The the length of a side of a

The
concept of a radical expression, or more specifically square and cube roots has
recorded history thought to date back to 1800 B.C in Egypt and Babylonia. The
idea of square and cube roots seems to have come from geometric
representations, specifically finding the length of a side of a square or cube
with a known area (Eves, 1990; Guilberg, 1997; Hopper, 1948; Katz, 1993).

According
to (Crisan 2012, Needham 1997), the concept of a radical carries with it many
subtleties that can cause confusion to even those well versed in Mathematics.
Radical expression can’t be easily understood; it requires attachment of
physical meaning to an irrational number. (Sirotic and Zazkis 2007) states that
studies shows that the majority of secondary mathematics students are unable to
interpret radicals geometrically, demonstrating that their number sense is very
weak.

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 According to, (Gomez & Buhlea 2009; Ozkan,
2011; Ozkan & Ozkan, 2012; Sirotic & Zaskis, 2007) Secondary
mathematics students often lack a geometric understanding of radical numbers,
as well as, the prerequisite algebraic skills required to concretely grasp the
concept of a radical expression. Like on the discussion of Gough (2007), there
is an argument that the vocabulary terms used when teaching square roots and
square numbers which can be confusing and detrimental to student’s
understanding. Since square root and square number are sounding similar phrases
that evoke images from our everyday English language use of those words then,
it could cause them to be confused. Square root and square number has opposite
process. Square number deals with squaring a number or multiplying it to itself
while, square root is the process of backward squaring or finding what has been
squared in order to know the answer. Therefore; if students were confused of
the English term itself they might think that both of it has the same process
or else they might be mistaken of square number as square root and square root
as square number. Park (2013) found that the confusion between the two concepts
was exacerbated by the similarity of terms.

Crisan
(2012), expresses her frustrations with the ambiguity of the square root symbol
and the vocabulary used to read the symbol, Ball (2008) and Roach et al (2004)
expresses similar frustration regarding the ambiguity for the vocabulary for
square root symbol and report student misconceptions regarding the principal
square root similar to those reported by Crisan (2008).

Many
students view the concept of a radical very abstractly and do not possess a
concrete, geometric understanding of what a radical actually means (Sirotic
& Zackis, 2007). Students are frequently taught to memorize properties and
shortcuts to work with radicals algebraically. Learning mathematics in this
manner can actually hinder a student’s number sense and aid the development of
misconceptions about radicals and related mathematical concepts such as
exponents, irrationality, and complex numbers (Gomez & Buhlea 2009; Ozkan,
2011; Ozkan & Ozkan, 2012). 

The
following study was conducted one year after Ozkan’s 2011 study. Ozkan and
Ozkan (2012) conducted a nearly identical study to OZkan (2011) the following
year with grade ten students from Istanbul regarding misconceptions with
radicals. Ozkan and Ozkan (2012) report similar student misconceptions and use
a nearly identical assessment as Ozkan (2011). Ozkan and Ozkan (2012) suggest
that teachers must first correct the misconceptions students have with the
prerequisite topics of operations, absolute value, and factorizations before
moving on the topic of radicals. To help resolve the misconceptions regarding
radical expressions which were prevalent in this study, Ozkan and Ozkan
suggests to spend more time teaching the concept of a radical, allow for students
to discover the algebraic rules of radicals rather than presenting properties
formally, and to use appropriate materials and technology to allow students to
develop the concept of a radical in a concrete way.

Gomez
and Buhlea (2009) state that the radical sign holds different interpretations
in algebra and arithmetic. The topic of radicals seems to be widely
misunderstood by many mathematics students and even some mathematics educators
(Ball, 2008; Crisan, 2008; Crisan, 2012; Gomez & Buhlea, 2009).  The concept of a radical has developed from
the ancient times of the Babylonians and Egyptians in 1800 B.C. and has picked
up a few subtle nuances along the way. For example, in 1800 B.C. concepts such
as negative numbers, irrationality, and complex numbers were not yet in
existence, so there was no need to worry about defining terms such as principal
square root or the rule. In the present age of high stakes testing and a wide
ranging mathematics curriculum, students are expected to grasp concepts such as
irrationality and complex numbers, along with several other terms and
subtleties which are inherent with radical expressions in just a few short
weeks.

Certain
concepts regarding radicals have taken almost 4,000 years to develop to where
they are today, so it is reasonable to expect that many students will
experience difficulty in grasping concepts such as irrationality in such a
short period of time. A possible reason for student misconceptions regarding
the topic of radicals may be the ambiguous and subtle nature of the radical
sign (Crisan, 2008; Crisan, 2012; Gomez & Buhlea, 2009).  Research studies suggest that mathematics
students commonly misuse properties when simplifying radical expressions,
especially properties which involve an absolute value (Huntley & Davis,
2008; Ozkan, 2011; Ozkan & Ozkan, 2012).

 Studies also suggest that students often
display misconceptions with geometric representations of radical expressions,
especially those which are irrational (Huntley & Davis, 2008; Sirotic &
Zazkis, 2007). Ozkan and Ozkan suggest to spend more time teaching the concept
of a radical, allow for students to discover the algebraic rules of radicals
rather than presenting properties formally, and to use appropriate materials
and technology to allow students to develop the concept of a radical in a
concrete way.

