This note addresses the issue as to which ceers can be realized by word problems of computably enumerable (or, simply, c.e.) structures (such as c.e. semigroups, groups, and rings), where being realized means to fall in the same reducibility degree (under the notion of reducibility for equivalence relations usually called “computable reducibility”), or in the same isomorphism type (with the isomorphism induced by a computable function), or in the same strong isomorphism type (with the isomorphism induced by a computable (...) permutation of the natural numbers). We observe, e.g., that every ceer is isomorphic to the word problem of some c.e. semigroup, but (answering a question of Gao and Gerdes) not every ceer is in the same reducibility degree of the word problem of some finitely presented semigroup, nor is it in the same reducibility degree of some non‐periodic semigroup. We also show that the ceer provided by provable equivalence of Peano Arithmetic is in the same strong isomorphism type as the word problem of some non‐commutative and non‐Boolean c.e. ring. (shrink)

Every science has its technical vocabulary, consisting in part of terms coined for explicit purposes and in part of words borrowed from ordinary discourse and used with greater or lesser degrees of precision. Words of the latter sort pose curious problems, some of them familiar to those historians of science concerned with, for example, what Galileo meant by forza and Newton by attraction. Indeed, analogous problems face any historian seeking to understand the older meanings of terms still in use today.

By theword problemfor some class of algebraic structures we mean the problem of determining, given a finite setEof equations between words and an additional equationx=y, whetherx=ymust hold in all structures satisfying each member ofE. In 1947 Post [P] showed the word problem for semigroups to be undecidable. This result was strengthened in 1950 by Turing, who showed the word problem to be undecidable forcancellation semigroups,i.e. semigroups satisfying thecancellation propertyNovikov [N] eventually showed the word (...) class='Hi'>problem for groups to be undecidable.In 1966 Gurevich [G] showed the word problem to be undecidable forfinitesemigroups. However, this result on finite structures has not been extended to cancellation semigroups or groups; indeed it is easy to see that a finite cancellation semigroup is a group, so both questions are the same. We do not here settle the word problem for finite groups, but we do show that the word problem is undecidable for finite semigroups with zero satisfying an approximation to the cancellation property. (shrink)

We have two polynomial time results for the uniform word problem for a quasivariety Q: The uniform word problem for Q can be solved in polynomial time iff one can find a certain congruence on finite partial algebras in polynomial time. Let Q* be the relational class determined by Q. If any universal Horn class between the universal closure S and the weak embedding closure S̄ of Q* is finitely axiomatizable then the uniform word (...) class='Hi'>problem for Q is solvable in polynomial time. This covers Skolem's 1920 solution to the uniform word problem for lattices and Evans' 1953 applications of the weak embeddability property for finite partial V algebras. (shrink)

Two experiments were conducted to investigate children’s interpretations of standard arithmetic word problems and the factors that influence their interpretations. In Experiment 1, children were required to solve a series of problems and then to draw and select pictures that represented the problems’ structures. Solution performance was found to vary systematically with the nature of the representations drawn and chosen. The crucial determinant of solution success was the interpretation a child assigned to certain phrases used in the problems. In (...) Experiment 2, solution and drawing accuracy were found to be significantly improved by rewording problems to avoid ambiguous linguistic forms. Together, these results imply that (a) word-problem solution errors are caused by misinterpretations of certain verbal expressions commonly used in problem texts, and (b) these misinterpretations are the result of missing or inadequate mappings of these verbal expressions to partwhole knowledge. (shrink)

Word problems are notoriously difficult to solve. We suggest that much of the difficulty children experience with word problems can be attributed to difficulty in comprehending abstract or ambiguous language. We tested this hypothesis by (1) requiring children to recall problems either before or after solving them, (2) requiring them to generate f'mal questions to incomplete word problems, and (3) modeling performance pattems using a computer simulation. Solution performance was found to be systematically related to recall and (...) question generation performance. Correct solutions were associated with accurate recall of the problem structure and with appropriate question generation. Solution "errors" were found to be correct solutions to miscomprehended problems. Word problems that contained abstract or ambiguous language tended to be miscomprehended more often than those using simpler language, and there was a great deal of systematicity in the way these problems were miscomprehended. Solution error pattems were successfully simulated by manipulating a computer model’s language comprehension strategies, as opposed to its knowledge of logical set relations. o was.. (shrink)

A pseudo-natural algorithm for the word problem of a finitely presented group is an algorithm which not only tells us whether or not a word w equals 1 in the group but also gives a derivation of 1 from w when w equals 1. In [13], [14] Madlener and Otto show that, if we measure complexity of a primitive recursive algorithm by its level in the Grzegorczyk hierarchy, there are groups in which a pseudo-natural algorithm is arbitrarily (...) more complicated than an algorithm which simply solves the word problem. In a given group the lowest degree of complexity that can be realised by a pseudo-natural algorithm is essentially the derivational complexity of that group. Thus the result separates the derivational complexity of the word problem of a finitely presented group from its intrinsic complexity. The proof given in [13] involves the construction of a finitely presented group G from a Turing machine T such that the intrinsic complexity of the word problem for G reflects the complexity of the halting problem of T, while the derivational complexity of the word problem for G reflects the runtime complexity of T. The proof of one of the crucial lemmas in [13] is only sketched, and part of the purpose of this paper is to give the full details of this proof. We will also obtain a variant of their proof, using modular machines rather than Turing machines. As for several other results, this simplifies the proofs considerably. MSC: 03D40, 20F10. (shrink)

