score and easy to administer but guessing the answer has a 50% chance of success. Another consideration, when the test item is false it is hard to know whether UNIT IV • ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES j 1 2 5

the student really knows the correct response and why. One possible to improve the quality of True or False test is to ask the student to prov explanation for the incorrect items or rewrite the items correctly. 2. Multiple Choice Test According to Bailey (1998), this type of test is commonly utiliz teachers, schools, and assessment organizations for the following reasor a. Fast, easy, and economical to score. Machines can be used in scoring b. This measure can be scored objectively, thus give an impression of most fair and/or more reliable than other forms of tests. c. Compared with True or False test, the Multiple Choice test reduce chances of learners guessing the correct items. There are many critiques on the use of Multiple Choice test. The most con critique is the test items of multiple choice is effective only in testing the low of cognitive skills like recalling of previously memorized knowledge, while item: demand higher-order thinking skills such as analyzing and synthesizing are diffic produce (Simonson, 2000; Bailey, 1998). The other critiques on the use of mu choice test are the guessing may considerable but with unknown effect in the scores, the test severely restricts what can be tested, difficult to write successful it backwash may be harmful, and cheating is highly possible (Bailey, 1998). 3. Essay

An essay is an effective assessment tool because the answer is fie and measures higher-order learning skills—written communication organization of ideas. However, it is not a practical measure because difficult and time-consuming to score an essay output. Another issue is subjectivity in scoring, hence creating a rubric is necessary to evaluate output (Simonson et al., 2000). The rubric is a "criteria-rating scale" w gives the teachers a tool that allows them to track student performance, teacher has an option to create, adapt, or adopt rubrics depending on 1 instructional needs. 4. Short-answerTest

In a short-answer test, the items are written either as a direct ques requiring the learner fill in a word, phrase, or statement in which a space been left blank for a brief written answer. The question needs to be pre<

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otherwise, the items that are open to interpretations allow learners to fill in the blanks with any possible information.

A sse ss Answer the following questions to verbalize your understanding of assessment. 1. What are the advantages of using traditional assessment?

2. Which traditional assessment tool do you usually see used in mathematics classes? Why do you think is it popular in assessing mathematics learning?

Challenge The following questions will challenge your critical thinking skills as they raise issues in assessing mathematics learning. 1. After knowing the existing issues on the utilization, construction, and purpose of traditional assessments, do you think these types of tests helps to achieve the goals of mathematics education for life-long learning?

UNIT IV • ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 1 2 7

2. There are teachers saying that constructing test items requires technical competence that each other should have. What are your plans to equip yourself of this competence in constructing test items?

H arn ess The best learning in mathematics happens when the instruction and assessment is an opportunity to learn both the concept and skills (Silva, 2009). Choose a topic in mathematics Grades 1 to 3. Then write a three-item multiple choice test to measure both the concepts and skills in your chosen topic. This activity will be part of the learning portfolio which you will compile at the end of this module. Topic:_____________________________________________________ Grade level:________________ Multiple-choice test: 1.

________________________________

d. *

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Summary Traditional assessments like most paper-and-pen tests that we know are objective, valid, and reliable ways to assess learning. They are still relevant despite the presence of the contemporary authentic assessments.

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Objective Construct a performance task in mathematics

Introduction The criticisms of the traditional assessment measures and the new focus of learning standards of acquiring the complex and essential skills needed in today's society pushed the need to rethink the criteria and nature of the learning assessment. The proposal was to use open-ended problems, hands-on problems, computer simulations of real-world problems, and the use of portfolio in learners' work. These types of measures are called authentic assessment where learners are asked to perform real-world tasks, and the criteria are based on actual performance in the field of work (Wiggins, 1989; Archbald & Newman, 1988). Authentic assessment is also known as the "performance assessment" or "performance task" where the students must complete real-life activities e.g., preparing memo or policy recommendation, which involves reviewing and evaluating a series of documents. Performance assessment measures the demonstrated ability to interpret, analyze, and synthesize information (Silva, 2009).

