1. THE DEVELOPMENT
OF TEACHING AND
LEARNING
MATHEMATICS
GLOBALLY AND
LOCALLY
Theories and Principles
Servino, Edel A.
Bron, Irene B.
Delatado, Mary Joy
Ibo, Jessica
Penero, Mary Jonabelle
Villareal, Antonette

2. Mathematics Education
- In contemporary education, mathematics education is the practice of teaching and
learning mathematics, along with the associated scholarly research.
- Researchers in mathematics education are primarily concerned with the tools, methods
and approaches that facilitate practice or the study of practice.
History
- Elementary mathematics was part of the education system in most ancient civilisations,
including Ancient Greece, the Roman empire, Vedic society and ancient Egypt. In most
cases, a formal education was only available to male children with a sufficiently high
status, wealth or caste.
- In Plato's division of the liberal arts into the trivium and the quadrivium, the quadrivium
included the mathematical fields of arithmetic and geometry.
- The first mathematics textbooks to be written in English and French were published by
Robert Recorde, beginning with The Grounde of Artes in 1540.
- In the Renaissance, the academic status of mathematics declined, because it was strongly
associated with trade and commerce.
- This trend was somewhat reversed in the seventeenth century, with the University of
Aberdeen creating a Mathematics Chair in 1613, followed by the Chair in Geometry
being set up in University of Oxford in 1619 and the Lucasian Chair of Mathematics
being established by the University of Cambridge in 1662.
- In the 18th and 19th centuries, the industrial revolution led to an enormous increase in
urban populations. Basic numeracy skills, such as the ability to tell the time, count money
and carry out simple arithmetic, became essential in this new urban lifestyle.
- By the twentieth century, mathematics was part of the core curriculum in all developed
countries.
Objectives
At different times and in different cultures and countries, mathematics education has attempted
to achieve a variety of different objectives. These objectives have included:
• The teaching of basic numeracy skills to all pupils
• The teaching of practical mathematics (arithmetic, elementary algebra, plane and solid
geometry, trigonometry) to most pupils, to equip them to follow a trade or craft
• The teaching of abstract mathematical concepts (such as set and function) at an early age
• The teaching of selected areas of mathematics (such as Euclidean geometry) as an
example of an axiomatic system and a model of deductive reasoning

3. • The teaching of selected areas of mathematics (such as calculus) as an example of the
intellectual achievements of the modern world
• The teaching of advanced mathematics to those pupils who wish to follow a career in
Science, Technology, Engineering, and Mathematics (STEM) fields.
• The teaching of heuristics and other problem-solving strategies to solve non-routine
problems.
Methods
• Conventional approach - the gradual and systematic guiding through the hierarchy of
mathematical notions, ideas and techniques. Starts with arithmetic and is followed by
Euclidean geometry and elementary algebra taught concurrently.
• Classical education - the teaching of mathematics within the quadrivium, part of the
classical education curriculum of the Middle Ages, which was typically based on Euclid's
Elements taught as a paradigm of deductive reasoning.
• Rote learning - the teaching of mathematical results, definitions and concepts by
repetition and memorization typically without meaning or supported by mathematical
reasoning. A derisory term is drill and kill. In traditional education, rote learning is used
to teach multiplication tables, definitions, formulas, and other aspects of mathematics.
• Exercises - the reinforcement of mathematical skills by completing large numbers of
exercises of a similar type, such as adding vulgar fractions or solving quadratic equations.
• Problem solving - the cultivation of mathematical ingenuity, creativity and heuristic
thinking by setting students open-ended, unusual, and sometimes unsolved problems.
Problem solving is used as a means to build new mathematical knowledge, typically by
building on students' prior understandings.
• New Math - a method of teaching mathematics which focuses on abstract concepts such
as set theory, functions and bases other than ten.
• Historical method - teaching the development of mathematics within an historical, social
and cultural context. Provides more human interest than the conventional approach.
• Standards-based mathematics - a vision for pre-college mathematics education in the

4. US and Canada, focused on deepening student understanding of mathematical ideas and
procedures, and formalized by the National Council of Teachers of Mathematics which
created the Principles and Standards for School Mathematics.
1. Transfer of Learning
- Transfer of learning deals with transferring one's knowledge and skills from one problem-
solving situation to another. You need to know about transfer of learning in order to help
increase the transfer of learning that you and your students achieve.
- For example, suppose that when you were a child and learning to tie your shoes, all of
your shoes had brown, cotton shoelaces. You mastered tying brown, cotton shoelaces.
Then you got new shoes. The new shoes were a little bigger, and they had white, nylon
shoe laces. The chances are that you had no trouble in transferring your shoe-tying skills
to the new larger shoes with the different shoelaces.
Near Transfer
- A nearly similar problem or task is automatically solved with little or no conscious
thought.
In recent years, Salomon & Perkins (1988) developed the low-road/high-road theory on transfer
of learning and proven to be a more fruitful theory.
Low-road transfer
- refers to developing some knowledge/skill to a high level of automaticity. It usually
requires a great deal of practice in varying settings. Shoe tying, keyboarding, steering a
car, and single-digit arithmetic facts are examples of areas in which such automaticity can
be achieved and is quite useful.
High-road transfer
- involves: cognitive understanding; purposeful and conscious analysis; mindfulness; and
application of strategies that cut across disciplines.
- In high-road transfer, there is deliberate mindful abstraction of an idea that can transfer,
and then conscious and deliberate application of the idea when faced by a problem where
the idea may be useful.
2. General Learning Theory
Benjamin Bloom
- is probably best known for his 1956 "taxonomy."
- he also did seminal work on student learning through different methods such as tutoring, peer

5. tutoring, mastery learning, and so on.
Knowledge: Observe and recall information knowledge of dates, events, places
• know major ideas, mastery of basic subject matter
verbs:
• list, define, tell, describe, identify, show, label, collect, examine, tabulate, quote, name,
who, when, where
Comprehension
• understand information grasp meaning translate knowledge to a new context interpret
facts, compare, contrast order, group, infer causes predict consequences
verbs:
• summarize, describe, interpret, contrast, predict, associate, distinguish, estimate,
differentiate, discuss, extend
Application
• use information, use methods, concepts, theories in new situations solve problems; use
required skills or knowledge
verbs:
• apply, demonstrate, calculate, complete, illustrate, show, solve, examine, modify, relate,
change, classify, experiment, discover
Analysis
• see patterns, organize the parts, recognize hidden meanings, identify components
verbs:
• analyze, separate, order, explain, connect, classify, arrange,divide, compare, select,
explain, infer
Synthesis
• use old ideas to create new ones, generalize from given facts relate knowledge from
several areas, predict, draw conclusions
verbs:
• combine, integrate, modify, rearrange, substitute, plan, create, design, invent,
• what is it?, compose, formulate, prepare, generalize, rewrite
Evaluation
• compare/discriminate between ideas, assess value of theories, make choices based on
argument, verify value of evidence, recognize subjectivity
verbs:
• assess, decide, rank, grade, test, measure, recommend, convince, select, judge, explain,

