This document summarizes research on teaching students with disabilities to solve mathematics story problems. It discusses how previous research has focused on basic math skills but not higher-level problem solving. The study analyzed "precurrent skills", or behaviors that increase the effectiveness of solving a problem, like identifying values and operations. Researchers taught two students with disabilities four precurrent responses in sequence. Results showed the teaching was effective at increasing correct use of precurrent behaviors on new problems, and this led to more correct problem solutions.
CONCEPTUAL APPROACH AND SOLVING WORD PROBLEM INVOLVING MULTIPLICATION OF WHOL...WayneRavi
This study was conducted to determine the effect of conceptual approach on solving word problems involving multiplication of whole numbers as well as addition and subtraction. The study was carried out in Tambongon Elementary School to Fourty-one Grade Two students. Descriptive statistics (mean & SD), paired-sample T-test and ETA2 were used as tools in the analysis of data. Results revealed that there was a significant difference on the pretest and post test scores of conceptual approach. Further, conceptual approach has large effect.
An Exploratory Study Of A Story Problem Assessment Understanding Children S ...Angie Miller
This study explored how students use their number sense when solving story problems by analyzing their strategies and reasoning. Researchers interviewed three 8- and 9-year-old students and had them solve story problems. Through qualitative coding, the researchers analyzed how the students' number sense was reflected in their problem-solving approaches and explanations. The results provided insights into how students coordinate different number sense constructs and showed flexibility in problem-solving. The study aimed to understand number sense in a holistic way rather than by assessing individual constructs in isolation.
The Effect of the Concrete-Representational-Abstract Mathematical Sequence o...Janet Van Heck
This document describes a study that examined the effects of using the concrete-representational-abstract (CRA) teaching sequence with explicit instruction to teach addition skills from 0 to 9 to kindergarten students struggling in math. The CRA sequence begins by using manipulatives, then representations like pictures, before moving to abstract problems. Three kindergarten students identified as needing math support through RTI screening were given scripted CRA lessons and their test scores were measured. The study found the CRA sequence improved students' conceptual understanding and performance on addition facts from 0 to 9 when delivered with explicit instruction.
08 test ideatets.JPGLiterature review on learning dSilvaGraf83
08 test idea/tets.JPG
Literature review on learning disability
Nature of problem
There are certain cases in children learning problems that instigate effective control over the barrier of learning new things. This study is based on the arithmetic interventions that are related to the linguistic approach. It individualizes the aspects of quick adaptability and the learning process with the orientation of the task (Allison, 2021).
The group of kindergarten students with collected data provided a comprehensive picture of the many facets of learning impairments. The term "problem solving" has been connected with the process of learning new things and comprehending facts, which has resulted in the exposure of several interventions (Christy, 2021). It demonstrates skill in resolving mathematical issues using language interventions and useful directions.
Subjects
The assessment of learning disability is judged with the kindergarten to the sixth-grade students. That included both male and female students randomly assessed. The sessions for problem-solving were of 50 minutes with a total of 34 sessions. The meta-analysis has instilled the perspective of dynamic range in subject choice with more task-oriented mathematic problems structured to resolve learning disabilities (Jennifer, 2021). There were 29 group-design studies and 10 single-subject studies (39 totals) were considered in the assessment of the meta-analysis
Procedures
Elements of educational possibilities exemplify the authoritative touch in a framed format. Continuous research to achieve a learning goal has resulted in psychological success and an authoritative demeanour. Students are likely to be relevant for the competent structure of the learning process. The usual models are instructed for people to learn with mathematical problems and stringent regulations. Every organization looks to the example of rules to understand the power of continual review. Recognizing the sluggish update in self-learning, the whole projects are framed by quality validation of the work and other refreshing information. This article contains the capabilities that are ready to survey students with innovation-related sections of development and typical cycles for moral assessment of program.
The author has conducted a diverse selection of meta-analyses of optimized studies on WPS. It assists the students with math disabilities or learning disabilities. The subjects were identified as MD if their scores fell below the 25th percentile on an average math test. The representation techniques are determined from the computer-assisted design of the test with significant changes in meta-analysis (Jennifer, 2021). The effectiveness of the procedure has instructional components with skill modeling and group identification. It promotes the various instructional barriers for students using WPS. The study manipulations were assorted that there were no significant differences in learning disability of stude ...
This literature review summarizes a meta-analysis of 18 studies on word-problem-solving interventions for elementary students with learning disabilities. The meta-analysis found that overall, word-problem-solving interventions had a significant positive effect (effect size of 1.08) on students' word-problem-solving accuracy. Interventions that included peer interaction and explicit transfer instruction yielded particularly large effects. Intensive interventions delivered in 50-minute sessions over 34 total sessions for students in Grade 3, regardless of instructional setting, produced the largest effect sizes. The findings support developing and implementing high-quality, evidence-based Tier 1 instruction in classrooms before providing more intensive individualized interventions.
A Comparison Of Two Instructional Approaches On Mathematical Word Problem Sol...Nat Rice
This study compared the effects of two instructional strategies for teaching mathematical word problems to middle school students with learning disabilities: 1) an explicit schema-based problem solving strategy (SBI) and 2) a traditional general heuristic instructional strategy (TI). Twenty-two students were randomly assigned to receive instruction using one of the two strategies. Results indicated that students who received the SBI outperformed those who received the TI on post-tests, maintenance tests 1-2 weeks later, and follow-up tests 3 weeks to 3 months later. The SBI group also did better in solving transfer problems. Additionally, the SBI group's performance exceeded that of typically achieving 6th grade peers. The study aimed to extend research on schema
REALITY – BASED INSTRUCTION AND SOLVING WORD PROBLEMS INVOLVING SUBTRACTIONWayneRavi
This study was conducted to determine the effect of reality based on the solving word problems involving subtraction. Descriptive-Comparative research design using paired sample T-test was used to utilized in the study. The study was carried out in Tibungol Elementary School to Fifty student of Grade Three section 1. Results revealed that there was a significant difference on the pretest and post test scores of pupils in reality based approach. Further, the reality based approach is effective in improving the performance of student.
An Arithmetic Verbal Problem Solving Model For Learning Disabled StudentsSandra Long
This study examined an instructional approach for teaching arithmetic word problem solving to 60 fifth and sixth grade students with learning disabilities. The experimental group received direct instruction twice a week for six weeks using a systematic problem solving model involving scripted steps and prompt cards. This group significantly outperformed the control group on a post-test measuring different problem types. The findings suggest that learning disabled students can improve their problem solving skills through explicit instruction in strategies and the use of calculators.
CONCEPTUAL APPROACH AND SOLVING WORD PROBLEM INVOLVING MULTIPLICATION OF WHOL...WayneRavi
This study was conducted to determine the effect of conceptual approach on solving word problems involving multiplication of whole numbers as well as addition and subtraction. The study was carried out in Tambongon Elementary School to Fourty-one Grade Two students. Descriptive statistics (mean & SD), paired-sample T-test and ETA2 were used as tools in the analysis of data. Results revealed that there was a significant difference on the pretest and post test scores of conceptual approach. Further, conceptual approach has large effect.
An Exploratory Study Of A Story Problem Assessment Understanding Children S ...Angie Miller
This study explored how students use their number sense when solving story problems by analyzing their strategies and reasoning. Researchers interviewed three 8- and 9-year-old students and had them solve story problems. Through qualitative coding, the researchers analyzed how the students' number sense was reflected in their problem-solving approaches and explanations. The results provided insights into how students coordinate different number sense constructs and showed flexibility in problem-solving. The study aimed to understand number sense in a holistic way rather than by assessing individual constructs in isolation.
