Theresearchers gathered the following related literature and studies with regardsto the performance of the students in solving integers. The collectedinformation were relevant, and have been analyzed by the researchers for betterunderstanding to those who will try to read or study the research. Brief History of Integers Integers is one of Mathematical unit,particularly in Algebra.
They are set of natural numbers and is not afraction which can be either positive (+), negative (-), or zero (0). Harris(2010) stated in her research that integers are early introduced and exploredduring Grades 3-5 through activities which extend the number line to the leftpast zero, exploring real world situations of temperature or bank accounts,where negative numbers are needed, and through literature connections thatintroduce the larger world of numbers to upper elementary students (p.18). InGrade 7, integers is introduced directly and taught in the succeeding years.
Since integers are one of thesignificant units of Mathematics for centuries, they have contributed a lot thedevelopment of different scientific and mathematical concepts. It is believedthat integers were one of the first numeric systems used by our ancestors forcounting but historical researches remarked it is not true, for fingers andobjects like pebbles and leaves served as an ancient calculator. The invention of integers can betraced back to historical Babylonian period about 4000 years ago. It was thengiven by the Babylonians to the Greeks, and the latter modified the system ofAlgebra and Integers.
In 250 A.D., “Diophantus of Alexandria”, one of theancient Greek philosophers, presented the concept of solving system or set ofequations using integers. This concept was applicable for finding solutions forboth single equation and set of equations involving one or more than oneunknown variables.
Yet the computed result was always in the form of wholenumbers. About 1700’s, Europeans made newimprovements in the field of Mathematics using integers. In 1795, MathematicianCarl Friedrich Gauss came out with a new concepts in his book Arithmetical Disquisitions. Years later,Scientist Albert Einstein used Gauss’ concepts in developing his famous Theoryof Relativity.
Meanwhile, Mathematician Leopold Kronecker proposed new conceptsabout integers which improved the previous theories about it. Integers Today Nowadays, integers have become part ofhumanity’s daily life. These set of natural numbers can be seen in any lifesituation: reading the thermometers and the weather’s temperature; countinggains and losses, debits and credits; measuring above and below sea-levels;tallying sport scoring (e.g.
hockey and football); determining history timeline(A.D. and B.C); and electroplating using charges are just some of the basicapplication of integers in daily life. In the field of science, integers aremostly associated in scientific formulas, particularly in chemistry andphysics. Integers are part of calculating speed, velocity and acceleration, andused to determine proper elevation of when and where to land or dock vehicular,vessels and machines. Understanding Integer Operations Integer Operations convey the basicmathematical operation: addition, subtraction, multiplication and division.AbsoluteValue.
Before discussing the integer operations, absolute value is introduced.This is a practice removing any negative sign in front of a number, and tothink of all numbers as positive. It includes a pair of the same number,separated by “?” (e.g. ?-1? = 1: ?8-3? = 5; ?3-8? = 5 wherein: 3-8 = -5, and?-5? = 5). Addition.Integerscan be both positive and negative, and in the rule of adding integers and thesum of the both rules in adding integers acquire the sign of the added biggeramount of number.
Rulenumber 1. Like signs (positive + positive, negative + negative) are addedaccordingly (e.g. +16+(+5) = 21; -16+(-5) = -16-5 = -21). Rule number 2. Likewise,unlike signs (positive + negative, negative + positive) are subtracted (e.g.16+ (-5) = 16-5 = 11; -16+5 = -11) Subtraction.
Unlikeaddition of integers, subtraction of integer is a bit complicated. Rule number 1. Subtracting a positiveinteger from a positive integer is a normal subtraction (e.g. 5-3 = 2)Rule number 2. Subtractinga positive integer from a negative integer starts at the negative number,counting backwards the additional amount subtracted (e.g. -5-3 = 8).
Rule number 3. Subtractinga negative integer from a negative integer turns the minus (-) sign followed bya negative sign into plus (+) sign. Instead of subtracting the negativenumbers, it is added since the operation became plus positive (e.g. -6—(-5) = -6+5 = -1).
