1.2.1 Patterns
Overview
Standard 1.2.1 Essential Understandings
First grade students create, extend and complete repeating patterns. 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). They recognize the same repeating pattern when it is represented using different elements (objects, sounds, movements).
First graders expand their work with patterns to include patterns of change. Patterns of change include growing patterns (•, ••, •••, ••••, ... ) and shrinking patterns (••••••, •••••, ••••, ...). First graders 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. First graders can identify a possible "rule" used to complete and extend a pattern.
All Standard Benchmarks - with codes
1.2.1.1 Create simple patterns using objects, pictures, numbers and rules. Identify possible rules to complete or extend patterns. Patterns may be repeating, growing or shrinking. Calculators can be used to create and explore patterns.
1.2.1.1 Create simple patterns using objects, pictures, numbers and rules. Identify possible rules to complete or extend patterns. Patterns may be repeating, growing or shrinking. Calculators can be used to create and explore patterns.
What should students know and be able to do [at a mastery level] related to these benchmarks:
- Describe and name repeating patterns
- Create repeating patterns using objects, pictures, and numbers.
- Create simple growing and shrinking patterns using objects, pictures, and numbers.
- Complete and extend repeating and growing/shrinking patterns
- Identifying pattern "rules".
- Represent counting patterns on a number line or hundreds chart.
Work from previous grades that supports this new learning includes:
- Recognize simple repeating patterns in everyday situations and throughout the curriculum.
- Identify repeating patterns, growing and shrinking patterns.
- Create, complete and extend simple repeating, and growing/shrinking patterns
- Represent a repeating pattern in more than one way.
- Describe patterns both verbally and pictorially.
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 and Common Errors
Students may think...
● patterns must repeat.
● pattern can't be made using number sequences.
● a repeating pattern can only involve two elements.
Vignette
In the Classroom
Vignette #1
Wrapping Paper Designs
Seeing and representing repeating patterns in more than one way is important as first graders build upon their previous pattern work in Kindergarten. In this activity, first graders are taking a pattern core and repeating it on a grid. New patterns emerge within the grid.
Prepare strips that are two, three, four, and five squares long and
several different size grids --- 3 x 3. 4 x 4, 5 x 5, 4 x 6, 5 x 7, etc.
"A new company that makes wrapping paper is moving into our town. The owner has decided to hire you and your partner to work as a design team to create patterns for a new kind of wrapping paper."
"You and your partner will choose a beginning pattern for your design. Beginning patterns may be two, three, four or five squares long. Here are some examples of a beginning pattern:
You may decide to partially color each square or fill it completely. Pictures or symbols may also be used."
"Talk with your partner about your ideas, experiment with some patterns, and choose one design to use. Next, you and your partner will color your design over and over on the different grids used by the company. Finally, you and your partner will write about your work."
When the students have decided on a design, they transfer their strip pattern to a grid. Always starting the pattern in the upper left hand square.
This three part pattern
creates these patterns on different width grids.
Uncover student thinking with these questions:
- What patterns do you see?
- Why does the same design look different on the different-sized grids?
- When do you get vertical stripes on your patterns?
- When do you get diagonal stripes?
- What happens to the 3-square design on the different width papers?
- What do you notice about the designs on the wrapping paper that is seven squares wide?
As the groups finish, encourage them to share their work and talk about their discoveries. Post all design sheets and ask students to compare and contrast them. Note that new patterns emerge on the grids and these new patterns depend on the size of the grid. Record their observations.
After analyzing the grid patterns, the students decide on the recommendations they would make to Ms.Stripe, the owner of the wrapping paper company.
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 two (8&10, 3&5, 6&8). The second group of cards has a difference of three (8&11, 3&6, 6&9).
Mr. Johnson shows the students a card with the number 8 and says, "On the back of this card, I've written another number." He turns the card over to show the students the number 10. Then he shows the students a second card with the number 13 on the front and 15 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 not big numbers like 100 or 500!"
