2.1.2A Addition & Subtraction
Overview
Standard 2.1.2 Essential Understandings
Second graders use their understanding of addition and subtraction to develop quick recall of basic facts with sums to 18 and the related subtraction facts. Quick recall of basic facts include the use of these mental strategies:
- counting on and counting back
- composing and decomposing numbers (12 is the same as 8 and 4 or 7 and 5, etc)
- making a ten ( 9 + 5 = 10 + 4)
- fact families ( 7 + 6 = 13, 6 + 7 =13, 13 - 7 = 6, 13 - 6 = 7)
- doubles and doubles plus/minus one
- commutative property ( 6 + 9 = 9 + 6 )
- associative property ( 5 + 8 = (3 + 2) + 8 = 3 + (2 + 8) = 3 + 10 = 13 )
Second graders develop, understand, explain and use accurate and generalizable methods to add and subtract two-digit whole numbers. Estimating sums and differences is also included in the work with addition and subtraction. They use mental strategies and algorithms based on place value and equality to solve problems. Strategies include:
- composing and decomposing numbers
- using expanded notation
- partial sums and differences ( 47 - 23 = (40 - 20) + (7 - 3) = 20 + 4 = 24)
- using the associative property of addition.
All Standard Benchmarks
2.1.2.1
Use strategies to generate addition and subtraction facts including making tens, fact families, doubles plus or minus one, counting on, counting back, and the commutative and associative properties. Use the relationship between addition and subtraction to generate basic facts.
2.1.2.2
Demonstrate fluency with basic addition facts and related subtraction facts.
2.1.2.3 Estimate sums and differences up to 100.
2.1.2.4
Use mental strategies and algorithms based on knowledge of place value to add and subtract two-digit numbers. Strategies may include decomposition, expanded notation, and partial sums and differences.
2.1.2.5
Solve real-world and mathematical addition and subtraction problems involving whole numbers with up to 2 digits.
2.1.2.6
Use addition and subtraction to create and obtain information from tables, bar graphs and tally charts.
Benchmark Group A
2.1.2.1 Use strategies to generate addition and subtraction facts including making tens, fact families, doubles plus or minus one, counting on, counting back, and the commutative and associative properties. Use the relationship between addition and subtraction to generate basic facts. 2.1.2.2 Demonstrate fluency with basic addition facts and related subtraction facts.
What students should know and be able to do [at a mastery level] related to these benchmarks:
- understand and use fact families
- quickly recall basic addition and subtraction facts up to and including 18
- understand, explain and use accurate and generalizable methods to add and subtract two two-digit whole numbers when solving problems
- select and apply appropriate methods to estimate sums and differences
Work from previous grades that supports this new learning include
- understand the concepts of addition and subtraction
- solve addition and subtraction problems using words, pictures, objects, length based models (connecting cubes), numerals and number lines
- add and subtract in part-part-total, adding to, taking away from, and comparing situations
- use of a variety of manipulatives/models to solve addition and subtraction problems in part-part-total, adding to, taking away from, and comparing situations
- compose and decompose numbers up to and including 12
- recognize the relationship between addition and subtraction.
NCTM Standards
Understand meanings of operations and how they relate to one another
Pre-K - 2 Expectations
- understand various meanings of addition and subtraction of whole numbers and the relationship between the two operations;
- understand the effects of adding and subtracting whole numbers;
- understand situations that entail multiplication and division, such as equal groupings of objects and sharing equally.
Compute fluently and make reasonable estimates.
Pre-K-2 Expectations:
- develop and use strategies for whole-number computations, with a focus on addition and subtraction;
- develop fluency with basic number combinations for addition and subtraction;
- use a variety of methods and tools to compute, including objects, mental computation, estimation, paper and pencil, and calculators.
Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them.
Pre-K-2 Expectations
- pose questions and gather data about themselves and their surroundings;
- sort and classify objects according to their attributes and organize data about the objects;
- represent data using concrete objects, pictures, and graphs.
