Pattern
Mhairi (Vi) Maeers and Rick Seaman
Faculty of Education
University of Regina

   "Pattern is all around us" and "the world is composed of many intricate patterns" are expressions we sometimes hear from students and colleagues. Pattern is certainly very much a part of our everyday lives, but it is the recognition and application of pattern which enables us to survive more than the fact that pattern 'exists'. One may even wonder if pattern does exist outside our personal interpretation, understanding and application of it. In this introduction to the latest issue of Ideas and Resources for Teachers of Mathematics we would like to introduce the concept of pattern through some ideas from the literature and through a recent personal 'pattern' experience.

Background

Mathematicians have been described as makers of patterns of ideas (Billstein, Libeskind & Lott, 1993, p. 4) and mathematics has been considered the classification and study of all possible patterns (Sawyer, 1963, in Orton, 1993, p. 8). Patterns created by one mathematician may well lay the groundwork for another (Polya has stated that Rene Descartes' ideas helped him with his work on problem solving). Searching for and identifying patterns has long been considered a very important problem solving heuristic. Determining patterns is crucial to police investigation, to household plumbing problems, to young children's daily routines. Anything 'out of the pattern,' for example taking a nap before lunch instead of after lunch can throw a young child's day askew. We tend to live by personally 'inflicted' patterns in order to survive the stress of daily life. Once something becomes a commonly accepted pattern, then it no longer needs to be calculated; one has established order out of what may appear chaotic--but we know now that chaos has an implied order of its own.
  The ability to perceive pattern is very much at the core of mathematical understanding. For the very young child, the ability to sort objects such that all the buttons are in one pile and all the pennies are in another demonstrates an early notion of pattern. Collections of objects, such as stamps, hockey cards, stickers and shells signify to the collector (and the observer) an understanding of what makes a stamp a stamp, or a shell a shell. In other words shells have certain unique properties that make them distinct from stamps, or stones etc. It is the identification of these unique properties or attributes that set objects apart in groups. In pre-kindergarten classrooms, kindergarten and the early grades it is quite common to find children working with sets of objects, sorting, classifying, ordering, lining them up in one-to-one correspondence, making bead strings, acting out patterns, and then using these same objects for early number development, early work with operations and with graphing. Working with sets and the attributes of sets can consume a large percentage of mathematical activity in the early grades. As Cathcart, Pothier, and Vance (1997) indicate "the recognition of patterns is a basic skill that enhances the development of mathematics concepts" (p. 63). They also state that concrete experiences with pattern should precede working with number patterns.
  The ability to determine, recognize, and apply pattern, if begun with relevant concrete experiences, will be easily transferred to working with number patterns (e.g., multiplication tables, equivalent fractions, commutative property, operational relationships), problem solving, such as looking for a pattern as a solution strategy, and into algebra which focuses on the ability to identify, extend, and create patterns with unknowns and variable expressions. Number pattern and mathematical relationships (such as missing addend problems) can be used to help children understand the concept of variables and the use of numbers instead of (or with) letters (e.g., 5 + x = 7).
  In algebra word problem-solving, the use of worked-out examples highlights by means of pattern recognition the underlying structures of problems. For example, if students study the worked-out examples of proportional reasoning questions whose surface features are all different, the time taken to study these worked-out examples would be less than solving the equivalent problems and the pattern with respect to the method of solution would be evident. This notion of surface feature and underlying (pattern) structures can be found in an article by Robins Shani and Richard Mayer (1993), "Schema Training in Analogical Reasoning."

Pattern Resources

The National Council of Teachers of Mathematics [NCTM] (1989) in The Curriculum and Evaluation Standards for School Mathematics, outline two standards for working with patterns. Standard 13 for grades K-4 (Patterns and Relationships) states that the "mathematics curriculum should include the study of patterns and relationships so that students can
  recognize, describe, extend, and create a wide variety of patterns;
  represent and describe mathematical relationships;
  explore the use of variables and open sentences to express relationships" (p. 60).
For more information about this standard please see the following website:

http://www.enc.org/reform/journals/ENC2280/nf_28060s13.htm

Standard 8 at the grades 5-8 level (Pattern and Functions) informs us that "the mathematics curriculum should include explorations of patterns and functions so that students can
  describe, extend, analyze, and create a wide variety of patterns;
  describe and represent relationships with tables, graphs, and rules;
  analyze functional relationships to explain how a change in one quantity results in a change in another;
  use patterns and functions to represent and solve problems" (p. 98).
For further information on this standard please see the following website:

http://www.enc.org/reform/journals/ENC2280/nf_28098s8.htm

There is no specific pattern standard at the grades 9-12 level, but Standards 5 and 6 found in the table of contents for the on-line Standards document indicate a need for a fundamental understanding and application of pattern. These standards can be located at:

http://www.enc.org/reform/journals/ENC2280/nf_280dtoc1.htm

From 1991-93 the NCTM published two Addenda Series documents which concentrated on Patterns (Kindergarten through Grade 6) and Patterns and Functions for Grades 5-8. Both of these documents present many practical activities in which the concept of pattern plays a major role and is used in application to other mathematical concepts.
  A delightful book which we recently discovered is entitled Making Patterns (1992), a Scholastic publication, written by Helen Pengelly. This book focuses on pattern and presents many practical pattern tasks for young children to engage in. Each task is accompanied by illustrations of children constructing the pattern activity, a running commentary on the process and product of children's work, and ideas for teacher observation and follow-up work. Pengelly writes, "the potential for experiences with pattern to develop mathematical understanding is profound. In fact, pattern can feature as a prominent and unifying characteristic of the curriculum" (p. 4).

