scholarly journals What we count dictates how we count: A tale of two encodings

2020 ◽  
Author(s):  
Hippolyte Gros ◽  
Jean- Pierre Thibaut ◽  
Emmanuel Sander
Keyword(s):  
Word Problems ◽  
Mathematical Knowledge ◽  
Direct Problem ◽  
Mathematical Reasoning ◽  
Arithmetic Problem ◽  
Validity Assessment ◽  
Mathematical Word Problems ◽  
Problem Comparison ◽  
Problem Solvability ◽  
Multiple Strategy

We argue that what we count has a crucial impact on how we count, to the extent that even adults may have difficulty using elementary mathematical notions in concrete situations. Specifically, we investigate how the use of certain types of quantities (durations, heights, number of floors) may emphasize the ordinality of the numbers featured in a problem, whereas other quantities (collections, weights, prices) may emphasize the cardinality of the depicted numerical situations. We suggest that this distinction leads to the construction of one of two possible encodings, either a cardinal or an ordinal representation. This difference should, in turn, constrain the way we approach problems, influencing our mathematical reasoning in multiple activities. This hypothesis is tested in six experiments (N = 916), using different versions of multiple-strategy arithmetic word problems. We show that the distinction between cardinal and ordinal quantities predicts problem sorting (Experiment 1), perception of similarity between problems (Experiment 2), direct problem comparison (Experiment 3), choice of a solving algorithm (Experiment 4), problem solvability estimation (Experiment 5) and solution validity assessment (Experiment 6). The results provide converging clues shedding light into the fundamental importance of the cardinal versus ordinal distinction on adults’ reasoning about numerical situations. Overall, we report multiple evidence that general, non-mathematical knowledge associated with the use of different quantities shapes adults’ encoding, recoding and solving of mathematical word problems. The implications regarding mathematical cognition and theories of arithmetic problem solving are discussed.

scholarly journals Are content effects out of sight? An eye-tracking study of arithmetic problem solving

2020 ◽  
Author(s):  
Hippolyte Gros ◽  
Emmanuel Sander ◽  
Jean- Pierre Thibaut
Keyword(s):  
Eye Movements ◽  
Problem Solving ◽  
Eye Tracking ◽  
Word Problem ◽  
Mathematical Knowledge ◽  
Arithmetic Problem ◽  
Assessment Task ◽  
Validity Assessment ◽  
Gaze Patterns ◽  
Arithmetic Word Problem

General, non-mathematical knowledge about the entities described in an arithmetic word problem may interfere with its encoding. We used behavioral and eye-tracking measures to investigate how the use of specific quantities may foster a cardinal representation of the numbers mentioned in a problem, whereas other quantities may favor an ordinal representation instead. We asked 50 pre-service teachers to complete a solution validity assessment task. We compared participants’ gaze patterns on isomorphic problems to gather insights into their encoded representations. On problems featuring cardinal quantities, we found that specific sentences describing elements relevant in a cardinal understanding of the problems but irrelevant otherwise were looked at longer and were the focus of a higher number of backward eye movements. Additionally, an increase in pupil dilation on correctly solved cardinal problems supported the idea that participants need to engage in a recoding process when facing semantic incongruence.

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.

scholarly journals ERROR ANALYSIS OF PRIMARY SIX PUPILS IN WORD PROBLEMS INVOLVING FRACTIONS

2018 ◽  
Vol 18 (1) ◽  
pp. 9
Author(s):  
Aliyu Alhaji Zakariyya ◽  
Adamu Barwa Beji ◽  
Unogwu Itodo
Keyword(s):  
Error Analysis ◽  
Diagnostic Test ◽  
Word Problems ◽  
Structured Interview ◽  
Transformation Process ◽  
Chi Square ◽  
Public And Private ◽  
Qualitative Research Design ◽  
Mathematical Word Problems ◽  
Primary School Pupils

The purpose of this study was to analyze the errors made by primary school pupils in solving mathematical word problems in fraction using Newman’s Error Analysis procedure. The study used a qualitative research design and collected data using a diagnostic test and interview. The population of the study was primary six pupils in both public and private schools in Minna metropolis of Niger State, Nigeria. The samples were 105 primary six (6) pupils, 61 males and 44 females. The instruments used in the study are (a) a set of paper and pencil test consisting of 12 theory questions on fractions and (b) structured interview based on Newman’s theorem. The diagnostic test was administered to the pupils. Structured interview adopted from Rohmah and Sutiarso was also used to identify at which level students’ errors occur. The type of error was based on Newman Error Hierarchy Model that includes reading, comprehension, transformation, process skill, and encoding error. The data were analysed using inferential statistics of Chi-square (

scholarly journals Associative and Strategic Study Approaches: Their Significant Relationships to Solving Meta Cognitive Problems of Mathematics Majors

2013 ◽  
Vol 1 (1) ◽  
Author(s):  
Marvin S. Daguplo
Keyword(s):  
Word Problems ◽  
Strong Relationship ◽  
Male And Female ◽  
Mathematics Majors ◽  
Classroom Activities ◽  
Significant Relationships ◽  
Significant Difference ◽  
Mathematical Word Problems ◽  
Strategic Study ◽  
Cognitive Problems

University students still found difficulties in working successfully mathematical word problems. Some researchers attributed this to students’ weak cognitive and abstract thinking. To address such problem, this study aimed to understand student’s cognitive approaches in processing mathematics information to determine students’ level of cognition and come up with classroom activities that enhance the desired approaches in processing mathematical information which influences learning. A total of thirty-seven students of SLSU – Tomas Oppus were the respondents of this descriptive-correlational study. A standardized Mathematics Information Processing Scale (Cronbach’s alpha coefficient=.89) was utilized to gather the data for this study. Statistical analysis revealed that there is no significant difference in the performance of male and female mathematics majors in solving word problems who performed at below average level. The same finding is found between male and female mathematics majors in their approaches in processing mathematics information when solving metacognitive problems, when doing deep-associative study, and when doing strategic study. Correlational analysis revealed that a strong relationship exists between solving metacognitive problems and doing associative and strategic study approaches. This study concludes that students who apply associative and strategic study methods perform well in solving meta-cognitive problems.Keywords: Mathematics, Processing Mathematics Information, Action Research, Philippines

The role of L1 and L2 reading comprehension in solving mathematical word problems

2005 ◽  
Vol 28 (1) ◽  
pp. 52-66 ◽  
Author(s):  
Emine Erktin ◽  
Ayse Akyel
Keyword(s):  
Second Language ◽  
Reading Comprehension ◽  
Private School ◽  
Word Problems ◽  
First Language ◽  
L2 Reading ◽  
Mathematical Word Problems ◽  
And Performance ◽  
And Mathematics ◽  
L1 And L2

Abstract Mathematics educators are concerned about students’ lack of ability to translate mathematical word problems into computable forms. Researchers argue that linguistic problems lie at the root of students’ difficulties with mathematical word problems. The issue becomes more complicated for bilingual students. It is argued that if students study mathematics in a second language they cannot be as successful as when they study in their first language. This study investigates the relationship between reading comprehension and performance on mathematics word problems in L1 and L2 for students learning English as a second language in a delayed partial immersion program. Data were collected from 250 Turkish students from Grade 8 of a private school in Istanbul through reading comprehension tests in L1 and L2 and an algebra word problems test prepared in L1 and L2. The results indicate a positive relationship between reading comprehension and mathematics performance. They also show that the students who participated in this study were not disadvantaged when they studied mathematics in English.

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