Problem Solving and the Common Core

Content

Readiness Skills for Problem Solving

The Committee on Early Childhood Mathematics has concluded that once children demonstrate counting skills including knowledge of number word list and cardinality, they begin to recognize relationships between quantities and develop counting strategies to solve simple word, or story, problems. (Cross, 2009) Despite the challenges that word problems pose for older students, young children actually appear to have greater success with problems posed in context of a story. This is especially true in problems that depict action or in the case of part-part-whole problems, but others (change, start unknown & comparison) are more difficult. At first children will attempt problems that they understand and can act out, either through role-play or by interacting with physical objects. The problems need to include numbers within their counting range. However, as young children have opportunities to apply counting strategies in real world context, they show advancement in their counting skills and develop more advanced strategies and begin to build a range of strategies for computation. (Van de Walle, 2013) Studies have shown children as young as preschool and kindergarten can successfully apply counting strategies to solve word problems involving addition, subtraction, and even multiplication or division. (Cross, 2009)

Problem Types

It is important that students develop an understanding of the operations of addition and subtraction within the wide range of situations where addition and subtraction can be applied. In Grade 1, The Common Core State Standards advocate that students solve problems of 3 types including change problems, part-part-whole problems and comparison problems. Change problems are categorized into two subtypes based on whether the initial quantity increases or decreases. Comparison problems are categorized into two subtypes based upon the difference between the two quantities being described as more or less than the other. These sub-types of problems are further divided according to position of the unknown quantity. This yields 6 types of change problems, and 6 types of comparison problems. There are only two types of part-part-whole problems, according as the whole is unknown, or a part is unknown. These 14 categories make up a taxonomy developed by mathematics educators, consisting in all of fourteen types of one-step addition and subtraction word problems. The complete taxonomy is provided as table I on page 88 of the Common Core State Standards (http://www.corestandards.org/the-standards/mathematics). A more detailed discussion is given below.

Research indicates that some of the 14 types are more challenging than others. I will discuss this issue more thoroughly as I discuss each type. I plan on systematically introducing the problem types, discussing and comparing the structure of the problems, then providing students opportunities to practice newly introduced problem types in a mixed format with problem types previously introduced. Once a problem type is introduced, I intend on practicing the problem type, giving students the opportunity to demonstrate consistent proficiency for minimum of 2 consecutive days in isolation and within mixed practice for minimum of 2 consecutive days before moving on. The pacing will be based upon student performance and individual learning needs. I will initially begin with number sets within 5, then based on student need, I will scaffold the number sets within 10 and within 20. However, I do not intend to slow the introduction of additional problem types until all students are able to work within the larger number sets, as we will continue to work on building these proficiencies outside of the context of word problems. I will discuss later and in more detail how I will make decisions about differentiating the number sets.

Change Problem Types

Change problems are dynamic, involving change over time. The change can be an increase or a decrease. Subtypes consist of change increase (aka add to) and change decrease (aka take from) problem types. These problems should be introduced and intermixed together in order to illustrate the inverse relationship between addition and subtraction. The physical action of joining and separating that occurs in change problems lends to acting out the problem, making these problems easier for students to comprehend and tackle. Change problems have been found to be the simplest problem type; therefore I plan on introducing change problems first.

Change Increase, Result Unknown & Change Decrease, Result Unknown

Change problems with result unknown have been found to be the easiest for children to deal with. The Common Core State Standards advocate that change increase, result unknown and change decrease, result unknown problems be introduced and performed to mastery within 10 during Kindergarten. Since, many of my students enter the year performing below grade level expectations, this will be my starting point. I want to ensure my students have ample experience with this basic problem type. 

Change increase, result unknown problem example: 3 + 2 = __

3 children are playing tag at recess. 2 more children come to play. How many children are playing tag now?

Change decrease, result unknown problem example: 5 - 1= __

5 children were playing tag at recess. 1 child went inside. How many children are playing now?

Change Increase, Change Unknown and Change Decrease, Change Unknown

As I see students demonstrating proficiently solving change increase, result unknown and change decrease, result unknown within number sets to 5, I will introduce change increase, change unknown problem types, followed by the introduction of change decrease, change unknown problem types.  

Change increase, change unknown problem example: 3 + __= 5

3 children are playing tag at recess. Some more children come to play. There are 5 children playing now. How many children joined the game?

Change decrease, change unknown problem example: 5 - __= 3

5 children were playing tag at recess. Some children quit playing. There are 3 children playing now. How many children quit?

