Annotated Bibliography for 2011 Book by Moursund and Albrecht





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This IAE-pedia page contains an annotated version of the bibliography in the following book:

Moursund, Dave and Albrecht, Bob (2011). Using math games and word problems to increase the math maturity of K-8 students. Salem, OR: The Math Learning Center.


 * This book was created for preservice and inservice teachers with the goal of improving the informal and formal math education of preK-8 students. The authors emphasize using simple, inexpensive games to provide students with learning environments that help to increase their levels of math maturity.

The book is a mixture of theory and practice. It contains a careful analysis of a small number of games and links to a large number of games and related resources.


 * The book has a strong emphasis on students learning to develop and test strategies. This is in marked contrast with a common approach to learning about strategies in which students are expected to memorize (and, accept without question) strategies to solve or help solve a variety of problems.


 * The book is 222 pages in length and is available via electronic download for $10. See ordering information and some sample at http://www.mathlearningcenter.org/node/6948.

More information about the book and access to some parts of the content are available at http://iae-pedia.org/IAE_Progress_Report.

The Annotated Bibliography is available under available under Creative Commons Attribution-Share Alike 3.0.

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Asgari, Mahboubeh; & Kaufman (2004). Relationships among computer games, fantasy, and learning. Retrieved 1/22/2010 from http://www.ierg.net/confs/2004/Proceedings/Asgari_Kaufman.pdf.

Computer games can give the opportunity to learners to explore their imagination comfortably (Millians, 1999). Using fantasies, mental images and non-real situations in computer games, can stimulate learners’ behavior (Vockell, 2004). To make learning motivating and appealing for learners, one way is to present the material to them either in an imaginary context which is familiar to them or in a fantasy context which is emotionally appealing for the learner (Malone and Lepper, 1987). Creating such environments that stimulate learners to become absorbed in a fantasy world can motivate and engage them in activities (Cordova, 1993). Empirical research suggests that embedding material in a fantasy context can enhance learning more than a generic, decontextualized environment (Cordova, 1993; Garris et al. cites from Druckman, 1995).

This paper focuses on the relationship between fantasy and learning in computer-based instructional games. Since learning is believed to be one of play benefits which is related to factors such as increased motivation (Rieber, 2001), and computer games are reported to increase motivation, we first review the features that make such games 2 motivational. Among those features, we will focus on ‘fantasy’. The features ‘curiosity’ and ‘goal’ will also be explained, as these two game characteristics are also related to our discussion. Finally, we report what previous research has shown in regard with presenting instructional materials in fantasy contexts and their effects on learning.

Bentsen, Todd (11 Apr 2009). Adult brain processes fractions 'effortlessly.' Medical News Today. Retrieved 9/19/09 from http://www.medicalnewstoday.com/articles/145587.php.

Quoting from the article:

Although fractions are thought to be a difficult mathematical concept to learn, the adult brain encodes them automatically without conscious thought, according to new research in the April 8 issue of The Journal of Neuroscience.

"Fractions are often considered a major stumbling block in math education," said Daniel Ansari, PhD, at the University of Western Ontario in Canada, an expert on numerical cognition in children and adults who was not affiliated with the study. "This new study challenges the notion that children must undergo a qualitative shift in order to understand fractions and use them in calculations. The findings instead suggest that fractions are built upon the system that is employed to represent basic numerical magnitude in the brain," Ansari said.

Brown, John Seely; Collins, Allan; and Duguid, Paul (1989). Situated cognition and the culture of learning. Educational Researcher. Retrieved 1/23/2010 from http://www.sociallifeofinformation.com/Situated_Learning.htm. Quoting the Abstract of this article:

Many teaching practices implicitly assume that conceptual knowledge can be abstracted from the situations in which it is learned and used. This article argues that this assumption inevitably limits the effectiveness of such practices. Drawing on recent research into cognition as it is manifest in everyday activity, the authors argue that knowledge is situated, being in part a product of the activity, context, and culture in which it is developed and used. They discuss how this view of knowledge affects our understanding of learning, and they note that conventional schooling too often ignores the influence of school culture on what is learned in school. As an alternative to conventional practices, they propose "cognitive apprenticeship" (Collins, Brown, & Newman, in press), which honors the situated nature of knowledge. They examine two examples of mathematics instruction that exhibit certain key features of this approach to teaching.

Brown, Stephen (1997). Thinking like a mathematician. For the Learning of Mathematics, 1997, 32, pp. 36-38. Retrieved 4/18/2010 from http://mumnet.easyquestion.net/sibrown/sib008.htm. Quoting from the article:

In the present era, it [thinking like a mathematician] places greater emphasis on problem solving heuristics -- focusing on students' ability to make use of a wide array of inductive and deductive skills as they operate on incomplete knowledge.

One of the most interesting ways of depicting the concept of "thinking like a mathematician" was proposed between the advent of the "new math" and its present mutation. In For the Learning of Mathematics in 1982, David Wheeler recalls and further develops some thoughts he had recorded earlier on the concept of mathematizing XE &quot;mathematize&quot;.

Wheeler suggests that it is "more useful to know how to mathematize than to know a lot of mathematics," and he wonders why "the majority of teachers [do] not encourage their students to 'function like a mathematician'."

