5.4.1 Data Analysis

Grade: 
5
Subject:
Math
Strand:
Data Analysis & Probability
Standard 5.4.1

Display and interpret data; determine mean, median and range.

Benchmark: 5.4.1.1 Mean, Median & Range

Know and use the definitions of the mean, median and range of a set of data. Know how to use a spreadsheet to find the mean, median and range of a data set. Understand that the mean is a "leveling out" of data.

For example: The set of numbers 1, 1, 4, 6 has mean 3. It can be leveled by taking one unit from the 4 and three units from the 6 and adding them to the 1s, making four 3's.

Benchmark: 5.4.1.2 Display & Interpret Data

Create and analyze double-bar graphs and line graphs by applying understanding of whole numbers, fractions and decimals. Know how to create spreadsheet tables and graphs to display data.

Overview

Big Ideas and Essential Understandings 
Standard 5.4.1 Essential Understandings

Fifth graders use technology to create spreadsheet tables and graphs. The types of data displays that can be created using spreadsheet software include frequency tables, bar graphs, double bar graphs, number line plots, and line graphs. Fifth graders interpret and analyze data containing fractions, whole numbers and decimals. They describe data in terms of mean, median and range. They understand how to find the mean by "leveling out" the data. They are able to use their understanding of the operations of addition and division to explain how this process is reflected in the formula for calculating the mean.

All Standard Benchmarks

5.4.1.1 
Know and use the definitions of the mean, median and range of a set of data. Know how to use a spreadsheet to find the mean, median and range of a data set. Understand that the mean is a "leveling out" of data.

5.4.1.2 
Create and analyze double-bar graphs and line graphs by applying understanding of whole numbers, fractions and decimals. Know how to create spreadsheet tables and graphs to display data.

Benchmark Cluster 

5.4.1.1 
Know and use the definitions of the mean, median and range of a set of data. Know how to use a spreadsheet to find the mean, median and range of a data set. Understand that the mean is a "leveling out" of data.

5.4.1.2 
Create and analyze double-bar graphs and line graphs by applying understanding of whole numbers, fractions and decimals. Know how to create spreadsheet tables and graphs to display data.

What students should know and be able to do [at a mastery level] related to these benchmarks:
  • Find the mean, median, and range of a data set.
  • Demonstrate understanding that the mean is a "leveling out" of data.
  • Create and analyze double-bar graphs and line graphs.
  • Use technology to create spreadsheet tables and graphs to organize and display data.
  • Adjust the scale of the graph to accommodate rational numbers.
Work from previous grades that supports this new learning includes:
  • Collect, organize, and display data.
  • Interpret data using frequency tables, bar graphs, picture graphs and number line plots having a variety of scales.
  • Use appropriate titles, labels and units.
  • Use tables, bar graphs, time-lines and Venn diagrams to display data sets.
  • Work with data involving fractions or decimals.
  • Understand that spreadsheet tables and graphs can be used to display data.
  • Use information in a data display to answer questions.
Correlations 

NCTM Standards

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

Grades 3-5 Expectations:

  • 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

Grades 3-5 Expectations:

  • 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.

Common Core State Standards

Represent and interpret data.
  • 5.MD.2. Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally.

Misconceptions

Student Misconceptions 
Student Misconceptions and Common Errors

Students may think...

  • the terms mean and median can be used interchangeably.
  • a number that is repeated in a data set only has to be used once when determining the mean or the median.

 

Resources

Instructional Notes 
Teacher Notes
  • Students may need support in further development of previously studied concepts and skills.
  • Many students memorize and use the formula for the mean without understanding what the mean represents. Activities in which the mean is found without relying on a formula can strengthen students' conceptual understanding of the mean. Consider approaching the mean as a "leveling out" of the data. For example, in determining the mean of the numbers 10, 4, 8, 3, and 5, tiles (or cubes) can be used to make stacks corresponding to the numbers as shown below.

levelling out - colored tiles

The tiles can then be moved to "level out" the stacks so they are all the same height. An example of how the tiles might be moved is shown below along with a recording of the moves.

