Teaching Tools and Instructional Materials in Mathematics

INSTRUMENTATION IN
TEACHING MATHEMATICS
Teaching Tools and Instructional Materials in
Mathematics
by
Mary Alyssa A. Cacha
BSEd – Math IV

MANIPULATIVES IN MATHEMATICS
What are manipulatives?
These are concrete models which involve or integrate
mathematics concepts appealing to several senses.
Manipulatives are physical objects that are used as
teaching tools to engage students in the hands-on
learning of mathematics. They can be used to
introduce, practice, or remediate a concept.

 Why use manipulatives?
 enable students to explore concepts at the first, or concrete, level of
understanding.
 concretize the abstract concepts of Math
 contribute to the development of well-grounded, interconnected
understandings of mathematical ideas.
 students can more easily remember what they did and explain what
they were thinking when they used manipulatives to solve a problem
 make students’ retention of learning longer since more senses are
involved

 Why use manipulatives?
It help students learn:
to relate real-world situations to
math symbols
to work together cooperatively
in solving problems
to discuss math ideas and
concepts
to verbalize students’ math
learning and thinking
to make presentations in front of
a large group
to explore alternative ways to
solve problems
to symbolize math problems in
different ways
to solve problems without
following teachers’ directions

 How to evaluate learning using manipulatives?
Demonstration of skills
Making of outputs
Concept development and understanding
Students’ interaction and sharing of Math concepts

 What are some techniques in evaluating learning using manipulatives?
Observe students working cooperatively and individually.
Let students discuss their Math ideas by using manipulatives.
Ask “how” and “why” questions rather than “what”.
Let students write the solution or process than just providing the
answer.

EXAMPLES OF
MANIPULATIVES AND
THEIR PEDAGOGICAL USES

EXAMPLES OF MANIPULATIVES AND THEIR
PEDAGIGICAL USES
 Geometry
Tangrams
Geoboard
Platonic and Archimedean
Solids
Graphing Calculator
Protractor
Compass
 Trigonometry
Protractor
Clinometer
 Numbers and Number Sense
Fraction Bars

 What is a tangram?
 A traditional Chinese puzzle made of a square divided into seven pieces (one
parallelogram, one square and five triangles) that can be arranged to match
particular designs.
 What are its pedagogical uses?
 Boosts the visual-spatial skills of the students.
 Helps develop the capacity to determine the whole from its randomly scattered parts
 Helps classify shapes
 Helps learners learn geometric terms and develop stronger problem solving skills
 Develops spatial rotation skills
 Helps students acquire a precise vocabulary for manipulating shapes (e.g. flip, rotate)
Tangram

 What is a geoboard?
It is a math manipulative used to support early geometric,
measurement and numeracy concepts. A geo-board is a square
board with pegs that students attach rubber bands to.
Used to explore basic concepts in plane geometry such as
perimeter, area, and the characteristics and properties of
polygons.
Geoboard

Platonic and Archimedian Solids

 What are solid figures?
 Solid figures are three dimensional figures which occupy space. It occupies
some volume in it. Geometric solids have height as its third dimension as an
extension to two-dimensional figures.
 Platonic solids are solid figures whose surfaces are regular convex polygons.
 Archimedean solids are solid figures whose surfaces are made up of a
combination of two or more regular convex polygons.
 What are their pedagogical uses?
 Used to explore basic concepts in solid geometry such as volume and the
characteristics and properties of solid figures.
 Used to identify the number of vertices, edges, and faces of solid figures.
Platonic and Archimedean Solids

 What is a graphing calculator?
 Is a handheld computer that is capable of plotting graphs, solving
simultaneous equations, and performing other tasks with variables.
 Used in interpreting graphs.
 Used in graphing conic sections – parabola, ellipse, hyperbola, and circle.
 Improves students' operational skills and problem-solving skills
 Helps students to be more flexible in working with real data.
Graphing Calculator

Protractor
 What is a protractor?
A protractor is a measuring instrument, typically made of
transparent plastic or glass, for measuring angles.
Used in constructing and measuring angles
Used in measuring angles of geometric shapes and polygons.

Compass
 What is a compass?
 It is a technical drawing instrument which are usually made of
metal or plastic, and consist of two parts connected by a hinge
which can be adjusted to allow the changing of the radius of
the circle drawn. Typically one part has a spike at its end, and
the other part a pencil, or sometimes a pen.
Used in constructing or inscribing arcs and circles
Accompanied with a straightedge, it can also be used in
constructing lines, angles, and geometric figures.

Clinometer
 What is a clinometer?
It is an instrument for measuring angles of slope (or tilt), elevation
or depression of an object with respect to gravity.
Used in presenting trigonometric applications – angle of
elevation, angle of depression, and trigonometric functions.
Used in demonstrating how trigonometry can be used to find the
height of a structure or high object

Protractor
 What is a protractor?
A protractor is a measuring instrument, typically made of
transparent plastic or glass, for measuring angles.
Used in constructing and measuring angles
Used in discussing circular functions – degrees and radians.

