Keywords: Geometry, Puzzles, Problem Solving
Ref: Tom1
Author(s): Rulf, Benjamin
Date: 1998
Title: A Geometric Puzzle That Leads to Fibonacci Sequences
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 91, Number 1, Pages 21-23
Reviewer: Tom
Date of Review: February 12, 2002
The article begins with the introduction of a tricky geometry problem: we have four shapes (two triangles and two quadrilaterals) that when put together can form a square (one side length equal to x + y). However, when rearranged into a rectangle (sides lengths of y and 2y+x), we get a small sliver surrounding what would be a diagonal in the rectangle where there is an empty space. The idea of the puzzle is to show that we cannot always rely so heavily on instruments of measurement; instead we should rely more on the mathematics behind such a situation. As it turns out, by changing the values of x and y so that the difference in area between the square and rectangle approaches zero, we see that the ratio y/x also approaches the golden ratio or "divine proportion" (approx. = 1.618). Now that we have discovered that the golden ratio is the key here, it is easy to see that as we find larger number pairings that approach the golden ratio, we are also finding the fibonacci numbers. From here the author goes on to prove two things. First he proves that given any two initial positive integers, using the fibonacci formulas, the difference in the areas of the two figures remains constant - regardless of how large your numbers become. The constant difference is dependent on your chosen initial numbers. The author then proves that the ratio of growing (yet consecutive) fibonacci numbers has a limit of the golden ratio.
I found this article to be very good. The puzzle that all this originated from is a fun to introduce to middle school students and above, as they assume that everything will stay the same yet everything is not as perfect as it once seemed when you rearrange the pieces. I enjoyed how the author continued to explore mathematically what was going on behind the geometrical confusion. Especially impressing was the fact that the fibonacci numbers were at the heart of it all, as well as the golden ratio. Plus the idea of having a constant difference of area which was dependent on the initial fibonacci values was very intriguing.
Keywords: Technology
Ref: Tom2
Author(s): Joyner, Jeane; Rich, Wendy
Date: 2002
Title: Using Interactive Web Sites to Enhance Mathematics
Learning
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Volume 8, Number 8, Pages 380-383
Reviewer: Tom
Date of Review: February 14, 2002
This article involves integrating the use of interactive web sites into the classroom. The article is separated into different parts, each of which refers to a specific grade range: pre-kindergarten through 2nd grade, grades 3 through 5, and grades 6 through 8. The grade band for grades 9 through 12 is unfortunately not included. The hope of the this program is to encourage students to focus on decision making, reflection, reasoning, and problem solving in almost every area of mathematics. The article provides an internet link for electronic examples of technological activities for the classroom: standards.nctm.org/document/eexamples/index.htm. For each of the grade bands the article provides an example of what can be found at the web address listed above for that given grade band. For pre-K through 2nd grade, the program focuses on developing geometrical understanding and spatial skills through puzzle problems with tangrams. More topics for this grade band include pattern geometry, measurement, number relationships, and estimation. For grades 3 through 5, the example given deals with students' understanding of functions and representation of change over time. More topics for this grade band include geometry, data, algebra, and number operations. For grades 6 through 8, the example provided involves comparing the properties of the mean and the median through the use of technology. More topics for this grade band include geometry, algebra, and statistics.
The second portion of this article focuses on another program that can also be found on the web. The name of this one is "Illuminations". This program can be found at: illuminations.nctm.org/index2.html. For each of the grade bands, the program offers professional development activities, investigation for students, reflections on teaching, selected web resources, internet-based lesson plans, and the document "Principles and Standards" by the NCTM. There are also links for professional development, teacher study groups, and individual teacher use in classrooms.
Both of these websites appear to have quite a bit of valuable information, especially the last one which appears to be more geared towards a helping a math teacher run the classroom than perhaps providing activities for students on the web (although there is a little of that as well). Although the first article does not provide any examples of electronic problems for high school level students (for whatever reason), on the web site there is in fact an area designated for them. Not too surprising, but important nonetheless, is that for each of the programs the web site tells the reader which of the math standards and/or principles are being emphasized. In general, this article does not go into too much detail for either of the websites, which I am little disappointed about. But the websites themselves are pretty intuitive, so that might explain part of the reason for that. The authors do say that both of these websites are very well done, however, and that they should prove to be valuable resources.
