Solving word problems in free algebras using complexity functions

2006 ◽  
pp. 476-495 ◽  
Author(s):  
Alex Pelin ◽  
Jean H. Gallier
Keyword(s):  
Word Problems ◽  
Free Algebras

Out Subword-Free Languages and Its Subclasses

2016 ◽  
Vol 27 (03) ◽  
pp. 305-326 ◽  
Author(s):  
Jing Tian ◽  
Yong Shao ◽  
Xianzhong Zhao
Keyword(s):  
Partial Order ◽  
Word Problems ◽  
Finite Alphabet ◽  
Free Algebras ◽  
Independent Subset ◽  
Algebraic Structures ◽  
Ordered Semigroups ◽  
Binary Operations ◽  
Free Language

Let [Formula: see text] be a finite alphabet and [Formula: see text] the set of all words over [Formula: see text]. A subword-free language (also known as a hypercode) is an independent subset of [Formula: see text] with respect to the embedding order (denoted by [Formula: see text]) on [Formula: see text]. In this paper we introduce three subsets of the partial order [Formula: see text], and study three subclasses of languages defined by these subsets. They are out subword-free languages, left subword-free languages and right subword-free languages. The properties of these languages are established for determining their combinatorial and algebraic structures. By equipping them with two binary operations, respectively, all these classes of languages form semilattice-ordered semigroups. It is shown that they are freely generated by [Formula: see text] in three subvarieties of semilattice-ordered semigroups, respectively. It is also shown that the word problems for these free algebras are solvable.

Schema-Based Instruction Implemented Under Routine Conditions

2019 ◽  
Author(s):  
Corey Peltier ◽  
Mindy E Lingo ◽  
Faye Autry-Schreffler ◽  
Malarie Deardorff ◽  
Leslie Mathews ◽  
...  
Keyword(s):  
Problem Solving ◽  
Word Problems ◽  
Mathematical Problem ◽  
Functional Relation ◽  
Probe Design ◽  
Resource Room ◽  
Group Settings ◽  
Intervention Protocol ◽  
Prior Literature ◽  
Based Instruction

Students identified with a specific learning disability (SLD) experience difficulty with mathematical problem solving. One specific intervention identified as a promising practice for students with a SLD is schema-based instruction (SBI). The current projects aimed to tests the efficacy of SBI under routine conditions. This extends prior literature by (a) using a teacher as the implementer, (b) allowing flexibility in the intervention protocol, (c) condensing the duration of intervention sessions, and (d) providing instruction in small group settings. In addition, we examined student problem solving performance on word problems requiring two-steps and combined schema structures. We used a multiple-probe design across three groups of fifth-grade participants (n = 7) receiving supplemental instruction in a resource room setting. Results indicated a functional relation between SBI and problem-solving performance for all students on simple structure word problems, with the magnitude of effects varying across cases. The NAP, Tau, and BC-SMD effect sizes were used to quantify effects. Implications were discussed in regard to systematic replication and conditions that may impact fidelity.

scholarly journals Linking teachers’ solution strategies to their performance on fraction word problems

2021 ◽  
Vol 101 ◽  
pp. 103314
Author(s):  
Yasemin Copur-Gencturk ◽  
Tenzin Doleck
Keyword(s):  
Word Problems ◽  
Solution Strategies

scholarly journals Arithmetic Word Problems Revisited: Cognitive Processes and Academic Performance in Secondary School

2021 ◽  
Vol 11 (4) ◽  
pp. 155
Author(s):  
Gonzalo Duque de Blas ◽  
Isabel Gómez-Veiga ◽  
Juan A. García-Madruga
Keyword(s):  
Academic Performance ◽  
Word Problems ◽  
Spatial Reasoning ◽  
Deductive Reasoning ◽  
Executive Processes ◽  
Mathematical Operation ◽  
Visual Spatial ◽  
And Mathematics ◽  
The One ◽  
Memory Resources