- describe parts of the data and the set of data as a whole to determine what the data show.
Select and use appropriate statistical methods to analyze data.
Pre-K - 2 Expectations
- describe parts of the data and the set of data as a whole to determine what the data show.
Common Core State Standards
Represent and solve problems involving addition and subtraction.
2.OA.1. Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1
Add and subtract within 20.
2.OA.2. Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers.
Work with equal groups of objects to gain foundations for multiplication.
2.OA.3. Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.
2.OA.4. Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.
Use place value understanding and properties of operations to add and subtract.
2.NBT.5. Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
2.NBT.6. Add up to four two-digit numbers using strategies based on place value and properties of operations.
2.NBT.7. Add and subtract within 1000, 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. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
2.NBT.8. Mentally add 10 or 100 to a given number 100-900, and mentally subtract 10 or 100 from a given number 100-900.
2.NBT.9. Explain why addition and subtraction strategies work, using place value and the properties of operations.
Represent and interpret data.
2.MD.9. Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units.
1.MD.10. Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems1 using information presented in a bar graph.
Misconceptions
Student Misconceptions and Common Errors
Students may think...
- knowing basic facts is about memorizing rather than being able to derive an unknown fact from a known fact or using other thinking strategies.
- every fact is an isolated piece of knowledge. They may not recognize and understand the commutative property therefore seeing 5 + 7 and 7 + 5 as two separate facts to be learned.
- they automatically add when solving problems that are not embedded in a context.
- the number sentences in a fact family are isolated facts. Students may not recognize the relationships between addition and subtraction.
- estimation means the "correct" answer.
- the only correct format for a problem is a + b = c or a - b = c, not recognizing that it can also be c = a + b or c = a - b.
- a number can be decomposed in only one way. For example, they see 47 as 40 + 7, but not 30 + 10 + 7.
- the equation = 90 - 13 is read: 13 minus 90 equals and will put - 67 (negative 67) as the solution.
Vignette
In the Classroom
Building Doubles Using Ten Frames
Student Materials: two blank ten-frames, twenty two-color counters or ten each of two different colors
Teacher Materials: 2 each of the following:
If children are unfamiliar with ten-frames, start with:
Ms. G.: What do you notice about our new tools?
Alaina: There are two rows.
Ms. G.: What do you notice about the rows?
Alaina: There are five squares in each row.
Tony: That makes 10.
Ms. G.: These tools are called "Ten-Frames."
Aiyana: Because it looks like picture frames and there are 10 of them!
Ms. G.: Nice connection. We are going to use these tools today to help us learn about doubles facts. Any idea on how we could do that?
Carlos: We could put five on the top row and five on the bottom row because 5 and 5 makes ten.
Ms. G.: Any other ideas?
Rotho: Well, we have two ten-frames, so we could put some counters on one and the same number of counters on the other.
Ms. G.: Tell me more . . .
Rotho: Well, to do doubles you have to have two of the same number.
Ms. G.: Show us an example on your ten frames - then we'll copy you. There is a convention we are going to follow. What does "convention" mean?
Mason: Something that mathematicians have agreed to do.
Ms. G.: In this case, the convention is that we start at the left side and fill up the top row, then start on the left in the second row. Aiyana, please show us your example on your ten frames, and remember to start with the top row. (Aiyana put 7 on one ten-frame and 7 on the other.
Folks, please set up the same problem on your ten frames.
Imraan, please come up and write the equation for this doubles fact.
(He wrote: 7 + 7 = 14.)
Ms. G.: Imraan, how did you know Aiyana put out fourteen chips on her ten-frames?
Imraan: Because there are 5 on the top row of both ten-frames, so that's 10, and then 2 on the bottom rows, that's 4 more, so it's 14.
Ms. G.: Thanks, Imraan. That was a clear explanation. Did anyone see it a different way?
Bryan: On my ten-frames I took 3 from one ten-frame to fill up the other ten-frame. Then I had a full ten-frame and one with 4 left on it. So I knew it was 14.