Brown, Stephen I. (n.d.). Towards humanistic mathematics education. Retrieved 4/18/2010 from http://mumnet.easyquestion.net/sibrown/sib003.htm. Quoting from the introduction to the article:

Humanistic mathematics education? What is it? Those who have experienced mathematics as a depersonalized, uncontextualized, non-controversial and asocial form of knowledge might very well consider the expression humanistic mathematics education to be the epitome of an oxymoron. While the concept of humanistic mathematics education can better be understood as a collection of family resemblances—to use a Wittgenstein construct -- than as a sharply defined concept, it is best appreciated as a reaction to the world view suggested above.

The concept of humanistic mathematics education represents an internationally evolving major paradigm shift in its view of mathematics and of education. There have however been harbingers of the shift in various forms in different countries over the past half-century.

What are the disciplines and perspectives that one might include in an effort to understand the concept of humanistic mathematics education? In fact that question requires considerable reconstruction before it can be understood, no less answered. It requires an awareness of the concept of humanism and human nature and their long history; an understanding of what is meant by education and how that concept has evolved as well; and of course an appreciation of what we mean by mathematics itself.

Bruer, John T. (1999; fifth printing). Schools for thought: A science of learning in the classroom. Cambridge. MA: The MIT Press. Learn more about Bruer at http://www.jsmf.org/about/bruer-biography.htm.

Bruer is a cognitive neuroscientist who is Head of the James S. McDonnell Foundation. Quoting from the Website, “John's latest book The Myth of the First Three Years: A New Understanding of Early Brain Development and Lifelong Learning debunks many popular beliefs about the all-or-nothing effects of early experience on a child's brain and development urging parents and decision-makers to consider for themselves the evidence for lifelong learning opportunities.”

Calvin, Duif (n.d.). How do you read chess notation? Retrieved 6/23/08 from http://www.jaderiver.com/chess/notate.html.

Standard chess notation is often called algebraic chess notation. It can be used to keep a written record of a game. Quoting from the Website: “Algebraic is now the official format for FIDE (the international Chess Federation), as well as many national federations like the USCF (US Chess Federation). It tends to lead to fewer score sheet errors, and can be more easily read by players from different countries.”

Cathcart, et al. (2002). The van Hiele levels of geometric thinking. Learning Mathematics in Elementary and Middle School. Pages 282-283 retrieved 7/23/2010 from http://education.uncc.edu/droyster/courses/spring04/vanHeile.htm. Quoting from the document:

Dina and Pierre van Hiele are two Dutch educators who were concerned about the difficulties that their students were having in geometry. This concern motivated their research aimed at understanding students’ levels of geometric thinking to determine the kinds of instruction that can best help students.

The five levels that are described below are not age-dependent, but, instead, are related more to the experiences students have had. The levels are sequential; that is, students must pass through the levels in order as their understanding increases. The descriptions of the levels are in terms of “students” – and remember that we are all students in some sense.

Commons, M.L. and Richards, Francis Asbury (2002). Organizing components into combinations: How stage transition works. Journal of Adult Development. 9(3), 159-177. Retrieved 6/18/09 from http://www.tiac.net/~commons/Commons&Richards04282004.htm.

This paper presents details on a 15-stage Piagetian-type cognitive developmental scale. Quoting the Abstract:

This paper investigates the nature of transition between stages. The Model of Hierarchical Complexity of tasks leads to a quantal notion of stage, and therefore delineates the nature of stage transition. Piaget’s dialectical model of stage change was extended and precisely specified. Transition behavior was shown to consist of alternations in previous-stage behavior. As transition proceeded, the alternations increased in rate until the previous stage behaviors were “smashed” together. Once the smashed-together pieces became coordinated, new-stage behavior could be said to have formed. Because stage transition is quantal, individuals can only change performance by whole stage. We reviewed theories of the specific means by which new-stage behavior may be acquired and the emotions and personalities associated with steps in transition. Examples of transitional performances were. Contemporary challenges in the society increasingly call for transition to post-formal and post-conventional responses on the part of both individuals and institutions as the example illustrate.

Commons, M. L. XE &quot;Commons, Michael&quot; & Richards, F. A.  XE &quot;Richards, Francis Asbury&quot;  (2002). Four postformal stages. In J. Demick (Ed.), Handbook of adult development. New York: Plenum. Retrieved 6/18/09 from http://www.tiac.net/~commons/Four%20Postformal%20Stages.html. Quoting from this document:

The term "postformal" has come to refer to various stage characterizations of behavior that are more complex than those behaviors found in Piaget's last stage—formal operations—and generally seen only in adults. Commons and Richards (1984a, 1894b) and Fischer (1980), among others, posited that such behaviors follow a single sequence, no matter the domain of the task e.g., social, interpersonal, moral, political, scientific, and so on.

Commons, M. L., Miller, P. M., Goodheart, E. A., & Danaher-Gilpin, D. (2005). Hierarchical complexity scoring system (HCSS): How to score anything. Retrieved 5/5/09 from http://www.tiac.net/~commons/Scoring%20Manual.htm.

Crace, John (1/24/2006). Children are less able than they used to be. The Guardian. Retrieved 6/21/09 from http://www.guardian.co.uk/education/2006/jan/24/schools.uk. Quoting from the article:

New research funded by the Economic and Social Research Council (ESRC) and conducted by Michael Shayer, professor of applied psychology at King's College, University of London, concludes that 11- and 12-year-old children in year 7 are "now on average between two and three years behind where they were 15 years ago", in terms of cognitive and conceptual development.