Solving arithmetic word problems is a complex task that requires individuals to activate their working memory resources, as well as the correct performance of the underlying executive processes involved in order to inhibit semantic biases or superficial responses caused by the problem’s statement. This paper describes a study carried out with 135 students of Secondary Obligatory Education, each of whom solved 5 verbal arithmetic problems: 2 consistent problems, whose mathematical operation (add/subtract) and the verbal statement of the problem coincide, and 3 inconsistent problems, whose required operation is the inverse of the one suggested by the verbal term(s). Measures of reading comprehension, visual–spatial reasoning and deductive reasoning were also obtained. The results show the relationship between arithmetic problems and cognitive measures, as well as the ability of these problems to predict academic performance. Regression analyses confirmed that arithmetic word problems were the only measure with significant power of association with academic achievement in both History/Geography (β = 0.25) and Mathematics (β = 0.23).

scholarly journals Students’ suspension of sense making in problem solving

ZDM ◽  
2021 ◽  
Author(s):  
Gemma Carotenuto ◽  
Pietro Di Martino ◽  
Marta Lemmi
Keyword(s):  
Problem Solving ◽  
Qualitative Study ◽  
Word Problem ◽  
Word Problems ◽  
Mathematical Problem ◽  
Critical Issue ◽  
Mathematical Problem Solving ◽  
Sense Making ◽  
Mathematical Problems

AbstractResearch on mathematical problem solving has a long tradition: retracing its fascinating story sheds light on its intricacies and, therefore, on its needs. When we analyze this impressive literature, a critical issue emerges clearly, namely, the presence of words and expressions having many and sometimes opposite meanings. Significant examples are the terms ‘realistic’ and ‘modeling’ associated with word problems in school. Understanding how these terms are used is important in research, because this issue relates to the design of several studies and to the interpretation of a large number of phenomena, such as the well-known phenomenon of students’ suspension of sense making when they solve mathematical problems. In order to deepen our understanding of this phenomenon, we describe a large empirical and qualitative study focused on the effects of variations in the presentation (text, picture, format) of word problems on students’ approaches to these problems. The results of our study show that the phenomenon of suspension of sense making is more precisely a phenomenon of activation of alternative kinds of sense making: the different kinds of active sense making appear to be strongly affected by the presentation of the word problem.

scholarly journals Strategic competence for multistep fraction word problems: an overlooked aspect of mathematical knowledge for teaching

2021 ◽  
Author(s):  
Yasemin Copur-Gencturk ◽  
Tenzin Doleck
Keyword(s):  
Word Problem ◽  
Word Problems ◽  
Mathematical Knowledge ◽  
Teaching And Learning ◽  
Mathematics Learning ◽  
Mathematical Knowledge For Teaching ◽  
Important Indicator ◽  
Strategic Competence ◽  
The Usa ◽  
Valid Solution

AbstractPrior work on teachers’ mathematical knowledge has contributed to our understanding of the important role of teachers’ knowledge in teaching and learning. However, one aspect of teachers’ mathematical knowledge has received little attention: strategic competence for word problems. Adapting from one of the most comprehensive characterizations of mathematics learning (NRC, 2001), we argue that teachers’ mathematical knowledge also includes strategic competence, which consists of devising a valid solution strategy, mathematizing the problem (i.e., choosing particular strategies and presentations to translate the word problem into mathematical expressions), and arriving at a correct answer (executing a solution) for a word problem. By examining the responses of 350 fourth- and fifth-grade teachers in the USA to four multistep fraction word problems, we were able to explore manifestations of teachers’ strategic competence for word problems. Findings indicate that teachers’ strategic competence was closely related to whether they devised a valid strategy. Further, how teachers dealt with known and unknown quantities in their mathematization of word problems was an important indicator of their strategic competence. Teachers with strong strategic competence used algebraic notations or pictorial representations and dealt with unknown quantities more frequently in their solution methods than did teachers with weak strategic competence. The results of this study provide evidence for the critical nature of strategic competence as another dimension needed to understand and describe teachers’ mathematical knowledge.

Question Design Affects Students' Sense‐Making on Mathematics Word Problems

2021 ◽  
Vol 45 (4) ◽  
Author(s):  
Patrick K. Kirkland ◽  
Nicole M. McNeil
Keyword(s):  
Word Problems ◽  
Sense Making ◽  
Mathematics Word Problems
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