The sentence “3 girls + 2 boys = 3 kids + 2 kids” is not a number sentence.

First, let’s recall the what a number sentence is.  For that, we have to briefly describe numerical expression. (See “What is a variable? Part II: Expressions.”)  A numerical expression is an algebraic expression involving only numbers (no variables) that evaluates to a single number.  More generally, it is an expression that denotes a single quantity (see “What is a Quantity“).

A number sentence is a statement of equality between two numerical expressions.

In mathematics, a number sentence is a complete thought and must be either true or false. That means that both numerical expressions must be valid expressions (not 3/0, for example) and that the statement can be clearly and unequivocally assigned a truth value.  Undetermined sentences, even if they contain numbers, are not number sentences.

Back to: 3 girls + 2 boys = 3 kids + 2 kids.

This sentence is undetermined and cannot be given a truth value of either true or false.  To see why, ask, “Given 3 kids and 2 kids, would I know exactly which kids are girls and which are boys?”  It may be that, of the 3 kids and 2 kids, 4 of them are girls and 1 is a boy.  I don’t know whether “3 kids + 2 kids” means “4 girls + 1 boy” or “3 girls + 2 boys,” so I can’t tell whether the sentence is true or false.  The use of the = symbol in a number sentence should always be used for definitive statements about numbers or quantities—that all equations involving only numbers or quantities have well-defined truth or false values.  By starting with this premise in 1st grade, and maintaining the same level of precision in grades 1-8, students develop one coherent picture of the use of the = symbol from the very beginning.

Maybe you read the previous paragraph and registered the thought, “Well, a word problem gives context to the sentence.”  This is absolutely true.  But the beauty of a mathematical number sentence involving the = symbol is that it is always true or always false regardless of any surrounding context.  We are trying to build that sense of beauty and absolute resoluteness in the minds of our students as they progress through the elementary grades.  The binary true-false nature of number sentences is crucial for students to make sense out of the meaning of a solution set in algebra—in fact, much of algebra basically rests on the meaning and use of the = symbol and true-false number sentences.

Still not convinced?  Here is a pedagogical reason to reconsider for first graders: too much is going on in that (non-number) sentence.  If we ignore the main point above for the moment, and say that “we are just renaming units from boys and girls to kids,” then, mathematically, the sentence above is like the following (non-1st grade) true number sentence,

3 x (7/1) + 2 x 700% = 3 sevens + 2 sevens,

since 7/1 and 700% are two ways to name seven.  In this number sentence you can see that there is a lot to digest since you have to recall several definitions at the same time.  While the boys-and-girls sentence is much simpler, we should be careful to view the original sentence through the eyes of a first grader and how they might perceive it (which may be more like the more complicated sentence for us).  However, that doesn’t mean you can’t have a discussion about number sentences with first graders.  Excellent first grade teachers have told me that first graders can handle a discussion about the difference between a sentences with numbers and a number sentence, and I believe them.

CHANNEL: Engineering School Mathematics