Keywords: Problem Solving, Games,
Ref: Kueffer1
Author(s): Tanton, James
Date: 1999
Title: A Half-Dozen Mathematical Activities to Try with
Friends
Journal or Publisher: Math Horizons
Volume, Issue, Pages: September 1999
Reviewer: Kueffer
Date of Review: 3-16-00
This article is exactly about what the title says. A few puzzles are
create with real world scenarios, and matematical problem solving skills
can be use to help solve them. Problems such as tying to figure out the
length of a bicycle when only given a picte of the front and rear wheel
tire tracks, or the smallest number of cuts it would take to cut a 4 x 4 x
4 block of tofu or cheese into 64 smaller cubes? The ideas of the article
came from a college math club in which students think up interesting
problems, and then in groups, they see who can solve them. relating this
to mathematics teaching, problems such as these are things in which you
can use in the classroom either if they directly relate to what you are
teaching, or even just extra credit problems. Many times, random puzzles
like these are just fun things for students to do, and being based around
math, they encourage problem solving and thinking skills.
Keywords: Research, ,
Ref: Kueffer2
Author(s): NCTM editor
Date: 1999
Title: U.S. Students' Math Scores Vary Widely
Journal or Publisher: National Council of Teachers of
Mathematics
Volume, Issue, Pages: Volume 35, Issue 7, pp.1 & 5
Reviewer: Kueffer
Date of Review: 3-22-00
In an international report provided by the Organization for Economic Cooperation and Developement, the U.S. math students were found to have a greater variance of math scores than many other countries. It was given that this variance may be caused by the many different regions and areas U.S. students attend class in, or the fact that the U.S. doesn't track students into different schools according to their abilities like other countries do. The report stated that a large part of the variance in the U.S. is more likely due to the influence of home life.
Another factored they considered in determining the roots of the variance problem, was by looking at teachers salaries. Teachers in the U.S., even though they make more money relative to teachers in other countries, when U.S. salaries were compared to the Gross Domestic Product per capita, which incorporates standard of living, the U.S. ranked behind half of the countries surveyed.
The article stasted that the theories given are not ones to panic about,
but rather to inform. Being conscious of how we can become a better
education system as a whole, is always important. The theories given
definitely have topics for us to look into as math teachers, and it was
good to read that issues such as these are being discussed.
Keywords: Teaching Strategies, ,
Ref: Kueffer3
Author(s): Lappan, Glenda
Date: 1999
Title: "Mathematics for All" Must Include High-Ability and
Highly Motivated Students
Journal or Publisher: National Council of Teachers of
Mathematics News Bulletin
Volume, Issue, Pages: Volume 35, Issue 8, p.3
Reviewer: Kueffer
Date of Review: 3-22-00
"We speak often about providing rich opportunities for disadvantaged students. But among the students we have in our mathematics programs are some that have either high abilities or high interest, or both." This is a quote Glenda Lappan gives in her article. It is the basis to her argument. She feels that many times we let students who have hihg math intelligence "breeze" through our programs without ever being challenged. She feels we need to set up a system that encourages all ability levels to work hard.
She gives examples like setting up math clubs, giving students extra
resources to rea about math, offer additional courses, or provide seminars
that student can attend as examples to offer higher ability students. She
believes that many high school students have the potential to learn about
very challenging math topics if they are offered them. One magazine she
mentioned specifically, is called Quantum. It is a science and
mathematics orientated magazine that provides many different insights
towards math problems. She feels it is a great resource for students to
be exposed to.
Keywords: Probability, ,
Ref: Kueffer4
Author(s): Schilling, Mark
Date: 1999
Title: Aliens, Asteroids, and Astronomical Odds
Journal or Publisher: Math Horizons
Volume, Issue, Pages: November 1999, pp.26-28
Reviewer: Kueffer
Date of Review: 3-22-00
This article introduces a unique concept related to probability. In the article, the author ssupports an idea which shows that it is equally likely for a person to be killed by an extraterrestrial object craching into the earth, such as a meteroid, and being killed in a plane crash. (The only qualification is that the person killed in the airplane is a person who flies at least 6 times every 6 months) The author also shows problems that incoporate probability in gene testing, the probability of life on another planet, and the lottery.
The article is more for pleasure than it is for mathematical reference,
but it did provide some insight. It referred a little to the ignorance of
non-mathematicians but also to people who like to read about word
problems. By using topics such as asteroids and plane crashes, it is a
topic which many people think about, and some may be frightened by.
Because of this, the article stimulates more minds than just mathematical
ones.
