Instrumentation in Mathematics TNESC

INSTRUMENTATION
IN
MATHEMATICS
TNESC

Before the Ancient Greeks:
 Egyptians and Babylonians (c. 2000 BC):
Knowledge comes from “papyri”
Rhind Papyrus

•Main source: Plimpton 322
•Sexagesimal (base-sixty) originated with
ancient Sumerians (2000s BC), transmitted
to Babylonians … still used —for
measuring time, angles, and geographic
coordinates

• Thales (624-548 BC)
• Pythagoras of Samos (ca. 580 - 500 BC)
• Zeno: paradoxes of the infinite
• 410- 355 BC- Eudoxus of Cnidus (theory of proportion)
• Appolonius (262-190): conics/astronomy
• Archimedes (c. 287-212 BC)

Ptolemy (AD 83–
c.168), Roman
Egypt
• Almagest: comprehensive treatise on
geocentric astronomy
• Link from Greek to Islamic to European
science

Al-Khwārizmī (780-850), Persia
• Algebra, (c. 820): first
book on the systematic
solution of linear and
quadratic equations.
• He is considered as
the father of algebra:
• Algorithm: westernized
version of his name.

Leonardo of Pisa (c. 1170 – c.
1250) aka Fibonacci
• Brought Hindu-Arabic
numeral system to Europe
through the publication of
his Book of Calculation, the
Liber Abaci.
• Fibonacci numbers,
constructed as an example
in the Liber Abaci.

• illegitimate child of Fazio Cardano, a friend of
Leonardo da Vinci.
• He published the solutions to the cubic and quartic
equations in his 1545 book Ars Magna.
• The solution to one particular case of the cubic, x3 +
ax = b (in modern notation), was communicated to him
by Niccolò Fontana Tartaglia (who later claimed that
Cardano had sworn not to reveal it, and engaged
Cardano in a decade-long fight), and the quartic was
solved by Cardano's student Lodovico Ferrari.
Cardano, 1501 —1576)

• Popularized use of the (Stevin’s) decimal point.
• Logarithms: opposite of powers
• made calculations by hand much easier and quicker, opened the way to
many later scientific advances.
• “MirificiLogarithmorumCanonisDescriptio,” contained 57 pages of
explanatory matter and 90 of tables,
• facilitated advances in astronomy and physics
John Napier (1550 –1617)

• “Father of Modern Science”
• Proposed a falling body in a vacuum
would fall with uniform acceleration
•Was found "vehemently suspect of
heresy", in supporting Copernican
heliocentric theory … and that one
may hold and defend an opinion as
probable after it has been declared
contrary to Holy Scripture.
Galileo Galilei
(1564-1642)

Developed “Cartesian
geometry” :
 uses algebra to describe
geometry.
Invented the notation
using superscripts to show
the powers or exponents,
for example the 2 used in
x2 to indicate squaring.
René Descartes (1596
1650

 important contributions to the construction of mechanical
calculators, the study of fluids, clarified concepts of
pressure and.
 wrote in defense of the scientific method.
 Helped create two new areas of mathematical research:
projective geometry (at 16) and probability theory

• conservation of momentum
• built the first "practical" reflecting
telescope
• developed a theory of color based on
observation that a prism decomposes
white light into a visible spectrum.
• In mathematics:
• development of the calculus.
• demonstrated the generalised binomial
theorem, developed the so-called
"Newton's method" for approximating
the zeroes of a function....
Sir Isaac Newton (1643 – 1727)

•important discoveries in
calculus…graph theory.
•introduced much of modern
mathematical terminology
and notation, particularly for
mathematical analysis,
•renowned for his work in
mechanics, optics, and
astronomy.

Invented or developed a
broad range of fundamental
ideas, in invariant theory,
the axiomatization of
geometry, and with the
notion of Hilbert space
David Hilbert (1862 –1943)

•famous for having founded “information
theory” in 1948.
•digital computer and digital circuit
design theory in 1937
•Demonstrated that electrical
application of Boolean algebra could
construct and resolve any logical,
numerical relationship.
•It has been claimed that this was the
most important master's thesis of all
time.
Claude Shannon
(1916 –2001)]

Theano was the wife of Pythagoras. She
and her two daughters carried on the
Pythagorean School after the death of
Pythagoras.
She wrote treatises on mathematics,
physics, medicine, and child psychology.
Her most important work was the principle
of the “Golden Mean.”

Hypatia
Hypatia was the daughter of Theon, who
was considered one of the most educated
men in Alexandria, Egypt.
Hypatia was known more for the work she
did in mathematics than in astronomy,
primarily for her work on the ideas of
conic sections introduced by Apollonius.
Home

Caroline Herschel
Her first experience in mathematics was
her catalogue of nebulae.
She calculated the positions of her
brother's and her own discoveries and
amassed them into a publication.
One interesting fact is that Caroline never
learned her multiplication tables.

Sophie Germain
She is best known for her work in number
theory.
Her work in the theory of elasticity is also
very important to mathematics.

Emilie du Chatelet
Among her greatest achievements were
her “Institutions du physique” and the
translation of Newton's “Principia”, which
was published after her death along with a
“Preface historique” by Voltaire.
Emilie du Châtelet was one of many
women whose contributions have helped
shape the course of mathematics

Methods of
Teaching
Mathematics

Model Method
• is a visual way of picturing a situation. Instead of
forming simultaneous equations and solving for the
variables, model building involves using blocks or
boxes to solve the problem. The power of using
models can be best illustrated by problems, often
involving fractions, ratios or percentages, which
appear difficult but if models can be drawn to show
the situation, the solution becomes clearer,
sometimes even obvious.

Socratic Method
• Teaching by asking instead of telling.
• Involves asking a series of questions until a contradiction
emerges invalidating the initial assumption.
• Socratic irony is the position that the inquisitor takes that
he knows nothing while leading the questioning.

