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Keywords: Geometry, Measurement, Trigonometry
Ref: Andrew1
Author(s): Maxwell, Sheryl A.
Year of Publication: 2006
Title: Measuring Tremendous Trees: Discovery in Action
Journal or Publisher: Mathematics: Teaching in the Middle School
Volume, Issue, Pages: Vol. 12, No. 3
Reviewer: Andrew
Date of Review: February 14, 2007

This article describes a geometry activity which allows students to use their knowledge of triangles outside of the classroom. The goal of this activity is for students to determine the heights of large trees using their knowledge of geometry.

Students begin by reviewing geometric concepts and terms. Some things that should be discussed include properties of right triangles, the names of the sides (legs, hypotenuse), the sum of the angles, etc. Specifically, triangles with two 45 degree angles should be discussed, as they are an important part of the activity.

After the review of geometry, students start the activity. Students are in groups of 2-4 for this activity. The materials needed for this activity are a clinometer, a calculator, and a 100-foot reel tape. A clinometer is made from a protractor, a drinking straw, a small weight, and a piece of dental floss; the drinking straw is taped to the straight edge of the protractor, and the weight is tied to the string which is tied to the protractor. One student walked backward from the tree, looking through the drinking straw to the top of the tree. Another student walked alongside while noting the angle of the clinometer (using the weight and the dental floss as a guide). Once the angle reaches 45 degrees, the student stops walking, and the distance to the tree is measured. The students can then use rules of trigonometry, while making sure to adjust for the distance from the ground to the their eyes, to find the height of the tree. Students repeat the process from all sides of the tree to see if their results are consistent.

Throughout the process, students are asked open-ended questions, helping them to see the mathematical concepts driving this activity. After finishing, students are asked to reflect on the activity. They are asked to consider the mathematical ideas that are part of the activity, factors that affect the accuracy of their measurements, and possible applications of this activity to other problems.

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Keywords: Teaching Strategies, Communications
Ref: Andrew3
Author(s): Reinhart, Steven C.
Year of publication : 2000
Title: Never Say Anything a Kid Can't Say!
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 5, No. 8, pp. 478-483
Reviewer: Andrew
Date of Review: February 28, 2007

This article focused on teaching strategies and communication with students. Reinhart begins with a description of his early years of teaching. Despite preparing wonderful lessons, explaining concepts clearly, and answering questions thoroughly, Reinhart was not getting through to his students. Eventually, Reinhart came to the conclusion that teacher-directed instruction was not working; he decided that explanations needed to come from his students rather than from him.

Reinhart discusses a number of ways to create an atmosphere in which students feel comfortable participating. One strategy that he offers is planning questions to ask students prior to the start of a lesson. When asking students questions, teachers should remember to ask open-ended questions, avoid questions that only require recalling facts, allow students time to think, and never say anything a kid can’t say. Additionally, teachers should be aware that most students will be uncomfortable answering questions in front of their classmates. Consequently, teachers should communicate that wrong answers are helpful to the class. They should also never use questions to embarrass or punish students. Finally, Reinhart discusses some valuable strategies for encouraging discussion. He introduces a think-pair-share strategy, which requires students to work individually first, then to work in small groups, and finally to share their findings with the class.

I found that this article offered a number of valuable question-asking strategies. It helped me realize the importance of allowing students to come to their own conclusions.

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Keywords: Assessment...
Ref: Andrew4
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: pages 205-220
Reviewer: Andrew
Date of Review: March 7, 2007

The purpose of this article, which focuses on strategies for ordering fractions, is to illustrate the importance of understanding student thinking and problem solving strategies. The article focuses on three students who were given a series of problems to solve. For some of the problems, they were allowed to use paper, pencil, and a calculator. For other problems, they were asked to solve the problems mentally. Additionally, the students were required to describe, either in writing or verbally, how they arrived at their answer.

The three students used three separate strategies for ordering fractions. One student used percentages, one used procedural strategies (common denominators), and one used conceptual strategies (pieces of a whole). Each student was fairly successful in ordering fractions when they were allowed to use paper, pencil, and calculator. When students were interviewed and were asked to order fractions without finding exact answers, their results varied. Ben, who had been using percentages, used the ineffective strategy of comparing the whole numbers in the fraction (8/10 is bigger than 5/6 because the numbers are bigger). Kevin, who found common denominators, and Natalie, who used conceptual comparing strategies, were both successful in this portion of the test. However, when given problems involving addition and subtraction of integers, Kevin and Ben struggled, while Natalie was again successful.

This article brings up a number of good points. First of all, procedural skills without conceptual knowledge do not illustrate an understanding of concepts, as illustrated by Kevin. It also shows that students may get the right answer despite wrong thinking. Finally, this article demonstrates the importance of assessing student thinking rather than simply the right answer.

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Keywords: Geometry, Algebra
Ref: Andrew5
Author(s): Foletta, Gina ;Zbiek, Rose Mary
Year of publication :
Title: All in the Family
Journal or Publisher:
Volume, Issue, Pages: http://illuminations.nctm.org/LessonDetail.aspx?id=L619
Reviewer: Andrew
Date of Review: March 15, 2007

In this lesson, students examine the relationship between perimeter, side length, area, and diagonal length in squares. This lesson involves both geometry and algebra. To start the lesson, students are asked to get into groups and think of a graph with “side length” on the horizontal axis and “perimeter” on the vertical axis. The groups then report to the class how they thought the graph would look. Next, the each group will begin working with a computer applet that shows relationships among several measures related to squares. Students will explore as many relationships as time allows. Before the end of class, the groups will fill out a “Graph Results” chart and discuss their results as a whole. Students should be asked to explain which graphs are linear, which are not, and why.

I feel that this lesson is a great combination of geometric and algebraic subject matter. Students will better understand the relationships between various characteristics of squares. The lesson will also teach students about functions, variables, and slope. Additionally, the lesson exposes students to new forms of technology.

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Keywords: Curriculum
Ref: Andrew6
Author(s): Lappan, Glenda; Fey, James T.; Fitzgerald, William M.; Friel, Susan N.; Phillips, Elizabeth Difanis
Year of publication : 1998
Title: Accentuate the Negative: Introduction
Journal or Publisher: Dale Seymour Publications
Volume, Issue, Pages: pp. 1a - 4a
Reviewer: Andrew
Date of Review: April 11, 2007

The introduction to Accentuate the Negative provides an overview of the curriculum outlined in this book. The book is divided into 5 portions called “investigations,” and each investigation focuses on positive and negative integers. The goal of this curriculum is for students to understand the meaning of positive and negative numbers. The introduction also contains a list of specific goals for students and a summary of the goals of each investigation. In addition, the introduction provides helpful information relating to lesson planning, such as materials needed, uses of technology, pacing charts, and assessment options.

I was interested by the introduction to this book/curriculum. I am surprised that a 7th grade curriculum would be this basic, although I understand that negative numbers is a hard concept to grasp. I think that the list of goals is helpful to a teacher, especially for long-term planning. The author provided specific examples of how to explain negative addition and subtraction; I was surprised to find this amount of detail in the introduction. After reading the introduction, I’m unsure of whether I would like using this book.

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Keywords: Problem Solving, Games
Ref: Andrew7
Author(s): Fennell, Francis (Skip)
Year of publication : 1983
Title: The Agenda in Action
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: pp. 33-41
Reviewer: Andrew
Date of Review: April 4, 2007

Although identifying appropriate problem solving activities can be difficult, problem solving is a necessity at the primary level (grades K-3). It is true that computational skills such as counting, place value, addition, and subtraction are very important, but students should also be participate in activities that require them to apply these skills. This article emphasizes the importance of problem solving at the primary level and provides a number of strategies for implementing problem solving into curriculum.