Keywords: Statistics, ,
Ref: Kueffer5
Author(s): Nally, Michael T., Sommers, Paul M.
Date: 1998
Title: Striking Back: The Baseball Fan Boycott of 1995
Journal or Publisher: Journal of Recreational Mathematics
Volume, Issue, Pages: Volume 29, Number 3, pp.184-188
Reviewer: Kueffer
Date of Review: 3-22-00
This article takes a unique look at the use of statistics. Through statistics, the researcher looked at the relationship between the 1994 strike in Major League Baseball and the attendance to baseball games in 1995. The authors looked at how significantly the strike caused attendance to change the next year.
Using t-tests on the attendance data, the reasearchers came up with a claim that most stadiums attendance was significantly different from 1994 to 1995. Different in such that 1995 attendance was much lower. By compiling data from all of the teams, the statisticians were able to provide results on what teams dropped in attendance the most and which were least effected. Also from their tests, the researchers concluded that the strike was a significant event that impacted by fewer fans attending games in 1995.
If you have taken a statistics course, this article is very easy to
follow, and fairly interesting to see the apllication of statistics to the
real world. It shows a unique way that statistics is used, and also
shows how statistic compiling is sometimes a good way to prove a
generalization or assumption of an event in the real world.
Keywords: Probability, ,
Ref: Kueffer6
Author(s): Bay, Jennifer, Reys, Robert, Simms, Ken, Taylor,
P. Mark
Date: 2000
Title: Bingo Games: Turning Student Intuitions into
INvestigations in Probability and Number Sense
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93 No. 3, March 00, pp.200-206
Reviewer: Kueffer
Date of Review: 4-3-00
This is a very interesting article on probability. It compares using probability to show the odds of winning bingo such as, theoretically, what game will be won in fewer numbers called, a straight line with the free space, or a straight line with out. Or the prbability of getting four corners versus postage stamp.
The atricle also displays further how probability is understood. Probabilities are shown as fractions or decimals, and the article highlights a few things to be careful about. Such as comparing 0.000000232 to 0.000004114. Is this easier to distiguish between scientific notation of comparing 2.32 x 10^(-7) to 4.114 x 10^(-6). The article goes further in showing how this which were derived from fractions actually reduces to the ratio 71:4.
Viewing this article from a teacher's standpoint, the emphasis seemed to
be to show all of the variation of answers a mathematical problem can be
viewed from. Using probability techniques to get answers, and then
comparing the answers. The article also goes further into probablity
theory of combinations and permutations while continuing the trend of
using all posible methods of solving the problem.
Keywords: Teaching Strategies, ,
Ref: Kueffer7
Author(s): Misener, Jeff
Date: 2000
Title: Blake's Slope-Intercept Surprise
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93 No. 3, March 2000, pp.242-243
Reviewer: Kueffer
Date of Review: 4-3-00
This article was written by a teacher and a new teaching method he uses in his class. The method has to deal with finding point-intercept forms of a line. The teacher taught how to find the equation of a line in what he thought was a traditional method in mathematics. After he taught his lesson, one of his students approached him with a new method that he had discovered. The method worked, and the following year, the teacher used the students method to teach about point-intercept form. From this lesson, he received great feedback from the students.
The teacher had discovered a new way to teach a lesson, and it was
directly from the mind of a student. Because the student was able to set
up a system that was easy for himself to understand as an eight grader,
other eighth grade students were able to relate to the method and
understand it better than the "traditional" way of learning it.
I feel every teacher is going to run into similar situations like this
teacher did. A good teacher is going to actualy be able to take the
feedback from students and find out what teaching methods really work.
Year after year will be a continual process of adapting new methods that
work better and better. Involving the students ideas adds to the
curriculum of the class.
Keywords: Problem Solving, ,
Ref: Kueffer8
Author(s): Devlin, Keith
Date: 1998
Title: Move Over Fermat, Now It's Time for Beal's Problem
Journal or Publisher: Math Horizons
Volume, Issue, Pages: Februrary
Reviewer: Kueffer
Date of Review: 4-9-00
This article talks about a topic that can be included in the motivation to get people to understand math. Most of the time, people want math to relate to things or be applicable to real world situations. There is another side to math different than this. Some people pursue math just for the challenge and desire to prove things.
In his article, "Move over Fermat, Now It's time for Beal's Problem," Keith Devlin writes about the "good" that can come out of liking math for the sake of problem solving. He explains about a problem that a person conjectured that relates to Fermat's last theorem, and the person who created the problem is offereing money to the one who can prove the statement.
This event is a good example of how math can be taken as a fun topic.
