1.1.1A Place Value
Overview
Standard 1.1.1 Essential Understandings
First graders read, write and represent whole numbers up to 120. Representations for numbers include numerals, pictures, tally marks, number lines, addition and subtraction and manipulatives (including base ten blocks). Formal work with place value begins as students describe two-digit numbers in terms of tens and ones; i.e. 37 can be represented as 37 ones or as 3 tens and 7 ones or as 2 tens and 17 ones. First graders use place value knowledge to compare and order numbers up to 120 and to find a number that is 10 more or 10 less than a given two-digit number. They describe the relative magnitude of numbers using words such as equal to, more than, less than, fewer than, or about the same as.
All Standard Benchmarks
1.1.1.1 Use place value to describe whole numbers between 10 and 100 in terms of groups of tens and ones. For example: Recognize the numbers 11 to 19 as one group of ten and a particular number of one
1.1.1.2 Read, write and represent whole numbers up to 120. Representations may include numerals, addition and subtraction, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks
1.1.1.3 Count, with and without objects, forward and backward from any given number up to 120.
1.1.1.4 Find a number that is 10 more or 10 less than a given number. For example: Using a hundred grid, find the number that is 10 more than 27.
1.1.1.5 Compare and order whole numbers up to 120.
1.1.1.6 Use words to describe the relative size of numbers. For example: Use the words equal to, not equal to, more than, less than, fewer than, is about, and is nearly to describe numbers.
1.1.1.7 Use counting and comparison skills to create and analyze bar graphs and tally charts.
For example: Make a bar graph of students' birthday months and count to compare the number in each month.
Benchmark Group A
1.1.1.1 Use place value to describe whole numbers between 10 and 100 in terms of groups of tens and ones. For example: Recognize the numbers 11 to 19 as one group of ten and a particular number of ones.
1.1.1.4 Find a number that is 10 more or 10 less than a given number. For example: Using a hundred grid, find the number that is 10 more than 27.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- describe numbers less than 100 in terms of tens and ones;
- represent two-digit numbers in more than one way using place value models;
- find a number that is 10 more than a given one- or two-digit number;
- find a number that is 10 less than a given two-digit number.
Work from previous grades that supports this new learning includes:
- finding numbers that are one more and one less than a given number (1-20);
- comparing and ordering numbers up to 20;
- reading, writing and representing numbers up to 31 using pictures, objects, connecting cubes, etc.
NCTM Standards
Understand numbers, ways of representing numbers, relationships among numbers, and number systems.
Pre-K - 2 Expectations:
- count with understanding and recognize "how many" in sets of objects;
- use multiple models to develop initial understandings of place value and the base-ten number system;
- develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections;
- develop a sense of whole numbers and represent and use them in flexible ways, including relating, composing, and decomposing numbers;
- connect number words and numerals to the quantities they represent, using various physical models and representations.
Common Core State Standards
Extend the counting sequence.
1.NBT.1. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.
Understand place value.
1.NBT.2. Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:
1.NBT.2a. 10 can be thought of as a bundle of ten ones - called a "ten."
1.NBT.2b. The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.
1.NBT.2c. The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).
1.NBT.3. Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.
Represent and interpret data.
1.MD.4. Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.
Misconceptions
Students may think...
- there is only one way to represent a whole number in terms of tens and ones. For example, the number 75 can only be represented using 7 tens and 5 ones. Seventy-five can also be represented using 75 ones, 5 tens and 25 ones, 2 tens and 55 ones, etc. There are many ways to represent a number.
- of 4 bundles of ten as 4 single units instead of 4 "tens" or 40 ones.
- they have to decompose groups of ten into ten ones in order to determine how many.
Vignette
In the Classroom
In this vignette, Mr. Dahlberg is working with a small group of students assessing place value understanding.
Mr. Dahlberg gives two students, Henry and Lucy, a large number of popsicle sticks. He asks each student to bundle their sticks into groups of ten.
Henry: I have 4 tens and 8 ones.
Mr. Dahlberg: Hmmm. I wonder if that is the same as 48 sticks? (Henry thinks about the question. Mr. Dahlberg notices that Henry seems unsure whether the representation of 4 tens and 8 ones is equal to forty-eight.)
Lucy: I think Henry's sticks are the same as 48.
Henry: I'm not sure. I need to count them. (Henry unbundles the sticks and begins counting by ones.)
Lucy: I have 3 tens and 6 ones. That's thirty-six.
Mr. Dahlberg: How do you know that is true, Lucy?
Lucy: Look (Points to bundles of ten). Ten, twenty, thirty, now (points to single sticks) thirty-one, thirty-two thirty-three, thirty-four, thirty-five, thirty-six.