"It's a staggering result," admits Shayer, whose findings will be published next year in the British Journal of Educational Psychology. "Before the project started, I rather expected to find that children had improved developmentally. This would have been in line with the Flynn effect on intelligence tests, which shows that children's IQ levels improve at such a steady rate that the norm of 100 has to be recalibrated every 15 years or so. But the figures just don't lie. We had a sample of over 10,000 children and the results have been checked, rechecked and peer reviewed."

Devlin, Keith (2000). The math gene: How mathematical thinking evolved and why numbers are like gossip. Basic Books. Learn more about Devlin at http://www.stanford.edu/~kdevlin/.

The primary thesis of this book is that our innate language skills form the basis of gaining number sense, numerical abilities, and algorithmic abilities. Thus, the intact human brain is quite capable of learning math.

Dewar, Gwen (2008). In search of the smart preschool board game: What studies reveal about the link between games and math skills. Parenting Science. Retrieved 10/5/09 from http://www.parentingscience.com/preschool-board-game-math.html.

This article reports on various research projects done using board games and young children. Quoting from the article:

There is compelling evidence that certain kinds of board games boost preschool math skills. And these early skills are strongly predictive of math achievement scores later in life (Duncan et al 2008).

For instance, consider the research of Geetha Ramani and Robert Siegler (2008).

Ramani and Siegler asked preschoolers (average age: 4 years, 9 months) to name all the board games they had ever played.

The more board games that a kid named, the better his performance in four areas:

• Numeral identification

• Counting

• Number line estimation (in which a child is asked to mark the location of a number on a line)

• Numerical magnitude comparison (in which a child is asked to choose the greater of two numbers)

The four bulleted items are all important aspects of a combination of growing math knowledge and skills, and growing math maturity.

Dupuis, Mary Ed. and Merchant, Linda, Ed. (1993). Reading across the curriculum: A research report for teachers. Retrieved 12/30/09 from http://www.eric.ed.gov/ERICWebPortal/recordDetail?accno=ED350597. Quoting from the Website:

Focusing on grades 4-12, this book supplies content area teachers with the information they need to function as reading and writing teachers within their subject/academic discipline. Chapters in the book usually begin with a summary or overview, showing the major concerns and unique features of language use in that area. Some of the chapters in the book have extensive bibliographies of research and/or teaching techniques germane to the subject.

Fermandez, Alvaro (10/18/08). Training, attention and emotional self-regulation—interview with Michael Posner XE &quot;Posner, Michael&quot;. Sharp Brains. Retrieved 11/9/09 from http://www.sharpbrains.com/blog/2008/10/18/training-attention-and-emotional-self-regulation-interview-with-michael-posner/. Quoting from the article:

Michael I. Posner is a prominent scientist in the field of cognitive neuroscience. He is currently an emeritus professor of neuroscience at the University of Oregon (Department of Psychology, Institute of Cognitive and Decision Sciences). In August 2008, the International Union of Psychological Science made him the first recipient of the Dogan Prize “in recognition of a contribution that represents a major advance in psychology by a scholar or team of scholars of high international reputation.”

[When asked to summarize some of his research, Posner responded:]

I would emphasize that we human beings can regulate our thoughts, emotions, and actions to a greater degree than other primates. For example, we can choose to pass up an immediate reward for a larger, delayed reward.

We can plan ahead, resist distractions, be goal-oriented. These human characteristics appear to depend upon what we often call “self-regulation.” What is exciting these days is that progress in neuroimaging XE &quot;neuroimaging&quot; and in genetics make it possible to think about self-regulation in terms of specific brain-based networks.

[When asked about his work in attention Posner responded:]

I have been interested in how the attention system develops in infancy and early childhood.

One of our major findings, thanks to neuroimaging, is that there is not one single “attention”, but three separate functions of attention with three separate underlying brain networks: alerting, orienting, and executive attention.

1) Alerting: helps us maintain an Alert State.

2) Orienting: focuses our senses on the information we want. For example, you are now listening to my voice.

3) Executive Attention: regulates a variety of networks, such as emotional responses and sensory information. This is critical for most other skills, and clearly correlated with academic performance. It is distributed in frontal lobes and the cingulate gyrus XE &quot;cingulate gyrus&quot;.

The development of executive attention can be easily observed both by questionnaire and cognitive tasks after about age 3–4, when parents can identify the ability of their children to regulate their emotions and control their behavior in accord with social demands.

Geary, David C. (2007). An evolutionary perspective on learning disability in mathematics. Developmental Neuropsychology. Retrieved 9/18/09 from http://web.missouri.edu/~gearyd/articles_math.htm. Quoting from this paper:

When viewed from the lens of evolution and human cultural history, it is not a coincidence that public schools are a recent phenomenon and emerge only in societies in which technological, scientific, commercial (e.g., banking, interest) and other evolutionarily-novel advances influence one’s ability to function in the society (Geary, 2002, 2007). From this perspective, one goal of academic learning is to acquire knowledge that is important for social or occupational functioning in the culture in which schools are situated, and learning disabilities (LD) represent impediments to the learning of one or several aspects of this culturally-important knowledge. It terms of understanding the brain and cognitive systems that support academic learning and contribute to learning disabilities, evolutionary and historical perspectives may not be necessary, but may nonetheless provide a means to approach these issues from different levels of analysis. I illustrate this approach for MLD. I begin in the first section with an organizing frame for approaching the task of decomposing the relation between evolved brain and cognitive systems and school-based learning and learning disability (LD). In the second section, I present a distinction between potentially evolved biologically-primary cognitive abilities and biologically-secondary abilities that emerge largely as a result of schooling (Geary, 1995), including an overview of primary mathematics. In the third section, I outline some of the cognitive and brain mechanisms that may be involved in modifying primary systems to create secondary abilities, and in the fourth section I provide examples of potential the sources of MLD based on the framework presented in the first section.