6. discriminate, support, conclude, compare
3. Situated Learning
 “To Situate”- means to involve other learners, the environment, and the activities to
create meaning
Situated Learning
- Learning is a function of the activity, context and culture in which it occurs.
- Knowledge and skills are learned in the contexts that reflect how knowledge is obtained and
applied in everyday situations.
- A means for relating subject matter to the needs and concerns of learners.
- Learning is essentially a matter of creating meaning from the real activities of daily living.
 Social interaction
- Critical component of situated learning.
- Learners become involved in a “community of practice” which embodies certain
beliefs and behaviors to be acquired.
Four Major Premises Guiding the Development of Classroom Activities (Anderson, Reder
and Simon, 1996)
1. Learning is grounded in the actions of everyday situations;
2. Knowledge is acquired situationally and transfers only to similar situations;
3. Learning is the result of a social process, encompassing ways of thinking, perceiving, problem
solving and interacting in addition to declarative and procedural knowledge; and
4. Learning is not separated from the world of action but exists in robust, complex, social
environment made up of actors, actions and situations.
Situated learning uses cooperative and participative teaching methods as the means of acquiring
knowledge.
Knowledge is obtained by the processes described (Lave, 1997) as Way In and Practice.
 Way in – a period of observation in which a learner watches a master and makes a first
attempt at solving a problem.
 Practice- is refining and perfecting the use of acquired knowledge.

7. 4. Constructivism
 Constructivism
- Programs aim to have students construct their own knowledge through their own process
of reasoning.
 Teachers
- Pose their problems and encourage students to think deeply about possible solutions.
- Promote making connections to other ideas within mathematics and other disciplines.
- Ask students to furnish proof/ explanations for their work.
- Use different representations of mathematical ideas to foster students’ greater
understanding.
 Students
Are expected to:
- Solve problems
- Apply mathematics to real-world situations; and
- Expand on what they already know
Constructivist approach to thinking about mathematics education.
1. People are born with innate ability to deal with small integers and to make comparative
estimates of larger numbers.
2. The human brain has components that can adapt to learning and using mathematics.
3. Humans vary considerably in their innate mathematical abilities or intelligence.
4. The mathematical environments that children grow up vary tremendously.
5. thus, when we combine nature and nurture, by the time children enter kindergarten, they
have tremendously varying levels of mathematical knowledge, skills, and interests.
6. Even though we offer a somewhat standardized curriculum to young students, that actual
curriculum, instruction, assessment, engagement of intrinsic and extrinsic motivation, and
so on varies considerably.
7. thus, there are huge differences among the mathematical knowledge and skill levels of
students at any particular grade level or in any particular math course
8. thus, mathematics curriculum, instruction, and assessment needs to appropriately take
into consideration these differences.
Educational Trends:
1. The transition of the teachers’ role from “sage on the stage” to “guide on the side”
2. Teaching “higher order” skills such as problem-solving, reasoning, and reflection
3. Enabling learners to learn how to learn;
4. More open-ended evaluation of learning outcomes;
5. And, of course, cooperative and collaborative learning skills

8. 5. Cooperative Learning
• is an approach to organizing classroom activities into academic and social learning
experiences. Students must work in groups to complete tasks collectively.
5 Elements
Positive interdependence
• Students must fully participate and put forth effort within their group
• Each group member has a task/role/responsibility therefore must believe that they are
responsible for their learning and that of their group
Face-to-Face Promotive Interaction
• Each group member has a task/role/responsibility therefore must believe that they are
responsible for their learning and that of their group
• Students explain to one another what they have or are learning and assist one another
with understanding and completion of assignments
Individual Accountability
• Each student must demonstrate master of the content being studied
• Each student is accountable for their learning and work, therefore eliminating “social
loafing”
Social Skills
• Social skills that must be taught in order for successful cooperative learning to occur
• Skills include effective communication, interpersonal and group skills
– Leadership
– Decision-making
– Trust-building
– Communication
– Conflict-management skills
Group Processing
• Every so often groups must assess their effectiveness and decide how it can be improved
Challenges
• Need to prepare extra materials for class use
• Fear of the loss of content coverage
• Do not trusts students to acquire knowledge by themselves
• Lacks of familiarity with cooperative learning methods
• Students lack the skills to work in group

9. VARIOUS MATHEMATICIANS AND EDUCATORS THAT CONTRIBUTED TO
THE DEVELOPMENT OF TEACHING AND LEARNING MATHEMATICS
GLOBALLY AND LOCALLY
Mathematics Educators
1. Euclid (fl. 300 BC), Ancient Greek, author of The Elements
2. Tatyana Alexeyevna Afanasyeva (1876–1964), Dutch/Russian mathematician who
advocated the use of visual aids and examples for introductory courses in geometry for high
school students [16]
3. Robert Lee Moore (1882–1974), American mathematician, originator of the Moore method
4. George Pólya (1887–1985), Hungarian mathematician, author of How to Solve It
5. Georges Cuisenaire (1891–1976), Belgian primary school teacher who invented Cuisenaire
rods
6. Hans Freudenthal (1905–1990), Dutch mathematician who had a profound impact on Dutch
education and founded the Freudenthal Institute for Science and Mathematics Education in
1971
7. Toru Kumon (1914–1995), Japanese, originator of the Kumon method, based on mastery
through exercise
8. Pierre van Hiele and Dina van Hiele-Geldof, Dutch educators (1930s - 1950s) who
proposed a theory of how children learn geometry (1957), which eventually became very
influential worldwide
9. Robert Parris Moses (1935-), founder of the nationwide US Algebra project
10. Robert & Ellen Kaplan (about 1930/40s-), authors of Nothing That Is, The Art of the
Infinite: The Pleasures of Mathematics, and Chances Are: Adventures in Probability (by
Michael Kaplan and Ellen Kaplan).
10 Famous Filipino Mathematicians and Physicists
The names of the 10 famous Filipino Mathematicians are listed randomly below. Let’s
get to know them.
1. Raymundo Favila: He was elected as Academician in 1979. He is one of the people who
initiated mathematics in the Philippines. He had extensive contributions to the progression of
mathematics in the country.
2. Amador Muriel: This mathematician was known due to his significant works and
contributions to theoretical physics. He has made new kinetic equation which is essential for
discovering problems on a statistical method that is non-equilibrium.
3. Bienvenido F. Nebres: Dr. Nebres contributed much to the development of higher
mathematics teaching in the nation being the president of Mathematical Society of the
Philippines for years. He has successfully published 15 documents about pure mathematics and
mathematics education.

