Paul Pearson Reviews, Spring 2000

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Keywords: Curriculum, ,
Ref: Pearson1
Author(s): NCTM, Carnegie Learning
Date: 2000
Title: cd496_Exemplary and Promising Mathematics Programs: Cognitive Tutor
Journal or Publisher: NCTM
Volume, Issue, Pages: http://www.enc.org/ed/exemplary/cd496/496_4.htm
Reviewer: Pearson
Date of Review: April 17, 2000

http://www.enc.org/ed/exemplary/cd496/496_4.htm

Exemplary and Promising Mathematics Programs

Cognitive Tutor Algebra

This program is one of ten exemplary and promising mathematics programs cited by the Eisenhower National Clearinghouse. The "Cognitive Tutor Algebra" program integrates technology in the 7th - 12th grade classrooms. It is based on research done at Carnegie-Mellon University and the NCTM standards. With the focus on depth of understanding instead of breadth, the program concentrates on real-life applications through problem solving and using multiple learning styles. Students in this program boasted 50-100% higher scores than students in traditional Algebra I courses on two project-developed, problem-solving tests. Students who went through this program also had less computer anxiety than the control group. The total cost of this program, including software, is $25,000. Though it is noted that the program is particularly appropriate for the urban underachiever, the cost of the program is almost prohibitively high. Nice features of this program are its emphasis on making connections, reasoning, problem solving, using technology, active hands-on learning, self-pacing, and instant feedback.

The use of computers to enable students to know and work on their strenghts and weaknesses is beneficial for all students. Interaction, meaningful activities, real-life applications, and incorporation of many different viewpoints and types of learning make this a successful program.

I would like to see this program in action, or at least a sample of the curriculum to be able to see what makes it different from other curricula. I went to the site http://www.carnegielearning.com to learn more about the project, and found that "Carnegie Learning's Cognitive Tutors are based on Dr. John R. Anderson's ACT theory of learning and performance (Anderson, 1983; 1989; 1990; 1993; 1995; 1998). The theory distinguishes between tacit performance knowledge and static verbalized knowledge. According to ACT, performance knowledge can only be learned by doing, not by listening or watching. ACT theory quantifies such performance knowledge via a series of `if-then' production rules that associate internal goals and external perceptual cues with new internal goals and external actions." I was dissapointed not to be able to see some examples of the curriculum on their site. I think that people who don't offer samples of their work, or even the possibility to receive samples of their work, are being unfair to teachers by asking them to put their own money into a textbook and curriculum (or the school district's money). I wish I could get a trial version of this curriculum, but it wasn't offered anywhere. So, whether or not the curriculum is as great as it claims to be will still be a mystery to me. Their objectives and goals seem to be very thought inspiring and in line with what the new NCTM standards are going to be. Additionally, the activities that they suggest seem to be thought provoking, but without examples, the actual content of the curriculum still remains vague.

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Keywords: Issues, Statistics, Probability
Ref: Pearson1
Author(s): Dewdney, A. K.
Date: 1993
Title: 200% of Nothing
Journal or Publisher: John Wiley and Sons, Inc., New York
Volume, Issue, Pages: pp. 1 - 182
Reviewer: Pearson
Date of Review: April 22, 2000

This book was written to combat abuses of mathematics which also abuse us. The focus of this book is on the most typical abuses of mathematics with examples drawn from the real world, the areas of life where abuse is particularly rampant, and what the reader can do to become one of the numerate people of society. People's innumeracy is not only costing us money, property, and freedom of choice, but it is also indicative of a society which cannot think for itself since "[n]umeracy concerns thought itself" (2).

Through his use of "math detectives"---people who have sent him examples of math abuse---Dewdney has collected may examples of math abuse, among which is the especially egregious claim that by switching to a metal halide light bulb fixture you can save 200% on you energy bill. The problem with faulty mathematical conclusions such as this is that it can be very hard to figure out where they went wrong. Number numbness can set in when exaggerated claims such as this go overlooked. He cites faulty applications of set theory as another example of a public which can use flawed logic in gathering data. Another common error he cites is called "filtering," and it occurs when we omit a crucial number from a mathematical calculation. If a certain treatment has cured 50 people, is it advisable to use it if you don't know how many people it hasn't cured? A neat little term that the author has coined---ratiocinitis---accurately describes how most people fail to deal correctly with ratios and percents, usually by overlooking what quantity the ratio is multiplied by. Additionally, people don't have a good sense of order of magnitude and can make frequent errors because of it. Exploiting the dimensions of a problem is a particularly frequent error (or advertising trick) in society today.

According to Mark Twain, there are "lies, damned lies, and statistics"; Dewdney believes that statistics are probably the most casually overlooked sets of mathematical concepts. Inaccurate sample sizes and sample trashing abound in statistics. Detemporizing statistics, such as the American Cancer Society did with its 1 in 9 probability that women will get breast cancer over a lifetime, into a smaller time window can also be very misleading. In general, people don't grasp the concept of having a probability "compounded" over many years, and more specifically, people don't realize the variability in something that has a "compound" probability. Statistically significant results are often claimed by people who have done a statistical study incorrectly. Procedures for obtaining a sample must be done in a way that is accurate, otherwise the entire set of statistics is inaccurate. All of these types of abuses exploit our innumeracy.

Advertising using numbers can be particularly misleading. Prices, product descriptions, and quasi-surveys are frequently manipulated to the advantage of advertisement. Frequently information that is crucial to understanding is left out of advertisements, and hence they are not 100% true when they do leave out critical information. Most often only one statistic will be reported instead of all of the relevent statistics, and the consumer is easily misled by not pausing to think of all the possibilities. Claims of very accurate results and too many significant digits are all too commonly accepted as a part of daily life.

Dewdney also points out how people fail to see that some events, such as rolling dice, are truly independent events, that is, one event doesn't have anything to do with any of the previous results of a particular experiment. "Hot" lottery numbers which have shown up in lotteries lately are included in this category of events which are independent but are not seen as independent by everyone in society. Belief in the law of creeping chance (for example, that a slot machine will pay off only after a long streak of not paying off) is regarded by too many people as true. The probability does not go up the longer you wait, especially with independent events such as these. In the long run, the casino will get all of your money, regardless of how much money you start off with. Interestingly enough, Dewdney uses the same example that Paulos did in the three curtain game similar to "Let's make a deal."

Perpetrators of mathematical inaccuracies make many common errors, many of which come about as a result of not asking qualitative questions about seemingly simple statistics. People are too easily swayed by statistics which may seem to be true, at least superficially. It doesn't take much to make someone overestimate or underestimate the value of something. Most people don't realize or understand all of the assumptions that go into statistics or probability and they are therefore easily duped by things which may have the guise of being reasonable. In fact, we do many mathematical things every day. For instance, when we look at a clock and plan our day, we are using calculations. The familiarity and logic inherent in this is often overlooked. I really liked all of the examples in this book, and I liked Dewdney's approach to the problem of innumeracy better than Paulos's approach. Merely pointing out the defecits and allowing me, the reader, to draw my own conclusions is much more satisfying than trying to make more generalized sweeping statements such as those found in Paulos's book. In comparing the two books, I would have to say that I liked this one better because it didn't come across as being preachy (if that is even a true word). Furthermore, I like that his claims are centered around cases of math abuse, not simply innumeracy. As I see it, a person who is mathematically literate could abuse math without truly being mathematically illiterate. The term math abuse doesn't bring with it the debilitating stigma that innumeracy brings with it. After all, we all make mathematical errors. The difference between being innumerate and simply a abuser of math is that the innumerate person perpetuates and doesn't learn from their mistakes, where as an abuser of math might still have a chance at not being completely stupid. That's how I see it. Overall, I like that this book aims at pointing out some of the more common errors instead of indicting us of already having committed most of them (even though previous misconceptions may still exist!).

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Keywords: Equity, Issues,
Ref: Pearson2
Author(s): Trentacosta, Janet; Kenney, Margaret
Date: 1997
Title: Multicultural and Gender Equity In the Mathematics Classroom
Journal or Publisher: The National Council of Teachers of Mathematics, Inc.
Volume, Issue, Pages: pp. 1 - 248
Reviewer: Pearson
Date of Review: April 22, 2000

Diversity issues in mathematics education need to be addressed so that all students may achieve their full potential, regardless of race, ethnicity, gender, or socioeconomic status. Furthermore, as this is an NCTM publication, the Standards need to be implemented in such a way that promote equity for all in the classroom. Tracking should be eliminated so that there is a level playing field for all, and so that there will be a support system for all students. Personell in schools should be encouraging all students---especially those of African American, Hispanic, Native American, and females---to become interested in math. In alignment with the current trend toward multicultural education, the NCTM supports efforts to introduce higher-level concepts from an appropriate cultural context. Students should be encouraged to construct their own understanding of mathematical concepts, and one way of doing this is having them actively involved in a context that they are familiar with and which stimulates their thoughts. It will be important for teachers to examine exactly how they feel about who can learn mathematics in their classroom, and it will be more important for them to change their beliefs. Professional development, changing pedagogical practices, and encouraging cooperative learning are three things that will be crucial in the change to a multicultural mathematics.

Concentrating on what the mathematics curriculum is centered around can have a large impact on the diversity of mathematics education, says Marilyn Frankenstein. The focus of mathematics education, especially in diverse school settings, should be focused on providing the resources necessary for all students to succeed in mathematics and science. All people can learn mathematics and science, given the proper resources, enough time, and hard work.

Malloy points out that for African American students holistic experiences, using elaborate verbal and motor skills, and interaction with the environment affect cognition, attitude, behavior and personality. Analytical approaches which start by breaking a problem down into parts may not work as well for them. Steps should be taken to make classrooms culturally responsive.

Lubienski reminds us that lower socioeconomic status can affect the way in which students perceive things such as advertisements which contain statistics, and how data are collected. Lower SES students tended to be very skeptical of the results of surveys because of how they were done whereas middle or high SES students assumed that the methods employed to collect the data were done correctly. Higher SES students tend to deal with the data at hand and be critical of it, but not critical of the situation in which it arose.

Multicultural education needs to also address the issue of supporting a variety of languages in schools. Things that can be done to support the effective use of language in mathematics are using mixed ability groups, having students journal, having students share their ideas in small groups before presenting to the class, have students restate a word problem in their own words, and calling on students with language difficulties or fewer experiences before calling on others. All students should be asked quesions which develop mathematical power such as "How did you get your answer?" or "Did anyone get that answer in a different way?" or "Tell me what you're thinking."

African American students are reluctant to take mathematics courses for a variety of reasons: their peers aren't in them, they don't want their GPA to drop, they don't want the challenge, or they are scared. Teachers need to turn these students interests around so that students are able to focus on math and find it interesting instead of something that is creating learned helplessness for them. A critical factor that is often overlooked is getting parents interested in having their children doing well; the parents who place value on mathematics and nag their children to get their math homework done or, better yet, take an interest in helping their children with math is very important. Students aren't necessarily aware of how important math is for them, so we should not be content to let them make a decision not to take math when that is clearly not in their best interest. If you read two articles from this book, please read 6 and 7.

The history of mathematics draws on a diverse array of cultures all of which approach mathematics in different ways. The discovery of pi, although not universal, occurred in many different cultures in many different ways. The history of the mathematicians themselves can be used to create interesting stories which fascinate students and give them things with which they can identify. For example, Johannes Kepler was crippled by smallpox and lived in poverty, yet he was able to unravel many of the mysteries of the universe in 18 years. Many different theories of math can be seen in the variety of number base systems across the world. Examples such as these can give students with diverse background somethings to latch onto, indentify with, and be inspired by.

There are no universal algorithms for solving problems, and in fact, not even any universal symbols that mathematicians use. The Japanese algorithm for subtraction involves adding the "nines complement" and then adding the extra digit to the ones place (see page 166). Giving students the opportunity to learn a new method or use one that is familiar to them can have a large impact on how they view math.

Effective mathematics education for girls can be augmented in a variety of ways: encouraging group work, ignoring girls "small talk" as long as work is being done, studying famous female mathematicians, encouraging girls to be proactive about their math education, providing hands on spatial activities over time, keeping expectations high, and encouraging discussion among students. Actually, it would be interesting to see what kind of dynamic an all female math class would create in terms of sharing and learning of mathematics. There is a laundry list of things that teachers can do as professionals on page 193 to better educate all students in a gender fair classroom. Among these suggestions are providing silent reflection, encouraging females in non-traditional tasks, holding high expectations, and using gender fair language.

There are many ways that the classroom environment can be enhanced by multicultural education, many of which have implications of reaching out to the family and community for a better understanding of what the needs of a particular group are. A diverse mathematical curriculum is not one that is watered down, I feel it is one that encourages multiple viewpoints, a diverse collection of algorithms and knowledge, and one that makes students take more than a casual interest in mathematics. A diverse mathematical curriculum needs to start at a young age and continue (yes, continue) throughout a students education so that a student doesn't feel left out in the cold during some academic years. A continuous effort is needed in order for this to happen. The views expressed in this book will, unfortunately, be read by too few people and perhaps blown off by the rest of the people. Mathematics is a subject that has been handed down in the same way for years and years here in America. Changing the status quo is tough, especially with such a long standing tradition which is rife with cultural bias through its exclusion of other viewpoints. We mathematics teachers need to realize that we are teaching people, not mathematics, and that the only way we can change the future is to realize this now. I wish that this NCTM publication were integrated with the Standards so that people would see it alongside or incorporated with the Standards instead of some small part of a fractious, amorphous mass of information. Why is this issue not addressed within the all of the Standards?

