Teaching math in the intermediate grades by ctu_sanfran

H o t 'k ^14’1'i-

A Course Module for

Teaching Math in the Intermediate Grades Riza C. G u sa n o M ark Zoel 3. M asan g kay La d y A n g e la M. R o ce n a M elanie C. U nida Authors G reg Tab ios P aw ile n Coordinator

TEACH Series

OUTCOMESBASED

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Jnit I: THE MATHEMATICS CURRICULUM IN THE INTERMEDIATE GRADES Lesson 1: Mathematics in the Intermediate Grades

Lesson 2: Mathematics Curriculum in the Intermediate Grades

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

Unit II: INST rI j C T IC ^ Lesson 4: The Teaching Cycle

Lesson 5: Things to Consider in Planning Instruction in Mathematics in the Intermediate Grades Lesson 6: Instructional Planning Model

23 29

Unit III: INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES Lesson 7: Problem-Solving

Lesson 8: Inductive Learning

Lesson 9: Concept Attainment

Lesson 10: Mathematical Investigation

Lesson 11: Design Thinking

Lesson 12: Game-based Learning

Lesson 13: Use of Manipulatives

Lesson 14: Values Integration

Lesson 15: Collaboration

Lesson 16: Teaching by Asking

104

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Unit IV: ASSESSM EN T STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES Lesson 17: Assessing Learning Lesson 18: Traditional Assessment Lesson 19: Authentic Assessment Lesson 20: Designing Learning Portfolios References Index

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 selfreflection. •

Harness

Requires the readers to engage in an activity that harnesses creativity.

O b je c tiv e To understand the purpose of learning mathematics in the intermediate grades

Introduction Mathematics as a subject has a unique nature that demands a special and distinct approach to make learning interesting, challenging, and fun for the learners. This unique nature of mathematics must be learned and understood by mathematics teachers.

Think In the Philippines, mathematics in the intermediate levels includes five content areas: Numbers and Number Sense, Geometry, Patterns and Algebra, Measurement, and Statistics and Probability. The contents and topics are sequentially arranged with each topic being a prerequisite. It is therefore imperative to understand each topic and acquire the skills for every topic to avoid gaps and future difficulties. For example, in the elementary levels, the skills in the operation on whole numbers must be learned first before the operation on decimals and fractions. Knowing that the five content areas are just part of the whole discipline, the questions now are: what is the purpose of learning these standards in the intermediate levels? What is the purpose of learning whole number up to 10, 000, 000? What is the purpose of learning to measure the area, perimeter, circumference, surface area, and volume of two- to three-dimensional objects? What is the purpose of learning to collect and present data in tables, bars, and pie graphs? These learning standards in the mathematics curriculum, in intermediate levels in particular, are part of the whole mathematics education program because it has roles in achieving the goals of mathematics education—to acquire the skills needed to be analytic, critical, and a problem solver in real life. Moreover, they are necessary prerequisites to higher level of mathematics. For instance, learning the linear equations in algebra is more than representing mathematical problems symbolically and finding the value of an unknown 2 I

TEACHING MATH IN THE INTERMEDIATE GRADES

.ariable, it is finding patterns and predicting certain behaviors or phenomena, then to 'eatizing that a certain cause will lead to a specific result. To relate the graph of equations :o business supply and demand, then Mathematics lessons in the intermediate grades should be leading to this kind of realization for the learners. Learning mathematics is more than getting good grades. It must be applied oeyond the walls of the classroom. The main goal of mathematics education is to develop lifelong skills so that the students will be ready to interact with the real world. Therefore, it is a challenge for the mathematics teacher to make the mathematics Sesson as real as real-life situations and for the learners to acquire the skills such as critical thinking, analytical thinking, and problem-solving.

Experience "I am not good at math." "I fear attending my math class." "There is an upcoming math test, I am stressed out!" The above are few statements given by students who experience math anxiety. Math anxiety is fear, tension, or stress associated with mathematics usually due to repetitive failures. The development of mathematics skills begins in the primary and intermediate levels, so when repeated failures and disappointments happen in these levels, the mathematics anxiety begins to manifest at the intermediate grades. If not addressed, it will have a definite influence on their future performances, future choices, and decisions in mathematics. By Grade 7, when they enter junior high school, the learners have already a fix, solid mental models of mathematics learning. With their experiences in the elementary levels, the learners by Grade 7 are vocal in saying: "Mathematics is difficult." It is therefore important that the students' mind-set toward mathematics be addressed in the elementary levels.

Unit 1 * THE MATHEMATICS CURRICULUM IN THE INTERMEDIATE GRADES | 3

Assess Answer the following question to verbalize your understanding of teaching mathematics in the intermediate grades. Why is it important to learn mathematics in the intermediate grades? Cite some experiences to support your answer.

Challenge The following questions will practice your reflective-thinking skills. As you will learn later, it is important for teachers to develop these skills as they evaluate their lessons. Have you experienced mathematics anxiety? If not, do you know someone who did? Describe your experience below. Focus on how you viewed math, math class, and your math teacher during the times when you had mathematics anxiety.

4 | TEACHING MATH !N THE INTERMEDIATE GRADES

Harness The following activity will require you to interact with students in the intermediate evels. This experience will give you a broader understanding of the learners in this evel and will also enhance your communication skills with them. This activity will be oart of the learning portfolio that you will compile at the end of this module. 1. Survey at least five students in Grades 4, 5, or 6. Ask them the following questions: Are you afraid of math? Why or why not? Record their responses in the table below. Are you afraid of Math? — lllnfRl!

Why/Why not?

Student 1

Student 2

Student 3

Student 4

Student 5

Unit I • THE MATHEMATICS CURRICULUM IN THE INTERMEDIATE GRADES |

2. Based on the students' responses in #1, suggest a classroom setup (including classroom rules) that will help reduce math anxiety among the students.

Sum m ary Learning math in the intermediate grades is important because it provides the necessary prerequisites to learning a higher-level of mathematics. Many students develop math anxiety in these levels, so it is crucial that teachers present math in a way that does not elicit fear.

6 | TEACHING MATH iN THE INTERMEDIATE GRADES

O b jectiv e To understand the features of the Philippine mathematics curriculum and the learning standards for Grades 4 to 6

Introduction The mathematics curriculum framework of the Philippines put critical thinking and problem-solving skills as the goals of learning and teaching mathematics. The following lesson will give you a deeper understanding of this curriculum that is currently implemented in the country.

Think The figure below presents the framework of the mathematics curriculum in the Philippines.

K to 12 BASIC EDUCATION CURRICULUM Unit I • THE MATHEMATICS CURRICULUM IN THE INTERMEDIATE GRADES |

Critical thinking and problem-solving are the goals across the levels in each topic of the 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, and making connections) are important that prior learning/attaining prerequisite skills is always considered. Moreover, mathematics in the context of real-life situations is always the main consideration in designing mathematics activities. Mathematics education in the Philippines contains five general contents: Numbers and Number Sense, Measurement, Geometry, Patterns and Algebra, and Statistics and Probability. These general contents are the same across levels, from Kinder to Grade 10. The key stage standards for the intermediate grades are shown below. KEY STAGE STANDARDS

At the end of Grade 6, the learner demonstrates understanding and appreciation of key concepts and skills involving numbers and number sense (whole numbers, number theory, fractions, decimals, ratio and proportion, percent, and integers); measurement (time, speed, perimeter, circumference and area of plane figures, volume and surface area of solid/space figures, temperature and meter reading); geometry (parallel and perpendicular lines, angles, triangles, quadrilaterals, polygons, circles, and solid figures); patterns and algebra (continuous and repeating patterns, number sentences, sequences, and simple equations); statistics and probability (bar graphs, line graphs and pie graphs, simple experiment, and experimental probability) as applied—using appropriate technology— in critical thinking, problem solving, reasoning, communicating, making connections, representations, and decisions in real life. For better understanding, let us look at the standards per grade of the intermediate levels.

GRADE 4

8 I

The learner demonstrates understanding and appreciation of key concepts and skills involving numbers and number sense (whole numbers up to 100 000, multiplication and division of whole numbers, order of operations, factors and multiples, addition and subtraction of fractions, and basic concepts of decimals including money); geometry (lines, angles, triangles, and quadrilaterals); patterns and algebra (continuous and repeating patterns and nu^r^ sentences); measurement (time, perimeter, area, and volume); a statistics and probability (tables, bar graphs, and simple experimer as applied—using appropriate technology—in critical thinking, problem solving, reasoning, communicating, making connections, representations, and decisions in real life.

TEACHING MATH IN THE INTERMEDIATE GRADES

GRADE 5

The learner demonstrates understanding and appreciation of key concepts and skills involving numbers and number sense (whole numbers up to 10 000 000, order of operations, factors and multiples, fractions and decimals including money, ratio and proportion, percent); geometry (polygons, circles, solid figures); patterns and algebra (sequence and number sentences); measurement (time, circumference, area, volume, and temperature); and statistics and probability (tables, line graphs and experimental probability) as applied—using appropriate technology—in critical thinking, problem solving, reasoning, communicating, making connections, representations, and decision in real life.

GRADE 6

The learner demonstrates understanding and appreciation of key concepts and skills involving numbers and number sense (divisibility, order of operations, fractions and decimals including money, ratio and proportion, percent, integers); geometry (plane and solid figures); patterns and algebra (sequence, expression, and equation); measurement (rate, speed, area, surface area, volume, and meter reading); and statistics and probability (tables, pie graphs, and experimental and theoretical probability) as applied—using appropriate technology—in critical thinking, problem solving, reasoning, communicating, making corrections, representations, and decisions in real life.

Notice that there is a spiraling progression design in the curriculum standards. Spiral progression ensures seamless integration of content standards. Each content and topic is a piece of the overall curricular landscape. Hence, learning each mathematics content is fundamental because each is related to the previous content and a prerequisite to the next higher one. Moreover, a misconception of concept and skills means a gap or discord in the whole mathematics curriculum.

Experience Study the K—12 mathematics curriculum. What key components do you notice? The mathematics curriculum is not simply a list of competencies. It is logically arranged and organized. For the teachers' reference, the content standards, the performance standards, and the learning competencies are explicitly stated. See the following example:

Unit 1 • THE MATHEMATICS CURRICULUM IN THE INTERMEDIATE GRADES I 9

Grade 6 - FIRST QUARTERS Numbers and Number Sense

demonstrates understanding of the four fundamental operations involving fractions and decimals.

is able to apply the four fundamental operations involving fractions and decimals in mathematical problems and reallife situations.

adds and subtracts simple fractions and mixed numbers without or with regrouping,

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.

