The five major themes of discrete mathematics, as discussed in the K-12 Overview, are systematic listing, counting, and reasoning; discrete mathematical modeling using graphs (networks) and trees; iterative (that is, repetitive) patterns and processes; organizing and processing information; and following and devising lists of instructions, called "algorithms," and using them to find the best solution to real-world problems.

Despite their formidable titles, these five themes can be addressed with activities at the K-2 grade level which involve purposeful play and simple analysis. Indeed, teachers will discover that many activities they already are using in their classrooms reflect these themes. These five themes are discussed in the paragraphs below.

Activities involving systematic listing, counting, and reasoning can be done very concretely at the K-2 grade level. For example, dressing cardboard teddy bears with different outfits becomes a mathematical activity when the task is to make a list of all possible outfits and count them; pictured on the right are the six outfits that can be arranged using one of two types of shirts and one of three types of shorts. Similarly, playing any game involving choices becomes a mathematical activity when children reflect on the moves they make in the game.

An important discrete mathematical model is that of a network or graph, which consists of dots and lines joining the dots; the dots are often called vertices (vertex is the singular) and the lines are often called edges. (This is different from other mathematical uses of the term "graph.") The two terms "network" and "graph" are used interchangeably for this concept. An example of a graph with seven vertices and twelve edges is given below. You can think of the vertices of this graph as islands in a river and the edges as bridges. You can also think of them as buildings and roads, or houses and telephone cables, or people and handshakes; wherever a collection of things are joined by connectors, the mathematical model used is that of a network or graph. At the K-2 level, children can recognize graphs and use life-size models of graphs in various ways. For example, a large version of this graph, or any other graph, can be "drawn" on the floor using paper plates as vertices and masking tape as edges. Children might select two "islands" and find a way to go from one island to the other island by crossing exactly four "bridges." (This can be done for any two islands in this graph, but not necessarily in another graph.)

Children can recognize and work with repetitive patterns and processes involving numbers and shapes, using objects in the classroom and in the world around them. For example, children at the K-2 level can create (and decorate) a pattern of triangles or squares (as pictured here) that cover a section of the floor (this is called a "tessellation"), or start with a number and repeatedly add three, or use clapping and movement to simulate rhythmic patterns.

Children at the K-2 grade levels should investigate ways of sorting items according to attributes like color, shape, or size, and ways of arranging data into charts, tables, and family trees. For example, they can sort attribute blocks or stuffed animals by color or kind, as in the diagram, and can count the number of children who have birthdays in each month by organizing themselves into birthday-month groups.

Finally, at the K-2 grade levels, children should be able to follow and describe simple procedures and determine and discuss what is the best solution to a problem. For example, they should be able to follow a prescribed route from the classroom to another room in the school (as pictured below) and to compare various alternate routes, and in the second grade should determine the shortest path from one site to another on a map laid out on the classroom floor.

Two important resources on discrete mathematics for teachers at all levels are the 1991 NCTM Yearbook Discrete Mathematics Across the Curriculum K-12 and the 1997 DIMACS Volume Discrete Mathematics in the Schools: Making An Impact. Another important resource for K-2 teachers is This Is MEGA-Mathematics!

The cumulative progress indicators for grade 4 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in kindergarten and grades 1 and 2.

Experiences will be such that all students in grades K-2:

1. Explore a variety of puzzles, games, and counting problems.

• Students use teddy bear cutouts with, for example, shirts of two colors and shorts of three colors, and decide how many different outfits can be made by making a list of all possibilities and arranging them systematically. (See illustration in K-2 Overview.)

• Students use paper faces or Mr. Potato Head type models to create a "regular face" given a nose, mouth, and a pair of eyes. Then they use another pair of eyes, then another nose, and then another mouth (or other parts) and explore and record the number of faces that can be made after each additional part has been included.

• Students read A Three Hat Day and then try to create as many different hats as possible with three hats, a feather, a flower, and a ribbon as decoration. Students count the different hats they've made and discuss their answers.

• Students count the number of squares of each size (1 x 1, 2 x 2, 3 x 3) that they can find on the square grid below. They can be challenged to find the numbers of small squares of each size on a larger square or rectangular grid.
  

