Katie's Article Reviews, 2007

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Keywords: Teaching Strategies, Assessment.
Ref: Katie1
Author(s): Pierce, Rebecca L.; Adams, Cheryll M.
Year of publication : 2005
Title: Using Tiered Lessons in Mathematics
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 11, No. 3, pp. 144-149
Reviewer: Katie
Date of Review: February 19, 2007

The focus of this article is differentiated instruction and how it can be incorporated into the classroom on multiple levels. By use of the CIRCLE MAP (Creating an Integrated Response for Challenging Learners Equitably: A Model by Adams and Pierce), differentiation can be incorporated through classroom management techniques, anchoring activities, instructional strategies, and assessments.

The idea behind differentiated instruction is that students within the classroom have various interests, learning styles, and abilities. These differences create the necessity for teachers to re-evaluate their teaching methods to ensure that the material is accessible to all students. One way this can be made possible is through flexible grouping. Small groups make it easier for teachers to meet all students’ individual needs by allowing students to work together and consult one another while the teacher is aiding other students.

Anchoring activities provide ways for students who complete the assigned material quickly to continue expanding their knowledge while other students finish their assigned work. These activities are not intended to be completed by the whole class, but they do help to eliminate the distractions that can arise from some students finishing ahead of others.

Assessment can be differentiated similar to the ways in which both instruction and activities are differentiated. Assessment should also be based off of the lessons that are given on a certain subject, so there should be different assessments for each of the different groups that have been created in the classroom.

This article does a very thorough job of outlining numerous ways in which classrooms can become more accommodating for students with different talents. The suggestions it offers for classroom management, instruction, activities, and assessment allow me to better understand the challenges that are present for students in their everyday academic lives. It also illustrates that for teachers to accomplish this, they will need the support of other teachers and support staff, and that these are resources that should be utilized to ensure the success of differentiation in the classroom.

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Keywords: Teaching Strategies
Ref: Katie3
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, www.nctm.org
Reviewer: Katie
Date of Review: February 28, 2007

This article provided me with a nice dose of reality. The author very simply and honestly stated that his students were not learning to their full potential. I think that the majority of teachers would feel uncomfortable saying this, and even more so, I believe many teachers would be unwilling to put their students, classroom, and teaching methods through the type of radical change displayed by Reinhart. What is portrayed in the article is an understanding that although you as the teacher may be presenting the material in a very "learnable" fashion, it is not guaranteed that the students will comprehend it. The changes made by Reinhart provide a tremendous example of keeping the students as the number 1 priority in the classroom.

I found some of the changes mentioned in Reinhart's article to be rather surprising. The idea that students should paraphrase each other is new to me in the context of a math classroom. I can see how a teacher would want to rephrase and clarify what a student says, but now I see by doing that, the teacher allows other students to not pay attention, and also undermines the importance of what the student said.

Another idea unfamiliar to me is whole group discussions regarding specific problems. When I think of math classes in which I have been a student, I recall daily lectures and notes regarding the content being covered; I do not have any recollection of large group discussions. Reading this article, however, helps me to understand the benefits of having students talk to each other about problems, and the value of always asking one more question.

Overall, I think the article did a great job of forcing teachers to take a closer look at their students and classroom, carefully suggesting that there is always room for improvement.

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Keywords: Research , Problem Solving, Manipulatives
Ref: Katie4
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: Pp. 205-220
Reviewer: Katie
Date of Review: March 7, 2007

This article helped me gain a much better insight as to the processes students use when figuring out answers to problems such as these. I have become quite used to solving problems without putting much thought into how I obtain the answer; this article helped me see how detrimental that can be to learning.

One area in which this article supports my personal learning style is through the importance of visual representation. One strategy I have always relied upon is the creation of visual aids to assist my understanding of certain problems. Along with this procedural type of support, the article also recommends doing activities to promote conceptual thinking. Conceptual thinking is a tool that must be worked with over time, something that students will develop at their own pace. As teachers, it is our job to recognize the level at which our students are understanding and build them up from there.

The last point of this article is probably the most important: getting students to be aware of how they are thinking and how they are getting their answers. It is wonderful if students get the correct answers, but it is more valuable to their learning if they can understand how they obtained those answers. Once they are able to recognize the ways in which they uncover correct answers, the more aware they will become of their learning.

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Keywords: Games, Problem Solving
Ref: Katie5
Author(s):
Year of publication :
Title: Petals Around the Rose
Journal or Publisher:
Volume, Issue, Pages: http://illuminations.nctm.org
Reviewer: Katie
Date of Review: March 16, 2007

This game/lesson is one that I am unsure if I would assign to students younger than middle school. I have played it before and became so frustrated when I could not determine the rule that I put it down and walked away- this was just a few months ago. In it, 5 dice are rolled. After the roll, a person must determine the number of petals (hence the name "Petals Around the Rose." The teacher is not allowed to tell students the rule, and anyone who figures it out is also sworn to secrecy. I would not want students to feel bad about not figuring out the rule and in turn, feeling bad about their own mathemaical skills.

