4.2.1 Input-Output Rules
Create and use input-output rules involving addition, subtraction, multiplication and division to solve problems in various contexts. Record the inputs and outputs in a chart or table.
For example: If the rule is "multiply by 3 and add 4," record the outputs for given inputs in a table.
Another example: A student is given these three arrangements of dots:
Identify a pattern that is consistent with these figures, create an input-output rule that describes the pattern, and use the rule to find the number of dots in the 10^{th} figure.
Overview
Standard 4.2.1 Essential Understandings
Fourth graders further develop their understanding of a function as they work with input-output rules involving more than one operation. They create, complete and extend input-output tables. Fourth graders describe patterns in given input-output situations/tables and find a rule. Given a rule, fourth 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. Fourth graders use input-output rules involving addition, subtraction and multiplication to solve problems in various contexts.
All Standard Benchmarks
4.2.1.1
Create and use input-output rules involving addition, subtraction, multiplication and division to solve problems in various contexts. Record the inputs and outputs in a chart or table.
4.2.1.1
Create and use input-output rules involving addition, subtraction, multiplication and division to solve problems in various contexts. Record the inputs and outputs in a chart or table.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Create a chart or table organizing a list of inputs and outputs.
- Recognize relationships in input-output tables involving more than one operation.
- Use a given "rule" involving more than one operation to create an input-output table.
- Identify the output when given the corresponding input.
- Identify the input when given the corresponding output.
- Identify and apply the "rule" involving more than one operation in completing an input-output table.
- Use addition, subtraction, multiplication and division in identifying input-output "rules."
- Use information in input-output tables to solve problems.
Work from previous grades that supports this new learning includes:
- Understand input-output situations that can be described with a single operation.
Find the output when given an input and the rule.
Find the input when given an output and the rule.
- Use single-operation input-output rules to represent patterns and relationships and to solve real-world and mathematical problems.
NCTM Standards
Understand patterns, relations, and functions
Grades 3-5 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 3-5 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 3-5 Expectations:
- Model problem situations with objects and use representations such as graphs, tables, and equation to draw conclusions.
Analyze change in various contexts
3-5 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
Generate and analyze patterns.
4.OA.5. Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.
Misconceptions
Student Misconceptions and Common Errors
Students may think...
- the relationships between input values and the relationships between output values are the most important when trying to describe an input-output rule.
- an input-output rule can be determined using only one input-output pair.
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- Students need to see input-output tables in both vertical and horizontal orientation.
- Use pattern blocks, toothpicks, square tiles, etc., to create growing patterns that can then be represented in an input-output table. Find the rule for the patterns.
For example,
- Students may have difficulty applying a given rule to generate a pattern due to computational errors rather than a failure to comprehend the rule.
- Students may have more difficulty finding rules for patterns that do not have a constant change.
- Students need to see and use input-output tables that do not have sequential inputs.
- Provide experiences with input-output tables that have gaps and/or unknowns in input and/or output columns.
- Emphasize the relationship between the input and output.
- Students should be able to prove that a "suggested rule"(incorrect rule) does not fit the input-output situation.
- Students should be able to describe rules in a variety ways.
- Multiple representations of input-output situations should include verbal descriptions, tables, equations, models or drawings.
- Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting StartedWhat 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
Using a context of chairs around square tables, students will be exposed to three different linear patterns in this lesson. The patterns vary slightly from situation to situation, and the third situation allows students to determine a solution in multiple ways, in the end leading to an intuitive understanding of perimeter.
Present the following situation to students:
At Pal-a-Table, a new restaurant in town, there are 24 square tables. One chair is placed on each side of a table. How many customers can be seated at this restaurant?
Show an arrangement of one table with four chairs. If your room contains large square tables at which students work in groups, use them as a demonstration. If not, you can draw a picture on the chalkboard, or you can use pattern blocks or other transparent manipulatives on the overhead projector.
