Keywords: Arithmetic, Connections, Number and Operation
Ref: Laura7
Author(s): Wilson, Patricia S.
Date: 2001
Title: Zero: A Special Case
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 6, Issue 5, Page 300
Reviewer: Laura
Date of Review: 5/17/03
This article describes the way in which a group of middle school students effectively learns about the number zero. It desribes the lesson they learn about its properties, origins, and devolopment. There is a strong historical element to the lesson as well, since they are learning about the importance of where mathematical concepts come from. Views from the Greeks, Romans, and such figures as Pathagoris are covered.
Also included in this article are a number of really effective worksheet examples and web resources. These allow students to practice the concepts and exercise the knowledge they have gained regarding zero.
I really enjoyed reading about an example a lesson devoted to the number zero. It is such a unique element of mathematics, yet seems to have very little personalize attention in curriculum. The concept of nothing is pointed out, but rarely elaborated on. This article did a nice job of illustrating ways to clarify the idea of nothing is the world, equations, and practice.
The only thing I wonder is whether or not the lesson illustrated in this article should be taught to students at an earlier age. Middle school ensures, for the most part, that students will be able to grasp the concepts presented in this lesson, but I can't help but think that those concepts are more important than just to be touched on in the early days of arithmetic. Dare teachers try and teach the properties of zero to elementary school students? I'm just not sure if it will help clarify or complicate the lessons learned at that age. In hindsight, I want to say clarify, but I hardly remember what it was like to learn math when I was in grade school. Perhaps my opinion is biased now that I have a deeper understanding of the subject.
Keywords: Curriculum, Number and Operation, Representations
Ref: Laura9
Author(s): Reys, Barabara J.; et. all
Date: 1991
Title: Developing Number Sense
Journal or Publisher: Curriculum and Evaluation Standards for School Mathematics
Volume, Issue, Pages: Fifth Printing
Reviewer: Laura
Date of Review: 5/17/03
This book provides curriculum that will integrate number sense into the mathematical education of young people beginning in early grade school. It offers ways in which students can be taught to understand multiple representations for numbers, analyze the way that various algorithms work and relate that to the numbers involved and the meaning of the algorithm, know the way numbers are used in every day life and what they mean, etc. It also explains the importance that students learn this as it will effect their future mathematical education.
Through a variety of activities offered, this book offers many ways to enhance students' understanding of numbers and "number sense" in grade school so that they will have an advantage in understanding the role of numbers in their future studies. Also offered is how this information is evaluated and how an educator can be confident that their students have actually attained number sense.
When I first read the title of this book, I thought it sounded a bit odd. What exactly is number sense? When it was made clear to me in the book's forward, I realized that it was something I feel that I have, but do not recall having a specific lesson or unit in math classes to cover the topic. It was slowly but surely obtained through my years of mathematical study.
The idea of having a book as part of curriculum that specifically approaches this topic is a good one in my opinion. Perhaps math lessons I had in the past would have been even more effective had I been taught a strong foundation in number sense rather than merely learn that 2 + 2 = 4 without learning precisely what two and four are or that there are numerous ways for them to be expressed and described.
I think that this book offers a great way to integrate "number sense" into the math lessons of grade school students so that they will get the most out of their future lessons inthe subject.
Keywords: Curriculum, Geometry...
Ref: Laura10
Author(s): Lott, Johnny W.
Date: 2001
Title: Navigation Through Geometry in Grades 9-12
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages:
Reviewer: Laura
Date of Review: 5/17/03
This book begins by clearly laying out the standards for geometry in grades nine through twelve that are to be met and how this book will help that happen. Then, it has a series of chapters that begin with a brief presentation of information to aid the students in their learning, and a number of exploration problems to solidify it.
The problems offered in each chapter provide students with goals to be achieved, and hints on having that happen. The key is that the students will be exploring the solutions, and once those solutions are found, they, along with the concept at hand, will be far more memorable.
I honestly believe that I would have enjoyed studying geometry a lot more had I learned from a curriculum more like this one. What I remember of it back in 8th grade was a lot of proofs, theorems, and tedious angle measurements and congruence proofs. Perhaps if I had been told to figure out for myself what characteristics make two triangles similar rather than have it laid out for me, the entire process would have been a bit more exciting. All in all, I really like the way that geometry is presented in this text, as well as the way that the standards are layed out so clearly that teachers can be confident that they are being met.
Keywords: Curriculum......
Ref: Laura11
Author(s): Burkhardt, Hugh
Date: 2001
Title: The Emperor's Old Clothes, or How the World Sees It...
