1. Lesson 3 Constructivist Theory in Teaching Mathematics in the Primary Grades
Objectives
Demonstrate understanding and appreciation of the constructivist learning theory
Determine how the constructivist learning theory is applied in teaching mathematics in the early grades
Introduction
The constructivist learning theory states that learning is an active process of creating meaning from
different experiences. In other words, students learn best by trying to make sense of something on their
own with the teacher as a guide. DepEd (2016) specifically noted constructivist theory as the backbone
of the curriculum. According to DepEd, knowledge is constructed when the learner is able to draw ideas
from his/her own experiences and connect them to new ideas.
In this lesson, you will learn about the constructivist learning theory and how it is applied in teaching
mathematics in the primary grades.
Think
Constructivism was conceptualized by educational theorist Jean Piaget. Do you remember him from
your psychology classes? Piaget believed that young children learn by doing, constructing knowledge
from experiences rather than from adults telling them about their world. According to Piaget, and
others who practice what is known as constructivist education, the method most likely to truly educate
students is the one in which they experience their world. Constructivism is appropriately so applied in
teaching mathematics since math is a cumulative, vertically structured discipline. One learns new math
by building on the math that has been previously learned.
Brooks & Brooks (1993) listed the following characteristics of constructivist teaching.
1. Constructivist teachers invite student questions and ideas.
2. Constructivist teachers accept and encourage students' invented ideas.
3. Constructivist teachers encourage student's leadership, cooperation, seeking information, and
the presentation of the ideas.
4. Constructivist teachers modify their instructional strategies in the process of teaching based
upon students; thought, experience and or interests.
5. Constructivist teachers use printed materials as well as experts to get more information.
6. Constructivist teachers encourage free discussions by way of new ideas inviting student
questions and answers.
7. Constructivist teachers encourage or invite students' predictions of the causes and effects in
relation to particular cases and events.
8. Constructivist teachers help students to test their own ideas.
9. Constructivist teachers invite students' ideas before the student is presented with the ideas and
instructional materials.
10. Constructivist teachers encourage students to challenge the concepts and ideas of others.
11. Constructivist teachers use cooperative teaching strategies through student interactions and
respect, sharing ideas, and learning tasks.
12. Constructivist teachers encourage students to respect and use other people's ideas.
Experience
So how is a constructivist classroom different from a traditional classroom? In the constructivist
classroom, the focus shifts from the teacher to the students. The classroom is no longer a place where
the students are seen as empty vessels to be filled by the teacher. In a constructivist classroom, the
2. students are actively involved in their own learning. The teacher functions as a facilitator who guides,
prompts, and helps students to develop and assess their own understanding.
The table below compares the traditional classroom to the constructivist one. Notice differences in the
foci of the curricula and the roles of teachers and students.
Traditional Classroom Constructivist Classroom
Curriculum Curriculum begins with the
parts of the whole, emphasizing
basic skills.
Curriculum emphasizes big
concepts, beginning with the
whole and expanding to include
the parts.
Teacher's role Teachers disseminate
information to students;
students are recipients of
knowledge.
Teachers have a dialogue with
students, helping students
construct their own knowledge.
Student's role Students work primarily alone. Students work primarily in
groups.
Assess
Answer the following questions to verbalize your understanding of the constructivist learning theory.
1. What is the constructivist theory? Explain it in your own words.
Constructivist theory, to put it simply, is a teaching strategy that acknowledge that each
student’s understanding and knowledge are founded on their own “real life” experiences. The
statement that “reality is determined by the experiences of the learner” has been used to
express it by philosophers and others.
2. Expound why the constructivist theory is applicable in teaching mathematics.
The use of constructivist teaching techniques in mathematics classes enables students to go
beyond direct instruction, create meaningful context for understanding the material, and take
control of the learning process as an active participant instead of a passive observer.
Challenge
How well do you understand the constructivist learning theory? Consider the following scenarios and
answer the questions that follow.
Scenario 1
A teacher told the students, "Four glasses of water will fill this pitcher."
Scenario 2
A teacher provides a glass and lets the children pour water into the pitcher. They are learning how much
water it takes to fill the pitcher. In which scenario do you think will the students learn better? Why do
you think so?
