# K.1.1B More/Less, Compare & Order

K
Subject:
Math
Strand:
Number & Operation
Standard K.1.1

Understand the relationship between quantities and whole numbers up to 31.

Benchmark: K.1.1.4 One More or One Less

Find a number that is 1 more or 1 less than a given number.

Benchmark: K.1.1.5 Compare & Order Numbers

Compare and order whole numbers, with and without objects, from 0 to 20.

For example: Put the number cards 7, 3, 19 and 12 in numerical order.

## Overview

Big Ideas and Essential Understandings

All Standard Benchmarks - with codes

K.1.1.1      Recognize that a number can be used to represent how many objects are in  a set or to represent the position of an object in a sequence.

K.1.1.2      Read, write, and represent whole numbers from 0 to at least 31. Representations may include numerals, pictures, real objects and picture graphs, spoken words, and manipulatives such as connecting cubes.

K.1.1.3      Count, with and without objects, forward and backward to at least 20.

K.1.1.4      Find a number that is 1 more or 1 less than a given number.

K.1.1.5      Compare and order whole numbers, with and without objects, from 0 to 20.

Essential Understandings

Developing an understanding of numbers, ways of representing numbers, and relationships among numbers is essential for Kindergarten students.

Research suggests that children learn number word sequences, the reading of numerals and the concept of quantity separately even though they are connected in instructional activities.  Counting means so much more than reciting the words and writing the numerals for the counting sequence: 1, 2, 3, 4, ... When successfully counting a group of objects, children give a unique number name to each object (one-to-one correspondence) and then recognize that the last number represents the quantity of objects in the group (cardinality). In addition, they will also successfully show a group of objects to represent a given number (up to 31).

Kindergartners expand their conceptual understanding of counting to include the idea that, when counting, the next number name refers to a quantity that is one larger. This understanding leads to the comparing and ordering of numbers with and without objects.

Kindergartners will use multiple representations to represent numbers. These representations include spoken words, numerals, pictures, real objects and manipulatives.

Benchmark Cluster

Benchmark K1.1.4

Find a number that is 1 more or 1 less than a given number.

Benchmark K1.1.5

Compare and order whole numbers, with and without objects, from 0 to 20.

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

• use matching or counting strategies to identify whether the number of objects in one set is greater than, less than, or equal to the number of objects in another set.
• find a number that is one more or one less than a given number without counting from one. If asked, "What is one more than 7?" A kindergartner should respond 8 automatically without having to count 1,2,3...6,7...so 8.
• compare and order numbers from 0 -20 with and without objects
• use the language of more than, greater than, as many as, less/ than, fewer than when comparing numbers.

Work from previous grades that supports this new learning includes:

• counting objects.
• representing the number in a group or set with a numeral.
• an intuitive sense of more and less. They can pick out the big truck and the small truck. They also know that they would rather have 3 cookies, than just 1.
• arranging and ordering things in their world.

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.

Common Core State Standards

Know number names and the count sequence.

K.CC.1. Count to 100 by ones and by tens.

K.CC.2. Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

K.CC.3. Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).

Count to tell the number of objects.

K.CC.4. Understand the relationship between numbers and quantities; connect counting to cardinality.

K.CC.4a.  When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.

K.CC.4b.  Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted.

K.CC.4c.  Understand that each successive number name refers to a quantity that is one larger.

K.CC.5. Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.

Compare numbers.

K.CC.6. Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.1

K.CC.7. Compare two numbers between 1 and 10 presented as written numerals.

## Misconceptions

Student Misconceptions

Student Misconceptions and Common Errors:

Students may think...

• less and more mean the same thing.
• they only have to use one digit when ordering two-digit numbers.
• that a set of three large objects is more than a set of three small objects.

## Vignette

In the Classroom

More

The game of More is a simple card game like the game of War. Kindergarten students play in pairs using a deck of playing cards, 1-10 (ace equals one).  It is important the cards have objects, representing the numeral, that can be touched and counted. Cups labeled with the numbers 1-10 will also be used to hold cards already played.

In this vignette, Ms. Burkhardt is facilitating a game with Susie and Julia while assessing their understanding of quantity and the concepts of more and less. This partner game was previously introduced to the class.

Ms. Burkhardt deals out all the cards face down to Julia and Susie. The girls have their cards in front of them.

Ms. Burkhardt: Julia, you turn over a card, and Susie, you do too.  Then we will see which is more.  Julia turns over a six of clubs and Susie turns over a five of hearts.

Ms. Burkhardt:  Which is more?

Both girls: Six.