Assess Many 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 because they serve as the skeleton of the mathematics curriculum. In your own understanding, explain the difference among content standards, performance standards, and learning competencies. What is the importance of each?

Challenge The following question will challenge your research and reasoning skills. It was discussed that the Philippine math curriculum is primarily concerned with critical thinking and problem-solving skills. Why do you think this is so? Research on the importance of these skills and synthesize your learning on the next page.

| TEACHING MATH IN THE INTERMEDIATE GRADES

teaching Intermediate

Demonstrate understanding and appreciation of the constructivist learning theory -Determine how the constructivist learning theory is applied in teaching mathematics : early grades

luction DepEd (2016) specifically noted constructivist theory as the backbone of the nculum. According to DepEd, knowledge is constructed when the learner is able rraw ideas from his/her own experiences and connect them to new ideas. In this jn, you will learn about the constructivist learning theory and how it is applied in sching mathematics in the intermediate 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 constructing knowledge from experiences rather than from adults telling them about the world. According to Piaget and others who practice constructivist education, the method that is the most likely to educate the students is the one in which they experience their world. Constructivism is appropriately applied in teaching mathematics since math is a cumulative and vertically structured discipline. One learns new math by building on the math that has been previously learned. Constructivist learning is described as follows: • Learning builds on the learner's prior knowledge and the approach is a constructive process. • Learner involves in the processes to ensure self-regulated and self-directed process.

Unit I • THE MATHEMATICS CURRICULUM IN THE INTERMEDIATE GRADES | 1 3

• Learning is grounded in the context of the learners and fundamentally social process. Interaction and communication are open and basic elements of learning process. • Learning is more than the acquisition of knowledge. It is collaborative, involves interaction and enculturation with community of practitioners. Collaboration with experts is basic. • The learning processes do not only require cognitive but also motivational and emotional domains.

Experience In a constructivist mathematics class, knowledge is constructed by the learners. To teach is not to explain, not to lecture, not to transfer mathematical knowledge; instead, teaching is to create situations that allow the learners to form the mental construction. The following are some recommendations on how to apply constructivism in teaching mathematics: • pose problems that is relevant to the learners; • use big concepts than segmented or disjoint topics. It invites the learners to participate irrespective of learning styles and dispositions; • create situations that will reveal the learner's point of view. The teacher must create opportunities for this to occur and must be willing to listen to the learner's reasoning and thinking processes; and • use authentic assessments, which includes interaction between the teacher and learner and learner and peer.

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

I TEACHING MATH IN THE INTERMEDIATE GRADES

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

Challenge There is no perfect theory. The following questions will challenge your critical thinking skills as they raise criticisms on the constructivist learning theory. 1. What do you think could be the possible challenges in using constructivism in teaching mathematics?

2. What other learning theories could be implemented in teaching math that could complement the down sides of constructivism?

Unit I • THE MATHEMATICS CURRICULUM IN THE INTERMEDIATE GRADES | 15

Harness The next activity will expose you to an actual mathematics class. You will do numerous classroom observations throughout this module. In this activity, 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 suggest ways to improve the learning activities. This activity will be part of the learning portfolio that you will compile at the end of this module. Observe a Grade 6 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.

Sum m ary The constructivist learning theory states that learning takes place when we build on what the students already know. Moreover, it is student-centered, allowing the students to take ownership of their own learning.

16 i TEACHING MATH IN THE INTERMEDIATE GRADES

UNIT II

INSTRUCTIONAL PLANNING In this unit, you will learn about how to plan, develop, and execute lessons in ■ ;:-'ematics for the primary grades. You will go over the learning cycle, the things to sider in lesson planning, and the different instructional planning models.

O b je c tiv e

Demonstrate an understanding and appreciation of the instructional p

Introduction

The work of a teacher does not start and end in teaching per se

process is not a linear activity that starts with planning and ends with t

it is a cycle of repeating stages until the students acquire an under; targeted concepts and skills. You may think of the teaching cycle as a through the same process over and over again, but each time with a objective and a better understanding of what it means to learn and teac

T h in k

There are many models of the teaching cycle that various e

developed through out the years. However, all models boil down 1

stages: (1) identify objectives, (2) plan instruction, (3) implement plai

understanding, (5) reflect on teaching, and (6) assess learning and ref The cycle that involves these stages is illustrated below. Identify objective

Assess learning and reflect on the results

Plan instructions

/ Reflect on teaching

Check for understanding

Im plem ent plan

The Teaching Cycle 18

1 TEACHING MATH IN THE INTERMEDIATE GRADES

L p , —e ngure. What do you observe? Do you now get the idea of the teaching H k i a cycle? The following describes each stage of the learning cycle. ■ r-T fy objectives ■■H=t knowledge and/or skills do the students need to learn? You must be lec oy the content standards, performance standards, and the learning iz-rencies that are found in the curriculum guide. Ha- instruction phhat strategies must be implemented for the students to achieve the e ~ .e s targeted in the previous stage? In planning instruction, it is important lyo u have mastered the content of the lesson that you are about to teach. It is p beneficial to be familiar with your students-what they know, how they learn, t 'o u will learn more about instructional planning in the next chapter. —plement plan I This is the stage where you conduct the learning activities that you have lecared during the planning stage. A word of advice: even though you have refully and delicately planned for the lesson, you must be flexible with the cisible changes that you need to accommodate. How will you know whether ■3ige is needed? Read on to the next stage.

L Check for understanding Teaching is about helping students learn. During the implementation of the

rsson plan, you must every now and then check whether the students have

rderstood what you have covered so far. Facial reactions and verbal cues help in

ssessing whether or not the students can move on to another concept or skill. If not,

ou might need to give a more elaborate explanation, more examples, or whatever

ou think is needed based on the students' reactions. This stage also makes use of

;rmative assessments that you will learn more about in Chapter 17. . Reflect on teaching You must evaluate every teaching period that you finished. Were the objectives chieved? Were the implemented strategies effective? How can instruction be

nproved? Your answers to the last two questions will give you insight on how

3 improve instruction the next time you teach the same lesson. However, if your nswer in the first question is no, i.e., the objectives were not met, then you eed to plan again. What do you need to do differently in order to achieve the bjectives? Unit il • INSTRUCTIONAL PLANNING I 1

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

Experience The following is a narrative of how a teacher might experience the t 1. Identify objectives Teacher Gina identified "multiplication of whole numbers up to the goal of her next lesson. 2. Plan instruction Teacher Gina thought it is best to apply a constructivist approac students learn techniques in multiplying whole numbers. She plar that incorporates the problem-solving strategy. 3. Implement plan The class went on smoothly. The activities that Teacher Gina pi successfully done by her students. 4. Check for understanding To make sure that her students understood the lesson, Teachei three-item exercise as an exit pass. 5. Reflect on teaching

Based on the exit pass, Teacher Gina found out that many of the s

difficulty multiplying numbers that involve the digit 8. So, she decidec

your-error activity the next day for the students to realize their miste planned to give a short drill on skip counting by 8.

I TEACHING MATH IN THE INTERMEDIATE GRADES

Lissess learning and reflect on the results "eacher Gina, later on, gave a multiplication quiz. Ninety percent of the students

peec She planned to give remedial exercises to those who failed. This teaching ice taught Teacher Gina that the students can discover concepts on their own. ch k «er, they must still be guided by a teacher because misconceptions may arise.

rs.ver the following questions to verbalize your understanding of the teaching cycle. . - which stage/s of the teaching cycle are the students involved? Explain.

I A/hich stage/s of the teaching cycle requires the teacher to reflect about teaching =nd learning? Explain.

ialleng e 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 j

Aside from classroom observations, many things can be learned from convers; with other teachers in the field. The following activity will require you to inte math teachers and summarize what you learn from them in a diagram. This ac will be part of the learning portfolio that you will compile at the end of this mod 1. Interview two mathematics teachers. Ask them about the stages of the tea cycle that they follow. Then, create a diagram illustrating their common ansv

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

Sum m ary Teaching involves a repetitive cycle of defining objectives, planning implementing instruction, assessing learning, and reflecting on teaching and lea Each part of the cycle provides a better understanding of what it means to teac' learn mathematics and so should result in better instruction in the next repetr :i the cycle. 2 2 I TEACHING MATH !N THE INTERMEDIATE GRADES

O b jective Demonstrate understanding and appreciation of the things to consider in planning -struction for mathematics in the intermediate grades

Introduction In education, planning refers to the designing and preparation of learning activities -'or the students. In lesson planning, teachers thoughtfully contemplate about the esson 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 ^ar the objectives were met. This lesson enumerates the things to consider in planning nstruction for mathematics in the intermediate grades.

Think There are five important elements in lesson planning that you need to consider— the 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 that 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 content 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!

Unit II • INSTRUCTIONAL PLANNING | 2 3

2. Objectives Before you begin planning, you need to know what specific kni skills you want your students to develop during the lesson or unit. Te focus too much on knowledge and forget about developing skills, whic term are more important than knowing mere facts. So, in planning you always consider both knowledge and skills. 3. Students Get to know your students—where they came from, what their i what they already know, their learning style, attention span, specia These will all help you determine your students' needs. Remember th to prepare your lessons with all your students in mind and that yoi should be to meet their needs and offer them enabling environme their preferred way. Knowing your students will also help you build r them, which is important if you want your students to be freely sharing with you and their classmates. Another important consideration that needs serious attention i especially mathematics, is the students' mind-set. You may havf considered—lesson mastery, focused objectives, and comprehensive un< of students—but still find that the lesson is not coming through the sti may be because the students have closed their doors toward math. \ children have come to believe that math is difficult and they can ne\at it. This is called a fixed mind-set. Students with a fixed mind-set t their math skills cannot be improved, which results in underperform subject. Reasons for a fixed mind-set include influence from adults ' math, previous unpleasant experience in math class, and others. Yoi teacher is to develop students with a growth mind-set. Students wit mind-set believe that they can be better at math. They know that tl are not wasted and that they can learn even in their failures. Many st proven that students who have a growth mind-set perform better in : those who have a fixed mind-set. So, in planning your lesson, you mu how to encourage a growth mind-set in class. 4. Learning environment

Aside from the physical environment where the learning takes | also important to consider the social and emotional learning environrr class. You need to make sure that you promote a positive environment 2 4 I TEACHING MATH IN THE INTERMEDIATE GRADES

students are motivated and are supportive of each other's growth. The students must feel safe to express their thinking without fear of being embarrassed because of mistakes or different views. Most importantly, you must create an atmosphere where the 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 is available in school? These are some of the questions that you can reflect on.

ixperience The next activity will delve into the experiences of math teachers and will give you isights on effective lesson planning. Interview three experienced intermediate grades (Grades 4, 5, or 6) mathematics teachers. Ask them 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.