The teacher provided a more thorough explanation of scenario 2 than scenario 1 did. Not all
kids grasp things we say right away, therefore we should clarify things to them more clearly so
they can understand. A strict and hurried learning environment may not be suitable for all
students. As a result, a discrepancy between their actual ability and performance level
develops, leading to the label of "slow learners" for these kids.
3. Harness
The following activity will direct your observation skills to the teaching style of the teacher. Note that
this is not an activity to criticize the teacher. The purpose is for you to develop keen observation skills on
teaching styles implemented in the classroom, and later on, suggest ways to improve the learning
activities. This activity will be part of the learning portfolio which you will compile at the end of this
module.
Observe a Grade 3 mathematics class. Did the lesson develop in a constructivist way? If yes, describe the
part of the lesson that followed constructivism. Otherwise, explain how you would revise the lesson in
order to facilitate a constructivist lesson.
When I observed a Grade 3 math class, the teacher encouraged the students to create
examples of situations that they would resolve using the lesson they were studying, allowing
the lesson to progress in a constructivist manner.
Summary
The constructivist learning theory states that learning takes place when we build on what students
already know. Moreover, it is student-centered, allowing the students to take ownership of their own
learning.
4. INSTRUCTIONAL PLANNING
In this unit, you will learn about how to plan, develop, and execute lessons in mathematics for the
primary grades. You will go over the learning cycle, the things to consider in lesson planning, and the
different instructional planning models.
Lesson 4 The Teaching Cycle
Objectives
Demonstrate an understanding and appreciation of the instructional planning cycle
Introduction
The work of a teacher does not start and end in teaching per se. The teaching process is not a linear
activity that starts with planning and ends with testing. Instead, it is a cycle of repeating stages until the
students acquire an understanding of the targeted concepts and skills. You may think of the teaching
cycle as a spring-you go through the same process over and over again, but each time with a more
informed objective and a better understanding of what it means to learn and teach mathematics.
Think
There are many models of the teaching cycle that various educators have developed over the years.
However, all models boil down to six common stages: (1) identify objectives, (2) plan instruction, (3)
implement plan, (4) check for understanding, (5) reflect on teaching, and (6) assess learning and reflect
on results.
The cycle that involves these stages is illustrated below. Assess learning and reflect on the results
Identify objectives reflect on teaching Check for understanding Implement plan The Teaching Cycle
Study the figure. What do you observe? Do you now get the idea of the teaching process as a cycle? The
following describes each stage of the learning cycle.
1. Identify objectives
What knowledge and/or skills do the students need to learn? You must be guided by the content
standards, performance standards, and the learning competencies that are found in the curriculum
guide.
2. Plan instruction what strategies must be implemented for the students to achieve the objectives
targeted in the previous stage? In planning instruction, it is important that you have mastered
the content of the lesson that you are about to teach. It is also beneficial to be familiar with your
students-what they know, how they learn, etc. You will learn more about instructional planning
in the next chapter.
3. Implement plan this is the stage where you conduct the learning activities that you have
prepared during the planning stage. A word of advice-even though you have carefully and
delicately planned for the lesson, you must be flexible with the possible changes that you need
to accommodate. How will you know whether change is needed? Read on to the next stage.
4. Check for understanding Teaching is about helping students learn. During the implementation of
the lesson plan, you must every now and then check whether the students have understood
what you have covered so far. Facial reactions and verbal cues help in assessing whether or not
the students can move on to another concept or skill. If not, you might need to give a more
5. elaborate explanation, more examples, or whatever you think is needed based on the students'
reactions. This stage also makes use of formative assessments which you will learn more about
in Chapter 17.
5. Reflect on teaching you must evaluate every teaching period that you finished. Were the
objectives achieved? Were the implemented strategies effective? How can instruction be
improved? Your answers to the last two questions will give you an insight on how to improve
instruction the next time you teach the same lesson. However, if your answer in the first
question is no, i.e., the objectives were not met, then you need to plan again. What do you need
to do differently in order to achieve the objectives?
6. Assess learning and reflect on the results
This stage gives you a concrete measure of what the students have learned. In math, this is usually
through a paper-and-pen examination. However, some authentic assessments may also be implemented
as you will learn in the later chapters of this book. Take note that this stage does not end in assessing
learning. You need to reflect on the results. What can you learn about student learning and teaching
practice based on the results? After assessment and reflection; you will once again identify the next
learning goals and so the cycle continues.
Experience
The following is a narrative of how a teacher might experience the teaching cycle.