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Keywords: Problem Solving, Activities,
Ref: Pearson3
Author(s): Mason, John; Burton, Leone; Stacey, Kaye
Date: 1985
Title: Thinking Mathematically
Journal or Publisher: Addison-Wesley Publishing Company
Volume, Issue, Pages: pp. 1 - 208
Reviewer: Pearson
Date of Review: April 22, 2000

Thinking mathematically is a mind stretching approach at solving problems. We need to realize that we are all capable of mathematical thinking, mathematical thinking can be improved through thought and reflection, mathematical thinking is engendered by not being able to find a solution easily, mathematical thinking is supported by an atmosphere of questioning, challenging and reflecting, and mathematical thinking helps in understanding the world and yourself.

This book walks the reader through the mathematical process of discovering solutions to problems. It encourages starting with specific examples, looking for patterns, making conjectures, and finding justifications for conjectures. Things that are encouraged in this process are writing down all of your thoughts, writing out what you are trying to do, and exploring how you feel about the problem. Often if we can clarify what it is that we are looking for, we are able to be systematic and analytical in our approach. As we are ready for generalizing our solutions, we need to ask ourselves why certain things are true, and whether or not they are always true. Keywords involved in this process---words which should be written down next to the things they represent---are stuck (describe why), aha (followed by try, maybe, or but why?), check (see if it is accurate), and reflect (write down the key ideas and why they are important).

There are three phases of work which go into solving any math problem. The first phase is the entry phase. Too often people attack a problem with a full scale assault right away. This can be counterproductive. The entry phase is a time to explore a few options and to try to decide what the best way to attack the problem is. Reading over the problem a few times is a good start in the right direction. Asking yourself what you know is another good first step, as is asking what it is that I want, and what can be introduced to make the problem easier. The second phase is the attack phase, and it is where specializing and generalizing take place. It is also where the states of stuck and aha occur most frequently. The authors encourage looking upon the attack phase without tension or judgment. The final phase in problem solving is the review phase in which the resolution is checked, key ideas are reflected upon, and what is learned is extended to a wider context.

Being stuck is a healthy state, the authors contend, because you can learn from it. It is often most difficult to recognize and to accept that you are really stuck. Writing down that you are stuck can change your focus so that you are not incapacitated by the things you think you cannot do. Plus, when you do get that long awaited aha, it will have some true significance. Conjecturing and finding patterns can aid in problem solving during the attack phase. Conjectures should be formulated, checked, then scrutinized and verified. Seeking why and explaining why are an important part of problem solving.

Real learning only takes place over a period of time when the learner has had time to process and reflect on what has been learned. Provoking and sustaining mathematical thinking is not easy, but it can be made easier by associating it with a process that can alleviate some of the stress involved. I think that this book does an excellent job of setting up a process to follow when doing math (even if it is not the definitive process, it is still a good one). It doesn't shy away from addressing the issue of mathematical anxiety, but it takes on the task of conquering it through the difficult process of learning. I like that it doesn't try to start off with really simple examples. I like that it takes on problems with a variety of skill levels needed at various times---the presentation of the problems is more like the real world than it is in math textbooks. Most of all, I like that the approach to the problems is one of continued exploration with an emphasis on writing down the process of discovery as you go along. In all of my math courses I would have loved it if I were able to write on my assignment things like "STUCK!" and "AHA!" to announce where I had troubles on the assignment. It would mean messier work, it would mean that students would have to be willing to show what their weaknesses are, and it would mean more work for the teacher, but I think that this is one lesson that everyone (especially teachers) should take away from this book. All too often we think that the presentation of a math problem means that we should be able to solve it, usually in about as much time as it takes to read the problem if we consider ourselves "smart" (whatever that means). This misconception sets us up for failure every time we attempt something, because it would be hard to do all the math problems we are faced with as quickly as we would like to be able to do them. Furthermore, this book stresses that problem solving is a process of discovery, not merely a repetition of what is previously learned and seemingly easy. Having a conceptualization of problem solving such as this can lessen anxiety about what it means to problem solve and do math. Stressing that problem solving is not easy, it is a process, and that it can have very enlightening solutions should be a part of every math classroom.

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Keywords: Curriculum, ,
Ref: Pearson4
Author(s): NCTM
Date: 2000
Title: cd496_Exemplary & Promising Mathematics Programs:College Preparatory Mathematics
Journal or Publisher: http://www.enc.org/ed/exemplary/cd496/496_7.htm
Volume, Issue, Pages:
Reviewer: Pearson
Date of Review: April 22, 2000

Concentrating on six or seven core ideas, this program integrates algebra and geometry into a four-year program. With each year's objectives focusing on math as a process, students are taken through the normal Algebra I, Geometry, Algebra II, and Mathematical Analysis/Pre-Calculus sequence. Some of the materials are now available in Spanish. Materials are provided for students to self-evaluate their work. Students who have taken CPM have scored statistically significantly higher on tests than non-CPM students in Algebra and Geometry. Aligned with NCTM standards, this program develops mastery of a few important concepts. Textbooks are $21 softcover, $38 hardcover, $45 for the teacher's edition, and costs for teacher training programs are approximately $375-$750 per teacher. A list of topics covered is also on this site. One of the benefits that this program has clearly shown is that students in this program do succeed and go on to take higher level math than before. One school went from offering no calculus classes to offering ca lculus courses to 92% of the students in the program. It is nice that the website for CPM is kept up to date and has advice and helpful hints for teachers. This website can be found at http://www.cpm.org/. The list of topics covered in the program has many interesting titles such as "the poison weed" or "going camping: 3D and circles". The materials for this program are easily obtained by following the information on the web, but they will cost the teacher $20. I can't understand why they don't give away preview and evaluation copies of such materials. Sampler materials are free, but if I were a teacher, I would want to see the entire set of things which I would potentially be teaching my students. There is a lot of documentation on the cpm.org sit e. I am particularly intrigued by the idea that they cite that "recent brain research shows that long-term learning occurs best when students are engaged in puzzling through problems to create their own solutions" as one of the goals behind this project. I am an advocate of this belief, because I feel that for people to truly overlearn the material, they need to create their own personal meanings, interpretations, and understandings of the material---all of which are stimulated by searching for solution s to puzzles.

Their goals are very noteworthy because they are not frilly or fancy. Developing independent problem-solvers who can think for themselves, and students who master basic math skills are the focus of the program, and how can one go wrong with a simple focu s like that? Equally encouraging are the statistics which show that students who are in the CPM program take more math courses than their counterparts in non-CPM schools. I think that this is important because it speaks to the fact that students are bec oming more interested in math and are viewing math as a fun topic. The question needs to be raised, however, that CPM may not meet the needs of all students. CPM seems to be aimed at getting students to take more pure math courses such as Calculus, and may leave out students who will need more applied math for the jobs that they will do after graduation. CPM is a great program for college bound students; Scholastic Aptitude Test scores have improved using the CPM program, high ability atudents have be nefited from taking CPM Classes, and CPM teaches for long-term learning.

Personally, I was disappointed to see some selective advertising going on for this program. I noticed that they said that TIMSS scores were above average for fourth grade but fail to mention the disappointing scores at higher grade levels. Despite some things that may or may not seem like oversights, CPM is a curriculum which appears to be on the cutting edge and willing to stay that way. I was amazed to see quarterly reports on their website at cpm.org about their curriculum, and their willingness to document research and statistics relating to their program.

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Keywords: Curriculum, ,
Ref: Pearson5
Author(s): NCTM
Date: 2000
Title: cd496_Exemplary & Promising Mathematics Programs:Cognitive Tutor^TM Algebra
Journal or Publisher:
Volume, Issue, Pages: http://www.enc.org/ed/exemplary/cd496/496_4.htm
Reviewer: Pearson
Date of Review: April 22, 2000

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Keywords: Research, ,
Ref: Pearson6
Author(s): Battista, Michael
Date: 2000
Title: The Mathematical Miseducation of America's Youth
Journal or Publisher: Kappan
Volume, Issue, Pages: http://www.pdkintl.org/kappan/kbat9902.htm
Reviewer: Pearson
Date of Review: April 22, 2000

This article focuses on the reform movement of mathematics education from the perspective of scientific research on mathematics learning. The author, Battista, disdains current trends in mathematics education that follow in the traditional teaching methods. School mathematics is an endless sequence of memorizing and forgetting facts and procedures that make little sense to students, according to Battista. I would have to agree that much of the mathematics I observed this spring at Northfield High School was presented in this traditional way. Calling our current mathematics education system a "traditional system of miseducation" that is doing little to counteract the "long-term hidden illness that gradually incapacitates its victims". Battista claims that even the bright students are being miseducated in our current education system, since they are memorizing algorithms instead of learning the mathematical concepts which are so important for anything more than a superficial understanding. He is an advocate of having students formulate and test the validity of their own mathematical ideas---methods of making mathematics meaningful to students. Inherent in this constructivist view of teaching mathematics is his belief that students should be able to convince themselves and their peers of the validity of their solutions. In talking about the NCTM standards, he stresses that problem solving, reasoning, justifying ideas, and learning new ideas independently are the critical skills for Americans now. As an example of what he means, he states that most students have learned how to "invert and multiply" when there is a fraction in the denominator of a fraction. However, he points out, students who understand what division means (that is, they know the definition of division) will be able to apply their reasoning to numerous real world situations which may not present themselves in the form of a fraction. Additionally, he feels students should know the "basic number facts" because such knowledge is essential for mental computation, estimation, performance of computational procedures, and problem solving. Number facts should illuminate the abstract for students. The real world should be what illuminates the textbook.

The point of this article, unfortunately, is not made clear until half way through; scientific constructivism is a learning method in which abstraction, reflection, and learning are engendered by exporing the abstract. Abstraction, he claims, is the critical mechanism that enables the mind to construct mental entities which can be used to reason about "mathematical realities." Reflection is necessary in this process in order to be able to solidify learning in the mind. He is not advocating that we begin teaching students more abstract math right now, rather he believes that a gradual formalization of the concepts of the real world ought to build upon burgeoning abstract concepts. If I may draw an analogy to English class, Battista is suggesting that students be taught where to put commas in sentences by finding the natural breaks in the language and then reinforcing this idea through use and formalization as a procedure. The rule has to make sense, and then it can become a rule that is worthwhile and useful. The whole point of constructivist mathematics teaching is to have "students derive these symbolic procedures through personally meaningful manipulation of quantities," so that "their knowledge of the procedures becomes semantically rich in its connection to their reasoning about quantities. It is no longer inert and strictly syntactical. Research clearly shows that such `construction-focused' mathematics instruction produces more powerful mathematical thinkers."

I think that constructivist teaching methods are most important in math. Mathematical systems usually start with a set of axioms and postulates and go from there. In other words they start with a seed and grow into a plant. In order to understand what the plant is like, we could try to pick off all of its leaves and understand each one of them, or we could start with the fundamental concepts and see where they branch out and create stems with living, vital leaves on them. Rote memorization is a lot like picking the leaves off the trees in that the learning which occurs is fleeting and lasts only as long as a leaf does that is taken from the plant. If we start with the core concepts of mathematics and see where it is that they branch out, we are really looking at a constructivist model for teaching mathematics. As TIMSS indicates, we need to start teaching the core concepts---even if it means that we need to dig a little and get our hands dirty in order to see the roots of the plant. Rote memorization may be the easiest quick fix to the perceived problem of a deficit of substantial mathematical thinking among American students, but it most certainly is not the best way to go about things if we are interested in the overall health of the plant. As Battista points out, the American public does not always have the correct perspective on the overall health of the plant, and he encourages professional mathematics educators to implement constructivist curriculum decisions. Although I don't agree with his rather extreme view that the public should not be allowed to have its input about mathematics curriculum in the political arena, I do agree that the time for true mathematics reform---one spearheaded by professional mathematics educators who act responsibly and responsively---is now. Students don't learn math to the fullest extent with rote memorization; we need to change the status quo and explore constructivist mathematics learning.

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Keywords: Inquiry, ,
Ref: Pearson7
Author(s): Cipra, Barry; Zorn, Paul
Date: 1994
Title: What's Happening in the Mathematical Sciences
Journal or Publisher: American Mathematical Society
Volume, Issue, Pages: Volume 2, pp. 1 - 51
Reviewer: Pearson
Date of Review: April 22, 2000

In this issue of "What's Happening in the Mathematical Sciences" some very exciting mathematical discoveries are talked about. It is truly fascinating to be able to read about how Andrew Wiles proved Fermat's Last Theorem and actually understand on a superficial level why the proof works. It's very interesting to think that somehow the simple equation of Fermat's Last Theorem is a consequence of theories of large classes of elliptic curves being modular. It is also amazing to me that the theories which Wiles developed in trying to prove Fermat's Last Theorem have advanced the topic of Number Theory as much as they have. Number theory, after all, must be one of the oldest subjects in math.

The second article on knot theory makes connections among knot theory, molecular biology, and quantum field theory. I did not know that knots can be represented by different Alexander polynomials, and that knots with different Alexander polynomials must be different knots. Mathematicians are concerned with four dimensional knot models (the fourth dimension is time) and they've been making some great breakthroughs on old conjectures.

A very interesting wave phenomenon, known as solitary waves or compact waves, are characterized by a very nonlinear behavior: when the amplitude of a wave increses, so does the speed. Mathematicians are interested in the behavior of these abnormal compact waves because they leave trails of small ripples behind them when they meet.

Scientists in Troy, New York are testing new methods of being able to see what is happening in the tissues of our bodies, and their methods are centered around electrical current conductivity and permittivity. Their new technology depends on a lot of linear algebra and a knowledge of where the sensors are in the body, since inaccurate results can occur when the technology isn't adjusted for the medium it is sending signals through. The technology is called impedance imaging, and it may offer the fastest, most reliable pictures of what is going on inside our bodies to date.

Wavelets is a new discovery in applied mathematics that is uniting people from many different scientific fields. Wavelet theory takes the basic premise of Fourier analysis, that a function can be broken down into mathematically simple components, and makes it "simpler" by not limiting its simple components to just the trigonometric functions (which undulate infinitely in both directions along the "time" axis). Wavelet theory has applications in being able to compress images and sound waves in real time (or as close to real time as computers can get). Some examples of the uses of wavelet theory are encoding fingerprint data, restoring old recordings by taking out the noise in them, and analyzing electrocardiogram information.

