Helping Students Make Sense of Algebraic Expressions: The Candy Shop

2002 ◽  
Vol 7 (9) ◽  
pp. 492-497
Author(s):  
Diana Underwood Gregg ◽  
Erna Yackel
Keyword(s):  
Algebraic Thinking ◽  
Instructional Sequence ◽  
Algebraic Expression ◽  
Instructional Approaches ◽  
Algebra For All ◽  
Algebraic Expressions ◽  
Experience Difficulty

Increasing emphasis on “Algebra for all” (NCTM 1997a, 1997b) compels educators to identify and address fundamental ideas that build the foundations for algebraic thinking and reasoning. Identifying these foundational concepts and developing appropriate instructional approaches are the focuses of our work. One area in which students often experience difficulty is adding and subtracting algebraic expressions. Although students may be able to memorize a procedure, such as “distribute the negative” when subtracting algebraic expressions, they are often unable to make sense of this procedure. Our work suggests that part of students' difficulty in this area is that they do not conceptualize an algebraic expression as a composite unit. In the paragraphs below, we explain what is meant by composite units and how this construct helped frame our development of an instructional sequence to help students make sense of, and find meaning in, algebraic expressions and operations on algebraic expressions.

scholarly journals Examination of Semantic Structures Used by Teacher Candidates to Transform Algebraic Expressions into Verbal Problems

2021 ◽  
Vol 4 (3) ◽  
Author(s):  
Ayten Pinar Bal ◽  
Keyword(s):  
Teacher Candidates ◽  
Quantitative Research ◽  
Algebraic Expression ◽  
Study Group ◽  
State University ◽  
Algebraic Expressions ◽  
Purposeful Sampling ◽  
Division Problems ◽  
Addition And Subtraction ◽  
Research Types

The aim of this study is to examine the semantic structures used by mathematics teacher candidates to transform algebraic expressions into verbal problems. The research is a descriptive study in the survey model, which is one of the quantitative research types. The study group of the research consists of 165 teacher candidates studying in the primary school mathematics teaching department of a state university in the south of Turkey in the 2019-2020 academic years. 73.2% of the teacher candidates in the study group are female and 26.8% are male. Criterion sampling method, one of the purposeful sampling methods, was used in the selection of teacher candidates in the study group. While the Algebraic Expression Questionnaire Form was used as the data collection tool, the evaluation rubric of verbal problems was used in the analysis of the data. As a result of the research, it has been revealed that pre-service teachers are more successful in transforming algebraic expressions into verbal problems, but they have problems in creating problems with algebraic expressions that make up systems of equations. Again in the study, it was concluded that pre-service teachers used addition and subtraction problems more than multiplication and division problems. On the other hand, when the problems in the type of addition and subtraction are examined in the study, in the type of combining and separating; It has been concluded that the category of equal groups is mostly used in the problems of multiplication and division.

Teacher to Teacher: Building Algebraic Expressions: A Physical Model

1997 ◽  
Vol 2 (4) ◽  
pp. 238-242
Author(s):  
Anne C. Patterson
Keyword(s):  
Elementary School ◽  
Middle School Students ◽  
School Experiences ◽  
Algebraic Thinking ◽  
Physical Models ◽  
Mathematical Language ◽  
School Students ◽  
Student Activity ◽  
Algebraic Expressions ◽  
The Universe

Simple physical models can help middle school students move naturally from their limited world of numbers into the universe of algebraic thinking (NCTM 1989). According to Bruner (1995, 333), mathematics should feel like something a child already knows rather than something totally unfamiliar. He states that the end product of mathematics should not be formalism with a premature emphasis on mathematical language but the confidence a child gains from realizing that mathematics is something that he or she has been thinking right along. This two-part lesson, complete with a step-by-step student activity log, allows students to build on their elementary school experiences of measuring in nonstandard units—disguised here as variables—and thus draw their own blueprint of algebra.

Creating Connections: Promoting Algebraic Thinking with Concrete Models

2001 ◽  
Vol 7 (1) ◽  
pp. 20-25
Author(s):  
Michaele F. Chappell ◽  
Marilyn E. Strutchens
Keyword(s):  
Natural Extension ◽  
Eighth Graders ◽  
Algebraic Thinking ◽  
The Other ◽  
Seventh Graders ◽  
Science Study ◽  
Algebra For All ◽  
The Third ◽  
Algebraic Concepts ◽  
Mathematics And Science

The recent “Algebra for all” era has meant “the best of times and the worst of times” in many middle schools. At one extreme, many adolescents delight in the opportunity to study algebra or algebraic thinking and perform well in this course of study. At the other extreme, too many adolescents encounter serious challenges as they delve into fundamental ideas that make up this essential mathematical subject. Instead of viewing algebra as a natural extension of their arithmetic experiences, significant numbers of adolescents do not connect algebraic concepts with previously learned ideas. For instance, data from the Third International Mathematics and Science Study (TIMSS) showed that at the international level, only 47 percent of the seventh graders and only 58 percent of the eighth graders were able to recognize that m + m + m + m was equivalent to 4m (Beaton et al. 1996).

