4.2.2 Number Sentences
Understand how to interpret number sentences involving multiplication, division and unknowns. Use realworld situations involving multiplication or division to represent number sentences.
For example: The number sentence a × b = 60 can be represented by the situation in which chairs are being arranged in equal rows and the total number of chairs is 60.
Use multiplication, division and unknowns to represent a given problem situation using a number sentence. Use number sense, properties of multiplication, and the relationship between multiplication and division to find values for the unknowns that make the number sentences true.
For example: If $84 is to be shared equally among a group of children, the amount of money each child receives can be determined using the number sentence 84 ÷ n = d.
Another example: Find values of the unknowns that make each number sentence true:
12 × m = 36
s = 256 ÷ t.
Overview
Standard 4.2.2 Essential Understandings
Fourth graders use number sentences to represent real world situations involving multiplication, division and unknowns. They create a situation that can be represented by a given number sentence involving multiplication, division and unknowns. For instance, arranging 24 children into different sized groupings can be represented in the following number sentences, 24 ÷ A = B or 24 = A x B.
They find values for unknowns that make number sentences true using the properties of multiplication and the relationship between multiplication and division.
All Standard Benchmarks  with codes
4.2.2.1
Understand how to interpret number sentences involving multiplication, division and unknowns. Use realworld situations involving division to represent number sentences.
4.2.2.2
Use multiplication, division and unknowns to represent a given problem situation using a number sentence. Use number sense, properties of multiplication, and the relationship between multiplication and division to find values for the unknowns that make the number sentences true.
4.2.2.1
Understand how to interpret number sentences involving multiplication, division and unknowns. Use realworld situations involving division to represent number sentences.
4.2.2.2
Use multiplication, division and unknowns to represent a given problem situation using a number sentence. Use number sense, properties of multiplication, and the relationship between multiplication and division to find values for the unknowns that make the number sentences true.
What students should know and be able to do [at a mastery level] related to these benchmarks:
 Write a number sentence representing a realworld situation involving multiplication and division and unknowns.
 Write a realworld situation for a given number sentence involving multiplication, division and unknowns.
 Find values for unknowns that make number sentences true using the properties of multiplication and the relationship between multiplication and division.
Work from previous grades that supports this new learning includes:
 Understand the equal sign.
 Write number sentences for realworld situations involving addition and subtraction.
 Create realworld situations for given number sentences involving multiplication and division basic facts.
 Find unknowns in order to make number sentences involving multiplication and division basic facts.
 Mastery of multiplication and division basic facts (current grade level).
NCTM Standards
Represent and analyze mathematical situations and structures using algebraic symbols
Grades 35 Expectations:
 identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers;
 identify such properties as commutativity, associativity, and distributivity and use them to compute with whole numbers;
 express mathematical relationships using equations.
 Use mathematical models to represent and understand quantitative relationships.
Grades 35 Expectations:
 Model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions. (NCTM, online algebra standards.)
Common Core State Standards:
Use the four operations with whole numbers to solve problems.
4.OA.1. Interpret a multiplication equation as a comparison; e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.
4.OA.2. Multiply or divide to solve word problems involving multiplicative comparison; e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.1
4.OA.3. Solve multistep word problems posed with whole numbers and having wholenumber answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Misconceptions
Student Misconceptions and Common Errors
Students may think...
 there are rules that determine which number a letter stands for. For example, e = 5 because e is the fifth letter of the alphabet or y = 4 because y was 4 in the last number sentence.
 that a letter always has one specific value.
 different letters always represent different numbers.
 an equal sign means "and the answer is." In this way, when they see an equal sign, they want to carry out the operation preceding it. For example, when asked what the △ represents in the equation 4 x 3 = △ x 2 they say 12. They need to think of the equal sign as meaning "is the same as."
 the commutative property applies to division; e.g., 12/3 = 4 is the same as 3/12 = 4.
