Emily's Article Reviews, 2007

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Keywords: Activities, Algebra, Geometry
Ref: Emily1
Author(s): Taber, Susan
Year of publication : 2005
Title: The Mathematics of Alice's Adventures in Wonderland
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 11 No. 4
Reviewer: Emily
Date of Review: February 15, 2007

Taber's article gives a brief background on how the book Alice's Adventures in Wonderland came to be, and then describes ways in which the book may be used to attack particular mathematical topics. First, Taber talks about how Alice is able to grow and shrink by drinking a potion or eating a cake. She ties this idea into an activity for teaching multiplication by rational numbers less than 1. Ultimately, the goal of this activity is for students to see that when they multiply by a number less than 1, the get a smaller number than the original. Next, Taber uses the Cheshire Cat's habit of appearing and disappearing in pieces to examine additive and multiplicative change, as well as when something can be represented mathematically and when it cannot be represented mathematically. Taber also uses the idea that (most of the time) Alice's proportions do not change when she changes size to talk about proportional and nonproportional transformations. She then goes on to extend this idea to transformations on the Cartesian plane. Finally, Taber uses Alice's "mistakes" to approach logical reasoning in a mathematical setting.

I found this article to be both interesting and frustrating at the same time. Perhaps I am just an odd example, but I have never actually read Alice's Adventures in Wonderland, and I am not sure I could name many middle schoolers who have. From reading this article I saw that there are many interesting activities that get at some math concepts that are often difficult for students, but I also felt a little lost during much of the article because I did not have a solid frame of reference. I think these activities could prove to be very valuable in a classroom, but only if EVERY student has read Alice's Adventures in Wonderland. Ultimately, I think this article can serve well as a springboard for thinking about what other works of literature (that are possibly more read) include math concepts and problems. The idea of using a well-known story to approach a mathematical concept seems like it would be interesting and intriguing.

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Keywords: Teaching Strategies
Ref: Emily2
Author(s): Reinhart, Steven C.
Year of publication : 2000
Title: Never Say Anything a Kid Can Say
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: vol 5, no 8: pp 478-483
Reviewer: Emily
Date of Review: March 1, 2007

This article focuses on the transformation of one teacher’s instructional strategies. Steven Reinhart is a middle school math teacher in Wisconsin who previously taught clearly and thoroughly, though not effectively from the front of the classroom. Over a series of years he began to change his teaching strategies one by one. He moved from teaching in front of the class where only he was learning, to a style of teaching in which students ultimately teach themselves and each other.

The main focus in this article was questioning strategies. The five questioning strategies that Reinhart discussed were: never say anything a kid can’t say, ask good questions, use more process questions, replace lectures with sets of questions, and be patient. Reinhart stressed the importance of not only asking quality questions that guide student thinking, but to give all students enough time to think through their thoughts to arrive at an answer.

Additionally, this article focused on ways to include and encourage more discussion in a math classroom. By incorporating think-pair-share, small groups, and large groups, as well as requiring every student to contribute questions and answers, lines of communication can be opened. Then, students become responsible for their own understanding and learning.

Overall, this article makes a lot of sense to me. When you are able to get students to take responsibility for their understanding, and really get them to ask good questions, the learning process becomes so much more enjoyable for you and for them. I also really appreciated the portion of the article on “wait time”. It is important that we give students a chance to really think through things, rather than always calling on the first hand that shoots in the air.

I suppose my one reservation about this article is simply how do you get through an entire year’s curriculum using this sort of approach? From all my experience in a classroom, anything involving group work and discussion takes longer, and the school year is already rushed enough as it is. I think that this, just like every other method of instruction, must be used in conjunction with other more direct methods in order to find a balance that allows students to truly learn while also getting through all they are supposed to cover.

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Keywords: Number and Operation, Research , Assessment
Ref: Emily4
Author(s): Cramer, Kathleen; Wyberg, Terry
Year of publication :
Title: When Getting the Right Answer is Not Always Enough: Connecting How Students Order Fractions and Estimate Sums and Differences
Journal or Publisher: The Learning of Mathematics
Volume, Issue, Pages: p. 205-220
Reviewer: Emily
Date of Review: March 7, 2007

This article focused on the research investigating the strategies students use when ordering fractions, as well as when asked to estimate the sums and differences in fractions. The bulk of this article followed the strategies of three fifth grade students who were part of a district wide written test, as well as chosen to participate in an interview.

The four strategies for ordering fractions that were discussed in this article were finding a common denominator, converting to percent, comparing to benchmarks (1/2, 1, etc), and cross-multiplication. One major finding of this study was that students could possess a successful strategy for ordering fractions but still be unable to estimate sums and differences. In general, students who had a more conceptual understanding of fractions, rather than just a procedure to follow, were more successful at the tasks presented to them. However, the vast majority of students, while able to correctly order fractions in isolation, are not able to use that skill to estimate sums and differences.