• Students work in groups to figure out the rules of addition and placement that are used to pass from one row to the next in the diagram below, and use these rules to find the numbers in the next few rows.

  1
  1 1
  1 2 1
  1 3 3 1
  1 4 6 4 1

  In this diagram, called Pascal's triangle, each number is the sum of the two numbers that are above it, to its left and right; the numbers on the left and right edges are all 1.

• Students cut out five "coins" labeled 1 cent, 2 cents, 4 cents, 8 cents, and 16 cents. For each number in the counting sequence 1, 2, 3, 4, 5, ... (as far as is appropriate for a particular group ofstudents), students determine how to obtain that amount of money using a combination of different coins.

• Students play simple games and discuss why they make the moves they do. For example, two students divide a six-piece domino set (with 0-0, 0-1, 0-2, 1-1, 1-2, and 2-2) and take turns placing dominoes so that dominoes which touch have the same numbers and so that all six dominoes are used in the chain.

2. Use networks and tree diagrams to represent everyday situations.

• Students find a way of getting from one island to another, in the graph described in the K-2 Overview laid out on the classroom floor with masking tape, by crossing exactly four bridges. They make their own graphs, naming each of the islands, and make a "from-to" list of islands for which they have found a four-bridge-route. (Note: it may not always be possible to find four-bridge-routes.)

• Students count the number of edges at each vertex (called the degree of the vertex) of a network and construct graphs where all vertices have the same degree, or where all the vertices have one of two specified degrees.

• On a pattern of islands and bridges laid out on the floor, students try to find a way of visiting each island exactly once; they can leave colored markers to keep track of islands already visited. Note that for some patterns this may not be possible! Students can be challenged to find a way of visiting each island exactly once which returns them to their starting point. Similar activities can be found in Inside, Outside, Loops, and Lines by Herbert Kohl.

• Students create a map with make-believe countries (see example below), and color the maps so that countries which are next to each other have different colors. How many colors were used? Could it be done with fewer colors? with four colors? with three colors? with two colors? A number of interesting map coloring ideas can be found in Inside, Outside, Loops and Lines by Herbert Kohl.
  

3. Identify and investigate sequences and patterns found in nature, art, and music.

• Students use a calculator to create a sequence of ten numbers starting with zero, each of which is three more than the previous one; on some calculators, this can be done bypressing 0 + 3 = = = . . . , where = is pressed ten times. As they proceed, they count one 3, two 3s, three 3s, etc.

• Students "tessellate" the plane, by using groups of squares or triangles (for example, from sets of pattern blocks) to completely cover a sheet of paper without overlapping; they record their patterns by tracing around the blocks on a sheet of paper and coloring the shapes.

• Students listen to or read Grandfather Tang's Story by Ann Tompert and then use tangrams to make the shape-changing fox fairies as the story progresses. Students are then encouraged to do a retelling of the story with tangrams or to invent their own tangram characters and stories.

• Students read The Cat in the Hat or Green Eggs and Ham by Dr. Seuss and identify the pattern of events in the book. Students could create their own books with similar patterns.

• Students collect leaves and note the patterns of the veins. They look at how the veins branch off on each side of the center vein and observe that their branches are smaller copies of the original vein pattern. Students collect feathers, ferns, Queen Anne's lace, broccoli, or cauliflower and note in each case how the pattern of the original is repeated in miniature in each of its branches or clusters.

• Students listen for rhythmic patterns in musical selections and use clapping, instruments, and movement to simulate those patterns.

• Students take a "patterns walk" through the school, searching for patterns in the bricks, the play equipment, the shapes in the classrooms, the number sequences of classrooms, the floors and ceilings, etc.; the purpose of this activity is to create an awareness of all the patterns around them.

4. Investigate ways to represent and classify data according to attributes, such as shape or color, and relationships, and discuss the purpose and usefulness of such classification.

• Students sort themselves by month of birth, and then within each group by height or birth date. (Other sorting activities can be found in Mathematics Their Way, by Mary Baratta-Lorton.)

• Each student is given a card with a different number on it. Students line up in a row and put the numbers in numerical order by exchanging cards, one at a time, with adjacent children. (After practice, this can be accomplished without talking.)