Rule number 4. Subtractinga negative integer from a positive integer results changing the minus (-) signfollowed by a negative sign into plus sign. Thus, instead of subtracting anegative, adding takes place like a simple addition problem (e.g.
5-(-3) = 5+3= 8). Multiplication.Multiplying positive and negative integers has far less rules than adding orsubtracting positive and negative integers. Rulenumber 1.
A positive integer multiplied to a positive integer results to apositive product (e.g. 4*3 = 12).
Rulenumber 2. A positive integer multiplied to a negative integer results to anegative product (e.g. 4*-3 = -12) Rulenumber 3. A negative integer multiplied to a negative integer results to apositive product (e.
g. -4*-3 = 12) Division. Dividing positive andnegative numbers follows the same rules in multiplying integers.
Rulenumber 1. A positive integer divided by a positive integer results apositive quotient (e.g. 12/3 = 4). This is the kind of normal division alwaysdone.
Rulenumber 2. A positive integer divided by a negative integer results to anegative quotient (e.g. 12/-3 = -4). Rulenumber 3. A negative integer divided by a negative number results to apositive quotient (e.g. -12/-3 = 4).
Evaluation in Students’ Understanding inIntegers Before introducing integer concepts tostudents in upper elementary grades, it is important to consider thefoundations that have been laid in grades K-2 in order to connect with priorlearning about numbers and operations with whole numbers (Harris, 2010). Theseare done through teaching ordinal and cardinal properties of numbers and servedas a prerequisite for students of Kindergarten in understanding integers(p.16). At the age of 4-7, the children also showed the beginning ability tocombine positive and negative amounts (Davidson 1992 as cited in Harris 2010).Children are typically first introduced to negative integers in upperelementary grades, using two informal semantic models. These models are helpfulbecause they are direct examples of the properties of integers (Schwarz, Kohn,& Resnick, 1993, 1994 as cited in Harris 2010).
In grades 5-8, integers areearly introduced and explored (p.18) and taught accordingly during grades 7 and8 (p.21). However, students struggle with this foundational concept,particularly around negative numbers (Ryan & Williams, 2007 as cited inHarris 2010). Many students enter high school levelwith severe gaps in their concepts and skills in mathematics. One of thesebasic foundational knowledge and skills is the integers, a necessaryprerequisite skill to solve equations. Performing operations on integersinvolves signs of the numbers and the signs of required operation. This makesstudents get confused and struggle when asked to perform operations on integers(Muñoz, 2010).
In the study conducted by Vlasis (2010), he indicated thatstudents’ misconceptions about integers caused difficulties in algebra.Research shows that negative numbers create difficulties for students as theytry to make sense of them (Gallardo, 2002; Gallardo & Romero, 1999; Peled,Mukhopadhyay & Resnick, 1989).In additionto the findings, students’ attitude towards Mathematics including theoperation of integers are also studied.
According to Albayrak (2000); Akgün(2002); Ba?ar, Ünal & ?n (2002); Umay (1996); Yenilmez & Duman(2008): the general opinion concluded was most students exhibit feelings ofanxiety and fear of mathematics courses not only in Turkey, but throughout theworld. This attitude is define by Neale (1969) as “a total measure of liking ordisliking of mathematics, a tendency to engage in or avoid mathematicalactivities, a belief that one is good or bad at mathematics and a belief thatmathematics is useful or useless” (Neale, 1969 as cited in Alkan, Güzel, , 2004). It may be concluded that this is why because students haddifficulties and misconceptions about the subjects of operations, word problemsand producing a model with integers (Melemezo?lu 2005), which is also revealedby Avcu and Durmaz (2011) in their study that students had difficulties aboutordering the integers and doing operations with integers, especially insubtracting integers (Harris 2010). With regards to this, this study seeksnot only to determine the level of performance of the selected participants butalso to help the problem deflate.
In the study conducted by Rubin, Marcelino,Mortel and Lapinid (2014), they found out that 80% of their participantsanswered in the interview that they had difficulty in handling signs ofintegers particularly in subtraction of integers What difficult experiencehave you had in performing operations of integers?. This finding confirmed theforementioned stated difficulties of students in solving integers in the senseof negative operation.