Geary, David C. and four other authors. (July/August 2007). Cognitive mechanisms underlying achievement deficits in children with mathematical learning disability. Child Development. Retrieved 4/9/09. To access this paper go to http://web.missouri.edu/~gearyd/articles_math.htm, find the paper in the list of papers, and click on its link. Quoting from this paper:

Using strict and lenient mathematics achievement cutoff scores to define a learning disability, respective groups of children who are math disabled (MLD, n=15) and low achieving (LA, n=44) were identified. These groups and a group of typically achieving (TA, n=46) children were administered a battery of mathematical cognition, working memory, and speed of processing measures (M=6 years). The children with MLD showed deficits across all math cognition tasks, many of which were partially or fully mediated by working memory or speed of processing. Compared with the TA group, the LA children were less fluent in processing numerical information and knew fewer addition facts. Implications for defining MLD and identifying underlying cognitive deficits are discussed.

GMU (n.d.). George Mason University’s XE &quot;George Mason University&quot; online resources for developmental psychology XE &quot;developmental psychology&quot;. Retrieved 12/29/09 from http://classweb.gmu.edu/awinsler/ordp/index.html.

Provides links to an extensive set of documents useful in studying the field of Developmental Psychology.

Glbblguy XE &quot;Glbblguy&quot; (n.d.). 1 marshmallow or 2? A study on the benefits of delayed gratification. Retrieved 9/8/09 from http://www.gatherlittlebylittle.com/2008/03/1-marshmallow-or-2-a-study-on-the-benefits-of-delayed-gratification/.

This blog entry contains one person’s insights into the gratification ideas in the marshmallow research. It makes the observation that a great many of the people who have become millionaires exhibited the ability of working toward gratification that would occur far in the future. The article contains the following example quoted from Money Savey Generation: http://www.gatherlittlebylittle.com/2008/03/1-marshmallow-or-2-a-study-on-the-benefits-of-delayed-gratification/

At age 12 you decide not to buy soda or extra snacks – either during the school week or on weekends or vacations. You save $4.00 a day. You put $4 a day in a savings vehicle such as a long-term IRA CD at five percent annual interest and leave it alone. At age 67, your savings is:

(a) $1,159 (b) $25,355 (c) $80,352 (d) $427,025

Answer: (d), or $427,025. Note that $80,352 is from the daily deposits and the remaining $346,673 is interest! [Comment by David Moursund: This is an example of very long term delayed gratification.]

Graham, Charles and Plucker, Jonathan (2002). The Flynn effect. Human intelligence. Retrieved 1/10/2010 from http://www.indiana.edu/~intell/flynneffect.shtml. Quoting from the Website:

In his study of IQ tests scores for different populations over the past sixty years, James R. Flynn discovered that IQ scores increased from one generation to the next for all of the countries for which data existed (Flynn, 1994). This interesting phenomena has been called "the Flynn Effect." Many of the questions about why this effect occurs have not yet been answered by researchers. This site attempts to explain the issues involved in a way that will better help you to understand the Flynn Effect. It also provides references for further inquiry.

H to O
Huitt, W. & Hummel, J. (2003). Piaget’s theory of cognitive development. Educational Psychology Interactive. Valdosta, GA: Valdosta State University. Retrieved 12/9/09 from http://www.edpsycinteractive.org/topics/cogsys/piaget.html.

This article provides an introduction to Piaget’s 4-stage theory of cognitive development and the role this theory provides in constructivist learning. In listing the four stages, the article indicates:

“4. Formal operational stage (Adolescence and adulthood). In this stage, intelligence is demonstrated through the logical use of symbols related to abstract concepts. Early in the period there is a return to egocentric thought. Only 35% of high school graduates in industrialized countries obtain formal operations; many people do not think formally during adulthood.” [Bold added for emphasis.]

Kaiser Family Foundation (1/20/2010). Generation M2: Media in the Lives of 8- to 18-Year-Olds. Retrieved 5/3/2010 from http://www.kff.org/entmedia/mh012010pkg.cfm. Quoting from the research report:

A national survey by the Kaiser Family Foundation found that with technology allowing nearly 24-hour media access as children and teens go about their daily lives, the amount of time young people spend with entertainment media has risen dramatically, especially among minority youth. Today, 8-18 year-olds devote an average of 7 hours and 38 minutes (7:38) to using entertainment media across a typical day (more than 53 hours a week). And because they spend so much of that time 'media multitasking' (using more than one medium at a time), they actually manage to pack a total of 10 hours and 45 minutes (10:45) worth of media content into those 7½ hours.

Generation M2: Media in the Lives of 8- to 18-Year-Olds is the third in a series of large-scale, nationally representative surveys by the Foundation about young people's media use. It includes data from all three waves of the study (1999, 2004, and 2009), and is among the largest and most comprehensive publicly available sources of information about media use among American youth.

Lehrer, Jonah (5/18/09). Don't! The Secret of Self Control. The New Yorker. Retrieved 5/3/2010 from http://www.newyorker.com/reporting/2009/05/18/090518fa_fact_lehrer?currentPage=all.