Advantages
 involves discussion
 students can actively engage with their knowledge instead of simply
memorizing or retaining it
 students can exchange opinions and ideas, develop excellent speaking
and communication skills
 The Socratic Method is a fun yet educational way to teach your
students how to make use of their knowledge.
 The Socratic Method also teaches students how to think critically,
accept others' opinions or viewpoints, and apply their knowledge to
the real world and to other forms of knowledge.

Lecture Method
• The teacher has a great responsibility to guide the thinking
of the students and so he must make himself intelligible to
them. Unlike other methods where motivations can come
from subsequent activities, in the lecture, students
interest depends largely on the teacher.
• Getting the attention.
• Comprehension by the class is the measure of success

Deductive Method
• The teacher tells or shows directly what
he/she wants to teach. This is also referred to
as direct instruction.
Example 1:
Find a2 X a10 = ?
Solution:
General : am X an = am+n
Particular: a2 X a10 = a2+10 = a12

Inductive method
• method of solving a problem from particular to general
• in this we first take a few examples and then generalize
• it is a method of constructing formula with the help of
sufficient number of concrete examples

Critical Thinking
Creative Thinking
Collaborating
Communicating
Learning Skills

Project Method
• This methods aims to bring practically designed experience
into the classroom. Often conducted over a period of
three to six months, the projects give students an
opportunity to work in a team environment and apply
theory learned in the classroom. There are some parts of
the curriculum in which students are necessarily
dependent on the teacher and others in which they can
work more independently.

The term "21st-century skills" is generally used
to refer to certain core competencies such as
collaboration, digital literacy, critical thinking,
and problem-solving that advocates believe
schools need to teach to help students thrive in
today's world.

COLLABORATION
 the action of working with someone to produce or create something.
TEAMWORK
The process of working collaboratively with
a group of people in order to achieve a goal.
CREATIVITY
Mental characteristic that allows a person to think
outside of the box, which results in innovative or
different approaches to a particular task.

Imagination, also called the faculty of imagining, is the
ability to form new images and sensations in the mind that
are not perceived through senses such as sight, hearing, or
other senses.
CRITICAL THINKING
the objective analysis and evaluation of an issue in order
to form a judgment
PROBLEM SOLVING
process of working through details of
a problem to reach a solution.

The 21st century skills are a set of abilities that
students need to develop in order to succeed in
the information age.
The Partnership for 21st Century Skills lists three
types:
What are 21st Century Skills?

Information Literacy
Media Literacy
Technology Literacy
Literacy Skills

Flexibility
Initiative
Social Skills
Productivity
Leadership
Life Skills

PRINCPLES AND STANDARD IN
TEACHING MATHEMATICS

1. Equity Principle
Excellence in mathematics education
requires equity—high expectations and strong
support for all students.
2. Curriculum Principle
A curriculum is more than a collection
of activities: it must be coherent, focused on
important mathematics, and well articulated
across the grades.

3. Teaching Principle
Effective mathematics teaching requires
understanding what students know and need
to learn and then challenging and supporting
them to learn it well.
4. Learning Principle
Students must learn mathematics with
understanding, actively building new
knowledge from experience and prior
knowledge.

5. Assessment Principle
Assessment should support the learning
of important mathematics and furnish useful
information to both teachers and students.
6. Technology Principle
Technology is essential in teaching and
learning mathematics; it influences the
mathematics that is taught and enhances
students' learning.

• Content Standard
The five Content Standards each encompass
specific expectations, organized by grade bands:
1. Number and Operations
Instructional programs from prekindergarten through grade
12 should enable all students to:
 Understand numbers, ways of representing
numbers, relationships among numbers, and
number systems.
 Understand meanings of operations and how they
relate to one another.
 Compute fluently and make reasonable estimates.

2. Algebra
Instructional programs from prekindergarten
through grade 12 should enable all students to:
 Understand patterns, relations, and functions
 Represent and analyze mathematical situations and
structures using algebraic symbols
 Use mathematical models to represent and
understand quantitative relationships
 Analyze change in various contexts

3. Geometry
 Analyze characteristics and properties of two-
and three-dimensional geometric shapes and
develop mathematical arguments about geometric
relationships
 Specify locations and describe spatial
relationships using coordinate geometry and
other representational systems
 Apply transformations and use symmetry to
analyze mathematical situations
 Use visualization , spatial reasoning, and
geometric modeling to solve problems

4. Measurement
Instructional programs from
prekindergarten through grade 12 should
enable all students to:
Understand measurable attributes of
objects and the units, systems, and
processes of measurement
Apply appropriate techniques, tools, and
formulas to determine measurements.

5. Data Analysis and Probability
 Formulate questions that can be addressed with
data and collect, organize, and display relevant
data to answer them
 Select and use appropriate statistical methods to
analyze data
 Develop and evaluate inferences and predictions
that are based on dataUnderstand and apply basic
concepts of probability

Process Standard
The five Process Standards are described through
examples that demonstrate what each standard looks
like and what the teacher's role is in achieving it:
1. Problem Solving
Instructional programs from
prekindergarten through grade 12 should enable
all students to:
build new mathematical knowledge through problem
solving;
 solve problems that arise in mathematics and in other
contexts;
apply and adapt a variety of appropriate strategies to
solve problems;
 monitor and reflect on the process of mathematical
problem solving.

2. Reasoning and Proof.
 recognize reasoning and proof as fundamental
aspects of mathematics
 make and investigate mathematical conjec- tures;
 develop and evaluate mathematical arguments
and proofs;
 select and use various types of reasoning and
methods of proof.

