This article gives an example of a successful problem solving session during a student teacher's day in the classroom. The basic premise is that a question or problem is posed, in this case "Find all rectangles with integral dimensions whose area and perimeter are numerically equal," and from here the problem solving begins.
First, as a class, the question was deciphered to determine exactly what question was being asked. Second, groups of 3 or 4 students were formed. Third, one group would periodically be asked to present and discuss ideas with the entire class. Fourth, each group was asked to make conclusions regarding the question. Any method of solving was encouraged and brought into the whole of the class for discussion. Fifth, the teacher asked if graphs could be used to illustrate the conclusions formed. The final results were to be presented to the class the next day.
It seems this form of teaching is more real-life oriented in how real people answer real life problems. Because all methods of problem solving are encouraged, no one is wrong and there is no 'right' way to solve the problem. Working collectively allows the students to determine together which methods work better and whether or not these methods could be useful in the future.
Keywords: History
Ref: Andrew2
Author(s): Gray, S. I. B.
Date: 2000
Title: Mathematics in the Age of Jane Austen: Essential Skills
of 1800
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(8), 670-679
Reviewer: Andrew
Date of Review: 2/12/02
This article discusses two primary mathematical texts used in the early 1800's for teenage ladies and gents and one book for upper level mathematics. Interestingly, different books were used for different the different sexes. However, the math taught in both the ladies' and gents' books was the same or very similar. The ladies mathematics book contained much more history, culture, literature, etc. in its 619 problems. The gents math book simply stated a mathematical problem without an introduction or set-up or reason for it's needing to be known. Both books taught the basic calculations of addition, subtraction, multiplication, and division. It was also of importance to learn the weights, measures, and money tables. The third book for upper level mathematics dealt with the binomial theorem, solving equations, solving square roots of polynomials, and logarithms, among other mathematical concepts.
If one is wanting to gain a deeper understanding of where math was in the 1800's this would be a good source to look at, if for nothing else the bibliography at the end. The article, however, is not too practical, with the exception of a few 'interesting facts' to amuse a class.
Keywords: Activities, Algebra
Ref: Andrew3
Author(s): Winter, Mary Jean; Carlson, Ronald J.
Date: 2000
Title: Liquid Assets: Increasing Students' Mathematical Capital
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(3), 172-175
Reviewer: Andrew
Date of Review: 2/19/02
This is an activity for teaching linear function, slope, intercept, and dependent and independent variables. The basic activity is that of recording the height of water in a glass of water while taking invariant sips from the glass. The students record both the number of the sip and the height and graph the data on graphing paper or a graphing calculator. Then, a linear function is found describing the change in height. Variation can be used in the experiment by changing the size and shape of glass or the types of sips taken. Different worksheets are provided in the article for printing and using. Comparisons can be made between different groups and individuals work.
This is an excellent and simple hands-on activity. I recommend using for grades 5-12. Adaptation would be necessary depending on the level or specific instruction needed. This activity could be used for quadratic functions as well.
Keywords: Puzzles, Problem Solving
Ref: Andrew4
Author(s): Howe, Roger
Date: 2002
Title: Hermione Granger's Solution
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 95(2), 86-89
Reviewer: Andrew
Date of Review: 2/21/02
This article provides the information for determining the solution to a problem given in Harry Potter and the Sorcerer’s Stone, by J.K. Rowling. The problem involves a small axiomatic system to solve the combination or order of letter or potions or bottles, etc. The axiomatic system involves four clues and from these clues the reader is to determine which bottle contains the good potion.
While this article did not show a way in which the lesson could be conducted
to students, a lesson plan could be derived from the article. It would
be good for combinitorics, problem solving, axiomatic systems, or logic
and reasoning. Similar and more complex problems could be formed from this.
Keywords: Research , Algebra
Ref: Andrew5
Author(s): Smith III, John P.; Phillips, Elizabeth A.
Date: 2000
Title: Listening to Middle School Students' Algebraic Thinking
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 6(3), 156-161
Reviewer: Andrew
Date of Review: 2/21/02
This research article looks at the way 8th grade students think about and solve problems. The focus is on the Connected Mathematics Project which is a math curriculum and how it prepared students to understand more complex algebraic analysis from linear functions and rate of change to the beginning of an understanding of exponential and quadratic relationships. The article is interesting because it shows the many diverse ways in which students think about and solve problems. The different thought processes ranged from algebraic analysis to looking at a table to looking at a graph to looking at a single variable to competency in calculator usage. The article also gives a few good algebra problems.
I recommend this article for anyone struggling with and trying to gain
an understanding of the different problem solving methods that students
use and also the misconceptions many students may have.
Keywords: Teaching Strategies, Curriculum
Ref: Andrew6
Author(s): Erisckson, Dianne K.
