K.2.1 Patterns

Grade: 
K
Subject:
Math
Strand:
Algebra
Standard K.2.1

Recognize, create, complete, and extend patterns.

Benchmark: K.2.1.1 Patterns

Identify, create, complete, and extend simple patterns using shape, color, size, number, sounds and movements. Patterns may be repeating, growing or shrinking such as ABB, ABB, ABB or ●,●●,●●●.

Overview

Big Ideas and Essential Understandings 

Essential Understandings

Kindergarten students recognize and identify simple repeating patterns using concrete materials, sounds, and actions. They identify and label repeating patterns as AB (red, blue, red, blue), ABB (red, blue, blue, red, blue, blue),  ABC ( red, blue, green, red, blue, green), etc.  They are able to name the pattern elements and recognize the core (the shortest string of elements that repeat over and over in the pattern).

In addition, they recognize the same repeating pattern when it is represented using different elements (objects, sounds, movements).  They are able to create, extend and complete repeating patterns.

Kindergarten students expand their work with patterns to include patterns of change.  Patterns of change include growing patterns (•, ••, •••, •••• ...) and shrinking patterns (••••••, •••••, •••• ...).  Kindergartners are able to describe the change ("getting bigger by one" or "getting smaller by one") and can extend the pattern of change.  They describe these patterns as growing or shrinking.

All Standard Benchmarks - with codes

K.2.1.1      Identify, create, complete, and extend simple patterns using shape, color, size, number, sounds and movements. Patterns may be repeating, growing or shrinking such as ABB, ABB, ABB or ●,●●,●●●.

Benchmark Cluster 

Benchmark Group A

K.2.1.1 - Identify, create, complete, and extend simple patterns using shape, color, size, number, sounds and movements. Patterns may be repeating, growing or shrinking such as ABB, ABB, ABB or ●,●●,●●●.

What should students know and be able to do [at a mastery level] related to these benchmarks?

  • Recognize patterns in everyday situations and throughout the curriculum
  • Represent a given repeating pattern (AB, ABC, etc.) in more than one way
  • Identify and label a repeating pattern as AB, ABB, ABC, etc.
  • Recognize the core of a repeating pattern
  • Describe a pattern both verbally and pictorially
  • Create and represent variety of patterns and explain the pattern
  • Extend repeating, growing and shrinking patterns and explain thinking
  • Complete repeating, growing and shrinking patterns and explain thinking

Work from previous grades that supports this new learning includes: 

  •  Display awareness of colors, size, and numbers to be able to create a pattern
  • Have a natural curiosity about people and things in their world
  • Notice similarities and differences in objects
  • Enjoy repeating predictable sequences in books and rhymes
  • Can extend simple rhythmic patterns in songs and chants

 

Correlations 

NCTM Standards

Understand patterns, relations, and functions

Pre-K - 2 Expectations:

  • sort, classify, and order objects by size, number, and other properties;
  • recognize, describe, and extend patterns such as sequences of sounds and shapes or simple numeric patterns and translate from one representation to another;
  • analyze how both repeating and growing patterns are generated.

Common Core State Standards:  None

 

Misconceptions

Student Misconceptions 

Students may think...

  • only repeating patterns are patterns.
  • a pattern only involves two elements.

 

Vignette

Vignette 1

Using Patterns to Make a Macaroni Necklace

Kindergartners in Mrs. Hanson's class apply their knowledge of pattern while making a macaroni necklace.  This activity provides an opportunity to assess the children's understanding of pattern and number as they make their necklaces.  During the lesson, Mrs. Hanson will refer to the pattern core. The core of a repeating pattern is the smallest group of elements that repeat.

macaroni necklace - repeating pattern

Mrs. Hanson asked the students to share what they know about patterns.

Student A:   Something that goes over and over and over.

Mrs. Hanson:  Can you give me an example?

Student A:   Like the flag --red, white, red, white.  Red and white keep going over and over.

American Flag

Student B:  That is ABAB.

Ms. Hanson:   What else do you know about the red, white, red, white pattern on the flag?

Student A:  It is a repeating pattern.

Mrs. Hanson:  What part repeats in the red, white, red, white pattern on the flag?

