2.1.1B Rounding Numbers
Round numbers up to the nearest 10 and 100 and round numbers down to the nearest 10 and 100.
For example: If there are 17 students in the class and granola bars come 10 to a box, you need to buy 20 bars (2 boxes) in order to have enough bars for everyone.
Overview
Standard 2.1.1 Essential Understandings
Second graders read, write and represent whole numbers up to 1000. Representations may include numerals, addition, subtraction, multiplication, words, pictures, tally marks, number lines and manipulatives. They build on their previous understanding of tens and ones to include an understanding that ten tens can be grouped to make a hundred and ten hundreds can be grouped to make a thousand. They describe numbers up to 1000 in terms of hundreds, tens and ones, and can find the number that is ten more/less and a hundred more/less than a given three digit number. Comparing and ordering of whole numbers is now expanded to include numbers to 1000. Second graders develop and use an understanding of rounding by rounding up or down to the nearest 10 or 100.
All Standard Benchmarks - with codes
2.1.1.1
Read, write and represent whole numbers up to 1000. Representations may include numerals, addition, subtraction, multiplication, words, pictures, tally marks, number lines and manipulatives, such as bundles of sticks and base 10 blocks.
2.1.1.2
Use place value to describe whole numbers between 10 and 1000 in terms of groups of hundreds, tens and ones. Know that 100 is ten groups of 10, and 1000 is ten groups of 100.
2.1.1.3 Find 10 more or 10 less than any given three-digit number. Find 100 more or 100 less than any given three-digit number.
2.1.1.4
Round numbers up to the nearest 10 and 100 and round numbers down to the nearest 10 and 100.
2.1.1.5
Compare and order whole numbers up to 1000.
2.1.1.4
Round numbers up to the nearest 10 and 100 and round numbers down to the nearest 10 and 100.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- Round any two- or three-digit number up or down to the nearest 10.
- Round any three-digit number up or down to the nearest 100.
Work from previous grades that supports this new learning includes:
- Use place value to describe whole numbers between 10 and 100 in terms of groups of tens and ones (56 is 5 tens and 6 ones).
- Read, write and represent whole numbers up to 120.
- Count forward and backward from any given number up to 120.
- Compare and order whole numbers up to 120.
- Describe the relative size of numbers; e.g., equal to, not equal to, more than, less than, fewer than.
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 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.
- Understand and represent commonly used fractions, such as ¼, ⅓, and ½.
Common Core State Standards
Understand place value:
2.OA.1. Understand that the three digits of a three-digit number represent amounts of hundreds, tens, and ones; e.g., 706 equals 7 hundreds, 0 tens, and 6 ones. Understand the following as special cases:
2.OA.1b. The numbers 100, 200, 300, 400, 500, 600, 700, 800, 900 refer to one, two, three, four, five, six, seven, eight, or nine hundreds (and 0 tens and 0 ones).
2.OA.2. Count within 1000; skip-count by 5s, 10s, and 100s.
2.OA.1a. 100 can be thought of as a bundle of ten tens - called a "hundred."
2.OA.3. Read and write numbers to 1000 using base-ten numerals, number names, and expanded form.
2.OA.4. Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
Misconceptions
Student Misconceptions and Common Errors
Students may think...
- they can round to some other multiple of 10 or 100 when they don't understand the concept of "nearest" multiple of 10 or "nearest" multiple of 100.
- rounding is guessing.
- rounding a two-digit number to the nearest ten means replacing the digit in the one's place with a zero. For example, 57 rounded to the nearest ten is 50.
- to "round down" means to lower the target digit in the original number. They would mistakenly think that the tens in 53 need to be rounded "down" to 40 because the 3 in the one's place means to "round down," as opposed to leaving it at 50. This misconception can be avoided by having the children list the two multiples (of ten in this case) between which the given number falls. 53 is between 50 and 60, written like this: 50 53 60. Is the number 53 closer to 50 or 60 (can use a number line or rounding number grid to determine)? It's closer to 50, so we round down to 50 from 53.
Vignette
In the Classroom
Marilyn Burns shares a thoughtful approach to teaching the concept of rounding. She begins by talking with students about "friendly" numbers. In her words, "They're numbers that are easy to think about." She then asks a variety of questions.
