Common Errors in Secondary Mathematics

Common Errors Committed by the 

Students in Secondary Mathematics 

Errors that students often make in doing secondary mathematics during their practice and during the examinations and their remedial measures are well explained here stp by step. 

Some Common Errors in Mathematics

Teachers may often observe that there are certain errors which are frequently committed by the students. In order to help students in correcting such errors, it is imperative that the teachers understand the reasons behind such mistakes. The common errors that students commit may be categorized into two types

• Conceptual errors
• Procedural errors.

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Conceptual Errors

A teacher has to keep in mind that knowledge is not transferred. It is constructed through one's experiences. The students construct knowledge though various experiences, both inside and outside the classroom. Sometimes, flawed reasoning or lack of proper understanding may lead to the development of some pre-conceived notions. These misconceptions or naive theories may lead to errors. Such misconceptions arise when the content is not properly transacted. 

A conceptual error occurs when a student does not understand the underlying idea, principle, or concept behind a topic. Unlike a procedural error, which is just a mistake in steps or method, a conceptual error shows that the basic meaning or reasoning is not clear.

Conceptual errors happen when the foundation is weak, definitions are not clear, or connections between concepts are not fully understood. 

Identifying and correcting these errors is important because once the concept becomes clear, the student can solve many problems correctly, Only if such conceptual errors are recognized in time and the reasons behind these are figured out, can they be clarified and the resulting mistakes corrected. Understanding the justification behind a procedure is a prerequisite for conceptual understanding. Conceptual knowledge includes the decision regarding which procedure is to be applied.

Following are some common conceptual errors that students often commit:

1. We know that    Many students will say   Remember that division by 0 is not defined. Here is a very good example of the kind of absurd result that we may arrive at if we divide by 0.

Consider the algebraic identity:

                                           

Put x = y, we obtain, 

 
 

 

 

We have managed to prove 1 = 2. This is, clearly, an absurd result. Surely, we have made some mistakes somewhere. We have divided both sides of equation (3) by x – x, which is actually 0 and we have taken 0/0 as 1. Remember that you can’t divide by 0.

2.  2(a + b) = 2a + 2b

Due to this statement students get an impression that everything works like this. Here is whole list , in which this does not work.

Incorrect	Correct
(a + b)2 = a2 + b2	(a + b)2 = a2 + b2  + 2ab
sin(x + y) = sinx + siny	sin(x + y) = sinx cosy + cosx siny

3.  Improper use of square root

Students seem to be under misconception that   which is incorrect.

The proper working is as under

x² – 4 = 0 

⇒  x² = 4 

⇒ 

⇒    

4. Improper use of power of trigonometric functions

Incorrect	Correct
sinnx = sinxn	sinnx = (sinx)n

5.  Incorrect use of trigonometric results:

Incorrect	Correct
cos85o = sin(90o – 5o) = sin5o	cos85o = cos(90o – 5o) = sin5o  or cos85o = sin(90o – 85o) = sin5o

6. Incorrect solution of quadratic equation

Solve : 3x² = x

Incorrect	Correct
3x2 = x ⇒ 3x = 1 ⇒ x = 1/3 Here students divide both side by x without realising that x could be 0 also	3x2 = x ⇒ 3x2 – x = 0 ⇒ x(3x - 1) = 0 Either x = 0 or 3x – 1 = 0  x = 0, 1/3 Both values of x satisfy the given equation; hence both are the solution of quadratic equation.

Procedural Errors

Procedural knowledge means the knowledge regarding how to do something. This can also be understood as mathematical skills. Rectification of procedural errors requires practice and feedback.

A procedural error occurs when the correct steps or method for solving a problem are not followed properly.

It does not mean that the student does not understand the concept; instead, it shows that the student made a mistake while carrying out the procedure.

These errors usually happen when a student applies the wrong formula, skips a step, performs the steps in the wrong order, substitutes values incorrectly, or simplifies inaccurately. 

