Making High School Mathematics Real and Relevant : Current Trends and Pedagogies

Making High School Mathematics Real and Relevant : Current Trends and

PedagogiesPrepared by

Harold John D. Culala

Far Eastern University

Institute of Education

[email protected]

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Day OneSession One

Topics: Constructivist Based Teaching in Mathematics Why do you teach the way you do? Constructivism Theory Constructivist Teaching Strategies The Two Math Traditions in The Philippine

Setting

August 28-30, 2008 Making High School Mathematics Real and Relevant

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Learning Targets

At the end of this session, the participants should be able to: understand the concept of Constructivist Based

Teaching unveil the myth and the fact in teaching

mathematics reflect on the strategies they used in teaching

mathematics Appreciate constructivist based teaching


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Workshop #1

The participants will be grouped with 3 participants each group.

Make a 15-minute demo teaching on this topic : “Division of Fractions”

Make a lesson plan for the demo teaching. Lesson Plan will be presented after the demo

teaching


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Session Notes

Reflect:

Why do you teach the way you do?


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BELIEF STATEMENT A D1. Mastering facts or information principles and algorithms is critical in learning.2. Problems/Questions given to the students should be quickly solvable in a few steps.3. The teacher should share his own understandings of the concepts before inquiring about students’ own understanding.


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BELIEF STATEMENT A D4. The teacher should manage, control and direct learning by arranging the environment, selecting stimuli, reinforcing desired responses, and discouraging undesirable responses.5. Students are rational decision-makers capable of determining for themselves what is right and wrong6. The learner, not the teacher, should take control of the learning situation


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BELIEF STATEMENT A D7. Young students are capable of much higher levels of thinking than has been suggested traditionally.8. Period of uncertainty, conflict, confusion, surprise are a significant part of the learning process9. Learning is enhanced by activities which build upon and respect students’ experiences


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BELIEF STATEMENT A D10. Learning is enhanced by challenge within a supportive environment11. Teachers should provide instructional activities, which result in problematic situations for learners12. The role of the teacher is to transmit knowledge and verify that learners received this knowledge13. Teachers should recognize that what seem like the errors and confusions from an adult point of view are students’ expression of their understanding


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BELIEF STATEMENT A D14. Teachers should negotiate social norms with the students in order to develop a cooperative learning environment in which students can construct their knowledge.15. There is no mathematics if there is no computation16. Right answers are more important in mathematics than the ways in which you get them.17. Being able to memorize facts is critical in mathematical learning


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The Previous Situation…

“Myth about your class”

Think about the subject you liked most and the subject you like least in elementary or high

school…

And the subject is…


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Kapag maingay kayo,Magma-MATH tayo!!!

MATH is aform of

punishment!


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Trust me! Believe me!

The value of πis 3.1416.

Math seems to be aReligion…

You simply acceptthings by faith!


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Should we view Math as a set of rules or facts?

Or, should we view it as a set of inter related concepts and processes?

Consider the terms POINT and LINE of the Plane Euclidean Geometry

Recall that:A point has no dimension

A line has a dimensionAugust 28-30, 2008 Making High School Mathematics

Real and Relevant14




QUESTION:

How did you convince your students that a point has no dimension?

How come that a line consisting of dimensionless points has dimension?


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QUESTION:

When dividing fractions, why do you get the reciprocal of the second fraction and proceed to multiplying the two fractions?

How do you convert an improper fraction to mixed fraction or vice versa?


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Six Questions for Math Teachers Do students enjoy your math classes? Do they realize the importance of math in

their daily lives? Do they feel that they can solve problems if

they think? Has math class made them more confident

about their abilities? Do they like you?


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According to Prof Lee Pee Yee of NIE, Singapore

Teacher play a central role in the process of education. Given a good syllabus and a good textbook, but without a good teacher, then the subject may not be taught well.

If there is no good syllabus and no good textbooks but there is a good teacher, then the subject matter could be taught well.


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According to Prof Lee Pee Yee of NIE, Singapore

Having good teachers may not be a sufficient condition for good teaching but is definitely a necessary condition.

