5.1.1 Division
Divide multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms. Recognize that quotients can be represented in a variety of ways, including a whole number with a remainder, a fraction or mixed number, or a decimal.
For example: Dividing 153 by 7 can be used to convert the improper fraction $\frac{153}{7}$ to the mixed number $21\frac{6}{7}$.
Consider the context in which a problem is situated to select the most useful form of the quotient for the solution and use the context to interpret the quotient appropriately.
For example: If 77 amusement ride tickets are to be distributed equally among 4 children, each child will receive 19 tickets, and there will be one left over. If $77 is to be distributed equally among 4 children, each will receive $19.25, with nothing left over.
Estimate solutions to arithmetic problems in order to assess the reasonableness of results.
Solve real-world and mathematical problems requiring addition, subtraction, multiplication and division of multi-digit whole numbers. Use various strategies, including the inverse relationships between operations, the use of technology, and the context of the problem to assess the reasonableness of results.
For example: The calculation 117 ÷ 9 = 13 can be checked by multiplying 9 and 13.
Overview
Standard 5.1.1 Essential Understandings
Fifth grade students divide multi-digit numbers and solve real-world and mathematical problems using arithmetic. Students demonstrate their understanding of division by using various models including an area model and rectangular arrays. They recognize quotients can be represented in a variety of ways, including a whole number with a remainder, a fraction or mixed number, or a decimal. They understand that the context of a problem has an impact on both the quotient and the representation of the remainder.
For example:
- If 77 amusement ride tickets are to be distributed equally among 4 children, each child will receive 19 tickets, and there will be one left over.
- If $77 is to be distributed equally among 4 children, each will receive $19.25, with nothing left over.
- If 77 students attend the science fair, transportation will be by car with 4 students per car. In this case a quotient of 19 with remainder of one student will necessitate 20 cars to transport all the students.
Fifth graders develop an understanding of the standard algorithm by relating it to place value, other strategies, properties of operations such as the distributive property, and the relationship of division to multiplication. They use these understandings to develop efficient and generalizable procedures involving dividends with multi-digits.
Fifth graders solve real-world and mathematical problems involving multi-digit numbers using addition, subtraction, multiplication and division and assess the reasonableness of results.
All Standard Benchmarks
5.1.1.1
Divide multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms. Recognize that quotients can be represented in a variety of ways, including a whole number with a remainder, a fraction or mixed number, or a decimal.
5.1.1.2
Consider the context in which a problem is situated to select the most useful form of the quotient for the solution and use the context to interpret the quotient appropriately.
5.1.1.3
Estimate solutions to arithmetic problems in order to assess the reasonableness of results.
5.1.1.4
Solve real-world and mathematical problems requiring addition, subtraction, multiplication and division of multi-digit whole numbers. Use various strategies, including the inverse relationships between operations, the use of technology, and the context of the problem to assess the reasonableness of results.
5.1.1.1
Divide multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms. Recognize that quotients can be represented in a variety of ways, including a whole number with a remainder, a fraction or mixed number, or a decimal.
5.1.1.2
Consider the context in which a problem is situated to select the most useful form of the quotient for the solution and use the context to interpret the quotient appropriately.
5.1.1.3
Estimate solutions to arithmetic problems in order to assess the reasonableness of results.
5.1.1.4
Solve real-world and mathematical problems requiring addition, subtraction, multiplication and division of multi-digit whole numbers. Use various strategies, including the inverse relationships between operations, the use of technology, and the context of the problem to assess the reasonableness of results.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Understand and use the fraction bar as division notation.
- Understand and apply the standard algorithm for division.
- Represent the quotient in a variety of ways including whole number with a remainder, fraction or mixed number, or a decimal.
- Recognize that the context of a problem determines how to interpret quotients with remainders when solving real world problems.
- Estimate quotients and/or products in order to assess the reasonableness of their answer.
- Understand that the fraction bar represents division.
- Use various strategies, including the inverse relationship between operations, in solving problems.
Work from previous grades that supports this new learning includes:
- Subtraction skills
- Multiply two- and three-digit numbers by a two-digit number.
- Understand the inverse relationship between multiplication and division.
- Fluent in multiplication and division facts.
- Divide multi-digit numbers by 1-digit divisors with and without remainders.
- Estimation skills.
- Place value understanding- including expanded form.
- Understanding of division problem types; equal groups, area and array, multiplicative comparison, and combination.