Think Mathematics education aims to develop learners with critical and analytical thinking skills to solve real-life problems. Thus, mathematics classes must have tasks and activities the same with how the mathematicians use mathematics outside the classroom. How the students learn mathematics inside the classroom shall not be different on how they will use it outside the classroom.

Principles of Authentic Assessm ent 1. Authentic mathematics requires essential skills which can be measured by the ability to communicate and ask questions, to assimilate unfamiliar information, and to work cooperatively with the team—the mathematical skills for lifelong learning with the computer literacy. Part of mathematics literacy is the ability to learn and assimilate new information, hence there is a need for essential skills of 1 3 0 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

flexibility and adaptability. Related to communication is the ability of leaners to articulate what they understand and do not. Communication can be fostered in school if learners learn and use the language of mathematics, activities provide opportunities to make conjectures and reasons. Adaptability will be developed if learning provides multiple contexts that promote the value of mathematical interpretation in a variety of interrelated experiences. 2. In authentic assessment, the use of multiple types of measures is possible. For example, elementary levels may ask to find one data point, while intermediate levels the question could be to find the trends among multiple points in the data. And the most recommended question is to identify the multiple patterns or understanding the overall picture of the data to test the learners understanding of the deeper structure of the data. 3. Authentic assessment is built on the accuracy of the mathematical content and interdisciplinary integration. In geography, there are opportunities to use scaling, proportions, and ratio. In genetics, there are opportunities to apply statistics and probability. An interdisciplinary approach provides opportunities for different contexts. It promotes attitudes of inquiry, investigation, and sensitivity on the interrelatedness between contents and real world. 4. Authentic assessment measures the complete picture of learners' intellectual growth. It measures the various kinds of knowledge, measures either group or individual for different purposes. An authentic assessment is a combination of many measures. Small group situations may be useful to measure the ability to talk and listen, while individual assessment can be used to measure the ability to synthesize knowledge. 5. Authentic assessment uses the dynamic and adaptive form of feedback. This is also called scaffolding feedback where the learner can identify the skills to model and reflect and connect on their performances. Thus, assessment becomes learning opportunities and assessment aims to measure not only the actual performance but more important the potential. 6. Authentic assessment must take place in the context of the learning process. 7. It must consider both the learner and the situation in which the learner is assessed. 8. It must provide information on what the learner knows, what he/she does not know, and on the development of changes in such learning. UNIT iV • ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 131

9. Repeated measures of appropriate learning indicators must be made to obtain a clear picture of the learner's knowledge. 10. Indicators must include cognitive and conative abilities to capture different perspectives. 11. Authentic assessment will require instruments that provide in-depth perspectives on learning. The use of at least three different mediums in assessment to obtain the integrated picture of a learner. For example, the use of paper and pen test, video and computer used jointly to have an authentic understanding of the learner. Paper and pencil can use to measure the student's knowledge of facts, concepts, procedures, and text comprehension abilities. It can also be used to measure how well the students can critique the quality of other documents. Videos can assess communication, explanation, summarization, listening, argumentation, question asking, and answering skills. It can also assess how the student interacts during cooperative learning. Computers can be used to simulate realistic situations inside the classroom, can effectively track the process of learning and the learner's response to adaptive feedback. Computers can make possible the dynamic assessment of relevant criteria. 12. The purpose of assessment must be considered. If the assessment results will be used by the student or the teacher, then, the tool must be available in the classroom on a regular basis, which promotes the integration of instruction and assessment. This kind is called systemic approach of assessment, which is often used in the context of performance assessment.

Authentic Assessm ent Tools 1. Presentations, debate, exhibition, written reports, videotapes of performances, demonstrations, open-ended questions, computer simulation, hands-on execution of experiments, portfolios, and projects 2. In-depth evaluation in the context of problem-solving. It involves individual and cooperative problem-solving activities. Teachers must have scoring template to facilitate their task of assessing the learning. This project provides an example of how to examine both the individual and cooperative group problem solving activities, provides insights on how students form their hypotheses by comparing theirs with other hypotheses, and how to generalize concepts from one problem situation to another.