However, if we want good teaching, then we must have good teachers.


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Prof Lee’ recommendations:

Train teachers in content areas.

Train teachers in general pedagogy.

Make available to teachers useful resources.


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The Emerging Situation

What is mathematics?

What are the goals of mathematics teaching?


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Mathematics is

a study of patterns and relationships. a way of thinking. an art, characterized by order and internal

consistency. a language, using carefully defined terms and

symbols. a tool.

(Suydam, Reys, Lindquist 2001 and NCTM)


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Goals of Mathematics*

To value mathematics To reason mathematically To communicate mathematics To solve problems To develop confidence

*NCTM


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According to Prof Jin Akiyama, RIED, Tokyo, Japan

To make children good in math is to let them enjoy math.

Light the lamp of curiosity in children’s minds.


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QUESTION:

When dividing fractions, why do you get the reciprocal of the second fraction and proceed to multiplying the to fractions?

How do you convert an improper fraction to mixed fraction or vice versa?


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Mother has ten pieces of bananas. She distributed two bananas each for her children. How many children does she have?


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10 bananas 2 bananas = __________

10 thirds 2 thirds = __________





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There are 16 balls in the basket. You want to distribute them equally to 6 boxes. How many balls in each box?

18 balls 6 = __________

18 fifths 6 = __________







Allow them to experience the process of discovery with all 5 senses.


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Problems:

Is the sum of any three consecutive whole number divisible by 3?

Is the sum of any four consecutive whole number is divisible by 4?

Is the sum of any five consecutive whole number is divisible by 5?

Generalize these to any “n” consecutive whole numbers.


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Allow them to experience the process of discovery with all 5 senses.

Imagination is more important than knowledge


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Problem:

Without using Pythagorean Theorem, how can you show that the length of the diagonal of the square below is ?


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21

1






Mathematics, as the air, is invisible, and we can not live without it


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Which is a better buy? Why?

A liter of cooking oil costing P120 or a bottle of a 600 ml cooking oil costing P75?


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Mathematics, as the air, is invisible, and we can not live without it

Listen to the secrets of nature under the blue sky


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Numerical Strategy

39 + 28 = 40 + 27

37 – 18 = 39 – 20

Algebraic Generalization


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Teaching must give the learners the opportunity

for concrete, contextually meaningful experience through which they can

search for patterns, raise their own questions, and

construct their own models, concepts and

strategies.


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Problem:

Mr C has P4.10 consisting of 10c and 25c. If she has 6 more 10c than 25c, how many 25c coins does she have?

a. 10 b. 16 c. 20 d. 26


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Problem:

If an eight-inch square cake serves four people, how many twelve-inch square cake are needed to provide equivalent servings to eighteen people?


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Problem:

Gina bought a ring for P600 and sold it for P700, bought it back for P800 and sold it for P900. How much did Gina gain or lose in the ring-trading business?


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CONSTRUCTIVISM

A theory about knowledge and learning anchored on the following principles:

1. Learners construct understanding2. To understand something is to know

relationships3. All learning depends on prior knowledge


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Contrast between the SMT and IMT (Bernardo, 2000)

Dimension School Math Inquiry MathNature of Math Collection of facts and

proceduresModes of inquiry and problem solving

Learning Activities

Replicating and mastering specified mathematicalprocedures and operations

Exploring,conjecturing, arguing, proving, problem posing and collaborating

Classroom Interaction

Teacher-controlled, initiation – reply, evaluation pattern

Discussion and negotiations among students, and between students and teacherAugust 28-30, 2008 Making High School Mathematics

Real and Relevant41




Contrast between the SMT and IMT (Bernardo, 2000)


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Dimension School Math Inquiry Math

Roles of Teachers

Authority or transmitter of mathematical knowledge

Facilitator of students’ inquiry and learning

Indicators ofStudent Learning

Ability to follow procedural instructions toobtain the correct answer

Ability to explain and justifyactions on mathematical objects




What kind of learning opportunities do we offer our students?

TRUE or FALSE?1. Any number divided by itself is equal to 1.