This article presents a nice introduction to and overview of gratification and the marshmallow test research.

Logan, Robert K. (2000). The Sixth Language: Learning a Living in the Internet Age. NJ: The Blackburn Press. See also, http://osdir.com/ml/culture.internet.nettime-announce/2006-07/msg00008.html. Quoting from a product description given at Amazon.com:

The Sixth Language updates the work of Marshall McLuhan by applying his ideas to the communications revolution taking place due to digital information technology. Logan's work interweaves ideas which touch on language, education, work, social class, information technology and management theory. He establishes the theoretical background for his study with a succinct and very readable summary of McLuhan's ideas.

Logan develops a new theory of language by showing that a language is not merely a system of communication but also an information processing tool. He goes on to show that speech, writing, mathematics, science, computing and the Internet form an evolutionary chain of verbal languages. A new language evolved each time the informatic capacity of the previous set of languages was exhausted. Math and writing arose to deal with the information overload associated with economic transactions of the agriculture based city states of Sumer.

The schools that evolved to teach the new math and writing skills gave rise to scholarship and a new form of information overload ensued. Science or organized knowledge arose to deal with the new information glut created by the teachers in the newly established schools. Computing developed out of the need to cope with the information overload created by science and the Internet was the response to the info overload and need for greater connectivity of computing.

Louisiana Content Standards (n.d). Louisiana Content Standards for Programs Serving Four-Year Old Children. Retrieved 9/8/09 from http://doa.louisiana.gov/osr/lac/28v77/28v77.doc.

This document includes a 4-stage Piagetian-type math developmental scale for preschool age children.

Maier, Eugene (1976). Folk math. Retrieved 11/21/09 from http://iae-pedia.org/Folk_Math. Quoting from the article:

One of my first realizations that something was awry in school mathematics was the chasm that existed between people's everyday encounters with mathematics and what they learned in school. “Folk Math” addresses the nature of that gap and offers some suggestions for narrowing it.

Maier, Eugene XE &quot;Maier, Eugene&quot; (1985). Mathematics and visual thinking. Retrieved 2/23/2010 from http://iae-pedia.org/Mathematics_and_Visual_Thinking. Quoting from the article:

For our purposes, “visual thinking” shall mean at least three things: perceiving, imaging and portraying. Perceiving is becoming informed through the senses: through sight, hearing, touch, taste, smell, and also through kinesthesia, the sensation of boy movement and position. Imaging is experiencing a sense perception in our mind or body that, at the moment, is not a physical reality. Portraying is depicting a perception by a sketch, diagram, model, or other representation.

Whereas the previous quotations might suggest that some individuals are visual thinkers and others are not, everyone to a greater or lesser extent, engages in visual thinking. Our dreams attest to that. And visual thinking, when nurtured and developed, can play a significant role in the development of mathematical understanding and in the creative and insightful use of mathematics in other areas.

Mitchell, A. and Savill-Smith, C. (2004). The use of computer and video games for learning: A review of the literature. Learning and Skills Development Agency; Ultralab; m-learning. 93 page report. Retrieved 2/13/2010 from http://www.mlearning.org/docs/The%20use%20of%20computer%20and%20video%20games%20for%20learning.pdf. Quoting from the document—which is one of several good reports available on the listed site):

The use of recreational computer and video games, particularly by young people, is commonplace. It is often suggested that playing these games can also have educational value. This review synthesises the key messages from past research studies which have considered how and why both recreational and educational computer games have been used for learning and the impact of their use on young people. Other areas of investigation include young people's experiences and performances and preferences in using such games. Finally some recommendations are made concerning the planning and design of the future 'edugames'

Moniot, Robert K. (2/7/2007). The taxman game. Math Horizons. Retrieved 7/29/2010 from http://www.maa.org/mathhorizons/pdfs/feb_2007_Moniot.pdf. Quoting from the article:

Went to know how to beat the taxman? Legally, that is? Read on, and we will explore this cute little mathematical game.

The taxman game is a “golden oldie” computer game. It has been used as an exercise in introductory programming classes and in books on recreational computing since time immemorial. (Well, at least since the 1970s.)

Montana State University (2009). The language of mathematics. Retrieved 1/16/2010 from http://augustusmath.hypermart.net/. Quoting from the Website:

Algebra is written in a symbolic language that is designed to express mathematical thoughts. This website describes a text for a course that emphasizes language skills. Most math courses concentrate on what is said, this one concentrates on how is it is said so you can understand what it says. This course emphasizes all the ways that mathematics is used to express thoughts in algebra and higher-level classes. It will make you far better at math regardless of your current level.

Mathematical language skills include the abilities to read with comprehension, to express mathematical thoughts clearly, to reason logically, and to recognize and employ common patterns of mathematical thought.

Moursund, David (2006a). Computers in education for talented and gifted students: A book for elementary and middle school teachers. Eugene, OR: Information Age Education. Retrieved 5/4/09 from http://i-a-e.org/downloads/doc_download/13-computers-in-education-for-talented-and-gifted-students.html.

Moursund, David (2006b). Computational thinking and math maturity: Improving math education in K-8 schools. Eugene, OR: Information Age Education. Retrieved 5/6/09 from http://i-a-e.org/downloads/doc_download/3-computational-thinking-and-math-maturity-improving-math-education-in-k-8-schools.html.

