3.2.1 InputOutput Rules
Overview
Standard 3.2.1 Essential Understandings
Third graders develop a basic understanding of a function as they work with single operation inputoutput rules involving addition, subtraction and multiplication. They describe patterns in inputoutput situations and find the rule. Given a rule, third graders are able to find the output for a corresponding input and find an input for a corresponding output. They realize the value of the output varies depending on the value of the input. They solve problems in various contexts using the information on a simple inputoutput situation.
Benchmark Group A
3.2.1.1 Create, describe, and apply singleoperation inputoutput rules involving addition, subtraction and multiplication to solve problems in various contexts.
What students should know and be able to do [at a mastery level] related to these benchmarks:
 Determine the rule for an input/output situation.
 Determine the relationship between two sets of numbers in an input/output situation.
 Determine the output when given the corresponding input.
 Determine the input when given the corresponding output.
 Represent a relationship using a rule for an input/output situation.
 Complete an input/output situation with correct numbers and identify the rule.
 Extend the input/output situation by using the rule.
Work from previous grades that supports this new learning includes:
 Identify simple number patterns.
 Identify, describe, create, complete and extend repeating patterns.
 Identify, describe, create, complete and extend simple growth patterns.
 Identify, describe, create, complete and extend simple number patterns involving addition and subtraction or arrays of objects.
 Skip count.
 Solve problems involving patterns.
 Know basic addition and subtraction facts.
NCTM Standards
Understand patterns, relations, and functions
Grades 35 Expectations:
 describe, extend, and make generalizations about geometric and numeric patterns;
 represent and analyze patterns and functions, using words, tables, and graphs.
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;
 represent the idea of a variable as an unknown quantity using a letter or a symbol;
 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.
Analyze change in various contexts
Grades 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:
Solve problems involving the four operations, and identify and explain patterns in arithmetic.
 3.OA.9. Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
Misconceptions
Students may think...
 input values and output values are not related
 looking at only inputs or outputs will help them find the rule for an inputoutput situation
Resources
Teacher Notes
 Students may need support in further development of previously studied concepts and skills.
 Single operation rules involving addition, subtraction and multiplication are to be used in inputoutput situations.
 Students need to use words to describe the patterns they discover in input/output situations.
 All inputoutput situations must contain at least three given corresponding inputoutput values.
 Students need to see a variety of inputoutput situations
6 12 3 baseballs cost $15
8 16 8 baseballs cost $40
2 4 5 baseballs cost $25
? 20 7 baseballs cost ?
9 ? ? baseballs cost $50
 Students may have difficulty applying a rule to generate input and output values due to computational errors rather than a failure to comprehend the rule.
 Computation errors may interfere with rule identification.
 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)
1995
Additional Instructional Activities
 Draw a table on the board and ask: How many legs are on 1 ant? How many legs are on 2 ants? 3 ants? 5 ants? Fill out the table:
Number of ants  Number of legs 






Questions:
How are the number of ants related to the number of legs?
Is there a way to figure out the number of legs if you know the number of ants?
Is there a way to figure out the number of ants if you know the number of legs?
If there are 20 ants in the sugar bowl, can you determine how many legs there are? How?
Which numbers between 6 and 30 would be unlikely if you are finding the number of ant legs? Why?
As a writing activity: Write a story to go with the ant table, or make your own table, with labels and write a story that uses the information in the table to explore multiples.
 In/Out
Record an inputoutput situation on the board and have students complete the missing values and fill in three or more inputs and related outputs.
10 ⇨ 6
4 ⇨ 0
12 ⇨ 8
⇨ 4
15 ⇨
⇨
⇨
Questions:
Use words to write a rule for the input/output situation. Subtract 4 from the in to solve for or find the value of out.
Write a story to go with the table; e.g., you want to buy a marble for 4 cents, the in number shows how many pennies you have or how much money in cents you have, the out number shows how much you have left after you buy a marble.
If 10 is the in number, what is the out number? Are there any other numbers that work if 10 is in? Are there any in numbers that could have more than one out number? This table represents a function because for each in there is only one out that satisfies the equation.
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.
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.
Van de Walle, J., & Lovin, L. (2006). Teaching studentcentered mathematics grades 35. Boston, MA: Pearson Education.
Willoughby, S.S. (1997). Functions from kindergarten through sixth grade. Teaching children mathematics 3 (February 1997): 31418.
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
rule  a set of instructions that is applied to a situation over and over
input  a number that is acted on according to a rule
output  a number that is the result of a rule acted upon an input number
value  quantity or number in an inputoutput situation
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 inputoutput situations at the third grade level? How do student misconceptions interfere with mastery of these ideas?
What experiences do students need in order to develop an understanding of single operation inputoutput situations?
What kind of inputoutput situations should third graders experience? What real world situations could these inputoutput rules represent?
How would you know a student understands the relationships shown in an inputoutput situation? Are some relationships in an inputoutput situation more important as students develop algebraic reasoning?
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 a single operation inputoutput situation. 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 third 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.
O'Connell, S., & SanGiovanni, J. (2011). Mastering the basic math facts multiplication and division strategies, activities & interventions to move students beyond memorization. Portsmouth, NH: Heinemann
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.
Uittenbogaard, W., & Fosnot, C. (2008). Minilessons for early multiplication and division, Grades 34. Portsmouth, NH: Heinemann.
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., Ohanian, S., & Burns, M. (2002). Teaching arithmetic: Lessons for introducing division, Grades 34. Sausalito, CA: Math Solutions.
Willoughby, S. S. (1997). Functions from kindergarten through sixth grade. Teaching Children Mathematics, Feb., 31418.
Yeatts, K., Battista, M., Mayberry, S., Thompson, D., & Zawojewski, J. (2004). Navigating through problem solving and reasoning in grade 3. Reston, VA: National Council of Teachers of Mathematics.
Zemelman, S., Daniels, H., & Hyde, A. (2005). Best practices: Today's standards for teaching and learning in America's schools. Portsmouth, NH: Heinemann.
Assessment
 The table shows how much money the Crafty Craft Store can make by selling packages of beads.
Number of packages of beads sold  Money made 
1  $4 
3  $12 
5  $20 
7  $28 
9 