 What is a graphing calculator?
 Is a handheld computer that is capable of plotting graphs, solving
simultaneous equations, and performing other tasks with variables.
 Used in discussing trigonometric functions.
 Used in graphing trigonometric function – graphs of sine, cosine, and tangent.
 Improves students' operational skills and problem-solving skills
 Helps students to be more flexible in graphical approaches, and working with
real data
Graphing Calculator

Fraction Bars
 What are fraction bars?
 These are rectangular strips to represent different parts of the same whole.
They can be cut apart and manipulated to see how various parts can be
added together to make the whole or compare different fractional amounts
for equivalency.
 What are their pedagogical uses?
 Used in discussing fractions.
 Help students visualize and explore fraction relationships.
 Used in presenting fractions as a part of a whole.
 Used in showing equivalent fractions.
 Used in performing operations on fractions.

OTHER INSTRUCTIONAL
MATERIALS IN
MATHEMATICS

OTHER INSTRUCTIONAL MATERIALS IN
MATHEMATICS
 Geometry
 Golden Ratio Scrapbook
 Nails and String
 Tessellations
 Circle (Mandala)
 Booklet of Definitions, Postulates,
Corollaries, and Theorems
 Booklet of Mensuration
 Trigonometry
 Radian the Snowman
 Graph of Trigonometric Functions
 Linear Algebra
 Application of Linear Regression:
Story Book
 Problem Solving
 Compilation of Math Problems

Transformational Geometry:
Transformations of Number 6

Booklet of Definitions, Postulates, Corollaries,
and Theorems

Application of Linear Programming:
A Story Book

Combination: Chef Alyssa’s Menu

Compilation of Mathematical Problems

INTEGRATING
TECHNOLOGY IN
MATHEMATICS

INTEGRATING TECHNOLOGY IN MATHEMATICS
 What is technology in teaching?
 Why integrate technology in teaching Mathematics?
 Presentations
 Geometry Software by Divine Grace Cabahug
 Computer Algebra System by Marie Christine Regis
 Presentation Software by Mary Alyssa Cacha
 Spreadsheet by Mayend Dagunan
 Virtual Manipulatives by Mayend Dagunan
 Internet and Java Applets by Angerica Gellecania
 Movies and Videos in Mathematics Teaching by Miriam Grace Ceprado

 What is technology in teaching?
This refers to the set of ways which supports both teaching and
learning.
Technology infuses classrooms with digital learning tools, such as
computers and hand held devices;
Technology also has the power to transform teaching by
ushering in a new model of connected teaching.
This model links teachers to their students and to professional
content, resources, and systems to help them improve their own
instruction and personalize learning.
Integrating Technology in Mathematics

 Why integrate technology in teaching of Mathematics?
 To address the abstract nature of mathematics.
 To make math class discussions more engaging to the learners.
 To make math class discussions more interactive.
 To increase the learners’ participation and confidence.
 To link mathematics in real-life context.
 To present math lessons clearer.
Integrating Technology in Mathematics

Dynamic Geometry
Softwares
A computer program for
interactive creation and
manipulation of geometric
constructions

Help build a geometric
model of objects such as
points, lines, circle, etc
and their relationship to
one another

is a visualization of an
abstract model (of
geometric nature) and, in
particular, provides a visual
interface for its
manipulation

The user can manipulate the
model by moving some of
its parts, and the program
accordingly – and instantly
– changes the other parts, so
that the constraints are
preserved.

Typical Uses
graphical presentation of
geometry on the screen
exploring geometric
properties, testing hypotheses
and visualising complex data

Typical Uses
geometric reasoning
 illustrations in document
preparation
 illustrations for the Web
 libraries for geometric
programming.

Features
 2D or 3D
constructive richness
 easy to use interface and other
convenience-related issues
kinds and degrees of
dynamicism (incl. animation)

adaptability to specific domains
and needs
accepting text commands
 extensibility (through
programming)
observability of the
constructive dependencies (what,
how)

 reusability (of parts or
techniques)
• portability to foreign
environments: Web, general text
processing tools, document
preparation systems, automatic
provers

independence of operating
system and other elements of the
operating environment
 ability to talk to the user in
different languages;

Popular DGS
The Geometer’s Sketchpad
(1990s).
•This lets one publish Sketchpad sketches
on the Internet and interact with them
independently. The program can be
obtained for free, separately from
Sketchpad.

Popular DGS
GeoGebra
•is a typical modern DG system for planar
geometry.
•Features:
1. rich, well thought-out set of
constructive and other commands
2. easy to learn and use, ergonomic
interface w. r. t. command access,
object and attribute manipulation

Popular DGS
GeoGebra
3. all parts of the drawing are draggable
and always accessible for adding,
changing or deletion of attributes
4. output formats for different uses, such
as e. g. printable documents and Web

Computer Algebra
• relates to the use of machines, such as
computers, to manipulate mathematical
equations and expressions in symbolic form, as
opposed to manipulating the approximations of
specific numerical quantities represented by
those symbols.
• Such a system might be used for symbolic
integration or differentiation, substitution of one
expression into another, simplification of an
expression, etc.