However, this game does provide opportunities for students to practice their organizational strategies. There are numerous ways in which this information can be organized to aid students' comprehension- those who are visual and kinesthetic learners. Students have the opportunity to practice making tables, graphs, or any other charts they determine may help them out. This also provides a lesson in logical reasoning. The teacher gives them prompts via questions to guide their learning. From these prompts, students must use their critical thinking and reasoning skills to assist them in their investigation.

As I said before, the age and personality of my students would determine whether or not I would use this in my classroom. Although it does allow for many types of applications of other mathematical ideas, I believe there are other activities that would lead to the same results with less room for negative feelings.

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Keywords: Curriculum, Problem Solving, Manipulatives
Ref: Katie6
Author(s): Lappan, Glenda; Fey, James T.; Fitzgerald, William M.; Friel, Susan N.; Phillips, Elizabeth Difanis
Year of publication : 1997
Title: Accentuate the Negative: Integers
Journal or Publisher: Dale Seymour Publications
Volume, Issue, Pages: Teacher's Guide
Reviewer: Katie
Date of Review: March 26, 2007

I found the introduction to this textbook very informative regarding the methods of teaching the concept of positive and negative integers. This is an area where I believe students need to have manipulatives easily accessible in order to comprehend and apply the material, and the use of the number line and colored chips seem to be a valuable approach.

The examples given for addition and subtraction are methods with which I feel students of various ages and abilities will be able to follow along. Real-life Scenarios to follow along with these examples can be easily created, which will make learning that much easier.

The ways in which multiplication and division are portrayed make sense and the pattern nature make them relatively easy to follow. However, the real-life aspect of these two operations was not explored to the extent I would have expected given the ways in which addition and subtraction were examined. Using these approaches, students may be able to carry out required tasks, but I believe they may struggle with real-life applications.

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Keywords: Research , Problem Solving
Ref: Katie7
Author(s): Ma, Liping
Year of publication : 1999
Title: Knowing and Teaching Elementary Mathematics
Journal or Publisher: Lawrence Erlbaum Associates, Inc., Publishers
Volume, Issue, Pages: Chapter 4
Reviewer: Katie
Date of Review: April 3, 2007

This chapter provided a very interesting look at the differences in how mathematics is approached in the United States versus how it is approached in China. In the past few decades there has been much talk of the U.S. falling behind other countries when it comes to research and fields such as mathematics and science. This chapter highlights one specific instance in which it clarifies why this is happening.

The concept explored is the relationship between perimeter and area. Both teachers from the U.S. and China were approached by students who "discovered" that area increases when the perimeter increases. The responses given by U.S. teachers to the claim were portrayed as surprisingly uneducated. Many of the teachers could not even remember the formulas for either perimeter or area and needed to look them up in a book. Others automatically accepted the students' claim, never taking the time to try it or ask the students to elaborate. Teachers who did explore the claim relied very little on strategies that used mathematical thinking.

The responses of the Chinese teachers were drastically different. Although there were still a few who accepted the students' claim without objection, the majority based their careful response on mathematical strategies and thinking, even if not all achieved the correct answer. Main differences in the responses involved the Chinese teachers not consulting books, addressing the topic of area versus perimeter rather than the validity of the students' claim, and most importantly (in my opinion), the Chinese teachers demonstrated a better knowledge of mathematical (geometric) concepts.

I find the differences between the teachers astonishing. Chinese teachers typically have 4 years less training in mathematics, so it remains unclear to me why they have more well-developed thinking strategies rooted in math. I think this chapter serves a wake-up call to the United States, especially teachers. We need to make sure that our thinking becomes more connected with the subject matter that is being taught in order to remain on the same performance level with other countries. If that does not happen, the U.S. will undoubtedly be surpassed.

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Keywords: Issues
Ref: Katie8
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: Katie
Date of Review: April 11, 2007

This chapter really helped break down the possible reasons why students may not fully understand mathematical concepts (such as equality). It clarified how important it is that teachers are prepared for problems students may encounter and stressed the importance of allowing them to explore and struggle with those concepts. It is easy for those of us who are at a higher level of mathematics to forget the difficulties we ourselves encountered at a younger age. To return to that point in our learning provides us with an opportunity to reexamine those problems and allow students to create a deeper understanding within themselves.

I find it reassuring on some level to know that the majority of students approach problems regarding equality in 3 ways. Knowing this as the teacher, it is much easier to guide students along their paths of "equailty discovery" if I know where they are beginning their journey. Teachers have the power to tell students whatever they wish, but for students to learn, it is important that they are allowed to examine concepts on their own, guided by the teacher.

