Keywords: Teaching Strategies, Assessment, Standards
Ref: Katie1
Author(s): Murdock, T.B.
Date: 1999
Title: "Discouraging Cheating in Your Classroom
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92 (7), p.587-591
Reviewer: Katie
Date of Review: 2/10/00
This article discusses the problem of academic dishonesty, and provides strategies for combating this increasing problem. Studies have shown that cheating is more prevalent in math and science than in other disciplines, and that it becomes more common as students age. Factors that increase the likelihood of cheating include the "fear of failure, desire for a better grade, and pressure from parents to do well in school" (page 588). Higher instances of cheating are also linked to performance- focused assessment, grade-oriented students, and weak correlations between classroom sessions and assessment.
The article states that although cheating is a result of broad social situations, such as the large number of high school students with after-school jobs and less time to study, teachers can make adjustments in their assessment and feedback methods that will improve students' value of learning. First, it suggests that assessment practices should be clearly aligned with instruction, and clarifies that this does not mean giving only easy tests. In fact, more difficult questions may emphasize the strategies rather than the solution and deter cheating. "It is far easier to copy an answer than to copy someone's reasoning" (page 590). Making approximations of real-world examples into such problems not only reinforces the goal of learning, but students will be more likely to see the value of working toward a solution of a problem that is relevant to life outside of the classroom. In addition, teachers should use multiple forms of assessment so that students have multiple opportunities to demonstrate their mastery of certain concepts, and so that individual tests do not have extremely high stakes.
Finally, in giving feedback to students, the article discusses the importance of reviewing concepts and continuing the idea that what a student understands is more important than the grade he or s he receives. Teachers can do this by giving credit on tests not only for right answers but for correct techniques of problem- solving, and by writing comments and reviewing in class the information that will help students learn what they do not know.
I thought that this article did a good job of explaining concrete ways in which teachers can make their students take control of
their learning and shift the focus of their classrooms from
competitive atmospheres to ones in which individual progress and
concept mastery is more important than the grade received.
Keywords: Teaching Strategies, Issues,
Ref: Katie2
Author(s): Fiore, Greg
Date: 1999
Title: "Math-Abused Students: Are We Prepared to Teach Them?"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(5), p.403-405
Reviewer: Katie
Date of Review: 2/20/01
Math Anxiety is defined as "the panic, paralysis, and mental disorganization that arises among some people when they are required to solve a mathematical problem." According to this article, math anxiety results more from the way the subject matter is taught than from the actual subject matter itself. Thus, teachers have enormous power to prevent math anxiety, as well as to recognize its symptoms and create comfortable learning environments for students who suffer from this fear.
The article gives two in-depth case studies of women who suffer from math anxiety, and discusses effective pedagogical strategies for combating it. It reminds educators of the often underestimated value of encouraging comments, building student's confidence, and motivating students to learn.
I thought this article was most valuable for its reminder of the importance
of teaching toward understanding rather than towards the right answer. It
reinforces the idea of encouraging questions and collaboration not only
to facilitate understanding by some students, but to create a comfortable
learning environment in which all students are encouraged to succeed.
Keywords: Problem Solving, Activities, Teaching Strategies
Ref: Katie3
Author(s): Miller, Catherine M.
Date: 2000
Title: Student-Researched Problem-Solving Strategies
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93 (2), pg. 136-138
Reviewer: Katie
Date of Review: February 20, 2001
This article is a response to the National Council for Teachers' of Mathematics' call for Teachers to encourage their students to become independent problem solvers. Just as understanding a concept is ultimately more important that arriving at the correct answer, experiencing problem-solving strategies firsthand is more valuable that being given a list of such strategies.
The author describes her experiment, which can be used at any level from middle school to college, for enabling students to gain a better understanding of problem-solving strategies. In short, she gives students a worksheet of problems and asks them to choose friends or family of varying math ability to solve a problem. The students are to observe, record, and reflect on the differences between the strategies used to solve the problems. The teacher then facilitates a class discussion of the data, and compiles a master list of the results, which can be posted in the classroom for future reference. The author explains that such an experiment allows students to make sense of the strategies and thus increases the likelihood that they will be useful.
I thought this article brings up a really valid point that not only math
concepts are best learned through experimentation. I really appreciated
the concrete descriptions of each step of the successful completion of
this project, and am very interested to see what students would come up
with!
Keywords: Activities, Probability,
Ref: Katie4
Author(s): Szydlik, Jennifer E.