3. Communication
 organize and consolidate their mathematical
thinking through communication;
 communicate their mathematical thinking
coherently and clearly to peers, teachers, and
others;
 use the language of mathematics to express
mathematical ideas precisely

4. Connections
 recognize and use connections among
mathematical ideas;
 understand how mathematical ideas interconnect
and build on one another to produce a coherent
whole;
 recognize and apply mathematics in contexts
outside of mathematics.

5. Representation
 create and use representations to organize, record,
and communicate mathematical ideas;
 select, apply, and translate among mathematical
representations to solve problems;
 use representations to model and interpret
physical, social, and mathematical phenomena.

concepts, processes, procedures, and principles

Is a theory or a philosophy about
teaching and learning that
supports the notion that:
Learners must be independent
thinkers (cognitive)
Learners create their own
knowledge
Learners work in teams
Learning is active & student-
centered

Constructivism is the idea that
learning doesn’t just happen by the
traditional methods of teachers
standing in front of the class and
lecturing. Traditionally, teachers
present knowledge to passive
students who absorb it. No wonder
students are often bored

Jerome Brunner
Very influential psychologist
His concern with cognitive psychology “led
to a particular interest in the cognitive
development of the children and just what
the appropriate form of education might be”

Jean Piaget
Develop the cognitive learning
theory.
Felt children were “active
learners” who constructed new
knowledge
“as they moved through
different cognitive stages,
building on what they already
know”.

Lev Vygotsky
Developed the social cognitive
theory which “assert that culture is
the prime determinant of individual
development” because humans are
the only creatures to have created
cultures and therefore it effects our
learning development

John Dewey
Believed that learning should be
engaging to the students. They will
learn better if they are interested.

According to the National Council of
Teachers of Mathematics (NCTM),

Teachers should:
a. select the mathematics and
collaborative objectives to target for
instruction and cooperative learning
groups,
b. plan the math activity,
c. identify ways to promote the
elements of cooperative learning,
d. identify roles,
e. establish groups.

Euclid was a Greek mathematician, often referred to as the "Father of Geometry".
Euclid was born in 365 B.C. He went to school at Plato's academy in Athens,
Greece. He founded the university in Alexandria, Egypt. He taught there for the rest
of his life. One of his students was Archimedes.
Euclid was kind, fair, and patient. Once, when a boy asked what the point of learning
math was, Euclid gave him a coin and said, "He must make gain out of what he
learns." Another time, he was teaching a king. When the king asked if there was an
easier way to learn geometry Euclid said, "There is no royal road to geometry." Then
he sent the king to study.
In his time he was thought of as being too thorough. Now, in our time, we think he
wasn't thorough enough. Euclid died in 275 B.C.
Euclid's most famous work was the Elements. This series of books was used as a
center for teaching geometry for 2,000 years. It has been translated into Latin and
Arabic.
The Elements were divided into thirteen books, which subjects are as follows: Books
1-6= plane geometry, books 7-9= number theory, book 10= Eudoxus's theory of
irrational numbers, and books 11-13= solid geometry. More than 1,000 editions of
Elements have been published since 1482. Elements were popular until the 20th
century.
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Born: c. 365 BC
Birthplace: Alexandria, Egypt
Died: c. 275 BC
Location of death: Alexandria, Egypt
Cause of death: unspecified
Occupation: Mathematician, Educator
Nationality: Ancient Greece
Executive summary: Father of geometry
University: Plato's Academy, Athens, Greece
Teacher: Library of Alexandria, Alexandria,
Egypt Asteroid Namesake 4354
Euclides
Lunar Crater Euclid (7.4S, 29.5W,
11km dia, 700m height)
Eponyms Euclidean geometry
Slave-owners
Author of books: Elements (13 volumes)
Data (plane geometry)
On Divisions (geometry)
Optics (applied mathematics)
Phenomena (astronomy)

Dutch/Russian mathematician who
advocated the use of visual aids and
example for introductory courses in
geometry for high school students

(November 14, 1882 – October 4,
1974)

Born: November 14, 1882
Dallas, Texas
Died October 4, 1974 (aged 91)
Austin, Texas
Nationality American
Field Mathematics
Institutions University of Texas at Augustin
Alma Mater University of Chicago ((Ph.D.,1905)
Thesis Sets of Metrical Hypothesis
Robert Lee Moore
Robert Lee Moore

Robert Lee Moore
 (November 14, 1882 – October 4,
1974) was an American
mathematician, known for his work in
general topology and the Moore
method of teaching university
mathematics.

Born December 13, 1887
Budapest, Austria-Hungary
Died September 7, 1985(aged 97)
Palo Alto, California
Nationality Hungarian (–1918)
Swiss (1918–1947)
American (1947–his death)
Fields Mathematics
Institutions ETH Zürich
Stanford University
Alma mater Eötvös Loránd University
Doctoral advisor Lipót Fejér

His first job was to tutor Gregor the young
son of a baron. Gregor struggled due to his
lack of problem solving skills. Polya
(Reimer, 1995) spent hours and developed
a method of problem solving that would
work for Gregor as well as others in the
same situation. Polya (Long, 1996)
maintained that the skill of problem was
not an inborn quality but, something that
could be taught.

In 1945 he published the book How to Solve It
which quickly became his most prized publication. It
sold over one million copies and has been translated
into 17 languages. In this text he identifies four
basic principles

 Can you state the problems in you
own words?
What are you trying to find or do?
What are the unknowns?
What information do you obtain from the
problem?
What information, if any is missing and
not needed?

Look for a pattern
Remember related problems
Break the problem down into different
parts
Make a table
Make a diagram
Write an equation
Use a guess and check
Work backward
Identify a subgoal

Check the result in the
original problem. Does your
answer make sense? Is it
reasonable?
Determine whether there
is another method of finding
the solution..
If possible, determine
problems for which techniques

Implement the strategy in Step 2 and
perform the necessary math computations.
Check each step of the plan as you do it.
Keep an accurate record of your work.
Organize your work into easy to
understand visuals.
Double check your math work.

