# K.1.2 Operations

K
Subject:
Math
Strand:
Number & Operation
Standard K.1.2

Use objects and pictures to represent situations involving combining and separating.

Benchmark: K.1.2.1 Represent Addition & Subtraction

Use objects and draw pictures to find the sums and differences of numbers between 0 and 10.

Benchmark: K.1.2.2 Compose & Decompose

Compose and decompose numbers up to 10 with objects and pictures.

For example: A group of 7 objects can be decomposed as 5 and 2 objects, or 2 and 3 and 2, or 6 and 1.

## Overview

Big Ideas and Essential Understandings

As kindergarten children learn to count and develop number sense, they simultaneously build their understanding of combining (addition) and separating (subtraction).  At first, young children rely heavily on physical objects to represent numerical situations and relationships, and they use these objects to model the results of combining and separating.  They come to understand combining as putting together two sets or adding to a set.  They understand separating as taking a set apart or taking from a set.  Children use physical objects (counters, cubes, drawings) or fingers to directly model the action or relationships described in the problem.

All Standard Benchmarks

K.1.2.1      Use objects and draw pictures to find the sums and differences of numbers between 0 and 10.

K.1.2.2      Compose and decompose numbers up to 10 with objects and pictures.

Benchmark Cluster

Benchmark Cluster A

K.1.2.1 Use objects and draw pictures to find the sums and differences of numbers between 0 and 10.

K.1.2.2 Compose and decompose numbers up to 10 with objects and pictures.

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

• Find sums and differences for numbers between 0 and 10 by drawing pictures and using objects.
• Compose (create different combinations) and decompose (break apart numbers in different ways) numbers up to 10 with objects.  For example, 7 can be 3 cubes and 4 cubes or 5 cubes and 2 cubes, etc.  In addition, 3 cubes and 4 cubes make 7 in all, 8 cubes and 1 one cube make 9, etc.

Work from previous grades that supports this new learning includes:

• Use one-to-one correspondence when counting. (current grade level)
Correlations

### NCTM Standards

Understand numbers, ways of representing numbers, relationships among numbers, and number systems

Pre-K - 2 Expectations:

• count with understanding and recognize "how many" in sets of objects;
• use multiple models to develop initial understandings of place value and the base-ten number system;
• develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections;
• develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers;
• connect number words and numerals to the quantities they represent, using various physical models and representations;
• understand and represent commonly used fractions, such as 1/4, 1/3, 1/2.

Understand meanings of operations and how they relate to one another

Pre-K - 2 Expectations:

• understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations;
• understand the effects of adding and subtracting whole numbers;
• understand situations that entail multiplication and division, such as equal grouping of objects and sharing equally.

Common Core State Standards

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

• K.OA.1. Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
• K.OA.2. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
• K.OA.3. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
• K.OA.4. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
• K.OA.5. Fluently add and subtract within 5.

Work with numbers 11-19 to gain foundations for place value.

• K.NBT.1. Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (such as 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.

## Misconceptions

Student Misconceptions

Student Misconceptions and Common Errors:

Students may think...

• combining groups makes a new number that is not related to the numbers that were combined.
• the word more means combine.
• subtraction situations are only about "take away."

## Vignette

In the Classroom:

Composing Numbers Using Ten Frames

In this Kindergarten classroom, the teacher shows a ten-frame to students and asks them how many dots they see.  In this example, the colors on the ten-frame highlight the relationship of five and four to nine.

The Teacher showed the card for a few seconds.

Teacher:  How many dots did you see?

Amy:  I saw 9 dots

Teacher:  How did you know it was nine dots, Amy?

Amy:  I saw a line of 5 dots on the top, then I counted 4 more 6, 7, 8, 9 (demonstrated by holding up five fingers on one hand and counting four more on the other hand).

Teacher:  I heard you say that you started with five and then counted four more.  You used a counting on strategy.

Teacher:  Did anyone else see the dots the same way Amy saw them---a line of five and then counting four more 6, 7, 8, 9?

A few students indicated they had seen the dots in the same way as Amy.

Teacher:  Did anyone see the picture of dots in a different way?

David:  I saw four dots on the top and four dots on the bottom.  I know that is eight so I just knew one more is nine.

Teacher:  I heard you say that you used four dots on the top and four dots on the bottom to make eight and then went one more.

Teacher:  David, can you use the card to show us the four dots on the top and the four dots on the bottom?