Keywords: Technology, Activities, Problem Solving
Ref: Tom3
Author(s): Huinker, DeAnn
Date: 2002
Title: Calculators as Learning Tools for Young Children's
Explorations of Numbers
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Volume 8, Number 6, Pages 316-321
Reviewer: Tom
Date of Review: February 19, 2002
This article discusses how to use a calculator to help children explore numbers in a kindergarten classroom. The calculator is used to help the students expand their current knowledge and understanding of simple numbers. The teacher began by introducing the basic keys (numbers and operations). Much of the instruction was done using discovery learning. The article is broken up into a few sections centered on exploring numbers with a calculator: exploring numerals, counting, number magnitude, and number relationships. Each of these sections explains how the teacher started with a very basic idea, and then let the students expand it themselves and they would tell her what they figured out. The hope of the teacher and the author of this article is that the students' use of calculators this early on will give them a head start with learning of mathematical concepts/skills and problem solving.
I enjoyed this article. Allowing the younger-aged students to discover their way through elementary mathematics using a calculator is a great way to maintain (if not initiate) interest in the subject. The students also get perhaps their first chance at using technology (what the older kids get to use all the time!) with mathematics. I also agree with the article's final claim that the use of calculators this early on will give the students some idea of what lies ahead for them. My only concern with these students is this: what will happen when they first learn what multiplication really is? Will they work it through with pencil and paper, or will they immediately reach into their backpack and reach for the calculator?
Keywords: Activities, Algebra, Connections
Ref: Tom4
Author(s): Hines, Ellen
Date: 2002
Title: Exploring Functions with Dynamic Physical Models
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 7, Number 5, Pages 274 -278
Reviewer: Tom
Date of Review: February 21, 2002
This article by Ellen Hines focuses on helping students visualize algebraic functions' role in the real world and understand relationships between input and output. To do this, Hines uses what she calls "input-output machines". The particular model that she uses for this is called a "spool elevating system". Her goal is that this machine will concretely illustrate how two variables can change systematically. The article is broken in a few sections, including instructions for making a spool-elevating system, explorations with the spool-elevating system, and writing mathematical sentences. Hines points out that students "will need plenty of time to discuss and distinguish among variables and non-variables and to experience the systematic relationship between in the variables in the functions and spool system". There is also room for expansion within this activity: once the main idea is understood, the teacher can introduce larger and smaller spools - allowing students to compare and contrast results.
I really enjoyed this article. This teacher has some fantastic ideas.
I have always enjoyed activities in math that not only keep me out of my
desk and working with my hands, but also those activities that allow students
to make mathematical connections to the real-world. To me, these types
of activities always seem to inspire students and help them to understand
math outside the classroom in the real world (i.e. more connections to
other subject areas). Activities such as this that allows students to encounter
functions in a non-symbolic, real-world context are excellent.
Keywords: Problem Solving, Issues
Ref: Tom5
Author(s): Knuth, Eric J.
Date: 2002
Title: Fostering Mathematical Curiosity
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 95, Number 2, Pages 126-130
Reviewer: Tom
Date of Review: February 26, 2002
Knuth begins by pointing out that in most math books and classrooms today, too often the material is geared towards having the student obtain the “final answer” for a given problem. He, on the other hand, wants teachers and books to focus more on using problems as starting points for further mathematical exploration because “exploration lies at the heart of mathematics”. Knuth gives two examples of problems where is approach is emphasized. His hope is that “wondering and speculating associated with exploration may lead to important generalizations or to a deeper understanding of the original solution”. Knuth does not label “rephrasing the question” as being a complete waste – he does see some value in it. He simply wants to redirect the focus to something he feels to be more important.
I agree with Knuth for the most part. There needs to be less emphasis
on the “right” or “final answer”, and more emphasis on the exploration
of the problem. I also agree that only by doing this will students be able
to find more curiosity within mathematics – something which appears to
have dissipated over the years.