Toru Kumon (公文公 Kumon Tōru ,
March 26, 1914 – July 25, 1995)
Japanese mathematics educator
Born in Kōchi Prefecture, Japan
He graduated from the College of
Science at Osaka University with a
degree in mathematics and taught high
school mathematics in his home town of
Osaka
Asteroid 3569 Kumon is named after
him

During his 33-year career, he taught math at his
alma mater Tosa Junior/Senior High School, and
later at Sakuranomiya High School in Osaka City, as
well as at other schools
In 1958, he established the Osaka Institute of
Mathematics, which later became the Kumon
Institute of Education Co., Ltd in 1983

Toru Kumon died in Osaka on July 25, 1995 at the
age of 81, from pneumonia
He devoted the rest of his life to improving the
Kumon method and making it available to more
and more people around the world.

In 1954 in Japan, a grade 2 student by the name of
Takeshi scored poorly in a math test. His mother,
Teiko, asked her husband, Toru Kumon, to take a
closer look at their son’s school textbooks. Being a
high school math teacher, Toru Kumon thought
Takeshi's textbooks did not give children enough
practice to be confident in a topic. Mr. Kumon
decided to help his son by handwriting worksheets
for him to practice. This was the start of the
Kumon Method

By the time Takeshi
was in Year 6, he was
able to solve
differential and
integral calculus usual
ly seen in the final
years of high school.
This was the
beginning of the
Kumon Method of
Learning

As a result of Takeshi's progress, other parents
became interested in Kumon's ideas, and in 1956,
the first Kumon Center was opened
in Osaka, Japan
In 1958 the Kumon Institute of Education was
established in Osaka, Japan.

Half a century ago, the Kumon Method was born
out of a father' s love for his son
The late Chairman Toru Kumon developed the
prototype of the Kumon Method in 1954 while he
was a high school mathematics teacher. At the
time, Toru Kumon's wife, Teiko, asked him to take
a look at his eldest son Takeshi's second-grade
arithmetic studies because she was not satisfied
with the results of a test he had taken.

Toru Kumon wrote out calculation problems on
loose-leaf paper for Takeshi, and the materials
that he created from 1955 became the prototype
for today’s Kumon worksheets
Based on his experience as a high school teacher,
Toru Kumon knew that many senior high school
students had problems with their math studies
because of insufficient calculation skills
Therefore, he focused on developing Takeshi’s
calculation skills, and created materials that made
it possible for his son to learn independently

Kumon has two core programs, the Kumon Math
and Kumon Native Language Program (language
varies by country)
How Kumon works is that each student is given an
initial assessment of his or her abilities
Based on the results and the student's study skills,
a Kumon Instructor will create an individualized
study plan.

As a high school mathematics teacher, Mr. Kumon
understood that an understanding of calculus was
essential for Japanese university entrance exams
so in writing worksheets for his son, Mr. Kumon
focused on all the topics needed for a strong
understanding of calculus starting from the basics
of counting.

Level 6A: Counting numbers to 10, reading
numbers.
Level 5A: Reading numbers to 50, sequence of
numbers.
Level 4A: Reading numbers, writing numbers to
120.
Level 3A: Numbers up to 120, adding up to 3.
Level 2A: Adding up to 10.
Level A: Horizontal addition, Subtraction from
numbers up to 20.
Level B: Vertical addition and subtraction.
Level C: Basic multiplication, division.
Level D: Long multiplication, long division,
introduction to fractions.

Level E: Fractions
Level F: Four operations of fractions, decimals.
Level G: Positive/negative numbers, exponents,
Algebraic expressions, Single-Variable Equations
with 1-4 steps.
Level H: Transforming Equations,
Linear/simultaneous equations, inequalities,
algebraic functions and graphs, adding and
subtracting Monomials and Polynomials.
Level I: Factorization, square roots, quadratic
equations, Pythagorean theorem.
Level J: Algebra II.
Level K: Functions: Quadratic, fractional, irrational,
exponential.

Level L: Logarithms, basic limits, derivatives,
integrals, and its applications.
Level M: Trigonometry, straight lines, equation of
circles.
Level N: Loci, limits of functions, sequences,
differentiation.
Level O: Advanced differentiation, integration,
applications of calculus, differential equations.
Level X (elective level): Triangles, vectors,
matrices, probability, statistics.

The Kumon Native Language Programs are
designed to expose students to a broad range of
texts and develop the skill of reading
comprehension. A number of Kumon Centres also
use audio CDs to help students with
pronunciation. (Note: Levels vary slightly by
country

Level 7A: Look, Listen, Repeat.
Level 6A: Reciting Words with Pictures.
Level 5A: Letter Sounds.
Level 4A: Consonant Combinations and Vowel
Sounds.
Level 3A: Advanced Vowel Sounds & Advanced
Sounding Out.
Level 2A: Functions of Words (nouns, verbs,
adjectives), Reading Aloud.
Level AI: Structure of Simple Sentences.
Level AII: Sentence Structure, Sentence Topics,
Thought Sequence.
Level BI: Subject and Predicate.
Level BII: Comparing and Contrasting.

Level CI: Constructing Sentences.
Level CII: Organizing Information.
Level DI: Combining Sentences.
Level DII: Main Idea, Understanding Paragraphs.
Level EI: Clauses.
Level EII: Reason and Result.
Level FI: Referring Words, Interpreting Text.
Level FII: Concision, Analysis of & Recounting
Events from Paragraphs.
Level G: Point Making, Theme, Story Elements,
Summary.
Level H: Summation.
Level I: Persuasion.
Level J: Critical Reading.

Level K: Elements of Literature.
Level L: Interpretation.