Francis Fennell offers a number of problem solving strategies that can be implemented easily into primary level classrooms. One way for teachers to introduce problem solving is through informal oral questioning (How many kids in our class have blue shirts?). Fennell feels that having students answer questions in groups or as a class is an effective method of problem solving. Another method of problem solving is Questioning and Logic. An example of this would be a teacher picking a number between 1 and 100; students would then ask questions to help them guess the number. Another method is Pictures to Words. Students are asked to look at pictures and interpret them as word problems. For example, if a teacher shows a picture with 4 ducks and 1 moose, students my say that there are 5 animals in the picture (addition). The Math Drama is a method in which students act out various counting and operations problems. Teachers can also hand out newspapers, phone books, catalogs, etc., and have their students experiment with the numbers they find. Finally, having students work with calculators is a good tool for problem solving as well.

I found this article to be very helpful. It offered a number of problem solving strategies that seemed fun and interesting, especially for young children. I think this article is very helpful for grade school teachers.

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Keywords: Representations, Algebra
Ref: Andrew8
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: Andrew
Date of Review: April 11, 2007

This chapter on Equality focuses on student’s conception of the equal sign. It starts with a study of how grade school students answer an open number sentence, 8+4=_+5. A very low percentage of students were able to answer the question correctly, and the results got worse with age. The reason for this is because many students understand the equal sign as a command to do something, rather than a representation of the relationship between two numbers. For example, many students think that a true equation can only be written in the form “number + number = ___.” Clearly, this is a misconception that must be corrected. The chapter provides examples of good ways to establish the correct meaning of the equal sign. True/false questions are a good start. Teachers can prepare a list of equations and ask their students to identify the true statements. By doing this, teachers can identify exactly what kinds of equations students do not understand. Open number sentences are also an effective way of teaching the correct meaning of the equal sign. It is also important to choose problems that give a true representation of what the equal sign means. For example, it is not good to represent Jonny’s age by writing, “Jonny = 7.”

I think this is a very important issue in teaching. The fact that, in this particular study, students got worse with age is upsetting. Understanding the meaning of the equal sign is extremely important in algebra, which is a major part of almost all future math courses. I think this chapter provides good insight into a very important issue in teaching.

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Keywords: Algebra
Ref: Andrew9
Author(s): Usiskin, Zalman
Year of publication :
Title: Albebraic Thinking Grades K-12: Defining Algebraic Thinking and an Algebra Curriculum
Journal or Publisher:
Volume, Issue, Pages: Conceptions of School Algebra and Uses of Variables, pp. 7-13
Reviewer: Andrew
Date of Review: April 25, 2007

This article focused on the numerous applications of algebra in mathematics and how the definition of “variable” differs in each of these areas. Because of their many different forms and meanings, variables are often confusing for students. Since algebra is a main focus of middle and high school mathematics, it is important that teachers understand students’ difficulty in identifying the meaning of a variable.

The article identified four conceptions of algebra and the variable uses that coincide with them. One function of algebra is “generalized arithmetic.” Here variables are used to generalize a pattern. For example, “3+5=5+3” can be generalized as “a+b=b+a.” Algebra is also the study of the procedure for solving certain types of problems. For example, algebra would be used to solve the equation “5x+3=40.” In this case, the variable is an unknown or a constant value. Algebra can also be the study of relationships and quantities. An example of this is an equation in the form of y=mx+b. Here the variables can take on a number of meanings; they can be unknowns, constants, arguments, or parameters. Finally, algebra is also a study of structures. In this case, variables are just arbitrary marks on a paper that are used to manipulate and justify.

This article was valuable because it showed how many different ways variables can be defined or used. After many years of mathematics, I am comfortable with the many meanings of variables. However, it’s easy to forget what it was like to encounter variables for the first time in various applications of algebra. I think it is always important for teachers to consider the difficulty of certain concepts that seem simple after years of experience.

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Keywords: Number and Operation
Ref: Andrew10
Author(s):
Year of publication :
Title: Core-Plus Book 2: Multiplying Matrices
Journal or Publisher:
Volume, Issue, Pages: pp. 26-31
Reviewer: Andrew
Date of Review: April 25, 2007

The subject of this review is a chapter from a text book titled “Multiplying Matrices.” The chapter begins by introducing a real life problem about shoe trends. Students are provided with data about shoes that are currently popular and shoes that will be popular next year. Students are then asked to predict next year’s shoe sales for three brands of shoes. It is soon revealed that matrix multiplication is a helpful tool in solving this kind of problem.

The chapter continues with a series of examples involving matrix multiplication. The method for multiplying matrices is explained. Also, in each problem students are asked to explain what the numbers mean in their resulting matrix. For example, in the problem about shoe trends, the answer matrix is labeled “Buyers Next Year.” For one of the problems, I wasn’t sure if I had arrived at the right answer, but my answer made sense once I saw labeled matrix provided in the chapter. I found the emphasis on meaning to be very important.

I think that this chapter does an adequate job of explaining matrix multiplication. It provides examples with meaning rather than just pure computations. I think they could have done a better job of explaining the actual process of multiplying matrices (which entries are multiplied, which are added).

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Keywords: Algebra, Curriculum,
Ref: Andrew11
Author(s): Edwards, Edgar L. Jr.
Year of publication : 1990
Title: Algebra for Everyone
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages:
Reviewer: Andrew
Date of Review: May 3, 2007

This book contains a collection of essays from experts in mathematics education. They argue that changes in curriculum must be made to teach algebra more effectively. First of all, algebra must be taught on a broader scale; students need to encounter algebra long before they reach the class titled, ?Algebra.? Secondly, they feel that traditional American curriculum is too focused on computational skills. According to this book, American students are much less proficient in algebra than other prominent nations. They argue that a shift away from the computational focus will result in much more skilled algebra students.

One major theme of this book is that teachers should change their ideas of what skills students need. Rather than spending year after year insisting that students learn algorithms for computing basic math skills by hand, teachers should stress students? ability to reason and solve problems. This would involve a shift in the activities that students at all levels of school math. It?s not necessary for a student to be completely proficient in computational skills before they are able to encounter other kinds of problems. Students may benefit from learning computational skills by working through problem solving activities; these activities force students to think about what they are computing rather than just memorizing rules. In addition to learning about computation, students should encounter exercises involving number sense, tables, graphs, calculators, mathematical language, real world situations, physical representation with pictures and manipulatives, patterns, relationships, and functions.

I agree with much of what is said in this book. Although I feel that computational skills are important in school math, basic computational skills are less important after middle school. Very few times have I had to add, subtract, divide, or multiply by hand since I?ve been in college. Why not focus more on skills that students will need in later levels of mathematics, such as reasoning and problem solving. In my opinion, students can sometimes get so concerned by computation errors that they have trouble focusing on other concepts. I also think that diversifying the mathematics curriculum results in students gaining a better understanding of what they are actually doing when they multiply, divide, add, or subtract.

Keywords: Teaching Strategies, Assessment
Ref: Anna1
Author(s): Danielson, Christopher; Luke, Michele
Year of Publication: 2006
Title: If I Only Had One Question: Partner Quizzes in Middle School Mathematics
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol.12 No.4, pgs. 206-213
Reviewer: Anna
Date of Review: February 13, 2007

In order for Partner quizzes to work correctly and efficiently, guidelines must be set. The two authors have created guidelines that enhance student work, collaboration and support within the pairs. Lastly, the authors present students' work for insight into the thought process of each pair; the method used to solve the problem, what mistakes they may have made and/or what concepts each pair was skilled in or understood.

I thought this was an insightful article into different methods to use for assessment in the classroom. The authors' use of guidlines during the quizzes was essential in the consistancy across quizzes. This, in turn, created a more accurate assessment of the understanding of each student. Also, their guidelines created an environment of collaborative learning/working for each pair of students. For example, the students were only allowed to ask one question on the entire quiz. Both the students were to agree on the question they were to ask the teacher. Knowing that they only were allowed one question, the students were encouraged to think and talk through the problem.

Taking quizzes in partners is not only helpful to the students but to the teacher as well. For instance, if a student gets an answer wrong on an indivdual test, one can see they do not understand the problem (unless it's a arithmetic error). If a partner test is used and a problem is wrong, not one of the students but both students do not understand the problem. If the students can not figure out the problem in pairs, the teacher may need to go back and re-do problems similar to it to ensure understanding of all students in the classroom.