Mr. Dahlberg: Good for you Lucy. You did a nice job explaining what you were thinking. I wonder if we can help Henry figure out how many sticks he has? Henry, have you finished counting your sticks?
Henry: I got forty-eight. Henry's sticks are no longer grouped into bundles of 10s.
Mr. Dahlberg notes that Henry can identify groups of tens and ones, but cannot count tens and ones. He only sees the bundle of ten as "ten ones." When asked how many he has, he reverts to counting all the sticks by ones. Mr. Dahlberg plans to give Henry opportunities to make collections of tens using different materials and counting by tens with the materials.
For now, Mr. Dahlberg wants to see what happens if Henry regroups the sticks with teacher support.
Mr. Dahlberg: Henry, can you take all forty-eight sticks and make bundles of ten? Mr. Dahlberg knows that Henry was previously able to successfully bundle the sticks.
Henry bundles the first group of ten sticks.
Mr. Dahlberg: Henry, how many sticks are in the bundle?
Henry: Ten.
Mr. Dahlberg: How do you know?
Henry: I counted ten and made the group with a rubber band.
Henry bundles another group of ten and Mr. Dahlberg clarifies that this new bundle of ten contains ten sticks. This continues as Henry makes the other two bundles of ten.
When he finishes he has four tens and eight ones. Henry helps Mr. Dahlberg understand that another group of ten cannot be made as there are only eight sticks.
Mr. Dahlberg: Henry, how many sticks are in this bundle? (pointing to a bundle of ten sticks)
Henry: Ten sticks.
Mr. Dahlberg: How do you know, Henry?
Henry: I counted ten sticks and made a group. There are ten sticks in that bundle.
Mr. Dahlberg points to the other three bundles of ten and asks Henry how many sticks are in each.
Henry says there are ten sticks in each bundle.
Mr. Dahlberg: Henry, do you know any way to count groups that have exactly ten in them?
Henry: Count by ten.
Lucy starts 10, 20, 30, and Henry joins in counting.
Mr. Dahlberg: Henry, you said these bundles had ten sticks in them and that you can count by ten to count groups that have exactly ten.
Henry: 10, 20, 30, 40 (pointing to each bundle of ten as he counts):
Mr. Dahlberg: How many sticks in all, Henry?
Henry: 40
Mr. Dahlberg: How many bundles of ten are there, Henry?
Henry: 4 bundles
Mr. Dahlberg: How many sticks are in 4 bundles Henry?
Henry: 40
Lucy: Don't forget these---pointing to the eight ones.
Mr. Dahlberg: Henry, you have four bundles of sticks. That is 40 sticks. What about these?
Henry: I can count them. 1, 2, 3, 4, 5, 6, 7, 8. There are eight.
Mr. Dahlberg: Henry, I see four bundles of ten and eight more. How can we count all of them?
Lucy: Let's do tens first and then the ones.
Lucy begins and Henry mirrors her counting. 10, 20, 30, 40, 41, 42, 43, 44, 45, 46, 47, 48.
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- Students should clearly grasp the value of digits according to their place. The digit "1" in the ones place has a value of one while a "1" in the tens place has a value of ten.
- Saying the number that is ten more than a given number does not always mean that the student understands the quantity of 10 more. Having students demonstrate the meaning of ten more or less than a given number using place value materials helps make the concept explicit.
- Counting 10 more and 10 less on the decade (20, 30, 40,...) should come before instruction on counting 10 more and 10 less off the decade (23, 33, 43,...).
- Organizing numbers in different ways can help highlight patterns in the base ten number system. For example, in the arrangement below, moving left to right across each row highlights the ten more relationship among two-digit numbers.
61 71 81 91
62 72 82 92
63 73 83 93
64 74 84 94
65 75 85 95
66 76 86 96
67 77 87 97
68 78 88 98
69 79 89 99
70 80 90 100
- To encourage flexible thinking with numbers and understanding of place value, students should be able to use manipulatives (i.e., bundles of sticks or connecting cubes) to make groups of ten, and also to break apart groups of ten to make ones.
- Students may reverse the numerals in a number when reading and writing numbers.
- 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)
NCTM Illuminations
Click on the button: Count By 10s. Students will help the alien spaceship move cows into corrals by counting. Students must make as many groups of 10 cows as possible and identify how many cows (ones) are left. This activity helps children learn grouping and place value.
Other Instructional Activities
Ten More and Ten Less (adapted from Wright, Stanger, Stafford, Martland, 2006) Contact B. Hagelberger for complete reference.