The Effect of the Concrete-Representational-Abstract Mathematical Sequence o...Janet Van Heck
This document describes a study that examined the effects of using the concrete-representational-abstract (CRA) teaching sequence with explicit instruction to teach addition skills from 0 to 9 to kindergarten students struggling in math. The CRA sequence begins by using manipulatives, then representations like pictures, before moving to abstract problems. Three kindergarten students identified as needing math support through RTI screening were given scripted CRA lessons and their test scores were measured. The study found the CRA sequence improved students' conceptual understanding and performance on addition facts from 0 to 9 when delivered with explicit instruction.
08 test ideatets.JPGLiterature review on learning dSilvaGraf83
08 test idea/tets.JPG
Literature review on learning disability
Nature of problem
There are certain cases in children learning problems that instigate effective control over the barrier of learning new things. This study is based on the arithmetic interventions that are related to the linguistic approach. It individualizes the aspects of quick adaptability and the learning process with the orientation of the task (Allison, 2021).
The group of kindergarten students with collected data provided a comprehensive picture of the many facets of learning impairments. The term "problem solving" has been connected with the process of learning new things and comprehending facts, which has resulted in the exposure of several interventions (Christy, 2021). It demonstrates skill in resolving mathematical issues using language interventions and useful directions.
Subjects
The assessment of learning disability is judged with the kindergarten to the sixth-grade students. That included both male and female students randomly assessed. The sessions for problem-solving were of 50 minutes with a total of 34 sessions. The meta-analysis has instilled the perspective of dynamic range in subject choice with more task-oriented mathematic problems structured to resolve learning disabilities (Jennifer, 2021). There were 29 group-design studies and 10 single-subject studies (39 totals) were considered in the assessment of the meta-analysis
Procedures
Elements of educational possibilities exemplify the authoritative touch in a framed format. Continuous research to achieve a learning goal has resulted in psychological success and an authoritative demeanour. Students are likely to be relevant for the competent structure of the learning process. The usual models are instructed for people to learn with mathematical problems and stringent regulations. Every organization looks to the example of rules to understand the power of continual review. Recognizing the sluggish update in self-learning, the whole projects are framed by quality validation of the work and other refreshing information. This article contains the capabilities that are ready to survey students with innovation-related sections of development and typical cycles for moral assessment of program.
The author has conducted a diverse selection of meta-analyses of optimized studies on WPS. It assists the students with math disabilities or learning disabilities. The subjects were identified as MD if their scores fell below the 25th percentile on an average math test. The representation techniques are determined from the computer-assisted design of the test with significant changes in meta-analysis (Jennifer, 2021). The effectiveness of the procedure has instructional components with skill modeling and group identification. It promotes the various instructional barriers for students using WPS. The study manipulations were assorted that there were no significant differences in learning disability of stude ...
This literature review summarizes a meta-analysis of 18 studies on word-problem-solving interventions for elementary students with learning disabilities. The meta-analysis found that overall, word-problem-solving interventions had a significant positive effect (effect size of 1.08) on students' word-problem-solving accuracy. Interventions that included peer interaction and explicit transfer instruction yielded particularly large effects. Intensive interventions delivered in 50-minute sessions over 34 total sessions for students in Grade 3, regardless of instructional setting, produced the largest effect sizes. The findings support developing and implementing high-quality, evidence-based Tier 1 instruction in classrooms before providing more intensive individualized interventions.
A Comparison Of Two Instructional Approaches On Mathematical Word Problem Sol...Nat Rice
This study compared the effects of two instructional strategies for teaching mathematical word problems to middle school students with learning disabilities: 1) an explicit schema-based problem solving strategy (SBI) and 2) a traditional general heuristic instructional strategy (TI). Twenty-two students were randomly assigned to receive instruction using one of the two strategies. Results indicated that students who received the SBI outperformed those who received the TI on post-tests, maintenance tests 1-2 weeks later, and follow-up tests 3 weeks to 3 months later. The SBI group also did better in solving transfer problems. Additionally, the SBI group's performance exceeded that of typically achieving 6th grade peers. The study aimed to extend research on schema
REALITY – BASED INSTRUCTION AND SOLVING WORD PROBLEMS INVOLVING SUBTRACTIONWayneRavi
This study was conducted to determine the effect of reality based on the solving word problems involving subtraction. Descriptive-Comparative research design using paired sample T-test was used to utilized in the study. The study was carried out in Tibungol Elementary School to Fifty student of Grade Three section 1. Results revealed that there was a significant difference on the pretest and post test scores of pupils in reality based approach. Further, the reality based approach is effective in improving the performance of student.
An Arithmetic Verbal Problem Solving Model For Learning Disabled StudentsSandra Long
This study examined an instructional approach for teaching arithmetic word problem solving to 60 fifth and sixth grade students with learning disabilities. The experimental group received direct instruction twice a week for six weeks using a systematic problem solving model involving scripted steps and prompt cards. This group significantly outperformed the control group on a post-test measuring different problem types. The findings suggest that learning disabled students can improve their problem solving skills through explicit instruction in strategies and the use of calculators.
Assessing Algebraic Solving Ability A Theoretical FrameworkAmy Cernava
This document presents a theoretical framework for assessing algebraic solving ability. It discusses three phases of algebraic processes: 1) investigating patterns in numerical data, 2) representing patterns in tables and equations, and 3) interpreting and applying equations to new situations. It also reviews different approaches for developing algebraic solving ability in students, such as game-based learning, building conceptual understanding of variables/functions, and representing problems in multiple ways. The theoretical framework is based on these algebraic processes and how they align with levels of the SOLO model for assessing student understanding.
An Exploration Of Mathematical Problem-Solving ProcessesBrandi Gonzales
The study explored the problem-solving processes used by 10th grade students in solving mathematical problems. It found that mathematics achievement, representing students' conceptual knowledge, accounted for 50% of the variance in problem-solving success. The use of heuristic strategies accounted for an additional 13% of the variance. The study identified strategies that were specific to certain problems versus more general strategies. It also identified two clusters of students - one based on their use of different heuristic strategies, and another based on their use of trial-and-error and equations. Overall, the study indicated that students who used a wide range of strategies and techniques were better able to solve problems.
Investigating and remediating gender difference in mathematics performance am...Alexander Decker
This study investigated gender differences in mathematics performance among dyslexic and dyscalculic learners in junior secondary schools in Sokoto State, Nigeria. A sample of 827 students, including 423 males and 404 females, was given pre- and post-tests to measure mathematics achievement before and after a 12-week remedial instruction program. The results showed that the experimental groups who received the remedial instruction improved their mean test scores by 54.49%, while the control groups who did not receive the instruction only improved by 0.3%. The findings indicated that the remedial program was effective for dyslexic and dyscalculic students and that there was no significant gender difference in academic performance on the pre
Alternative Representations In Algebraic Problem Solving- When ArTye Rausch
- The document discusses a study that tested whether symbolic or graphical representations provide different advantages for solving algebraic problems, depending on the type of problem.
- The study presented college students with problems involving either computation of a single function or interpretation that required comparing multiple functions, using either symbolic equations or graphical representations.
- Graphical representation was found to facilitate problem solving overall. Problems requiring comparison of slopes were more difficult when presented symbolically rather than graphically.
