Paul Goldenberg

Welcome to “Delving Deeper,” a new department in the Mathematics Teacher. The concept for this department has been evolving for several years, but the basic goal is to provide teachers with a forum for their own mathematical investigations. Do you have an idea in your mind or on your computer that is based on one of your mathematical explorations? Here is a chance to get your idea off your hard drive and into print. Submissions of up to ten single-spaced pages are welcomed. Articles written jointly with others (including students or mathematicians) are encouraged.

From the first time that we used such inter-active geometry programs as The Geometer's Sketchpad and Cabri Geometry, we were intrigued by their potential to help students develop ways of thinking that underlie calculus and analysis (Cuoco, Goldenberg, and Mark 1995; Cuoco and Goldenberg 1997). One theme in this approach has been to look at optimization problems as geometrically defined functions, for example, the sum of the distances from a point to three fixed points. In an interactive geometry world such as Sketchpad or Cabri, the user does not need to impose coordinates on the plane and specify the distances algebraically at the outset. He or she may define the function directly as a geometric relationship; manipulate its variable (the movable point); and observe, numerically or geometrically, the value (sum of distances) that results. The algebraic step is also valuable; both geometric and algebraic interpretations lead to important insights. But the geometric step is often a particularly productive starting place for generating the ideas that one may want to revisit algebraically.

In a recent “Sound Off” in Mathematics Teacher, Robert Reys and Rustin Reys (2009) contrasted two curricular approaches, what they called “subjectbased” and “integrated.” They came down heavily in favor of the latter, arguing that many of the difficulties that students have with high school mathematics are consequences of the subject–based organization.

Although it is necessary to infuse courses and curricula with modern content, what is even more important is to give students the tools they will need in order to use, understand, and even make mathematics that does not yet exist. A curriculum organized around habits of mind tries to close the gap between what the users and makers of mathematics do and what they say (Cuoco, Goldenberg, and Mark 1996, p. 376).

Elementary school teachers are in the unique and difficult position of managing just about everything, and often try to integrate their activities, finding ways to draw math lessons from literature, science, art, or lunch-money collection, finding ways to draw language lessons ...

This glossary was written for the purpose of providing a quick and concise yet accurate description of the primitives and special words and characters of the March 18, 1975 PDP 11 implementation of the LOGO languge. Many entries include references to other related words and/or examples of the use of the primitive being described, but this is not intended to replace the functions of a good manual. For a more detailed and comprehensive description of the language, see the LOGO MANUAL, LOGO MEMO 7. The description of each LOGO word includes the work, itself, any arguments that the word may require, the "type" of word it is, abbreviated and alternate forms of the work, if any, and a definition correct as the date of this glossary. Word tupe is described on the first page and an example of the formatt of the entries is given below. In the appendix to this glossary are sections about 1) LOGO words that take a variable number of inputs, 2) infix operators, 3) editing characters, 4) special characters, 5) special names, 6) decimal ascii code and corresponding characters

Natural language helps express mathematical thinking and contexts. Conventional mathematical notation (CMN) best suits expressions and equations. Each is essential; each also has limitations, especially for learners. Our research studies how programming can be a advantageous third language that can also help restore mathematical connections that are hidden by topic‐centred curricula. Restoring opportunities for surprise and delight reclaims mathematics' creative nature. Studies of children's use of language in mathematics and their programming behaviours guide our iterative design/redesign of mathematical microworlds in which students, ages 7–11, use programming in their regular school lessons as a language for learning mathematics. Though driven by mathematics, not coding, the microworlds develop the programming over time so that it continues to support children's developing mathematical ideas. This paper briefly describes microworlds EDC has tested with well over 400 7‐to‐8‐year‐olds in school, and others tested (or about to be tested) with over 200 8‐to‐11‐year‐olds. Our challenge was to satisfy schools' topical orientation and fit easily within regular classroom study but use and foreshadow other mathematical learning to remove the siloes. The design/redesign research and evaluation is exploratory, without formal methodology. We are also more formally studying effects on children's learning. That ongoing study is not reported here. Practitioner notesWhat is already known Active learning—doing—supports learning. Collaborative learning—doing together—supports learning. Classroom discourse—focused, relevant discussion, not just listening—supports learning. Clear articulation of one's thinking, even just to oneself, helps develop that thinking. What this paper adds The common languages we use for classroom mathematics—natural language for conveying the meaning and context of mathematical situations and for explaining our reasoning; and the formal (written) language of conventional mathematical notation, the symbols we use in mathematical expressions and equations—are both essential but each presents hurdles that necessitate the other. Yet, even together, they are insufficient especially for young learners. Programming, appropriately designed and used, can be the third language that both reduces barriers and provides the missing expressive and creative capabilities children need. Appropriate design for use in regular mathematics classrooms requires making key mathematical content obvious, strong and the ‘driver’ of the activities, and requires reducing tech ‘overhead’ to near zero. Continued usefulness across the grades requires developing children's sophistication and knowledge with the language; the powerful ways that children rapidly acquire facility with (natural) language provides guidance for ways they can learn a formal language as well. Implications for policy and/or practice Mathematics teaching can take advantage of the ways children learn through experimentation and attention to the results, and of the ways children use their language brain even for mathematics. In particular, programming—in microworlds driven by the mathematical content, designed to minimise distraction and overhead, open to exploration and discovery en route to focused aims, and in which children self‐evaluate—can allow clear articulation of thought, experimentation with immediate feedback. As it aids the mathematics, it also builds computational thinking and satisfies schools' increasing concerns to broaden access to ideas of computer science.