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Keywords: Teaching Strategies
Ref: Anna2
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, pgs. 478-83
Reviewer: Anna
Date of Review: February 28, 2007

Teachers must educate themselves on strategies to help their students in the classroom. Reinhart introduces a few ideas for successful change in the classroom: never say anything a kid can say, i.e. don't give kids the answer when they have the abilities to figure them out; ask good questions to stimulate thinking; use more process questions (open ended) then produce questions (yes, no); replace lectures with sets of questions; be patient.

Reinhart ends the article with more strategies to make the students gain confidence, engage in thinking and learning and feel comfortable in the classroom around their peers. Most of these ideas come from the fact that the students are middle schoolers and are still struggling with self-confidence and other issues dealing with their peers. A teachers should recognize these characteristics in students and be sure not to single a student out for a wrong answer, or intentionally make them embarressed.

I like Reinhart's ideas in the articles. I liked that he has compiled these ideas from years of teaching and also from colleagues. The ideas are consistent with asking the question, "how can I change my classroom so that the students are learning more, learning better?" Rather than, "What is wrong with these students, how can I get them to understand me?" Teachers must always be prepared to change their learning style in the classroom especially since all students have different learning styles. Also, understanding students' emotions and insecurities in the middle school can help a teacher find better methods to teach with.

These ideas are good but how do you implement them? Reinhart explains a few of the methods of involving students in the classroom such as the think-pair-share strategy. One of the biggest obsticles of a beginning teacher are the lack of ideas and strategies that older teachers have acquired from colleagues and from their own experiences. Without these ideas and teaching strategies, how much more difficult is it to create a classroom that has all the elements of a conducive learning environment for all students?

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Keywords: Problem Solving, Measurement, Curriculum
Ref: Anna4
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:
Reviewer: Anna
Date of Review: March 6, 2007

I thought this article was imformative on the different strategies students can use to solve problems. For me, without reading this article, do not even realize the methods I use to solve problems such as these because it comes so automatically. It is good to realize that students can use these different strategies to help them understand fractions. Also, I think it is a really good strategy to ask students how they got an answer instead of just accepting the answer when it could have just been a guess.

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Keywords: Problem Solving
Ref: Anna5
Author(s): Schmid, Doug
Year of publication : 2000-2007
Title: Illuminations: Arithme-Tic-Toc
Journal or Publisher:
Volume, Issue, Pages: http://illuminations.nctm.org/LessonDetail.aspx?id=L671
Reviewer: Anna
Date of Review: March 18, 2007

I really liked the real-world application that was used at the beginning of the lesson. It is a good introduction to use to get students to start thinking in terms other then 10. I think it is important for students to explore the patterns and connections that exist between modulus charts. The charts are a key element to understanding how and why modulus work. This lesson is not too creative but mods are not an easy concept to grasp at first. Letting students discover modulus through charts will help them grasp the information.

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Keywords: Teaching Strategies, Activities, Manipulatives
Ref: Anna6
Author(s): Lappan, Glenda; Fey, James T.; Fitzgerald, William M.; Friel, Susan N.; Phillips, Elizabeth Difanis
Year of publication : 1998
Title: Accentuate the Negative
Journal or Publisher: Dale Seymour Publications
Volume, Issue, Pages: pgs. 1a-1g
Reviewer: Anna
Date of Review: March 20, 2007

I think this is a very useful resource for a teacher. There are many concepts in mathematics that I can do, without thinking, even though I might not understand WHY it works. A teacher must understand why it works in order to teach the concept well. I am also a big fan of number lines and other manipulatives when working with students who do not fully understand a concept. Also with hands on activities, students can explore and discover patterns that might suggest why a concept works. Having a student reach conclusions as to why adding a smaller positive number to a larger negative number for example, will equal a negative number is a lot more useful to them then the teacher telling them the answer.

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Keywords: Manipulatives
Ref: Anna7
Author(s): Jackson, Robert L.; Prigge,Glenn R.
Year of publication : 1976
Title: Measurement In School Mathematics: Manipulative Devices for Elementary School Measurement Activities
Journal or Publisher: The National Council of Teachers of Mathematics
Volume, Issue, Pages: 187-209
Reviewer: Anna
Date of Review: April 4, 2007

The authors make a good point at the beginning of the chapter. They state that manipulatives should be used to allow self discovery by the students but students should always see the point in using manipulatives. They should be used for a purpose in the classroom, particularly to enhance learning.

I thought that this chapter was interesting because there are so many different resources a teacher can use to enhance learning in the classroom. The authors just mentioned a few examples of how you might use each manipulative but there are clearly more ways you could incorporate them into a lesson. Teachers should never be afraid to "steal" ideas from colleagues or books and I think the authors do a good job of illustrating that here by stating illustrations, descriptions, companies and prices that pertain to manipulatives that work well in the classroom.

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Keywords: Number and Operation
Ref: Anna8
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: Anna
Date of Review: April 11, 2007

The common misconceptions of an open ended problem are: the answer ALWAYS comes after the equal sign; all the numbers must be used to compute the answer, no matter what side of the equal sign they are on; extending the problem using another equal sign and expressing the equal sign as a relation between numbers, which is correct. Challenging students on their understanding and conceptions of the equal sign is productive to realize their mistakes.

The fact that students' understanding of the equal sign decreases from grade 1 to 6 is very shocking. It is very interesting to see the different methods students use to solve a simple open ended problem and their rationale behind their answers. The article brings up a few good points about developing the meaning of equality in students. First, students must be challenged in their conceptions to be able to explore the different methods of solving. Also, teachers must be careful in representing what an equal sign means. Teachers wording and notion should always emphasize that the equal sign signifies a relation between two numbers, and avoid anything that does not do this. If more time is given to students to explore what an equal sign means, as well as teacher awareness of demonstrating the proper definition of what an equal sign means, hopefully students will grasp the concept. Students' understanding of the equal sign should not be decreasing from grades 1-6, not fully understanding the concept of an equal sign is only going to hurt them in future mathematics.

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Keywords: Algebra, Curriculum
Ref: Anna9
Author(s): Usiskin, Zalman
Year of publication :
Title: Algebraic Thinking, Grades K-12: Defining Algebraic Thinking and an Algebra Curriculum
Journal or Publisher:
Volume, Issue, Pages: Conceptions of School Algebra and Uses of Variables, Pgs 7-13
Reviewer: Anna
Date of Review: April 19, 2007

I thought this article was interesting. I never thought about the different conceptions of algebra, more specifically the meanings that variables can have in algebra, and how these different meanings affect the handling of them. With increased technology, it is easier for algebra to be used to explore variables, instead of a means to answer a question. I believe this gives more area for variety in the classroom and perhaps more interesting and meaningful for the students as well. With new algebra techniques, more modern teaching techniques can be developed in the classroom.

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Keywords: Curriculum
Ref: Anna10
Author(s):
Year of publication :
Title: Multiplying Matrices
Journal or Publisher: Core Plus
Volume, Issue, Pages: pgs 26-35
Reviewer: Anna
Date of Review: April 25, 2007

I don't really know how I feel about this lesson. I think it is a very good idea to illustrate that multiplying two matrices together can give useful results that can be used in everyday life. I believe the lesson before this one dealt with addition and subtraction within matrices. There is hardly any introducation to multiplying matrices in this lesson. While going through the problems given in the lesson, I questioned whether a student who had never seen this material before, could understand HOW to multiply two matrices together. There are only a few scattered reasons as to what works and what doesn't work when working with matrices. I think the lesson could use examples showing how to multiply matrices together, this way if a student gets stuck or doesn't understand, they have some reference to look back to.

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Keywords: Statistics
Ref: Anna11
Author(s): Ma, Liping
Year of publication : 1999
Title: Knowing and Teaching Elementary Mathematics
Journal or Publisher:
Volume, Issue, Pages:
Reviewer: Anna
Date of Review: May 2, 2007

The first chapter dealt with subtraction a regrouping. The chapter is split into 3 sections: the US teachers' approach, the Chinese teachers approach and a discussion which talked about procedural and conceptual understanding for teachers. The author interviewed 23 American teachers, as well as a similar number of Chinese teachers. The findings reported that most US teachers focuses on the procedure of computing. Individual teachers were quoted stating what they taught and why they taught it. Manipulatives were also discussed, where most teachers misused them and did not convey any conceptual understanding. 14 % of Chinese teachers, on the other hand, held procedurally directed ideas. Main conclusions of this chapter revealed that 77% of American teachers and 14% of Chinese teachers displayed limited knowledge of the algorithm needed to solve this subtraction with regrouping.