Fusion in relational structures and the verification of monadic second-order properties

2002 ◽  
Vol 12 (2) ◽  
pp. 203-235 ◽  
Author(s):  
B. COURCELLE ◽  
J. A. MAKOWSKY
Keyword(s):  
Second Order ◽  
Algebraic Expression ◽  
Order Property ◽  
Unary Predicate ◽  
Free Graph ◽  
Relational Structures ◽  
Clear Cut ◽  
Algebraic Expressions ◽  
Common Framework ◽  
Fusion Operation

Relational structures offer a common framework for handling graphs and hypergraphs of various kinds. Operations like disjoint union, the creation of new relations by means of quantifier-free formulas, and relabellings of relations make it possible to denote them using algebraic expressions. It is known that every monadic second-order property of a structure is verifiable in time proportional to the size of such an algebraic expression defining it. We prove here that this result remains true if we also use in these algebraic expressions a fusion operation that fuses all elements of the domain satisfying some unary predicate. The value mapping from these algebraic expressions to the structures they denote is a monadic second-order definable transduction, which means that the structure is definable inside the tree representing the algebraic expression by monadic second-order formulas. It follows (by using results of other articles) that, with this fusion operation, we cannot generate more graph families, but we can generate them with less unary auxiliary predicates. We also obtain clear-cut characterizations of Vertex Replacement and Hyperedge Replacement context-free graph grammars in terms of four types of operations, amongst which is the fusion of vertices satisfying a specified predicate.

scholarly journals Examination of Semantic Structures Used by Teacher Candidates to Transform Algebraic Expressions into Verbal Problems

2021 ◽  
Author(s):  
Ayten Pinar Bal
Keyword(s):  
Teacher Candidates ◽  
Quantitative Research ◽  
Algebraic Expression ◽  
Study Group ◽  
State University ◽  
Algebraic Expressions ◽  
Purposeful Sampling ◽  
Division Problems ◽  
Addition And Subtraction ◽  
Research Types

The aim of this study is to examine the semantic structures used by mathematics teacher candidates to transform algebraic expressions into verbal problems. The research is a descriptive study in the survey model, which is one of the quantitative research types. The study group of the research consists of 165 teacher candidates studying in the primary school mathematics teaching department of a state university in the south of Turkey in the 2019-2020 academic years. 73.2% of the teacher candidates in the study group are female and 26.8% are male. Criterion sampling method, one of the purposeful sampling methods, was used in the selection of teacher candidates in the study group. While the Algebraic Expression Questionnaire Form was used as the data collection tool, the evaluation rubric of verbal problems was used in the analysis of the data. As a result of the research, it has been revealed that pre-service teachers are more successful in transforming algebraic expressions into verbal problems, but they have problems in creating problems with algebraic expressions that make up systems of equations. Again in the study, it was concluded that pre-service teachers used addition and subtraction problems more than multiplication and division problems. On the other hand, when the problems in the type of addition and subtraction are examined in the study, in the type of combining and separating; It has been concluded that the category of equal groups is mostly used in the problems of multiplication and division.

Algebra for All: To Encourage “Algebra for All,” Start an Algebra Network

1998 ◽  
Vol 91 (4) ◽  
pp. 282-365
Author(s):  
Suzanne Davis ◽  
Denisse R. Thompson
Keyword(s):  
Professional Development ◽  
High School Teachers ◽  
Position Statement ◽  
Algebraic Thinking ◽  
National Council ◽  
School Teachers ◽  
Algebra For All ◽  
Curriculum Changes ◽  
Enrollment Patterns ◽  
K 12

In 1994, the National Council of Teachers of Mathematics issued a position statement titled “Algebra for Everyone … More Than a Change in Enrollment Patterns” (NCTM 1994). Success in addressing this call will involve changes on many fronts. Although curriculum changes will be an essential ingredient, teachers at all levels will need to assume responsibility for fostering algebraic thinking. This article describes the efforts in one large county in the Southeast to provide professional development and encourage dialogue among elementary, middle, and high school teachers about issues related to algebraic thinking. This dialogue is essential in helping teachers appreciate their role in the development of algebra as a K–12 concept. By constantly weaving algebraic ideas into the curriculum at all levels, the district's teachers hope to give students the necessary background to pursue successfully the formal study of algebra.

