Overview
Session Summary
This session focuses on the operations of addition and subtraction in the Investigations curriculum for Kindergarten through second grade. Participants consider how students work on important aspects of solving story problems: making sense of the action of the problem, solving the problem in ways that make sense, and recording their strategies so that others can see how they solved the problem. They examine the different strategies students use to solve addition and subtraction problems and how those strategies develop over time. They learn about the work students do with number composition, addition combinations and place value, and how these are building blocks for the development of strong, efficient strategies for solving addition and subtraction problems.
This session is part one of two sessions about addition and subtraction in the Investigations curriculum. This session focuses on addition and subtraction in Kindergarten through second grade. We will look at how students work on story problems, the strategies students use to solve addition and subtraction problems, and the work students do with decomposing numbers and place value that helps them develop efficient strategies for solving addition and subtraction problems. In the second session on addition and subtraction you will look at the work that students do in grades three through five of Investigations.
Mathematics in this session
- Visualize, retell and model the action of addition and subtraction situations
- Examine strategies for solving addition and subtraction story problems
- Decompose numbers in different ways
- Work with place value, including breaking numbers up into tens and ones
- Consider how students develop fluency with two-addend combinations up to 10 + 10
Common Core Alignment
The content in this session is aligned to the following Common Core State Standards and Math Practices:MATH STANDARDS
Operations and Algebraic Thinking
- Understand addition as putting together and adding to, and understand subtractions as taking apart and taking from (K)
- Represent and solve problems involving addition and subtraction (1 and 2)
- Understand and apply properties of operations and the relationship between addition and subtraction (1)
- Add and subtract within 20 (1 and 2)
- Work with addition and subtraction equations (1)
- Work with equal groups of objects to gain foundations for multiplication (2)
K.OA.1-5
1.OA.1-8
2.OA.1, 2
Number and Operations in Base Ten
- Work with numbers 11-19 to gain foundations for place value (K)
- Extend the counting sequence (1)
- Understand place value (1 and 2)
- Use place value understanding and properties of operations to add and subtract (1 and 2)
K.NBT.1
1.NBT.2abc
1.NBT.4-6
2.NBT.1ab, 2.NBT.5-9
2.MD.5
MATH PRACTICES
MP1: Make sense of problems and persevere in solving them.
MP5: Use appropriate tools strategically
MP6: Attend to precision
You may also wish to review the full set of Common Core State Standards in Math and/or videos of the Math Practices enacted in Investigations.
Readings
Story Problems in Kindergarten, from Unit KU2 – Counting and Comparing (Grade K)
Three Approaches to Story Problems, from Unit IU3 – Solving Story Problems (Grade 1)
Adding With Stickers, from Unit 2U6 – How Many Tens? How Many Ones? (Grade 2)
Place Value in Grade 2, from Unit 2U6 - How Many Tens? How Many Ones? (Grade 2)
The Role of Games in Investigations
The readings above are all published in Russell, S.J.; Economopoulos, K.; Wittenberg, L.; et al. Investigations in Number, Data, and Space®, Second Edition. Glenview: Pearson, 2012.
Additional Resources
Teacher Note in Implementing Investigations in Grade 5:“Racial and Linguistic Diversity in the Classroom: What Does Equity Mean in Today’s Classroom?”
Case in Implementing Investigations in Grade 5,“What’s the “Difference”?” A fifth grade teacher writes about the issue of language in mathematics for ELL students.
Talk and paper about mathematics, the CCSS and language by Judith Moschkovich, University of California, Santa Cruz
Listen as Professor Moschkovich talks about teaching math for conceptual understanding with ELL students. ELLs may not speak like mathematicians right away, but teachers can build on the mathematical reasoning they're already doing.
Dr. Moschkovich’s paper (downloadable from the website):
This paper makes recommendations for developing mathematics instruction for English Language Learners (ELLs) aligned with the Common Core State Standards. The recommendations can guide teachers, curriculum developers, and teacher educators as they develop their own ways of supporting mathematical reasoning and sense-making for ELLs.
Research Brief based on Judith Moschkovich’s Using Two Languages When Learning Mathematics, in Educational Studies in Mathematics (2007).
Story Problems
In this introductory activity you are asked to solve a problem mentally. After you have solved it, write down your steps (on paper or electronically) to remember them.