Three experiments explored the effects of word problem cueing on debiasing versions of the bat-and-ball problem. In the experimental condition order, participants solved a simpler isomorphic version of the problem prior to solving a standard version that, critically, had the same item-and-dollar amounts. Conversely, in the control condition order, participants solved the standard version prior to solving the isomorph. Across the first 2 experiments, participants cued with the isomorph were more likely to correctly solve the standard (...) version of the problem. In the third experiment, this cueing effect was not replicated, but importantly cued participants were more likely to correctly solve the standard version of the problem with other item-and-dollar amounts, indicating near transfer. Far transfer was evident in faster response times to other problems in the Cognitive Reflection Test following practice. Along with providing converging evidence for the cueing account, these results are consistent with recent attentional debiasing findings. (shrink)

Mathematics word problems provide students with an opportunity to apply what they are learning in their mathematics classes to the world around them. However, students often neglect their knowledge of the world and provide nonsensical responses (e.g., they may answer that a school needs 12.5 buses for a field trip). This study examined if the question design of word problems affects students' mindset in ways that affect subsequent sense‐making. The hypothesis was that rewriting standard word problems to (...) introduce inherent uncertainty about the result would be beneficial to student performance and sense‐making because it requires students to reason explicitly about the context described in the problem. Middle school students (N = 229) were randomly assigned to one of three conditions. In the standard textbook condition, students solved a set of six word problems taken from current textbooks. In the modified yes/no condition, students solved the same six problems rewritten so the solution helped answer a “yes” or “no” question. In the disfluency control condition, students solved the standard problems each rewritten in a variety of fonts to make them look unusual. After solving the six problems in their assigned condition, all students solved the same three “problematic” problems designed to assess sense‐making. Contrary to predictions, results showed that students in the modified yes/no condition solved the fewest problems correctly in their assigned condition problem set. However, consistent with predictions, they subsequently demonstrated more sense‐making on the three problematic problems. Results suggest that standard textbook word problems may be able to be rewritten in ways that mitigate a “senseless” mindset. (shrink)

Understanding arithmetic word problems involves a complex interaction of text comprehension and mathematical processes. This article presents a computer simulation designed to capture the working memory demands required in “bottomup” comprehension of arithmetic word problems. The simulation's sentence‐level parser and text integration component reflect the importance of processing the problem from its original natural language presentation. Children's probability of solution was analyzed in exploratory regression analyses as a function of the simulation's sentence‐level and text integration processes. Working (...) memory variables measuring the combined effects of concepts to remember and text integration inferences account for a significant proportion of variance in children's solution probabilities across the first four grade levels (K‐3). Consistent with previous results from others, which highlighted the significance of small changes in problem wording, the simulation offers a process‐oriented perspective as to why natural language presentation constrains the comprehension of mathematical relationships. (shrink)

The practice of mathematical word problem is ubiquitous and thought to impact academic achievement. However, the underlying neural mechanisms are still poorly understood. In this study, we investigate how lexical consistency of word problem description is modulated in adults' brain responses during word problem solution. Using functional magnetic resonance imaging methods, we examined compare word problems that included relational statements, such as “A dumpling costs 9 dollars. A wonton is 2 dollars less than (...) a dumpling. How much does a wonton cost?” and manipulated lexical consistency and problem operation. We found a consistency by operation interaction in the widespread fronto-insular-parietal activations, including the anterior insula, dorsoanterior cingulate cortex, middle frontal gyrus, and intraparietal sulcus, such that inconsistent problems engaged stronger activations than consistent problems for addition, whereas the consistency effect was inverse for subtraction. Critically, these results were more salient in the less successful problem solvers than their more successful peers. Our study is the first to demonstrate that lexical consistency effects on arithmetic neural networks are modulated during reading word problem that required distinct arithmetic operations. More broadly, our study has strong potentials to add linkage between neuroscience and education by remediating deficits and enhance instruction design in the school curriculum. (shrink)

ABSTRACTIn a bilingual educational setting, even when mathematical word problems are presented in one’s first language, students may still perform poorly if cognitive constraints such as working me...

The central question underlying this study was whether metacognition training could enhance the two metacognition components—knowledge and skills—and the mathematical problem-solving capacities of normal children in grade 3. We also investigated whether metacognitive training had a differential effect according to the children's mathematics level. A total of 48 participants took part in this study, divided into an experimental and a control group, each subdivided into a lower and a normal achievers group. The training programme took an interactive approach in (...) accordance with Schraw's (1998) recommendation and was carried out over five training sessions. Results indicated that children in the training group had significantly higher post-test metacognitive knowledge, metacognitive skills, and mathematical problem-solving scores. In addition, metacognitive training was particularly beneficial to the low achievers. Thus metacognitive training enabled the low achievers to make progress and solve the same number of problems on the post-test as the normal achievers solved on the pre-test. (shrink)