K.1.1A Numbers, Representation & Counting
Recognize that a number can be used to represent how many objects are in a set or to represent the position of an object in a sequence.
For example: Count students standing in a circle and count the same students after they take their seats. Recognize that this rearrangement does not change the total number, but may change the order in which students are counted.
Read, write, and represent whole numbers from 0 to at least 31. Representations may include numerals, pictures, real objects and picture graphs, spoken words, and manipulatives such as connecting cubes.
For example: Represent the number of students taking hot lunch with tally marks.
Count, with and without objects, forward and backward to at least 20.
Overview
Developing an understanding of numbers, ways of representing numbers, and relationships among numbers is essential for Kindergarten students.
Research suggests that children learn number word sequences, the reading of numerals and the concept of quantity separately even though they are connected in instructional activities. Counting means so much more than reciting the words and writing the numerals for the counting sequence: 1, 2, 3, 4, ... When successfully counting a group of objects, children give a unique number name to each object (one-to-one correspondence) and then recognize that the last number represents the quantity of objects in the group (cardinality). In addition, they will also successfully show a group of objects to represent a given number (up to 31).
Kindergartners expand their conceptual understanding of counting to include the idea that, when counting, the next number name refers to a quantity that is one larger. This understanding leads to the comparing and ordering of numbers with and without objects.
Kindergartners will use multiple representations to represent numbers. These representations include spoken words, numerals, pictures, real objects and manipulatives.
All Standard Benchmarks - with codes
K.1.1.1 Recognize that a number can be used to represent how many objects are in a set or to represent the position of an object in a sequence.
K.1.1.2 Read, write, and represent whole numbers from 0 to at least 31. Representations may include numerals, pictures, real objects and picture graphs, spoken words, and manipulatives such as connecting cubes.
K.1.1.3 Count, with and without objects, forward and backward to at least 20.
K.1.1.4 Find a number that is 1 more or 1 less than a given number.
K.1.1.5 Compare and order whole numbers, with and without objects, from 0 to 20.
Benchmark K.1.1.1
Recognize that a number can be used to represent how many objects are in a set or to represent the position of an object in a sequence.
Benchmark K.1.1.2
Read, write, and represent whole numbers from 0 to at least 31. Representations may include numerals, pictures, real objects and picture graphs, spoken words, and manipulatives such as connecting cubes.
Benchmark K.1.1.3
Count, with and without objects, forward and backward to at least 20.
What students should know and be able to do related to these benchmarks at a mastery level.
- understand the relationship between numbers and quantities.
- connect counting (number word sequence) to cardinality (number of items in a set is represented by a number word). For example, if there are 8 counters on the desk and child is asked, "How many counters are there?" a child that has made connections from counting to cardinality will understand that the last number he/she says is the number of counters in the set.
- understand that when counting forward, the numbers are one larger and that when counting backwards, the numbers are one smaller.
- read, write and represent whole numbers to at least 31.
- count forward and backward from at least 20.
- understand ordinal numbers and their use when talking about being first or fourth in line, or the fifth book.
Work from previous grades that supports this new learning includes:
- extend the appropriate number of fingers when someone asks, "How old are you?"
- spoken vocabulary includes some number words.
- informally use relationships of less than and more than.
- have a sense of counting but may not be successful counters
Misconceptions
Students may think...
- saying any sequence of numbers means they are counting. For example, 1,2,4,6, 9, 5.
- there are more big objects than small objects even though the quantity in the groups is the same.
- when saying a number like thirteen they need to touch two objects, one for each syllable.
- they can recount objects or skip objects when counting objects in a group or a set.
- any numeral can represent the number of objects in a given set.
- the teen numbers are written in reverse order. For example, when hearing fourteen they might record 41 because they heard four first.
- a group of objects that looks bigger has more objects.
- ordinal numbers are not connected to counting numbers.
Vignette
In the Classroom
Ms. Sharpe is sharing collections of objects with groups of Kindergarten students. The sorting boxes have been part of mathematics instruction since the first day of school. During the lesson, Ms. Sharpe wanted to use the terms set/s and group/s interchangeably to facilitate the transition from student's natural language to the language of mathematics.
"Come and have a look at what I have in my box today," announced Ms. Sharpe to her group. Ms. Sharpe's sorting boxes were quite popular. The students tipped the contents out onto the table. There were some books, erasers, bears, and crayons. The children sorted the items and described their sets.
"Have a look at the objects we have on the table and let's see what we can find out."
The children first sorted the objects into 4 groups: a group of bears, a group of books, a group of crayons, and a group of erasers. They focused on the quantity in each group.
"We have as many books as crayons."
"And we have as many books as erasers."