Date: 1999
Title: A Problem-Based Approach to Mathematics Instruction
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(6), 516-520
Reviewer: Andrew
Date of Review: 2/26/02
This article shows how to put into practice the problem-based instructional approach recommended by the NCTM, ways of being successful, and issues that may arise. The article gives two good examples of problems; the first is finding the area of the shaded region of a square where the top left quarter of the square is shaded and then the top-left quarter of the bottom-right quarter is shaded and so on in sequence. The second problem is simply charting a road trip and determining the expenses, rates of consumption and of time. An outline of a lesson plan for problem-based instruction is given. Many additional suggestions are given such as how to run large-group discussion or small-group activities. Two websites suggested for finding good problems are: www.edc.org/mcc/ and forum.swarthmore.edu/students.
This is a good pro-NCTM article with good suggested problems and websites.
The outlay of the article is fairly general, but encouraging for those
teachers hoping to implement problem-solving, discovery type lessons into
the classroom.
Keywords: Algebra, Activities
Ref: Andrew7
Author(s): Phillips, Elizabeth
Date: 1991
Title: Graphs and Functions as Patterns
Journal or Publisher: The National Council of Teachers of Mathematics,
Inc.
Volume, Issue, Pages: Patterns and Functions, 55-57
Reviewer: Andrew
Date of Review: 2/28/02
This activity is focused for the middle grades, discovering slope, rate of change, and the relation between graphs and algebra (linear equations). The basic scenario is the general problem of Terry and Pat have different running speeds. The slower of the two starts ahead of the faster. Who wins the race? Using a chart the class can determine who wins, hopefully in groups. Then, it can be shown how to plot the data on a graph with x-axis as time and y-axis as distance. Have the students find the ratio of distance over time for a couple of points on Terry’s line. This ratio is the slope which is the same as Terry’s speed. Steepness of lines can be talked about, as well as negative slope.
This is a great discovery activity. I remember doing this exact problem
in the eighth grade and feeling satisfied after understanding the math
behind the problem. Extensions are rather simple to construct from this
problem, and assessment should not be a problem.
Keywords: Statistics, Connections
Ref: Andrew8
Author(s): Quinn, Robert J.; Tomlinson, Stephen
Date: 1999
Title: Randon Variables: Simulations and Surprising Connections
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(1), 4-9
Reviewer: Andrew
Date of Review: 2/5/02
This activity is good for advanced high school students. Connections are made between statistics, probability, and geometric sequences. The activity is a simulation using coins, dice, and ten-sided dice. Conjectures are made determining the number of flips to get a head, then for the number of rolls to get a one for both a six-sided die and a ten-sided die. In groups, 10 trials are taken for the coin tosses, then for the dice. The class is then encouraged to find the theoretical probabilities of predicting the likelihood of getting a head on one flip or getting a one after 3 rolls of the dice. It can then be shown that it is more likely to get a one on the first roll than it is on the remaining rolls. Also, that the probability of getting a one or a head with an infinite number of rolls is 1. A geometric series can be used to determine the sum of all the probabilities.
This activity will hook the high schools students and great math is
being learned. Good connections are being made among different mathematics
and group work can be exploited for maximum benefit.
Keywords: Assessment, Tests
Ref: Andrew9
Author(s): Romagnano, Lew
Date: 2001
Title: The Myth of Objectivity in Mathematics Assessment
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 94(1), 31-37
Reviewer: Andrew
Date of Review: 3/7/02
This article deals with objective versus subjective grading and its correlation to traditional versus alternative testing, which this author seeks to disprove. He feels that all testing is in some form subjective, for it reflects the graders interest in what particular part of a problem the student is learning. He says that strict scoring rubrics are needed for open ended questioning so that the grading is as objective as possible, or at least consistent. He looks at different examples from a teacher-made quiz and the advanced placement calculus test and the SAT-I test. From here he gets his ideas for a needed rubric and specific probing questions designed to get to the meat of what a student knows and does not know.
This article is somewhat relevant, but I wouldn’t read it again, for
while it seeks to convince the teacher of needing more emphasis on rubric
it gives very little insight into how to go about doing this, other than
things that we should not do.
Keywords: Teaching Strategies, Technology
Ref: Andrew10
Author(s): Hale, Patricia
Date: 2000
Title: Kinematics and Graphs: Students' Difficulties and
CBLs
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(5)
Reviewer: Andrew
Date of Review: 3/11/02
This article focuses on the misconceptions people have when interpreting kinematics graphs, the most frequent being how to differentiate between when to give the point or the slope at the point on a graph of velocity or distance or even acceleration. The suggested emphasis to help students better understand how to interpret graphs are: Emphasizing conceptual learning versus procedural, emphasizing the relation between mathematical ideas and real situations, and encouraging discussion among students and between students and teacher.