An Investigation Of The Look-Ask-Pick Mnemonic To Improve Fraction SkillsSarah Morrow
The document describes a study that evaluated the effects of the Look-Ask-Pick (LAP) mnemonic on the fraction addition and subtraction skills of 3 sixth-grade students in general education. The LAP mnemonic teaches students to categorize fraction problems into 3 types and solve them accordingly. The study used a multiple baseline design to teach students the LAP strategy and measure its impact on their percentage of fraction problems solved correctly and digits correct per minute. Results showed increases in both measures for all students, and gains were maintained after 3 weeks. The study extends previous research on the LAP strategy by targeting general education students and including a fluency measure.
KEYWORD APPROACH AND SOLVING WORD PROBLEMS INVOLVING ADDITION OF WHOLE NUMBER WayneRavi
This study was conducted to determine the effect of keywords approach on solving word problems involving addition of whole numbers with sums up to 99,999 including money following the steps in problem solving. Results revealed that the pre test score was low; the post test score was high. There was a significant difference on the pre test and post test scores of keywords approach. Further, the keywords approach has large effect on the solving word problems involving addition of whole number.
Modelling the relationship between mathematical reasoning ability and mathema...Alexander Decker
This document discusses a study that examined the relationship between mathematical reasoning ability and attainment in mathematics.
The study involved 240 students who completed tests of mathematical reasoning ability and attainment. Structural regression modeling showed that four measures of mathematical reasoning ability (class, variable, order, and classification) predicted success on a test of mathematics attainment.
The findings suggest that developing students' mathematical reasoning ability could help improve their attainment in mathematics. Teachers should implement intervention programs to strengthen students' ability to classify, recognize variables, identify orders, and make classifications, which may ultimately boost mathematics learning.
READING COMPREHENSION AND PROBLEM SOLVING SKILLS OF GRADE SEVENSTUDENTS: A MI...AJHSSR Journal
ABSTRACT: The purpose of this study was to determine the relationship between the extent of students‟
reading comprehension and problem solving skills and identify teaching strategies that would address the
problem in teaching problem solving in Mathematics. The research utilized mixed explanatory design. The
subject consists of 189 grade 7 students who were part of the general section enrolled at Davao City National
High School. Purposive sampling was used in identifying the respondents taking the reading comprehension test
and problem solving test while random sampling was used in identifying participants for the key informant
interview. The result of the study revealed that students reading comprehension and problem solving skills were
at developing level. Moreover, reading comprehension skill was a predictor of problem solving skill. This
means that students‟ problem solving skill is dependent on their reading skills. Results also showed from the
conducted focus group discussion that students gave importance to vocabulary and main idea in learning
problem solving. Furthermore, using differentiated instruction was the identified best teaching strategy to
understand problem solving.
1) The purpose of this study was to examine the relationship between visual static models and students' written solutions to fraction problems using a large sample of student work.
2) The results indicate that common student errors relate to how students interpret the given model or their own model of the situation. Students' flexibility with visual models is related to successful written solutions.
3) Researchers hypothesize that exposure to varied mathematical representations influences students' ability to flexibly use static visual representations. Students need a solid understanding of real-world situations to successfully create and interpret visual models.
This document summarizes a study that explored the metacognitive processes of 33 prospective mathematics teachers when solving a non-routine math problem individually. The study found that very few students were able to successfully monitor their cognitive processes during problem solving. Specifically, most students had difficulty monitoring their comprehension of the problem, representing the problem conditions correctly, and detecting and correcting errors. The majority of students misunderstood the problem and used an incorrect strategy of calculating the mean value, resulting in incorrect answers.
A Review Of The Literature Fraction Instruction For Struggling Learners In M...Kaela Johnson
This document reviews literature on fraction instruction for struggling learners in mathematics. It finds that three interventions shown to be effective for improving math outcomes for struggling learners - graduated sequence, strategy instruction, and direct instruction - were also effective for teaching fractions. Explicit instruction was also identified as necessary for improving student performance with fractions. However, the review highlighted the lack of research on effective instructional practices specifically for fractions with struggling learners.
1) The document discusses Cognitively Guided Instruction (CGI), which uses students' intuitive understanding of math concepts as the basis for instruction. Research shows students intuitively understand the structures of multiplication, division, and word problems and can solve them using strategies like repeated addition or subtraction.
2) The three basic problem types for multiplication and division are identified based on which quantities (total objects, number of groups, objects in each group) are known/unknown. Students' strategies depend on discerning these structures.
3) A clinical interview with a student showed he relied on known math facts to solve most problems quickly. For problems requiring modeling, his reasoning was sometimes flawed. Developing multiple strategies could help him better
The Effect of Problem-Solving Instructional Strategies on Students’ Learning ...iosrjce
This study investigated the use of problems-solving and its effect on student achievement in the mole
concept. Ninety six (96) senior secondary II students were randomly selected form Demonstration Secondary
School, College of Education Azare. The instrument for data collection was 30-item chemistry achievement test
(CAT). The instrument was validated and its reliability determined to be 0.81. Two research questions and two
hypotheses guided the study. The data collected were analyzed using mean and standard deviation to answer the
research questions, while t-test statistics was used to answer the hypotheses at 0.05 level of significance. The
results revealed that student taught using problem-solving performed significantly better than those taught
through lecture method. From the findings chemistry teachers are encouraged to attend seminars/workshops on
problem -solving in order to facilitate the teaching and learning of chemistry in schools.
An ICT Environment To Assess And Support Students Mathematical Problem-Solvi...Vicki Cristol
1) The study used an ICT environment to assess and support 24 fourth-grade students' problem-solving performance on non-routine word problems.
2) The ICT environment allowed students to work in pairs, producing and experimenting with solutions while their dialogues and actions were analyzed.
3) The study aimed to better understand students' problem-solving processes and how the ICT environment could improve their problem-solving skills, though pre- and post-tests did not conclusively show its impact on performance.
Assessing Algebraic Solving Ability Of Form Four StudentsSteven Wallach
This study aimed to assess Form Four students' algebraic solving ability using linear equations across four content domains: linear patterns, direct variation, concepts of functions, and arithmetic sequences. Students took a pencil-and-paper test consisting of items assessing different levels of algebraic understanding based on the SOLO model. Clinical interviews were also conducted. Results showed that 62% of students had less than 50% probability of success at the relational level. Most students were at the unistructural or multistructural levels. High-ability students could identify patterns and relationships between variables more easily than low-ability students. The study provided evidence that the SOLO model is useful for assessing algebraic solving ability in upper secondary school.
Assessing The Internal Dynamics Of Mathematical Problem Solving In Small GroupsHannah Baker
This study developed an assessment system to examine the problem-solving behaviors and perceptions of students working in small groups on mathematical problems. To assess behaviors, a Group Problem Solving Assessment Instrument was created that categorized student actions as either "talking about the problem" (related to metacognitive processes) or "doing the problem" (related to cognitive processes). It also included categories for watching/listening and being off-task. Videotapes of student groups were analyzed using this instrument to understand the cognitive and metacognitive processes used. To assess perceptions, stimulated-recall interviews were conducted with individual students to understand their views of themselves and their group members. The assessment system aimed to provide insight into the internal dynamics that influence problem
An Inquiry Into The Spontaneous Transfer Of Problem-Solving SkillKaren Benoit
This document summarizes a study that explored the minimum conditions necessary to facilitate the spontaneous transfer of problem-solving skills to unfamiliar contexts. The study presented subjects with logically identical selection tasks that differed in familiarity. In pretests, only 10.5% could solve unfamiliar tasks, versus 57.3% for familiar tasks. The study assessed the effects of prior exposure to familiar scenarios, repeat opportunities, and feedback on later performance with unfamiliar scenarios. Overall, the interventions were not effective, indicating the threshold for spontaneous transfer may be higher than the interventions provided.