Keywords: Technology, Curriculum
Ref: Tom6
Author(s): Smith, Alma Coggs
Date: 2002
Title: Natural Chemistry with Technology
Journal or Publisher: NEA Today
Volume, Issue, Pages: Volume 20, Number 5, Page 7
Reviewer: Tom
Date of Review: February 28, 2002
Smith is essentially describing her experience in evolving from a classroom where very little technology was used to a classroom where technology is a primary resource. She first enrolled in the Intel Teach to the Future program, which, as she describes it, sounds like it would be very helpful for all teachers – not only for science teachers. The program even “requires” that any teacher that passes through the 40-hour training process (in addition to approximately 20 hours of take-home activities) must also train an additional 13 teachers when he or she returns to his or her school. From the sound of it, the program allows for any subject to be taught using this technology. And although the technology has helped Smith in her planning and teaching strategies, the part about this program that she finds the most beneficial is how much more interested her students are in the subject matter now that they are using technology in the classroom. To quote Smith, “My students are now willing to spend more time studying a given research topic. They’re reading more and enjoying it. I think that they’re more creative now that they’ve learned some technological tricks.” Clearly Smith feels that the Intel Teach to the Future program has done but help her with classroom management and curriculum, in addition to stimulating her students’ interest in the subject. More information on the Intel Teach to the Future program can be found at the following web address:
www.intel.com/education/teach
This was a really good article. According to one of the statistics mentioned
in the article, only 33 percent of all teachers feel comfortable using
technology in their classrooms. Hopefully teachers reading this who don’t
feel comfortable, or even think that technology could be incorporated into
the classroom’s activities. Especially with how many students these days
have access to computers, continuing to use computers as a major resource
in the classroom can only help since the students have such familiarity
with them.
Keywords: Connections, Curriculum, Standards
Ref: Tom7
Author(s): Froelich, Gary W.
Date: 2002
Title: Connecting Mathematics
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages:
Reviewer: Tom
Date of Review: March 5, 2002
This book is a compliment to math teachers for the classroom. It provides worksheets to help in student development which cover different areas in mathematics with “connections” to the real world being the focus. These five areas include connecting with functions, connecting with matrices, connections with data analysis, building and using connections in reasoning, building and using connections in problem solving. Each of these chapters provides pedagogical explanations for each of the topics inside the chapter, and also provide suggestions for methods of teaching the particular subjects. Each of the subjects, worksheets, and activities appear to have a focus on problem solving – for the individual and the group (small or large) – as well as showing connections between the mathematics involved and the “real world”. The book does admit that there is always more that a teacher can do to take things further than this particular book does, but that its intention was to provide a standard (perhaps a minimum expectation) for teachers.
I found this book to be very well laid out. I have always agreed with
educators that one of the most effective ways to learn is to make connections
with current knowledge and conceptions of the world. Needless to say, this
publication does plenty of that. One statement in the “final comments”
section of this book that I especially was pleased to see was that they
encourage teachers to reward students for solving problems in more than
one way. This is always important, if no other reason than it prevents
students from falling into the “there is always one way to work through
a problem and one right answer” trap.
Keywords: Connections, Curriculum
Ref: Tom8
Author(s): Martinez, Joseph G. R.
Date: 2002
Title: Building Conceptual Bridges From Arithmetic to Algebra
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 7, Number 6,Pages 326-331
Reviewer: Tom
Date of Review: March 7, 2002
The article begins by explaining common problems that students have when first introduced to algebra. Most students that have problems with algebra for a while have trouble "making the first step towards problem solving: reading and interpreting the problem". Compared to many countries in the world, the US lags behind in algebra skills. The author makes the claim that when students say they cannot about dealing with "X's", this suggests their inability to make connections between algebraic expressions and arithmetic symbols and operations. The author continues to say that when making the transition from numbers and operations into algebra one must be very specific. Also important is once the teacher has introduced variables and basic algebra to the students, he or she should then continue to switch back and forth between algebraic and numerical expressions so that the students have the chance of seeing as many connections between the two areas as possible. The author believes in the thinking of the NCTM in that teachers need to emphasize how "...mathematical ideas interconnect and build on one another to produce a coherent whole".
Martinez makes a strong point with his article. Even in my limited experience in working with students as a tutor or teacher's assistant, I know that one of the best ways to help a student having a trouble with algebra is to start with basic numerical expressions and build up to algebraic expressions. Then, as the author states, it is important to continue switching back and forth between the two so that the student can see the differences and the CONNECTIONS. Just like everyone else, I am concerned that our students aren't doing as well in math as the rest of the world's students are. But I hope that our eagerness to improve our methods of teaching is not because we want to improve our students' standing when compared to the rest of the world, but more because we want our students to enjoy and have a better understanding of mathematics as a whole.
