Stephanie's Article Reviews, 2007

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Keywords: Geometry, Activities
Ref: Stephanie1
Author(s): Adams, Thomasenia Lott; Aslan-Tutak, Fatma
Year of publication : 2005/2006
Title: Serving Up Sierpinkski!
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 11, Issue 5, pages 248-253
Reviewer: Stephanie
Date of Review: February 15, 2007

This article begins by providing the reader with background information on Sierpinski and describing how one can construct a Sierpinski triangle. Next the authors explain fractals, which are shapes that are self-similar because they appear identical at different magnifications. The authors then describe how the area of the Sierpinski triangle is found and how the Sierpinski triangle can be used to demonstrate the concept of infinity to students. The following two pages consist of two student activity sheets and solutions for teachers.

One thing I enjoyed about this article is the background information on Sierpinski. It says, “Researching his life would be a good way to integrate mathematics and the social sciences,”(248). Also, it mentions that there are two stamps that honor Sierpinski. This reminds the reader of the relationship between mathematics and other academic subjects as well as the fact that mathematicians have made a huge impact on society.

I really enjoyed reading about fractals and think that students would benefit from learning about fractals. They are very interesting, both visually and mathematically; especially when one examines their area and perimeter. I think students would be interested in leaning about how fractals can be seen in the physical world, i.e. coastlines (the area of a country is finite but the coastline is infinite).

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Keywords: Teaching Strategies, Planning
 Ref: Stephanie3
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, pages 478-483, April 2000
Reviewer: Stephanie
Date of Review: February 28, 2007

I thought this article was very valuable; the author offers many useful pieces of advice to future math teachers. I really like that he made a commitment to change 10 percent of his teaching every year. I think this is a very good idea because as more and more research is conducted to discover the best approach to teaching math, educators are learning that techniques used in the past may not be the most effective way of teaching math. As math educators we owe it to our students to constantly modify our teaching methods so that our students get the most out of our lessons.

Another thing I really liked about the article was the point the author made on teachers’ responses to students answers. If a teacher responds to excitedly to one student’s response, other students might be too intimidated to follow. If a teacher responds negatively, that student might be discouraged from participating again in the future. The author says that teachers need to encourage more discussion and move on to the next comment. By doing this students build on each other’s ideas and everyone feels as though they contributed to learning. Also, this promotes all students to participate.

Lastly, I really liked the think-pair-share strategy because this approach results in students thinking individually about a topic and benefiting from other students’ insights.

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Keywords: Research
Ref: Stephanie4
Author(s): Cramer, Kathleen; Wyberg, Terry
Year of publication :
Title: When Getting the Right Answer is Not Always Enough
Journal or Publisher:
Volume, Issue, Pages: The Learning of Mathematics
Reviewer: Stephanie
Date of Review: March 6, 2007

This article points out that a student’s thought process is just as important as his or her final answer. The article mentions some strategies that are incorrect but lead to correct answers. One such strategy involves students identifying fractions with larger numbers, such as ¾, as being greater than fractions with smaller numbers, such as ½. While the answer is correct, the approach is incorrect and if the student tries to use that approach on a different question, he or she will not necessarily find the correct answer. I think this is an interesting point and something that teachers need to keep in mind when teaching fractions.

Also I feel Ben’s strategies for ordering fractions on the written test and in the interview are interesting. During the written test he converted the fractions to percentages using a calculator but during the interview he used whole-number thinking; he said that because 4/15 involves larger numbers, it is greater than 4/10. I think that it is easy for a student to pull out a calculator and convert fractions to percentages; students should be introduced to this approach only after they have successfully mastered the other approaches to ordering fractions. I think it is very interesting that students tend to rely on whole-number thinking when they lack mental representations for fractions.

Another thing I found attention-grabbing is the fact that Kevin, who often found common denominators, regressed to using the whole-number strategy, adding numerators and denominators when the interviewer asked him for more of an estimate. It would probably be beneficial for Kevin to review fractions using direct modeling and then mental images.

