# An Account of 20 Common Math Misconceptions And How to Curb Them (PART-3)

- May 18, 2016
- Sudipto Das

This is the continued part of this topic after Part-1 and Part-2.

**11. Mixed fractions are always greater than Improper fractions**

Many students have this misconception purely because of the whole number part contained in mixed fraction. Give them a problem involving the two fractions:

1st fraction-

2nd fraction= **10/5**

Ask them which one is greater? You’ll receive many answers where the mixed fraction (1st) is considered to be a greater fraction purely because of the presence of whole number. This misconception can be curbed by telling them to convert both fractions to a similar form while comparing between them. They can be converted mixed form or improper form but they have to keep in mind during the final comparison that “both” the fractions should be in the “same form”. In the above problem, we want to make the comparison on the basis of improper fractions.

1st fraction in improper form= 9/5

2nd fraction is already in improper form and doesn’t need any conversion. 2nd fraction= 10/5

Hence, it can be said that 2nd fraction is greater than the 1st one.

**12. Misunderstanding the statement that “Zero is just a placeholder in decimals”. **

Students are taught this concept that zero is just used as a placeholder in decimals. But students misinterpret it in a certain manner. The teacher means that zero becomes a placeholder only if there’s a non-zero integer at the end of a decimal number after which this placeholder zero is to be inserted or if there’s a non-zero integer at the beginning of a number before which this place-holder zero is to be inserted. An example should make this matter clearer from your point of view. Let’s pick a number say, 65.067.

By “zero as a placeholder”, the teacher means that 65.067=065.067=65.0670 etc.

Students misinterpret it thinking that 65.067= 65.67. This is wrong. You should clarify this statement clearly with citing relevant examples (like the one provided above) to clear his confusion.

**13. The length of the diagonal of a square is of the same as its side**

This is a common notion of many students particularly when the figure involved is a square. Generally, this misconception doesn’t appear if the figure involved is a rectangle. This misconception is particularly associated with a square. A simple way to cub this notion is to draw a square (should be drawn to scale) and ask the students to measure the diagonal from one point to another. Then ask them to measure the length of one side of the square and compare the length of the diagonal to the length of that side. Your students should understand this and would not repeat the same mistake twice.

**14. Divisions are commutative**

It’s a little bit similar to the point 9 (in PART-2) where the culprit involved was subtraction. Students have the belief that 3÷2= 2÷3. It’s a wrong concept. Of course, you know this. The best way, to curb the notion is to make them solve the sum in this way:

3÷2= 3/2= 1.5.

2÷3= 2/3= 0.66.

Now they’ll know that those two are not the same thing and divisions are non-commutative. This example will be enough to curb that wrong notion. They would definitely not repeat such mistakes again in future.

**15. In divisions, divisor is always smaller than the dividend**

It’s a pretty common misconception. If a problem like 4÷8 is given, you should not be astonished to get some answers like “2” from some students. Those students actually know the division process but are confused with the divisor and the dividend. They have the concept that the bigger number should always be divided with a smaller number. Try to clarify the meaning of the divisor and the dividend clearly with the students to curb that notion. Generally, these students also exhibit the wrong notion that divisions are non-commutative. To curb that notion, refer to the previous point (14).

**16. A solution of a math problem should not contain any operator symbols**

A common notion of students is that the final result of a math problem should always come in a single compact number. This is partially correct and there are a number of exceptions. A simple example can be any algebraic problem. Let’s consider a problem like the following one:

3x+6

What will be the answer? You’ll find a number of students answering it like “3x+6=9”, a clear effect of this misconception. Your students should know the clear difference between constants and variables in such type of problems. Clarify it clearly with your students so that they don’t repeat the mistake again in future and the misconception’s curbed. This problem generally arises in middle school levels, specially when algebra is introduced to students.

**17. Adding the numerators and the denominators separately in fractions**

This is a very common misconception of primary school students and is specially associated with fractions. Some of them develop the conception of adding the numerators and the denominators separately. An example is:

3/4 + 7/9 = 10/13.

This problem is easily curbed by making them aware of the proper methods of adding fractions. The same misconception is also there with fractional subtractions. The same clarifications can be done there as well to solve this wrong notion.

**18. Misconception on powers**

Some students have a wrong notion associated with powers. Say for example s/he is given problem like this one 6^2. You might get the answer as 12 and not 36 which is the correct one. That’s wrong notion of powers in which the student multiplies the number with its power itself. Clarify the meaning of powers clearly with the students that powers means the number of time of occurrence of the same number. For example:

2^4= 2*2*2*2= 16.

6^2= 36.

### This will surely curb that misconception of the student.

**19. Misconception on algebraic substitutions**

It’s a very common problem. We are explaining this to you with an image that’ll provide you with a better idea on this subject matter. Refer to this image below.

The student in the above image has simply ignored the multiplication operator or doesn’t know that 6y means (6*y). Please clarify with your students the fact that if no operator is provided in such cases, it automatically means there’s a multiplication operator in between the variable and the constant. 6y means 6*y, hence the answer should be “6*3=18”.

**20. Misconception on division of whole number by fractions**

Students face this problem when they are given a problem like this one:

40 ÷1/2.

You’ll see many answers as 20. The problem here is mainly overgeneralization of previous experience resulting in a faulty result. We are not counting out careless mistakes but the main problem is overgeneralization. Clarify your students the fact that they should look closely to the problem again and again and avoid doing the calculation in a hurry. They should calculate it in a step-by-step manner and not mentally, at least at the start of lessons until they become comfortable with these sums.

40÷1/2= 40*2= 80.

Misconceptions are problems that affect a student’s career in the later stage of his/her education. These misconceptions are the main problem for which the students loses interest in mathematics because in higher education teachers rarely correct these basic misconceptions of that particular student and jumps to a conclusion that s/he is not working hard enough. These misconceptions should therefore be removed at an early stage so that the students feel comfortable with math problems in the later stage of his career. Hope you had a good read of these 20 misconceptions highlighted in 3 parts. That concludes our article.

### Sudipto Das

Sudipto writes technical and educational content periodically for wizert.com and backs it up with extensive research and relevant examples. He's an avid reader and a tech enthusiast at the same time with a little bit of “Arsenal Football Club” thrown in as well. He's got a B.Tech in Electronics and Instrumentation.

Follow him on twitter @SudiptoDas1993

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