Words and Dehn's Word Problem

2012 ◽  
pp. 105-119
Author(s):  
John Meier
Keyword(s):  
2019 ◽  
Vol 36 (2) ◽  
pp. 142-156
Author(s):  
Lynn S. Fuchs ◽  
Douglas Fuchs ◽  
Pamela M. Seethaler ◽  
Caitlin Craddock

ZDM ◽  
2021 ◽  
Author(s):  
Gemma Carotenuto ◽  
Pietro Di Martino ◽  
Marta Lemmi

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.


Author(s):  
Yasemin Copur-Gencturk ◽  
Tenzin Doleck

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.


1986 ◽  
Vol 28 (2) ◽  
pp. 161-179 ◽  
Author(s):  
J. A. Gerhard ◽  
Mario Petrich

The word problem for free bands (idempotent semigroups) was solved by Green and Rees [4] in an early paper. They also established certain properties of the free band. This was followed by McLean [6] who provided a general structure theory for bands with some indication as to the structure of the free band. Since then a great many papers have appeared dealing with various aspects of the topic of bands and their varieties. A different solution of the word problem for free bands was recently given by Siekmann and Szabó [9]. For a discussion of bands, see the books [5] and [8].


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