10. 4. Tito A. Mijares: This doctor performs studies in relation to multi-variety hypothesis and
analysis. These were published in the Annals of Mathematical Statistics, a global journal.
5. Gregorio Y. Zara: He became famous for his two-way television telephone and the Zara
Effect or the electrical kinetic resistance.
6. Casimiro del Rosario: He was honored with the Presidential Award in 1965 for his
excellent works in physics, astronomy, and meteorology. His workings on soft x-rays made him
well-known.
7. Dr. Melecio S. Magno: He made researches on the absorption and fluorescence
spectroscopy of rare-earth crystals and how gravitation is affected by typhoons.
8. Apolinario D. Nazarea: He played important roles to the theories on biophysics and
recombinant biotechnology. His own conceptual framework on the structure of RNA/DNA
investigation is also included.
9. Eduardo Padlan: He was elected as Academician in 2003. He has significant work on
humanized antibodies which have possible applications in the healing of different diseases
including cancer.
10. Jose A. Marasigan: He is a multi-awarded professor here and abroad. He has received
awards like Young Mathematician Grant of the International Mathematical Union (IMU) to the
International Congress of Mathematicians (Finland) and the Outstanding Young Scientist Award
from the National Academy of Science and Technology. He also initiated the Program of
Excellence in Mathematics for mathematically gifted high school students in Ateneo de Manila.
They are the pillars of mathematics in the Philippines. You can try searching for more
information about them through a Filipino/Pilipino dictionary.
References:
http://www.denznet.com/amazing/10-famous-filipino-mathematicians/
http://www.ejmste.com/v3n1/EJMSTEv3n1_Zakaria%26Iksan.pdf
http://en.wikipedia.org/wiki/Cooperative_learning
http://en.wikipedia.org/wiki/Mathematics_education

11. Contributions of Various Mathematicians and Educators to the Development of Teaching
and Learning Mathematics Locally:
 The Mathematics Teachers Association of the Philippines (MTAP)
- An organization of Mathematics teachers working together to promote excellence in
Mathematics education.
- First organized in 1976 by Fr. Wallance G. Campbell, S.J. at the Ateneo de Manila
University
- MTAP has honed the mathematical skills of promising students through its Math
Competition.
-MTAP's other programs includes
 Scholarship grants leading to Master of Science in Teaching for selected members
2.
 intensive summer training programs for Math teachers
 tutorial programs for students
 Conduct of mastery/ inventory tests for teachers.
 Mathematics Trainers’ Guild
-Dr. Simon L. Chua- the president of MTG. His primary task is to improve and provide
quality education and open training among mathematics teachers and students
Top 10 Filipino Mathematicians:
1. Raymundo Favila- He made a new kinetic equation and studied geometric inequalities
and differential equations with applications to stratifiable congruences, among other
things. Favila has also helped write algebra and trigonometry textbooks.
2. Dr. Melecio S. Magno- has researched rare-earth crystals, how typhoons affect
atmospheric ozone distribution, gravitation, radiation in the atmosphere, sky luminosity,
and the philosophy of science. He has co-written the physics textbook University Physics
used at the University of the Philippines.
3. Jose A. Marasigan- He was instrumental in establishing the Philippine Mathematical
Olympiad and developing the Program for Excellence in Mathematics. He designed a
program for high school students in Ateneo de Manila who are mathematically gifted.

12. 4. Tito A. Mijares-He conducted studies on multi-variety hypothesis and analysis and his
results were published in the Annals of Mathematical Statistics. He manages the
statistical system in the Philippines as Executive Director of the National Census and
Statistics Office and Deputy Director-General of the National Economic Development
Authority.
5. Amador Muriel-He was one of the founders of the Quantum Theory of Turbulence and
has made other significant contributions in theoretical physics. He also studied stellar
dynamics and discovered that self-gravitation can make a system of structures in one
dimension.
6. Apolinario D. Nazarea- He also contributed to the design of synthetic vaccines. He was
elected as an Academician in 1990 mostly because of this work.
7. Bienvenido F. Nebres- He was one of the founders of the Consortium of Manila
universities that developed PhD programs in mathematics, chemistry, and physics. The
Consortium has become the center of a network of schools all over the Philippines
8. Eduardo Padlan-He was elected as an Academician in 2003, partially for his work on
antibodies that may have applications in the healing of certain diseases including cancer.
He has Ph. D in Biophysics and has 14 patents on the use of antibodies
9. Casimiro del Rosario-He has performed superior work in the fields of meteorology,
physics, and astronomy. He became well known for his work on soft X-rays. Other work
was done on radioactive radiation on Euglena, the different wavelengths of ultraviolet
light, and electrical discharges in a vacuum.
10. Gregorio Y. Zara- He designed a microscope that has a collapsible stage and helped on
the design of the Marex X-10 robot. He also invented an airplane engine that ran on
alcohol and contributed to new designs of producing solar energy.
Contributions of Mathematics Educators
The following are some of the people who have had a significant influence on the
teaching of mathematics at various periods in history.
1. Euclid of Alexandria
- was a Greek mathematician, often referred to as the "Father of Geometry". He was
active in Alexandria during the reign of Ptolemy I (323–283 BC).

13. - author of The Elements
Elements is one of his most influential works in the history of mathematics,
serving as the main textbook for teaching mathematics (especially geometry) from the t
ime of its publication until the late 19th or early 20th century. In the Elements, Euclid
deduced the principles of what is now called Euclidean geometry from a small set
of axioms.
- Euclid also wrote works onperspective, conic sections, spherical geometry, number
theory and rigor.
Other works
In addition to the Elements, at least five works of Euclid have survived to the present day. They
follow the same logical structure as Elements, with definitions and proved propositions.
• Data deals with the nature and implications of "given" information in geometrical
problems; the subject matter is closely related to the first four books of the Elements.
• On Divisions of Figures, which survives only partially in Arabic translation,
concerns the division of geometrical figures into two or more equal parts or into parts in
given ratios. It is similar to a third century AD work by Heron of Alexandria.
• Catoptrics, which concerns the mathematical theory of mirrors, particularly the
images formed in plane and spherical concave mirrors.
• Phaenomena, a treatise on spherical astronomy, survives in Greek; it is quite
similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC.
• Optics is the earliest surviving Greek treatise on perspective.
In its definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which
emanate from the eye.
Other works are credibly attributed to Euclid, but have been lost.
• Conics was a work on conic sections that was later extended by Apollonius of
Perga into his famous work on the subject.
• Porisms might have been an outgrowth of Euclid's work with conic sections, but
the exact meaning of the title is controversial.

14. • Pseudaria, or Book of Fallacies, was an elementary text about errors
in reasoning.
• Surface Loci concerned either loci (sets of points) on surfaces or loci which were
themselves surfaces; under the latter interpretation, it has been hypothesized that the work
might have dealt with quadric surfaces.
• Several works on mechanics are attributed to Euclid by Arabic sources.
2. Tatyana Alexeyevna Afanasyeva
- Dutch/Russian mathematician who advocated the use of visual aids and examples for
introductory courses in geometry for high school students
- Tatyana collaborated closely with her husband, most famously on their classic review
of thestatistical mechanics of Boltzmann. She published many papers on various topics
such asrandomness and entropy, and teaching geometry to children.
3. Robert Lee Moore
- was an American mathematician, known for his work in general topology and
the Moore method of teaching university mathematics.
- originator of the Moore method
4. George Pólya
- Hungarian mathematician, author of How to Solve It
He was a professor of mathematics from 1914 to 1940 at ETH Zürich in Switzerland
and from 1940 to 1953 at Stanford University carrying on as Stanford Professor
Emeritus the rest of his life and career.
- He worked on a great variety of mathematical topics, including series, number
theory,mathematical analysis, geometry, algebra, combinatorics, and probability.
5. Georges Cuisenaire
- (1891–1976), Belgian primary school teacher who invented Cuisenaire rods
6. Hans Freudenthal
- was aDutch mathematician. He made substantial contributions to algebraic
topology and also took an interest in literature, philosophy, history and
mathematics education.
- nFreudenthal focused on elementary mathematics education.