Date: 2000
Title: Photographs and Committees: Activities that Help Students Discover Permutations and Combinations
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93 (2) pg.93 -95
Reviewer: Katie
Date of Review: February 21, 2001
Two guiding principles form the basis for the National Council for Teachers of Mathematics' goals for math curriculum. "First, activities should grow out of problem situations; and second, learning occurs through active as well as passive involvement" (93). This article's purpose is to detail a sequence of problem situations that allow students to better understand permutations, combinations, and relationships between the two.
One example of the activities discussed is the photograph problem. This exercise requires students to find the number of ways to arrange four people in a line from a photograph. (Different orders are counted as different photographs). The problem can then be done with a different number of people in the photograph. Students then generalize their answers and make conjectures in class discussion. The teacher facilitates the discussion by encouraging further explanation and exploration, but does not give the answers.
The photograph problem and the two other activities described in the article is a way for students to develop mathematical reasoning and a concrete understanding of the formulas that they derive. The author also suggests that students complete a write-up on the problem of their choice. In doing this, they describe the process, discuss the strategies used to solve the problem, and present logical mathematical reasoning for their solutions. Thus, students will not only have a clearer understanding of permutations and combinations, but will "experience mathematics as problem solving, reasoning, and communication" (95).
The article does a good job of providing educators with a creative
introduction of what can be a confusing topic for students. I appreciated
the fact that it specifies multiple approaches to carrying out the activity, and
that it gives concrete suggestions as to how teachers can be effective facilitators
of the process.
Keywords: Issues, Equity,
Ref: Katie5
Author(s): Jackson, Carol D.; Leffingwell, R. Jon
Date: 1999
Title: "The Role of Instructors in Creating Math Anxiety in Students from Kindergarten through College."
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92 (7), p. 583-586
Reviewer: Katie
Date of Review: 1999
As the title suggests, this article presents the results of a study conducted to determine the role of instructors in creating math anxiety in students. One hundred fifty seven students responded to the prompt "discuss your worst or most challenging mathematics classroom experience from kindergarten through college." Researchers found three clusters of grade levels in which mathematics anxiety was particularly likely to take root: Elementary (particularly 3 and 4), Secondary (particularly 9-11), and College (particularly freshman year). After identifying these clusters, the researchers studied various causes of math anxiety at each of these levels.
The article then details these causes in order to make instructors aware of the impact of their actions. Most of the causes fall under the following categories: insensitive and uncaring instructors, hostile instructor behavior, gender bias, and unrealistic expectations. The article discusses concrete ways in which these behaviors are manifested at various levels.
I feel that this article is important for all educators to read in order to understand
the degree to which even small actions can affect students' perception of themselves and
the subject matter. Although the instructors are only one factor in causing math anxiety,
it is important that they are able to create a positive learning environment that encourages
all students to succeed.
Keywords: Algebra, Teaching Strategies,
Ref: Katie6
Author(s): Kinzel, Margaret Tatem
Date: 1999
Title: Understanding Algebraic Notation from the Students' Perspective
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92 (5) p. 436-442
Reviewer: Katie
Date of Review: 4 March 2001
In this article, the author examines the idea that students do not always understand algebraic notation in the way the instructor intended. Clearing up such misconceptions is an important step in improving students algebra skills as well as their ability to use mature mathematical reasoning. The first problem she discusses is students' difficulty of connecting symbols with the proper referent; in other words, students have a hard time comprehending the exact meaning of a variable, and especially with viewing expressions as objects. Research has shown that the two major contributing factors to this problem are the physical resemblance of the notation to the quantities and the explicit relationship between quantities.
The article also presents two strategies that have successfully developed students' ability to master the new system of notation. First, studies suggest that instructors "make symbolizing an explicit aspect of students' problem-solving activity" (439). This allows instructors to better understand their students' thought processes, and simultaneously encourages students to use reasoning to understand the notation. Second, instructors should be particularly thorough in developing the concept of a variable. This includes explicitly instructing students to get into the habit of identifying and labeling all variables.
This article does a thorough job of reminding instructors that seemingly uncomplicated
symbols may initially be difficult for students to grasp. It reminds educators to be clear
about what is being represented, and provides concrete strategies that instructors may
implement in order to effectively introduce such notation and clear up confusion regarding the
symbol and its referent.
