Bethany's Article Reviews, 2006



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Keywords: Standards, Equity, Curriculum
Ref: Bethany1
Author(s): Schoenfeld, Alan
Date: 2002
Title: Making Mathematics Work for All Children: Issues of Standards, Testing, and Equity
Journal or Publisher: Educational Researcher
Volume, Issue, Pages: Vol. 31 No. 1 (January-February 2002), pp. 13-25
Reviewer: Bethany
Date of Review: 2/16/2006

The article focuses itself on the following statement: mathematical literacy should be a goal for all students. Alan H. Schoenfeld discusses that even if this is the goal we have in mind as educators, it is not necessarily being achieved. He notes that disproportionate numbers of poor, African American, Latino, and Native American students drop out of mathematics and perform "below standard" on tests of mathematical competency, which puts them at a disadvantage upon entering the workplace. He praises the vision statement introduced in 1989 when NCTM's issued Curriculum and Evaluation Standards for School Mathematics and notes that this document led to curriculum reform in the classroom to help decrease the "performance gaps." But, he also points out criticism of the standards; the standards themselves tended to be "long on the direction and short on detail." Schoenfeld again supports the 2000 publication of Principles and Standards especially since after a decade, better curriculum to support the equitable vision statement is more readily available. Principles and Standards demands high expectations and strong support for all students which Schoenfeld sees as an absolute requirement.

The rest of the article goes on to discuss what effect the implementation of the principles and standards has on students, particularly those who were previously underserved. Schoenfeld seems to think the outlook is hopeful. In the past 10 years, the "performance gap" has been lessened and "minority" students' test scores have improved; from this research, there is encouraging evidence that this forward movement in curricula will continue to bring positive improvements.

Finally Schoenfeld touches on what can be done to more fairly assess student knowledge and performance. He continually reiterates the principle of equity throughout his article: every student deserves to be challenged and supported in mathematics.

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Keywords: Problem Solving, Technology, Connections
Ref: Bethany2
Author(s): Erbas, A. Kursat, Ledford, Sarah D., Hawley-Orrill, Chandra, Polly, Drew
Date: 2005
Title: Promoting Problem-Solving Across Geometry and Algebra by Using Technology
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 98 No. 9 (May 2005), pp. 599-603
Reviewer: Bethany
Date of Review: 2/20/06

This article presented the argument that multiple technologies can create multiple representations of a problem at hand. Utilizing these technologies, students can be creative in their problem-solving techniques and not be weighed down by heavy computation. Also, by using different technologies to create different representations, students can draw connections between branches of mathematics and see how they overlap. The article presents an open-ended problem for students to solve that makes use of both algebra and geometry. The authors show how The Geometer's Sketchpad can be used to draw diagrams to represent a situation. Due to the dynamic nature of the software, students are encouraged to find multiple solutions to the problem. The authors also show how students can make use of a spreadsheet to show numerous solutions to the problem at hand. After students observe that multiple solutions are possible, they are encouraged to come up with a general solution. While teachers could show students how to come to this general solution right away, the exploration led by the two technologies described allows students to explore on their own and draw connections between algebra and geometry before coming to a conclusion. After mentioning how the graphing calculator can also be used, the authors conclude, stating that multiple technologies can highlight different aspects of the problem solving process as well as urge students to form and test their own conjectures in order to come to a solution.

This article provided an excellent example of how useful multiple technologies can be. After reading this article, I would definitely want to integrate the technologies described by the authors into my classroom. This technology could be used at the middle school or high school level to help students problem solve more easily and creatively.

