Computation and Word-Problem Errors Among Grade 4 and 5 Students With Mathematics Difficulty

Analyzing the word-problem performance and strategies of students experiencing mathematics difficulty

Word problems require students to read a language-based problem, identify necessary information to answer a prompt, and perform calculation(s) to develop a problem solution. Solving word problems proves particularly challenging for students with mathematics difficulties because skill in reading, interpretation of language, and mathematics are required for word-problem proficiency. We examined whether two versions of a word-problem intervention increased students’ understanding of three word-problem language features: naming a superordinate category, identifying irrelevant information, and providing a word-problem label. At pre- and posttest, 145 3rd-grade students solved word problems and answered questions about word-problem language. Students who participated in the word-problem interventions demonstrated improvement on identifying irrelevant information and providing word-problem labels over students in the business-as-usual condition. We did not identify group differences related to naming a superordinate category. These results suggest the importance of explicit teaching of language comprehension features within word-problem intervention. Subscribe to LDMJ

There is disagreement with regard to the transfer effects of learning through varied worked examples. Ross(1989)found that multiple examples should be made very similar to each other; even a small difference in the surface feature could make the learner pay more attention to solving the problem. Gentner (2003) discovered that comparing examples with the same structure could prompt the learners' transition to problem solving. Holyoak (1987) pointed out that deeper structural differences had an impact on the transfer. Based on these researches, the present study hypothesized that the variability of the surface feature of two examples could have some effect on the near transfer of pupils' word-problem solving, and the proper variability of the structural feature of the two examples could have an impact on the far transfer of pupils' word-problem solving. A total of 210 second-grade students from a primary school were selected using a pretest and were divided into six experimental groups and a control group. Each group learnt a word problem example. Then, the pupils of three experimental groups learned one kind of word problem example that varied from the first example with regard to the surface feature: the pupils in the first group learned a number and substance varied example; the pupils in the second group learned a story varied example; and those in the third group learned an expression varied example. The pupils in the other three experimental groups learned another kind of worked example that varied from the first problem with regard to the structural feature. The first group learned a rule varied example, the second group learned a rule repeated example; the third group, learned a rule of speed composed example. The pupils in the control group did not learn any worked example. Finally, all the pupils of the six experimental groups and the control group were given a test with 15 word problems. Three of them varied from the first example with regard to the surface feature, three of them varied from the first example with regard to the structural feature, and the others varied from the first example with regard to the surface and structural features. The experimenters recorded the pupils' test scores. The results were as follows. First, the pupils in the control group who learned the first word problem eyample could solve the word problems whose example varied with regard to the surface feature, but could not solve the other word problems. Second, the pupils in the three experimental groups who learned the word problem example that varied with regard to the surface feature could solve the word problems that varied from the first example with regard to the surface feature, but could not solve the other word problems. Third, a different effect was observed for the pupils in the three experimental groups who learned the word problem example that varied with regard to the structural feature. The pupils who learned the rule repeated example solved more word problems than the pupils in the other groups. The pupils who learned the rule varied example solved more word problems than those who learned the rule of speed composed example. The pupils who learned the rule of speed composed example solved the fewest number of word problems. The variability of the surface feature of the two examples could have some impact on the near transfer of pupils' word-problem solving. The rule repeated example had the best effect on the far transfer of the pupils' word-problem solving. The rule varied example had a more positive effect than the rule of speed composed example on the pupils' word-problem solving far transfer. Finally, the rule of speed composed example had the poorest effect on the far transfer of the pupils' word-problem solving.

Students with mathematics difficulties and learning disabilities (LD) typically struggle with solving word problems. These students often lack knowledge about efficient, cognitive strategies to utilize when solving word problems. Cognitive strategy instruction has been shown to be effective in teaching struggling students how to solve word problems that employ specific word problem types. The cognitive strategy, Math Scene Investigator (MSI), is an example of a cognitive strategy for word problem solving. The MSI strategy described in this paper is suitable for primary-level students with mathematics difficulties and LD. Instructional steps are provided along with an example of an interactive lesson.

It is necessary to map the results of previous studies to conduct new research. The mapping is to obtain information about the novelty of the research to be carried out. Then, it needs a mapping of what has been done by previous researchers. This study aims to map the studies that have been carried out around the world on difficulties in learning mathematics. This research is literature review research with the type of mapping. Data consist of 1,000 research articles indexed by Google Scholar from 2013 to 2022, which were collected with Publish or Perish software using the keywords difficulties in mathematics education. Mapping analysis was carried out using VOSviewer software that images were then interpreted. Mapping is done on learning difficulties in mathematics, difficulty in learning mathematics seen from the abstract with a minimum occurrence of 15 and 10, difficulty in learning mathematics is related to mathematical learning difficulty, difficulty in learning mathematics if it is seen as a relationship if mathematics is applied in studying science, difficulty in learning mathematics is related to the application of realistic mathematics education, and difficulties in learning mathematics related to instruction. The results showed that difficulties in learning mathematics focused on mathematical concepts, applying mathematics in science, word problems, unclear instructions, limited time, and students' mathematics anxiety and ability.