1. Identify objectives
Teacher Gina identified "multiplication of whole numbers up to two digits" as the goal of her next
lesson.
2. Plan instruction Teacher Gina thought it is best to apply a constructivist approach to help her
students learn techniques in multiplying whole numbers. She planned a lesson which
incorporates the problem-solving strategy.
3. Implement plan
The class went on smoothly. The activities that Teacher Gina prepared were successfully done by her
students.
4. Check for understanding
To make that her students understood the lesson, Teacher Gina gave a three-item exercise as an exit
pass.
5. Reflect on teaching Based on the exit pass, Teacher Gina found out that many of the students
have difficulty multiplying numbers that involve the digit 8. So, she decided to do a find-your-
error activity the next day for the students to realize their mistakes. She also planned to give a
short drill on skip counting by 8.
6. Assess learning and reflect on the results
Teacher Gina, later on, gave a multiplication quiz. Ninety percent of the students passed. She planned to
give remedial exercises to those who failed. This teaching cycle taught Teacher Gina that students can
discover concepts on their own. However, they must still be guided by a teacher because
misconceptions may arise.
6. Assess
Answer the following questions to verbalize your understanding of the teaching cycle.
1. In which stage/s of the teaching cycle are the students involved? Explain.
Identifying Objectives because I this stage you will know what knowledge or skills do the
students need to learn that we need to be guided by the content standards, performance
standards, and the learning competencies that we need to the curriculum guide.
2. Which stage/s of the teaching cycle requires the teacher to reflect on teaching and learning?
Explain.
Check for Understanding why, because in this stage you will know what student learn during
the implementation of the lesson plan, you will know what students learn about your lesson
plan and what students earn knowledge.
Challenge
The next question will challenge your reasoning skill. What do you think is the most important stage of
the learning cycle? Why do you think so?
For me, being present in class and contributing are crucial components of the learning process.
As you read about the learning cycle, I believe it makes it simpler for you to remember the
lessons you've already learned and to refresh your memory of those lessons that will be useful
for future reference.
Harness
Aside from classroom observations, a lot can be learned from conversations with teachers in the field.
The following activity will require you to interview math teachers and summarize what you learn from
them in a diagram. This activity will be part of the learning portfolio which you will compile at the end of
this module.
1. Interview two mathematics teachers. Ask them about the stages of the teaching cycle that they
follow. Then, create a diagram illustrating their common answers.
Teacher 1
Learn
Decide
Interpret
Challenge
Anticipate
Teacher 2
Concrete experience
Active experimentation
Reflective observation
Abstract
conceptualization
7. 2. How is the diagram you created in #1 similar or different from the cycle that was presented in
this lesson?
The teaching cycle is a continually repeated process that effective teachers engage in to assess
students' requirements, prepare education, give instruction, evaluate results, and then
reassess students' needs. For seasoned educators, a lot of this is carried out internally. The
diagram I created for Teaching 1 shows how they employed a similar teaching cycle and an
approach to instructional design that capitalizes on what is known about how people learn.
Many teachers use various teaching cycles to produce positive results and guarantee that new
learning takes place. Children can learn how to better respect and care for others around
them by being taught about and seeing lifecycles illustrated. This can be demonstrated in a
variety of ways and aid youngsters in understanding and coping with birth, death, and other
recurring changes and growth in their environment.
Summary
Teaching involves a repetitive cycle of defining objectives, planning and implementing instruction,
assessing learning, and reflecting on teaching and learning. Each part of the cycle provides a better
understanding of what it means to teach and learn mathematics and should result in better instruction
in the next repetition of the cycle.
8. Lesson 5
Things to Consider in Planning Instruction in Mathematics in the Primary Grades
Objectives
Demonstrate understanding and appreciation of the things to consider in planning instruction in
mathematics in the primary grades
Introduction
In education, planning refers to the designing and preparation of learning activities for students. In
lesson planning, the teachers thoughtfully contemplate about the lesson objectives, the activities that
will meet these objectives, the sequence of those activities, the materials needed, how long each
activity might take, how the class would be managed during those activities, and the evaluation method
to assess how far the objectives were met. This lesson enumerates the things to consider in planning
instruction in mathematics in the primary grades.
Think
There are five important elements in lesson planning that you need to consider- content, objectives,
students, learning environment, and availability of resources.