Following are some of the common procedural errors that the students commit:

A). Many a time, students decide that parentheses are not needed at certain steps. They fail to understand the importance of parentheses.

Example 1: Square of 3x.

Incorrect	Correct
(3x)2 = 3x2	(3x)2 = (3)2 (x)2 = 9x2
Here students are failed to follow the procedure of BODMAS

Example 2: Square of -2

Incorrect	Correct
-22 = - 4	(-2)2 = 4

Example 3: Solve for x:      

Incorrect	Correct
⇒ -5x + 15 = 6 ⇒ -5x = -9 ⇒ x = 9/5	⇒ -5x - 45 = 6 ⇒ x = - 51/5

B) Improper Distribution

Example 1: Simplify: 3(2x² - 6)

Incorrect	Correct
3(2x2 - 6) = 6x2 - 6	3(2x2 - 6) = 6x2 - 18

Example 2: Simplify: 2(3x-5)²

Incorrect	Correct
2(3x-5)2 =(6x-10)2                = 36x2 - 120x + 100	2(3x-5)2 = 2(9x2 – 30x + 25)               = 18x2 – 60x + 50

C) Error in balancing the equation: 

Example: Simplify:   

Incorrect	Correct

D) The following mistake largely on the part of authors and the mistake is carried over the students.

Incorrect	Correct
Area of the rectangle having length and breadth 6cm and 5cm respectively is 6 х 5 = 30 cm2	Area of the rectangle having length and breadth 6cm and 5cm respectively is 6cm x 5cm = 30 cm2

Area of a rectangle has to be expressed in some square units. However, it is to be understood that though conceptual and procedural knowledge are often discussed as distincts entries, they do not develop independently in mathematics and, in fact, lie on a continuum, which often makes them hard to distinguish.

Common Errors in Finding Squares and Square Roots

1) In right angled triangle ABC right angled at C, if tanA = 1, find SinA

Incorrect	Correct
BC = AC = K	BC = AC = K

2) Find the square of (7x - 3).

Incorrect	Correct
(7x – 3)2 = 7x2 – 32                 = 7x2 – 9	(7x – 3)2 = (7x)2 - 2× 7x × 3– 32                 = 49x2 – 42x + 9

3) For what value of k the quadratic equation: (k + 4)x² + (k + 1)x + 1 = 0 has equal roots ?

Incorrect	Correct
Since the given equation has equal roots. ∴ Discriminant = 0 ⇒ b2 – 4ac = 0 ⇒ (k + 1)2 - 4× (k + 4) × 1 = 0 ⇒ k2 + 1 - 4k + 4 = 0 ⇒ k2  - 5k + k + 5 = 0 ⇒ k(k – 5) + 1(k – 5) = 0 ⇒ (k – 5) (k + 1) = 0 K = 5 or k = -1	Since the given equation has equal roots. ∴ Discriminant = 0 ⇒ b2 – 4ac = 0 ⇒ (k + 1)2 - 4× (k + 4) × 1 = 0 ⇒ k2 + 2k + 1 - 4k - 16 = 0 ⇒ k2  - 2k - 15 = 0 ⇒ k2  - 5k + 3k - 15 = 0 ⇒ k(k – 5) + 3(k – 5) = 0 ⇒ (k – 5) (k + 3) = 0 ⇒ k = 5 or k = - 3

(4) Factorise :  

Incorrect	Correct

Common errors committed by the students in Similarity of Triangles

S. No.	Common Errors	Reasons	Remedial Measures
1	Sometime students used ASA rule	Connot differentiate congruency and similarity rules	Congruency rules are: SSS, SAS, ASA, AAS, RHS Similarity Rules are: SSS, SAS, AAA, AA
2	AB = DE BC = EF AC = DF	Cannot understand the congruency and similarity rule properly.	Two triangles are congruent then corresponding angles and sides are equal. Two triangles are similar then corresponding angles are equal and sides are proportional.
3	If DE || BC then	Cannot differentiate between similarity and Basic Proportionality theorem	DE || BC, then first prove that △ADE ~ △ABC

 

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