kajian tentang quadratic expression and equation in math kbsm
DESCRIPTION
Task yang dibuat untuk kursus SXEX1101 Perancangan Kurikulum dalam Matematikuntuk maklimat lanjut boleh hubingi [email protected]TRANSCRIPT
http://prosacksblog.blogspot.com [ ]
ReSeaRCH oF Quadratic Expression and Equation in Mathematics KBSM Syllabus
| MUHAMAD ILI NURHAYAT BIN HASSAN 1
http://prosacksblog.blogspot.com [ ]
ASSIGNMENTS1. Choose one topic from the secondary school mathematics textbook.
2. Carry out the following activities:
a) Discuss the basic structure of the topic.
b) Suggest activities that used to be added to the topic (chapter) so that it
satisfy (10 – 20 pages) the constructivist approach to teaching and
learning.
c) Suggest activities that need to be added to the topic (chapter) so that is
satisfy the integrated approach to teaching and learning the topic.
d) Suggest ways of improving the usage of teaching in the topic.
3. Write your reflection concerning the activities you have done.
| MUHAMAD ILI NURHAYAT BIN HASSAN 2
http://prosacksblog.blogspot.com [ ]
INTRODUCTION
First and foremost I would like to express my deepest appreciation to my
SXEP1101 Lecturer, Prof. Nik Aziz Nik Pa for giving us such a wonderful assignment.
Besides, I also would like to thank my course mates, my friends and my family for
helping me and supporting me in doing and completing the assignment.
This assignment actually to me is quite heavy at first sight since we were told to
submit it in 3kg. To me it is quite big task and taking lot of my precious time and since
I’m also not quite understanding about what the needs of this task. However after
referring to the lectures, lecture notes, friends and internet; I finally understand what is
the proposed of this assignment.
After doing some researched, I make a conclusion to the second topic in form of
of our mathematic syllabus which is: Quadratic Expression and Equation since this topic
is one of the important topics that had been learn and taught from the secondary school
and still continued until now. Finding that this topic is one of the important of Algebra
and the Algebra had been included into our mathematic education syllabus for
secondary school since form one until form five states that this Algebra topic is
something big that has to be known for us as a student.
Apart for that, I hope my research here can provide some information that could
help us an educator to apply it in our teaching and learning process (P&P) to make our
education process happened more effective thus uttering our future generation to
become more reliable to our nation and country as proposed in Falsafah Pendidikan
Kebangsaan(FPK).
| MUHAMAD ILI NURHAYAT BIN HASSAN 3
http://prosacksblog.blogspot.com [ ]
THE HISTORY BEHIND QUADRATIC EXPRESSION AND EQUATION
This is the quadratic expression, as it has been taught to most of us since in school:
ax2 + bx + c = 0
which gives the formula solution to a generic quadratic equation in the form:
x1,2=(-b/2a) ± (1/2a)(b2-4ac)1/2
The development, or derivation, of a mathematical idea is usually as logical, deducible
and rectilinear as possible. This brings about the common notion that its historical
development is similarly as continuous, logical and rectilinear: one mathematician
picking up an idea where another mathematician left it.
Using the quadratic expression as an example, it will be shown that the historical
development of mathematics is not at all rectilinear. Instead, parallel developments,
interconnections and confluences can be found, which - to complicate this stuff even
further - are also interrelated with social, cultural, political and religious matters.
The quadratic expression has been derived in the course of a few millennia to its current
form, which had been taught in all school in Malaysia and all around the world. So, let
flash back a little story about this quadratic expression and equation’s history.
| MUHAMAD ILI NURHAYAT BIN HASSAN 4
http://prosacksblog.blogspot.com [ ]
The Original Problem 2000(or so)BC
Egyptian, Chinese and Babylonian engineers were really smart people - they knew how
the area of a square scales with the length of its side. They knew that it's possible to
store nine times more bales of hay if the side of the square loft is tripled. They also
found out how to calculate the area of more complex designs like rectangles and T-
shapes and so on. However, they didn't know how to calculate the sides of the shapes -
the length of the sides, starting from a given area - which was often what their clients
really needed. And so, this is the original problem: a certain shape1 must be scaled with
a total area, and in the end what's needed is lengths of the sides, or walls to make a
working floor plan.
1500BC The Beginnings - Egypt
The first aspect that finally led to the quadratic equation was the recognition that it is
connected to a very pragmatic problem, which in its turn demanded a 'quick and dirty'
solution. We have to note, in this context, that Egyptian mathematics did not know
equations and numbers like we do nowadays; it is instead descriptive, rhetorical and
sometimes very hard to follow. It is known that the Egyptian Wiseman (engineers,
scribes and priests) were aware of this shortcoming - but they came up with a way to
circumvent this problem: instead of learning an operation, or a formula that could
calculate the sides from the area, they calculated the area for all possible sides and
shapes of squares and rectangles and made a look-up table. This method works much
like we learn the multiplication tables by heart in school instead of doing the operation
proper.
1 For example: the floor-plan of a T-shaped temple with a square patio on a L-shaped lot
| MUHAMAD ILI NURHAYAT BIN HASSAN 5
http://prosacksblog.blogspot.com [ ]
So, if someone wanted a loft with a certain shape and a certain capacity to store bales
of papyrus, the engineer would go to his table and find the most fitting design. The
engineers did not have time to calculate all shapes and sides to make their own table.
Instead, the table they used was a reproduction of a master look-up table. The copyists
did not know if the stuff they were copying made sense or not as they didn't know
anything about maths. So, obviously, sometimes errors crept in, and copies of the
copies were known to be less trustworthy2. These tables still exist, and it is possible to
see where errors crept in during the copying of the documents.
400 BCE The Next Step - Babylon and China
The Egyptian method worked fine, but a more general solution - without the need for
tables - seemed desirable. That's where the Babylonian geeks come into play.
Babylonian maths had a big advantage over the one used in Egypt, namely they used a
number-system that is pretty much like the one we use today, albeit on a hexagesimal
basis, or base-60. Addition and multiplication were a lot easier to perform with this
system, so the engineers around 1000 BC could always double-check the values in
their tables. By 400 BC they found a more general method called 'completing the
square' to solve generic problems involving areas. There are no indications that these
people used a specific mathematical procedure to find out the solutions, so probably
some educated guessing was involved. Around the same time, or a bit later, this
method also appears in Chinese documents. The Chinese, like the Egyptians, also did
not use a numeric system, but a double checking of simple mathematical operations
was made astonishingly easy by the widespread use of the abacus
2 Imagine a multiplication table with a typo (8 x 7 = 57), and you learn that by heart!
| MUHAMAD ILI NURHAYAT BIN HASSAN 6
http://prosacksblog.blogspot.com [ ]
300BC Geometry - Hellenistic Mediterranean Area
The first attempts to find a more general formula to solve quadratic equations can be
tracked back to geometry (and trigonometry) top-bananas Pythagoras (500 BC in
Croton, Italy) and Euclid (300 BC in Alexandria, Egypt), who used a strictly geometric
approach, and found a general procedure to solve the quadratic equation. Pythagoras
noted that the ratios between the area of a square and the respective length of the side
- the square root - were not always integer, but he refused to allow for proportions other
than rational. Euclid went even further and found out that this proportion might also not
be rational. He concluded that irrational numbers exist.
Euclid's opus Elements covered more or less all the mathematics needed for technical
applications from a theoretical point of view. However, it didn't use the same notation
with formulas and numbers like we use nowadays. For that reason it was not possible
to calculate the square root of any number by hand, in order to obtain a good
approximation for the exact value of the root, which is what the architects and engineers
were after. Because all (theoretically relevant at least) maths seemed to be
complete3 but otherwise useless, the many wars occurring in Europe, and also the early
Middle Ages turned the mathematical world in Europe silent until the 13th Century. In
this period mathematics also suffered a big shift, going from a pragmatic science to a
more mystical, philosophical discipline.
3 Euclidean geometry, for example, was only expanded recently in the late 19th Century!
| MUHAMAD ILI NURHAYAT BIN HASSAN 7
http://prosacksblog.blogspot.com [ ]
700AD All Numbers – India
Hindu mathematics has used the decimal system (the one we use) at least since
600AD. One of the most important influences on Hindu mathematics was that it was
widely used in commerce. The average Hindu merchant was pretty fast in simple maths.
If someone had a debt the numbers would be negative, if someone had a credit the
numbers would be positive. Also, if someone had neither credit, nor debt, the numbers
would add up to zero. Zero is an important number in the history of mathematics, and its
relatively late appearance is due to the fact that many cultures had difficulty of
conceiving 'nothing'. The concept of 'nothing', like in 'shunya', the void, or the concept of
'equilibrium', was already anchored in Hindu culture.
Around 700AD the general solution for the quadratic equation, this time using numbers,
was devised by a Hindu mathematician called Brahmagupta, who, among other things,
used irrational numbers; he also recognised two roots in the solution. The final,
complete solution as we know it today came around 1100AD, by another Hindu
mathematician called Baskhara4. Baskhara was the first to recognise that any positive
number has two square roots.
4 In fact the quadratic formula is known in some countries, like Brazil, by the name of 'Baskhara's Formula'.
| MUHAMAD ILI NURHAYAT BIN HASSAN 8
http://prosacksblog.blogspot.com [ ]
820AD Powerful Islamic Science - Persia
Around 820AD, near Baghdad, Mohammad bin Musa Al-Khwarismi, a famous Islamic
mathematician5 who knew Hindu mathematics, also derived the quadratic equation. The
algebra used by him was entirely rhetorical, and he rejected negative solutions. This
particular derivation of the quadratic formula was brought to Europe by Jewish
mathematician/astronomer Abraham bar Hiyya (whose Latinised name is Savasorda)
who lived in Barcelona around 1100.
1500AD Renaissance - Europe
With the Renaissance in Europe, academic attention came back to original
mathematical problems. By 1545 Girolamo Cardano, who was a typical Renaissance
scientist (ie, interested in alchemy, occultism and suchlike), and one of the best
algebraists of his time, compiled the works related to the quadratic equations - that is,
he blended Al-Khwarismi's solution with the Euclidean geometry. He was possibly not
the first or only one, but the most famous. In his (mainly rhetorical) works he allows for
the existence of complex, or imaginary numbers - that is, roots of negative numbers. At
the end of the 16th Century the mathematical notation and symbolism was introduced
by amateur-mathematician François Viète, in France. In 1637, when René Descartes
published La Géométrie, modern Mathematics was born, and the quadratic formula has
adopted the form we know today.
5 His name lives on in the English word 'algorithm' ('Khwa' mutated to 'Go' and the 's' mutated to 'th'.).
| MUHAMAD ILI NURHAYAT BIN HASSAN 9
http://prosacksblog.blogspot.com [ ]
RAW MATERIAL OF QUADRATIC EXPRESSION AND EQUATION
Quadratic Expression and Equation is one element in algebra and the algebra topic
has been taught and introduced in the early level of primary school especially in Form 1, 2
and 3 but as a basic information and foundation for secondary students to go further after
this. In Form 1, this topic was introduced in Chapter 7 which is “Algebraic Expression” and
this topic continues in Form 2 which is “Algebraic Expression II”. Other than that, there are
other topics that are expands form of algebra which quite related to Quadratic Expression
and Equation and has been taught in Form 2 which is a continuity about this kind of topic
which is “Linear Equation”. Meanwhile in form 3, this topic is touched more deeply in several
topics which are “Algebraic Expression III”, “Algebraic Formulae”, “Linear Equation II”, and
also “Linear Inequalities”.
Since this topic also part and an expand from Algebra, which also quite related to
each other, let we go through about their previous topic that had been taught in Form 1, 2
and 3 first before we elaborate more about this Quadratic Expression and Equation topic as
an introduction to this topic.
| MUHAMAD ILI NURHAYAT BIN HASSAN 10
Figure 1
http://prosacksblog.blogspot.com [ ]
ALGEBRAIC EXPRESSION
From the Algebraic Expression I in Form 1, we were introduced about unknown, what is
unknown and its concept (see figure 1). After that we were introduced it in term, its principle
and concept. For example, we were informed that an algebraic term is written as 3x not
x3; a number, example 8 is also a term; x2
is
a term and in 7p: The coefficient of p is 7.
Then, the algebraic expression’s concept
was introduced such as “4s + 8s = 12s”; and
“3k + 4 + 6k –3 = 9 k + 1 ” . Then, in
Algebraic Expression ll in Form 2, we were
introduced about the computation involving
addition, substraction, multiplication and division of
more than 2 terms.
Other than that, the students also were teach to perform computations involving algebraic
expressions.
e.g:
| MUHAMAD ILI NURHAYAT BIN HASSAN 11
http://prosacksblog.blogspot.com [ ]
Then, in Form 3, the Algebraic Expression III is stressing about the concept of
brackets; expanding the brackets “e.g: (a± b)(a± b) = ( a± b)²”; factorization of algebraic
expressions to solve problems “e.g: ab– ac = a(b– c)” and also to perform computation
on algebraic fraction.