- Understand properties.
NCTM Standards
Understand meanings of operations and how they relate to one another
Grades 3 - 5 Expectations
- understand various meanings of multiplication and division;
- understand the effects of multiplying and dividing whole numbers;
- identify and use relationships between operations, such as division as the inverse of multiplication, to solve problems;
- understand and use properties of operations, such as the distributivity of multiplication over addition.
Compute fluently and make reasonable estimates
Grades 3 - 5 Expectations
- develop fluency with basic number combinations for multiplication and division and use these combinations to mentally compute related problems, such as 30 $\square$ 50;
- develop fluency in adding, subtracting, multiplying, and dividing whole numbers;
- develop and use strategies to estimate the results of whole-number computations and to judge the reasonableness of such results;
- develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students' experience;
- use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals;
- select appropriate methods and tools for computing with whole numbers from among mental computation, estimation, calculators, and paper and pencil according to the context and nature of the computation and use the selected method or tool.
Common Core State Standards
Perform operations with multi-digit whole numbers and with decimals to hundredths.
5.NBT.5. Fluently multiply multi-digit whole numbers using the standard algorithm.
5.NBT.6. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
5.NBT.7. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
Misconceptions
Student Misconceptions and Common Errors
Students may think...
- division makes smaller. For example, when you divide, the answer (quotient) is smaller than the starting amount (dividend). While this is true when dividing a whole number by a smaller whole number, it is not true when the divisor is greater than the dividend.
- division means only "fair shares" or "how many in a group." Students need experiences with division problems involving measurement division or "how many groups."
- the standard algorithm for division is a set of steps to be memorized.
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- While the idea of teaching "key words" might be appealing it does not support student comprehension of word problems and can lead to incorrect solutions. When students listen to a story problem and then try to "act out" what is happening in the problem, it is easier to see what mathematics might be involved.
- Teachers should be aware of statements such as all multiplication situations "make bigger" and division situations "make smaller" so that students don't generalize this as an absolute. This misconception often continues when students begin to work with rational numbers.
- Students who can correctly solve a division problem using the standard algorithm may still not understand the concept of division.
- Teach multiple strategies for solving division problems including the standard algorithm. Only teaching students the standard division algorithm, which is often taught without attention to place value, may lead teachers to conclude that students understand division if students can reproduce the procedure.
- When the standard algorithm is taught as "steps to follow," it works against students' number sense and place value understanding as it treats the dividend as a set of digits rather than as a number or quantity.
- MCA III test specifications indicate dividends may not be more than 4 digits and divisors may not be more than 2 digits. Fraction remainders are not required in lowest terms. Multiplication is limited to no more than three-digit numbers by no more than three-digit numbers.
- Teach all division problem types
- Consider the problem, Peter has 36 cookies. These cookies fill ¾ of a bag. How many cookies does he need to fill a whole bag? (This is an example of a partitive division problem with a fraction divisor). Students have a hard time understanding fraction division if measurement division interpretations are not included in whole number division instruction.
- Equal Groups
- Partitive Division-the total number of objects is partitioned into a specified number of groups.
- Measurement Division-the total number of objects is measured out into groups of a certain size.
- Area and Array-this involves finding one of the dimensions of a rectangular region or of a rectangular array (or arrangement) when one of the dimensions and the total area or the total number of objects in the arrangement is given.
- Multiplicative Comparison- involves comparing two quantities multiplicatively; i.e., expressing one quantity in terms of how many times it is when compared to another; e.g., Eric ate twice as many cookies as Patrick did. So the task is to find the second quantity being compared.
- Combination- also known as Cartesian products, involve different combinations that can be made from sets of objects such as how many outfits can be made from two blouses and three pairs of slacks. In the case of combination division problems, the total number of combinations is known as is the number of one of the elements of the being combined. The task is to find the number of the second element being combined.
- Teach all forms of remainder interpretation. If students are presented with a non-contextual whole number division problem, the remainder can be expressed as "R" + a whole number, a fraction, or a decimal. When students are presented with a contextual division problem they must interpret the context of the problem and handle the remainder in a way that is appropriate to that context. The following examples from Van de Walle (2007) illustrate different remainder scenarios:
- The remainder remains as a quantity left over. For example: you have 30 pieces of candy to share fairly with 7 children. How many pieces of candy will each child receive? Answer: 4 pieces of candy and 2 left over.