Keywords: Standards, Communication, Teaching Strategies
Ref: Katie7
Author(s): Knuth, Eric; Peressini, Dominic
Date: 2001
Title: Unpacking the Nature of Discourse in Mathematics Classrooms
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 6 (5), p.320 - 325
Reviewer: Katie
Date of Review: 6 March 2001
The call for meaningful mathematics discourse in classrooms is evident in the NCTM's Principles and Standards for School Mathematics (2000). This article is the result of a longitudinal professional development project conducted to increase educators' ability to foster such meaningful discourse in their classrooms. It provides a framework for examining discourse, and a description of varying types of discourse that arise as students work to solve problems.
The article states that all discourse is either univocal or dialogical. Univocal discourse conveys meaning; the listener receives an intended message from the speaker and discourse is finished. Dialogical discourse, on the other hand, generates meaning; the listener uses the initial message as a "thinking device" which generates further discussion and reflection. The article then uses vignettes to compare the role of the instructor and the effectiveness of both methods in the same classroom setting. Although the vignettes appear very similar on the surface, the authors draw our attention to the effects of key differences.
By detailing these differences and their effects, the article
clearly communicates the importance of the way in which one poses
questions. In univocal discourse, the teacher seemed to be
letting the students think for themselves. However, she was
guiding them toward her solution instead of encouraging them to
arrive at their own. In the dialogical discourse, she uses ideas
as thinking devices intended to generate further discussion and
reflection.
Keywords: Communication, Teaching Strategies,
Ref: Katie8
Author(s): Thompson, Denisse R.; Rubenstein, Rheta N.
Date: 2000
Title: Learning Mathematics Vocabulary: Potential Pitfalls and Instructional Strategies
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93 (7), p. 568 - 573
Reviewer: Katie
Date of Review: 6 March 2001
We often think of mathematics as a language. Yet those fluent in that language often forget that not everyone is familiar with even the most basic mathematical terminology. This article explains that the vocabulary of mathematics is vital for three reasons. First, it is the medium through which we teach. Also, it is through language that students reason and build understanding. Finally, language is the means by which educators assess their students' understanding. Because a clear understanding of mathematical terminology is necessary for mathematical achievement, this article presents several strategies through which educators can facilitate the development of a strong vocabulary.
The authors remind educators of the importance of building a concept and then attaching vocabulary to that concept. More specifically, students should "own" the language. Such ownership is a result of extensive oral practice with the notation. Written explanations are also important; students strengthen their mathematical reasoning as well as their ability to clearly communicate their thought processes. Teachers can also make connections between a term and its referent through visual displays or charts that display the relationships among new terms. Similarly, making models allows students to kinesthetically explore the properties of new concepts. Finally, the article talks about building bridges between mathematical words and everyday language in order to create memorable connections in the students' minds.
I think this article is very valuable - it give teachers a lot of good ideas as to how to effectively
develop students' math vocabularies!
Keywords: Assessment, Equity, Tests
Ref: Katie9
Author(s): Romagnano, Lew
Date: 2001
Title: The Myth of Objectivity in Mathematics Assessment
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 94 (1), p. 31 - 37
Reviewer: Katie
Date of Review: 8 March 2001
As both a math and a history major, I have always valued the way that my math professors grade "objectively," and expressed frustration with the subjective evaluations of the social sciences. This article shatters such beliefs, and effectively explains the important elements of just assessment.
The author explains that there exists no form of truly objective evaluation, and supports this claim by having the reader "evaluate" students answers on a teacher-made quiz, the AP calculus exam, and the SAT. In doing this Romagnano analyzes each of the possible interpretations of what a student who gave a certain answer actually knows. In the example of the teacher-made quiz, the student came very close to a right answer. However, the author points out that the teacher is unable to determine whether the had a clear understanding of the concept and had made careless algebraic errors, or if that student had tried unsuccessfully to memorize an algorithm. It is therefore difficult to assign a score.
In contrast, the AP calculus exam has a very specific grading rubric, and the author argues that such a system results in scores that have little relevance to the students' ability to carry out tasks. The author also criticizes the SAT for trying to use only one tool to measure knowledge, and describes the subjective influences on standardized tests.
The author suggests that meaningful classroom assessment should draw out
the information one wishes to test. Also, students should understand the exact
guidelines that accompany each method of assessment. This article provides
unique insight on different forms of evaluation, and stresses the need for
consistency and meaningful evaluation.
Keywords: Equity, Curriculum, Standards
Ref: Katie10
Author(s): Choike, James R.
Date: 2000
Title: Teaching Strategies for "Algebra for All"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93 (7) p. 556-560
Reviewer: Katie
Date of Review: 8 March 2001
How can we teach algebra to all students? This question reflects the movement towards math for all, overcoming the misconception that only some students are skilled at math and working to develop all students ability to think mathematically. James Choike, in his article, discusses several ways to avoid focusing on "making it through the book," and instead focus on strategies that help students to conceptualize the most important themes of algebra.