Moursund, David (2006c). Brief introduction to educational implications of artificial intelligence. Eugene, OR: Information Age Education. Retrieved 1/10/2010 from http://i-a-e.org/downloads/doc_download/6-introduction-to-educational-implications-of-artificial-intelligence.html.

Moursund, David (n.d.). Communicating in the language of mathematics. Retrieved 11/11/09 from http://iae-pedia.org/Communicating_in_the_Language_of_Mathematics. Quoting from the Wiki document:

Communication in math involves making use of processes of reading, writing, speaking, listening, and thinking as one communicates with one's self, other people, computers, books, and other aids to the storage, retrieval, and use of the collected mathematical knowledge of the world. Current precollege math education systems have substantial room for improvement in helping students learn to communicate effectively in the "language" of mathematics.

Moursund, David (2008). Introduction to using games in education: A guide for teachers and parents. Eugene, OR: Information Age Education. Retrieved 1/12/2010 from http://i-a-e.org/downloads/doc_download/19-introduction-to-using-games-in-education-a-guide-for-teachers-and-parents.html.

This book is written for people who are interested in helping children learn through games and learn about games. The intended audience includes teachers, parents and grandparents, and all others who want to learn more about how games can be effectively used in education. Special emphasis is given to roles of games in a formal school setting.

An appendix contains an extensive list of problem-solving strategies that are amenable teaching strategies useful in high-road transfer of learning.

Moursund, David (2010). Two brains are better than one. Retrieved 1/10/2010 from http://iae-pedia.org/Two_Brains_Are_Better_Than_One.

NCTM (2000). Principles and Standards for School Mathematics. National Council of Teachers of Mathematics. Reston, VA: NCTM. Retrieved 11/9/09 from http://www.nctm.org/standards/default.aspx?id=58. Quoting from the Preface:

Principles and Standards for School Mathematics is intended to be a resource and guide for all who make decisions that affect the mathematics education of students in prekindergarten through grade 12. The recommendations in it are grounded in the belief that all students should learn important mathematical concepts and processes with understanding. Principles and Standards makes an argument for the importance of such understanding and describes ways students can attain it.

NCTM (2006). Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics. National Council of Teachers of Mathematics. Reston, VA: NCTM. Retrieved 11/9/09 from http://www.nctm.org/standards/content.aspx?id=270. Quoting from the Website

Curriculum Focal Points are the most important mathematical topics for each grade level. They comprise related ideas, concepts, skills, and procedures that form the foundation for understanding and using mathematics and lasting learning. Curriculum Focal Points have been integral in the revision of many state math standards for Pre-K through grade 8.

NCTM (2009). Focus in high school mathematics: Reasoning and sense making. National Council of Teachers of Mathematics. Reston, VA: NCTM. Quoting from the document:

This publication advocates that all high school mathematics programs focus on reasoning and sense making. The audience for this publication is intended to be everyone involved in decisions regarding high school mathematics programs, including formal decision makers within the system; people charged with implementing those decisions; and other stakeholders affected by, and involved in, those decisions.

NCTM Illuminations (n.d.). The factor game. Retrieved 7/29/2010 from http://illuminations.nctm.org/LessonDetail.aspx?ID=L253. The game can be played online at the site http://illuminations.nctm.org/ActivityDetail.aspx?ID=12. Quoting from the first-mentioned Website:

The Factor Game engages students in a friendly contest in which winning strategies involve distinguishing between numbers with many factors and numbers with few factors. Students are then guided through an analysis of game strategies and introduced to the definitions of prime and composite numbers.

The Factor Game is a two-person game in which players find factors of numbers on a game board. To play, one person selects a number and colors it. The second person colors all the proper factors of the first person's number. The roles are switched and the play continues till there are no numbers remaining with uncolored factors. Each person adds up the numbers they've colored. The winner is the person with the largest total.

Ojose, Bobby (2008). Applying Piaget’s theory of cognitive development to mathematics instruction. The Mathematics Educator. Retrieved 4/17/2010 from http://math.coe.uga.edu/tme/issues/v18n1/v18n1_Ojose.pdf. Quoting from the paper:

The approach of this article will be to provide a brief discussion of Piaget’s underlying assumptions regarding the stages of development. Each stage will be described and characterized, highlighting the stage appropriate mathematics techniques that help lay a solid foundation for future mathematics learning. The conclusion will incorporate general implications of the knowledge of stages of development for mathematics instruction.

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Perkins, David N. and Salomon, Gavriel (September 2, 1992). Transfer of Learning: Contribution to the International Encyclopedia of Education. Second Edition. Oxford, England: Pergamon Press. Retrieved 1/23/2010 from http://learnweb.harvard.edu/alps/thinking/docs/traencyn.htm.

Salomon and Perkins have developed the high-road/low-road theory of transfer of learning. The article listed here provides a good overview of the domain of transfer of learning and how to teach transfer. It also contains an extensive bibliography, so it is a good starting point if you want to study the research on transfer of learning.

Perkins, David (Fall 1993). Teaching for understanding. American Educator: The Professional Journal of the American Federation of Teachers. Retrieved 1/23/2010 from http://www.exploratorium.edu/IFI/resources/workshops/teachingforunderstanding.html. Quoting from the article:

The first two examples happen to reflect the work of teachers collaborating with my colleagues and me in studies of teaching for understanding. The second two are drawn from an increasingly rich and varied literature. Anyone alert to current trends in teaching practice will not be surprised. These cases illustrate a growing effort to engage students more deeply and thoughtfully in subject-matter learning. Connections are sought between students' lives and the subject matter, between principles and practice, between the past and the present. Students are asked to think through concepts and situations, rather than memorize and give back on the quiz.

