Keywords: Technology, Activities,
Ref: Darren1
Author(s): Fernandez, Maria L.
Date: 2001
Title: Graphical Transformations and Calculator Greeting Cards
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 94, No. 2, pp. 106-110
Reviewer: Darren
Date of Review: 2-12-01
This article describes how the use of graphing calculators can help students learn to better visualize graphs and connect an algebraic expression for a function to it’s graph. Maria Fernandez introduces the idea of creating greeting cards on graphing calculators. By creating pictures on calculators students learn linear transformations and how to manipulate graphs. She explains that this method is better than syntactic descriptions of linear transformations that encourages memorization of rules without understanding. She also explains how she noticed an improvement of students understanding of these concepts after assigning the greeting card project. Her students were better able to explain transformations in connection with algebraic representations.
I really think this article provides a great teaching tool. It not only describes the benefits of the method but it also provides instructions on how to make pictures on the calculator as well as custom design ideas. This is definitely an activity that I would try in my classroom.
Keywords: Teaching Strategies, Equity,
Ref: Darren2
Author(s): Bishop, Alan J.
Date: 2001
Title: What Values Do You Teach When You Teach Mathematics?
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Vol. 7, No. 6, pp. 346-349
Reviewer: Darren
Date of Review: 2-19-01
The article I read forces a teacher to consider carefully what they are saying to their class and what values they are displaying while they teach. It makes one think about how what they say affects students and what messages students are receiving underneath what is being said. It talks about the Values in Mathematics Project, which examines teachers’ awareness of what values they teach in their math instruction. What I found most interesting was that the article listed questions that a teacher should as him/herself when making decisions ranging from choosing textbooks to assigning homework to grouping students in class. This all comes from the idea that people put values on mathematical practices and symbols. It discusses the values that are associated with western mathematics and explains how different cultures have different values.
I think this is a very useful article because it causes you to think more deeply about what you are teaching and how it is affecting students. It is strong evidence in favor of reflective teaching because if you reflect on how students react to what you are teaching and how you teach it you will be better prepared to know the best way to teach. I really enjoyed the sample questions that teachers should think about when making choices and also incorporating the values into lessons. This kind of article really makes you think about every word and how it could affect people with value systems different from your own.
Keywords: Issues, Teaching Strategies,
Ref: Darren3
Author(s): Fiore, Greg
Date: 1999
Title: Math-Abused Students: Are We
Prepared to Teach Them?
Journal or Publisher: Mathematics
Teacher
Volume, Issue, Pages: Vol 92 No. 5 pp.
403-405
Reviewer: Darren
Date of Review: 2-22-01
In this article, Greg Fiore talks about math anxiety, its causes, and solutions for teachers. He begins by introducing an idea called math abuse and offers it as a cause for math anxiety. He uses two former students as examples of victims of math abuse. One of them had been forced to stand at the chalkboard for several hours until she solved a long division problem in third grade. The other had been slapped and ridiculed by her father for not understanding a math problem, also in third grade. The first step towards a solution for this teacher was assigning a math and me assignment where students would write about their experiences with math. Through this type of assignment a teacher can learn what type of baggage students bring to a math classroom. Fiore reinforced that math anxiety is not a result of the content but of teaching styles. He encouraged teaching so students understand to reduce anxiety. He also suggests assigning group work. Most importantly he stresses the importance of encouragement. Students respond to rewards.
Although this article provided nothing earth
shattering as far as suggestions for teaching, I
think it can be helpful to look at an article
like this on a regular basis to remind us to
think about what students bring to the classroom
and how we can help them learn more effectively.
At first I laughed at the idea of math abuse. It
seemed silly to me until the examples were
given. I can now see more clearly how a student
could be nervous about math or any subject for
that matter.
