Quick Reads: Another Good Idea: Integers Made Easy: Just Walk It Off

2007 ◽  
Vol 13 (2) ◽  
pp. 118-121
Author(s):  
Julie Nurnberger-Haag
Keyword(s):  
Number Line ◽  
Similar Process ◽  
Learning Modalities ◽  
Number Lines ◽  
Addition And Subtraction ◽  
Brain Based Learning

Walk It Off is a multisensory method that I developed to teach students how to multiply and divide as well as add and subtract integers. In my experience, this method makes these processes much more effective, efficient, and entertaining than other approaches. Students have the opportunity to use the Process Standards while exploring a topic that is often taught strictly as algorithms without understanding. In addition to having visual, oral, and aural characteristics, Walk It Off is kinesthetic, because students literally walk off problems on number lines. The multiple learning modalities that this method uses are compatible with brain-based learning. Students become actively involved in doing calculations as they physically act out problems using the underlying concepts of integers. Although many books and lessons already use number lines to demonstrate addition and subtraction in various ways, the Walk It Off method makes it possible to use a number line for multiplication and division of integers as well. Using integers becomes easy, because students learn only two slightly different processes for the two basic groups of operations: one process for addition and subtraction and a similar process for multiplication and division.

Integer Target: Using a Game to Model Integer Addition and Subtraction

2007 ◽  
Vol 12 (7) ◽  
pp. 388-392
Author(s):  
Jerry Burkhart
Keyword(s):  
Small Groups ◽  
Number Line ◽  
Target Number ◽  
Number Lines ◽  
Addition And Subtraction

Imagine a classroom where students are gathered in small groups, working with number lines and cards marked with integers. The students have chosen a “target number” on the number line and are deep in discussion, trying to find ways to make the sum of the integers on their cards match this number. There is a deck from which they draw, discard, or exchange cards. They also give, take, or trade cards with one another.

A Proposed Instructional Theory for Integer Addition and Subtraction

2012 ◽  
Vol 43 (4) ◽  
pp. 428-464 ◽  
Author(s):  
Michelle Stephan ◽  
Didem Akyuz
Keyword(s):  
Number Line ◽  
Teaching Experiment ◽  
Learning Trajectory ◽  
Realistic Mathematics Education ◽  
Instructional Theory ◽  
Education Theory ◽  
Addition And Subtraction ◽  
Support Students ◽  
Grade Classroom ◽  
Viable Model

This article presents the results of a 7th-grade classroom teaching experiment that supported students' understanding of integer addition and subtraction. The experiment was conducted to test and revise a hypothetical learning trajectory so as to propose a potential instructional theory for integer addition and subtraction. The instructional sequence, which was based on a financial context, was designed using the Realistic Mathematics Education theory. Additionally, an empty, vertical number line (VNL) is posited as a potentially viable model to support students' organizing their addition and subtraction strategies. Particular emphasis is placed on the mathematical practices that were established in this setting. These practices indicate that students can successfully draw on their experiences with assets, debts, and net worths to create meaning for integer addition and subtraction.

scholarly journals A Study of Errors and Misconceptions in the Learning of Addition and Subtraction of Directed Numbers in Grade 8

SAGE Open ◽  
2016 ◽  
Vol 6 (4) ◽  
pp. 215824401667137 ◽  
Author(s):  
Judah Paul Makonye ◽  
Josiah Fakude
Keyword(s):  
Conceptual Understanding ◽  
Teaching And Learning ◽  
Study Data ◽  
Negative Numbers ◽  
Teaching And Learning Mathematics ◽  
Learning Mathematics ◽  
Number Lines ◽  
Addition And Subtraction ◽  
Logical Errors ◽  
Grade 8

The study focused on the errors and misconceptions that learners manifest in the addition and subtraction of directed numbers. Skemp’s notions of relational and instrumental understanding of mathematics and Sfard’s participation and acquisition metaphors of learning mathematics informed the study. Data were collected from 35 Grade 8 learners’ exercise book responses to directed numbers tasks as well as through interviews. Content analysis was based on Kilpatrick et al.’s strands of mathematical proficiency. The findings were as follows: 83.3% of learners have misconceptions, 16.7% have procedural errors, 67% have strategic errors, and 28.6% have logical errors on addition and subtraction of directed numbers. The sources of the errors seemed to be lack of reference to mediating artifacts such as number lines or other real contextual situations when learning to deal with directed numbers. Learners seemed obsessed with positive numbers and addition operation frames—the first number ideas they encountered in school. They could not easily accommodate negative numbers or the subtraction operation involving negative integers. Another stumbling block seemed to be poor proficiency in English, which is the language of teaching and learning mathematics. The study recommends that building conceptual understanding on directed numbers and operations on them must be encouraged through use of multirepresentations and other contexts meaningful to learners. For that reason, we urge delayed use of calculators.

