Comorbidity in Reading Comprehension and Word-Problem Solving Difficulties: Exploring Shared Risk Factors and Their Impact on Language Minority Learners

The purpose of this study was threefold: to examine unique and shared risk factors of comorbidity for reading comprehension and word-problem solving difficulties, to explore whether language minority (LM) learners are at increased risk of what we refer to as higher order comorbidity (reading comprehension and word-problem solving difficulties), and to examine the profiles of at-risk LM learners compared with at-risk non-LM learners. At-risk (LM n = 70; non-LM n = 89) and not-at-risk (LM n = 44; non-LM n = 114) students were evaluated on foundational academic (word reading, calculation), behavioral (behavioral attention), cognitive (working memory, processing speed, nonverbal reasoning), and language (vocabulary, listening comprehension) measures in English. Results indicated listening comprehension was the only shared risk factor for higher order comorbidity. Furthermore, LM learners were 3 times more likely to be identified as at-risk compared with non-LM learners. Finally, among at-risk learners, no differences were found on cognitive dimensions by language status, but LM learners had lower reading and listening comprehension skills than non-LM learners, with a relative advantage in behavioral attention. Results have implications for understanding higher order comorbidity and for developing methods to identify and intervene with higher order comorbidity among the growing population of LM learners.

Computational complexity of the word problem in modal algebras

Приводится доказательство $\mathrm{PSPACE}$-полноты проблемы равенства слов в классе всех нуль-порождённых модальных алгебр, или, эквивалентно, проблемы равенства константных слов в классе всех модальных алгебр. Также рассматривается вопрос о сложности равенства слов в произвольном многообразии модальных алгебр. Доказывается, что уже проблема равенства константных слов в многообразии модальных алгебр может быть сколь угодно трудной (включая как классы сложности, так и степени неразрешимости). Показано, как построить соответствующие многообразия. The paper deals with the word problem for modal algebras. It is proved that, for the variety of all modal algebras, the word problem is $\mathrm{PSPACE}$-complete if only constant modal terms or only $0$-generated modal algebras are considered. We also consider the word problem for different varieties of modal algebras. It is proved that the word problem for a variety of modal algebras can be $C$-hard, for any complexity class or unsolvability degree $C$ containing a $C$-complete problem. It is shown how to construct such varieties.

Proses Berpikir Siswa SMA dalam Menyelesaikan Soal Cerita Berdasarkan Teori Pemrosesan Informasi

<p><strong>Abstract:</strong> This study uses a descriptive exploratory approach because it aims to describe the thought processes of students in solving word problems based on information processing theory and involving high school students in grade XI. Research instruments in the form of interview guidelines and question instruments. Students are asked to complete the given word problem then interviewed to confirm their thought processes based on information processing theories, namely attention, perception, rehearsal, retrieval and encoding. Subjects were chosen based on the completeness of aspects of the thought process and suggestions from the teacher. The results showed that all subjects carried out all processes but in different ways.</p><strong>Abstrak:</strong><em> </em>Penelitian ini menggunakan pendekatan deskriptif eksploratif karena bertujuan untuk mendeskripsikan proses berpikir siswa dalam menyelesaikan soal cerita berdasarkan teori pemrosesan informasi dan melibatkan siswa kelas XI SMA. Instrumen penelitian berupa pedoman wawancara dan instrumen soal. Siswa diminta untuk menyelesaikan soal cerita yang diberikan kemudian diwawancarai untuk mengonfirmasi proses berpikirnya berdasarkan teori pemrosesan informasi, yaitu <em>attention</em>, <em>perception</em>, <em>rehearsal</em>, <em>retrieval</em> dan <em>encoding</em>. Subjek dipilih berdasarkan aspek kelengkapan, aspek proses berpikir, dan saran dari guru. Hasil penelitian menunjukkan bahwa semua subjek melakukan semua proses, namun dengan cara yang berbeda-beda.

NONSTANDARD MATH WORD PROBLEMS AND ANALYSIS OF THE PARTIAL STAGES OF ITS SOLUTION

Choosing the right strategy is an important condition to successfully solve math problems. Research studies often present individual types of strategies more or less separately. This study aims to determine student solutions of selected word problems in the whole context of the solution process. In this context, such nonstandard word problems combine verbal formulation and the character of nonstandard problems (impossible to be solved using an algorithm). In order to get an overall picture of the stages of word problem solution, an analysis of solving a given word problem was conducted among 171 respondents aged 10-11. The analysis was conducted in compliance with partial steps of word problem processing, as the solving of the problem was viewed from a wider perspective. The student’s reaction to the problem, working with the given information, individual forms of solution, and answer formation were recorded. In order to have a more complex idea and possibility to compare, the chosen way of solving the problem was also presented to a selected sample of 26 teachers. Available solutions were analyzed, and there were sought ways how the solution was assessed by the teachers based on selected parameters. Especially their meta-cognitive estimation of the correctness of their own solution was subject to scrutiny. Despite the fact that the respondents chose different strategies of solution (graphic, arithmetical, using judgment, etc.), it appears that the success rate of solving the given nonstandard word problem was very low. Thus, it is necessary to implement such word problems into standard math lessons, also within pre-graduate teacher preparation. Keywords: mathematics teaching, primary school mathematics, problem-solving, prospective teachers, word problem

Applying an Individual Word-Problem Intervention to a Small-Group Setting: A Pilot Study’s Evidence of Improved Word-Problem Performance for Students Experiencing Mathematics Difficulty

The purpose of this pilot study was to determine whether positive results from a word-problem intervention implemented one-to-one contributed to similar outcomes when implemented in small groups of three to four students. Third-grade students experiencing mathematics difficulty ( n = 76) were randomly assigned to word-problem intervention ( n = 56) or business-as-usual comparison ( n = 20). Intervention occurred for 13 weeks, 3 times per week, 30 min per session. Multilevel models revealed the intervention condition significantly outperformed the BaU on a proximal word-problem outcome, corroborating results from our prior individual intervention. When comparing student performance in the individual versus small-group intervention, findings suggest students received added benefit from the individual intervention. The word-problem intervention successfully translated to a small-group setting, which holds important implications for educators working with students in supplemental, targeted, or Tier-2 mathematics intervention settings.