This research study module published by NCETM was developed by Anne Watson based on the paper Growth Points in Understanding of Function published in Mathematics Education Research Journal.
Learning Intervention and Student Performance in Solving Word ProblemWayneRavi
This study was conducted to determine the relationship of learning intervention on the performance level in solving word problem involving addition of whole numbers including money with sums through millions and billions without and with regrouping following the steps in problem solving. Specifically, it sought to find out if learning intervention significantly contribute in the performance of student in solving word problem. Descriptive-Correlation was used to utilized in the study.
Buy-Custom-Essays-Online.Com Review Revieweal - Top Writing ServicesSara Alvarez
The document provides instructions for requesting writing assistance from HelpWriting.net. It outlines a 5-step process: 1) Create an account with an email and password. 2) Complete a 10-minute order form providing instructions, sources, and deadline. 3) Review bids from writers and select one based on qualifications. 4) Review the completed paper and authorize payment if satisfied. 5) Request revisions until fully satisfied, and the company guarantees original, high-quality work or a full refund.
Research Paper Executive Summary Q. How Do I WrSara Alvarez
1. The play Cyrano de Bergerac by Edmond Rostand tells the story of Cyrano, who has a large nose and is unable to reveal his love for Roxane. He instead helps the handsome but dim Christian woo her.
2. Both Christian and Cyrano sacrifice their own needs and desires to help Roxane find love. Christian dies in battle, realizing Roxane truly loves Cyrano.
3. Cyrano also hides his feelings and eventually dies, teaching that true love requires sacrificing oneself for another's happiness.
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Assessing Algebraic Solving Ability A Theoretical FrameworkAmy Cernava
This document presents a theoretical framework for assessing algebraic solving ability. It discusses three phases of algebraic processes: 1) investigating patterns in numerical data, 2) representing patterns in tables and equations, and 3) interpreting and applying equations to new situations. It also reviews different approaches for developing algebraic solving ability in students, such as game-based learning, building conceptual understanding of variables/functions, and representing problems in multiple ways. The theoretical framework is based on these algebraic processes and how they align with levels of the SOLO model for assessing student understanding.
An Exploration Of Mathematical Problem-Solving ProcessesBrandi Gonzales
The study explored the problem-solving processes used by 10th grade students in solving mathematical problems. It found that mathematics achievement, representing students' conceptual knowledge, accounted for 50% of the variance in problem-solving success. The use of heuristic strategies accounted for an additional 13% of the variance. The study identified strategies that were specific to certain problems versus more general strategies. It also identified two clusters of students - one based on their use of different heuristic strategies, and another based on their use of trial-and-error and equations. Overall, the study indicated that students who used a wide range of strategies and techniques were better able to solve problems.
Investigating and remediating gender difference in mathematics performance am...Alexander Decker
This study investigated gender differences in mathematics performance among dyslexic and dyscalculic learners in junior secondary schools in Sokoto State, Nigeria. A sample of 827 students, including 423 males and 404 females, was given pre- and post-tests to measure mathematics achievement before and after a 12-week remedial instruction program. The results showed that the experimental groups who received the remedial instruction improved their mean test scores by 54.49%, while the control groups who did not receive the instruction only improved by 0.3%. The findings indicated that the remedial program was effective for dyslexic and dyscalculic students and that there was no significant gender difference in academic performance on the pre
Alternative Representations In Algebraic Problem Solving- When ArTye Rausch
- The document discusses a study that tested whether symbolic or graphical representations provide different advantages for solving algebraic problems, depending on the type of problem.
- The study presented college students with problems involving either computation of a single function or interpretation that required comparing multiple functions, using either symbolic equations or graphical representations.
- Graphical representation was found to facilitate problem solving overall. Problems requiring comparison of slopes were more difficult when presented symbolically rather than graphically.
An Investigation Of The Look-Ask-Pick Mnemonic To Improve Fraction SkillsSarah Morrow
The document describes a study that evaluated the effects of the Look-Ask-Pick (LAP) mnemonic on the fraction addition and subtraction skills of 3 sixth-grade students in general education. The LAP mnemonic teaches students to categorize fraction problems into 3 types and solve them accordingly. The study used a multiple baseline design to teach students the LAP strategy and measure its impact on their percentage of fraction problems solved correctly and digits correct per minute. Results showed increases in both measures for all students, and gains were maintained after 3 weeks. The study extends previous research on the LAP strategy by targeting general education students and including a fluency measure.
KEYWORD APPROACH AND SOLVING WORD PROBLEMS INVOLVING ADDITION OF WHOLE NUMBER WayneRavi
This study was conducted to determine the effect of keywords approach on solving word problems involving addition of whole numbers with sums up to 99,999 including money following the steps in problem solving. Results revealed that the pre test score was low; the post test score was high. There was a significant difference on the pre test and post test scores of keywords approach. Further, the keywords approach has large effect on the solving word problems involving addition of whole number.
Modelling the relationship between mathematical reasoning ability and mathema...Alexander Decker
This document discusses a study that examined the relationship between mathematical reasoning ability and attainment in mathematics.
The study involved 240 students who completed tests of mathematical reasoning ability and attainment. Structural regression modeling showed that four measures of mathematical reasoning ability (class, variable, order, and classification) predicted success on a test of mathematics attainment.
The findings suggest that developing students' mathematical reasoning ability could help improve their attainment in mathematics. Teachers should implement intervention programs to strengthen students' ability to classify, recognize variables, identify orders, and make classifications, which may ultimately boost mathematics learning.
READING COMPREHENSION AND PROBLEM SOLVING SKILLS OF GRADE SEVENSTUDENTS: A MI...AJHSSR Journal
ABSTRACT: The purpose of this study was to determine the relationship between the extent of students‟
reading comprehension and problem solving skills and identify teaching strategies that would address the
problem in teaching problem solving in Mathematics. The research utilized mixed explanatory design. The
subject consists of 189 grade 7 students who were part of the general section enrolled at Davao City National
High School. Purposive sampling was used in identifying the respondents taking the reading comprehension test
and problem solving test while random sampling was used in identifying participants for the key informant
interview. The result of the study revealed that students reading comprehension and problem solving skills were
at developing level. Moreover, reading comprehension skill was a predictor of problem solving skill. This
means that students‟ problem solving skill is dependent on their reading skills. Results also showed from the
conducted focus group discussion that students gave importance to vocabulary and main idea in learning
problem solving. Furthermore, using differentiated instruction was the identified best teaching strategy to
understand problem solving.
1) The purpose of this study was to examine the relationship between visual static models and students' written solutions to fraction problems using a large sample of student work.
2) The results indicate that common student errors relate to how students interpret the given model or their own model of the situation. Students' flexibility with visual models is related to successful written solutions.
3) Researchers hypothesize that exposure to varied mathematical representations influences students' ability to flexibly use static visual representations. Students need a solid understanding of real-world situations to successfully create and interpret visual models.
This document summarizes a study that explored the metacognitive processes of 33 prospective mathematics teachers when solving a non-routine math problem individually. The study found that very few students were able to successfully monitor their cognitive processes during problem solving. Specifically, most students had difficulty monitoring their comprehension of the problem, representing the problem conditions correctly, and detecting and correcting errors. The majority of students misunderstood the problem and used an incorrect strategy of calculating the mean value, resulting in incorrect answers.
