5.2.1 Representing Change
Create and use rules, tables, spreadsheets and graphs to describe patterns of change and solve problems.
For example: An endoftheyear party for 5^{th} grade costs $100 to rent the room and $4.50 for each student. Know how to use a spreadsheet to create an inputoutput table that records the total cost of the party for any number of students between 90 and 150.
Use a rule or table to represent ordered pairs of positive integers and graph these ordered pairs on a coordinate system.
Overview
Standard 5.2.1 Essential Understandings
Fifth graders build on previous work with inputoutput rules and tables by representing patterns of change using ordered pairs (input value, corresponding output value) and points on a coordinate system. Fifth graders understand that any given point in a coordinate system represents an ordered pair. In addition, fifth graders will use spreadsheets to represent patterns of change. They solve realworld and mathematical problems using information from tables, graphs and/or spreadsheets describing a pattern of change.
All Standard Benchmarks
5.2.1.1 ..Create and use rules, tables, spreadsheets and graphs to describe patterns of change and solve problems.
For example: An endoftheyear party for 5th grade costs $100 to rent the room and $4.50 for each student. Know how to use a spreadsheet to create an inputoutput table that records the total cost of the party for any number of students between 90 and 150.
5.2.1.2 Use a rule or table to represent ordered pairs of positive integers and graph these ordered pairs on a coordinate system.
Benchmark Group A
5.2.1.1 Create and use rules, tables, spreadsheets and graphs to describe patterns of change and solve problems.
For example: An endoftheyear party for 5th grade costs $100 to rent the room and $4.50 for each student. Know how to use a spreadsheet to create an inputoutput table that records the total cost of the party for any number of students between 90 and 150.
5.2.1.2 Use a rule or table to represent ordered pairs of positive integers and graph these ordered pairs on a coordinate system.
What students should know and be able to do [at a mastery level] related to these benchmarks:
 Create tables, spreadsheets and coordinate graphs to represent patterns of change.
 Create a rule for a pattern of change and represent it in more than one way.
 Describe a given pattern of change using a rule.
 Represent a given pattern of change in more than one way using rules, tables, spreadsheets, and coordinate graphs.
 Use a rule or a table to represent ordered pairs of positive integers (input, output).
 Graph ordered pairs of positive integers on a coordinate system.
 Solve problems based on a pattern of change that has been represented using tables, spreadsheets, rules and/or the coordinate system.
Work from previous grades that supports this new learning includes:
 Recognize and describe patterns of change involving more than one operation.
 Record inputs and outputs in a chart or a table.
 Create inputoutput rules involving addition, subtraction, multiplication and division.
 Use inputoutput rules to solve realworld and mathematical problems.
 Use addition, subtraction, multiplication and division to solve problems.
NCTM Standards
Understand patterns, relations, and functions
Grade 35 Expectations
 describe, extend, and make generalizations about geometric and numeric patterns;
 represent and analyze patterns and functions, using words, tables, and graphs.
Analyze change in various contexts
Grade 35 Expectations
 investigate how a change in one variable relates to a change in a second variable;
 identify and describe situations with constant or varying rates of change and compare them.
Common Core State Standards
Analyze patterns and relationships.
5.OA.3. Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule "Add 3" and the starting number 0, and given the rule "Add 6" and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
Graph points on the coordinate plane to solve realworld and mathematical problems.
5.G.1. Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., xaxis and xcoordinate, yaxis and ycoordinate).
5.G.2. Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation.
Misconceptions
Students may think...
 examining one set of inputoutput values is enough to describe a pattern or determine a rule.
 the only way to describe is a pattern is using the change between outputs rather than the change from an input to its corresponding output.
 they cannot describe a pattern of change when input values are not consecutive.
 points on a coordinate graph are not related to an inputoutput table.
 rules describing patterns of change involve only addition and subtraction.
Resources
Teacher Notes
Students may need support in further development of previously studied concepts and skills.
Students may have difficulty applying a given rule to generate a pattern because of computational errors rather than a failure to comprehend the rule.
Students may have more difficulty with patterns that do not have a constant rate of change.
Use multiple representations such as tables, graphs, spreadsheets, rules and numerical expressions for a particular inputoutput situation. Discuss similarities and differences among the representations and the advantages/disadvantages of each.
Students need to see patterns of change in a variety of contexts. For example, block patterns, number patterns, input/output patterns, and realworld situations.
Students need many opportunities to build and describe patterns of change. This helps them describe change by representing what they see happening as the pattern continues. They will then be better able to justify their reasoning about patterns.
For example,
 Provide experiences with patterns generated from or embedded in real world contexts.
 Progress from simple patterns to more complex patterns such as those involving a change that is not constant.
 Expect students to communicate their thinking and reasoning about patterns of change. The written rule is not enough.
 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)
NCTM Illuminations
Students investigate the number of chairs that can be placed around an arrangement of square tables. Three related problems in this lesson yield different linear relationships for students to discover.
A puzzle involving five dice and a nonstandard pattern is used to promote problemsolving skills.