ALGEBRAIC FORMULAE
Meanwhile the “Algebraic Formula” is the continuing of algebraic expression which had
been teaches in Form 1, 2 and 3. This topic more stress on the basic understanding
about the concept of variables and constant which include determination of variables or
constant from a given situation and vice versa. It also stress on understanding the
concept of formulae to solve the problems based on a given statement and situation.
Here, the specified variables is express as the subject of formula involving basic
operations of +,-,x,÷; powers or roots; and also the combination of basic operations and
power or roots.
A. Normal Algebraic formulae
a2 - b2 = ( a + b )( a - b)
( a + b )2 = a2 + 2ab + b2
(a - b)2 = a2 - 2ab + b2
| MUHAMAD ILI NURHAYAT BIN HASSAN 12
http://prosacksblog.blogspot.com [ ]
B. Change the subject
| MUHAMAD ILI NURHAYAT BIN HASSAN 13
http://prosacksblog.blogspot.com [ ]
C. Finding the variables.
Example:
Given thatW = 3a –12
b, find (a) W when a = 2 and b = 4,
(b) b when a = 3 andW = 5.
Solution:
W= 3 a–12
b
(a)Substitute a = 2 and b = 4 into the formula W = 3(2) –12¿4) = 6 – 2 = 4
(b) 5 = 3(3) –12
b
12b= 9 – 5 = 4 × 2 = 8
D. Solving problems
Example:
The entrance fees to a museum are RM4 for an adult and RM3 for a child. If, on a
certain day, p adults and q children visited the museum, write a formula for the total
collection of that day.
Solution:
Let A = Total collection (in RM)
A = (Number of adults × 4) +(Number of children × 3)
| MUHAMAD ILI NURHAYAT BIN HASSAN 14
http://prosacksblog.blogspot.com [ ]
A = 4p + 3q
LINEAR EQUATION
Meanwhile linear equation is a further study of algebraic which involves unknown and
equation. Here, the concept of equality was introduced which are ‘=’ and ‘≠’. And the
discussion included:
1. Solve linear equation = Finding the value of the unknown which satisfies the
equation.
2. The solution of the equation is also known as the root of the equation.
3. A linear equation in one unknown has only one root.
4. To determine whether a given value is a solution of an equation, substitute the
value into the equation. If the sum of the left hand side (LHS) = sum of right hand
side (RHS), then the given value is a solution.
There are 4 different forms of linear equation as follow:
| MUHAMAD ILI NURHAYAT BIN HASSAN 15
http://prosacksblog.blogspot.com [ ]
Example
a) If a =b then b =a.
e.g: 2+3 = 4+1 then 4+1 = 2+3
b) If a =b and b =c, then a =c.
e.g: 4+5 = 2+7, then 2+7 = 3+6, then 4+5= 3+6
It’s also stress on the use and the concept of linear equations in one unknown, discuss
why given algebraic terms and expressions are linear, identifying the linear terms, and
also recognizing the linear algebraic terms and linear algebraic expression.
e.g: 3x,xy,x² are the linear term.
x + 3 = 5, x - 2y = 7 are linear equations.
x + 3 = 5is linear equation in one unknown.
2x + 3, x - 2y, xy + 2, x² - 1, 2x + 3, x - 2y are linear expressions.
Other than that is about solving Linear Equations in One Unknown involving combined
operations of +, -, x,÷
Steps:
1. Work on the bracket first, if there is any.
| MUHAMAD ILI NURHAYAT BIN HASSAN 16
http://prosacksblog.blogspot.com [ ]
2. Group the terms with the unknown on the left hand side of the equation while the
numbers on the right side.
3. Solve the equation using combined operations.
4. Check your solution by substituting the value into the original equation.
Examples:
Given that 2y + 11 = -5, calculate the value of y.
Solutions:
7 - ( x + 1) = - 4x
7 – x – 1 = - 4x
-x + 4x = -7 + 1
3x = - 6
x = -2
This topic continue in Form 3 where students are also exposed to solve a situation by
forming a linear equation from a given statements or problem and vice versa.
LINEAR INEQUALITIES
Linear inequalities is part of Algebra which almost same as linear equality except that it
emphasis on the use of everyday situation that had been illustrate in the symbol of “>”
which read as “greater than”, “<” read as “less than”, “≥” read as “greater than or equal
to”, and “≤” read as “less than or equal to”.
The process of learning included understand the use and the concept of inequalities
which is identify the relationship of “>”, “<”, “≥” and “≤” between two given numbers and
based on situation, understand and use the concept of linear inequalities in one unknown.
For example x > 0 and x < 2. So, the value of x is 1.
Example:
| MUHAMAD ILI NURHAYAT BIN HASSAN 17
http://prosacksblog.blogspot.com [ ]
Linear inequality on number line
Simultaneous inequalities on number line
| MUHAMAD ILI NURHAYAT BIN HASSAN 18
http://prosacksblog.blogspot.com [ ]
Other than that, it’s also stress on how to perform the computations, understanding the
concept of simultaneous linear inequalities involving addition, subtraction, multiplication
and division on linear inequalities.
Example: Solve the inequality
(x + 2) / 3 - 2 / 5 < (-x - 1) / 3 - 1 / 6
Solution:
Given
(x + 2) / 3 - 2 / 5 < (-x - 1) / 3 - 1 / 6
| MUHAMAD ILI NURHAYAT BIN HASSAN 19
http://prosacksblog.blogspot.com [ ]
Multiply all terms by 30, the LCD
30*(x + 2) /3 - 30*2 / 5 < 30(-x - 1) / 3 - 30*1 / 6
simplify
10(x + 2) - 6*2 < 10(-x - 1) - 5
Multiply factors and group like terms
10x + 20 - 12 < -10x - 10 - 5
10x + 8 < -10x -15
Subtract 8 to both sides and simplify
10x < -10x - 23
Add 10x to both sides and simplify
20x < -23
Divide both sides by 20
x < -23 / 20
Conclusion
The solution set consists of all real numbers in the interval
(- infinity, -23/20).
QUADRATIC EXPRESSION AND EQUATION
A quadratic expression is a mathematical expression that involves at least one
variable raised to the power of two and no variables raised to any higher powers -
quadratic expressions are second degree polynomials. When only one variable has
been raised to the second power, the graphical representation of the quadratic equation
is always a parabola1. Despite the single graphical representation, there are three
| MUHAMAD ILI NURHAYAT BIN HASSAN 20
http://prosacksblog.blogspot.com [ ]
different common ways to represent parabolas symbolically in mathematical notation.
The form:
y=a(x-h)2+k
Quadratic Expression and Expression is generally preferred above the others, as
the point (h,k) is the vertex of the graph and the value 'a' tells how the graph changes
with respect to the x- and y-axes (that is, how wide the parabola gets, how quickly).
There are two other parabolic forms, however. The form:
y=ax2+bx+c
which is called the 'general form' of a parabola, and from an equation with this form, all
the other forms are usually derived. The third form is known as 'factorised form', and is
symbolically represented by:
y=(ax+b)(cx+d)
This topic has been introduced in chapter 2 in form 4 and is the chosen topic for my
task. Basically all of the main elements in this topic is almost same and related to the
previous topics of Algebra. In this topic, there are a few things that had been introduced:
1. Understanding the concept of Quadratic Expression
| MUHAMAD ILI NURHAYAT BIN HASSAN 21
http://prosacksblog.blogspot.com [ ]
a. Realizing what is quadratic and reforming it.
Example:
ax²+bx+c
Above expression is quadratic expression since:
- the higher power is 2
- It has only one unknown
b. Forming quadratic expression from situation.
Example6:
A school hall measuring (3 + x) m long and (4 + x ) m wide is going to be carpeted.
Calculate the area of carpet needed.
Solution:
Area = length × width = (3 + x) × (4 + x) = (12 + 7x + x²) m²
x² + 7x + 12 is a quadratic expression.
2. Factorization of Quadratic Expression.
i. Factorize quadratic expression of the form ax² + c and ax² + bx
Example: 3x²+15 = 3(x² + 5)
ii. Factorize quadratic expressions of the form px² - q, where p and q are perfect
squares.
6 Example from text book form 4 for in the topic 2.1 quadratic expression and equations example 4
| MUHAMAD ILI NURHAYAT BIN HASSAN 22
http://prosacksblog.blogspot.com [ ]
Example: px² - q = a²x² - b² = (ax + b) (ax – b)
4p² - 25 = (2p + 5) (2p – 5)
iii. Factorize quadratic expressions of the forms ax² + bx + c where a = 1, and b and
c are not equal to zero.
iv. Example: 3x² + 7x – 6 = (3x – 2) ( x + 3)
3. Understanding the concept of Quadratic equation.
a. Recognizing quadratic equation and forming its general equation.
e.g: ax²+bx+c=0
b. Forming quadratic equation from situation.
Example7
A car is travelling at a constant speed of (2x + 3) km/h in (x+1) hours. If it has travelled a
distance 0f 45 km, form a quadratic speed equation in terms of x.
Solution: Distance = speed × time = (2x+3) (x+1) = 2x² + 5x + 3
The expression for the distance is 2x² + 5x + 3
The quadratic equation is 2x² + 5x + 3 = 45
2x² + 5x – 42 = 0
4. Understanding the concept of roots to solve problems.
Example8:
Encik Ramli’s age is 8 times his son’s age. Three years ago, the product of their age was 74
years. How old is Encik Ramli and his son now?
7 Example 12 topic 2.3 from text book form 4 8 Example 16 chapter 2.4 Quadratic Expression and Equation in text book Form 4
| MUHAMAD ILI NURHAYAT BIN HASSAN 23
http://prosacksblog.blogspot.com [ ]
Solution:
i. Read and understand the problems.
Encik Ramli’s age = 8 times his son’s age.
3 years ago: Encik Ramli’s age × His son’s age = 74
ii. Device a plan
Assuming his son’s age is x and Encik Ramli’s age is 8x,
3 Years ago: His son’s age = x-3
Encik Ramli’s age = 8x – 3
(x – 3)(8x – 3) = 74
iii. Carry out the plan
Factorize the equation to get the value of x.
(x – 3) ( 8x – 3) = 74
8x² - 3x -24x + 9 = 74
8x² - 27x – 65 = 0
(8x – 5) ( 8x + 13) = 0
8 = 5 or - 138
Encik Ramli’s son’s age = 5 years old
Encik Ramli’s age = 8× 5= 40 years old
iv. Check the solution
(5 – 3)((8)(5) – 3 ) = (2)(37) = 74
BASIC OPERATION OF QUADRATIC EXPRESSION AND EQUATION
Quadratic Expression and Equation are any algebra unit which has only one
variables and their maximum power of variables is 2 which exist in its general m is
| MUHAMAD ILI NURHAYAT BIN HASSAN 24
http://prosacksblog.blogspot.com [ ]
ax2+bx+c for quadratic expression and for quadratic equation where a,
b, c are coefficients and a ≠ 0.
Note that if a=0, then the equation would simply be a linear equation, not quadratic.
Examples
x² + 2x = 4 is a quadratic since it may be rewritten in the form ax² + bx + c = 0 by
applying the Addition Property of Equality and subtracting 4 from both sides of =.
(2 + x)(3 – x) = 0 is a quadratic since it may be rewritten in the form ax2 + bx + c = 0 by
applying the Distributive Property to multiply out all terms and then combining like
terms.
x² - 3 = 0 is a quadratic since it has the form ax² + bx + c = 0 with b=0 in this case.
3x² – 2/x + 4 = 0 is not a quadratic since it has the term 2/x. The term 2/x is the same as
2x-1, and quadratics do not have x raised to any power other than 1 or 2.
Just remember:
Quadratics always have an x² term, possibly an x-term, and possibly a constant term! If
your equation has an x² term or will have an x² term. After multiplying out, it may be a
quadratic, provided the other terms fit the form.
There are number of operations that can be applied to modify quadratic expression and
expression which is addition, subtraction, multiplication, and division.
| MUHAMAD ILI NURHAYAT BIN HASSAN 25
http://prosacksblog.blogspot.com [ ]
ADDITION AND SUBTRACTION OF QUADRATIC EXPRESSION AND EQUATION9
Recall that algebraic expression that is a number, a variable, or a product or
quotient of numbers and variables is called terms. Examples of terms are:
7 a -2b -47
y² 0.7ab -5/w
Two or more terms that contain the same variable or variables with
corresponding variables having the same exponents are called like terms or similar
terms. For example, the following pairs are like terms.
6k and k 5x² and -7x² 9ab and 0.4ab92
x²y and -113
x²y
Two terms are unlike terms when they contain different variables, or the same
variables with different exponents. For example, the following pairs are unlike terms.