Kevlin shares that doing math isn't just for application, but sometimes
just for the excitement of solving a problem.
Keywords: Research, ,
Ref: Kueffer9
Author(s): Hausperger, Deanna, Kennedy, Steve
Date: 1998
Title: Is an REU for You?
Journal or Publisher: Math Horizons
Volume, Issue, Pages: February 1998
Reviewer: Kueffer
Date of Review: 4-9-00
This article highlighted the benefits to extended learning in math. The extended learning is an REU (Research Experiences for Undergraduates). The article was composed of four students summaries of their experience in an REU program over summers. The problems they were told to solve, the stipend they were granted for their work, the professor to student ratio, and outside academic activities they participated in were all included in the summaries.
Overall, the students gave their first insight on what it is like to be a
research mathematician. Most replied that the experience was unique and
man times entertaining. I feel it was a good article just to show the
diversity of mathematics and what a mathematics education can lead
to.
Keywords: Teaching Strategies, Curriculum,
Ref: Kueffer10
Author(s): Chancellor, Dinah, Childs, Kimberly M.,
Schielack, Jane F.
Date: February 2000
Title: Designing Questions to Encourage Children's
Mathematical Thinking
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Volume 6, Number 6, pp.398-402
Reviewer: Kueffer
Date of Review: 4-10-00
This article emphasizes a big part of the "new curriculum" for mathematics. It talks about how a teacher's questioning in a math class is what can really make a difference. This article breaks questioning down into four types of questions a teacher can ask. Questions about modeling (many times though the use of manipulaives), logical analysis, inference, optimization, and abstraction.
It is through teacher guidance that students encounter these ways fo mathematical thinking. It is a teacher's job to create questions in which the student must think mathematically to arrive at a solution to the problem.
This article gives a few examples of how these types of questions can be
posed.
Keywords: Teaching Strategies, Connections,
Ref: Kueffer11
Author(s): Chapman, Susan
Date: February 2000
Title: The M.O.O.K. Book: Students Author a Book about
Mathematics
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Volume 6, Number 6, pp.388-390
Reviewer: Kueffer
Date of Review: 4-11-00
A teacher shares a teaching strategy in this article. The teacher is upset about the fact that there is such a great amount of encouragement for young students to read books, but so little encouragement to learn math. To combine these two issues, the teacher creates a book she calls M.O.O.K. for Math Offers Outstanding Knowlege. This book is authored only by the students. Through the year as the students encounter math problems, riddles, puzzles, concepts, etc. outside the classroom they are encouraged to write their problem in the M.O.O.K. book so other students can try the problem.
The teacher has explained how this strategy has made her students more
excited about learning mathematics. It allows the students to think
mathematically and relate their discoveries to a unified source. It is
just another idea of how we can get students to start thinking
mathematically at a young age and enjoy it. Just like we encourage kids
to read, we should encourage them to problem solve.
Keywords: Activities, Teaching Strategies,
Ref: Kueffer12
Author(s): Gaglione, Jeffrey T.
Date: April 2000
Title: Relay Review
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 4, pp.282-283
Reviewer: Kueffer
Date of Review: 4-25-00
In the article "Relay Review," a teacher shares a teaching strategy. It has to do with reviewing a unit or a specific amount of material covered in a certain time period. The strategy this teacher uses is to create a competition. At the end of a unit, the teacher divides teams up into groups of 3-5 students. Each group is then given 10 note cards with different math problems on each. The way the competition works is that each team has one person write and try to solve the problem on the board with the team members helping. When the answer is found, the chalk is passed to the next person in the group. The first team done with all 10 cards wins.
The teacher felt this strategy is a good idea, because it touches many aspects. It reviews ideas covered recently in class, it teaches students to work together, all students are engaged at one time, and the friendly competition makes it enjoyable.
This strategy does provide an environment that some students will flourish
in. I feel it is a goo method to at least try out in a classroom, and
potentially is could be very successful with the right group of students.
Keywords: Geometry, ,
Ref: Kueffer13
Author(s): Nissen, Phillip
Date: April 2000
Title: A Geometry Solution from Multiple Perspectives
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 4, pp.324-327
Reviewer: Kueffer
Date of Review: 4-25-00
This article takes a geometry problem and solves it with 4 different methods; through a synthetic approach, a coordinate approach, a vector approach, and a transformation approach. A few of the solutions the author gave, were approaches that I had never seen before, and many of them were a little bit challenging to follow.
The authors point was to show the inportance of viewing and accepting math in a variety of forms. He emphasized that students should be shown that geometry problems can be solved in many different ways and still arrive at the same answer. Solutions may vary in difficulty, but each method is correct in its own sense.