- The remainder is partitioned into fractions. For example: each jar holds 8 ounces of liquid. If there are 46 ounces in the pitcher, how many jars will that be? Answer: 5 and 6/8 jars (partitioned as a fraction).
- The remainder is discarded, leaving a smaller whole number answer. For example: the rope is 25 feet long. How many 7-foot jump ropes can be made? Answer: 3 jump ropes (4 ft discarded).
- The remainder can "force" the answer to the next highest whole number. For example: the ferry can hold 8 cars. How many trips will it have to make to carry 25 cars across the river? Answer: 4 trips (forced to the next whole number).
- The answer is rounded to the nearest whole number for an approximate result. For example: six children are planning to share a bag of 50 pieces of bubble gum. About how many pieces will each child get? Answer: About 8 pieces for each child (rounded, approximate result).
- 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)
Annenberg - using area model for division interactive: (pending permission)
Duncan, N., Geer, C., Huinker, D., Leutzinger, L., Rathmell, E., & Thompson, C. (2007). Navigating through number and operations 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.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.) Boston, MA: Allyn & Bacon.
Van de Walle, J. & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.
Wickett, M., & Burns, M. (2003). Teaching arithmetic: Lessons for extending division, grades 4-5. Sausolito, CA: Math Solutions.
divisible: when a number can be divided by another number with no remainder. For example, 21 is divisible by 3 because 21 divided by 3 equals 7 with a remainder of zero.
"Vocabulary literally is the
key tool for thinking."
Ruby Payne
Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions. Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.
Learning vocabulary in the mathematics classroom is contingent upon the following:
Integration: Connecting new vocabulary to prior knowledge and previously learned vocabulary. The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.
Repetition: Using the word or concept many times during the learning process and connecting the word or concept with its meaning. The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.
Meaningful Use: Multiple and varied opportunities to use the words in context. These opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems. Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.
Strategies for vocabulary development
Students do not learn vocabulary words by memorizing and practicing definitions.
The following strategies keep vocabulary visible and accessible during instruction.
Mathematics Word Bank: Each unit of study should have word banks visible during instruction. Words and corresponding definitions are added to the word bank as the need arises. Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.
Labeled pictures and charts: Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.
Frayer Model: The Frayer Model connects words, definitions, examples and non-examples.
Example/Non-example Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words. Teachers should use these during the instructional process to engage student in thinking about the meaning of words.
Vocabulary Strips: Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.
word | definition | illustration |
Encouraging students to verbalize thinking by drawing, talking, and writing, increases opportunities to use the mathematics vocabulary words in context.
Additional Resources for Vocabulary Development
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks.
What are the key ideas related to multi-digit division at the fifth grade level? How do student misconceptions interfere with mastery of these ideas?
How would you know a student understands multi-digit division? Is a correct answer enough?
What experiences do students need in order to develop an understanding of multi-digit division?
What do fifth graders need to know about remainders?
What strategies should a student be able to use when solving a multi-digit division problem?
When checking for student understanding of multi-division, 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 multi-digit division at the fifth 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 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. (2^{nd} ed.). Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.
Fosnot, C., & Dolk, M. (2002). Young mathematicians at work: Multiplication and division. Portsmouth, NH: Heinemann.
Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Otto, A., Caldwell, J., Wallus Hancock, S., & Zbiek, R.(2011). Developing essential understanding of multiplication and division for teaching mathematics in grades 3 - 5. Reston, VA.: National Council of Teachers of Mathematics.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J. (2009). Focus in grade 3, teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions prek-grade 5: From misunderstanding to deep understanding. Portsmouth, NH: Heinemann.
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Bresser, R., Melanese, K., & Sphar, C. (2008). Supporting English language learners in math class, grades k-2. Sausalito, CA: Math Solutions Publications.
Burns, Marilyn. (2007). About teaching mathematics: A k-8 resource (3rd ed.). Sausalito, CA: Math Solutions Publications.
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Caldera, C. (2005). Houghton Mifflin math and English language learners. Boston, MA: Houghton Mifflin Company.
Carpenter, T., Fennema, E., Franke, M., Levi, L., & Empson, S. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Chapin, S., & Johnson, A. (2006). Math matters: Understanding the math you teach, grades K-8. (2nd ed.). Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.
Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Dougherty, B., Flores, A., Louis, E., & Sophian, C. (2010). Developing essential understanding of number & numeration pre-k-grade 2. Reston, VA: National Council of Teachers of Mathematics.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 2 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. Reston, VA: National Council of Teachers of Mathematics.
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.
Reeves, D. (2007). Ahead of the curve: The power of assessment to transform teaching and learning. Indiana: Solution Tree Press.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., ... & Zbiek, R. M. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student-centered mathematics grades K-3. Boston, MA: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Wickett, M., & Burns, M. (2003). Teaching arithmetic: Lessons for extending division, grades 4-5. Sausalito, CA: Math Solutions.
Assessment
- Divide. 2,564÷8
A. 32 r4
B. 308
C. 320 1/5
D. 320.5
Solution: D
Benchmark 5.1.1.1
MCA III Item Sampler
- Jan has 500 pieces of paper. She prints as many copies as possible of a 16 page report. How many pieces of paper are left?
A. 4
B. 9
C. 25
D. 31
Solution: A
Benchmark 5.1.1.2
MCA III Item Sampler
- 37 x ____ = 703.
What is value of 37 x _____ ?
Solution:
Benchmark:
TIMMS 1995 Released Items, Grade 4
Problem Solving: Interpreting Remainders
- Ms. Xiong wants to buy a pencil for each of her 32 students. The pencils come in boxes of 12. How many boxes does Ms. Xiong need to buy?
1. Which of the following could you do to solve the problem?
a. You could find 32 ÷ 12 = 8 R2.
b. You could find 32 ÷ 12= 2 R8
c. You could find 32 - 12 = 20.
2. How should you use the remainder to solve the problem?
a. Use it to round up the quotient, so the answer is 1 more box.
b. Drop the remainder
c. Use the answer and include the remainder
Solution to 1: b.
Solution to 2: a.
Benchmark: 5.1.1.1
Differentiation
- Structure consistent computational fluency activities utilizing physical models such as number lines and manipulatives to help reconstruct facts when needed.
- Actively engage students in learning situations that focus on both concept (meaning of division) and skill development (division). Explicit systematic instruction that includes opportunities for students to ask and answer questions and think aloud when making decisions while solving problems.
- Instructional settings should include direct instruction work, whole group and small group situations that are well structured and have clear expectations and the use of technology.
- Use vocabulary graphic organizers such as the Frayer model to emphasize words such as divisor, dividend, quotient and remainder and division symbols.
- Carefully connect prior knowledge (subtraction and multiplication understanding and skills) to new learning of division.
- Pose meaningful problems set in familiar situations.
- Incorporate visual models such as the bar model to emphasize the relationships between numbers.
Concrete - Representational - Abstract Instructional Approach
(Adapted from The Access Center: Improving Access for All K-8 Students)
The Concrete-Representational-Abstract Instructional Approach (CRA) is a research-based instructional strategy that has proven effective in enhancing the mathematics performance of students who struggle with mathematics.
The CRA approach is based on three stages during the learning process:
Concrete - Representational - Abstract
The Concrete Stage is the doing stage. The concrete stage is the most critical in terms of developing conceptual understanding of mathematical skills and concepts. At this stage, teachers use manipulatives to model mathematical concepts. The physical act of touching and moving manipulatives enables students to experience the mathematical concept at a concrete level. Research shows that students who use concrete materials develop more precise and comprehensive mental representations, understand and apply mathematical concepts, and are more motivated and on-task. Manipulatives must be selected based upon connections to the mathematical concept and the students' developmental level.
The Representational Stage is the drawing stage. Mathematical concepts are represented using pictures or drawings of the manipulatives previously used at the Concrete Stage. Students move to this level after they have successfully used concrete materials to demonstrate conceptual understanding and solve problems. They are moving from a concrete level of understanding toward an abstract level of understanding when drawing or using pictures to represent their thinking. Students continue exploring the mathematical concept at this level while teachers are asking questions to elicit student thinking and understanding.
The Abstract Stage is the symbolic stage. Teachers model mathematical concepts using numbers and mathematical symbols. Operation symbols are used to represent addition, subtraction, multiplication and division. Some students may not make a clean transfer to this level. They will work with some symbols and some pictures as they build abstract understanding. Moving to the abstract level too quickly causes many student errors. Practice at the abstract level will not lead to increased understanding unless students have a foundation based upon concrete and pictorial representations.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5. Sausalito, CA: Math Solutions.
Murray, M., & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3-5. Boston, MA: Pearson Education.
