D.EL.ED.
CPS -03
MATHEMATICS
What is Vygotsky's ZPD?
ZPD (Zone of Proximal
Development) is the difference between a learner's ability to learn
independently and with guidance. It represents the area of collaborative
learning.
- Inadequate Support Materials
- Solution:
- Unprepared Teachers Get Stuck
Descending Thinking: Applying a general principle to a specific
example, such as "all birds have wings → penguins have wings."
Continuous Classes: Classes that have interrelated limits, such as
10-20, 20-30.
- Learning through Experience
Method for Teaching the
Concept of "The Sum of the Angles in a Triangle is 180°":
- Materials: A piece of paper, ruler,
protractor, scissors.
- Steps:
- Assessment: Monitor continuous
development.
- Behavioral Objectives Actions:
- Identify,
- Analyze,
- Solve,
- Explain.
- Building Self-Confidence
Difficulties:
- Difference Between Rectangular Prism
and Cube:
- All edges of a rectangular prism may
be unequal, all edges of a cube are equal.
- Faces of a rectangular prism are
rectangular, faces of a cube are square.
Ascending Thinking: Reaching general conclusions from specific
examples.
Example of Ascending
Thinking: Measuring the
angles of several triangles to show that the sum of all triangles' angles is
180°.
- Presentation:
- Write the differences between a unit
plan and lesson plan.
- A unit plan outlines a large topic,
while a lesson plan provides detailed instructions for a specific class.
- Cutting a Paper Circle into 4 Parts
to Explain ¼ How to
teach the concept of subtraction using teaching materials? Can be
demonstrated using counters or blocks. For example, if you take away 2
from 5 blocks, 3 remain, visibly illustrating subtraction.
Measures of Central
Tendency:
- Mean,
- Median,
- Mode.
- Using Math Games and Puzzles Write two psychological signs
of math anxiety.
- Avoiding math, 2. Excessive worry
during tests.
- Math Lab and Support Materials
Regulatory Objectives in
Math Education:
- Following mathematical steps
accurately,
- Maintaining patience and discipline
in problem-solving.
Examples of Intellectual
Objectives in Math Education: For
example, students will solve problems with reasoning, enhancing their
analytical skills.
Problems and Solutions in
Math Education: Problem: Write
two uses of computers in math education.
- Learning through interactive
software, 2. Data analysis and graph creation.
Functional Objectives of
Math Education: Students
will be able to calculate percentages in daily life, such as discounts or
interest.
Write two ways to
increase interest in learning math.
- Using games or everyday examples, 2.
Organizing competitions or rewards.
Example of Ascending
Thinking in Math: Reaching
general formulas from specific examples, such as inferring multiplication rules
by looking at products of several numbers.
Reasons for Falling
Behind in Math:
- Basic weaknesses,
- Math anxiety.
Two limitations of Direct
Instruction Method in Math:
- Time-consuming,
- Difficult comprehension for complex
concepts.
Two causes of
misunderstanding in Math:
- Inadequate explanation or rapid
teaching, 2. Lack of real-life examples.
Principles for Selecting
Teaching Aids in Math Learning: Unique
Nature of Mathematics:
- Abstraction,
- Logic,
- Universality,
- Sequencing.
Give a definition of
Mathematics: Mathematics is the
science of numbers, shapes, quantities, and reasoning, which expresses ideas
through symbols. It provides a framework for problem-solving and analysis.
Write four utilities of
Mathematics:
- Development of reasoning, 2.
Problem-solving, 3. Financial calculations, 4. Application in science and
technology.
Two characteristics of
Mathematics:
- Logical dependence – Each concept in
math is based on reasoning and proof.
- Universality – The formulas and rules
of math apply everywhere, e.g., 2+2=4 is the same in all cultures.
Five ways to increase
interest in Mathematics: How to
eliminate misconceptions in Mathematics?
- Through hands-on activities, 2.
Providing discussion and corrective feedback.
How to identify
misconceptions in Mathematics? Misconceptions
can be identified by allowing students to explain or solve problems.
Misunderstandings can also be determined through discussions and Q&A
sessions.
What is the basis of
mathematical representation?
- The basis of mathematical
representation is logical organization, clarity, and the use of symbols.