David indicates the first four dots blue dots in the top row and all four red dots in the bottom row.

Teacher:  Where was "the one more".

David indicates the last dot in the top row.

Teacher:  David used a double--four and four and then one more.  Did anyone else use this strategy for finding the total number of dots?

David was the only one who used this strategy.

Teacher:  Did anyone see the picture of dots in a different way?

Juan:  If the ten-frame is full, it would be 10.  But there's one missing so I know it's one less - that's 9- and I didn't have to count every dot.

Teacher:  I heard Juan say he thought about the frame being filled and that means ten dots.  Since there is one open space on this ten frame Juan thought of one less than ten.

Teacher:  Did anyone else use this strategy to find the number of dots?

In this vignette, the teacher focused on identifying the strategies students used.  These strategies will be extended as students work with addition and subtraction of numbers beyond ten.  Having students share their thinking with each other encourages flexible thinking and encourages the use of more than one strategy.

Extending the Activity:

This activity can be modified in many ways.  The ten-frame shown, with five blue on the top and four red on the bottom, draws particular attention to the 5+ pattern.  Additional cards could be made for 5+1, 5+2, 5+3 and 5+5.

When working on doubles, the ten-frame may look like this:

When students are able to successfully identify how many dots they see and offer a strategy for finding how many in all, use a ten frame with dots of the same color.  This requires students to organize the dots in their own way.

As this activity is repeated, symbolic representations can be used to illustrate student thinking.  Note:  The teacher is doing this recording as a way to represent mathematical thinking.  Students are seeing these representations but are not required to record their thinking in these ways.

Possible representations for the ten-frame in the vignette include:

## Resources

Instructional Notes

Teacher Notes

• Students may need support in further development of previously studied concepts and skills.
• Students should compose and decompose numbers to 5 prior to composing and decomposing numbers from 6-10.
• Showing an image for 1-2 seconds and asking students what they saw helps them develop mental images of numbers. These mental images allow Kindergartners to move beyond "count by one" strategies when solving problems.

• When finding sums and differences, students should be solving real world problems. The context of the problem helps students choose a solution strategy.
• When finding sums and differences, Kindergarten students should experience the following types of combining and separating problems (based on the work of Carpenter, 1999, p.12). Note: Teachers are connecting an equation to a story problem but students should not be required to do so.

Combining Problems

 Result Unknown There are five dogs in a yard. Three more dogs walked into the yard. How many dogs are in the yard? 5 + 3 = ___ Change Unknown There are five dogs in the yard. Some more dogs walked into the yard. There are now eight dogs in the yard. How many dogs came into the yard? 5 + ___ = 8 Start Unknown There were some dogs in the yard. Three dogs came into the yard. There are now eight dogs in the yard. How many dogs were in the yard to start with? ___ + 3 = 8

Separating Problems

 Result Unknown There were eight dogs in the yard. Five dogs ran out of the yard. How many dogs are still in the yard? 8 - 5 = __ Change Unknown There were eight dogs in the yard. Some ran away. Three dogs are still in the yard. How many dogs ran away? 8 - ___ = 3 Start Unknown There were some dogs in the yard. Five dogs ran away. Now there are three dogs in the yard. How many dogs were in the yard to start with? ___ - 5 = 3

• While the idea of teaching "key words" might be appealing, it does not support student comprehension of real world situations involving combining and separating and may lead to incorrect strategies. Asking students to describe what is happening in a real world situation will help them comprehend the situation.
• To prevent the student misconception that subtraction only means take away, use the words minus or subtract when referring to the subtraction symbol or a separating situation. Reinforce this language when students describe their strategies.
• Modeling word problems is critical as students develop an understanding of operations and the relationships that exist between and among those operations. The bar model is an effective tool that can be used to represent the relationships in a variety of problem solving situations.

Mathematics Best Practice tab in the Resources section of the Minnesota

Instructional Resources

NCTM Illuminations

Thinking about numbers using frames of 5 can be a helpful way to learn basic number facts.  The four games that can be played with this applet help to develop counting and addition skills.

Thinking about numbers using frames of 10 can be a helpful way to learn basic number facts.  The four games that can be played with this applet help to develop counting and addition skills.

