By Arindam Nag.

Elementary Algebra is generalized form of arithmetic. It provides a language to represent problems and functions. Algebraic thinking is also one of the first forms of abstract thinking that students develop in mathematics. It is most critical to understand what mental model your child has established and then look to mould and correct that model giving different examples. Lets look at some of the common gotchas of algebraic learning.

# Fear of the unknown *x*

In Algebra, we use a mixture of numerical digits (constants) and alphabets (variables). For many children, after having done arithmetic, they get pretty confused with the introduction of alphabets in mathematical expressions. Also, with * x* being the favorite letter for a variable, it is even more confusing since it is the symbol used for multiplication. In computer programming parlance,

*is overloaded.*

**x**The easiest way to introduce alphabets in arithmetic is to use them as placeholders while solving word problems. Consider the following problem: *5 bananas cost 20 rupees. What is the cost of each banana?* Such type of problem is introduced quite early, as soon as children learn multiplication and division. You can start with using a sketch of a banana to represent cost of one banana. You can show that instead of writing *5 x Cost of one banana = 20*,

they can draw a picture of a banana. Then they can replace that picture with an alphabet, say **b** to represent the banana. So, now they have a language to represent the problem to say

**5 x b = 20**

This is the beginning of Algebra. Algebraic thinking can be introduced by asking them what happens if the cost of 5 banans increases to 30 rupees and then 40 rupees.

# Algebraic expressions to represent functions

In the above example, the letter* b *really represented one value. The essence of Algebra is that an alphabet is used to represent a variable quantity which can take on different values and thus represent a function. Using Age related problems is a great way to explain this concept. Start by asking your child to express the following statement as an algebraic expression :

*Father is 30 years older than the son*. How would you write this?

Age of son : x

Age of father : x + 30

Then proceed by asking what happens to the father’s age as the son’s age increases. This will build a mental model of a function. Then one can easily introduce the concept of an equation. When the father is 50 years old what is the age of son?* *

x + 30 = 50Solve for x.

**Misconception about subtracting variables**

Another common mistake that almost all children make is dealing with expressions like this.

What is

5x – x?

Most children will answer that as 5, especially when the problem is asked verbally. This is because of two reasons. First reason is that they forget **x** is really a shorthand for **1x**. The other reason is that children have a mental model of the above expression as:

5(x – x)

rather than

(5 – 1)x

This can also be explained with an interesting age problem. The father’s age is 5 times the son’s age. The difference in their current ages is 20. What are their ages?

5x – x = 20

Solve for x.

# Conclusion

Remember that Algebra is beginning of abstract thinking in mathematics. Hence it is very important that we understand what is the model they have in their mind and then help shape that model. Most important is to remove the fear of use of alphabets in mathematical expressions. Make it fun by using puzzles and then inculcate the understanding of function representation. Then, definitions around like terms, coefficients and others will naturally follow.

# References:

1. Making Meaning in Algebra Examining Students’ Understandings and Misconceptions — David Foster http://library.msri.org/books/Book53/files/12foster.pdf

2. On the learning of Algebra — H. Wu. http://math.berkeley.edu/~wu/algebra1.pdf

**About the Author:**

Arindam Nag is a Founder and Director at Learnhive; he has over 15 years experience in information technology. He worked for over 13 years at one of the world’s leading investment banks, Goldman Sachs in various engineering and technology leadership roles. He has a Masters in Computer Engineering from the University of Texas at Austin. He is a recipient of National Talent Search Scholarship. He is a staunch believer in using technology to bridge the gap in the educational divide that exists not just in India but the world over.

**About Learnhive: **

Learnhive is a leading provider of technology based learning solutions for K12 students, parents, and tutors. Our goal is to make curriculum based learning more effective and fun. We specialize in providing after school learning solutions for students and parents. Our flagship product, Personal Concept Tutor™ gives students the flexibility to learn concepts at their own pace using a wide variety of materials and resources suited for their individual needs. Our technology is compatible with multiple device formats such as desktops, laptops, tablets and smart phones to make learning more fun, interactive and available to students anywhere and at anytime. Signup for free to access the learning materials (lessons and exercises).