15. - In the 1970s, his single-handed intervention prevented the Netherlands from
following the worldwide trend of "`new math"'.
- He was also a fervent critic of one of the first international school achievement
studies.
- In 1971 he founded the IOWO at Utrecht University, which after his death was
renamed Freudenthal Institute, the current Freudenthal institute for science and
mathematics education.
7. Toru Kumon
- was a Japanese mathematics educator
-In 1954, Kumon began to teach his oldest son, who was doing poorly in mathematics
in primary school, and developed what later became known as the Kumon method.
- This method involves repetition of key mathematics skills, such
as addition, subtraction, multiplication, and division, until mastery is reached. Students
then progress to studying the next mathematical topic. Kumon defined mastery as being
able to get an excellent score on the material in the time given, which is intended to benefit
students in all their studies. Kumon strongly emphasised the concepts of time andaccuracy.
- As a result of the method, other parents became interested in Kumon's ideas, and in
1956, the first Kumon Center was opened in Osaka,Japan.
- In 1958, Toru Kumon founded the Kumon Institute of Education, which set the
standards for the Kumon Centers that began to open around the world. The Institute
continues today to focus on individual study to help each student reach his or her full
potential. The underlying belief behind the Kumon Method is that, given the right kind
of materials and the right support, any child is capable of learning anything. At any
time, there are more than 4 million Kumon students worldwide, and since 1956, more
than 19 million students have enrolled in Kumon Centers worldwide.
8. Pierre van Hiele and Dina van Hiele-Geldof,
- Dutch educators (1930s - 1950s) who proposed a theory of how children learn
geometry (1957), which eventually became very influential worldwide

16. - Van Hiele model is a theory that describes how students learn geometry. The theory
originated in 1957 in the doctoral dissertations of Dina van Hiele-Geldof and Pierre van
Hiele (wife and husband) at Utrecht University, in the Netherlands.
9.Robert Parris Moses
- is an American, Harvard-trained educator who was a leader in the 1960s Civil Rights
Movement and later founded the nationwide U.S. Algebra project.
- In 1982 he received a MacArthur Fellowship, and used the money to create
the Algebra Project, a foundation devoted to improving minority education in math.
Moses taught math for a time at Lanier High School in Jackson, Mississippi, and used
the school as a laboratory schoolfor Algebra Project methods.
10.Robert & Ellen Kaplan
- (about 1930/40s-), authors of Nothing That Is, The Art of the Infinite: The Pleasures of
Mathematics, and Chances Are: Adventures in Probability (by Michael Kaplan and
Ellen Kaplan).
Filipino Mathematics Teachers and Their Contribution
The following people all taught mathematics at some stage in their lives, although they are better
known for other things:
Charles Lutwidge Dodgson
• was an English author, mathematician, logician, Anglican deacon and photographer.
Mathematical Findings and Work
Within the academic discipline of mathematics, Dodgson worked primarily in the fields of
geometry, matrix algebra, mathematical logic and recreational mathematics, producing
nearly a dozen books which he signed with his real name. Dodgson also developed new ideas in
the study of elections (e.g., Dodgson's method) and committees; some of this work was not
published until well after his death. He worked as a mathematics tutor at Oxford, an occupation
that gave him some financial security.

17. Mathematical works
• A Syllabus of Plane Algebraic Geometry(1860)
• The Fifth Book of Euclid Treated Algebraically(1858 and 1868)
• An Elementary Treatise on Determinants, With Their Application to Simultaneous Linear
Equations and Algebraic Equations
• Euclid and his Modern Rivals(1879), both literary and mathematical in style
• Symbolic Logic Part I
• Symbolic Logic Part II(published posthumously)
• The Alphabet Cipher(1868)
• The Game of Logic
• Some Popular Fallacies about Vivisection
• Curiosa Mathematica I(1888)
• Curiosa Mathematica II(1892)
• The Theory of Committees and Elections, collected, edited, analysed, and published in
1958, by Duncan Black
Thomas Andrew Lehrer
- born April 9, 1928) is an American singer-songwriter,satirist, pianist, and mathematician. He
haslectured on mathematics and musical theater. Lehrer is best known for the pithy, humorous
songs that he recorded in the 1950s and 1960s.
His work often parodies popular song forms, such as in "The Elements", where he sets the names
of the chemical elements to the tune of the "Major-General's Song" from Gilbert and
Sullivan's Pirates of Penzance. Lehrer's earlier work typically dealt with non-topical subject
matter and was noted for its black humor, seen in songs such as "Poisoning Pigeons in the Park".
In the 1960s, he produced a number of songs dealing with social and political issues of the day,
particularly when he wrote for the U.S. version of the television show That Was The Week That
Was.
Mathematical publications
The American Mathematical Society database lists Lehrer as co-author of two papers:

18. • RE Fagen & TA Lehrer, "Random walks with restraining barrier as applied to the biased
binary counter", Journal of the Society for Industrial Applied Mathematics, vol. 6, pp. 1–14
(March 1958) MR0094856
• T Austin, R Fagen, T Lehrer, W Penney, "The distribution of the number of locally
maximal elements in a random sample",Annals of Mathematical Statistics vol. 28,
pp. 786–790 (1957) MR0091251
Georg Joachim de Porris,
– also known as Rheticus (16 February 1514 – 4 December 1574), was a mathematician,
cartographer, navigational-instrument maker, medical practitioner, and teacher. He is perhaps
best known for his trigonometric tables and as Nicolaus Copernicus's sole pupil. He facilitated
the publication of his master's De revolutionibus orbium coelestium (On the Revolutions of the
Heavenly Spheres).
Trigonometry
For much of his life, Rheticus displayed a passion for the study of triangles, the branch of
mathematics now called trigonometry. In 1542 he had the trigonometric sections of Copernicus'
De revolutiobis published separately under the title De lateribus et angulis triangulorum (On the
Sides and Angles of Triangles). In 1551 Rheticus produced a tract titled Canon of the Science of
Triangles, the first publication of six-function trigonometric tables (although the word
trigonometry was not yet coined). This pamphlet was to be an introduction to Rheticus' greatest
work, a full set of tables to be used in angular astronomical measurements.
At his death, the Science of Triangles was still unfinished. However, paralleling his own
relationship with Copernicus, Rheticus had acquired a student who devoted himself to
completing his teacher's work.Valentin Otto oversaw the hand computation of approximately
100,000 ratios to at least ten decimal places. When completed in 1596, the volume,Opus
palatinum de triangulus, filled nearly 1,500 pages. Its tables were accurate enough to be used in
astronomical computation into the early twentieth century.
Works
• Narratio prima de libris revolutionum Copernici(1540)

19. • Tabula chorographica auff Preussen und etliche umbliegende lender(1541)
• De lateribus et angulis triangulorum(with Copernicus; 1542)
• Ephemerides novae(1550)
• Canon doctrinae triangulorum(1551)
MATH and LITERATURES
"... Using mathematics to tell stories and using stories to explain mathematics are two sides of
the same coin. They join what should never have been separated: the scientist's and the artist's
ways of uncovering truths about the world."(Frucht, xii)
LITERATURE
- It stirs our imaginations and emotions, making ideas more enjoyable and memorable.
-It enlivens what many people see as the isolating abstractness of mathematics (cf.
Midgley, 1-39).
-It also elicits expressions of feeling, increasing our insight about joys and frustrations in
studying math.
Different way of using math in literatures
(1) Call on math to illuminate a theory
(e.g., Dostoyevsky, and Tolstoy, and Austen);
(2) Be inspired by mathematical themes to create a work of art based on the themes
(e.g., Doxiadis, Growney, Lem,Reese, and Upson);
(3) Poke fun at typical experiences in learning math or at mathematicians
(e.g., Dodgson,Leacock and Russell);
(4) want to produce an educational work
(e.g., Enzensberger); or
(5) want to write theimagined life of an intriguing mathematician
(e.g., Petsinis).
Advantages of Literature in Teaching Mathematics
 Provide a context or model for an activity with mathematical content.
 Introduce manipulatives that will be used in varied ways (not necessarily as inthe story).
 Inspire a creative mathematics experience for children.