Student C:  I know, I know.  The red, white repeats.  It keeps going.

Mrs. Hanson then uses the words "pattern core" to describe the red white in the flag pattern.

Mrs.  Hanson asks if all patterns are repeating patterns.

Student D:  No.  Some patterns grow bigger.

Mrs. Hanson:  Can someone give an example of a pattern that gets bigger?  A growing pattern.

Student E: When I build towers with the blocks I build a tower with one block and then a tower with two blocks, then three blocks and keep going. 

Mrs. Hanson:  Can you show us the pattern with the blocks.

Student E builds four towers - 1 block, 2 blocks high, 3 blocks high, 4 blocks high.

Mrs. Hanson:  How are the towers changing in this growing pattern?

Student D:  They are one block bigger.

Mrs. Hanson:  What is one block bigger?

Student D:  The towers.  Each one is one block bigger going up.

Mrs. Hanson:  Can we use this information to build the next tower in this pattern?

Student A:  We need five blocks. 

Student B:  We have to go 1 more than four and that is five.

The pattern is continued with the next tower.

Mrs. Hanson:  Think of everything you know about patterns. Some are repeating like the red white red white pattern on the flag and some are growing patterns like the tower pattern.  Today you will be creating a necklace with a pattern.  You can decide what kind of pattern you want in your necklace.  It might be a repeating or it could be a growing pattern.  After you have made your necklace please make a drawing of it on blank paper.

The kindergartners began working with pre-cut lengths of yarn and each child selected two different colors of macaroni.  As they worked, Mrs. Hanson checked with individual students asking them to explain their pattern choice or to justify the placement of a particular piece of macaroni.

After the necklaces were finished and recorded the kindergartners began to share.

Malorie:  I used green and yellow.  My pattern is green, green, yellow, yellow, green, green, yellow, yellow, green, green, yellow...

Malorie's necklace pattern

Ms. Hanson:  What name can you give your pattern, Malorie?

Malorie: It is AABB AABB.

Ms. Hanson: What part of your pattern is A?  What part is B?

Malorie: The A is green macaroni and the yellow macaroni is the B part.

Mrs. Hanson: Is this a growing pattern?

Malorie:  It is repeating green-green-yellow-yellow-green-green-yellow-yellow...

Mrs. Hanson: What part of Malorie's pattern repeats?  What is the pattern core?

Isaac: I know, it is red-red-green-green. 

Kahlid: The AABB part.

Mariah: The AABB part is the pattern ...Ohhhhhh ...the core...pattern core.

Mrs. Hanson wrote the following on the board.

AABB             green-green-yellow-yellow

Mrs. Hanson:  Does anyone have a pattern that is like Malorie's in some way?

Phoebe:  I used two colors in my pattern just like Malorie. But my pattern is red-green-red-green.

Mrs. Hanson:  Is your pattern a repeating pattern or a growing pattern?

Phoebe:  It is repeating...red-green-red-green.

Aniah:  It's an ABABABAB pattern.

Phoebe:  I know because the red is A and the green is B.

Phoebe's necklace pattern

Mrs. Hanson added a description of Phoebe's pattern on the board.

AABB             green-green-yellow-yellow                 AB      red-green-red-green

Mrs. Hanson asked the kindergartners to raise their hands if their pattern was like Malorie's AABB pattern or like Phoebe's ABAB pattern. Several hands went up.

Mrs. Hanson:   Drew, would show us your necklace?  (Drew held up his necklace drawing.)

Mrs. Hanson:   Boys and girls look at the pattern in Drew's necklace.  Is it an AABB or an AB pattern?  How do you know?

Class:  It's ABABAB.

Sarah:  It is ABAB because it goes yellow-green-yellow-green-yellow-green.

Mrs. Hanson:  I want to add Drew's pattern to the board.  Where should I put it?

Drew:   It goes there (pointing to the AB on the board).

Mrs. Hanson adds the description of Drew's pattern to the board.

AABB             green-green-yellow-yellow                 AB      red-green-red-green

                                                                            AB      yellow-green-yellow-green

Students examine other necklace patterns, identify other AB patterns, and, after some discussion, Mrs. Hanson adds each to the board.