2. Any number raised to a zero exponent is 1.

3. Zero (0) divided by any number is 0.


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1nn

0 1n

0 0n




What kind of learning opportunities do we offer our students?


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TRUE or FALSE?1. Parallel lines do not meet however far they are produced.2. is factorable.3. The cube root of 8 is 2.4. 9 + 6 = 3 can not hold true5. Multiplication is always commutative.

2 25x





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TRUE or FALSE?1. Any number divided by itself is equal to 1.

FALSE. If n=0, 0/0 = 1

2. Any number raised to a zero exponent is 1.FALSE. If n=0,

3. Zero (0) divided by any number is 0.FALSE. If n=0, 0/0 = 1

1nn

0 1n

0 0n

00 1





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TRUE or FALSE?

1. Parallel lines do not meet however far they are produced.FALSE. In Projective Geometry, parallel lines meet at a

point called the ideal point.

2. is factorable.TRUE.The factors are (x+5i) (x-5i)

3. The cube root of 8 is 2.FALSE. The other cube root is

2 25x

1 3




4. 9 + 6 = 3 can not hold trueFALSE.9 + 6 = 3 (mod 12)

5. Multiplication is always commutative.FALSE.Multiplication of matrices is always

commutative.


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Remember!

To avoid confusion,ALWAYS define the DOMAIN.


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(Observation in a 3rd year HS class)

T: Ok, class, who can give a question for everyone to solve? (Students raised hands. She called the a student)

S1: Singapore has a population of 300,000. If ½% are students, then how many are students.

S2: It is ½% of 300,000 which is 150,000

S1: That’s correctAugust 28-30, 2008 Making High School Mathematics

Real and Relevant49




S3: Classmates. I have a question. Isn’t 1% is greater than ½%?

(complete silence)

S3: Because 1% of 300,000 is 3000. So the answer to the problem must be less than 3000.

S2: (went to the board) Let me show you my working. (Wrote on the board: 300,000 x 0.50 = 150,000)

S3: Which is greater 1% or ½%?


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(Discussions were made by students)

T: Ok class, since you can not arrive at a decision, can we just discuss this on Monday? We now continue our lesson.

(And the teacher continued the activities reflected in her lesson plan)


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I taught my dog how to whistle

WHAT WE VALUE IN STUDENTS’ PERFORMACE





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I said I taught him,I didn’t say he learnt

it!

I’m sorry but I didn’tHear him whistle!

Could there be teachingwhen there is NO

Learning?




Issues regarding Assessment

In a math test consisting of 10 problems:Daisy was able to finish 10, but got only 5

correct answers.Ruth was able to finish only 4 problems, but

got them all correct.Q: Who performed better?

How will you help Daisy?How will you help Ruth?


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Ability to explain and justify actions on mathematical objects (Inquiry Math Tradition)Sample observation: During the problem posing act:

S1: My garden is in a form of square with sides 5 cm. What is the perimeter?

S2: How big is your garden?(S1 looked at his teacher wondering why S2 asked

that question)T: Yes, how big is your garden? Could you imagine

how big is your garden?August 28-30, 2008 Making High School Mathematics

Real and Relevant55




Ability to follow procedural instructions to obtain the correct answers (School Math Inquiry)(As an assignment, the teacher asked the students to

measure the dimensions of their during table at home and compute for the areas)

S1: Our table has length 7cm and width 5cm. Its area is 35 cm.

T: Very good!S2: Ma’am ang liit naman ng table nila!T: Wala tayo magagawa kung maliit ang table nila.


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Rule of Thumb!

PRINCIPLE 1 : Pose problems that are or will be relevant to the students

PRINCIPLE 2 : Structure learning around essential conceptsPRINCIPLE 3 : Be aware that students’ points of view are

windows to their reasoning.PRINCIPLE 4 : Adopt curriculum to address students’

suppositions and development.PRINCIPLE 5 : Assess student learning in the context of

learning.