20.  Pose an interesting problem.
 Prepare for a mathematics concept or skill.
 Develop or explain a mathematics concept or skill.
 Review a mathematics concept or skill.
Components of Mathematical content
1. Accuracy
2. Visual and Verbal Appeal
3. Connections
4. Audience
5. “Wow” Factor
Literary samples
The Symbolic Logic of Murder by John Reese
"... adjusts Boolean algebra , of an admittedly elementary order, to the requirements of popular
fiction." (Fadiman, Fantasia ..., 223.)
The solution to the murder depends on facility with negations, unions and intersections!
Young Archimedes by Aldous Huxley
- young hero combines a loving proficiency in music with an extraordinary ability in
math.
*As the story unfolds, we encounter both geometric and algebraic proofs of the Pythagorean
Theorem!
Star, Bright by Mark Clifton
two aspects:
o the problem of rearing a genius, and
o mathematical activities.
--- Star, a three-year old child, invents a Moebius strip and also figures out a way to teleport
herself into 4-dimensional space and to travel backwards and forwards in time.
Arcadia by Tom Stoppard
Although the 19 th century heroine (aged 13) of Arcadia fails to solve Fermat's Last
Theorem, she does anticipate the 20th century topics of chaos and iteration.
Proof by David Auburn
- a play about genius and love, considers the probability that a young woman could have
authored a path-breaking proof.

21. The Law by Robert Coates
-focuses on insights into human behavior, and the important role of statistics. It motivates
discussion of the meaning of the familiarly cited "law of averages," the various types of
averages,
The Brothers Karamazov
(an excerpt )
-it shows Dostoyevsky=s use of the new mathematical ideas to his philosophy.
War and Peace, another Russian novel by Tolstoy
*Tolstoy's theory is that history needs to be analyzed mathematically and statistically: not
as discrete incidents, but (in a reference to calculus) as a continual process.
Tolstoy also use Achilles and the Tortoise to show that history cannot be analyzed as a series of
discrete vignettes.
In addition, Tolstoy provides an example of the use of ratio and linear equations to clarify how
the disadvantaged (such as the Russians) can win battles against more advantaged (such as the
French) if they have enough spirit and energy.
Emma by Jane Austen
- it alludes to the ratio M/A, based on the 18th century philosophy of Francis Hutcheson,
who believed that the ratio measured "virtue, " where A is perfect virtue and M is attained
virtue.
The Extraordinary Hotel or the Thousand and First Journey of Ion the Quiet by Stanislaw Lem's
- the story goes on to many other possible scenarios, illuminating beautifully many
properties of infinite sets.
A
" Mathematician's Nightmare“ by JoAnne Growney
- seems on the surface to be about decision-making in pricing and shopping, but it is an excellent
depiction for a student or lay reader of the Collatz Conjecture, a famous unsolved problem.
"My Dance is Mathematics,“
- poem about Emmy Noether
I
"f a woman's dance / is mathematics,/ must she dance alone?“
- The relationship to mathematics is usually seen in the content of the poem, but may also
be a matter of structure.

22. OTHER EXAMPLES
Sorting
Strega Nona by Tomie De Paola
Noodles by Sarah Weeks
Counting
How Many Snails? by Paul Giganti, Jr.
Who Took the Cookies from the Cookie Jar? by Bonnie Lass
Addition/Subtraction
Time
The Very Hungry Caterpillar by Eric Carle
The Grouchy Ladybug by Eric Carle
Ten Sly Piranhas by William Wise
Mouse Count by Ellen Stoll Walsh
Fractions
Eating Fractions by Bruce McMillan
Lunch with Cat and Dog by Rozanne Williams
Measurement
How Big is a Foot? by Rolf Myllar
Inch by Inch by Leo Lionni
Money
Bennie's Pennies by Pat Brisson
Research studies on the use of teaching and learning aids in math.
“Teaching and Learning Mathematics using Research”
-Dr.Terry Bergeson
Four key ingredients
• The students trying to learn mathematics
• The teachers trying to teach mathematics
• The content of mathematics and its organization into a curriculum
• The pedagogical models for presenting and experiencing this mathematical content
Advantages of Research in Math Education
• It can inform us. • It can create reflection and discussion.
• It can educate us. • It can challenge what we currently do as
• It can answer questions. educators
• It can prompt new questions. • It can clarify educational situations
• It can help make educational decisions and educational policy
• It can confuse situations
• It can focus on everything but your situation
• It can be hidden by its own publication style.

23. STUDIES CONDUCTED ON THE RESEARCH:
RESEARCH IN NUMBER SENSE
1. Number and Numeration
2. 2. Estimation
RESEARCH ON MEASUREMENT
1. Attributes and Dimensions
2. Approximation and Precision
O Difference between estimation and approximation
3. Systems and tools
O Measurements strategies
RESEARCH ON GEOMETRIC SENSE
O Define shape
O Characteristic of different shape
O 3-D environment
O Relationships/ Transformation
 The research conducted implies that manipulative materials are good teaching aids in
teaching mathematics.
ROLES AND IMPACT OF USING MANIPULATIVES
O Increase mathematical achievement
O Students’ attitude towards mathematics are improved
O Help students understand mathematical concepts and processes
O Increase students” flexibility of thinking
O Tool to solve new mathematical problem
O Reduce students” anxiety
Note:
 Manipulative need to be selected and used carefully.
 Students do not discover or understand math concepts simply by manipulative concrete
materials.
 Math teachers need assistance on selecting appropriate manipulative materials.
Mistaken beliefs about manipulative materials