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Keywords: Algebra, Problem Solving
Ref: Bethany3
Author(s): Nathan, Mitchell J., Koedinger, Kenneth R.
Date: 2000
Title: Moving Beyond Teacher's Intuitive Beliefs About Algebraic Learning
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 93 No. 3 (March 2000) pp. 218-223
Reviewer: Bethany
Date of Review: 2/23/06

The authors of this article aimed to uncover the misconceptions teachers and textbook writers have regarding word problems in algebra. They took notice of the fact that teachers seem to regard algebra story problems as difficult for students, textbooks tend to put story problems at the end of problem sets, and students all too often dread them. So, in order to dispel myths about student performance in algebra, the authors decided to investigate. They had teachers rank 6 versions of essentially the same mathematical problem from easiest to hardest. The majority of the teachers labelled the "algebra story problem" form as the hardest; in general teachers labelled story problems as harder than symbolic problems and algebraic problems as harder than arithmetic problems. Then the authors had students complete similar problems of the same 6 forms. As teachers predicted, students performed better on arithmetic problems than algebraic problems, but contrary to teacher predictions, students performed better on algebra story problems than on symbolic algebra problems.

When it comes to algebraic story problems, teachers tend to think students go through this solving process: translate the word problem into symbols, then solve the symbolic problem. What the authors found, is that students don't necessarily solve in this way, rather they often tend to use informal, intuitive methods to solve algebraic word problems. Teachers can learn from what students have to offer; they should build on the more natural, intuitive solving skills students possess, skills they tend to develop before they develop symbolic algebraic skills.

This article provides some vital information to teachers of beginning algebra. From the authors investigation, teachers should build on the natural solving skills students possess, before introducing the formal symbolic algebra.

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Keywords: Teaching Strategies, Communication, Problem Solving
Ref: Bethany4
Author(s): Artzt, Alice F.
Date: 1999
Title: Cooperative Learning in Mathematics Teacher Education
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 92, No. 1
Reviewer: Bethany
Date of Review: 2/26/2006

The author of this article stresses that new teachers must be challenged to understand the necessity for new models of teaching and must be ready to effectively implement techniques such as cooperative learning. Researchers have documented positive experiences and positive effects of the use of small groups in mathematics classrooms; new teachers, then, need to know how to effectively include cooperative learning in their teaching. The purpose of this article is to describe how a cooperative learning activity was used in a college math-teacher-education course to enable new teachers to experience, learn about, and reflect on the intricacies, complexities and values of effective cooperative learning strategies. The teacher of this math-education course designed a cooperative learning activity centered around an unfamiliar, yet attainable math problem. Students in this class broke into groups and spent some time working together on the multi-faceted problem. Afterwards, they reflected on their experience. The second half of this article focuses on the insight these students gained about effective group-work that strives to ensure mutual interdependence and individual accountability.

I gained a lot of insight from reading what these students had to say about cooperative learning. This article is a great resource for anyone who wants tips to designing and implementing effective cooperative learning activities.

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Keywords: Activities, Games
Ref: Bethany5
Author(s): Devaney, Robert L.
Date: 2004
Title: Fractal Patterns and Chaos Games
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 98, No. 4 (Nov. 2004) pp.228-233
Reviewer: Bethany
Date of Review: 3/1/2006

This article outlines an activity for middle school or high school students that provides an introduction to contemporary mathematics, specifically to the topic of fractals. The activity is centered around the classic chaos game, in this case resulting in the Sierpinski Triangle. The instructions are as follows: (1) draw a triangle and label one vertex blue, one vertex red, and one vertex green (2) also color a die with two sides blue, two sides red, and two sides green (3) pick any point on the triangle and roll the die (4) depending on the color on top, move the designated point half the distance to the similarly colored vertex (5) repeat this process and record where the generated points are landing. After numerous trials, students will start to see the pattern that is emerging. This chaos game as an approach to fractals provides teachers with an opportunity to help students comprehend the geometry of affine transformations. After students see the outcome of the classic chaos game, questions will likely come to mind: what if we change the number of vertices? What if we change the scaling factor? Etc. This sort of exploration is a great introduction to fractals. One can even go further and ask students to look at a completed fractal and try to determine the "rules" of the chaos game needed to produce that fractal.