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For your own reflection as a teacher: Why do we have to spend our time and energy

attending workshops such as this? Are we not yet satisfied with our usual classroom

teaching practice in mathematic? Are we not yet confident with our content

knowledge? Look at our students’ performance in both locally

and internationally administered exams. How did the perform in math sub-component of

NEAT, NSAT? TIMSS?August 28-30, 2008 Making High School Mathematics

Real and Relevant58




Synthesis of Workshop #1

Do you see any difference between your usual classroom practices and the constructivist based teaching?

How do you think the Constructivist Based Teaching will be beneficial for you? For your students?

In the nature of your students, what are the problems you think you might face in doing the constructivist based teaching?


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Day OneSession Two

Topics: (STATISTICS) Using Graphs to Organize Data and

Gathering Data Measures of Central Tendency Pinoy Idol Investigators


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Learning Targets:

At the end of the session, the participants should be able to: use different types of graphs to organize and to

present data create survey forms for gathering data to be

examined using measures central tendency appreciate different methods in presenting

lessons in statistics


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Session Notes

What is statistics?

To best understand what statistics is all about, consider the following illustration


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Illustration

A researcher wishes to investigate the hypoglycemic (blood sugar lowering) effect of a new herbal supplement on diabetic patients. Instead, he selects 100 persons with diabetes and give them the supplement. After observing with proper scientific controls, the researcher infers that the herbal supplement will have a similar effect on all diabetic persons.


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Illustration

A pollster wishes to determine the political strength of a certain candidate running in a national election. It is no possible to ask each of the approximately 15 million registered voters about his preference for candidate. Instead, the pollster uses 1000 registered voters and infer from this smaller subset of registered voters the political strength of the candidate.


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Illustration

To determine the life span of light bulbs from the production line, the person in charge of quality control selects a sample of 50 light bulbs, test each one of them and infers the average life span of a light bulb from the sample.


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For the above illustration, we observe the following common characteristics:

Each involves the measurement of something which cannot predict with absolute certainty. Thus, one cannot tell for sure how a randomly chosen diabetic patient will react to the supplement or whether a randomly selected registered voter will choose the candidate in question or not, or how long a randomly chosen light bulb will last.


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Each involves a much smaller portion of the entire collection of persons/items to be studied. The entire collection constitutes the population; the portion or subset of the population selected is called the sample.

Each involve a collection of measurement (data), one measurement observation for each element of the sample.


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Each has a similar goal to make a conclusion (inference) about the population based on data obtained from a sample.

Statistics is the branch of mathematics that involves collecting, organizing,

summarizing, and presenting and drawing general conclusions from data.


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How to obtain unbiased samples? Random Sampling – each item in a population has an

equal chance of being drawn Systematic Sampling – is one in which every nth item on a

list or production line is drawn, the 1st item is determined using random sampling.

Cluster Sampling – population divided into groups of units (clusters) according to some criteria

Stratified Sampling – dividing the population depending on characteristics called stratifying factors

Convenience Sampling – a.k.a. grad=b or opportunity sampling


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Using Graphs to Organize Data

Commonly used graphs to present and organize data

Histogram Frequency Polygon Line graph Bar graph Circle graph (a.k.a. Pie charts) Pictograph


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Histogram

Used to illustrate frequency distribution

HG is a graph observed values on its horizontal axis and frequencies on its vertical axis. A bar is constructed above each observed value, indicating the frequency of that value


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Frequency Polygon a.k.a broken – line graph Line graphs with scales as

those of the histogram, that is the horizontal scale indicates observed values and the vertical scales indicates frequency.

A dot is placed at the corresponding frequency above each of the observed values and then the dots are connected with straight line object


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Line Graph

Line graphs compare two variables. Each variable is plotted along an axis . A line graph has a vertical axis and a horizontal axis.

They show trends in data clearly, meaning that they visibly show how one variable is affected by the other as it increases or decreases.


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Bar Graph

Bar graphs are used to display data in a similar way to line graphs. However, rather than using a point on a plane to define a value, a bar graph uses a horizontal or vertical rectangular bar that levels off at the appropriate level.

They clearly show trends in data, meaning that they show how one variable is affected as the other rises or falls.


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Circle Graph

Often used to compare part of one or more components of the whole to the whole

Very different from the other three types of graphs that we've looked at. They don't use a set of axes to plot points. In addition, they don't work with the same type of data that the other three work with. Pie charts are display percentages. Therefore, they are used to compare different parts of the same whole.