A Review Of The Literature Fraction Instruction For Struggling Learners In M...Kaela Johnson
This document reviews literature on fraction instruction for struggling learners in mathematics. It finds that three interventions shown to be effective for improving math outcomes for struggling learners - graduated sequence, strategy instruction, and direct instruction - were also effective for teaching fractions. Explicit instruction was also identified as necessary for improving student performance with fractions. However, the review highlighted the lack of research on effective instructional practices specifically for fractions with struggling learners.
1) The document discusses Cognitively Guided Instruction (CGI), which uses students' intuitive understanding of math concepts as the basis for instruction. Research shows students intuitively understand the structures of multiplication, division, and word problems and can solve them using strategies like repeated addition or subtraction.
2) The three basic problem types for multiplication and division are identified based on which quantities (total objects, number of groups, objects in each group) are known/unknown. Students' strategies depend on discerning these structures.
3) A clinical interview with a student showed he relied on known math facts to solve most problems quickly. For problems requiring modeling, his reasoning was sometimes flawed. Developing multiple strategies could help him better
The Effect of Problem-Solving Instructional Strategies on Students’ Learning ...iosrjce
This study investigated the use of problems-solving and its effect on student achievement in the mole
concept. Ninety six (96) senior secondary II students were randomly selected form Demonstration Secondary
School, College of Education Azare. The instrument for data collection was 30-item chemistry achievement test
(CAT). The instrument was validated and its reliability determined to be 0.81. Two research questions and two
hypotheses guided the study. The data collected were analyzed using mean and standard deviation to answer the
research questions, while t-test statistics was used to answer the hypotheses at 0.05 level of significance. The
results revealed that student taught using problem-solving performed significantly better than those taught
through lecture method. From the findings chemistry teachers are encouraged to attend seminars/workshops on
problem -solving in order to facilitate the teaching and learning of chemistry in schools.
An ICT Environment To Assess And Support Students Mathematical Problem-Solvi...Vicki Cristol
1) The study used an ICT environment to assess and support 24 fourth-grade students' problem-solving performance on non-routine word problems.
2) The ICT environment allowed students to work in pairs, producing and experimenting with solutions while their dialogues and actions were analyzed.
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Analysis Of Precurrent Skills In Solving Mathematics Story Problems
1. 21
JOURNAL OF APPLIED BEHAVIOR ANALYSIS 2003, 36, 21–33 NUMBER 1 (SPRING 2003)
ANALYSIS OF PRECURRENT SKILLS IN
SOLVING MATHEMATICS STORY PROBLEMS
NANCY A. NEEF
THE OHIO STATE UNIVERSITY
DIANE E. NELLES
OAKLAND SCHOOL DISTRICT, OAKLAND, MI
BRIAN A. IWATA
THE UNIVERSITY OF FLORIDA
AND
TERRY J. PAGE
BANCROFT, INC., HADDONFIELD, NJ
We conducted an analysis of precurrent skills (responses that increase the effectiveness of
a subsequent or ‘‘current’’ behavior in obtaining a reinforcer) to facilitate the solution of
arithmetic word (story) problems. Two students with developmental disabilities were
taught four precurrent responses (identifying the initial value, change value, operation,
and resulting value) in a sequential manner. Results of a multiple baseline design across
behaviors showed that the teaching procedures were effective in increasing correct per-
formance of each of the precurrent behaviors with untaught problems during probes and
that once the precurrent behaviors were established, the number of correct problem
solutions increased.
DESCRIPTORS: precurrent behavior, problem solving, mathematics, story prob-
lems, developmental disabilities
The principal goal of mathematics in-
struction is to teach students to solve prac-
tical problems (National Council of Teachers
of Mathematics, 1989, 2000). Toward that
end, teachers often present tasks to students
in the form of word (story) problems to il-
lustrate a wide range of everyday activities
(e.g., cooking, shopping, budgeting, time
management) that require competence in
basic mathematics skills. Nevertheless, large
numbers of students of all ages fail to dem-
onstrate grade-level proficiency in solving
Address correspondence and requests for reprints to
Nancy A. Neef, PAES, College of Education, The
Ohio State University, 1945 N. High St., Columbus,
Ohio 43210 (e-mail: neef.2@osu.edu) or to Brian A.
Iwata, Department of Psychology, University of Florida,
Gainesville, Florida 32611 (e-mail: iwata@ufl.edu).
story problems (Cawley, Parmar, Foley,
Salmon, & Roy, 2001; National Assessment
of Educational Progress, 1992). Students
with disabilities have performed particularly
poorly on these tasks, which are important
to achieving goals of successful employment,
independent living, and successful integra-
tion into school and community settings
(Butler, Miller, Lee, & Pierce, 2001).
Solving story problems is often difficult
because it requires both reading comprehen-
sion and mathematics skills as well as the
ability to transform words and numbers into
the appropriate operations. Research sug-
gests that difficulties in discriminating the
correct operation, the order of operation
(when placement of the unknown within the
problem differs), and extraneous informa-
2. 22 NANCY A. NEEF et al.
tion, as well as problems with computational
speed, are factors commonly associated with
poor performance in solving story problems
(Zentall & Ferkis, 1993). Most behavioral
research on mathematics instruction for stu-
dents with disabilities has emphasized basic
foundation skills (e.g., counting, number
recognition, matching number words and
numerals, concepts of quantity) and math-
ematics facts (accuracy and fluency in com-
putation) that are necessary but insufficient
for higher level problem solving (e.g., Harp-
er, Mallette, Maheady, Bentley, & Moore,
1995; Horton, Lovitt, & White, 1992; Lin,
Podell, & Tournaki-Reid, 1994; Mattingly
& Bott, 1990; Miller, Hall, & Heward,
1995; Whalen, Schuster, & Hemmeter,
1996; Wood, Frank, & Wacker, 1998;
Young, Baker, & Martin, 1990). By contrast,
relatively few experimental investigations
have examined procedures for teaching sto-
ry-problem solving, especially to students
with moderate to severe disabilities.
A common instructional strategy involves
the use of manipulatives, in which a student
uses objects (e.g., blocks) to represent parts
of the problem-solving equation and then
adopts a ‘‘counting all’’ strategy for addition
problems and a ‘‘separating from’’ or
‘‘matching’’ strategy for subtraction prob-
lems, depending on whether the operation
involves a change, comparison, or combi-
nation. Although manipulatives have been
shown to enhance performance on story-
problem tasks (e.g., Marsh & Cooke, 1996),
they are limited aids because they do not
ensure discrimination of the appropriate
strategy for solving a particular type of prob-
lem, and are not readily applied to problems
involving either large quantities or abstract
concepts such as distance or time.
Several investigators have used cue cards
or checklists outlining problem-solving steps
to teach students with mild disabilities to
solve addition and subtraction story prob-
lems. For example, Cassel and Reid (1996)
taught students to use a cue card with the
mnemonics FAST DRAW (corresponding to
the steps of finding and highlighting the
question, asking what the parts of the prob-
lem are and what information is given, set-
ting up the equation, tying down the sign of
using addition or subtraction, discovering
the sign, reading the number problem, an-
swering the number problem, and writing
the answer), which were then translated into
a checklist, to solve story problems. Results
showed that students’ ability to generate cor-
rect equations for story problems improved
following training. The effects of the pro-
cedure on correct solutions, however, were
not clearly demonstrated, and prompting
use of a strategy does not ensure discrimi-
nation of the relevant concepts and opera-
tions necessary to solve a problem.