What are some examples of numbers that you think are friendly?
Typically students choose 1, 5, 10, and 100; older students include 25 and 1,000. I ask them to explain the reasons for their choices. I point out that 1, 10, 100, and 1,000 are landmark numbers for our number system, and we discuss why.
Which do you think is friendlier, seventy-three or seventy?
Typically students agree that seventy is friendlier. "You get to seventy by counting by tens," Alan told me, "but you can't get to seventy-three that way." We count by tens--ten, twenty, thirty, forty, and so on--and I identify these friendly numbers as multiples of ten.
What's a friendly number that's close to fifty-seven? One hundred thirteen?
Four hundred eighty-seven?
Multiple answers are possible for questions like these, so I'm sure to have students explain their reasoning. For example, Robert thought that 113 was close to 100, but Lindsay argued that it was closer to 110, which was also a friendly number. I pointed out that if you were thinking about the hundreds part of the number, then 113 was closer to 100 than to 200, but if you were thinking about the tens part, then 113 was closer to 110 than 120.
Then, for individual assignments, I give students "closer to " problems, choosing numbers that are appropriate for students. Then I talk more with the class about rounding numbers to the closest ten or the closest hundred. Following are examples:
Is 163 closer to 160 or 165?
Is 163 closer to 160 or 170?
Is 163 closer to 100 or 200?
Is 163 closer to 150 or 200?
Mental Math Problems
I use regular classroom routines to present problems to the class that encourage them to round numbers and make estimates. We talk about both their solutions and how they reasoned. I phrase some questions so that they have one right answer. For example:
Eighteen children bought milk for lunch. Did we collect more or less than $10?
If everyone in the class took out four books from the library, would we take out
more or less than one hundred books altogether?
I phrase other questions so more than one right answer is possible.
For example: I measured the hallway and it is 94 feet plus some more inches. About how long is it? Some children think that 94 feet is a good estimate, some say 95 feet, and others vote for 100 feet.
Some questions suggest longer investigations. For example:
About how much do you think our class spends for milk in a week?
About how much does our year's supply of pencils cost?
About how many buttons do we have altogether?
About how many letters are there altogether in our first names? Our last names?
Our first and last names together?
Connecting to the Language of Rounding
Once students are comfortable answering questions like the ones in the previous section, I address rounding directly. One way I've done this is to write on the board a number in the hundreds: for example, 138. Then I ask two questions:
What friendly number that ends in zero is closest to 138?
What friendly number that ends in two zeros is closest to 138?
I repeat this with other numbers between 100 and 900.
Then I introduce the typical language of rounding. I tell them that "rounding a number" means "finding a friendly number that is close to your original number." I explain how "ends in a zero" relates to tens while "ends in two zeros" relates to hundreds. To give them practice, I ask the same kinds of questions I did before, this time rephrasing to use the standard language of rounding (Burns, 2007, p.381 - 383).
Resources
Teacher Notes
- Students may need support in further development of previously studied concepts and skills.
- There are ten possible digits for any place in the base ten number system.
When making the decision to round up or down, the first five digits (0,1,2,3,4) round down and the last five digits (5,6,7,8,9) round up.
- The number line provides the basis for conceptual understanding when rounding numbers. When rounding numbers, the scale of the number line should represent the position being rounded to. This will either be 300 or 400, since those are the two consecutive hundreds before and after 368. Draw a number line, subdivide it as much as necessary, and determine whether 368 is closer to 300 or 400. Since 368 is closer to 400, this number should be rounded to 400.
- To round to the nearest 10, have the students locate a given number on a rounding number grid and determine the multiples of 10 between which the given number falls. They can count the hops to the smaller multiple of 10 and the larger multiple of 10 to determine which is closer. Discuss how "5" is in the middle of the two multiples of 10 so mathematicians have decided to round up to the larger multiple of 10 when 5 is in the one's place. With repeated experience using the rounding grid, we want children to recognize that when 1, 2, 3, or 4 appears in the one's place, we round "down" to the lower multiple of 10. When 5, 6, 7, 8, or 9 appears in the one's place we round up to the next multiple of 10. When children discover this and make the generalization for themselves, they will remember it better.
