Reflecting on the What and Why of Whole Number Arithmetic: A Commentary on Chapter 5

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.

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Semantic alignment across whole-number arithmetic and rational numbers: evidence from a Russian perspective

Thinking & Reasoning ◽

10.1080/13546783.2017.1374307 ◽

2017 ◽

Vol 24 (2) ◽

pp. 198-220 ◽

Cited By ~ 1

Author(s):

Yulia A. Tyumeneva ◽

Galina Larina ◽

Ekaterina Alexandrova ◽

Melissa DeWolf ◽

Miriam Bassok ◽

...

Keyword(s):

Rational Numbers ◽

Whole Number ◽

Semantic Alignment ◽

Whole Number Arithmetic

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Tradition in Whole Number Arithmetic

New ICMI Study Series - Building the Foundation: Whole Numbers in the Primary Grades ◽

10.1007/978-3-319-63555-2_15 ◽

2018 ◽

pp. 343-373 ◽

Cited By ~ 2

Author(s):

Ferdinando Arzarello ◽

Nadia Azrou ◽

Maria G. Bartolini Bussi ◽

Sarah Inés González de Lora Sued ◽

Xu Hua Sun ◽

...

Keyword(s):

Whole Number ◽

Whole Number Arithmetic

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How to Teach and Assess Whole Number Arithmetic: A Commentary on Chapter 11

New ICMI Study Series - Building the Foundation: Whole Numbers in the Primary Grades ◽

10.1007/978-3-319-63555-2_12 ◽

2018 ◽

pp. 287-298

Author(s):

Claire Margolinas

Keyword(s):

Whole Number ◽

Whole Number Arithmetic

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Math Anxiety: Personal, Educational, and Cognitive Consequences

Current Directions in Psychological Science ◽

10.1111/1467-8721.00196 ◽

2002 ◽

Vol 11 (5) ◽

pp. 181-185 ◽

Cited By ~ 396

Author(s):

Mark H. Ashcraft

Keyword(s):

Cognitive Processing ◽

Career Paths ◽

Math Anxiety ◽

Brain Activity ◽

Teaching Styles ◽

Ongoing Activity ◽

Cognitive Components ◽

Whole Number ◽

On Line ◽

Whole Number Arithmetic

Highly math-anxious individuals are characterized by a strong tendency to avoid math, which ultimately undercuts their math competence and forecloses important career paths. But timed, on-line tests reveal math-anxiety effects on whole-number arithmetic problems (e.g., 46 + 27), whereas achievement tests show no competence differences. Math anxiety disrupts cognitive processing by compromising ongoing activity in working memory. Although the causes of math anxiety are undetermined, some teaching styles are implicated as risk factors. We need research on the origins of math anxiety and on its “signature” in brain activity, to examine both its emotional and its cognitive components.

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Patterns in the Multiplication Table

The Arithmetic Teacher ◽

10.5951/at.32.7.0036 ◽

1985 ◽

Vol 32 (7) ◽

pp. 36-40

Author(s):

Cornelia C. Tierney

Keyword(s):

Eighth Graders ◽

Multiplication Table ◽

Similar Amount ◽

Concerted Effort ◽

Great Range ◽

Whole Process ◽

Whole Number ◽

Whole Number Arithmetic ◽

Number Facts ◽

Better Than

In elementary school, great importance is placed on memorization of number facts. In teaching fifth through eighth graders, 1 have assumed that most of my students had made a concerted effort to memorize facts in earlier grades. I have observed. however, that children who have had a similar amount of practice have a great range of recall. A few students complete tests of 100 multiplication or division facts perfectly in less than three minutes, whereas others are made miserable by the whole process. They skip many problems, look around the room to compare their progress with that of other students, and finally give up with few correct answers. Although those who have memorized the facts do better than others at whole-number arithmetic, they do not always do well in work with fractions.

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The Sleep of Reason Produces Monsters: How and When Biased Input Shapes Mathematics Learning

Annual Review of Developmental Psychology ◽

10.1146/annurev-devpsych-041620-031544 ◽

2020 ◽

Vol 2 (1) ◽

pp. 413-435

Author(s):

Robert S. Siegler ◽

Soo-hyun Im ◽

Lauren K. Schiller ◽

Jing Tian ◽

David W. Braithwaite

Keyword(s):

Computer Simulations ◽

Conceptual Knowledge ◽

Mathematics Learning ◽

Arithmetic Operations ◽

Mathematical Performance ◽

Children’S Learning ◽

Whole Number ◽

Whole Number Arithmetic ◽

Children's Learning

Children's failure to reason often leads to their mathematical performance being shaped by spurious associations from problem input and overgeneralization of inapplicable procedures rather than by whether answers and procedures make sense. In particular, imbalanced distributions of problems, particularly in textbooks, lead children to create spurious associations between arithmetic operations and the numbers they combine; when conceptual knowledge is absent, these spurious associations contribute to the implausible answers, flawed strategies, and violations of principles characteristic of children's mathematics in many areas. To illustrate mechanisms that create flawed strategies in some areas but not others, we contrast computer simulations of fraction and whole number arithmetic. Most of their mechanisms are similar, but the model of fraction arithmetic lacks conceptual knowledge that precludes strategies that violate basic mathematical principles. Presentingbalanced problem distributions and inculcating conceptual knowledge for distinguishing flawed from legitimate strategies are promising means for improving children's learning.

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