The second chapter dealt with multiplying multidigit numbers. Again, American teachers and Chinese teachers were compared and contrasted in this chapter. Some of the American teachers admitted not knowing why a zero must be added before the second digit multiplication, but because "it was the rule". The American teachers and Chinese teachers both differed in explanations as to why students made mistakes in these types of problems. Again, American teachers showed procedural knowledge while Chinese teachers showed conceptual knowledge.

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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

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

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

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

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

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

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

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

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

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

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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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 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

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

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

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

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

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

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

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

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

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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Keywords: Curriculum, Standards, Teaching Strategies
Ref: Leanne1
Author(s): Britton, Kristine; Johannes, Jennifer
Year of publication : 2003
Title: Portfolios and a Backward Approach to Assessment
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 9, Issue No. 2, Pages 70-71
Reviewer: Leanne
Date of Review: February 19, 2007

The two decided to stick with the Mathematics in Context (MiC) curriculum currently employed in Rice Lake and used Wiggins and McTighe’s Understanding by Design book to help them create a standards-based unit which incorporated backward design.

In general, standards were identified first, then assessments were created, and finally the decision on how to teach a concept was made. The article gives examples of chosen assessments and assignments the teachers used. Periodically, students were assessed on their understanding of standards, and were given the chance to add their own comments and evidence of progress to the assessments. In Jennifer’s classroom, students were given the responsibility of showing that standards had been met. At the end of the unit, students brought their portfolios home, and a student/parent reflection sheet was filled out at home and returned to the school.

The article concludes with some reflections. Kristine and Jennifer found that student responsibility was increased and that parents valued the additional communication with their students and the school. Overall this method showed a more clear picture of what the students had learned than grades did. Shortcomings of the method include: portfolios take a lot of time and effort, parents sometimes don’t understand standards, standard-based grading does not always show when improvement has been made, and the overall system still boils results down to a single letter grade.

I liked that this article was written by the two teachers who had experienced portfolios and “backward design” first-hand. They were able to give a detailed account of the process they went through in order to implement this in the classroom, and this would make it much easier for another teacher to follow the process. I was a bit skeptical of some of the claims of this approach. I do believe that responsibility could certainly have increased for some students, and that some parents would appreciate the additional contact. However, I felt the article talked about the good things about this approach and only skimmed over the negative aspects at the end. It’s hard to feel that an accurate picture of the situation was given. I certainly think that valuable information and ideas were presented in this article, and that I would like to incorporate some aspects, especially backward design, into my own classroom. However, I don’t think the picture is always as rosy as it seems here.

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Keywords: Teaching Strategies
Ref: Leanne3
Author(s): Reinhart, Steven
Year of publication : 2000
Title: Never Say Anything a Kid Can Say
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 5, Number 8, Pages 478-483
Reviewer: Leanne
Date of Review: February 28, 2007

Reinhart soon had a very different way of teaching, and he shares in this article some of the strategies he has found to be most effective. He focuses primarily on questioning strategies, because he belkieves it is important to get students to the point where they can explain conceepts clearly. These strategies include having a plan, sharing with students the reasons for asking questions, making a safe environment for students to answer questions in, not judging responses, never taking only one right answer, and making participation mandatory.

I really enjoyed this article. It brought up many points that I agreed with and have found to be very important through my own experience. For example, I liked the idea about making the classroom environment one in which students feel comfortable stating wrong answers, sharing only partly formed thoughts, and simply asking questions. I know that in my math classrooms in school I often would not raise my hand because I didn't want to be wrong or look stupid. This affected my ability to learn and understand, and probably prevent other students from learning too. Another idea that stood out to me was having students explain both when they get the right answer and when they don't understand. It is true that students shut off if they feel they can based on the teacher's response. Asking them questions keeps them more engaged and helps them to learn more deeply and in a new way.

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Keywords: Number and Operation, Manipulatives, Assessment
Ref: Leanne4
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: 205--220
Reviewer: Leanne
Date of Review: March 7, 2007

The article explains how children's ability to order fractions does not necessarily imply that thye understand fractions. Many students incorrectly use whole-number thinking, some think procedurally, and some conceptually. The article emphasizes the need to use manipulatives in order to truly teach fractions and help students think about them in a variety of ways.

This article was very enlightening regarding students' thought processes regarding problems with fractions. For me, it certainly got the point across that, as a teacher, I need to make sure I know how and why my students are coming up with answers, rather than just be concerned with whether or not they get answers correct. The emphasis on manipulatives is something I am coming to agree with more and more, both through my own experience and realization of what I did not understand as a student, and through observation of students in my field experience.

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Keywords: Teaching Strategies, Manipulatives, Number and Operation
Ref: Leanne5
Author(s): Lappan, Glenda; Fey, James; Fitzgerald, William; Friel, Susn; Phillips, Elizabeth
Year of publication : 1998
Title: Accentuate the Negative: Integers
Journal or Publisher: Dale Seymour Publications
Volume, Issue, Pages: pages 1a-1g
Reviewer: Leanne
Date of Review: April 2, 2007

Next, the introduction points out particular things that students find "difficult about integers and operations on integers," and states that these things can be approached through the observation of patterns. It shows how the addition of integers can be modelled with number lines and chips. Then it goes on to show the same for the subtraction of integers, the multiplication of integers, and the division of integers.

The introduction concludes by connecting this integer unit to others that come before and after it, explaining the reasons Accentuate the Negative was created, and summarizing the 5 investigations that are in the book (Extending the Number Line, Adding Integers, Subtracting Integers, Multiplying and Dividing Integers, and Coordinate Grids.

I learned quite a bit from reading this introduction. It actually helped me visualize adding negatives in a way that I don't remember ever having done before, and also made me think about patterns in a different way. I think that much of this information will be valuable for teaching about integers. The use of manipulatives is very important for this topic, particularly if we want students to understand the why and how behind operations. I also think it is a good idea to include application to coordinate grids immediately following introduction to integers; students should see the ways supposedly different concepts in math connect to each other.

The connection to other units at the end was a nice touch, which would be helpful if I were using this book to teach. Overall, I think this introduction was very helpful, and I have already found it useful to me as a teacher.

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Keywords: Teaching Strategies, Manipulatives, Connections
Ref: Leanne6
Author(s): Ma, Liping
Year of publication : 1999
Title: Knowing and Teaching Elementary Mathematics
Journal or Publisher: Lawrence Erlbaum Associates, Publishers
Volume, Issue, Pages: pages 1-27
Reviewer: Leanne
Date of Review: April 7, 2007

Next, the article discusses US teachers’ approaches to teaching subtraction with regrouping as compared to Chinese teachers’ approaches. The research study which this chapter focuses on found that a majority of US teachers “focused on the procedure of computing” when teaching this topic. In general, the teachers described the step of taking a 1 from the tens place as “borrowing,” a term that is mathematically inaccurate. A minority of US teachers expected students to understand the rationale behind the “taking” and “changing” that occur. Manipulatives were used by most, but usually not in a way that helped students understand the topic conceptually.

In contrast, a majority of Chinese teachers explained subtraction with regrouping as “decomposing a higher unit value,” which is a mathematically accurate way of teaching the topic, and came after their teaching of “composing a higher unit value” in addition. They generally taught multiple ways of regrouping. Chinese teachers by and large discussed subtraction with regrouping in terms of a larger package of knowledge, and believed certain topics were necessary for students to learn before they could understand this one. Manipulatives were used less by Chinese teacher, but when they were used, a discussion often took place afterwards; this was not present in US teaching.

The article then goes on to discuss the importance of making connections in mathematics, and that a teacher’s understanding of the structure of a subject greatly influences the way they teach it and therefore how students understand it. The chapter ends with a summary.