Materials: 110 popsicle sticks/straws, bundled into groups of 10 and single sticks/straws for each student/pair of students and the teacher
12 x 18 sheet of paper/foam for screening (covering materials during the activity)
Show students one bundle. How many sticks are in this bundle? Students may need to count the sticks/straws or take the bundle apart in order to count the sticks/straws. Establish that there are 10 sticks in each bundle.
- Ten More
1. Using sticks/straws show 3 bundles and 2 sticks. How many tens are there? How many ones are there? How many sticks are there? Record 32. Add another bundle. How many sticks are there? Record 42 and state, 42 is 10 more than 32.
32
42 42 is ten more than 32
Continue to add another bundle of ten asking, How many tens are there? How many ones are there? How many sticks are there? Continue the recording as each group of ten is added.
32
42 42 is ten more than 32
52 52 is ten more than 42
62 62 is ten more than 52
...
92 92 is ten more than 82
2. Remove one bundle. How many sticks are there now? Remove another ten. How many sticks are there now? Continue until zero bundles and 2 sticks. Occasionally ask, How many tens would that be?
- Ten Less Adaptation
- Start the activity with nine bundles and 2 sticks. Record 92 on the board. In turn, remove one bundle of ten. How many tens are there? How many ones are there? How many sticks are there? Record after each bundle of ten is removed.
92
82 82 is ten less than 92
72 72 is ten less than 82
62 62 is ten less than 72
...
2 2 is ten less than 12
Briefly display and screen (cover) two bundles and four sticks. How many sticks are there under the screen? Place another bundle under the screen. How many now? Continue to six bundles and 4 ones. How many sticks are there under the screen? Check to see if you are correct. Place another bundle under the screen. How many now? Continue to 10 bundles and 4 ones. How many sticks now? How many bundles? Check to see if you are correct.
2. Using nine bundles and four sticks, display for students and then screen (cover). Remove one bundle. How many sticks are there now? Remove another bundle. How many sticks are there now? Continue to six bundles. How many now? Check to see if you are correct. Remove one bundle. How many sticks are there now? How many bundles are there now? Continue to zero bundles and 4 sticks.
Scroll to the bottom of this site and click on PDF Base Ten Deck. The site also includes directions and variations of the activity. Who Has? is an activity that will require students to use their knowledge of groups of 10s and 1s by connecting a pictorial representation to a numeral representation.
Additional Instructional Resources
Baratta-Lorton, M. (1994). Mathematics their way. Menlo Park, CA: Addison-Wesley Publishing Co.
Burns, M. (2005). The importance of making sense of numbers: Teaching number sense. Sausalito, CA: Math Solutions.
Burns, M., Tank, B. (1988) A collection of math lessons: From grades 1 through 3, Sausalito, CA: Math Solutions.
Cavanagh, M., Dacey, L., Findell, C., Greenes, C., Sheffield, L., & Small, M. (2001). Navigating through number and operations in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics.
Confer, C. (2005). Teaching number sense: Grade 1. 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. & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.
whole numbers: The numbers 0, 1, 2, 3, 4, ...
place value: The value a digit has because of its place in a number; for example, 52 is 5 tens and 2 ones.
tens: A group of ten units
ones: A single unit
"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 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.
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 place value at the first grade level? How do student misconceptions interfere with mastery of these ideas?
- What models should a student be able to make when representing 84, using tens and ones?
- When checking for student understanding of place value at the first grade level, what should teachers...
listen for in student conversations?
look for in student work?
ask during classroom discussions?
- How can teachers assess student learning related to these benchmarks?
- Examine student work related to a task involving place value at the first grade level. What evidence do you need to say a student is proficient? Using three pieces of student work, determine what student understanding is observed through the work.
- How are these benchmarks related to other benchmarks at the first 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.
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
Fosnot, C., & Dolk, M. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann.
Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 1: 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.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Wright, R., Stanger, G., Stafford, A., & Martland, J. (2006). Teaching number in the classroom with 4-8 year-olds. London: Paul Chapman Publishing.
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Baratta-Lorton, M. (1994). Mathematics their way. Menlo Park, CA: Addison-Wesley Publishing Co.
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, M. (Ed.). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
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.
Conklin, M. (2010). It makes sense! Using ten-frames to build number sense, grades k-2, Sausalito, CA: Math Solutions Press.
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.
Felux, C., & Snowdy, P. (Eds.). (2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Fosnot, C., & Dolk, M. (2001) Young mathematicians at work: Constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann.
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.
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.
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. & Lovin, L. (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.
Wright, R., Martland, J., Stafford, A., & Stanger, G. (2006). Teaching number (2nd ed). London: Sage Publishing.