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: 36 ÷ 6 = m, 3 x ∆ = 21, 6 = m ÷ 3, t = 6 x 4.
 Fourth graders need opportunities writing number stories that match number sentences before they write number stories for number sentences involving unknowns.
 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.
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if...?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work? Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part ...
(Adapted from They're Counting on Us, California Mathematics Council, 1995)
NCTM Illuminations
 Use the following link to develop understanding of number relationships and multiplication properties. http://illuminations.nctm.org/ActivityDetail.aspx?id=26 Set the beginning equality to a desired value and ask students to provide new expressions. After there is sufficient student work, discuss patterns and properties students used because all expressions are equal (the initial value of 10 is provided in this example). You can demonstrate strategies using any part of the equation table. In the example below the distributive property is seen on the left side of the bottom two entries [2x(2+3)= 2x2+2x3]; the commutative property is illustrated as 5x2 and 2x5 both on the left. Also, point out that in the equation 111=100 each value on the left is reduced by one to make the expression on the right; this relationship can help students develop mental math skills. If students do not add specific items, you may wish to contribute to the equations with 10/1 or 10 + 10x0 to focus on other properties. This works well on a Smartboard as a class warmup:
 Creating Number Sentences
List: 6, 7, 8, +, , = and the example: 6+78=5
Students must write number sentences for as many different numbers as they can, using the given numbers and symbols only once. Properties of numbers should be explored, such as the commutative property, using the number sentences students create.
Choose 2 of your number sentences and write a story about each of them making a note of which part of the expression is unknown in the story, or otherwise including a reason the number sentence describes the story.
Example: 6 students were on the swings and 7 students joined them from lunch, when Mr. D called in his class and 8 of the students left; how many students are still on the swings (this is the unknown in this story problem)? [Note: we cannot tell which students of Mr. D are part of the lunch group or how many of the original students were in Mr. D's class, but we can infer that 8 students from the swings were in Mr. D's class. Questions that cannot be explicitly answered also provide excellent discussion.
As a center, provide an = and have students draw 3 digits and 2 operations from a cloth bag and construct equations. Alternatively, provide '<' and/or '>' and proceed as above to construct inequalities. Students should consider and record at least one story either before or after constructing their number sentences.
Additional Instructional Resources
Cuevas, G., & Yeatts, K. (2001). Navigating through algebra in grades 35. 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 35. Boston, MA: Pearson Education.
Wickett, M., Kharas, K., & Burns, M. (2002). Grades 35 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.
equivalent: having the same value
variable: a symbol that can be replaced by a number in a formula, expression or number sentence
"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.
MeaningfuUse: 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 fourth grade level? How do student misconceptions interfere with mastery of these ideas?
What kind of number sentences should fourth graders see related to equality in an instructional setting? Do fourth graders understand that the amounts are equal on each side of the equal sign? (working with number sentences such as
21 ____ = 12 + 5)
Write a set of number sentences you could use with fourth graders in exploring their understanding of equality. Which are the most challenging for third 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 fourth grade level? How do student misconceptions interfere with mastery of these ideas?
What kind of equations should fourth graders experience when identifying an unknown in an equation?
Write a set of equations you could use with fourth graders in exploring their understanding of identifying unknowns in equation. Which are the most challenging for fourth 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 in equations. 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 fourth grade level?
Materials  suggest articles and books for book study with PLC.
Professional Learning Community Resources
Bamberger, H., Oberdorf, C., & SchultzFerrell, K. (2010). Math misconceptions prekgrade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Chapin, S., and 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. (2002). Young mathematicians at work: Multiplication and division. 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.
Otto, A., Caldwell, J., Wallus Hancock, S., & Zbiek, R.(2011). Developing essential understanding of multiplication and division for teaching mathematics in grades 3  5. Reston, VA.: National Council of Teachers of Mathematics.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K5. Sausalito. CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J. (2009). Focus in grade 3, teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Bamberger, H., Oberdorf, C., & SchultzFerrell, K. (2010). Math misconceptions prekgrade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Bender, W. (2009). Differentiating math instruction: Strategies 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, Marilyn. (2007). About teaching mathematics: A k8 resource (3rd ed.). Sausalito, CA: Math Solutions Publications.