3x and 4y 5x² and 5x 9ab and 0.4a83
x²y and 47
xy
And for Quadratic expression and equation, they only included one term of
variable with the maximum power of two only. To add or subtract these terms, we use
the distributive property of multiplication over addition and subtraction.
9x - 2x = (9 + 2)x = 11x
18y² - 5y² - (18 – 5)y² = 13y²
Since the distributive properties is true for any numbers of terms, we can express
the sum or difference of any number of like terms as a single terms.
Recall that when like terms are added:
9 Taken from http://www.babylon.k12.ny.us/PDF/integrated_algebra/Chapter05.pdf
| MUHAMAD ILI NURHAYAT BIN HASSAN 26
http://prosacksblog.blogspot.com [ ]
1. The sum or difference has the same variable or variables as the original terms.
2. The numerical coefficient of the sum or difference is the sum or difference of the
numerical coefficients of the terms that were added.
The sum of unlike terms cannot be expressed as a single term. For example, the sum of
2x and 3 cannot be written as a single term but is written 2x±3y.
EXAMPLE
Express the difference (4x² + 2x- 3) - (2x² - 5x - 3) in simplest form.
Solution
How to Proceed
i. Write the subtraction problem: (4x² + 2x - 3) - (2x² - 5x – 3)
ii. To subtract, add the opposite (4x² + 2x - 3) + (2x² + 5x + 3)
of the polynomial to be subtracted:
iii. Use the commutative and (4x² - 2x²) + (2x + 5x) + (-3 + 3)
associative properties to group like
terms:
iv. Add like terms: 2x² + 7x + 0
2x² + 7x
Answer: 2x² + 7x
| MUHAMAD ILI NURHAYAT BIN HASSAN 27
http://prosacksblog.blogspot.com [ ]
MULTIPLICATION OF QUADRATIC EXPRESSION AND EQUATION
Multiplying a Monomial by a Monomial
We know that the commutative property of multiplication makes it possible to
arrange the factors of a product in any order and that the associative property of
multiplication makes it possible to group the factors in any combination. For example:
(3x)(7x) = (3)(7)(x)(x) = (3 . 7)(x . x) = 2x²
(-2x)(+5x) = (2)(x)(+5)(x ) = [(2)(+5)] [(x)(x)] = 10x²
In the preceding examples, the factors may be rearranged and grouped mentally.
Procedure
To multiply a monomial by a monomial:
i. Use the commutative and associative properties to rearrange and group the
factors.This may be done mentally.
ii. Multiply the numerical coefficients.
iii. Multiply powers with the same base by adding exponents.
iv. Multiply the products obtained in Steps 2 and 3 and any other variable factors
by writing them with no sign between them.
| MUHAMAD ILI NURHAYAT BIN HASSAN 28
http://prosacksblog.blogspot.com [ ]
EXAMPLE
Represent the area of a rectangle whose length is 3x and whose width is 2x.
Solution
How to Proceed
(1) Write the area formula: A = lw
(2) Substitute the values of l and w: = (3x)(2x)
(3) Perform the multiplication: = (3.2)(x. x)
= 6x²
Multiplying a Polynomial by a Monomial
The distributive property of multiplication over addition is used to multiply a
polynomial by a monomial. Therefore,
a(b + c) = ab + ac
x(4x + 3) = x(4x) + x(3)
= 4x² + 3x
This result can be illustrated geometrically. Let us separate a rectangle, whose length is
4x + 3 and whose width is x, into two smaller rectangles such that the length of one
rectangle is 4x and the length of the other is 3.
Since the area of the largest rectangle is equal to the sum of the areas of the two smaller
rectangles:
x(4x + 3) = x(4x) + x(3) = 4x² + 3x
| MUHAMAD ILI NURHAYAT BIN HASSAN 29
http://prosacksblog.blogspot.com [ ]
Procedure
To multiply a polynomial by monomial, use the distributive property:
Multiply each term of the polynomial by the monomial and write the result as the sum of
these products.
Multiplication and Grouping Symbols
When an algebraic expression involves grouping symbols such as parentheses,
we follow the general order of operations and perform operations with algebraic terms.
In the example at the right, first simplify 8y - 2(7y - 4y) + 5
the expression within parentheses: 8y - 2(3y) + 5
Next, multiply: 8y - 6y + 5
Finally, combine like terms by addition 2y + 5
or subtraction:
In many expressions, however, the terms within parentheses cannot be combined
because they are unlike terms. When this happens, we use the distributive property to
clear parentheses and then follow the order of multiplying before adding.
Here, clear the parentheses by using the 3 + 7(2x + 3)
distributive property: 3 + 7(2x) + 7(3)
Next, multiply: 3 + 14x + 21
Finally, combine like terms by addition: 24 + 14x
| MUHAMAD ILI NURHAYAT BIN HASSAN 30
http://prosacksblog.blogspot.com [ ]
The multiplicative identity property states that a = 1.a. By using this property, we
can say that 5 + (2x - 3) = 5 + 1(2x - 3) and then follow the procedures shown above.
5 + (2x - 3) = 5 + 1(2x - 3) = 5 + 1(2x) - 1(3) = 5 + 2x - 3 = 2 + 2x
Also, since -a = -1.a, we can use this property to simplify expressions in which a
parentheses is preceded by a negative sign:
6y - (9 - 7y) = 6y - 1(9 - 7y) = 6y - 1(9) - 1(7y) = 6y - 9 - 7y = 13y - 9
Multiplying polynomial
As discussed in previous, to find the product (x + 4)(a), we use the distributive
property of multiplication over addition:
(x + 4)(a) = x(a) + 4(a)
Now, let us use this property to find the product of two binomials, for example,
(x + 4)(x + 3).
(x + 4)(a) = x(a) + 4(a)
(x + 4)(x + 3) = x(x + 3) + 4(x + 3)
= x² + 3x + 4x + 12
= x² + 7x + 12
| MUHAMAD ILI NURHAYAT BIN HASSAN 31
http://prosacksblog.blogspot.com [ ]
This result can also be illustrated geometrically
In general, for all a, b, c, and d :
(a + b) (c + d) = a(c + d) + b(c + d)
= ac + ad + bc + bd
Notice that each term of the first polynomial multiplies each term of the second.
At the right is a convenient vertical arrangement of proceeding multiplication,
similar to the arrangement used in arithmetic
multiplication. Note that multiplication is done
from left to right.
The world FOIL serves as a convenient way
to remember the steps necessary to multiply two binomials.
| MUHAMAD ILI NURHAYAT BIN HASSAN 32
http://prosacksblog.blogspot.com [ ]
EXAMPLE
Dividing by a Monomial
We know that
We can rewrite this equality interchanging the left and right members.
Using this relationship, we can write:
| MUHAMAD ILI NURHAYAT BIN HASSAN 33
http://prosacksblog.blogspot.com [ ]
Procedure
To divide a monomial by a monomial:
i. Divide the numerical coefficients.
ii. When variable factors are powers of the same base, divide by subtracting
exponents.
iii. Multiply the quotients from steps 1 and 2.
If the area of a rectangle is 42 and its length is 6, we can find its width by dividing the
area, 42, by the length, 6. Thus, 42 ÷ 6 = 7, which is the width.
Similarly, if the area of a rectangle is represented by 42x and its length by 6x, we can
find its width by dividing the area, 42x, by the length, 6x:
42x² ÷ 6x = 7x
Therefore, the width can be represented by 7x.
Dividing by a Binomial
When we divide 736 by 32, we use repeated subtraction of multiples of 32 to
| MUHAMAD ILI NURHAYAT BIN HASSAN 34
http://prosacksblog.blogspot.com [ ]
determine how many times 32 is contained in 736. To divide a polynomial by a
binomial, we will use a similar procedure to divide x²+ 6x + 8 by x + 2.
How to Proceed
i. Write the usual division form:
ii. Divide the first term of the dividend by the
first term of the divisor to obtain the first
term of the quotient:
iii. Multiply the whole divisor by the first term of
the quotient. Write each term of the product
under the like term of the dividend:
iv. Subtract and bring down the next term of
the dividend to obtain a new dividend:
v. Divide the first term of the new dividend by
the first term of the divisor to obtain the next
term of the quotient:
vi. Repeat steps (3) and (4), multiplying the
whole divisor by the new term of the quotient. Subtract this product from
the new dividend. Here the remainder is zero and the division is complete:
| MUHAMAD ILI NURHAYAT BIN HASSAN 35
http://prosacksblog.blogspot.com [ ]
The division can be checked by multiplying the quotient by the divisor to obtain the
dividend:
(x + 4)(x + 2) = x(x + 2) + 4(x + 2)
= x² + 2x + 4x + 8
= x² + 6x + 8
EXAMPLE
Divide 5s + 6s² - 6 by 2s + 3 and check.
| MUHAMAD ILI NURHAYAT BIN HASSAN 36
http://prosacksblog.blogspot.com [ ]
FACTORING THE QUADRATIC EXPRESSION AND EQUATION
Other than that, quadratic expression and equation with real coefficients can
have none, one or two distinct real roots. To find them, we have a few methods to solve
which are:
Factoring
The easiest way to solve a quadratic equation is to solve by factoring, if possible.
Here are the steps to solve a quadratic by factoring:
1. Put the equation in the standard form (ax2+bx+c=0)
2. Factor the equation (find two numbers that will not only multiply to equal the
constant term "c", but also add up to equal "b", the coefficient on the x-term).
3. Set each of the two binomial expressions equal to cero.
4. Solve each of the equations.
5. Check you answer.
| MUHAMAD ILI NURHAYAT BIN HASSAN 37
http://prosacksblog.blogspot.com [ ]
Solve by factoring x2+2x = 15
1. Put the equation in the standard form: x2+2x-15=0
2. We require to numbers that multiply together to give -15 and add together to give -2.
x2+2x-15=(x-3)(x+5).
3. Set each of the two binomial expressions equal to cero.
x2+2x-15 = 0
(x-3)(x+5) =0
4. Solve each equation:
x-3=0, x=3
x+5=0, x=-5
When the leading coefficient (the number on the x2 term) is not 1, the first step in
factoring will be to multiply "a" and "c"; then we'll need to find factors of the product "ac"
that add up to "b".
Solve by factoring 2x2+4x-6 = 0
We need to find factors of 12 (ac=2·(-6)=-12) that add up to +4.
We will use the pair "-2 and 6".
Draw a two-by-two grid, putting the first term in the upper left-hand corner and the last
term in the lower right-hand corner:
2x2
-6
Take the factors –2 and 6 and put them, complete with their signs and variables, in the
diagonal corners:
2x2 -2x
6x -6
| MUHAMAD ILI NURHAYAT BIN HASSAN 38
http://prosacksblog.blogspot.com [ ]
Factor the rows and columns:
2x -2
2x 2x2 -2x
6 6x -6
Then, 2x2+4x-6 = (2x-2)(2x+6)
We will find the solutions of the equations by solving each equation:
2x-2=0, x=1
2x+6=0, x=-3
Sometimes you cannot find integer factors that work, then this quadratic is said to be
"unfactorable over the integers" or "prime". On these cases, you must try to solve the
equation using another method.
Extracting Square Roots
Extracting square roots is a very easy way to solve quadratics, provided the equation is
in the correct form.
Basically, Extracting Square Roots allows you to rewrite x² = k as x = ±√k, where k is
some real number. Algebraically, we are taking square roots of both sides of the
equation as shown below and inserting the ± to account for both a positive and negative
case. Note that the squared quantity must be isolated on one side of = before you can
extract the square roots.
Example: Solve x² = 9 by extracting square roots
| MUHAMAD ILI NURHAYAT BIN HASSAN 39
http://prosacksblog.blogspot.com [ ]
Example: Solve (2x – 5) ² + 5 = 3
(2x – 5) ² + 5 = 3 Given
(2x – 5) ² = -2 Addition Property of Equality used to add –5 to both
sides
√ (2x – 5) ² = ±√(-2) Extract Square Roots
2x – 5 = ± i√2 Simplify Radicals and Apply Definition of “i”
2x = 5 ± i√2 Addition Property of Equality
x = (5 ± i√2) / 2 Division Property of Equality
Completing The Square
This method of solving quadratic equations is straightforward, but requires a specific
sequence of steps. Here is the procedure:
Example: Solve 3x² + 4x – 7 = 0 by Completing The Square
1. Isolate the x² and x-terms on one side of = by applying the Addition Property of
Equality.
3x² + 4x = 7
2. Apply the Division Property of Equality to divide all terms on both sides by the
coefficient on x².
(3x2)/3 + (4x)/3 = 7/3
x² + (4/3)x = 7/3 Note: Steps 1 and 2 may be done in either
order.
