UNIT 2 | UNDERLYING PRINCIPLES AND STRATEGIES
Lesson 2.1 – Constructivism in Mathematics Teaching
Constructivism
Since the 1980s there has been a growing
acceptance of constructivist theories of learning.
Central to these theories is the claim that students
learn by active meaning-making and by involvement
their own learning processes.
These learner activities can be visible, such as
inactive problem solving, or not, such as when a
learner is listening intently to understand an
explanation. In each case, the learner is making
sense of the mathematical situation presented, and
drawing upon previously learned knowledge, skills
understandings to do so.
in
and
It was the influence of the great Swiss psychologist Jean Piaget which established constructivism
as a leading theory of learning mathematics. The Ideas about Mathematics Education
Piaget’s works had a great influence on mathematics education. Piaget’s ideas were considered a new
field in mathematics education, which encouraged the development of cognitive knowledge.
According to Piaget’s theory of cognitive development, it is not possible to provide information that
will be immediately understood and used, but the students themselves build their knowledge as a result
of assimilation and accommodation processes.
Following Glasersfeld (1987), perceiving is always an active making rather than a passive receiving. He
claimed that knowledge is not passively received but actively built up by the cognizing subject and
knowledge is the result of a self-organized cognitive process.
The Russian psychologist Vygotsky emphasizes the importance of the social environment in cognitive
development as well as culture and people as the most important factors in the development of an
individual. He proved that an individual cannot develop without interacting with the environment.
Five Aspects of Constructivism Regard to Mathematics
1. Learning is a process of interaction between what we know and what we still need to learn.
2. Learning is a social process – “is not in leads, but in relations between people.
3. Learning is a situational process, i.e., “participation in certain social and cultural circumstances”;
4. Learning is a metacognitive process that “includes the understanding of the skills and strategies
that enable successful resolutions of the problems, and to use these skills and strategies to learn
effectively.
5. Learning is based on the students’ activity and autonomy.
According to the theory of constructivism, the study process includes the following components (Parsons,
Hinson, & Sardo-Brow, 2002):
a. A student creates knowledge himself with the meaning that they have to find.
b. This means understanding both the whole and individual parts. Understanding interconnections
must be found in the learning process.
c. Motivation is an essential learning indicator.
d. Self-directed learning is a process.
Remember!
Educational environments should be structured to cause students to develop more powerful
thinking teaching and classroom environments change if you accept that students must construct their
knowledge (Clements, Battista, 2009). Students’ differences also have to be taken into account.
Lesson 2.2 – Teaching for Understanding in Mathematics Teaching
An understanding can never be “covered” if it is to be understood. Wiggins and McTighe (2005, p. 229)
The National Council of Teachers of Mathematics (NCTM, 2000) identifies the process standards
of problem-solving, reasoning and proof, representation, communication, and connections as ways to
think about how students should engage in learning mathematics content as they develop both procedural
fluency and conceptual understanding.
Students engaged in the process of problem-solving build mathematical knowledge and
understanding by grappling with and solving genuine problems as opposed to completing routine
exercises.
They use reasoning and proof to make sense of mathematical tasks and concepts and to develop,
justify, and evaluate mathematical arguments and solutions.
Students create and use representations (e.g., diagrams, graphs, symbols, and manipulatives) to
reason through problems.
They also engage in communication as they explain their ideas and reasoning verbally, in writing,
and through representations.
Students develop and use connections between mathematical ideas as they learn new
mathematical concepts and procedures.
FIVE STRANDS OF MATHEMATICAL PROFICIENCY IN UNDERSTANDING MATHEMATICS
EIGHT STANDARDS FOR MATHEMATICAL PRACTICE
1. Make sense of problems and persevere in solving them.
- To make sense of problems, students need to learn how to analyze the given information, the
parameters, and the relationships in a problem so that they can understand the situation and
identify possible ways to solve it.
2. Reason abstractly and quantitatively
- This practice involves students' reasoning with quantities and their relationships in problem
situations.
3. Construct viable arguments and critique the reasoning of others
- This practice emphasizes the importance of students’ using mathematical reasoning to - justify
their ideas and solutions, including being able to recognize and use counterexamples.
4. Model with mathematics
- This practice encourages students to use the mathematics they know to solve problems from
everyday life.
