Foundational Understanding For Mathematical Growth

Here is an article written by one of our instructional specialist, Kim Bowen, who won the Presidential Awards for Excellence in Mathematics and Science Teaching in 2014. We hope you enjoy!

What mathematical ideas are fundamental to understanding addition and subtraction?

Mathematics success with the concept of addition and subtraction is directly impacted by the early experiences children have with numbers. The prerequisite ideas fundamental for children to be successful with addition and subtraction are number sense and computational fluency. Children build addition and subtraction fluency by learning relationships between number combinations to ten. This foundation is crucial for children to be successful when working with larger numbers. If children are deprived of building fluency to ten, and prematurely add and subtract large numbers, they compromise solid understanding and tend to rely on memorized rules, procedures, and counting to solve problems. To understand the concept of addition and subtraction to ten, children must first be competent in counting, recognizing small quantities instantly, and understanding the relationships between numbers.

Perhaps most basic to the development of addition and subtraction to ten is becoming a competent counter. Counting is finding out “how many.” Children generally enter school able to recite the forward counting sequence to ten or beyond, but do not understand counting. To develop a competency in counting correctly, children must demonstrate correct rote sequence, one-to-one correspondence, cardinality (understand the last number identifies how many in a group), be able to keep track of an unorganized pile of objects, remember the quantity counted, know one more and one less than a number without counting, and make connections between the symbols, names, and quantities they represent.

Children progress through predictable, sequential, developmental stages when learning to count. Being mindful as to where children are in their development of counting allows for instructional adjustments. For example, a kindergarten child may display all the skills of being a competent counter with quantities up to “7,” while a classmate may display mastery of quantities above “12.” A classroom teacher’s instructional design must be sensitive and flexible in order to meet the progressing needs of all learners. When counting is experienced in a variety of settings, using tasks that are adjustable, children’s proficiency in counting larger quantities grows.

In addition to becoming a competent counter, children must also be able to identify smaller quantities within larger quantities without counting. This concept is called “subitizing.” A child’s development of number sense is dependent on their ability to subitize. The ability to subitize groups to “5” leads to a stronger sense of numbers to “10.” When children instantly recognize dot patterns on a die, they have perceptual subitizing. Conceptual subitizing paves the way for developing addition and subtraction concepts beyond ten, as children recognize smaller groups of dots within larger quantities to find the total. A child’s ability to instantly recognize dot displays with large quantities is impacted by the spatial arrangement of the dots. In order for this skill to develop, it must be nurtured and practiced with children through Number Talks. A Number Talk is used in a daily routine to promote accuracy, efficiency and flexibility with numbers, while building computational fluency using number relationships.

Finally, knowing number relationships is essential for a child’s future skill in addition and subtraction. Knowing number relationships requires children to understand the relative size of numbers and its differences. When children compare numbers of isolated amounts, such as “3 is more than 1” they are comparing two separate sets. The goal is to be able to tell how many more or how many less one number is when compared to another. The concept of adding “1 more” is

typically learned first because children are more familiar with the forward counting sequence. Taking “1 away,” or counting backwards, is more challenging due to the difficulty children have thinking about what came before rather than what came after. Children show evidence of understanding number relationships when they change one number to another and tell how many more were added or how many were taken away without having to count. Children’s ability to understand number relationships is influenced by the size of and the difference between the numbers with which they work. For example, children may know to add “2” to “4” to get “6” but not yet know to add “2” to “5” to get “7.” This is because smaller amounts were not recognized within the larger quantities. Understanding number relationships to ten has a proven impact on a child’s future ease with addition and subtraction. When children comfortably manipulate all the parts of numbers to ten, they build a solid foundation for solving equations to twenty and beyond.

Why are number relationships important for students to learn and how does this relate to more complex concepts that students will encounter as they continue through school?

Understanding number relationships is essential for future work in mathematics; specifically in algebra. Success in algebra is based on knowing and describing relationships. As children advance through elementary grades their work with place value takes precedence over their work to continue developing number relationships. However, continual development of number relationships should not be separated from children’s work with place value. Ironically, the intricacy associated with understanding place value goes back to counting. Children must first recognize place value as organizing quantities into groups of tens and ones and be able to count the groups as single units and extras to know “how many.” When learning about place value children must hold two concepts in their brain concurrently. They must recognize that ten is both one group and ten ones. As their knowledge of place value deepens, children begin to grasp the structure of numbers as tens and ones. Children who understand the relationship of numbers to ten and the structure of beginning place value concepts can solve 56 + 26, verbally and/or on paper, effortlessly. For example, children will break apart “26” into “20” and “6,” add “56 + 20” to get “76”, break apart the “6” into “4” and “2,” use the “4” to create another group of ten by adding it to the “6” in “56” and add the “2” extras for a total of “82.” While this may appear to be “taking the long road” to solving an equation, it is a true indication of understanding place value, computation and the development of number sense. The ability to mentally manipulate numbers and use landmarks by creating combinations of (and later multiples of) 10 to calculate and solve problems, is the defining skill of successful math learners.

What are the misconceptions or misunderstandings that students typically have with regard to this topic or concept.

Children’s understanding of addition and subtraction is compromised by misunderstanding the processes represented by the symbols as well as the use of timed tests and computerized drill and practice programs that are used to reinforce basic fact mastery. When children fail to master basic facts, it is assumed they need more drill. This implies children learn what is represented by the symbol simply by working with the symbol. An example of this is the equal sign. Children think of the equal sign as a symbol to write an answer. Yet it means “is the same as.” Using calculators before the concept is truly understood reinforces this misconception. Intended to be used as a tool to help children be more efficient with math processes, when “=” is pressed an answer is provided.

Separating the processes of addition and subtraction leads to confusion between the plus and minus signs as well as regrouping work in place value. Questions such as “Do I have to borrow?” or “Do I need to carry the 1?” confirm children are performing memorized procedures rather than looking for number relationships. Children’s correct use of mathematical language does not necessarily demonstrate understanding. Misunderstandings of math language hinder children’s grasp of subtraction when working with place value. For example, when solving 52-26, children will take “2” away from “6” because “6” is more than “2.” This is associated with the commonly used language “take the smaller number away from the larger number,” leading to incorrect answers and interference’s when negative numbers are taught. The only difference between first graders and third graders and their lack of number sense is the size of the numbers with which they work. Teaching memorized processes and procedures to get quick right answers on timed tests overlooks the need to first develop conceptual understanding of number relationships. Furthermore, this omission may be the factor that weakens the ability to layer complex sequences for higher math learning. A strong foundation in number relationships to 10 is essential if children are to be successful in future mathematics.

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