Keywords: Assessment, Standards, Geometry
Ref: Tom9
Author(s): Carroll, William M.
Date: 1998
Title: Middle School Students' Reasoning About Geometric
Situations
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 3, Number 6, Pages 398 - 403
Reviewer: Tom
Date of Review: March 12, 2002
Carroll seems to be providing an example with this article of his. He is emphasizing the importance of having students not only think through their problems in their head, but also explain their reasoning on paper. He hopes that by writing out an explanation for a given problem, the student can help him or herself see things more clearly in addition to showing the teacher exactly what her or she knows. To help get the teacher started with this sort of approach for assessment, Carroll has provided three examples that a teacher can use in the classroom for assessing a student’s ability more accurately, all of which can be used for individual or group activities. He also wants to emphasize the importance of introducing geometry prior to high school so that the students are more familiar with it. According to Carroll, studies have shown that high school students are not well prepared for formal geometry classes.
I found Carroll’s article to be very well done. His ideas for questioning
the students more often and getting themselves to explain their answers
more thoroughly than in the past can only help teachers and students. For
teachers, it helps because they will have a better understanding of what
the students are struggling with or what the teacher needs to be better
at explaining. For students, it helps because they become more familiar
with a more formal thought process, as opposed to simply circling the correct
answer.
Keywords: Problem Solving, Assessment, Communication
Ref: Tom10
Author(s): Ross, Sharon R.
Date: 2002
Title: Place Value: Problem Solving and Written Assessment
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Volume 8, Number 7, Pages 419 - 423
Reviewer: Tom
Date of Review: March 14, 2002
Ross' article is a summary of a study performed on about 50 or 60 elementary school students. The goal of the study was to get an idea of how many children have difficulty understanding the value and meaning of place values and numbers (or digit correspondence as Ross calls it), and figuring out what exercises, lessons, and learning environments would help to improve understanding. The initial tests, which assessed the students' understanding of place value, were not encouraging, as only 19 percent of the students were successful. The teachers involved in this study, the instructors, presented the information to students through lessons that emphasized group and class discussions while also making sure that the lessons were still inquiry (or discovery) based. There are three lessons that the article focuses on: one involving area of basic two-dimensional figures, one with volume, and one with measuring quantities (specifically tires on cars). Since the lessons were inquiry based and also performed mostly in small groups, the students worked together to try to figure out (for example) the number of cars that could be supplied with wheels when there were 26 wheels available. The results at the end of the study were much more encouraging, since 70 percent of all students were successful during the second assessment.
I remember having little difficulty with math through 8th grade, so
I initially had difficulty understanding how students could have so much
trouble understanding digit correspondence. But after reading the article
things are clearer to me now. Not only that, but the activities and lessons
that the teachers and students were working through would have been useful
for me anyway since they build social skills, problem solving skills, good
number sense, and better understanding of estimation early on in a student's
education.
Keywords: Calculus, Connections, Teaching Strategies
Ref: Tom11
Author(s): Sherfinski, John
Date: 2002
Title: A Multilayered Maximum-Minimum Problem
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 95, Number 3, Pages 218 - 220
Reviewer: Tom
Date of Review: March 19, 2002
Teacher John Sherfinski approaches the first few days of his calculus class a little differently than most teachers. Instead of handing out a syllabus or establishing class rules on the first day of class, he begins the first day of the new semester with “the lake problem”. The idea of the problem is to get students involved in the class material immediately. Since a teacher’s time with the students is limited as it is, it makes sense to do it this way. Something that Sherfinski especially enjoys about using this particular problem (it’s laid out in detail in the article) is that is that there are so many things that a teacher can do to change the problem so that it can be analyzed again and again with different topics. The way that he analyzes the problem is interesting too because he asks only a few questions but the class spends a lot of time looking into the answers. Sherfinski believes that “investigating fewer questions in depth is more beneficial than racing through as many problems as possible”.
I really like Sherfinski’s idea of introducing one problem right away
and returning to it throughout the semester. It allows the students to
get familiar with a problem and yet each time they come back to it they
learn something new about it. I think that repeatedly coming back to a
problem like that allows for better student understanding because they
are making connections with previous knowledge.