• Also called Symbolic computation or algebraic
computation,symbolic manipulation, symbolic
processing, symbolic mathematics, or
symbolic algebra, but these terms also refer to
non-computational manipulation
Computer Algebra

What are Computer
Algebra Systems?

Some mathematics tools can input
symbolic expressions, but output
only numbers or graphs. CAS
technology, however, can also
output symbolic mathematical
expressions.
Researchers recommend that
teachers use CAS features to
focus on concepts, personalize the
curricular sequence to fit student
needs, and emphasize meaningful
mathematical tasks.
Computer Algebra System
(CAS)

(CAS)
is a software program that
allows computation over
mathematical expressions in a
way which is similar to the
traditional manual
computations of
mathematicians and scientists.
https://www.youtube.com/watch?v=i2Jkosioz5U

• Is a software package
having capabilities for
–Numerical computations
–Symbolic computations
–Graphical computations
Computer Algebra System (CAS)

• is a a software program that
facilitates symbolic
mathematics.
• The core functionality of a
CAS is manipulation of
mathematical expressions in
symbolic form.

• Computer algebra systems
may be divided into two
classes: specialized and
general-purpose.
• The specialized ones are
devoted to a specific part of
mathematics, such as
number theory, group
theory, or teaching of
elementary mathematics.

• General-purpose computer
algebra systems aim to be useful
to a user working in any scientific
field that requires manipulation of
mathematical expressions. To be
useful, a general-purpose
computer algebra system must
include various features such as:
• a user interface allowing to enter
and display mathematical
formulas,
• a programming language and an
interpreter (the result of a
computation has commonly an
unpredictable form and an
unpredictable size; therefore user
intervention is frequently needed),

• a simplifier, which is a rewrite
system for simplifying
mathematics formulas,
• a memory manager, including a
garbage collector, needed by
the huge size of the
intermediate data, which may
appear during a computation,
• an arbitrary-precision
arithmetic, needed by the huge
size of the integers that may
occur,
• a large library of mathematical
algorithms.

Additional capabilities
Many also include:
• a programming language,
allowing users to implement
their own algorithms
• arbitrary-precision numeric
operations
• exact integer arithmetic and
number theory functionality

• display of mathematical
expressions in two-
dimensional mathematical
form, often using typesetting
systems similar to TeX (see
also Prettyprint)
• plotting graphs and
parametric plots of functions
in two and three dimensions,
and animating them
• drawing charts and
diagrams

• APIs for linking it on an
external program such as a
database, or using in a
programming language to use
the computer algebra system
• string manipulation such as
matching and searching
• add-ons for use in applied
mathematics such as physics,
bioinformatics, computational
chemistry and packages for
physical computation

Some include:
• graphic production and editing
such as computer generated
imagery and signal processing
as image processing
• sound synthesis
• Some computer algebra systems
focus on a specific area of
application; these are typically
developed in academia and are
free. They can be inefficient for
numeric operations compared to
numeric systems.

(CAS)
Mathematics used in computer
algebra systems
• Symbolic integration
• Gröbner basis
• Greatest common divisor
• Polynomial factorization
• Risch algorithm
• Cylindrical algebraic
decomposition

• Cantor–Zassenhaus algorithm
• Padé approximant
• Schwartz–Zippel lemma and testing polynomial identities
• Chinese remainder theorem
• Gaussian elimination
• Diophantine equations

• Axiom
• Cadabra
• CoCoA-4
• CoCoA-5
• Derive
• DataMelt (DMelt)
• Erable (aka ALGB)
• Fermat
• FORM
• FriCAS
• GAP
• GiNaC
• KANT/KASH
• Macaulay2
• Macsyma
• Magma
• Magnus
• Maple
• Mathcad
• Mathematica
• Mathics
• Mathomatic
• Maxima
• MuMATH
• MuPAD
• OpenAxiom
• PARI/GP
• Reduce
• Scilab
• SageMath
• SINGULAR
• SMath Studio
• Symbolic Math
Toolbox (MATLAB)
• SymPy
• TI-Nspire CAS
(Computer
Software)
• Wolfram Alpha
• Xcas/Giac
• Yacas

PRESENTATION SOFTWARES IN
MATHEMATICS TEACHING

What are presentation softwares?
These are software packages
used to display or show
information in the form of a
slideshow.

What are its major functions?
An editor that allows text to be inserted and
formatted.
A method for inserting and manipulating graphic
images.
A slideshow system to display the content.

Why use presentation software in
teaching Mathematics?
To address the abstract nature of mathematics.
To make math class discussions more engaging to the
learners.
To make math class discussions more interactive.
To increase the learners’ participation and confidence.
To link mathematics in real-life context.
To present math lessons clearer.