School Assembly Problem:
475 students attended an assembly at the Mitchell school. 89 students left early to go on a field trip. How many students were still at the assembly?
What strategy did you use to solve the problem? Can you think of a different approach than the one you used?
Once you have brainstormed one or more strategies, click the Show link to review other possible solutions.
Some possible solutions include:
NOTE: The open number line is a tool* used to solve and/or record an addition or subtraction strategy.
What did you need to know or understand about addition or subtraction or about numbers in order to solve this problem using these strategies?
Note: We want 4th and 5th graders to have a few different strategies they can efficiently and flexibly use to solve addition and subtraction problems.
In order to solve problems efficiently and flexibly, students need to develop building blocks of computation. This work begins in the early grades.
Story Problems
Beginning in Kindergarten, students solve addition and subtraction story problems. This work on addition and subtraction story problems continues throughout the grades and is an essential part of the Investigations curriculum. In this activity you will learn about the focus of the story problem work students do in Investigations. You will view a video and sort student work according to the approaches students used to solve some story problems.
The video below illustrates how a teacher might introduce a story problem to a group of students.
This is a story problem students might solve in first grade:
Last night I picked up 12 pencils from the floor. I put 4 of the pencils in the pencil box. How many were left in my hand?
When students are introduced to a problem they are asked to visualize what happened in the story, retell the story and think about whether there are going to be more or fewer pencils at the end of the story.
As students work on story problems, the focus is on students making sense of the action of different types of addition and subtraction problems and developing strategies to solve them. Listen to the comments in the VoiceThread below for additional insight on the focus in helping students solve story problems.
EQUITY MOMENT
Story problems can be particularly challenging for students whose first language isn’t English. Brainstorm different ways to support English language learners. Some ideas you touch on could include: acting out stories, using manipulatives, reviewing new vocabulary, using pairs or small groups to explain the story, and having students write their own stories.
You may also wish to explore the following optional resources to learn more about supporting ELL students in working with story problems:
- Teacher Note in Implementing Investigations in Grade 5 :Racial and Linguistic Diversity in the Classroom: What Does Equity Mean in Today’s Classroom?
- Case in Implementing Investigations in Grade 5 ,“What’s the “Difference”?” A fifth grade teacher writes about the issue of language in mathematics for ELL students.
- Talk and paper about mathematics, the CCSS and language by Judith Moschkovich, University of California, Santa Cruz
- Listen as Professor Moschkovich talks about teaching math for conceptual understanding with ELL students. ELLs may not speak like mathematicians right away, but teachers can build on the mathematical reasoning they're already doing.
- Dr. Moschkovich’s paper (downloadable from the website):
Mathematics, the Common Core, and Language: Recommendations for Mathematics Instruction for ELs Aligned with the Common Core
This paper makes recommendations for developing mathematics instruction for English Language Learners (ELLs) aligned with the Common Core State Standards. The recommendations can guide teachers, curriculum developers, and teacher educators as they develop their own ways of supporting mathematical reasoning and sense-making for ELLs. - Research Brief based on Judith Moschkovich’s Using Two Languages When Learning Mathematics, in Educational Studies in Mathematics (2007).
Three Approaches to Solving Story Problems
When students solve problems in ways that make sense to them, they use a few different approaches. These approaches are based on each student’s understanding of number and of the specific operation.
Young students commonly take one of three approaches to solving a story problem:
- counting all
- counting on or back, or
- numerical reasoning.
Learn more about these three approaches by clicking the Show link.
Counting All. Read aloud the description of counting all and emphasize that, in this approach, students count out each quantity in the problem. Illustrate counting all with the example of student work, beginning by reading aloud the problem, then talking about how the student solved it – the student counted out 12 crayons in order to draw them and then counted 4 as she crossed out 4 and then counted how many were left.
Counting On or Back. Read aloud the description of counting on or back. Emphasize that in this approach the student starts with one quantity and counts up or down from that quantity. The difference between the counting down and counting all approach when a student is subtracting might be confusing. It may help to compare the examples of student work for each approach. In the counting all example Leah counted out all 12 pencils first. In the counting back example Bruce did not count out 12 first, instead he simply started at 12.
Numerical Reasoning. Read aloud the description of numerical reasoning. Teachers often call this “using something you know.” Give an example of numerical reasoning (“I know that 5 + 5 is 10 so 6 + 5 is one more so it’s 11”). Look together at the student work and talk briefly about what reasoning Lyle is using.