Keywords: Activities, Algebra,
Ref: Darren4
Author(s): Goetz, Albert
Date: 1999
Title: Smokey The Bear Takes
Algebra
Journal or Publisher: Mathematics
Teacher
Volume, Issue, Pages: Vol. 92, No. 7,
pp. 596-604
Reviewer: Darren
Date of Review: 2-28-01
Smokey the Bear Takes Algebra is an interesting activity that combines some environmental science with algebra. It uses the National Fire Danger Rating System (NFDRS) to teach students about forest fire danger as well as show them how to manipulate graphs. Students learn about factors influencing the possibility of a fire starting and spreading based on the conditions. Also there is an interesting story about the origin of Smokey the Bear. It comes with four worksheets for the activity. On the first worksheet students are asked to speculate on the conditions influencing the incidence of fires and why it is important to try to predict fires. The second worksheet gives an example of a rating system and asks students to graph it and explain what is shown in the graph. The third worksheet requires recent local data on fire danger indexes. It also provides a sample rating system in which students are to use the local data they found to explain a graph. The final worksheet asks students to explain relationships they found in the activity.
I think this is an interesting activity for
students as a real world example of some basic
algebra and graphing. It does a good job of
working through the lesson and asking for higher
level thinking along the way. Each worksheet
requires a little more thinking and the final one
asks good questions to help students sum it all
up. It could have more or less interest based on
what region of the country you are teaching but
if you've heard of Smokey, it can be fun.
Keywords: Problem Solving, Teaching
Strategies,
Ref: Darren5
Author(s): Kelly, Janet A.
Date: 1999
Title: Improving Problem Solving
Through Drawings
Journal or Publisher: Teaching
Children Mathematics
Volume, Issue, Pages: Vol. 6, No. 1,
pp. 48-50
Reviewer: Darren
Date of Review: 2-28-01
This article addresses the problems that many students have with problem solving by proposing a solution. Most students learn problem solving by the algorithm approach with emphasis on rules, and procedures. Most students can learn the procedures but some have trouble applying it to problems. A group of teachers was asked to meet for a couple hours once a week and discuss problem solving and the idea of using drawings to assist students. The teachers were asked to introduce this strategy to their students. At first the students had trouble visualizing the problems so the teacher would give them problems with a picture included. Gradually, the students moved to drawing their own pictures to creating their own problems with pictures. Eventually students were given problems with no visual cues and asked to draw a picture and solve the problem. All of the teachers that participated accepted the idea of using visualization and plan to use it regularly in their classrooms. There was also a general increase in the enthusiasm for teaching mathematics. Students also seemed to like the strategy.
I have always been a visual learner so this is
a strategy that I use often. When I tutor I tell
my tutee to draw a picture before they do any
calculations if possible. I really think it
helps to see what the question is asking and how
to go about solving the problem. I will
definitely use this strategy in my classroom.
Keywords: Discrete, Geometry, Activities
Ref: Darren6
Author(s): Simmt, Elaine and
Davis, Brent
Date: 1998
Title: Fractal Cards: A Space for
Exploration in Geometry and Discrete Mathematics
Journal or Publisher: Mathematics
Teacher
Volume, Issue, Pages: Vol 91, No. 2,
pp. 102-108
Reviewer: Darren
Date of Review: 4-2-01
This article demonstrates the process of creating a fractal-card and discusses the connections to mathematics. The authors were drawn to this activity because of the aesthetic quality of the cards as well as the powerful math that is involved. The visual image of the cards as well as the process by which it is created provide an opportunity for students to develop a stronger understanding of ideas like iteration and self-similarity. On of the best parts about this activity is that the mathematical concepts come out through observation as well as in-depth inquiry. One of the connections that comes out of this activity is observing the number of cells at each stage of cutting. Also, the surface area of the cells provides an interesting pattern when looked at stage-by-stage. At a higher level, students can explore connections to discrete math. They are asked to generalize about what happens at the nth cut or the nth stage and can use limits and series to describe what is happening.
I think this is a good activity but it will
be most effective for students at higher levels
of math. It can be difficult to see the
connections that are made to some higher level
concepts. There is definitely something
attractive and intriguing about the appearance of
these cards. I can see that motivated students
would appreciate this activity and want to
explore further.