I really enjoyed reading this article. I feel it is valuable for anyone who is going into teaching or who is currently teaching.

I found very compelling the argument asserting the importance of teaching for conceptual understanding along with teaching for procedural understanding. I learned some new things about math just from reading about this; I was never taught (or at least not taught well enough to remember) many of the conceptual aspects of this topic. Especially as a person who is slow to make connects that are not clearly stated, I agree that the “why” of everything behind mathematics needs to be emphasized in the classroom. Helping students see the reasons behind the shortcuts before they even learn the shortcuts leads to a much more complete understanding of math and allows students to learn more, and to learn more quickly, later on.

I feel that the term “decomposing a unit of higher value” describes subtraction with regrouping well. From what I read in this chapter, and from my own experience with many people’s lack of understanding of basic mathematical concepts, I think the idea of “composing” would be an effective way to approach the general concept of regrouping. Another idea I found useful from this chapter was “subtraction within 20”; it makes sense to establish a foundation by mastering “subtraction within 20,” and from there move one to do subtraction with higher numbers.

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Keywords: Algebra, Representations, Teaching Strategies
Ref: Leanne8
Author(s): Carpenter, Thomas; Franke, Megan Loef; Levi, Linda
Year of publication :  2003
Title: Thinking Mathematically
Journal or Publisher:
Volume, Issue, Pages: Chapter 2: Equality, Pages 9-24
Reviewer: Leanne
Date of Review: April 13, 2007

The research study found five typical conceptions held by elementary students as to what the equal sign means. These are “the answer comes next,” “use all the numbers,” “extend the problem,” and two relational views. The first relational view is to find the missing number by calculating the sum on one side and getting the other to match. The second is to notice the differences between the numbers on each side, and find the answer without having to calculation, based on the relationships between the numbers.

The article goes on to explain that using true/false sentences is an effective way to teach children what the equal sign, by convention, means. There are 4 “benchmarks” that children may pass through as they learn the meaning of the equal sign, the fourth and desired being the second relational view mentioned above.

Next, the article gives some examples of how NOT to use the equal sign, and explains the importance of teaching how the equal sign is a convention. It explains how teaching about the equal sign can transition to algebra and how children’s misconceptions about the equal sign may have come about. Finally, the article ends by encouraging us to believe in kids’ potential to understand concepts like these, even at young ages, and it gives some challenges for teachers to use in their classrooms.

I found this article valuable. I did not realize that students had such misconceptions about the equal sign, even into mid and late elementary grades. Very likely, this is an issue that will come up in teaching any grade, well beyond elementary school.

I found especially useful the idea of using true/false sentences in order to explain what the equal sign really means. It is a good idea in general to teach students to pay attention to whether a number sentence is true or not. Also, I feel the article made a good point about where it is not a good idea to use equal signs. I had not given much thought to the fact that those are instances where the equal sign is not accurate.

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Keywords: Algebra, Curriculum, Manipulatives
Ref: Leanne9
Author(s): Usiskin, Zalman
Year of publication :
Title: Algebra and Uses of Variables
Journal or Publisher: Algebraic Thinking, Grades K-12
Volume, Issue, Pages: Pages 7-13
Reviewer: Leanne
Date of Review: April 18, 2007

It goes on to state that the two most important issues in algebra currently are whether or not students should be required to do manipulative skills by hand, and the question of the role of functions and when they should be introduced. The purposes of algebra, the article says, “are determined by, or are related to, different conceptions of algebra, which correlate with the different relative importance given to various uses of variables.”

These conceptions are 1. algebra as generalized arithmetic (and variable as pattern generalizer), 2. algebra as a study of procedures for solving certain kinds of problems (variables as unknowns or constants), 3. algebra as the study of relationships among quantities (variables as arguments and parameters), and 4. algebra as the study of structures (variables as arbitrary marks on paper). Algebra is used in all these different ways, and all should be considered when determining how to present the curriculum.

The article concludes by giving a summary of how variable is used in computer science, and by giving a summary of how the different conceptions of algebra are related to the use of variables. It emphasizes the many different uses of algebra and its importance in the modern world.

This article was intriguing. I have not thought a great deal about the different roles that variables play in algebra; I do not recall them ever being explained so explicitly in any of my schooling. It is so true that variables represent a variety of things and that our understanding of this influences the way we understand algebra. This, in turn, will influence the teaching of algebra to students.

After reading this article, I am interested to know more about what this all means practically for classroom teaching. Cleary, this should influence how we teach algebra, but the article did not say much about specific practical implications.

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Keywords: Teaching Strategies
Ref: Leanne11
Author(s): Johnson, David
Year of publication : 1994
Title: Motivation Counts: Teaching Techniques that Work
Journal or Publisher: Dale Seymour Publications
Volume, Issue, Pages:
Reviewer: Leanne
Date of Review: May 9, 2007

The first chapter explains how the classroom routine itself can do a lot to motivate students. Johnson emphasizes three main routines that increase motivation in the classroom. They are teaching by walking around (TBWA), having a “desk-top code” that is enforced, and not having shouted answers. He states the importance of starting class right at the bell, not doing tasks like taking role and answering individual questions.

The second chapter talks about motivating students through good questioning techniques. Here Johnson emphasizes directing good quality questions at the entire class. He talks about the importance of pausing after questions, not over-praising students, and emphasizing mistakes as a natural part of learning.

In the third chapter, Johnson discusses how to make homework and tests meaningful. Homework should be unrelated to student’s behavior, and should not be vague or optional. Tests should not be threatening or surprising to students, and teachers should teach students how to prepare for tests.

The fourth chapter talks about helping students understand abstract concepts. In this chapter, Johnson emphasizes the importance of spending enough time on concepts that give students difficulty. He shares about the importance of helping students think about numbers and problems unconventionally, through the use of counterexamples.

In the fifth chapter, Johnson discusses problem-solving and the need to place problems in real-world contexts for students to truly understand the math they are doing. In the sixth, he gives some examples of “questions and problems that motivate.”

I really enjoyed reading this text. Johnson clearly has given a great deal of thought to how to teach mathematics. His ideas make a lot of sense, unconventional as they might be. Some things that stood out to me as I read this book were the importance of starting at the bell with students ready to go; the need to turn isolated equations and expressions into word problems, even for things like simplifying; the idea of teaching students not just what to study but how to study, and the importance of using students responses as a chance to take the discussion deeper instead of just offering praise. I found many other useful ideas, but these ones were particularly impressive to me.

I think that this is a valuable book for anyone who works with students in mathematics, and I would (and have) recommend it to others in the field of mathematics education.

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Keywords: Activities, Measurement,
Ref: Linnea1
Author(s): Bombaugh, Ruth; Jefferys, Lynn
Year of publication : 2006
Title: Body Data
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: volume 11, number 8, pages 378-383
Reviewer: Linnea
Date of Review: February 19, 2007

Students are assigned small groups at the beginning of the year, and will stay with these groups for the entire project. Within the group, they have specific roles, but basically they keep a running measurement list of their own heights. They track growth, learn to be precise in measurements, observe trends, compare individual data to a norm set, and set up and use spreadsheets. Then they make predictions about how much they or others will grow in the next month and check to see how accurate their predictions were.

Several points brought up in this article seemed especially good to me. I agree with the authors’ statement that when students are dealing with data relating to them personally (their own height) they are more curious about the results and have more of a vested interest in doing each step carefully and correctly. They are more likely to proceed with care and not rush through the steps so that their data will be accurate. For middle schoolers especially, then, this seems like a good subject matter to be using in a project meant to introduce and develop measurement, recording, and analysis skills.

I also think it would be interesting to try this lab-style project with middle schoolers because, as the authors point out, students are growing rapidly at that stage, and the results might be very interesting to both the teacher and the students themselves. According to the authors, sixth or seventh grade girls are often taller in September, but boys are often taller by June. I think it would be fun to have the data tracking this progression for students to see.