scholarly journals Yngre elevers uppfattningar av det matematiska i algebraiska uttryck

2021 ◽  
Vol 9 (1) ◽  
Author(s):  
Sanna Wettergren ◽  
Inger Eriksson ◽  
Torbjörn Tambour
Keyword(s):  
Full Text ◽  
Ways Of Knowing ◽  
Algebraic Expression ◽  
Research Project ◽  
Algebraic Expressions ◽  
Preschool Class ◽  
Critical Aspects

Det övergripande syftet med denna artikel är att analysera och beskriva yngre elevers uppfattningar av det matematiska i ett algebraiskt uttryck och utifrån det diskutera vad som kan utgöra kritiska aspekter för utvecklandet av mera kvalificerade uppfattningar. Artikeln bygger på data från ett forskningsprojekt där elever i förskoleklass, årskurs 1 och årskurs 4 intervjuades med syfte att analysera de aktuella elevernas kvalitativt skilda sätt att uppfatta det matematiska i algebraiska uttryck. Intervjuerna analyserades fenomenografiskt. Studiens resultat visar tre kvalitativt skilda kategorier av yngre elevers uppfattningar av det matematiska i algebraiska uttryck. Det matematiska i ett algebraiskt uttryck erfars som ”något som kan och bör räknas ut”, ”något som beskriver en relation mellan komponenter” och ”något som representerar en situation”. Vidare identifierades tre kritiska aspekter i relation till kategorierna. De kritiska aspekter som ger eleverna möjlighet att kvalificera sina uppfattningar för att utveckla ett mer komplext kunnande av algebraiska uttryck är att kunna urskilja att 1) ett uttryck består av olika komponenter som har olika funktioner, 2) en och samma variabel i ett uttryck har samma värde och 3) värdet på en variabel i ett uttryck bestäms relationellt. Att urskilja sådana kritiska aspekter kan hjälpa eleverna att kvalificera sitt kunnande. Således måste de kritiska aspekterna beaktas vid utformningen av undervisningen. Abstract in English The overall purpose of this article is to analyze and describe younger students’ conceptions of or ways of experiencing the mathematics in an algebraic expression and to discuss what can be critical aspects for the development of more qualified conceptions. The article is based on data from a research project where students in preschool class, Grade 1 and Grade 4 were interviewed with the aim of analyzing the students' qualitatively different ways of experiencing the mathematics in algebraic expressions. The interviews were analyzed with phenomenography. The results show three qualitatively different categories of younger students’ conceptions of the mathematics in algebraic expressions. The mathematics in an algebraic expression is experienced as ”something that can and should be calculated", ”something that describes a relationship between components”, and ”something that represents a situation”. Furthermore, three so-called critical aspects the students need to discern were identified in relation to the categories 1) an expression consists of different components that have different functions, 2) one and the same variable in an expression has the same value and 3) the value of a variable in an expression is determined relationally. Discerning such critical aspects may help the students to qualify their ways of knowing. Thus, the critical aspects need to be considered in the design of teaching. FULL TEXT IN SWEDISH.

Analyzing the Potential Benefit of Microcomputer Use for Teaching Figurative Language

1992 ◽  
Vol 1 (2) ◽  
pp. 36-43 ◽  
Author(s):  
Marilyn A. Nippold ◽  
Ilsa E. Schwarz ◽  
Molly Lewis
Keyword(s):  
Language Learning ◽  
Children And Adolescents ◽  
Standardized Tests ◽  
Figurative Language ◽  
Language Intervention ◽  
School Age Children ◽  
Educational Materials ◽  
Language Learning Disabilities ◽  
Promising Application ◽  
Experience Difficulty

Microcomputers offer the potential for increasing the effectiveness of language intervention for school-age children and adolescents who have language-learning disabilities. One promising application is in the treatment of students who experience difficulty comprehending figurative expressions, an aspect of language that occurs frequently in both spoken and written contexts. Although software is available to teach figurative language to children and adolescents, it is our feeling that improvements are needed in the existing programs. Software should be reviewed carefully before it is used with students, just as standardized tests and other clinical and educational materials are routinely scrutinized before use. In this article, four microcomputer programs are described and evaluated. Suggestions are then offered for the development of new types of software to teach figurative language.

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