Wright, R., Stanger, G., Stafford, A., & Martland, J. (2006). Teaching number in the classroom with 4-8 year-olds. London: Paul Chapman Publishing.
Assessment
1. Using base ten blocks, show the number 54 in more than one way.
Solution: Shows 54 in more than one way: 5 tens and 4 ones, 54 ones, 4 tens and 14 ones, etc.
Benchmark: 1.1.1.1
2. Write the number.
Solution: 73, 38
Benchmark: 1.1.1.1
3. Write the number that is ten more.
10 more than 73 is_______ 10 more than 38 is_______ 10 more than 4 is_______
Solution: 83, 48, 14
Benchmark: 1.1.1.4
4. Write the number that is ten less.
10 less than 41 is _____ 10 less than 87 is _____ 10 less than 25 is_______
Solution: 31, 77, 15
Benchmark: 1.1.1.4
Differentiation
Struggling Students
Struggling students need to use base ten materials throughout instruction. Representing numbers with base ten materials is critical as struggling students connect two digit numbers to groups of tens and ones.
Understanding that ten ones are called a "ten" is essential in the understanding of place value. Struggling students need many experiences building a ten from single cubes and decomposing a "ten" into ten ones.
Students should build a model of a given number, say 36, using base ten blocks.
A group of ten can be added to illustrate ten more or a group of ten could be removed illustrating ten less than the starting number.
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 k-2. 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 k-3. Boston, MA: Pearson Education.
- Using base ten blocks and other place value materials to represent numbers provides a common experience and helps English Language Learners understand the language of the base ten system. It is easier for students to explain their thinking when they have models to support their language.
- Place value language can be confusing. Connecting manipulatives to appropriate language is an important part of the learning process. Snap cubes, for example, help students see ten ones become one ten as they are connected.
- Students need to see numbers represented in more than one using a variety of place value materials.
- Students should build a model of a given number; say 36, using base ten blocks.
- A group of ten can be added to illustrate ten more or a group of ten could be removed illustrating ten less than the starting number.
Word banks need to be part of the student learning environment in every mathematics unit of study.
- Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words count, first, second, third, etc.
- Sentence Frames
Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions. Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.
Sample sentence frames related to these benchmarks:
The number _________ has ___________ tens and ___________ ones. |
Ten more/less than ______________ is ________________. |
There are _________ tens and _________ ones. That is the same as ________. |
_____________ is ten more/less than___________________. |
- 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 k-2. Sausalito, CA: Math Solutions Publications.
Extending the Learning
- Students who quickly see and understand that the number 43, for example, can be represented using 4 tens and 3 ones, and as 43 ones, should be challenged to find all possible representations of 43 using tens and ones. These representations include 2 tens and 23 ones, 1 ten and 33 ones, 3 tens and 13 ones. It might be easy to ask these students to start representing three-digit numbers but it is more powerful for them to find all possible representations of a 2 digit number.
- Base Ten Riddles
Students can solve and write their own base ten riddles. For example,
I have 14 ones and 3 tens. Who am I?
I have 16 ones. I am between 50 and 60. Who am I?
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 k-2. 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 ... |
describing numbers using place value vocabulary...tens and ones, ten more, ten less, one more, one less. | translating between the numeric representation of place value, the pictorial and manipulative based representations of a number. |
representing numbers in more than one way, using place value models. For example, 43 can be 43 ones, 4 tens and 3 ones, 3 tens and 13 ones, etc. | asking questions that elicit students' thinking and reasoning as students work representing numbers with manipulatives using place value models. |
communicating their mathematical understanding orally, pictorially and in writing. This includes sharing with each other how they build representations of numbers with manipulatives. | using correct place value vocabulary. This includes translating between a student's natural language and the language of place value...tens and ones. The three in 34 means something different than the three in 63 even though the numeral is the same. |
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 edition. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). (2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
For Administrators
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Parent Resources
Mathematics handbooks to be used as home references:
Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Helping your child learn mathematics
Provides activities for children in preschool through grade 5
What should I look for in the mathematics program in my child's school?
A Guide for Parents developed by SciMathMN
Help Your Children Make Sense of Math
Ask the right questions
In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.
Good questions, and good listening, will help children make sense of the mathematics, build 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
Read-Aloud Books
Out for the count by Kathryn Cave
One hundred hungry ants by Elinor J. Pinczes
The king's commissioner by Aileen Friedman
Sir Cumference and all the king's tens by Cindy Neuschwander
Related Frameworks
Framework Feedback
Love this Framework? Have thoughts on how to improve it? We want to hear your Feedback.
Give us your feedback