What is Patterning?
Through a study of current and relevant literature, algebra is a generalization of the ideas of arithmetic where unknown values and variables can be found to solve problems (Cox, 2003). This is a broad statement which encapsulates mathematics as whole; as "mathematics is the study of patterns and relationships" (Reys et al, 2012, p. 144). Algebraic thinking begins before school and continues right through high school (Cox, 2003). Ormond (2012) suggests that the early years are vitally important in developing pre algebraic and underlying arithmetic ideas of algebra. Through the development of foundational skills in algebraic thinking a students functional thinking in the real world situation grows (Wilkie, 2014).
The earliest stage of early algebraic concept is patterning. Creating constructing and describing patterns require problem-solving skills and constitute an important part of mathematics learning. Patterns can be based on geometric attributes (shapes, symmetry), relational attributes ( sequence, function), pphysical attributes (colour, size, texture, number) or affective attributes (like, happiness) (Reys et al, 2012).
There are four types of patterning (Van de Walle & Lovin, 2006):
Repeating patterns – Have a core (the shortest string of the pattern) which is repeated and continued over and over. These patterns usually use concrete materials, shapes, colours or sounds. They are then explained using some form of symbolism (eg. the alphabet) to represent the structure of the pattern. This is the beginning of algebraic reasoning.
The earliest stage of early algebraic concept is patterning. Creating constructing and describing patterns require problem-solving skills and constitute an important part of mathematics learning. Patterns can be based on geometric attributes (shapes, symmetry), relational attributes ( sequence, function), pphysical attributes (colour, size, texture, number) or affective attributes (like, happiness) (Reys et al, 2012).
There are four types of patterning (Van de Walle & Lovin, 2006):
Repeating patterns – Have a core (the shortest string of the pattern) which is repeated and continued over and over. These patterns usually use concrete materials, shapes, colours or sounds. They are then explained using some form of symbolism (eg. the alphabet) to represent the structure of the pattern. This is the beginning of algebraic reasoning.
Growing Patterns – These patterns involve a progression from step to step, with each new step related to the previous one. Recursive relationships are when the patterns changes from step to step and functional relationships is when there is a rule that determines the number of elements in a step.
Number Patterns – A string of numbers that follow a rule for determining how the string continues. This includes skip counting or odd/even numbers.
Relations/Functions – A relation is a correspondence between two sets of things (eg. the height of a bean plant). A function is a relationship in which each element is uniquely associated with another element (eg. algebraic equations and graphs).
In the Australian Curriculum, Mathematics calls for students to model, generalise and justify when they are learning about patterns (Reys et al., 2012). Through modelling students are asked to model and show the teacher a pattern or give them a problem where they have to work out the solution by using a pattern . Through justifying, students work on the problem and try to describe the pattern and their thinking. Having children justify their answers or their approaches to problems can help them understand the mathematics and gain confidence in their knowledge and skills. Finally, it is important that student are able to generalise and find a rule that generates the pattern.
Our content descriptor focus on Number Patterns formed by skip counting. Reys et al (2012) state that counting is facilitated by the counting process. Therefore skip counting follows a rule determined by the pattern, for example counting by 2's, 5's or any other value. The starting point and direction of skip counting is optional. Skip counting is a key development toward multiplication and division. Cox (2003) states that "patterns serve as the cornerstone of algebraic thinking" (p. 15). Hence, patterns enter into everyday life through skip counting; counting chocolate, sport scoring, counting money, etc. These everyday situations require skip counting that is linear and non-linear. Thus it is equally important learn to skip count both ways.
Skip counting like all mathematic concepts are open to complications and difficulties when trying to develop the concept of patterning and skip counting. These are a few examples found throughout the relevant literature (Bobis, Mulligan & Lowrie, 2013; Reys et al., 2012).
- Students often use symmetry alternate colours or finish their pattern with the starting unit rather then with a unit of repeat.
- Students may find it difficult to grasp that a pattern is made up of units of repeat, especially in regards to skip counting.