Ms. G.: Thanks for sharing. Erin, can you explain Bryan's strategy to us again?
Erin: He filled up one whole ten-frame and then looked at the leftovers because 10 is a friendly number for adding.
Ms. G.: Nice explaining, Erin. How are these two strategies alike?
Alicia: Well, both of them made 10 and then added 4 more.
Ms. G.: Tell me more . . .
Alicia: Imraan looked at the 5s in the top rows and made a 10. Bryan moved chips to one ten-frame so he had two rows of 5 on one ten frame. Then 4 more on the other frame.
Ms. G.: Thank you, Alicia. How were the two strategies different?
Juan: Well, Bryan moved his chips. Imraan kind of just put them together in his head.
Ms. G.: Thanks, Juan. Did anyone look at it a different way? (no response) Ok, would someone please read what Imraan has written on the board?
Zamariay: Seven plus seven equals fourteen.
Ms. G.: How else can we read that number sentence?
Gordon: Seven plus seven is the same as fourteen.
Ms. G.: Another way?
Amaiya: Seven and seven more makes fourteen.
Ms. G.: I'm proud of your flexible thinking!
The class worked through two more examples together and students shared their strategies.
For the double, 4 + 4 = , one child moved four from one ten frame to the other and noted the top row was filled making five and then went three more to make eight. 4 + 4 = 8
For the double, 9 + 9 = , another child moved one chip to the top ten frame to make a ten and then noticed that there were two empty spaces on the other ten frame. If both ten frames were filled it would be twenty but there are two empty spaces so that means eighteen. Twenty go back two - nineteen, eighteen, 9 + 9 = 18.
Similar activities can be done with doubles +/- 1, doubles +/- 2
adding 8 or 9 to another number
and using the" make a ten" strategy.
Resources
- Students may need support in further development of previously studied concepts and skills.
- Basic fact knowledge is not about memorization. It is about being able to derive an unknown fact from a known fact or use other thinking strategies. "All of the facts are conceptually related. You can figure out new or unknown facts from those you already know." (Van de Walle & Lovin, 2006, p.94)
- Timed tests are not an effective strategy for fact acquisition. ". . . asking students to demonstrate this knowledge [basic facts] within an arbitrary time limit may actually interfere with their learning." (Seeley, 2009, pg.93)
- Students need to use a variety of strategies to develop fluency with basic facts. These strategies include:
doubles (4 + 4; 7 + 7; etc.)
doubles plus 1 (7 + 8 think 7 + 7 + 1)
doubles plus 2 (5 + 7 think 5 + 5 + 2 OR double the number in between: 6 + 6, because you're taking one off the 7 - leaving a 6, and adding that one to the five, creating a 6, resulting in 6 + 6)
counting on - when adding 1, 2, or 3 (5 + 3, start with the larger number 5, and count on three hops: 6, 7, 8
counting back - when subtracting 1, 2, or 3
fact families: 7 + 5 = 12, 5 + 7 = 12, 12 - 7 = 5, 12 - 5 = 7
adding 9 (4 + 9 think 4 + 10 - 1 = 14 - 1 = 13)
adding 8 (4 + 8 think 4 + 10 - 2 = 14 - 2 = 12)
make a ten
relating addition to subtraction and subtraction to addition: (11 - 5 = 6 because 5 + 6 = 11 and 13 - 6 = 7 because 6 + 7 = 13).
- Number Talks are a great way to keep strategies visible in the classroom. During a Number Talk, students are given three to five related problems and asked to think about strategies they would use to find the answer. Second graders might, for example, look at these sets of problems.
The Making a Ten strategy is the focus of this set of problems.
8 + 2 8 + 4 8 + 7 8 + 6 |
The Using a Double strategy is the focus of this set of problems.
5 + 5 5 + 6 7 + 5 5 + 8 |
Implementing Number Talks Helpful Hints
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.
- Keep the commutative property and associative property visible and explicit during instruction. Students do not make the connection unless these properties are identified as they are being used.
- Second graders need constant reminders to read left to right on either side of the equal sign.
