scholarly journals Second and fifth graders’ use of knowledge-pieces and knowledge-structures when solving integer addition problems

2021 ◽  
Vol 7 (2) ◽  
pp. 82-103
Author(s):  
Mahtob Aqazade ◽  
Laura Bofferding
Keyword(s):  
Conceptual Change ◽  
Fifth Graders ◽  
Knowledge Structure ◽  
Second Graders ◽  
Negative Sign ◽  
Solution Strategies ◽  
Grade Levels ◽  
Whole Number ◽  
Use Of Knowledge ◽  
Number Knowledge

In this study, we explored second and fifth graders’ noticing of negative signs and incorporation of them into their strategies when solving integer addition problems. Fifty-one out of 102 second graders and 90 out of 102 fifth graders read or used negative signs at least once across the 11 problems. Among second graders, one of their most common strategies was subtracting numbers using their absolute values, which aligned with students’ whole number knowledge-pieces and knowledge-structure. They sometimes preserved the order of numbers and changed the placement of the negative sign (e.g., −9 + 2 as 9 – 2) and sometimes did the opposite (e.g., −1 + 8 as 8 – 1). Among fifth graders, one of the most common strategies reflected use of integer knowledge-pieces within a whole-number knowledge-structure, as they added numbers’ absolute values using whole number addition and appended the negative sign to their total. For both grade levels, the order of the numerals, the location of the negative signs, and also the numbers’ absolute values in the problems played a role in students’ strategies used. Fifth graders’ greater strategy variability often reflected strategic use of the meanings of the minus sign. Our findings provide insights into students’ problem interpretation and solution strategies for integer addition problems and supports a blended theory of conceptual change. Adding to prior findings, we found that entrenchment of previously learned patterns can be useful in unlikely ways, which should be taken up in instruction.

Precocity and Acceleration

2003 ◽  
Vol 17 (1) ◽  
pp. 55-58 ◽  
Author(s):  
John F. Feldhusen
Keyword(s):  
Fifth Graders ◽  
Second Graders ◽  
Reading Curriculum ◽  
Advanced Material ◽  
Grade Levels ◽  
Formal Instruction

Precocious kids have the motivation and the ability to surge ahead of what is normative for their ages and grade levels. Many first and second graders learn to read at levels in the reading curriculum that are typical for third to fifth graders. Some surge ahead in math too hut not as many as get ahead in reading. To get ahead precocious kids need access to advanced material, and that is easier to do in reading than in math. Brief contact without formal instruction often is all precocious kids need to learn and master advanced material.

Mediation or Production Deficiency in Disabled Readers?

1980 ◽  
Vol 50 (2) ◽  
pp. 519-530
Author(s):  
Lauren Leslie
Keyword(s):  
Short Term Memory ◽  
Fifth Graders ◽  
Memory Capacity ◽  
Second Graders ◽  
Short Term ◽  
Term Memory ◽  
Memory Processing ◽  
Overt Rehearsal ◽  
Covert Rehearsal ◽  
Disabled Readers

Deficiencies in disabled readers’ short-term memory processing were studied. A deficit in memory capacity versus susceptibility to interference was investigated by examining performance over trials. A mediation versus production deficiency in memory processing was examined by testing the effect of instructions for rehearsal on performance of average and disabled readers in Grades 2 and 5. Contrary to prior research, facilitative effects of rehearsal instructions on second graders’ memory were found only on Trial 1. Fifth graders’ memory was adversely affected by overt rehearsal. Requiring children to rehearse overtly at a set rate may account for the results. A second study examined effects of covert rehearsal on the memory of average and disabled readers in Grade 2 over trials. Facilitative effects of covert rehearsal were shown when data of children who spontaneously rehearsed were removed. A deficiency in production by second graders was supported. Disabled readers who did not rehearse were more susceptible to interference.

