Handling the decimal dilemma
Krittika Hazra
“The decimal point moves to the right as many places as there are zeros in the factor. (shifts to the left in case of division).”1 We find this sentence in standard textbooks dealing with multiplication of decimal numbers with power of tens under the topic, ‘decimal’.
However, we do not spend any time explaining the reason behind such ‘movements’ and the scope of generalization as a deductive outcome. Consequently, we observe that students try to memorize the imagery rather than comprehend the mathematical logic.
The practise of memorization without the provision of reason is further elaborated by providing the students with thumb rules like:
“MR. Deepak Lal
(M – Multiplication
R – Right
Deepak – Division
Lal – Left)”2
A Khan Academy video lecture, on the other hand, in an effort to reduce the erroneous calculation of problems related to decimal refers to some normative character of mathematical operations and delivers, “…why while dividing the point shifts to the left and while multiplying it shifts to the right. Because after dividing I should be getting a smaller value, hence shifts to the left and vice-versa.”3
Approach – Excerpts from my own teaching experiences
Not satisfied with this relatively simple explanation in the textbook, I explored different approaches that made mathematical sense and came up with the following.
Approach 1: I assigned the following task to my students of class 5 – “Multiply the following numbers with 10 and place them in the place value chart according to their places before and after multiplication.”
i) 3
ii) 33
iii) 333
iv) 0.8
v) 0.88
vi) 0.888
Here is a specimen
| Thousands | Hundreds | Tens | Units | Tenths | Hundredths | Thousandths |
|---|---|---|---|---|---|---|
| 3 | ||||||
| 3 | 0 | |||||
| 3 | 3 | |||||
| 3 | 3 | 0 | ||||
| 3 | 3 | 3 | ||||
| 3 | 3 | 3 | 0 | |||
| 0 | 8 | |||||
| 8 | 0 | |||||
| 0 | 8 | 8 | ||||
| 8 | 8 | 0 | ||||
| 0 | 8 | 8 | 8 | |||
| 8 | 8 | 8 | 0 |
Then I worked out an extension of the earlier idea and assigned the following task: “Multiply different numbers with 100, 1000, and so on.”
While going through these series of tasks most students made the following observation: “Each of the numbers shifts towards the left of the place value chart in accordance with the powers of ten.”
So without giving any rule or memorization trick to them, the students were able to find the pattern themselves and were able to apply it to other problems.
Approach 2
Abacus can also be used as an assisting teaching and learning material for classroom teaching of the decimal. I performed the multiplications of decimal numbers with 10 with the help of the abacus and demonstrated the shift of the numbers (as will be apparent by the shifts of the columns of beads). Then I provided them with the following sheet to figure out the multiplication process by themselves:4
Here also by understanding the idea of multiplication of numbers with powers of 10 the students were asked to generalize any decimal number using the abacus in groups of 5 or 6 and noting down their observation. The students were able to find the decimal multiplying pattern very quickly when multiplying by powers of ten.
Some environmental difficulties
The primary and probably the biggest hurdle to acquiring knowledge is the learning that occurs outside school. Before citing an example, I must reiterate that when the child encounters opposite views from the school teacher and the home teacher, it creates confusion around her/his idea of mathematical thinking itself. While teaching the introduction of multiplication of decimal with powers of ten, I started with the learning materials and methodology as described in approach 1. A few students came up with problem solving tactics similar to the tactics prescribed in the above text. To complete the journey from mundane muggings to actualization of concepts, I had to make the children unlearn what they had acquired in homeschooling. I used the following example which I believe may be suitable for others also in such a similar situation: consider a tree, rooted firmly to the earth, consider an observer on a train that is moving past the tree, who is looking at the tree. From the perspective of the observer the position of the tree will change fast. But actually the tree is constant (decimal point) and the train moves past it (the number).
In the pursuit of learning, students should also learn how to score in tests. However to do so, we often ignore the criticality of comprehension and place certain mediocre achievements higher above the potential of the growing mind. It is true that some mathematical concepts are delivered better in the future years of a child’s pursuance of higher studies in mathematics. But withholding the delivery of a concept at this stage not only undermines the capability of the child but also may undermine the child’s belief in his/her capabilities.
I therefore strongly argue in favour of deliverance of actual concepts of decimals in schools so that the child learns the right kind of mathematics from an early age and also comprehends the actual meaning.
References
- Junior Mathematics by Bharati Publication
- New Enjoying Mathematics by Ashalata Badami, Oxford University Press
- https://www.khanacademy.org/math/cc-fifth-grade-math/cc-5th-place-value-decimals-top/cc-5th-mult-powers-of-10/v/dividing-decimals-2-1
- Manual teaching aids for primary schools; Developed with Shri P.K Srinivasan, Mathematics Education Consultant, Madras – Published: NCERT
The author is a mathematics educator working in The Future Foundation School, Kolkata. She specializes in middle school mathematics teaching and has been working on extending the scope and understanding of mathematical skills to improve life skills in children. She can be reached at krittika.hazra7@gmail.com.