Fixated in unfamiliar territory: Mapping estimates across typical and atypical number lines

2019 ◽  
Vol 73 (2) ◽  
pp. 279-294
Author(s):  
Sabrina Michelle Di Lonardo ◽  
Matthew G Huebner ◽  
Katherine Newman ◽  
Jo-Anne LeFevre
Keyword(s):  
Numerical Range ◽  
Number Line ◽  
Strategy Use ◽  
Reverse Direction ◽  
Tracking Data ◽  
Spatial Processes ◽  
Number Lines ◽  
Number Line Estimation ◽  
The Right ◽  
Typical Range

Adults ( N = 72) estimated the location of target numbers on number lines that varied in numerical range (i.e., typical range 0–10,000 or atypical range 0–7,000) and spatial orientation (i.e., the 0 endpoint on the left [traditional] or on the right [reversed]). Eye-tracking data were used to assess strategy use. Participants made meaningful first fixations on the line, with fixations occurring around the origin for low target numbers and around the midpoint and endpoint for high target numbers. On traditional direction number lines, participants used left-to-right scanning and showed a leftward bias; these effects were reduced for the reverse direction number lines. Participants made fixations around the midpoint for both ranges but were less accurate when estimating target numbers around the midpoint on the 7,000-range number line. Thus, participants are using the internal benchmark (i.e., midpoint) to guide estimates on atypical range number lines, but they have difficulty calculating the midpoint, leading to less accurate estimates. In summary, both range and direction influenced strategy use and accuracy, suggesting that both numerical and spatial processes influence number line estimation.

A physical model for teaching multiplication of integers

1968 ◽  
Vol 15 (6) ◽  
pp. 525-528
Author(s):  
Warren H. Hill
Keyword(s):  
Positive Integer ◽  
Physical Model ◽  
Number Line ◽  
Negative Integer ◽  
Physical Models ◽  
Addition And Subtraction ◽  
Positive Integers ◽  
Partial Success

One of the pedagogical pitfalls involved in teaching the multiplication of integers is the scarcity of physical models that can be used to illustrate their product. This problem becomes even more acute if a phyiscal model that makes use of a number line is desired. If the students have previously encountered the operations of addition and subtraction of integers using a number line, then the desirability of developing the operation of multiplication in a similar setting is obvious. Many of the more common physical models that can be used to represent the product of positive and negative integers are unsuitable for interpretation on a number line. Partial success is possible in the cases of the product of two positive integers and the product of a positive and a negative integer. But, how does one illustrate that the product of two negative integers is equal to a positive integer?

scholarly journals Not toeing the number line for simple arithmetic: Two large-n conceptual replications of Mathieu et al. (Cognition, 2016, Experiment 1)

2021 ◽  
Vol 7 (3) ◽  
pp. 248-258 ◽  
Author(s):  
Jamie I. D. Campbell ◽  
Yalin Chen ◽  
Maham Azhar
Keyword(s):  
Number Line ◽  
Operation Condition ◽  
Single Digit ◽  
Simple Arithmetic ◽  
Position Effects ◽  
Large N ◽  
Addition And Subtraction ◽  
Mixed Operation ◽  
The Right ◽  
The University

