Theconcept of a radical expression, or more specifically square and cube roots hasrecorded history thought to date back to 1800 B.C in Egypt and Babylonia. Theidea of square and cube roots seems to have come from geometricrepresentations, specifically finding the length of a side of a square or cubewith a known area (Eves, 1990; Guilberg, 1997; Hopper, 1948; Katz, 1993).Accordingto (Crisan 2012, Needham 1997), the concept of a radical carries with it manysubtleties that can cause confusion to even those well versed in Mathematics.Radical expression can’t be easily understood; it requires attachment ofphysical meaning to an irrational number. (Sirotic and Zazkis 2007) states thatstudies shows that the majority of secondary mathematics students are unable tointerpret radicals geometrically, demonstrating that their number sense is veryweak.
According to, (Gomez & Buhlea 2009; Ozkan,2011; Ozkan & Ozkan, 2012; Sirotic & Zaskis, 2007) Secondarymathematics students often lack a geometric understanding of radical numbers,as well as, the prerequisite algebraic skills required to concretely grasp theconcept of a radical expression. Like on the discussion of Gough (2007), thereis an argument that the vocabulary terms used when teaching square roots andsquare numbers which can be confusing and detrimental to student’sunderstanding. Since square root and square number are sounding similar phrasesthat 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 oppositeprocess. Square number deals with squaring a number or multiplying it to itselfwhile, square root is the process of backward squaring or finding what has beensquared in order to know the answer. Therefore; if students were confused ofthe English term itself they might think that both of it has the same processor else they might be mistaken of square number as square root and square rootas square number.
Park (2013) found that the confusion between the two conceptswas exacerbated by the similarity of terms. Crisan(2012), expresses her frustrations with the ambiguity of the square root symboland the vocabulary used to read the symbol, Ball (2008) and Roach et al (2004)expresses similar frustration regarding the ambiguity for the vocabulary forsquare root symbol and report student misconceptions regarding the principalsquare root similar to those reported by Crisan (2008). Manystudents view the concept of a radical very abstractly and do not possess aconcrete, geometric understanding of what a radical actually means (Sirotic& Zackis, 2007).
Students are frequently taught to memorize properties andshortcuts to work with radicals algebraically. Learning mathematics in thismanner can actually hinder a student’s number sense and aid the development ofmisconceptions about radicals and related mathematical concepts such asexponents, irrationality, and complex numbers (Gomez & Buhlea 2009; Ozkan,2011; Ozkan & Ozkan, 2012). Thefollowing study was conducted one year after Ozkan’s 2011 study. Ozkan andOzkan (2012) conducted a nearly identical study to OZkan (2011) the followingyear with grade ten students from Istanbul regarding misconceptions withradicals. Ozkan and Ozkan (2012) report similar student misconceptions and usea nearly identical assessment as Ozkan (2011). Ozkan and Ozkan (2012) suggestthat teachers must first correct the misconceptions students have with theprerequisite topics of operations, absolute value, and factorizations beforemoving on the topic of radicals. To help resolve the misconceptions regardingradical expressions which were prevalent in this study, Ozkan and Ozkansuggests to spend more time teaching the concept of a radical, allow for studentsto discover the algebraic rules of radicals rather than presenting propertiesformally, and to use appropriate materials and technology to allow students todevelop the concept of a radical in a concrete way. Gomezand Buhlea (2009) state that the radical sign holds different interpretationsin algebra and arithmetic.
The topic of radicals seems to be widelymisunderstood 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 fromthe ancient times of the Babylonians and Egyptians in 1800 B.C. and has pickedup a few subtle nuances along the way. For example, in 1800 B.C.
concepts suchas negative numbers, irrationality, and complex numbers were not yet inexistence, so there was no need to worry about defining terms such as principalsquare root or the rule. In the present age of high stakes testing and a wideranging mathematics curriculum, students are expected to grasp concepts such asirrationality and complex numbers, along with several other terms andsubtleties which are inherent with radical expressions in just a few shortweeks. Certainconcepts regarding radicals have taken almost 4,000 years to develop to wherethey are today, so it is reasonable to expect that many students willexperience difficulty in grasping concepts such as irrationality in such ashort period of time.
A possible reason for student misconceptions regardingthe topic of radicals may be the ambiguous and subtle nature of the radicalsign (Crisan, 2008; Crisan, 2012; Gomez & Buhlea, 2009). Research studies suggest that mathematicsstudents 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 oftendisplay 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 conceptof a radical, allow for students to discover the algebraic rules of radicalsrather than presenting properties formally, and to use appropriate materialsand technology to allow students to develop the concept of a radical in aconcrete way. Dugopolski(2009) and Martin-Gay (2001), introduce rules for exponents first, and then userules for exponents to define radicals and derive the product and quotientrules for radicals.
From here,Dugopolski and Martin-Gay proceed with an introduction to radicals which issimilar to the first two textbooks in this review. Dugopolski and Martin-Gay first provideexamples of how to use the radical product and quotient rules to simplifyradicals, followed by a demonstration of how the distributive law can be usedto combine like radical terms. this review used an algebraic approach tointroduce the concept of radicals and there was no mention of a geometricinterpretation for radicals (Blitzer, 2010; Dugopolski, 2009; Johnson &Steffensen 1994, Martin-Gay, 2001). Itappears that many educators also rely on an algebraic approach when introducingradicals to students (Ozkan & Ozkan, 2012). For many mathematics educators, radicals are one of the hardest topicsto teach to students because there are many subtleties which students, and evensome educators, have trouble understanding (Crisan, 2008; Crisan, 2012).