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Keywords: Activities, Geometry, Problem Solving
Ref: Stephanie5
Author(s):
Year of publication :
Title: Illuminations Marco Polo, "Cubes Everywhere"
Journal or Publisher:
Volume, Issue, Pages: http://illuminations.nctm.org/Lessons.aspx
Reviewer: Stephanie
Date of Review: March 15, 2007

This lesson is very useful for teachers who want their students to learn about cubes by using spatial thinking. It starts out by showing a map of a river and a shoreline. On the shoreline there is a steeple, water tower, and lighthouse. After the map there are six pictures and the teacher is supposed to ask the students to determine the order the six pictures were taken in while the boat moves along the shoreline. By doing this, students imagine how different shapes look from different angles.

I think this lesson is very well-designed because its visual characteristic will appeal to many students. Also I think that it is a good beginning to exploring cubes. Although the lesson is fun and enlightening, there are a few things I do not like about the lesson. First, it never says which direction the boat is traveling, which makes it quite confusing. If the direction the ship is traveling in were listed, the lesson would be much more understandable. Second, I think the worksheet following the activity is too long; this can easily be fixed by assigning it or working on it for more than one day. Overall, I think the lesson would be very enjoyable.

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Keywords: Manipulatives, Teaching Strategies
Ref: Stephanie6
Author(s): Lappen, Glenda; Fey, James T; Fitzgerald, William M; Friel, Susan N; Phillips, Elizabeth Difanis.
Year of publication : 2002
Title: Accentuate the Negative
Journal or Publisher: Prentice Hall
Volume, Issue, Pages: Pages 1a-1j
Reviewer: Stephanie
Date of Review: March 22, 2007

In the overview to the Connected Mathematics Program the author talks about how students sometime develop technical mathematical skills but they do not know how to apply their knowledge to solve problems. It is important that students can integrate mathematical knowledge so that they are able to relate their knowledge to real-world situations. The authors mention that students often get confused when multiplying negative numbers and they suggest using two different colored chips (such as red and black) to represent negative and positive numbers. This is a good suggestion because the use of manipulatives helps students visualize abstract concepts. Since many other topics in math build off the concept of positive and negative numbers it is important that students recognize their properties.

The authors then talk about how big ideas in positive and negative integers relate to mathematical concepts students learned previously and concepts students will learn in the future. It is important to keep this in mind because if a teacher knows a specific concept will be important in future math, he or she can emphasize it until his or her students have mastered the concept. The overview ends with materials needed, technology needed, and an assessment summary. One of the listed assessments is the notebook/journal. The authors describe the notebook/journal as a safe place where students can try out their thinking. I think this assessment is a good way for a teacher to find out what a student is thinking.

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Keywords: Number and Operation
Ref: Stephanie8
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: Stephanie
Date of Review: April 10, 2007

This article focuses on the fact that the majority of students view “=” as a sign telling the reader to perform a certain operation rather than a representation of a relationship between numbers. I think it is interesting that on page 10 Lucy says that the 5 in the equation is only there to confuse students and that the teacher sometimes includes extra pieces of information, which are not necessary, in order to trick students.

Another thing I think is interesting is the fact that true/false questions encourage students to examine conceptions of the meaning of the equal sign. This is uncommon because the general rule is that teachers should not ask their students yes/no questions. The examples the author gives on page 16 show how asking true/false questions can guide students in their understanding of the equal sign. I especially like the idea of including a zero in the number sequence such as the following: 9+5=14+0. This allows students to adjust to the use of two or more terms on both sides of the equal sign.

Lastly, I think the table on page 20 is very enlightening because it points out examples where people use the equal sign incorrectly. By doing so they are confusing students; the kids do not understand that it represents a relationship between numbers. It is important for teachers to realize that they are confusing students so that they can stop using it the equal sign incorrectly.