Therefore, the Kumon Method has been
welcomed into communities around the world
with widely differing cultures, values, and
educational systems*total enrollments for all
subjects (as of March 2016)
1958: Kumon is established
1974: First steps of overseas expansion
1985: Increasing numbers of Kumon students
around the world
2009: Kumon as a global brand as we move
toward the next 50 years of our history
2014: Instructors around the world learn
from children

Born on 23 January 1935 in New York City

 Moses grew up in a housing project in Harlem. He attended
Stuyvesant High School, an elite public school, and won a
scholarship to Hamilton College in Clinton, New York.
• He earned a master’s degree in philosophy in 1957 from
Harvard University, and was working toward his
doctorate when he was forced to leave because of the
death of his mother and the hospitalization of his
father.
• Moses returned to New York and became a mathematics
teacher at Horace Mann School a prestigious private high
school.

 In the 1990s, the Algebra Project students learned to think
and speak mathematically through tackling problems that
arose in their daily lives.
• In 12 years the program helped more than 10,000
students master fundamental algebraic skills in cities
across the country.
• In 1992, Moses returned to Mississippi to start the
Delta Algebra Project. Moses would later tell the New
York Times. ” But this time, we’re organizing around
literacy-not just reading and writing, but mathematical
literacy… Now math literacy holds the key.”

 His first involvement came with the Southern Christian
Leadership Conference (SCLC) where he organized a youth
march in Atlanta to promote integrated education.
• In 1960 Moses joined the Student Nonviolent
Coordinating Committee (SNCC) and two years later
became strategic coordinator and project director with
the newly formed Council of Federated Organizations
(COFO) which worked in Mississippi.
• In 1963 Moses led the voter registration campaign in the
Freedom Summer movement. The following year he helped
form the Mississippi Freedom Democratic Party which
tried to replace the segregationist-dominated Mississippi
Democratic Party delegation at the 1964 Democratic
National Convention.

In regards to entertainment, Moses served as an influential
figure in the construction of two World’s Fairs. Shea Stadium
and Lincoln Center.

During his decades, Robert Moses brought:

Pierre Van Hiele
and
Dianna Van Hiele

Biography of Pierre Van Hiele
Van Hiele was famous for his theory that describes how students
learn geometry, he was born in 1909 and died November 1,2010.
This theory came about in 1957 when he got his doctoral at
Utrecht University in Netherlands. He was also a publisher, he
published a book titled Structure and Insight in 1986 which
further describe his theory. The theory came about by two Dutch
educators, Diana Van Hiele-Gelof and Pierre Van Hiele (wife and
husband).

Critical examination of how students
learn based on Van Hiele’s theory
Based on Diana Van Hiele-Gelof and Pierre Van Hiele
theory there are five levels to describe how students
learn or understand geometry. These are:
-Level 0: Visual
-Level 1: Description
-Level 2: Relational
-Level 3: Deductive
-Level 4: Rigor

Example of how to demonstrate the
thinking process that children use in
learning math base on his theory
Van Hiele strongly believed that using his theory in
Geometry it would improve the student learning. For
example in geometry at the visual level the teacher
could draw some triangles on the board, so the
students would know what a triangle looks like.

How Hiele ‘s theory contribute to
Mathematics education and it application
to the Jamaican classroom?
This theory contributes greatly to Mathematics Education
since it is a Geometry theory and most students find
Geometry difficult. This theory can be applied through five
phases.
Phase 1- (Information/ Inquiry): At this stage teacher
introduce a new idea and allow student to work with it.
This new idea is normally easier to understand than the
original but it means the same. So students get a better
understanding. Example: Alternate angles are equal but
she could say ‘Z’ angles are equal. So students will
understand easier.

Phase 2-(Guided or Direct Orientation): At this stage
teacher give lots of work to students for practice so
they get aquatinted with the concepts and learn it
well.
Phase 3-(Explication): At this stage teacher told
students to, in their own word describe what they
learn using mathematical terms. Example: Reflection
writing.
Phase 4-(Free Orientation): This is where teacher
allow students to apply relationships they learn to
solve harder problems. Example: They learn from a
cxc and allow using the same principles learn to solve
questions from a cape book.

Phase 5-(Integration): This is where students reflect on what
they learn and find easier way to do what they learn.
These phases when perform will build geometry students
understanding therefore build better students including
Jamaica.

A wife of Robert kaplan
She is also an
author

The nothing that is: A natural history of
zero
Originally published- January 1,1999
The ‘invention’ of zero made arithmetic
infinitely easer- try to doing division in
roman numerals- and it now forms part
of the binary code which powers all our
computers…

Originally published- January 4, 2011
 A squared equals c squared. It sounds
simple, doesn’t it? Yet this familiar
expression opens a gateway into the
riotous garden of mathematics, and
send us on a journey of exploration in…

A Filipino mathematician. He has his
Ph D. from the University of California,
Berkeley from 1939 under the supervision
of Pauline Sperry, and had his career at
the University of the Philippines in
Manila.[3] Dr. Raymundo Favila was elected
as Academician of the National Academy
of Science and Technology in 1979

He was one of those who initiated
mathematics in the Philippines. He
contributed extensively to the progression
of mathematics and the mathematics
learning in the country. He has made
fundamental studies such as on stratifiable
congruences and geometric inequalities.
Dr. Favila has also co-authored textbooks
in algebra and trigonometry.

The Filipino Legend
Dr. Amado Muriel was recognized
because of his important works and
marvelous contributions to the field of
theoretical physics, in particular, his
advancement of theoretical apparatus to
clarify turbulence. His new kinetic
equation is valuable for discovering
essential problems of non-equilibrium
statistical procedure.

Dr. Muriel discovered the
accurate and estimated solutions for
the performance of a two-level
system, which were considered by
his peers as a revolutionary
contribution to a quantum Turing
machine, now a rising field in
quantum computing. Also, in his
studies on stellar dynamics, he has
recognized realistically that self-
gravitation alone is adequate to
create a hierarchy of structures in
one dimension.