- Fact Triangles are tools used to help build mental arithmetic skills. Fact Triangles are effective for helping students memorize facts because of their emphasis on relationships in a fact family.
A fact family is a collection of four related facts linking two inverse operations.
For instance, the following four equations represent the fact family relating 5, 6, and 11 with addition and subtraction.
5 + 6 = 11 11 - 5 = 6 6 + 5 = 11 11 - 6 = 5
To use the Fact Triangles,
ask students to write or say the four equations that represent the fact family for the numbers on the card;
cover one of the numbers and ask students to identify the missing number for the fact family. Once the missing number has been identified, students can write all four equations for the fact family.
Download Fact Triangles for Addition and Subtraction here.
(insert Addition Fact triangles)
- 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
Ten Frame
Thinking about numbers using frames of 10 can be a helpful way to learn basic number facts. The four games that can be played with this applet help to develop counting and addition skills. Be sure to check the box for "add" to practice basic fact addition.
Let's Learn Those Facts: Some Special Sums
Students practice doubles and doubles-plus-one addition facts. They record their current level of mastery of the addition facts on their personal addition charts.
Additional Instructional Activities
Make a Ten
Using a deck of cards (jacks, queens, and kings removed), hold one card for all to see and ask, "How many to make ten?" If you are holding a 3, the answer would be 7. This activity develops the complements of ten, those two numbers that add up to ten. Once students are proficient with Make a Ten the game can be changed to Make a Twenty. In Make a Twenty students rely on the compliments of ten. If you are holding a 3 students might think 7 more to make ten and ten more to make twenty which means 17 is needed to make twenty.
Using Who Has? Decks to Practice Basic Facts
Once students have developed conceptual understanding of the basic operations, they need to develop fluency with the facts. One way to include daily practice and motivate students to master basic facts is through the use of the Who Has? card decks. Scroll down to find the Addition Deck, Doubles Deck, and more at the following link: http://mathwire.com/whohas/whohas.html
I have 0.
Who has 6 + 6 ? | I have 17.
Who has 4 + 5 ? |
I have 12.
Who has 8 + 9 ? | I have 9.
Who has 8 + 8? |
Additional Instructional Resources
Conklin, M. (2010). It makes sense! Using ten-frames to build number sense, grades k-2, Sausalito, CA: Math Solutions Press.
Karp, K., Caldwell, J., Zbiek, R. M., & Bay-Williams, J. (2011). Developing essential understanding of addition and subtraction for teaching mathematics in pre-k-grade 2. Reston, VA: National Council of Teachers of Mathematics.
O'Connell, S., & SanGiovanni, J. (2011). Mastering the basic math facts in addition and subtraction, strategies, activities & interventions to move students beyond memorization. Portsmouth, NH: Heinemann.
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.
Sutton, Kim. (2011). 10 block schedule for math fact fluency: 1st and 2nd grade. 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: Pearson Education.
addend: number being added to another number. In 6 + 3 = 9, 6 and 3 are addends.
associative property of addition: when adding three or more addends, the way the numbers are grouped will not change the sum (7 + 8 + 3 has the same sum as 7 + 3 + 8 which is "easier" to add, because 7 + 3 = 10 and then adding 8 to 10)
basic facts: addition and subtraction facts that involve single digit numbers For example, 4 + 5 = 9 and 15 - 7 = 8.
commutative property of addition: when adding two numbers, changing the order of the numbers does not change the sum (4 + 9 = 9 + 4)
counting down: subtraction strategy that can be used to subtract 1, 2 or 3 from another number. In finding 10 - 2, one would count down two from 10, saying 9, 8.
counting on: addition strategy that can be used to add 1, 2, or 3 to another number. In thinking about 7 + 2, one would count up two from 7, saying 8, 9.
decompose: break numbers apart into useful chunks (8 + 5, break 8 into 5 + 3 because 5 + 5 is an "easy" fact that makes 10, and then add 3 to 10 to get 13)
difference: the answer resulting from subtraction In the subtraction fact 16 - 9 = 7, the difference is 7.