ELL students often struggle with computational skills when embedded in a word problem.
Teachers may have to adjust sentence complexity, problem context, and response complexity.
Teachers need to demonstrate and model the use of manipulatives (place value blocks) or pictorials such as bar and area models, use technology such as division applets, use cooperative groups or partners for word problems, and encourage students to retell word problems and use graphic organizers.
- Word banks need to be part of the student learning environment in every mathematics unit of study. Refer to these throughout instruction.
- Use vocabulary graphic organizers such as the Frayer model (see below) to emphasize vocabulary words count, first, second, third, etc.
- Sentence Frames
Math sentence frames provide support that English Language Learners need in order to fully participate in math discussions. Sentence frames provide appropriate sentence structure models, increase the likelihood of responses using content vocabulary, help students to conceptualize words and build confidence in English Language Learners.
Sample sentence frames related to these benchmarks:
The remainder is _________. This means _____________________________________. |
In this problem the quotient needs to be expressed as a ________ because __________. |
- 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.
Compare and contrast division algorithms from around the world. Determine how each of the algorithms work and select your personal favorite.
Write partitive and measurement division problems.
Additional Resources
Bender, W. (2009). Differentiating math instruction: Strategies that work for k-8 classrooms! Thousand Oaks, CA: Corwin Press.
Dacey, L., & Lynch, J. (2007). Math for all: Differentiating instruction grades 3-5. Sausalito, CA: Math Solutions.
Murray, M. & Jorgensen, J. (2007). The differentiated math classroom: A guide for teachers k-8. Portsmouth, NH: Heinemann.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Parents/Admin
Administrative/Peer Classroom Observation
Students are... | Teachers are... |
dividing multi-digit numbers using efficient and generalizable methods, including the standard algorithm. | introducing both "fair shares" and "measurement" division (including all the different types of problems they encompass) through word problems using "Think Aloud" to model mathematical reasoning and division vocabulary, using place value blocks, or visual models such as the area modal and bar model, computational thinking and justification. |
recognizing remainders can be represented in a variety of ways, including a whole number with a remainder, a fraction, a mixed number or a decimal. | providing contexts for division that require a variety of representations of remainders. |
sharing solutions, asking questions, and sharing insights/understandings of meaning of division. | asking students to compare and contrast student solution strategies and evaluate them for understanding and efficiency. |
interpreting remainders within the context of the problem. | keeping student focus on the interpretation of the remainder in finding the correct solution. |
solving real-world and mathematical problems in ddition, subtraction, multiplication and division using a variety of strategies. | highlighting inverse relationships between operations. |
What should I look for in the mathematics classroom?
(Adapted from SciMathMN,1997)
What are students doing?
- Working in groups to make conjectures and solve problems.
- Solving real-world problems, not just practicing a collection of isolated skills.
- Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
- Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
- Recognizing and connecting mathematical ideas.
- Justifying their thinking and explaining different ways to solve a problem.
What are teachers doing?
- Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
- Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
- Connecting new mathematical concepts to previously learned ideas.
- Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
- Selecting appropriate activities and materials to support the learning of every student.
- Working with other teachers to make connections between disciplines to show how math is related to other subjects.
- Using assessments to uncover student thinking in order to guide instruction and assess understanding.
What should I look for in the mathematics classroom?
(Adapted from SciMathMN,1997)
What are students doing?
- Working in groups to make conjectures and solve problems.
- Solving real-world problems, not just practicing a collection of isolated skills.
- Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
- Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
- Recognizing and connecting mathematical ideas.
- Justifying their thinking and explaining different ways to solve a problem.
What are teachers doing?
- Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
- Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
- Connecting new mathematical concepts to previously learned ideas.
- Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
- Selecting appropriate activities and materials to support the learning of every student.
- Working with other teachers to make connections between disciplines to show how math is related to other subjects.
- Using assessments to uncover student thinking in order to guide instruction and assess understanding.
Additional Resources
For Mathematics Coaches
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach: Grades k-8. (2nd ed.). Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
For Administrators
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Parent Resources
Mathematics handbooks to be used as home references:
Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Helping your child learn mathematics
Provides activities for children in preschool through grade 5
What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN
Help Your Children Make Sense of Math
Ask the right questions
In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.
Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if . . . ?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to...?
Can you make a prediction?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work?
Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part...
Adapted from They're counting on us, California Mathematics Council, 1995.