This activity would be wonderful to implement in an enrichment program. It seems like it would be a lot of fun for students and it allows students to learn about contemporary math, which is something students often don't get to work with.




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Keywords: Representations, Standards
Ref: Bethany6
Author(s): Garner, Amanda S., Preston, Ronald V.
Date: 2003
Title: Representation as a Vehicle for Solving and Communication
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 9, No. 1 (September 2003) pp.
Reviewer: Bethany
Date of Review: 3/7/06

This article displayed unabashed support the for the PSSM Standard, "Representations." Its authors describe representations in mathematics as tools that are vital for recording, analyzing, solving, and communicating mathematical data, problems, and ideas. Moreover, representations act as our "language of math" and allow us to experience abstract notions with physical materials or drawings. The authors describe an activity implemented in a middle school classroom which allowed students to use whichever representation worked best for them. The problem dealt with determining the most cost efficient location to have a class party; students were asked to decide from three possible places, which setting they would pick and be able to explain why. Students explained using formal and informal language, tables, graphs (line and bar), equations, patterns, and hand gestures. Since students were given the option to choose whichever representation they wished to use, they chose ones that they were most comfortable with.

From this article I was reminded of the variety of ways students learn and experience math. While sometimes it's a good idea to allow students to choose the representation they use, it's also a good idea to require students to experience using a variety of representations, as some representations work better than others for certain types of problems. This article does a great job of explaining the importance of the representations standard.


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Keywords: Planning, Algebra, Puzzles
Ref: Bethany7
Author(s): Lanius, Cynthia
Date: N/A
Title: "Fun With Calendars"
Journal or Publisher: http://math.rice.edu/~lanius/Lessons/cnotes.html
Volume, Issue, Pages: N/A
Reviewer: Bethany
Date of Review: 3/9/06

The objective for this lesson plan is for students to review topics of assigning variables, solving simple linear equations, and factoring through an interactive calendar puzzle activity. The "hook" consists of the following "trick" that is presented by the teacher: "Take any calendar. Tell your friend to choose 4 days that form a square. Your friend should tell you only the sum of the four days, and you can tell her what the four days are." After a few baffling examples of how this is true, students will say, "How did you do that?" Answer "I used algebra." Now, it is the students' job to figure out how this puzzle works. With a hint here and there, students will eventually solve the puzzle: Let's call the first number n. Then you know that the next number would be n + 1 and the next number would be n + 7 and the next number would be n + 8. We had our friend add up the four numbers, so let's add our four numbers: n + n + 1 + n + 7 + n + 8. All we need to do is use some algebraic manipulation, solve for n and find values for n+1, n+7, and n+8. Then students will work with each other on some more examples with partners. Finally, as a way of assessing understanding and extending the topic, students are asked to create their own variation of the calendar puzzle.

I thought this lesson idea was extremely innovative. My own lesson plan surrounding this activity would be a bit more detailed and go a little further, but on the whole, this lesson plan had most everything one would require for a good plan.

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Keywords: Geometry, Curriculum, Representations
Ref: Bethany8
Author(s): Coxford, Jr., Arthur F.
Date: 1991
Title: Geometry from Multiple Perspectives
Journal or Publisher: NCTM Addenda
Volume, Issue, Pages: N/A
Reviewer: Bethany
Date of Review: 4/6/2006

As its title would suggest, this book published by NCTM, stressed the exploration of geometric topics from multiple perspectives. A few things about this book stood out to me: (1) There was heavy emphasis on using technology to visualize geometric topics. It is certainly true that when integrated thoughtfully, technology can help bring students to a deeper understanding of the geometry. Technology such as Cabri or Geometry Sketchpad is interactive, allowing students to create visual representations of the problem or principle they are studying. (2) The text stressed the use of multiple representation of the mathematics. For example, geometric transformations can be depicted in many ways: using matrices, graphing on a coordinate plane, describing distances through algebra, using vectors, etc. This text realized the way geometry works with other topics in mathematics. (3) This book provided many useful handouts for classroom activities and enrichment topics. It suggested topics that many would deem too advanced for high school students such as Frieze patterns, tessellations, the chaos game, and fractals. But, in doing so, the text stressed that students should not miss out on the opportunity to study advanced topics in math. This text would be a useful supplement to any high school geometry class.