Ideas can be easily conveyed through graphs, formulas, or illustrations.
What is the nature of
mathematical language?
- It is concise, logical, and
symbol-dependent. Ideas are expressed through formulas, definitions, and
rules.
Example of mathematical
reasoning:
- "All integers are either even or
odd. 4 is even, so it is not odd." This is a simple mathematical
reasoning.
How can mathematical
communication be done?
- Mathematical communication can be
conducted through images, formulas, symbols, or language. Students can
express their thoughts by solving problems, discussing, or creating
models.
- Group Work Assignment:
Interquartile Range:
The interquartile range (Q3−Q1), which shows the spread of the data. - Four purposes and differences of four
types of assessment:
Objectives: - How will you show the relationship
between pressure and volume through a diagram?
A graph can be drawn with pressure on the x-axis and volume on the y-axis. According to Boyle's Law, pressure increases as volume decreases. - Increase thinking and analytical
skills.
- Provide students the opportunity to
discover on their own.
- Students are asked to draw multiple
shapes of triangles on paper.
- Life-related and relevant.
- What do you mean by 'data handling'?
- It is the process of collecting,
organizing, analyzing, and presenting data. For instance, making
conclusions from survey data.
Data Collection Methods:
2.
Surveys,
3.
Observation,
4.
Experiments,
5.
Interviews.
- Skill Assessment:
- It can be observed that three angles
together form a straight line—meaning 180°.
Conclusion:
- In this method, students grasp the
concept through hands-on exploration.
- Practical applicability (real-life).
Norm-referenced
assessment:
- This assesses the student's skill
based on pre-defined criteria, such as a passing score of 50%.
What is a
criterion-referenced test?
- It is an assessment that measures
skill according to pre-defined criteria, such as a final examination for
grade determination.
- Specific Objective:
Discrete Variable:
A variable that takes only integer values, such as the number of family members.
What is constructivism?
Application in mathematics:
- Definition: Constructivism is a learning
theory that states learners construct new ideas based on their prior
knowledge and experiences.
- Application in Mathematics: Constructivist methods enable
students to learn through discovery. For example, geometry formulas are
understood through hands-on activities.
Characteristics of
constructivism:
- Student-centered learning,
- Knowledge is self-constructed.
What is the difference
between measurement and assessment?
- Measurement is the expression in
numbers (e.g., scores), while assessment is the interpretation (e.g.,
skill levels).
- Measurement: Verification of data through
numbers.
- Examination: Verification of knowledge
within a specific timeframe.
Rules and methods for
experimental work:
- Rules:
- Adequate materials must be
available,
- Selection based on lesson
objectives.
- Reward and Recognition:
- Students are asked to cut and pair
sticks together.
- Lack of relevant examples.
Characteristics and
examples of project-based learning:
- Characteristics: How will you explain the
concepts of time and distance through football?
Ask to measure the speed of a player. For instance, if a player runs 50 meters in 10 seconds, speed = 50/10 = 5 m/s. - Age and grade appropriateness.
Examples of external
activities:
- Learning about measurement and space
allocation through planting saplings in a garden.
Difference between bar
graphs and histograms:
- Categories are discrete in bar
graphs, whereas they are continuous in histograms.
- Bar graphs have gaps, while
histograms do not.
- Incorporate real-life experience:
- Coordinate real-life experiences.
- Connect with real-world examples:
List two strategies for teaching mathematics using real-life examples: - Teaching percentages through
budgeting,
- Teaching fractions through
measurements in cooking.
- Utilize real-life examples
School external activities and real-life situations:
Example: Learning addition and subtraction through shopping in the market.
Policies:
- Abstract concepts.
What is the importance of
skewness measurement?
- It indicates the asymmetry of data
distribution. Positive skewness shows a longer tail on the right, while
negative skewness shows a longer tail on the left.
Measures of skewness:
- Skewness,
- Karl Pearson's measure,
- The quantile method.
Difference between a
circle and a sphere:
- A circle is two-dimensional, having
only area and circumference. A sphere is three-dimensional, having volume
and surface area. A circle lies in a plane, while a sphere is a solid
object.
Teaching using circular
charts:
Example: Drawing a circular chart with students' favorite fruits in
a class—e.g., 10 students like mangoes, 5 like jackfruit, etc.
