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.

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.