More in depth explanations and additional student work illustrating each approach can be found in Teacher Note, Three Approaches to Story Problems.
EQUITY MOMENT
Studies suggest that there seem to be gender differences in problem solving strategies. Carr and Jessup1 found that girls were more likely to use overt strategies (counting on fingers or with counters), while boys were more likely to use retrieval (from memory) to solve addition and subtraction problems. In the group sessions, all children were more likely to use covert strategies, e.g., mental calculation and retrieval. What ideas do you have about how to support all students in being flexible problem solvers?
Student Work
Looking at Student Work is an integral part of the work in a mathematics classroom – we assess students to inform our teaching. We encourage you to use this DIET protocol as a way of listening to student thinking in what they have communicated in writing. Rather than rush to evaluate, think about these questions: What do you see on the paper? What is it that the student knows and understands? What does the student not yet understand? How can you help – what are your next steps to help the student understand or to extend their thinking?
In this sorting activity, you will organize each sample of student work into one of three different bins based on how the student has approached the problem. Make sure you are logged into the Moodle system, and try it out!
The strategies students use in later grades to solve problems with larger numbers grow out of the strategies students are using in the early grades. You will examine strategies that students use to solve addition problems with larger numbers at the end of this session and the strategies students use for solving subtraction problems with larger numbers in the Addition and Subtraction, part 2, session.
After you have completed the Sorting activity, click on the Show link to read other teachers' observations about the student work.
- Student D drew 8 large stars and then a set of small stars to get to 14. Counting all.
- Student E starts at 8 and then counts up from 8 to get to the total of 14. She then figures out how many she added on. Counting on.
- Student F knows that 12 is made up of 10 and 2 and also knows that if you add up parts from both numbers and then combine them you will reach the total. Numerical Reasoning.
1Carr, Martha and Donna Jessup. (1997). “Gender Differences in First Grade Mathematics Strategy Use: Social and Metacognitive Influences.” Journal of Educational Psychology, Vol. 89 (no. 2), 318-328.
EQUITY MOMENT
When looking at student work, teachers often use the adjective “better” when comparing and assessing student work. This is a subjective description; instead, we are trying to look for levels of mathematical efficiency. All students should be expected to achieve at high levels and to master more efficient strategies. One way to think about high expectations for all students is to: 1) appreciate and acknowledge where a student currently is and 2) figure out where you want the student to go and how you can help him/her get there.
Number Composition and Facts
In this activity you will practice using two games focused on number composition and discuss the mathematics in the games. You will also look at examples of how students use addition cards and think about how the games help students develop fluency with addition combinations.
In Kindergarten through second grade, students work a lot on number composition. For example, in first grade when students solve a problem about 12 crayons - some are red, some are blue - they may find out that 12 is composed of 6 and 6, but it can also be 5 and 7. In a game called Ten Plus students see that 8 + 4 or 12 is equivalent to 10 + 2. When Today’s Number is 12 they might break it into three parts: 4 + 4 + 4.
Working on number composition encourages flexibility with numbers, helps students learn addition and subtraction facts and exposes them to ideas about place value.
How Many Am I Hiding? and Make Ten
You will now view sample “hands” of two games that focus on number composition: How Many Am I Hiding? - a 1st grade game, and Make 10 – a game that is in both 1st and 2nd grades (a version of this game called Make 6 is in Kindergarten). Games are an integral part of the Investigations curriculum. When students play games multiple times they can deepen their understanding of numbers and the operations and practice computation.
As students compose and decompose numbers they are learning about number composition and the operations of addition and subtraction. As they play games like these they are also beginning to learn addition combinations or facts. For example, after students play Make 10 multiple times they often remember many of the combinations that make 10: 1 + 9, 2 + 8, 5 + 5, etc.
In 2nd grade students also use addition cards to work towards becoming more fluent with addition facts. They learn addition facts in categories (all of the doubles combinations, all of the 10s combinations, etc.). As they work with the addition cards for a category they also do activities that focus on those particular combinations (e.g., playing Make Ten while practicing the 10s combinations). They use the addition cards to think about what combinations they already know and to practice those they don’t know. To help practice the combinations they find difficult they write clues to help them remember them.