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Pictograph

A special bar graph Symbols replace the

bars from the bar graph


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Measures of Central Tendency of an Ungrouped Data Mean (a.k.a. arithmetic mean, average) The mean of a set of n numbers is the sum of the n

numbers divided by n.Mathematically speaking,

or; xi is the ith item value and n is the number or

itemsAugust 28-30, 2008 Making High School Mathematics

Real and Relevant77

1 2 3 ... nx x x xxn

ixx

n




Measures of Central Tendency of an Ungrouped Data Median – is the midpoint of the distribution. It

is a number where half of the values or observations are smaller and the other half are bigger.

Mathematically speaking


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12 2

2

n nx xMe

1

2nxMe

If n is evenIf n is odd




Measures of Central Tendency of an Ungrouped Data Mode – the values that occur most

frequently. It can be easily determined if the data is

organized in a FDT.


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Workshop #2Pinoy Idol Investigators


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Toni Piolo Kyla Josh Arnee




PROCEDURE

Groups will be composed of 3 members Each group will develop a survey that allows them

to scientifically analyze the popularity of five "Idol" contestants.

Examine survey data by analyzing the mean, median and modes.

Develop three charts to represent their data. (MEAN, MEDIAN, MODE Graphs)

Use their survey data to predict who will win the competition.


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Procedure

Group members will give the survey to the other groups. Surveys must be conducted in Five minutes or less

Groups should pay careful attention to survey results. These results form the foundation of their analysis, so they need to methodically track the incoming information.

The results will be presented in the class.


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Guide questions in constructing a survey sheet How do you quantify popularity on a survey? Should you ask people who they vote for? Should you ask people to list their favorite

contestants? Should you ask people to rank their favorite

contestants? Which current contestants have been in the bottom

three? What predictions are "Idol" experts making?


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Guide questions in constructing graphs Based on your survey data, who are the

most popular "Idol" contestants? Who do you predict will win the competition?


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How do you find the activity? Have you encountered any problems while

doing the activity? In your classroom, do you think this kind of

activity will be useful for your students?


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Day TwoSession One

Topics: Distance Formula – Workshop #3 Financial Investments– Workshop #4 Problem Solving Strategies


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Learning Outcomes

At the end of the sessions, the participants are expected to: experience different strategies in presenting

specific lessons in Advanced Algebra. discuss the different ways to solve Problem

Solving in Advanced Algebra be familiar with different interactive games/drills

in specific lessons in Advanced Algebra appreciate new techniques in Advanced Algebra


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Workshop #3 – Distance Formula “A Story of My Neighborhood” Materials

Styrophore Push pins Yarn 2 pairs of dice (1 pair white color, 1 pair colored) Calculator (optional)


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Remember: Workshop #3

Before giving this activity, students should have the idea of how to get the distance of two points on a number line and in the Cartesian Plane.

The teacher should have derived the distance formula in the class


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2 22 1 2 1( ) ( )d x x y y




Procedure

Directions: to allow for negative coordinates, have the white dice represent positive values and have the colored dice represent negative values. Place the two pair of dice in a bag.

To start the game, shake the bag and draw one die. Roll the die. This value represents the x-coordinate. Draw a second die. Roll the die. This value represents the y-coordinate. These values will be the coordinates of the starting point called HOME. Mark the starting point on the plane using a push pin.


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Procedure

Have each student shake the bag, draw a die, and roll a number to represent his/her x-coordinate. Do the same to obtain the student's y-coordinate. Remember, if a colored die is drawn, the coordinate is negative. This ordered pair will represent the student's position on the grid.

Each student is to determine the distance of his/her coordinate from the HOME coordinate.


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Procedure

Draw another die until you plot the following establishments. Hibuk Hibuk Elementary School Hibuk Hibuk Wet and Dry Public Market Aling Puring’s Sari-Sari Store Municipal Hall HH General Hospital HH People’s Park

Make a story out of your neighborhood. Present it in class. Use the concept of Distance Between Two Points.