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3. Use of open-ended questions will provide opportunities for learners to think for themselves and express their ideas. Communication is fostered as well as writing tasks. This is also an opportunity to measure learner's misconceptions and reasoning abilities. 4. De Lange (1987) designed mathematical problem situations composed of multiple items with varying levels of difficulty. There are five tasks: a timed written task, two-stage tasks, a take-home examination, an essay task, and an oral task. This is a multifaceted evaluation of a learner. Stage one includes open-ended and essay questions. These items are scored and returned to the students. In stage two, students are provided with their scores in stage one, they are also allowed to take again the stage one tests at home as long as they accomplish them within the agreed time. The final assessment includes the scores of the stage one and stage two tests. Students can learn from their mistakes in the previous stages and from feedback regarding their mistakes. The assessment process becomes interactive and helps assists the students in reaching their potential. 5. Portfolio assessment is also a recommended form of authentic or performancebased assessment. However, there is a caution in creating guidelines on how to score the portfolios because of the existence of multiple audiences. 6. Projects is an example of authentic assessment (Simonson et al, 2000). This can be made individually or as a group. The project can possess authenticity, real-life related concepts, and prior experience of the learners. Any type of method that displays what student know about a certain topic can be used, (i.e., development of plans, research proposals, multimedia presentations is considered a project). Problem-based learning requires learners to use their problem-solving skills to respond to a given situation. For example, a scenario can be presented and the learners either as a group or individual will be asked to provide strategies or solutions. The learners may provide their findings in various forms like multimedia presentations, role-play, or written report. According to Elliot (1995), to increase the effectiveness of performance or authentic assessment, teachers must pay attention to the following details: •

Select assessment tasks that are clearly aligned or connected to what has been taught.

•

Involve the learner in the formulation of scoring criteria for the assessment task and share the final criteria prior to working on the task.

UNIT IV « ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 1 3 3

9. Rfg <o,

id explain, if necessary, the clear statements of learning standards er models of acceptable/ best performances prior to engagement

^nttasks.

^■din the importance of completing the self-assessment tool in improving their performances. • Provide examples of interpreting the student's performances by comparing it to learning standards that are developmental^ appropriate or compare it to other student's performances.

Experience There are many forms of authentic assessment to ensure collaborative effort, interaction, and active participation of learners. The Department of Education recommends the use of GRASPS framework in giving and assessing performance tasks. GRASPS stands for: Goal

The statement of problem or challenge to be solved

Role

Explains the scenario, role of students, and what are they being asked to do

Audience

Who they need to convince; who they present the output/outcome or propose solution to the problems; the audience is not limited to the teacher

Situation

Provides the context of the situation and any additional factors affecting the situation

Product, Performance, Purpose

Explains the product or performance that needs to be created, its purpose

Standards and Criteria for Success

Explains the learning standards that must be met and each criterion of evaluating the product/ performance and how the output will be judged by the audience

Following is an example of a performance task for Grade 3 Statistics and Probability. 134 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

1. Statistics and Probability Learning Standards for Grade 3 Content Standard Performance Standards The learner demonstrates The learner is able to understanding of bar graphs create and interpret simple and outcomes of an event representations of data using the terms sure, likely, (tables and single bar graphs) equally likely, unlikely, and and describe outcomes of impossible to happen. familiar events using the terms sure, likely, equally likely, unlikely, and impossible to happen.

Learning Competencies Collects data on one variable using existing records Sorts, classifies, and organizes data in tabular form and presents this into a vertical or horizontal bar graph Infers and interprets data presented in different kinds of bar graphs (vertical/horizontal) Solves routine and non-routine problems using data presented in a single-bar graph

2. GRASP Goal

The parents-teachers council of an elementary school agreed to have an educational field trip for the students per grade level. In Grade 3, there are data needed before making decisions: •

How many students in the whole Grade 3 population will be joining the field trip?

• Given the proposed educational places, choose only three that they will visit for a one-day educational field trip. •

Propose a time, sequence in visiting the three places (based on the result of the previous data).