Keywords: Connections, Discrete,
Ref: Darren7
Author(s): Anderson, Corey
Date: 1998
Title: Fibonacci and Pascal
Together Again: Pattern Exploration in the
Fibonacci Sequence
Journal or Publisher: Mathematics
Teacher
Volume, Issue, Pages: Vol. 91, No. 3,
pp. 250-253
Reviewer: Darren
Date of Review: 4-2-01
The author uses exploration of the Fibonacci sequence to make a connection to Pascal's Triangle. The patterns are not obvious to people who may not be experienced with the Fibonacci sequence and Pascal's Triangle. The math involved is not too difficult for high school students but they must have a desire to explore in-depth math. The author begins with noticing that every 5th Fibonacci number is divisible by 5. He develops his observation into noticing that every nth Fibonacci number is divisible by Fn. By looking at Fn/n, he develops a pattern that generates a sum that involves diagonals of Pascal's Triangle. He then goes on to make a connection to the Lucas Numbers. He uses the base of the power used to expand Fn/n with the numbers from Pascal's Triangle. Finally he describes how, in an advanced high school math course, students can generate a formal generalization involving the Lucas Numbers.
This could be a very interesting lesson for
an advanced math class that is exploring
sequences. The first few connections that the
author makes are clear once he explains them but
they would not be obvious and most high school
students would not be expected to come up with
them without guidance. Some of the later
connections he makes are pretty advanced and
require some experience with discrete
mathematics. The connections he makes are very
interesting to me given the fact that I am
studying discrete math and have experience with
the involved topics.
Keywords: Connections, ,
Ref: Darren8
Author(s): Cuoco, Al
Date: 1998
Title: What I Wish I Had Known
About Mathematics When I Started Teaching
Journal or Publisher: Mathematics
Teacher
Volume, Issue, Pages: Vol. 91, No. 5,
pp. 372-374
Reviewer: Darren
Date of Review: 4-2-01
This article includes three wishes that the author has about what he should have known before he started teaching. His first wish is that undergraduate courses would connect with secondary curriculum. In his experience, he saw few connections from his college courses to the courses he took and taught at the high school level. He states two ways to help prospective teachers make these connections. The first is to use problems from secondary curricula as springboards to advanced mathematics. This helps teachers ot make connections to the material that they will be teaching by taking problems that they know and working in-depth on them. The other way is to use "metaproblems" to develop or apply undergraduate mathematics questions like, "how do I generate integer-sided triangles with a 60 degree angle?" help teachers develop a stronger understanding of the concepts they will teach. His second wish is that prospective teachers will develop the basic results that underlie high school math. Instead o
This is a very interesting article that
prospective teachers should definitely look at.
It makes me think about how I am connecting what
I am learning to my previous learning. The
author's wishes do not only apply to prospective
teachers. These are valuable connections that
all math majors should make.
Keywords: Arithmetic, ,
Ref: Darren9
Author(s): Crossfield, Don
Date: 1997
Title: (Naturally) Numbers Are Fun
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 90, No. 2, pp. 92-95
Reviewer: Darren
Date of Review: 04-11-01
The author of this article used groups of numbers to teach his students number sense. He didn't exactly tell them that they were learning number sense. He defined four groups of numbers; pink, blue, red, and green numbers. The pink numbers are better known as prime numbers, the blues are powers of 2, the greens are perfect cubes, and the red numbers are perfect squares. He then asked his students questions beginning with "what is the next number in the sequence?" and moving up to finding sums and products of numbers and relationships between the groups. The questions become more and more in-depth as the students learn more about the groups of numbers. Eventually students were asked to approximate the values of irrational numbers from the lists of numbers. The author noticed a considerable improvement in his students in number sense.
I think this teacher has a very interesting approach to number sense and I believe that it is very effective. What was not clear to me in the article was whether or not he moved them from referring to the groups as a color toward referring to them based on their relationships. I think students need to know what makes numbers prime rather than just knowing that they come from the pink group. I would like to think that the author has covered this problem. This would seem to me to be an activity for middle school students. It would help them develop a strong grasp of relationships between numbers and help them take that number sense further at the high school level.