Polya, George (1969). The goals of mathematical education. Mathematically Sane. Retrieved11/17/09 from http://mathematicallysane.com/analysis/polya.asp. This is a talk that Polya gave to a group of inservice and preservice math education students. Quoting from the Website:

Polya (1887-1985) was a distinguished mathematician and professor at Stanford University who made important contributions to probability theory, number theory, the theory of functions, and the calculus of variations. He was the author of the classic works How to Solve It, Mathematics and Plausible Reasoning, and Mathematical Discovery, which encouraged students to become thoughtful and independent problem solvers. He was an honorary member of the Hungarian Academy, the London Mathematical Society, and the Swiss Mathematical Society, and a member of the (American) National Academy of Sciences, the American Academy of Arts and Sciences, and the California Mathematics Council, as well as a corresponding member of the Academie des Sciences in Paris.

Prensky, Marc (2001). Chapter 5 of the book: Digital game-based learning. NY: McGraw-Hill. Retrieved 1/19/2010 from http://www.marcprensky.com/writing/Prensky%20-%20Digital%20Game-Based%20Learning-Ch5.pdf. Some other free chapters can be located by an Internet search of Prensky “Digital game-based learning.” Quoting from the chapter:

Computer and videogames are potentially the most engaging pastime in the history of mankind. This is due, in my view, to a combination of twelve elements:

1. Games are a form of fun. That gives us enjoyment and pleasure.

2. Games are form of play. That gives us intense and passionate involvement.

3. Games have rules. That gives us structure.

4. Games have goals. That gives us motivation.

5. Games are interactive. That gives us doing.

6. Games are adaptive. That gives us flow.

7. Games have outcomes and feedback. That gives us learning.

8. Games have win states. That gives us ego gratification.

9. Games have conflict/competition/challenge/opposition. That gives us adrenaline.

10. Games have problem solving. That sparks our creativity.

11. Games have interaction. That gives us social groups.

12. Games have representation and story. That gives us emotion.

Prensky, Marc (n.d.). Social impact games. Entertaining games with non-entertainment goals (a.k.a Serious Games). Retrieved 1/19/2010 from http://www.socialimpactgames.com/. For a Prensky video and other excellent resources see http://www.marcprensky.com/writing/. Quoting from the site:

The goal of this site is to catalog the growing number of "serious games" (i.e. video and computer games whose primary purpose is something other than to entertain.)

It should help:

1. people who want to locate serious games to find them, 2. people who want to create serious games to see what others have done.

Prensky, Marc (2002). What kids learn that's POSITIVE from playing video games. Retrieved 1/1/2010 from http://articles.smashits.com/articles/family/69663/what-kids-learn-that-s-positive-from-playing-video-games.html. Quoting from the article:

For whenever one plays a game, and whatever game one plays, learning happens constantly, whether the players want it to, and are aware of it, or not. And the players are learning “about life,” which is one of the great positive consequences of all game playing. This learning takes place, continuously, and simultaneously in every game, every time o

But we do need to pay some attention in order to analyze how and what players learn.

The first thing we need to pay attention to is the difference between a games’ “surface” messages, as presented in its in its graphics, audio and text (what is commonly called its “content”) and a game’s underlying messages and required skills. I am not an apologist for all the content in computer games, but that “surface” content is all most critics ever see of a much richer experience. The fact is that in every game, a great deal of useful learning goes on in addition to, or even despite the game’s surface content, whatever that may be. This huge amount of powerful, positive learning is almost universally ignored by critics, parents and educators alike.

Project Zero (n.d.). Project Zero at Harvard Graduate School of Education. Retrieved 12/2/09 from http://pzweb.harvard.edu/. Quoting from the Website:

Project Zero's mission is to understand and enhance learning, thinking, and creativity in the arts, as well as humanistic and scientific disciplines, at the individual and institutional levels.

Ratey, John and Hagerman, Eric (2008). Spark: The Revolutionary New Science of Exercise and the Brain. NY: Little Brown and Company. Quoting from the publisher:

Did you know you can beat stress, lift your mood, fight memory loss, sharpen your intellect, and function better than ever simply by elevating your heart rate and breaking a sweat? The evidence is incontrovertible: Aerobic exercise physically remodels our brains for peak performance.

In Spark, John J. Ratey, M.D., embarks upon a fascinating and entertaining journey through the mind-body connection, presenting startling research to prove that exercise is truly our best defense against everything from depression to ADD to addiction to aggression to menopause to Alzheimer's. Filled with amazing case studies (such as the revolutionary fitness program in Naperville, Illinois, which has put this school district of 19,000 kids first in the world of science test scores), Spark is the first book to explore comprehensively the connection between exercise and the brain. It will change forever the way you think about your morning run---or, for that matter, simply the way you think.