Unit li • INSTRUCTIONAL PLANNING j

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 the content, objectives, students, learning environment, and availability of resources are the essential considerations in planning a lesson.

2. Sketch an infographic about the difference between growth mind-set and fixed mind-set.

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

| TEACHING MATH IN THE SNTERMEDIATE GRADES

2. Why do you think having a fixed mind-set is a setback in learning? Can you think of specific examples when you were a student and tended to have a fixed mind pattern?

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 :iscussed in the next chapter) while focusing on how the content, objectives, students, earning environment, and availability of resources were given attention to in the plan. ~his activity will be part of the learning portfolio that you will compile at the end of :nis module. J

Borrow a lesson plan from an intermediate grade mathematics teacher. Give specific examples from his/her lesson plan wherein you saw the conscious consideration for the content, objectives, students, learning environment, and availability of resources.

Unit IS * INSTRUCTIONAL PLANNING j 2 7

Content

Objectives

Students

Learning Environment

Availability of Resources

Sum m ary 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 | TEACHING MATH IN THE INTERMEDIATE GRADES

O b jective 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 models in the Philippines and their common features.

Thirtk 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 5 Es model. ADIDAS stands for Activity, Discussion, /nput, 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, talk about their experiences during the activity. Here, the questioning skill 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.

Unit li • INSTRUCTIONAL PLANNING |

Deepening. Here, the teacher asks questions that will engage the students to critical and creative thinking. Nonroutine mathematical problems or real-life word problems may be given. The purpose is to give the students the opportunity to deepen 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 5 Es. The 5 Es are Engage, Explore, Explain, Elaborate, and Evaluate. Engage. This part activates the students' prior knowledge and engages them with 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 the students seeing patterns that will help them describe the new concept in their own words. Elaborate. The Elaborate part of the lesson allows the students to expand their understanding of the concept by applying the concept that they have learned to solve mathematical problems. Evaluate. The last part of the 5 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.

3 0 | TEACHING MATH IN THE INTERMEDIATE GRADES

E x p e rie n c e Aside from the components of whatever instructional planning model, an istructional plan also reflects basic information about the lesson like prerequisite oowledge and skills, time allotment, materials needed, etc. Below is a sample Template of a lesson plan.

Topic:--------------------------------------------------------- i------Subject:_____________________ Grade Level: _ __________ __ Duration:__________________ Objectives At the end of the session, the student will be able to: 1_________________________________________________________________________ 2________________________________________________________________

3__________________________________________________________________________________ Prerequisite Concepts/Skills: • ______________________________________________________________________________________ _ _________ _ • ________________________________ _____

•

_________________________________________

__ ____________________________________________________________

New Concepts/Skills: 1_________________________________________________________________________________________________________________________ 2 ________________________________________________________________

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

Materials:

Unit H * INSTRUCTIONAL PLANNING j

References: • _____

Lesson Proper: Activity

Duration (Number of Minutes)

Teacher's Role

r ■ H P H W . L

Students' Role

7 7 7 ::

Assess 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 5 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

•

1 TEACHING MATH IN THE INTERMEDIATE GRADES

2. Explain the matching you did in #1.

Challenge Even though ADIDAS and 5 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 5 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 think you will have a hard time adjusting? Explain.

Unit il

• INSTRUCTIONAL PLANNING | 33

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 5 Es Model. This activity will be part of the learning portfolio that you will compile at the end of this module. 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

G MATH IN THE INTERMEDIATE GRADES

b. 5 Es

Engage

Explore

■

Explain

Elaborate

Evaluate 1

2. Are there components of the ADIDAS/5 Es Model that were not reflected in the lesson plan? If you are to fill in these missing parts, what would you write?

Unit II

• INSTRUCTIONAL PLANNING I 3 5

Le 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 c tiv e Plan a lesson that uses a problem-solving strategy

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

Think The problem-solving strategy involves students being challenged to collaborative solve real-world math problems that they have not yet previously encountered. It student-centered and promotes critical and creative-thinking skills, problem-solvin abilities, and communication skills. The integral part of this strategy is the time givei to the students to struggle with the problem, and its beauty is in the varied solution 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 the students are expected to answer them by applying the concept or skills that have 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 has 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 that can be 38

| TEACHING MATH IN THE INTERMEDIATE GRADES

sc ved using the target concept of the lesson at hand but can also be answered using :'eviously learned knowledge and skills. How you present the problem also matters especially for the elementary grades. : is not always helpful to introduce the problem by posting it on the board; doing s may intimidate some of the students and reading and comprehension skills may itervene. Instead, it is suggested to narrate the problem in a storytelling manner to e-igage the learners. Encourage the students to imagine the scenario and allow them :o clarify information if they find some details confusing. Showing drawings or real -bjects 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, t 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 to 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 don't. 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 INTERMEDIATE GRADES |

Experience Study the following lesson plan. Take note that the plan only shows development of the lesson that involves the problem-solving strategy; other parts are not included. Topic: Division of fractions Grade level: 5 Target learning competency: By the end of the lesson, the learners will be able to divide a whole number by a fraction. Prerequisite knowledge and skills: 1. Fraction as part of a whole 2. Fraction as repeated subtraction 3. Division of decimals 4. Multiplication of fractions 5. Reciprocal Presentation of the problem:

I have 6 liters of milk. I will transfer the milk in to glasses. Each glass can hold | liter. How many glasses can I fill? Present the problem above in a narrative approach that will engage the students. See example below. "Have you heard of feeding programs? What do you know about feeding programs? Who benefits in feeding programs? Have you participated in such? How many children usually participate? What kind of food is usually served in feeding programs? Last weekend, I volunteered in a small feeding program. I got to meet some Grade 5 children like you. I was in charge of distributing milk. We brought with us 6 liters of milk. My task was to pour the milk into glasses. If my estimate is right, each glass approximately contained | liter of milk. The milk we brought was just enough for all the children in the area. How many children do you think we served during the feeding program?" Generation of solutions: The 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 4 0 I TEACHING MATH IN THE INTERMEDIATE GRADES

no one right solution. They may do calculations or draw; any solution is welcome as long as they can explain why they did such. Watch out: The students should realize that the given is a division problem. Their discussions with their partner/groupmates should revolve around figuring out how many | s would "fit in" 6. Give 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) Illustration

s aa! Use of decimals

— = 0.4 5 6 t 0.4

II II 60 4 15 4160 4_

N=15 glasses

20 20

Unit Hi * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 4 1

(3) Division as repeated subtraction

15 4_ 5

2_ 5

_ “

_2_ 5

Guide the students to realize that the different solutions all lead to the same answer, which is 15—there are 15 children who were served during the feeding program. 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. Lead the students to realize that 6 -s-1 is equal to 15.

(See the continuation of this plan in Lesson 10) The next part of the lesson involves the students discovering the algorithm for dividing a whole number by a fraction. This applies another teaching strategy that will be discussed in Lesson 8.

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? 42 I TEACHING MATH IN THE INTERMEDIATE GRADES

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

Challenge As you may have noticed in the sample lesson plan, the problem is basically a usual division problem that you may see in common mathematics assessments. It is not even a nonroutine word problem. However, the students' thinking and creativity are challenged because (always remember) it is the first time that they will encounter such a problem. So, the task of the teacher is to present the problem in an engaging way. Do the following to practice this important task. 1. Browse the DepEd mathematics curriculum guide. Choose a topic from Grades 4 to 6. Write the topic and grade level below.

Unit I!!

• INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES j 4 3

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

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

2. The possible solutions in the sample lesson plan are written by real Grade 5 students. So be convinced that Filipino students are actually capable of thinking! As would-be teachers, your task is to anticipate such possible solutions. How? Consult with experienced teachers! Show your problem to some teachers and ask them how they think the students would answer if they are given the problem for the first time. You may also ask the children themselves. Talk to your nephew/ niece, godchild, neighbor, etc. Share with them your problem and have them explain to you how they think they can solve it. Write in the boxes three of the possible solutions that you have gathered.

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Possible Solution 1

Possible Solution 2

Possible Solution 3 i

Unit 111 • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES |

Harness Write a lesson plan that makes use of the problem-solving strategy. Use the topic, problem, and possible solutions that you have answered in Challenge. This activity will be part of the learning portfolio that 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 to:

Prerequisite knowledge and skills:

Presentation of the problem:

Generation of solutions:

j TEACHING MATH IN THE INTERMEDIATE GRADES

Processing of solutions:

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 the students in mathematical discourse.

Unit 11! • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 4 7

Objective Plan a lesson that allows the students to inductively learn a concept

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

Think The inductive learning strategy, sometimes called discovery learning, is based on the principle of induction. Induction means to derive a concept by showing that if it is true to some cases, then it is true for all. This is in contrast to deduction where a concept is established by logically proving that it is true based on generally known facts. The inductive method in teaching is commonly described as "specific to general," "concrete to abstract," or "examples to formula." Whereas the vice versa is used to describe the deductive method. In an inductive learning lesson, the teachers design and facilitate activities that guide the learners in discovering a rule. Activities may involve comparing and contrasting, grouping and labeling, or finding patterns. In mathematics classes, the learners engage in inductive learning when they observe examples and then, later on, generalize a rule or formula based on the examples. There are four processes that the students go through when given an inductive learning activity: (1) observe, (2) hypothesize, (3) collect evidence, and (4) generalize.

Observe Children love looking for patterns. When given many examples, it is natural for them to look for similarities and assume rules. So, the key is to give them examples to observe. These examples must be well-thought-of so that the students would eventually arrive at a complete rule. For instance, if you want your students to discover

| TEACHING MATH IN THE INTERMEDIATE GRADES

~e rule in multiplying decimal numbers, it is better to use the examples in set B than -Dse in set A so that the students' observations would focus on the "placement" of ~e decimal point.