Guide Questions:
Is there a way to start my lesson in a visual way? Could I use an
animation or video that links my math lesson to a real-life
context?
Would an interactive element consolidate my students’ learning or
just confuse matters?
Would the interactivity improve the engagement of pupils who
sometimes lack focus in the lessons?
What software could I use to create the activity or element I
want?

Examples of
Presentation Softwares

Microsoft PowerPoint
Features:
•Embed and edit video within a slide
•Embed audio or voice over your PowerPoint presentation
•Add bookmarks to media files to pause or enhance media at
designated points
•Microsoft-designed themes and animations to bring your slides to life

Pros:
• User-friendly - relatively intuitive design and layout
• Comes with Microsoft Suite, so likely to already be at your fingertips (versus
other programs that you might have to create accounts for, etc.)
• The new PowerPoint 2013 will allow you to create a Microsoft Live account so
that you can store your presentations in the cloud and work on them
anywhere
Cons:
• Operating system-specific; viewers must have Microsoft Office or a program
that can read Microsoft files to view show
• Linear design for presentations limits the conceptual capabilities for
presentations on non-linear subjects

KeyNote
Features
•Built in narration tool
•Powerful tools for adding and editing graphics and other media files
•Apple-designed themes and animations to bring your slides to life
•Keynote app for iPad and iPhone has surprisingly similar functionality
and ease-of-use as the software itself

Pros:
• Intuitively similar to PowerPoint - linear, slide format
• Integration with mobile Apple devices - Keynote Remote on iPad or iPhone
allows you to control your presentation from the palm of your hand
• Easy format conversions - import PowerPoint slides into Keynote, and vice
versa; save your Keynote in other formats such as a Quicktime movie or PDF
Cons:
• When importing PowerPoint slides and vice versa, some features, such as
particular fonts, may not translate exactly due to the differences in the
programs
• As an Apple product, Keynote is not available for PCs

Prezi
Features
•Better depicts the complexity and interrelatedness of material;
contrasted with the linearity of PowerPoint or Keynote
Pros:
•Better displays complex, non-linear ideas
•Done properly, Prezis tend to be very visually appealing
•Web-based - not specific to an operating system and able to be edited
from any computer with internet access

Cons:
•Not nearly as intuitive to use
•Less easy to import audio/video/graphics
•Transitions, especially the zooming features, can cause
queasiness for viewers

SPREADSHEET
PREPARED BY: Mayend A. Dagunan BSEd-Math III

EXCELComputerized worksheet
standard feature of an electronic
spreadsheet which uses cells that are
represented in rows and columns
designed to perform arithmetic
operations.
calculate numeric information such
as budgets, income, expenses,
scientific, and statistical data.

 Active Cell - The thick-bordered cell where you
can enter numbers or formulas.
 Cell - The space at the intersection of a row and a
column.
 Cell Address - the location of a cell on the
spreadsheet identified by the column letter and
row number.
 Cell Range - a group or block of cells in a
spreadsheet. Operations can be performed on a
range of cells.

 Column - The vertical divisions in a spreadsheet
that are named with an alphabetical letter.
 Column Heading - The lettered box at the top of a
column in a spreadsheet, used to highlight an
entire column.
 Data Entry Bar - The bar at the top of the
spreadsheet used to enter data into the selected
cell; data will appear in cell after pressing
Enter/Return or tab.
 Fill Series - To copy the contents of one cell to
another range of cells

 Formula Bar or Data Entry Bar--the bar at the top
of the spreadsheet used to enter data into the cell
that you have selected
 Graph/Chart - a visual representation of numerical
data; can present complex relationships clearly,
and make trends and patterns identifiable.
 Grid - Blank setup of rows and columns in a
spreadsheet on which data is to be entered
 Gridlines - The vertical and horizontal lines
displayed in the body of a spreadsheet; the
intersections of the gridlines forms cells

 Legend - Text that explains the meaning of colors and
patterns used in a graph/char sometimes called the
key.
 Range/Series - One or more cells in a sequence. For
example, a range could be an entire row or column,
or multiple rows or columns. Identify a range by using
the beginning and ending cell address (A1:A8).
 Row - The horizontal divisions in a spreadsheet
named with a number.
 Descending Order - Organizing or sorting information
in order from largest to smallest, Z-A, or 9-1.

 Row Heading - The numbered box at the
beginning of a row in a spreadsheet, used
to highlight an entire row.
 Sort - Arranging information in a specific
order (usually ascending and descending)
 Ascending Order - Organizing or sorting
information in order from smallest to largest, or A-
Z or 1-9.

 Label - The term given to the data entered as text in a spreadsheet.
 Values - The term give to the data entered as numbers in a spreadsheet.
 Formulas - Mathematical equation consisting of numbers, other cell
designators, and symbols for mathematical operations. The result of the
formula is displayed in the cell, the formula is displayed in the data entry bar.
Formulas are calculated using the normal algebraic rules concerning order of
operations.