• Students draw stick figures of members of their family and arrange them in order of size.

• Students sort stuffed animals in various ways and explain why they sorted them as they did. Students can use Tabletop, Jr. software to sort characters according to a variety of attributes.

• Using attribute blocks, buttons, or other objects with clearly distinguishable attributes such as color, size, and shape, students develop a sequence of objects where each differs from the previous one in only one attribute. Tabletop, Jr. software can also be used to create such sequences of objects.

• Students use two Hula Hoops (or large circles drawn on paper so that a part of their interiors overlap) to assist in sorting attribute blocks or other objects according to two characteristics. For example, given a collection of objects of different colors and shapes, students are asked to place them so that all red items go inside hoop #1 and all others go on the outside, and so that all square items go inside hoop #2 and all others go on the outside. What items should be placed in the overlap of the two hoops? What is inside only the first hoop? What is outside both hoops?
  

  This is an example of a Venn diagram. Students can also use Venn diagrams to organize the similarities and differences between the information in two stories by placing all features of the first story in hoop #1 and all features of the second story in hoop #2, with common features in the overlap of the two hoops. A similar activity can be found in the Shapetown lesson that is described in the First Four Standards of this Framework. Tabletop, Jr. software allows students to arrange and sort data, and to explore these concepts easily.

5. Follow, devise, and describe practical lists of instructions.

• Students follow directions for a trip within the classroom - for example, students are asked where they would end up if they started at a given spot facing in a certain direction, took three steps forward, turned left, took two steps forward, turned right, and moved forward three more steps.

• Students follow oral directions for going from the classroom to the lunchroom, and represent these directions with a diagram. (See K-2 Overview for a sample diagram.)

• Students agree on a procedure for filling a box with rectangular blocks. For example, a box with dimensions 4"x4"x5" can be filled with 10 blocks of dimensions 1"x2"x4". (Linking cubes can be used to create the rectangular blocks.)

• Students explore the question of finding the shortest route from school to home on a diagram like the one pictured below, laid out on the floor using masking tape, where students place a number of counters on each line segment to represent the length of that segment. (The shortest route will depend on the placement of the counters; what appears to be the most direct route may not be the shortest.)
  

• Students find a way through a simple maze. They discuss the different paths they took and their reasons for doing so.

• Students use Logo software to give the turtle precise instructions for movement in specified directions.

References

Baratta-Lorton, Mary. Mathematics Their Way. Menlo Park, CA: Addison Wesley, 1993.

Casey, Nancy, and Mike Fellows. This is MEGA-Mathematics! - Stories and Activities for Mathematical Thinking, Problem-Solving, and Communication. Los Alamos, CA: Los Alamos National Laboratories, 1993. (A version is available online at http://www.c3.lanl.gov/mega-math)

Geringer, Laura. A Three Hat Day. New York: Harper Row Junior Books, 1987.

Kenney, M. J., Ed. Discrete Mathematics Across the Curriculum K-12. 1991 Yearbook of the National Council of Teachers of Mathematics (NCTM). Reston, VA: 1991.

Kohl, Herbert. Insides, Outsides, Loops, and Lines. New York: W. H. Freeman, 1995.

Murphy, Pat. By Nature's Design. San Francisco, CA: Chronicle Books, 1993.

Rosenstein, J. G., D. Franzblau, and F. Roberts, Eds. Discrete Mathematics in the Schools: Making an Impact. Proceedings of a 1992 DIMACS Conference on "Discrete Mathematics in the Schools." DIMACS Series on Discrete Mathematics and Theoretical Computer Science. Providence, RI: American Mathematical Society (AMS), 1997. (Available online from this chapter in http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/.)

Seuss, Dr. Cat in the Hat. Boston, MA: Houghton Mifflin, 1957.

Seuss, Dr. Green Eggs and Ham. Random House.

Tompert, Ann. Grandfather Tang's Story. Crown Publishing, 1990.

Software

Logo. Many versions of Logo are commercially available.

Tabletop, Jr. Broderbund Software. TERC.

On-Line Resources

http://dimacs.rutgers.edu/archive/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.