Ms. Burkhardt: How do you know that six is more than five?

Susie uses the fingers on one hand to show five. After she matches her five fingers to the clubs,  she explains there is one extra club.  Julia agrees and explains, "When you count you go past five to get to six.  One, two, three, four, five, six.  So six is more."

Ms. Burkhardt : Yes, six is more than five.  I am going to place both (five and six) cards in the cup labeled six since six was the largest number.

Julia then turns over a ten of diamonds.

Before Susie can turn over her card, Ms. Burkhardt asks, "Susie, do you think you have a card that will be more than 10?"

Julia: No, because ten is the biggest in all the cards.

Susie:  I think.........maybe????

Even though Susie knows only the cards 1-10 are included in the game she has not related this information to the magnitude of the numbers being compared. Ms. Burkhardt uses this opportunity to help Susie relate each of the numbers 1-9  to 10.

Ms. Burkhardt:  What cards are we using for the game?

Susie says, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10 as Ms. Burkhardt writes the numbers on the board.

Ms. Burkhardt:  Is one is more than ten?

Susie: No.

Ms. Burkhardt: Is two more than 10?

Susie: No.

This continues until all numbers have been compared to ten.

Mrs. Burkhardt:  Susie, are any of the numbers we are using in the game more than ten?

Susie:  No, ten is the biggest.

Ms. Burkhardt:  Is ten always the largest number when we play games?

Julia:  No, just in this game because it is the largest of 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.

That's all the cards in the game.

Susie then turns over a six of diamonds and says, "Ten is more than six."

Ms. Burkhardt places both cards in the cup labeled 10.

Once all the cards have been played the contents in the cups are examined to review the concept of more. In the ten cup were ten and six, ten and four, ten and two, and ten and five. After looking at each pair of cards, the girls reiterated that ten is more than six, ten is more than four, etc.

Susie: Hey, the one cup has no cards!

Julia: Because one is the littlest.  It can't beat anybody.

Julia: (continues exploring the cups) The four cup doesn't have any cards either.

Ms. Burkhardt:   Could it?

Julia: It could have because four is more than three.

Susie: And two and one!  How come it doesn't? She looks in the other cups.

Susie:   Oh, look. Here's a four.  It's in the five cup.

Julia:  I get it, five is more than four. I think all the fours are with bigger numbers than four.

Susie:  "Next time, the fours might be with other numbers."

Note:   Kindergartners could use number cards for the game but would need access to objects in order to represent the numbers if they were unsure of which was more. This game can be varied to focus on the concept of less.

(adapted from Fosnot, C., Dolk, M. Young mathematicians at work, Constructing number sense, addition, and subtraction. Heinemann, 2001.pp 39-40.)

## Resources

Instructional Notes
• Students may need support in further development of previously studied concepts and skills.
• The concept of less tends to be more difficult for kindergartners than the concept of more. When asking "Which is more?" questions, immediately follow with "Which is less?" in order to make connections from a less familiar term and concept to a better known idea.
• Kindergartners may be using the words "greater than" and "less than" but are not using the symbols < and >.
• It is easier for Kindergartners to order cards with numbers on them than to order numbers written on a single sheet of paper. The physical act of moving number cards allows them to easily rearrange when order numbers.
• Kindergartners need to order numbers using a number path. The numbers 0, 5 and 10 become benchmark numbers for Kindergartners. When using a number path, students are asked to determine where a particular number belongs rather than just filling in the numbers in counting order. For example, Where does the number 4 go on this number path? Does the number 3 belong in this space (pointing to the empty space next to 0)? The relationship of the numbers 1, 2, 3, 4, to 0 and 5 is highlighted when Kindergartners have to order them on the following number path:
 0 5

Kindergartners should also order numbers on a number path from 0 to 10

 0 5 10
• 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?

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

Instructional Activities

Stack It!

Materials: Connecting cubes or other connecting cubes

Instructions: Give children a variety of directions requiring them to use the concepts of more or less. For example...

Build a stack that is 1 more than _____.

Build a stack that is 1 less than _____.

Counting Pots

Materials: containers with dot labels; blocks or other objects

Instructions: After the children count the dots on the label of the container, they are to count out a set with one more than the number of dots on the label and place the blocks in the container.  The activity can be changed to place 1 less than the number on the label into the container.

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

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.

Conklin, M. (2010). It makes sense! Using ten-frames to build number sense, grades k-2, Sausalito, CA:  Math Solutions Press.

Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA:  Math Solutions.

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, John A., & Lovin, LouAnn H. (2006) Teaching student-centered mathematics grades K-3. Boston: Pearson Education.