Another approach involves teaching stu-
dents to translate problems into equations
whose components are inserted into parts of
a diagram. Diagrams have been used to sup-
plement explicit teaching of steps of a prob-
lem-solving chain in the context of comput-
er programs (Jaspers & van Lieshout, 1994)
or direct instruction (e.g., Jitendra & Hoff,
1996). Jaspers and van Lieshout, for exam-
ple, compared text analysis, external model-
ing, a combination of the two procedures,
and a control (practice) procedure on the
ability of students classified as educable
mentally retarded to solve story problems.
Text analysis involved a four-step procedure
consisting of reading the problem, determin-
ing the question being asked, locating the
relevant sets, and producing the numerical
answer. External modeling involved a five-
step procedure consisting of reading the
problem, representing the first set with
squares, locating the second set on the com-
puter screen, representing the second set
with squares, and locating the answer set on
the screen. Results showed that students in
the external-modeling group performed bet-
ter than those in the other three groups
3. 23
PRECURRENT PROBLEM SOLVING
when they were allowed to use diagrams to
represent the word problems, but that the
text-analysis group performed better on pa-
per-and-pencil tests. Using a similar ap-
proach, Jitendra and Hoff taught students
via direct instruction to use diagrams to
transform story problems into equation
form. This procedure was effective in teach-
ing 3 students with learning disabilities to
solve simple addition and subtraction story
problems when they were provided with a
cue card stating the steps to solve the prob-
lems.
From a behavior-analytic perspective,
problem solving involves a precurrent oper-
ant contingency, which functions ‘‘mainly to
make subsequent behavior more effective,’’
whereas a current operant may be character-
ized as the solution (‘‘effective behavior . . .
a response which is likely to be reinforced’’)
that is made more probable by the precur-
rent contingencies (Skinner, 1968, pp. 120,
124). Precurrent behaviors may need to be
taught, but once they are established, they
can be maintained by their ultimate effect
on the current behavior that directly pro-
duces the reinforcer (Skinner, 1968). Parsons
(1976), for example, examined the effects of
precurrent behaviors on the solution of
quantity-matching problems (circling a sub-
set of comparison stimuli equal to the num-
ber of symbols comprising the sample) by 5
preschool children. The precurrent behaviors
consisted of overtly counting and marking
the symbols as they were vocally enumerat-
ed. Results showed that before precurrent
behaviors were taught and when they were
prohibited, continuous differential reinforce-
ment of the current behavior (terminal so-
lutions) failed to produce accurate problem
solving. Once established and when allowed
to occur, precurrent behaviors resulted in ac-
curate current behavior (problem solution),
and both precurrent and current behaviors
were maintained when reinforcement was
contingent only on the latter.
Analysis of precurrent behaviors has im-
plications for students who have difficulty
with story problems. First, as illustrated by
Parsons (1976), effective precurrent behav-
iors may not develop when instruction fo-
cuses only on the completion of a complex
sequence of responses. As Skinner (1968)
pointed out, differential reinforcement of so-
lutions alone without access to the variables
that control them ‘‘does not teach; it simply
selects those who learn without being
taught’’ (pp. 118–119). Second, instruction
on aspects of a problem might occasion be-
haviors that are irrelevant to, or that fail to
control, problem solution. Thus, cognitive
strategies such as those that prompt ‘‘self-
reflection’’ in the form of ‘‘What did you
notice about how you did your work?’’ (Na-
glieri & Gottling, 1995) or instructions to
‘‘read, paraphrase, visualize, hypothesize, es-
timate, compute, and check’’ (Montague,
1992) may not be effective with students
who lack precurrent skills that relate directly
to problem solution. Third, mastery of com-
ponent precurrent behaviors, albeit neces-
sary, is not always sufficient for problem
solving (Mayfield & Chase, 2002). In ad-
dition to learning how to apply a mathe-
matical operation such as addition and sub-
traction, for example, students must also dis-
criminate when to apply it to different prob-
lems. Finally, precurrent responses that are
covert may be too weak to occasion the so-
lution behavior (Skinner, 1957) and, because
they are unobservable, are not available for
correction if faulty. It is not surprising,
therefore, that teacher knowledge of individ-
ual problem-solving skills was found to be a
strong predictor of math achievement, and
that teachers who lacked such information
were more likely to rely simply on checking
and monitoring student work (Peterson,
Carpenter, & Fennema, 1989).
In the present study, we examined the ef-
fects of teaching overt precurrent behaviors
(identifying the initial value, change value,
4. 24 NANCY A. NEEF et al.
operation, and ending value) on the current
operant of solving mathematics story prob-
lems with 2 students with developmental
disabilities. The precurrent behaviors were
designed to facilitate problem solving when
the components were arranged in varying se-
quences within the story problems, when the
operation consisted of either addition or
subtraction, and when the unknown value
was in different positions in the resulting
equations.
METHOD
Participants and Setting
Two individuals enrolled in an education-
al program for students with developmental
disabilities participated. Saul and Lucien
were 19 and 23 years old and had IQs of
46 and 72, respectively, as measured by the
Wechsler Adult Intelligence Scale. Prerequi-
site skills for inclusion in the study consisted
of the ability (a) to compute addition and
subtraction numeric equations in which the
sums were 10 or less and with the unknown
in any position (e.g., 2 1 6 5 ?, ? 2 4 5
3), and (b) to read the vocabulary words
used in the word problems (second-grade
level). These skills were assessed informally
(math achievement scores were unavailable).
All training and testing sessions were con-
ducted individually in a classroom.
Stimulus Materials
The general equations A 1 B 5 C or A 2
B 5 C were used to generate word problems
indicating whether an individual gave away
or received objects. One of the variables (A,
B, or C) was unknown, and the task was to
solve for the unknown. All problems repre-
sented variations of six specific equations
generated by changing the operation to be
performed and the location of the unknown
variable. Each of the equations also was pre-
sented in two different sequences, yielding a
total of 12 problem sequences. The rationale
for using these different sequences was based
on the assumption that generative skill ac-
quisition would be enhanced through ex-
posure to a variety of problem formats. A
list of the equations and word sequences is
presented in Table 1.
Testing and training materials consisted of
a series of worksheets, each containing five
different word problems drawn from the 12
basic sequences. The problems involved ad-
dition and subtraction in which the sums
were 10 or less. Words used in the problems
were drawn from 10 proper names, 20 verbs,
and 20 object nouns. These problem se-
quences, numeric values, and words were
combined in different ways to formulate a
sufficiently large pool of problems such that
neither student was exposed to the same
problem more than once during the study.
An example of a typical worksheet is shown
in Figure 1 (formulas are shown for illustra-
tive purposes and were not included on the
worksheet).
Each problem was divided into five com-
ponent parts. The initial set (1) was deter-
mined by words indicating the number of
objects in the person’s possession at the out-
set. Verbs stating which objects were either
added or subtracted specified the change set
(2), and these same verbs indicated the op-
eration (3) to be performed. A phrase de-
noting the final number of objects in the
person’s possession was the resulting set (4).
One set (initial, change, or resulting) was
unknown and provided the question of the
problem. The answer to that question
formed the solution (5). Given the word
problem, ‘‘If Sam began with 7 pens and
ended up with 5 pens, how many did he
give away?’’ as an example, the various com-
ponents were as follows:
Initial set: Sam began with 7 pens.
Change set: How many did he give away?