Pattern Websites

Border Pattern Gallery:

Grade 1 Pattern Activity:

Meteorology--Does Weather Happen Randomly:

The Language of Mathematics:

L-systems--Fractals:

What is a Fractal?

The Golden mean:

Penrose Tilings and the Golden Mean:

Origami Tessellations:

Pattern Pals:

3-D Modeling, Fabrication and Illustration:

Tessellations Project (and links to other tessellation sites):

Puzzles Involving the Fibonacci Series:

Teaching Strategic Skills:

The World of M.C. Escher:

Classification of Patterns:

You can view many more pattern sites at the Math Forum--> Steve's Dump at http://forum.swarthmore.edu/~steve/. Simply click on "Quick Search" and enter the word pattern. The sites listed above are ones that we visited and found useful as a result of this search. At our own Math Central, there are a number of resources that focus on pattern. They can be found in the Resource Room/Keyword/Pattern.

A Pattern Project

Pattern can be interpreted through all disciplines and can connect all disciplines, an idea similar to the last sentence in the above quote by Pengelly. Just recently a group of five elementary education professors worked with 60 third year students on a semester-long curriculum integration project. The focus of study was PATTERN and our task was to: (1) determine how pattern could be represented and interpreted through each discipline [mathematics (Vi Maeers and Liz Cooper), language arts (Carol Fulton), social studies (Kathryn McNaughton), and the arts (Nancy Browne)], (2) demonstrate to our students our collective perspectives on pattern (through a Hyperstudio presentation), (3) take our students through a variety of pattern-focused activities within each subject area, and (4) work with groups of children in a local elementary school to help them understand different ways of thinking about pattern.
  At the beginning of our project (September, 1997), as each member of the curriculum project team tried to define pattern it became clear that each of us had a different definition and indeed quite different ways of thinking about pattern within our separate subject areas. This is in agreement with Tahta, 1992, who stated that the word pattern was used differently in different contexts. Dictionary definitions did not help us reach a consensus. We decided to develop a 'composite' of our definitions and present them to students within the framework of a multimedia (Hyperstudio) project. As each of us worked on this project, collaborating constantly with at least one other member of the team, and talking through our examples to understand how our examples illustrated our sense of pattern within our subject area, we actually began to agree on some big ideas surrounding pattern and extensions from those big ideas.
  We began with a statement like "pattern extends from a 'design' such that its composite parts can be distinguished as separate from its surroundings; a unity in time, shape, place." This statement was then illustrated through a mathematical example, a social studies one, a language arts one, and an arts one. Our next big idea, "pattern is a design altered by transformation (translation, rotation, reflection)," was also illustrated by examples from each subject area. Another big idea was that "pattern can be an extension of an original design, a replication of it, or a recursion of it." Our last big idea was that "pattern can be a disruption of an original design" but implicit in this idea is the need to be able to distinguish the original design to be able to perceive the disruption. This last big idea leads us into Chaos Theory and Fractals--an interesting and challenging notion to illustrate in each subject area.
  The big ideas outlined above were identified and examined within each subject area first and then, by extrapolation, across the subjects. Our intent was for our students to understand these big ideas and how they could be represented in each of the above subject areas. The mathematics part of the project highlighted tessellations and Escher patterns, social studies highlighted where and how people live in different parts of the world, language arts focused on story and word patterns, and arts education focused on patterns in song and action. Each part of the Hyperstudio presentation highlighted a slightly different focus, but the big ideas were the same. Our Hyperstudio project will soon be available on Math Central--so watch for it. Our project and this introduction close with the following words:
   "An ability to see pattern in our world provides us a base to predict when things do and could change, provides us with a foundation of (and for) order, and provides us with an ability to see boundaries in space, place and time. We use our ability to perceive pattern to enable us to make sense of things in our world, to understand society better."
We hope that as you read this latest issue of Ideas and Resources for Teachers of Mathematics you will find suggestions for activities at different grade levels that enable both you and your students to explore both the process and product of pattern.

References
Billstein, R., Libeskind, S., and Lott, J.W. (1993). 
 A Problem Solving Approach to Mathematics for Elementary School Teachers. Don Mills, ON: Addison-Wesley.
Cathcart, W. G., Pothier, Y., and Vance, J. H. (1997). 
 Learning Mathematics in Elementary School. Scarborough, ON: Prentice Hall, Allyn and Bacon Canada.
National Council of Teachers of Mathematics (1989). 
 The Curriculum and Evaluation Standards for School Mathematics. Reston, VA: NCTM.
National Council of Teachers of Mathematics (1991). 
 The Curriculum and Evaluation Standards for School Mathematics, Addenda Series--Patterns and Functions. Reston, VA: NCTM.
National Council of Teachers of Mathematics (1992). 
 The Curriculum and Evaluation Standards for School Mathematics, Addenda Series--Patterns. Reston, VA: NCTM.
Orton, J. (1993). 
 What is pattern? Mathematics in School, 22, 2, pp. 8-10.
Pengelly, H. (1992). 
 Making Patterns. Toronto, ON: Ashton Scholastic.
Robins, S. and Mayer, R. E. (1993). 
 Schema training in analogical reasoning. Journal of Educational Psychology, 85, 3, pp. 529-538.
Tahta, D. G. (1991) 
 A universal activity. Strategies, 1, 3.


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