Dr.Melecio Magnowas born onMay 24,1920.
Dr.Melecio S. Magno was Chairman of NSDBfrom
1976 to 1981. Heobtained a BSin Mining
Engineering, and MS in PhysicsfromtheUniversity
of the Philippines.

His interests spanned
seismology, metrology, meteorology, physics and
held administrative positions in these fields in UP.
He was appointed Minister of NSDB from 1978 to
1981. He later became President of the Science
Foundation of the Philippines, Vice President of the
National Academy of Science and Technology, and
President of the National Research Council of the
Philippines.

He pushed for the establishment of the National
Academy of Science and Technology.
Under Dr. Magno, the NSDB pursued mission-
oriented R&D programs in different sectors.
Dr. Magno has researched on the assimilation and
fluorescence spectroscopy of crystals, particularly
rare-earth crystals; effects of typhoons on the
allotment of gravitation; and the idea of science. He
also a co-author of a Textbook in physics at the
University of the Philippines named University
Physics.

Dr. Apolinario Nazarea made significant role
to the theories on biophysics and recombinant
biotechnology including his own conceptual
framework on the structure of RNA/DNA
investigation. Born on October 11, 1940.

On the turning point of the country's technical
course, his part in the expansion of biophysics and
biotechnology is both essential and well timed. With his
worldwide-cited systematic work and hypothetical
expertise, he has laid the groundwork for the design of
artificial vaccines on a sounder molecular beginning.

A native of Lipa, Batangas,
Zara finished primary
schooling at Lipa
Elementary School, where
he graduated as
valedictorian in 1918. In
1922, he again graduated
valedictorian in Batangas
High School, an accolade
which warranted him a
grant to study abroad

In the middle of his first
semester, he finally got
the scholarship when his
rival got sick and died
abroad. Dr. Zara then
enrolled at the
Massachusetts Institute of
Technology (MIT) in the
United States, and
graduated with a degree
of BS in Mechanical
Engineering in 1926.

 he obtained a Master of Science in
Engineering (Aeronautical Engineering)
at the University of Michigan, USA,
graduating summa cum laude.
Zara then sailed to France to take up
advanced studies in physics at the
Sorbonne University in Paris. In 1930
he again graduated summa cum laude
with a degree of Doctor of Science in
Physics, with "Tres Honorable," the
highest honor conferred to graduate
students. Zara was the first Filipino
given that honor. Madam Marie Curie
was given the same accolade for her
discovery of radiu

 became chief of the aeronautical division of the
DPWC.
1936, he was assistant director and chief
aeronautical engineer in the Bureau of
Aeronautics of the Department of National
Defense.
For 21 years, he was director of aeronautical
board, a position he held and confirmed by the
Congress of the Philippines up to 1952.
Considered expert in the Field, he was chosen
to be the technical editor of Aviation Monthly
and at various times, he worked as vice
chairman and acting chairman of the National
Science Development Board, where a number
of Science projects were impetus.

The Zara Effect – He discovered the physical law of electrical
kinetic resistance called the Zara effect (around 1930)[3]
He improved methods of producing solar energy including
creating new designs for a solar water heater (SolarSorber);
A sun stove, and a solar battery (1960s);
Invented a propeller-cutting machine (1952);
He designed a microscope with a collapsible stage;
helped design the robot Marex X-10;
Invented the two-way television telephone or videophone
(1955) patented as a "photo phone signal separator network";
Invented an airplane engine that ran on plain alcohol as fuel
(1952);

Name: Padlan,
Eduardo A.
Specialization: Ph.D.
Biophysics
Division: Chemical,
Mathematical and
Physical Sciences

Dr. Padlan was elected as
Academician on 2003 and born on
August 31, 1940. His latest work on
humanized antibodies has possible
applications in the healing of different
diseases as well as cancer. He has
fourteen approved and awaiting patents
on diverse part and uses of antibodies.

Honors and award received
-Concepcion Dadufalza Award for
Distinguished Achievement, University of the
Philippines, 2007
-Severino & Paz Koh Lectureship in Science,
Philippine-American Academy of Science and
Engineering, 2008
Research involvement
-Research Interests
-Molecular Immunology, Protein Structure &
Function

Aside from doing research in
geometry, Dr. Marasigan served as President
of the Mathematical Society of the
Philippines for two terms. His many
achievements include pushing for the
formulation of Policies and Standards for
Mathematics, approved by the government
in 1989. For ten years, he served as chair of
the Ateneo Math Department and headed
the Math and Operations Research Division
of the National Research Council of the
Philippines.

According to a study
Douglas Clements and
Julia Saramel (2006),
“there’s an overlap
between language and
math” and success in one
area reinforces the other.

Math requires
“precision in language.”,
so talking about math
helps kids increase their
vocabularies.

 What may be less
obvious is that many
mathematical concepts
are embedded in
children’s stories…

Literature is an effective tool
for mathematics instruction
because it:
•incorporates stories into the
teaching and learning of
mathematics
•introduces math concepts
and contexts in a motivating
manner

•acts as a source for
generating problems and
building problem solving
skills
•helps build a conceptual
understanding of math skills
through illustrations

By Joseph LaCoste and
Mikala Smith
Goldilocks and the Three
Bears is simply one of
many modern
interpretations of Robert
Southey's original.

Robert Southey
August 12, 1774 in Bristol –
March 21, 1843 in London) was an
English poet of the Romantic school,
one of the so-called "Lake Poets",
and Poet Laureate for 30 years from
1813 to his death in 1843. Although
his fame has long been eclipsed by
that of his contemporaries and
friends
William Wordsworth and Samuel
Taylor Coleridge, Southey's verse still
enjoys some popularity.

the mathematical
principal of ordering
correspondences
between ordered sets
patterning
classifications and
cause and effect thinking

“Such concepts
can be applied to
simultaneous
comprehension of
math and
literature.”