 Resizing columns and rows - To quickly re-size a
column/row so that you can see all the contents within the
cells, place your mouse on the border between cell
headers until your cursor has arrows, then click and drag
the divider to make it wider. Data in a cell that is too small
may display ##### in the cell. To remove the error, resize
the cell.
 Deleting row/columns - Click on row or column heading
which will highlight the entire row/column and then delete.
 Add data using fill series

 How to calculate using formulas
 How to create a graph from spreadsheet
 How to insert a new worksheet
 How to rename a worksheet
 How to Merge cells
 How to format cells - Identifying the type of data
found in the cells - Examples: show how time or
dates will be displayed or how many decimal
places will be displayed.

 All formulas begin with an equal (=) sign and must have operands and
operators.
 Spreadsheets can also use shortcut formulas, known as FUNCTIONS such as
SUM, AVG and DIV

 =B4+B5 To find the sum of the contents of cells B4 and B5
 =B4-B5 To find the difference between the contents of
cells B4 and B5
 =B4*B5 To multiply cells B4 and B5
 =B4/B5 To divide cell B4 by B5
 =SUM(B4:B10) To add cells B4, B5, B6, B7, B8, B9, and
B10 (Note: The colon between the cell address represents
a range)
 =AVG(B4:B10) To calculate the average of B4, B5, B6, B7,
B8, B9, and B10

Spreadsheets are an important, powerful and
versatile business tool, and can provide and store
valuable information. Spreadsheets can hold as little
or as much information as necessary. Some
spreadsheet programs can also work together with
other programs, such as word processing and
presentation software.

“ an interactive, Web-based visual
representation of a dynamic object that
presents opportunities for constructing
mathematical knowledge”

Virtual manipulatives allow teachers to
allow for efficient use of multiple
representations and to provide concrete models
of abstract mathematical concepts for learners
of mathematics

Effective Use of Virtual Manipulatives
• teachers must have an understanding of how to
use representations and how to structure a
mathematics lesson where students use
technology
• teachers must also be comfortable with technology
and be prepared

EFFECTS OF STUDENTS' ATTITUDE
ON THEIR ACADEMIC
PERFORMANCE IN MATHEMATICS

FINDINGS
Students' Confidence
• Students strongly agreed that they are sure that they can
learn Mathematics
Usefulness of the Subject
• The participants agreed taht Mathematics can help them.

Students' Perception towards the Attitude of Teacher
• Mathematics teacher had encouraged them to study more
in Mathematics
Students' Attitude towards Mathematics
• the use of CAI(computer aided instruction) has the
greatest impact on students' attitude

SUMMARY OF STUDENTS' PERFORMANCE
• Out of 104 participants, 73 of them got an average
performance in Mathematics

CONCLUSION
• The use of CAI room is a helpful tool in developing
positive attitude in Mathematics among students
• The result of the performance of grade 7 students in the
researchers made questionnaire in Mathematics is
average
• There is a moderate positive correlation between the
students' attitude and academic performance in
Mathematics. The students' attitude has a direct
relationship on their academic performance in
Mathematics.

RECOMMENDATION
• That the students enhance their confidence by allowing
one's self to learn through engaging onself in answering
practice exercises and worksheets in Mathematics
• That the students experience more practical applications
involving themselves in computer games which are
related to Mathematics
• That the students understand instructions and keep in
mind the information and rules for the computer activity
• That the students take Mathematics subject seriously for
them to improve their academic performance

to the teachers....
• It is necessary for teachers to gain awareness of the
attitudes of their students for them to better understand
their learners
• They actively observe and listen to the students as they
engage in mathematical explorations
• Teachers ensure that students have the needed
knowledge and skill for any computer activity
• Evaluate students' achievement by testing in the specific
expected outcomes

ARTIE OUNCES SODA JERK
http://mrnussbaum.com/soda-play/
Objectives:
• to convert the units in volume
• to produce the the correct order of the customer

INTERNET
 “information superhighway”
 the original name was ARPANET
• ARPA – Advanced Research Projects Agency
• to create network that would allow scientists to share
information on military and scientific research

 the thousands of interconnected networks were called
an Inter-Net- Network (internet / network of networks)

APPLET
 is a special kind of program that is transmitted over the
internet and automatically executed by the java-
compatible web browser
 helped in moving some user interactive programs from
server to client
 improves the usability of the web application

JAVA
 a programming language for the Web which can
be downloaded by any computer
 influenced internet by simplifying the web
programming and inventing applets which
expanded the scope of internet

 addressed two other important issues of
internet:
•security
-there are restrictions on what can be done in applets
-they can use only a subset of all the functions supported by
java

• portability
- since internet is comprised of many different types
of computers and operating systems, it is
important for the programs to run all these systems
- is achieved by using Bytecode in Java

BASIC JAVAAPPLET EXAMPLES
 Interactive/ complex Java applet animations
• watchful eyes
• sliding puzzle
•randomly blinking text
 Providing services over the WWW
• temperature conversion
• calculator

 Controllable information display
•Weather statistics
 Toy demo applets
• the classic “Hello World!”
• Editable text- jumps with mouse clicks
• Simple graphics: Display a diagonal line
• a simple game , etc.