3. Take ½ of the coefficient on x. Square this product. Add this square to both sides
using the Addition Property of Equality. In this case, we take ½ of 4/3 which is
| MUHAMAD ILI NURHAYAT BIN HASSAN 40
http://prosacksblog.blogspot.com [ ]
(1/2)•(4/3) = 4/6. Square 4/6 to get (4/6) •(4/6) = 16/36 = 4/9 when reduced. Add
4/9 to both sides to get
x² + (4/3)x + 4/9 = 7/3 + 4/9
x² + (4/3)x + 4/9 = 21/9 + 4/9 multiply 7/3 by 3/3 to get common denominator
x² + (4/3)x + 4/9 = 25/9 add fractions
4. Factor the left side.
Note: It will always factor as (x ± the square root of what you added) ²
(x + 2/3) ² = 25/9
5. Solve by extracting square roots.
√ (x + 2/3) ² = ±√(25/9) Extract Square Roots
x + 2/3 = ±5/3 Simplify Radicals
x = -2/3 ± 5/3 Addition Property of Equality
This results in two answers: x = -2/3 + 5/3 = 3/3 = 1 and x = -2/3 – 5/3 = -7/3
You may have noticed that we solved this same problem earlier in a much easier
fashion by factoring! So why learn this method of extracting square roots? Answer: This
method is used in higher levels of math (like calculus) to perform similar or identical
equation rearrangements. Also, we need this method to justify and derive the Quadratic
Formula.
| MUHAMAD ILI NURHAYAT BIN HASSAN 41
http://prosacksblog.blogspot.com [ ]
Using The Quadratic Formula
Solving a quadratic equation that is in the form ax² + bx + c = 0 only involves plugging
a, b, and c into the formula.
Example: Solve (x + 3)2 = x – 2
(x + 3)2 = x – 2 Given
(x + 3)(x + 3) = x – 2 Rewrite
x² + 6x + 9 = x – 2 Multiply out with Distributive Property, Combine Like
Terms
| MUHAMAD ILI NURHAYAT BIN HASSAN 42
http://prosacksblog.blogspot.com [ ]
x² + 5x + 11 = 0 Addition Property of Equality - add 2, add –x to both
sides
Plug a=1, b=5, c =11 from 1x2 + 5x + 11 = 0 into the Quadratic Formula to get
which simplifies to
after we simplify the radical and rewrite √(-19) as (√19) • i by applying the
definition of i.
CONSTRUCTIVIST APPROACHES IN TEACHING AND LEARNING QUADRATIC
EXPRESSION AND EQUATION
INTRODUCTION
Constructivism is one of the modern approaches in teaching and learning which
applied both to learning theory and to epistemology which both of these stress on how
people learn, and to the nature of knowledge10. The main principle of constructivist
learning is that people construct their own understanding of the world, and in turn their
own knowledge.
10 Constructivism asserts two main principles whose applications have far-reaching consequences for the study of cognitive development and learning as well as for the practice of teaching, psychotherapy, and interpersonal management in general. The two principles are (1) knowledge is mot passively received but actively built up by the experiential world, not the discovery of ontological reality." International Encyclopedia of Education. "Constructivism In Education," 1987.
| MUHAMAD ILI NURHAYAT BIN HASSAN 43
http://prosacksblog.blogspot.com [ ]
We can see in our country, Malaysia, a typical teaching and learning process in
mathematics involves presenting the concepts followed by reinforcement and
enrichment exercises. Here rote learning style discourages the mind to think
inquisitively and eventually mathematics will be perceived as a boring subject.
To minimize this, the teaching and learning process must involve engaging
activities which include active participation from the students and relate to real-life
situations which then will construct their thinking skills in mathematics subject especially
Quadratic Expression and Equation
All in all, the aim of constructivist learning is that the teacher-defined
goal to achieve an output, where the task practice environment is designed
to reveal the knowledge and concepts to be learned; feedback is given by
the environment in relation to the extent to which learners’ actions achieved
the intended goal. According to Audrey Gray, the characteristics of a constructivist
classroom are as follows:
the learners are actively involved
the environment is democratic
the activities are interactive and student-centered
the teacher facilitates a process of learning in which students are
encouraged to be responsible and autonomous
ACTIVITIES TO BE ADDED
| MUHAMAD ILI NURHAYAT BIN HASSAN 44
http://prosacksblog.blogspot.com [ ]
From the model of Constructivist Learning Environments, there are about six
models of teaching and learning that can be bring forward to satisfy the teaching and
learning process which are:
1. Question/Case/Problem/Project
2. Related Cases
3. Information Resources
4. Cognitive (Knowledge Construction) Tools
5. Conversation and Collaboration Tools
6. Social/ Contextual Support.
1. Question/Case/Problem/Project
The focus is the question or issue, the case, the problem, or the project that
learners attempt to solve or resolve. It constitutes the learning goal. The quadratic
problem as examples or applications of the concepts and principles previously taught.
| MUHAMAD ILI NURHAYAT BIN HASSAN 45
http://prosacksblog.blogspot.com [ ]
Students learn domain content in order to solve the problem, rather than solving the
problem to apply the learning.
Constructive Learning Environments at this part stress on question/issue-based,
case-based, project-based, or problem-based learning. Question- or issue-based
learning begins with a question with uncertain or controversial. For example, what is
quadratic? In case-based learning, students acquire knowledge and requisite thinking
skills by studying cases and preparing case summaries or diagnoses. For example,
quadratic is a part of algebra and had been learn at form 1, 2 and 3 which can be a
guide to know the quadratic, how about refer it back to know about it more?
Project-based learning focuses on relatively long-term, integrated units of
instruction where learners focus on complex projects consisting of multiple cases. For
example, what are quadratic and where it’s come from? They debate ideas, plan and
conduct experiments, and communicate their findings11. Meanwhile, problem-based
learning integrates courses at a curricular level, requiring learners to self-direct their
learning while solving numerous cases across a curriculum. For example, x²+2 is part of
quadratic. Why?
Basically, case-, project-, and problem-based learning represent a set of
complexity, but all share the same assumptions about active, constructive, and
authentic learning. By doing this way, it can support constructivist approaches in
learning where the students learn to question the reason of every single thing that
happened and then construct their own knowledge and answer by finding it.
It is important to provide interesting, relevant, and engaging problems to solve.
The problem should not be overly prescribed. Rather, it should be ill-defined or ill-
structured, so that some aspects of the problem are emergent and definable by the
learners. Ill-structured problem have unstated goals and constraints and have multiple
11 Proposed by Krajcik, Blumenfeld, Marx, & Soloway, 1994
| MUHAMAD ILI NURHAYAT BIN HASSAN 46
http://prosacksblog.blogspot.com [ ]
solutions, solution paths, or no solutions at all. We need to decide and guided them if
the students possess prerequisite knowledge or capabilities for working on the problem
that we identify.
Other than that, the problems constructivist model generated three integrated
components which are;
1. The problem context
2. The problem representation or simulation, and
3. The problem manipulation space.
2. Related Cases
The students basically lack most in experiences. This lack is especially critical
when trying to solve problems. So, it is important to provide access to a set of related
experiences that novice students can refer to. The primary purpose of describing related
cases is to assist learners in understanding the issues implicit in the problem
representation. Besides, understanding any problem requires experiencing it and
constructing mental models of it. Related cases in constructivist learning environments
support learning in at least two ways: by scaffolding memory and by representing
complexity.
Usually, the lessons that we understand the best are those in which we have
been most involved and invested the greatest amount of effort. Related cases can
scaffold memory by providing representations of experiences that learners have not
had. They cannot replace learners’ involvement, but they can provide referents for
comparison. When humans first encounter a situation or problem, they naturally first
check their memory for similar cases that they may have solved previously12. If they can
12 Proposed by Polya, 1957
| MUHAMAD ILI NURHAYAT BIN HASSAN 47
http://prosacksblog.blogspot.com [ ]
recall a similar case, they try to map the previous experiences and its lessons onto the
current problem. If the goals or conditions match, they apply their previous lesson. By
resenting related cases in learning environments, you are providing the learners with a
set of experiences to compare to the current problem or issue. Learners retrieve from
related cases advice on how to succeed, pitfalls that may cause failure, what worked or
didn’t work, and why it didn’t. They adapt the explanation to fit the current problem and
that way of learning is called constructivism learning.
Related cases also help to represent complexity in constructivist learning process
by providing multiple perspectives, themes, or interpretations to the problems or issues
being examined by the learners. Usually, this model stress on the conceptual
interrelatedness of ideas and their interconnectedness by providing multiple
interpretations of content and by using multiple, related cases to convey the multiple
perspectives on most problems. In order to enhance cognitive flexibility, it is important
that related cases provided a variety of viewpoints and perspectives on the case or
project being solved.
For example, while introducing the quadratic, how about related it to the previous
topic that the students had learned. Linear equation for example, is quite similar to
quadratic especially is its general form and their properties. How about co-relate this to
the quadratic to make the student understand more easily.
EXAMPLE
In general, the graph of a quadratic equation
| MUHAMAD ILI NURHAYAT BIN HASSAN 48
http://prosacksblog.blogspot.com [ ]
y = ax2 + bx + c is a parabola.
If a > 0, then the parabola has a minimum point and it opens upwards (U-shaped) eg.
y = x2 + 2x − 3
If a < 0, then the parabola has a maximum point and it opens downwards (n-shaped)
eg.
y = -2x2 + 5x + 3
| MUHAMAD ILI NURHAYAT BIN HASSAN 49
http://prosacksblog.blogspot.com [ ]
And all of this quadratic equation can be related to the previous topic which had been
learning in form to which is linear equation.
Intercept Form of a Straight Line: ax + by = c
In general, linear equation is often a straight line and is written in the form ax + by = c.
One way we can sketch this is by finding the x- and y-intercepts and then joining those
intercepts.
ax + by = c
by = -ax +c
y = -ab
x + cb
let ab
= m and cb
= C
hence, y = mx + c
Slope-Intercept Form of a Straight Line: y = mx + c
If the slope (also known as gradient) of a line is m, and the -intercept is c, then the
equation of the line is written: y = mx + c
Example:
The line y = 2x + 6 has slope m = 2 and y-intercept c = 6.
| MUHAMAD ILI NURHAYAT BIN HASSAN 50
http://prosacksblog.blogspot.com [ ]
By showing the previous topic that quite
relate and almost same to the quadratic,
the students can easily understanding
their new knowledge by constructing the
idea from the previous knowledge.
3. Information Resources
In order to investigate problems, learners need information about the problem in
order to construct their mental models and formulate hypotheses that drive the
manipulation of the problem space. So, when using constructivist approaches in
learning, we should determine what kinds of information the learner will need in order to
understand the problem. Rich sources of information are an essential part of
constructivist learning and since nowadays, we were provide by information technology
to get information and reference, it should provide learner-selectable information just-in-
time with the guidance of us, teacher.
So, what we need to do to make sure that information makes sense not only in
the context of learning quadratic only, but only in solving problems and application. So,
determine what information learners need to interpret the problem. Maybe we naturally
used the text book and syllabus in our lesson. How about using other relevant
information banks and repositories that can be linked to the environment? These may
include text documents, graphics, sound resources, video, and animations that are
appropriate for helping learners comprehend the problem and its principles. And since
now, technology is one of the easiest paths to gain knowledge. The World Wide Web
(WWW) for example, is the default storage medium, as powerful new plug-ins enables
users to access multimedia resources from the net. However make sure that we guide
the students in using it since WWW resources may provide unlimited information that
| MUHAMAD ILI NURHAYAT BIN HASSAN 51
http://prosacksblog.blogspot.com [ ]
unnecessary for our students. Actually, our textbook also included the information of
website that can be used as a reference and addition information resource.
EXAMPLE
Below are example of the website information of Quadratic Expression and Equation
that had been included in the Mathematics form 4 KBSM text book.
| MUHAMAD ILI NURHAYAT BIN HASSAN 52
http://prosacksblog.blogspot.com [ ]
4. Cognitive (Knowledge Construction) Tools
| MUHAMAD ILI NURHAYAT BIN HASSAN 53
http://www.analyzemath.com/Algebra2/Tutorials.html
http://prosacksblog.blogspot.com [ ]
Cognitive tools are generalizable computer tools that are intended to engage and
facilitate cognitive processing13. They are intellectual devices that are used to visualize
(represent), organize, automate or supplant information processing. Some cognitive
tools replace thinking while others engage learners in generative processing of
information that would not occur without the tool. Cognitive tools support learners in a
variety of cognitive processing tasks. For example, visualization tools help learners to
construct those mental images and visualize activities. Numerous visualization tools
provide reasoning-congruent representations that enable learners to reason about
objects that behave and interact14. Examples include the graphical proof tree
representation in the Geometry Tutor15; the Weather Visualizer (colorizes climatological
patterns); the Climate Watcher (colorizes climatological variable).
