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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 as conceptual errors and procedural errors.


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. The teachers may sometimes fail to explain the concept behind a particular procedure and this may lead to faulty understanding on the part of learners. Teachers need to understand and address any such misconceptions that students might have. It is essential to address these because they are difficult to remove and addressing them calls for a lot of effort.

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:

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

 equation

 equation

 equation

 equation

 equation

 equation

  sin(x + y) = sinx + siny

 sin(x + y) = sinx cosy + cosx siny 

Improper use of square root

Students seem to be under misconception that equation  which is incorrect.

The proper working is as under

x2 – 4 = 0 

⇒  x2 = 4 

⇒ equation

⇒  equation  

* Improper use of power of trigonometric functions

Incorrect

Correct

sinnx = sinxn

sinnx = (sinx)n


* Incorrect use of trigonometric results:

Incorrect

Correct

cos85o = sin(90o – 5o) = sin5o 

cos85o = cos(90o – 5o) = sin5o  or

cos85o = sin(90o – 85o) = sin5o 

*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.

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

(-2)2 -22 = - 4

(-2)2 = 4

Example 3: Solve for x:     equation 

Incorrect

Correct

 equation

equation 

equation 

⇒ -5x + 15 = 6

⇒ -5x = -9

⇒ x = 9/5

 equation 

equation 

equation 

equation 
⇒ -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:  equation 

Incorrect

Correct

 equation

 equation


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

 equation 

BC = AC = K

equation 

equation 

equation 

equation

 equation 

BC = AC = K

equation 

equation 

equation 


equation

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) 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

equation

Cannot differentiate between similarity and Basic Proportionality theorem

DE || BC, then first prove that

△ADE ~ △ABC 

equation



 
THANKS FOR YOUR VISIT 
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Comments

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