“Using math related children's
literature can help children
realize the variety of situations
in which people use
mathematics for real purposes.”
Professor David Whitin,
Wayne State University

Language Concepts
and Skills that can be
embedded into
Mathematics
activities:

-Increasing vocabulary
-Cause & Effect
-Matching to sample
-Same & different
-Articulation skills
-Answering “Wh” questions
-Calendar concepts
-Labeling
objects/pictures/symbols
- Answering yes/no questions

-Sorting by color, shape, size
-Classifying/Categorizing
-Using 1-1 correspondence
-Counting - Sequencing
-Number recognition
-Identifying functions of
objects/pictures/symbols
-Increasing Mean Length of
Utterance
- Comparisons (more/less;
large/small; long/short)

-Ordinal numbers
- Recognizing shapes
-Following a pattern
- Problem solving
-Money skills
-Time concepts
-Utilizing AAC device
-Making predictions
-Following a task analysis
- Comparing and Contrasting

 Making predictions….like making
estimates before solving math
problems
 Writing things down in graphic
organizers can help reinforce reading
comprehension, while writing things
down in a similar way in math can
help them “develop, cement and extend
understanding.”

 When responding to literature
children’s writing is never
identical, there are several ways to
come up with a “right answer”.
The same idea holds true for
Mathematics. Teachers should
“encourage different methods for
reasoning, solving problems and
presenting solutions.”

In conclusion, the
significance of developing
children’s English through
math skills is very obvious.
It is in the early grades that
lifelong foundations are
formed for skills these
important subject areas.

Confidence and interest
play a large role in learning as
well. Finally, as Clement and
Sarama (2006) state about
children’s literature and math,
“by connecting the two areas
children also build a far deeper
understanding of each.”

4 PARTS
1. Visual Learners- Students
prefer the use of images,
maps and graphic organizers
to access and understand
new information.
Neil Fleming, 2006

2. Auditory Learners- best
understand new content
through listening and speaking.
3. Read and Write Learners-
students learn best through
words.

4. Kinesthetic Learners-
Students that are hands-on
learners and learn best
through figuring things out by
hand.

(Howard Gardner)
1. Verbal Linguistic- People
who possess this learning
style learn best through
reading, writing, listening, and
speaking.

2. Logical/Mathematical- Those
who exhibits this kind of
intelligence learn by
classifying, and thinking
abstractly about patterns.
3. Visual-These people learn
by drawing looking at the
picture.

4. Auditory- Students who are
music smart learn using rhythm
or melody.
5.Bodily Kinesthetic- Body smart
individuals learn best through
touch and movement.
6.Interpersonal- Those who are
people smart relating to others.

7. Intrapersonal- Learn best by
working alone.
8.Naturalistic- Working with
nature

Teaching and
Learning
Mathematics using
Research

What you mean by Mathematics education?
 In contemporary education,
Mathematics Education is the
practice of teaching and learning
mathematics, along with the
associated scholarly research.

What is the meaning of Research in Mathematics
Education?
Researches in Mathematics education primarily
concerned with the tools, methods and approaches
that facilitate practice or the study of practice.
Pedagogy of Mathematics has developed into an
extensive field of study, with its own concepts,
theories, methods, national and international
organizations, conferences and literature.

Objectives
The teaching and learning 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 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.
The teaching of heuristic and other problem solving
strategies to solve non-routine problems.

Need and its Importance
1. Continuing math research is important because incredibly useful
concepts like cryptography, calculus, image and signal processing
to continue to come from mathematics and are helping people
solve real-world problems.
- This “math as tool” is absolutely true and probably the easiest way to
go about making the case for math research.
- It’s a long-term project.
- We don’t know exactly what will come out next, or when, but if we
follow the trend of “useful tools”.
- We trust that math will continue to produce for society.
- Mathematics is omnipresent in the exact science.
- Mathematics is basic stuff that has been known for decades or
centuries.

2. Continuing math research is important because it
is beautiful. It is an art form, and more than that,
an ancient and collaborative art form,
performed by an entire community. Seen in this
light it is one of the crowning achievements of
our civilization.
- compare mathematics research directly with
some other fields like philosophy or even writing or
music.
- Our existence informs us on the most basic
questions surrounding what it means to be human.

3.Continuing math research is important because it
trains people to think abstractly and to have a
skeptical mindset.
- Mathematicians properly trained are psyched to
hear a mistake pointed out their argument because
it signifies process. There is no shame in being
wrong.
- It is an inevitable part of the process of learning.

The following results are examples of some of the current
findings in the field of mathematics education:
Important results
Conceptual
Understanding
Formative Assessment
Home work
Students with
difficulties
Algebraic reasoning

Conclusion
At the current stage of research in mathematics education,
its main contribution to practice may be to raise teacher
awareness and deepen teacher understanding of the
complicated nature of mathematical knowledge, knowing,
and learning.
Reading and discussing research articles, contributed to
the teachers learning, in general, that students construct
their knowledge.
The mini-study made this theoretical idea more specific,
concrete and relevant for the teachers. They learned what
the constructivist view might mean in a practical context.

This instructional material is made for
students to :
• easily review on the basic concepts on
fractions
• identify the basic skills in using
fractions
• solve algebraic operations with fractions
and for mastery of any problems
involving fractions.

• Basic operation on fractions
• Solving algebraic equations involving fractions
• Solving word problems involving fractions

students to master :
the rules in solving basic operations on
integers (the laws of signed numbers)
Solving problems on integers.

ADDITION
To add a positive on the number line, move to the
right, towards the larger numbers. To add a negative
on a number line you move to the left.
Simple rule
Rule for adding integers with different signs:
Subtract the absolute values of the numbers and the use the
sign of the larger absolute value.