I liked how many suggestions for directions to go with this project were given in the article, but I also appreciate how easy it would be to choose some parts to focus on and not others (depending on the student make-up of the class I was teaching). There is a lot of room for tweaking the project idea. Overall, it could be made a very useful and interesting activity!

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Keywords: Teaching Strategies
Ref: Linnea3
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, Number 8 National Council of Teachers of Mathematics, Inc. www.nctm.org
Reviewer: Linnea
Date of Review: February 28, 2007

I also liked that the author came across as being very down-to-earth and realistic. He acknowledges that it will be difficult to switch your teaching patterns, especially if you have never experienced this type of teaching or learning. To this, he adds that it will only confuse the students if you completely change your teaching style all of a sudden. It is better to gradually incorporate some of these tactics, and by the end of a school year or after a few years you will be following his suggestions without even thinking twice about it. He also points out that it will be uncomfortable for a while since it is not the type of teaching that you might be used to. For example, it will be hard not to offer the answer to a question if no student is volunteering it, especially since a “class discussion” format is unusual for a math classroom. But once you and the students get used to it and understand the expectations (such as the understanding that you will not just give them the answers; they will have to work with you so that you can guide them to discoveries) it will work wonderfully!

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Keywords: Assessment
 Ref: Linnea4
Author(s): Cramer, Kathleen; Wyberg, Terry
Year of publication : 2000?
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. 205-218
Reviewer: Linnea
Date of Review: March 7, 2007

I liked how the students were made more real in the article by having names and having their exact words of their explanation to some of the problems included so that we got a sense for what each of the three fifth graders was like. As the reader, this aspect made me more engaged in the reading.

I also like that the exact questions asked of the students are included in the article along with their answers. Having the question there as the “interview item” is a nice guide for me as a future teacher thinking about questioning skills and how I would best be clear in my question so that I do in fact ask what I am intending to ask.

One point that I think is interesting is that the algorithms are not the most valuable tool for students if they do not yet have the foundation of mental picture abilities. The article said, “Even though Kevin had a correct procedure [common denominators] for ordering fractions, his way of knowing did not provide whim with the type of understanding needed for more complex number-sense tasks.” So even though he knows which numbers to multiply together and compare, he does not necessarily know what mathematics is behind the operation or why his method works.

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Keywords: Activities, Algebra
Ref: Linnea4
Author(s): Nelson, Joanne
Year of publication : 2007
Title: "Escape from the Tomb" lesson
Journal or Publisher: NCTM: Illuminations
Volume, Issue, Pages: http://illuminations.nctm.org/LessonDetail.aspx?id=L698
Reviewer: Linnea
Date of Review: March 15, 2007

I really liked the set-up of the lesson. It is very hands-on, very interactive, and will keep students’ attention. It would definitely be a good break from the traditional types of lessons in math classrooms. The directions on the student worksheet pages are very clear, and lead them through the process in a logical, step-by-step way so that they can use data they have already found in developing further hypotheses.

I could see different variations of the activity working well, too. For example, it could be combined with a science class if done at the beginning of the year in the context of “how to perform an experiment and gather data” and then the data analysis part could happen in the math classroom in the context of “how to represent data in graph form, analyze the graph, write equations, etc.” Or, I think this could be an appropriate lesson for older middle-schoolers, too, like 8th graders. They might not be able to do the final step of solving the systems algebraically, but if they have had experience with linear functions, they should be able to do everything leading up to that very last question.

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Keywords: Curriculum
Ref: Linnea6
Author(s): Lappan, Glenda; Fey, James T; Fitzgerald, William M; Friel, susan N; Phillips, Elizabeth Difanis
Year of publication : 1998
Title: Overview of Accentuate the Negative
Journal or Publisher: Connected Mathematics - Accentuate the Negative, Dale Seymour Publications
Volume, Issue, Pages: pages 1a-1j
Reviewer: Linnea
Date of Review: March 21, 2007

I would like a bit more background on the book and the curriculum series as a whole, though. Is this book intended to be spread over the whole school year for 7th graders, for example? In that case, how do they cover all of the material for 7th grade math that is not related to operations with negative integers? And how do students retain knowledge and material from one year to the next if each year’s math topic is so specialized and they do not allow students to review by making connections to other recent material?

I liked the way the overview was written, though. Its examples and explanation of the different models of adding, subtracting, multiplying, and dividing using chips, for example, make it clear for the teacher to know what the book authors are intending.

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Keywords: Curriculum, Keyword 2
Ref: Linnea7
Author(s): Cain, Ralph W.; Carry, L. Ray; Lamb, Charles E.
Year of publication : 1985
Title: "Mathematics in Secondary Schools: Four Points of View"
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: a chapter in The Secondary School Mathematics Curriculum - 1985 Yearbook. pp.22-28
Reviewer: Linnea
Date of Review: April 4, 2007

I liked having the differences in these curriculum focuses pointed out explicitly. The descriptions were concise and easy to understand, complete with a chart to compare specific characteristics. I personally like the idea of the Conceptual Mathematics program best, I think, because it targets the majority of the students (whereas Pure Mathematics is only meant for the top 10% of the student population) and because I think that Comprehension is a major priority for high school math. If students can compute an answer but do not know why their answer is right, they will not be able to use their math skills. And applied math is nice, but can be taught in physics class or in college, whereas high school students really still need to focus on comprehending their algebra/geometry/calculus/etc.

I thought the content of the article was interesting, but the book was written in 1985 so I'm sure the debates about focuses in secondary math classrooms have developed far beyond what they were 22 years ago. I wouldn't recommend basing any opinions for our future classroom curriculums on an article that old. Furthermore, the authors seemed really pompous (which, at the time, I'm sure this information was very valuable and applicable, but it seems funny to be reading in 2007 what seems like someone in 1985 is saying is all the best research and findings). It was nice of them also to make sure and point out that we math teachers should not base our curriculum choices just on this reasearch but that we should also make sure to assess the school's student population and student needs. (Thanks for that. This was the point where I was thinking, "Wow. I feel like St. Olaf's education department has done well... hopefully nobody here would ever have chosen a curriculum without taking students into consideration!")

So, if you're curious, get the book from Martha and read about the 4 approaches to math. If you're really just interested in developing secondary math curriculums, don't bother reading this.

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Keywords: Algebra
Ref: Linnea8
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: Linnea
Date of Review: April 10, 2007

I found it interesting that the chapter pointed out and spent so much time on the idea that our use of the equals sign is not necessarily the only meaning it could have, but that it is an established convention. Mathematicians and people with strong understandings of the system of math all agree on the use of the equals sign to mean what we understand it to mean, but our students might have other constructed ideas of it. Thus, it is our job as teachers to help them to understand the accepted conventions so that they are working within the same framework as the rest of us!

I liked how the chapter was written with very concrete student examples, so that the reader had a good example in mind of what type of scenario the author was referring to. It was an interesting chapter, easy to understand and interesting to read and think about.

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Keywords: Algebra
Ref: Linnea9
Author(s): Usiskin, Zalman
Year of publication : 1988 (?)
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: p.7-13
Reviewer: Linnea
Date of Review: April 21, 2007

I think this is an interesting question, and it drives the method of teaching for many curriculums, starting at a very early age. Since kids can begin to understand algebra in early elementary grades and variables can be introduced just as early, the way a curriculum or an individual teacher presents the idea of variables can form students’ basis of understanding algebra for a long time afterwards! The functions approach, which the article says is becoming more popular as a key method of teaching algebra (whereas it used to be left until about Algebra 2 as a type of example instead of a way to think about more basic algebra), is also debated.

I thought the article was well-written and interesting, but I would be curious to compare different curriculums directly, so that I could personally see the differences in thinking and teaching about variables. It would also be fascinating to compare two 8th grade classes, for example, in two different school districts which had chosen different approaches to teaching about variables and see if they had noticeably different understandings or problem-solving styles.

