4.4.1 Data Analysis

4
Subject:
Math
Strand:
Data Analysis & Probability
Standard 4.4.1

Collect, organize, display and interpret data, including data collected over a period of time and data represented by fractions and decimals.

Benchmark: 4.4.1.1 Collect & Interpret Data

Use tables, bar graphs, timelines and Venn diagrams to display data sets. The data may include fractions or decimals. Understand that spreadsheet tables and graphs can be used to display data.

Overview

Big Ideas and Essential Understandings

Standard 4.4.1 Essential Understandings

Fourth graders collect, display and interpret data sets that may include fractions and decimals. They expand their work with data displays to include timelines, Venn diagrams, spreadsheets and graphs.

Benchmark Cluster

4.4.1.1 Use tables, bar graphs, time lines, and Venn diagrams to display data sets. The data may include fractions or decimals. Understand that spreadsheet tables and graphs can be used to display data.

What students should know and be able to do [at a mastery level] related to these benchmarks:

• Create, read, and analyze tables
• Create, read, and analyze bar graphs
• Create, read, and analyze time lines
• Create, read, and analyze Venn diagrams
• Understand decimal and fraction data
• Understand relationships and differences between tables, graphs, time lines, and Venn diagrams
• Recognize applications of spreadsheet tables and graphs

Work from previous grades that supports this new learning includes:

• Formulate questions and collect data.
• Organize and classify data from surveys and questionnaires using a tally chart or frequency table.
• Understand the concept of scale and key in a data representation.
• Display data using:
•
• frequency tables,
• bar graphs,
• picture graphs (keys not exceeding 5),
• number line plots with scale increments not exceeding 5.
• understand that appropriate titles, labels, units, and keys are needed on data representations so the information can be interpreted correctly.
• describe parts of the data and the set of data as a whole to determine what the data show.
• use the information on a graph to answer questions.
Correlations

NCTM Standards

Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.

• design investigations to address a question and consider how data-collection methods affect the nature of the data set;
• collect data using observations, surveys, and experiments;
• represent data using tables and graphs such as line plots, bar graphs, and line graphs;
• recognize the differences in representing categorical and numerical data.

Select and use appropriate statistical methods to analyze data

• describe the shape and important features of a set of data and compare related data sets, with an emphasis on how the data are distributed;
• use measures of center, focusing on the median, and understand what each does and does not indicate about the data set;
• compare different representations of the same data and evaluate how well each representation shows important aspects of the data.

(NCTM online data analysis and probability standards)

Common Core State Standards

Represent and interpret data.

• 4.MD.4. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. For example, from a line plot find and interpret the difference in length between the longest and shortest specimens in an insect collection.

Misconceptions

Student Misconceptions

Students may think...

• the scale is automatically one often overlook the scale or key in a data display.
• the scale of bar graphs and time lines is not important.
• all data displays are equally effective in representing any given data set.
• they can randomly choose a data representation for a set of data.
• data sets only include whole numbers.
• once data is represented graphically, alternate forms are not useful.
• all sections of a Venn diagram must have data.
• the area outside of the circles of a Venn diagram is not part of the Venn diagram.
• labels are not always needed for graphic representations.

Resources

Instructional Notes

Teacher Notes

• Students may need support in further development of previously studied concepts and skills.
• MCA III test specifications indicate fraction denominators are limited to 2, 3, 4, 5, 6, 8, 10, and 12 in a data set.
• MCA III test specifications indicate decimals are limited to hundredths in data sets.
• When displaying data in a Venn diagram, students need to use a data set with values that will not be included within the defined regions of the diagram.

The data analysis process includes the following:

• Formulate a question (teacher guided)
• With teacher direction, children formulate questions in conjunction with lessons on counting, measurement, numbers, and patterns.
• Make a plan for data collection
• Collect data
• Organize data and select a data representation.
• Represent data using an appropriate representation. Data representations include:

Charts - include a title, categories and category labels and data.

Tally Charts - include a title, categories, category labels and data

Picture Graphs (also called pictographs) - should be seen both vertically and horizontally and include a title, key, categories, category labels and data.

Bar Graphs - should be drawn both vertically and horizontally and include a title, scale, scale label, categories, category labels and data.

Frequency Tables - should be drawn both vertically and horizontally and  include a title, categories, category labels and data counts.

Number Line Plots - use a segment of the number line. Unlike bar graphs, number line plots display values having no data points, but do  include a title, scale, scale label and data.

Venn Diagrams

• Analyze data. When posing questions about collected and represented data, refer to the initial question and use language that keeps the focus on the meaning of the data categories.  For example: What do you know when looking at the graph, chart or table? Guide student responses so that they are related to the original question, not to the height of the bars on the graph. Students will look for patterns and draw conclusions based on the data.
• Which category shows the greatest, least number or any given number of responses to the original question?  What does this tell us?
• Students will focus on the physical characteristic of the data representation and need the language to describe what this means when answering the original question.  Note:  A question such as, "Which column has the most?" does not connect the data representation to the meaning of the collected data.
• Asking questions that require students to compare two or more categories (with reference to the original question) involves a higher level of thinking.
• Reflect on the original question.  Does the data we collected answer the original question?  What else do you wonder about?
• 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...?

Why did you...?

What assumptions are you making?

What did you try that did not work? Has the question been answered?