"Look, they are the same --bears and books and crayons and erasers."
"Tell us more," said Ms. Sharpe.
"There are four bears and four books and four crayons and four erasers. They are all four....the same."
"There are four groups and everyone has one book, one bear, one crayon, and one eraser."
As the children organized the objects, they were counting and also talking about the number of objects in each group/set. Organizing the objects so that each group/set had one of each of the different objects, allowed the students to see there were equal amounts of books, crayons, erasers and bears.
Ms. Sharpe then produced an envelope with some paper clips and coins inside. The children sorted these and matched them one to one with the other groups.
"We have as many paper clips as books," said Sarah, placing a paper clip on top of each book to show what she meant.
"And we have as many erasers as coins." The erasers and coins were arranged to show groups of one eraser and one coin.
Ms. Sharpe decided to widen the inquiry and give the students an opportunity to represent "four." "Jose, can you bring me another group of objects with the same count as the groups here on the table? Have a look around the room and see what you can find."
Jose returned with some calculators. As he put each one down in turn, he said, "This calculator matches this eraser" as he touched each item. When all the erasers were matched, he announced, "I have more calculators than I need." He returned them to the other table.
Ms. Sharpe intervened, saying, "All the sets match. There are as many here as here," she said touching the sets/groups as she spoke. Let's count how many are in each set/group. After counting to four, she says, "When we have a set that matches any of these sets, we know we have 'four' in the set. Let's display all our sets of four."
The children helped Ms. Sharpe make a chart. She stressed that "4" was the numeral and "four" was the name. They would see a lot of these symbols in the future. "Now that we have made our chart. I want you all to find a set/group of four things and bring it over here to the display."
|
4
• • • •
• • • •
four
Four
|
The children collected many groups/sets of four and placed them next to the chart. Sometimes they brought too many things and other times not enough. Each time they were encouraged to check their set by matching with the four erasers or coins or other objects. Their display included large balls and small blocks. The number four had become a rich experience for Ms. Sharpe's class, as would other numbers.
Resources
- Students may need support in further development of previously studied concepts and skills.
- Kindergartners need a variety of experiences exploring number in many classroom contexts. For example, "How many children are in your group? "How many books are on this shelf?" "How many children are wearing a red shirt today?" "How many girls/boys are in our class?"
- Kindergartners learning the skill of counting should be given sets of objects that can be easily moved or pictures of objects that can be marked as they are counted.
- Some Kindergartners associate quantity with the physical size of an object. Kindergartners should count several sets where the number of objects is the same, but the objects are different in size.
- It is important for Kindergartners to know that they can start to count with any object in the group and still make an accurate count.
- Kindergartners should be counting a set of objects more than once. After each successful count, rearrange the objects for recounting.
- It is important to begin the oral counting sequence at many different numbers, not just 1 or 20. For example, from 12 to 21 is often easier than saying the number sequence from 18 to 21. Counting from 20 to 15 is easier than from 13-8.
- Kindergartners may appear to understand one-to-one correspondence when working with numbers less than ten but are unable to apply this concept when working with larger numbers.
- Kindergartners are more successful writing numerals when they see patterns.
0 1 2 3 4 . . .
10 11 12 13 14 . . .
20 21 22 23 24 . . .
- Number vs. Numeral: Number is the idea or concept that identifies a given quantity. A numeral is the written symbol used to represent the quantity.
Questioning
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
In this game, students must make matches of equivalent representations. Representations include dot patterns, ten-frames, numerals, and numeral words.
Oh, no! Okta and his friends need help. Help rescue them by transporting them to a safe ocean. How fast can you transport the Oktas? Use your counting skills to save as many as you can before the timer runs out
Additional Instructional Activities
Teddy Bear March
Using a collection of teddy bear counters or other colored objects, ask students to line up the teddy bears in a specific order. For example, "Put a red bear first, a blue bear second, ..." Once the teddy bear counters are in line, ask questions which focus on the ordinal positions. "Which color bear is third? What position is the green bear in? Point to the fourth bear. Put finger next to the first bear."
Count and Rearrange
Give a student a set of objects to count. Then ask, "How many are there?" Next, rearrange the set or cover the set with a piece of paper. Ask again, "How many are there?" If the child answers correctly without recounting, you can infer that he/she has connected cardinality to a set regardless of its arrangement. Asking the student, "How did you know it was the same?" will give further insight into the child's thinking.
Additional Instructional Resources
Baratta-Lorton, M. (1994). Mathematics their way. Menlo Park, CA: Addison-Wesley Publishing Co.
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: Kindergarten. 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.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.
Richardson, K. (1999). Developing number concepts: Counting, comparing, and pattern. White Plains, New York: Dale Seymour Publications.