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Keywords: Curriculum
Ref: Linnea10
Author(s): author(s) of Core-Plus curriculum: Arthur F. Coxford, et al.
Year of publication : 1999 (?)
Title: Lesson 2: Multiplying Matrices
Journal or Publisher: Everyday Learning Corporation
Volume, Issue, Pages: from Core-Plus Book 2, pg. 26-35
Reviewer: Linnea
Date of Review: April 25, 2007

Without having Lesson 1 of this chapter to read, where the students learned about adding or subtracting two matrices (entry by entry) but the first paragraph of this lesson references that and emphasizes that these processes would be used in contexts such as taking inventory of different products in a store. I think this is good to remind students of the uses of mathematical ideas (especially when they seem so abstract, like numbers put into a matrix). It would be interesting to see if the set-up and philosophy behind Lesson 1 was the same as that of Lesson 2 – leading the students through a series of tasks increasing in difficulty and progressing until the goal (matrix multiplication in the case of Lesson 2) is reached. I get the feeling this is the way the whole curriculum might be organized.

As a student, I think I would like the textbook. It is user-friendly and each step is explained very clearly. As a teacher, I also like the textbook. Since the lesson is so applied to the examples, it forces students to read and think about the example situations. (With many books that I have had, I am able just to read the explanation and skip over the example problems, and when I do that, I do not understand as well.) I wonder how most teachers who use this curriculum organize their lessons and lesson plans. Do they teach the material as the section in the book is written, or do they leave that as a student reference and teach in a more traditional way of telling the rules right away and then doing examples (instead of having the students come to a conclusion and understanding after working through examples) or do they use the same style that the book did? I think I would try to use a similar style to the book, but maybe use different real-life situation examples so that students have more exposure to applications of the matrix multiplication. I’d like to know more about the Core-Plus curriculum and teachers’ thoughts about it.

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Keywords: Management
Ref: Linnea11
Author(s): Johnson, David R.
Year of publication : 1994
Title: Motivation Counts: Teaching Techniques that Work
Journal or Publisher: Dale Seymour Publications
Volume, Issue, Pages:
Reviewer: Linnea
Date of Review: May 2, 2007

I liked his idea of motivation needing to start immediately at the sound of the bell. Students need to be engaged this early in the class period so that they understand they are there to learn and so that time is not wasted on menial, time-consuming secretarial tasks such as passing back papers, taking attendance, answering individual questions about missed homework or make-up quizzes, etc. I think it would be fun to use his idea of an ACT or SAT review practice question as an opening question, assuming it also served either as practice for the current class material or review of past material.

The book again has a good section about questioning, and it was cool to read that Johnson prefers the wording, ?What questions do you have?? over the wording, ?Does anyone have any questions?? because we had just discussed this in class on Tuesday and come to the same conclusion! I also liked his metaphor of a math teacher as a band director, so that when you ?wave your arms in the air,? EVERYONE responds and is held responsible for their participation.

Overall, it was a good book with lots of great suggestions and I recommend it.

Keywords: Curriculum, Planning, Teaching Strategies
Ref: Michael1
Author(s): Tarr, James E.; Reys, Barbara J.; Barker, David D.; Billstein, Rick
Year of Publication: 2006
Title: Selecting High-Quality Mathematics Textbooks
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Vol. 12, No. 1, p. 50-54
Reviewer: Michael
Date of Review:  February 12, 2007

For content emphasis, the authors note that a wide range of topics are included in many textbooks in order to align with diverse state and district curriculum requirements. However, they note the importance of selecting texts that continue to develop skills, rather than being redundant, and provide students with contextual purposes for learning mathematics.

For instructional focus, the authors discuss the importance of a textbook's providing of problems, activities, and investigations that can engage students and lead them to seek out mathematical ends to these problems. Quality textbooks should also provide the means to connect new ideas to prior knowledge.

For teacher support, the authors highlight the importance of textbooks that offer teachers insight into engaging students in mathematics, as well as providing a clear educational/instructional path. Quality textbooks should also provide ideas for applying activities to a diverse student population and give the teacher appropriate assessment resources.

I found this article to be both valuable and interesting because I have wondered what the process is that a teacher goes through in choosing a textbook. Therefore, through this article I now have a guide for how these decisions should be made. Were I able to cut apart this issue, I would keep the purple summary boxes for future reference, as they provide sets of questions that a teacher should ask him- or herself during this selection process. A textbook that is strong in these three aforementioned areas should alleviate some stress from a teacher, as it provides a strong support for both the teacher and student.

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Keywords: Teaching Strategies, Planning, Communications
Ref: Michael3
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, p. 478-483
Reviewer: Michael
Date of Review: February 28, 2007

What should be taught? How should it be taught? Should the classroom be teacher-centered or student-centered? These sorts of questions posed themselves throughout the article. As the title of the article states, teachers should “never say anything a kid can say.” We should let the thoughts and words that are floating around be the students’ thoughts and words whenever possible. Students that verbally question and reason are students that are showing engagement and understanding. By being more of a questioner, the teacher puts students in the driver’s seat in the classroom. By asking for more than just “correct answers,” a teacher forces his or her students to try to gain an understanding of their level of understanding.

Participation is the key to the student-centered classroom that Mr. Reinhart is attempting to build. His “think, pair, share” strategy is based on trying not to overwhelm students (especially middle school children) with the pressure of sharing their thoughts with a room full of their peers. It encourages students to work at a comfortable pace, and it allows them to build support behind any presentations they make. Beyond this strategy, however, Reinhart highlights several other quality teaching tactics that encourage participation. From having students use hand signals during large group discussions, to taking the pencil out of his hand during individual assistance, this article is laced with tips that math teachers, both current and future, should take to heart.

I found Mr. Reinhart’s article to be valuable because of its practicality. As a future teacher, I often find myself asking the questions that Reinhart has posed answers to. While he notes that it is nearly impossible to use everything he suggests at all times, his ideas lend perspective, and potential guidelines, to the idea of developing as a teacher, which is a development that hopefully continues through all of our careers.

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Keywords: Assessment, Connections
Ref: Michael4
Author(s): Cramer, Kathleen; Wyberg, Terry
Year of publication : 200?
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: Michael
Date of Review: March 7, 2007

When comparing fractions, it is possible for students to use either conceptual or procedural strategies. A student’s choice of method may play a role in how he or she thinks about the size of fractions, which therefore affects his or her way of operating on these fractions. The nice thing about the three students mentioned is that they all offer a different perspective on how others may approach working with fractions.

Kevin showed a preference for finding common denominators when ordering fractions. When doing fraction estimation, his common denominator method produced exact answers, but defeated the purpose of estimation. When pushed to estimate, he struggled, reverting to whole number strategies.

Ben used a percent strategy to order the fractions, converting via calculator. When the calculator was removed, he used a difference perspective, which is far too inaccurate. He also reverts to a whole number strategy for sums of fractions.

Natalie approached the ordering problems by visualizing “pieces” that connect fractions to a more concrete model. This would qualify as a conceptual strategy. Natalie was able to use this understanding to make sounder sum estimates than Kevin and Ben.

Perhaps the most interesting/valuable part of the article was the set of questions at the end. While we can identify student preferences, as above, and their strengths and weaknesses, we do not get anywhere with this if we don’t ask and attempt to answer these questions. We notice that Natalie showed the strongest grasp of fraction values. We also notice, however, that there are times and places for Kevin and Ben’s methods, and the best case scenario would be students who are equipped with all of these fraction tools.

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Keywords: Algebra, Activities
Ref: Michael5
Author(s):
Year of publication :
Title: Trout Pond Population
Journal or Publisher: NCTM
Volume, Issue, Pages: illuminations.nctm.org/LessonDetail.aspx?ID=L476
Reviewer: Michael
Date of Review: March 16, 2007

There is an activity sheet that goes along with the lesson that I believe I would encourage the students to use if this was their first contact with recursion and iteration. On this sheet there is a table that asks the students to find the number of trout for each year, 1-25, hoping they can thus see the pattern. I think this table would be useful in getting students to see that a table would be a very logical way to organize (and attack) this problem, and hopefully as this topic was discussed further, or reviewed later on, the students would come back to the idea of using a table to help them solve recursion and iteration problems.