As students study phenomena, it is important that they articulate their
understanding of the phenomena. Modeling tools provide knowledge representation
formalisms that constrain the ways that learners think about, analyze, and organize
phenomena and provide an environment for encoding their understanding of those
phenomena. For example, creating a knowledge database or a semantic network
requires learners to articulate the range of semantic relationships among the concepts
that comprise the knowledge that they had learn in class. Modeling tools help learners
to answer “what do I know” and “what does it mean” questions. As a teacher who apply
constructivist learning, we must decide when learners need to articulate what they know
and which formalism will best support their understanding.
13 Proposed by Kommers, Jonassen, & Mayes, 199214 Proposed by Merrill, Reiser, Bekkalaar, & Hamid, 199215 Proposed by Anderson, Boyle, & Yost, 1986
| MUHAMAD ILI NURHAYAT BIN HASSAN 54
http://prosacksblog.blogspot.com [ ]
Complex systems contain interactive and interdependent components. In order to
represent the dynamic relationships in a system, learners can use dynamic modeling
tools for building simulations of those systems and processes and testing them.
Programs like Stella use a simple set of building blocks to construct a map of a process.
Learners supply equations that represent causal, contingent, and variable relationships
between the variables identified on the map. Having modeled the system, Stella
enables learners to test the model and observe the output of the system in graphs,
tables, or animations. At the run level, students can change the variable values to test
the effects of parts of a system on the other. Modeling phenomena may also call on
dynamic modeling tools, such as Model IT16 which scaffolds the use of mathematics by
providing a range of qualitative relationships that describe the quantitative relationships
between the factors or allowing them to enter a table of values that they have collected.
Picture of Stella program
16 Proposed by Spitulnik, Studer, Finkel, Gustafson, & Soloway, 1995
| MUHAMAD ILI NURHAYAT BIN HASSAN 55
http://prosacksblog.blogspot.com [ ]
In many environments, performing repetitive, algorithmic tasks can rob cognitive
resources from more intensive, higher-order cognitive tasks that need to be performed.
Therefore, constructivist learning should automate algorithmic tasks in order to offload
the cognitive responsibility for their performance. For example, using simple software
like Microsoft office; Microsoft excels especially, we already provided with the
component-component to solve mathematics problem including quadratic problem. So,
it is depends to us whether to use it or not. Most forms of testing in constructivist
learning should be automated so that learners can simply call for test results. And even
generic tools such as calculators or database shells may be embedded to help learners
organize the information that they collect.
Most of this way of learning provide note taking facilities to offload memorization
tasks. Identify in the activity structures those tasks with which learners are facile and
may distract reasoning processes and try to find a tools which supports that
performance.
| MUHAMAD ILI NURHAYAT BIN HASSAN 56
http://prosacksblog.blogspot.com [ ]
Another cognitive tool support interactive learning is GeoGebra. This software is
quite famous in all over the educational space since it’s provide the most simple
interactive learning ways to help students understanding the process of learning. It’s
constructions can be made with points, vectors, segments, lines, polygons, conic
sections, and functions. All of them can be changed dynamically afterwards. Elements
can be entered and modified directly on screen, or through the Input Bar. GeoGebra
has the ability to use variables for numbers, vectors and points, find derivatives and
integrals of functions and has a full complement of commands like Root or Extremum.
Teachers can use GeoGebra to make conjectures and prove geometric theorems.
Below are the example of GeoGebra for quadratic.
EXAMPLE OF GEOGEBRA
| MUHAMAD ILI NURHAYAT BIN HASSAN 57
http://prosacksblog.blogspot.com [ ]
Using Geogebra is one of the most simple way which help students to understand the
quadratic effectively. For example, as an introduction of quadratic, we just need to
explain a little bit briefing about the quadratic. Then, we can introduce this software to
help student’s visual in understanding Quadratic. The software of GeoGebra will explain
the rest. What the meaning of a, b and c in the general form of quadratic which is
ax²+bx+c and what happened if each of the value in a, b and c changing.
5. Conversation and Collaboration Tools
Learning most naturally occurs not in isolation but by teams of people working
together to solve problems. As teacher, we should provide access to shared information
and shared knowledge building tools to help learners to collaboratively construct socially
shared knowledge.
Problems are solved when groups work toward developing a common conception
of the problem, so their energies can be focused on solving it. Conversations may be
supported by discourse communities, knowledge-building communities, and
communities of learners.
Scardamalia and Bereiter (1996) argue that schools inhibit, rather than support,
knowledge building by focusing on individual student abilities and learning. In
knowledge building communities, the goal is to support students to "actively and
strategically pursue learning as a goal (Scardamalia, Bereiter, & Lamon, 1994, 201). To
enable students to focus on knowledge construction as a primary goal, Computer
Supported Intentional Learning
| MUHAMAD ILI NURHAYAT BIN HASSAN 58
http://prosacksblog.blogspot.com [ ]
By doing this approaches, it enable students to produce knowledge databases so
that their knowledge can "be objectified, represented in an overt form so that it could be
evaluated, examined for gaps and inadequacies, added to, revised, and reformulated".
Since this method provide a medium for storing, organizing, and reformulating the ideas
that are contributed by each of the members of the community. The knowledge base
represents the synthesis of their thinking, something they own and for which they can
be proud.
We sometimes think that our students maybe know each other but actually, not
all students are close to each other. So, by doing this activity, we may can collaboration
among groups of students. As teacher, we should encourage conversations about the
problems between the students to make them participates actively. Students write notes
to the teacher and to each other about questions, topics, or problems that arise.
Textualizing discourse among students makes their ideas appear to be as important as
each other’s and the instructor’s comments (Slatin, 1992). When students and teacher
collaborate, they share the same goal which is to make the process of learning
happened effectively.
| MUHAMAD ILI NURHAYAT BIN HASSAN 59
http://prosacksblog.blogspot.com [ ]
6. Social/Contextual Support
Most of the process of learning failed to achieve the target is because of poor
implementation. Frequently, our education system tried to implement their innovation
without considering important physical, organizational, and cultural aspects to the
students or learners and environment. Actually, all the innovation, and implement that
need to do need support not just only from the teachers, but everyone along it. As
teacher, it is necessary to provide the learning aids which show our support in teaching
something. The environments of studies also need to reconstruct to make sure it fulfills
the optimal need of process of learning. For the school administrator, it is compulsory to
make sure that the utilities of school and class is in perfect condition. And for the family,
their support is really necessary to make sure that the educators feel that their effort is
worthy.
EXAMPLE ARRANGEMENT OF CLASS
Figure 1: The dance-floor seating chart in all its finery
| MUHAMAD ILI NURHAYAT BIN HASSAN 60
http://prosacksblog.blogspot.com [ ]
Figure 2: The runway-model seating chart — effective but underrated.
Figure 3: The independent-nation-state seating chart.
| MUHAMAD ILI NURHAYAT BIN HASSAN 61
http://prosacksblog.blogspot.com [ ]
Figure 4: The Battleship seating chart.
| MUHAMAD ILI NURHAYAT BIN HASSAN 62
http://prosacksblog.blogspot.com [ ]
INTEGRATED APPROACHES IN TEACHING AND LEARNING QUADRATIC
EXPRESSION AND EQUATION
INTRODUCTION
To satisfy the integrated approaches in teaching and learning process, there are
several things that need to be emphasis upon student’s thinking, active learning,
discovery learning, and interest in Mathematics. Some think that’s, the student-centered
mathematical classrooms are now considered to be more effective in learning
mathematics than the teacher-centered traditional classrooms. These ways was
supported since it requires following characteristics:
1. Emphasize on processes as well as results.
2. Encourage and support various levels of oral and written mathematical
communication.
3. Encourage and empower leadership and authority shared with students, and
4. Encourage reflective mathematical practice with thinking mathematically.
Some suggested problem solving as an approach to learning mathematics. In
this method, the teaching of a mathematical topic begins with a problem situation that
embodies key aspects of the topic, and mathematical concepts and techniques are
developed as reasonable responses to reasonable problems.
And nowadays, since technology had also been introduce in this path, our
education system also prefer the use of computer as a tool for teaching mathematics
because of the availability of appropriate numerical, graphic, and symbolic capability of
software. In using technology, some of them have revealed that by using the computer
| MUHAMAD ILI NURHAYAT BIN HASSAN 63
http://prosacksblog.blogspot.com [ ]
as a tool for performing the mathematical procedure, the students can be provided with
an opportunity and time to work on real-life problems in mathematics. However, some of
them agree that there should be less emphasis on algebraic manipulation skills and
more emphasis on underlying concepts and mathematical thinking.
Actually, there are some problems that had been faced by our students in the
beginning lesson of secondary mathematics which is the students are more interested
in the use of mathematics and their basic operation than its justification or proofs of
theorems. Because of this problem, our mathematics had been reorganized into two
types which is basic mathematic and additional mathematics.
This action was done since there are some other problems that had been
discovered from most of our students which are:
1. Are mathematically unprepared — they have gaps in mathematical knowledge
and understanding, poor recall and retention of mathematical knowledge.
2. Have perception of having rusty math skills.
3. Have a short attention span, poor attitude and lack of motivation.
4. Have a lack of meta-cognitive math skill.
5. Are passive listeners and have mental blocks.
6. Have not enough time and show poor attendance patterns.
7. Have a keen desire to have a passing grade with very little efforts.
Such diverse students with variety of their abilities and backgrounds are
constrained by learned or acquired behavior patterns that inhibit advanced learning and
the traditional teaching method of chalk-talk-homework- exam does not work for such
students.
| MUHAMAD ILI NURHAYAT BIN HASSAN 64
http://prosacksblog.blogspot.com [ ]
IMPORTANT FEATURES OF INTEGRATED APPROACH
An integrated approach to teaching and learning process in Malaysia nowadays
had been offers a revolution approach that is different from the traditional approach of
chalk-talk-homework-exam. In this approach, the students were given more stress on
the following ten points:
i. Conceptual understanding rather than only computations.
ii. Relational understanding rather than just instrumental understanding.
iii. Exploring patterns and relationships rather than just memorizing formulas.
iv. Variety of pedagogical strategies rather than just chalk and talk.
v. Variety of non-traditional assessments rather than just traditional tests/exams.
vi. Effective and meaningful learning rather than just learning for test/exams.
vii. Listening (hearing, interpreting) to students’ thinking rather than only telling
(speaking, explaining).
viii. Cooperative learning rather than just individualistic learning.
ix. Making sense of mathematics using real life applications rather than just
explaining abstract concepts.
x. Helping students to develop an appreciation of the power of mathematics rather
than a negative view of math.
For interactive and discussion-based teaching, new material is introduced either
with a class discussion or via teacher-made worksheets. The teacher poses problems
and questions for discussions or investigations. During group work, the teacher
observes group interactions and their individual working on the computer or using paper
| MUHAMAD ILI NURHAYAT BIN HASSAN 65
http://prosacksblog.blogspot.com [ ]
and pencil. After completion of group work, there is whole-class discussion and the
teacher serves as a facilitator.
To create mathematical culture in the classroom we emphasize four key features:
i. Various levels of math communication.
ii. Processes as well as results.
iii. Leadership and authority shared with the students, and
iv. Reflective mathematical practice with thinking mathematically.
These four elements are chosen because it is user-friendly, easy to learn and it
supports most of the concepts needed for the freshman and sophomore levels of
college mathematics. All computers in our computer labs are equipped with the latest
features and there are enough computers available to provide each student with at least
a computer.
For group work, the class is divided into small groups of two or three members
each. The students are allowed to choose their group members. During the group work,
the teacher observes the group interactions and individual contributions. While
observing the groups, the teacher checks their work, makes corrections, answers
questions and provides motivation. In the class discussion that follows group work, the
teacher generally serves as a facilitator.
Use the small class which is generally kept between 16 and 20 of students and
use a combination of qualitative and quantitative instruments in gathering data. The
students’ performance is assessed by observations of student’s presentations and
discussions, students’ worksheets, out-of-class assignments, pop-quizzes, class tests,
and midterm and final examinations.
| MUHAMAD ILI NURHAYAT BIN HASSAN 66
http://prosacksblog.blogspot.com [ ]
The evaluation of the effectiveness of the teaching style is done through formal
assessments such as quizzes and tests, students’ and teachers’ weekly logs, and
informal surveys and peer evaluations.
EXAMPLES HIGHLIGHTING INTEGRATED APPROACH
In this section, we present a variety of our classroom activities and examples that
highlight our integrated approach to teaching and learning mathematics.
Conceptual understanding rather than only computations
Conceptual understanding is essential in teaching and learning mathematics; it
influences the mathematics that is taught and enhances a student’s learning.
Mathematics is not just about computation, but understanding and defining the concept
and problem and solving it by computations. Here is an example that we often use in
our Quadratic Expression and Equation:
Example
What do we mean by a root of a quadratic? Find the roots of this quadratic and explain
briefly about what the meaning of it.
x² + 2x − 8
Answer
1.Root of quadratic is a solution to the quadratic equation.
