1.2.2 Number Sentences
Overview
Standard 1.2.2 Essential Understandings
First graders build on their previous work with the composition and decomposition of numbers by writing number sentences to represent a real world or mathematical situation involving addition and subtraction. In addition, they will write a real world problem to represent a given number sentence.
Work with number sentences continues as first graders determine if a number sentence is true or false. For example, is 5 + 3 = 8 true or false? They are also able to write their own true number sentences and false number sentences.
First graders begin their work with variables by determining an unknown in a number sentence. These unknowns are found in varying positions in the number sentences.
For example, 5 + [math]\square[/math] = 8, 5 = [math]\square[/math]  3, 3 + 5 = ∆ .
All Standard Benchmarks  with codes
1.2.2.1
Represent realworld situations involving addition and subtraction basic facts, using objects and number sentences.
1.2.2.2
Determine if equations involving addition and subtraction are true.
1.2.2.3
Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as:
2 + 4 = ∆
3 + ∆ = 7
5 = ∆  3.
1.2.2.4
Use addition or subtraction basic facts to represent a given problem situation using a number sentence.
1.2.2.1
Represent realworld situations involving addition and subtraction basic facts, using objects and number sentences.
1.2.2.2
Determine if equations involving addition and subtraction are true.
1.2.2.3
Use number sense and models of addition and subtraction, such as objects and number lines, to identify the missing number in an equation such as:
2 + 4 = ∆
3 + ∆ = 7
5 = ∆  3.
1.2.2.4
Use addition or subtraction basic facts to represent a given problem situation using a number sentence.
What students should know and be able to do [at a mastery level] related to these benchmarks
 know the equal sign means "the same as."
 find the number that makes a number sentence true with the unknown in any position:
3 + 6 = ∆ and ∆ = 3 + 6
3 + ∆ = 9 and 9 = 3 + ∆
∆ + 6 = 9 and 9 = ∆ + 6
9  3 = ∆ and ∆ = 9  3
9  ∆ = 6 and 6 = 9  ∆
∆  3 = 6 and 6 = ∆  3
 model the above types of equations using manipulatives and number lines.
 create equations to match story problems.
 create story problems to match equations.
 determine the truth value of equations involving addition and subtraction.
Work from previous grades that supports this new learning includes:
 compose and decompose numbers to twelve.
 solve addition and subtraction problems to 10 using counters and other visible materials such as fingers and 10frames.
NCTM Standards
Use mathematical models to represent and understand quantitative relationships.
PreK2 Expectations
 Model situations that involve the addition and subtraction of whole numbers, using objects, pictures, and symbols.
Common Core State Standards
Represent and solve problems involving addition and subtraction.
1.OA.1. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions; e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
Work with addition and subtraction equations.
1.OA.7. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8  1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.
1.OA.8. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 +? = 11, 5 = _  3, 6 + 6 = _.
Misconceptions
Student Misconceptions and Common Errors
Students may think...
 the equal sign means the answer.
 when they see two numbers they should combine them.
 the only correct format for a problem is a + b = c or a  b = c, not recognizing that it can also be c = a + b or c = a  b or a + b = m + n.
 ∆ = 10  3 is read: 3 minus 10 equals ∆.
 they can completely ignore an unknown in an equation.
Vignette
In the Classroom
First graders in Mr Xiong's class are determining if number sentences are true.
Mr. Xiong writes 3 + 5 = 8 on the board.
Mr. Xiong: I want you to think about this number sentence. Is this number sentence true or false? Skyler, what do you think?
Skyler: I think it's true.
Mr. Xiong: Why do you think it's true?
Skyler: Well, look. I have five here (flashes five fingers) and three here (flashes three fingers). That's eight! So it's true.
Mr. Xiong: Brie, I noticed that you were agreeing with Skyler. Why do you think it's true?
Brie: I started at the five and counted three more, 5  6..7..8..so I know it's true.
Mr. Xiong writes 8 = 3 + 5 on the board.
Mr. Xiong:. What about this number sentence? True or false? He looks around the room and sees some puzzled faces. Alan, what do you think?
Alan: It's false. But you wrote it wrong, Mr. Xiong. Are you trying to trick us? You can't write it that way. The equal sign goes at the end...and so it's false because 8 + 3 isn't 5, it's 11. Look at the other one, that's how you write it.
Mr. Xiong: Let's look at the first equation. Alan, can you read it for us?
Mr. Xiong points to 3 + 5 = 8.
Alan: Three plus five equals eight.
Mr. Xiong: Alan, can you read the problem another way?
Alan: Pauses and thinks for a moment. Three and five make eight.
Mr. Xiong: Okay...good. How about another way? Could anybody read it another way?
Beth: Three plus five is the same as eight.
Mr. Xiong: Good. Now, let's take a look at this problem. Tia, will you read this number sentence.
Mr Xiong points to 8 = 3 + 5.
Tia: Eight equals three plus five.
Mr. Xiong asks the class to read the number sentence with Tia as he points to each part of the number sentence.
Brie: I can read it another way. Eight is the same as three plus five.
Mr. Xiong: Points to 3 + 5 = 8. Three plus five is the same as eight.