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What are the advantages and disadvantages of using this kind of activity?

Do you think that this activity will help to motivate your students? In what way?

In your classroom, do you think that this kind of strategy/activity will be effective? Why?


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Remember: Workshop #4

The teacher could either give this activity before presenting the lesson or during the practice exercises. Financial Transaction Simple and Compound Interest

The teacher should explain clearly the importance of Financial Transaction in their daily lives.

However, in this workshop, this is given after the presentation of the lesson in the class.


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Workshop #4

“My Business is My Business!” Participants will be grouped in two. The task of each group is to create their own

business of any nature (i.e. food chain, clothes, etc.)


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Procedure

All the group are given P1.5 Million to start their own business. All the starting money should used.

Their business should have the following: Name and logo of the business Nature of the business List of all the products and corresponding amount List of all the expenses that you used in starting a

business. Salary of the employees Rental and others


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Procedure

You will also include the projected gains of the company every year. 50% of the gains per year will be placed in a bank with a

compound interest of 7.2% annually. How much will be your money in the back after 15 years without withdrawing anything from it.

The other 20% on the first year will be placed on a stock market with a simple interest of 10.6%. How much will be your money after 25 years?


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Procedure

Each group will present their business in class and will justify their computation


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If you are in the placed of your students, do you think that this kind of activity will be effective to them? Why?

How do you think this activity can help the learning process of your student?

Can these kind of activity (Role Playing Activity) be used to determine the mastery of the students in a particular lesson?


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Day TwoSession Two

Session Notes Problem Solving is one of the basic problems

of students in High School around the world? It is also the hardest part for the teacher to teach.

WHY????


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Myth and Fact in Problem Solving Students thought that problem solving is difficulty One – Step Problem Prevails in the end of lessons

practicing a specific operation Many school books don’t have enough GOOD word

problems Teachers are afraid of word problems so they skip

them


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Students thought that problem solving is difficulty

Students always thought that math is difficult. Students thought that way because they

don’t understand the reason why they need to answer such questions.

Teachers are not fully equipped in teaching problem solving

Understanding the language makes the problem solving difficult


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One – Step Problem Prevails in the end of lessons practicing a specific operation

You see this in all elementary grades. Kids are practicing perhaps multi-digit multiplication, perhaps borrowing in subtraction, perhaps dividing decimals. After the calculation problems come some word problems, which oddly enough are solved by using the exact operation just practiced!

It extends beyond the lessons on the four operations, too. Haven't you ever noticed it: if the lesson is about topic X, then the word problems are about the topic X too!


103




One – Step Problem Prevails in the end of lessons practicing a specific operation

When kids are exposed to such lessons over and over again, they figure it out that it's mentally less demanding to not even read the problem too carefully. Why bother? Just take the two numbers and divide (or multiply, or add, or subtract) them and that's it.

How to avoid it? In the end of such calculation lessons, if you want word problems, mix them up so that not all are solved by the operation you just studied. Or, give students a bunch of short word problems for the purpose of NOT finding the answers but to find what operation(s) are needed to get the answer.


104




Many school books don’t have enough GOOD word problems

Some school books doesn’t really give much emphasis on worded problems. Most of the worded problems have not thought much of the authors. Consider this question:

A boy and a girl collected 24 nuts. The boy collected two times as many nuts as the girl. How many did each collect?


105




Teachers are afraid of word problems so they skip them

It is not because teachers doesn’t know how to do it, but some of them they don’t know how to teach them.

No matter how good the teacher is, if he can’t transmit that goodness, then it is just like having 1 billion pesos but you don’t know how to use it.


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Purpose of Word Problems

prepare children for real life develop children's logical and abstract thinking and

mental discipline to motivate students


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Tips in Teaching Problem Solving in Mathematics Model a useful problem-solving method.

Problem solving can be difficult and sometimes tedious. Show students by your example how to be patient and persistent and how to follow a structured method, such as Woods’ model described here. Articulate your method as you use it so students see the connections.


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Tips in Teaching Problem Solving in Mathematics Teach within a specific context.