 Some “real” applets:
• The ripple effect applet
• A proper game
 More advance examples:
•A bouncing ball animation, using threads
•An event monitor

THE USE OF JAVA
APPLET IN
MATHEMATICS

•Applets to generate examples:
Rather than one or two pictures
of acute and obtuse angles the
terminal side sweeps through
angles from 0 to 180 degrees. You
will find that even people who are
quite familiar with acute and
obtuse angles will patiently watch
the applet as the angles change.

•Here is another applet that
provides examples, this time of
triangles. When the applet runs
the sides and angles of the
triangle change size as the
student watches. We can use
this applet to practice
estimation.

•Applets for simple
multiple choice
questions: We can use
applets to generate
simple yes/no or multiple
choice questions so that
students can get
immediate reassurance
that they have
understood a concept.

• Applets to generate data: Applets can
be used as quick ways to generate data
for students to analyze. There are
times when it is desirable for students
to analyze data using graphs and
calculators and to make conjectures as
to what the explanation for the data is.
One could, for instance, have students
construct right angled triangles and
measure the sides.

• Applets to show a sequence of
steps: Applets can guide a
student through a sequence of
steps with the student
performing the activities at each
step as the applet runs. The
applet below tries to convince
the student that the sum of the
measures of the angles in a
triangle is 180 degrees by having
him or her cut up a triangle and
rearrange the angles to form a
180 degree angle.

•Applets that show animated
picture proofs: A popular use
of applets is in animated
picture proofs. Here is an
applet that gives a picture
proof of the Pythagorean
Theorem. At this point the blue
triangles are being moved to
new positions.

•Applets with mathematical puzzles: An applet can be in the form of a
mathematical puzzle. Students are challenged to explain how the applet
works. If the level of difficulty of the puzzle is appropriate the students
can extract the underlying mathematical concepts. The applet below was
written for an undergraduate discrete structures class. It tiles a
deficient 2 n by 2 n grid using right trominos. It is a good introduction to
the principle of mathematical induction. It also helps students to develop
problem solving skills.

•Applets that are the center of a course:
The final applet I will present is an example of what I call a "theme applet". This is the
most ambitious type of project where a whole course revolves around an applet that
appears and reappears in different contexts. In this example the applet presents an
"overhead photograph" of a forest. Green dots are healthy trees and red ones are
diseased trees. The light green patches are grass. The applet is meant for an
elementary statistics class. The version of the applet I show takes a sample, gives you
the statistics and you (the student) are expected to estimate confidence intervals for
the proportion of the trees that are diseased.

 for Internet users about appropriate use of technology
and online engagement to provide content, links, and
conventions that increase popularity (hits, clicks, and views)
based on social value.

HOW MOVIES/VIDEOS HELP IN THE
TEACHING - LEARNING PROCESS?
•FACILITATING THINKING AND PROBLEM SOLVING
• SHEPARD AND COOPER (1982) AND MAYER AND GALLINI (1990) MADE THE
CONNECTION BETWEEN VISUAL CLUES, THE MEMORY PROCESS, AND THE RECALL
OF NEW KNOWLEDGE. ALLAM (2006) OBSERVES THAT THE CREATIVE CHALLENGE
OF USING MOVING IMAGES AND SOUND TO COMMUNICATE A TOPIC INDEED
ENGAGING AND INSIGHTFUL, BUT ADDS THAT IT ALSO ENABLES STUDENTS TO
ACQUIRE A RANGE OF TRANSFERABLE SKILLS IN ADDITION TO FILMMAKING
ITSELF. THESE INCLUDE RESEARCH SKILLS, COLLABORATIVE WORKING, PROBLEM
SOLVING, TECHNOLOGY, AND ORGANISATIONAL SKILLS. (BIJNENS, N.D.)

•ASSISTING WITH MASTERY LEARNING
• IN SOME CASES, VIDEO CAN BE AS GOOD AS AN INSTRUCTOR IN
COMMUNICATING FACTS OR DEMONSTRATING PROCEDURES TO ASSIST IN
MASTERY LEARNING WHERE A STUDENT CAN VIEW COMPLEX CLINICAL OR
MECHANICAL PROCEDURES AS MANY TIMES AS THEY NEED TO.
FURTHERMORE, THE INTERACTIVE FEATURES OF MODERN WEB-BASED
MEDIA PLAYERS CAN BE USED TO PROMOTE ‘ACTIVE VIEWING’ APPROACHES
WITH STUDENTS (GALBRAITH, 2004)