This chapter argues for fidelity to mathematics in the preparation of elementary teachers by immersing them in doing mathematics—at their knowledge level, of course—by giving them experiences that are true to its investigative, logical, problem-solving, and system-building, curiosity-inspiring nature. Learning to think like a mathematician is very different from learning the facts and procedures that result from that thinking. Mathematics includes both. Doing mathematics—knowing what mathematics is—requires knowledge, but it is not about that knowledge; it is about what you can figure out using that knowledge. Courses for teachers that focus on explaining facts and procedures and how they work miss the heart of mathematics, the thinking. Teachers will naturally perpetuate that message. By contrast, doing mathematics, centered in the discipline of elementary teaching and its contents, cannot help but also include its key facts, affording a more faithful view of the discipline.

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How people see the world, even how they research it, is influenced by beliefs. Some beliefs are conscious and the result of research, or at least amenable to research. Others are largely invisible. They may feel like “common knowledge” (though myth, not knowledge), unrecognized premises that are part of the surrounding culture. As we will explain, people also hold ideas in both a detailed form and in a thumbnail image and may not notice when they are using the low-resolution image in place of the full picture. In either case, unrecognized myths about how young learners develop mathematical ideas naturally or with instruction are insidious in that they persist unconsciously and so sway research and practice without being examined rigorously. People are naturally oblivious to the ramifications of unrecognized premises (myths) until they encounter an anomaly that cannot be explained without reexamining those premises. Like all disciplines, mathematics education is shaped and constraine...

How you think about a phenomenon certainly influences how you create a program to model it. The main point of this essay is that the influence goes both ways: creating programs influences how you think. The programs we are talking about are not just the ones we write for a computer. Programs can be implemented on a computer or with physical devices or in your mind. The implementation can bring your ideas to life. Often, though, the implementation and the ideas develop in tandem, each acting as a mirror on the other. We describe an example of how programming and mathematics come together to inform and shape our interpretation of a classical result in mathematics: Euclid's algorithm that finds the greatest common divisor of two integers.

Inevitably, reading is one of the requirements to be undergone. To improve the performance and quality, someone needs to have something new every day. It will suggest you to have more inspirations, then. However, the needs of inspirations will make you searching for some sources. Even from the other people experience, internet, and many books. Books and internet are the recommended media to help you improving your quality and performance.

IN THIS ARTICLE, WE GIVE AN EXAMPLE OF WHAT “delving deeper” might mean with respect to standard, rather ordinary high school problems. The purpose is to illustrate the mathematical depth that is potentially present, even in simple problems. We use what we call an extended analysis of a problem, which is an analysis from a mature mathematical perspective, with careful attention paid to mathematical reasoning and to using good mathematical habits of mind. (See Cuoco, Goldenberg, and Mark 1996.) Our intent is to foster powerful ways of thinking that are characteristic of mathematics and science.

Elementary school teachers are in the unique and difficult position of managing just about everything, and often try to integrate their activities, finding ways to draw math lessons from literature, science, art, or lunch-money collection, finding ways to draw language lessons ...

Abstract 1. Discusses the use of computer software as an aid in teaching graphing functions to mathematics students. It is noted that students often make significant misinterpretations of what they see in graphic representations of functions. To interpret graphs correctly, ...

This chapter argues for fidelity to mathematics in the preparation of elementary teachers by immersing them in doing mathematics—at their knowledge level, of course—by giving them experiences that are true to its investigative, logical, problem-solving, and system-building, curiosity-inspiring nature. Learning to think like a mathematician is very different from learning the facts and procedures that result from that thinking. Mathematics includes both. Doing mathematics—knowing what mathematics is—requires knowledge, but it is not about that knowledge; it is about what you can figure out using that knowledge. Courses for teachers that focus on explaining facts and procedures and how they work miss the heart of mathematics, the thinking. Teachers will naturally perpetuate that message. By contrast, doing mathematics, centered in the discipline of elementary teaching and its contents, cannot help but also include its key facts, affording a more faithful view of the discipline.

... The first part takes up the grammar of sentences, poems, and stories -syntax, semantics ... E. Paul Goldenberg is Researcher and Developer at Education Development Center, Inc., Newton ... and former Director of the Computer Center at Lincoln-Sudbury Regional High School. ...

Abstract: Building coherence in the development of mathematical ideas across the grades is key to improving students' mathematical learning in the United States. Knowing the mathematical experiences, understanding, skills, and habits of mind that students bring to ...

This paper reports on a collaboration between curriculum developers, classroom teachers, researchers, and education administrators to develop and test curriculum materials that integrate computational thinking into elementary grades science and mathematics instruction. It discusses different levels of integration, provides an example, and shares questions and challenges that have grown out of this work.