1. Content
Research the subject matter that you will be teaching. You should consult the curriculum and teaching
guides published by DepEd. Aside from books, you can also visit websites which will give you
information relevant to your subject area. You should master the contents of your lesson before you
teach it. Remember, you cannot give what you do not have. Moreover, you would not want to teach
wrong contents to the students. It is easier to learn than to unlearn; it is difficult to take back wrong
contents that have already been taught. You have a big responsibility as a teacher-master your content!
2. Objectives
Before you begin planning, you need to know what specific knowledge and skills you want your students
to develop during the lesson or unit.
Teachers often focus too much on knowledge, forgetting about developing skills which in the long term
are more important than knowing mere facts. So, in planning your instruction, always consider both
knowledge and skills.
3. Students
Get to know your students-where they came from, what their interests are, what they already know,
their learning style, attention span, special needs, etc. These will all help you determine your students'
needs. Remember that you need to prepare your lessons with all your students in mind and that your
main goal should be to meet their needs and offer them enabling environments to learn their preferred
way. Knowing your students will also help you build rapport with them which is important if you want
your students to be freely sharing their ideas with you and their classmates. Another important
consideration that needs serious attention in teaching, especially mathematics, is the students' mindset.
You may have all things considered-lesson mastery, focused objectives, and comprehensive
understanding of students-and yet still find that the lesson is not coming through the students. This may
be because the students have closed their doors toward math. Many school children have come to
believe that math is difficult and they can never be good at it. This is called a fixed mindset. Students
with fixed mindsets believe that their math skills cannot be improved, which results in
underperformance in the subject. Reasons for fixed mindset include influence from adults who dislike
math, previous unpleasant experience in math class, and others. Your goal as a teacher is to develop
students with growth mindset. Students with growth mindsets believe that they can be better at math.
They know that their efforts are not wasted and that they can learn even in their failures. Many studies
9. have proven that students who have a growth mindset perform better in school than those who have a
fixed mindset. So, in planning your lesson, you must consider how to encourage growth mindset in class.
4. Learning environment
Aside from the physical environment where the learning takes place, it is also important to consider the
social and emotional learning environment of the class. You need to make sure that you promote a
positive environment where students are motivated and are supportive of each other's growth. The
students must feel safe to express their thinking, without fear of being embarrassed because of mistakes
or different views. Most importantly, you must create an atmosphere where students are open to
learning through the activities you prepared and interactions with their classmates.
5. Availability of resources Take into consideration the instructional materials that you will be
needing before you write your lesson plan. Is a blackboard available? If not, can you improvise?
Are there specific manipulatives that you need? Where can you get them? Can you make them
instead? Do you need technology resources? Have you checked whether your devices are
compatible with what are available in school? These are some of the questions that you can
reflect on.
Experience
The next activity will delve into the experiences of math teachers and will give you insights on effective
lesson planning. Interview three experienced primary grades (Grades 1, 2, or 3) mathematics teachers.
Ask him/her the following question: "If you were to give a piece of advice about lesson planning to your
rookie teacher self, what would it be?" What are common about their responses? Write them down
below.
Assess
Answer the following questions to verbalize your understanding of the things to consider when planning
instruction in mathematics.
1. In addition to what has been discussed, explain why content, objectives, students, learning
environment, and availability of resources are the essential considerations in planning a lesson.
These elements will help them learn during the lesson. Content should be considered in lesson
planning as it guides the teacher on what to teach the students. An objective tells the
students what the lesson is about. This refers to the students’ ability to provide the resources
needed for the lesson.
2. Sketch an infographic about the difference of growth mindset and fixed mindset.
10. Challenge
The following questions will challenge your reasoning and critical thinking skills. I will also initiate
reflection on the kind of mathematical mindset you had as a student
1. Why is it important to be in consultation with the curriculum guide when planning instruction?
2. Why do you think is having a fixed mindset a setback in learning? Can you think of specific
examples when you were a student and had a tendency of having 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 discussed in the next chapter) while
focusing on how content, objectives, and students were given attention to in the plan. This activity will
be part of the learning portfolio which you will compile at the end of this module. Borrow a lesson plan
from a primary grade mathematics teacher. Give specific examples in his/her lesson plan wherein you
saw the conscious consideration for content, objectives, students, learning environment, and availability
of resources.