Brief Reports: Fraction Knowledge in Preschool Children

1988 ◽  
Vol 19 (2) ◽  
pp. 175-180
Author(s):  
Robert P. Hunting ◽  
Christopher F. Sharpley
Keyword(s):  
Preschool Children ◽  
School Mathematics ◽  
The Core ◽  
Whole Number ◽  
Whole Number Arithmetic ◽  
Number Knowledge

Much school mathematics is devoted to teaching concepts and procedures based on those units that form the core of whole number arithmetic (ones, tens, hundreds, etc.). But other topics such as fractions and decimals demand a new and extended understanding of units and their relationships. Behr, Wachsmuth, Post, and Lesh (1984) and Streefland (1984) have noted how children's whole number ideas interfere with their efforts to learn fractions. Hunting (1986) suggested that a reason children seem to have difficulty learning stable and appropriate meanings for fractions is that instruction on fractions, if delayed too long, allows whole number knowledge to become the predominant scheme to which fraction language and symbolism is then related.

Students' Use of Part-Whole and Direct Comparison Strategies for Comparing Partitioned Rectangles

1995 ◽  
Vol 26 (1) ◽  
pp. 2-19
Author(s):  
Barbara E. Armstrong ◽  
Carol Novillis Larson
Keyword(s):  
Rational Number ◽  
Grade Level ◽  
Clinical Interview ◽  
Eighth Grade ◽  
Grade Levels ◽  
Comparison Strategy ◽  
Fractional Terms ◽  
Number Knowledge

Thirty-six students, twelve each at the fourth-, sixth-, and eighth-grade levels, were asked to solve 21 comparison-of-area tasks during a clinical interview. The areas were partitioned and shaded so that the tasks could be solved by using rational number knowledge. The types of strategies students used to compare the areas were identified and classified into categories. Initially, fractional terms and symbols were not introduced into the tasks in order to compare the types of strategies students used without and with symbols introduced. Most students tended to ignore the part-whole relationships inherent in the tasks and used a Direct Comparison strategy when they compared the areas. The use of a Part-Whole strategy increased with the introduction of fractional terms and symbols, especially at the eighth-grade level.

Bridging the Gap: Using Interactive Computer Tools to Build Fraction Schemes

2002 ◽  
Vol 8 (6) ◽  
pp. 356-361
Author(s):  
John Olive
Keyword(s):  
Mathematics Instruction ◽  
Elementary Grades ◽  
Mathematical Development ◽  
Computer Tools ◽  
Whole Number ◽  
Fraction Concepts ◽  
Teaching Fractions ◽  
Number Knowledge

Teaching fractions has been a complex and largely unsuccessful aspect of mathematics instruction in the elementary grades for many years. Students' understanding of fraction concepts is a big stumbling block in their mathematical development. Some researchers have pointed to children's whole-number knowledge as interfering with, or creating a barrier to, their understanding of fractions (Behr et al. 1984; Streefland 1993; Lamon 1999). This article illustrates an approach to constructing fraction concepts that builds on children's whole-number knowledge using specially designed computer tools. This approach can help children make connections between whole-number multiplication and their notion of a fraction as a part of a whole, thus bridging the gap between whole-number and fraction knowledge.

scholarly journals Linking inhibitory control to math achievement via comparison of conflicting decimal numbers

2020 ◽  
Author(s):  
Linsah Coulanges ◽  
Roberto A. Abreu-Mendoza ◽  
Sashank Varma ◽  
Melina Uncapher ◽  
Adam Gazzaley ◽  
...  
Keyword(s):  
Working Memory ◽  
Executive Functions ◽  
Inhibitory Control ◽  
Cognitive Flexibility ◽  
Math Achievement ◽  
Academic Learning ◽  
Whole Number ◽  
Calculation Skills ◽  
The Relationship ◽  
Number Knowledge

