Monday, March 19, 2007

Our Learning Journey

We decided to research the topic of patterns in the primary classroom for several reasons. We thought that children would enjoy experimenting with patterns using different colors and blocks, that planning for these types of lessons would be enjoyable for us as teachers, and we thought that patterns must be an important topic because it is so common in our everyday lives.

Throughout our research we were surprised at how much we actually learned. We found so many ways to incorporate patterns into primary classroom instruction and that patterns were such an important aspect of the math curriculum. Patterns range from such simple arrangements of color to number fact families (including addition & subtraction) to open number sentences (in multiplication).

We also gained the necessary experience in using the mathematics curriculum guides, as this is our first mathematics course we have taken in the education program. We became familiar with age-appropriate strategies for each of the primary grades and how the concepts are built-upon each year. Another valuable aspect of our experience is that we learned how to use the wilson web, for example, and other methods of finding scholarly journals and articles to guide us on our way. We had not realized prior to this assignment just how many valuable resources there are for teachers right at our fingertips.

Although we gained many answers during our research, there are still some questions unanswered and new ones that we have formed throughout our journey. They include:
- Where did the concept of a "pattern" originate? Do we need this information to provide to our students?
- What are some more types of accomodations that we can make for struggling or more advanced students?
- What are some more cross-curricular activities in which patterns can be involved?
- How does one know that they have taught to the standard so that their students can build upon their pattern knowledge in the future?
- There are lots of ideas about how to teach patterns, but what are the most effective methods?
- What is the most effective assessment technique for assessing your students' understanding of patterns?

Kindergarten Activities with Patterns

Here are some activities that may interest all of you kindergarten teachers!

Karen Economopoulos
What Comes Next? The Mathematics of Pattern in Kindergarten
Teaching Children Mathematics 5 no4 230-3 D '98

When students first encounter patterns they may be seeing only that red comes after blue and blue comes after red. At some point they must make the connection that red-blue is a fundamental unit of the pattern. Being able to generalize about a pattern and use known information to predict unknown information is truly the powerful aspect of patterns. To generalize and predict, students must move from looking at a pattern as a sequence of "what comes next" to analyzing the structure of the pattern, that is, seeing that it is made of repeating units. Consider the following two activities to try with your students.

"What comes here?" is an activity in which students use information that they already know about a pattern to predict unknown information. In this activity students build a pattern using color tiles or interlocking cubes. They cover the last four or five elements in their pattern with small paper cups. Students take turns pointing to a cup and guessing the color of the tile underneath. In this game, the focus of the activity shifts from what comes next to what comes here as they use known information to predict unknown information. As teachers played this game, they predicted how their students might decide on the hidden colors. All agreed that a range of responses would be received, from random guesses to predictions based on the given information. Teachers were very interested in observing how students would determine the hidden color. Would they need to begin with the first cube in the sequence and "read" the pattern in order from left to right, or would they do what many of them did and say, "I looked at the last color, and that was green; so in my mind I just went green-blue-green and knew that the next one had to be blue." Although these two strategies seem similar, the first one may suggest that students are focusing on "what comes after what," whereas in the second, students seem to be moving toward thinking about green-blue as the smallest repeating unit in the pattern. The teachers became interested in not only whether students could do this activity but also how they would approach the task. They realized that they would know how their students were thinking about the task only if in addition to their observations, they asked students to explain their thinking. For these teachers this activity was not only something new to try but also an introduction to a new idea in patterning--the importance of identifying the repeating unit in a pattern and using that unit to predict what comes here.

Pattern Towers - Same or Different?
In addition to the "What comes here?" game, teachers were asked to consider how a tower built from alternating green-orange-green-orange cubes was similar to a tower built from blue-yellow-blue-yellow cubes. They quickly saw that even though the towers were composed of different colors, their basic composition was the same--that both towers were made up of repeating units with the same structure. Many teachers predicted that for young students, color would be an overwhelming attribute that would make these towers seem very different, so that blue-yellow would not in any way be considered the same as green-orange. Although this prediction was prevalent, most every teacher also recognized the thinking that could be developed with continued exposure to such an activity. They realized that through these kinds of comparison activities, children would learn to be careful observers of the structure that characterizes a pattern. The teachers' curiosity about this activity seemed to stem from their own interest in thinking about these simple pattern towers in a new and different way. For these teachers, the experience of exploring a mathematical idea for themselves illuminated some important mathematics for them, which in turn led to questions about their work with children and enthusiasm about exploring some new ideas with their students. This experience emphasized to us all the importance of exploring mathematics ourselves as a way to make decisions about teaching. In both of these activities, students work with the idea that a pattern is predictable and that it is made up of units that repeat. By looking at the unit of a repeating pattern, one is able to make generalizations that can help in predicting unknown information. To do this sort of work, it became clearer to teachers that although students need a great deal of varied experience with copying and extending patterns and with creating their own patterns, they also need opportunities to describe their patterns to others and to consider how patterns are the same and different.