- This is helpful for understanding
statistics.
What mathematical
concepts can be taught from rainfall?
- Average rainfall can be understood
through statistics, represented in graphs, or understood through
probability. For instance, analyzing monthly rainfall data.
Goals of mathematics
education based on practical and structural value:
- Use of teaching aids for teaching
fractions: Example: What is the cognitive goal of
teaching fractions?
The main goal of learning fractions is to understand the relationship between parts and the whole. It helps in division, comparison, and measurement in real life.
Teaching the concept of
division using teaching aids:
- Show using fruits or counters. For
example, if 10 fruits are divided among 2 people, each receives 5 fruits.
- Linguistic complexity.
- Fear and disinterest.
When to use Venn
diagrams?
- Venn diagrams are used to show the
relationship or common elements between two or more sets.
List two differences
between models and charts:
- A model is three-dimensional and can
be held (e.g., geometric shapes), while a chart is two-dimensional and
shows data (e.g., pie chart).
When to use the median as
a central tendency?
- When there are extreme values in the
data that affect the mean, the median serves as a better indicator.
- Assessment of mentality:
- Teaching fractions through money and
coins using a manipulative.
Outcome: Abstract concepts are materialized.
Difference between
evaluation and assessment:
- Evaluation is outcome-based, while
assessment is process-based.
- Grades are given in evaluation, while
feedback is provided in assessment.
- Evaluation: Value determination.
Define evaluation:
Evaluation is the measurement of students' knowledge, skills, and understanding.
It identifies improvements and weaknesses in learning.
- Logical thinking and problem-solving
(structural).
Barriers to
communication:
- Complexity of technical language,
- Students' fear or hesitation.
- Report preparation.
How to explain the relationship
of triangle sides using sticks?
- Demonstrate with three sticks of
different lengths that a triangle can only be formed when the sum of any
two sides is greater than the third side.
Difficulties with
illustrations:
- Complex data is hard to understand
easily,
- Incorrect scaling or labeling can
lead to misinterpretation.
Identify misconceptions
in the volume of a sphere:
- Giving students the chance to explain
will show whether they are misunderstanding the application of the formula
(length × width × height).
- Observation by the teacher.
Methods for effective
implementation: 5. Use
of teaching aids:
- Student-centered.
How to introduce real
numbers to students?
Real numbers can be written in the form p/q, where q≠0. This can be explained
on a number line or through fractions.
- Student participation.
- Students’ real-life relevance.
What are teaching
materials? Explain with examples:
- Definition: Objects that facilitate the
teaching process and make it engaging.
Examples: Cubes, manipulatives, protractors, flashcards, etc.
Application in
mathematics: Clarifying the
concept of solids using 3D models.
How to show the product
of numbers and their factors graphically?
A rectangular area can be divided using x and 1 to show that the product is
always 1.
- Active participation.
Importance of
active-based methods:
- Increases student participation,
- Leads to lasting learning.
Cooperative learning:
Students work in groups to acquire knowledge, such as through group projects.
- Skills for time management,
measurement, and financial calculations, etc.
- Time-bound.
Difficulties in
problem-solving:
- Time-consuming,
- Complex for some students.
- Finding problem-solving methods on
their own.
Advantages of
problem-solving:
- Enhances creativity,
- Teaches real applications,
- Improves reasoning skills,
- Boosts confidence.
- Problem-based approach.
Definition of the
problem-solving method and its advantages and disadvantages:
- Definition: Students acquire knowledge
through a process of solving a problem.
Advantages:
- Influence of social culture:
- Local measurement systems (e.g.,
hand, feet),
- Traditional mathematical games (e.g.,
dice).
Resources: Educational materials, such as geometric boxes,
software, or manipulatives.
- Easily accessible and low cost.
- Opportunity for discussion among
peers to form concepts.
- Collaborative work.
Example:
Maintaining expenses and revenues of a store (application of arithmetic).
- Observation of social behavior.
Main difference:
If the score increases by 5, what will happen to the mean and median?
Both the mean and median will increase by 5 since every value changes.
Drawing polygons on the
histogram:
A simple line can be drawn holding the midpoints of each category.