AABB      green-green-yellow-yellow                        AB      red-green-red-green

                                                                            AB      yellow-green-yellow-green    

                                                                            AB      yellow-red-yellow-red

                                                                             AB      green-yellow-green-yellow.

                                                                             AB      red-green-red green

Kevin:   I have a different pattern.  (Kevin shows his necklace to the class.)

                   Kevin's necklace pattern

Mrs. Hanson:   What colors did you use in your necklace, Kevin?

Kevin:  I have red and green.  

Alicia:   It goes green-red-red-red-green-red-red-red-green-red-red-red.

Mrs. Hanson:  How can we use letters to describe Kevin's pattern?

Jake:   It goes green-red-red-red-green-red-red-red that is the same as ABBBABBB.

The class agreed and Mrs. Hanson added ABBB to the descriptions of patterns on the board.

 

Vignette 2

Number Patterns

Mr. Johnson has prepared two groups of cards for his students.  All the cards have numbers on the front and back.  In the first group of cards the difference in the number on the front and the back of the cards is one (8&9, 3&4, 6&7). 

Mr. Johnson shows the students a card with the number 8.  He says, "On the back of this card, I've written another number."  He turns the card over to show the students the number 9.  Then he shows the students a second card with the number 13 on the front and 14 on the back.  He continues to show the class a few more cards. 

Mr. Johnson:  What do you notice about the numbers on these cards?"

Kendra: "They are all numbers!"

Mr. Johnson:  "What did you notice about the numbers, Kendra?"

Kendra: "Ummm...well they were counting."

Mr. Johnson: "What do you mean by 'they were counting'?"

Kendra:  "If 6 was on the front the number on the back would be 7 because it comes after 6 when you count 1, 2, 3, 4, 5, 6, 7."

Mr. Johnson:  "Did anyone notice anything else?  Did you think about it differently from Kendra?" 

Jack: "The numbers are neighbors;   it's like this...when you showed us the 8 then it's 9."

Mr. Johnson: "Do we all agree that the pattern is the number after or get one more?  Is that the same thing?"

The class agrees.

Mr. Johnson: "Now, I'm going to show you another group of cards. Think about the numbers.  Don't say anything out loud ....but touch your nose if you notice a pattern."

Mr. Johnson shows a card with 8 on the front and 7 on the back.  The next card has 4 on the front and 3 on the back.  He starts seeing fingers on noses, but decides to show a few more cards. 

Mr. Johnson: "This card has the number 11 on the front.  What number do you think will be on the back?  Turn and tell your neighbor.  Be sure to explain how you knew what number would be on the back."

Mr. Johnson listens to the children's conversations and makes a mental note of the students that were able to find and explain the pattern.

Mr. Johnson: "Who would like to share what you and your partner discussed?" 

Mr. Johnson: "Emma and Tariq, what did you find?"

Emma: "It was different from the last cards.  It wasn't getting bigger it was getting smaller."

Tariq:  "They went backwards like counting 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0".

Jack:  "The numbers are still neighbors but going the other way.   Jack used a number line to explain what he meant."

Isaac:  "The number on the back is one less than the number on the front."

Mr. Johnson continued showing the remaining cards as students made predictions and checked to see if the pattern was true for all the cards. 

Resources

Instructional Notes 

Teacher Notes

  • Students may need support in further development of previously studied concepts and skills
  • Students should be encouraged to look for patterns in their surroundings.
  • Kindergartners should use a variety of objects to create, extend and complete repeating, growing and shrinking patterns.
  • The counting sequence is one of the first number patterns Kindergartners experience.  Encourage them to explain what is happening in this pattern.
  • Kindergarten students need to see the same pattern represented in more than one way.
  • Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication.  A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started

What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?

While Working

How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if...?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Why did you...?
What assumptions are you making?

Reflecting about the Solution

How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work? Has the question been answered?
Can the explanation be made clearer?

Responding (helps clarify and extend their thinking)

Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part ...