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Keywords: Geometry, Standards
Ref: Bethany9
Author(s): Day, Roger, Kelley, Paul, Krussel, Libby, Lott, Johnny W., Hirstein, James
Date: 2001
Title: Navigating Through Geometry
Journal or Publisher: NCTM
Volume, Issue, Pages:
Reviewer: Bethany
Date of Review: April 11, 2006

In the preface to this book, that authors stress the importance of approaching geometry through a transformational lens; transformations provide the study of geometry with a functional basis that lends itself to extensions to algebra, statistics, and calculus. Also, viewing geometry through this lens easily allows the integration of technology into the classroom.

There are four main chapters in this book and the first is called "Transforming Our World," and as the title would suggest it focus upon transformations (translations, reflections, rotations, dilations) by using multiple representations and ideas (sketches, coordinates, vectors, function notation and matrices. Advanced topics such as tessellations are also introduced.

The second chapter is called "The Geometry of Location and Map Making." This chapter introduces some very unique and somewhat advanced math ideas that are rarely found in a geometry course such as how GPS works, projections on a sphere, and the basics of cartography. This chapter takes into consideration the literal meaning of geometry: earth measure.

Chapter 3 is called "Multiple Dimensions of Similarity" and it introduces on the idea of similarity in a real world context before moving on to more abstract examples. Dilations are introduced as well as topics like "field of vision" and "multiple transformations." Finally, chapter 4 is called "Visualizing Limits in Our World." It presents geometric approaches to sequences and series and natural extensions of the patterns that students begin to examine in early grades. Sequences and series are presented, as well as infinite sums and fractals. There are also blackline masters available for copying in the last section of this book.

I think this book provides a fresh view of geometry that allows students to delve into interesting and advanced topics in geometry. Students are capable of so much, we simply need to challenge them and then can certainly explore topics in higher mathematics.




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Keywords: Assessment, Teaching Strategies
Ref: Bethany10
Author(s): Lapp, Douglas A, Manouchehri, Azita
Date: 2003
Title: Unveiling Student Understaning: The Role of Questioning in Instruction
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 96 No.8 pp.562-566
Reviewer: Bethany
Date of Review: 4/12/06

This article explained the role of questioning in the classroom and what teachers need to do and think about in order to make their questioning a valuable part of the teaching and learning process. The authors write, "questions are central to the type of learning that takes place in the classroom; teachers' questions control students' learning because they focus students' attention on specific features of the concepts that they explore in class." So, questioning techniques can make or break student learning. So, how can we learn to ask good questions? The authors suggest a few things to think about. First of all, it is important to think about your purpose for asking a particular question. With this purpose in mind, you can go on to decide on the content and form of the question. Closed form questions are helpful when seeking a particular answer, while open form questions are aimed at promoting a description, strategy, or opinion. The authors suggest that teachers take time to plan for successful questioning. Teachers should identify the main concepts of their lesson and then ask themselves questions regarding the objectives and strategies they hope to present and use. From this reflection, teachers will be more adequately prepared to question students effectively.