• Lesson 1 - Building Numbers to Five:
• In this lesson, students make groups of zero to 5 objects, connect number names to the groups, compose and decompose numbers, and use numerals to record the size of a group.  Visual, auditory, and kinesthetic activities are used to help students begin to acquire a sense of number.
• Lesson 2 - Writing Numerals to Five:
• As students construct groups of a given size, recognize the number in the group, and record that number in numerals, they learn the number words through 5 in order (namely, to rote count), and develop the ability to count rationally.
•  Lesson 3 - Building Sets of Six:
• In this lesson, students construct sets of six, compare them with sets of a size up to six objects, and write the numeral 6.  They also show a set of six on a "10" Frame and on a recording chart.
• Lesson 4 - Building Sets of
• Students construct and identify sets of seven objects.  They compare sets of up to seven items, and record a set of seven in chart form.
• Lesson 5 - Building Sets of Eight:
• Students explore the number 8.  They make and decompose sets of eight, write the numeral 8, and compare sets of up to eight objects.
•  Lesson 6 - Building Sets of Nine:
• Students construct sets of up to nine items, write the numeral 9, and record nine on a chart.  They also play a game that requires identifying sets of up to nine objects.
•  Lesson 7 - Building Sets of Ten
• Students explore sets of up to 10 items and practice writing the numbers 0 through 10.  Students count back from 10, identify sets of up to 10 objects, and record 10 on a chart.  They also construct and decompose sets of up to 10 items.
•  Lesson 8 - Wrapping Up the Unit
• Students review this unit by creating, decomposing, and comparing sets of zero to 10 objects and by writing the cardinal number for each set.

In this unit, students make groups of 10 to 20 objects, connect number names to the groups, use place value concepts, compose and decompose numbers, and use numerals to record the size of a group.  Visual, auditory, and kinesthetic activities are included in each lesson.  This unit is most appropriate for students typically in the first grade.

Other Instructional Activities

Composing and Decomposing Numbers

Kindergartners combine the number of fingers on two hands.  Click on the hands in the menu bar and set the viewing timer to five seconds as you begin play.

Beads are displayed for a few seconds, students click to tell how many beads they saw. Click on the beads in the menu bar and set the viewing timer to five seconds.

• Flash fingers, dice or a ten frame to students and have them answer questions like: What did you see?  How did you see it?  How many are there altogether?  The flash time is adjustable.

Finger Patterns

• Partitioning Numbers 3 to 10: Place your hands on your head. Make 3 on your fingers.  Can you make three another way?  Check to see if you are correct.  Make 4 on your fingers. How many ways can you make 4? Similarly for 5, 6, ...10
• Partitioning 10 fingers: Put your hands out in front.  Show me 10 fingers.  Put down one finger.  How many does that leave?  What does 9 and 1 make?  Do the same with 8 and 2; 7 and 3; so on.

Wright, R., Martland, J., Stafford, A., & Stanger, G. (2007). Teaching number (2nd Ed.) Sage Publishing, 2007.

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

Confer, C. (2005). Teaching number sense: Kindergarten. Sausalito, CA: Math Solutions.

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

New Vocabulary

compose:  combine numbers to make another number (6 and 2 make 8, 5 and 3 make 8)

decompose:  break numbers apart (break 8 into 6 and 2, or 4 and 4, or 1 and 7

difference:  the answer resulting from subtraction.  In 8 - 3 = 5, 5 is the difference.

same:  having equal amounts

subtract:  separate a quantity from another

sum:  answer resulting from addition.  In 3 + 5 = 8, 8 is the sum.

ten-frame:  a  2 x 5 grid

"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 Use:    Multiple and varied opportunities to use the words in context.

These 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.

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

What are the key ideas related to composing and decomposing numbers at the kindergarten level?  How do student misconceptions interfere with mastery of these ideas?

How is counting related to composing and decomposing numbers?

What experiences do students need in order to successfully compose and decompose numbers?

Examine student work related to a task involving composing and/or decomposing numbers.  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.

What are the key ideas related to finding sums and differences at the kindergarten level?  How do student misconceptions interfere with mastery of these ideas?

What experiences do students need in order to find sums and differences successfully?

What types of problems would kindergarten students solve when finding sums and differences?

What strategies would kindergarten students use when finding sums and differences?

Examine student work related to a task involving sums and differences.  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 can teachers assess student learning related to these benchmarks?

How are these benchmarks related to other benchmarks at the kindergarten level?

Professional Learning Community Resources

Caldwell, J., Karp, K., Bay-Williams, J., Rathmell, E., & Zbiek, R. (2011). Developing essential understanding of addition and subtraction for teaching mathematics in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics.