Keywords: Activities, Measurement, Connections
Ref: Tom12
Author(s): Hynes, Mary Ellen; Dixon, Julie K.; Adams, Thomasenia
Lott
Date: 2002
Title: Rubber-band Rockets
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Volume 8, Number 7, Pages 390 - 395
Reviewer: Tom
Date of Review: April 4, 2002
This article is a guide for a hands-on activity for K-6 age students. The students will enjoy it because they get out of their seats, and the teacher will enjoy it because the students are learning good mathematics and finding/seeing connections. The basic idea for the activity is for the students to perform a few trials where they each shoot rubber bands using a protractor with a ruler taped to it as the "gun". The students then measure the distances that the rubber bands went, how far they pulled the rubber-bands back before shooting them, and then the angle at which they shot the rubber bands. Additionally, the students will also be analyzing their data by finding the mean, mode, median, and range of their data. After that the teacher has them interpreting what these new numbers represent. The authors also lay out some general objectives for of the K-2 and 3-6 grade bands. The main idea of this entire activity seems to be investigation.
This activity seems like it would be very valuable for all students.
Many kids that age enjoy playing with rubber bands anyway, so when given
the chance to play with them as part of an assignment, my guess is that
they would have a lot of fun with it. Little do they know that the teacher
has had them learning important mathematics the entire time.
Keywords: Research , Technology, Algebra
Ref: Tom13
Author(s): Heid, M. Kathleen; Hollebrands, Karen F.; Iseri,
Linda W
Date: 2002
Title: Reasoning and Justification, with Examples from Technological
Environments
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 95, Number 3, Pages 210 - 216
Reviewer: Tom
Date of Review: April 4, 2002
This article focuses on how two students each worked through a math problem using both technology (a calculator in both cases) and their minds. Both students, although the authors admit they are each above average students, showed versatility in switching from using reasoning and justification in their minds to using the calculator for help. The apparent goal of this article is that teachers will recognize the results of recent research: "...teachers must understand the ways in which students conceive of evidence and proof and the effects of technology on those understandings." Once teachers can achieve this, they will gain immeasurable power in the classroom. They will be able to ask different questions and receive different answers from students in their attempts to justify their own results. The authors believe that the correct use of technology in the classroom can be invaluable in the classroom when aiming for high-level thinking.
The past few articles that I have read on technology in the classroom
have been very reassuring. It sounds to me like teachers today who use
calculators in the classroom are not using it for the same plug-n-chug
approaches that many of my teachers did. Instead they are using calculators
primarily for explorations, investigations, and high level thinking. The
authors do a good job explaining the recent research on technology in the
classroom and the approach that teachers should take when incorporating
it.
Keywords: Technology
Ref: Tom14
Author(s): Math Medics
Date: 2002
Title: S.O.S. MATHematics
Journal or Publisher: Math Medics
Volume, Issue, Pages: website
Reviewer: Tom
Date of Review: April 17, 2002
This is a popular web site run by three college professors with Ph.D.’s in mathematics. This is geared to be an automated tutoring system or computer program of some sort. You type in the subject that you need help with, and a few links pop up. By following these you get explanations similar to those you might find in a textbook. After each explanation of a topic, there are a few practice problems that a student can follow to try gaining a better understanding or more practice. You obtain the answer to your specific problem clicking on a link. The problems are geared for upper level high school and college students, covering the following subjects: Algebra, Trigonometry, Complex Variables, Calculus, Differential EQ, and Matrix Algebra. If you are reading the explanations and trying out the problems but you are still having difficulty, then there is a message board that you can post your questions on. Here your problems can be addressed by the webmasters (the three math professors) or anyone that wants to respond.
This seems to be a very valuable website for students of high level
mathematics. The explanations that I saw were detailed and the problems
that they provided for practice seem to be relevant to the materials.