There is an interesting article on soap bubbles and how they form. They tend to be objects with maximal volume and minimal surface area, though not in any particularly "regular" shape---especially when there is more than one bubble involved. Mathematicians are also interested in higher dimensional soap bubbles (theoretical bubbles?).

An article on error correcting codes points out how important coding theory is for computers; a 25 MHz processor which only makes an error once every billion calculations will make mistakes an average of 90 times per hour. That is a lot of mistakes (especially when we look at how many computers there are and how many hours they are used!). Error correcting codes, to be the most effective, need to be nonlinear. This will be the challenge for the next few years.

I find it fascinating that pure math topics such as wavelets could have so many applications. I find it amazing that a fingerprint image that takes up 10 Megabytes of space can be compresses at a ratio of 26:1 using wavelets so that it only takes up around 500 Kilobytes of storage space. With the ethernet pushing speeds up into the stratosphere, this should make police departments across the country rejoice over the new technology. Equally intriguing is the proof of Fermat's Last Theorem which has baffled mathematicians for many years. I would have loved to read things like this in high school, but do you think that any high school library would ever order something like this? I was surprised to read in a few of these articles that high school students had been on the research teams of a few of these projects. Another interesting thing that I liked about this publication is that it is written for a relatively mathematically literate audience but not written in the all too formal language of mathematics prose. I like being able to think of a knot as a tangled up extension cord with the two ends plugged together; the conceptualization is every bit as accurate as any definition of a knot, but it is so much easier to conceive. I wish that all math publications were written at this level so that the reader who is a novice can take some interesting insights away from the reading and the expert can get the content which they want out of the reading.

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Keywords: Resoning, Teaching Strategies,
Ref: Pearson8
Author(s): Costa, Arthur; Lowery, Lawrence
Date: 1989
Title: Techniques for Teaching Thinking
Journal or Publisher: Midwest Publications
Volume, Issue, Pages: pp. 1 - 105
Reviewer: Pearson
Date of Review: April 22, 2000

The goal of this book is to enhance the thinking skills of students through better teaching techniques. Better thinking skills are directly correlated with better test scores on any kind of test. The teacher's power to shape students' thinking is awesome. Teacher behaviors, perceptions, and attitudes greatly influence their students' achievements and thinking abilities (among a multitude of other things). Children's brains are hard wired to learn from a stimulating and complex environment, and they will only develop if they are stimulated enough. The best way to stimulate real world interaction for children is to increase your verbal interaction with them; this means that they shouldn't be left to hours of TV and video games (no matter how stimulating they are, they are definitely not authentic and they don't engender genuine interaction). Specific things that teachers can do to icrease the thinking that goes on in their classrooms are structuring the classroom environment, use directions and questions to prompt old thinking and stimulate new thinking, and respond to students in a way that maintains or extends student thinking. Modeling thought processes, selecting proper content, and focusing on thought processes is also par for the course when trying to augment student thinking.

Structuring the classroom for thinking means having clear written and verbal instructions, a stuctured time period which matches energy levels of students, and effective grouping of students. Preparing students for the lesson and explaining things from multiple perspectives are among some of the best ways to enhance clariy of instruction, as is simply taking the time to present your thoughts as clearly as possible. The classroom environment should be structured so that it is promoting active learners, not passive ones. Structuring the environment so that students have some part of the locus of control of the classroom can aid in developing learners who seek higher-level creative thinking.

Teachers should also be aware of the level of questions that they ask (as in the levels on Bloom's taxonomy). Questions should be varied, convergent, divergent, sometimes aimed at recalling previously learned material, and aimed at a variety of tasks. The output desired of the students should be implicitly or explicitly made clear by how the question is phrased. Teacher responses to student answers should extend student thinking and encourage them to try different options if they are incorrect. Closed responses such as criticizing and praising don't engender further thought; open responses such as silence, accepting, clarifying, or providing information tend to engender more thought on the topic from all students. I like their suggestion that praise is not the solution for everything; learners need to be able to find something that is inherently satisfying in their work before they will acquire skills which they value themselves. Chapter three of this book should be taught in Ed 330, in my opinion.

Using the proper words when posing a question can enhance learning. Actions such as compare, predict, classify, analyze, hypothesize, and conclude ask for more specific responses than more blunt verbs do. Asking thoughtful questions also puts the responsibility on the student for their action; saying "What must you remember to do?" instead of "Remember to write your names on the top of the paper" provides temporal cues that the student can use instead of a simple reminder which may be easily forgotten.

Inherent in teaching thinking is teaching metacognition, or teaching how to think about thinking. The difference in the power of thinking that is made possible by metacognition is like the difference between being told that a 3-4-5 triangle is a right triangle and being told the Pythagorean Theorem and how to use it. Learners often get stuck on the "How to use it" part of thinking when they actually possess the tools necessary to do the thought processes. Strategy planning, checking answers, and quesiton generating are among the many processes that can be taught in order to enable students to apply them to any number of many general situations.

How do we know when students are becoming better thinkers? The data collected to answer these questions is both observational as well as qualitative. Increased thinking skills will show up in understanding and careful examination of problems as well as behavioral traits such as decreased impulsivity, persistence, checking for accuracy and precision, flexibility of thinking, metacognition, and going beyond the limits of the problem to incorporate a variety of sources of information. In short, students will start to show intelligent, inquisitive behavior.

I think that techniques for teaching thinking is a very well researched book and it should be taught to Ed 330 classes instead of simply trying to memorize Bloom's taxonomy. Bloom's taxonomy is great, but it doesn't have the practical suggestions that this book does. There are a lot of informal assessments going on in a classroom, as well as a lot of informal questions and answers, and this book addresses these situations whereas Bloom's taxonomy doesn't. Chapter 3, as I said above, is an absolute necessity for any teacher, but all of the chapters in this book are very worthwhile. The thing that this book does so well is offer suggestions of ways to get your students thinking without really even getting them to know that their thinking because they are stimulated by their own curiosity. The suggestion that we rethink how we pose statements so that they are questions which the learner has to process is not brilliant, but it is something that we should all learn to do because it would make such a huge difference in shaping students thoughts. I really liked that the emphasis of this book was on shaping students into responsible thinkers. This is a really great book. Read it. Put it under your pillow at night and read it whenever you can't sleep. It's not revolutionary, but it reinforces what should be done.

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Keywords: Assessment, Research,
Ref: Pearson9
Author(s): Gardner, Howard
Date: 1993
Title: Multiple Intelligences
Journal or Publisher: BasicBooks, a division of HarperCollins
Volume, Issue, Pages: pp. 1 - 304
Reviewer: Pearson
Date of Review: April 23, 2000

Howard Garnder didn't start off anticipating that his multiple intelligences theory would be anything important or anything of interest to the general public. But his Piagetian theories were quickly latched onto by educators everywhere and expanded upon. Gardner's secret to success is inherent in the name of his theory: multiple (meaning many) intelligences (a psychology buzzword). It moves away from unidimensional IQ tests such as the standard IQ tests and more specialized IQ tests such as the SAT. The evolving set of multiple intelligences (MIs) are musical intelligence, bodily-kinesthetic intelligence, logical-mathematical intelligence, linguistic intelligence, spatial intelligence, interpersonal intelligence, and intrapersonal intelligence. One important aspect of Gardner's Piagetian perspective is that he views these intelligences to be developmental stages. Children should be exposed to a wide variety of experiences in hopes that many of them become "crystallizing experiences"---ones that create a sudden and lasting impact on the way something is perceived. It is also important that these experiences come at the right time or stage of development. The best way to do this is to expose an individual to a sufficiently complex situation which may require the use of several intelligences. This has implications for assessment; traditional paper and pencil tests won't be very effective at assessing complex situations where multiple intelligences are involved.

It is interesting to hear Gardner speak of intelligences as things that can vary, and indeed can be developed or delayed depending on the stimuli. His theory is much like the Suzuki violin method theory in that he believes that if people are exposed to a multitude of factors that develop a certain intelligence, then they will begin to develop that intelligence. What implications does this have for people taking IQ tests or the SAT? Well, depending on whether or not they study, it could have a huge impact. He goes on to define intelligences as a biophysical potential which is not to be confused with being gifted or a prodigy. He describes rather rigidly set time frames for developmental advances which may or may not be correct. At least he uses these stages of development as ways of describing various stages of intelligence (i.e. being gifted or a prodigy).

As an extension of his theories, Gardner stresses that students need practical intelligence for school (PIFS). Their practical concerns for success in school are centered around finding ways that students can integrate both academic knowledge about particular subjects and practical knowledge about the world and themselves. This kind of practical knowledge intervention should certainly be presented by middle school, argues Gardner. Getting students at this age to elaborate on responses, use different strategies and resources, see themselves as learners, and become active learners are all goals of this stage of development. More disciplined inquiry should be stressed in high school. Thinking within a domain, reflecting on aspects of the domain, discovering ways of perceiving the domain, and ways of approaching work should be stressed within the context of many different subject domains at this age level.

Because Gardner's theory is based on potential intelligences as well as ones that already manifest themselves, assessment of such intelligences can be complicated and not very straighforward. The emphasis on finding new ways of assessment in place of the old methods of testing will be central to the advancement of the theory of multiple intelligences. Portfolios and ways of demonstrating understanding will be critical tools in moving away from the one-dimensional and programmatical tests of today. Implications that this has for teaching is that teaching will have to develop multiple windows or ways of presenting information to people so that they can be exposed to, and develop, multiple intelligences.

With a dearth of practical suggestions for enacting the multiple intelligences theory, especially in the logical-mathematical category, this book leaves something to be desired from a practical standpoint. Everything about this book is great in theory, but I would be willing to be that most people would have to see it in action before it would really start to make sense to them. This book will change your perspective on what really is valid to teach and how these valid things should be taught, so it does have merit for the average teacher. I'm just afraid that in trying to put a multiple intelligences framework around an old curriculum is not going to work very well, and, in fact, may lead to a watered-down curriculum. It would be nice if Gardner were to publish some sort of resource book with tidbits of his theories annotated throughout that teachers could use as a practical guide to get them started using multiple intelligences. The problems I forsee are people who are throwing in multiple intelligence lessons sporadically with little preparation and little success. I think that with something such as this theory, there is a process of shaping and forming students into different learners. This, I believe, is something that cannot be undertaken as a quick fix. I don't say this to discourage anyone from trying to use multiple intelligences, but I do want them to realize that they may be opening Pandora's box without being prepared for what may come out.

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Keywords: Inquiry, Problem Solving,
Ref: Pearson10
Author(s): Mathematical Association of America; Albers, Donald
Date: 1994
Title: Math Horizons
Journal or Publisher: Mathematical Association of America
Volume, Issue, Pages: Spring 1994, pp. 1 - 28
Reviewer: Pearson
Date of Review: April 23, 2000

John Horton Conway is a fascinating, diverse mathematician. He is especially gifted at doing lightning quick calculations such as telling you what day of the week December 4, 1602 fell on. (The answer is Saturday, and Conway can do 10 such dates in under 20 seconds.) I like his approach to mathematics; he likes to have several problems to think about on hand at any one time and he prefers that some of them be "simple" problems which he can probe into with great depth. He has an "intense need to make things simple." It is funny, almost comical, that this great mathematician recalls that when he plays board games (things which he likes to analyze) with his wife or daughter, he rarely wins despite great concentration. I also like his guilt-free attitude that reminds me so much of Richard Feynman, a famous Physicist. He is not ashamed to take on simple problems in great depth, and obviously for him this pays large dividends in his exploration of his creative genius.

Long Run Predictions is an interesting article on winning and losing streaks in seemingly random data. "On August 18, 1913, at the famous Monte Carlo casino in Monaco, black came up 26 times in a row on a roulette wheel." Was this roulette whell rigged? Probably not, points out the author Mark Schilling. Although such an occurrence has only a 1 in 68,411,592 chance of occurring in a specified set of 26 trials, it is not altogether totally unlikely that this result has occurred on some wheel at some time given the large number of trials that occur every day. The author points out that with streaks in sports, all the hype could just be that, only hype. We are trained to recognize patterns and so when we see one we attach significance to it, whether or not it is truly significant or not.

An interesting interview of many math majors at various colleges yields a very interesting set of perspectives on math. Being able to communicate, read, and make connections in math were among the most valuable skills they feel they have developed over their nascent math careers. It is interesting to find out how many of them went into math only after trying other majors, only to find out that they enjoyed their math courses most of all (although, couldn't this be said for people who have tried a math major and switched to another major?). Most of them mentioned having good experiences with math from their secondary teachers and then coming to college where they absolutely loved math.

There is a rather comical section about mathematicians in the movies which tries to pin down how mathematicians are portrayed in Hollywood. You can read this section at your own risk, it isn't very thought provoking, but it is kind of funny to read.

Mary Schilling has an article in here on tips for job interviews. This would be a good article to copy and distribute to the class, because we'll all be looking for a good job someday (preferably something other than selling shoes or flipping burgers). The article is very informative and has a few salient checklists of information which could turn out to be invaluable.

There are many problems and brain teasers in this magazine, and my favorite one in this issue is the one about cutting an equilateral triangle into four pieces (on page 27). What can I say? I love geometry. I really enjoy reading magazines like this which are written for a wide variety of mathematicians and even non-mathematicians (in the sense that they might not be math majors). The funniest calculus problem ever invented has to be on page 5 (it encorporates a triangular trough, a lighthouse, a ladder, and Newton gathering pebbles on the beach). Some of the number theoretic problems would be fun to do in the secondary education setting. It would be great if high schools could afford to spend money on these kinds of periodicals which might inspire students to become interested in math, especially since this journal is very focused on showing the human side of mathematics instead of the technical and dry side of mathematics. It would be neat if this journal had more student input from undergraduates and graduates so that people at the high school level could maybe see themselves being mathematicians in a few years.

