Developing Algebraic Thinking through Pattern Exploration

2006 ◽  
Vol 11 (9) ◽  
pp. 428-433 ◽  
Author(s):  
Lesley Lee ◽  
Viktor Freiman
Keyword(s):  
Middle School ◽  
Middle Grades ◽  
Algebraic Thinking ◽  
School Mathematics ◽  
Scientific Disciplines ◽  
Algebraic Language ◽  
Principles And Standards ◽  
Relationship Of ◽  
The Relationship ◽  
Repeating Patterns

Pattern exploration is A pivotal activity in all mathematics, indeed in all the scientific disciplines. Children who are attempting to express perceived patterns mathematically are in an excellent position to learn algebraic language and engage in algebraic activity. Principles and Standards for School Mathematics (NCTM 2000) acknowledges the relationship of pattern exploration and algebraic thinking by placing pattern work within the Algebra strand. Yet one can undertake considerable pattern exploration without engaging students in any algebraic thinking whatsoever and teachers may, themselves, be unclear about how patterns can be used to further algebraic thinking. Work with repeating patterns in the early grades, or teaching patterns as a “topic” in the middle grades, may not foster the development of algebraic thinking in students. In this article, we will address this question: How can teachers exploit pattern work to further algebraic thinking and introduce the formal study of algebra in middle school?

Concentration: Connecting Fractions, Decimals, & Percents

2000 ◽  
Vol 5 (5) ◽  
pp. 324-328
Author(s):  
Elizabeth S. Sweeney ◽  
Robert J. Quinn
Keyword(s):  
Middle School ◽  
Middle Grades ◽  
Middle School Mathematics ◽  
School Mathematics ◽  
Innovative Method ◽  
The Relationship ◽  
The Way

Fractions, Decimals, and Percents are often included in discussions about middle school mathematics. Unfortunately, these discussions also include groans of dissatisfaction, stemming from the lack of success that teachers often have in teaching these concepts. Many students fail to see the relationship among fractions, decimals, and percents. As one student put it, a decimal is “a thing that makes numbers even more confusing,” whereas another characterized a percent as “the way teachers give you points.” The fact that these topics are typically taught in isolation is the main source of dissatisfaction. Often, the only connection mentioned by textbooks is a cursory discussion of conversions. This article describes one innovative method that can help middle-grades students become more flexible in their ability to represent fractions, decimals, and percents, an outcome recommended by the NCTM's Standards.

Short Takes: Another Good Idea: Pasta Symmetry

2007 ◽  
Vol 12 (9) ◽  
pp. 516-517
Author(s):  
Tara Windle
Keyword(s):  
Middle School ◽  
Middle School Students ◽  
Middle Grades ◽  
Assessment Tool ◽  
Rotational Symmetry ◽  
Authentic Assessment ◽  
School Mathematics ◽  
School Students ◽  
Mathematical Ideas ◽  
Principles And Standards

Students enjoy the chance to be creative, especially those in the middle grades. Teachers can channel that creative energy into an authentic assessment tool that students will love. Principles and Standards for School Mathematics states that students in middle school are expected to “apply transformations and use symmetry to analyze mathematical situations” (p. 232). Our students have also been challenged to “recognize and apply mathematics in contexts outside of mathematics” (p. 274) and to “create and use representations to organize, record, and communicate mathematical ideas” (p. 280). Using card-stock paper, glue, gold spray paint (optional), and as many varieties of pasta as I could find, I gave my sixthgrade middle school students the opportunity to convince me that they understood the concepts of reflectional and/or rotational symmetry while creating a unique piece of art.

University-Based Teacher Preparation and Middle Grades Teacher Effectiveness

2016 ◽  
Vol 68 (1) ◽  
pp. 102-116 ◽  
Author(s):  
Courtney Preston
Keyword(s):  
Teacher Preparation ◽  
Teacher Effectiveness ◽  
Middle Grades ◽  
Structural Features ◽  
Program Improvement ◽  
Teacher Preparation Programs ◽  
Preparation Programs ◽  
Achievement Gains ◽  
Relationship Of ◽  
The Relationship

For over two decades, there have been calls to assess the relationship of the features of teacher preparation programs to teacher effectiveness, to provide guidance for program improvement. At the middle grades level, theory suggests that coursework in educational psychology is particularly important for teacher effectiveness. Using 4 years of data from 15 middle grades teacher preparation programs, this study estimates the relationship of their structural features, that is required elements of coursework and fieldwork, to student achievement gains in math and English/Language Arts. Findings suggest that few requirements are positively associated with achievement gains.

Soundoff: Geometry: A Remedy for the Malaise of Middle School Mathematics

1989 ◽  
Vol 82 (9) ◽  
pp. 678-680
Author(s):  
Alfred S. Posamentier
Keyword(s):  
Middle School ◽  
Middle Grades ◽  
Mathematics Curriculum ◽  
Middle School Mathematics ◽  
School Mathematics ◽  
School Level ◽  
Mathematics Educators ◽  
Arithmetic Teaching ◽  
Middle School Level

Many mathematics educators perceive that the weakest part of the precollege mathematics curriculum is at the middle school level, more specifically, the years immediately preceding the study of algebra. It seems that in the middle grades the development of mathematics has been put into a “holding pattern.” A quick glance at the curriculum for seventh and eighth grades—or in some cases sixth and seventh gradesshows that much arithmetic is still being taught. Haven't we, or shouldn't we have, completed teaching arithmetic in the previous five or six years? Indeed, how much arithmetic teaching do we need to do in an age of ever-improving calculators (Heid 1988)? Very often students greet a unit in these grades with the now famous comment, “Oh, I had this already.” “Sure,” thinks the teacher, “you may have had it, but have you learned it?” It is clear to many educators that these middle grades are key to turning a student “on” to or “off” from mathematics.