When all students understand how chairs are placed, ask, "If there were 24 tables in a room, how many chairs would be needed?" Depending on students' understanding of multiplication, they may immediately realize that the answer is 24 × 4 = 96. If not, work with the class to complete a table, as follows:
Tables | Chairs |
1 | 4 |
2 | 8 |
3 | 12 |
4 | 16 |
5 | 20 |
From this table, students should realize that the number of chairs is equal to four times the number of tables. Alternatively, they might recognize that each time a table is added, four chairs are added. If there are some students who use each approach, this is a good opportunity to reinforce the connection between multiplication and repeated addition. That is,
2 × 4 | = | 4 + 4 |
3 × 4 | = | 4 + 4 + 4 |
4 × 4 | = | 4 + 4 + 4 + 4 |
5 × 4 | = | 4 + 4 + 4 + 4 + 4 |
and so on.
The following link provides a unit made up of 5 lessons:
lesson 1 explores patterns using pattern blocks and pattern core
lesson 2 explores multiplication and hundreds chart patterns
lesson 3 explores growing patterns and Pascal's Triangle
lesson 4 explores numeric patterns, Fibonacci numbers, and tables to organize information
lesson 5 explores creation, description, and analysis of patterns using tables and charts.
Additional Instructional Resources
Cuevas, G., & Yeatts, K. (2001). Navigating through algebra in grades 3-5. 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., & Bay-Williams, J. (2010). Elementary and middle school mathematics: teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.
Wickett, M., Kharas, K., & Burns, M. (2002). Grades 3-5 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications
input-output table
"Vocabulary literally is the
key tool for thinking."
Ruby Payne
Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions. Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.
Learning vocabulary in the mathematics classroom is contingent upon the following:
Integration: Connecting new vocabulary to prior knowledge and previously learned vocabulary. The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.
Repetition: Using the word or concept many times during the learning process and connecting the word or concept with its meaning. The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.
Meaningful Use: Multiple and varied opportunities to use the words in context. These opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems. Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.
Strategies for vocabulary development
Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.
Mathematics Word Bank: Each unit of study should have word banks visible during instruction. Words and corresponding definitions are added to the word bank as the need arises. Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.
Labeled pictures and charts: Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.
Frayer Model: The Frayer Model connects words, definitions, examples and non-examples.
Example/Non-example Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words. Teachers should use these during the instructional process to engage student in thinking about the meaning of words.
Vocabulary Strips: Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.
word | definition | illustration |
Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.
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 input-input rules at the fourth grade level? How do student misconceptions interfere with mastery of these ideas?
What type of input-output table relationships are the easily seen and described by students? Write an example of a pattern that can be easily described by students when looking at its representation in an input-output table.
How would you know a student understands the relationships shown in an input-output table? Are some relationships in an input-output table more important as students develop algebraic reasoning?
What representations should a student be able to make for a given pattern or relationship in a problem solving situation?
What are some examples of patterns that are easily explored using input-output tables.? What makes some patterns more difficult to analyze when using input-output tables?
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 input-output situation. What evidence do you need to say a student is proficient? Using three pieces of 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?
Professional Learning Community Resources
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Chapin, S., and Johnson, A. (2006). Math matters: Understanding the math you teach, grades K-8. (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 K-6). 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, K-6. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
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 K-5. 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., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.
Burns, Marilyn. (2007). About teaching mathematics: A k-8 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 K-8. (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 K-6). Sausalito, CA: Math Solutions.
Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration pre-k-grade 2. Reston, VA: National Council of Teachers of Mathematics.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in 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, K-6. Portsmouth, NH: Heinemann.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. 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., Bay-Williams, 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 student-centered mathematics grades K-3. 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 3-5 lessons for algebraic thinking. Sausalito, CA: Math Solutions Publications.
Assessment
Solution: B
Benchmark: 4.2.1.1
MCA III item sampler
Performance Assessments (Adapted from Minnesota 2007 Mathematics Standards document examples):
- If the rule is "multiply by 3 and add 4," record the outputs for given inputs in a table.
Sample Answer:
X | Y |
0 | 4 |
1 | 7 |
2 | 1 |
Note: Differentiate by providing table and some cells, or requiring the use of specific values for X.