In the following section, a variety ofstrategies for teaching the concept of radicals are explored.Thereare two overlapping styles of articles written regarding square numbers andsquare roots with respect to mathematics education. One style consists ofarticles published in professional journals written primarily for teachers,that give suggestions or instruction on how to teach students square roots orexponents more effectively (Boomer, 1969; Edmonds, 1970; Goodman & Bernard,1979; Kamins, 1969; Thompson, 1992).
These articles often focus on an intuitiveapproach to exponents and square roots or give tips and tricks for teaching.The majority of these articles emphasise square roots rather than squarenumbers, or focus on exponents in general, often with the specific purpose ofextensions for explaining rational exponents or negative exponents. Somearticles discuss only the zero exponent, possibly the most confusing aspect ofexponents for students (Astin, 1984; Bernard, 1982). There also exists aselection of articles from a time when scientific calculators were not widelyavailable or used, that give various algorithms for calculating the square root(Chow & Lin, 1981; Edge, 1979). From these articles, it is clear that thenature of exponential growth and the relationship between whole numberexponents and rational exponents are the aspects of exponents that teachers aremost concerned with when teaching.
Thesecond style of work comprises scholarly papers that research students’learning and understanding. However, the vast majority of these articles focuson much higher-level mathematical topics than square numbers or square rootsthat cause more obvious problems for students. These topics include generalexponents involving negative, rational or the zero exponent (Pitta-Pantazi,Christou & Zachariades, 2007; Sastre & Mullet, 1998; Vinner, 1977),exponential growth (Brown, 2005; Ebersbach & Wilkening, 2007), andexponential functions (Confrey & Smith, 1994; Confrey & Smith, 1995).Of particular interest is Sastre and Mullet’s (1998) work that shows that whenestimating the magnitude of an exponential expression, students usedmathematical models that included both the base of the expression and theexponent. AlthoughGough lamented the lack of a simple ‘forwards’ concept and definition for thesquare root, the lack a simple precise definition given to students for squarenumbers may also be lamented. Pólya points out “Technical terms in mathematicsare 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 formaldefinition is acceptable and natural. The role of definitions has not beenstudied with particular respect to square numbers or square roots, but it isclear that the similarity of the terms square number and square root may be anobstacle for students. There’s a suggestion that the lack of clear and concisedefinitions of square numbers and square roots given to students, will also bean obstacle for students to overcome when attempting to solve square numberproblems. Zazkis and Leikin (2008) point out that “that mathematical conceptsrather often have several equivalent definitions and the choice of a definitionuseful 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 inlieu of a precise definition, and so many students may be left without a cleardefinition for square numbers. Square numbers are not regularly spaced alongthe number line; the difference between two subsequent square numbers getslarger as the square numbers themselves get larger. Studentsoften have small square numbers memorized, and may not be attentive to theunderlying organization of the distribution throughout the natural numbers.
Therole of representation dealing with numbers, rather than algebraic or geometricor other mathematical representations, has been well documented with respect toprime numbers (Zazkis & Liljedahl, 2004; Zazkis, 2005), irrational numbers,(Zazkis & Sirotic, 2004; Zazkis, 2005), divisibility and primefactorization (Zazkis & Campbell, 1996; Zazkis, 2008). Therole of definitions has not been studied with particular respect to squarenumbers or square roots, but it is clear that the similarity of the termssquare number and square root may be an obstacle for students. Erica Hiebertsuggest that the lack of clear and concise definitions of square numbers andsquare roots given to students, will also be an obstacle for students toovercome when attempting to solve square number problems.
Geometrystudents are typically exposed to working with radicals first using thePythagorean theorem, which we use to introduce the topic, and second usingsimilar triangles, in particular working with 30-60-90º and 45º righttriangles. Advanced algebra students initially need a review of simplifyingradicals when they are exposed to working with the quadratic formula, and theydefinitely need one by the time they get to exponential models and logarithms.Even trigonometry students may benefit from a review as they learn the variousexact values of the sine and cosine functions. That is how much importantradicals is in human’s day to day life. Even radicals were helpful to peopledue to its application in real life still, it carries confusion since, itconcerns many topics and gaining knowledge. Inreality you can’t easily understand this topic without even understanding thebasic operations of integer, without understanding reciprocals, without knowingabout fractions and many more. In addition to that, students oftenmisunderstood or encountered misconception in English terms.
When studentsencountered square number and square root then, they might also think that bothof it has the same process in order to get the correct answer since, they’rehaving almost the same phrase or else they might be confused and have eachother meaning for the wrong term which will lead them to the wrong answer. Whena teacher asked a student to search for the meaning of radicals he/she mightalso get the wrong answer since, there’s also a radical term in science. Knowing the fact that most Filipino’s used toknow their own language rather than English then, they might have a hard timeunderstanding radicals. There are lots of rules to consider in radicals thatmight also hinder the students to learn or, hinder the students to understandit due to the confusion in the topic.