24. -Jackson(1979)
1. Almost all manipulative can be used to teach any mathematical concept.
2. It simplify students’ learning of math.
3. Good math teaching always include manipulative.
4. The number of manipulative is positively correlated to the amount of learning that occur
5. There is a multipurpose manipulative
6. It is more useful in primary grades that in the upper grades.
7. It is more useful with low-ability students than high-ability students.
 The use of concrete manipulative do not seem as effective in promoting algebraic
understanding.
 Manipulative help students at all grade levels conceptualize geometric shapes and their
properties.
Suggestions in using Manipulative
1) Use it frequently and throughout the instructional program
2) It should be used in conjunction with other learning aids.
3) It should be used by students in a manner consistent with the mathematical content
4) used with learning activities that are exploratory and deductive in approach
5) Simplest and yet ideal
6) Used with activities that include that symbolic recording of results and ideas
Research study of material use in teaching and learning in mathematics
Abstract
The introduction of laptops in the teaching of mathematics and science in English under
the Teaching and Learning of Science and Mathematics in English Programme (Pengajaran dan
Pembelajaran Sains dan Matematik dalam Bahasa Inggeris, PPSMI) has been implemented by
the Ministry
Education since 2003. The preliminary observations found that teachers are not fully
utilising these facilities in their teaching. A survey was conducted to study the barriers preventing
the integration and adoption of information and communication technology (ICT) in teaching
mathematics. Six major barriers were identified: lack of time in the school schedule for projects
involving ICT, insufficient teacher training opportunities for ICT projects, inadequate technical

25. support for these projects, lack of knowledge about ways to integrate ICT to enhance the
curriculum, difficulty in integrating and using different ICT tools in a single lesson and
unavailability of resources at home for the students to access the necessary educational materials.
To overcome some of these barriers, this paper proposes an e-portal for teaching mathematics.
The e-portal consists of two modules: a resource repository and a lesson planner. The resource
repository is a collection of mathematical tools, a question bank and resources in digital form
that can be used for other teaching and learning mathematics. The lesson planner is a user
friendly tool that can integrate resources from the repository for lesson planning.
Digital Teaching Aids Make Mathematics Fun
"Students are increasingly living in two worlds: the world of the classroom and the real
world... and the two are growing farther apart," cautions Chronis Kynigos, a researcher at the
Research Academic Computer Technology Institute (RACTI) and director of the
Educational Technology Lab at the University of Athens.
Working in the EU-funded ReMath project, the team developed new teaching aids, in
the form of software tools known as Dynamic Digital Artefacts (DDAs), and a comprehensive
set of Pedagogical Plans for teachers to use within the guidelines of national education curricula.
A specific set of six Dynamic Digital Artefacts (DDAs) was designed and developed
during the ReMath Project. They have been selected in order to reasonably reflect the existing
diversity of representations provided by ICT tools
Examples of Program use inDDA’s
AlNuSet
- the building of a microworld consisting of an Algebraic Line and Algebraic manipulator
component for visual representation of geometrical and symbolic manipulation of number
sets,
MoPiX
- a tool for programming games and animations with equations,
MaLT
- an extension to the ‘Machine-lab’ authoring system for interactive virtual reality scenes
to include a mathematical scripting mechanism and a set of programmable and
mathematical controllers (such as variation tools and vectors) for manipulating virtual
objects, their properties and relations between them in small-scale 3d spaces,

26. Cruislet
- an extension to the ‘Cruiser’ G.I.S. and geographic space navigator to include a
mathematical scripting mechanism and custom mathematical user interface controls for
vector-driven navigation in 3d large-scale spaces.
A Study on the Use of ICT in Mathematics Teaching
Chong Chee Keong, Sharaf Horani & Jacob Daniel
Faculty of Information Technology
Multimedia University, 63100 Cyberjaya
Selangor Darul Ehsan, Malaysia
Abstract
The introduction of laptops in the teaching of mathematics and science in English under
the Teaching and Learning of Science and Mathematics in English Programme (Pengajaran dan
Pembelajaran Sains dan Matematik dalam Bahasa Inggeris, PPSMI) has been implemented by
the Ministry of Education since 2003. The preliminary observations found that teachers are not
fully utilisingthese facilities in their teaching.
A survey was conducted to study the barriers preventing the integration and adoption of
information and communication technology (ICT) in teaching mathematics.
Six major barriers were identified:
1. lack of time in the school schedule for projects involving ICT
2. insufficient teacher training opportunities for ICT projects,
3. inadequate technical support for these projects,
4. lack of knowledge about ways to integrate ICT to enhance the curriculum,
5. difficulty in integrating and using different ICT tools in a single lesson and;
6. unavailability of resources at home for the students to access the necessary educational
materials.
To overcome some of these barriers, this paper proposes an e-portal for teaching
mathematics. The e-portal consists of two modules: a resource repository and a lesson planner.
The resource repository is a collection of mathematical tools, a question bank and other

27. resources in digital form that can be used for teaching and learning mathematics. The lesson
planner is a user friendly tool that can integrate resources from the repository for lesson
planning.
METHODOLOGY
This research deployed a survey method to investigate the use of ICT and the barriers of
integrating ICT into the teaching of mathematics. The survey was carried out during a
mathematics in-service course conducted by the State Education Department. Before the
commencement of the survey, the respondents were given a briefing on the purpose of the
survey.
A total of 111 responses was received and they were analyzed using the SPSS statistical
package. A questionnaire was adapted from the Teacher Technology Survey by the American
Institute for Research (AIR, 1998).
The questionnaire was divided into seven areas:
(A) the teacher’s profile,
(B) how teachers use ICT,
(C) professional development activities,
(D) the teacher’s ICT experience,
(E) the level of use in ICT,
(F) the barriers faced by teachers and
(G) the proposed solution.

28. ORIGAMI
Origami is a Japanese compound word which means “paper folding”. It is used to
describe craft made from folded paper in Japan as well as pieces originating in other regions,
since so many people associate folded paper crafts with Japan in particular. Individual origami
pieces can vary widely in size and design, from simple folded boxes to ornate creatures made by
joining several different sheets of paper. Many young people learn origami in school, and some
people continue to practice this craft into adulthood.
The art of paper folding actually originated in China around the first century CE. The
Chinese referred to their folded paper crafts as zhe zhi, and monks brought the tradition with
them to Japan when they visited in the sixth century. The Japanese quickly took to paper folding
as a pastime, developing a number of traditional folds, shapes, and styles, many of which were
considered fortuitous for particular occasions or life events. The crane is a particularly famous
lucky origami shape.
Highlights in Origami History
100 AD
Paper-making originated in China by Ts'ai Lun, a servant of the Chinese emperor. The art of
paper folding began shortly after.
600 AD
Paper-making spread to Japan where origami really took off.
800-1100AD
Origami was introduced to the West (Spain) by the Moors who made geometric origami models.

29. 1797
Hiden Senbazuro Orikata, the oldest origami book for amusement in the world is published.
Translated it means "The Secret of One Thousand Cranes Origami".
1845
Kan no mado (Window on Midwinter)-The first published collection of origami models which
included the frog base
1900
Origami spread to England and the United States
1935
Akira Yoshizawa developed his set of symbols used for origami instructions.
1960
Sadako and One Thousand Cranes was published by Eleanor Coerr and is linked with the
origami crane and the international peace movement.
2000
International Peace Project-An international project which is engaging communities in
collaborative activities to promote peace, non-violence and tolerance - A Million Paper
Cranes for Peace by the Year 2000!
Folding a single piece:
The actual purpose of this fold is just to
give you a reference to make the next
Start with a 1.5-inch square of paper: two folds.
Make a precise and creased fold lengthwise.
Dividing the square in half.