This article reminds teachers of keeping an open mind on students work.
You may be teaching one method, but that does not mean another method is
wrong, just different. The important part of each solution is that the
students are thinking mathematically and coming up with solutions.
Keywords: Geometry, Activities,
Ref: Kueffer14
Author(s): Urso, Josephine; Welchman, Rosamond
Date: April 2000
Title: Midpoint Shapes
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Volume 6, Number 8, pp.506-509
Reviewer: Kueffer
Date of Review: 5-4-00
The authors of this article describe an activity for students in the elementary grades. The activity deals with basic shapes: squares, rectangles, and triangles. It is a lesson given in the form of a lab where students are given specific geometric shapes cut out on paper, and they discover the properties (length, width, area) through different folds in the paper and connecting midpoints of the sides of the geometric shape. Students are guided to conclusions such as, "...the midpoint shape for squares (shape constructed by connecting the midpoints of the sides of the square) is a square and that the area of the midpoit shape of a square is half the area of the midpoint shape."
The entire lesson is directed through the students comments. The students are given a task, and then told to report their findings. The teacher does not tell them if they are right or wrong right their on the spot, but rather he/she writes the student's comments on the board or has the student write their method on the board. Then the teacher has the rest of the class analyze the students work.
The authors of this article gives this lesson as an example of a mathemtical investigation. They want the reader to understand what a math problem consists of; a multidimesional content, an open ended question with different methods to solve, an exploration to find a solution, centered on a theme... Everything relates to having the student discover and having each other analyze their peers work. I could not find anywhere in the leson where the teacher taught, but rather highlighted the facts that the students discovered.
I like the approach these authors took. It is similar to most of the new
curriculums where the students are responsible for discovering the math
concepts. At the elementary level, I see this lesson as a good example of
how students can learn math, and it has the potential to cover many
different math concpets, just depending on the classroom environment.
Keywords: Games, Probability, Statistics
Ref: Kueffer15
Author(s): Watters, Deborah
Date: May 2000
Title: Basketball Math
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Vol. 6, No. 9, pp.556-559
Reviewer: Kueffer
Date of Review: 5-10-00
This article makes good use of the fact that there are many different ways to look at numbers. This particular article dealt with teaching kids math though analyzing numbers related to basketball. Ideas such as free throw, 2-point shots, 3-pointer, a player's field-goal percentage, a players freethrow percentage, etc. Students can take a concept like shooting percentage and free throw percentage and see trends such as free throw percentages on average are higher than field-goal percentages. The students can then analyze why.
Or the students can take a player's shooting percentage, say 67.5%, and answer questions like, if that player took ten shots, how many is he/she expected to make? Reported in the article was the concept that elementary kids were able to pick up on these trends.
The article also reported on the results of the students taking data by shooting their own baskets. They shot ten free-throws, ten 2-point shots, and ten 3-point shots and recorded their data. From this data, the students were able to analyze their own shooting percentages like they had done with data that was just given to them.
By getting the students involved in an activity, the interest level of the assignment was much higher than usual. The discussion of the students calculations also had more input because the students were more interested in sharing their analyzing methods because they enjoyed the basketball activity.
And the fun didn't stop here, but more math games were created from the topic of a basketball court; finding the area of the court, how far the freethrow lines are from the baselines...
All of the activities were math orientated, but also connected to real life. One could even extend the idea of promoting college thinking in young students by analyzing statistics from college basketball.
This article is only another method of showing the potential of what math can be!
Keywords: Geometry, ,
Ref: Kueffer16
Author(s): Fox, Thomas
Date: May 2000
Title: Implications of Research on Children's Understanding
of Geometry
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Vol. 6, No. 9, pp.572-576
Reviewer: Kueffer
Date of Review: 5-10-00
This article deals with research on children's understanding of geometry. During the research children were asked questions like, given these figures what are some similarities, or students were given different cut out shapes and told to organize them in groups based on their shape...answers such as both have four sides, or both look like triangles...
The research was done to see if students look at shapes as resemblance to other shapes, or look at shapes by their different characteristics. ie. it looks like a triangle, or it has 3 sides and 3 angles.
Results were varied among different students, but it was suggested that
this is a good task to see the ways students are thinking. And it also
will tell a teacher which students may need to be taught how to look at
geometry from a different perspective also.
Keywords: Standards, ,
Ref: Kueffer17
Author(s): Lappan, Glenda
Date: April 2000
Title: We have our Principles and Standards: What Now?