Role

Your section is in charge of collecting data that will support the plans, decisions, and proposal. The whole section will be divided into three groups. Each group will collect data and answer one unique question from the problem statement. You need to collect, present, and explain the data.

Audience

Each group will present the outputs of the activity to the class with the principal, other Grade 3 teachers and some parent officers who are involved in the planning.

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Situation

Prior to gathering the data, you have to prepare a question that aims to seek the answer to your respective question. The prepared question has to be submitted for approval of your section adviser. Two weeks will be allotted to do all the preparations, data gathering of all Grade 3 students, and to make presentations of the data. You may consult your section adviser or any one of your teachers within the conduct of the activities (scaffolding or adaptive feedback). After two weeks, the finding and recommendations will be presented to the audience.

Product, Performance, Purpose

The outcomes of the activities are the data necessary to answer the posed questions and recommendations for the educational field trip of Grade 3 students.

Standards and Criteria for Success

• The learning standards involve in this activity is presented in a separate table below. These learning standards will also be presented to the whole class prior at the outset. • The criteria to evaluate the students' output will be clarity of the questions to be asked among the Grade 3 students, creativity and accuracy of the data presentation, accuracy of the data interpretation, and feasibility of the answer or proposal. The rubrics to be used will be on the next table.

3. Rubrics on the evaluation of the GRASPS output Criteria

Excellent (10 points)

Clarity of activity questions

Only one or two statements need improvement

Creativity and accuracy of the data presentation

The single bar graph has no numerical error; uses other labels or colors to emphasize the data per bar graph

Developing to Excellent (7 points) Three to four statements need improvement

Beginner (5 points) Five or more statements need improvement

With three or more With one to two errors in the bar errors in the single bar graph; uses labels graph or colors but not quite successful

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Excellent (10 points)

Criteria

Developing to Excellent (7 points) With one to two mistakes

Accuracy of the data interpretation

No mistake in the interpretation of bar graph

Feasibility of the answer/proposal

The proposal is 100% The proposal feasible and based on has some minor the presented data recommendations

Beginner (5 points) With three or more mistakes

The proposal is not feasible

Assess Answer the following to verbalize your understanding of authentic assessment. How do performance tasks assess the achievement of 21st century goals?

C h a lle n g e The following questions will challenge your critical thinking skills about the use of authentic assessment. 1. What elements of the authentic assessment measure makes it better than traditional assessment? Why?

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* ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES | 137

2. What are the possible challenges in the use of any form of authentic assessment? Why?

HARNESS The following activity will guide you in creating an authentic assessment and a rubric in the GRASPS framework to grade it. This activity will be part of the learning portfolio which you will compile at the end of this module. 1. Choose a topic in Grades 1-3 mathematics. You will develop a performance task in this topic. Grade level: Content Standards

Performance Standards

1 3 8 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

Learning Competencies

2. Develop a performance task to assess student learning in your chosen topic. Keep the principles of authentic assessment in mind. Goal

Role

Audience

Situation

Product, Performance, Purpose

Standards and Criteria for Success

UNIT IV • ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES I 1 3 9

Write a rubric to evaluate the GRASP output Criteria

Developing to

Excellent ( _points)

Excellent ( _ points)

Beginner ( _points)

■■■ g

■ ■ §g|1 ■ Summary Authentic assessment addresses the concerns that traditional assessments cannot answer. In the Philippines, authentic assessment is encouraged through the use of the GRASPS framework. 1 4 0 | TEACHING MATHEMATICS IN THE PRIMARY GRADES

Lesson 20 « ”V ' $

ing Learning Portfolios ■ ■

Objective Create a learning portfolio in mathematics

Introduction Portfolio assessment is a detailed, unique, and personalized evaluation of what the learners know and can do. A portfolio is a collection of pieces of evidence of efforts, learnings, development, growth, and achievement. It emphasizes a learner's milestone on his/her development of concepts and skills. It contains not only outputs and works-in-progress but also reflections on the learner's strengths and progress toward the learning goals.