Students learn combinations best by using strategies and not just simply rote memorization. If students use their understanding of numbers and number relationships to work on combinations then they have ways to figure out combinations when they can’t remember them.
Refer to Teacher Note, Learning the Addition Combinations, part 1 and part 2 for more information on how students learn the combinations.
EQUITY MOMENT
Are there some students that typically do better in math when activities are framed in terms of competition? Cooperation? Does this break down along gender lines? In what ways can you help students feel more comfortable in various situations? What management techniques do (or can) you employ?
In past workshops, many participants have said that typically boys engage more readily when the activities involve competition and girls “do better” when activities involve cooperation. However, many participants have said that over the last few years, girls have become more competitive in classrooms.
Some students naturally work more slowly than others. What role does speed play in how students approach and complete math problems?
What implication does this have for classrooms? Does the person who completes an activity faster “win”? Do you notice issues of speed breaking down along gender lines or other variables (including confidence and experience)?
How can you provide opportunities for students working at various paces to work at their pace while also being participatory in whole group work and discussions?
Supporting Addition Strategies
In this activity you will solve an addition problem in the Sticker Station context and look at students’ strategies for solving the same problem. Then you will solve another problem using one of the strategies the students used.
Work on number composition includes work on breaking numbers into tens and ones. This is a part of learning about place value. Earlier we saw an example from the game Ten Plus in which students find an equivalent expression with one of the addends being ten. In grade 2 students are introduced to a sticker station context to work on ideas about place value and the way numbers are structured using tens and ones. This context is revisited in grade 3.
Sticker Station
In the Sticker Station context, there is a store that sells stickers and the stickers are sold in different ways: single stickers, strips of 10 stickers, and pages of 100 stickers (which students are introduced to later). Students in second and third grade solve a variety of problems based in this context of the Sticker Station.
Students begin by finding all the different ways someone can have 46 stickers. What is one way you can have 46 stickers? How many strips of ten? How many singles? What is a different way?
Here are two different ways to make 46 stickers using sticker notation. The lines stand for one strip of ten and each dot stands for a single sticker.
Students find out that, for example, the number 46 is not only 4 tens and 6 ones but also 3 tens and 16 ones.
Note: While the visual representation of the stickers is nearly the same as the Base Ten Blocks, the field test for the revision of Investigations showed that second graders are more likely to connect with the specific context of stickers.
Next, please solve the following problem mentally in any way that works for you.
Franco had 66 car stickers. Jake gave him 52 car stickers. How many car stickers does Franco have now?
View the three screencasts below to see several options for how this problem can be solved.
Students’ Strategies
Earlier we saw how students solved problems using counting all, counting on, and numerical reasoning. As students gain a better understanding of the structure of the base ten number system, they apply this knowledge to develop strategies for solving addition and subtraction problems.
Annotate Student Work
Holly and Henry used the strategy of breaking up both numbers into tens and ones. In Investigations this strategy is called adding tens and ones or adding by place. Holly added the tens and ones separately while Henry added the tens together and then added on the ones from one number and then the ones from the other number.
Both Simon and Chen used the strategy of keeping one number whole and breaking the other number into parts. In Investigations this strategy is called keeping one number whole. Simon and Chen broke up the 52 in different ways – Chen broke it up into 5 tens and a 2 and Simon broke it up into 4, 40 and 8.
There are other strategies that students may use to solve addition problems but these two are the main ones.
Solving Another Problem
Finally, solve the following problem using either an adding by place strategy or a keeping one number whole strategy.
48 + 37
Note how you solved the problem. Which strategy did you use?
Discussion
How do the activities you experienced in this session help students develop the building blocks of computation? Give some examples. How do you think development of these building blocks will help students in later grades efficiently and flexibly solve addition and subtraction problems?
Key Learning
- In order to solve problems efficiently and flexibly students need to develop the building blocks of computation.
- As students work on story problems, the focus is on students making sense of the action in the problems and on them developing strategies that make sense to them to solve the problems.
- Young students commonly use one of three approaches to solving a story problem: counting all, counting on or back, or numerical reasoning (using numbers they know).
- Working on number composition encourages flexibility with numbers, helps students learn addition and subtraction facts and exposes them to ideas about place value.
- Students learn combinations best by using strategies and not just simply rote memorization.
This session is aligned with the following K-2 Units in Investigations.LINK TO 6A HO