Operation: Give away (subtraction).
Resulting set: Ended up with 5 pens.
5. 25
PRECURRENT PROBLEM SOLVING
Table 1
Equations and Sequences Used to Generate Word Problems
Equation Sample word sequence
A 1 B 5 ? 1. If (name) started out with A objects and was given B objects, how many did
he or she end up with? (A 1 B 5 ?)
2. How many objects did (name) end up with if he or she started out with A
objects and was given B objects? (? 5 A 1 B)
A 2 B 5 ? 3. If (name) started out with A objects and gave away B objects, how many did
he or she end up with? (A 2 B 5 ?)
4. How many objects did (name) end up with if he or she started out with A
objects and gave away B objects? (? 5 A 2 B)
A 1 ? 5 C 5. If (name) started out with A objects and ended up with C objects, how many
was he or she given? (A 1 ? 5 C)
6. How many objects was (name) given if he or she started out with A objects and
ended up with C objects? (C 5 A 1 ?)
A 2 ? 5 C 7. If (name) started out with A objects and ended up with C objects, how many
did he or she give away? (A 2 ? 5 C)
8. How many objects did (name) give away if he or she started out with A objects
and ended up with C objects? (C 5 A 2 ?)
? 1 B 5 C 9. If (name) was given B objects and ended up with C objects, how many objects
did he or she start out with? (? 1 B 5 C)
10. How many objects did (name) start out with if he or she was given B objects
and ended up with C objects? (C 5 ? 1 B)
? 2 B 5 C 11. If (name) gave away B objects and ended up with C objects, how many objects
did he or she start out with? (? 2 B 5 C)
12. How many objects did (name) start out with if he or she gave away B objects
and ended up with C objects? (C 5 ? 2 B)
Solution (unknown): How many did he
give away?
Experimental Design
A multiple baseline across behaviors de-
sign (Baer, Wolf, & Risley, 1968) was used
to evaluate the effects of training on stu-
dents’ problem-solving skills. Following the
collection of baseline probe data on each of
the five component skills (initial set, change
set, operation, resulting set, solution), train-
ing was begun on the first component and
proceeded as necessary across the other four.
Procedure
Probe sessions. Probes were administered
during baseline and following the comple-
tion of training on each of the separate pre-
current problem-solving components. A
probe consisted of 10 word problems (two
worksheets). The student was requested to
‘‘Read each problem aloud. Find the answers
and put your work here [as the trainer point-
ed to the boxes and circles].’’ Correct re-
sponses were defined as having the correct
numbers in the appropriate boxes for known
quantities, an X above the appropriate box
for the unknown, the correct operation sym-
bol in the circle, and the correct solution in
the box for the unknown. Other responses
(or no responses) were considered incorrect.
Students were allowed 20 min to complete
each probe, during which intermittent praise
was delivered for on-task behavior indepen-
dent of the occurrence of precurrent or cur-
rent behaviors.
Training. Following the completion of
baseline probes, students were taught to
identify the five components of a word prob-
lem in the sequence described previously.
The prompts and definitions of correct and
incorrect responses for each of the five com-
ponents are described in Table 2. Training
on each component was preceded by a dem-
6. 26 NANCY A. NEEF et al.
Figure 1. Example of a typical worksheet. Formulas (in bold) are included only for illustrative purposes
and were not part of the worksheet.
onstration with five practice problems. The
trainer read each problem aloud, presented
the verbal prompt (e.g., ‘‘How many prob-
lems did [name] start out with?’’), and de-
scribed and modeled the correct response.
Training trials then began. Each trial con-
sisted of one complete word problem, and
10 trials (excluding remedial trials) were pre-
sented each session. One to two sessions
were conducted per day. Correct responses
were followed by praise. When an error oc-
curred on either the component being
trained or any previously trained compo-
nent, the trainer modeled the correct re-
sponse directly on the worksheet using a dif-
ferent-colored pen. A remedial trial using a
different problem of the same type was then
presented. This sequence was repeated for
7. 27
PRECURRENT PROBLEM SOLVING
Table 2
Prompts, Correct Responses, and Incorrect Responses for the Five Problem Components
Initial set
Prompt
Correct
Incorrect
‘‘How many objects did (name) start out with?’’
Appropriate words underlined; number in first box if known or X over box if unknown
Incorrect underline, number, or X; no response in 10 s
Change set
Prompt
Correct
Incorrect
‘‘What happened next?’’
Appropriate words underlined; number in second box if known or X over box if unknown
Incorrect underline, number, or X; no response in 10 s
Operation
Prompt
Correct
Incorrect
‘‘Was that number added or subtracted from the first number?’’
Finger placed under words indicating the operation; correct symbol in circle
Incorrect pointing or symbol; no response in 10 s
Resulting set
Prompt
Correct
Incorrect
‘‘How many objects did (name) end up with?’’
Appropriate words underlined; number in third box if known or X over box if unknown
Incorrect underline, number, or X; no response in 10 s
Solution
Prompt
Correct
Incorrect
A question pertaining to the unknown, as in ‘‘How many objects did (name)
(start out with, end up with, get, lose, etc.)?’’
Correct answer placed in box with the unknown (indicated by X)
Incorrect answer; no response in 10 s
subsequent errors on remedial trials. A cor-
rect response on a remedial trial produced
the next training trial.
During instruction on the initial, change,
and resulting sets, three training phases typ-
ically were conducted unless the participant
displayed consistent correct responding be-
fore the planned training phase was initiat-
ed. First, problems were presented in which
the component being trained was always a
known quantity, and responses were preced-
ed by a specific trainer prompt. Next, prob-
lems were presented in which the compo-
nent being trained was randomly unknown
(i.e., in approximately half of the problems,
that component constituted the solution).
Finally, problems were presented in which
the component being trained was randomly
unknown, and no prompt was provided.
During training of the operation, only two
phases were used—prompted and un-
prompted—because one of the components
always was unknown. The criteria for ad-
vancing from one training phase to the next
were 100% correct responses on all trained
components (including previously trained
components) for one session under prompt-
ed conditions and 100% correct responses
for two consecutive sessions under un-
prompted conditions. When a student met
the unprompted training criterion for a giv-
en component, a probe session was con-
ducted and training was begun on the next
component.
Interobserver Agreement
A second observer independently scored
students’ written responses for both training
and probe sessions at least once during base-
line and each training phase. Interobserver
agreement was calculated by dividing the
number of agreements on each component
part of a problem by the agreements plus
disagreements and multiplying by 100%.
Mean agreement scores were 99.9% and
99.5% for probe and training sessions, re-
spectively.
8. 28 NANCY A. NEEF et al.
Figure 2. The number of correct responses across training phases for each precurrent component.
RESULTS
Figure 2 shows the training data for each
student, expressed as the number of correct
responses during training phases for each
precurrent component. Both students re-
quired fewer sessions to reach criterion as
training progressed across components. Saul
reached mastery criterion in 26 sessions for
the initial set, in 35 sessions for the change
set, in 17 sessions for the operation, and in
two sessions for the resulting set. Lucien met
criterion in 28 sessions for the initial set, in
16 sessions for the change set, and in seven
sessions for the operation.