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Keywords: Inquiry, Problem Solving,
Ref: Pearson11
Author(s): Mathematical Association of America; Albers, Donald
Date: 1995
Title: Math Horizons
Journal or Publisher: Mathematical Association of America
Volume, Issue, Pages: Spring 1995, pp. 1 - 36
Reviewer: Pearson
Date of Review: April 23, 2000

There is an interesting article by Joel Chan, an undergraduate at the University of Toronto, about cryptography and cracking the famous code that locked up the message "The magic words are squeamish ossifrage." The nature of the keys (the codes which lock up the message and keep it safe) is that they are the product of two very large prime numbers, say p and q (somewhere in the vicinity of 600 digits), and then randomly choose a number e less than the product pq such that gcd( (p-1)(q-1), e )=1. Then they do some modular arithmetic with these numbers and end up with some very random, difficult to guess keys to secure the message.

There is an article about whether or not mathematics is something which could receive a patent in this issue. Since things that haven't even been invented yet are patentable, one might think that something like a Theorem is patentable. However, since mathematics involves ideas rather than machinery, this is very unlikely. It is likely, however, that algorithms for computers become a mathematical object that are patentable. Algorithms receive this special attention since they are something that can truly have an economic benefit, presumably an economic benefit for someone other than the creator. Mathematicians such as Andrew Wiles will be immortalized and make ample money off of their mathematical creations simply because they possess a rare quality that cannot be transferred to another person. An algorithm, on the other hand, could be transferrable and profitable, hence it could receive a patent.

Persi Diaconis is a professor of Magic (oops, actually Mathematics) at Harvard. This colorful character was a vagabond magician in his adolescence, much to the chagrin of his parents who had primed him to be a virtuoso violinist by having him study music at Joulliard as a youth. By chance, he was lucky enough to get to know Martin Gardner during his vagabond magic years, and Gardner's recommendation is what eventually got this magician out of City College in New York and into Harvard to study statistics. Wow. What a diverse youth this person was. Ultimately it was Gardner's recommendation to a friend at Harvard that they allow this seemingly undereducated person to study here because he had invented 2 of what Gardner considered to be the 5 most original and intriguing card tricks known. This is a very interesting article which is a fun read.

There is an interesting article on how computers use the binary system to compute numbers. The article briefly explains how place values are different in binary, and that for ease of decoding, hexadecimal is often used in programming. Over half of the article is on how computers and other machines that man has made up in the past deal with negative numbers. From some machines, if you subtract 1 from 0 you will get a display full of nines. This should intuitively make sense in a modular arithmetic sense, as well as being obvious when we realize that the string of nines is a finite string, so one plus this finite string of nines will be zero.

The article by Paul Davis explains what it is like for mathematicians to work in industry, especially in settings where group work is extensive. Mathematicians, more than Physicists and Chemists, need to demonstrate to the company why they are a vital part of the company. Part of this is bridging the communications gaps among fellow workers. But describing the situation and clarifying solutions isn't enough. Mathematicians need to carve out a market niche for themselves by being willing to help out others so that they, too, can become skilled at the routine mathematical portions of their jobs which you may find rather trivial. This trading of skills goes both ways, though, and mathematicians should be eager to barter for skills from other scientists. The article stresses that solutions to real world problems can often be messy and very involved, and that shying away from reaching at least an approximate answer to a problem could be detrimental.

There is an inspirational article by Thomas Harms about being a new math teacher. He routinely runs into people who don't see how math is useful in everyday life. Rather bluntly, when he hears about people who claim to not even be able to balance their checkbook, he gives them the phone number of their local bank and asks them if they've got something about math that they'd like to talk about. I would have to agree. People that I know see math as totally useless outside of arithmetic. Granted, they are mostly joking when they make remarks like this, but I have my doubts. People truly don't realize how much math is at work around them because they leave the math to the mathematicians. Harms combats this sort of apathy every day in his classroom by being enthusiastic about math. It is not enough just to be excited about the math that is being taught, but math teachers need to be excited about how the math relates to who they are. "The gains will be both yours and those whose lives you touch."

I was happy to see that there was an article in this issue which was submitted by an undergraduate student. Equally encouraging is seeing high school students submitting solutions to the problems posed in the magazine. I like that this magazine focuses on developing mathematicians, not just on interesting math problems or what's wierd and new in math. It really fills a hole left in the system between relatively inexperienced mathematicians and ones who have a lot of mathematical savvy. It's just nice to see a magazine devoted to math which uses sentences that you might actually be interested in reading for some other reason than to gain some factual information. It is also nice to see candid articles about math which will actually tell students things like a "harsh reality is the absence of a safe haven for mathematics outside of academia" (18). It's a good thing that we will all be part of academia in one way or another.

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Keywords: Inquiry, Research,
Ref: Pearson12
Author(s): Cipra, Barry
Date: 1993
Title: What's Happening in the Mathematical Sciences
Journal or Publisher: American Mathematical Society
Volume, Issue, Pages: Volume 1, pp. 1 - 50
Reviewer: Pearson
Date of Review: April 23, 2000

In reading the article "Equations Come to Life in Mathematical Biology" I begin thinking to myself that this must not be real article. (Just Kidding.) All kidding aside, I have doubts about our biologists (our meaning St. Olaf Biologists). I had a St. Olaf biology major ask me what an inverse square relationship was. I explained it to this person and was a little dismayed to find out that the words "inverse square" didn't previously have any meaning. This student was trying to calculate their body mass index (BMI), and when using the formula was getting results inconsistent with what some of the examples were. I asked if the units were in metric. No. I rattled off the metric conversions from memory, becoming rather astounded that these things didn't occur to this student. The student finally got the answer, but I was left with many questions. OK, all of that aside, let's dig into the article. The article is about how mathematics is involved in things like CAT scans, nuclear magnetic resonance imaging (NMR), and positron tomography. Mathematical models of vital organs such as the heart can also be used to to lead to improved surgical techniques, improved artificial valves, and many other things. The models are not real, so they have the advantage of being "relatively" cheap (when compared to the value of a human life) and easy to manipulate. Currently it takes analyzing around 3 billion pieces of data, but with mathematics and compression formulas such as wavelets, who knows how much this may be compressed by.

The article on computer insights on proofs is interesting. It points out that computers are not infallible as we may believe them to be. We program them, they might malfunction, any number of things may go awry. Errors are very hard to detect. Solutions, especially when the computer can output something meaningful like a graph of a solution, can be checked with reasonable assurance of being correct.

At Washington University an interesting conjecture about being able to hear the shape of a drum was shown to have a counterexample. A drum is any two dimensional figure which has "tension" at its edges. The question of whether or not two completely different shaped drums, polygons, could sound alike. The answer is yes. It is tantalizing to think that this problem which has been around for more than 50 years was solved with little more than linear algebra and that the solution fits on a postcard. Many other important discoveries were made along with this discovery about drums, mostly dealing with the precision and accuracy assumptions that applied mathematicians can make.

Computers are time and environment savers. "Environmentally Sound Mathematics" talks about how computer models save time and money in environmental projects because they are capable of doing very sophisticated mathematics very quickly and without expensive trials in the field. For environmentalists, the theoretical field trials available through computer simulation save time, energy, and money, as well as allowing scientists to do multiple tests in the time that one field trial would take.

There is a very interesting article about higher dimensional math in which the seemingly obvious analogs from lower dimensions may not hold and be true. Geometric intuition doesn't hold in higher dimensions all of the time.

Spheres, no matter how badly they become distorted, will have infinitely many geodesics (the analogues of great circles on a perfect sphere) which go around it. Interestingly, this discovery is the result of some odd combinations of things: a mathematician from Germany, a mathematician from Evanston, Illinois, differential geometry and dynamical systems. These very different subjects---one is concerned with objects that are fixed whereas the other deals with objects that change---seem to have connections that make solving this interesting problem possible.

Another interesting application of computers in mathematics is in producing crystals. At the geometry center at the University of Minnesota, these kinds of computations are going on in order to see who can produce the most beautiful crystals. Crystals, suprisingly, are probably the best things we have to model dendrites (parts of neurons in our brains) right now. Other novel things such as representing music with the repeated recursive application of simple rules, much like the idea of a seed crystal generating the entire crystal.

When I read something like this I wonder about who else reads it. It is truly fascinating stuff described at a level that most anybody could understand, and I find it most fascinating to use as a springboard which launches me into futher inquiry of the vast field of mathematics. Even though a publication like this seems watered down at times, because it doesn't go into the complexities of all of the math behind the theories presented, I think that it serves a great purpose. I really wish that I could hand the article on mathematical modeling to my biology friends and say, "You should read this stuff on mathematical modeling of the bodies organs and the environment. It is worthwhile stuff, and it should make you consider taking some more math." I'd like to see the looks on their faces before and after reading the article. While some of the interesting results in this publication seem trivial, you never know who will be reading and decide that this is related to what they're dealing with in a particular problem. I like that this publication can help span the gaps between the mathematicians and the other scientists.

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Keywords: Inquiry, Research,
Ref: Pearson13
Author(s): Cipra, Barry; Zorn, Paul
Date: 1995
Title: What's Happening in the Mathematical Sciences
Journal or Publisher: American Mathematical Society
Volume, Issue, Pages: Volume 3, pp. 1 - 112
Reviewer: Pearson
Date of Review: April 23, 2000

The article that receives the utmost attention in this issue is the one on Andrew Wiles proving Fermat's Last Theorem. Wiles was able to fill the gaps in his earlier proof of the Taniyama-Shimura conjecture of which Fermat's Last Theorem is a direct consequence. Astonishingly, describing what wasn't working with his old proof gave him the insight he needed to see that he would have to "glue" together Hecke rings in order to take a different approach to the problem and eventually solve it. It is also interesting to learn that Fermat has made other, false conjectures, such as the one about Fermat prime numbers of the form 2^k+1. Not all numbers of this form are prime.

Interesting analogues exist between particle physics and geometry such as the idea that every conservation law is equivalent to a symmetry of a geometrical figure. When things are view from multiple perspectives such as this, physics and mathematics are complementary. The article goes into the fascinating topic of compact manifolds after giving an explanation of why the cartesian plane is not compact.

The next article really blew my hair back. It proposes using DNA to do calculations. Yes, that's right, DNA. These scientists have already used biotechnology to solve an abstract problem that has nothing to do with biochemistry. The primary advantage that DNA has is its remarkable ability to do parallel processing in contrast to the serial computations of the computer hardware we have today. The researchers point out that even if DNA computing doesn't pan out, important advances in parallel processing will have occurred and been advanced.

An interesting article about floating point decimals and tiny errors having large effects when compounded is described in the next article which tells of how a mathematician found the error with the Intel Pentium chip. The point of the article is that when an error correcting algorithm goes wrong, many decimal places of accuracy can be lost.

"Computational Fluid Dynamics---Verging on Turbulence" is an interesting discussion of how fluids move. For example, fluids flow fastest in the middle of a cylindrical pipe but do not flow at all along the edges. Turbulent systems such as fluid flow are fascinating to study because they have nonlinear equations, which we don't fully understand yet, which govern their behavior. Meteorologists as well as pilots have interest in the applications of studies such as this. Currently computers are being used to model and solve differential equations to any desired degree of accuracy. Could improvements with approximations such as these lead to breakthroughs?

"Are Algebraists Simple Minded?" is a very interesting article that describes a 15,000 page proof written by hundreds of algebraists over a span of 30 years. The so called "Enormous Theorem" is so hard to follow that nobody has ever read the entire thing. As is the case with many large theorems (many of which are classification theorems such as the Enormous Theorem), revisions are needed, and powerful new tools are being devised each day to make classifications easier. The four varieties of simple groups are cyclic, alternating, Lie-type, and sporadic. Mathematicians believe that they have found all of the 26 sporadic groups, one of which contains 10⁵³ elements. Hopes of the revision group are that by straightening out the old threads of the proof, more generalities will be found and the theorem will have a less bloated proof.

I read the third volume of this series because it really makes me excited to hear about such real world applications of math. As a novice mathematics student in high school I could see myself looking into these real world applications in depth in my science classes for projects and other activities. Though reading at this level seems rather timid in comparision to graduate level textbooks in math, for a high school student writing like this could have a large impact on their future decisions, possibly influencing them into mathematics instead of another science. The depth is just great enough to be enlightening but not so much that it becomes frightening. Journals like this would help to introduce high school students to what math is really like at higher levels, and what sort of topics in math are ones that are researched. I think that one of our problems in producing more students that are interested in mathematics is the perceived difficulty and abstractness of the subject. Good writing such as this could easily dispel some of those myths. Kids should be exposed to writings like this even if they are over their heads, because it gives them an idea of what the possibilities are which they might pursue.

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Keywords: Issues, Curriculum,
Ref: Pearson14
Author(s): Alder, Henry; Cicero, Joseph; Dossey, John; Enneking, Marjorie; Guy, Richard; Hight, Donald; Layton, Katherine; Meserve, Bruce; Young, Gail; Bushaw, Don
Date: 1983
Title: Recommendations on the Mathematical Preparation of Teachers (Committee on the Undergraduate Program in Mathematics)
Journal or Publisher: American Mathematical Society
Volume, Issue, Pages: Volume 2, pp. 1 - 76
Reviewer: Pearson
Date of Review: April 23, 2000

This book focuses on what sort of education teachers of mathematics should receive before being given the rubber stamp seal of approval and told to find a job. The book starts off by differentiating among the various teaching levels: elementary, middle school, and secondary. Accordingly, the number of mathematics courses needed for secondary teachers is more than the number of courses for middle school teachers which is in turn more than for the elementary school teachers. The focus of these courses should be on maintaining enthusiasm in mathematics and clearly communicating knowledge of mathematics. Interestingly, the recommendations for secondary teachers include courses in Number Theory, the History of Mathematics, Mathematical Modeling and Applications, and Mathematics Appreciation. These are courses that are not required of St. Olaf Math Education Majors.