The article was rather short and had a few good points, but overall
the article gave rather basic information.
Keywords: Games, Problem Solving
Ref: Andrew11
Author(s): Verhille, Charles; Blake, Rick
Date: 1982
Title: Activities for Active Learning and Teaching
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: 39-43
Reviewer: Andrew
Date of Review: 3/14/02
This article focuses on a game called the Peg Game. The purpose is to find patterns and develop problem solving strategies. The board has 11 holes with five black in the farthest right holes, and five white in the farthest left holes. The goal of the game is to interchange the black and white pegs by either moving on space of jumping one peg, where the black pegs can only move left and the white pegs can only move right.
Many patterns can be shown to result from this activity. I think more
applications could come from this game, but the article does not reflect
on these applications. This is a good activity nonetheless especially if
the sole purpose of a lesson is to teach good problem solving and pattern
finding techniques.
Keywords: Connections
Ref: Andrew12
Author(s): FASE productions
Date: 1991
Title: Math: who needs it?!
Journal or Publisher: FASE productions
Volume, Issue, Pages: Video
Reviewer: Andrew
Date of Review: 4/4/02
This particular movie is focused on getting students excited about math and giving them some applications of mathematics in the world. The mathematics discussed is, in general, the engineering aspect of mathematics. The video looks at sound, roller coasters, agriculture, water technology, profit maximization, and other things. By interviewing stars of Hollywood and having them discuss their notions of mathematics, the video hopes to encourage students to work hard in school and learn math because it is important.
While the video is alright, it is 10 years old and the Hollywood stars may be unfamiliar to some students, especially the younger ones. The video might be more useful if shown in snipits to introduce certain topics. Since each application in the video is only a couple minutes long it might be fun to show this as long as the video is forwarded to the appropriate place.
Keywords: Algebra, Teaching Strategies
Ref: Andrew13
Author(s): James R. Choike
Date: 2000
Title: Teaching Strategies for "Algebra for All"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(7), 556-560
Reviewer: Andrew
Date of Review: 4/18/02
This article gives pointers on how to teach the big ideas of algebra to all students. The basic themes Choike discusses are as follows: take away confusing numbers that hinder the thinking of big ideas such as using $10 instead of $12.36; ensure that students understand problems where ambiguity may not even relate to the math part of the problem; emphasize multiple representations of a problem including words, tables, graphs, and symbols; use common themes and settings in problems, such as always using a turtle and a hare for rate problems, which allows for students not to get confused in a changed setting; do not begin a year with weeks of review which only reminds students of how frustrating certain material was; use review problems at the beginning of class to revisit old material and assess where students are; use learning by discovery approaches; learn to recognize correct thinking in students even when it may be incomplete or lacking in closure; form lessons around the interests of students; establish a safe learning environment for students.
While many of these ideas seem rather straightforward and obvious, there
are some very good ideas that should definitely be used. The two that stand
out for me are using simpler numbers and not worrying about review at the
beginning of the year.
Keywords: Assessment
Ref: Andrew14
Author(s): Coates, Diane
Date: 1995
Title: SCORE webpage, www.frontiernet.net/~dcoates/altass.htm
Journal or Publisher: SCORE Assessment Resources
Volume, Issue, Pages:
Reviewer: Andrew
Date of Review: 4/18/02
This article is unfortunately out of date, even though it was written in 1995. Some ideas that were quite new were introduced in this paper, including portfolio. The premise behind the paper is that the U.S. wants to be number one in math and science by the year 2000. Well, this just isn’t the case. However, it is because of the ideas in the paper that are being put into place that mathematics in the U.S. is changing and students are learning more. Assessment was seen as the key idea behind change in that assessment should help the learner learn and the teacher teach. All the standards have this as the focus for assessment. Another good idea is doing panel assessments where students have an opportunity through writing, speaking, and visual aides to correspond with a panel of judges to show the extent of their knowledge of a particular area of interest. Another huge idea is merely using larger projects, even in groups, to assess the students’ learning.
This article is probably not worth reading because there is so much
more current and useful literature out there to read and get your hands
on. The webpage, however, may be a good starting point if wanting to find
more information.
Keywords: Problem Solving, Discrete, Connections
Ref: Andrew15
Author(s): Gardner, Martin
Date: 1978
Title: aha!
Journal or Publisher: Scientific American
Volume, Issue, Pages: all pages
Reviewer: Andrew
Date of Review: 4/25/02
This book is great! If one is seeking to learn more problem solving tecniques, find more problems, learn more history about problems, find cartoons relating to problems, to find fun math for parties, or looking for a good read, this would be a great book to check out. I feel like I am selling this book, but in all honesty it is a great book with easy reading for even the "I’m no good at math" person. They break the math up into four sections, even though each section cannot truly be contained within that particular section. The sections are combinatorial, geometry, number, logic, procedural, and word. The math is explained in an interesting way with history and connections to other math and other subjects being made.
