The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students

Dr. Ansell and Dr. Pagliaro were available from 1/14/08 until 2/13/08 to answer questions and share ideas concerning their research and its implications for parents of children who are deaf/hard of hearing, their teachers and other professionals who work with them.

You are encouraged to read the research summary below and review the attached discussion.

Abstract: This study determines the relative difficulty and associated strategy use of arithmetic (addition and subtraction) story problems when presented in American Sign Language to primary level (K-3) deaf and hard-of-hearing students. Results showed that deaf and hard-of-hearing students may consider and respond to arithmetic story problems differently than their hearing peers, with the critical dimension in problem difficulty being based on the operation typically used to solve the problem, not the story within the problem. The types of strategies used by the students supported the order of problem difficulty. The visual–spatial nature of the problem presentation appeared not to assist the deaf and hard-of-hearing students in solving the problems. Factors that may have contributed to this pattern of problem difficulty are discussed so that educators can better align mathematics instruction to the thinking of the deaf child.

Paragraph #1: Most adults categorize arithmetic story problems (word problems) based on the operation they would use to solve the problem: addition if the answer requires combining two sets, or subtraction if the answer requires finding the difference between two sets. Thus, adults would see story problems as belonging to just two types. If we look at the story described within the problem (the situation), however, the problems can be categorized differently. For example, let’s look at the following problem:

Bob wants 8 worms. He found 5 already.

How many more worms does he need to find?

Because the answer is the difference between the amount of worms Bob wants (the end total – 8) and the amount Bob started with (5), the problem would typically be identified by adults as a subtraction problem. The story, however, reveals that the situation actually involves adding more to the first set (5) in order to reach a total amount (8). Like a movie scene, Bob has 5 worms and continues to look for more, adding worms to what he already has until he reaches his goal of 8 worms. Thus, from the point of view of the actor in the story (Bob), this is an addition situation; one then might solve the problem by counting forward from the given initial amount to the total at the end, keeping track of how much is added. The answer would be the amount added (3).

Paragraph #2: By starting with situations to which children can relate and encouraging children to focus on the situation involved, story problems can be a bridge from children’s informal knowledge and experience to the more formal mathematics they learn in school. Children will begin to see mathematics not only as a tool to problem solving, but also as a sense-making activity. They will also begin to develop skills associated with good problem solvers including curiosity, objectivity, flexibility, persistence, and decisiveness. When instead we impose the adult viewpoint on children, mathematics may not make sense causing children to possibly develop an approach to problem solving that relies on procedures generated or given to them from outside of themselves. They would not see themselves as problem solvers.

Paragraph #3: Our study of K-3 deaf/hard-of-hearing students found that, overall, the children focused on the operation for solving a problem without necessarily considering the story that the problem describes. The difficulty of a problem for the students, for example, was based on whether they had to add or subtract (addition problems being easier for them than subtraction problems), not in the complexity of the situation (the amount of inference needed to figure out the answer). However, when we looked within this group at the more successful problem solvers (those children who correctly solved more than half of the problems given [half the children in the study]), we found that these children did tend to follow the story, as shown by the strategies they used. For example, in the “Bob’s worms” problem given earlier, these successful problem solvers tended to find the difference in the problem not by subtracting, but by adding on, as described above.

Paragraph #4: In addition to following the story or situation within the problem, these successful problem solvers were flexible in their strategy choice. Strategies used by children to solve problems in general have been classified into three broad categories based on their level of abstractness (facts being the most abstract). The successful problem solvers in our study were likely to use more concrete strategies (using manipulatives as representatives of objects or as markers for counting) with more difficult problems and less concrete strategies for less difficult problems. This relation between problem difficulty and type of strategy used indicates that these children are aware of their own understanding and use it to plan and adjust as they problem solve; that is, their approach to problem solving is thoughtful.

Paragraph #5 (Summary): In summary, our study showed that success is linked to, among other things, the ability to flexibly use a variety of strategies depending on the problem and to focus on the story situation. Further, our study shows that deaf/hard-of-hearing children can be successful with problem solving (half of the children in the sample were able to solve more than half of the six problems presented to them); however, given the fact that half were also not as successful (could not solve at least four of the six problems), it also indicates a critical need for attention to this area. Below are some suggestions for teachers and parents to help their students/children to become successful problem solvers.

Paragraph #6 (Suggestions):

For young children, frequent opportunities to solve story problems of various types can form the foundation for the more complex problem solving later in school as well as in “real-life” outside of school. It is critical then that children have access to a variety of problem-solving opportunities.

Create “problems” within daily activities for children.

During meal or snack time, purposefully give a child less napkins than is needed requiring him/her to recognize the deficiency and ask for more. Ask the child to tell you how many more napkins (or cups, cookies, etc.) are needed.

Build towers of blocks with a child and compare the two in height and/or number of blocks.

Have older children help with cooking and preparing meals converting fractions in recipes, for example, to accommodate larger or smaller parties; or help teens to prepare and keep a budget via an Excel sheet on the computer.

Center problems around a book that you’re reading together to integrate math and literacy

Have children create their own problems for you and/or other children to solve so that they come to understand the structure of story problems

Be sure to have children explain their thinking in how they solved the problems. This will allow them not only to organize their thoughts, but also to show others that there are a variety of solution strategies that can address one problem and that this variety is valued. Discuss these strategies without discrediting any of them. Children will choose the strategy most efficient (most meaningful) to them.