2.By factorize the equation, the root of the quadratic are −4 and 2. For, we can factor that
quadratic as
(x + 4)(x − 2).
| MUHAMAD ILI NURHAYAT BIN HASSAN 67
http://prosacksblog.blogspot.com [ ]
3. Now, if x = −4, then the first factor will be 0. While if x = 2, the second factor will be
0. But if any factor is 0, then the entire product will be 0. That is, if x = −4 or 2, then
x² + 2x − 8 = 0.
Therefore, −4 and 2 are the solutions to the quadratic equation. They are the roots of
that quadratic.
Relational understanding rather than just instrumental understanding .
New material may be introduced either with a class discussion or via teacher-made
worksheets. Rational understanding can be very handy in motivating students for class
discussion. For example, before we give more detail about types of factorization of
quadratic, make a group work to research about it and followed by class discussion is
found very useful in students’ learning new material.
Example
Giving the equation is
x2 + 6x – 7 = 0
Discuss and show how to solve this equation by completing the square.
Solution
| MUHAMAD ILI NURHAYAT BIN HASSAN 68
http://prosacksblog.blogspot.com [ ]
Learning via problem solving in developing concepts
It is a well-known fact that students learn math more meaningfully when they can make
sense of what they are talking about and they can connect ideas or skills they learn in
new situations, in real-life and in other subjects. For example, the following worksheet
followed by classroom discussion helped our students in understanding exponential
functions.
Example
A ball is thrown straight up. Its height, h(in metres), after t seconds is give by
h= -5t² + 10t + 2
To the nearest tenth of a second, when is the ball 6m above the ground? Explain why
there are two answers.
Solution
Formula Given:
h= -5t² + 10t + 2
| MUHAMAD ILI NURHAYAT BIN HASSAN 69
http://prosacksblog.blogspot.com [ ]
They tell you h=6, find t
So,
-5t² + 10t + 2= h
-5t²+10t+2 = 6
-5t²+10t-4=0
Use the Quadratic Equation on the above formula.
Therefore x, or t = 0.552786 or t=1.44721
Rounding to the nearest tenth of a second, t = 0.6 and 1.4
Why are there 2 values for t?
Because the ball hits the 6m mark on the way up (at t=0.6) and then hits the 6m mark
on the way down (at t=1.4)
Learning for discovering facts
Sometimes, mathematics is not just only about logic. There are also some parts that we
can relate it to the facts. It is because, in quadratic, we are using it in studying the effect
and course of something that happened around us. For the above example, we are
finding the time taken for the ball to reach at the height 6cm. This thing actually not
about logic only, but it is also about fact that we have found by using logic or calculation.
And for the quadratic, it is the fact that the roots it related to the value of x when the
other axis is zero.
| MUHAMAD ILI NURHAYAT BIN HASSAN 70
http://prosacksblog.blogspot.com [ ]
Example
Use Derive and sketch the graphs of the following functions. Observe changes in
graphs as you draw. How do these graphs differ? In what ways are they similar?
a) y = (x − 2)(x − 3) and y = x – 3
b) y = (x − 2)(x − 3) and y = (x − 2)2(x − 3)
c) y = x(x − 2)(x − 3) and y = x²(x − 2)2 (x − 3)2
Using problem solving approach for sound pedagogical reasons
Problem solving is designed as a process by which an individual uses previously
acquired knowledge, skills and understanding to satisfy the demands of an unfamiliar
situation. The situation must synthesize what she or he has learned and apply it to new
and different situations17. In our integrated approach, we generally select those
problems that can;
a) engage students in mathematical discussion,
b) promote mathematical thinking,
c) focus on the development of both cognitive and meta-cognitive strategies, and
d) wherever possible, help students to learn math through problem solving.
Our main strategy of problem solving is small group work followed by class discussion.
Also, we encourage students to use Polya’s 4-step approach to problem solving as
outlined in the below figure.
17 Proposed by Krulik & Rudnick in 1989
| MUHAMAD ILI NURHAYAT BIN HASSAN 71
http://prosacksblog.blogspot.com [ ]
The following problem is from our Developmental Mathematics, a remedial course,
which is not a credit course, but is a prerequisite for the Introductory Algebra course
especially quadratic.
Example
If one of the roots of the quadratic equation
x2 + mx + 24 = 0
is 1.5, then what is the value of m?
Solution
We know that the product of the roots of a quadratic equation ax2 + bx + c = 0 is
In the given equation, x2 + mx + 24 = 0, the product of the roots = = 24.
The question states that one of the roots of this equation = 1.5
If x1 and x2 are the roots of the given quadratic equation and let x1 = 1.5
| MUHAMAD ILI NURHAYAT BIN HASSAN 72
http://prosacksblog.blogspot.com [ ]
Therefore, x2 = = 16.
In the given equation, m is the co-efficient of the x term.
We know that the sum of the roots of the quadratic equation
ax2 + bx + c = 0 is = -m
Sum of the roots = 16 + 1.5 = 17 = -17.5.
Therefore, the value of m = -17.5
Using JAT Approach by Mathematical Connections
Mathematics makes sense and is easier to remember and apply when students can
connect new knowledge to existing knowledge in meaningful ways. We have found that
by using “Just At Time” (JAT) approach by mathematical connections helps students to
meaningfully recall pre-requisite concepts and skills and connect them with new
knowledge. For instance, we use the following example with several follow-up activities
as ‘Warm-Up Problem’ (adapted from Choike (2000)) for our Introductory Algebra
students.
Example
Use the factorization method for solving x2 + 19x + 18=0
Solution:
x2 + 19x + 18
| MUHAMAD ILI NURHAYAT BIN HASSAN 73
http://prosacksblog.blogspot.com [ ]
x2 +x + 18x + 18 = 0
(x2 + x) + (18x + 18) = 0
x(x + 1) + 18(x + 1) = 0
(x + 1)(x + 18) = 0
x + 1=0 and x + 18 = 0
x + 1 – 1 = 0 - 1 and x + 18 – 18 = 0 -18
x = -1 and x = -18
Therefore, x1,2 = -1,-18
Using Open Approach
In this approach we select problems that exemplify a diversity of approaches to solving
a problem or multiple correct answers. There are three aspects of the approach: open
process, open-end product, and open problem formulation18. The main purpose of an
open approach is to make mathematics alive and relevant and help students to develop
divergent thinking.
Example
1. Graph the following functions:
i. y = x
ii. y = x²
iii. y = x² - x - 2
18 Refer to Becker & Shimada on 1997
| MUHAMAD ILI NURHAYAT BIN HASSAN 74
http://prosacksblog.blogspot.com [ ]
iv. y = −x
v. y = −x²
vi. y = −x² + 2x + 3
2. Write as many properties and difference as you can see from the functions that
you had see above.
Solution
i. y = x
| MUHAMAD ILI NURHAYAT BIN HASSAN 75
http://prosacksblog.blogspot.com [ ]
ii. y = x²
iii. y = x² - x - 2
iv. y = -x
| MUHAMAD ILI NURHAYAT BIN HASSAN 76
http://prosacksblog.blogspot.com [ ]
v. y = −x² + 2x + 3
Example
Describe how to solve a quadratic equation and give a story problem to illustrate. Solve
your story problem by using as many methods of solving a quadratic equation as
possible.
Writing about Mathematics
The simple exercise of writing an explanation of how a problem was solved not only
helps to clarify a student’s thinking but also may provide other students with fresh
insights gained from viewing the problem from a new perspective.
Example 1
Make up a word problem with a variable having a quadratic relationship.
| MUHAMAD ILI NURHAYAT BIN HASSAN 77
http://prosacksblog.blogspot.com [ ]
Find a quadratic formula for this variable. Try your formula for two hypothetical
situations.
Example 2
One of your friends in College Algebra sends you an email and asks you to explain how
to graph a quadratic formula and equation. Starting with quadratic equation, write a
clear instruction to help her with her problem. Can you generalize this to same root
quadratic, different root quadratic, and no root quadratic? Consider cases when a or b
or c in quadratic equation of ax² + bx + c is exchange.
Example 3
Explain the difference between a quadratic equation and a quadratic expression.
Using puzzles and interesting examples for motivation
There are many websites and books that give variety of mathematical puzzles and
games and some of them may be used as motivation prior to teaching a new concept or
skills. The example below gives a well-known puzzle followed by motivation for some
quadratic concepts.
Example
| MUHAMAD ILI NURHAYAT BIN HASSAN 78
http://prosacksblog.blogspot.com [ ]
Forward Tracking:
Make a puzzle similar to Example above and explain the mathematical logic you used.
Using worksheets for understanding problems and improving writing
The following example and accompanying worksheet show how a worksheet can serve
as a medium that can permit students to follow sequence of steps in learning math.
Example
Below are some example of quadratic equation worksheet.
| MUHAMAD ILI NURHAYAT BIN HASSAN 79
http://prosacksblog.blogspot.com [ ]
| MUHAMAD ILI NURHAYAT BIN HASSAN 80
http://prosacksblog.blogspot.com [ ]
| MUHAMAD ILI NURHAYAT BIN HASSAN 81
http://prosacksblog.blogspot.com [ ]
SUGGESTED APPROACHES IN TEACHING AND LEARNING QUADRATIC
EXPRESSION AND EQUATION
INTRODUCTION
Algebra is central to proficiency in mathematics and Quadratic Expression and
Equation is part of Algebra. The development and understanding of algebraic concepts
and skills, especially in the learning of quadratic expressions and equations are of
fundamental importance in mathematics, thus can raise students’ mathematical
understanding’s skills. Yet students generally have difficulties in algebraic manipulation,
and in formulating quadratic equations to solve problems. These difficulties are
encountered by students around the world formerly and in Malaysia specially.
We can see the proof by looking our secondary school mathematics’ syllabus
where the students are only introduced by the very basic information about the
Quadratic Expression and Equation. At first, they were introduced by the basic concept
of Quadratic Expression and Equation, their basic operation, and then the very basic
problem about the quadratic related to real life. By doing this way, students basically
only got the very basic elements and the very surface about the quadratic. How can the
students master the quadratic if they only exposed by only this basic thing? Isn’t it?
To me, there are something that needs to be added to satisfy the teaching and
learning process of Quadratic Expression and Equation especially in putting the higher
level of understanding in the learning and teaching process. For me, I will use the latest
learning and teaching approaches in my teaching which is cognitive approaches since
this approach is one of the latest approaches that had been introduced and this
approaches basically overlapped and almost same in certain part of ideas and principle
such as behaviorist and constructivist.
| MUHAMAD ILI NURHAYAT BIN HASSAN 82
http://prosacksblog.blogspot.com [ ]
This cognitive approach can be applied according to the situation and condition
regarding the part of the subtopic in the quadratic especially by integrating the variety
types of learning theory which included discovery learning, active learning, meaningful
verbal learning and others can make the teaching and learning process happened
effectively.
THE COGNITIVE APPROACHES IN TEACHING AND LEARNING
QUADRATIC EXPRESSION AND EQUATION
INTRODUCTION
As its name implies, the cognitive approach deals with mental processes like
memory and problem solving. By emphasizing mental processes, it places itself in
opposition to behaviorism, which largely ignores mental processes. Yet, in many ways
the development of the cognitive approach, in the early decades of the 20th century, is
intertwined with the behaviorist approach.
Behaviorist, cognitivist, and constructivist ideas and principles overlap in many
areas. Therefore, classifying some theories fit in more than one classification and
different sources classify the theories in different ways. For example, in some sources
Jerome Bruner‘s Discovery Learning Theory is classified as cognitive and not
developmental. In other sources, Bruner is deemed developmental (Driscoll,
2005/2007). In still other sources, Bruner is considered constructivist (Learning Theories
Knowledgebase, 2009). In addition, Albert Bandura is often classified as a behaviorist;
however, Bandura, himself, claimed that he was never a behaviorist.
| MUHAMAD ILI NURHAYAT BIN HASSAN 83
http://prosacksblog.blogspot.com [ ]
All in all, basically, cognitive approaches are devided into two parts which is
cognitive cognitive approaches and cognitive developments approaches. Cognitive
cognitive approaches including:
i. Social Cognitive Theory (Social Learning Theory) by Bandura which focused on
observational learning and self-efficacy19.
ii. Information Processing Theories by various theorists where the computer was
seen as a metaphor for the mind.
iii. Assimilation Theory (Meaningful Learning) by Ausubel which focused on
reception learning; he noted that the learner was active and thus he differentiated
between rote and meaningful learning. Ausubel also stressed the importance of
the advance organizer.
Meanwhile, the cognitive development is divided into three part which are:
i. Genetic Epistemology by Piaget.
- Piaget believed that experience with the environment affected knowledge
acquisition.