Journal or Publisher: Mathematics Education Dialogues
Volume, Issue, Pages: Issue #8, January 2001
Reviewer: Laura
Date of Review: 5/17/03
This article is Hugh Burkhardt simply stating his perspective on integrated curiculum and the debate surrounding it. He points out that he thinks it is silly for people to view it as an issue. Having seen math teaching styles from all over the world, he claims that nowhere other than the US would it be so broken up into one year of a subject, one year of another, and so on. It should all flow more smoothly together or, be integrated.
He illustrates the problems surrounding one year curriculum "chunks" such as summer amnesia, the lack of usability when knowledge is missing from the flow of curricula, and the lack of connections between various years such as Algebra I and Geometry. He points out the way that integrated curriculum simply makes the learning process more smooth and valuable as it connects all that the students need to learn and makes it more memorable and usable.
I think that Burkhardt makes some excellent points about the benefits of integrated curriculum, however he fails to give traditional curriculum enough credit for what it accomplishes. The phenomenon of summer amnesia that he points out, did not seem to be a problem from what I recall of classes I had in the past. In fact, having an entire summer to digest the year of Algebra or whatever subject I had just had, helped me to make the main concepts sink in and lose the unnecessary details that would often bog me down as a student. Of course this is just my experience, but I still think that there are more positive aspects to traditional curricula than he acknowledges.
I also think that the one year of Algebra, one year of Geometry, etc. that he refers to as "one year chunks" of curriculum have their positive points. I recognize the lack of coherency, but by having the separation, I think it can make the material covered in each year more obtainable and less confusing as curriculum that blends or integrates them runs the risk of being.
In this article, Burkhardt illustrates the strengths of integrated curriculum well, but I think his argument would have been more effective had he given older textbook series and curricula more credit for their strong points.
Keywords: Connections, Communication...
Ref: Laura8
Author(s): Bell, E.T.
Date:
Title: Men of Mathematics
Journal or Publisher: Simon and Schuster, Ney York
Volume, Issue, Pages: Pages 1-55
Reviewer: Laura
Date of Review: 5/17/03
This book offers a very accessable walk through of the major topics in math as well as the history behind them. Bell does a really nice and clear job of articulating geometry from the Greeks, Calculus thanks to Newton, as well as theories of statistics and probability laws that evolve out of devoloped math.
Also present in this book is a lot of emphasis on the philosophy of math and why it devoloped as it did throughout history. Between the history of the thought and actions that went into developing math, the philosophy involved, and all the concepts covered, Bell does a really nice job of articulating a subject that is not always well liked, in a way that people who normally do not enjoy it can understand and appreciated.
I have to admit, when I first began reading this book, I was a little bit put off by the title. I'm really big on making math a subject considered to be for both genders, not just men. However, once I got past that I was really able to appreciate this book. Bell does an amazing job of explaining mathematical concepts, their history, and the philosophy behind them in such a way that laymen of the math world can understand and enjoy. I was fascinated by the description of the way that Greeks developed geometry and had secret societies to discuss it. Having detested geometry in the past, I can't help but wonder if I would enjoy it more having learned more about its history.
Also, the fact of the matter is that this book looks a lot at the history of math, and historically, it primarily was devoloped by men, or they were the ones credited. Perhaps it is appropriate to title a book, "Men of Mathematics," when it does, in fact, focus on the men that devolop the topic. It doesn't necessarily imply women can't be a part of the subject. They just aren't discussed in the book is all.
My point,(that I am not reaching smoothly at all) is that this is a great book with a fresh look at math from a historical and philosophical point of view that teachers would be well advised to share with a high school class as it enhances appreciation for subjects previously, currently, and to be studied.
Keywords: Connections, Measurement, Problem Solving
Ref: Laura1
Author(s): Jones, Graham A., Thornton, Carol A., McGehe, Carol A., Colba, David
Date: 1995
Title: Rich Problems--Big Payoffs
Journal or Publisher: Mathematics Teaching in the Middle School (NCTM)
Volume, Issue, Pages: Vol. I, No. 7
Reviewer: Laura
Date of Review: 3/25/03
In this article, David Colba takes a mathematical problem provided by his friend that is an actual architect to his 8th grade students. There was a hotel being designed in which each room opened onto a walkway overlooking a central atrium area which would be a rectangular shape. There would be a protective brass railing surrounding the edges of the overlook on each floor. With 650 feet of railing available for each floor, the students needed to figure out the dimensions of the railing to maximize the area of the scenic view of the atrium below.
Initially, the students declared that the answer was to divide 650 by four, and square the answer resulting in 26,406.25 feet. However, when asked if they were certain that this would provide the maximum area, no one was sure. Colba then had his students break up into small groups to further discuss the problem.