In this activity the computer makes up a mystery operation, and you have to figure out what the operation is. You give the computer two numbers to calculate, and it gives the answer. This is your first piece of evidence. Need another clue? Then gather more evidence. Keep entering pairs of numbers for the computer to calculate until you think you know what's going on.
In this lesson students will review plotting points and labeling axis. Students generate a set of random points all located within the first quadrant. Students will plot and connect the points and then create a short story that could describe the graph. Students must ensure that the graph is labeled correctly and that someone could recreate their graph from their story.
Additional Instructional Resources
Blanton, M. (2008). Algebra and the elementary classroom, transforming thinking, transforming practice. Portsmouth, NH: Heinemann.
Carpenter, T., Franke, M., & Levi, L. ( 2003). Thinking mathematically integrating arithmetic & algebra in elementary school. Portsmouth, NH: Heinemann
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
New Vocabulary
"Vocabulary literally is the key tool for thinking."
Ruby Payne
ordered pair: A pair of numbers used to find or define the position of a point on a plane or a grid. The first number describes the horizontal position and the second pair describes the vertical position.
coordinate graph: A coordinate graph has two perpendicular lines, or axes, labeled with numbers and called number lines. The horizontal axis is called the xaxis. The vertical axis is called the yaxis. The point where the x and y axes intersect is called the origin.
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 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 patterns of change at the fifth grade level? How do student misconceptions interfere with mastery of these ideas?
What experiences do students need in order to develop an understanding of patterns of change?
What kind of patterns of change should fifth graders experience? What real world situations could these patterns represent?
What representations should a fifth grader construct for a given pattern of change?
What errors do students make when using coordinate systems to represent a pattern of change?
How would you know a student understands the relationships shown on a coordinate graph?
When checking for student understanding, what should teachers
 listen for in student conversations?
 look for in student work?
 ask during classroom discussions?
How can teachers assess student learning related to these benchmarks?
Examine student work related to a task involving a pattern of change. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.
How are these benchmarks related to other benchmarks at the fifth 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.
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.
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.
Wickett, M., Kharas, K., & Burns, M. (2002). Grades 35 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.
Assessment
 At a movie store, Erin pays a monthly fee and is charged for each movie she rents. The table shows the monthly cost when Erin rents different numbers of movies.
Monthly Cost
Number of Movies 
Total Cost (dollars) 
6 
33 
8 
39 
10 
45 
How much is the monthly fee that Erin pays?
A. $3
B.$6
C.$15
D.$18
Solution: C $15
Benchmark 5.2.1.1
MCA III Item Sampler
 Three points are shown on a grid.
Solution: D
Benchmark 5.2.1.2.
MCA III Item Sampler
 TIMSS (1995)
Here is a number pattern.
100, 1, 99, 2, 98, __, __, __,
What three numbers should go in the boxes?
a 3, 97, 4
b 4, 97, 5
c 97, 3, 96
d 97, 4, 96
Solution: a. 3, 97, 4
TIMSS Released Item
Benchmark:
Differentiation
The move from concrete to representation to abstract is effective with students with math difficulties. Students should begin with visual representations of a growing pattern with manipulatives. Students describe what they see happening as they are building the pattern, and the teacher models their description using a numeric expression. Students then progress to representing a growing pattern by writing their own numeric expressions collecting this information in a table.
Student Think Alouds are an effective tool for students with math difficulties as they help students make meaning of their mathematics. Teachers are more aware of student thinking when students are asked to think aloud.
For example:
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.
Explaining an understanding of patterns is critical in student understanding. Students should use a combination of objects, words, drawings, and numeric expressions when describing patterns. For example,
 Word banks need to be part of the student learning environment in every mathematics unit of study. Refer to these throughout instruction.
 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 input is _____________________. The output is __________________________. 
I see this pattern:________________________________________________________. 
The ordered pair for this point is ___________________________________________. 
 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.
Figure This. Poses the question, would you rather be paid $20 per day for a week or $2 the first day double the previous days amount for the next day?
An effective strategy to solve this is to generate an input/output table of each pay rate for comparison.
$20 per day table:
in 
out 
1 
20 
2 
40 
$2 doubling per day table:
in 
out 
1 
2 
2 
4 
Solution to problem is below, taken from here.
Complete Solution: If you are paid $20 per day for seven days, the you earn $20 × 7 or $140. If you are paid $2 the first day and your salary doubles every day for the next six days, then you earn $2 + $4 + $8 + $16 + $32 + $64 + $128, or $254. The second scheme earns you more money by the end of the week. 

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.
Parents/Admin
Administrative/Peer Classroom Observation
Students are ... 
Teachers are ... 
finding missing parts of the growing numerical and geometric patterns and justifying their solutions when asked. Students then supply a rule for the pattern. 
giving examples of different patterns of change using patterns using objects, tables, and graphs. Asking students to justify their thinking. 
building patterns of change using objects and describing these patterns with numeric expressions. 
translating student thinking into numeric expressions. 
listening, sharing, comparing, and asking questions about patterns of change and their representations. 
listening for misconceptions and asking clarifying questions. 
representing ordered pairs from an input/output table in a coordinate system. 
encouraging students to generalize and generate a rule and support their thinking. 
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.
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.