Using Clock Arithmetic to Teach Algebra Concepts

2005 ◽  
Vol 11 (2) ◽  
pp. 104-109
Author(s):  
Elizabeth M. Brown ◽  
Elizabeth Jones
Keyword(s):  
Middle School Students ◽  
Number System ◽  
Abstract Concepts ◽  
Real Numbers ◽  
School Students ◽  
Multiplicative Inverse ◽  
Finite Systems ◽  
Solving Equations ◽  
Original Number ◽  
Addition And Subtraction

Learning algebra concepts can be difficult for middle school students. One reason may be because we work in only one number system, the set of real numbers. Students have only one frame of reference to provide examples of abstract concepts, such as the additive and multiplicative identities, additive and multiplicative inverses, and connections among the operations. These concepts are essential in solving equations. For example, we can think of an equation like 3x + 4 = 7 in the following way: Begin with a number, multiply it by 3, and add 4. If the answer is 7, what number did we start with? To solve this type of equation algebraically, we just undo what was done to the original number, that is, we add -4, the additive inverse of 4, and multiply the result by 1/3, the multiplicative inverse of 3. We can also think of this as subtracting 4 and dividing by 3, because addition and subtraction are inverse operations as are multiplication and division. The ideas of inverse operations and inverse elements are, therefore, central to algebra. Many students understand these ideas well enough to do simple problems like 3x + 4 = 7 but get confused with more difficult problems, such as 3(x + 6) + 2 = 5x + 5, where it is not as easy to see the order in which to undo these operations. We use finite systems to help students understand these key concepts in algebra, including additive and multiplicative identities, additive and multiplicative inverses, closure, and the relationships between addition and subtraction and multiplication and division.

Techland

2019 ◽  
pp. 761-780
Author(s):  
Michelina Occhioni
Keyword(s):  
Middle School ◽  
Middle School Students ◽  
Learning Environment ◽  
Virtual Worlds ◽  
Learning Objects ◽  
Public Access ◽  
Abstract Concepts ◽  
School Students ◽  
The World ◽  
Learning Sets

This work describes Techland, an OpenSim-based VW developed and owned by the author for educational purposes for middle school students. Techland is a group of islands focused on math, and science, where explore various approaches to VW, in different and mutually not exclusive learning sets and degrees of immersion. The main objectives are to simplify abstract concepts by using scripted 3D learning objects and to engage students offering a way to go beyond the classroom as the only learning environment. So Techland can be considered a giant 3D book. Activities range from viewing the contents of the world on an IWB, as a support for the teacher's lesson, to logging on the student's avatars into the world for exploration, producing machinima videos and collaborating together. Most islands are public access and reachable from other similar virtual worlds by the HG system, so other schools could take advantage from its contents or they could share projects. Techland has got the proper requirements to will become a meeting place between students and teachers who want to find new approaches to scientific matters.

Abacus Interactive

2018 ◽  
Vol 23 (5) ◽  
pp. 288-297
Author(s):  
Ellen Robinson ◽  
Xiaowen Cui ◽  
Nama Namakshi ◽  
Hiroko K. Warshauer ◽  
Sonalee Bhattacharyya ◽  
...  
Keyword(s):  
Teaching And Learning ◽  
Number System ◽  
Place Value ◽  
Elementary Grades ◽  
School Students ◽  
Learning Standards ◽  
Mathematical Properties ◽  
Hands On ◽  
Addition And Subtraction ◽  
Expanded Form

Calculators are often efficient in finding the answer to an addition or subtraction problem, but they do not reveal the process by which the answer is obtained. Developing students' fluency in addition and subtraction using strategies and algorithms based on place value, composing and decomposing numbers in base 10, and reading and writing numerals in expanded form are important teaching and learning standards not only for the elementary grades but for middle school students as well (NCTM 2000; CCSSI 2010; TEA 2015). We introduced the Chinese abacus to our students as a hands-on tool to illuminate the meaning of a number in expanded form in terms of place values and to strengthen students' conceptual understanding of the standard algorithms of addition and subtraction. “Students' understanding of the base 10 number system is deepened as they come to understand its multiplicative structure” (NCTM 2000, p. 143). This activity will let students explore the mathematical properties of the base 10 system in a creative and interactive way. Students develop a deeper meaning of why the standard algorithms work and how they relate to a number in expanded form. This activity is best suited for elementary and middle grades.