30. Unfold the paper and lay it flat.
Take the bottom edge of the paper and fold it to the center crease. Then spin the paper
180 degrees and do the same.
Unfold the paper and lay it flat.
Take the bottom-right corner of the paper and fold it into a triangle so that the left side of the
paper now lies on top of the second fold you made.
Leave that folded, spin the paper 180 degrees and make the same fold.
Now, take the bottom-right corner of the paper and make another needle nose-type fold.

31. That means bringing the fold that you just made to lie exactly on top of the second fold you
made.
Then rotate the paper 180 degrees and make the same fold.
Another "needlenose" type fold.
Now is the time to remake the second and third folds you made:

32. Now, take the bottom-left corner of the
paper and fold it so that what was the Rotate the paper 180 degrees and repeat. A
left edge of the paper now lies on top parallelogram! Now, you must tuck
of the top edge of the paper, in that large triangle fold into the
producing a triangle, like this: paper. . Here is what I mean:
Then rotate the paper 180 degrees and tuck Now flip the paper over and rotate it so
in the other fold, resulting in: that it looks like this:
Fold the bottom point of the paper straight Then rotate the paper 180 degrees and
up to meet another vertex of the repeat, producing this:
parallelogram,

33. Now you need to give the paper a bend in the middle. You will end up with this:
THE BASIC UNIT
Making models:
1. The cube. The easiest to construct, it takes 6 pieces.
2. The stellated octahedron. Takes 12 pieces.
3. The icosahedron. (A stellated icosahedron.) Takes 30 pieces.
4. The stellated truncated icosahedron. Takes 270 pieces.
Model construction
A piece has two sharp corners and two pockets, which allow them to interlock.

34. Here are two pieces placed to illustrate And here they are locked together, corner
this: in pocket:
Here is a third piece, placed over the first two:

35. And here the third piece is locked in:
There is a free corner and free pocket that can be locked together. Doing so necessitates forming
the three pieces into a three-dimension configuration that I call a peak:
REFERENCES:
http://nuwen.net/
library.thinkquest.org/5402/history.html
en.wikipedia.org/wiki/History_of_origami

36. Solving Quadratic Equations
Using Quadratic Formula
and
TI – 84 Plus
(Graphing Calculator)

37. THE QUADRATIC FORMULA
Entering a calculation
Use the Quadratic Formula to solve the quadratic equation
3x2 + 5x + 2 = 0
1. Press 3 STO > ALPHA [A] (above MATH) to store the coefficient of the x2 term.
2. Press ALPHA [:]( above .). the colon allows you to enter more than one instruction
on a line
3. Press 5 STO > ALPHA [B] (above APPS) to store the coefficient of the X term.
Press ALPHA [:] to enter a new instruction on the same line. Press 2 STO>
ALPHA [C] (above PRGM) to store the constant.
3 → 𝐴: 5 → 𝐵: 2 → 𝐶
4. Press ENTER to store the values to the variables A, B, and C.
3 → 𝐴: 5 → 𝐵: 2 → 𝐶
5. Press ( ( ) (-) ALPHA [B] + 2nd [√] ALPHA [B] x2 – 4 ALPHA [A] ALPHA [C] ) )
÷ ( 2 ALPHA [A] ) to enter the expression for one of the solutions for the quadratic
formula,
−𝒃 ± √𝒃 𝟐 − 𝟒𝒂𝒄
𝟐𝒂
( -B+√ (B2 −4AC) )/(2A)
6. Press ENTER to find one solution for the equation 3x2 + 5x + 2 = 0
( -B+√ (B2 −4AC) )/(2A)
-.6666666667
Converting to a Fraction
You can show a solution as a fraction
1. Press MATH to display the MATH menu
2. Press 1 to select 1:> Frac from the MATH menu.
When you press 1, Ans>Frac is displayed on the home screen. Ans is a variable that
contains the last calculated answer.
3. Press ENTER to convert the result to a fraction.
To save the keystrokes, you can recall the last expression you entered, and then edit it
for a new calculation.

38. 4. Press 2nd [ENTRY] (above ENTER) to recall the fraction conversion entry, and then
press 2nd [ENTRY] again to recall the quadratic formula expression
−𝒃 + √𝒃 𝟐 − 𝟒𝒂𝒄
𝟐𝒂
5. Press ^ to move the cursor onto the + sign in the formula. Press – to edit the quadratic
formula expression to become
−𝒃 + √𝒃 𝟐 − 𝟒𝒂𝒄
𝟐𝒂
6. Press ENTER to find the other solution for the quadratic equation
3x2 + 5x + 2 = 0

39. NeoCube
Neocube magnets
 are small high-energy sphere magnet that allows you to create and recreate an endless
number of different patterns and shapes. Neocube magnets are very strong because are made of
neodymium iron boron material and it is pretty fun to play with it. Has 216 pieces of magnets.
It is not important where you will buy the NeoCube. If it will be in some country as
Canada, India, Mexico, South Africa, Australia, Hong Kong, UK, Ireland or in some city as
London, Delhi, Dublin. You can buy it from local retail stores or order it on an internet shop.
Bulk NeoCube you can buy from wholesaler or from factory. From different brands as
BuckyBalls, Nanodots, Zen Magnets you will get different packing of magnets. But the magnets
are always made in China. Original source of neodymium magnets.
Features of Neocube Magnets
 Neocube is the future of puzzles.
 Dual-brain hemisphere stimulation.
 Gaming.
 Stress relief.
 Boredome busting.
Most common on the market are Neocube magnets made of neodymium N35. Nickel coating
with the diameter size 5mm.
 neocube diameter size : 4.8mm, 5mm, 6mm, 7mm, 8mm
 neocube colors : nickel, black, silver, gold, blue, red
 neocube grade : N35, N35, N40, N42, N45, N48, N52
 neocube coating : Ni-Cu-Ni, Ni-Cu-Ni-Cr / nickel, copper, chrome
Different size, color and grade of the material means also a different price.

40. Warnings
 This product is not designed or intended for children under the age of fourteen.
 This product contains small parts that may be harmful or fatal if swallowed.
Consult a doctor immediately if this occurs.
 This product contains magnets. Magnets sticking together or becoming attached to a
metallic object inside the human body can cause serious or fatal injury. Seek immediate
medical help if the magnets are swallowed or inhaled.
 The NeoCube or any of the spheres should never be put in the mouth, ears, nose, or any
other bodily orifice.
 The strong magnets in the NeoCubeTM can damage or destroy some electronic devices.
Therefore, it should never be put close to or directly in contact with electronic products.
Strong magnets can even damage electronic medical devices. Therefore the NeoCubeTM
should never be handled, used by, or brought near anyone with a pacemaker or other
electronic medical device.
 Strong magnets can also damage or destroy information stored magnetically. Some
examples of these are: credit card strips, floppy disks and hard disks. Therefore the
NeoCubeTM should not be put close to or directly in contact with any type of
magnetically stored data.
 Never attempt to burn the NeoCubeTM.
 If the metallic coating around the spheres breaks down, discontinue use. This is
precautionary. The NdFeB material which is the magnetic material in the NeoCubeTM is a
relatively new material, and long term effects of direct skin exposure are therefore
unknown, although there have been no studies which indicate that it is in any way
transdermally toxic.
 This product is not intended to treat, diagnose or cure any diseases.
 This product contains small balls.
.