- His four stages of development detail how humans develop cognitively.
ii. Sociocultural Theory by Vygotsky
- Vygotsky’s Zone of Proximal Development (ZPD) details the difference
between what a learner can do independently and what the leaner can do
with help; independent learning may not take place, but scaffolded learning
can.
iii. Discovery Learning by Bruner which describes representational stages, and
emphasizes exploring the environment.
19 Proposed by Zeldin, Britner, & Pajares in 2008
| MUHAMAD ILI NURHAYAT BIN HASSAN 84
http://prosacksblog.blogspot.com [ ]
Below are the diagram about the Cognitive Cognitive Approaches and Development
Cognitive Approaches.
Figure 1: The Diagram of Cognitive Cognitive Approaches
| MUHAMAD ILI NURHAYAT BIN HASSAN 85
http://prosacksblog.blogspot.com [ ]
Figure 2: The Diagram of Cognitive Cognitive Approaches
| MUHAMAD ILI NURHAYAT BIN HASSAN 86
http://prosacksblog.blogspot.com [ ]
SOCIAL LEARNING THEORY
The social learning theory of Bandura emphasizes the importance of observing
and modeling the behaviors, attitudes, and emotional reactions of others. Bandura
states that "Learning would be exceedingly laborious, not to mention hazardous, if
people had to rely solely on the effects of their own actions to inform them what to do.
Fortunately, most human behavior is learned observationally through modeling:
from observing others one forms an idea of how new behaviors are performed, and on
later occasions this coded information serves as a guide for action. Social learning
theory explains human behavior in terms of continuous reciprocal interaction between
cognitive, behavioral, an environmental influences. The component processes
underlying observational learning are:
1. Attention, including modeled events (distinctiveness, affective valence,
complexity, prevalence, functional value) and observer characteristics (sensory
capacities, arousal level, perceptual set, past reinforcement).
2. Retention, including symbolic coding, cognitive organization, symbolic rehearsal,
motor rehearsal)
3. Motor Reproduction, including physical capabilities, self-observation of
reproduction, accuracy of feedback, and
4. Motivation, including external, vicarious and self reinforcement.
Because it encompasses attention, memory and motivation, social learning theory
spans both cognitive and behavioral frameworks. Bandura's theory improves upon the
strictly behavioral interpretation of modeling provided by Miller & Dollard
1941). Bandura’s work is related to the theories of Vygotsky and Lave which also
emphasize the central role of social learning.
| MUHAMAD ILI NURHAYAT BIN HASSAN 87
http://prosacksblog.blogspot.com [ ]
Scope and application
In the learning process of Quadratic Expression and Equation, social learning
theory can be apply extensively by converting the topic of learning into labels or images
results which is better retention than simply explaining by words. By using technology
for example GeoGebra, it can attract student’s attention to focus on the technology
tools, and by using this software, it can help the student’s understanding by visualizing
what we want to explain about quadratic. At the same time, polish their understanding
by including some example about the application of quadratic in the real life since it can
adopt a modeled behavior if it results in outcomes they value and if the model is similar
to the observer and has admired status and the behavior has functional value.
Example
Below are the example of how to use GeoGebra in introducing the quadratic equation
and relate it to the real life.
Water Fountain and the Parabola
Embed a water fountain picture under the x-y system. Give some random values to a, b,
c, and futher graph the quadratic function water(x)=ax2+bx+c. Manipulate a, b, c so that
the graph of water(x) fits the water stream. If necessary, change the increment step for
a, b, c for accurate fitting. Explain what is the actually the relation of a, b and c to the
graph and correlate it.
| MUHAMAD ILI NURHAYAT BIN HASSAN 88
http://prosacksblog.blogspot.com [ ]
After fitting the curve, place a point on the graph and draw a line tangent to the curve at
P. Drag P along the curve and observe the changes in the slope of the tangent
line. What does the slope of that line mean in a physical sense? Also, please try to
explain why the water stream behaves like a parabola? Does it look like that same on
the Moon? How do you interpret the horizontal displacement of the curve?
| MUHAMAD ILI NURHAYAT BIN HASSAN 89
http://prosacksblog.blogspot.com [ ]
INFORMATION PROCESSING
Information processing includes theories that focus on the structure and function
of mental processing. They focus on these structures and functions within specific
contexts and environments. In other words, information processing theories focus on
how people pay attention to events occurring within the environment, how they encode
that information by relating it to knowledge currently stored in memory, how the new
information is stored and finally, how that information is later retrieved when needed.
Human Information Processing can metaphorically be compared to computer
processing. In a computer, information is entered using an input device such as a
keyboard or a scanner. In humans, these input devices could be compared to the ears,
eyes and all other senses. The computer takes the inputed information, organizes it in
specific locations and saves it until further use. The human mind does this as well when
it makes connections and organizes information so that it can be recalled later when
needed.
The working memory or short term memory can be compared with a computers
Central Processing Unit. In a computer, the CPU acts as the brain of the computer.
Working memory is the area where we think about the information presented to us and
process it in specific ways. Once the information has been processed and rehearsed, it
then moves on to long term memory. In a computer, information is stored on hard
drives, memory sticks and CDs. Computers can demonstrate the information they have
through displays on the screen or on printed paper. Humans demonstrate their
knowledge by acting in everyday life; walking, talking and doing
| MUHAMAD ILI NURHAYAT BIN HASSAN 90
http://prosacksblog.blogspot.com [ ]
Scope and application
Principal Example
1. Gain the students' attention
Use cues to signal when you are ready to
begin.
Move around the room and use voice
inflections.
2. Bring to mind relevant prior knowledge
Review previous day's lesson.
Have a discussion about previously covered
content.
3. Point out important information Provide handouts.
Write on the board or use transparencies.
4. Present information in an organized manner
Show a logical sequence to concepts and
skills.
Go from simple to complex when presenting
new material.
5. Show students how to organize (chunk) related information
Present information in categories.
Teach inductive reasoning
| MUHAMAD ILI NURHAYAT BIN HASSAN 91
http://prosacksblog.blogspot.com [ ]
6. Provide opportunities for students to elaborate on new information
Connect new information to something already
known.
Look for similarities and differences among
concepts.
7. Show students how to use coding when memorizing lists
Make up silly sentence with first letter of each
word in the list.
Use mental imagery techniques such as the
keyword method.
8. Provide for repetition of learning
State important principles several times in
different ways during presentation of
information (STM).
Have items on each day's lesson from previous
lesson (LTM).
Schedule presiodic reviews of previously
learned concepts and skills (LTM).
Give exercise
9. Provide opportunities for overlearning of fundamental concepts and skills
Use daily drills for arithmetic facts.
Play form of trivial pursuit with content related
to class.
| MUHAMAD ILI NURHAYAT BIN HASSAN 92
http://prosacksblog.blogspot.com [ ]
MEANINGFUL LEARNING
Ausubel's theory is concerned with how individuals learn large amounts of
meaningful material from verbal/textual presentations in a school setting. According
to Ausubel, learning is based upon the kinds of superordinate, representational, and
combinatorial processes that occur during the reception of information. A primary
process in learning is subsumption in which new material is related to relevant ideas in
the existing cognitive structure on a substantive, non-verbatim basis. Cognitive
structures represent the residue of all learning experiences; forgetting occurs because
certain details get integrated and lose their individual identity.
A major instructional mechanism proposed by Ausubel is the use of advance
organizers. Ausubel emphasizes that advance organizers are different from overviews
and summaries which simply emphasize key ideas and are presented at the same level
of abstraction and generality as the rest of the material. Organizers act as a subsuming
bridge between new learning material and existing related ideas.
Scope and Application
Basically this theory stress in creating a meaningful verbal learning and prevent
rote learning among students. From what we can see in our syllabus, basically our
syllabus are tendency to support students in rote learning since it just only introduce by
just a basic and simple exercise and example with almost the same way of problem and
solution. So, how about revolution a little bit approaches in our teaching by:
| MUHAMAD ILI NURHAYAT BIN HASSAN 93
http://prosacksblog.blogspot.com [ ]
1. Derivative Subsumption
- Is a situation in which the new information that an individual learns is an
instance or example of a concept that has already been learned.
- Introduce the previous topic that had been learn in the previous that related to
the topic such as “Algebraic Expression”, “Algebraic Formulae”, “Linear
Equation”, and also “Linear Inequalities” which had been taught in form 1, 2 and 3
as a briefing.
2. Correlative Subsumption
- Is a situation in which the new material that the individual learns is an
elaboration or extension or alteration of what has already been learned.
- Correlate the previous topic that had been learn into the quadratic such as
quadratic expression is ax² + bx + c meanwhile linear expression is ax + b.
3. Superordinate Learning
- Is a situation in which the new information to be learned is a concept that
relates known examples of a concept. In this case an individual is able to
provide examples of a concept but does not know the concept itself.
- Now, the students may be able to correlate what is the meaning of a, b and c
in the quadratic expression ax² + bx + c after refer to the function of a and b in
ax + b of linear expression.
4. Combinatorial Learning
- It is describes as a process by which the new idea is derived from another
idea that is neither higher nor lower in the hierarch which is similar to learning
by analogy.
| MUHAMAD ILI NURHAYAT BIN HASSAN 94
http://prosacksblog.blogspot.com [ ]
- Students now understanding deeply about quadratic expression and all the
basic concept that related to the quadratic and can correlate it to the real life.
GENETIC EPISTEMOLOGY
Genetic epistemology is a study of the origins of knowledge (epistemology),
which was established by Jean Piaget.
The goal of genetic epistemology is to link the validity of knowledge to the model
of its construction. In other words, it shows that the method in which the knowledge was
obtained/created affects the validity of that knowledge. For example, our direct
experience with gravity makes our knowledge of it more valid than our indirect
experience with black holes or in Quadratic is how can an unknown be connected to an
equation.
Genetic epistemology also explains the process of how the process of
intelligence growth:-
1. Assimilation
o which occurs when the perception of a new event or object occurs to
the learner in an existing schema and is usually used in the context of
self motivation.
2. Accommodation
o one accommodates the experiences according to the outcome of the
tasks.
3. Equilibration
o encompasses both assimilation and accommodation as the learner
changes their way of thinking in order to arrive at a correct or different
answer. This is the upper level of development.
| MUHAMAD ILI NURHAYAT BIN HASSAN 95
http://prosacksblog.blogspot.com [ ]
Scope and Application
By using this approach, teachers can guide the students by following the process of
intelligence growth which is assimilation, accommodation and equilibration.
1. Assimilation
- Using the environment which relate to the use of quadratic such as the Paris
Eiffel Tower or just a Penang Bridge or just as simple U bridge as a main
focus of study.
2. Accommodation
- Correlate the bridge to the quadratic such as in the picture below and explain
what the meaning of quadratic and it’s concept after that to polish student’s
thinking.
| MUHAMAD ILI NURHAYAT BIN HASSAN 96
http://prosacksblog.blogspot.com [ ]
3. Equilibration
- Give the students chance to think and regenerate their own knowledge and
ask them what they had understands from the learning. Guide them to make
sure they got the right information
SOCIOCULTURAL THEORY
Current conceptualizations of sociocultural theory draw heavily on the work of
Vygotsky (1986). This sociocultural perspective has profound implications for teaching,
schooling, and education. A key feature of this emergent view of human development is
that higher order functions develop out of social interaction.
| MUHAMAD ILI NURHAYAT BIN HASSAN 97
http://prosacksblog.blogspot.com [ ]
Vygotsky’s theory lies the understanding of human cognition and learning as
social and cultural rather than individual phenomena. The sociocultural approach are
items in the culture such as computers, books, and traditions that teach children about
the expectations of the group. By participating in the cultural events and using the tools
of the society, human learns what is important in his culture.
This approach rely on three aspects since understanding of human cognition and
learning as social and cultural rather than individual phenomena is what it’s stress on:
1. Zone of proximal development (ZPD)
- Which explain about the distance between what a person can do with and
without help
2. Symbolic tools
- Intellectual tools such as language which is the basic elements of
socialization.
3. Scaffolding
- Metaphor to describe and explain the role of teachers or more knowledgeable
peers in guiding student’s learning and development.
Scope and Application:
This approach basically stress on the sociocultural theory of learning process
which means the learning process happened through communication whether between
the students, between the students and the teachers or between the environments.
| MUHAMAD ILI NURHAYAT BIN HASSAN 98
http://prosacksblog.blogspot.com [ ]
In teaching and learning process of quadratic expression and equation, the study
can continue by:
1. Using the scaffolding.
- Arrange the lesson to become more systematic and effective by doing a
lesson plan that enhances understanding in the important concept of
quadratic
2. Activate the Zone of Proximal Development(ZPD)
- Arrange some group work containing a various level of intelligent students in
each group where the students can depend on each other in doing the
learning process.
3. Naturalized the students using the symbolic tools
- Courage the students to participate actively in group work by giving the
variety of Quadratic problems and question to make them discuss between
each other.
