FOUNDATIONS OF SUBTRACTION
This book is just a book away from creating history - popular genre books for each arithmetical operation. The 'Foundations of addition' and Foundations of multiplication' are already published. The 'Foundations of division' is the next.
This book, 'Foundations of subtraction', presents a peerless 300-page 'story' of discovering the most varied arithmetical operation. However, it must be read after duly investing time and attention in the addition and multiplication books.
Here are a couple of excerpts from the book:
Lesson: Knowing how to learn subtraction
School math curricula place teaching of subtraction just after the teaching of addition. It follows the belief that subtraction is the opposite of addition. While it is true that there are 'opposite' features between addition and subtraction (we discussed them in an earlier Learning Outcome), the two can only be learnt in quite different ways. For example, subtraction is varied and far more nuanced. There are significant dissimilarities between addition and subtraction beyond what may be classified as 'opposites.'
Subtraction may not be the second arithmetical operation to be taught. The skills and support required for teaching and learning subtraction are significantly different. A majority of teachers also lack the nuanced understanding needed to teach subtraction.
Lesson: Preparing children to learn subtraction
Subtraction is intuitive and just about as natural as addition. However, addition is fairly simple to observe and state (for example, things being added are all in distinct quantities). Subtraction is identified in more varied situations. It is more nuanced and needs better comprehension of situations to be identified and mathematically expressed as subtraction. As already mentioned, there is more planning and preparation needed to teach and learn subtraction. It may be the most complex of the four operations, as far as visualization is concerned.
It would be easier for children to connect with the varied situations which are mathematically expressed as subtraction if the following aspects are taken care of in the teaching of subtraction:
First, appreciating the limitations of counting when it comes to subtraction.
In situations involving addition, counting is always a simple direct alternative – counting is tiring, time-consuming, etc. but counting is still an option as a real alternative to situations involving addition to know the sum of given quantities/numbers.
It is hard for children to realise that counting and subtraction have a more complex relationship than the relationship between addition and counting. It is not possible to get the difference in quantities by counting the quantities. For example, we cannot distinguish the number of books in two stacks of books by counting the books in both the stacks! We will have to use subtraction in such situations.
Second, physical segregation of quantities in subtraction situations is unlike how it is in addition:
A. The quantities in subtraction situations may or may not be segregated to start with. For example, if there are 12 crayons and you take 3 out of it to colour, the 3 crayons taken away are not distinct (they are part of the 12 crayons). In addition, the quantities to be added, the addends, are all distinctively identifiable, well segregated. If we add 3 chairs and 4 chairs, the 3 chairs and the 4 chairs are always distinctly visible.
This shift from the certain need for segregation of quantities to be added to accepting unsegregated quantities in subtraction is quite a new learning experience.
B. Unlike addition, subtraction makes two quantities out of one – the initial quantity is split into two – what is taken away and what is left. In addition, two quantities become one, the sum.
Interestingly, all these above are not valid for all subtraction situations.
The book is a must read for everyone. Math needs a rebooting for us all, and this is one such book.
Finding out the difference between all kinds of things may be the most common thought and task we indulge in every day. Here is a sample:
- The time left to get ready for the school bus, set out for office.
- The time left for the math, or the games, periods to end, the time left to head back home.
- The money expected to be returned after a payment against a due amount.
- The time left to solve a math test.
- The number of bananas left if you bought a dozen and ate four of them.
- More weight a lift can carry if it can carry 850Kg and you weigh 40Kg.
- The number of slices of pizza left if there were 8 slices, and you ate 1.
- The distance left to run after running 15m in a 100m race.
- The amount of water you need to fill a 1 litre bottle that is 700 ml full.
- The number of bananas needed to balance a weighing scale that has 2 similar bananas on one side and 5 bananas of the same kind and size on the other.
Finding the amount of difference is far too frequent, varied, useful, and essential.
In each of the examples, there are two things which are being compared, for example:
- The time left to get ready for the school bus, set out for office.
In this situation, the two things being compared are time – the current time and the time to leave for school/office - The time left for the math, or the games, periods to end, the time left to head back home In this situation, the two things being compared are time – the current time and the time for the period end, or the time to start the head back home
- The cash expected to be returned after a payment against a due amount In this situation, the two things being compared are amounts of money – the amount of payment, and the amount due
- The time left to solve a math test In this situation, the two things being compared are time – the elapsed time and the test duration time.
- The number of bananas left if you bought a dozen and ate four of them In this situation, the two things being compared are quantities of bananas – the number of bananas bought (dozen) and the number of bananas eaten (4).
- More weight a lift can carry if it can carry 850Kg and you weigh 40Kg In this situation, the two things being compared are weights – your weight and the maximum weight that can be safely carried by the lift.
- The number of slices of pizza left if there were 8 slices and you ate 1. In this situation, the two things being compared are the number of pizza slices – the total number of pizza slices and the number of pizza slices eaten.
- The distance left to run after running 15 m in a 100 m race In this situation, the two things being compared are lengths on track – the distance you’ve run and the total length of the race.
- The amount of water you need to fill a 1 litre bottle that is 700 ml full In this situation, the two things being compared are volumes – the total volume of the bottle and the volume of water already in the bottle.
- The number of bananas needed to balance a scale that has 2 similar bananas on one side and 5 bananas of the same kind and size on the other In this situation, the two things being compared are the weights of bananas – the bananas on one side of the balance and the bananas on the other side of the balance
A common thread through the examples is that both the quantities are comparable. They have the same ‘unit of counting’ or can be made of the same kind (such as time in an hour can be expressed as time in minutes or days). The difference can be found out only among similar things. For example, we cannot compare the current time to the current date to find out the time left for a school period to end.
Subtraction may be the more common arithmetical situation. However, jumping the gun, it may just be stated that operationally, we express many subtraction situations as equivalent addition expressions.
Summary
Subtraction is the mathematical equivalent of the more common quantification process of finding ‘what is left’ in far too many situations than we can imagine.