In this example, two blue tiles were moved to the stack of yellow tiles. The new totals for each stack are recorded below the original totals in order to  keep track of the thinking.

colered tiles II

At this point, the stacks are not level so another move is needed.

Next, 2 red tiles are moved to the green stack, and the new totals are recorded.

 colo0red tiles III

The stacks are not level so another move is needed.

Finally, a blue tile is moved to each of the green and purple stacks. The new totals are then recorded.

colored tiles IV

Since all stacks are now the same height the data has been leveled out.  This leveling leads to the determination that the mean of 10, 4, 8, 3 and 5 is 6.

This understanding of mean, along with conceptual understanding of the operations of addition and division can be used to relate the leveling out process to the formula for finding the mean.  In other words, the mean can be viewed as combining all the tiles (10 + 4 + 8 + 3 + 5 = 30) and then placing them into 5 groups so that each group has the same number of tiles.

If the stacks are not all level after all possible moves have been made, students realize that the extra tiles need to be broken into equal parts in order to level the stacks.  For example, if the data set above had been 12, 4, 8, 3 and 5 the result of the leveling might have been as shown below.

colered tiles V

Students who understand leveling are able to determine that one-fifth of each of the extra blue tiles needs to go to each stack. Each stack now has 6 and tiles resulting in a mean of  6 .

  • Students should have access to tiles to demonstrate understanding of the "leveling out" of the data that occurs when finding the mean. They should also have ways of recording their thinking as they move the tiles to level the data.
  • Students need to know how to find various data measures, but memorizing formulas or procedures is not the focus. Students need to understand the concepts behind these measures, in order to make sense of the data. 
  • Assist students with developing appropriate scales since they are no longer working with just whole numbers. Be sure to provide them opportunities to determine these themselves and not always provide a graph with predetermined and labeled scales.
  • 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, which include: 
Represent data using an appropriate representation
Analyze Data

When posing questions about collected and represented data refer 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 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.

Ask questions that require students to compare two or more categories in a data set.
Ask questions that require students to compare two or more sets of data.

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?

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).

Instructional Resources 
NCTM Illuminations
  • Accessing and Investigating Population data This cluster of Internet Mathematics Excursions describes activities in which students can use census data available on the Web to examine questions about population. Working on such activities, students also formulate their own questions and use the mathematics they are studying to address these questions. They can propose and justify conclusions that are based on data and design further studies on the basis of conclusions or predictions
  • Dealing with Data in the Elementary School. This project-based unit on statistics furnishes a vehicle for problem solving through real-world data collection and analysis. Students use the mean, mode and median to analyze their data and use graphs to represent their findings.
Additional Instructional Resources

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 

double bar graph:  a graph that uses pairs of bars to compare and show the relationship between two sets of data.

line graph:  a line graph shows information using line segments. Data values are plotted as ordered pairs on a coordinate grid and connected using line segments.

mean:  A measure of center in a set of numerical data, computed by adding the values in a list and then dividing by the number of values in the list.

median:  A measure of center in a set of numerical data. The median of a list of values is the value appearing at the center of a sorted version of the list-or the mean of the two central values, if the list contains an even number of values. Example: For the data set {2, 3, 6, 7, 10, 12, 14, 15, 22, 90}, the median is 11.

range:  A measure of variation in a set of numerical data. The range is the difference between the maximum and minimum values of the data set. Example: For the data set {1, 3, 6, 7, 10, 12, 14, 15, 22, 120}, the range is 119 (120 - 1 = 119).

"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.

Frayer Model

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.

example /non -example chart

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.

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 fifth grade level? How do student misconceptions interfere with mastery of these ideas?

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

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

What type of data should be represented in a line graph? What will be challenging for fifth graders when using a line graph as a data display?

What strategies will help fifth graders understand the concepts mean, median and range?  What strategies might fifth graders use when finding the mean, median and range of a data set?

What common errors do fifth graders make when finding the mean, median and range for a data set?  How can these errors be corrected?

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 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 fifth 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.

Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. 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

  • Anya listed the prices of meals on a menu.

$14.85   $10.75   $8.50   $12.45   $9.20

What is the mean price of the meals?

A. $6.35
B. $8.50
C. $10.75
D. $11.15

Solution:          D. $11.15
Benchmark      5.4.1.1.
MCA III Item Sampler

●     Maria recorded the heights of 2 plants for 4 weeks.

Plant Growth

How much did plant 2 grow from week 1 to week 2?

A. 1 cm
B. 1 1/2 cm
C. 2 cm
D. 4 1/2 cm

Solution:  B. 1 ½ cm
Benchmark 5.4.1.2.                                                                
MCA III item sampler

  • Given a group of five numbers [4, 10, 4, 5, 2], find the median, range, and mean. Solution: 4; 10-2=8; 2+4+4+5+10=25, 25/5=5]


If another number is included in this group and the mean goes down, what might the number be?
Solution: any number less than 5
If another number is included in this group and the mean goes up, what might the number be?
Solution: any number greater than 5

Benchmark: 5.4.1.1:

  • Create two different data sets having the same range, the same median but different maximum values.
    Solution:    will vary
    Benchmark: 5.4.1.1
  • Create a set of data where the mean is greater than the median.  List both the mean and median.
    Solution:    will vary
    Benchmark: 5.4.1.1
  • Create a dot plot for a set of data having eleven data values where the median is eight and there are more data values at ten than at eight.
    Solution:    will vary
    Benchmark: 5.4.1.1

Differentiation

Struggling Learners 

Many students memorize and use the formula for the mean without understanding what the mean represents. Activities in which the mean is found without relying on a formula can strengthen students' conceptual understanding of the mean. Consider approaching the mean as a "leveling out" of the data. For example, in determining the mean of the numbers 10, 4, 8, 3, and 5, tiles (or cubes) can be used to make stacks corresponding to the numbers as shown below.

levelling out - colored tiles

 

The tiles can then be moved to "level out" the stacks so they are all the same height. An example of how the tiles might be moved is shown below along with a recording of the moves.

In this example, two blue tiles were moved to the stack of yellow tiles. The new totals for each stack are recorded below the original totals in order to keep track of the thinking.

colered tiles II

At this point, the stacks are not level so another move is needed.

Next, 2 red tiles are moved to the green stack, and the new totals are recorded.

 colored tiles III

The stacks are not level so another move is needed.

Finally, a blue tile is moved to each of the green and purple stacks. The new totals are then recorded.

colored tiles IV

Since all stacks are now the same height the data has been leveled out.  This leveling leads to the determination that the mean of 10, 4, 8, 3 and 5 is 6.

This understanding of mean, along with conceptual understanding of the operations of addition and division can be used to relate the leveling out process to the formula for finding the mean.  In other words, leveling out can be viewed as combining all the tiles (10 + 4 + 8 + 3 + 5 = 30) and then placing them into 5 groups so that each group has the same number of tiles.

If the stacks are not all level after all possible moves have been made, students realize that the extra tiles need to be broken into equal parts in order to level the stacks.  For example, if the data set above had been 12, 4, 8, 3 and 5 the result of the leveling might have been as shown below.

colered tiles V

Students who understand leveling are able to determine that one-fifth of each of the extra blue tiles needs to go to each stack. Each stack now has 6 and  tiles resulting in a mean of  6 .

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. 

CRA Triangle

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.

English Language Learners 

Many students memorize and use the formula for the mean without understanding what the mean represents. Activities in which the mean is found without relying on a formula can strengthen students' conceptual understanding of the mean. Consider approaching the mean as a "leveling out" of the data. For example, in determining the mean of the numbers 10, 4, 8, 3, and 5, tiles (or cubes) can be used to make stacks corresponding to the numbers as shown below.

levelling out - colored tiles

The tiles can then be moved to "level out" the stacks so they are all the same height. An example of how the tiles might be moved is shown below along with a recording of the moves.