A n C\J X

0.6

1.2

6 x 2 = 12

1.8 x 0.3 = 0.54

0.6 x 2 = 1.2

0.21 x 1.4 = 0.294

0.6 x 0.2 = 0.12

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. In our example, the hypothesis, "place the decimal ooint according to the number of decimal places of the factors" may be considered the same as, "from the whole number product, move the decimal point to the left according to the number of decimal places of the factors."

Collect Evidence Here, the students test their hypothesis by applying their hypothesis to other examples. If there is 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. For example, instead of saying "the number of digits to the right of the decimal point," lead the students to say, "the number of decimal places." Doing this would develop the students' mathematical vocabulary and therefore their overall mathematical communication skills.

Unit III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES |

Experience The continuation of the plan in Lesson 10 is shown below. The lesson was cut at the point when the students have generated many different solutions to 6

Applying the Different Techniques Instruct the students to solve the following using any of the techniques that their classmates shared in dividing a whole number by a fraction.

Other examples may be given but consider that one of the possible solutions is illustration so use small values. Also, include examples of dividing by a unit fraction; it will be useful in the discussion later. If time is limited, group the students into five and each group will answer one example. Each technique discussed must be used by at least one group. Move around while the students are working. Make sure to clarify confusion and correct misconceptions about the techniques, if there are any, because the class discussion that will follow will focus on the discovery of rules. Observe Write the examples with answers on the board (including the first one). 1 1 1 3 2 2 4-5-—= 123-f-—= 62-s-—= 103-5-—= 44-f—= 66-5-—= 15 3 2 5 4 3 5

Ask the students about their experiences as they solve. Lead them to realize that their techniques are creative ways of solving the problem, but they are not time-efficient. This should motivate them to discover a shortcut. Give some time for the students to observe the examples. The fast learners may become too excited to share their hypotheses but don't allow them to. The goal is for all the students to have the "Aha!" moment. Hypothesize The struggling students may not see the pattern right away. Help them by focusing their attention on the unit fraction divisors first. Call on some students to explain their hypotheses. After each explanation, ask who has the same hypothesis.

I TEACHING MATH IN THE INTERMEDIATE GRADES

Challenge The following activity will engage you in identifying mathematical concepts that can be taught using the inductive learning strategy. 1. Browse the DepEd mathematics curriculum for Grades 4 to 6. 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 that you can use in class to allow discovery. What were your considerations in choosing your examples?

TEACHING MATH IN THE INTERMEDIATE GRADES

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

Observe b ■

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

' v-' f ----- _ ---------- ------------------------------------------------------------------ ---------- --------------

Hypothesize

— ---------------------------------- ----------- — V

' IIIIIIIII!l i l l l l l l l

Unit Ml » INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES I 5 3

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, the students observe, hypothesize, collect evidence, and generalize.

I TEACHING MATH IN THE INTERMEDIATE GRADES

O b je ctiv e Plan a lesson that applies concept attainment strategy

Introduction The inductive learning strategy in the previous lesson is most useful in discovering 'jle s. In mathematics, the students do not only study rules; they also need to 'emember and understand many definitions of terms. For better retention, it is best ;or the 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 the 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. Non-examples are given to confirm their guesses.

Listing of common attributes List down the common attributes given by the students. This may be done as a whole class or by pairs or triads first. Some listed attributes may be later crossed out as the listing of examples and non-examples go on.

Unit HI • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES j 5 5

Adding student-given examples Ask the 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.

Experience Study the lesson plan below that 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: Prism Grade level: 4 Target learning competency: By the end of the lesson, the learners will be able to define a prism, give examples and non-examples of prisms and identify whether a given is a prism or not. Prerequisite knowledge and skills: Definitions of: 1. Parallel 2. Polygon 3. Circle 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. Give the following examples by batch. 56

I TEACHING MATH IN THE INTERMEDIATE GRADES

Non-examples

Examples

Batch 1

Cereals box Chalk box

Cone Ball

Single Toblerone box Combined Toblerone boxes forming a hexagonal prism

Egyptian pyramid replica

Egyptian pyramid cut at a cross-section

12-sided die (dodecahedron)

Pentagonal prism prototype

Oblique pentagonal prism prototype Can

The following 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. Common Attributes

Batch 1

The faces must be squares or rectangles.

Most faces are squares or rectangles, two opposite faces may be any other shape.

The two opposite faces must be congruent.

The two identical faces are connected by rectangles.

The two identical faces are connected by parallelograms.

The identical faces must be polygons.

In between batches, ask the students to look around the room and give examples of what they think are prisms. Classify the student-given objects as examples or non-examples of prisms. Defining the mathematical term: Lead the students to agree that prisms are solid figures with the following properties: with polygonal faces, two of which are parallel and congruent and the rest are parallelograms connecting them. Checking of understanding: Show 10 architectural structures around the world and let the students identify each as prism or not.

Unit Mi » INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 5 7

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.

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 Grade 4,5, or 6 mathematics teacher. Ask him/her what mathematical term the students have a hard time remembering or find confusing. List three mathematical terms and explain why each term is difficult to remember. Term 1 :_______________________________________

Term 2:

5 8 I TEACHING MATH IN THE INTERMEDIATE GRADES

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? Tprm-

— Examples

Non-examples

Considerations:

Harness

\ » Write a lesson plan that allows the students to discover the meaning of a

mathematical term. Use the topic you answered in Assess. This activity will be part of the learning portfolio that 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 to:

Unit HI

• INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 59

Prerequisite knowledge and skills:

Identifying common attributes based on examples and non-examples:

Defining the mathematical term:

Checking of understanding:

I TEACHING MATH IN THE INTERMEDIATE GRADES

Summary Mathematics is considered a language with its own set of jargons. Mathematical rerms can also be defined through discovery by applying the concept attainment strategy. Concept attainment involves presentation of examples and non-examples, sting of common attributes, adding student-given examples, definingthe mathematical term, and checking of understanding.

Unit ill • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES j

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

Introduction Contemporary leaders in mathematics education revolutionized the goal of mathematics teaching and learning from a passive transfer of knowledge to an active process where 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 that involves not only problem solving, but equally importantly, problem posing as well. In this strategy, the word "investigation" does not refer to the process that may occur when solving a close-ended problem but an activity in itself that promotes independent 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 of the students might attempt to 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. The students may or may not choose to find the total number of high-fives. Some students may want to investigate a more general case where they would want to know how many high-fives

I TEACHING MATH !N THE INTERMEDIATE GRADES

there would be given a certain number of children. Some may want to find out how ~iany high-fives there would be if instead of once, the children would high-five each :ther 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 -ot specified by the teacher; the students have the freedom to choose any goal to oursue. In problem-solving, the students are encouraged to think outside the box; in •nathematical 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: the (1) problemposing, (2) conjecturing, and (3) justifying conjectures. In the problem-solving 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 not the 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.

Unit HI • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 6 3

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. Close-ended problem: There are 24 animals in a farm. Some are cows and the rest are chicken. There are 60 animal legs in all. How many cows and how many chickens are there? Investigative task: There are 24 animals in a farm. Some are cows and the rest are chicken. Investigate. Possible student-generated problems: 1. How many cows and how many chickens could there be? 2. What is the possible total number of animal legs? 3. Given any total number of animals, what is the ratio of the number of cow legs to the number of chicken legs?

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.

I TEACHING MATH IN THE INTERMEDIATE GRADES

2. In what ways does mathematical investigation help develop the 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 undergo mathematical investigation. 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 tricks: 854 ~458 396 +693 1089

Unit Hi • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 6 5

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 Grades 4 to 6. Transform it to an investigative task then list down the possible problems that the students could pose given the task. This activity will be part of the learning portfolio that you will compile at the end of this module.

Close-ended problem:

....... . _

I TEACHING MATH IN THE INTERMEDIATE GRADES

Possible student-generated problems:

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

Unit Ml • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES

iff ;■ :

O b je c tiv e Execute the Empathize, Define, Ideate, Prototype, and Test stages of the designthinking process

Introduction The students find learning mathematics most engaging when they are involved in a thinking process that results in an output that can be applied to 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 the 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)

Empathize The goal of design thinking is for the 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 68

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developmental stages, it is not natural for children in the intermediate grades to be empathetic toward others. It is a common observation by teachers that the students 3t these levels often do not realize that their actions affect others. So, applying design thinking in the classroom gives the children opportunity to cultivate empathy, at the same time, their skills for problem-solving and empathy are being developed.

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 the 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, sticky tapes, recycled plastics, and so on. Once a prototype is created, they test it or allow a user to test it, and then make improvements, or possibly overhaul the design, depending on their observations and the feedback of the user. In these stages, it is important to emphasize that it is totally fine to fail at the first attempt of prototyping. The trial-anderror aspect of the design-thinking process is glorified because it is believed that the students learn many things through their failures. Even though a physical product is the expected output of design thinking, it should be emphasized that going through the process is what is more important because it is where the learning takes place. Unit IM * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES |

Experience Following is a template of a sample worksheet that will guide the students through the design-thinking process. This is a simplified version of Stanford's sample template. In this example, the students are to create a project about their recess time experience.

Your challenge is to redesign your school's recess time.

: I ■

Empathize Observe your classmates, teachers, and canteen managers during recess time. Take time to casually interview some of them about their usual recess experience. Notes from your observation:

Notes from your interview:

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

I TEACHING MATH IN THE INTERMEDIATE GRADES

Ideate

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

Idea 2:

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 the members agree that the necessary ideas are integrated in to the solution. Sketch your group's unified solution below. Unified Solution

Unit ISI • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES

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 ruler, compass, and/or measuring tape. Test Share your prototype with a user. Write your observations below. What worked:

What can be improved:

v "

More ideas:

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

I TEACHING MATH IN THE INTERMEDIATE GRADES

C hallen g e Many teachers are not informed about design thinking because it is a new, if not :ne newest, strategy in teaching mathematics. 1. Make an infographic about design thinking for the teachers who have not yet heard about it. Sketch a draft of your infographic below.

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?

Unit III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 7 3

H arness 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 that you will compile at the end of this module.

Your challenge is to redesign your school's lunch experience. Empathize Observe the students, your co-teachers, and canteen managers during lunch time. Take time to casually interview some of them about their usual lunch experience. Notes from your observation:

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

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insight | TEACHING MATH IN THE INTERMEDIATE GRADES

ideate

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

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

Prototype Create a prototype of your unified solution. Unit III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 7 5

Test Share your prototype with a user. Write your observations below. What worked:

A T

What can be improved:

More ideas:

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 they themselves identified.