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Keywords: Equity/Diversity, ,
Ref: Bethany11
Author(s): Gilbert, Melissa C.
Date: 2001
Title: Applying the Equity Principle
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 7 No. 1 pp. 18-19,36
Reviewer: Bethany
Date of Review: 4/18/2006

This article, entitled "Applying the Equity Principle" outlines some basic techniques teachers can use to ensure equity (particularly gender equity) in the classroom. While the strategies the authors describe seem like common sense, the article is a good reminder of these simple, effective techniques. It isn't a bad ideal to have cards with a student's name on each; these cards can be used ensure that every student is called on during the hour. Cooperative learning groups can help to ensure that each student participates in the learning. (The "jigsaw" technique seems like a particularly effective idea. Well-assigned group roles can also do the trick.) In order to combat gender stereotypes, it isn't a bad idea to have a bulletin board about women in science and math. It is a simple way to get the gender equity message across. Moreover, it is important to empower your students and give them an active role in discovering the mathematics.

While basic, I found this article to be a good reminder.

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Keywords: Assessment
Ref: Bethany12
Author(s): Mary L. Crowley
Date: 1993
Title: Student Mathematics Portfolio: More Than A Display Case
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: October 1993
Reviewer: Bethany
Date of Review: April 30, 2006

This article stresses the value in portfolio assessment. The NCTM standards even say, "Records of students' progress should be more than a set of numerical grades and checklists; they can include brief notes or samples of students' work." Very simply, a mathematics portfolio is a collection of student work. It can contain a student's best work or a combination of work that displays student improvement. Either way, the concrete examples contained in a student portfolio can "show the teacher or parent the student's performance in more detail than would an abstract number or letter grade."

The portfolio can be useful for assessment and evaluation. So, what goes into a portfolio? This depends largely on the portfolio's purpose, the age of the students, and the types of assessment activities used in the class. Ideas of portfolio items include: journal entries, a math autobiography, math research, several solutions to a challenging problem, an elegant proof, student-formulated problems, a book review, group projects, photos of presentations, etc.

When it comes to organization and evaluation it is important to be clear and specific when communicating your expectations to your students and their parents. Evaluating a portfolio seems a bit daunting, but with a detailed rubric, the ordeal can be somewhat simplified.

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Keywords: Assessment
Ref: Bethany13
Author(s): Clarke, Doug, Wilson, Linda
Date: 1994
Title: Valuing What We See
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: October 1994
Reviewer: Bethany
Date of Review: April 30, 2006

This article reminds teachers that continual in-class observation can be an extremely useful form of assessment. The authors give a few good ideas of how to keep track of these observations:

One documentation approach is to construct a grid with students' names down one side and the "focus aspects" across the top. "These focus aspects could be mathematical concepts, processes, dispositional factors, or a mixture." Then teachers can simply place a check mark beside a student's name as a particular insight or behavior is observed.

Software has been created to record observations. Each student has a barcode by his or her name and each "focus aspect" has a barcode. Teachers can scan a student's name and then the behavior or insight. The information can then be downloaded to a computer.

Rather than depending on a list of predetermined behaviors and skills, a grid with students names down the left and simply space next to each name for comments can be a good way to assess during class. No matter how the observation is done, it can be very useful in getting to know your students and their strengths and weaknesses.


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Keywords: Assessment
Ref: Bethany14
Author(s): Miller, L. Diane
Date: 1992
Title: Begin Mathematics Class with Writing
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: May 1992
Reviewer: Bethany
Date of Review: April 30, 2006

This article opens, "first impressions often influence the development of our attitudes toward, and decisions about people, places, and things." I think we all can see the truth in this statement, so it is only appropriate to say the same thing about the beginning of a math class. As Miller writes, "the first few minutes of a class often influence the success of the class." So, how can we make these first few minutes valuable and interesting to students? L. Diane Miller offers one option: writing.

The few minutes between classes can be hectic and the first few minutes of class can be equally jam-packed with necessary tasks like roll call, returning papers, etc. "Exemplary teachers do not waste class time." One way to get students started right away is to develop a routine where students enter the room, sit down, and get started on some sort of constructive activity. The use of writing prompts each day could be just the ticket. "Prompts elicit from students a written response to a specific question or problem." This sort of valuable writing activity can be the perfect introduction/transition to math class. It is also a useful form of "informal" assessment. The writing prompts can ask students to dig up old knowledge or to try their hands at something new. Either way, students responses can give you a good idea of what students know and don't know.