(Adapted from They're Counting on Us, California Mathematics Council, 1995)

Instructional Resources 

NCTM Illuminations

Calculating Patterns

In this sequence of lessons, students represent patterns in different ways. They solve problems; make, explain, and defend conjectures; make generalizations; and extend and clarify their knowledge. Lesson 3 and Lesson 4 are good for patterning with Kindergarten students.

Additional Instructional Resources     

Baratta-Lorton, M. (1994). Mathematics their way. Menlo Park, CA: Addison-Wesley Publishing Co.

Greenes, C., Cavanagh, M., Dacey, L., Findell, C., & Small, M. (2001). Navigating through algebra in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics.

Richardson, K. (1999). Developing number concepts counting, comparing, and pattern. White Plains, New York: Dale Seymour Publications.

Small, M. (2009).  Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.)  Boston, MA:  Allyn & Bacon.

Van de Walle, J., & Lovin, L. (2006).  Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

Von Rotz, L., & Burns, M. (2002). Grades k-2 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.

New Vocabulary 

between: in the space separating (two points, objects, etc.)

first: number one, the beginning one; an ordinal number

last:  coming after all others; being the only one remaining

next:  immediately following; nearest or adjacent in place or position

pattern:  repeated design or recurring sequence

"Vocabulary literally is the

 key tool for thinking."

Ruby Payne

Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions.  Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.

Learning vocabulary in the mathematics classroom is contingent upon the following:

Integration:   Connecting new vocabulary to prior knowledge and previously learned vocabulary.  The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.

Repetition:    Using the word or concept many times during the learning process and connecting the word or concept with its meaning.  The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.

Meaningful    Multiple and varied opportunities to use the words in context.  These

Use:              opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems.  Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.

Strategies for vocabulary development

Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.

Mathematics Word Bank:  Each unit of study should have word banks visible during instruction.  Words and corresponding definitions are added to the word bank as the need arises.  Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.

Labeled pictures and charts:  Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.

Frayer Model: The Frayer Model connects words, definitions, examples and non-examples.

Frayer model

Example/Non-example Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words.  Teachers should use these during the instructional process to engage student in thinking about the meaning of words.

Example/ Non Example chart

Vocabulary Strips:  Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.

word

definition

illustration

Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.

Additional Resources for Vocabulary Development

Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.

Sammons, L. (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

Professional Learning Communities 

Professional Learning Commnities

Reflection - Critical Questions regarding the teaching and learning of these benchmarks:

  • What are the key ideas related to understanding patterns at the kindergarten level?  How do student misconceptions interfere with mastery of these ideas?
  • What kind of patterns should kindergarten students experience in an instructional setting?
  • What experiences do students need in order to develop an understanding of patterns? 
  • Give several examples of repeating patterns that kindergarten students should encounter in the classroom.  How might this repeating pattern be represented? What materials could be used?  How could this pattern be represented using only blue cubes?
  • How might kindergartners represent an ABB pattern?  What materials or actions might they use?  How would they describe each pattern if they had to describe it in more than one way?
  • Give several examples of growing and or shrinking patterns that kindergarten students should encounter in the classroom.  What is the change in the pattern? How would a kindergarten child describe this change?
  • When checking for student understanding of pattern, what should teachers:
  • listen for in student conversations?
  • look for in student work?
  • ask during classroom discussions?
  • How can teachers assess student learning related to these benchmarks?
  • Examine student work related to a task involving patterns.  What evidence do you need to say a student is proficient?   Using three pieces of student work, determine what student understanding is observed through the work.
  • How are these benchmarks related to other benchmarks at the kindergarten level?

Professional Learning Community Resources

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding.  Portsmouth, NH: Heinemann.

Chapin, S. and Johnson, A. (2006).  Math matters: Understanding the math you teach, grades K-8, 2nd Edition. Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions:  Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions

Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics,

K-6. Portsmouth, NH: Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA:  National Council of Teachers of Mathematics.

Sammons, L. (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

References 

References

Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding.  Portsmouth, NH: Heinemann.

Baratta-Lorton, M. (1979). Mathematics their way. Menlo Park, CA: Addison-Wesley Publishing Co.

Bender, W. (2009). Differentiating math instruction, Strategies that work for k-8 classrooms! Thousand Oaks, CA. Corwin Press.

Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Burns, M. (Eds). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA:  Math Solutions.

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Chapin, S. and Johnson, A. (2006).  Math matters: Understanding the math you teach, grades K-8 (2nd Ed.).  Sausalito, CA: Math Solutions Press.

Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions:  Using math talk to help students learn; Grades K-6. Sausalito, CA: Math Solutions.

Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2. Sausalito, CA: Math  Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.

Felux, C., & Snowdy, P. (Eds.). (2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.

Fuson, K., Clements, D., & Beckmann, S. (2010).  Focus in kindergarten: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

Greenes, C., Cavanagh, M., Dacey, L., Findell, C., & Small, M. (2001). Navigating through algebra in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics.

Hyde, Arthur. (2006). Comprehending math: Adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH:  Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA:  National Council of Teachers of Mathematics.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA:  National Council of Teachers of Mathematics.

Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.

Murray, M. & Jorgensen, J. (2007).  The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics.  Reston, VA: NCTM.

Richardson, K. (1999). Developing number concepts counting, comparing, and pattern. White Plains, New York: Dale Seymour Publications.

Sammons, L. (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., & Zbiek, R. M., (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence.  Reston, VA:  NCTM.

Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.

Small, M. (2009).  Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.)  Boston, MA:  Allyn & Bacon.

Van de Walle, J., & Lovin, L. (2006).  Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

Von Rotz, L., & Burns, M. (2002). Grades k-2 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.

Assessment

1.      Using different objects, make each pattern in two ways.

pattern

Solution:  Successfully creates each pattern in two ways using different materials.

Benchmark: K.2.1.1

 

2.      Extend these patterns.

Extend these patterns -shapes

Solution:  Correctly extends the patterns.

Benchmark:  K.2.1.1

 

3.      Extend these patterns.

Extend these patterns - colored blocks

Solution:  Correctly extends the patterns.

Benchmark:  K.2.1.1

 

4.      Make a growing pattern.

Solution:  Correctly makes a growing pattern.

Benchmark:  K.2.1.1

 

5.      Make a shrinking pattern.

Solution:  Correctly makes a shrinking pattern.

Benchmark:  K.2.1.1

Differentiation

Struggling Learners 

 

Students work with simple repeating patterns using concrete materials. They need to copy patterns and then describe them before creating their own.

Struggling learners need to have limited attribute choices when first making patterns.  For example, only red, yellow, and blue cubes should be used as students copy and then create their own repeating patterns.

Concrete - Representational - Abstract Instructional Approach

(Adapted from The Access Center: Improving Access for All K-8 Students,

The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.

The CRA approach is based on three stages during the learning process:

              Concrete     -     Representational     -     Abstract

The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts.  At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level.  Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task.   Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.

The Representational Stage is the drawing stage.  Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems.  They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking.  Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.

The Abstract Stage is the symbolic stage.  Teachers model mathematical concepts using numbers and mathematical symbols.  Operation symbols are used to represent addition, subtraction, multiplication and division.  Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding.  Moving to the abstract level too quickly causes many student errors.   Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations. 

Concrete Triangle

Additional Resources

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA. Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2. Sausalito, CA: Math Solutions.

Murray, M., & Jorgensen, J. (2007).  The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.

Small, M. (2009).  Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.)  Boston, MA:  Allyn & Bacon.

Van de Walle, J., & Lovin, L.  (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.

English Language Learners 
  • Physical objects representing different repeating patterns (AB, AB,  ABC, etc.) will help students label and identify repeating patterns.
  • Word banks need to be part of the student learning environment in every mathematics unit of study.
  • Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words such as: first, second, third, next, after, before, last, repeating patterns, growing patterns, and shrinking patterns.

Frayer model

  • Sentence Frames

Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions.  Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.

Sample sentence frames related to these benchmarks:

This is a ________________ pattern.  It has _________ and __________.

This is a repeating pattern.  It has _______ and ________. 

This repeating pattern is called a(n) __________ pattern.

I know this is a pattern because ________________________________.

  • When assessing the math skills of an ELL student it is important to determine if the student has difficulty with the math concept or with the language used to describe the concept and conceptual understanding. 