I thought this was a rather interesting article to read. Personally, I find it very intriguing to see how children's minds work and in essence get inside their thought process. This article did a good job at offering insight into advances in the teaching of fractions, as well as areas that most students still need a lot of work in. I strongly believe that it is imperative for all students to be able to accurately estimate sums and differences in fractions, because it is a skill that is used no matter who you are or what you do.

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Keywords: Activities, Algebra, Measurement
Ref: Emily5
Author(s): Chandler, Kristen
Year of publication :
Title: NCTM Illuminations--"Constant Dimensions"
Journal or Publisher:
Volume, Issue, Pages: http://illuminations.nctm.org/LessonDetail.aspx?id=L572
Reviewer: Emily
Date of Review: March 14, 2007

This lesson plan guides middle school students through an exploration activity regarding the ratio of length to width in a rectangle. Students use a traditional method (centimeters or inches) to first measure both the length and width of the rectangle on their worksheet. Then, they use alternative methods of measurement such as pennies, pencils, erasers, etc. to measure the same rectangle. Ultimately, students graph the length of the rectangle and the width of the rectangle in different units and should see a linear relationship between all of the points, with the slope being the ratio between the length and width, regardless of the measurement tool used.

I thought this was a very creative, hands on way to allow middle school students to explore a particular property of rectangles--specifically, regardless of what is used to measure the length and width the ratio between the two remains the same. It also seemed like a good activity to get students thinking a little more abstractly than just simply knowing formulas such as length times width equals area. I wonder, however, just how obvious the relationship in this activity would be to students. Most of the middle school students I have worked with lately don't have a real strong understand of what slope is, so that might make this exploration a little bit more difficult.

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Keywords:
Ref: Emily6
Author(s): Fey, James T.; Fitzgerald, William M.; Friel, Susan N.; Lappan, Glenda; Phillips, Elizabeth Difanis
Year of publication : 1998
Title: Accentuate the Negative: Integers
Journal or Publisher: Dale Seymour Publications
Volume, Issue, Pages: Overview pp. 1a-1j
Reviewer: Emily
Date of Review: March 20, 2007

The overview of this book provides teachers with a basic look at what students will be investigating and how they will be investigating it. The two ideas of using number lines and using two-colored chips are discussed as the main methods of investigation. Additionally, the overview talks about how this particular unit fits in with the rest of the Connected Mathematics Curriculum. This introduction also details the materials in this book, the materials needed besides this book, and the approximate number of days/lessons needed to cover each investigation.

I liked how the overview gives teachers a glimpse into what will be going on so that you have an idea of where you're headed. Also, the overview does a nice job of giving clear examples and step-by-step procedures. With regards to the number line idea, "moving to the right" for adding postive integers and "moving to the left" for adding negative integers, I have also seen this done with life-size number lines--using tape on the floor, you mark out the units, and then students stand at the integer they are starting at. For example, to do 5+-7, students would stand at the postive five and then face the negative integers, because they are going to be adding a negative number. Then, they walk 7 steps forward (walking forward is addition, walking backward is subtraction). I thought it was a nice way to get real kinesthetic learners invovled.

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Keywords: Technology, Teaching Strategies, Technology
Ref: Emily7
Author(s): Bitter, Gary G.; Hatfield, Mary M.
Year of publication : 1992
Title: Calculators in Mathematics Education
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: "Implementing Calculators in Middle School Mathematics: Impact on Teaching and Learning" pp. 200-207
Reviewer: Emily
Date of Review: April 3, 2007

This article is concerned with a study conducted in Arizona in 1988. The middle school chosen consistently lagged behind on standardized test scores, had a turn-over of almost one-third of its students yearly, and had a majority of its students qualify for Title 1 services.

At the beginning of the 1988-1989 school year, the district bought enough TI Explorer calculators for every student to have the use of one on a daily basis. The students were allowed to use these calculators in class, on tests, and at home.

The study found that students performed significantly better on three of the mathematics subtests on the Iowa Tests of Basic Skills after having the use of the calculators for the school year. Interestingly, the performance of girls improved more drastically than that of the boys. This article then goes on to describe how a similar plan can be put into action in any district, focusing on the responsibilities of the administration, teachers, students, and parents.

I thought this was an interesting, if outdated, article. I was surprised by the drastic improvement that occurred over one year simply by having access to calculators. Honestly, I would have thought the students' performance on basic skills tests would decrease, because they would become dependent on the calculators, but that was not the case in this district. I think that anyone planning to teach math, particularly in elementary and middle school, must wrestle with the question of how much to use calculators.

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Keywords: Algebra, Number and Operation, Teaching Strategies
Ref: Emily8
Author(s): Carpenter, Thomas; Franke, Megan Loef; Levi, Linda
Year of publication : 2003
Title: Thinking Mathematically. Chapter 2: Equality
Journal or Publisher: Heinemann Books
Volume, Issue, Pages: pages 8-24
Reviewer: Emily
Date of Review: April 11, 2007

This chapter focused on students' understanding of the equal sign. One commonly held misconception is that the equal sign is meant as an action marker, that students must do something when they see it. However, this chapter emphasized how the equal sign is a relational concept between two different things.