doubles: adding the same number to itself (7 + 7; 8 + 8; 3 + 3)
doubles plus 1: adding consecutive counting numbers (5 + 6; 6 + 7)
doubles plus 2: adding counting numbers that are separated by one number on a number line (5 + 7; 6 + 8; 2 + 4
fact family: series of four related facts involving three numbers, two addition facts, two subtraction facts (5 + 8 = 13; 8 + 5 = 13; 13 - 5 = 8; 13 - 8 = 5). Fact families involving doubles have one addition fact and one subtraction fact (6 + 6 = 12; 12 - 6 = 6).
recompose: to combine parts of a number to make a whole number usually done after decomposing a number. For example, when solving 9 + 6 it helps to decompose the addends to 9 + 1 + 5 and then recompose as 10 + 5 = 15.
sum: answer resulting from addition. The sum of 4 + 2 is 6.
"Vocabulary literally is the
key tool for thinking."
Ruby Payne
Mathematics vocabulary words describe mathematical relationships and concepts and cannot be understood by simply practicing definitions. Students need to have experiences communicating ideas using these words to explain, support, and justify their thinking.
Learning vocabulary in the mathematics classroom is contingent upon the following:
Integration: Connecting new vocabulary to prior knowledge and previously learned vocabulary. The brain seeks connections and ways to make meaning which occurs when accessing prior knowledge.
Repetition: Using the word or concept many times during the learning process and connecting the word or concept with its meaning. The role of the teacher is to provide experiences that will guarantee connections are made between mathematical concepts, relationships, and corresponding vocabulary words.
Meaningful Multiple and varied opportunities to use the words in context. These
Use: opportunities occur when students explain their thinking, ask clarifying questions, write about mathematics, and think aloud when solving problems. Teachers should be constantly probing student thinking in order to determine if students are connecting mathematics concepts and relationships with appropriate mathematics vocabulary.
Strategies for vocabulary development
Students do not learn vocabulary words by memorizing and practicing definitions. The following strategies keep vocabulary visible and accessible during instruction.
Mathematics Word Bank: Each unit of study should have word banks visible during instruction. Words and corresponding definitions are added to the word bank as the need arises. Students refer to word banks when communicating mathematical ideas which leads to greater understanding and application of words in context.
Labeled pictures and charts: Diagrams that are labeled provide opportunities for students to anchor their thinking as they develop conceptual understanding and increase opportunities for student learning.
Frayer Model: The Frayer Model connects words, definitions, examples and non-examples.
Example/Non-example Charts: This graphic organizer allows students to reason about mathematical relationships as they develop conceptual understanding of mathematics vocabulary words. Teachers should use these during the instructional process to engage student in thinking about the meaning of words.
Vocabulary Strips: Vocabulary strips give students a way to organize critical information about mathematics vocabulary words.
word | definition | illustration |
Encouraging students to verbalize thinking by drawing, talking, and writing increases opportunities to use the mathematics vocabulary words in context.
Additional Resources for Vocabulary Development
Murray, M. (2004). Teaching mathematics vocabulary in context. Portsmouth, NH: Heinemann.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Reflection - Critical Questions regarding the teaching and learning of these benchmarks:
What are the key ideas related to basic fact understanding and fluency 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 basic addition facts and the related subtraction facts?
What strategies should students have in their "math toolbox" when developing basic fact fluency?
When checking for student understanding of basic facts, what should teachers
- look for in student work?
- listen for in student conversations?
- ask during classroom discussions?
Examine student work related to a task involving basic addition and related subtraction facts. 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.
What are the key ideas related to basic fact understanding and fluency at the second grade level? How do student misconceptions interfere with mastery of these ideas?
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
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Caldwell, J., Karp, K., Bay-Williams, J., Rathmell, E., & Zbiek, R. (2011). Developing essential understanding of addition and subtraction for teaching mathematics in prekindergarten-grade 2. Reston, VA: National Council of Teachers of Mathematics
Carpenter, T. P., Fennema, E., Franke, M., Loef, L. L., & Empson, S. B. (1999). Children's mathematics cognitively guided instruction. Portsmouth, NH: Heinemann.