Seeing Pattern as Structure: Connection to Later Work in Mathematics
Although we should not expect early childhood students to understand fully the complexities of pattern, we can engage them in activities that involve thinking about the predictability and the structure of patterns - that patterns are composed of units that repeat. Young students' work with these patterning ideas will connect significantly to their later work in mathematics. Because our number system is built on a system of patterns and predictability, students must be able not only to identify the patterns that they see but also to give reasons and evidence for why the patterns exist. For example, as students begin to examine closely the sequence of numbers, they soon recognize that many patterns and relationships are evident. When students count by fives, they quickly notice a pattern in the counting numbers of 5, 10, 15, 20, 25, 30, 35, and so on, and that in this counting sequence, the ones digit alternates 5, 0, 5, 0. As they examine this pattern more closely, they can explain that it is not just coincidental but is in fact linked to the base-ten structure of the number system and that every group of five that is added makes either half a group of 10 (5, 15, 25) or a whole group of 10 (10, 20, 30). Students may then generalize that any number that is a multiple of 5 will end in either a 0 or a 5. From this knowledge they can then predict the types of answers that they will get when they multiply a number by 5.Most early primary teachers may never encounter students using their knowledge of pattern in these ways. In fact, understanding and explaining the pattern of counting by fives may seem quite far away from pattern trains and predicting "What comes here?" However, for students to use patterns in powerful ways, they must build an understanding of what patterns are, beyond finding what comes next. Likewise, although the work of kindergarten may seem to be clear and simple, when one group of kindergarten teachers engaged in mathematics together and considered the work of their students, they began to think more deeply about an area of their curriculum that they had always taken for granted. They returned to their classrooms with new insights about a once familiar topic - insights that allowed them to further their students' exploration of ideas about pattern.

Patterns All Around

This journal provided some interesting ideas for a variety of pattern activities that would be great to use in the primary classroom.

Paula Filliman
Patterns All Around
Teaching Children Mathematics 5 no5 282-3 Ja '99

Examine the dot patterns on dominos containing 1 to 6 dots. Pretend that you are a domino manufacturer and that you need to make dominos with 7 through 9 dots. Where will you place the extra dot from 6 to 7? Use "dots" from a paper punch, stick-on dots, or counters to make patterns for 7, 8, and 9 dots. The dots must fit into the same size of square as the 1-through-6 dot pattern. How would you arrange 10, 11, and 12 dots?

Place five counters of one color on a ten-frame, and fill the remaining spaces with another color. Can you make a number sentence to show this sum? Try other ways to fill the ten-frame with two colors. How many ways did you find? How can you tell if you have found them all?
Use two colors of Unifix cubes to show all the ways to make the number 5, such as using four blues and one yellow. Draw pictures of each combination; for example, draw four blues on top of one yellow. Try to make all your pictures the same size. How are the pictures alike? Different? What patterns do you see?

Ask your parents to help you find (1) a game, such as Sorry! or Candyland, with a board that has moves from space to space along a path and (2) a deck of number cards. Choose a sum, and remove all cards greater than that sum. For example, if you choose 9, remove the 10 and higher cards. During your turn, if you draw a 1, you must say the number that goes with 1, which is 8, to make the sum of 9. Move 8 spaces. Would you rather draw a high or low number?

Find at least twenty nickels and a hundreds chart. Place one nickel on the hundreds chart on each number you say as you count by 5s. What does your pattern look like on the chart? Can you describe where the nickels are placed? Which odd numbers are covered by a nickel? Which even numbers?

