CHAPTER ELEVEN
COMMON ERRORS IN MATHEMATICS
11.1 Objectives
At the end of this chapter, you should be able to:
i. identify and explain all types of commonerrors in mathematics
ii. give at least one example of each type of error
iii. know the Semantic and Syntactic Errors in some concepts
11.2 Introduction
Learning will be facilitated What is stated and proved is unambiguous and transparent. The art of communication, or the transfer of ideas from one mind to another, is a major determinant of the effectiveness of education. Gilbert Highet emphasized the need of clear communication in education, stating, "Let him (teacher) be good at communication, and even if he is a mediocre scholar, he may be an exceptional teacher." In another meaning, communication refers to the relationship between students and teachers. Studies often demonstrate a strong association between communication and good teaching-learning. Teachers of mathematics should avoid semantic and syntactic mistakes that negatively impact teaching-learning in mathematics; hence, each topic has its own technical language and terminology. As a field of study, mathematics appears to be primarily concerned with the precision of its solutions and has little to do with its medium of education. Inekwe (2003) discovered that if a child's language proficiency imposes a linguistic handicap, the expected consequence of what is taught would always fall short of the input. Clear grammatical structure during classroom instruction contributes to mathematical language comprehension. Examining the mathematical response scripts of students at all levels reveals at least eight types of mistakes. These include syntactic, semantic, pragmatic errors (Ahmad, 2010). Some other errors are errors in mechanistic understanding, cultural inhibition, premature approximation, misreading and computational errors. Some of these errors are in the nature of mathematics while other is due to undue emphasis or lack of emphasis placed by the teacher during the teaching process.
11.3 Types of Common Errors
Examination of students’ answer scripts all levels mathematics reveals at least eight types of errors. These include syntactic, semantic, pragmatic errors (Ahmad, 2010). Some other errors are errors in mechanistic understanding, cultural inhibition, premature approximation, misreading and computational errors. Some of these errors are in the nature of mathematics while other is due to undue emphasis or lack of emphasis placed by the teacher during the teaching process.
11.3.1 Syntactic Error
It is an error in the relationship between a symbol and other symbols in mathematical expression. For example, consider these expressions a. p = 3q-5 b. p = p3-5 the two expressions contain the symbols p, q, =, -, 5, 3, but have different meanings in expression. 3 and q in first expression means 3 multiply by q while in second expression the relation is that of exponential or power.
11.3.2 Semantic Error
Semantic error is an error as a result of not taking into consideration what a symbol stands for or represent, in a mathematical expression. What a symbol stands for in an expression is known as Referent. Example, given that 7/x find the value’ x” the variable cannot be zero. As student who recorded that 7÷x=7 or 0 that student committed the error of semantic because division by zero has no meaning.
11.3.3 Cultural Inhibition Error
This is an error committed by the student due to past experience, which prevent that student from responding correctly to new situation. For instance, a pupil in primary two was taught by their basic science teacher that half is rabi( using Vernacular Hausa) and rabi is part of whole. In later, years this pupil was given a diagram to identify half. Student looking shaded area in figure and concluded that it is half, which is wrong.
11.3.4 Pragmatic Error
It is an error committed by student through individual bias, emphasis or connotations placed on the symbol by individual learner of mathematics. For example, if a student has understood that the equation y=ax2+bx+c is quadratic equation, whenever given q=py2+c, the student will no longer see that it is also quadratic equation becausesince the teacher mainly used an equation in x.
11.3.5 Understanding Error
It is an error committed by many students given wrong interpretation to word problems. Example: The mathematics teacher asked question that there are three times many pupils (P) as there are teachers (T). Put down this statement in correct mathematics expression. This problem was forwarded to ten students in 100 level. Majority of the students obtained the result as 3P=T, only two students responded correctly as P=3T
11.3.6 Premature Approximation Error
This is an error committed by learners of mathematics where a student approximates a value obtained in a part of problem and used the approximated value in working other parts of the given problem. For example, given that the radius of the earth is 7400 km and pie=3.142 with two points P(500N, 400W) and Q(500N,100 W). Compute:
i. the radius of the parallel of latitude 500N
ii. the distance PQ along the circle of latitude. Correct your answer to one decimal place
Solution: if the value of r is as 7400cos30 = 6408.4 or 6408, the student approximated r=6408 and work out arc PQ=30/360×2×3×3.142×6408. This provided a wrong answer as 10,067.0 but if use correct value as 6408.4, the answer will read as10068.2
11.3.7 Mechanistic Understanding Error
This is type of error that can be committed by student for not enable to transfer knowledge to another mathematics process. For instance, student has knowledge of sum of square difference as X2-Y2= (X+Y) (X-Y), when given a problem as 2252-152, the student went on to use long multiplication thereby committed error through long multiplication.