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.
Fosnot, C., Dolk, M. (2001). Young mathematicians at work: Constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann.
Fuson, K., Clements, D., & Beckmann, S. (2009). Focus in grade 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.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for elementary school teachers. 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.
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.
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Burns, M. (Ed.). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach, grades k-8. (2nd ed.) Sausalito, CA: Math Solutions Press.
Chapin, S., O'Connor, C., & Canavan Anderson, N. (2009). Classroom discussions: Using math talk to help students learn (Grades K-6). Sausalito, CA: Math Solutions.
Conklin, M. (2010). It makes sense! Using ten-frames to build number sense, grades k-2. Sausalito, CA: Math Solutions Press.
Dacey, L., & Salemi, R. (2007). Math for all: Differentiating instruction k-2. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, D.C.: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). (2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Fosnot, C., & Dolk, M. (2001).Young mathematicians at work constructing number sense, addition, and subtraction. Portsmouth, NH: Heinemann.
Fuson, K., Clements, D., & Beckmann, S.. (2009). Focus in grade 2 teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics.
Greenes, C., Larson, M., Leiva, M. A., Shaw, J. M., Stiff, L., ...& Yeatts, K. (2005). Houghton Mifflin Math: Grade 2. Boston, MA: Houghton Mifflin Company.
Hyde, Arthur. (2006). Comprehending math adapting reading strategies to teach mathematics, K-6. Portsmouth, NH: Heinemann.
Kamii, C., & Dominick, A. (1998). In Morrow, L. J., & Kenney, M. J. (Eds.). The teaching and learning of algorithms in School Mathematics (pp. 130 - 140). Reston, VA: National Council of Teachers of Mathematics.
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, Miki. (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.
O'Connell, S., & SanGiovanni, J. (2011). Mastering the basic math facts in addition and subtraction, strategies, activities & interventions to move students beyond memorization. Portsmouth, NH: Heinemann.
Parrish, S. (2010). Number talks: Helping children build mental math and computation strategies grades K-5. Sausalito. CA: Math Solutions.
Sammons, L. (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
Schielack, J., Charles, R., Clements, D., Duckett, P., Fennell, F., Lewandowski, ...& 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. (September, 2010). Beyond One Right Answer. Educational Leadership, 60 (1), p. 29 - 32.
Small, M. (2009). Good questions: Great ways to differentiate mathematics instruction. New York, NY: Teachers College Press.
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. (7th ed.). Boston, MA: Allyn & Bacon.
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades k-3. Boston, MA: Pearson Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
Assessment
- Write the fact family for 5, 7, and 12.
Solution: 5 + 7 = 12, 7 + 5 = 12, 12 - 7 = 5, 12 - 5 = 7
Benchmark: 2.1.2.1
- Explain 3 different ways to find the answer to 9 + 7.
Solution: Add 7 + 10 then subtract 1 = 16.
Double the number in between: 8 + 8 = 16.
Take one from the 7 and give it to the 9 making 10 + 6 = 16.
Decompose the 9 into 7 + 2 then double 7 and add the remaining 2:
7 + (7 + 2) = (7 + 7) + 2 = 16.
Count up 7 from 9
Benchmark: 2.1.2.1
- Imraan had 18 marbles. He lost some and now he has 9. How many marbles did Imraan lose?
Solution: Imraan lost 9 marbles
Benchmark: 2.1.2.1 and 2.1.2.2
Differentiation
Using a variety of manipulatives, pictorial representations, verbal descriptions of strategies will help them construct basic fact knowledge.
Each strategy must be explicitly labeled during instruction.
Practicing one strategy at a time is better than working on multiple strategies simultaneously. Repeated exposure to thinking strategies is necessary in the learning cycle.