Change Increase, Start Unknown & Change Decrease, Start Unknown

Change increase, start unknown & change decrease, start unknown are the most complex problem of the change class. The Common Core State Standards advocate for the introduction of change increase start unknown & change decrease start unknown in the 2^nd grade. Consequently, this problem type will be introduced at a later time.

Change increase, start unknown problem example: __ + 2 = 4

Some children were playing tag at recess. 2 more children came to play. There are 4 children playing now. How many children were playing tag to start?

Change decrease, start unknown problem example: __ - 3 = 7

Some children were playing tag at recess. 3 children quit playing. Now there are 7 children playing. How many children were playing tag at first?

Part-Part-Whole Problems

Part-Part-Whole problems, sometimes known as collection problems, are problems in which some quantity or collection of objects is made up of two parts. This class, though similar to change problems, is slightly more difficult for young children. These problems have been found difficult for students because there is no physical action, making them less transparent and harder to model for young children. There are two types of part-part-whole problems, part-part-whole with an unknown whole or total and part-part-whole with an unknown part or addend. I intend first to introduce part-part-whole, total unknown problems, then part-part-whole, part unknown problems.

Part-Part-Whole, Total Unknown

In part-part-whole, total unknown situations, two quantities or parts are put together to make a third quantity.

Part-part-whole, total unknown problem example:

There are 2 girls and 3 boys playing tag. How many children are playing tag?

Part-Part-Whole, Part Unknown

In part-part-whole, part unknown situations, a total quantity is taken apart to make two quantities.

Part-part-whole, part unknown problem examples:

18 children are playing tag. 8 children are from Mrs. Brulia’s class. The rest are from Ms. Loftas’ class. How many of Ms. Loftas’ children are playing?

10 children are playing tag. There are some boys and 2 girls playing. How many boys are playing?

Comparison Problems

Comparison problems are the most challenging of the 3 types of one-step addition and subtraction word problems. There
are three types of compare problems: difference unknown, larger unknown, and smaller unknown. Compare problems involve relationships between quantities. Children are asked to determine the difference between the smaller and larger set. This third set or value is a non-specific subset of the larger set and tells the difference between the two amounts or how many more/less. Comparison situations typically do not involve a physical action, making it difficult for students to act out or model the problems. Moreover, comparison problems include language that is complex for young children, especially those facing language delays. The following terms are commonly used to describe the relationships present in the problems: fewer, less than, more, bigger, greater, longer, shorter, older, younger, heavier, lighter, etc. The comparison problems can each be posed in two ways, one uses words indicating greater and one uses words indicating lesser. This language can be consistent with the needed operation or inconsistent. See the examples below for more discussion of consistent and inconsistent language.

The Common Core State Standards calls for the introduction of addition and subtraction comparison problems in first grade. However, prior to the introduction of comparison problems students are exposed to comparison situations. These situations require students to visually compare sets to determine which is more than, less than, or equal to based on perception of length and density of the sets. In addition, students utilize match and unitizing count strategies to determine which set is more than, less than, or equal to other sets. The ability to apply comparative language and the ability to identify the relational situations between quantities are necessary components of solving the comparison problems. Many of my students have language delays and this is an area of weakness. I intend to spend time addressing these precursor skills to ensure they have the underlying foundation to promote success with these problem types.

Comparison, difference unknown

When solving compare problems, students demonstrate greater ease navigating comparison, difference unknown problems utilizing the terms more and bigger than. I will introduce these problems, then intermix comparison, difference unknown problems utilizing fewer and less than.

Comparison, difference unknown problem example:

8 students from Mrs. Brulia’s class are playing tag. 14 students are playing from Ms. Loftas’ class. How many more students from Ms. Loftas’ class are playing than students from Mrs. Brulia’s class?

Comparison, bigger unknown

When students are attempting compare with a bigger unknown problems stating the difference with the term more is simpler for students because the relationship between the quantities and operation are consistent.

Comparison, bigger unknown, consistent language (more, greater, larger) problem example:

Students are playing tag at recess. 8 students are from Mrs. Brulia’s class. Ms. Loftas has 6 more students playing from her class than are playing from Mrs. Brulia’s class. How many of Ms. Loftas’ students are playing tag?

Comparison, bigger unknown, inconsistent language (less, fewer) problem example:

Students are playing tag at recess. 8 students are from Mrs. Brulia’s class. Mrs. Brulia has 6 fewer students playing from her class than are playing from Ms. Loftas’ class. How many of Ms. Loftas’ students are playing tag?

