Mathematical Errors in Textbooks

The Misuse of the Equal Sign in Elementary School and Middle School Division

Consider a typical Division with Remainder  lesson. There is sometimes a subtle misuse of the equal sign. For instance, we may see an equation such as 9,317 ÷ 95 = 98 R7. But what does that mean? Let us first consider equivalence relations before picking apart this matter of using 98 R7 to represent the answer to a division problem.

Equality is an equivalence relations. An equivalence relation R on a set is defined as follows. Let a, b, c belong to a set S. Then, we define an equivalence relation R on S if the following properties hold:

11.      aRa (Reflexive Property)

22.     aRb implies bRa (Symmetry Property)

33.    If aRb and bRc, then aRc (Transitive Property).

Now, let R be equality. Since middle school math involves rational numbers, let a, b, c be rational numbers. Then

11'.    a = a (Reflexive Property)

22'.     a = b implies b = a (Symmetry Property)

33'.  If a = b and b = c, then a = c (Transitive Property).

Now, let us return to our division problem.

9,317 ÷ 95 = 98 R7.

So, let a = 9,317 ÷ 95 = 98 R7. Now, using the same representation of an answer we have that 

           987 ÷ 10 = 98 R7. 

Let

          987 ÷ 10 = 98 R7 = b. 

Then, by the Transitive Property of Equality,

           a = 98 R7, 

and we just decided that 

            98 R7 = b, 

in the previous line. Then, by the Transitive Property of Equality, we have that a = b. But in actuality, 9,317 ÷ 95 = = a; and 987 ÷ 10 =  = b, showing

a ≠ b. Now, we have that a = b and a ≠ b which is a contradiction.

This issue of the meaning of equality became a teaching point in this author’s 6^th grade math class. The advice for the students was to use an arrow (à) instead of an equal sign when they got a remainder in solving a division problem. They would later learn how to complete the division process to get a decimal fraction instead a remainder. So, instead of, for instance, 987 ÷ 10 à 98 R7, students would write 987 ÷ 10 = 98.7, after pursuing a solution process. The problem with the Go Math textbook is a problem of interpretation of the Common Core state standards (CCSS). New York’s CCSS 6.NS. B.2 states that students should be able to, “[f]luently divide multi-digit numbers using the standard algorithm”