- A student is given these three arrangements of dots:
Identify a pattern that is consistent with these figures, create an input-output rule that describes the pattern, and use the rule to find the number of dots in the 10th figure.
Sample Answer:
Students may draw the next few figures and recognize that the number of dots is a multiplication problem where the rectangle has a side that is one more than the other side.
There are 10*11 = 110 dots in the tenth figure.
Benchmark: 4.2.1.1
Differentiation
- Use pattern blocks, tiles, and other manipulatives to create growing patterns that can be represented in an input-output table. Describing the relationship between inputs and outputs is important as fourth graders develop algebraic thinking.
For example,
Concrete - Representational - Abstract Instructional Approach
(Adapted from The Access Center: Improving Access for All K-8 Students,)
The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.
The CRA approach is based on three stages during the learning process:
Concrete - Representational - Abstract
The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts. At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level. Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task. Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.
The Representational Stage is the drawing stage. Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems. They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking. Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.
The Abstract Stage is the symbolic stage. Teachers model mathematical concepts using numbers and mathematical symbols. Operation symbols are used to represent addition, subtraction, multiplication and division. Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding. Moving to the abstract level too quickly causes many student errors. Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5. Sausalito, CA: Math Solutions.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.).. Boston, MA: Allyn & Bacon.
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.
- Use pattern blocks, tiles, and other manipulatives to create growing patterns that can be represented in an input-output table. Describing the relationship between inputs and outputs is important as fourth graders develop algebraic thinking.
For example,
- 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 _______________________ the output is __________________________. |
I know the rule is _________________ because ________________________________. |
If the output is __________________ the input was ______________________________. |
- 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 3-5. Sausalito, CA: Math Solutions Publications.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA.: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5. Sausalito, CA: Math Solutions.
Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Parents/Admin
Administrative/Peer Classroom Observation
Students are . . . | Teachers are . . . |
using input-output tables to represent patterns and relationships. | modeling the use of input-output tables as a representation of a pattern or "rule." |
extending a given growing pattern with manipulatives and describing the rule. | providing hands-on experiences with growing patterns. |
searching for patterns within input-output tables. | using the terms input and output when describing patterns and relationships. |
describing the patterns found in input-output tables. | keeping the functional relationship (input to output) the focus of classroom discussions. |
finding the rule for a given input-output table. | scaffolding student experiences with one and two step rules for increased student success. |
finding missing "input"' and "output" values in input-output tables. | providing varied activities for understanding input/output relationships. |
What should I look for in the mathematics classroom?
(Adapted from SciMathMN,1997)
What are students doing?
- Working in groups to make conjectures and solve problems.
- Solving real-world problems, not just practicing a collection of isolated skills.
- Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
- Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
- Recognizing and connecting mathematical ideas.
- Justifying their thinking and explaining different ways to solve a problem.
What are teachers doing?
- Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
- Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
- Connecting new mathematical concepts to previously learned ideas.
- Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
- Selecting appropriate activities and materials to support the learning of every student.
- Working with other teachers to make connections between disciplines to show how math is related to other subjects.
- Using assessments to uncover student thinking in order to guide instruction and assess understanding.
What should I look for in the mathematics classroom?
(Adapted from SciMathMN,1997)
What are students doing?
- Working in groups to make conjectures and solve problems.
- Solving real-world problems, not just practicing a collection of isolated skills.
- Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
- Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
- Recognizing and connecting mathematical ideas.
- Justifying their thinking and explaining different ways to solve a problem.
What are teachers doing?
- Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
- Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
- Connecting new mathematical concepts to previously learned ideas.
- Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
- Selecting appropriate activities and materials to support the learning of every student.
- Working with other teachers to make connections between disciplines to show how math is related to other subjects.
- Using assessments to uncover student thinking in order to guide instruction and assess understanding.
Additional Resources
For Mathematics Coaches
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach: Grades k-8. (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 self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if . . . ?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to ...?
Can you make a prediction?
Why did you...?
What assumptions are you making?
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.