Definitely read this book and get it on your shelf!
Keywords: Statistics, ,
Ref: Andrew16
Author(s): Shaughnessy, J. Michael; Pfannkuch, Maxine
Date: 2002
Title: How Faithful is Old Faithful?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 95(4), 252-259
Reviewer: Andrew
Date of Review: 5/7/02
This article explains, with a good example, what statistical thinking is all about and how it can and should be implemented into the classroom. The article, while focused more towards high school statistics courses, definitely gives great ideas and good teaching techniques for math teachers and probably teachers in general. The data set used comes from Old Faithful geyser wait times which leads to much good classroom discussion. The data set for 16 days is given at the end of the article, which would be good to have even if the article was of no use beyond that. The article does however have good questions and good examples of statistical thinking. Variation and transnumeration, the process of connecting statistical models with the real system, are emphasized as probably the most important aspects of statistical thinking.
I would definitely recommend reading this article as it is good for the teacher's knowledge and also has good teaching ideas.
Keywords: Connections, Number Theory, Activities
Ref: Andrew17
Author(s): Devlin, Keith
Date: 2002
Title: Numbers in the Garden and Geometry in the Jungle
Journal or Publisher: Mathematics Teaching in the Middle
School
Volume, Issue, Pages: 7(8), 422-425
Reviewer: Andrew
Date of Review: 5/07/02
This article explains, in general, how math can be seen in nature. Devlin discusses the Fibonacci numbers and relates them to trees and flowers and even spots on a leopard. His hope is to encourage teachers to seek out connections between math and nature so that students will stay engaged in learning number. The article gives some good ideas that may launch more research into different areas of interest. The article also discusses a PBS series on video that may help some teachers integrate interesting math/nature into the classroom.
This article is a good launch pad for further inquiry. While the article most likely doesn't give enough information in full, it does wet the appetite. If anything it would be a good article for students to read.
Keywords: Geometry, Problem Solving, Proof
Ref: Andrew18
Author(s): Pandiscio, Eric A.
Date: 2002
Title: Alternative Geometric Constructions: Promoting
Mathematical Reasoning
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 95(1), 32-36
Reviewer: Andrew
Date of Review: 5/8/02
This article discusses different methods of discovering geometric relationships. It takes the idea of using a straight edge and compass and expands the idea to using different tools for construction, namely a 3X5 index card, a Mira (a reflective window), and a TESE (constructs parallel lines). By forcing students to use only one of these tools and having them form different constructions, they use and discover different geometric properties. The article gives three examples using each of the three tools and walks the reader through the process.
This article has great ideas for problem solving and discovering geometric properties. All examples and ideas could be extended and could definitely be used in the classroom.
Keywords: Issues, Standards, Keyword 3, Optional...
Ref: Andrew19
Author(s): NEA Today Online
Date: 2002
Title: Inside Scoop: Math Wars
Journal or Publisher: NEA Today
Volume, Issue, Pages: www.nea.org/neatoday/0105/scoop.html
Reviewer: Andrew
Date of Review: 5/8/02
This article, found on the NEA website, outlines the basic battle between traditional math curriculum and the newer NCTM standards. The emphasis here is between rote memorization and the understanding of the mathematics. The argument is that by understanding why algorithms work the way they do, less time will need to be spent on practicing the memorization. On the flip side, since formulas for certain procedures are already known, our students should not be wasting there time on discovering the procedures on there own. By giving students the formulas, they can hopefully get to the next level quicker. While the article is good at outlining the debate, it is quite evident that they favor the NCTM standards, and rightly so.
This article would be good for non-math and non-teacher people to understand the debate between the traditional math curriculum and the new Integrated curriculum.
Keywords: Communication, Problem Solving,
Ref: Andrew20
Author(s): Burns, Marilyn
Date: 1985
Title: The Role of Questioning
Journal or Publisher: Arithmetic Teacher
Volume, Issue, Pages: 32(6), 14-16
Reviewer: Andrew
Date of Review: 5/8/02
This article, though from 1985, is good for getting at the importance of questioning to develop mathematical thinking. It gives examples to help teachers elicit classroom discussion about mathematics. It stresses the importance of open ended questions and doing math, not to 'get the answer,' but to develop thinking that will enable good problem solving skills down the road. The questions we should be asking students are ones such as: what do you think, why do you think that, how come _____'s answer doesn't make sense, what if____, is there another way, and can you convince me?
I would recommend this article as well as the subsequent articles in this journal. There is good research discussed into the way we think about math and good teaching ideas for getting students to delve deeply into mathematical thinking.