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O b je ctiv e 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 a game-based learning strategy takes advantage of children's natural love for play to lead them toward complex problem-solving.

Think 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 the 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, the students naturally develop strategic and creative thinking and problem-solving skills. Games that require the 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 value. According to Koay Phong Lee (1996) in his article "The Use of Mathematical Games in

Unit 111 • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES I 7 7

Teaching Primary Mathematics," a game that has instructional value has the following characteristics: 1. the game has two or more opposing teams; 2. 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; 3. there is a set of rules that govern decision-making; 4. the rules are based on mathematical ideas; and 5. the game ends when the goal is reached. The fourth 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, (3) observe, assess, and process the students' understanding, and (3) work with the students who need additional help.

E x p e rie n c e The following game is a modification of a mathematical game developed by Koay Phong Lee (1996) to introduce the addition of fractions.

Topic: Addition of fractions Materials {for each pair of players): 1 .1 spinner

l/2_xfl/8 j V / l/ 4 j

2 .1 game board

| TEACHING MATH IN THE INTERMEDIATE GRADES

3.2

sets of fraction pieces, different colored set for each player (rectangles, squares,

triangles, and smaller squares representing 1 ,1 /2 ,1 /4 , and 1/8 respectively)

10 Pcs

m 10 Pcs

2 Pcs Mechanics: 1. Players take turns to spin. 2. After each spin, the player picks fraction piece(s) that represent(s) the fraction indicated on the spinner and adds it on the game board. 3. A player may exchange two or more of his/her adjacent fraction pieces on the game board for an equivalent fraction piece. 4. A fraction piece placed on the game card cannot be moved unless for "exchanging" purposes. 5. When the game board has been completely covered, the player whose fraction pieces cover up a larger area is the winner. 6. In case of a tie, the player who used less fraction pieces is the winner. In the game, allowing the students to exchange fraction pieces caters the idea of adding fractions. For a more meaningful discussion, the students may be required to record their moves (what fraction turned in each spin, which fraction pieces he/she used, etc.).

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

Unit III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES

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

C h a lle n g e 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 the 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?

TEACHING MATH IN THE INTERMEDIATE GRADES

Harness Develop a mathematical game that has instructional value following the :5Tiplate below. This activity will be part of the learning portfolio that you will compile r :he end of this module.

Topic:----------------------------------------------Materials:

Mechanics:

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

Unit 11! • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 8 1

O b je c tiv e Develop a manipulative to aid mathematical instruction

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

Think Manipulatives are concrete objects like blocks, tiles, and geometric figures that the students can interact with (touch and move) in order to develop a conceptual understanding of mathematics concepts. The use of manipulatives is not at all new; manipulatives have helped people learn mathematics since ancient times. For example, 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 of 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 the students believing that there are two different worlds of mathematics—the manipulative and the symbolic. It is therefore important that the teacher carefully plan on how to integrate manipulatives in classroom discussion in such a way that there is a smooth transition

8 2 I TEACHING MATH IN THE INTERMEDIATE GRADES

■'om concrete to abstract. The following are some guidelines for using manipulatives in :ne 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 of the 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.

Unit SSI * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES

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Experience The following is an example of how a geoboard is used to reinforce the understanding of different kinds of triangles. This activity may be done after the students have learned the kinds of triangles according to angle measure (acute, right, and obtuse) and according to side length (scalene, isosceles, and equilateral). Sample of student works are also presented.

Materials: 1. Geoboards 2 .Rubber bands Instructions: 1. Make an image with an equilateral triangle, an isosceles triangle, and a scalene triangle.

2. Make an image with an acute triangle, a right triangle, and an obtuse triangle. 3. Is it possible to have the following triangles? If it is possible, come up with an example using the geoboard and a rubber band. Otherwise, discuss why you think it is impossible to create such a triangle. a. Equilateral acute b. Equilateral right c. Equilateral obtuse d. Isosceles acute e. Isosceles right f. Isosceles obtuse g. Scalene acute h. Scalene right i. Scalene obtuse

| TEACHING MATH IN THE INTERMEDIATE GRADES

Sample responses: 1. A pair of students created a boat. The body of the boat is an isosceles triangle. The sails are scalene (left) and equilateral (right) triangles.

2. A pair of students created a satellite using acute (stand), right (dish), and obtuse (middle portion) triangles.

A sse ss Answer the following questions to verbalize your understanding of the use of manipulatives in mathematical instruction. 1. Give an analogy that involves the use of manipulatives and the use of games in classroom instruction. manipulatives:________________________ ; games:-------------------------

Unit III

» INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES I 8 5

2. The first guideline in using manipulatives is to let the students play with the material. Why do you think is this so?

C h a lle n g e The following activity will challenge your skills in relating with how students think. 1. Befriend some children and ask them to do #3 in the instructions in Experience. If you do not have a geoboard, improvise one using a thick carton and push pins. t

2. Record the children's discussion below.

3. What can you tell about the student's understanding of the different types of triangles based on their discussion?

I TEACHING MATH !N THE INTERMEDIATE GRADES

Harness Invent your own manipulative for a topic of your choice. Draw your design and label important parts of it. Then, explain how to use your invention. This activity will oe part of the learning portfolio that you will compile at the end of this module. Topic:__________________________________________________ Name of manipulative:__________________________________________________ Design:

Instructions:

■: X

;

■

/ 'v'i

i n . in ■ ■

■

.• ■;

. ::: : ■

■ >

2^?

Su m m ary Manipulatives have long been used to facilitate learning mathematics. Although many manipulatives are available in the market, nothing beats the manipulatives created by a teacher who knows exactly what his/her students need.

Unit III * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES

O b je c tiv e Plan a lesson in which values education can be incorporated into existing mathematics curricula

Introduction Mathematics has been conceived mainly as a tool for solving real-life situations through mathematical modeling. Since math is often remembered for its practical use, teachers would often capitalize on this aspect to make the learners see its relevance to their lives. In this lesson, you will explore an alternative way to make the teaching of math meaningful and engaging for the learners.

Think Integrating Math Into Other Subject Areas Integrating mathematics into the curriculum can be quite challenging and rigorous. However, math is connected to many disciplines and should not be isolated 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 Doctor Benjamin Bloom classified three domains of educational learning: cognitive, affective, and psychomotor. In the formal classroom set-up, the bulk of the teacher's lesson planning focuses on the cognitive and psychomotor aspects of the teachinglearning process. The third domain, which is the affective domain, is often overlooked. The affective domain includes the manner in which we deal with things emotionally, such as feelings, values, appreciation, motivations and attitude (Kratwohl, 1964). This particular domain, when tapped during the learning process, can really make the 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. 88

I TEACHING MATH IN THE INTERMEDIATE GRADES

Values Integration and Retention of Information Associating values or standards of behavior with mathematical concepts can serve 35

a source of motivation for the students. Values integration will help the students

get life lessons through math. If the students find a learning material engaging and meaningful, then they will ask for more (since curiosity will start to kick in). Curiosity s the force behind lifelong learning!

E xp e rie n ce 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. Whole Numbers

Reflection Questions Connect the lesson to the concept of 'wholeness'— comprising the full quantity, the start of forming a complete and harmonious whole, and the state of being unbroken and undamaged. 1. What were your experiences in the past that make you feel "whole" or "complete"? 2. Bullying can make a person's heart broken/not whole. Cite a specific event in the past in which you or your classmate has 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 hearts?

2. 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?

Unit Mi • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES

3. Factors and

Associate this with the idea of organizing things.

Multiples

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?

4. Addition and Subtraction of Fractions

Relate this to the idea that most of us tend to be attracted to people who are similar to us. 1. What are the qualities would you like your friends to have? Do you also possess these qualities? 2. Reflect on the saying: "Opposites attract." Do you believe in this saying? 3. Should you listen to the opinion of a person that is not like 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.

5. Geometry (Triangles) 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, and you have to make sure that you have a strong support group. 6. Patterns

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?

9 0 | TEACHING MATH IN THE INTERMEDIATE GRADES

7. Data Presentation

Give examples where the use of a bar graph or a pictograph is 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. "MASSIVE INCREASE IN HOUSE PRICES THIS YEAR!"

Average House Price {£)

What can you say about the first graph? Second graph? Which graph presents a more accurate reflection on the increase in house prices? 2. Which graph is misleading? Why? N um b e r o f singles sold

Num ber o f singles sold

200, 000

100,000

L _ ___ IliiM il 1995

1996

1997

1998

Unit til - INSTRUCTIONAL STRATEGIES f:OR MATHEMATICS IN THE INTERMEDIATE GRADES | 91

3. Which graph is misleading? Why?

8. Area

In mathematics, the whole is equal to the sum of its parts (Area Addition Postulate). Try to deviate from this postulate, and let the students reflect on the statement: "The whole is greater than the sum of its parts." For example, a laptop is made up of many parts that do nothing by themselves but when these parts are combined, the final product performs an astounding function. "Alone we can do so little; together we can do so much."

9. 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 many trials. Increasing the number of trials means gaining new insights/ perspectives. These new insights will make you a better person.

10. Solving Equations

An equation is like a weighing scale. The right-hand side and the left-hand side are balanced because they represent the same quantity. Why is it important to keep a balance between studying and relaxation?

TEACHING MATH IN THE INTERMEDIATE GRADES

In our society, it is really vital to educate people in the traditional values of our country. There is a growing demand for teachers to deliberately teach values, through settinjg a good example and discussing/processing moral issues with the learners. It is :nere^ fore crucial to the formation of the students that you deliberately use an eclectic nix o f methods to convey the important values that the students must uphold.

Ass e ss 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 4 to 6. Write the topic and grade level below.

How would you inject values into 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.

Unit III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 9 3

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

C h a lle n g e 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 4 to 6 do you think this strategy is moist appropriate? Why do you think so?

9 4 I TEACHING MATH IN THE INTERMEDIATE GRADES

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 that you will compile at the end of this module. Consider this situation. A student consulted with you and raised the following points. "Hi, Teacher! Our lesson on Solving Equations is not that hard. But why do we study something that we probably will never use in life? If I buy candy in a store, do I need to solve for 'x' before getting the candy that I want?" 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: Solving for the Unknown in an Equation 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 that teachers need do is to be intentional about it and reflect on ways to inject values in their lessons.