Keywords: Equity, Assessment, Standards
Ref: Tom15
Author(s): North Central Regional Educational Laboratory
Date: 2002
Title: Ensuring Equity and Excellence in Mathematics
Journal or Publisher: North Central Regional Educational Laboratory
Volume, Issue, Pages: website
Reviewer: Tom
Date of Review: April 17, 2002
This is a web site run by the North Central Regional Educational Laboratory. It appears to be an argumentative web site striving for equality of gender and race in the mathematics classroom. They have three primary goals: (1) All students will have equitable access to challenging and meaningful mathematics learning and achievement. (2) Teachers will promote and model a belief in the importance of diversity, excellence, and high-quality mathematics instruction in their work with students, colleagues, and the community. (3) Administrators, school board members, parents, and other members of the school community will support and model a belief in the importance of equity and excellence in mathematics education. The web site briefly addresses the opposing views that others have, and promptly shoots them down and/or discredits them. There is also a small section devoted to the recognition that there are pitfalls when it comes to implementation of equality. They realize that many parents and teachers have different views of what exactly “equality” means. But they quickly try to address this by providing their own standard definition of equality in the classroom. There are several links and contacts of well-known foundations (the National Science Foundation for example) which strongly support this movement.
I agree with the message and purpose of this web site one hundred percent.
At the very least, since we are all (going to be) mathematics teachers,
we need to be following the standards set by the NCTM: important mathematics
needs to be experienced by ALL students.
BR>
Keywords: Games, Discrete, Connections
Ref: Tom16
Author(s): Johnson, Donovan A.
Date: 1973
Title: Games for Learning Mathematics
Journal or Publisher: Published by J. Weston Walch
Volume, Issue, Pages:
Reviewer: Tom
Date of Review: April 30, 2002
One of the first things mentioned by the author of this book is that this publication was not designed – by any means – to be used as the centerpiece for a course in mathematics. It is, however, an excellent supplement. Johnson explains how this book should be used as an educational tool for student in mathematics courses and also how (why) it is valuable on so many levels for students of mathematics. The games in this book cover a number of subjects in mathematics with an emphasis on algebra, geometry, and arithmetic. One should not believe that those are the only skills and topics that this tool focuses on – many of the games suggested by this book can be adapted and adjusted to fit any subject. These games are geared so that the students improve their study habits, develop desirable (positive and enthusiastic) attitudes towards mathematics, develop a stronger sense of creativity with mathematics, improve their problems solving skills, work on working together with others on mathematics, and develop better and more advanced mathematical ideas.
I love this book! This is definitely something that I would like to have and incorporate into my classroom. Not only are many of these games taken from what kids play anyway, but the author (Johnson) also explains why each of the activities is important to fostering more advanced mathematical thinking and how it does it. These games are fun, intriguing and – most importantly – they focus on good mathematics.
Keywords: Problem Solving, Statistics,
Ref: Tom17
Author(s): Shaughnessy, J. Michael; Pfannkuch, Maxine
Date: 2002
Title: How Faithful is Old Faithful?
Journal or Publisher: NCTM: Mathematics Teacher
Volume, Issue, Pages: Volume 95, Number 4, Pages 252 - 259
Reviewer: Tom
Date of Review: May 5, 2002
This article focuses on an activity where students try to predict when the geyser Old Faithful will erupt next. After listing out the data, the students graph their results. The authors say that many students initially claim your best chance of predicting a time at which Old Faithful will erupt next is to find the mean or median time at which it does and that is when you should expect it. However, this problem is designed to make the students think more than simply finding the mean and median: it makes them consider the variability of time at which the geyser erupts. The authors introduce the concept of “statistical thinking”, whose fundamental processes are apparently much different from those in mathematics. The authors and their associates have identified five elements that are essential to statistical thinking: (1) recognition of the need for data, (2) transnumeration, (3) consideration of variation, (4) reasoning with statistical models, and (5) integrating the statistical and contextual. The first three are not usually too troublesome for most students, but the last two are. Students have difficulty making to the transition from calculating results and determining exactly what they mean. The teacher is encouraged to speak to the students and ask them questions (starting out basic and getting more and more complex) so that the students have some direction for the problem.
I enjoyed this article, and I think it is a great activity for students
when the teacher decides to expand students’ understandings of
statistical analysis. It incorporates graphical analysis in addition to
the data crunching, and it focuses on the transition from these two methods
of analysis to interpretation.