Include the language of mathematics in your everyday conversations with children to get them used to and comfortable with their meanings. This is not to suggest a focus on key words (e.g., “altogether means addition”), however. In fact, we would suggest NOT emphasizing key words. Focus instead on making the situation more explicit. For example, if the situation takes place over time, you might use temporal words/signs (instead of “How many birds are there altogether,” you might say, “How many birds are there now?”)

Reference:
Ellen Ansell and Claudia M. Pagliaro
The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students
J. Deaf Stud. Deaf Educ., Spring 2006; 11: 153 - 170.

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing StudentsDr. Ansell and Dr. Pagliaroto answer questions and share ideas concerning their research and its implications for parents of children who are deaf/hard of hearing, their teachers and other professionals who work with them.wereavailable from 1/14/08 until 2/13/08Abstract: This study determines the relative difficulty and associated strategy use of arithmetic (addition and subtraction) story problems when presented in American Sign Language to primary level (K-3) deaf and hard-of-hearing students. Results showed that deaf and hard-of-hearing students may consider and respond to arithmetic story problems differently than their hearing peers, with the critical dimension in problem difficulty being based on the operation typically used to solve the problem, not the story within the problem. The types of strategies used by the students supported the order of problem difficulty. The visual–spatial nature of the problem presentation appeared not to assist the deaf and hard-of-hearing students in solving the problems. Factors that may have contributed to this pattern of problem difficulty are discussed so that educators can better align mathematics instruction to the thinking of the deaf child.Paragraph #1:Most adults categorize arithmetic story problems (word problems) based on the operation they would use to solve the problem: addition if the answer requires combining two sets, or subtraction if the answer requires finding the difference between two sets. Thus, adults would see story problems as belonging to just two types. If we look at the story described within the problem (the situation), however, the problems can be categorized differently. For example, let’s look at the following problem:Because the answer is the difference between the amount of worms Bob wants (the end total – 8) and the amount Bob started with (5), the problem would typically be identified by adults as a subtraction problem. The story, however, reveals that the situation actually involves adding more to the first set (5) in order to reach a total amount (8). Like a movie scene, Bob has 5 worms and continues to look for more, adding worms to what he already has until he reaches his goal of 8 worms. Thus, from the point of view of the actor in the story (Bob), this is an addition situation; one then might solve the problem by counting forward from the given initial amount to the total at the end, keeping track of how much is added. The answer would be the amount added (3).

Paragraph #2:By starting with situations to which children can relate and encouraging children to focus on the situation involved, story problems can be a bridge from children’s informal knowledge and experience to the more formal mathematics they learn in school. Children will begin to see mathematics not only as a tool to problem solving, but also as a sense-making activity. They will also begin to develop skills associated with good problem solvers including curiosity, objectivity, flexibility, persistence, and decisiveness. When instead we impose the adult viewpoint on children, mathematics may not make sense causing children to possibly develop an approach to problem solving that relies on procedures generated or given to them from outside ofthemselves. They would not seethemselvesas problem solvers.Paragraph #3:Our study of K-3 deaf/hard-of-hearing students found that, overall, the children focused on the operation for solving a problem without necessarily considering the story that the problem describes. The difficulty of a problem for the students, for example, was based on whether they had to add or subtract (addition problems being easier for them than subtraction problems), not in the complexity of the situation (the amount of inference needed to figure out the answer). However, when we looked within this group at the more successful problem solvers (those children who correctly solved more than half of the problems given [half the children in the study]), we found that these children did tend to follow the story, as shown by the strategies they used. For example, in the “Bob’s worms” problem given earlier, these successful problem solvers tended to find the difference in the problem not by subtracting, but by adding on, as described above.Paragraph #4:In addition to following the story or situation within the problem, these successful problem solvers were flexible in their strategy choice. Strategies used by children to solve problems in general have been classified into three broad categories based on their level of abstractness (facts being the most abstract). The successful problem solvers in our study were likely to use more concrete strategies (using manipulatives as representatives of objects or as markers for counting) with more difficult problems and less concrete strategies for less difficult problems. This relation between problem difficulty and type of strategy used indicates that these children are aware of their own understanding and use it to plan and adjust as they problem solve; that is, their approach to problem solving is thoughtful.Paragraph #5 (Summary):In summary, our study showed that success is linked to, among other things, the ability to flexibly use a variety of strategies depending on the problem and to focus on the story situation. Further, our study shows that deaf/hard-of-hearing children can be successful with problem solving (half of the children in the sample were able to solve more than half of the six problems presented to them); however, given the fact that half were also not as successful (could not solve at least four of the six problems), it also indicates a critical need for attention to this area. Below are some suggestions for teachers and parents to help their students/children to become successful problem solvers.Paragraph #6 (Suggestions):Reference:Ellen Ansell and Claudia M. Pagliaro

The Relative Difficulty of Signed Arithmetic Story Problems for Primary Level Deaf and Hard-of-Hearing Students

J. Deaf Stud. Deaf Educ., Spring 2006; 11: 153 - 170.