Keywords: History, Geometry, Algebra
Ref: Darren10
Author(s): Allaire, Patricia R., and Bradley, Robert E.
Date: 2001
Title: Geometric Approaches to Quadratic Equations from Other
Times and Places
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 94, No 4, pp308-313, 319
Reviewer: Darren
Date of Review: 5-23-01
Before symbolic manipulation was used to solve quadratic problems, mathematicians had to use other methods. Often, geometric approaches were used. This article provides some excellent examples of geometric approaches to quadratic problems from history. First, the branch of mathematics called Geometrical Algebra is introduced. In this branch, quantities (known or unknown) are represented by objects, usually line segments. Products are represented by areas of rectangles. This provides a nice introduction to the multiplication of binomials and to the reason that FOIL works. Since algebra only deals with positive quantities, we have to be careful about how quadratic problems are set up. We have an equation of the form x^2+bx+c=0 and there are 5 cases in which b and c are always positive. (1)x^2=bx (2)x^2=c (3)x^2=bx+c (4)x^2+c=bx (5)x^2+bx=c. The article then goes on to describe geometric methods for solving each of these cases. Some interesting methods are presented including methods by Pythagoreus.
All in all I think that this article could provide to be pretty useful in
the classroom. Unfortunately, time is limited and it is not likely that there
will be time to explore all of these methods with students. I think many of
these methods can provide useful to students to visualize quadratic problems.
Also, some interesting history of math is mixed in that students might find
interesting. I could see myself using this as a group presentation activity.
The students could explore the findings of these mathematicians from the past
and then present their findings to the class.
Keywords: Curriculum, Standards, Equity
Ref: Darren11
Author(s): Meiring, Steven P., et al.
Date: 1992
Title: Curriculum and Evaluation Standards for School
Mathematics: A Core Curriculum
Journal or Publisher: The National Council of Teacher of
Mathematics
Volume, Issue, Pages: Addenda Series for grades 9-12
Reviewer: Darren
Date of Review: 5-23-01
This book is written in an effort to help teachers with instructional ideas that will support the standards and the implementation of a core curriculum. It provides teachers with tips on how to implement a core curriculum as well as reasons for why it is important. It talks about a need to increase the amount of "Newtonian" mathematics because of the increase in demand for real-world applications for math. Also, it talks about how learning is changing in mathematics and how learning can be more involved with doing. The book gives an example of a cooperative learning activity with dart throwing, a couple of practical application to matrices, as well as crossover curriculum models. A crossover curriculum consists of two classes with the same syllabus but are intended for students with different learning needs. An example might be college bound students and students that are not necessarily college bound.
I think this book is a very useful tool for teachers. It is a great effort
by the NCTM to assist teachers in implementing the standards. I especially
like the crossover curriculum section. It provides some great examples of how
the same material can be taught to different groups of students with different
learning needs. I also like the dedication of the NCTM to making math
meaningful for students. It shows in this publication.
Keywords: Assessment, ,
Ref: Darren12
Author(s): Long, Vena M.
Date: 2000
Title: Anatomy of an Assessment
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 93, No 4, pp.346-348
Reviewer: Darren
Date of Review: 4-23-01
This article addresses the importance of assessment and making sure it addresses meaningful mathematics. "If the mathematics is not significant, correct, and relevant to the required curriculum, then the assessment item is worthless." The math should be worthwhile and realistic. The assessment should allow teachers to infer about students' mathematical knowledge, understanding, and thinking processes. This is why it is important that the assessment is based on the mathematical content that the teacher is trying to verify. The article emphasizes a strong base of relevant mathematics and rich context for the math. There is a nice example of an assessment question with sample answers. The article evaluates the problem in accordance with the author's ideas about what assessment should be.
I have found very few articles on assessment in my recent search. I am glad
I found this one. I like it because it emphasizes the importance of teaching
math in a meaningful context. It is also helpful for looking at how students
reason and how we can get the most out of assessment.