Figure 3 shows probe results for both stu-
dents across baseline and training conditions
9. 29
PRECURRENT PROBLEM SOLVING
Figure 3. The number of correct responses during mathematics probes across baseline and posttraining
conditions.
for each of the five problem-solving com-
ponents. Data are expressed as number of
correct responses. The results show (a) an
increase in the number of correct responses
on probes following instruction on the re-
spective precurrent (problem-solving) com-
ponent and (b) an increase in the number
of correct solutions (current operants) after
the precurrent responses for all components
had been established. Saul’s mean numbers
of correct precurrent responses (of 10 pos-
sible) during baseline and posttraining
10. 30 NANCY A. NEEF et al.
probes, respectively, were 4.5 and 9.6 for the
initial set, 2.2 and 9.5 for the change set,
2.0 and 9.0 for the operation, and 0.2 and
9.0 for the resulting set. His correct solu-
tions increased from a mean of 1.2 to 8.0
after the precurrent responses in all compo-
nents had been established. Lucien’s mean
numbers of correct precurrent responses dur-
ing baseline and posttraining probes, respec-
tively, were 5.8 and 9.3 for the initial set,
3.9 and 9.5 for the change set, and 5.8 and
10.0 for the operation. After the precurrent
responses in those three components had
been established, generalization occurred to
the resulting set without further training;
correct responses increased from a mean of
1.4 to 10. His correct responses on the un-
trained solution set increased from a mean
of 3.1 to 10 after the component precurrent
behaviors had been established.
DISCUSSION
Results of this study showed that teaching
students with developmental disabilities a set
of precurrent behaviors consisting of the
identification of each of the component
parts of a story problem resulted in the gen-
erative solution of problems. As such, the
study represents one of the few experimental
investigations of a systematic procedure to
teach generative story problem-solving skills
to students with developmental disabilities,
addressing a principal goal of mathematics
instruction.
As discussed by Skinner (1953, 1966,
1968, 1969), problem solving involves an
interresponse relation in which the emission
of precurrent operants makes the solution
(current operant) more probable. In the
present case, the number of correct solutions
increased for both students after the succes-
sion of precurrent responses had been estab-
lished for each component. The precurrent
contingencies can be interpreted as a re-
sponse chain, but in situations in which it is
possible for the current behavior to occur
and be reinforced without the emission of
precurrents, they can nevertheless function
to alter the probability of reinforcement
(Polson & Parsons, 1994).
Although contingent praise was used dur-
ing training sessions to promote the acqui-
sition of the precurrent behaviors, it is pos-
sible that reinforcement derived from prob-
lem solution may have served to maintain
both the precurrent and current responses
during probes (when contingent praise was
not used). In other words, the precurrent be-
haviors may have been indirectly reinforced
by the solution they prompted. This is con-
sistent with the results of investigations by
Parsons (1976) and Parsons, Taylor, and
Joyce (1981), in which precurrent collateral
behavior was maintained when reinforce-
ment was made contingent on a subsequent
current operant. Alternatively, the precurrent
and current responses may have been main-
tained during the probe session because
praise was intermittently delivered for on-
task behavior (rather than for the precurrent
or current responses) and the participants
may not have discriminated the different
contingencies in effect between the training
and probe sessions.
The acquisition of effective precurrent be-
haviors may have been facilitated by the pro-
gressive training sequence that required
demonstration of the target as well as all pre-
viously trained components. Mayfield and
Chase (2002), for example, found that the
rate and accuracy of algebra problem solving
and novel application were higher for college
students who received instruction that in-
corporated cumulative practice on a mix of
previously trained component skills (rules)
than for students who received instruction
that incorporated simple practice on one
component at a time or one session of extra
practice. The authors argued that the jux-
taposition of different kinds of problems in
the cumulative review enabled learning of
11. 31
PRECURRENT PROBLEM SOLVING
multiple discriminations for rule application
necessary for problem solving with novel
rule combinations.
Nevertheless, it is possible that training in
the present study may have required less
time without the known prompted and un-
known prompted phases. The relatively large
number of sessions needed to reach criterion
during the random unprompted phase sug-
gests that performance may not have been
facilitated by the two preceding training
phases, but the design did not allow that
determination. Similarly, it may be that
teaching all components simultaneously
rather than sequentially would be more ef-
ficient.
Our results are limited in several addi-
tional respects. First, because of time con-
straints, we were unable to examine the ex-
tent to which the effects were maintained
over an extended period of time. Second, a
more complete analysis of the function of
the behaviors as precurrents would require
a demonstration that correct solutions de-
crease when the precurrent behaviors were
prevented, or that those behaviors were not
maintained when they were not followed by
a correct solution. Parsons (1976) and Par-
sons et al. (1981), for example, showed dec-
rements in terminal symbol counting and
matching-to-sample behaviors, respectively,
when collateral precurrent behaviors were
prohibited. It would be difficult, however,
to use such preparations in the context of
the present study. For example, unlike in
the Parsons et al. investigation, the precur-
rent behaviors could not be prevented be-
cause, once learned, they may become co-
vert (Skinner, 1976); indeed, that would be
a desirable outcome. In addition, to teach
behaviors that are ineffective in producing
the current operant seemed counterproduc-
tive to the purpose of our applied investi-
gation.
Our purpose was to extend the concept
of precurrent operants examined in basic re-
search (Parsons, 1976; Parsons et al., 1981;
Polson & Parsons, 1994; Torgrud & Hol-
born, 1989) to our own problem-solving ef-
forts in developing effective procedures for
teaching an educationally significant skill
that has proven to be difficult for students
with developmental disabilities. The proce-
dure was efficient in that the identification
of the basic elements and operations in-
volved in a problem, regardless of their po-
sition or sequence, enabled solution of all
types of problems within that class. Thus,
the procedure enabled students to respond
to problems that require a novel synthesis of
responses in the presence of a novel stimu-
lus, consistent with Becker, Engelmann, and
Thomas’ (1975) definition of problem solv-
ing.
Our results were limited to simple addi-
tion and subtraction story problems and did
not encompass all the categories and sub-
types of problems represented in various tax-
onomies (e.g., Riley & Greeno, 1988).
However, the procedures could easily be ap-
plied to other types of problems and oper-
ations. Efficiency might be improved by ap-
plying a general case strategy that systemat-
ically samples the range of stimulus and re-
sponse variations in the universe of possible
problems (Endo, 2001).
Because story problem-solving skills have
been infrequently addressed in behavioral re-
search, some have suggested that they are be-
yond the reach of a behavioral analysis. But-
ler et al. (2001), for example, recently char-
acterized the current problem-solving stan-
dards for mathematics curricula as
demonstrating ‘‘a shift from a behaviorist ap-
proach of teaching rote learning of facts and
procedures to a constructivist approach’’ (p.
20). Extensions of the current approach in
subsequent research may make more prom-
inent the potential contributions of behavior
analysis to the acquisition of complex prob-
lem-solving skills of all types.
12. 32 NANCY A. NEEF et al.
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Received September 12, 2001
Final acceptance November 4, 2002
Action Editor, Iser DeLeon
STUDY QUESTIONS
1. What are the differences between precurrent- and current-operant behaviors? Provide an
original example that illustrates the interaction of these behaviors.
2. Explain the importance of a precurrent analysis to the understanding and improvement of
problem-solving behavior.
3. Describe the five components into which word problems were divided.
4. How were correct responses defined?
5. What independent variables were included in the training procedures used to establish the
precurrent responses?
6. Summarize the results of the study with respect to acquisition of (a) precurrent and (b)
current responses.
7. How were the results of the current study consistent with Skinner’s conceptualization of
problem solving?
8. What useful information might be gained by including a condition in which participants
received reinforcement following correct solutions in the absence of any precurrent training?
Questions prepared by Stephen North and Jessica Thomason, The University of Florida
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