5. Use appropriate tools strategically.
- Students should become familiar with a variety of problem‐solving tools and they should learn
to choose which ones are most appropriate for a given situation.
6. Attend to precision
- In communicating ideas to others, students must learn to be explicit about their reasoning.
7. Look for and make use of the structure.
- Students who look for and recognize a pattern or structure can experience a shift in their
perspective or understanding.
8. Look for and express regularity in repeated reasoning
-
Encourage students to step back and reflect on any regularity that occurs to help them develop
a general idea or method to identify shortcuts.
Note that learning these mathematical practices and developing understanding takes time. So, the
common notion of simply and quickly “covering the material” is problematic. Understanding is an end
goal—that is, it is developed over time by incorporating the process standards and mathematical practices
and striving toward mathematical proficiency.
Lesson 2.3 – Dale’s Cone of Experience
“The cone is a visual analogy, and like all analogies, it does not bear an exact and detailed relationship to
the complex elements it represents.” -Edgar Dale
What is the Cone of Experience?
• First introduced in Dale’s 1946 book,
Audio-Visual Methods in teaching.
•
•
The elements of the Cone of
Experience are the 2M’s of
instruction namely the media and
material. It guides the teachers in
choosing the kind of instructional
materials to teach.
Designed to “show the progression
of learning experiences” (Dale
(1969) o. 108) from the concrete to
the abstract.
Concrete vs Abstract Learning
Influences on the Cone of Experience
•
•
•
Hoban, Hoban & Zisman’s Visual Media Graph
- The value of educational technology is based on its degree of realism
Jerome Bruner’s Theory of Instruction
- Three levels in the learning process
- Enactive-direct experience
- Iconic- representation of experience
- Symbolic- words of visual symbols
The process of learning must begin with concrete experiences and move toward the abstract if
mastery is to be obtained.
Mis-conceptions of the Cone
 All teaching /learning must move from the bottom to the top of the Cone.
 One kind of experience on the Cone is more useful than another.
 More emphasis should be put on the bottom levels of the Cone.
 The upper level of the Cone is for older students while the lower levels are for younger students.
 It overemphasizes the use of instructional media.
ENACTIVE
→ Refers to the direct experiences or encounters with what is.
→ This is life on the raw, rich, and unedited.
→ They form the basis for all other learning experiences.
→ Example: (Actual swimming lesson)
Direct Purposeful Experiences
-
“First-hand experiences”
Have direct participation in the outcome
Use of all our senses
Contrived Experience
-
Models and mock-ups
“Editing of reality”
Necessary when real experience cannot be used or is too complicated.
Examples: Conducting election of class and school officers and Mock-up of a clock.
Dramatized Experiences
-
“Reconstructed experiences”
Can be used to simplify an event or idea to its most important parts.
Divided into two categories
Acting (Role playing)-actual participation
Observing-watching a dramatization take place
Other forms: Plays, puppets, pageant, pantomime, tableau.
ICONIC EXPERIENCES ON THE CONE
→ Progressively moving toward greater use of imagination
→ Successful use in a classroom depends on how much imaginative involvement the method can
illicit from students.
Demonstrations
-
A visualized explanation of an important face, idea, or process by the use of: photographs,
drawings, films, displays, and guided, motions.
Showing how things are done (how to play the piano)
Visualized explanation of an important fact, idea, or process
Demonstrations are a great mixture of concrete hands-on application and more abstract verbal
explanation
Study/Field Trips
-
Watch people do things in real situations.
Observe an event that is unavailable in the classroom
There are the excursions, educational trips, and visits conducted or observe an event that is
unavailable with in the classroom.
Exhibits
-
These are displays to be seen by spectators.
-
May consist of working models, charts, and posters.
-
Sometimes are “for your eyes only” more visual.
Two types:
1. Ready Made (Museum, career fair)
2. Home- made (classroom project, national history day competition)
Educational Television and Motion Pictures
Recordings, Radio, and Still Pictures
-
Can often be understood by those who cannot read
Helpful to students who cannot deal with the motion or pace of a real event or television.
These are visual or auditory devices which may be used by an individual or a group.
Examples: Time Life Magazines, listening to old radio broadcasts, listening to music
SYMBOLIC
→ Refers to the use of words or printed materials which no longer resemble the object under study.