SUBTRACTION
To subtract a positive number, move to the left on the
number line. This is the same thing that happens when we
add a negative number.
Subtract a negative number we need to move to the right.
Simple Rule:
KEEP the first number the same. CHANGE the
subtracting to adding. Then CHANGE the sign
of the second number

MULTIPLICATION AND DIVISION
Multiplying is really just showing repeated adding. To add 2 three times. 2 + 2
+ 2 = 6

SIMPLER RULES
Rule #1:If the signs are the same, the answer is positive.
Examples:
Rule #2:If the signs are different, the answer is negative.
Dividing integers are the same as the rules for multiplying integers.
Remember that dividing is the opposite of multiplying. So we can
use the same rules to solve.
Rule #1:If the signs are the same, the answer is positive.
Rule #2:If the signs are different, the answer is negative.

• Introduction of integers
• Basic operations on integers
• Solving algebraic equations

This instructional material is made
for the learners to:
better understand ways of algebraic
thinking and the concepts of
Algebra.

Each tiles represents to a certain variable/ constant
x2
x
1
2X2 + 2x+ 3

• Concepts on Algebra
 basic operations on signed numbers
 Simple substitution
 Solving equations
 Distributive property
 Representing polynomials
 Basic operations on polynomials
 Factoring polynomials
 Completing the square
• Geometric figures on square and parallelogram

This instructional material will help the
learners :
be introduced with the concepts of plane figures
to master the skill in solving areas and perimeter
of plane figures.

This instructional material will help the learners :
be introduced with the concepts of plane figures
and its characteristics
to use concrete material on finding the area and
perimeter of plane figures
to master the skill in solving areas and perimeter
of plane figures

OBJECTIVES
the students to:
solve for the area and circumference of
a circle
identify the relationship between a
circle and a parallelogram.

• Concept of a circle; area and perimeter
• Relationship of a parallelogram and a
circle
• Fraction
• Division of numbers

 Define Perimeter and Area.
 Illustrate the formulas on
finding the perimeter and
area of plane figures.
 Find the perimeter and area
of common plane figures.

PERIMETER
The perimeter of any polygon is the sum of
the measures of the line segments that
form its sides. OR SIMPLY, the
measurement of the distance around any
plane figure.
Perimeter is measured in linear units.

The perimeter P of a triangle with sides of lengths a, b,
and c is given by the formula
P = a + b + c
a
b
c

The perimeter P of a square with all sides of
length s is given by the formula
P = 4s
s
s
s
s

The perimeter P of a rectangle with length l and width
w is given by the formula
P = 2L + 2W
W
L
W
L

AREA
The amount of plane surface covered
by a polygon is called its area. Area
is measured in square units.

The area of a rectangle is the length of its
base times the length of its height.
A = bh
HEIGHT
BASE

The area of a parallelogram is the length of its base
times the length of its height.
A = bh
Why?
Any parallelogram can be redrawn as a rectangle
without losing area.
BASE
HEIGHT

The area of a triangle is one-half of the length of its base times the
length of its height.
A = ½bh
Why?
Any triangle can be doubled to make a parallelogram.
HEIGHT
BASE

Remember for a trapezoid, there are two parallel sides, and they
are both bases.
The area of a trapezoid is the length of its height times one-half of
the sum of the lengths of the bases.
A = ½(b1 + b2)h
Why?
Red Triangle = ½ b1h
Blue Triangle = ½ b2h
Any trapezoid can be
divided into 2 triangles.
HEIGHT
BASE 2
BASE 1

The area of a kite is related to its diagonals.
Every kite can be divided into two congruent
triangles.
The base of each triangle
is one of the diagonals.
The height is half of the
other one.
A = 2(½•½d1d2)
A = ½D1D2
d1
d2

Diameter
d=2r
Circumference
C=2πr
Area
A=πr2
diameter

Rectangle P = 2l + 2w A = bh
Square P = 2l + 2w A = bh
Triangle P = side + side
+ side
A = ½ bh
Parallelogram P = 2l + 2w A = bh
Trapezoid P = 2l + 2w
Circles C = 2∏r A = ∏r²
1 2
1
( )
2
A b b h 

SURFACE AREA AND
VOLUME OF SOLID
FIGURES

SURFACE AREA
the amount of
paper you’ll need
to wrap the shape
VOLUME
the number of
cubic units
contained in the
solid.

SURFACE AREA
Total surface area:
6 (side) ² or 6(s) ²
Lateral surface area:
4(side)² or 4 (s) ²
CUBE

VOLUME
CUBE/SQUARE PRISM
V = s²H
The product of its
height H and the
area of its base s².

SURFACE AREA
Total surface area:
2(lb+bh +lh)
Lateral surface area:
2(l+b)h
bl
h
RECTANGULAR
PRISM

VOLUME
V = lwh
The product of
its length ,
width/base and
height
w
l
h

SURFACE AREA
Curved surface area
2 π rh
+
area of the circle
2 π r2
0r
Total surface area: πrh
+2 π r2
=2 π r(h+r)
CYLINDER

VOLUME
V = Bh
V= πr²h
The product of its
base (πr²) and
height (h)
h
b

SURFACE AREA
SA = ½ lp + B
Where l is the Slant
Height and
p is the perimeter and
B is the area of the Base
TRIANGULAR PRISM

VOLUME
(1/3) Area of the Base x
height
Or
(1/3) Bh
Or
1/3 x Volume of a Prism
b
h

SURFACE AREA
Total surface area of cone:
π r(s+r)
Lateral surface area of
cone-
π rs
CONE

VOLUME
V = ⅓Bh
V= ⅓ πr²h
where B is the area of the
base and h is the height of
the cone.
(1/3 the area of a cylinder)

Instrumentation in Mathematics TNESC

Instrumentation in Mathematics TNESC

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