- Give them chance to share what they had understand to the others by giving
them group presentation to make them become more reliable and confident in
learning something new and challenging.
Example
Example of the simple group work task:
| MUHAMAD ILI NURHAYAT BIN HASSAN 99
http://prosacksblog.blogspot.com [ ]
Give them time to answer the question and ask them to prepare an explanation of the
answer to share with all the class.
DISCOVERY LEARNING
Discovery learning is an inquiry-based, constructivist learning theory that takes
place in problem solving situations where the learner draws on his or her own past
experience and existing knowledge to discover facts and relationships and new truths to
be learned.
| MUHAMAD ILI NURHAYAT BIN HASSAN 100
http://prosacksblog.blogspot.com [ ]
Students interact with the world by exploring and manipulating objects, wrestling
with questions and controversies, or performing experiments. As a result, students may
be more likely to remember concepts and knowledge discovered on their own. Models
that are based upon discovery learning model include: guided discovery, problem-based
learning, simulation-based learning, case-based learning, incidental learning, among
others.
Proponents of this theory believe that discovery learning has many advantages,
including:
encourages active engagement
promotes motivation
promotes autonomy, responsibility, independence
the development of creativity and problem solving skills.
a tailored learning experience
Scope and Application:
GUIDED DISCOVERY LEARNING
| MUHAMAD ILI NURHAYAT BIN HASSAN 101
http://prosacksblog.blogspot.com [ ]
Guided discovery learning is a constructivist instructional design model that
combines principles from discovery learning and sometimes radical constructivism with
principles from cognitivist instructional design theory.
In here, teachers just play the role by giving the hint about the topic, giving
something that interesting and attract the students to get interest to learn about the
topic. The teacher also give some guided about the topic by giving reference that can
be use as a guide so that the students did not lost.
Example
The first page of the chapter Quadratics Expression and Equation in Mathematics
KBSM for 4 is one of the best reference to attract students’ interests, provided guided
links about the topic and the focus of the topic which can be seen as guided tool to
support the guided discovery learning.(see picture at the next pages)
| MUHAMAD ILI NURHAYAT BIN HASSAN 102
Information related to real life
Picture to attract interest
http://prosacksblog.blogspot.com [ ]
PROBLEM BASED LEARNING (PBL)
| MUHAMAD ILI NURHAYAT BIN HASSAN 103
http://prosacksblog.blogspot.com [ ]
Problem-based learning (PBL) is a student-centered instructional strategy in
which students collaboratively solve problems and reflect on their experiences.
Characteristics of PBL are:
Learning is driven by challenging, open-ended, ill-defined and ill-structured
problems.
Students generally work in collaborative groups.
Teachers take on the role as "facilitators" of learning.
In Quadratic Expression and Equation topic, this technique can be applied by
giving the students variety of question and problems from the basic concept until the
application and advance part of quadratic. This will cause them to find the answer and
at the same time, learning the topic independently. The teachers just be a reference or
a guidance when it is in need only and all the students all the learning process.
CASE-BASED LEARNING
| MUHAMAD ILI NURHAYAT BIN HASSAN 104
http://prosacksblog.blogspot.com [ ]
Case-based learning (CBL) is an instructional design model that is a variant
of project-oriented learning. It is popular in business and law schools. CBL in a narrow
sense is quite similar to to problem-based learning, but it may also be more open ended
as in our definition of project-based learning.
Case-based learning (CBL) features a learner-centered, collaboration and
cooperation between the participants, discussion of specific situations, typically real-
world examples and also questions with no single right answer.
The process occurs by students engaged with the characters and circumstances
of the story, identifies problems as they perceive it, connect the meaning of the story to
their own lives, bring their own background knowledge and principles, raise points and
questions, and defend their positions and formulate strategies to analyze the data and
generate possible solutions.
Meanwhile, the teachers functioning as facilitators which encourages exploration
of the case and consideration of the characters' actions in light of their own decisions.
The learning process in quadratic occurs by giving the complex problems written
to stimulate classroom discussion and collaborative analysis which involves the
interactive, student-centered exploration of realistic and specific situations.
| MUHAMAD ILI NURHAYAT BIN HASSAN 105
http://prosacksblog.blogspot.com [ ]
INCIDENTAL LEARNING
Incidental learning is unintentional or unplanned learning that results from other
activities. It occurs often in the workplace and when using computers, in the process of
completing tasks. It happens in many ways: through observation, repetition, social
interaction, and problem solving by watching or talking to colleagues or experts about
tasks from mistakes, assumptions, beliefs, and attributions or from being forced to
accept or adapt to situations. This "natural" way of learning has characteristics of what
is considered most effective in formal learning situations: it is situated, contextual, and
social.
In learning the quadratic expression and equation, teachers can also included
pop quiz, interactive exercise or maybe use the cognitive tools such as computer games
since nowadays, we can get so many types of educational games by the tip of our
finger. Below are some example of interactive games that can be used as incidental
learning process in teaching quadratic expression and equation.
Example
| MUHAMAD ILI NURHAYAT BIN HASSAN 106
http://prosacksblog.blogspot.com [ ]
1. Quadratics Equation Matching Games
| MUHAMAD ILI NURHAYAT BIN HASSAN 107
http://prosacksblog.blogspot.com [ ]
2. Bug Match Games
3. Quadratic Equation Hang Man
| MUHAMAD ILI NURHAYAT BIN HASSAN 108
http://prosacksblog.blogspot.com [ ]
4. Study Stack
| MUHAMAD ILI NURHAYAT BIN HASSAN 109
http://prosacksblog.blogspot.com [ ]
REFLECTION ABOUT THIS ASSIGNMENT
I look upon this assignment as opportunities. It will give me a chance to talk
about the Quadratic Expression and Equation that I had learn in my secondary school
and still learning till now, which is an excellent way to enhance my knowledge. My
efforts will go more smoothly and be more successful if my approach these assignments
the same way I should approach doing mathematics; methodically, and with a spirit of
play and discovery. The information which follows is intended to make assignments
more manageable for me, and to give me an edge in doing the best papers possible.
When doing this assignment, I concentrate first on the topic itself and save
formatting and proofing till last – but then be thorough. In an academic paper, searching
for what I want to say can take time, and then saying it clearly can take several
attempts. The biggest part of my grade comes from organizing my ideas and presenting
them clearly. The perfect grade is apt to go to that paper which is free of glaring spelling
and grammar errors, is in the proper format, and presents its material clearly, supporting
it well from the sources. Other than that, this assignment also force me to look back all
the theory of education that I had learn in the previous as a reference and basic material
to use in this task.
I can foresee that objective of all of this assignment is to promote grow through
self-awareness and self-research. Besides, this assignment is designed to foster
communication among the students with students and students with lecturer. By
sharing information, we are perceptiveness, and insightfulness as well as the efforts that
have been made to improve, learn, and grow. Many misconceptions are avoided when
the student lecturer communicates this information to each others.
| MUHAMAD ILI NURHAYAT BIN HASSAN 110
http://prosacksblog.blogspot.com [ ]
This assignment helps me to understand certain concepts of quadratics and help
me to build certain skills. I also try to understand the process of the specific problem.
Classify problems in the assignment by problem type. I try to figure out why I missed the
ones my friends did instead of just working toward the answer
It helps me to build on a concept or skill I did not posses before. Before doing this
assignment, I look back over the assignment questions many times and try to explain to
myself what the assignment was about, what each kind of problem was asking, how I
got the answers and what the answers tell me. This process will help me understand the
material and will help me discover what I don’t understand.
I noticed three things that I have never had to do for a math assignment before.
These goals may take me a while to get used to and/or get good at. They include
defining quadratics basic element, quadratics raw materials, basic operation and its
manipulation and so on. When I was asking the question, I can understand better about
the questions demand. We were never allowed to stray too far off topic. As a result of
this I am a little apprehensive of this assignment but hope that I will learn quickly. As a
saying goes “Be patient with the process. You are right in noting wholesale changes in
expectations. Give yourself time to adjust.”
In the process of completing this assignment, rarely before have I ever written
narrations of my discoveries and my teaching techniques. I believe that if I am capable
of explaining my actions in writing, then I would definitely be able to understand what I
was doing since writing and explaining could be considered as my weaknesses.
| MUHAMAD ILI NURHAYAT BIN HASSAN 111
http://prosacksblog.blogspot.com [ ]
But even more important is the question of what skills will really prepare today’s
students for the future. Surely the next decades will be ones of rapid change where old
answers don’t always work, where employers demand communication and human
relations skills as well as the ability to think incisively and imagine creative solutions to
unforeseen problems. Many of today’s computer applications offer poor preparation for
such abilities.
The future also will favor those who have learned how to learn, who can respond
flexibly and creatively to challenges and master new skills. At the moment, the computer
is a shallow and pedantic companion for such a journey. We should think long and
carefully about whether our purpose is to be trendy or to prepare students to be
intelligent, reasoning human beings whose skills extend far beyond droid-like button
clicking.
As a result, I can say that the approach in this assignment offer possibilities for a
real interaction to take place between the students. This might provide information
which normally would not be available and that can be used to help students develop
our mathematical understanding. I hope that my analysis will contribute to my
understanding of the characteristics of this approach and its implications for
mathematics educations.
| MUHAMAD ILI NURHAYAT BIN HASSAN 112
http://prosacksblog.blogspot.com [ ]
I now believe that straight teaching Quadratics should be replaced by what is
appropriate to student’s settings. I can no longer ignore the inefficiency of the traditional
teaching method. I feel that communication between the mathematics and mathematics
education communities is vital to enriching the teaching of Algebra. In fact research
results do not always have an instantaneous impact on the teaching and learning of
Algebra especially in the quadratic topics. Rather, they show ways to understand
student learning that can later be used as a means of improving instruction.
| MUHAMAD ILI NURHAYAT BIN HASSAN 113
http://prosacksblog.blogspot.com [ ]
REFERENCE
An Approach to Curriculum Design, (2010, April), Institute of Education University of
London (IOE), http://www.lkl.ac.uk/ltu/files/publications/Laurillard-
An_Approach_to_Curriculum_Design-WIP.pdf
Brooks, J.G., & Brooks, M.G. (1999). "In search of understanding: The case for
constructivist classrooms." Alexandria, VA: Association for Supervision and
Curriculum Development.
Chee, S.N, Hamzah Sahrom, & Thiam C.L (2005) “Mathematic KBSM Form .4” KDEB
ANZAGAIN .
Constructivism and the 5Es, Miami Museum of Science, 2001
http://www.miamisci.org/ph/lpintro5e.html
ERIC Clearinghouse for Science Mathematics and Environmental Education, 2003
(ERIC). http://www.ericdigests.org/2004-3/views.html
Ernest, P. (1996). Varieties of constructivism: A framework for Comparison. In L.P.
Steffe, P. Nesher, P. Cobb, G.A Goldin, and B. Greer (Eds.), "Theories of
mathematical learning." Nahwah, NJ: Lawrence Erlbaum.
Good, R.G., Wandersee, J.H., & St. Julien, J. (1993). Cautionary notes on the appeal of
the new "ism" (constructivism) in science education. In K. Tobin (Ed.), "The
practice of constructivism in science education." Hillsdale, NJ: Lawrence
Erlbaum.
| MUHAMAD ILI NURHAYAT BIN HASSAN 114
http://prosacksblog.blogspot.com [ ]
Huraian Sukatan Pelajaran Matematik Tingkatan 4 KBSM, 2006, Pusat Perkembangan
Kurikulum Kementerian Pendidikan Malaysia(PPK),
http://www.scribd.com/doc/493831/Matematik-Tingkatan-4
Inquiry Education Information for the Muzeum Education, 1996 (Institutet for Inquiry).
http://www.exploratorium.edu/IFI/resources/constructivistlearning.html
Integrated Approaches to Teaching and Learning in the Senior Secondary School,
2008, Western Australian Certificate of Education(WACE),
http://www.curriculum.wa.edu.au/internet/_Documents/General/Guidelines+for+in
tegrated+approaches+to+teaching+and+learning+in+senior+secondary+school+
V5.pdf
Lakenheath, Can the Quadratic Formula Kiss-off Completing the Square, Pat Ballew
United Kingdom, http://pballew.net/quadform.doc
Mathematics Education: Constructivism in the Classroom,1994-2010. (The Math Forum)
http://mathforum.org/mathed/constructivism.html
Packer, J. & Bain, J. (1978). Cognitive style and student-teachers compatibility. Journal
of Educational Psychology, 70, 864 - 871.
Von Glaserfeld, E. (1993). Questions and answers about radical constructivism. In K.
Tobin (Ed.), "The practice of constructivism in science education." Hillsdale, NJ:
Lawrence Erlbaum.
| MUHAMAD ILI NURHAYAT BIN HASSAN 115