Can the explanation be made clearer?

Responding (helps clarify and extend their thinking)

Tell me more.

Can you explain it in a different way?

Is there another possibility or strategy that would work?

Is there a more efficient strategy?

Help me understand this part ...

(Adapted from They're Counting on Us, California Mathematics Council, 1995)

Instructional Resources

Instructional Resources

NCTM  Illuminations

Students estimate, experiment, and display data based on the number of breaths they take.

• A Tale of Two Stories Unit

Lesson 1: Pigging out

Students use the story of The Three Little Pigs and activities to understand the regions in a 3 circle Venn diagram.

Lesson 2: If the shoe fits...

Students use Venn diagrams to compare two fairy tales and understand attributes

• Picture This Unit

Students use a survey to collect data and create a bar graph. Content of survey investigates pictures as representations of addition and subtraction story problems.

Students use categorical data to create a bar graph

Students take a survey and use the collected data to create a bar graph. Includes use of Bar Grapher online tool.

Students create a survey and use the collected data to create a bar graph. Includes use of Bar Grapher online tool.

• Bar graphs, and other graphs, can be created with student data using Data Grapher. Instructions are included on the site.

Bereska, C., Bolster, C., Bolster, L., & Scheaffer, R. (1998). Exploring statistics in the elementary grades: Book 1, grades K-6.  Parsippany, NJ: Dale Seymour Publications.

Bereska, C., Bolster, C., Bolster, L., & Scheaffer, R. (1999). Exploring statistics in the elementary grades: Book 2, grades 4-8. Parsippany, NJ: Dale Seymour Publications.

Chapin, S., Koziol, A., MacPherson, J., & Rezba, C. (2007). Navigating through data analysis and probability 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.

New Vocabulary

"Vocabulary literally is the

key tool for thinking."

Ruby Payne

New Vocabulary

table: an organized way to list data, usually in rows and columns

Venn diagram: a graphic organizer for comparing two or three attributes or concepts

time line: a number line that provides a list of events in history in order

survey: a list of questions given to a sample of people

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 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.

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.

Professional Learning Communities

Reflection - Critical Questions regarding the teaching and learning of these benchmarks

What are the key ideas related to data analysis at the fourth grade level? How do student misconceptions interfere with mastery of these ideas?

What experiences do students need in order to develop an understanding of data analysis?

What representations will fourth graders use when representing a data set?

What difficulties might students have when representing data using a Venn Diagram?

When checking for student understanding, what should teachers

• listen for in student conversations?
• look for in student work?

Examine student work related to a task involving data analysis. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.

How can teachers assess student learning related to these benchmarks?

How are these benchmarks related to other benchmarks at the fourth grade level?

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. (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.

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.

References

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.

Assessment

1.  A student creates a time line for a history project.  Which shows a timeline?

Solution:  D. (Values listed in increasing order and spaced as they would appear on a number line.)

Benchmark:  4.4.1.1

MCA III Item Sampler

Solution:  B. (Both 53 and 47 are closest to 50; 43 is closer to 40.)

Benchmark:  4.4.1.1

MCAII item sampler.

Differentiation

Struggling Learners

Initially, use real objects and yarn to make a Venn diagram.  Students can see the attributes of the objects as they place them in sections on the Venn diagram.  In addition, if representing objects that are big and red, students can see that something that is both big and red cannot be in two categories at once and the intersecting regions of a Venn diagram make sense.

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.

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.

English Language Learners

Initially, use real objects and yarn to make a Venn diagram.  Students can see the attributes of the objects as they place them in sections on the Venn diagram.  In addition, if representing objects that are big and red, students can see that something that is both big and red cannot be in categories at once and the intersecting regions of a Venn diagram make sense.

• 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:

 I use a spreadsheet to show _______________________________________________.
 A Venn diagram shows ____________________________________________________.
• 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.

Bresser, R., Melanese, K., & Sphar, C. (2008).  Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.

Extending the Learning

Illuminations Lesson Plans

This unit investigates more complex bar graphs and has students creating different graphic representations as well as asking questions and predicting and summarizing data.

This unit investigates Internet searches, more complex bar graphs, spreadsheets (lesson 3 Spreadsheets and Census Data), and has students creating different graphic representations as well as asking questions and summarizing data.

Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.

Bereska, C., Bolster, C., Bolster, L., & Scheaffer, R. (1998). Exploring statistics in the elementary grades: Book 1, grades K-6. Parsippany, NJ: Dale Seymour Publications.

Bereska, C., Bolster, C., Bolster, L., & Scheaffer, R. (1999). Exploring statistics in the elementary grades: Book 2, grades 4-8.. Parsippany, NJ: Dale Seymour Publications.

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.

Classroom Observation

 Students are... Teachers are... generating or gathering data and displaying data involving decimals and fractions. guiding students in the use of data involvingdecimals,  fractions and appropriate data displays. using spreadsheets and tables to find patterns in data. checking student understanding of spreadsheets and accuracy in the data representation. analyzing data with their classmates; communicating information in data displays. asking open-ended questions like:How do you know?Will that always be the case?What are the similarities and differences between...?What questions could you ask that could be answered with this data? displaying data in multiple ways. asking for similarities and differences between data displays and the appropriate use of each.

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.

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.

Parents

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.

What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN

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?

What did you try that did not work?

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.