→ Example the word whale. Upon reading or hearing the word whale, the learner can form a
mental image of it.
Visual Symbols
-
It no longer involves reproducing real situations
Chalkboards and overhead projectors are the most widely used media
Helps students see an idea, event, or process
Examples; Chalkboard, flat maps, diagrams and charts Verbal Symbols
They are not like the objects or ideas for which stand. They usually do not contain visual clues to
their meaning.
Two types:
-
1. Written words- more abstract
2. Spoken words- less abstract
Lesson 2.4 – Various Constructivism Strategies in Teaching Mathematics
Repetition
A simple strategy teachers can use to improve math skills is repetition. By repeating and reviewing
previous formulas, lessons, and information, students are better able to comprehend concepts at a faster
rate.
According to Professor W. Stephen Wilson from Johns Hopkins University, the core concepts of basic
math must be mastered before students can move into more advanced study. Repetition is a simple tool
that makes it easier for students to master the concepts without wasting time. According to the University
of Minnesota, daily re-looping or reviews will bring the previous lesson back into the spotlight and allow
teachers to build on those previous skills.
Timed Testing
When teachers are moving beyond the simple concepts of numbers into addition, subtraction,
multiplication, and division, it is important to incorporate timed tests that review the previous class or
several classes.
Taking a short test and then grading the test in class will help teachers assess student understanding.
When the test shows that students are answering more questions correctly within the time period,
teachers are able to determine that students have mastered the basic skills.
Pair Work
Mathematics is not limited to learning from a textbook, lessons, or testing strategy. Students have different
learning styles and need to have lessons that help improve all styles of learning to get the best results.
Group work is a simple strategy that allows students to work and problem-solve with a buddy. When a
teacher has provided the basic instruction, it’s helpful to split the class into pairs or groups to work on
problems.
Since the pairs are working as a team, the students can discuss the problems and work together to solve
the issues. The goal of pair work is to teach students critical thinking skills that are necessary for future
math problems and real life.
Manipulating Tools
The use of blocks, fruits, balls, or other manipulation tools helps students learn the basics of place
value, addition, subtraction, and other areas of basic math. According to Kate Nonesuch on the National
Adult Learning Database of Canada, manipulation tools help slow down the process of problem-solving
so that students can fully understand the information.
Manipulation tools make it easier for students to learn and understand basic skills. These are ideal when
students learn best through hands-on experience and building, rather than traditional lessons and
repetition.
Math Games
Reinforcing the information learned in class is not always the easiest task for teachers, but math games
provide the opportunity to make the lesson interesting and encourage students to remember the concepts.
Depending on the class size, computer availability, and the lesson being taught, games can vary. Teachers
can use computer games for particular skills or can opt to use class games to make the lesson more fun.
Teachers should be sure to incorporate a strategy into games to help students learn the material.
Math skills are an important part of life. To offer students the most help, teachers need to incorporate
several strategies to give students the opportunity for future growth.
NAME: _____________________________
Date:___________
Score:__________
PSTM: QUIZ 2: UNDERLYING PRINCIPLES AND STRATEGIES
I. Multiple Choice Questions (15 items). Direction: Choose the letter of the correct answer. Write
your answer on the space provided.
_______1. Who is known for establishing constructivism as a leading theory in learning mathematics?
a. Vygotsky
b. Piaget
c. Bruner
d. Glasersfeld
_______2. According to Piaget, knowledge is built through which processes?
a. Listening and observing
b. Imitation and repetition
c. Assimilation and accommodation
d. Memorization and practice
_______3. Which of the following emphasizes the importance of the social environment in cognitive
development?