In general, the K-8 teachers need more math preparation. They need more variety, more breadth of knowledge, and more problem solving. Problem solving strategies should be developed among these teachers so that they can impart effective problem solving techniques upon their students as well as create situations which will engender student generated problem solving techniques. Specific examples of the topics that teachers should teach at this level are given, as well as how many course hours of preparation are advised for each topic. Interestingly, geometry concepts are suggested to be taught as part of this teacher preparation. I can't think of an elementary teacher which I had that consistently did anything with geometry as a mathematical topic. Probability and statistics are also recommended. Wow. Why didn't I ever get any teachers that were willing to do these things with me? A seemingly useless computer requirement is also suggested (I say that it is useless because I have a hard time imagining an elementary school teacher trying to teach students how to program a computer, which is what this course suggests). The expectations are written out in a very textbook-like manner. Very dry and very boring. I hope that the actual courses can breathe some life into the mathematical concepts described.

Interestingly, the middle school courses stress developing skills in students that will prepare them for topics in higher level math courses such as calculus. One example is estimating the area of irregular and regular regions. Wouldn't it be fascinating to teach kids Riemann sums without telling them that they were doing Riemann sums? There isn't anything particularly abstruse about Riemann sums, even though there is a little bit of mathematical sophistication that would need to be developed to put them to good use. Other such topics that they mention would be included in today's reformed curricula. All in all, the middle school teachers do not get an adequate description of what it is that they are supposed to do for their students who are developing.

The high school preparation guidelines suggest a lot of applied math and not very much of the more theoretical topics in math. This may be a boon for people looking to teach for many years, but it may also hinder people who only plan on teaching for a while and would like to go on to study more math. Besides, by limiting high school teachers to such a large number of required courses, the resulting teachers will not have a breadth of knowledge in all areas of mathematics as they might if they were allowed to take more courses as "electives." Teachers at this level should be taught mathematical correctness, especially in phrasing (why isn't this a goal mentioned for the other teachers?), and making experiential relations for their students. Again, the computer requirement (practically asking teachers to learn how to program) is unnecessary, even though it is a nice thought.

This short guide is definitely outdated in many ways. Basically, the guidelines mention only the content of the math to be covered and nothing else. The new Standards by the NCTM supercede these guidelines by the MAA and do a much better job of describing not only the "what" of teaching but the "how" of teaching. It would be nice if the Standards were addressed to us in a separate manual so that we knew exactly what was expected of us, but the Professional standards do a reasonable job of covering most of the bases. I found it interesting that the MAA felt it necessary to spell out what basically amount to course descriptions when most textbooks are basically the same in the content which they cover and the basic content which is taught in courses tends to be centered around what is in the textbook. What is still unclear is what the MAA expects us to do with the skills that they list in our "course descriptions" as I'm going to call them. Are Poisson distributions necessary to prepare a teacher to teach elementary discrete probabilities? Are we expected to teach the topics that we have learned? What are we to get out of courses that teach us more than we could ever use when teaching in the classroom? These are questions I would like to ask the people at the MAA who wrote this document.

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Keywords: Problem Solving, ,
Ref: Pearson15
Author(s): Gilbert, George; Krusemeyer, Mark; Larson, Loren
Date: 1993
Title: The Wohascum County Problem Book
Journal or Publisher: Mathematical Association of America
Volume, Issue, Pages: pp. 1 - 233
Reviewer: Pearson
Date of Review: April 23, 2000

This problem solving book contains around 30 pages of problems and approximately 190 pages devoted to their solutions. The problems are at an advanced high school level and would require that students have already taken or be taking calculus. The soluti ons presented are very thorough and rigorous. Multiple solutions are often presented, and hints and broader perspectives are given along with the solutions. With 130 problems to choose from, this would be a great resource book for challenging brain teas ers for high school students.

There are some very thought provoking problems about derivatives such as fiding all of the lines tangent to two particular parabolas and then trying to find the generalized solution for all parabolas of a certain kind. There are many geometry problems wh ich students could apply their calculus skills to in this book. Unfortunately, there are many solutions which require linear algebra, a topic that most students won't have had in high school. There are many problems involving limits and series in this b ook, and it would be great if we could get students to work through more limits problems. Limits tend to get overlooked and only thrown into the curriculum only where necessary, so a special unit or two dealing with limits as an extension of a topic or s omething that is concurrent with a topic would be really fun. There are many fun limit problems in this book.

One nice thing about this book is that it lists all of the prerequisite knowledge needed for every problem, so that when you're looking for an interesting trigonometry problem for your students, all you have to do is look for trigonometry in the appendix and select a problem that you like. Not only is this convenient, but it can help you find increasingly difficult brain teasers for your kids since the questions are supposedly arranged in order from least difficult to most difficult. Best of all, though, is that if you get your students interested in problem solving such as this they will read thoroughly explained solutions to problems and learn how to write better solutions themselves.

High school teachers should definitely look into buying this problem solving book because it could contain the problems that inspire students to apply their knowledge and learn to love math. The authors are wittty enough to make the problems have a fun t wist for each one, and the soluitons are well written, well presented, and indexed well. There are many problem books out there, but I would venture to say that few of them have devoted themselves as wholeheartedly to clear, cogent solutions as this one.

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Keywords: Resoning, ,
Ref: Pearson16
Author(s): NCTM
Date: 1999
Title: Developing Mathematical Reasoning in Grades K-12
Journal or Publisher: NCTM
Volume, Issue, Pages: pp. 1 - 285
Reviewer: Pearson
Date of Review: April 23, 2000

Mathematical reasoning is the fundamental tool for understanding abstraction in math. Russell believes that mathematical reasoning is about "the development, justification, and use of mathematical generalizations" which stresses the interconnected nature of the topics in mathematics. Young children need to develop a solid sense of the abstract, which means trusting that number concepts exist independently of concrete representations. Flaws in mathematical reasoning should be explored for the benefit of everyone in the class.

Malloy believes that mathematics must promote and value the varied and valid ways that students make sense of mathematics. A mixture of discovery learning and lecture should be focused on creating learners who can learn well from both techniques.

Lyn English suggests that students be encouraged to develop appropriate analogues for mathematical concepts. An example of this is using blocks grouped in ones, tens, hundreds, etc. to represent the place value system in our number system. Word problems should use a variety of wordings for the same operation.

Sternberg emphasizes that students need to develop practical skills as well as analytical skills. Practical applications of math need more emphasis and we need to change our assessments to reflect this emphasis. Identifying the problem, formulating a st rategy for the problem, representing information about a problem, and time allocation skills should be an inherent part of developing mathematical reasoning as a process.

Artzt is an advocate of small group problem solving in order to bring about matematical reasoning. Students who are confronted with thier own or someone else's ideas about a math problem are forced to use mathematical reasoning to figure out if the solut ion is correct. Creating an evironment that is rich in problem solving will also create one which uses mathematical reasoning.

Greenes and Findell suggest many methods of making algebraic reasoning skills inherent in prealgebra courses. They stress the problem solving can have many solution paths, and that the important skills which are developed along the way are being able to communicate mathematically and learn how to justify solutions. In other words, just being able to get the answer is not nearly enough.

Krulik and Rudnick stress that critical and creative thinking skills should be developed concurrently with mathematical reasoning skills. Asking, "What's another way?" is one easy way to induce such thinking. Exploring other paths to solutions or asking what would be different with different initial conditions is vital for developing further exploration.

Peressini and Webb think that multiple performance assessments involving criteria such as foundational knowledge, solution processes, and communication should be used in schools today. The idea is to look at an assessment package which contains many different skills which are inherent in the Standards. Guided instructional practices need to be developed by any teacher who is looking to implement such an assessment program.

Peggy House explored 13 different ways of solving one problem that she posed to her students. Different perspectives on this problem have led to the diverse approaches, many of which are unique. Open ended problems like this can be used as ways of broadening (diverging) in the curriculum without getting spread to thin in terms of content coverage.

Epp stresses that language development, particularly the development of a mathematical language that uses logic, is one part of mathematical reasoning that often is overlooked. Teachers need to be aware that their students may not be having trouble with the math, but with how to read what the math is saying. Everyday language and mathematical language are different. Basically, Epp wants statements with quantifiers to be learned so that students know exactly what a mathematical statement means.

Robert Gerver believes that developing the skills to write a math journal will have a beneficial effect on students reasoning skills because it will force them to try to be clear in their reasoning. Additionally, students will learn how to start with a problem and explore beyond its boundaries. The justifications for why things are the way they are do not have to be written in ultra formal language; however, they should be directed toward cogent answers.

Tate and Johnson argue for changing the policies which govern mathematics education, mainly the policy of tracking and other such things that put some students at a disadvantage. Teachers need to have a paradigm shift in the way that they think about their students, too. Knowledge of inequalities should not be dismissed when it causes a teacher cognitive dissonance. Teachers should work to find ways for all of their students to succeed.

Boldt and Levine's article on Mathematics Inquiry would be a good source of ideas for teachers who are interested in doing projects in which the responsibility is on the students to do the math. Stressing cooperative work on projects, students are able to take on projects that seem to be too difficult for them to solve, but they find that they can solve them.

There is an excellent article by George W. Bright which is aimed at helping elemenatary and middle-grades preservice teachers understand and develop mathematical reasoning. The crux of the article is that these prospective teachers need to learn many different perspectives on teaching math in order to be able to effectively examine student's thinking---especially on problems which are not familiar to the preservice teacher.

The last article, by Lynn Steen, asks us to evaluate how necessary the goal of developing mathematical reasoning is. After all, he points out, graduate students in mathematics are probably among the least well-prepared people when it comes to trying to solve real-world problems involving math. This should enable us to realize that mathematical reasoning has a lot to do with the set-up and conditions of a problem, and not as much to do with the mathematics itself. Questions about how necessary learning how to write proofs also comes up when talking about mathematical reasoning. I think that, at the very least, students should be exposed to informal proofs in which they are forced to at least go through the steps of an argument. Proof writing is not so much about being able to write perfect proofs at the secondary school level, it should be used as a formal means of assessing mathematical reasoning. We shouldn't be concerned if the first time a student writes a proof that they forget to make epsilon a small positive number, we should be concerned that they start to follow the reasons why we want epsilon to be small. The twenty questions that Lynn Steen poses are hard ones to answer, all of them except for the last question which asks, "Is it too late?" No. It is never too late. Learning is lifelong and shouldn't be discouraged at any time. Formal reasoning skills have to start somewhere, and just because they didn't start with informal reasoning in Kindergarten, that doesn't mean that they can't all be learned between 8^th grade and graduation. I think that mathematical reasoning is of paramount importance and should be (as it is) a common thread in all of the Standards.

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Keywords: Standards, ,
Ref: Pearson17
Author(s): NCTM
Date: 2000
Title: Principles and Standards for School Mathematics
Journal or Publisher: NCTM
Volume, Issue, Pages: pp. 1 - 22
Reviewer: Pearson
Date of Review: April 23, 2000

I am pleased to see that the first principle listed for school mathematics is equity. I am glad to see that they are trying to raise the bar for all students. It only makes sense to start off the document on sound principles, that "all children can learn mathematics when they have access to high-quality mathematics instruction." The second principle, which is the curriculum principle, is focused on creating a curriculum which is a cohesive unit across many classes and grades that has topics which are built on cumulative knowledge. The curriculum should be focused on mathematics that is important for students to learn. The third principle is the teaching principle, and it requires that teachers have a solid understanding of what is important for their students to know and supporting them in the quest to learn that information. Teachers need to continually improve and learn in order to get an accurate sense of what is important to be taught. The learning principle is the next one, and it stresses the importance of creating and active learning environment which active learners utilize to the fullest extent possible. Passive learning such as lecture should be interspersed with active learning to benefit students with different learning styles. The assessment principle is focused on changing from simple tests to multiple assessments which allow students a variety of ways to demonstrate their abilities. Procedural skills as well as understanding should be stressed in the assessment process. The technology principle is there so that students can have the opportunity to enhance their mathematical thinking through using technology in an engaging way (other than playing Z-pong!). Technology should be used to advance understanding of concepts, not as a replacement for teaching concepts.

Among the standards are number and operations, algebra, geometry, measurement, data analysis and probability, problem solving, reasoning and proof, communication, connections, and representation. Computational fluency, estimation skills, and sense of number are the basis for a solid mathematical background. Algebraic topics, whether implicitly using reasoning or explicitly using variables, are also part of the broad base of knowledge needed for mathematics. Geometrical sense should be developed through manipulations of concrete objects and through computer enhanced lessons. Measurement is a practical, pervasive skill in everyday life that needs to be encoroporated into all levels of mathematics. Being able to reason statistically will be a goal for students so that they can be mathematically literate citizens of a world filled with statistics. Problem solving as well as reasoning and proof should be the focus of all math classes in order to prepare students for more authentic uses of their mathematical knowledge. Connections and communication should be brought about by a structured curriculum and a structured learning environment. Clarity of thought will be paramount for these two skills. Finally, using representations to model real-world situations and interpret phenomena.

The standards recommend at least an hour of math every day for elementary students, much of this time being spent on concrete examples of more generalizable situations. Calculators, interestingly enough, are encouraged for exploration of numbers at this age. As students progress, word problems should become more complex and require more steps in order to be solved. The middle grades should be a time where students gain a sense that they are competent at mathematics. Middle grades teachers should be prepared with a thorough background in mathematics so that they can get the most out of their students. High school teachers should have an ambitious focus on applied mathematics as well as pure mathematics. High school students should take at least three years of foundational courses in addition to material that extends the math taught in these grades. The standards, in order to be effective, will need the support of local school districts, parents, students, teachers, administrators, and the public. The goal of the standards is to act as a catalyst for change.