The relationship between executive functions (EF) and academic achievement is well-established, but leveraging this insight to improve educational outcomes remains elusive. Here, we propose a framework for relating the role of specific EF on specific precursor skills that support later academic learning. Specifically, we hypothesize that executive functions contribute to general math skills both directly – supporting the online execution of problem solving strategies – and indirectly – supporting the acquisition of precursor mathematical content. We test this hypothesis by examining the contribution of inhibitory control on processing rational numbers pairs which conflict with individual’s prior whole number knowledge and on general math knowledge. In 97 college students (79 female, age = 20.63 years), we collected three measures of EF: working memory (backwards spatial span), inhibition (color-word Stroop) and cognitive flexibility (task switching), and timed and untimed standardized measures of math achievement. Our target precursor skill was a decimals comparison task where correct responses were inconsistent with prior whole number knowledge (e.g. 0.27 vs. 0.9). Participants performed worse on these trials relative to the consistent decimals pairs (e.g. 0.2 vs. 0.87). Individual differences on incongruent Stroop trials predicted performance on inconsistent decimal comparisons, which in turn predicted performance on both timed and untimed measures of math achievement. With respect to relating inhibitory control to math achievement, incongruent Stroop performance was an independent predictor of untimed calculation skills after accounting for age, working memory and cognitive flexibility. Finally, we found that inconsistent decimals performance partially mediated the relationship between inhibition and untimed math achievement, consistent with the hypothesis that mathematical precursor skills can explain the relationships between executive functions and academic outcomes, making them promising targets for intervention.

An Application of Configural Frequency Analysis

2013 ◽  
Vol 72 (2) ◽  
pp. 61-70
Author(s):  
Li Zhang ◽  
Ziqiang Xin ◽  
Cody Ding ◽  
Chongde Lin
Keyword(s):  
General Population ◽  
Frequency Analysis ◽  
Fifth Graders ◽  
Second Graders ◽  
Not Given ◽  
Class Inclusion ◽  
Configural Frequency Analysis ◽  
Grade 5 ◽  
Grade 3 ◽  
Number Of Classes

Development of class reasoning was investigated using configural frequency analysis (CFA). We administered class inclusion, vicariant inclusion, and law of duality tasks to a sample of 540 Chinese second through fifth graders. In each task, children were asked to compare two classes and make a choice from four alternative answers while the number of classes was not given. Results showed that (1) children’s performance on both class inclusion and vicariant inclusion tasks improved significantly from Grade 2 to Grade 3 and from Grade 3 to Grade 4, but children did not tend to give correct answers to class inclusion items until Grade 4 and to vicariant inclusion items until Grade 5; (2) children from Grades 2 to 5 performed poorly on the law of duality task, but fifth graders were more likely to respond correctly than the general population; and (3) second graders tended to give wrong answers such as “equal number” and “not sure.” A discussion of the development of class reasoning followed.

Matching Design Tasks to Knowledge-Based Software Tools: When Intuition Does Not Suffice

1999 ◽  
Author(s):  
R. Amen ◽  
I. Rask ◽  
S. Sunnersjö
Keyword(s):  
Engineering Design ◽  
Design Process ◽  
Knowledge Structure ◽  
Software Tools ◽  
Specific Design ◽  
Design Task ◽  
Solution Strategies ◽  
Knowledge Based ◽  
Operational Life ◽  
Matching Design

Abstract Choosing methods and tools to automate engineering design tasks has farreaching implications. A system that is first established for a limited task may later require being scaled up and maintained for many years of operational life. As the system grows, the company’s reliance on the system increases and the need for a stable platform to start from becomes critical. The system must be tailored to the specific design process and knowledge structure at hand. The purpose of this work is to present a structured methodology to classify design task(s) and to give guidelines for how this definition should be linked to suitable solution strategies and computerised tools.

Mixed Numbers Made Easy: Building and Converting Mixed Numbers and Improper Fractions

2007 ◽  
Vol 13 (9) ◽  
pp. 488-492
Author(s):  
Chris Neumer
Keyword(s):  
Fifth Graders ◽  
Computer Programs ◽  
Fifth Grade ◽  
Teaching Approaches ◽  
New Approach ◽  
Whole Number

After several years of observing and interacting with children, I have noticed their struggles with converting mixed numbers and improper fractions. I have tried several types of manipulatives, computer programs, pictures, and algorithm-based activity sheets. Even after being exposed to all these teaching approaches, many students still struggle with these representations. The concept of a mixed number—a whole number paired with a fraction—really boggles students' minds. More so, the idea of an improper fraction—a fraction with a numerator that is greater than or equal to the denominator—is virtually impossible for some fifth graders to grasp. They cannot connect their written algorithms to the concept or explain how they derived their answers. The reason for rewriting their answer in a different form was too abstract for them. Through direct work with my fifth-grade classes over the years, I have developed a new approach that produces great results and a deeper understanding of mixed numbers, improper fractions, and their relationship to each other.

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