Draw a number line from 0 through 12. Connect the points at either end, 0 and 12, by drawing an arc. Move to the numbers 1 and 11, and draw another arc. What is the sum of each pair? Continue to connect all the pairs of numbers following this pattern. Are any numbers left unmatched? If you connect an unmatched number to itself, what is the sum? Do this activity on a number line from 0 through 15. What differences do you see? What would happen on a number line from 3 through 15?

Use cubes, dots, or other counters to build two-row rectangular arrays, and observe the pattern of the numbers as you build each successive array. The first array is two rows, with one in each row. Add one counter to each row for a two-by-two array. Continue to build two-row arrays, and name the numbers. What kind are they? Can you put seven counters in a two-row array? Eleven counters? Repeat for three-row rectangular arrays.

Cut several ten-frames from grid paper. Use one ten-frame and count by 2s, recording one number in each square of the frame. What numbers are in the first and sixth squares? Compare the second with the seventh square. What do you notice? Complete separate ten-frames for the multiples of 4, 6, and 8. Are the patterns similar? Is the pattern the same if you count by odd numbers?

Patterns with Quilting

Here is a great activity about quilting that can be incorporated into the primary classrooms to help your students learn about patterns.

Shapes and Patterns with Quilting Math
Instructor (New York, N.Y.: 1999) 110 no6 54-5 Mr 2001

Designing Squares
As your students create their own dynamic and colorful quilt squares, they will build on important geometry concepts. Discuss with your students the way quilt designers combine familiar elements from their lives with ideas from their own imaginations to create new patterns. Give each student a copy of the grid pattern. Talk about shapes that might make interesting patterns. While children are working, circulate through the room and encourage them to talk to you about their design process. Invite them to show you what elements of quilt design, such as symmetry, color, and balance, they are incorporating into their designs. When they are finished, display the squares individually or use them to make a collaborative quilt!

Making a Collaborative Album Quilt
With an album quilt, each square is unique. Make a paper album quilt that will celebrate the uniqueness of your class! Ask students to use their square designs as a pattern for their final quilt squares. Each student will trace the various colored shapes that make up his or her pattern onto brightly colored construction paper. When all of the pieces have been created, the child will assemble them to match the pattern from the reproducible and glue them in place on a piece of construction paper. After the individual blocks are completed, introduce the method of using strips to outline each square. Cut strips of paper (2-inches wide and as long as your finished square). Paste the blocks between the strips, and then add a contrasting border to complete the project. Your students will be amazed at the beauty and complexity of your class album quilt.

Color Tile Patterns
Tiling with colored manipulatives is a great way to explore geometry concepts such as symmetry, adjacency, and shape. Your students can use colored tiles, rainbow cubes, or one-inch squares of construction paper to make tile patterns. Model a basic AB pattern with the tiles. After completing one row of five or six tiles, continue the pattern on the next two rows. Ask students what they see. Invite them to notice the resulting diagonal and vertical patterns. Then ask students to work in pairs or groups on their own tile patterns, first with two colors, and then extending, as they feel comfortable, to three and four colors. Encourage them to explore other combinations. They'll enjoy and learn from replicating each other's patterns and inventing new ones!

Sunday, March 18, 2007

Additional Ideas from Scholarly Journals

We have found a few ideas from journals that we felt would be a good idea to incorporate into your classroom while learning about patterns! Ideas from these journals will be added to the blog periodically.

Michael Naylor
Do You See a Pattern?
Teaching PreK-8 36 no6 38-9 Mr 2006

Digit circles
A digit circle is a circle with digits 0-9 equally spaced around the outside. Fabulous patterns can be created while giving your kids practice with number operations. Make up some sheets that have several digit circles on them.

Addition circles (Grades 1-2)
Ask your students to choose an "adding number," say 3. Starting with 3, add 3 to get 6. On the digit circle, draw a line segment from 3 to 6. Now add 3 again to get 9, and make another line segment from 6 to 9. Adding 3 gives 12, but look only at the number in the ones place, 2, and connect from the 9 to the 2. Continue in this manner until you get back to the starting point. Have your students try these with all digits from 0-9, and then compare the designs to look for similarities and differences. There are some striking patterns that emerge; for example, the designs are identical for pairs that add to 10, so 1 and 9 make the same design, as do 2 and 8, 3 and 7 and 4 and 6. The designs are created in the opposite direction, though. Ask your students to come up with ideas as to why this might be. One way to think about it is that adding 3 gives the same last digit as subtracting 7 and vice versa.