L. Diane Miller goes on to give examples of writing prompts for several different topics in math. I think this article is most useful for that fact.



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Keywords: Technology, Geometry
Ref: Bethany15
Author(s): Zheng,Tingyao
Date: 2002
Title: Do Mathematics With Interactive Geometry Software
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 95, No. 7 pp. 492-497
Reviewer: Bethany
Date of Review: 5/1/06

You could say that the main objective of mathematics is to teach students to think critically. The author of this article believes this and values the importance of problem solving skills in mathematics. In order to discuss mathematical topics, it is necessary to make “generalizations.” These generalizations can be formulas, theorems, or proofs and it is important present these generalizations in a way that promotes student learning. One way to do this is to let students explore and uncover these generalizations for on their own. Consider, for example, the Pythagorean Theorem. Using a program like Geometer’s Sketchpad, students can discover this theorem by seeing patterns in right triangles. This technology allows students to look at numerous examples of right triangles without being overwhelmed with dozens of computations. Geometer’s Sketchpad will record side length, angle measure, and sine, cosine and tangent values if you program it to do so. So, you can “pull on” a leg of the triangle and observe what happens to the recorded values.

This article gives another idea for an activity using Geometer’s Sketchpad. Technology, when used appropriately, can enhance the geometry classroom.




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Keywords: Research , Representations, Teaching Strategies
Ref: Bethany16
Author(s): Walmsley, Angela L.E.; Hckman, Aaron
Date: 2006
Title: A Study of Note-Taking and Its Impact on Student Perception of Use in a Geometry Classroom"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 99, No. 9, pp. 614-621
Reviewer: Bethany
Date of Review: May 16, 2006

This article describes an informal research study on note-taking that was conducted in a Geometry classroom. The authors warn that too typically teachers insist on note-taking strategies that bind and inhibit the curiosity and creativity of students. This sort of note-talking often involves copying "stuff" from the board, including definitions, theorems, processes, etc. The purpose of this study was to test the "effectiveness" of this traditional type of note-taking against two alternatives. The hope was that maybe these alternatives would engage students and sustain their attention better than the old way.

The first alternative consisted of a three-column note sheet; at the top, there is space to fill in “Today’s Question,” and the three columns are labeled “the concepts,” “the examples,” and “background information.” The aim here is to make the note-taking process more thoughtful and meaningful, by connecting it all together.

The second alternative was a “mini-textbook” assignment. Students were given a task to, at the end of the unit, produce their own “mini-textbook.” The teacher would require a list of concepts to be included/explained with 2 examples for each. The aim here was to give students a reason to take careful notes during class. (Also, the list of concepts would be a helpful guide for note-taking.

Student response was positive for both alternatives but most preferred the columnar method. I found this article to be very helpful and interesting. The article also included actual templates for the note-taking strategies described, as well as a sample scoring rubric for the “mini-textbook” assignment.


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Keywords: Technology, Representations, Activities
Ref: Bethany17
Author(s): Moyer, Todd. O.
Date: 2006
Title: Non-Geometry Mathematics and The Geometer's Sketchpad
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 99, No.7, pp. 490-495
Reviewer: Bethany
Date of Review: May 16, 2006

This article describes some of the benefits of using The Geometer’s Sketchpad (GSP) in even the non-geometry classroom. What makes this software to exceptional is its dynamic, “movie-like” nature. GSP can be used to graph many functions at once and does a quicker, better job than a graphing calculator. The author gives several examples of activities that take advantage of the dynamic nature of this software. One such example is the following: students can discover properties of the sine curve represented by the function, f(x)=Asin(Bx+C)+D. That is, students can simply plug in values for A, B, C, and D and quickly see the effect. GSP can also be used to visualize and further understand other topics including the composition of two functions and function inverses. What makes this article so helpful is that it includes actual step-by-step instructions for carrying out these activities using GSP. When used carefully and clearly, technology can greatly enhance student understanding of topics like these.