10 

If the pattern in the table continues, how much money can the store make by selling 9 boxes? _________________
By selling 10 boxes? _____________
Adapted from: NCTM. (2005).Grades 35 Mathematics Assessment Sampler. Reston, VA: NCTM.
Solution: 9 boxes = $36, 10 boxes = $40
Benchmark: 3.2.1.1
 A table is shown.
Input Output
2 12
4 24
8 48
What is the output number when the input number is 12?
A. 2
B. 60
C. 72
D. 96
Solution: C 72
Benchmark: 3.2.1.1
MCA III Item Sampler
Differentiation
Struggling Learners
Start with simple number patterns and rules for students to explore. Realworld contexts will help students make sense of a situation. InputOutput charts are more challenging because of the lack of context. Help students apply skip counting skills in inputoutput situations.
People  Legs  Input  Output 
1  2  1  6 
2  4  2  12 
3  6  3  18 
8  ?  4  ? 
?  20  ?  30 
Ask questions: What is the relationship between the number of people and the number of legs? What patterns do you see?
Using the inputoutput information: What patterns do you see? How might the input and output values be related?
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 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 k3. Boston, MA: Pearson Education.
Van de Walle, J. & Lovin, L. (2006). Teaching studentcentered mathematics grades 35. Boston, MA: Pearson Education.
The terms input and output might be confusing. Realworld contexts help students make sense of an inputoutput situation. InputOutput tables are more challenging because of the lack of a context. Make the relationship between an input and the corresponding output as visible as possible.
People  Legs  Input  Output 
1  2  1  6 
2  4  2  12 
3  6  3  18 
8  ?  4  ? 
?  20  ?  30 
Ask questions: What is the relationship between the number of people and the number of legs? What patterns do you see?
Model the language of the relationships in inputoutput situations and encourage students to describe the relationships. For example,
 The number of legs depends on how many people there are.
 If there are 3 people, then there will be 6 legs.
 If there are 10 legs, then there are 5 people.
Using the inputoutput information, what patterns do you see? How might the input and output values be related?
Help students understand that rules are explicit and connected with each input/output situation.
 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:
If the input is ___________ then the output is ______________. 
The rule is _____________ 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.
Play "Guess My Rule" games. These are games in which a student gives examples of some "rule," and others try to discover the rule. After giving a few inputoutput examples (in writing) that follow the rule, other students take turns silently adding other inputoutputs they think follow the rules. If the added inputoutput follows the "secret rule" the student leader nods yes; if incorrect, the leader nods no and the values are erased from the board. After several correct inputoutput values are added the student players can describe the rule using words and a "rule."
For example, if doubling was the "secret rule", the leader could write two pairs of numbers that follow the rule.
6 ⇨ 12
2 ⇨ 4
7 ⇨ 14
Players would add other inputoutput values and the leader would let players know if the values followed the rule.
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 ... 
creating inputoutput rules involving addition, subtraction and multiplication.  asking questions to clarify student understanding of the inputoutput situations, and modeling the creation of an inputoutput table. 
describing and completing the pattern for an inputoutput situation and determine the rule.  providing inputoutput situations for students that contain at least three given corresponding inputoutput values.

determining the rule for a given inputoutput situation.  asking questions about the patterns in an inputoutput table. Focusing students on the relationship between a given input and a corresponding output. 
applying a rule to find an output value when given an input value and finding an input value when given an output value.  providing single operation input output rules and some related inputoutput values. 
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
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
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