Keywords: Connections, ,
Ref: Darren12
Author(s): Craine, Timothy V., Rubenstein, Rheta N.
Date: 2000
Title: Traveling Toward Proof
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 93, No 4, pp.289-291
Reviewer: Darren
Date of Review: 4-23-01
This article takes an flight schedule for an imaginary airline (Aristotlo Air) and applies it to proof writing. Students must start in one city and take connecting flights to other cities. This is analogous to proofs because you start with the given (starting city) and you must take steps that lead you to the desired conclusion (destination). This connection leads nicely to flow proofs and two column proofs. Some important things to remember: You must start with the given (starting city) and move to what is being proved (destination), A statement (city) cannot be skipped, and a reason (flight must join any to statements (cities) you want to connect. This analogy also helps to show that there may be more than one way to prove something. If and only if statements can be seen as two way flights as opposed to a one way implication.
I really like this analogy for writing proofs. I am always looking for
connections between math and real life. There are so many ways to connect the
airline example to proofs. I think it would really help students develop good
proof writing habits.
Keywords: Connections, Curriculum,
Ref: Darren13
Author(s): Froelich, Gary W., et al.
Date: 1991
Title: Curriculm and Evaluation Standards for School Mathematics:
Connecting Mathematics
Journal or Publisher: National Council of Teachers of
Mathematics
Volume, Issue, Pages: Addenda Series grades 9-12
Reviewer: Darren
Date of Review: 4-23-01
This book provides teachers with examples of how they can make connections in their classrooms. It provides a couple of nice real-world connection with functions involving plumbers and pipe cleaners. There are also some nice connections with matrices. Data analysis, reasoning, and problem solving are also areas where they make some strong real-world connections. All of the activities are practical classroom activities that require little or no additional materials.
I like the fact that the NCTM has provided specific examples of connections
in mathematics. It shows strong commitment to improving math education when
they go beyond establishing the standards and providing ways to implement
them. The activities in this book seem very worthwhile and they are definitely
some activities that I could see myself using.
Keywords: History, Curriculum, Research
Ref: Darren14
Author(s): Marshall, Gerald L.; Rich, Beverly S.
Date: 2000
Title: The Role of History in the Mathematics Classroom
Journal or Publisher:
Volume, Issue, Pages: Vol. 93,
Reviewer: Darren
Date of Review:
Keywords: History, Curriculum, Research
Ref: Darren14
Author(s): Marshall, Gerald L.; Rich, Beverly S.
Date: 2000
Title: The Role of History in the Mathematics Classroom
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 8, pp.704-706
Reviewer: Darren
Date of Review: 5-29-01
This article emphasizes the increasing interest in using history in the mathematics class. There is little research as to the actual impact on students but the studies that this article sited showed a strong opinion toward using history to motivate students and add meaning to mathematical content. The article talks about the NCTM's standards and the call for teaching for meaning and understanding in the mathematics class. "For a complete understanding, mathematics must be considered in the context of time and place in which it developed."
This article provides some strong opinions and arguments in favor of using history in math. I like the idea of making connections to where the math came from. I really do think that students will find more meaning in the material if they learn about the origins of what they are learning.
Keywords: Problem Solving, Technology,
Ref: Darren15
Author(s): Santos-Trigo, Manuel; Diaz-Barriga, Eugenio
Date: 2000
Title: Posing Questions from Proposed Problems: Using Technology to Enhance Mathematical Problem Solving
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol.93, No.7, pp.578-580
Reviewer: Darren
Date of Review: 5-29-01
This article takes a problem posed in one of the calendars from The Mathematics Teacher and gives a different approach to solving it. It involves two circles being cut from a 9cm by 12cm piece of paper. The question is "What is the maximum possible radius of these circles?" The authors use Cabri-Geometry II to construct the problem and explore the relationship between the position of the circles and the radius of the circles.