I am curious as to whether or not the NCTM plans on revising the Professional Standards. I really am looking forward to seeing the new Standards to see if their goals were accomplished. Are the topics truly connected and do they include methods that incorporate techniques for diverse learners? How much will the new Standards impact how mathematics really is taught in the classroom---especially in places where a lack of money may inhibit schools from trying out new curricula. Who will be there to make sure that the Standards go into effect across the country? Will the MAA and the AMS be behind the new standards? Will the Standards be implemented in states that have standardized graduation tests? These are some of the questions I would like to have the answers to, but I honestly do not know where to look for them or who could answer them.

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Keywords: Inquiry, ,
Ref: Pearson18
Author(s): Mathematical Association of America
Date: 2000
Title: Math Horizons
Journal or Publisher: Mathematical Association of America
Volume, Issue, Pages: April, 2000, pp. 1 - 36
Reviewer: Pearson
Date of Review: April 24, 2000

Ingrid Daubchies of Princeton University is a very unique mathematician. Growing up in Belgium, she was fortunate to go to an all girls school and eventually to the University of Brussels where she would study physics. She always had a diverse background in many areas, and it was her ability to draw on knowledge from several different sciences that enabled her to pull together concepts from different fields to form what is known as wavelet theory. Wavelet theory is an extension of Fourier analysis, and it may be the secret to such new technology as digital TV, and real time wave analysis.

There is an interesting article on a nontransitive dice game involving three sets of dice, any two of which can beat the other one (as long as the correct selection is made). These puzzles are really brain teasers from the great Martin Gardner.

The excerpt from Philibert Schogt's "The Wild Numbers" has to be some of the most fascinating math fiction that I have ever read. I think it is hilarious that this mathematician, who is very intense about trying to prove a theorem, gets out of bed in the middle of the night to work on his proof and his wife becomes so jealous that she leaves him. It really is interesting fiction, if you like an easy read.

GIMPS (or the Great Internet Prime Search for those of you not familiar with it) is searching for new world records. Anyone who finds the first ten-million-digit prime will win a large sum of money. The article explains what Mersenne primes are and a little bit about the history of primes. Mersenne primes and Fermat primes would make an interesting topic for introducing kids to some number theory.

There are some other novelty articles in this issue that are interesting. I think that the best thing about this magazine is its intended audience. I think that students in high school should be exposed to some of the real findings in math through magazines like this one. I think that if students had a better idea of what mathematicians really do, they might be more interested in math and in becoming a mathematician. Being exposed to things like set theory, different notation, and more powerful ideas at work would certainly be inspiring to some students.

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Keywords: Activities, ,
Ref: Pearson19
Author(s): National Council of Teachers of Mathematics
Date: 2000
Title: Teaching Children Mathematics
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: Volume 6, Number 7, March 2000, pp. 428 - 477
Reviewer: Pearson
Date of Review: April 24, 2000

What can we expect from research? Research can tell us what we ought to try doing, but it cannot tell us what our values are. Do we value students who understand and invent methods for solving problems, or do we value the standard math curriculum which does little more than teach students how to sit still at times? We should work on getting our learning goals in line with the math skills being taught, the examples being worked on, and the problems worked on in class. Like the Standards suggest, we need to get our goals in line with what we are doing in the classroom. Research cannot give specific things that will work miracles in classrooms, it has its limitations. However, research can be invaluable in determining which teacher practices will help students reach goals.

As a symbol of peace on the day of the Nobel Peace Prize, West Chester, Pennsylvania schools made a large dove out of milk jugs placed on the ground. Students were asked how many jugs they estimated would be needed to create the design. Students underestimated the actual number of jugs by quite a bit.

There is a section about various problem solving activities that have been done in elementary classrooms. The problems dealt with identifying sample spaces and combinations, and were, overall, interesting for the students to do because the math problems came about as a spin off of a story from a story book. This seems like a perfectly natural way to introduce math into any classroom.

An article about Standards 2000 describes how the new Standards ask for more qualitative math and more reasoning, posing more open ended questions to students. The new standards will focus on the process of discovering math rather than just the content.

There is an interesting lesson plan idea which suggests presenting the data to the kids first and having them reason which is the most likely question that goes along with it. This inferential kind of mathematical reasoning is important for students because they will be asked to use it in the real world. Most of the ideas in this magazine are creative twists on familiar problems. It is encouraging to see that teachers are asking questions which are open ended and that they're not looking for just one right answer.

The most interesting article in this magazine is one by Diana Steele about "Enthusiastic Voices from Young Mathematicians." She noticed that in one classroom she was observing, "students made conjectures and presented their points of view to convince others of their validity" (464). They were already acting like real mathematicians. Best of all, these students are all participating in the discovery of these new ideas. The teacher is able to redirect the student ideas so that they become interesting, connected, real-world examples and problems. The teacher gives students the opportunity to rethink the answers to their original problems (when does this ever happen in a traditional classroom setting!!!). Reflecting on and extending student knowledge is not the only thing that the teacher does, the teacher is also able to give direction and meaning to conjectures and justifications for proofs. The amazing thing about this classroom is that students were internalizing the attributes of a mathematician at a very young age, and hopefully these attributes will carry over into the future.

I was a little disappointed in the small amount of theoretical discussion about how to teach math to children, but I can see why the teachers who read this magazine might prefer to have examples which they can work with. Some things in the magazine seemed very gratuitous, such as the peace day article which seemed to be about three pages too long for the amount of actual content that was discussed, but mostly there were very good examples of some very good things going on in math. The article about young mathematicians was exciting to read, but being a secondary education teacher, some of the suggestions seemed like they would be very hard to do simply because of the expectations placed on teachers by parents and the school administration. If there were some way to make high school math an experience to be guided through without the restrictions of standardized tests and more specific preparations for tests, this would be the way to teach all general education math courses. General education students are so bored that if we gave them something interesting to do, they might not even realize that they were doing work. I was surprised that there wasn't more content in this magazine for the middle grades.

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Keywords: Activities, ,
Ref: Pearson20
Author(s): National Council of Teachers of Mathematics
Date: 2000
Title: Mathematics Teacher
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: Volume 93, Number 2, February 2000, pp. 81 - 166
Reviewer: Pearson
Date of Review: April 24, 2000

An interesting activity relating group theory to the Chinese Zodiak is among the first of the articles in this magazine. It is a bit complicated because it uses the actual Chinese characters for some things.

There is an interesting activity which explores why Paul Bunyan could not have truly lived. It is an exercise in discovering proportionality, and it prompts the students to think of whether or not Paul Bunyan's bones could have supported the additional volume and weight of his body. I like that this magazine has sample materials right in the magazine, ready to be copied and used as a lesson.

There is a lesson focusing on making connections between slope and understanding what slope means as a rate of change. Lessons like this should be taught in middle school, in my opinion, since they can be beneficial to students who take physics in ninth grade. The lesson forces the students to be responsible for discovering the meaning of what slope is by how the questions are posed. Thinking critically and being able to write out explanations are among the ways that students can demonstrate that they understand the concept of slope. (Note: the lesson is only dealing with linear equations.)

There is a cool media clips section where real world problems found in newspapers are explored. This is a cool idea, and I think that it is mathematics version of "current events" trivia games found in newspapers. What if students embarked on math problems found in the newspaper during homeroom? Would they become more inquisitive mathematicians?

There is an interesting math calendar with a problem for every day of the school week. Some of them are interesting, but many of them would be ignored by students in upper level math. It would be nice if there was more variety or more word problems instead of number trick problems.

Anyone interested in getting journals started in their math classes should check out the article on page 132. A valuable set of criteria and suggestions are listed for possible journals that kids produce. Allow students to make multiple tries at solving one thing that they are stuck on in their journal. Often the struggle can be very rewarding.

I really liked that the magazine has a calendar full of math problems for students to do, but how many of these calendars actually get posted? I don't think that I've ever seen one in any classroom. It would be a really cool idea to have a calendar---like those ones you can buy at Christmas---with the dates hidden and a small piece of chocolate for the first person who solves the hidden problem of the day. I would've gone to my math classroom before school started if that had been the case (not so much for the chocolate, but to hang out with other math people). If it takes a few bags of Hershey's Kisses to get students motivated to do math in the morning, why not? As with the magazine "Teaching Children Mathematics," I wish that this magazine had more articles about research and theory, but this is just a personal preference.

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Keywords: Activities, ,
Ref: Pearson21
Author(s): National Council of Teachers of Mathematics
Date: 2000
Title: Mathematics Teacher
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: Volume 93, Number 3, March 2000, pp. 167 - 264
Reviewer: Pearson
Date of Review: April 24, 2000

The magazine starts off with an article about rollercoasters. I am not a big fan of rollercoasters. I think that the mathematics and physics used on projects which deal with rollercoasters should not be used in curriculum. I am against this because I really dislike being obligated to go to an amusement park, and I wouldn't want to force my students to go, either. There is real math to be found there, but I won't have my students go to an amusement park, even if the makers of some new fangled data collection device offer to pay the way for my class. I just don't think that the amount of real learning that results from such trips is worth the time spent (and the money spent) on the trip. This is all my own opinion, though.

There is a very cool explanation of Einstein's theory of relativity on page 194. The only thing I don't like about the article is that it doesn't talk about hyperbolic geometry, one of the most important parts of his theory. The article presents interesting ways of explaining how Einstein's theory of relativity works and should definitely be passed on to the physics teachers.

The next activity is an interesting one which explores some of the probability of winning at Bingo. This would be a fun, hands on way to explain some concepts of probability such as reduced sample spaces.

There is an article in here about finding roots of a problem, and all I have to say about it is that it is a shame that the methods for finding roots are no longer taught in every classroom (we truly have become dependent on calculators).

There is an article about having students develop their own solution methods for algebra problems. I would love to see this sort of approach implemented in the classroom. In the general math classroom that I observed, students were totally dependent upon the teacher for the methods to solve problems. Even if students came up with incorrect methods for solving problems, wouldn't that be a good thing because we as teachers would be able to see what mistakes are being made, what assumptions are being made, etc.? As it is right now, though, changing a class of traditionally taught students to do this would lead to complete chaos. As it is right now, students go crazy within one minute. But these are worthy skills to be developed. The article is on page 218 and well worth taking the time to read.

An article on determining an understanding of R² coefficients takes a very standard approach at explaining what R² is and how it is calculated, but if I were to teach this topic I would use a constructivist method devised to get students to find some way to relate data points to the mean, hoping that they would come up with something analagous to R². The article also takes a very calculator-oriented approach which I think can be very detrimental in statistics since the calculator can do all of the work for you even if you don't understand what is going on.

Again, I like that this magazine is such a rich source of teaching resources, but I wish that it dealt with more of the issues involved with teaching mathematics. It would be nice to see what other teachers are thinking about the new mathematics Standards in greater detail. I wish that mathematics teaching wasn't so focused on technology. I like using technology, but I think that students can get into the habit of using it as a crutch in place of doing real mathematical thinking. Case in point is the article about R² that I reviewed. We need to develop the basics before using the calculator, in my opinion.

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Keywords: Standards, ,
Ref: Pearson22
Author(s): Children, Families, and Learning
Date: 2000
Title: Minnesota's Graduation Standards
Journal or Publisher: http://cfl.state.mn.us/GRAD/gradhom.htm
Volume, Issue, Pages:
Reviewer: Pearson
Date of Review: April 30, 2000

This is part of the Children, Families, and Learning department of the state of Minnesota. The current debate is about Minnesota's High Standards. Minnesota High Standards are more commonly known as the Profile of learning. On this site are very quick links to the results of all of the tests. The menu system that is set up allows you to look at individual districts or all of the districts and choose among a variety of age levels. The graduation standards now in place require results on tests, and not just the completion of the required classes in order to graduate. The Basic standards are the minimum standards set up. Schools are responsible for providing help to students who have trouble passing the basic standards tests. The High Standards define what students should know, understand, and be able to do to demonstrate a high level of achievement. There are 24 High Standards which students must complete. In order to graduate, students must complete both the Basic Standards and the High Standards. For mathematics, students have several options for fulfilling their High Standards (there are more than 24 High Standards from which to choose). These High Standard requirements deal with mathematical applications such as solving problems by applying mathematics and are are follows:

One Required: Space, Shape & Measurement
Choose 1: Discrete Mathematics or Chance & Data Analysis
Choose 1: Algebraic Patterns or Technical Applications

The tests include both multiple choice and short answer questions. The mathematics test has story problems which can be solved by applying shape, space, and measurement, number sense, chance, and data handling.

There is an interesting new program affiliated with the Minnesota Graduation Standards, and it is the scholars of distinction program. The students who earn this distinction have demonstrated their expert ability in a specific subject or area.

For students with learning disabilites there are many interesting suggestions on the website for helping them pass the requirements. I think that the standards ought to make exceptions for students who are truly learning disabled, or at least be willing to make more accomodations so that they have a reasonable chance at passing. What if graduating from high school is a really big deal for these students and the new standards take the fulfillment of that dream away from them. I know what the standards are trying to do; however, setting the bar at the same height for everyone doesn't always work in my opinion. If I had a student I know could not concentrate well because of something beyond their control, I would make accomodations so that they could learn.