Points to 8 = 3 + 5. Eight is the same as three plus five.
Does it matter where we put the equal sign, at the beginning or at the end?
Skyler: It doesn't matter. It's just like turning things around, but it means the same thing.
Mr. Xiong used Alan's response as a springboard for the conversation about the meaning of the equal sign.
Resources
Teacher Notes
 Students may need support in further development of previously studied concepts and skills.
 Provide opportunities to find an unknown in number sentences with the unknown in various locations. For example: 2 + 4 = ∆, 3 + ∆ = 7, 5 = ∆  3, ∆ = 6 + 4,
 6 + 3 = ∆ + 2.
 The number sentence 10  ∆ = 6 can be read ten minus some number equals six. When first asked to find an unknown students need help understanding the symbolic language.
 First graders will apply what they know about composing and decomposing number and the operations of addition and subtraction to help them find the unknown. First graders should initially work with combinations of six or less in order to make the partwhole relationship more visible when finding an unknown.
 A foundational algebraic idea is the equality relationship, which is represented by the equal sign. Students must learn that the equal sign means a balanced relationship. Rather than saying "the answer is," use "is the same as" to help children develop a sense of equality.
 Use concrete materials to explore concepts of equality and inequality.
 Use concrete materials to explore and describe number relationships expressed in openended number sentences (e.g. [math]\square+\square=7[/math])
 The equal sign can be confusing when students only see it as "what comes before the answer." Number sentences should be presented with the = sign at the beginning, middle, or end of a problem.
 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. For more information on
the bar model, read Modeling Word Problems in the Mathematics Best Practice tab in the Resources section of the Minnesota Mathematics Framework.
Questioning
Good questions, and good listening, will help children make sense of the mathematics, build selfconfidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if...?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work? Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part ...
(Adapted from They're Counting on Us, California Mathematics Council, 1995)
Instructional Resources
Activities
 True or False
Asking first graders to determine if a number sentence is true or false is important as first graders develop a sense of equality. It is important to use number sentences with "easy numbers." Number sentences such as 5 = 5 and number sentences which are false (3 + 5 = 10) challenge student thinking. It is also important for first graders to consider the following types of number sentences:
6 = 4 + 2, 3 = 10  7, 4 + 2 = 2 + 4, 5 + 3 = 5 + 3.
The following number sentences might be used the first time first graders are asked to determine if a number sentence is true or false.
True or False?
3 + 2 = 5
2 + 3 = 5
5 = 3 + 2
2 + 3 = 4
5 = 5
5 = 3 + 2
5 = 3 + 3
2 + 3 = 2 + 3
2 + 3 = 2  3
Additional Instructional Resources
Greenes, C., Cavanagh, M., Dacey, L., Findell, C., & Small, M. (2001). Navigating through algebra in prekindergartengrade 2. Reston, VA: National Council of Teachers of Mathematics.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., & BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.
Van de Walle, J. & Lovin, L. (2006). Teaching studentcentered mathematics grades k3. Boston, MA: Pearson Education.
Von Rotz, L., & Burns, M. (2002). Grades k2 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.
number sentence: mathematical sentence written in numerals and mathematical symbols
addition: to join two or more numbers to get one number (called the sum or total)
subtraction: an operation that gives the difference between two numbers. Subtraction can be used to compare two numbers, or to find out how much is left after some is taken away
equal: having the same amount or value, the symbol is =
"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 nonexamples.
Example/Nonexample 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 an understanding of equality at the first grade level? How do student misconceptions interfere with mastery of these ideas?
What kind of number sentences should first graders see related to equality in an instructional setting?
Write a set of number sentences you could use with first graders in exploring their understanding of equality. Which are the most challenging for first graders?
When checking for student understanding, what should teachers
 listen for in student conversations?
 look for in student work?
 ask during classroom discussions?
Examine student work related to a task involving equality. 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 identifying an unknown in an equation at the first grade level? How do student misconceptions interfere with mastery of these ideas?
What kind of equations should first graders experience when identifying an unknown in an equation?
Write a set of equations you could use with first graders in exploring their understanding of identifying unknowns in equation. Which are the most challenging for first graders?
When checking for student understanding, what should teachers
 listen for in student conversations?
 look for in student work?
 ask during classroom discussions?
Examine student work related to a task involving identifying unknowns. 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 first grade level?
Professional Learning Community Resources
Bamberger, H., Oberdorf, C., & SchultzFerrell, K. (2010). Math misconceptions prekgrade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Carpenter, T., Franke, M., & Levi, L. ( 2003). Thinking mathematically integrating arithmetic & algebra in elementary school, Portsmouth, NH: Heinemann.
Chapin, S., & Johnson, A. (2006). Math matters, understanding the math you teach, grades k8. (2^{nd} 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 K6). Sausalito, CA: Math Solutions.
Fosnot, C., & Dolk, M. (2010). Young mathematicians at work constructing algebra. Portsmouth, NH: Heinemann.
Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K6. 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 comprehensionusing literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Bamberger, H., Oberdorf, C., & SchultzFerrell, K. (2010). Math misconceptions prek grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Bender, W. (2009). Differentiating math instructionStrategies that work for k8 classrooms! Thousand Oaks, CA.: Corwin Press.