The purpose of this study is to investigate the effectiveness of two teaching methods of word problems, one based on mathematical modeling learning(ML) and the other on traditional learning(TL). Additionally, the influence of mathematical modeling learning in word problem solving behavior, application ability of real world experiences in word problem solving and the beliefs of word problem solving will be examined. The results of this study were as follows: First, as to word problem solving behavior, there was a significant difference between the two groups. This mean that the ML was effective for word problem solving behavior. Second, all of the students in the ML group and the TL group had a strong tendency to exclude real world knowledge and sense-making when solving word problems during the pre-test. but A significant difference appeared between the two groups during post-test. classroom culture improvement efforts. Third, mathematical modeling learning(ML) was effective for improvement of traditional beliefs about word problems. Fourth, mathematical modeling learning(ML) exerted more influence on mathematically strong and average students and a positive effect to mathematically weak students. High and average-level students tended to benefit from mathematical modeling learning(ML) more than their low-level peers. This difference was caused by less involvement from low-level students in group assignments and whole-class discussions. While using the mathematical modeling learning method, elementary students were able to build various models about problem situations, justify, and elaborate models by discussions and comparisons from each other. This proves that elementary students could participate in mathematical modeling activities via word problems, it results form the use of more authentic tasks, small group activities and whole-class discussions, exclusion of teacher's direct intervention, and classroom culture improvement efforts. The conclusions drawn from the results obtained in this study are as follows: First, mathematical modeling learning(ML) can become an effective method, guiding word problem solving behavior from the direct translation approach(DTA) based on numbers and key words without understanding about problem situations to the meaningful based approach(MBA) building rich models for problem situations. Second, mathematical modeling learning(ML) will contribute attitudes considering real world situations in solving word problems. Mathematical modeling activities for word problems can help elementary students to understand relations between word problems and the real world. It will be also help them to develop the ability to look at the real world mathematically. Third, mathematical modeling learning(ML) will contribute to the development of positive beliefs for mathematics and word problem solving. Word problem teaching focused on just mathematical operations can't develop proper beliefs for mathematics and word problem solving. Mathematical modeling learning(ML) for word problems provide elementary students the opportunity to understand the real world mathematically, and it increases students' modeling abilities. Futhermore, it is a very useful method of reforming the current problems of word problem teaching and learning. Therefore, word problems in school mathematics should be replaced by more authentic ones and modeling activities should be introduced early in elementary school eduction, which would help change the perceptions about word problem teaching.

Many students have more difficulty in solving word problems than numerical problems in school mathematics. It is because solving word problems requires linguistic knowledge as well as mathematical knowledge. Recently neuroimaging technique attempts to investigate areas of human brain for mathematical and linguistic thinking. This research aimed at analyzing the relative power spectrum of EEG activities and the degree of cognitive load of 12 adults during performing numerical and word problems. The results showed the theta band tended to more increase when solving word problems than numerical problems, the alpha band tended to increase from solving numerical problems to word problems, the beta band tended to be activated in more brain areas when solving word problems than numerical problems, and the gamma band tended to increase in the frontal area for both numerical and word problems. The EEG analysis revealed that the activation pattern of the human brain for numerical problems was different from the activation pattern for word problems. This neurological approach can be used to find the solution for difficulties of learning mathematics.

Students in the elementary grades often experience difficulty setting up and solving word problems. Using an equation to represent the structure of the problem serves as an effective tool for solving word problems, but students may require specific pre-algebraic reasoning instruction about the equal sign as a relational symbol to set up and solve such equations successfully. We identified students with mathematics difficulty (n = 138) from a sample of 916 third-grade students. We randomly assigned students to a word-problem intervention with a pre-algebraic reasoning component, a word-problem intervention without pre-algebraic reasoning, or the business-as-usual. Students in the 2 active intervention conditions participated in 45 individual sessions and learned about 3 additive word-problem schemas. Students who received word-problem intervention with a pre-algebraic reasoning component demonstrated improved nonstandard equation solving, equal sign understanding, and word-problem solving compared to students in the other two conditions.