Choike developed these strategies while leading the College Board's Equity 2000 program and professional development activities with other teachers and students. The strategies include: focusing on the "big ideas," using simple numbers to avoid initial confusion with word problems, giving specific and unambiguous examples, emphasizing alternate modes of representation (i.e. graphs, tables, symbols), using consistent examples, teaching by discovery, recognizing correct thinking even when it is incomplete, molding lessons around the interests of individual students, and maintaining a safe learning environment.
He offers elaborations on each of these strategies - they offer a concrete way to teach "out
of the box." Because these ideas are new to many educators, I think this is a valuable tool
in modifying classroom technique!
Keywords: Geometry, Activities,
Ref: Katie12
Author(s): Lornell,Randi; Westerberg, Judy
Date: 1999
Title: Fractals in High School: Exploring a New Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92 (3), 260 - 265
Reviewer: Katie
Date of Review: 4 April 2001
This article was written by two teachers who have successfully incorporated a unit on fractal geometry into their high school geometry courses. They explain the importance of giving students an opportunity to use this new approach to explore traditional topics, as well as to provide tools with which students are able to connect mathematics and the natural world. Examples of such traditional topics include those of surface area, volume, and perimeter; these ideas and more can be explored using Koch snowflakes and Mandelbrot and Cantor sets. Fractals can be tied to the natural world in multiple areas, including the identification of liver cancer and the containment on liver cancer as well as their presence in everything from plant life to fashion design!
The article only includes an explanatory discussion of the history of fractal geometry, as well as a clear definition
and information about the significance of fractals. The body of the article provides four in-depth classroom activities
that teachers can use to facilitate the exploration of fractals as well as their properties. Because the authors have
experience with each activity in the classroom, they are able to provide helpful instructional guidelines.
This article is a comprehensive introduction to teaching fractal geometry, and a valuable aid to incorporating such
a lesson into high school geometry curriculum!
Keywords: Geometry, Algebra, Connections
Ref: Katie13
Author(s): Allaire, Pttricia R.; 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 (4), 308 - 313
Reviewer: Katie
Date of Review: 21 April 2001
Students often complain that mathematics is boring and rarely changes. Ironically, it is the history in this article that offers fresh approaches to basic algebraic and geometric concepts and offers many new ways a teacher could present "ordinary" material to his or her class. This article focuses on quadratic equations with one unknown and provides a variety of ways in which such problems were solved before mathematicians adopted the regular system of symbolic notation and manipulation. They introduce geometrical algebra techniques (using geometric concepts to illustrate algebraic concepts) that were widely used in ancient Babylonia, classical Greece, medieval Arabia, and early modern Europe to solve quadratic problems.
When observing at a middle school the other day, I actually watched a
teacher lead her students through one of these problems. It was extremely
effective in practice, and this article expands on a relatively simple
example to include much more complex geometric concepts; it therefore would
facilitate not only a deeper understanding of quadratics, but also an
increased understanding of the connections between several geometric and
algebraic concepts. This article is also strong in that variations of
the given problems could be used with students at different levels;
it provides challenges for those who have a better grasp on the concepts
as well as clear illustrations and problems for students who have a
more difficult time understanding the concepts.
Keywords: Discrete, Games,
Ref: Katie11
Author(s): Gannon, Gerald E.; Martelli, Mario U.
Date: 2001
Title: Discrete Dynamical Systems Meet the Classic Monkey and the Bananas Problem
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol 94 (4), 299-301
Reviewer: Katie
Date of Review: 19 April 2001
This article introduces the familiar Monkey-and-the-Bananas problem for those who have not encountered the problem before. It then works teachers through two solutions; the first is specific and the second, more general. The general solution allows the authors to introduce a unique way of looking at the mathematics behind the problem by using the idea of discrete dynamical systems; this approach provides a much easier solution to the general case as well as an introduction to this newer branch of mathematics.
The problem, in short, deals with three shipwrecked sailors who have only a pile of bananas on which to survive. Each distrusts the other and so they keep dividing the bananas, hiding their share, and feeding the remainder to the monkeys. The strength of this article lies not in its short explanation of using guess and check or other common methods, but in its clear explanation of how to recognize the pattern and use composition of functions by writing each function in terms of its fixed points. The activity would be most useful for teachers incorporating the ideas of the standards into high school level math - I think it would be a good thing to have!