Journal or Publisher: NCTM News Bulletin
Volume, Issue, Pages: Vol. 36, Issue 9, p.3
Reviewer: Kueffer
Date of Review: 5-11-00
In her "President's Message," in the April 2000 issue if the NCTM bulletin, Glenda Lappan raises a few good issues dealing with the new Principles and Standards. Her first statement that caught my attention was, "These Standards are visionary. They are a statement of where we want to be." This is true. We should always be striving for a new level of achievement. Society is still evolving and so should schools curriculum. A good example would be that technology is such an emphasis in the new Standards. Technology progresses in the world, and so our schools should adapt and teach students with the use of technological tools.
"Principles and Standards offers you (the school) a way to make your school's efforts to improve mathematics teaching and learning more focus and coherent." This is the main idea see the Standards represent. Math is very diverse subject, and it is important that we have people who research for three years to produce a booklet in which every math teacher can reference to to find the important aspects of math in todays world.
The final quote I want to comment on is, "Our programs must meet the
general
population's growing need for quantitative literacy and other skills." We
don't
need to set up a system where many more people are going to be on tract to
a Ph.D.
in mathematics, but we need to recognize the fact that the critical
thinking
skills involved in math are very necessary to survive in today's society.
Keywords: Algebra, ,
Ref: Kueffer18
Author(s): Crawford, Ann; Scott, William E.
Date: February 2000
Title: Making Sense of Slope
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 2, pp.114-118
Reviewer: Kueffer
Date of Review: 5-11-00
"Making sense of slope." This article is dedicated to givig ideas on how to show students that the slope of a line is actually a measure of rate of change. The article gives examples such as produce's cost per pound of food and earnings per hours of work. The article also gives an example of how t incorporate a y-intercept into the equations. The example they gave was the amount a band charges to play music; a $100 base fee, with $2.50 per person that attends the concert. The $100 term creates the y-intercpet in the algebraic equation.
The article continues suggesting different sources where a teacher can find data where a linear function can be related to the data; Internet, almanacs, newspapers, magazines, etc. There are statistics many places that can be analyzed, and with technology, students can apply concepts such as linear regression to creat lines and understand slope.
There are a few good examples in this article, but I feel the main thing I
got out of reading it, is the fact that we need to teach students what
slope actually means. It is not just the term associated with
(y2-y1)/(x2-x1).
Keywords: Curriculum, ,
Ref: Kueffer19
Author(s): Williams, Nancy; Wynne, Brian
Date: February 2000
Title: Journal Writing in the Mathematics Classroom: A
Beginner's Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No. 2, pp.132-135
Reviewer: Kueffer
Date of Review: 5-11-00
This article dealt with two teachers introduction of mathematiclal journals to their classes. The journal assignment was a one page journal entry that allowed the students to reflect on a question dealt with in class. The initial reaction of both the teachers and the students was that the journals was an added workload to each week, but as the when the students assessed the journal program they said that even though it was a burden they think it helped them undertand math better.
The teachers felt that the immediate feedback that they were able to give to the students on their journal was worth while to the students and the teachers efforts. They said it takes a while to get the system going in a class, but as soon as a pattern is developed, journal writing enhanced the learning in their class.
From the teachers I read about, they recommended for all math teachers to try this. They prewarned us that the program may work well for some classes, but not others. But in the end, the purpose of getting students to think more about what they were learning in math class was achieved.
Keywords: Algebra, Connections,
Ref: Kueffer20
Author(s): McGlone, Chris; Nieberle, Gary M.
Date: May 2000
Title: Using Hooke's Law to Explore Linear Functions
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93, No.5, pp.391-398
Reviewer: Kueffer
Date of Review: 5-15-00
This article gives an example of an activity a math class can do to discover the properties of linear functions. The lab that is given is about Hooke's law (y=(1/k)x) and how it relates to linear functions. Through creating a lab with slinky springs data is collected and graphed, and from the data collected a linear function is determined from a best-fit line.
I feel the real learning in this lab takes place upon analyzing the data and graphs. I cannot decide if the time used to collect the data is worthwhile, or if the students will learn more just by being told where the data comes from. I feel that more higher ended questions could be asked if the lab time was cut out of this activity. It is hard to judge if the hands-on activity is removed from the lesson, if the student will be able to connect with the data, but I believe that more learning would take place if the students were guided through analyzing the graphs.
A very good part of this lab is that is gives an example of how to fit a best regression line with a graphing calculator. The regression line tool on a calculator is a very useful function in many different areas of math and science, and familiarizing the technique with students is a good idea.