I really appreciate the author's idea of viewing equality as a relationship between two or more things. That is an idea that can be applied to numerous concepts within mathematics. When implemented at a young age, it will undoubtedly make future mathematical concepts easier to comprehend. Also, it would seem reasonable to assume that when students use the same ideas over and over again, they will become more familiar with and comfortable approaching new math topics in the classroom.

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Keywords: Algebra
Ref: Katie9
Author(s): Usiskin, Zalman
Year of publication : After 1985
Title: Conceptions of School Algebra and Uses of Variables
Journal or Publisher:
Volume, Issue, Pages:
Reviewer: Katie
Date of Review: April 23, 2007

This article sparked my thinking of algebra and caused me to examine the ways in which I define and use it. If asked very simply, what is the difference between arithmetic and algebra, I'm not entirely sure I would be able to give a concise answer. That is a point I think is essential to this article. Algebra has been presented in so many different ways, in a number of different classes (levels), it is difficult to have just one definition.

The two issues brought up by Usiskin are very prevalent in the schools today. Many teachers, myself included, struggle with the idea of students knowing how to perform certain computations by hand versus only knowing how to do them on a calculator or other piece of technology. The other issue regarding when certain mathematical topics are introduced to students can be cause for concern when there are some students performing high above grade level while others are still far below. If this trend continues, the performance gap could easily remain as is, if not grow. However, is it fair to not challenge those students who are ready?

By breaking algebra up into four dofferent conceptions, Usiskin helps create a way for teachers to differentiate algebraic instruction in the classroom, while ensuring that all students are being exposed to it. By doing this, teachers are making it easier for students to build onto their current schemas to advance to the next level of algebraic learning.

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Keywords: Teaching Strategies, Connections
Ref: Katie10
Author(s):
Year of publication :
Title: Lesson Two: Multiplying Matrices
Journal or Publisher: Core Plus 2: Unit One: Matrix Models
Volume, Issue, Pages:
Reviewer: Katie
Date of Review: April 26, 2007

This lesson was a great way to use matrix multiplication in a real life scenario. Matrix operations can sometimes seem pointless if they are not put into a context to which students can relate. Most students have experienced shoe-buying, uniform-wearing, or working a certain job at some point in their lives, so the contest is familiar. Even if those may not be the most interesting topics, the business sides of the problems will help to engage other students.

The lesson also promotes both individual and group work. Math can sometimes be unexciting for students because teachers do not include group work into the daily plan. However, if students know that there will normally be time for group work, but that they are also expected to work individually, their engagement and participation in the lesson will typically improve.

The only potential problem I see with this lesson is that the actual way in which matrix multiplication is carried out is never clearly stated within the lesson. I realize the importance of exploration and how it helps students grasp topics, but there are some students who need to actually see the formula in order to understand it. In that respect, I think the exploration could be a little more thorough.

Overall, the lesson provides many opportunities for students to test out their knowledge. There are many chances to work alone and in groups, and the way in which the lesson is structured, there are many opportunities for the teacher to check for understanding and take advantage of teachable moments. To go along with this, I would maybe have the class generate a formula sheet, possibly including examples, that would remain visible in the room, so even if they could not remember how to do it, they could refer to the sheet and refresh their memories. This would be a book I would consider using, if for no other reason than because it seems to give practical applications that the students will be able to understand.

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Keywords: Algebra,
Ref: Katie11
Author(s): Driscoll, Mark
Year of publication : 1999
Title: Fostering Algebraic Thinking: A Guide for Teachers Grades 6-10
Journal or Publisher: Heinemann
Volume, Issue, Pages: Chapters 1 & 3
Reviewer: Katie
Date of Review: May 3, 2007

The first chapter of Driscoll's book focuses on two aspects of algebraic thinking, thinking about functions and how they work and thinking about the impact a system's structure has on calculations. Driscoll believes that these two aspects can be facilitated by focusing on three frames of mind: doing-undoing, building rules to represent functions, and abstracting from computation. It is Driscoll's belief that these three habits of mind can be learned when teachers utilize questioning within the classroom.

In all subject areas, questioning is vital to students' understanding of the content; math is no exception. Driscoll's research has led him to find that in order to be effective, teachers' questions need to have intention and context. Without either of these two factors being incorporated, the algebraic potential of the classroom activities will go unnoticed. Driscoll proceeds to give multiple sample problems and questions that follow his guidelines.

In chapter three, Driscoll addresses the issue of whether students really understand the material and the algebraic relationships that exist, or if they merely recognize patterns. Here again, he focuses on maintaining those algebraic habits of mind discussed in Chapter 1. With sample problems and questions filling the remainder of the chapter, Driscoll allows anyone reading this book to gain numerous ideas on how to approach and teach algebra in the classroom while ensuring that students really understand what they are doing.



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