T h in k Portfolio assessment is an example of authentic and non-traditional assessment of learning. The use of portfolio assessment is an answer to the need for continuous assessment in the course of day-to-day instruction that traditional assessment like standardized testing cannot address. The portfolio assessment can measure a variety of skills that is not measurable by single testing of traditional assessment. The portfolio can be in written, oral, and graphics outputs set and developed by learners themselves. These outputs have some degree of quality that cannot be measured by traditional test. A portfolio develops awareness of one's own learning. Knowing the criteria of the content and assessment, learner can always refer to these in each stage to verify the progress in achieving the set of objectives and goals. Further, it also aims to develop independent and active learning. Portfolio assessment can address the heterogeneous groupings of learners since part of the objectives is to exhibit the unique and personal effort, development, and growth of each learner. This flexibility is also a way to provide opportunities to demonstrate their abilities in a personal preferential manner. Implicitly, engaging in learning portfolio promotes social interaction between learner to teacher and learner to other learners. Additional interaction is between UNIT IV • ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE PRIMARY GRADES I 141

• ^nxess Portfolio demonstrates a ^cets or phases of the learning process, hence the arrangement is based on the learner's stages of metacognitive processing. s portfolio contains reflective journals, think logs, and other related evidence. • Showcase Portfolio is the kind that shows only the best of the students' outputs and products. • Evaluation Portfolio includes some work that had not previously been submitted. • Class Portfolio contains a student grade and evaluative assessment of the student by the teacher. • Ideal Portfolio contains all the work a student has completed. In deciding the type of portfolio, the teacher needs to consider the level of the :c..rse, the age of the student, and the portfolio will be used and evaluated.

Essential Characteristics of Portfolio Assessm ent 1. Portfolio is an assessment that is done together by the learners and the teacher. The teacher guides the learner from planning, execution, and evaluation of contents of learning portfolios, hence the interaction and discourseare important elements of the process. They together formulate the objectives, which is based on the learning standards. The teacher shall assist the learner in choosing the contents or items to be included in the portfolio based on the objectives, but the learner has the final say on the selection because the portfolio is supposed to be representing the unique and personal preferences of the learner. 2. The portfolio should be an opportunity to exhibit the samples of work or outputs which shows the growth, development, and achievement over time. In this purpose, the learner shall reflect or do self-assessment of his/her own work to identify the strength and weakness, so that the weaknesses become improvement goals. 3. The criteria for selecting and assessing the portfolio especially the contents must be clear both to the teachers and students at the outset of the process. The set of agreed criteria can be referred by the learner in each step of the process to avoid inclusion of unessential components and to avoid resorting to what is only available at the time. So that necessary planning in each step of the process of portfolio development can be done especially by the learner.

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Experience The table below presents the basic elements of learning portfolio which can be given to students as a guide in creating a portfolio of their own. The Basic Elements of a Learning Portfolio Cover Letter

This part contains the "About the Author," brief context of what the portfolio shows about the progress of development of skills, abilities, and learnings. It summarizes the evidence of progress and learnings.

Table of Contents with numbered pages Entries (both the Cores and Optional • The Core items are the mandatory items based on the assessment criteria. Items) • Cores are the items that need to be • The Optional items will be included to show included the uniqueness of the learner's outputs, it may be the "best pieces of work" or "the • Optional items are those based on trouble" or the "less successful" with learner's choice respective explanation, but ultimately shall be based on the objectives set during the initial stage. Dates on all entries

This is to document the proof of development or growth over time.

Drafts of initial oral and written outputs The first drafts ad the corrected versions—to highlight the changes, identify improvement, and the revised versions and explain the context Reflections

The reflections can be at different stages in the learning process, subjective because it depends on the learners' unique experiences.

For each item in the core and optional Provide a rationale for why the item is was included as evidence of growth, development, entries and learning. What did the learner learn from it? What went well for the learner? What needs to improve? What were the problem areas? What is the feeling of the learner on his/her performance? 144

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Assess Verbalize your understanding of the portfolio assessment.

1. What are the essential characteristics of a learning portfolio?

2. How does each part of a portfolio contribute to the assessment of student learning?

Challenge The following questions will challenge your critical thinking skills as they raise issues about the use of portfolio is assessing learning. 1. Does the portfolio assessment contribute to a better appreciation of mathematics? How about its contribution to mathematics achievement? Share your thoughts below.