Teach problem-solving skills in the context in which they will be used. Use real-life problems in explanations, examples, and exams. Do not teach problem solving as an independent, abstract skill


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Tips in Teaching Problem Solving in Mathematics Help students understand the problem.

In order to solve problems, students need to define the end goal. This step is crucial to successful learning of problem-solving skills. If you succeed at helping students answer the questions “what?” and “why?”, finding the answer to “how?” will be easier.


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Tips in Teaching Problem Solving in Mathematics Take enough time.

When planning a lecture/tutorial, budget enough time for: understanding the problem and defining the goal, both individually and as a class; dealing with questions from you and your students; making, finding, and fixing mistakes; and solving entire problems in a single session.


111




Tips in Teaching Problem Solving in Mathematics Ask questions and make suggestions.

Ask students to predict “what would happen if…” or explain why something happened. This will help them to develop analytical and deductive thinking skills. Also, ask questions and make suggestions about strategies to encourage students to reflect on the problem-solving strategies that they use.


112




Tips in Teaching Problem Solving in Mathematics Recommend them to do visuals or drawings.

In this way, it will be easier for the students to comprehend what does the problem is really trying to say.


113




Day TwoSession Two

Topics: Constructing a good and valid word problems

/ problem solving questions


114




Session Notes

How to construct a good and valid problem solving???


115




Workshop #5

Constructing Problem Solving Questions. Construct ten (10) problem solving questions on

different topics in only 3 disciplines: Algebra, Trigonometry, and Geometry.

Problem solving questions with figures are also accepted.

This will be presented during the session individually to be critiqued by the other members of the group.


116




Things to remember in constructing a problem solving question

An effective problem must first engage students' interest, and motivate them to probe for deeper understanding of the concepts being introduced. It should relate the subject to the real world, so that students have a stake in solving the problem. If at all possible, the problem should be placed in a context with which students are familiar.


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Good problems require students to make decisions or judgments based on facts, information, logic and/or rationalization. Students should be required to justify all decisions and reasoning based on the principles being learned. Problems should require students to define what assumptions are needed (and why), what information is relevant, and/or what steps or procedures are required in order to solve the problem.


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The content objectives of the topic should be incorporated into the problems, connecting previous knowledge to new concepts, and connecting new knowledge to concepts in other courses and/or disciplines.


119





The problem's questions should challenge students to develop higher order thinking skills, moving them beyond Bloom's (1956) lower cognitive levels of knowledge and comprehension to the higher Bloom levels, where they analyze, synthesize and evaluate.


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Did you encounter any problems while doing this activity? Why do you think it became a problem to you? How did you overcome it?

In your classes, how often do you give problem solving questions?

How important to know how to construct good problem solving question?


121




Day ThreeSession One

Topics: Trigonometry Angle of Elevation and Depression

Circle Function Permutation


122




Learning Outcomes

At the end of the sessions, the participants are expected to: experience different strategies in presenting

specific lessons in Trigonometry. be familiar with different interactive games/drills

in specific lessons in Trigonometry appreciate new techniques in teaching

Trigonometry


123




Session Notes

Discovery Learning as a part of Constructivist Based Teaching

What is Discovery Learning? How is this applicable in your math classes? What benefits will your student achieve in

these kinds of methods?


124




Discovery Learning

Discovery Learning provides students with opportunties to develop hypotheses to answer questions and can contribute to the development of a lifelong love of learning. Students propose issues or problems, gather data and observations to develop hypotheses, confirm or refine their hypotheses, and explain or prove their problems.


125




Discovery Learning

Discovery Learning is an inquiry-based learning method.

There is an intimate and necessary relation between the processes of actual experience and education. (Dewey)

Discovery learning takes place most notably in problem solving situations where the learner draws on his own experience and prior knowledge to discover the truths that are to be learned (D. Clark, 1999)


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Workshop #7

Angle of Elevation and Depression The Clinometer A clinometer is an instrument used in measuring

slope. Most of the times, this is being used by Foresters to measure the height of a tree or how far a tree form each other.