• INSPIRING AND ENGAGING STUDENTS
MORE RECENTLY, WILLMOT ET AL (2012) SHOW THAT THERE IS STRONG
EVIDENCE THAT DIGITAL VIDEO REPORTING CAN INSPIRE AND ENGAGE STUDENTS
WHEN INCORPORATED INTO STUDENT-CENTRED LEARNING ACTIVITIES THROUGH:
• INCREASED STUDENT MOTIVATION
• ENHANCED LEARNING EXPERIENCE
• DEVELOPMENT POTENTIAL FOR DEEPER LEARNING OF THE SUBJECT DEVELOPMENT
POTENTIAL FOR DEEPER LEARNING OF THE SUBJECT DEVELOPMENT POTENTIAL
FOR DEEPER LEARNING OF THE SUBJECT
• ENHANCED TEAM WORKING AND COMMUNICATION SKILLSLEARNING RESOURCES
FOR FUTURE COHORTS TO USE

• AUTHENTIC LEARNING OPPORTUNITIES
• THE WORK OF KEARNEY AND COLLEAGUES SHOW THE BENEFITS OF USING
VIDEO TO PRODUCE AUTHENTIC LEARNING OPPORTUNITIES FOR
STUDENTS (KEARNEY AND CAMPBELL 2010; KEARNEY AND SCHUCK, 2006),
AND HOW ‘IVIDEOS’ ENCOURAGE ACADEMIC RIGOUR FROM AN ADVOCACY,
RESEARCH BASED PERSPECTIVE.

•MEETS ADDITIONAL LEARNING STYLES
• PRESENTING INFORMATION IN NUMEROUS WAYS CAN BE THE KEY
TO HELPING STUDENTS UNDERSTAND TOPICS. FOR EXAMPLE,
HAVING STUDENTS WATCH THE MOVIE SEPARATE BUT EQUAL CAN
HELP THEM UNDERSTAND THE REASON BEHIND THE COURT
CASE BROWN V. BOARD OF EDUCATIONBEYOND JUST WHAT THEY
CAN READ IN A TEXTBOOK OR LISTEN TO IN A LECTURE.

•PROVIDE TEACHABLE MOMENTS:
•SOMETIMES A MOVIE CAN INCLUDE MOMENTS THAT GO BEYOND
WHAT YOU ARE TEACHING IN A LESSON AND ALLOW YOU TO
HIGHLIGHT OTHER IMPORTANT TOPICS. FOR EXAMPLE, THE
MOVIE GANDHI CAN PROVIDE YOU WITH THE ABILITY TO
DISCUSS WORLD RELIGIONS, IMPERIALISM, NON-VIOLENT PROTEST,
PERSONAL FREEDOMS, RIGHTS AND RESPONSIBILITIES, GENDER
RELATIONS, INDIA AS A COUNTRY, AND SO MUCH MORE.

•CAN BE A GOOD THING TO DO ON DAYS WHERE STUDENTS
WOULD BE UNFOCUSED:
• IT IS A FACT OF DAY-TO-DAY TEACHING THAT THERE WILL BE DAYS WHEN
STUDENTS WILL BE FOCUSED MORE ON THEIR HOMECOMING DANCE AND GAME
THAT NIGHT OR THE HOLIDAY THAT STARTS THE NEXT DAY THAN ON THE
TOPIC OF THE DAY. WHILE THIS IS NOT AN EXCUSE TO SHOW A NON-
EDUCATIONAL MOVIE, THIS COULD BE A GOOD TIME TO WATCH SOMETHING
ON THE TOPIC YOU ARE TEACHING.

DRAWBACKS OF USING FILMS & VIDEOS
• MOVIES CAN BE DISTRACTING.
• IT IS IMPORTANT THAT YOU COMPLETELY WATCH AND KNOW EVERYTHING ABOUT THE
MOVIE YOU ARE SHOWING SO THAT IT DOES NOT LEAD TO UNWANTED
CONVERSATIONS AND SITUATIONS. FOR EXAMPLE, YOU MIGHT HAVE WATCHED A MOVIE
NUMEROUS TIMES AT HOME BUT ONLY WHEN WATCHING IT WITH A CLASSROOM FULL
OF STUDENTS WILL THOSE CURSE WORDS THAT YOU DIDN'T NOTICE TRULY STAND OUT
OR AN OFF-COLOR JOKE REAR ITS HEAD CAUSING STUDENTS TO LAUGH AND TALK
AMONGST THEMSELVES.

• MAY TAKE TOO MUCH TIME
• MOVIES CAN SOMETIMES BE VERY LONG. I TAUGHT AT A SCHOOL DISTRICT WHERE IT
WAS THE POLICY TO WATCH SCHINDLER'S LIST WITH EVERY 10TH GRADE CLASS (WITH
THEIR PARENT'S PERMISSION OF COURSE). THIS TOOK AN ENTIRE WEEK OF CLASSROOM
TIME. EVEN A SHORT MOVIE CAN TAKE UP 2-3 DAYS OF CLASSROOM TIME. FURTHER, IT
CAN BE DIFFICULT IF DIFFERENT CLASSES HAVE TO START AND STOP AT DIFFERENT
SPOTS OF A MOVIE. A FIRE DRILL CAN REALLY MESS UP YOUR LESSON PLANS FOR THAT
WEEK SINCE ONE CLASS WILL NOT HAVE GOTTEN AS FAR IN THE MOVIE AS THE OTHERS.