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2. As a mathematics teacher, describe below the ideal portfolio for you. What are the characteristics of a portfolio that you wish your students to submit?

HARNESS The following activity will guide you in creating a rubric for checking a learning portfolio. This activity will be part of the learning portfolio, which you will compile at the end of this module. Based on your answer in Challenge #2, create a rubric for checking a learning portfolio in mathematics.

Criteria

Excellent ( _points)

■ r

Developing to Excellent ( _points)

Beginner ( _points)

■ Jr |

11 1 1

1 4 6 I TEACHING MATHEMATICS IN THE PRIMARY GRADES

Excellent

Criteria

( _points)

Developing to Excellent ( _points)

Beginner ( _points)

Su m m a ry The learning portfolio is one of the authentic assessments that DepEd recommends. A portfolio is a collection of pieces of evidence of efforts, learnings, development, growth, and achievements.

Final A ctiv ity Compile all your Harness outputs into a learning portfolio. Remember to put a date on each output. Your portfolio must also include the following: • A short reflection about each activity—how you felt about it, what challenges you faced in doing it, what you like most about doing it? What did you learn from the activity? • A final reflection about how you feel now that you have completed this module. What are the changes in your belief about teaching and learning math?

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Vthbald, D. A., & Newman, F. M. (1988). Beyond standardized testing: Assessing authentic academic achievement in the secondary school. Reston, VA: National Association of Secondary School Principals. Bailey, K. M. (1998). Learning about language assessment: dilemmas, decisionjs, and directions. US: Heinle & Heinle. Benjamin, R. (2012). The seven red herrings about standardized assessments in higher education. National Institute for Learning Outcomes Assessment. Occasional paper #15. National Institute for Learning Outcomes Assessment. Retrieved September 18, 2012 from http://learningoutcomesassessment.org/documents/ HerringPaperFINAL.pdf Brooks. J.G. and Brooks, M.G. (1993) In Search of Understanding: the Case for Constructivist Classrooms. Alexandria, VA: American Society for Curriculum Development. De Lange, J. (1987). Mathematics, insight and meaning. Utrecht: Freudenthal Institute. Department of Education (2016). K to 12 Curriculum Guide Mathematics. Pasig City, Philippines. Dikii, S. (2003). Assessment at a distance: traditional vs. alternative assessment. The Turkish Online Journal of Educational Technology, 2(3), 13-19. Elliott, S. N. (1995). Creating meaningful performance assessments. ERIC Digest E531. EDRS no: ED381985. Institute of Design at Stanford (2016). An Introduction to Design Thinking Process Guide. Retrieved from http://dschool.stanford.edu Krathwohl, D.R., Bloom,B.S. and Masia, B. B. (1964). Taxonomy of educational objectives, Book II. Affective domain. New York, NY. David McKay Company, Inc. Orton, A. and Frobisher, L.J. (1996). Insights into teaching mathematics. London: Cassell. Ryan, R. M., & Deci, E. L. (2004). An overview of self-determination theory: An organismic-dialectical perspective. In E. L., Deci & R. M. Ryan (Eds.), Handbook of Self-Determination Research (pp. 3-36). Rochester, NY: University of Rochester Press. Silva, E. (2009). Measuring 21st Century Skills. Phi Delta Kappan International, 90(9), 630-634.

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Articles inside

Lesson 13: Use of Manipulatives

Lesson 15: Collaboration

Lesson 11: Design Thinking

Lesson 12: Game-based Learning

Lesson 6: Instructional Planning Models

Lesson 4: The Teaching Cycle

Lesson 3: Constructivist Theory in Teaching Mathematics in the Primary Grades

Lesson 2: Mathematics Curriculum in the Primary Grades

Lesson 10: Mathematical Investigation

Lesson 8: Inductive Learning

Lesson 5: Things to Consider in Planning Instruction in Mathematics in the Primary Grades

Lesson 9: Concept Attainment