41. Some Objects formed by neocubes:
NEOCUBE SHAPES, patterns - unique magnet gadget toy
Neocube Magnet Ball - 216 Neo Cube Magnet Ball - China Cybercube .

42. PAPER SPINNER
A type of manipulative that can be used to teach about chance and random
choices.
How to Make a Spinner?
Things you’ll need
 Paper (printed)
 Markers (optional)
 Scissors (or just tear it)
 Creativity (for markers)
Steps:
1. Get some printed paper. (it also works with loose leaf notebook paper)
2. Fold the piece of paper in half vertically.
3. Cut down the crease.
4. Fold the two large rectangles in half vertically, so that they become long and skinny.
5. Fold the bottom corner of each rectangle to the right, so that it forms a triangle shape.
6. Repeat at the top, except this time, make sure the triangles are facing left.
7. Put one of the triangles (it should have the little triangles) facing vertically upward.
8. Put the other rectangle horizontally, facing down in space between the two triangles on the
other rectangle.
9. Fold the bottom triangle to the center, then fold the left triangle to the center, overlapping
the one you just folded. With the top triangle, fold to the center also.
10. Fold the right triangle so it overlaps the top triangle and make sure it goes under the
bottom triangle.
HOW IT IS USED?
Color the ¼ part of the paper spinner by RED and the ¾ by BLUE. Spin the paper spinner and
find out what color will be on top when it stops.

43. FOLDABLES
o FOLDABLES
 an artistic graphic organizer.
 This Foldable project is used to help teachers analyze data, sort the strengths and
weaknesses of their students and determine the question levels from a TAKS-Released
Test so that they can make informed decisions about instruction.
Example of Foldables
TYPES OF FOLDABLE
o A POCKET BOOK FOLDABLE
1. Fold a piece of 8 ½” x 11” paper in half horizontally
2. Open the folded paper and fold one of the long sides up two inches to form a pocket.
3. Glue the outer edges and the center (on the valley/crease) of the two inch fold with a
small amount of glue.

44. o A LAYERED LOOK BOOK FOLDABLE
1. Stack four sheets of paper (8 ½” x 11”) together, placing each consecutive sheet
around ¾ of an inch higher than the sheet in front of it.
2. Bring the bottom of both sheets upwards and align the edges so that all of the layers or
tabs are the same distance apart.
3. When all of the tabs are equal distance apart, fold the papers and crease well.
4. Open the papers and glue them together along the valley/center fold.
o A JOURNAL RESPONSE THREE QUARTER BOOK FOLDABLE
1. Fold a piece of 8 ½” x 11” paper in half horizontally
2. Fold it in half again horizontally.
3. Unfold the paper (just once so that it is still folded in half) and cut up (along the edge
of the paper at the center where you can see the crease) to the mountain top
4. Open flat, lift the left-hand tab. Cut the tab off at the top fold line.
o A STUDENT INTEREST BOUND BOOK FOLDABLE
1. Fold two pieces of ¼ sheet paper (4 ¼” x 5 ½”) separately in half horizontally
2. Place the folds side-by-side allowing 1/16” between the mountain tops. Mark both
folds 1” from the outer edges.
3. On one of the folded sheets, cut-up from the top and bottom edge to the marked spot
on both sides.
4. On the second folded sheet, start at one of the marked spots and cut out the fold
between the two marks. Do not cut into the fold too deeply, only shave it off.
5. Take the “cut-up” sheet and burrito it.
6. Place the burrito through the “cut out” sheet and then open the burrito up.
7. Fold the bound pages in half to form a book.

45. o A TWO-TAB POINT OF VIEW BOOK FOLDABLE
1. Fold a piece of (4 ¼” x 5 ½”) paper in half horizontally
2. Fold it in half again horizontally
3. Unfold the paper (just once so that it is still folded in half) and cut up the valley (along
the edge of the paper at the center where you can see the crease) to the mountain top.
o A THREE-TAB BOOK FOLDABLE
1. Fold a piece of (8 ½ x 11”) paper in half vertically
2. With the paper horizontal and the fold up, , fold the right side toward the center, trying
to cover one half of the paper. (Make a mark here, but do not crease the paper.)
3. Fold the left side over the right side to make a book with three folds.
4. Open the folded book. Place your hands between the two thicknesses of paper and cut
up the two valleys on one side only. This will form three tabs.
o A TWO-TAB BOOK FOLDABLE Z
1. Fold a piece of (4 ¼” x 5 ½”) paper in half vertically
2. Fold it in half again horizontally
3. Unfold the paper (just once so that it is still folded in half) and cut up the valley (along
the edge of the paper at the center where you can see the crease) to the mountain top.

46. MANIPULATIVES
‘Multiplying two Binomials using Teaching Manipulative”
SQUARE BASE TABLE MANIPULATIVE
- + INSTRUMENT
S
+ =3
- = 𝑥2
=x
=3 =1
These instruments can be used in multiplying two binomials.
for example:
(x-1)(x+3)
- + (x+1)(x-3)
= x2-
3x+1x-3
= x2-2x-3
-2x x2
-3
+ -

47. Pentominoes as Math Manipulative
Definition
 Use the 12 pentomino combinations to solve problems.
 Is a geometric pattern which is the basis of a number tiling patterns and puzzles.
 An arrangement of five identical squares in a plane, attached to one another edge to edge.
 Is a polymino composed of five congruent squares, connected long their edges (which
sometimes is said to be an orthogonal connection).
How it is done or constructed?
 Know that there are 12 pentominoes shape. They are named for the letters they represent:
F I L N P T U V W X Y Z. A pentomino is a shape composed of five congruent
squares connected by at least one side. Since there are twelve pentominoes made of five
squares each, pentomino puzzles are played on grids of 60 squares: 6 by 10, 5 by 12, 0r 3
by 20.
 Make your grids. Sketch them on one sheet of graph paper. Cut them out, then trace them
on two card stock. Go over the lines with a permanent marker to make a boarder, then cut
the grids out and set aside.
 Make your puzzle pieces. Sketch out one of each pentomino onto graph paper. Cut them
out and trace them onto card stock. Color the pentominoes. Try to use one color for each
piece if you have enough markers available. Otherwise, just make it as colorful as
possible. Then, cut out the pieces and set aside.
 Construct your folder, which will contain your puzzle. Open the folder and staple the
zipper bag to one side of it. Your grids and pieces will be stored in the bag when you are
not playing with your puzzle. Put the paper clip on the other side. This will be used to
hold which ever grid you are playing on at that time.
 Play pentominoes. Take a grid out of your bag and clip it onto the folder. Using your
pentomino pieces, fill the gried by leaving no empty spaces and overlapping no pieces.
Each grid size has several olutions, so enjoy fiding them all.

48. When to use?
Subject Tag: problem Solving involving geometry and Algebra graphing.
Pentominoes Shape
F I L N P T
U V W X
Y Z

53. Group1

54. Instruments for
Mathematics
Teaching
Manipulatives and other
Instructional Materials
GROUP3
Carlos, Aiza A.
Francisco, Ma. Salome V.
Gonzales, Karen C.
Habana, Sarah Mae
Laguna, Jan Rea O.
Poche, Michille
Baylon, Kevin