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Keywords: Algebra, Teaching Strategies, Keyword 3, Optional...
Ref: Bethany18
Author(s): Clausen, Mary C.
Date: 2005
Title: Did you 'code'?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 99, No. 4, pp. 260-263
Reviewer: Bethany
Date of Review: May 16, 2006

In this article, Mary C. Clausen describes an idea she came up with to help students solve algebraic equations. Symbolic algebra is not easy to grasp at first and this method of “coding” is a great idea to help students get past all the symbols and figure out how to solve equations. It is based on the basic method used to solve equations: do the opposite of what is “done” to the variable to “undo” it. Once all the operations on the variable are “undone,” the solution remains. Clausen simply has her students write down “in code” what is being done to the variable paired with must what be done to undo it. For example consider the following equation: x+3=0. Students would “code” by writing “A3S3” to say that 3 is being added to x, so 3 needs to be subtracted from both sides.

The article gives more detailed “rules” for coding that you could give to your students to explain the procedure for more complex equations. “Coding” can help students with special needs better understand algebra, as well remind all students why they are performing the operations they’re performing. Coding requires students to outline their thinking before completing a problem.

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Keywords: Assessment, Problem Solving, Keyword 3, Optional...
Ref: Bethany19
Author(s): Goetz, Albert
Date: 2005
Title: Using Open--Ended Problems for Assessment
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 99, No. 2005, pp. 12-17
Reviewer: Bethany
Date of Review: May 17, 2006

The author of this article starts out by pointing out an important point about assessment: we should assess what we value. Such a statement seems like common sense, but I wouldn’t be surprised if it is often forgotten. In his classroom, Albert Goetz values cooperative group activities based on open ended questions. Students work in groups on problems like this on at least a weekly basis, so it is obvious that this type of cooperative interaction is greatly valued by him. So, according to his statement about assessment, this type of group activity should be assessed. This work could very well be assessed by completing group worksheets in class or giving group presentations, but Goetz goes so far as to include group activity on tests. In this article he describes how he integrates group work on a final exam.

First of all, including a group question on a test can only happen when students are familiar and quite comfortable working in groups and understand that goals and expectations before them. For the final exam, he allows one fourth of the time and points to go toward a group question. He composes groups of 3-4 as heterogeneously as possible and students work together to respond to a multi-part open-ended question. (The example in this article is a data analysis problem.) I think this is a really innovative idea, and as long as students are fully comfortable working in groups and they understand the grading rubric they face. Goetz stresses the importance of a post-exam discussion if possible. BR>

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Keywords: Communications, Teaching Strategies, Keyword 3, Optional...
Ref: Bethany20
Author(s): Mason, Ralph, T.; McFeetors, P. Janelle
Date: 2002
Title: Interactive Writing in Mathematics Class: Getting Started
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 95, No. 7, pp. 532-536
Reviewer: Bethany
Date of Review:

The authors of this article discuss the value of interactive writing in the mathematics classroom. This is what is meant by “interactive”: teachers provide a prompt, students respond, teachers briefly reply. This type of writing can be the key to student reflection and teacher-student communication. Some ideas for writing prompts include the following: “How did you study for this test? Did your study habits pay off?” “Do you like what we’re studying in class this unit? Is it difficult? Easy?” “What’s one think you like about algebra?” Prompts can also be more along the line of problems. (i.e.- a story problem that requires students to write out their thinking, their work, and rationale for their solution.)

Writing can help students express feelings, concerns, triumphs, etc. in words in a non-threatening way; that is, students can write to respond, report, reflect, and relate. This article gives some good guidelines for incorporating writing into the math classroom.

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