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, John A., & Lovin, LouAnn H. (2006) Teaching student-centered mathematics grades K-3. Boston: Pearson Education.
Wright, R., Martland, J., Stafford, A., & Stanger, G. (2002). Teaching number: Advancing children's skills and strategies. London, England: Paul Chapman Publishing.
count: to find how many objects are in a group
counting number: a counting number (natural number) is a member of the set 1,2,3,4,5,6,...
first: The ordinal number matching the number one in a series of objects
second: The ordinal number matching the number two in a series of objects.
third: The ordinal number matching the number three in a series of objects.
"Vocabulary literally is the
key tool for thinking."
Ruby Payne
Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions. Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.
Learning vocabulary in the mathematics classroom is contingent upon the following:
Integration: Connecting new vocabulary to prior knowledge and previously learned vocabulary. The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.
Repetition: Using the word or concept many times during the learning process and connecting the word or concept with its meaning. The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.
Meaningful Use: Multiple and varied opportunities to use the words in context.
These opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems. Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.
Strategies for vocabulary development
Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.
Mathematics Word Bank: Each unit of study should have word banks visible during instruction. Words and corresponding definitions are added to the word bank as the need arises. Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.
Labeled pictures and charts: Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.
Frayer Model: The Frayer Model connects words, definitions, examples and non-examples.
Example/Non-example Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words. Teachers should use these during the instructional process to engage student in thinking about the meaning of words.
Vocabulary Strips: Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.
|
word |
definition |
illustration |
Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.
Additional Resources for Vocabulary Development
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Reflection - Critical Questions regarding classroom practice
What are the key ideas related to an understanding numbers and counting at the kindergarten level? How do student misconceptions interfere with mastery of these ideas?
- What does it mean to count successfully? How do you know when a kindergartner can count successfully?
- What experiences do students need in order to develop an understanding of number and counting?
- What is meant by equivalent representations of number? How can teachers help students understand equivalent representations for a number?
- What experiences do students need in order to develop an understanding of ordinal numbers and their use?
- When checking for student understanding of number, 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 counting and/or representing numbers. 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 kindergarten 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 Edition. 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.
Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in kindergarten 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., Martland, J., Stafford, A., & Stanger, G. (2002). Teaching number: Advancing children's skills and strategies. London, England: 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., 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.
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 Edition. 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.
Confer, C. (2005). Teaching number sense: Kindergarten. 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.
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 kindergarten 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: 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.
Richardson, K. (1999). Developing number concepts counting, comparing, and pattern. White Plains, New York: Dale Seymour Publications.
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., Trevino, E., & Zbiek, R. M., (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: National Council of Teachers of Mathematics.
SciMath Minnesota K-12 Mathematics Framework (1997).
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, John A. (2001). Elementary and middle school mathematics: Teaching developmentally. New York: Longman.
Van de Walle, John A., & Lovin, LouAnn H. (2006). Teaching student-centered mathematics grades K-3. Boston: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Wright, R., Martland, J., Stafford, A., & Stanger, G. (2002). Teaching number: Advancing children's skills and strategies. London, England: Paul Chapman Publishing.
Assessment
1. Representing a Whole Number
Show 18.
Solution: Child represents 18 with 18 objects or pictures.
Benchmark: K.1.1.2
2. Using Numbers to Represent a Set
Count the objects and write the number on the line that tells how many.
Solution. 12, 6
Benchmark: K.1.1.1
3. Performance Assessment
Count and Rearrange (K.1.1.1)
Give students a set of objects to count (cubes, teddy bears, counters, etc.). Then ask, "How many are there?" Next, rearrange the set or cover the set with paper. Ask again, "How many are there?" If the child does not need to count over, you can infer that they have connected cardinality to a set regardless of its arrangement. To get more information from the child ask, "How did you know it was the same?"
Solution: Correctly identifies number in the set and connects cardinality to the set.
Benchmark: K.1.1.1
4. Representing the Position of an Object in a Set
Ask students:
What color is the star that is first: ____________________
What color is the star that is fourth: ____________________
The green star is: ____________________
The blue star is: ____________________
first
Solution: red, blue, second, fourth
Benchmark: K.1.1.1
5. Forward Rote Counting (K.1.1.3)
- Teacher says, "Start at one and count forward until I say stop." Stop the student at 20.
- If the student is successful with the above task, ask him or her to start at a different point and count. For example, "Start at 9 and count forward until I say stop." Stop at 14. "Start at 11 and count forward until I say stop." Stop at 16.
Solution: Student correctly counts forward from 1 to 20, 9 to 14, and 11 to 16.