In this example, two blue tiles were moved to the stack of yellow tiles. The new totals for each stack are recorded below the original totals in order to keep track of the thinking.

colored tiles II

At this point, the stacks are not level so another move is needed.

Next, 2 red tiles are moved to the green stack, and the new totals are recorded.

 colored tiles III

The stacks are not level so another move is needed.

Finally, a blue tile is moved to each of the green and purple stacks. The new totals are then recorded.

colored tiles IV

Since all stacks are now the same height the data has been leveled out.  This leveling leads to the determination that the mean of 10, 4, 8, 3 and 5 is 6.

This understanding of mean, along with conceptual understanding of the operations of addition and division can be used to relate the leveling out process to the formula for finding the mean.  In other words, "leveling out" can be viewed as combining all the tiles (10 + 4 + 8 + 3 + 5 = 30) and then placing them into 5 groups so that each group has the same number of tiles.

If the stacks are not all level after all possible moves have been made, students realize that the extra tiles need to be broken into equal parts in order to level the stacks.  For example, if the data set above had been 12, 4, 8, 3 and 5 the result of the leveling might have been as shown below.

colored tiles V

 

Students who understand leveling are able to determine that one-fifth of each of the extra blue tiles needs to go to each stack. Each stack now has  6 and  tiles resulting in a mean of  6 .

  • 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.

Frayer Model

  • Sentence Frames

Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions.  Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.

Sample sentence frames related to these benchmarks:

The range is_________________.  That means ________________________.

The median is ________________.  That means _______________________.

The mean is __________________.  That means _______________________.

  • 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.

Extending the Learning 

Average 5th Grader. Adapted from 1997 MN Mathematics Framework.

What would you like to know about your classmates?

  • What does the average 5th grader look like?
  • What are his or her favorite sports? music? books? foods?
  • How does the average 5th grader spend her or his time?

What does "average" mean?

What data does your group need to gather in order to answer these questions?

Before beginning, certain decisions about conducting the data investigation have to be made by your group, including:

  • What is being investigated and why?
  • What data need to be collected?
  • How will the data be collected and from whom?
  • How will the data be organized and analyzed?
  • What comparisons will be made?
  • Who will find this data useful or interesting?

After you have examined the data, identify patterns. Use the evidence to formulate conjectures and theories. Also view the data from different perspectives in order to address other questions. When you have finished, prepare a detailed report of your group's investigation, including graphs and charts, as well as conjectures and ideas on how the investigation might be extended.

Assessment: Portfolio

Prepare a report describing your group's creation of a database, your use of a spreadsheet, and your analysis of selected data collected in your survey with accompanying charts, graphs, and other displays. Indicate how your project could be extended by asking some "What if..." questions, such as "What if we compared selected parts of these data?"  For example, compare class A with class B, compare homework time with TV time. How would the results differ?

Another look at the data

Separate the boys' and girls' data. Look at the male and female mean, male and female range, and male and female median. Are the female mean and median identical? Are the male mean and median identical? Why or why not?

Extend the investigation

Is our classroom representative of our school? city? state? Is it the same the world over? How could we explore these questions?

Have students find records of the size of people today and in bygone days. Could today's women wear the fashions we see in a museum? Could today's people fit comfortably on the Mayflower?  For example, could they stand up below decks without hitting their heads? A visit to a historical museum may take on a whole new dimension and lead to some new explorations.

Additional Resources

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.

Parents/Admin

Classroom Observation 

Administrative/Peer Classroom Observation

Students are...

Teachers are...

finding the mean of a data set by "leveling" the data.

providing manipulatives and guiding students as they level a data set.

describing data sets in terms of mean, median and mode.

clarifying student understanding of mean, median and range.

creating data sets having a given range, median or mean. 

asking students to explain their strategies for creating a data set when given a range, median or mean.

using technology to create spreadsheet tables and graphs

provide opportunities for students to explain how to use spreadsheet tables and graphs.

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.

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.

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.

Comments

writing frames

Check out the book: Write for Mathematics by Dr. Evelyn Rothstein (Corwin Press). She has amazing ideas for getting kids writing about mathematics and thinking much deeper.