Number Sense and Patterns/Impact of Higher Order Cognition

A student’s number sense includes the ability to understand both number meanings (as in the association of the digit 7 with seven items) and quantitative concepts (for example, that the symbol ‘+’ signifies ‘add’ which means ‘combine‘). Such understanding is promoted by a student’s ability to relate a concrete symbol to a concept or idea.

The awareness of basic math patterns is closely related to number sense. Pattern awareness involves an appreciation of basic rules and patterns that exist in math, knowing, for example, that multiplying a number by zero yields zero, while multiplying by 1 yields the original number.

Both number sense and the ability to make simple generalizations from patterns are aided by a student’s higher order thinking skills, including the ability to think on an abstract level.

Here are some strategies to develop and strengthen students’ use of math facts by building number sense and pattern awareness.

Helpful Hints

  • Encourage students to use a strategic approach for practicing and recalling math facts. For example, "I don‘t know 7 X 6, but I do know that 7 X 5 = 35, so one more 7 makes 42; or I know that 7 X 7 = 49, so one less 7 makes 42."
  • Assess and review students’ understanding of the commutative property of addition and multiplication to build awareness of number patterns.
  • The commutative property of addition tells us that regardless of the order in which the same numbers are combined, the sum is the same (e.g. 3 + 4 = 4 + 3).
  • Likewise, the commutative property of multiplication tells us that regardless of the order in which the same numbers are multiplied, the product remains the same (e.g. 7 X 3 = 3 X 7).
  • Grasping the commutative property not only enhances students‘ conceptual understanding in math, but provides them with the thrilling notion that such patterns or "tricks" make it easier to learn math, for if they know 3 X 5, they also know 5 X 3, etc., essentially cutting the number of math facts to ‘memorize‘ in half!
  • When teaching math facts, move students through a sequence of understanding: Concrete’ Semi-abstract‘ Abstract.
  • The concrete level of understanding involves using concrete objects or manipulables to learn a symbolic process. For example, to work out the problem 3 X 2, you might place three paper plates on the floor, put two blocks on each plate, and finally, count the number of blocks to arrive at the understanding that ‘3 groups of 2 blocks equals 6 blocks.‘
  • The semi-abstract level of understanding involves the use of pictures or drawings to represent numbers in the symbolic process. For example, you might work out the problem 3 X 2 by making marks or dots next to the numerals, so that the student links ‘3 groups of 2 dots equals 6 dots’ with the computation ‘3 times 2 = 6’.
  • The abstract level of understanding involves the use of numerals as representatives of the symbolic process, that is, using numbers only to solve the computational problem 3 X 2 = 6. In order to master the abundance of addition, subtraction, multiplication and division facts, students must progress to this abstract level.
  • Build students’ abilities with math facts through the use of number families. Number families are clusters of related math facts. For example, have students learn the facts 2+3=5, 3+2=5, 5-2=3, and 5-3=2 grouped together as a ‘family’. Encourage students to write problems horizontally and vertically, as well as to say them aloud.