Although students are often taught to look for keywords when solving word problems, this strategy is erroneous. It is especially problematic when students solve inconsistent word problems that include a relational term, such as more but are not solved with the assumed operation (e.g., addition). In this study, we analyzed 112 Grade 3 students’ constructed equations on four word problems that included the word more. We compared students with and without mathematics difficulty and disaggregated based on dual-language status. Most students constructed accurate equations for the two consistent word problems, but fewer constructed accurate equations for the two inconsistent word problems. Students with mathematics difficulty, particularly those who were also dual-language learners, had the lowest rates of accurate equations on the inconsistent word problems. This analysis reinforces previous calls by researchers to avoid the ineffective keywords strategy.

Dedication Foreword by Dan Domenech Preface Acknowledgments About the Authors Introduction: Why American Schools Must Move Toward Common Core State Standards Section I. Designing Competitive Curriculum for a Global Economy 1. Common Core State Standards: What Are They? Common Core State Standards: More Than Standards With an International Flavor A Quick Trip Around the World Moving Forward 2. The School Leader's Role in Making the Common Core State Standards Work Creating Guiding Ideas to Implement the Common Core State Standards Guiding Ideas and Evidence Getting to How: Operationalizing Your Guiding Ideas Identifying the Innovation Teaching the Change The Social and Emotional Literacy Components of the Common Core State Standards School Leaders and Teachers Partner to Help Students Inspiring Trust, Creativity, Transparency, and Success: A Framework for Loose Versus Tight Leadership Part I: Building Understanding About the Issue and Its Impact on Your Organization Part II: Moving From Understanding to Action Section II. Helping Communities Create a New Future 3. Designing Local Curriculum to Absorb the Common Core State Standards in English Real-World Applications Grades K-2, Phonological Awareness (One Topic) Grades K-2: Key Ideas and Details and Craft and Structure Grades 3-4: Fluency (One Topic) Grade 5: Reading With Fluency and Accuracy Grades 6-8: Key Ideas and Details (One Topic) Grades 9-10: English Language Arts Curriculum Grades 11-12: Reading and Writing 4. Designing Local Curriculum to Absorb the Common Core State Standards in Mathematics Comparison of State and Common Core State Math Standard Emphasis Common Core State Math Standards Kindergarten Common Core State Standard and Local Curriculum Math Standards Grade 1 Common Core State Standard and Local Curriculum Math Standards Grade 2 Common Core State Standard and Local Curriculum Math Standards Grade 5 Common Core State Standard and Local Curriculum Math Standards Six Domains of Instructional Strategies for All Students Purpose of the Math Common Core Grade 7 Common Core State Math Standards: Ratios and Proportional Relationships Grades 9-10 Common Core State Math Standards: The Real Number System Grade 10 Common Core State Math Standards for Geometry: Congruence Grade 12 Common Core State Math Standards for Statistics: Using Probability to Make Decisions 5. Challenges to the Implementation of Rigorous Common Core State Standards Testing Policies Do Not Improve Schools The Common Core State Standards: Opportunities for Equity and Excellence Section III. Helping Teachers Redefine Their Profession 6. How to Assess Mastery of the Common Core Curriculum State Standards Direct Instruction The Power of Formative Assessments Formative Assessment Examples From Real Schools Building Successful Schools Special Education and General Education 7. The Leadership Challenge: Creating Common Understandings of the Common Core State Standards The Role of School Leaders Envisioning Excellence We Get the Results We Design Schools to Achieve Vision or Nightmare? Enter the Common Core State Standards 8. Dialogue: Providing Opportunity So Facts Influence Opinions Creating Context for the Common Core State Standards Teachers Need Deep Understanding of the Common Core State Standards Identifying Gaps: Current Versus Desired Reality Section IV. Creating Systems That Accelerate Learning for All 9. Multicultural Issues That Teachers and School Leaders Must Face Beyond the Common Core State Standards Common Core State Standards Lift Us Beyond Difference What All Teachers and School Leaders Need to Know About Multiculturalism Grouping and Labeling Students Building a Multicultural Learning Community 10. An Effective Intervention in Schools That Improves Instruction and Learning Creating Sustainable Change in Struggling Schools Leaders Create the Context and Design for Change to Succeed Homework Student Work and Deficit Assumptions Focus Groups School Leaders Parent Involvement Students Changing Public Mental Models About School Reform 11. Public Policy: Helpful and Harmful References Index

In mathematics, the expectation to set up and solve word problems emerges as early as kindergarten; however, many students who experience mathematics difficulty (MD) and dual-language learners often present with specific challenges in this area. To investigate why these populations experience word-problem difficulty, we examined the word, problem solving and oral explanations of third-grade dual-language learners (DLLs; n = 40) and non-DLLs ( n = 40), all of whom were identified as experiencing MD. Students solved five additive word problems and provided oral explanations of their work, which were transcribed and coded for the number of words in each explanation, type of mathematics vocabulary terms used, inclusion of correct numbers in explanations, and descriptions of addition or subtraction. We identified no significant differences in word-problem scores between DLLs and non-DLLs with MD. For both DLLs and non-DLLs, students who answered problems correctly used more words in each explanation and used more mathematics vocabulary terms within their explanations. For incorrectly answered problems, the most common mistake for both DLLs and non-DLLs involved using the incorrect operation to solve the problem.