Unit III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES j 9 5

O bjective Design collaborative activities that will encourage involvement, interdependen and fair division of labor among the students

Introduction When transitioning from primary to intermediate grades, children develop a re; strong bond with one friend. Some child psychologists point out that it is easier some kids to relate to just one co-learner rather than socializing with a big gro at the same time. Teachers, however, can provide intermediate graders with ma opportunities for interactions. Within collaborating groups, children learn to 1 things out, conjecture, explore, justify, evaluate, and convince others of their findinj Collaborative tasks provide enriching opportunities for the learners to explore oth students' perspectives that may differ from their own. Thus, these can develop stronger sense of empathy among the students. Group activities, if facilitated carelessly, could waste classroom time. Because this, it is important for teachers like you to ensure that the group activities are caref-; designed and successfully implemented. This lesson aims to help you prepare, mor : : and process collaborative tasks in your classroom that will maximize your student capacity to socialize with each other.

T h in k

Vygotsky's Social Learning Theory Collaborative learning branches out from the zone of proximal development ZPI theory of Vygotsky.

Vygotsky defined the ZPD as follows: "The zone of proximal development is the distance between the a^ja developmental level as determined by independent problem solving and the eve 3 potential development as determined through problem solving under adult gu zzrm or in collaboration with more capable peers."

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TEACHING MATH IN THE INTERMEDIATE GRADES

In the ZPD, the learners are 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 ZPD, that is, he/she has 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 solve problems together. As the 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 the learners. Instead of passively accepting information from teachers, the learners discover new insights by cooperatively working with other learners. As mentioned earlier, teachers should be keen in selecting appropriate learning activities for the students. Listed below are some tips about preparing, monitoring, and processing collaborative tasks in your classroom that will maximize your students' 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 the 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 diverse, productive, active, and cohesive groups, then try dividing your class into groups with four to five members. • Decide how you will divide the class. Unit Hi • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES |

Will you group them based on proximity? Will you group them according to their preferences? The fastest way to group the students is to divide the class based on proximity. You might also want to randomly assign the students to groups by counting off and grouping them according to number. Another idea is to let the students get a chocolate from a basket of different chocolates and group them according to the flavor they choose. You may also strategically assign them to groups instead of randomly assigning them. Prepare a list with names vis-avis 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 the 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. The contract should also include the behaviors that you want them to avoid and the values that you want them to observe and uphold.

I TEACHING MATH IN THE INTERMEDIATE GRADES

• 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 your 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 in many 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 the 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 fora 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 the students make 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 perceptions about group work. Relate the points raised to your current lesson and the objectives of the activity.

Unit li! * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES

E x p e rie n c e The following are sample collaborative activities that you might want your learners to engage in.

Activity 1: BUZZ Group Buzz groups can be used from time to time because it is easy to implement. If you wish to include buzz groups in your sessions throughout the school year, establish all the expectations during the start of the school year. Let your students sign a contract that contains all the ground rules for this activity at the start of the school year. Reiterate the roles that each student is expected to perform, as the need arises. Topic: Area and Perimeter Objective: To discuss various solutions to a geometric problem involving area and/ or perimeter. Time frame: 15 minutes Task: The perimeter of the identified usable space in a certain room is found to be 108 meters. The space is roughly represented by the shaded region in the figure below. Given the indicated dimensions, find its area. (All intersecting lines are perpendicular to each other.)

Activity: Ask the students to solve the problem on their own. After five minutes, let them turn to their seatmates to discuss any problems in understanding or answering the problem. Ask questions like: 1. What are the things that you don't understand about the problem? 2. What are the possible solutions to the problem? 3. What method can be used to check if the solution method is correct? 100 I

f "'"ctPP iRSB

TEACHING MATH IN THE INTERMEDIATE GRADES k r . r■wn

After five minutes, reconvene as a class and have a general discussion in which students share ideas or questions that arose within their subgroups. Activity 2: 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: Operations on Decimals Objective: To work cooperatively with other students to ensure the attainment of your group goals Time frame: 55 minutes Task: Pretend you are a member of a committee that is in charge of organizing a math quiz bee for your school. In order to get financial assistance from the school, the board of trustees of your school wants your committee to present a project proposal containing all the projected expenses associated with the project. The committee head delegated the following tasks to his members. Student A: Compute the total cost of all the materials needed for the activity. Material |

Amount per quantity

__________ Quantity__________

11-

The teacher will give prices with decimals values. This will test if the students know how to multiply decimals. Student B: Compute the projected utility bills (water, electrical, cellphone bills).

The teacher will give a word problem involving the addition of decimals. Student C: Determine how many students are needed to attend the event in order to breakeven and to earn profit.

The teacher will give a word problem involving the conversion of fractions to decimals and multiplication of decimals. e.g., A survey is given to the entire student body. They were asked the question: "Are you going to the concert? If yes, will you bring a buddy to the concert?" The results of the survey are the following:

Unit III • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES j 1 0 1

Sections A, B, and C have 40 students each. % of each section will attend the concert. 8/5 of the total number of attendees will be joined by their buddy. How many students will attend the concert? Student D: Determine if the school has a sufficient number of school personnel available during the event.

A problem similar to the previous ones may be given. After completing each task, the group will now assemble to consolidate their work and verify the accuracy of their solutions. The group is expected to determine the total costs covering all the expenses for the said project.

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? 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?

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Challenge The i Snowball Technique is a way for the students to teach each other important concepts and information. The students begin by working individually. Next, they collabc irate with a partner. After that, partners form groups of four. Groups of four oin 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 Intermediate Mathe matics in which this collaborative activity can be used. Write down a sequence of incn ?asingly complex tasks that can be given in this activity.

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

Summiary Group >activities can foster collaboration when thoughtfully designed and carefully facilitatec I. Group size and composition are some of the considerations in designing group act ivities. It also helps to explain to the students why doing the activity by group is essenti a I in learning the lesson where it is applied. Unit ill * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 103

O bjective 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 d<afinitions and theorems, but rather he/she asks meaningful questions that lead the le.arners to the correct ideas. Also, the teacher gets to identify the 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 misunderstandi ngs of the learners. If a teacher is knowledgeable about the misunderstandings of thie learners, then the teacher will have the greatest understanding of his/her learners,. It is, therefore, necessary that teachers deliberately frame questions tl "lat will keep the class discussion moving. The goal of this strategy is to keep the lear ners' voices at the forefront of every classroom session. The challenge for you now i s to think of questions that you could ask that would get your students engaged.

Think Discussion vs Lecture In a discussion-based strategy, the teacher's role is to engage the le larners in a question-oriented dialogue. The teacher spends a significant amount of ■ time asking scaffolding questions to help the students understand an idea deeply. The interaction in this method leans on both the teachers' and the students' equal partici| pation. 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 amswered by a simple yes or no. Some questions would require the students to th in k more

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meaningfully. Asking 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 that 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 the 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 the students 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 the 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 the learners to give responses instantly. Give them 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.

Unit SII * INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES |

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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 the 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. The learners should practice directing questions not only to you but also to their co-learners. You should give the 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 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 the students for further participation.

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

Situation 1. You want to help a

Questioning Technique -What part/s of the problem is/are difficult to

student who gets stuck on a understand? problem.

-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

-What do you think about your classmate's answer?

to listen and respond to the

-How would you restate your classmate's reasoning?

answers of their classmates. -Did anyone get the same result but with a different solution method? -Why is your classmate's assumption correct/incorrect? -Would you tike to comment on any of the previous answers? 3. You want to lead your

- Does this formula always work? Why?

students to the right

-How do perimeter and area differ?

conjecture, definition, or

-How does the radius of a circle relate to the diameter of

generalization

the circle? -Will this solution method work if some conditions about the problem are changed? -Do you notice nay pattern? What can you conjecture 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 following 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.

Unit ill • INSTRUCTIONAL STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES

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2. Only the bright students are answering the questions. The challenged ones do not raise their hands.

3. When a student was asked to answer a question, he/she responded with an angry stare.

4. A student answered the question in a manner that is not comprehensible to the other students.

C h a lle n g e The following questions will challenge your understanding of the teaching-byasking 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?

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

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

Harness Imagine that you are going to teach the area formulas for quadrilaterals. Write a script containing all the scaffolding questions that you will ask to lead your students to the correct area formulas. This activity will be part of the learning portfolio that you will compile at the end of this module.

Unit III

♦INSTRUCTIONAL STRATEGIES

FOR MATHEMATICS IN THE INTERMEDIATE GRADES j

109

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

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UNIT IV

Assessm ent Strategies for M athem atics in the Interm ediate G rades I

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

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This unit discusses one of the most important aspects of teaching — assessment. In this unit you will learn how to assess student's learning for formative and grading purposes alike. You will also learn about the contemporary types of authentic assessments.

Ill

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 that should be within the K-12 Basic Education framework and aligned with the learning standards of the enhanced curriculum. Due to the need, DepEd issued the DepEd Order No. 8, s. 2015, which is the Policy Guidelines on Classroom Assessment for the K-12 to 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 the learners' progress in relation to learning standards including that of the development of 21st century skills, which is part of the new K-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 ZPD of Vygotsky (1978). In the center of the process is the nature of the learner. Assessment shall recognize the diversity of the learners inside the classroom, which requires multiple ways of assessment measures of their varying abilities, skills, and potentials. The ZPD assessment framework puts premium consideration on the recognition of the learner's ZPD at the heart of the assessment. A learner-centered assessment supports the learners' 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 to learning and teaching within a degree that is not difficult yet challenging for the learners. It facilitates the ultimate objectives of the 112

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<,-12 program in each learner that is to develop higher-order thinking and 21st rentury skills. From this view, there is unity between instruction and assessment, nstruction 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-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 the learners perform well in relation to :nese learning standards. There is a recommended type, component, period, and approach of assessment for each learning standard lifted from DepEd's policy guidelines.

CONTENT STANDARDS

PERFORMANCE STANDARDS

LEARNING COMPETENCIES

........................ .................................... mmmmmtmm

Diagnostic

Summative

Formative

Assessment "for" Learning

Assessment "of" Learning

Before the Lesson

During the Lesson

After the Lesson

Written Work

Performance Task

Quarterly Assessment

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Principles of Assessment 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.