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

New Vocabulary

less: Being a smaller number -Fewer

more: Greater in amount, number, or size

equal: is the same amount

compare: to examine in order to observe resemblances or differences

"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.

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

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

• What are the key ideas related to an understanding numbers and counting at the kindergarten level?  How do student misconceptions interfere with mastery of these ideas?
• What experiences do students need in order to develop an understanding of number and counting?
• When checking for student understanding of number, what should teachers
•
• listen for in student conversations?
• look for in student work?
• Examine student work related to a task involving counting and/or representing 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.
• 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

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.

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

Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010).  Developing essential understanding of number & numeration pre-k-grade 2. Reston, VA: National Council of Teachers of Mathematics.

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

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.

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

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

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.

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.

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

Conklin, M. (2010). It makes sense! Using ten-frames to build number sense, grades k-2, Sausalito, CA:  Math Solutions Press.

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.

Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010).  Developing essential understanding of number & numeration pre-k-grade 2. Reston, VA: National Council of Teachers of Mathematics.

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

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

Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA:  Math Solutions.

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 points for prekindergarten through grade 8 mathematics: A quest for coherence.  Reston, VA:  National Council of Teachers of Mathematics.

SciMath Minnesota K-12 Mathematics Framework (1997).

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, John A. (2001). Elementary and middle school mathematics: Teaching developmentally.  New York: Longman.

Van de Walle, John A., & Lovin, LouAnn H. (2006). Teaching student-centered mathematics grades K-3. Boston: 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

1.      Ordering Numbers

Randomly place number cards (cards from 1 to 10) on the table.

Direct the student to arrange cards left to right in increasing order.

"Now say the numbers as you point to them."

(Similarly order the cards from 11 to 15 and 16 to 20.)

Solution, Part 1:  1,2,3,4,5,6,7,8,9,10 and names them correctly

Benchmark:  K.1.1.5

2.      Ordering  Numbers

Using number cards (0-20),  lay three cards in front of the student.  Ask the

student to put the cards in order from the least to the greatest or greatest to least.

Solution:  student orders numbers correctly

Benchmark:  K.1.1.5

3.      Compare Numbers

Using number cards (0-20), lay the numbers 5, 2, and 9 in front of the student.  Ask, "Which number is the largest?  How do you know?  Which number is the smallest?  How do you know?"   Continue with other numbers such as, 13, 20, and 12.

I

Benchmark:  K.1.1.4

4.      One more/ One less

I am going to say a number. Tell me the number that is one more than the number I say.  For example, What is one more than 2?  Student: 3.

 Number Student Response 13 9 19 11

I am going to say a number.  Tell me the number that is one less than the number I say.  For example, What is one less than 2?  Student; 1.

 Number Student Response 11 7 20 9

Solution:  one more and one less than given number

Benchmark:  K.1.1.4

## Differentiation

Struggling Learners

Students will need to use objects along with number cards when determining more and less.  When physical models of numbers are used it is easier to match the objects in order to compare them.  Use the language of more and less throughout these experiences.

Finding one or one less than a number is easily illustrated when using objects.  Use the language of one more and one less throughout these related activities.

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., & Salemi, R. (2007). Math for all: Differentiating instruction 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.

English Language Learners

Develop the language of more and less as students work with objects representing numbers.  These objects can then be matched to help differentiate between the set with more and the set with less.

Finding one or one less than a number is easily illustrated when using objects.  Use the language of one more and one less throughout these related activities.

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.

• 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.

• ELL students need to see visual models with the numeral, word and quantity represented. By displaying visual models with the three aspects of number, they will make relationships between the numeral, word and quantity, rather than viewing them as three separate and different concepts.
• 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:

 7 is one more than _____________.

 5 is one less than ______________.

 __________ is one more than _____________.

 __________ one less than ______________.

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 compare and order groups of four or five numbers.

Expand activities using whole numbers beyond 20.

• find one more and one less
• compare number to find the largest and smallest
• order numbers--least to greatest and greatest to least

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

Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction 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

(table missing)

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.

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.

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

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?

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?

What did you try that did not work?

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

Activities:

●      Using a deck of cards (all face cards removed) and a collection of pennies, flip a card and have your child count out the same number of pennies as the card shows.  Later, have your child count out 1 more or 1 less than the number on the card.  Try with 2 more, 2 less, 3 more, 3 less.

●      Write the numbers 0-20 on pieces of paper. Lay out 4 of the numbers.  Have your child put the numbers in order from smallest to largest.  Ask questions such as: How did you know which was the smallest?  Biggest?