If you are interested in math specific High Standards, please visit:

http://cfl.state.mn.us/GRAD/highstds/HS-M.htm

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Keywords: Standards, Curriculum,
Ref: Pearson23
Author(s): SciMath Minnesota
Date: 2000
Title: SciMath Minnesota: Math in Minnesota
Journal or Publisher: http://www.scimathmn.org/math_in_mn/
Volume, Issue, Pages:
Reviewer: Pearson
Date of Review: April 30, 2000

If you want to know about what is going on with mathematics education in Minnesota, visit this site. It is well organized, there are links to almost everything about math in Minnesota, and the site seems to be very comprehensive. Definitely bookmark this site. The site is aimed at educating parents about what is going on with mathematics in Minnesota, so the descriptions of things on this site are clear and concise. There are links to all of the different curricula being tried as integrated mathematics, there are links to the Minnesota Graduation Standards, and there are links to various sites of interest: some articles from periodicals, reviews, and student achievement results on a wide variety of tests. The nice thing about this site is that it isn't overwhelmingly large and it links you to the most current things in mathematics education. I would say that it is impeccably laid out since it is easy to find things, summaries of off-site content are brief and meaningful, and it is easy to navigate through even though there are so many links off-site. This website is very true to the purpose of SciMath which are to create alignment among the many different groups interested in math, work with all of the people involved in shaping learning opportunities, and most importantly, focusing on policy, public awareness, and professional development. While it is not going to give you lesson plan ideas or give you links to everything that you might need in the future, this is a great site to visit if you are interested in topics dealing with curriculum, student achievement, educational research, and general pedagogical concerns.

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Keywords: History, Geometry,
Ref: Pearson24
Author(s): SciMath Minnesota
Date: 2000
Title: Cornell Theory Center Math and Science Gateway
Journal or Publisher: http://www.tc.cornell.edu/Edu/MathSciGateway/
Volume, Issue, Pages:
Reviewer: Pearson
Date of Review: April 30, 2000

On this website are many links to both science and math resources on the special Cornell server which is dedicated just to this project. On the mathematics page there are six subject links: general topics, geometry, fractals, history of mathematics, tables, constants and definitions, and mathematics software. Under general topics are everything from online flash cards to help in calculus. One very notable link that is an enormous resource for lesson plans is

http://archives.math.utk.edu/k12.html

In the geometry section there are only two links; one link is to the University of Minnesota's Geometry Center and the other link is to the Non-Euclidean Geometry site at Rice University. Under the fractal subject heading I would suggest visiting the link put together by NASA which has a link to http://www.shodor.org/master/fractal/ where there are some very cool java applets which can render fractals. The Mandelbrot explorer is also a cool site.

The history of mathematics has three links, one of which is biographies of women in mathematics. This could be a great resource for you in order to get women more interested in mathematics. The tables, constants, and definitions section is basically a storehouse of mathematical information. The section on mathematics software has freeware as well as commercial links. One valuable link which I found from this section was the Guide to Available Mathematical Software which was a keyword-searchable index and repository of all different kinds of mathematical software. The address is http://gams.nist.gov/.

I really appreciate people who put their time into making indecies such as this for the math population. I think that if we could get our students to start using these resources, we would have no trouble fostering their interest in mathematics. Kids would be easy to hook on things like fractals, and then once their hooked, you might be able to get them interested in a variety of other things which are math related and found on sites like this.

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Keywords: Research, Curriculum, Teaching Strategies
Ref: Pearson45
Author(s): Florida State University
Date: 2000
Title: Mathematics WWW Virtual Library: Florida State University
Journal or Publisher: http://euclid.math.fsu.edu/Science/math.html
Volume, Issue, Pages:
Reviewer: Pearson
Date of Review: April 30, 2000

Particularly useful at this site is the plethora of links found on the electronic journals page. The journals are from all over the world and contain a mixture of journals for use by people of all ability levels (though there are many more for undergraduates and graduates than anyone else). News groups, gopher servers, high school servers, and education are all linked from this page, as are links to software and finding people in the math community. The link to the page of high school servers would be a great resource for ideas and contacts around the country. There are at least 30 schools listed on this index. The education page has approximately 100 links to everything under the sun. The education page is an invaluable resource and its address is:

http://euclid.math.fsu.edu/Science/Education.html

The general resources page found at http://euclid.math.fsu.edu/Science/General.html is also a good resource for lesson plans, curricula, and links to such things as the Association for Women in Mathematics.

There are too many good things about this site to list only a few. It is gargantuan in size, well laid out, covers a lot of information that is out there, and has a search tool so that if you're looking for something specific, you can find it. If I were looking for something to do with mathematics, this would be the first place I would go. This collection of resources is better than any list of resources that a web search engine would produce. Make this page your first stop when looking for mathematics on the web.

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Keywords: Activities, ,
Ref: Pearson25
Author(s): Texas Instruments
Date: 2000
Title: The Home Page for Users of TI Calculators and Educational Solutions
Journal or Publisher: http://www.ti.com/calc/docs/calchome.html
Volume, Issue, Pages:
Reviewer: Pearson
Date of Review: April 30, 2000

There are two things which make this site great for teachers: the free downloads of software and other products, and the T³ lessons and summer institutes. It would be nice if more of the T³ lessons were formatted as lesson plans, but they are at least good outlines of what calculators can do for you in the classroom. The discussion groups cover a wide variety of topics related to the TI calculators and probably have already addressed any questions you might have. You can search the discussion lists using their search engine to speed things up. Free newsletters can be downloaded and viewed in PDF format. A listing of some very important web links for teachers can be found at http://www.ti.com/calc/start/links/tea-math-high.htm. I was surprised to find a section of their site devoted to calculator research. The resources on the calculator research page are only a works cited, which was disappointing, because I wanted to see some of the reseach online. I think it is nice that TI makes websites for parents to view, but I think that they made this portion of their site seem to commercial. If I were TI I would've marketed how the calculators are used more to the parents than TI did.

I like this site and I think that it provides a lot of support for TI users. Personally, I would like to have seen a much more prominent feature on linking calculators up to computers on this site. I know that it can be done, but it isn't one of the things that TI is pushing. I think that the integrating of technology as well as the support that TI provides is great.

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Keywords: Activities, ,
Ref: Pearson26
Author(s): Math.com
Date: 2000
Title: Math.com
Journal or Publisher: http://www.math.com
Volume, Issue, Pages:
Reviewer: Pearson
Date of Review: April 30, 2000

I was wondering why math should be a commercial site on the internet. >From the homepage I can see that there is some advertising, but nothing else appears commercial about this site, yet. One cool resource I found on this site was under the teacher section. It is an algebra equation worksheet generator which describes a bunch of different types of equations and all you have to do is select how many of that type of equation you want and click generate equations. The thing that does this is a neat little perl script. Once you've generated the worksheet, you can print it out and make copies. The worksheet generator can be found at http://www.math.com/students/worksheet/algebra_sp.htm.

In the teacher's lesson plan section of this site (http://www.math.com/teachers/centers/lesson%20plans.html) there are many off-site links to good resources for lesson plans. Under the math standards portion of this website are four links to the NCTM standards, the Putnam Valley School's guide to developing educational standards which has many links to different programs all over the country, implementing math standards which includes interviews with people from all over the US, and explorAsource which is a search engine tool that helps teachers find resources and education standards that address student's learning needs.

This site has a free math library and free homework help for students (I wonder what they'd do with some of my homework problems?). This site has lesson plans, classroom resources, career information, information about the standards, free stuff, a problem of the week, reference sources, practice worksheets, teacher centers, and links to tutoring services. One particularly useful link that could get students interested in mathematics is http://www.math.com/teachers/recreational.html. I suggest keeping students away from Conway's game of life, however, because it can become very addicting to the point of obsession.

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Keywords: Teaching Strategies, ,
Ref: Pearson27
Author(s): Swarthmore
Date: 2000
Title: Math Forum - Mathematics Education
Journal or Publisher: http://forum.swarthmore.edu/mathed/
Volume, Issue, Pages:
Reviewer: Pearson
Date of Review: April 30, 2000

This site has many links to issues relating to pedagogy, organizations and journals for math education, research in math education, teacher education and professional development, technology in math education, conferences with math education components, and general interest sites. I looked at the teacher education and professional development portion of this site. There are some links to pages which can put you in touch with organizations which can help you get a job. For those of us who are undecided about what exactly it is that we want to do when we graduate, the MAA has an invaluable link for you: http://www.maa.org/students/career.html.

There are links to state teaching requirements for all 50 states, articles on the job market for mathematicians, articles about different carrer profiles of people who recently graduated with some degree in mathematics, links to sites such as GreatTeacher.net which has job classifieds, email, discussion areas, and many other services. If you are not sure where you want to teach there are links for interim teaching jobs for organizations such as Teach America and the Peace Corps. There is also a link for the NCTM job classifieds here (I would have never known that NCTM had classifieds if I hadn't seen a link to them).

This is a great link for those of us who are undecided about what exactly we want to do when we graduate from St. Olaf. There are many other resources available on this site, too. To take the time to review them all take a lot of time. I would be willing to bet that the rest of the parts of this site are as well organized and have as many valuable resources as the part of this site that I specifically looked at.

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Keywords: Geometry, ,
Ref: Pearson28
Author(s): University of Minnesota
Date: 2000
Title: Welcome to the Geometry Center
Journal or Publisher: http://www.geom.umn.edu/
Volume, Issue, Pages:
Reviewer: Pearson
Date of Review: April 30, 2000

I have visited this site before, so I was shocked to see that the Geometry Center is closed and is no longer functioning. The site is still there as a repository, however. The Geometry center was aimed at undergraduate and graduate students, however, some of the materials were very accessible for advanced high school students. The Geometry Center had projects, multimedia documents, reference archives, downloadable software such as Geometer's Sketchpad, video productions, course materials, and resources for new teachers. A particularly useful page on this site is:

http://www.geom.umn.edu/DETEP/curric.html

because it has links to lessons, units, standards, and curricula. Other educational materials can also be found at http://www.geom.umn.edu/docs/education/ where there are interesting html guides to entire units on geometry. On the videos page there are links to videos about knot theory, turning a sphere inside out, and a video about the possible shapes of our universe. The reference archive contains many cool pictures of fractals that have been generated as well as many geometry formulas and facts.

It is too bad that such an invaluable resource is no longer in action. There are many interesting java applets on this site as well as many other valuable resources, so I hope that they archive it for several years to come.

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Keywords: Proof, ,
Ref: Pearson29
Author(s): Selby, A.
Date: 2000
Title: Appetizers and Lessons for Mathematics and Reason, Entrance Page
Journal or Publisher: http://www.cam.org/~aselby/lesson.html
Volume, Issue, Pages:
Reviewer: Pearson
Date of Review: April 30, 2000

When I first visited this site I headed right for the five key lessons found at http://www.cam.org/~aselby/five_key_lessons.htm. There were some interesting approaches and examples that could be used in class to develop informal logic in class, the concept of a variable, the concept of slope and change, and complex numbers. All of these links are very fun and mostly based on constructivist learning. The lessons are really tidbits of lessons from some books. The pedagogical focus of these lessons stresses constructing learning, developing multiple approaches to problems, developing communication skills, and cooperative learning. There are links to 23 other interesting sites that are discussion sites for mathematics on the web.

This is obviously a commercial site, the more I look at it. It has some very interesting explanations on a very limited number of topics. I would recommend looking at this page only if you're looking for something dealing with the topics that I have already mentioned.

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Keywords: Curriculum, Technology, Planning
Ref: Pearson30
Author(s): Various
Date: 2000
Title: Mathematics Archives WWW Server (UTK)
Journal or Publisher: http://archives.math.utk.edu/
Volume, Issue, Pages:
Reviewer: Pearson
Date of Review: April 30, 2000

This repository can put you quickly in touch with some of the best links out there for practically any mathematics subject. For example, I went right to the probability section, clicked on the birtday problem, and found a page which would not only let me do sample trials where I choose the number of people in the room, but had questions and explanations in addition to this online simulation. The best thing about this archive is that it searches other archives for you, so you get the maximum output for the minimum energy. There are links to downloadable software, teaching materials, search tools, as well as many other miscellaneous links. The specific teaching materials available include calculus resources online, contests and competitions, graphing calculator resources, teaching materials, visual calculus, and miscellaneous stuff. This is a great meta-index of many valuable resources on the web and I would highly recommend it as one of the first places to go when searching anything to do with mathematics on the web.

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Keywords: Puzzles, Activities,
Ref: Pearson31
Author(s): Interactive Mathematics Miscellany and Puzzles
Date: 2000
Title: Interactive Mathematics Miscellany and Puzzles
Journal or Publisher:
Volume, Issue, Pages: http://www.cut-the-knot.com/
Reviewer: Pearson
Date of Review: May 15, 2000

This site is dedicated to finding interesting problems from the past and solving them. There are many interesting examples here, plenty of brain teasers, and lots of problems to pull your hair out trying to solve. The puzzles are mostly ones like Polya would sove or did solve. These problems would be excellent ones to put on the overhead as 5 minute brain teasers or other similar things. The problems are all referenced in case you are interested in looking up the sources of them. The best thing about this site is that you can find all sorts of interesting problems to ask students and they're all free. You probably could not buy all of the puzzle books which these came from with the school's money, but you could certainly get on the internet and download them. It would be nice if they were ready to print (so that an exercise doesn't get cut off in the middle of a printed page), but beggars can't be choosers. The topics covered in these problems are games and puzzles, arithmetic and algebra, geometry, probability, proofs, math as language, and many other miscellaneous topics such as cut the knot, eye opener, analog gadgets, inventor's paradox, and things impossible. This would be a great site to visit if your school doesn't have very many resources, if you wanted to get kids interested in mathematics puzzles, or if you have a propinquity for solving brain teasers.

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Keywords: Calculus, Activities,
Ref: Pearson32
Author(s): Douglas N. Arnold
Date: 2000
Title: Calculus Graphics
Journal or Publisher:
Volume, Issue, Pages: http://www.math.psu.edu/dna/graphics.html
Reviewer: Pearson
Date of Review: May 15, 2000

This is a cool page that anyone teaching calculus should make use of in teaching. It would require that you have a projector hooked up to your computer and that you have an internet connection in your classroom, though. There are 11 different graphics images which are animated in both GIF format and Java format. The Java version enables you to start and stop the animations as you wish which would make for a great presentation. The topics covered are differentials and differences, computing the volume of water in a tipped glass, Archimedes calculation of pi, how the ball bounces, secants and tangents, zooming in on a tangent line, a trigonometric limit (i.e. lim_{x \to 0} \frac{\sin x}{x} = 1), the limit, a nowhere differentiable function, Euler's number, and the intersection of two cylinders. This is a must see and must use site for anyone teaching calculus. For some of the examples there are postscript files of worksheets and Mathematica files to be used in the exploration of what is going on.