Unit IV * ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 1 1 3

The results of formative assessment are not graded but it is important to keep documents of these to study the patterns of the learning demonstrated by the learners to prepare them in taking the summative assessment. 3. Assessment results must be used by teachers to help the 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 the learners. The main purpose of assessment is to improve learning, not increase anxiety among the learners.

E x p e rie n c e 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 is also not confined within the classroom because any interaction with the learner is opportunity to assess the learner's abilities. The United Nations Educational, Scientific and Cultural Organization (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 the student identify his/her strengths and weaknesses in order to learn from the assessment. Formative Assessment as "assessment for learning" 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 pieces of evidence about the learners' strengths, weaknesses, progress, and needs. The results of these will help teachers design instructional activities and make decisions so that it is suited to the learners' situations and needs. The pieces of 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 and hence are 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 the learners do well in or which areas do the\ 114

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need help with. This can be through feedback from anyone around them especially *rom the teacher or any individual who is considered more knowledgeable. Formative assessment should also be a learning opportunity that enables the learners to take 'esponsibility for their own learning. 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 the orientation of the learning goals are to determine the current status of learners or pieces of evidence of prior understanding1; next is to provide clear, specifics and on-time feedback2; next is to provide instructional corrections/adjustments based on the needs of the learners3; next is to move the earners close to the goals/learning standards4; next is to evaluate the learners' progress5; and lastly, again to provide feedback of the learners' status6. In a case where a learner is highly self-regulated, the process begins with the orientation of learning goals, next is to determine the status/prior understanding of learners, next is to provide feedback, next is to provide instructional corrections/ adjustment, next is to evaluate the student's progress, and lastly, the processes ends in the same step, which is to provide feedback after evaluation of the student's progress. 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 on. Parts of the Lesson

Purpose For the Learner

For the Teacher

1. Know what s/he knows about the topic/lesson

1. Get information about what the learner already knows and can do about the new lesson

2. Understand the purpose of the lesson and how to do well in the lesson Before Lesson

3. Identify ideas or concepts he/she misunderstands

2. Share learning intentions and success criteria to the learners

3. Determine 4. Identify barriers to misconceptions learning 4. Identify what hinders learning

Examples of Assessment Methods

1. Agree/Disagree activities 2. Games 3. Interviews 4. Inventories/ Checklists of skills (relevant to the topic in a learning area) 5. KWL activities (what I know, what I want to know, what I learned) 6. Open-ended questions 7. Practice exercises

Unit IV * ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 1 1 5

1. identify one's strengths and weaknesses

Lesson Proper

1. Multimedia presentations

2. Identify what 2. Identify barriers to hinders learning learning 3. Identify what 3. Identify factors facilitates learning that help him/her 4. Identify learning learn gaps 4. Know what he/she 5. Track learner knows and does not progress in know comparison 5. Monitor his/her to formative own progress assessment results prior to the lesson proper

1. Multimedia presentations 2. Observations 3. Other formative performance tasks (simple activities that can be drawn from a specific topic or lesson) 4. Quizzes (recorded but not graded) 5. Recitations 6. Simulation activities

6. To make decisions on whether to proceed with the next lesson, reteach, or provide for corrective measures or reinforcements

After Lesson

1. Tell and recognize whether he/she met learning objectives and success criteria

1. Assess whether learning objectives have been met for a specified duration

2. Seek support through remediation, enrichment, or other strategies

2. Remediate and/or enrich with appropriate strategies as needed 3. Evaluate whether learning intentions and success criteria have been met

1. Checklists 2. Discussion 3. Games 4. Performance tasks that emanate from the lesson objectives 5. Practice exercises 6. Short quizzes 7. Written work

Summative Assessment is the assessment of learning. This assessment is always given at the end of a unit or toward the end of a period because it aims to measure what the 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 are more beneficial for others than to the learners. 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 was 116

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deliberately designed to measure how well the students learned and were 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 the learners are ready to take the summative assessments. The summative assessment measures the different ways the learners use and apply all the relevant knowledge, understanding, and skills. The learners synthesize the knowledge, understanding, and skills during the summative assessment and the results will be used as bases for computing the grades. The summative assessment is in the form of a unit test and a quarterly test; it must be spaced properly over the quarter. It has three components, namely, 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 percentage for each component. Purpose

Written Work

• Assess the learners' understanding of concepts At the end of the topic or unit and application of skills in written form • Prepare the learners for quarterly assessment • This can be individual or collaborative over a period of time

• Provide opportunities for the learners to demonstrate and integrate their knowledge, understanding and skills about topic or concept learned to apply in real-life situations through Performance Tasks performance

At the end of the lesson about particular topic/ skill Several times within a quarter

• Provide opportunities for the learners to design and express their learning in diverse ways • Encourage the learners' inquiry, integration of knowledge, understanding and skills in various contexts beyond the assessment period Quarterly Assessment

Synthesize all the learning skills, concepts and values learned in a quarter

Once, every end of the quarter

The DepEd guidelines provide a list of assessment tools per learning area. Shown below is for mathematics. Unit IV • ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 1 1 7

Components Written Work (WW)

A. Unit/Chapter Tests 8. Written output

Performance Tasks (PT) A. Products 1. Diagrams

1. Data recording and analyses

2. Mathematical Investigatory projects

2. Geometric and statistical analyses

3. Models/Making models of geometric figures

3. Graphs, charts, or maps

4. Number representations

4. Problem sets 5. Surveys Math

B. Performance-based tasks 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

A ss e s s Answer the following questions to verbalize your understanding of assessment in mathematics. 1. What does the ZPD say about assessment?

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2. Fill in the Venn diagram belowto compare and contrast formative and summative assessments.

C h a lle n g e The following questions will test your critical thinking skills as it presents an issue in 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 pieces of literature support your thoughts?

Unit IV

• ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES

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Harness The following activity will require you to examine an actual assessment given to intermediate grade students. This activity will be part of the learning portfolio that you will compile at the end of this module. 1. Collect a summative assessment from a Grade 4, 5, or 6 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?

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

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O b jective Develop a traditional assessment tool

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

T h in k For so long, the most widely used measure to describe the learners' 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, uni-dimensional timed exercise that usually in multiple-choice or short-answer form. The traditional assessment measures are the most widely used measure of a student's learning and measure of success in educational goals, and 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 the student's experiences in the learning activities, as well as the student's motivation to complete such tests. Some issues are relative and apparent vis-a-vis comparison with the authentic assessment. Traditional assessment

Alternative/Authentic Assessment

One-shot test

Continuous, longitudinal assessment

Indirect test

Direct test

Absence of feedback to the learners

Feedback is part of the processes

Speed exams

Basically untimed exams Unit IV * ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES | 1 2 1

Decontextualized task

Personalized task

Norm-reference score interpretation

Criterion-reference score interpretation

Standardized test

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 a variety of tasks within a test (Benjamin, 2012). On the other hand, standardized test scores reflect only a single measure of a student's attribute, performance, or ability but fail to generalize other settings. At present, traditional assessment may have many critiques but it still has its 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 Assessment 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 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, the information you need to gather to make those decisions, and what methods are the most effective in i gathering that information.

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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. It is 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 learners' 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 lay person. The grading of the assessment is often related to the referencing of the assessment, and the grading and the referencing should be considered in tandem.

E x p e rie n c e 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 a students to make decisions and find out which of two potential responses is true. 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 the student really knows the correct response and why. One possible action to improve the Unit IV

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quality of True or False test is to ask the student to provide an explanation for the incorrect items or rewrite the items correctly. 2. Multiple-choice Test. This type of test is commonly utilized by teachers, schools, and assessment organizations for the following reasons (Bailey, 1998): a. Fast, easy, and economical to score. Machine can be used in scoring. b. This measure can be scored objectively, thus giving an impression of being most fair and/or more reliable than other form of test. c. As compared with True or False test, the multiple choice test reduces the chances of the learners guessing the correct items. There are many critiques on the use of multiple-choice test. The most common critique is that the test items of multiple choice are effective only in testing the low level of cognitive skills like recalling of previously memorized knowledge, while items that demand higher-order thinking skills such as analyzing and synthesizing are difficult to produce (Simonson, 2000; Bailey, 1998). The other critiques on the use of multiple-choice test are the guessing may be considerable, but with unknown effect in the test scores, the test severely restricts what can be tested, difficult to write successful items, backwash maybe harmful, and cheating is highly possible (Bailey, 1998). 3. Essay Essay is an effective assessment tool because the answer is flexible, and it measures higher-order learning skills—written communication and organization of ideas. However, it is not a practical measure because it is difficult and timeconsuming to score. Another issue is the subjectivity in scoring, hence creating a rubric is necessary to evaluate the output (Simonson et al., 2000). The rubric is a "criteria-rating scale" which gives the teachers a tool that allows them to track student performance. The teacher has an option to create or adopt a rubrics depending on their instructional needs. 4. Short-answer Test. In a short-answer test, the items are written either as a direct question requiring the learner to fill in a word or phrase or as a statement in which a space has been left blank for a learner to fill in a brief written answer. The question needs to be precise; otherwise, the items that are open to interpretations allow the learners to fill in the blanks with any possible information. 124

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

Harness 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 4 to 6. Then, write a three-item multiple-choice test to measure both the concepts and skills in your chosen topic. This activity will be a part of the learning portfolio that you will compile at the end of this module.

Topic:_______________ Grade level:________ Multiple-choice test: 1_________________

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S u m m a ry 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 the learners are asked to perform real-world tasks and the criteria are based on actual performance in the field of work (Wiggins, 1989; 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 (P21, 2019; 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 Assessment 1. Authentic mathematics requires essential skills that 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. A part of mathematics literacy is the ability to 128

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learn and assimilate new information; hence there is a need for essential skills of

flexibility and adaptability. Related to communication is the ability of the learners to articulate what they understand and do not. Communication can be fosterec in school if the learners learn and use the language of mathematics like activities that 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. The use of technology paired with appropriate questions. For example, for elementary levels, they may ask to find one data point, while for intermediate levels, the question could be to find the trends among multiple points in the data. 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 mathematical contents and interdisciplinary integration. In geography, there are opportunities to use scaling, proportions, and ratio. In genetics, there are opportunities to apply statistics and probability. Interdisciplinary approach provides opportunities of different contexts. It promotes attitudes of inquiry, investigation, and sensitivity on the interrelatedness between content and real world. 4. Authentic assessment measures the complete picture of the learners' intellectual growth. It measures the various kinds of knowledge and measures either group or individual for different purposes. An authentic assessment is the combination of more than one. 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 can reflect and connect on their performances. Thus, assessment becomes learning opportunities and assessment aims to measure not only the actual performance but, more importantly, 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.