The purpose of this study is to investigate to what extent the use of L2 in math tests influences bilingual education learners’ process of word problem solving in a mandatory secondary education school with Content and Language Integrated Learning (CLIL). The reading comprehension level of the students was analysed using a standards-based assessment and the questions used in Programme for International Student Assessment (PISA) tests. The word problems were selected according to the students’ level of reading-comprehension and mathematical competence. Leaners also had to answer a questionnaire, which was used to analyse if contextual factors were affecting mathematical performance in L2. To this end, the questionnaire included some questions related to the bilingual history of the students and their perception about solving word problems in English. Data were analysed through one-way or two-way ANOVA tests to find out which factors were relevant. Results show that solving word problems is not only affected by the use of L2, but that it also depends on the mathematical difficulty, irrespective of the students’ level of language proficiency. The findings, hence, imply that interaction between linguistic difficulty and mathematical complexity is at the centre of the issues affecting word problem solving.

Students’ ability in solving mathematics word problems has been considered low, especially caused by difficulties in understanding information provided. Schema can be a helpful tool for students to face such difficulties. The aim of this research is to analyze the effectiveness of schema in resolving difficulties based on linguistic aspects constructing word problem (multiple representation systems, vocabulary, and grammar and syntax). A mixed method study of embedded experimental model was conducted with 69 participants of 11th grade students. Quantitative analysis to determine the effectiveness of schema was done by testing whether mathematical linguistic difficulties of treatment group was significantly lower than control students. Mathematical linguistic difficulties in each aspect were analyzed from the indicator-based coding of students’ work on word problem test. Qualitative analysis was conducted by comparing experiment and control students’ work on post-test. Experiment students showed significantly lower mathematical linguistic difficulties in each aspect compared to control ones. In general, schema is proven to be effective in resolving difficulties in each aspect. Qualitatively, students being taught schema showed more attention upon the whole information provided and were more self-directed in determining solving steps.

For students with mathematics difficulties (MD), math word problem solving is especially challenging. The purpose of this study was to examine the effects of a problem-solving strategy, bar model drawing, on the mathematical problem-solving skills of students with MD. The study extended previous research that suggested that schematic-based instruction (SBI) and cognitive strategy instruction (CSI) delivered within an explicit instruction framework can be effective in teaching various math skills related to word problem solving. A multiple-baseline design replicated across groups was used to evaluate the effects of the intervention of bar model drawing on math problem-solving performance of students with MD. Student achievement was measured in terms of increased correct use of cognitive strategies and overall accuracy of math word problem solving. Results showed that bar modeling drawing is an effective strategy for increasing elementary students’ accuracy in solving math word problems and their ability to use cognitive strategies to solve the problems.

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and debates that have dominated the field. After a short introduction, we begin with research that has conceived word problems primarily as problems of comprehension, and we describe the various ways in which this complex comprehension process has been conceived theoretically as well as the empirical evidence supporting different theoretical models. Next we review research that has focused on strategies for actually solving the word problem. Strengths and weaknesses of informal and formal solution strategies—at various levels of learners’ mathematical development (i.e., arithmetic, algebra)—are discussed. Fourth, we address research that thinks of word problems as exercises in complex problem solving, requiring the use of cognitive strategies (heuristics) as well as metacognitive (or self-regulatory) strategies. The fifth section concerns the role of graphical representations in word problem solving. The complex and sometimes surprising results of research on representations—both self-made and externally provided ones—are summarized and discussed. As in many other domains of mathematics learning, word problem solving performance has been shown to be significantly associated with a number of general cognitive resources such as working memory capacity and inhibitory skills. Research focusing on the role of these general cognitive resources is reviewed afterwards. The seventh section discusses research that analyzes the complex relationship between (traditional) word problems and (genuine) mathematical modeling tasks. Generally, this research points to the gap between the artificial word problems learners encounter in their mathematics lessons, on the one hand, and the authentic mathematical modeling situations with which they are confronted in real life, on the other hand. Finally, we review research on the impact of three important elements of the teaching/learning environment on the development of learners’ word problem solving competence: textbooks, software, and teachers. It is shown how each of these three environmental elements may support or hinder the development of learners’ word problem solving competence. With this general overview of international research on the various perspectives on this complex and fascinating kind of mathematical problem, we set the scene for the empirical contributions on word problems that appear in this special issue.