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Keywords: Activities, Games,
Ref: Pearson33
Author(s): Fantastic Fractals
Date: 2000
Title: Fantastic Fractals
Journal or Publisher:
Volume, Issue, Pages: http://library.thinkquest.org/12740/netscape/index.html
Reviewer: Pearson
Date of Review: May 15, 2000

This site has fractal tutorials, fractal galleries, fractal landscapes, fractal music, downloads, and many other useful pages. Coming soon will be a section for teachers only which will have worksheets, hand-outs, and much more. This site is nice because it offers a variety of links for students of all ages and abilities. The gallery contains fractals generated by iterated function systems, L-system fractals, Julia Sets, Mandelbrot sets, and Henon maps. There is very cool free software called Fantastic Fractals available at this site. If you need a quick activity to do with your students in the computer lab, visit this site and have them take the fractal challenge or the fractal workshop online activities. The mathematics behind all of the fractals here is not totally explained (the site is accessible to almost anyone) but it gives you links to many other fractal sites on the web as well as a bibliography of printed materials. I like the multiple ability level approach that this site has taken so that everyone from the elementary level to the college level can learn about fractals. I wouldn't say that this is a comprehesive site that deals with fractals based on just the mathematics, but I think that they've done a good job of making a site that can be used by a variety of people for a variety of purposes. I would've liked to been able to see what kinds of teachers' resources they have.

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Keywords: Discrete, Games,
Ref: Pearson34
Author(s): MegaMath Text Menu
Date: 2000
Title: MegaMath Text Menu
Journal or Publisher:
Volume, Issue, Pages: http://www.c3.lanl.gov/mega-math/menu.html
Reviewer: Pearson
Date of Review: May 15, 2000

The most colorful math of all is a brief introduction to the four color theorem. The reading level is definitely something that advanced middle school kids could do, and possibly even advanced elementary age students. The site encourages students to experiment with many different possibilities and learn how to do proof by induction. The maps on this site could all be printed off and used in class very easily. Since there are four maps to choose from, students could work in groups, come to different conclusions, and discuss what they found.

There is a lesson in graph theory here, too. The vocabulary are all laid out, and this would be an appropriate lesson for high school age students. There is even a little section explaining how this all fits in with the Standards.

Also available on this site are ideas for lessons on knot theory, algorithms, the hotel infinity, finite state machines (which recognize patterns), and a play in which the protagonist has to figure out which students are lying and which students are telling the truth (students either lie or tell the truth, not both).

There are some interesting ideas here for lesson plans to do with students. They seem to be aligned with the NCTM standards, and if you don't think that they are, you can just click on the part of the lesson that says NCTM and there will be a justification of why this particular lesson is aligned. This is a small resource for a few interesting problems which can be used if you choose. The activities certainly seem to stimulate mathematical reasoning, and the play is downright confounding (a very good exercise in logic).

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Keywords: Geometry, Connections,
Ref: Pearson35
Author(s): Welcome to COOLMATH.COM by Karen!!
Date: 2000
Title: Welcome to COOLMATH.COM by Karen!!
Journal or Publisher:
Volume, Issue, Pages: http://www.coolmath.com/
Reviewer: Pearson
Date of Review: May 15, 2000

I wanted to search the web for something that kids would be interested in exploring. This site has a lot of colorful and fun graphics in addition to introducing some of the basic concepts of math. In the section that explains what a function is, the site wisely used colors to code the different information such as things in the domain and range. There are many picutres to illustrate the various ideas behind the concepts, and the minilessons are structured with a lot of dialogue to keep them fun. At the end of the lesson it encourages you to look up the definition of the thing that you were looking at and see if it makes sense now. There are some online games at this site, but I'm afraid that they aren't very educational. Some of the games are instructional (such as magic squares and the tower of Hanoi), but the majority of them are not.

The topics covered at coolmath.com are functions, damping functions, tesselations, interior angles of regular polygons, polyhedra gallery, limits, decibels, pythagorean identities, fractals, puzzles and numbers, how to succeed in math, and careers in math. There are many other links on this site. Overall, this site is interesting and a good source for ideas, but it is no substitute for math class. However, it's presentation of the material is very interesting and fun. I would definitely recommend this site to students and parents looking for a place to review some of the basic ideas in a variety of topics of mathematics.

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Keywords: Calculus, Geometry,
Ref: Pearson36
Author(s): Designing a Baseball Cover
Date: 2000
Title: Designing a Baseball Cover
Journal or Publisher:
Volume, Issue, Pages: http://www.mathsoft.com/asolve/baseball/baseball.html
Reviewer: Pearson
Date of Review: May 15, 2000

I decided to search for something fun and very mathematical and I found it. This site has the history of how the cover of a baseball was developed. Through the process of trial and error C. H. Jackson invented the current baseball cover and patented it in the 1860's. Jackson wanted two identical pieces of leather which, when sewn together, covered the baseball.

After this brief introduction to the problem, the site immediately goes into the mathematics of the problem. There are two methods to choose between when doing this design, and the one which is mathematically the easiest approaches the problem as if the "flats" (the flat pieces of leather) are on the ball and we're taking them off. The process walks through how to parametrize the seam of the ball using arc length and an inverse function with calculus thrown in to make it fun. The problem is very involved, but it is fascinating because of the mathematics behind it and because of the assumptions that one has to make in order to get the problem to work. If you have Mathcad Plus 6.0, there are files which accompany the process explained on this site. A revised version of this article can be found n The College Mathematics Journal, Vol. 29, No. 1 (January 1998).

I think this is one of the coolest math sites I have found. (But I may be biased, because I like both geometry and calculus.) I think that sites like this could be used in calculus class as a fun project or activity. This site is very mathematically rich, and I'm sure that it would interest a large number of students in any calculus class. If the students in calculus did a project dealing with this, it would be good to get them to present it to other math classes so that people might get interested in taking calculus instead of stopping studying math after a certain point.

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Keywords: Activities, ,
Ref: Pearson37
Author(s): Eric Weisstein's World of Mathematics
Date: 2000
Title: Eric Weisstein's World of Mathematics
Journal or Publisher:
Volume, Issue, Pages: http://www.mathworld.wolfram.com
Reviewer: Pearson
Date of Review: May 15, 2000

This site claims to be the world's most extensive mathematics resource with 9083 entries, 156312 cross-references, 3675 figures, 70 animated graphics, and 919 live Java applets. I have used this site for some time now, and I thoroughly enjoy using it every time. This site has a many great sites for student in middle school and above, although elementary teachers could use this as a resource for ideas which they could appropriately scale down and use with their students. The subjects covered in depth at this site are algebra, applied mathematics, calculus and analysis, discrete mathematics, foundations of mathematics, geometry, history, number theory, probability and statistics, recreational mathematics, and topology. This site has definitions with many examples in addition to a lot of cool stuff. This link, along with a few others, should be listed on the beginning of the year handouts to students and parents so that they can have access to such a great resource. The entries have references, and many of these references are from scholarly mathematics journals and have links to their articles. There are just so many cool resources here. For example, there are live 3D graphics, and links to many common as well as obscure projects. The presentation of the site is very readable; all of the mathematical equations have been LaTeXed and converted to small graphics. This site has some of the best mathematical typesetting I've seen on the web. You should definitely direct your students toward this site (I can forsee some students leaving their books in their lockers and just using this site as their homework resource). The best thing about a site like this is that if you can get a kid interested in this site, he or she might browse this math site for a while just gleaning interesting information the whole time instead of browsing the internet and looking at junk. I know that I would've used this site extensively if I knew about it in high school.

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Keywords: Calculus, ,
Ref: Pearson38
Author(s): Eric Mary Margaret Shoaf-Grubbs
Date: 1996
Title: Discovering Calculus with the Graphing Calculator
Journal or Publisher: John Wiley and Sons, Inc.
Volume, Issue, Pages: pp. 1 - 204
Reviewer: Pearson
Date of Review: May 15, 2000

I always wondered if there were books like this because my teachers never used them. Whenever we did calculator exercises, either we had to figure out how to use the calculator on our own or there would be a brief period of instruction explaining what and how to do a specific task. This book warns us of calculator rounding error, especially when there are many steps involved, such as when using Newton's method to approximate roots. Other errors such as overflow errors can be only dealt with by using a more powerful calculator or doing the problem by hand. One critical thing that teachers need to make students aware of is their choice of the size of the graphing window. Finding an appropriate part of the domain and range to graph, as well as making sure that the equation is input correctly, makes a huge difference in what kind of picture the students see. Students are encouraged to avoid false asymptotes by using dot mode instead of connected mode when graphing equations which they suspect may have an asymptote. Students also need to be made aware that built in features on calculators always use methods of approximation to find answers (that is, unless the algorithm used does happen to give an exact answer) and students should adjust their thinking accordingly. Some built in functions are downright misleading at times. Such is the case with the DrawInverse function on the TI-82. Other things which may or may not be intuitive were brought up, such as how the view of a [-10,10] by [-10,10] window appears on a non-square screen and how that affects what is really seen. There are benefits to having a calculator book like this, such as learning new things which you can do with your calculator. I had no idea that it was possible to define piecewise functions on TI calculators. The downside to a book like this is that it would be expesive to purchase for the whole class, and it would limit how you taught things if you used it all the time. This book is obviously meant to be a workbook for students to use, because the examples all have blank graph screens. I did not know that it was possible for older TI calculators to generate tables of values, either. This book, fortunately, asks students to sketch what they think the graph looks like before they actually graph it, but it does not ask the students to do this all of the time. This book would be an excellent resource to use as a teacher because, although it does not have lessons, it has many applications and might be a lot less expensive than going to a TI calculator workshop. This book focuses almost exclusively on graphing, but there are a few sections devoted to investigating series and sequences. The graphing part of the book focuses on derivatives, limits, integrals, parabolas, ellipses, parametric equations, polar equations, and hyperbolas.

I think that a book like this could be very useful for the teacher, especially if they are not familiar with graphing calculators. There are many examples and many good questions in this book which should not be overlooked. I think that if the teacher had a copy of this book, the class would benefit greatly. I don't think that it is necessary for every student in the class to have this book. The teacher could easily direct these activites verbally (instead of having a copy of this book for each student or making handouts) for most of the activities. There are many tricks to be learned from this book, and many features of the calculators that are often not exploited to their fullest. Additionally, if a teacher had a book like this, he or she would be able to better design test questions so that there is not only a variety, but also a differentiation between calculator skills and student skills.

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Keywords: Issues, ,
Ref: Pearson39
Author(s): Steen, Lynn (editor); Cobb, George; Denning, Peter; Dossey, John; et. al.
Date: 1997
Title: Why Numbers Count; Quantitative Literacy for Tomorrow's America
Journal or Publisher: New York, College Entrance Examination Board
Volume, Issue, Pages: pp. 1 - 194
Reviewer: Pearson
Date of Review: May 15, 2000

This book is a collection of essays which attempt to explain why "[n]umeracy is the new literacy of our age" (Steen xv). Citing that only one in ten U.S. adults can reliably solve problems that require two or more steps (xvi), and the ever increasing amount of vital information that involves quantities, the authors of these essays claim that innumeracy is a threat to society today and in the future. The authors, with their often contrasting opinions about what is important beyond the basics of math, represent many different parts of society: industry, academia, government, and education. Quantitative literacy, as the authors call it, keeps mathematical quantities in context, whereas the term mathematical literacy does not. This means that students need to learn to see not only the applications of the math, but also the conceptual structure which underlies the application.

One inspiring quote from Malcom is, "If you change the way mathematics is taught, you'll be surprised at who can learn mathematics" (xxv). The authors encourage putting problems in context, looking at real-world applications, decision making, and interpretations as means of attaining quantitative literacy.

Porter's article argues about social and historical implications of quantitative literacy. Wadsworth emphasizes that we should first teach the basics of math, and look at subsequent topics with different degrees of priority. She recognizes that as a society we "have not found ways to help the public move from mass opinion to public judgment" (Wadsworth 22). Gina Kolata, a bioligist and mathematician, cites how it is necessary to separate anecdotal evidence from statistically significant and pertinent evidence when reading articles written by journalists. She points out that we are often swept away by statistics which have quantity but not quality. She stresses that quantitative literacy is about knowing how to reason and think.

Shirley Malcom, a proponent of taking an individualized education for an entire classroom, believes we should maintain the same standards and present the same mathematics concepts but use very different examples, such as encoporating examples which would enhance the education of students planning to go to technical or vocational school. Iddo Gal, a professor, believes that education should integrate numeracy and literacy skills in order to meet the needs of everyday life in the real world. He argues that students need computational skills, decision-making skills, and interpretive skills today. In this debate, he reminds us, we should keep in perspective that we teach people, not mathematics.

An interesting article by Glena Price points out that we need develop mathematics across the curriculum. She makes the claim that quantitative literacy is one of the skills which will determine who will be successful and who will not. A large part of doing this is spreading mathematics across the curriculum, giving students the opportunities to develop quantitative literacy, and encouraging students who are risk aversive to take mathematics courses. These are not easy things to do. Building math across the curriculum involves teachers teaching other teachers. Encouraging risk aversive students involves creating opportunities for student success.

There are many powerful ideas expressed from a variety of sources in this book. This is not a sourcebook for problems which will foster quantitative literacy. I'm sure there are many books out there that claim to do just that. This book examines the issues which will need to change if we are to realize our goal of quantitative literacy. We need to do this to empower our citizens in this rapidly changing world.

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