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Caldera, C. (2005). Houghton Mifflin math and English language learners. Boston, MA: Houghton Mifflin Company.
Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.
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.
Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration prekgrade 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 grade 2 teaching with curriculum focal points. 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 classroom: A guide for teachers k8. Portsmouth, NH: Heinemann.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K5. Sausalito. CA: Math Solutions.
Reeves, D. (2007). Ahead of the curve: The power of assessment to transform teaching and learning. Indiana: Solution Tree Press.
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., BayWilliams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J. A., & Lovin, L. H. (2006). Teaching studentcentered mathematics grades K3. Boston, MA: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Van de Walle, J. & Lovin, L. (2006). Teaching studentcentered mathematics grades 35. Boston, MA: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: transforming mathematics lessons. Portsmouth, NH: Heinemann.
Wickett, M., Kharas, K., & Burns, M.. (2002). Grades 35 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.
Assessment
Solution: D.
Benchmark: 4.2.2.2
MCA III Item Sampler
Solution: C.
Benchmark: 4.2.2.1
MCA III Item Sampler
Performance Assessment Items ():
Benchmark 4.2.2.1
 Describe a situation which can be represented by the number sentence a × b = 60.
Sample Solution: The number sentence a × b = 60 can be represented by the situation in which chairs are being arranged in equal rows and the total number of chairs is 60.
Benchmark 4.2.2.1
Adapted from Minnesota 2007 Mathematics Standards document examples
 Write a number sentence that would determine the amount of money each child receives if $84 is to be shared equally among a group of children.
Sample Solution: The amount of money each child receives can be determined using the number sentence 84 ÷ n = d.
Benchmark: 4.2.2.1
Adapted from Minnesota 2007 Mathematics Standards document examples
 Find values of the unknowns or variables that make each number sentence true:
12 × m = 36
s = 256 ÷ t
Sample Solution: m=3; if t = 2, then s = 128
Benchmark: 4.2.2.1
Adapted from Minnesota 2007 Mathematics Standards document examples
Differentiation
Model equations involving unknowns with manipulatives.
Students should work with number sentences involving unknowns using multiplication and division basic facts they have mastered. This provides an opportunity for them to develop an understanding of writing and interpreting open number sentences which will, in turn, help them learn the facts they do not know.
Concrete  Representational  Abstract Instructional Approach
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 instruction: Strategies that work for k8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 35. 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.
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 35. Boston, MA: Pearson Education.
 Model equations involving unknowns with manipulatives using language to develop student understanding.
 Word banks need to be part of the student learning environment in every mathematics unit of study. Refer to these throughout instruction.
 Word 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:
The unknown is _________ because ________________________________________. 
The equation______________ represents the situation 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.
Provide opportunities to explore equations with more than one variable.
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 35. 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... 
interpreting number sentences involving multiplications and division and unknowns. 
asking students to justify their thinking when as they interpret number sentences. 
using variables to represent unknowns. 
evaluating student use of variables as representations of unknowns.. 
writing stories that can be described by given number sentences involving unknowns. 
presenting multiple representations of problems; e.g., 4 x n = 12, n x 4 = 12, as well as 12 = 4 x n, 4 x n = 3 x 8 and 12 = n x 4.

writing a number sentence to describe a multiplication or division situation involving unknowns. 
providing a variety of situations involving unknowns and the operations of multiplication and division. 
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 ed.). Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
For Administrators
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Parent Resources
Mathematics handbooks to be used as home references:
Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Helping your child learn mathematics
Provides activities for children in preschool through grade 5
What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN
Help Your Children Make Sense of Math
Ask the right questions
In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.
Good questions, and good listening, will help children make sense of the mathematics, build 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.