As Colba walked through his classroom from group to group, he found that there were various ways that the students were approaching the problem. Some chose numbers at random to explore the kinds of rectangular shapes that maximized the area, others elaborated on the 650 feet provided. The main struggle students had was understanding the meaning of maximum area, and ultimately, all found that a square figure would provide it. Colba provided an excellent hint for his students when he related the problem to their lesson from the previous day that involved tiles.
Colba then found the opportunity to bring technology to the lesson by guiding the students in finding a formula for the area of a four sided figure with a perimeter of 650 feet, and using their calculators to graph it and calculate various examples of outputs. He then had the students trace the graph on their calculators to find that the maximum area was, in fact, 24,406.25 square feet when the area was a square.
Next, Colba provided several variations of the same problem for the students to explore such as what if only three of the sides of the rectangular figure would have brass railing, or what if there was no brass railing where there were two elevators.
Ultimately, this problem proved to be rich as it allowed students to explore and take an active role in their learning, and approach it with three possible strategies: graphic, numeric, and geometric. It also provided excellent opportunity for use of technology in the classroom. All in all, this article gives an excellent example of the kind of problem students can work on to exercise communication in mathematics and taking an active role in learning. My personal response: One thing that I really like about this example of a problem to present to students is that it is an actual architectural problem being solved in the "real world." All too often kids raise the question of when they will ever use the math that they are learning. Rather then answer that question after teaching material, this article gives an example of offering material that originates in a real life situation.
One question that came to my mind as I read the article, was how much prior knowledge did the students have before exploring this problem? They clearly knew the basics of a perimeter and an area, but at one point, Colba points out that a group struggled with understanding the term, "maximum area." I wonder whether a lack of prior knowledge would be an advantage or disadvantage when providing a problem such as this. With less knowledge, there is, perhaps, a greater risk of students becoming lost, but also an increase in the number of possible discoveries or directions that they may take their exploration. The greatest question in my mind now is how much knowledge of a particular topic students ought to have before receiving a problem that allows them to explore it mathematically.
Keywords: Activities, Teaching Strategies,
Ref: Laura2
Author(s): Draper, Roni Jo
Date: 1997
Title: Active Learning in Mathematics: Desktop Teaching
Journal or Publisher: NCTM Mathematics Teacher
Volume, Issue, Pages: Vol. 90, No. 8, pages 622-625
Reviewer: Laura
Date of Review: 3/26/03
The article, "Active Learning in Mathematics: Desktop Teaching" by Roni Jo Draper, offers a very detailed description of a review activity at the end of a term that has students take an active role in their learning and ultimately make the classroom resemble a fair more than anything else.
Draper points out that the fourth goal of NCTM 1989 is for students to learn to communicate mathematically. The active student participation in both group and individual exploration during desktop teaching provides opportunities for them to discuss, question, listen, and summarize mathematically and achieve this goal.
She then offers a definition of active learning as taking an active role in learning rather than one that is passive and the teacher is simply the instructor. Desktop teaching gives students an opportunity to learn actively and ultimately is an active learning strategy.
The way desktop teaching is implemented is each student is assigned an individual topic to prepare a five to seven minute lesson on. A schedule is then set up so that each student will teach and learn from every other student in the class. Two students will have their teaching materials on their desktop at a time, as the other students take time to be taught each of their lessons. Generally, this takes two days when block scheduling is used, four when regular.
To introduce desktop teaching, Draper has her students brainstorm what makes a really good teacher, and what makes a not-so-good teacher. They then discuss the responses and students grow motivated to be good teachers. Next, the students receive their topics and are expected to prepare lessons outside of class but feel free to visit Draper with questions and receive help with ideas.
The students generally all do well with the desktop method of review. They are evaluated on how well their topic is taught, the presence and appropriateness of learning activities in their lesson, etc. The key is to achieve full class participation.
My personal response: I think desktop teaching sounds like a really neat way to review for an exam at the end of a unit or term in class. I like the idea of having each student receive a topic and teach it to his or her fellow classmates as it makes for a more memorable experience and more retained knowledge.
My only question is how to handle the situation in which a student's lesson does not quite accomplish the goal of reviewing their topic without making it obvious to the students and making for an embarrassing situation. Draper mentions that students do not receive time in class to work but have the option of seeking her for help. If students do not choose to see her and receive help and guidance, how does she monitor their work and make sure their lesson plan will accomplish the necessary goal of reviewing their topic? That would be my one hesitation in using desktop teaching in a class I teach.