Benchmark: K.1.1.3
6. Backward Rote Counting
- Teacher says, "Start at 20 and count backwards until I say stop." Let the student count back to 1.
- If the student is successful with the above task, ask him or her to start at a different point and count. For example, "Start at 15 and count backwards until I say stop." Stop at 9. "Start at 12 and count backwards until I say stop." Stop at 7.
- If the student is unsuccessful counting back from 20, ask them to start at 15 and count back to 1. If the student is unsuccessful from 15, ask the student to start at 10.
Solution: Student correctly counts backwards from 20 to 1, 15 to 9, and 12 to 7.
Benchmark: K.1.1.3
7. Connecting numbers with representations
Place four counters (cubes or other counting material) in front of the student.
- Ask, "How many counters are there?"
- Note how he or she counts. Does the student count from one? Can he or she subitize and know there are 4 without counting?
- Ask the student to write the numeral that shows how many are in the set.
- Continue with other numbers of cubes until all numbers 0 through 10 have been placed in front of the student. (Repeat this assessment for numbers from ten to thirty-one.)
Record student responses.
Solution: Correctly writes the numeral to represent the sets.
Benchmark: K.1.1.1 and K.1.1.3
Differentiation
Differentiation is about providing access to the mathematical content through multiple entry points and the ability to use a variety of strategies. Universal Design for Learning is one principled approach. Additional background on differentiation approaches can be found at the Edutopia website.
Emergent learners need many experiences counting with physical objects while making connections between quantity and verbal word sequences. Counting sequences can also be related to the numbers on the number line.
Emergent learners need to see many representations of the same number several times. For example, the number five could be seen as the numeral, five books, five pencils, the word five, five large balls, and five small blocks.
Reading and writing numerals may be strengthened by providing students with tactile experiences. For example, students could write numbers in fine sand, jello, finger paints, etc. .
Concrete - Representational - Abstract Instructional Approach
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.
Insert Image: Concrete triangle from image folder
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA.: Corwin Press.
Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction 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.
- 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.
- 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.
- ELL students need to see visual models with the numeral, word and quantity represented. By displaying visual models with the three aspects of number, they will make relationships between the numeral, word and quantity, rather than viewing them as three separate and different concepts.
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 counted_____________________________. |
|
I can count to ________. 1, 2, 3, ____________________. |
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.
Students can be asked to count larger collections of items (beyond 20). Pay attention to grouping strategies as students organize larger collections. Ask students to explain their grouping strategies and the reasons for selecting a particular grouping strategy.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA.: Corwin Press.
Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction 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: |
|
representing numbers with numerals, words and objects.. |
observing and asking questions: How many do you think there are? How many crayons are here? If I put them over here now how many would I have? Will you count for me backwards starting with the number 14? Will you count for me starting at the number 8? |
|
counting sets of objects and recounting when the objects have been rearranged |
providing opportunities for counting, with and without objects. |
|
counting forward and backward, with and without objects. |
asking students if they notice any patterns as they are counting. Directing students to start counting at various numbers, not always one. making connections between rote counting and quantity. |
|
using ordinal language (first, fourth etc.) to describe the position of an object in a sequence and positioning objects so they are first, fourth, etc. |
incorporating ordinal language as children go about their daily routines--lining up, getting supplies, etc. |
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.
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.
Activities
- Number Hunt! Look around your house for numbers. What numbers do you find? What are the numbers used for?
- Look for numbers on a calendar.
- Using a deck of cards (remove all face cards) and a collection of pennies, flip a card over and have your child count out the same number of pennies as the card shows.
- Using a 100s chart (with only 1-20 showing), show two different numbers to your child. Have your child name each one and tell you which number is more or which is less. Repeat the activity using three numbers.
- Say a number from 0-20 and have your child write the number.
Games:
Chutes and Ladders
There are many early number concepts that can be developed by playing Chutes and Ladders. Help your child to:
- recognize regular dot patterns. The game is played with a die 1-6. Help your child learn the dot patterns for 1, 2, 3, 4, 5, and 6 so they are not counting the dots each time.
- identify numbers to 100.
- count in short sequences to 100.
- know that each space needs its own count.
Ratuki
In this card game, build piles from 1 to 5. The numbers from one to five are represented using numbers, number words, tallies, dice patterns, and finger patterns.
Dominoes
There are many different games you can play with Dominoes. All the games are based on dot patterns and counting.
Hi Ho Cherry-O
Players read numbers and race to have ten cherries in their baskets.
Read Aloud Books
- One monkey too many by Jackie French Koller and Lynn Munsinger
- How many feet? How many tails? A book of math riddles by Marilyn Burns
- Ten black dots by Donald Crews