11.3.8 Misreading Error
This is an error committed through wrong copying of problem given. For example, 53+ 5-4- 5-4
A student in copying the problem as: 53+ 54- 54 the negative power, -4 was copied as 4
11.4 Semantic and Syntactic Difficult
Semantic and syntactic difficult that affect teaching and learning of mathematics will be discussed on Directed numbers, Number and Numeration, Geometry and Algebra.
According to Inekwe(2019) pointed out that communication by many teachers of mathematics for a signed number, example is -8“minus 8” where 8 is not being subtracted from any number. Naturally speaking, the statement is not proper since you can’t subtract things where nothing exists. But 8 – 4 is rightly reads “8 minus 4” which is naturally possible. Hence this is a syntactic problem. Another example on number line zero is demarcation: -3, -2, -1, 0, 1, 2, 3 between negative and positive integers as shown above. hardly do we refer to the numbers right of zero as plus 1, plus 2, plus 3, but integers left of zero are refer to minus 1, minus 2, minus 3.Which are incorrect? The difficulty encountered by many students in directed numbers may not detach from this semantic problem being communicated to them.
“Zero” is refers to non-positive and non-negative integer but being defined by some mathematics teachers as nothing and that in build in the mind of young immature ones on shaky mathematical foundation. And when such errors are not checked and corrected the receivers may pass it on to another generation and it cycle continue.
Other semantic difficulties in geometry and algebra are a plane is wrongly defined as a flat surface rather than a 2-demensional surface without thickness, while a straight line to some is the shortest distance between two points rather than a set of points which continue in opposite directions indefinitely. A line segment is a set of points with two end points and a ray is the set of points with one fixed points. Because some of mathematics teacher gives unclear classification between the definition of line segment and ray. Some defined “cycle” as space between two rays. A triangle is incorrectly refers to a figure bounded by 3 lines rather than a plane figure bounded by 3 line segments.
Algebraically too, instead of if X = 5 some say when X = 5 whereas when refers to time. Logically, speaking there is never a time when X will be equal to 5, being two different entities are alphabetic and the other numeric. Instead of removing and applying the bracket, mathematics teachers commonly communicate open and close the bracket. Also some mathematics teachers say X=3 as X is equal to 3 rather than X equal 3. All these are syntactic errors that affects majority of mathematics teachers and students alike. The phrase cross multiply a/b =x/y to obtain ay = bx, instead of multiply both side by the L.C.M of the denominators to obtain ay = bx. Hence teachers of mathematics are advised rather to avoid such mechanical terms like cross-multiply, which lead to erroneous application especially at the primary level.
Phonologically 2008 is pronounced by many, as 2008 as in book instead of 2 zero 8, symbol O (numeric zero) is mixed up with letter O (alphabetic). Also, the write up of numeric zero (0) is spherical while that of alphabet is circle in nature (O). Instead of transfer 1 to the tens column many say carry 1 and write down 5 as in case of 7 + 5 = 15. It is “3/7” is read as 3 over 7 instead of 7 divide 3. All these are linguistic problems that affect meaning of concepts and constructions of mathematical ideas adversely. Therefore, it becomes necessary to take steps to minimize these problems.
According to Ahmad (2010) communication is an act that aid improving teaching and learning of mathematics. It is processes involve the use of spoken mathematics words, signals, gesture, pictures, visual, displays and film. Mathematics teacher must possess adequate communication skills required for successful teaching and learning of mathematics. Effective communication skills in mathematics teaching involves the ability to deliver or teach a subject with simplicity of language, easy to understand presentation, stimulating interactive environment and a display of the mastery of the subject master. Kajuru and Popoola (2010) observed that teachers of mathematics who provide good classroom condition and mathematics instruction will improve the academic achievement of students.
Student Activity
1. What type of error involved in the following solutions?
a. 52=10
b. 6+5(a-3) =11(a-3)
c. X+X=X2
2. Four families have respectively 5, 4 and 8 children in their houses. What is the average
number of children per family? Solution = 5+4+8/3=17/3=5 2/3 what kind of error committed in this answer.
3. A bus travels at P km per hour for 1 hour and at Q km per hour for 4 hours. Find its average speed.
Solution: P = 4P/hr, = 4P = Q therefore Average speed = P + P2 / 4
Identify the error in this solution.
4. Construct answer with pragmatic, misreading and premature errors
11.5 Summary
The content of this chapter is summarized as follows:
§ Syntactic error- an error committed of relationship between a symbol and other symbols in mathematics expression.
§ Semantic error- is an error as a result of not taking into consideration what a symbol represent in a given mathematics expression.
§ Premature approximation error- is an error resulted from approximation of value obtained in a part of a question and uses the approximated value in working subsequent parts of the question.
§ Understanding error- wrong interpretation of word problems
§ Other types of common errors were explained.
References
Ahmad, M. (2010). Effect of mathematical language on students’ performance in mathematicsM.Ed Thesis Unpublished
Inekwe, O. I. (2019). Mathemaphobia, The Differential Derivative 43rd Inaugural Lecture
National Teachers’ Institute (2000). NCE/DLS Mathematics Cycle 2