There is some pretty strong technology used in the solution of this problem. Cabri can really make it easier to visualize a problem and move around the circles to see how the radius changes. This is something I would like to explore further and gain more experience with. There is some really good math here.
Keywords: Connections, Probability,
Ref: Darren16
Author(s): Pagni, David L.
Date: 2000
Title: Mathematical Connection-A Baseball Opportunity
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol.93, No.7, pp.552-553
Reviewer: Darren
Date of Review: 5-29-01
The author looks at playoff possibilities prior to the 1998 National League Championship Series in baseball. Three teams are competing for the final playoff spot with one game remaining for each team with none of the three playing each other. The author uses a tree diagram to show the probabilities of each team making the playoffs and what needs to happen for them to get there.
This is a nice connection to something that I have an interest in and has some pretty interesting math in it. It can definitely help students practice tree diagrams in probability as well as calculate probabilities.
Keywords: Calculus, ,
Ref: Darren17
Author(s): Lipp, Alan
Date: 2001
Title: Visualizing the Complex Roots of Quadratic and Cubic Equations
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 94, No. 5, pp. 410-413
Reviewer: Darren
Date of Review: 5-29-01
This article uses multidimensional graphing to show how complex roots can be seen. Students are often asked to believe that all quadratic equations have 2 roots. This article shows that this is true. By substituting a+bi for x in a quadratic equation you get 3 variables that can be graphed in 3-dimensions. Clear, colored graphs are included in the article to show the reader where the complex roots lie.
I never really thought about what complex roots would look like. I just accepted that they existed. The visual connection in this article is very powerful. This can be very useful for students to see what they are learning rather that just accepting what they are told about complex roots.
Keywords: Connections, Probability, Statistics
Ref: Darren18
Author(s): Masse, Leonard
Date: 2001
Title: The Possibility of Perfection
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 6, No.9, pp. 500-506
Reviewer: Darren
Date of Review: 5-29-01
The author of article takes the recent occurrence of perfect games in major league baseball and creates an exploration for students to calculate the probability of pitching a perfect game. Students are asked to hypothesize and make estimates and check their estimates against statistics from the history of baseball. They use estimates for batting averages and errors to calculate the probability of a perfect game and then change the estimates to see how the probability changes.
With my interest in baseball I think this is a really cool activity for students to explore. It combines so many important aspects of data gathering and probability and even includes a bit of statistics. It seems to me that this would be a very valuable and meaningful activity for students.
Keywords: Connections, Assessment, Research
Ref: Darren19
Author(s): Knuth, Eric J.
Date: 2000
Title: Understanding Connections Between Equations and Graphs
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 1, pp. 48-53
Reviewer: Darren
Date of Review: 5-29-01
The author discusses a study that was performed to see if students understand the connections between equations and graphs. Students were given one of 5 questions involving an equation (most linear) and graphs. The article discusses the answers of students and the discouraging evidence that shows that very few students are seeing the connections between equations and their graphs. Most knew how to solve the problem but when asked to look at it in a different way they were unable to see the graph as a way to see a solution.
This was a very interesting study. It shows the importance of making clear connections between mathematical concepts. It also provides a strong argument for the NCTM standards by showing the importance of understanding representations.
Keywords: Connections, Algebra,
Ref: Darren20
Author(s): Horton, Bob
Date: 2000
Title: Making Connections Between Sequences and Mathematical Models
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: Vol.93, No.5, pp.434-436
Reviewer: Darren
Date of Review: 5-29-01
This article uses linear and exponential models and spreadsheets to help students make connections between models and sequences. The activity includes 3 phases. The first phase involves arithmetic sequences and linear functions. The second involves geometric sequences and exponential functions. The final phase involves making sure that students have made connections. In the activity students are asked to write recursive and explicit formulas for the data involved as well as use spreadsheets to graph the data.
This article provides a nice visualization of sequences and graphs and their relationships. What I really like is the final phase where the teacher makes sure that the students have made connections. I also like that it is an exploration for students rather than the teacher showing the connections. Students can develop better understanding this way.