There is also a second part to this lesson that brings up the question of, "What would happen if we changed the above parameters for the trout pond?" By parameters, we mean the initial number of fish, the population decrease rate and the restocking number. In order to comfortably manage this exploratory lesson, I think I would (in a class of roughly 24-25) have the students work in pairs, so therefore each student has someone else to bat ideas around with if they're having any difficulty.

Also, when we move into the second phase (changing parameters), these pairings will be helpful because I can assign two pairs each to one of the six parameter change situations - lowering or raising each of the three parameters - and then have them meet in that group of about four to get their ideas together and perhaps even present what they believe would be the effects of their respective parameter changes.

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Keywords: Number and Operation, Representations
Ref: Michael6
Author(s): Lappan, Glenda; Fey, James T.; Fitzgerald, William M.; Friel, Susan N.; Phillips, Elizabeth Difanis
Year of publication :
Title:
Journal or Publisher:
Volume, Issue, Pages:
Reviewer: Michael
Date of Review: Select month Select day of month, 2007

The approach they take to gaining this understanding is one that is immersed in models. Number lines, chip boards, thermometers and graphs are all excellent tools for helping students to fully grasp operating on integers. One or more of these models can be used to express the answers to each of the above questions. Looking to page 1f, we see the mathematical and problem-solving goals of the "Accentuate" book. Among them are the questions I have raised, along with many other goals. The most important thing to see with these goals, however, is how they are all related to one another, all building on the same concepts.

I really enjoyed this overview of the "Accentuate" book, not only for the questions and goals it brings forth, but also for the examples that exist in this pre-text text. Some of these examples include: relating adding a negative and subtracting a positive integer, number line understanding of the comparative value of negative integers, opposite chips (+1 + -1 = 0), and understanding the multiplication of two negatives by looking at the pattern as one of the numbers decreases from a positive to a negative.

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Keywords: Statistics, Probability
Ref: Michael7
Author(s): Schielack, Jr., Vincent P.
Year of publication : 1995
Title: Baseball Cards, Collecting, and Mathematics
Journal or Publisher: Connecting Mathematics Across the Curriculum (NCTM)
Volume, Issue, Pages: p. 210-218
Reviewer: Michael
Date of Review: April 4, 2007

Why statistics? Well, on the back of most baseball cards you can find a player's career (and year-by-year) statistics in a variety of categories, including at-bats, hits, and batting average. Of course there are many other categories that are kept track of, but the author chose these three to analyze because of the fact that batting average is found by using a function (hits/at-bats) of the other two. Baseball cards can invite students to analyze calculations and create statistics of their own.

Why probability? Well, in collecting, many people attempt to get complete sets of a type of baseball cards (or any other item), or they attempt to get individual players. Since the cards that one buys in a pack can be considered random, things such as the Monte Carlo techniques and Expected Values become applicable. For example, we can determine the expected number of packs of cards needed to attain a complete set of x cards.

I found this article interesting because I spent a great deal of my youth earnings on sports cards. More seriously, the article appealed to me because I am always keeping my eyes open for real-life applications and interesting connections to use in my future classroom. This article highlighted the value of baseball and baseball cards as statistics and probability tools. Those are two more tools for my future teaching. (By the way, the Twins now lead, 1-0 through 1.5 innings... is there any mathematical value we can get from this??)

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Keywords: Representations, Issues
Ref: Michael8
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: Michael
Date of Review: April 11, 2007

Even when these mistakes are pointed out, children seem to have a great deal of difficulty departing from the early conceptions they have formed for what "=" means. What the authors of this article suggest is that, in order to break these habits (or to stop them from forming in the first place) teachers should invite their students to explore the meaning of = through true/false number sentences that are eventually replaced with open number sentences. Four benchmarks are set. The first is that students are simply able to discuss their thoughts on the meaning of =. Second, students are able to "accept as true some number sentence that is not of the form a+b=c." Third, students are able to recognize the equal relationship between the left and right sides of the equal sign. Fourth, is the comparison of mathematical expressions without actually performing the calculations (19).

I like any ideas about how to strengthen students conceptions of important mathematical ideas, and the equal sign is definitely one of those important ideas. There are too many student who struggle understanding the manipulation that takes place in algebra because they don't see that what you do to one side of an equation you must do to the other, since the equality of the sides must be preserved.

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Keywords: Algebra
Ref: Michael9
Author(s): Usiskin, Zalman
Year of publication :
Title: Conceptions of School Algebra and Uses of Variables
Journal or Publisher: Algebraic Thinking, Grades K-12
Volume, Issue, Pages: p. 7-13
Reviewer: Michael
Date of Review: April 21, 2007

The author looks at four different conceptions of algebra. The first, algebra as generalized arithmetic, is a means of looking at patterns. The second, algebra as a study of procedures, involves the use of variables in holding a place, either for unknowns or constants, that are to be solved for. The third, algebra as the study of relationships among quantities, looks at things like area, where L and W affect A, and all other functions (e.g. f(x)) where there is a relationship between an input and output. The fourth conception is algebra as the study of structures, which involves things like groups and rings and how the properties of algebra carry over to these structures.

Our concerns, as future teachers, lie in two fundamental algebra issues. The first is the extent to which algebraic manipulation by hand needs to be known. The advent of computer technology is the factor coming into play here. The second issue is about the timing of the introduction of the function to a student's mathematical world. Some see functions as a major vehicle of all algebra learning, and some see them as too advanced and confusing for students just beginning the study of algebraic ideas.

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Keywords: Activities, Problem Solving
Ref: Michael10
Author(s):
Year of publication :
Title: Lesson 2: Multiplying Matrices
Journal or Publisher:
Volume, Issue, Pages: p. 26-35
Reviewer: Michael
Date of Review: April 30, 2007

What I really liked about this lesson is that it was very thorough. There were something like 40 different questions asked throughout the lesson, and each of them probed a valuable area of understanding matrices. Because the questions were often introduced before the cold, hard process, the students were thus forced to bring in prior thoughts and understandings and apply them to the given problems. Matrix multiplication serves as a simplification, or a mathematical representation of the common strain of thought. Thus, it's easy to like the fact that the students were given the opportunity to "discover" matrix multiplication and how matrix dimensions relate.

One of the things that I didn't like about the lesson is that it is definitely too long to comfortably present in a regular (non-block) class period. With forty-some questions, there obviously wouldn't be a whole lot of time to really stop and think. However, over a two-day period, this lesson might be fine. The one other thing that I was somewhat uncomfortable with was the lack of description for a "rule" for matrix multiplication. I understand that this whole exploration creates the rule(s) for the students, but looking at some of the problems they were presented with, I could see being forced to take your exploration and apply it to unlabeled matrices as a little overwhelming. Some students, I'm sure, would like to have something a little more succinct in front of them as they are getting used to this new tool.

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Keywords: Algebra, Representations,
Ref: Michael11
Author(s): Steen, Lynn Arthur; Herbert, Kristen; Rosnick, Peter
Year of publication : 1999
Title: Algebraic Thinking
Journal or Publisher: NCTM
Volume, Issue, Pages: p. 49-51, p.123-128, p. 313-315
Reviewer: Michael
Date of Review: May 2, 2007

The second article I chose is titled, "Patterns as Tools for Algebraic Reasoning." This article was written with a problem as its framework, where there is a group of people needing to cross a river with a single boat, where the question is, "How many trips will it take to get everyone across?" This question is meant to probe student understanding of what is going on, and lead to a generalization for a group of X number of people. The author describes the students' investigation as a three-step process: pattern seeking, pattern recognition, and then generalization. It is the generalization that shows the students the power of algebraic thinking. Students learn to perceive patterns will obviously grow in their confidence of their own mathematical abilities.

The third article was titled, "Some Misconceptions concerning the Concept of Variable." The article discussed the "path of increasing abstraction," that a mathematics curriculum naturally follows (313). As things become more abstract, what is lost in translation is often what our symbols are actually being used to stand for. The idea of a reversed equation is discussed at great length, where we see student misconceptions about how to translate sentences into mathematical phrases (314). What we need to do is to make sure we protect the distinction between different ideas, and attempt to foster a greater understanding of what variables and equations are in our classrooms.

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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

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

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

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

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

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

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

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

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.

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