•MAY NOT BE COMPLETELY HISTORICALLY ACCURATE
•MOVIES OFTEN PLAY WITH HISTORICAL FACTS TO MAKE A BETTER
STORY. THEREFORE, IT IS IMPORTANT THAT YOU NOTICE AND
POINT OUT THE HISTORICAL INACCURACIES OR STUDENTS WILL
BELIEVE THAT THEY ARE TRUE. IF DONE PROPERLY, POINTING OUT
THE ISSUES WITH A MOVIE CAN PROVIDE YOU WITH TEACHABLE
MOMENTS FOR YOUR STUDENTS.

FERMAT’S ROOM
• THIS BRILLIANT MOVIE IS ESSENTIALLY ABOUT
MATH ONLY. A FEW FAMOUS MATH PUZZLES
APPEAR IN THIS MOVIE, WHERE 4
MATHEMATICIANS ARE TRAPPED IN A ROOM
WHERE THE WALLS SLOWLY CRUSH THEM.

THE NUMBER 23
• NUMEROLOGY WITH THE NUMBER 23

INFINITY
• ABAKUS SCENE AND EXPLANATION HOW TO
COMPUTE THE THIRD ROOTS

• TUTORIAL VIDEOS ABOUT MATH FROM BASIC ALGEBRA TO THE MOST COMPLEX TOPIC ABOUT MATH
• TH

KHAN ACADEMY
• ..DOWNLOADSGRAPHING TRIG
FUNCTIONS.MP4

OTHER VIDEOS
• MATHANTICS
• MATHBFF
• MATHHELP.COM
• MATH MEETING
• WORLD CENTER OF MATHEMATICS
• TEACHERTUBEMATH
• HARVARD MATH
• VIHART
• LEARN MATH TUTORIALS
• TECHMATH
• MASHUP MATH

Book References:
 de Jesus, Joy T.(2008).Java Programming by Example. Quezon City: TechFactors Inc.
 Wells, Dr. Dolores.(2009). Basic Computer Concepts. Singapore: Cengage Learning Asia
Pte Ltd.
 JemaDevelopmentGroup(2014)OfficeProductivity.Philippines:Jemma,Inc.

Electronic References
 https://www.teachervision.com/using-manipulatives
 https://www.thoughtco.com/using-a-geo-board-in-math-2312391
 https://www.mathsisfun.com/definitions/tangram.html
 http://math.tutorcircle.com/geometry/geometric-solids.html
 https://en.wikipedia.org/wiki/Graphing_calculator
 https://education.ti.com/en/resources/funding-and-
research/research/research_teachingandlearning
 https://en.wikipedia.org/wiki/Inclinometer
 https://teachingmahollitz.wordpress.com/2011/04/26/teaching-with-fraction-strips/
 http://mypages.valdosta.edu/plmoch/MATH2160/Spring%202007/05-
2_Fraction_Bar_Model.pdf

o http://www.uq.edu.au/teach/video-teach-learn/ped-benefits.html
o https://www.thoughtco.com/pros-and-cons-movies-in-class-7762
o https://www.youtube.com/results?search_query=mathematics+channel
o www.cs.stir.ac.uk/~sbj/examples/Java-examples
o www.java.meritcampus.com
o http://any2any.org/EP/1998/ATCMP016/paper.pdf
o https://officialnetiquette.blogspot.com/2013/03/internet-etiquette-definition.htmls
o https://computerskills4teachers.wikispaces.com/SPREADSHEET+TERMS+AND+SKILLS
o https://www.avidian.com/resources/how-to-use-excel
o https://www.youtube.com/watch?v=xOU_hL2_zBo

 http://www.educatorstechnology.com/2012/05/list-of-20-free-tools-for-teachers-to.html
 https://math.stackexchange.com/questions/2025688/presentation-software
 https://www.chachatelier.fr/latexit/
 http://www.jonathanleroux.org/software/iguanatex/
 https://www.teachthought.com/technology/15-presentation-tools-for-teachers-from-
edshelf/
 https://prezi.com/signup/plus/
 http://law.indiana.libguides.com/c.php?g=19807&p=112380
 http://www.edugains.ca/resourcesMath/CE/LessonsSupports/Manipulatives/Manipulatives
_FractionStrips&FractionTowers.pdf
 https://www.ed.gov/oii-news/use-technology-teaching-and-learning
Electronic References:

 http://www.thefullwiki.org/Symbolic_mathematics
 https://en.wikipedia.org/wiki/Computer_algebra_system
 https://www.slideshare.net/ssuser89667c/computer-algebrasystemmaple
 https://education.ti.com/sites/UK/downloads/pdf/Research%20Notes%20-
%20Why%20CAS.pdf
 http://www.math.tamu.edu/~dallen/papers/strategies.pdf
 http://ucsmp.uchicago.edu/resources/conferences/2012-03-01/
 https://www.atm.org.uk/write/MediaUploads/Shop%20Images/Look%20Inside/DNL037.pdf

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