We conducted two conceptual replications of Experiment 1 in Mathieu, Gourjon, Couderc, Thevenot, and Prado (2016, https://doi.org/10.1016/j.cognition.2015.10.002). They tested a sample of 34 French adults on mixed-operation blocks of single-digit addition (4 + 3) and subtraction (4 – 3) with the three problem elements (O1, +/-, O2) presented sequentially. Addition was 34 ms faster if O2 appeared 300 ms after the operation sign and displaced 5° to the right of central fixation, whereas subtraction was 19 ms faster when O2 was displaced to the left. Replication Experiment 1 (n = 74 recruited at the University of Saskatchewan) used the same non-zero addition and subtraction problems and trial event sequence as Mathieu et al., but participants completed blocks of pure addition and pure subtraction followed by the mixed-operation condition used by Mathieu et al. Addition RT showed a 32 ms advantage with O2 shifted rightward relative to leftward but only in mixed-operation blocks. There was no effect of O2 position on subtraction RT. Experiment 2 (n = 74) was the same except mixed-operation blocks occurred before the pure-operation blocks. There was an overall 13 ms advantage with O2 shifted right relative to leftward but no interaction with operation or with mixture (i.e., pure vs mixed operations). Nonetheless, the rightward RT advantage was statistically significant for both addition and subtraction only in mixed-operation blocks. Taken together with the robust effects of mixture in Experiment 1, the results suggest that O2 position effects in this paradigm might reflect task specific demands associated with mixed operations.

A Complete Model for Operations on Integers

1983 ◽  
Vol 30 (9) ◽  
pp. 26-31 ◽  
Author(s):  
Michael T. Battista
Keyword(s):  
Physical Model ◽  
Negative Charge ◽  
Number Line ◽  
Complete Model ◽  
Line Model ◽  
Teaching Student ◽  
Charge Model ◽  
Addition And Subtraction

Teaching student the four basic operations on the set of integers in a meaningful way is a difficult task. The task could be made easier, however, if there was a single physical model for the integers in which all four basic operations could be represented. The number-line model, though useful, has serious shortcomings and is incomplete. As an alternative, some author have suggested the “positive-negative charge” model (Frand and Granville 1978, Grady 1978), but they used this model only for representing addition and subtraction. In this article I will describe the “charge” model and demonstrate how it can be extended to all four operations on the set of integers.

Teacher To Teacher: Adding and Subtracting Integers on the Number Line

1993 ◽  
Vol 40 (7) ◽  
pp. 388-389
Author(s):  
Pamala Byrd Çemen
Keyword(s):  
Number Line ◽  
Addition And Subtraction

Various models have been proposed for teaching addition and subtraction of integers (e.g., Grady [1978]: Dirks [1984]; and Chang [1985]), including money, two-color tiles, and the number line.

Nondecimal Slide Rules- and Their Use in Modular Arithmetic

1971 ◽  
Vol 64 (5) ◽  
pp. 467-472
Author(s):  
Katye Oliver Sowell ◽  
Jon Phillip Mcguffey
Keyword(s):  
The Other ◽  
Modular Arithmetic ◽  
Number Lines ◽  
Addition And Subtraction

Addition and subtraction are quickly done by means of two number lines placed one above the other.

scholarly journals Strategy use on bounded and unbounded number lines in typically developing adults and adults with dyscalculia: An eye-tracking study

2018 ◽  
Vol 4 (2) ◽  
pp. 337-359 ◽  
Author(s):  
Fae Aimée van der Weijden ◽  
Erica Kamphorst ◽  
Robin Hella Willemsen ◽  
Evelyn H. Kroesbergen ◽  
Anne H. van Hoogmoed
Keyword(s):  
Eye Tracking ◽  
Number Sense ◽  
Number Line ◽  
Strategy Use ◽  
Typically Developing ◽  
Target Number ◽  
Gaze Patterns ◽  
Pure Number ◽  
Number Lines ◽  
New Strategies

Recent research suggests that bounded number line tasks, often used to measure number sense, measure proportion estimation instead of pure number estimation. The latter is thought to be measured in recently developed unbounded number line tasks. Children with dyscalculia use less mature strategies on unbounded number lines than typically developing children. In this qualitative study, we explored strategy use in bounded and unbounded number lines in adults with (N = 8) and without dyscalculia (N = 8). Our aim was to gain more detailed insights into strategy use. Differences in accuracy and strategy use between individuals with and without dyscalculia on both number lines may enhance our understanding of the underlying deficits in individuals with dyscalculia. We combined eye-tracking and Cued Retrospective Reporting (CRR) to identify strategies on a detailed level. Strategy use and performance were highly similar in adults with and without dyscalculia on both number lines, which implies that adults with dyscalculia may have partly overcome their deficits in number sense. New strategies and additional steps and tools used to solve number lines were identified, such as the use of the previous target number. We provide gaze patterns and descriptions of strategies that give important first insights into new strategies. These newly defined strategies give a more in-depth view on how individuals approach a number lines task, and these should be taken into account when studying number estimations, especially when using the unbounded number line.

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