“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.

A Look at the Development of Algebraic Thinking in Curriculum Focal Points

2007 ◽  
Vol 13 (5) ◽  
pp. 266-269
Author(s):  
Cathy Seeley ◽  
Jane F. Schielack
Keyword(s):  
High School ◽  
High School Students ◽  
The United States ◽  
Algebraic Thinking ◽  
School Mathematics ◽  
Algebraic Reasoning ◽  
School Students ◽  
Evaluation Standards ◽  
Algebraic Concepts ◽  
Principles And Standards

As more and more students are expected to study algebra in middle school or high school, much attention has been given to the development of algebraic reasoning. In Curriculum and Evaluation Standards for School Mathematics (1989), the NCTM advocated incorporating algebraic thinking from the early grades on. Principles and Standards for School Mathematics (2000) further strengthened this stand by highlighting a new Standard on Representation, a foundational tool for algebra. Whether high school students are expected to enroll in a course called agebra 1 or an integrated high school course that contains algebraic content, it is clear that the United States is placing more emphasis on preparing for the use of algebraic concepts and skills.

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?

Appendix Mathematical models of the knee

2015 ◽  
Author(s):  
John Goodfellow ◽  
John O'Connor ◽  
Hemant Pandit ◽  
Christopher Dodd ◽  
David Murray
Keyword(s):  
Clinical Research ◽  
Mathematical Models ◽  
Experimental Method ◽  
Experimental Measurement ◽  
Physical System ◽  
The Way

Mathematical models make it possible to calculate the values of quantities which are difficult or impossible to measure and provide insights which are not obtained from experiment alone. They are a necessary adjunct to the experimental method, but are not a common feature of biological or clinical research. A model is based on a series of assumptions or hypotheses about the way a physical system works. It is validated by comparing its predictions with independent experimental measurement. Reasonable validation then gives confidence in the assumptions on which the model is based and in the predictions of quantities which cannot be measured. The purpose in presenting our models here is to explain the differences between unloaded and loaded motion described in Chapter 3.

What influence will the new Principles and Standards for School Mathematics (2000) have?

2000 ◽  
Vol 31 (4) ◽  
pp. 394-395
Author(s):  
Judith T. Sowder
Keyword(s):  
Annual Meeting ◽  
Research Community ◽  
School Mathematics ◽  
The Public ◽  
Principles And Standards

The new NCTM Principles and Standards for School Mathematics (2000) were presented to the public with great fanfare at the NCTM Annual Meeting in Chicago in April of this year. The mood was celebratory, perhaps even more so than when the 1989 Standards were presented. How will these new Principles and Standards be accepted? What influence will they have? Are there messages here to which the research community ought to be attending?

scholarly journals Modelling ecological systems in a changing world

2012 ◽  
Vol 367 (1586) ◽  
pp. 181-190 ◽  
Author(s):  
Matthew R. Evans
Keyword(s):  
Environmental Change ◽  
Mathematical Models ◽  
Ecological Systems ◽  
Ecological Modelling ◽  
Causal Relationships ◽  
Future State ◽  
The World ◽  
The Future ◽  
The Impact ◽  
The Way

The world is changing at an unprecedented rate. In such a situation, we need to understand the nature of the change and to make predictions about the way in which it might affect systems of interest; often we may also wish to understand what might be done to mitigate the predicted effects. In ecology, we usually make such predictions (or forecasts) by making use of mathematical models that describe the system and projecting them into the future, under changed conditions. Approaches emphasizing the desirability of simple models with analytical tractability and those that use assumed causal relationships derived statistically from data currently dominate ecological modelling. Although such models are excellent at describing the way in which a system has behaved, they are poor at predicting its future state, especially in novel conditions. In order to address questions about the impact of environmental change, and to understand what, if any, action might be taken to ameliorate it, ecologists need to develop the ability to project models into novel, future conditions. This will require the development of models based on understanding the processes that result in a system behaving the way it does, rather than relying on a description of the system, as a whole, remaining valid indefinitely.

What is Standing in the Way of Middle School Mathematics Curriculum Reform?

1998 ◽  
Vol 30 (2) ◽  
pp. 42-48 ◽  
Author(s):  
Robert Reys ◽  
Barbara Reys ◽  
David Barnes ◽  
John Beem ◽  
Ira Papick
Keyword(s):  
Middle School ◽  
Curriculum Reform ◽  
Mathematics Curriculum ◽  
Middle School Mathematics ◽  
School Mathematics ◽  
School Mathematics Curriculum ◽  
The Way

Effective and Efficient Use of Math Writing Tasks

2018 ◽  
Vol 112 (2) ◽  
pp. 143-146 ◽  
Author(s):  
Matt M. Bixby
Keyword(s):  
School Mathematics ◽  
Clear Picture ◽  
Student Writing ◽  
National Council ◽  
Writing In Mathematics ◽  
Mathematics Class ◽  
Principles And Standards ◽  
The Many

Almost twenty years ago, the National Council of Teachers of Mathematics (NCTM) published Principles and Standards for School Mathematics (2000), which recommended that teachers should incorporate more writing into their math lessons, claiming that writing helps students “consolidate their thinking” (p. 402) by causing them to reflect on their work. In recent years, various studies point to the many benefits that can be gained by writing in mathematics class (e.g., O'Connell et al. 2005; Goldsby and Cozza 2002). Much research suggests that writing activities, if implemented effectively, can help students enjoy class more (Burns 2005) and can also help them deepen their understanding of the content (Baxter et al. 2002). In addition to benefiting students, student writing benefits teachers as well by providing a clear picture of what their students understand and even deepening understanding of the content for teachers themselves (Burns 2005; Pugalee 1997).

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