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.
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.
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.
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.
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.
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.
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.
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.
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).
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.