- Round to the Nearest 10 Grid
0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
17 |
18 |
19 |
20 |
20 |
21 |
22 |
23 |
24 |
25 |
26 |
27 |
28 |
29 |
30 |
30 |
31 |
32 |
33 |
34 |
35 |
36 |
37 |
38 |
39 |
40 |
40 |
41 |
42 |
43 |
44 |
45 |
46 |
47 |
48 |
49 |
50 |
50 |
51 |
52 |
53 |
54 |
55 |
56 |
57 |
58 |
59 |
60 |
60 |
61 |
62 |
63 |
64 |
65 |
66 |
67 |
68 |
69 |
70 |
70 |
71 |
72 |
73 |
74 |
75 |
76 |
77 |
78 |
79 |
80 |
80 |
81 |
82 |
83 |
84 |
85 |
86 |
87 |
88 |
89 |
90 |
90 |
91 |
92 |
93 |
94 |
95 |
96 |
97 |
98 |
99 |
100 |
100 |
101 |
102 |
103 |
104 |
105 |
106 |
107 |
108 |
109 |
110 |
- 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)
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Sutton, K. (2003). Place value with pizzazz games and activities for meaning in place value. Arcata, CA: Creative Mathematics.
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.
hundreds place: The third place to the left in a whole number. It tells you how many sets of one hundred are in the number. The number 539 has five hundreds.
ones place: the digit on the right end of a whole number. It tells you how many ones are in a number. The number 458 has 8 ones.
round: a way to estimate where the number is close to the original number and has the same number of digits, but ends in at least one zero when rounding to the nearest 10, and ends in at least two zeros when rounding to the nearest hundred
"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. Theseopportunities 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 an understanding of rounding at the second grade level? How do student misconceptions interfere with mastery of these ideas?
What experiences do students need in order to develop an understanding of rounding to the nearest ten or hundred?
When checking for student understanding of rounding, 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 rounding. 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 second grade level?
Professional Learning Community Resources
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.
Hyde, A. (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.
Bamberger, H., Oberdorf, C., & Schultz-Ferrell, K. (2010). Math misconceptions pre k-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.
Sammons, L., (2011). Building mathematical comprehension-Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, S., ... & Zbiek, R. M. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics: A quest for coherence. Reston, VA: NCTM.
Seeley, C. (2009). Faster isn't smarter-Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J. A., & Lovin, L. H. (2006). Teaching student-centered mathematics grades K-3. Boston, MA: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Assessment
- Fill in the blanks with the multiples of 10 that come before and after each number and then round to the nearest ten.
a. ______ 23 ______
b. ______ 87 ______
c. ______ 65 ______
d. ______ 44 ______
e. ______ 36 ______
Solution: a. 20 23 30
b. 80 87 90
c. 60 65 70
d. 40 44 50
e. 30 36 40
Benchmark: 2.1.1.4
- Round 63 to the nearest ten.
Solution: 60
Benchmark: 2.1.1.4
- Which number is 657 rounded to the nearest 100?
a. 650
b. 600
c. 700
d. 660
Solution: 700
Benchmark: 2.1.1.4
- True or False?
405 rounded to the nearest ten is 400.
Solution: False
Benchmark: 2.1.1.4
Differentiation
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.
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. A., & Lovin, L. H. (2006). Teaching student-centered mathematics grades K-3. Boston: Pearson Education.
Clarify the vocabulary word "round" - to find a friendlier number to use (not round like a circle).
Use the number line to make the concept of rounding to the nearest more visible.
- 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:
I rounded __________ to ___________ 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 k-2. Sausalito, CA: Math Solutions Publications.
Relate rounding to the compensation strategy used to solve addition and subtraction 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 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... |
using number lines and rounding number grids to explain rounding numbers to the nearest 10 and 100. |
facilitating the development of generalizations about strategies for rounding numbers. |
Explaining, justifying rounding up or down to the nearest 10 or 100. |
asking: Why? How do you know? Will that always be the case? |
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