Dugopolski
(2009) and Martin-Gay (2001), introduce rules for exponents first, and then use
rules for exponents to define radicals and derive the product and quotient
rules for radicals.  From here,
Dugopolski and Martin-Gay proceed with an introduction to radicals which is
similar to the first two textbooks in this review.  Dugopolski and Martin-Gay first provide
examples of how to use the radical product and quotient rules to simplify
radicals, followed by a demonstration of how the distributive law can be used
to combine like radical terms. this review used an algebraic approach to
introduce the concept of radicals and there was no mention of a geometric
interpretation for radicals (Blitzer, 2010; Dugopolski, 2009; Johnson &
Steffensen 1994, Martin-Gay, 2001).  It
appears that many educators also rely on an algebraic approach when introducing
radicals to students (Ozkan & Ozkan, 2012). 
For many mathematics educators, radicals are one of the hardest topics
to teach to students because there are many subtleties which students, and even
some educators, have trouble understanding (Crisan, 2008; Crisan, 2012).  In the following section, a variety of
strategies for teaching the concept of radicals are explored.

There
are two overlapping styles of articles written regarding square numbers and
square roots with respect to mathematics education. One style consists of
articles published in professional journals written primarily for teachers,
that give suggestions or instruction on how to teach students square roots or
exponents more effectively (Boomer, 1969; Edmonds, 1970; Goodman & Bernard,
1979; Kamins, 1969; Thompson, 1992). These articles often focus on an intuitive
approach to exponents and square roots or give tips and tricks for teaching.
The majority of these articles emphasise square roots rather than square
numbers, or focus on exponents in general, often with the specific purpose of
extensions for explaining rational exponents or negative exponents. Some
articles discuss only the zero exponent, possibly the most confusing aspect of
exponents for students (Astin, 1984; Bernard, 1982). There also exists a
selection of articles from a time when scientific calculators were not widely
available or used, that give various algorithms for calculating the square root
(Chow & Lin, 1981; Edge, 1979). From these articles, it is clear that the
nature of exponential growth and the relationship between whole number
exponents and rational exponents are the aspects of exponents that teachers are
most concerned with when teaching. 

The
second style of work comprises scholarly papers that research students’
learning and understanding. However, the vast majority of these articles focus
on much higher-level mathematical topics than square numbers or square roots
that cause more obvious problems for students. These topics include general
exponents involving negative, rational or the zero exponent (Pitta-Pantazi,
Christou & Zachariades, 2007; Sastre & Mullet, 1998; Vinner, 1977),
exponential growth (Brown, 2005; Ebersbach & Wilkening, 2007), and
exponential functions (Confrey & Smith, 1994; Confrey & Smith, 1995).
Of particular interest is Sastre and Mullet’s (1998) work that shows that when
estimating the magnitude of an exponential expression, students used
mathematical models that included both the base of the expression and the
exponent. 

Although
Gough lamented the lack of a simple ‘forwards’ concept and definition for the
square root, the lack a simple precise definition given to students for square
numbers may also be lamented. Pólya points out “Technical terms in mathematics
are of two kinds. Some are accepted as primitive terms and are not defined”
(1945, p. 85). Tall and Vinner (1981) also suggest that a lack of formal
definition is acceptable and natural. The role of definitions has not been
studied with particular respect to square numbers or square roots, but it is
clear that the similarity of the terms square number and square root may be an
obstacle for students. There’s a suggestion that the lack of clear and concise
definitions of square numbers and square roots given to students, will also be
an obstacle for students to overcome when attempting to solve square number
problems. Zazkis and Leikin (2008) point out that “that mathematical concepts
rather often have several equivalent definitions and the choice of a definition
useful for a particular task is determined by the context” (p. 135). However,
in high school, it is often pragmatic to give examples of square numbers in
lieu of a precise definition, and so many students may be left without a clear
definition for square numbers. Square numbers are not regularly spaced along
the number line; the difference between two subsequent square numbers gets
larger as the square numbers themselves get larger.

Students
often have small square numbers memorized, and may not be attentive to the
underlying organization of the distribution throughout the natural numbers. The
role of representation dealing with numbers, rather than algebraic or geometric
or other mathematical representations, has been well documented with respect to
prime numbers (Zazkis & Liljedahl, 2004; Zazkis, 2005), irrational numbers,
(Zazkis & Sirotic, 2004; Zazkis, 2005), divisibility and prime
factorization (Zazkis & Campbell, 1996; Zazkis, 2008).

The
role of definitions has not been studied with particular respect to square
numbers or square roots, but it is clear that the similarity of the terms
square number and square root may be an obstacle for students. Erica Hiebert
suggest that the lack of clear and concise definitions of square numbers and
square roots given to students, will also be an obstacle for students to
overcome when attempting to solve square number problems.  

Geometry
students are typically exposed to working with radicals first using the
Pythagorean theorem, which we use to introduce the topic, and second using
similar triangles, in particular working with 30-60-90º and 45º right
triangles. Advanced algebra students initially need a review of simplifying
radicals when they are exposed to working with the quadratic formula, and they
definitely need one by the time they get to exponential models and logarithms.
Even trigonometry students may benefit from a review as they learn the various
exact values of the sine and cosine functions. That is how much important
radicals is in human’s day to day life. Even radicals were helpful to people
due to its application in real life still, it carries confusion since, it
concerns many topics and gaining knowledge.

In
reality you can’t easily understand this topic without even understanding the
basic operations of integer, without understanding reciprocals, without knowing
about fractions and many more. In addition to that, students often
misunderstood or encountered misconception in English terms. When students
encountered square number and square root then, they might also think that both
of it has the same process in order to get the correct answer since, they’re
having almost the same phrase or else they might be confused and have each
other meaning for the wrong term which will lead them to the wrong answer. When
a teacher asked a student to search for the meaning of radicals he/she might
also get the wrong answer since, there’s also a radical term in science.

 Knowing the fact that most Filipino’s used to
know their own language rather than English then, they might have a hard time
understanding radicals. There are lots of rules to consider in radicals that
might also hinder the students to learn or, hinder the students to understand
it due to the confusion in the topic.

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