Multiplication Tables and Mathematical Rigor

PROBLEM:  How many vans does it take to transport 63 children on a field trip to a museum if each van carries 9 children?

Array 63

Power-Start Goal #2, knowing the multiplication tables from memory by the end of third grade positions a student well for the upward climb toward math proficiency by 8th grade.  Moreover, the  word “knowing” here is a big one meaning not just rote memorization but deep conceptual understanding.  A proficient student can use quick recall of the fact (7 x 9 = 63) and understanding of the relationship between multiplication and division to know that it will take 7 vans to transport the 63 children.

Memorizing the Multiplication Tables: Mathematical Rigor

World-class tests available today are smart tests.  Therefore, when a student answers a question correctly, the assessment system is programmed to follow up with a more difficult question on the same topic.  In my blog post just before this one (World-Class Test Question 2), the student was to determine the total number of trading cards in an array of 5 rows and 3 columns.  The multiplication fact needed, 5 × 3 = 15, is an relatively easy one. Even if the student can’t recall the answer of 15 instantly, the strategies of skip counting (5, 10, 15) or using hands and fingers could work.  So, a smart test might follow up a question involving the multiplication fact 5 × 3 = 15 with the more challenging problem solved by knowing quickly that  63 ÷ 9 = 7.

Due to the great amount of time and effort required to teach and learn all the multiplication facts, we may be lured into saying, “Why bother? Isn’t that what we have calculators for?”   However, knowing from memory the multiplication facts within 100 provides a good example of what is meant by “rigor” in mathematics at this level.   In general,  rigor goes hand in hand with deep understandings of math concepts, and thus, promotes flexible and varied problem-solving strategies.  Deep understanding and flexibility in thinking about the math fact 7 × 9 = 63 can, in future grades, release an unlimited flow of math knowledge.  Here’s a table of some of the possibilities.

WCTQ 3 7 x 9

Early Examples of Multiplication and Division

So remember that providing early examples of small-number multiplication and division scenarios is not acceleration or pushing children too far forward.  Rather, these experiences develop children’s innate reasoning abilities during the years they are ready and eager to use them.  Here are a few examples.

  • There are 4 children.  How many cookies are needed if each child is to receive 2 cookies?  Now, suppose there are 10 cookies and 4 children. Draw a picture (or have the child draw a picture) describing how the 10 cookies can be shared fairly.
  • How many rows on a bus with 4 seats in each row will be needed for 16 children?
  • If cards are dealt in a game, how many cards will each child receive if there are 3 children and 20 cards.  How many will be left over?



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