- Children may just be repeating a number sequence, rather then adding sets of groups to another group.
- Children may lose track of the pattern due to place value.
Relations/Functions – A relation is a correspondence between two sets of things (eg. the height of a bean plant). A function is a relationship in which each element is uniquely associated with another element (eg. algebraic equations and graphs).
In the Australian Curriculum, Mathematics calls for students to model, generalise and justify when they are learning about patterns (Reys et al., 2012). Through modelling students are asked to model and show the teacher a pattern or give them a problem where they have to work out the solution by using a pattern . Through justifying, students work on the problem and try to describe the pattern and their thinking. Having children justify their answers or their approaches to problems can help them understand the mathematics and gain confidence in their knowledge and skills. Finally, it is important that student are able to generalise and find a rule that generates the pattern.
Our content descriptor focus on Number Patterns formed by skip counting. Reys et al (2012) state that counting is facilitated by the counting process. Therefore skip counting follows a rule determined by the pattern, for example counting by 2's, 5's or any other value. The starting point and direction of skip counting is optional. Skip counting is a key development toward multiplication and division. Cox (2003) states that "patterns serve as the cornerstone of algebraic thinking" (p. 15). Hence, patterns enter into everyday life through skip counting; counting chocolate, sport scoring, counting money, etc. These everyday situations require skip counting that is linear and non-linear. Thus it is equally important learn to skip count both ways.
Skip counting like all mathematic concepts are open to complications and difficulties when trying to develop the concept of patterning and skip counting. These are a few examples found throughout the relevant literature (Bobis, Mulligan & Lowrie, 2013; Reys et al., 2012).
- Students often use symmetry alternate colours or finish their pattern with the starting unit rather then with a unit of repeat.
- Students may find it difficult to grasp that a pattern is made up of units of repeat, especially in regards to skip counting.
- Children may just be repeating a number sequence, rather then adding sets of groups to another group.
- Children may lose track of the pattern due to place value.
Prior Knowledge
An educator must not assume that all students come with this prior knowledge. They must use various forms of assessment to identify each student’s learner level. This is critical to being an effective educator.
In regards to the prior knowledge required for the concept of patterning and skip counting:
- Students come into year 1 (in regards to (ACMNA005)) with the ability to count to and from 20 at any starting point, subitise small groups and they are able to sort, classify familiar objects, then copy, continue and create patterns.
- Walle, Karp & Bay-Williams (2013) state that an understanding of place value past the ones column is essential for skip counting to move past the teens. Because counting by 1's, 2's, 5's or any other value must follow the rule in which numbers are formed. i.e. we count 20, 21, 22... 29, 30.
From this the students are transitioning their prior understanding of patterning to the new achievement standard for year 1.
- Investigate and describe number patterns formed by skip counting and patterns with objects (ACMNA018)
Key words of the content descriptor are to ‘investigate’ and ‘describe’. Therefore, it is our aim for the students to:
- understand the words investigate and describe
- be competent in investigating and describing patterns formed by skip counting and patterns with objects
In regards to the prior knowledge required for the concept of patterning and skip counting:
- Students come into year 1 (in regards to (ACMNA005)) with the ability to count to and from 20 at any starting point, subitise small groups and they are able to sort, classify familiar objects, then copy, continue and create patterns.
- Walle, Karp & Bay-Williams (2013) state that an understanding of place value past the ones column is essential for skip counting to move past the teens. Because counting by 1's, 2's, 5's or any other value must follow the rule in which numbers are formed. i.e. we count 20, 21, 22... 29, 30.
From this the students are transitioning their prior understanding of patterning to the new achievement standard for year 1.
- Investigate and describe number patterns formed by skip counting and patterns with objects (ACMNA018)
Key words of the content descriptor are to ‘investigate’ and ‘describe’. Therefore, it is our aim for the students to:
- understand the words investigate and describe
- be competent in investigating and describing patterns formed by skip counting and patterns with objects