Fact Triangles are tools used to help build mental arithmetic skills. They are effective for helping students memorize facts because of their emphasis on relationships in a fact family.
A fact family is a collection of four related facts linking two inverse operations. For instance, the following four equations represent the fact family relating 5, 6, and 11 with addition and subtraction.
5 + 6 = 11 11 - 5 = 6 6 + 5 = 11 11 - 6 = 5
To use the Fact Triangles:
- ask students to write or say the four equations that represent the fact family for the numbers on the card.
- cover one of the numbers and ask students to identify the missing number for the fact family. Once the missing number has been identified, students can write all four equations for the fact family.
Download Fact Triangles for Addition and Subtraction.
(insert Addition Fact triangles)
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 2-5. 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 k-3. Boston, MA: Pearson Education.
Create time for students to listen to other students explain their strategies and mathematical thinking. Allow students to share their mathematical thinking one-on-one to other students or to the teacher, rather than always in front of the large group.
Helping students access the words needed to describe their thinking is necessary as students work to communicate in the mathematics classroom. Labeling strategies is a critical part of this process.
- 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:
To find the sum of 5 + 6 ,I can use the double _________________. |
__________ and ___________ make ___________. |
- 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.
Students who have immediate recall of basic facts may not understand the relationship between and among basic facts. For example, they know 6 + 8 = 14 but may not realize the double 6 + 6 = 12 could be useful in finding the answer 6 + 8. Discovering the patterns and relationships that exist within and among the basic facts will further mathematical development.
Students can apply their knowledge of basic facts in solving related problems. For example, if 6 + 8 = 14, then 60 + 80 = ?, 600 + 800 = ?, 6,000 + 8,000 = ?. These equations are not being solved in isolation and only have meaning when connected to the related basic fact.
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 2-5. 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: |
using ten frames, number lines and counters to represent basic facts. | providing appropriate manipulatives (number grids, number lines, ten frames, counters) for exploring basic facts. |
using a variety of strategies to derive basic facts, including counting on, doubles, near doubles, and make-ten. | labeling strategies and facilitating classroom discussions about strategies and relationships between basic facts. |
recognizing and describing number relationships; for example, knowing 4 + 5 = 9, helps me know 9 - 4 = 5. | evaluating student strategies/reasoning for mathematical accuracy. There are often many strategies that could be used to derive an unknown fact. |
naming strategies that could be used to derive a given fact. For example, what fact could you use to help answer 8 + 7 = ? | providing opportunities for students to explain possible strategies that might be used |
engaging in games and activities to practice addition and subtraction strategies. | keeping strategies explicit in the classroom by listing new strategies as students develop them & giving strategies a name. |
explaining their thinking strategies. | Asking - Why? How do you know? How did you figure that out? Will that always work? |
What should I look for in the mathematics classroom? (Adapted from SciMathMN,1997)
What are students doing?
- Working in groups to make conjectures and solve problems.
- Solving real-world problems, not just practicing a collection of isolated skills.
- Representing mathematical ideas using concrete materials, pictures and symbols. Students know how and when to use tools such as blocks, scales, calculators, and computers.
- Communicating mathematical ideas to one another through examples, demonstrations, models, drawing, and logical arguments.
- Recognizing and connecting mathematical ideas.
- Justifying their thinking and explaining different ways to solve a problem.
What are teachers doing?
- Making student thinking the cornerstone of the learning process. This involves helping students organize, record, represent, and communicate their thinking.
- Challenging students to think deeply about problems and encouraging a variety of approaches to a solution.
- Connecting new mathematical concepts to previously learned ideas.
- Providing a safe classroom environment where ideas are freely shared, discussed and analyzed.
- Selecting appropriate activities and materials to support the learning of every student.
- Working with other teachers to make connections between disciplines to show how math is related to other subjects.
- Using assessments to uncover student thinking in order to guide instruction and assess understanding.
Additional Resources
For Mathematics Coaches
Chapin, S. and Johnson, A. (2006). Math matters: Understanding the math you teach: Grades k-8, 2nd edition. Sausalito, CA: Math Solutions.