Additional ELL Resources

Bresser, R., Melanese, K., & Sphar, C. (2008).  Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Extending the Learning 

Students can continue creating patterns using numbers and explaining the rule for patterns. For example, students create a pattern that uses 3 colors and is growing.

Children recognize and represent patterns in their classroom such as numbers on the number grids, word patterns in poetry, or patterns that involve skip counting like wheels on a tricycle; 3, 6, 9, etc.

Additional Resources

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA. Corwin Press.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k-2. Sausalito, CA: Math Solutions.

Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.

Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction.  New York, NY: Teachers College Press.

Parents/Admin

Classroom Observation 

Administrative/Peer Classroom Observation

What should I look for in the mathematics classroom? (adapted from SciMathMN, 1997)

What are students doing?

  • Working in groups to make conjectures and solve problems.
  • Solving real-world problems, not just practicing a collection of isolated skills.
  • Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
  • Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
  • Recognizing and connecting mathematical ideas.
  • Justifying their thinking and explaining different ways to solve a problem.

What are teachers doing?

  • Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
  • Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
  • Connecting new mathematical concepts to previously learned ideas.
  • Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
  • Selecting appropriate activities and materials to support the learning of every student.
  • Working with other teachers to make connections between disciplines to show how math is related to other subjects.
  • Using assessments to uncover student thinking in order to guide instruction and assess understanding.

Students are:  (descriptive list)         

Teachers are:  (descriptive list)       

identifying repeating and growing patterns.

modeling repeating patterns, calling attention to the pattern core as the part that repeats over and over in a repeating pattern.

identifying the elements in a repeating pattern and the pattern core in a repeating pattern.

drawing attention to the elements in a repeating pattern.  Asking questions:

What elements are being used in this pattern? What is the core of this repeating pattern?

creating, extending and completing

repeating patterns and explaining the pattern verbally and in writing.

provide a variety of patterning experiences using sounds, objects, movements, numbers, etc.  

identifying, creating, extending and completing growing/shrinking patterns and explaining the pattern.

providing experiences with growing and shrinking patterns. Asking questions: How could you describe this pattern? How can it be repeated? Extended?

modeling patterns with concrete objects, movements, actions.

encouraging students to identify patterns that are the same and different.  Asking questions: How are these two patterns alike? How are they different?

creating a given repeating pattern (ABB, ABC, etc.) using different materials

Modeling how to use materials to represent patterns--repeating, growing, shrinking.

 

Additional Resources

For Mathematics Coaches

Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach: Grades k-8, (2nd ed.)  Sausalito, CA: Math Solutions.

Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.

Felux, C., & Snowdy, P. (Eds.). (2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.

Sammons, L. (2011).  Building mathematical comprehension: Using literacy strategies to make meaning.  Huntington Beach, CA: Shell Education.

West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.

For Administrators

Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.

Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.

Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH:  Heinemann.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA:  National Council of Teachers of Mathematics.

Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.

Parents 

Parent Resources

Mathematics handbooks to be used as home references

Cavanagh, M. (2004). Math to know: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA:  Great Source Education Group, Inc.

Helping your child learn mathematics
     Provides activities for children in preschool through grade 5
     http://www2.ed.gov/parents/academic/help/math/part.html#p1

What should I look for in the mathematics program in my child's school?
      A Guide for Parents developed by SciMathMN
      http://www.scimathmn.org/sub/parents_mathclass.htm

Help Your Children Make Sense of Math

Ask the right questions

In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do. Good questions and good listening will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."

Getting Started

    What do you need to find out?
    What do you know now? How can you get the information? Where can you begin?
    What terms do you understand/not understand?
    What similar problems have you solved that would help?

While Working

How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if . . . ?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Can you make a prediction?
Why did you...?What assumptions are you making?

 Reflecting about the Solution

How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work?
Has the question been answered?
Can the explanation be made clearer?

Responding (helps clarify and extend their thinking)

Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part...

Adapted from They're counting on us, California Mathematics Council, 1995.

Read-a-Loud Books

Pattern Fish, by Trudy Harris.
Ten Little Rabbits, by Virginia Grossman.
Eight Hands Round: A Patchwork Alphabet, by Ann Whitford Paul.