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8. It must provide information on what the learner knows, what does not know, and on the development of changes in such learning. 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 media in assessment to obtain the integrated picture of a learner, for example, the use of paper-and-pen tests, videos, and computers 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 gauge how well the students critique the quality of other documents. Videos can assess communication, explanation, summarization, listening, argumentation, question asking, answering skills, and how the student interacts during cooperative learning. Computers can be used to simulate realistic situations inside the classroom, and they can effectively track the process of learning and the learners' response to adaptive feedback. Computers can make possible the dynamic assessment of relevant criteria. 12. The purpose of the 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 often used in the context of performance assessment.

Authentic Assessment 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 contexts of problem solving. It involves individual and cooperative problem-solving activities. Teachers must have a 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 the students form their hypotheses by comparing theirs with other hypotheses, and provides an understanding of how to generalize concepts from one problem situation to another. 1 3 0 I TEACHING MATH IN THE INTERMEDIATE GRADES

• Provide and explain, if necessary, the clear statements of learning standards and/or other models of acceptable/best performances prior to engagement on assessment tasks • Explain the importance of completing the self-assessment tool in improving their performances • Provide examples of interpreting the students' performances by comparing them to learning standards that are developmentally appropriate or comparing them to an other student's performance.

Experience There are many forms of authentic assessment to ensure collaborative effort, interaction, and active participation of the learners. DepEd 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, the role of the students, and what are they being asked to do

Audience

Who they need to convince, to who they present the output/ outcome or propose a 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

The following is an example of a performance task for Grade 6 Numbers and Number Sense. 1. Numbers and Number Sense Learning Standards for Grade 6 Grade 6-SEGOND QUARTER Numbers and Number Sense

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demonstrates understanding of order of operations, ratio and proportion, percent, exponents, and integers.

is able to apply knowledge of order of operations, ratio and proportion, percent, exponents, and integers in mathematical problems and reallife situations.

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26. expresses one value as a fraction of another given their ratio and vice versa. 27. finds how many times one value is as large as another given their ratio and vice versa.

28. defines and illustrates the meaning of ratio and proportion using concrete or pictorial models. 29. sets up proportions for groups of objects or numbers and for given situations. 31. solves problems involving direct proportion, partitive proportion, and inverse proportion in different contexts such as distance, rate, and time using appropriate strategies and tools. 32. creates problems involving ratio and proportion, with reasonable answers. 33. finds the percentage or rate or percent in a given problem. 2. GRASPS Goal

Design a personal budget out of your weekly allowances that will allow you to save a target amount of money.

Role

You are a financial analyst/beginner businessman who's planning to put up a certain business. In order to make this happen, you have to save your initial capital.

Audience

You will present your plan of a personal budget to your guardian/parent or to a person providing your finances.

Situation

You have to create a plan out of your personal budget with details on how much you have, how much you plan to spend, itemizing what you plan to spend your money on, and future plans to generate savings and income.

Product, Performance, Purpose

Plan with chart, graph, and time frame with calculations. Itemized lists of what you will be spending your money on reflecting your personal choices. In the end, reflect on the choices you did as regards to how much, and what you chose to purchase.

Standards and Criteria for Success

The proposal has to be clear and detailed. The reflection should indicate your personal choices and values.

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3. Rubrics on the evaluation of the GRASPS output

Criteria

Excellent (10 points)

Developing to Excellent (7 points)

Beginner (5 points)

Clarity of activity questions

1 to 2 statements are needed to improve

3 to 4 statements are needed to improve

5 and up statements are needed to improve

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

With 1 to 2 errors in the single bar graph; uses labels or colors but not quite successful

With 3 or higher number of errors in the bar graph

Accuracy of the data interpretation

No mistake in the interpretation of bar graph

With 1 to 2 mistakes With 3 or higher number of mistakes

Feasibility of the answer/ proposal

The proposal is 100% feasible and based on the presented data

The proposal has some minor recommendations

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?

Challenge The following questions will challenge your critical thinking skills about the use of authentic assessment.

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1. What elements of the authentic assessment measures make it bette- than traditional assessment? Why?

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 that you will compile at the end of this module. 1. Choose a topic in Grades 4-6 mathematics. You will develop a Performance Task in this topic. Grade level:____________ Content Standards

Performance Standards

Learning Competencies

;

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2. Develop a Performance Task to assess student learning in your chosen topic. Keep the principles of authentic assessment in mind.

Audience

Situation

Product, Performance, Purpose

Standards and Criteria for Success

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3. Write a rubric to evaluate the GRASPS output. Criteria

Excellent ( _ points)

Developing to Excellent ( ___points)

Beginner ( ___points)

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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, learning, development, growth, and achievement. It emphasizes a learner's milestone in his/her development of concepts and skills. It contains not only output and worksin-progress, but also reflections on the learner's strengths and progress toward the learning goals.

Think Portfolio assessment is an example of authentic and nontraditional 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 the learners themselves. These outputs have some degree of quality that cannot be measured by traditional tests. Portfolio develops awareness of one's own learning. Knowing the criteria of the content and assessment, the learner can always refer to these in each stage to verify the progress in achieving the set of objectives and goals. Furthermore, it also aims to develop independent and active learning. Portfolio assessment can address the heterogeneous groupings of the learners because 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.

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Implicitly, engaging in learning portfolio promotes social interaction between the learner and the teacher and the learner and other learners. An additional interaction is between the learner, the teacher, and the parent during the output presentation and feedbacking where the collaborative comments of the parent and teachers are important for future performances. The collaborative approach of portfolio assessment is an important element of the process. Since a learning portfolio is anchored on the theory of self-determination (Deci & Ryan, 2004), students have the liberty to design it according to their preference. This results to higher student engagement to the task which in turn improves learner motivation and achievement. Active engagement to exhibit pieces of evidence of growth can enhance an individual's sense of independence, competence, and self empowerment. These basic psychological needs, if satisfied together, can improve the learner's motivation, achievement, and future self-options.

Purposes of Learning Portfolio • Portfolio guides the learner and the teachers to set and establish goals aligned in the learning objectives. • The process of portfolio ensures the active participation of a learner and helps the learner to examine his/her growth and development over time. • The portfolio processes provide chances for self-evaluation and reflection. • Portfolio enhances the student's learning and current achievement and showcases and documents the development and growth in more a contextualized approach. • Portfolio can evaluate teaching effectiveness. Portfolio provides flexibility in curriculum and instruction planning because it highly considers the developmental domains of the learners and the contents of the subject matter. • Portfolio can help evaluate and improve the curriculum. • Portfolio reinforces hands-on and concrete experiences. • Portfolio can motivate parents and other stakeholders to become involved in the learner's evaluation plan.

Types of Learning Portfolio • Documentary Portfolio. This involves a collection of work over time, showing the growth and improvement reflecting the students' learning and identified outcomes. 140

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It is also called growth portfolio. The collection and exhibit of items can be based on specific educational goals or experiences of particular the learning area. • Process Portfolio. This demonstrates all facets or phases of the learning process, hence the arrangement is based on the learner's stages of metacognitive processing. This portfolio contains reflective journals, think logs, and other related pieces of evidence. • Showcase Portfolio. This is the kind that shows only the best of the students' output and products. • Evaluation Portfolio. This includes some work that was previously been submitted. • Class Portfolio. This contains a student grade and evaluative assessment of the student by the teacher. • Ideal Portfolio. This contains all the work a student has completed. In deciding the type of portfolio, the teacher needs to consider the level of the course, the age of the student, and the portfolio that will be used and evaluated.

Essential Characteristics of Portfolio Assessment 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 the contents of learning portfolios, hence the interaction and discourse are important elements of the process. Together, they formulate the objectives that are 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 output that show the student's 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 strengths and weaknesses so that the weaknesses can become improvement goals. 3. The criteria for selecting and assessing the portfolio especially the contents must be clear both to the teacher and students at the outset of the process. The set of agreed criteria can be referred to 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. Necessary planning in each step of the process of portfolio development can also be done especially by the learner. Unit SV • ASSESSMENT STRATEGIES FOR MATHEMATICS IN THE INTERMEDIATE GRADES

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

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

Table of Contents with numbered pages Entries (both the Core and Optional items) • Core items that are those necessary to be included • Optional items are those based on the learner's choice

• The Core items are the mandatory items based on the assessment criteria • The Optional items will be included to show the uniqueness of the learner's output, it may be the "best pieces of work" or "the trouble" or the "less successful" with a respective explanation but ultimately shall be based on the objectives set during initial stage.

Dates on all entries

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

Drafts of initial oral and written output and the revised versions

The first drafts and the corrected versions—to highlight the changes, identify the improvement, 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 entries

Provide a rationale for why the item was included as evidence of growth, development, 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?

Assess Verbalize your understanding of the portfolio assessment. 1. What are the essential characteristics of a learning portfolio?

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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 in assessing learning. 1. Does portfolio assessment contribute to better appreciation of mathematics? How about its contribution to mathematics achievement? Share your thoughts below.

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?

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H a rn ess The following activity will guide you in creating a rubric for checking a learning portfolio. This activity will be part of the learning portfolio that 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. Excellent ( ___points)

Criteria

Developing to Excellent ( ___points)

Beginner ( ___points)

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I l m H n e portfolio is one of the auAealic assessmentsflat DepEd recommends. is a collection of pieces of ewfcnrr of efforts, learnings, development, *■*. and achievement.

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Archbald, 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 high er 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 Con structivist Classrooms. Alexandria, VA: American Society for Curriculum Devel opment. De Lange, J. (1987). Mathematics, insight and meaning. Utrecht: Freudenthal Insti tute. 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 objec tives, 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. 147

Teaching math in the intermediate grades

Articles inside

Lesson 14: Values Integration

Lesson 15: Collaboration

Lesson 13: Use of Manipulatives

Lesson 12: Game-based Learning

Lesson 4: The Teaching Cycle

Lesson 7: Problem-Solving

Lesson 2: Mathematics Curriculum in the Intermediate Grades

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

Lesson 11: Design Thinking

Lesson 9: Concept Attainment

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

Lesson 10: Mathematical Investigation