Techland

2017 ◽  
pp. 407-426
Author(s):  
Michelina Occhioni
Keyword(s):  
Middle School ◽  
Middle School Students ◽  
Learning Environment ◽  
Virtual Worlds ◽  
Learning Objects ◽  
Public Access ◽  
Abstract Concepts ◽  
School Students ◽  
The World ◽  
Learning Sets

This work describes Techland, an OpenSim-based VW developed and owned by the author for educational purposes for middle school students. Techland is a group of islands focused on math, and science, where explore various approaches to VW, in different and mutually not exclusive learning sets and degrees of immersion. The main objectives are to simplify abstract concepts by using scripted 3D learning objects and to engage students offering a way to go beyond the classroom as the only learning environment. So Techland can be considered a giant 3D book. Activities range from viewing the contents of the world on an IWB, as a support for the teacher's lesson, to logging on the student's avatars into the world for exploration, producing machinima videos and collaborating together. Most islands are public access and reachable from other similar virtual worlds by the HG system, so other schools could take advantage from its contents or they could share projects. Techland has got the proper requirements to will become a meeting place between students and teachers who want to find new approaches to scientific matters.

scholarly journals A Learning Trajectory Study on How the Concept of Variable Is Constructed by Students

2020 ◽  
Vol 10 (1) ◽  
pp. 134
Author(s):  
Mustafa Çağrı Gürbüz ◽  
M. Emin Ozdemir
Keyword(s):  
Middle School Students ◽  
Qualitative Data ◽  
Teaching Experiment ◽  
Learning Trajectory ◽  
Learning Trajectories ◽  
School Students ◽  
Paired Samples ◽  
Mathematical Abstraction ◽  
Experiment Method ◽  
Solving Equations

The aim of this study was to examine 6th-grade students’ mathematical abstraction processes related to the concept of variable by using the teaching experiment method and to reveal their learning trajectories in the context of the RBC+C model. A teaching experiment was administered to a class of 29 middle school students for 3 weeks. Observations, interviews, and the Diagnostic Algebra Test were used as data collection instruments to reveal the students’ abstraction processes and determine their learning trajectories. Qualitative data were analyzed through content analysis, and qualitative data were analyzed through paired-samples t-test. The learning trajectories showed that only the students with good performance exhibited the “construction” action when using “variables as changing quantities,” but the “building-with” action when using the other types of variables. Mediocre students, however, needed teacher support to perform the building-with action in the process of abstraction of variables. The students’ written tips such as drawing arrows or deleting the variable show that it helps to learn how to replace a variable with a known value. This study shows that the development of thought on variables is embedded in the progression of the concept of variable as a changing quantity. Similar studies can be conducted for the use of variables in equations and for the understanding and interpretation of variables when solving equations.

Traumatic experiences and associated symptomatology in Asian American middle school students.

2017 ◽  
Vol 8 (3) ◽  
pp. 209-223
Author(s):  
Elizabeth Davies-Mercier ◽  
Michelle W. Woodbridge ◽  
W. Carl Sumi ◽  
S. Patrick Thornton ◽  
Katrina D. Roundfield ◽  
...  
Keyword(s):  
Middle School ◽  
Middle School Students ◽  
Asian American ◽  
Traumatic Experiences ◽  
School Students

Longitudinal Effects of School Climate on Middle School Students' Development

2014 ◽  
Author(s):  
Jennifer L. Engelland ◽  
Renee M. Tobin ◽  
Adena B. Meyers ◽  
Brenda J. Huber ◽  
W. Joel Schneider ◽  
...  
Keyword(s):  
Middle School ◽  
School Climate ◽  
Middle School Students ◽  
School Students ◽  
Longitudinal Effects

Perceived Social Support and Middle School Students' School Adjustment

2009 ◽  
Author(s):  
Ji-Geun Kim ◽  
Yejin Lee ◽  
Bo-Ra Song ◽  
Hyunah Lee ◽  
Jung Eun Hwang
Keyword(s):  
Social Support ◽  
Middle School ◽  
Middle School Students ◽  
Perceived Social Support ◽  
School Adjustment ◽  
School Students
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