Confer, C. (2005). Teaching number sense: Kindergarten. Sausalito, CA: Math Solutions.

Fosnot, Catherine., Dolk, Maarten. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Portsmouth, NH:  Heinemann.

Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in kindergarten: teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

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.

Wright, R., Martland, J., Stafford, A., & Stanger, G. (2002). Teaching number advancing children's skills and strategies. London, England: Paul Chapman Publishing.

References

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

Caldwell, J., Karp, K., Bay-Williams, J., Rathmell, E., & Zbiek, R. (2011). Developing essential understanding of addition and subtraction for teaching mathematics in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics

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

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

Confer, C. (2005). Teaching number sense: Kindergarten. Sausalito, CA:  Math Solutions.

Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 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.

Fosnot, Catherine., Dolk, Maarten. (2001). Young mathematicians at work, constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann.

Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in kindergarten teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.

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., & 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.

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.

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

Wright, R., Martland, J., Stafford, A., & Stanger, G. (2002). Teaching number advancing children's skills and strategies. London, England: Paul Chapman Publishing.

## Assessment

### Sample Assessment Items:

• There are 6 pieces of fruit in a basket. Some are apples and some are bananas.  How many of each could there be? Use pictures, numbers, or words to show your thinking.

Solution:  Solutions will vary; e.e., 5 and 1, 4 and 2, 3 and 3, 2 and 4, etc.

Benchmark:  K.1.2.2

• Using red and blue snap cubes, show seven in many ways.

Solution:  Solutions will vary; e.g., 6 and 1, 1 and 6, 2 and 5, 5 and 2, 3 and 4, etc.

Benchmark:  K.1.2.2

• Given a set of snap cubes (0-10), students are asked to separate them into groups. (K.1.2.2)

Solution:  Solutions will vary; e.g., 9 and 1, 1 and 9, 2 and 8, 8 and 2, 3 and 7, etc.

Benchmark:  K.1.2.2

• I. Combinations of 5

• How many red dots? __________
• How many blue dots? __________
• How many dots altogether? __________

• II. Use red and blue dots to show a different way to make 5.

• How many red dots? __________
• How many blue dots? __________
• How many dots altogether? __________

Solution Part I:  Red dots-2, Blue dots-3, 5 in all.

Solution Part II:  Solutions will vary.

Benchmark:  K.1.2.2

There are six ducks in the pond.  Four flew away.  How many ducks are now in the pond?

Solution:  Two ducks are in the pond.

Benchmark:  K.1.2.1

• There are three cats on the bed.  Five cats jump on the bed.  How many cats are on the bed?

Solution:  There are 8 cats on the bed.

Benchmark:  K.1.2.1

## Differentiation

Struggling Learners

Struggling Learners:

Struggling learners need many hands on experiences making combinations for numbers to ten.  Composing and decomposing numbers less than six should be the focus of instruction before working with numbers greater than five.  Students need to see the same number composed and decomposed in many ways.  Kindergarten students need to work with a variety of materials when composing and decomposing numbers.

For example,

• Using red and blue cubes, how many different groups of six can you make?
• Can you find the ten frames that have exactly six dots?
• Here are four cubes, how many more are needed to make six?
• Bead strings and bead racks are another way students can construct numbers.

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.

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
• The context of a problem may be unfamiliar.  Using pictures and drawings will help students make sense of the problem context.
• Using physical objects to represent numbers and concepts helps bridge the language gap in mathematics.
• 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

 I can make ___________   and  _____________ from _____________.
 I can put _____________  and  _______________ together to make _____________.

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.

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 should be given the task of finding all possible ways to compose/decompose a number using three, or more groups in the composition/decomposition.  They should be encouraged to record the results of composing and decomposing numbers using numbers and words.  Additionally, these students should record the number in each group and the total number in all.

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.

Classroom Observation

 Students are... Teachers are... building and taking apart numbers up to and including ten using a variety of materials and strategies. providing multiple experiences for composing and decomposing the same number. sharing and comparing their thinking with other students. recording ways students compose and decompose numbers. drawing pictures or using objects to illustrate solutions when finding sums and differences. providing a context for finding sums and differences. playing games which involve making  combinations of numbers up to 10. asking students to explain their thinking.

• For Mathematics Coaches

Chapin, S. and 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.

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.

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.

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?

• How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
• 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.