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting english language learners in math class, grades k2. 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., & Johnson, A. (2006). Math matters, understanding the math you teach, grades k8, (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 K6). Sausalito, CA: Math Solutions.
Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k2. 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, C., & Dolk, M. (2010). Young mathematicians at work constructing algebra. Portsmouth, NH: Heinemann.
Greenes, C., Cavanagh, M., Dacey, L., Findell, C., & Small, M. (2001). Navigating through algebra in prekindergartengrade 2. Reston, VA: National Council of Teachers of Mathematics.
Hyde, Arthur. (2006) Comprehending math adapting reading strategies to teach mathematics, K6. 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. (2004). Teaching Mathematics Vocabulary in Context. Portsmouth, NH: Heinemann.
Murray, M, & Jorgensen, J. (2007). The differentiated math classroomA guide for teachers k8. Portsmouth, NH: Heinemann.
Sammons, L. (2011). Building mathematical comprehensionUsing literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., ... & Zbiek, R. M. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: Aquest for coherence. Reston, VA: NCTM.
Seeley, C. (2009). Faster isn't smarterMessages 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., & BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.
Van de Walle, J. & Lovin, L. (2006). Teaching studentcentered mathematics grades k3. Boston, MA: Pearson Education.
Von Rotz, L., & Burns, M. (2002). Grades k2 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Assessment
 Write an equation to represent the following number story.
There are seven turtles in the pond.
Three more turtles go into the pond.
How many turtles are in the pond?
Solution: 7 + 3 = 10
Benchmark: 1.2.2.4
 Write a number story to match this equation.
7  2 = 5
Solution: Contexts will vary but will represent seven minus two equals five.
Benchmark: 1.2.2.4
 Look at the number sentences. Tell if they are true or false.
5  1 = 6
9 = 4 + 5
6 + 4 = 6 + 4
6 = 6
4 + 1 = 5 + 2
5 + 4 = 4 + 5
Solution: F, T, T, T, F, T
Benchmark: 1.2.2.2
 What is the missing number in each equation?
3 + 4 = ∆
10  ∆ = 6
8 = ∆ + 5
Solution: 7, 4, 3
Benchmark: 1.2.2.3
Differentiation
Students may need to use materials to find the unknown in open number sentences involving addition and subtraction. For example, when presented with 3 + ∆ = 7, a teacher may lay out three counters and ask how many more are needed to make seven.
Students may need to use materials in order to determine if a number sentence is true or false.
Concrete  Representational  Abstract Instructional Approach
(Adapted from The Access Center: Improving Access for All K8 Students)
The ConcreteRepresentationalAbstract Instructional Approach (CRA) is a researchbased 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 ontask. 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 instructionStrategies that work for k8 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 classroomA guide for teachers k8. 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., BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J. & Lovin, L. (2006). Teaching studentcentered mathematics grades k3. Boston, MA: Pearson Education.
Using materials to model the action in a story problem will help students write an equation which matches the problem.
 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 think this number sentence is _____________ because _______________. 
I used ___________________ to find the missing number. 
I know __________________________ is true/false 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 k2. Sausalito, CA: Math Solutions Publications.
Students expand their skills by writing true or false number sentences on slips of paper and placing them in a container. After drawing a slip of paper from the container, students determine if the number sentence is true or false.
Students write number stories for equations involving three addends having a sum less than or equal to twelve.
Students write their own number sentences involving unknowns and find the unknown in each other's number sentences.
What Number Goes Where?
Students are given a set of digits 09 on small cards. Each playing board has 10 missing numbers involving addition, subtraction or counting. Using each digit only once on the playing card, first graders fill in the missing numbers to make the number sentences true or to complete the counting sequence.
Additional Resources:
Bender, W. (2009). Differentiating math instructionStrategies that work for k8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades k2. Sausalito, CA: Math Solutions.
Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k8. 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 . . . 
determining if number sentences involving addition and subtraction are true or false and explaining their thinking.
 providing a sequence of number sentences that will challenge first graders' thinking about equality. Asking students to explain their thinking.

finding the unknowns in open number sentences involving addition and subtraction and explaining their thinking.
 providing open number sentences involving addition and subtraction with the unknown in different positions. Listening to students' ideas and strategies for finding an unknown. 
writing number stories to match given number sentences involving addition and subtraction.  providing students with number sentences for students to write number stories. Noting strategies used to solve number sentences. 
writing number sentences to match number stories  providing students with number stories for students to write number sentences. Providing materials to assist students. 
Students are trying to solve problems with the use of objects and number lines to identify the missing numbers.  Teachers are encouraging them to come up with ways to solve the problem. Using vocabulary that is needed for understanding. 
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 realworld 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 k8, 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
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 selfconfidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if . . . ?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to ...?
Can you make a prediction?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work?
Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part...
Adapted from They're counting on us, California Mathematics Council, 1995
Read Aloud Books:
Domino Addition by Lynette Long, Ph.D.
The Hershey's Kisses Addition Book by Jerry Pallotta
One More Bunny: Adding From One to Ten by Rick Walton
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