Schoenfeld, Alan H. (2004). The math wars. Educational Policy. V 18 n 1, January and March 2004. Retrieved 4/3/2010 from http://gse.berkeley.edu/faculty/AHSchoenfeld/Schoenfeld_MathWars.pdf. Quoting from the article:

During the 1990s, the teaching of mathematics became the subject of heated controversies known as the math wars. The immediate origins of the conflicts can be traced to the “reform” stimulated by the National Council of Teachers of Mathematics ’Curriculum and Evaluation Standards for School Mathematics. Traditionalists fear that reform-oriented, “standards-based” curricula are superficial and undermine classical mathematical values; reformers claim that such curricula reflect a deeper, richer view of mathematics than the traditional curriculum. An historical perspective reveals that the underlying issues being contested—Is mathematics for the elite or for the masses? Are there tensions between “excellence” and “equity”? Should mathematics be seen as a democratizing force or as a vehicle for maintaining the status quo?—are more than a century old.

Tall, David (1996). Can all children climb the same curriculum ladder? Retrieved 10/2/09 from http://www.warwick.ac.uk/staff/David.Tall/pdfs/dot1996c-cur-ladder-gresham.pdf. Here is the abstract for the article:

This presentation presents evidence that the way the human brain thinks about mathematics requires an ability to use symbols to represent both process and concept. The more successful use symbols in a conceptual way to be able to manipulate them mentally. The less successful attempt to learn how to do the processes but fail to develop techniques for thinking about mathematics through conceiving of the symbols as flexible mathematical objects. Hence the more successful have a system which helps them increase the power of their mathematical thought, but the less successful increasingly learn isolated techniques which do not fit together in a meaningful way and may cause the learner to reach a plateau beyond which learning in a particular context becomes difficult. [Bold added for emphasis.]

Tall, David (December 2000). Biological brain, mathematical mind & computational computers: How the computer can support mathematical thinking and learning. Retrieved 9/18/09 from http://www.tallfamily.co.uk/david/papers/biological-brain-math-mind.pdf.

This article contains a number of ideas and/or examples that relate to math maturity. Here is an example quoted from the article.

Consider, a ‘linear relationship’ between two variables. This might be expressed in a variety of ways:

• an equation in the form y=mx+c,

• a linear relation Ax+By+C=0,

• a line through two given points,

• a line with given slope through a given point,

• a straight-line graph,

• a table of values, etc.

Successful students [that is, more mathematically mature students] develop the idea of ‘linear relationship’ as a rich cognitive unit encompassing most of these links as a single entity.

Less successful [that is, less mathematically mature] carry around a ‘cognitive kit-bag’ of isolated tricks to carry out specific algorithms. Short-term success perhaps, long-term cognitive load and failure.

Van Hiele, Dina and Pierre (n.d.). The van Hiele Levels of Geometric Thinking. From Cathcart, et al. Learning Mathematics in Elementary and Middle Schools (2002). Retrieved 7/23/2010 from http://education.uncc.edu/droyster/courses/spring04/vanHeile.htm. Also see the next reference.

Provides a discussion of the 5-stage Piagetian scale the Van Hieles’ developed for Geometry as part of their 1957 doctoral dissertation work.

Van Hiele, P. H. (1959). Levels of mental development in geometry. Retrieved 12/9/09 from http://www.math.uiuc.edu/~castelln/VanHiele.pdf.

This short article contains a 5-point Geometry cognitive development scale and a brief discussion of its components.

Wolpert, Stuart (1/27/09). Is technology producing a decline in critical thinking and analysis? UCLA Newsroom. Retrieved 1/29/2009 from http://newsroom.ucla.edu/portal/ucla/is-technology-producing-a-decline-79127.aspx. Quoting from the report:

As technology has played a bigger role in our lives, our skills in critical thinking and analysis have declined, while our visual skills have improved, according to research by Patricia Greenfield, UCLA distinguished professor of psychology and director of the Children's Digital Media Center, Los Angeles.

Learners have changed as a result of their exposure to technology, says Greenfield, who analyzed more than 50 studies on learning and technology, including research on multi-tasking and the use of computers, the Internet and video games.

Reading for pleasure, which has declined among young people in recent decades, enhances thinking and engages the imagination in a way that visual media such as video games and television do not, Greenfield said.

…

Visual intelligence has been rising globally for 50 years, Greenfield said. In 1942, people's visual performance, as measured by a visual intelligence test known as Raven's Progressive Matrices, went steadily down with age and declined substantially from age 25 to 65. By 1992, there was a much less significant age-related disparity in visual intelligence, Greenfield said.

"In a 1992 study, visual IQ stayed almost flat from age 25 to 65," she said.

Yeung, Bernice (October 2009). Arithmetic underachievers overcome frustration to succeed. Math test scores soar if students are given the chance to struggle. Edutopia: The George Lucas Foundation. Retrieved 7/24/2010 from http://www.edutopia.org/math-underachieving-mathnext-rutgers-newark. Quoting from the article:

New Jersey teachers have found a surprising way to keep students engaged and successful: They let underachieving youngsters get frustrated by math.

While working with minority and low-income students at low-performing schools in Newark for the past seven years, researchers at Rutgers University have found that allowing students to struggle with challenging math problems can lead to dramatically improved achievement and test scores.

"We've found there is a healthy amount of frustration that's productive; there is a satisfaction after having struggled with it," says Roberta Schorr, associate professor in Rutgers University at Newark's Urban Education Department. Her group has also found that, though conventional wisdom says certain abilities are innate, a lot of kids' talents and abilities go unnoticed unless they are effectively challenged; the key is to do it in a nurturing environment.

"Most of the literature describes student engagement and motivation as having to do with their attitudes about math—whether they like it or not," Schorr say.

Authors of This Page
This page was created by David Moursund and Robert Albrecht.