Color Cube Patterns (Grades K-1)
Children in the lower grades are learning the counting sequence and how number names relate to quantities. They're not ready to consider patterns in place value concepts. Here's a pattern activity using colors to develop algebraic reasoning. Without allowing students to see your pattern, place colored cubes in a line, perhaps starting with red, blue, red, blue, etc. Place a cup over one cube, and ask students to guess what color is hidden underneath. Try various patterns, and try covering up more than one cube. Children must be able to recognize the pattern and mentally repeat it to find what's missing. Looking for patterns and thinking about unknown elements in a sequence provide important foundations for powerful mathematical reasoning.

Monday, March 5, 2007

Picture Books for Patterns

There are so many books available today with obvious patterns. These would be great for use in the classroom. We thought we would list a few below. Enjoy!

Allen, Pamela. Who Sank the Boat? Putnam, 1990. ISBN 0-698-20679-7
A cow, a pig, a sheep, and a mouse enter a boat from biggest to smallest. Each passenger tips the boat and causes it to sit lower in the water. The question is repeated and answered after each animal gets into the boat.
Brett, Jan. Town Mouse, Country Mouse. Putnam, 1994. ISBN 0-399-22622-2
This rendition of the common folktale contrasts two lifestyles and the pattern is clear.
Brown, Margaret Wise. Goodnight Moon. Illustrated by Clement Hurd. HarperCollins, 1947. ISBN 0-06-020706-X
This classic has been around so long that we tend to take it for granted, but the repetitive text that exactly fits the pictures makes it an ideal pattern book.
Carle, Eric. Rooster's Off to See the World. Picture Book Studios,1991. ISBN 0-88708-0421 The pattern of animals joining and leaving the procession is similar to that in many folktales. Carle, Eric. The Tiny Seed. Picture Book Studio, 1991. ISBN 0-88708-015-4
It's fall and the seeds are being blown along by the wind. One tiny seed survives to flower and scatter its seeds to the wind. Hopkinson, Deborah. Sweet Clara and the Freedom Quilt. Illustrated by James Ransome. Random, 1993. ISBN 0-679-82311-5
A young slave stitches a quilt with a map pattern that will lead her to freedom.

Hutchins, Pat. Rosie's Walk. Simon and Schuster, 1968. ISBN 0-02-745850-4
Rosie, the hen, takes a leisurely walk around the barnyard, not heeding the fox whom she foils at every turn. The illustrations are full of unusual patterns, and predicting what will happen next to the fox brings students to the plot's pattern.
Martin, Bill Jr and Archambault, John. Chicka Chicka Boom Boom. Illustrated by Lois Ehlert. Simon & Schuster, 1989. ISBN 0-671-67949-X
Animated letters climb the tree in alphabetical order. The pattern is in the rhythmic chant and in the alphabetical order.

Grade Three

Students will explore, recognize, represent, and apply patterns and relationships, both informally and formally.

Gaining on their knowledge from grades one and two, students will continue to learn about patterns in the third grade through recognizing the pattern implicit in our place value system; recognizing and creating geometric patterns; using and recognizing patterns in a multiplication table; recording a repeated addition pattern using a multiplicative notation; and recognizing the meaning of open sentences in multiplication.

Providing students with ones, tenths, and hundredths blocks is necessary during grade three to show the pattern of how 1 tenth equals ten, 10 tenths equals 100, etc. Have students create different letters or shapes to continue patterns such as "L" shapes, triangular shapes, or square shapes where each time the student adds more blocks to continue the pattern.

Remember to incorporate the "flips" of shapes in your patterns!

Provide your class with a multiplication chart so they can find patterns in the numbers horizontally, vertically, and diagonally, and that some rows are "doubles" of other rows. For example, Row 4 (0, 4, 8, 12, 16...) is double Row 2 (0, 2, 4, 6, 8...).

When providing students with repeated addition problems, students should be able to recognize that the addition problem can be represented with multiplication.
For example: 3+3+3+3+3 = 15 can also be written as 5x3 = 15.

Providing students with multiplication operations with a missing number value is also a useful way to get students to recognize patterns.