Donovan, S., & Bradford, J. (Eds). (2005). How students learn: Mathematics in the classroom. Washington, DC: National Academies Press.
Felux, C., & Snowdy, P. (Eds.). ( 2006). The math coach field guide: Charting your course. Sausalito, CA: Math Solutions.
Sammons, L., (2011). Building mathematical comprehension: Using literacy strategies to make meaning. Huntington Beach, CA: Shell Education.
West, L., & Staub, F. (2003). Content focused coaching: Transforming mathematics lessons. Portsmouth, NH: Heinemann.
For Administrators
Burns, M. (Ed). (1998). Leading the way: Principals and superintendents look at math instruction. Sausalito, CA: Math Solutions.
Kilpatrick, J., & Swafford, J. (Eds). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academies Press.
Leinwand, S. (2000). Sensible mathematics: A guide for school leaders. Portsmouth, NH: Heinemann.
Lester, F. (2010). Teaching and learning mathematics: Transforming research for school administrators. Reston, VA: National Council of Teachers of Mathematics.
Seeley, C. (2009). Faster isn't smarter: Messages about math teaching and learning in the 21st century. Sausalito, CA: Math Solutions.
Parent Resources
Mathematics handbooks to be used as home references:
Cavanagh, M. (2004). Math to Know: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc.
Cavanagh, M. (2006). Math to learn: A mathematics handbook. Wilmington, MA: Great Source Education Group, Inc
Helping your child learn mathematics
Provides activities for children in preschool through grade 5
What should I look for in the mathematics program in my child's school? A Guide for Parents developed by SciMathMN
Help Your Children Make Sense of Math
Ask the right questions
In helping children learn, one goal is to assist children in becoming critical and independent thinkers. You can help by asking questions that guide, without telling them what to do.
Good questions, and good listening, will help children make sense of the mathematics, build self-confidence and encourage mathematical thinking and communication. A good question opens up a problem and supports different ways of thinking about it. The best questions are those that cannot be answered with a "yes" or a "no."
Getting Started
What do you need to find out?
What do you know now? How can you get the information? Where can you begin?
What terms do you understand/not understand?
What similar problems have you solved that would help?
While Working
How can you organize the information?
Can you make a drawing (model) to explain your thinking? What are other possibilities?
What would happen if . . . ?
Can you describe an approach (strategy) you can use to solve this?
What do you need to do next?
Do you see any patterns or relationships that will help you solve this?
How does this relate to ...?
Can you make a prediction?
Why did you...?
What assumptions are you making?
Reflecting about the Solution
How do you know your solution (conclusion) is reasonable? How did you arrive at your answer?
How can you convince me your answer makes sense?
What did you try that did not work?
Has the question been answered?
Can the explanation be made clearer?
Responding (helps clarify and extend their thinking)
Tell me more.
Can you explain it in a different way?
Is there another possibility or strategy that would work?
Is there a more efficient strategy?
Help me understand this part...
Adapted from They're counting on us, California Mathematics Council, 1995
- Fact Triangles are tools used to help build mental arithmetic skills. Fact Triangles are effective for helping children memorize facts because of their emphasis on relationships in a fact family.
A fact family is a collection of four related facts linking two inverse operations.
For instance, the following four equations represent the fact family relating 5, 6, and 11 with addition and subtraction.
5 + 6 = 11 11 - 5 = 6 6 + 5 = 11 11 - 6 = 5
To use the Fact Triangles,
ask your child to write or say the four equations that represent the fact family for the numbers on the card
cover one of the numbers and ask your child to identify the missing number for the fact family. Once the missing number has been identified, your child can write all four equations for the fact family.
Download Fact Triangles for Addition and Subtraction.
(insert Addition Fact triangles)
Read-Aloud Books
Tang, G. (2007). Math tables too. Scholastic Press: New York.
Tang, G. (2002). Math for all seasons. Scholastic Press: New York, 2002.
Related Frameworks
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