Focused Strategies for Middle-Grades Mathematics Vocabulary Development

2007 ◽  
Vol 13 (4) ◽  
pp. 200-207
Author(s):  
Rheta N. Rubenstein
Keyword(s):  
Mathematics Education ◽  
Middle Grades ◽  
Mathematical Thinking ◽  
Vocabulary Development ◽  
School Mathematics ◽  
One Dimension ◽  
Mathematical Language ◽  
Mathematics Vocabulary ◽  
Principles And Standards

Principles and Standards for School Mathematics reminds us that communication is central to a broad range of goals in mathematics education (NCTM 2000). These goals include students' being able to (1) organize and consolidate mathematical thinking; (2) communicate coherently with teachers, peers, and others; (3) analyze and evaluate others' strategies; and (4) use language to express mathematics precisely. One part of communication is acquiring mathematical language and using it fluently. This article addresses learning vocabulary as one dimension of mathematics communication.

“I Can't Write All the Way to 100”: Recognizing Students' Emerging Algebraic Strategies

2007 ◽  
Vol 13 (5) ◽  
pp. 278-282
Author(s):  
Vanessa R. Pitts Bannister ◽  
Jesse L. M. Wilkins
Keyword(s):  
Mathematical Models ◽  
Algebraic Thinking ◽  
School Mathematics ◽  
Student Work ◽  
Algebraic Concepts ◽  
Principles And Standards ◽  
Active Exploration ◽  
The Way

In Principles and Standards for School Mathematics (NCTM 2000), understandings of patterns, relations, functions, mathematical models, and quantitative relationships are recognized as key facets of algebraic thinking. In essence, algebraic thinking “embodies the construction and representation of patterns of regularities, deliberate generalization, and most important, active exploration and conjecture” (Chambers 1994, p. 85). Algebraic thinking should function as a means of shifting from arithmetic concepts to algebraic concepts (Chappell 1997). This shift would have occurred if there exists reasoning about relationships between quantities, rather than the specific quantities themselves (Ferrini-Mundy, Lappan, and Phillips 1997; Yackel 1997). Research shows that this arithmetic to algebraic shift is difficult for students (Stacey and MacGregor 2000). Therefore, it is imperative to explore students' reasoning as they approach problems that elicit algebraic thinking. For this reason, we will present and discuss samples of student work regarding problems that promote algebraic thinking.

Lollipop Statistics

2003 ◽  
Vol 9 (1) ◽  
pp. 12-15
Author(s):  
Dianne S. Goldsby
Keyword(s):  
High School ◽  
Middle School ◽  
High School Students ◽  
School Mathematics ◽  
Constructivist Approach ◽  
School Students ◽  
Area Of Interest ◽  
Approach To Teaching ◽  
The 1950S ◽  
Principles And Standards

AS NCTM'S Principles and Standards for School Mathematics (2000) points out, students should work directly with data to understand the fundamentals of statistical ideas. Teachers should also introduce statistics in a way that will capture the attention of students of varying abilities and interests. The constructivist approach to teaching emphasizes the idea that students work better when presented with tasks that are meaningful and relevant; in other words, they expend energy on topics that interest them (Brahier 2000). One way to harness that energy in the classroom is to teach with music, an area of interest for most middle school and high school students. This article describes the use of the 1950s hit “Lollipop” (Ross and Dixon 1986), heard in the movie Stand by Me, as a launching point to introduce ideas of counting, working with frequency tables, and graphing data.

Spotlight on the Principles/Standards: Problem Solving in the Middle Grades

2000 ◽  
Vol 6 (2) ◽  
pp. 105-108
Author(s):  
Carol E. Malloy ◽  
D. Bruce Guild
Keyword(s):  
Problem Solving ◽  
Mathematical Knowledge ◽  
Middle Grades ◽  
Mathematics Curriculum ◽  
School Mathematics ◽  
Measurement Algebra ◽  
Principles And Standards ◽  
Mathematical Relationships

IN WHAT WAYS WOULD YOU LIKE YOUR middle-grades students to experience problem solving in the mathematics curriculum? Do you want the curriculum to capture the excitement of geometry and measurement, algebra, statistics, and number relationships? Do you want it to help students understand and build new mathematical knowledge and explore new mathematical relationships? Do you want the curriculum to be filled with opportunities for students to ponder, create, and critique arguments about mathematics? If this is your vision for your students, then you should be pleased with, and excited by, the Problem Solving Standard in Principles and Standards for School Mathematics (NCTM 2000).

Call for Manuscripts: Big Ideas of Algebra

2009 ◽  
Vol 15 (7) ◽  
pp. 429
Keyword(s):  
High School ◽  
Middle Grades ◽  
School Mathematics ◽  
Big Ideas ◽  
Solid Foundation ◽  
Principles And Standards

“By viewing algebra as a strand in the curriculum from prekindergarten on, teachers can help students build a solid foundation of understanding and experience as a preparation for moresophisticated work in algebra in the middle grades and high school” (NCTM Principles and Standards for School Mathematics, p. 37).

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