BR>
Keywords: Algebra, Connections, Calculus
Ref: Tom18
Author(s): Helfgott, Michel; Lutz, P. Michael
Date: 2002
Title: The Boat-and-Ambulance Problem Revisited
Journal or Publisher: NCTM: Mathematics Teacher
Volume, Issue, Pages: Volume 95, Number 4, Pages 270 - 274
Reviewer: Tom
Date of Review: May 5, 2002
This article describes an idea that a teacher can use for one of the “big problems” that students spend lots of time on to learn a lot of mathematics. This problem is written for a third year algebra class (or pre-calc), but it can be used later as well. There are a few important connections for students to make. The problem centers on concepts in physics, but to solve the problem the student will need to use algebra in attempt to model motion. Graphing calculators are used to view graphs of motion relative to time, so students can practice using technology in the process as well. However, the use of algebra and functions on paper is also emphasized. The connection with calculus here is that this is a max-min problem. The students in a calculus class would need to use the chain rule to get a solution. The connection with physics also includes the application of Snell’s law of refraction of light (which in turn uses trigonometry) in addition to modeling motion. Also important for teachers (and students): this problem has possibilities for extensions. There are possibilities for students to do a project on this sort of problem, or simply go into further analysis if some students finish early (change the rates of speed and distances and see what happens).
This sounds like a great problem that I would like to use in my advanced
algebra or calculus class. Integrating the knowledge that students have
from the sciences into mathematics is always important, since mathematics
is the backbone of the sciences. The only thing that I would suggest for
these authors, or for any teacher that would like to use this problem, is
that instead of having the possible extensions be optional for the problem,
have them be part of the problem. The extensions are not the heart and
soul of this problem, but changing the rates of speed and distances in this
problem will likely help all students gain a better understanding of them
and the problem (and those related to it) as a whole.
Keywords: Issues, Standards, Curriculum
Ref: Tom19
Author(s):
Date:
Title: Mathematically Sane: Promoting the Rational Reform of
Mathematics Education
Journal or Publisher:
Volume, Issue, Pages: http://www.mathematicallysane.com
Reviewer: Tom
Date of Review: May 12, 2002
This web site is the residence of those who support math education reform. The web site is broken up into four major parts: analysis, evidence, resources, and links. The analysis section contains arguments explaining the whole debate over math education reform, allowing the uninformed reader to understand exactly what all the fuss is about. There are subsections here looking at what mathematicians think about the debate and explanations of the different facets and sides of it as well. The evidence section reports results from testing, surveys, and research which give insights as to whether or not math education reform is needed, and whether or not we have been taking the right approach. The last two, resources and links, are fairly intuitive. They both contain links to the research done and also links both supporting and arguing against math education reform so that any reader is given the chance to fully understand the debate and decide for themselves how they feel.
I really think this web site is laid out well. It is really easy to get
around into different subjects quickly. Also, the authors of the web site
are not in your face like some web sites supporting a side of a debate can
be. They are very open-minded about the whole ordeal - often pointing out
and encouraging their
readers to also look at
Keywords: Issues, Standards, Curriculum
This web site is the residence of those who do not support math
education reform. The layout is pretty simple. It consists mostly of links
to articles published by journalists (non-mathematicians) who are against
math education reform. The web site, unlike
This web site could be laid out a little better. A front page with a table
of contents would their readers get through the material more easily. They
should also provide some links and resources pointing out research and
opinions that support education reform. Much of this web site comes off as
saying that they are right and there is no way around it. Clearly both
sides have pros and cons to them, so the authors should be willing to admit
this to their readers.
Keywords: Communication, Management, Teaching Strategies
This web page is devoted to teachers and anyone that is interested in
communicating with others on CL (collaborative learning). There are several
different headings under which a reader can explore: general discussion of
CL, teachers, students, administration, assessment issues, CL materials and
resources. Under each of these headings are many sub-topics for discussion.
This looks to be a very good resource for teachers that are looking to use
collaborative or cooperative learning in their classrooms. There are a wide
range of questions that are asked here and people provide their email
addresses. If you have a similar question or a response of your own, you
can contact these people for assistance.
Ref: Tom20
Author(s):
Date:
Title: 2 plus 2: The Home of Mathematically Correct
Journal or Publisher:
Volume, Issue, Pages: http://www.mathematicallycorrect.com/
Reviewer: Tom
Date of Review: May 12, 2002
Ref: Tom21
Author(s):
Date:
Title: Collaborative Learning Forum
Journal or Publisher: Math Forum @ Drexel
Volume, Issue, Pages:
http://mathforum.org/discussions/co-learn/topics-gen.html
Reviewer: Tom
Date of Review: May 12, 2002