Mr. Johnson: What did you notice about the numbers, Leo?"
Leo: "Ummm...well the number on the front was smaller than the number on the back.
You know 8 and 10. 8 is smaller than 10."
Class agrees with Leo.
Mr. Johnson: What else did you notice about the numbers on the cards?
Juan: The number on the back is two more than the number on the front.
Mr. Johnson: Can you say more about that, Juan?
Juan: So..if 8 is on the front then 10 is on the back because 8 and 2 more make 10.
If 13 is on the front then 15 is on the back because 13 get 2 more is 15."
Julia: That is like adding. 13 + 2 = 15.
Mr. Johnson: Let's see if Juan's idea works for all of my cards.
As Mr. Johnson shows the cards, the class verifies that Juan's idea works for each and every card.
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 11 on the back. The next card has 13 on the front and 16 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 to them 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 going 2 more because then the 8 card would have 10 on the back.
Tariq: The cards are add 3, three more!
Some classmates agreed but some were not sure.
Mr. Johnson continued showing the remaining cards. The students made predictions based on Tariq's "rule" and checked to make sure the pattern continued on all cards.
The class verified Tariq's rule of add three more.
Mr.Johnson: "We can't play anymore! There are no cards left."
Tatiana: I have an idea. We could start with the back of the cards.
Chase: Then it would be going back ....like minus.
The learning was extended as Mr. Johnson took the cards and repeated the activity using Chase's suggestion.
Resources
- Students may need support in further development of previously studied concepts and skills.
- Pattern activities should include using physical materials. This allows students to easily make changes when creating and extending patterns.
- First graders need to explore and describe pattern in the numbers when counting by five or ten.
- Help students focus on the "change" that is occurring from one term to the next in patterns of change (growing and shrinking patterns). Is the change the same from one term to the next? If yes, this is described as a constant change. First graders need experience with patterns involving a constant change.
- Students will recognize that change is often predictable and that patterns may help describe that change (e.g., temperature, seasons, schedules, routines, growth).
- Students analyze the structure of a repeating pattern by identifying the pattern core, the smallest number of elements that repeats.
- 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)
NCTM Illuminations
Creating, Describing, and Analyzing Patterns
In this cluster of activities, students use interactive math applets to learn about repeating and growing patterns. In the first part, students use an interactive square-pattern grid to create, describe, and analyze repeating patterns based on a two-square pattern unit. In the second part, repeating patterns with pattern units of three, four, and five squares are investigated. In Part 3, students use an interactive cube-sequencing applet to analyze repeating patterns of colored cubes. In Part 4, growing patterns of colored cubes are created, described, and analyzed, and compared to repeating patterns.
Additional Instructional Resources:
Baratta-Lorton, M. (1994). Mathematics their way. Menlo Park, CA: Addison-Wesley Publishing Co.
Coburn, T., Bushey, B., Holton, L., Latozas, D., Mortimer, D., & Shotwell, D. (1993). Patterns: Curriculum and evaluation standards for school mathematics Addenda Series. Grades K-8. 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.
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.
Westley, J., Franco, B., & Randolph, M. (1990). Windows on mathematics: Work time activities for young children: patterns. Palo Alto, CA: Creative Publications.
repeating patterns: repeated design or recurring sequence of elements (red, blue, red blue)
growing pattern: a pattern that grows or increases. For example, *, **, ***, **** ... or 2, 4, 6, 8...
shrinking pattern: a pattern that shrinks or decreases. For example, 100, 99, 98, 97... or **********, *********, ********...
"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.
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.
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.
Reflection - Critical questions regarding the teaching and learning of these benchmarks
- What are the key ideas related to place value at the first grade level? How do student misconceptions interfere with mastery of these ideas?
- What models should a student be able to make when representing 84, using tens and ones?
- When checking for student understanding of place value at the first grade level, 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 place value at the first grade level. 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 first grade 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 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.
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.