a. Piaget
b. Bruner
c. Glasersfeld
d. Vygotsky
_______4. The five aspects of constructivism include the following, EXCEPT:
a. Learning as a process of communication
b. Learning as a metacognitive process
c. Learning as a process of social reasons
d. Learning is a situational process
_______5. What is the primary purpose of Dale’s Cone of Experience?
a. To expand on the concrete and abstract learning of students
b. To emphasize on the information of different skills of students
c. To guide teachers in choosing instructional materials
d. To discourage the use of media
_______6. According to Glasersfeld, knowledge is:
a. Passively received
b. Actively built up
c. Fixed and unchanging
d. Developed through social environment
_______7. What type of learning involves using diagrams and charts to visualize concepts?
a. Enactive learning
b. Verbal learning
c. Iconic learning
d. Visual learning
_______8. Which strategy encourages the use of blocks or models to simplify math concepts?
a. Timed testing
b. Pair work
c. Manipulating tools d. Repetition
_______9. What is the term for direct experiences that involve all senses?
a. Contrived experiences
b. Symbolic experiences
c. Direct purposeful experiences
d. Dramatized experiences
_______10. Which is NOT part of the eight standards for mathematical practice?
a. Model with mathematics
b. Attend to precision
c. Make sense of problems
d. Reason abstractly and qualitatively
_______11. Which type of experience is closest to real life in Dale’s Cone of Experience?
a. Enactive experiences
b. Iconic experiences
c. Symbolic experiences
d. Abstract experiences
_______12. The Russian psychologist Vygotsky believed cognitive development occurs through:
a. Individual study
b. Active listening
c. Social interaction
d. Direct observation
_______13. What is the final level of Dale’s Cone of Experience?
a. Dramatized experiences
b. Contrived experiences
c. Iconic experiences
d. Symbolic experiences
_______14. Which strategy involves breaking down lessons into smaller, repeatable steps?
a. Pair work
b. Repetition
c. Timed testing
d. Math games
_______15. What type of knowledge results from students’ interaction with their environment?
a. Constructed knowledge
b. Passive knowledge
c. Predefined knowledge
d. Abstract knowledge
II. True/False Questions (10 items). Direction: Write "True" if the statement is correct and "False" if it
is not.
_______1.
_______2.
_______3.
_______4.
_______5.
Constructivism emphasizes active participation in the learning process.
Vygotsky’s theory claims that individual learning is independent of social interaction.
The Cone of Experience begins with abstract learning and progresses toward concrete learning.
Manipulation tools are most effective for students who learn through hands-on experiences.
Timed testing is not useful in assessing mastery of basic math skills.
_______6. Learning is a process of interaction between what we know and what we still need to learn.
_______7. Dale’s Cone of Experience only applies to older students.
_______8. Mathematical games help reinforce concepts learned in class.
_______9. Problem-solving involves only routine exercises.
_______10. Repetition aids in mastering core math concepts.
III. Matching Type (10 items). Direction: Match the terms in Column A with their corresponding
descriptions in Column B. Write the letter of the correct answer.
Column A
Column B
_______1. Constructivism
a. Building knowledge using previously learned concepts
_______2. Assimilation
b. Hands-on tools for understanding math concepts
_______3. Enactive Learning
c. Learning through direct experience
_______4. Mathematical Proficiency
d. Cognitive theory emphasizing active participation
_______5. Manipulation Tools
e. Developing fluency, reasoning, and problem-solving skills
_______6. Iconic Learning
f. Progressively moving toward imagination
_______7. Symbolic Learning
g. Involves written or verbal symbols
_______8. Vygotsky
h. Social and cultural factors in learning
_______9. Piaget
i. Cognitive development through assimilation
_______10.Direct Purposeful Experience j. Learning through first-hand engagement
IV. Fill-in-the-Blank Questions (5 items). Direction: Complete the following statements with the
correct answer.
1.
2.
3.
4.
5.
According to Piaget, students build their knowledge through ______ and ______ processes.
The theory that emphasizes the role of social interaction in learning is _______.
_______ refers to reconstructed experiences like plays and role-playing.
A _______ is a strategy where students work in pairs to solve math problems.
Dale’s Cone of Experience guides teachers in selecting appropriate _______ and _______ for
instruction.
Answer Key
I. Multiple Choice Questions
1. b
2. c
3. d
4. c
5. c
6. b
7. d
8. c
9. c
10. d
11. a
12. c
13. d
14. b
15. a
II. True/False Questions
1. True
2. False
3. False
4. True
5. False
6. True
7. False
8. True
9. False
10. True
III. Matching Type
1. d
2. a
3. c
4. e
5. b
6. f
7. g
8. h
9. i
10. j
IV. Fill-in-the-Blank Questions
36. assimilation, accommodation
37. Vygotsky’s theory
38. Dramatized experiences
39. pair work
40. media, materials