Comparison, smaller unknown

When students are attempting compare problems with a smaller unknown problems stating the difference with the term fewer is simpler for students because the relationship between the quantities and operation are consistent.

Comparison, smaller unknown, consistent language (fewer, less) problem example:

8 students from Mrs. Brulia’s class are playing tag. Ms. Loftas has 6 fewer students playing tag than Mrs. Brulia. How many students from Ms. Loftas’ class are playing tag?

Comparison, smaller unknown, inconsistent language (more, greater, bigger) problem example:

8 students from Mrs. Brulia’s class are playing tag. Mrs. Brulia has 6 more students playing tag than Ms. Loftas. How many students from Ms. Loftas’ class are playing tag?

Building Conceptual Understanding through the Use of Tiered Number Sets

Doug Clarke, a researcher in mathematics education, indicates children follow a developmental path as they build conceptual understandings and apply these understandings as strategies to solve addition and subtraction problems. He developed a framework that recognizes 6 key growth points or transitions in how students approach solving addition and subtraction problems. (Clarke, 2001) His trajectory indicates children demonstrate increasing conceptual understanding as they apply increasing more complex strategies.

I plan to utilize Clarke’s growth points to vary the difficulty of problems by altering the numbers used. Sequential in nature, the complexity of the number sets used in the problems need to be tiered to develop flexibility with the understanding of numbers and place value, expecting students to demonstrate fluency of facts to 5, fluency of facts to10, making ten, decomposing a number to create a 10, adding to 10 and decomposing teen numbers, demonstrating an understanding that it is comprised of one ten and varying ones (1-9). I will make use of various strategies including the use of manipulatives, count all, count on, and recompose solution strategies and developing early connections to the measurement model to build conceptual understanding. Each of these strategies is discussed in more detail later in this text.

Initially I will introduce problems including numbers to 5. This allows me to begin working with students on problem solving who may have a limited counting range. I will intentionally choose number sets to assess the students’ ability to decompose 5 into 4 and 1, 3 and 2, and 5 and 0. I will also utilize visual representations of number sets to determine if students are able to recognize these by sight or if they need to count out the items to determine the value of a set.

Number sets will be increased to include partner numbers to 10 with students demonstrating flexibility with numbers within 5 and counting ranges to 10 and beyond. I will begin choosing number sets including 5 as a partner number to encourage students to view 5 as a part of the larger number. I will be assessing students’ ability to decompose 6 into 5 and 1 more, 7 into 5 and 2 more, 8 into 5 and 3 more, 9 into 5 and 4 more and 10 as 2 sets of 5, noticing when given visual representations, if they rely on subitizing or cardinality. I then focus on utilizing number sets that form ten: (10, 0), (9, 1), (8, 2), (7, 3), (6, 4) and (5, 5).

Once students demonstrate flexibility and fluency with numbers within 10 and a counting range equal or greater than 20, I will increase the number sets to partner numbers to 20. I will utilize numbers to illustrate adding to 10 and decomposing teen numbers, demonstrating an understanding that a teen number is composed of one ten and some ones (1-9). Finally, I will choose number sets to foster decomposition and recomposition of numbers to form 10s within the solution strategy.

Addition/Subtraction Single-Digit Solution Strategies and Correlating Levels

Consistent with Clarke’s trajectory, the Common Core Progression Documents recognize 1^st grade students typically utilize three levels of single-digit solution strategies as they develop conceptual understanding in problem solving: direct modeling by counting all, counting on and derived facts or decomposition of numbers to convert to an easier problem. Children approach problems differently depending on the level of understanding they have developed.

Level 1. Direct Modeling by Counting All or Taking Away

Children represent situations or mathematical problems with objects, drawings, or fingers. They create a model for the situation by creating groups to represent the addends or by separating a total amount into groups. Then, counting the resulting total or group can give the answer.

Level 2. Counting On

Children at this level do not need to count or model each number in the situation. The child sees a “part”, understanding its value. So, the child will start at this number, omitting the count of this addend and beginning with the number word, count up. The child employs some way of keeping track of the count such as fingers, objects, mentally imaged objects, body motions or count words. For addition, the count is stopped when the amount of the remaining addend has been counted. The last number word is the total. For subtraction, the count is stopped when the total occurs in the count. The tracking technique is used to determine the value of an unknown addend.

Level 3. Derived Facts

A child at this level begins to use break-apart strategies to decompose and compose numbers to make an easier problem. Children decompose an addend and compose a part with another addend to convert to the values in the problem to form 10 or utilize embedded place value concepts.