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Remember

Before giving this activity the teacher should give its set induction (motivation) to the students by engaging them in learning through recalling prior knowledge of trigonometric ratios. At this point the student should have the idea on Trigonometric Ratios.

By using the DISCOVERY METHOD, give the students this activity. Students discover that a clinometer can be used to measure the angle of elevation and depression of objects


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Materials

Soda straw Protractor String weight (rock or metal washer) masking tape, tacks, graph paper, pencil


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Procedure

Students form groups. Teacher provides materials to build a clinometer The video clip to use the clinometer is shown (if not

applicable, then the teacher should give the procedures clearly)

Students use the clinometer to read the angles of elevation and angle of depression of various objects.


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A Clinometer


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Procedure

Construct a clinometer by taping a straw to the straight edge of a protractor. Attach string to the straw adjacent to the 0-degree measure on the protractor. Attach a weight to the other end of the string.

Working with a partner, practice sighting on an object of known height, such as the top of a door, clock, or other high object. Sight through the straw to the object. Your partner should read the number where the string lies against the protractor while you hold the straw steady. Subtract this number from 90 degrees to obtain the angle of elevation. Then have your partner measure the horizontal distance between the object sighted and the place you are standing. Finally, your partner should measure your eye level at a vertical distance from the floor.


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Procedure

Write your data on your worksheets. Get at least 3 objects that you will get the angle of

elevation and 3 objects for angle of depression. Solve using the concept of Angle of Elevation and

Depression. Share your observations in the class.


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How did you find this activity? What do you think is the importance of

Discovery Learning in the learning process of your students?

With the characteristics of your students, do you think they will appreciate this kind of activity? Why?


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Workshop #8

“Permutation People” This activity should be given to introduce the

topic Permutation.


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Materials

Pictures of people provided by the teacher 2 colored papers Stapler Scissors


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Procedure

Gather all of your "people, some colored paper to use as front and back covers, a stapler, and a pair of scissors.

Arrange the pictures in the order you desire and place one cover on top and one on the back.

Staple the left edge of your book firmly. Make sure that all of the papers are perfectly aligned.


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Procedure

Now simply cut across the two dotted lines. This will cut your people into three turnable

sections, the neck area, the body area, and the leg area.

Now for the fun. Explore all the different "Permutation People" that it is possible to make. Just turn the strips to mix and match legs, bodies, and heads. You'll be surprised how many are possible!


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Procedure

It may seem amazing, but there are 64 possible people that can be made from our four originals.

You see there are 4 different heads, 4 different bodies, and 4 different legs. So each one of those heads can be matched with each one of those bodies, which also can be matched with each one of those legs.

It turns out to be a multiplication problem,4 x 4 x 4 = 64. Math people say there are 64 "permutations" in this situation.

If we had drawn 5 original people, the math would be 5 x 5 x 5 = 125. That's 125 different people in our book! With just 6 original people the number of permutations goes up to 6 x 6 x 6 = 216.


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Procedure

After the activity and the introduction, proceed in presenting your lessons. Still give emphasis on the activity made. In this students should be able to relate the activity in solving permutations.


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Session Notes - Humanistic Mathematics The strategy used in motivating the students

is called Humanistic Mathematics.

What is Humanistic Mathematics?


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Humanistic Mathematics

This strategy involves methods for using visual imagery and art to introduce students to mathematical ideas. Polyhedra, modern art, computer graphics, and computer animations are illustrated in examples for classroom use.


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It attempts to explore the human side of mathematical thought and to guide students through mathematical ideas by the use of imagery, history, internet and computer tools to discover the beauty of mathematics


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Mathematics is a very creative subject but it gets embalmed in textbooks. The major premise of humanistic mathematics is that lack of motivation is the root cause of the literacy and attitude problems in the learning of mathematics.


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Synthesis - Workshop #8

What do you think is the edge of this strategy in other strategies in teaching mathematics?

How do you think this activity can benefit your students?

In your classroom, do you entertain the artistic sides of your students?

What do you think is the strength of this activity?


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Thank you very much!

See you soon!


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Date post:	13-Nov-2014
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Making High School Mathematics Real and Relevant : Current Trends and Pedagogies

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