Content
Objectives
Students
Learning environment
Availability of resources
Summary
Before writing a lesson, teachers are expected to thoughtfully contemplate on the objectives, review the
content, and get to know the learners. Doing these will help them plan a relevant and effective lesson
for the learners.
11. Lesson 6 Instructional Planning Models
Objectives
Demonstrate understanding and appreciation of the most commonly used instructional planning models
in the Philippines
Introduction
Now that you have learned the things to consider when planning instruction, you are ready to create
one yourself. Teachers usually plan lessons following a specific model. In this lesson, you will learn about
the two most commonly used instructional planning model in the Philippines and their common
features.
Think
There are many instructional planning models that mathematics educators have constructed, but the
two most widely used in the Philippines are the ADIDAS and the Five E’s models.
ADIDAS stands for Activity, Discussion, Input, Deepening, Activity, and Summary.
Activity. The lesson begins with an activity that will later facilitate a meaningful discussion about the
topic of the session. In other words, the activity introduces the topic to the students. This activity must
be motivating and engaging to catch the attention of the students.
Discussion. The lesson proceeds with the processing of the activity. In this part, the students, as
facilitated by the teacher, talks about their experiences during the activity. Here, the questioning skills of
the teacher is important because he/she must be able to direct the discussion toward the targeted
lesson.
Input. In a traditional classroom, the Input is where the teacher lectures. However, in a constructivist
classroom), this is the part where the students would share the concepts that they learned based on the
activity and the discussion. Nevertheless, no matter which learning theory is applied in the lesson, this is
the part where the concepts are clearly established.
Deepening. Here the teacher asks questions that will engage the students to critical and creative
thinking. No routine 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 Five E’s. The Five E’s are
Engage, Explore, Explain, Elaborate, and Evaluate.
Engage. This part activates the students' prior knowledge and engages them into new concepts by doing
short activities. The aim of this part is to arouse the students' curiosity.
12. Explore. In this part, the students are exposed to different experiences that will facilitate the discovery
of new concepts. Explore may involve observation exercises, simulations, or manipulations of
instructional materials. The goal here is for the students to discover something new.
Explain. Here the students explain what they have experienced in Explore. The role of the teacher is to
facilitate the discussion that should lead to students seeing patterns that will help them to describe the
new concept in their own words.
Elaborate. The Elaborate part of the lesson allows students to expand their understanding of the
concept by applying the concept that they have learned in solving mathematical problems.
Evaluate. The last part of the Five Es model, Evaluate, lets the teacher and the students evaluate their
learning. Though giving short exercises are usually the mode of evaluation, the teacher can be creative
by implementing other evaluation activities.
Experience
Aside from the components of whatever instructional planning model, an instructional plan also reflects
basic information about the lesson like prerequisite knowledge and skills, time allotment, materials
needed, etc. Below is a sample template of a lesson plan.
Topic:
Subject:
Grade Level:
Duration:
Objectives:
At the end of the session, the student will be able to:
Prerequisite Concepts/Skills:
New Concepts/Skills:
Materials:
References:
Lesson Proper
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 Five Es model?
Match the components of the two models to summarize the similarities that you saw.
• Activity • Engage
•Discussion • Explore
• Input • Explain
• Deepening • Elaborate
• Activity • Evaluate
•Synthesis
2. Explain the matching you did in #1.
13. Challenge
Even though ADIDAS and Five Es are commonly used, they also receive criticisms such as not being
applicable to some topics in math. The following questions will challenge your reasoning skills regarding
this issue of applicability of instructional planning models.
1. Do you think the ADIDAS or the Five Es model is applicable to planning any lesson in
mathematics? Explain your thought.
2. What if, in the school where you will be employed, a different instructional planning model is
used? Do you think you will have a hard time adjusting? Explain.
Harness
In this activity, you will be asked to refer to the lesson plan you previously studied in Chapter 5. This
time focus your analysis on the different components of the lesson plan in relation to the ADIDAS and
Five Es models. This activity will be part of the learning portfolio which you will compile at the end of
this module.
Refer to the lesson plan you collected in the previous chapter and do the following.
1. Extract parts of his/her lesson plan that exhibits the components of
a. ADIDAS
Activity
Discussion
Input
Deepening
Activity
Synthesis
b. Five E’s
Engage
Explore
Explain
Elaborate
Evaluate
2. Are there components of the ADIDAS/Five E’s model that were not reflected in this lesson plan?
If you are to fill in the missing parts, what would you write?