By using examples and student responses, this chapter showed how students often interpet an equal sign. Also, the author discussed how to move students through the four different benchmarks for equal signs by using true/false number sentances.

I thought this was a really interesting chapter. It's amazing to see how different students think about and approach problems. It is also nice to have specific benchmarks to help work students through so that you can really gauge the progress you are making. I was really impressed with the logic some of the students used in coming to their conclusion.

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Keywords: Algebra, Curriculum
Ref: Emily9
Author(s): Usiskin, Zalman
Year of publication :
Title: Algebraic Thinking Grades K-12
Journal or Publisher:
Volume, Issue, Pages: Conceptions of School Algebra and Uses of Variables
Reviewer: Emily
Date of Review: April 19, 2007

In this chapter, Zalman Usiskin examines the different strands of algebra and the evolution of teaching algebra in schools. The beginning of the chapter focuses on the difficulty of defining the term "algebra" and examines how algebra is used in middle school and high school comepared to college. It also begins to examine the meaning of a variable, and how its definition has also changed through time.

Usiskin also examines four important conceptions of algebra which "correlate with the different relative importance given to various uses of variables". The four conceptions are: algebra as generalized arithmatic, algebra as a study of procedures for solving certain kinds of problems, algebra as the study of relationships among quantities, and algebra as the study of structures.

I thought this article was intersting to read and had a lot of points I had never considered. I have loved algebra since middle school, and I haven't really thought about all the ways in which I use both basic and advanced algebra all the time. One of the things I really like about this article was its examination of "variables" and different ways to represent and use them.

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Keywords: Algebra, Curriculum
Ref: Emily10
Author(s): Coxford, Arthur, et al
Year of publication : 1999
Title: Multiplying Matrices from Contemporary Mathematics in Context
Journal or Publisher: Everyday Learning Corporation
Volume, Issue, Pages:
Reviewer: Emily
Date of Review: April 25, 2007

This chapter is from the Contemporary Mathematics in Context book, a part of the core-plus mathematics project. In a larger context, it is part of a moderately reformed textbook, as opposed to a traditional textbook. This particular series of math books seeks to incorporate different strains of math throughout, rather than focusing on a year of algebra, a year of geometry, and so on.

The concepts covered in this chapter are matrices, matrix multiplication, and applications of matrix multiplication. The teaching/instruction is done through investigations that are meant to be done in groups. The book guides students through a step-wise process, inserting vocabulary when necessary. One big focus of the core-plus project is getting students to see connections between what they are doing, what they have already done, and the real world. As such, there is a lot of implicit review and explicit applications.

I spent January observing teachers at St. Paul Central High School who used this book, and saw it used with various degrees of success. As with most things, the more motivated (and generally, that meant the more advanced) the students were, the more successful this book was. However, when the students did not really care and were not self-motivated, there was little learning done. I like how this particular curriculum focuses on applications and connections, but at times I feel like it sacrifices the amount of actual instruction. This book is hard to navigate if you are looking for a particular topic or idea, as it does not follow a necessarily sequential pattern. Overall, this book can be used very well, but it depends--in my opinion--a lot on the students who are using it.

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Keywords: Algebra, Curriculum, Teaching Strategies
Ref: Emily11
Author(s): Carpenteer, Thomas P.; Franke, Megan Loef; Levi, Linda
Year of publication : 2003
Title: Thinking Mathematically: Integrating Arithmatic and Algebra in Elementary School
Journal or Publisher: Heinemann
Volume, Issue, Pages: pp. 27-63
Reviewer: Emily
Date of Review: May 2, 2007

This book focuses on ways to integrate algebra in the teaching of basic arithmetic during the elmentary school years. I specifically looked at the sections titled, "Developing and Using Rational Thinking" and "Making Conjectures About Mathematics".

In the section on developing relational thinking, the authors examime how students at equality benchmarks three and four solve problems. The focus of the chapter is on how to move students effectively from benchmark three to benchmark four, without explicitly telling them how to do it. This section uses a particular students interview, as well as some class interviews with commentary. The real highlighted point is that examples/questions are the key--how you choose what to ask your students next will affect the direction in which their thinking goes.

The second section on making conjectures was actually really interesting. I was impressed with the level of responses the teachers recieved with regards to possible "rules" about arithmatic. In general, this section focused on how to use guided questioning to help the class, as a whole, come up with and refine their mathematical conjectures.

I really enjoy this book because of the inclusion of transcripts of actual student interviews. It is really amazing to see the way students respond to questions that are thrown at them. I would really like to observe an elementary math class in which a lot of these ideas are actually used. Overall, I think this book has a lot of valid points and suggestions on moving kids forward in mathematics.

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