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Keywords: Algebra
Ref: Stephanie9
Author(s): Usiskin, Zalman
Year of publication :
Title: Conceptions of School Algebra and Uses of Variables
Journal or Publisher: Algebraic Thinking, Grades K-12; Defining Algebraic Thinking and an Algebra Curriculum
Volume, Issue, Pages: Pages 7-13
Reviewer: Stephanie
Date of Review: April 19, 2007

This article mentions many different descriptions of algebra, two of which are a generalizer of patterns and a study of procedures. Depending on which definition one uses, he or she comes to different conclusions when looking at the problem 5x+3=40. According to the article, “Under the conception of algebra as a generalizer of patterns, we do not have unknowns. We generalize known relationships among numbers, and so we do not have even the feeling of unknowns. Under that conception, this problem is finished,” (page 9). This is really interesting because I would not have thought of this problem in that manner. This leads me to believe that my outlook on Algebra may be limited.

The article also mentions different notions of variables. Variables are constants/unknowns that represent a relation, pattern generalizers, arguments, and parameters. In the beginning it mentions equations such as 40=5x, sin x=(cos x)(tan x), and y=kx. I never realized that variables are so different when used in different contexts. I’m glad the authors point out the different uses of variables because it may help teachers understand why students may become confused at the concept of variables and algebra.

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Keywords: Curriculum
Ref: Stephanie10
Author(s):
Year of publication :
Title: Lesson 2, Multiplying Matrices (Unit 1, Matrix Models)
Journal or Publisher:
Volume, Issue, Pages: Core Plus, Book 2, pages 26-35
Reviewer: Stephanie
Date of Review: April 26, 2007

This lesson starts out by presenting readers with a situation that requires a matrix to represent the percentage of customers that bought Nike, Reebok, and Fila shoes this year who will buy Nike, Reebok, or Fila next year. The reader is told that 700 people bought Nike, 500 bought Reebok, and 400 bought Fila. Then the reader is asked to determine how many people will buy each of the three brands next year. I like the way the lesson is set up because it pushed me to do matrix multiplication without me even realizing I was doing matrix multiplication. The author gave a problem where the reader explores matrix multiplication and then the author gave the formal definition of matrix multiplication.

Another think I liked about this lesson is that after the brand switching matrix problem the author says “the way you have been multiplying matrices in this investigation is so useful that all calculators and software with matrix capability are designed with this kind of multiplication built in,”(29). This allows students to see a specific situation where matrices are applied and how useful this particular branch of mathematics is.

The lesson goes through more word problems involving matrices and ends with a checkpoint, which asks the reader to describe how to multiply two matrices, to give two reasons why it may not make sense to multiply two matrices, and whether the order of matrix multiplication matter. I like this because rather than giving students formulas and properties of matrices the students discover them on their own.

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Keywords: Communications, Activities
Ref: Stephanie11
Author(s): Artzt, Alice F.; Newman, Claire M.
Year of publication : 1997
Title: How to Use Cooperative Learning in the Mathematics Class
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages:
Reviewer: Stephanie
Date of Review: May 2, 2007

On the first page of How to Use cooperative Learning in the Mathematics Class the authors point out the fact that that many schools value individual accomplishments, which hinders cooperative learning. I think this is a really good point; by focusing on individual actions, schools may send a message to students that learning to work together effectively with others is unimportant. Students will benefit if they feel a sense of community and that their actions affect others; this can be achieved through cooperative learning.

The authors provide many creative ideas for math teachers to use in their classes. They recommend that teachers form heterogeneous groups and give groups time to think of a team name. According to them, ?heterogeneity appears to lead to positive academic and social outcomes? and creating a team name allows students to find common interests. Also, there are many activities at the end of the book that can be used in math classes and require cooperative learning. I noticed that many of the activities can be done individually, but doing them in a group allows students to split up the work and use each other for resources.

Another good point the authors make is that positive attitudes towards math are important for math students. If students are interested in the topic they are learning in class then they will be more likely to put in extra effort to completely understand the material. This is why cooperative learning is so important; group activities allow students to learn and have fun. This in turn causes students to have a positive attitude towards math, which encourages them to continue working hard.

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