Von Rotz, L., & Burns, M. (2002). Grades k-2 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K.. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Baratta-Lorton, M. (1994). 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. (Edt).(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 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
Coburn, T., Bushey, B., Holton, L., Latozas, D., Mortimer, D., & Shotwell, D. (1993). Patterns: Curriculum and evaluation standards for school mathematics Addenda Series. Grades K-8. Reston, VA: National Council of Teachers of Mathematics.
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.
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.
Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.
Murray, M.. (2004). Teaching mathematics vocabulary in context. 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., Trevino, E., & Zbiek, R. M., (2006). Curriculum focal foints 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 thinkng. Sausalito, CA: Math Solutions Publications.
Westley, J., Franco, B., & Randolph, M. (1990). Windows on mathematics: Work time activities for young children: patterns. Palo Alto, CA: Creative Publications.
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.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Assessment
- Recognize and Extend Patterns
Draw the following patterns. Ask children to describe how to continue the pattern.
Solution: triangle, circle, triangle, circle: square, circle, circle, square, circle, circle.
Benchmark 1.2.1.1
- Look at the pattern below. Identify the rule.
Solution: Indication that pattern increases by 2.
Benchmark: 1.2.1.1
- Follow the rule to complete the pattern.
Solution: 12, 9, 6, 3,
Benchmark: 1.2.1.1
- Create Patterns
Use pattern blocks or cubes to create a pattern. Draw and describe your pattern.
Solutions: Will vary
Benchmark: 1.2.1.1
Differentiation
Use of hands-on materials is very important as students create repeating and growing patterns. In repeating patterns, students need to focus on the pattern core, the smallest number of elements that repeats. In addition, they need to see the same repeating pattern in many ways. For example, an AB pattern could be represented as red, blue, red, blue; triangle, circle, triangle, circle; snap, clap, snap, clap.
Growing patterns are more challenging because there is a change from term to term. Students need guidance in looking for the change instead of focusing on the attributes of the materials used to create the pattern. A growth pattern using the rule of one more is a good starting point. For example, •, ••, •••, . . ..
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.
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 need to experience patterns using physical materials, hear the appropriate vocabulary used to describe the patterns and see written labels. Being able to move objects to copy, extend, complete and create a pattern provides an opportunity in which language can describe the actions and the resulting pattern.
- Growing patterns are more challenging because there is a change from term to term. Students need guidance in looking for the change instead of focusing on the attributes of the materials used to create the pattern. A growth pattern using the rule of one more is a good starting point. For example, •, ••, •••, . . .. Language plays an important part in explaining the pattern "rule".
- 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 count, first, second, third, etc.
- 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 is an ___________________ pattern. |
I can extend this pattern by _______________________________. |
The rule of this pattern is ____________. I know this 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.
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.
Students create patterns from a given set of objects, such as these 10 blocks.
Encourage first graders to create number patterns. They can identify the pattern rule for each other's patterns.
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
Administrative/Peer Classroom Observation
Students are... | Teachers are... |
creating repeating patterns with objects using the attributes of size, color, shape, sound, etc. | asking students to describe repeating patterns and label them as AB, ABC, ABB, etc |
identify growing and shrinking patterns, describing the change, a giving a possible "rule" for the pattern. | asking questions such as: What would you put next in your pattern? Should I put this next? Why or why not? Would you explain your pattern to me? What is the pattern rule? |
exploring and describing number patterns, including patterns on the hundreds chart | providing number patterns, increasing/decreasing by 1, 2 or 3. Asking students to describe number patterns |
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.
Additional Resources
For Mathematics Coaches
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach: Grades k-8, 2nd edition. 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.
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
What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN
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
Look for patterns in your home. Ask your child to describe these patterns.
Read-a-Loud Books:
The patchwork quilt by Valerie Flournoy
Pattern (math counts) by Henry Arthur Pluckrose
Pattern fish by Trudy Harris
Pattern bugs by Anne Canevari Green
Framework Feedback
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Comments
Complete and Extend
What does complete and extend look like in Kindergarten?