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Discovering the sum of 7 + 3


Let us start with our response to some everyday examples of 7+3 where our statement and action indicates that we know what ‘7 + 3’ is:

  1. We know that 7 men and 3 women are ‘7 people and 3 people’, or 10 people
  2. We know that 7 women and 3 infants are ‘7 women and 3 infants’, not 10 people
  3. We know that 7 kg rice packets and 3 half-kg rice packets will be 8 ½ kg of rice, not 10 kg
  4. We know that 7 years and 3 months is ‘7 years and 3 months’ (which is 7.25 years), not 10 years, or 10 months
  5. We know that 7 dozen erasers and 3 erasers are 87 erasers, there is nothing 10 about it
  6. We know that USD 7 billion and USD 3 are USD 7,000,000,003!

For, sure we know all that the above six statements involving ‘7 + 3’ are true, then what must ‘7 + 3’ be?
One thing must be clear from the above examples, ‘7 + 3’ is NOT necessarily 10. It can be 10, 8.5, 7.25, 87, 7000000003, or even ‘nothing’ (that is when we can’t add 7 + 3).
Indeed, 7 + 3 could be any quantity, 10 being just one possibility!
It is easy to see in the above examples that 7 + 3 will equal to 10 only when 7 and 3 have the same unit, as in ‘7 people and 3 people’, or 10 people! For the units ‘men’ and ‘women’ we can use the common or like unit ‘people’.
But what does school math education make of ‘7 + 3’? ‘7 + 3 = 10’, always! From Grade I onwards ‘7 + 3’ is 10.
The twin fallacies at the heart of this blunder in school math education are:

  1. Numbers are numbers, 7 is 7, only 7!
  2. We do not look at the units of the numbers being added when we add, so 7 + 3 = 10

The fact checks
We know better now. In this book we have come to appreciate that:

  1. Numbers come alive only when we have to express precise quantification; numbers express exact quantity of things. Thus, 7 by itself means nothing, unless it is expressed with a unit of counting. Thus, 7 can be USD 7 billion, 7 kg, 7 years, 7 men, 7 women, and anything else. It is similar for 3 as well.
  2. We know well that the mathematical operation of addition can happen only among quantities that have a like unit, i.e., the addends must be some quantity of same* things and must have the same unit.

Hence, the expression ‘7 + 3’ cannot be added till we clearly know the units of the addends, 7 and 3. And ‘7 + 3’ would be a meaningful expression in math only if 7 and 3 represent quantities of like things. Can you think of what could be the sum of adding 7 pencils and 3 erasers?
In fact, till Grade II, children must be encouraged to identify expressions such as ‘18 + 2’, ‘2 pens + 2’, or ‘3 + 4 erasers’, to be wrong, or incomplete to be able to add. Units are central to quantification; they are not dispensable. After all, 7 feet + 3 inches, is not equal to 10 feet or 10 inches.

From Grade III onwards, we should train children to assume that 7 and 3 are never without units, and the sum would be calculated as if the units of 7 and 3 are the same. However, the reality of ‘7 + 3’ being an incomplete statement, and ‘7 + 3 = 10’ being a wrong statement (because it may not be), does not change from Grade III onwards. We come to agree that we only avoid writing the units of numbers wherever we have the same unit of the addends. When the units are not the same, we should write the units. 
Grade III onwards, children can be taught to see numbers as abstract quantities too (and should be explained the benefits of abstract numbers and operations). This is convenient, faster, and accurate (7 + 3 = 10 is in fact true in infinite situations - 7 dogs + 3 dogs, 7 dining chairs and 3 sofa chairs, etc.). Besides, operations on numbers would get unnecessarily complicated if we always use numbers with units (it makes sense to remind that numbers always have unit of counting too, numerals do not have unit).
This aspect of quantities and numbers is also central to the issues of ‘word problems’ in math being a ‘nightmare’; but if we see quantities and operations this way, we can never go wrong in calculations.

Summary
The expression ‘7 + 3’ cannot be added till we clearly know the units of the addends, 7 and 3. 7 + 3 could be any quantity, 10 being just one possibility!

Additional details
Specifically, when asked to add ‘7 + 3’, children need to ask - “7 of what?” and “3 of what?” 7 + 3 could refer to a collection of 7 marbles in one pouch and 3 marbles in the other pouch, 7 tables in one room and 3 tables in another, 7 balls in the red basket and 3 balls in the green basket, etc.
Similarly, what could be the common description for dogs and cats?


7 dogs added to 3 cats 7 pets added to 3 pets make 10 pets

And for cars and motorcycles?


7 cars added to 3 motorcycles 7 vehicles added to 3 vehicles make 10 vehicles

*While we use the word ‘same’, rather than ‘similar’, to define the things being counted because all the things being counted can hardly ever be exactly the same as the chosen/typical unit; if we are counting chairs, the weight of the chairs counted as ‘1 chair’ is not relevant.
The sum of numbers is dependent on the unit of the addends.



Excerpted from the book ‘Foundations of Addition (Mathematics as a language)’ by Sandeep Srivastava and Saloni Srivastava