Anna's Article Reviews, 2007

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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

This article discusses issues involving partner quizzes; what they are, how they work and examples of students' work during partner quizzes. The article suggests that partner quizzes show the value of working and discussing problems in pairs. Students are able to work together to formulate successful ways of solving problems. Partner quizzes also create a environment that involves collaboration and support by their fellow classmates.

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

This article talks about the types of changes a teacher can make in their classroom to make it a more stimulating environment for the students. Teachers need to step out of their comfort zone and change their teaching techniques to how their students will learn the best. Teachers must adapt to the students, not have the students adapt to them. Reinhart gives examples from his own teaching experience, what worked well and what didn't work well, to introduce teaching strategies to try in the classroom.

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

This article focused on the types of methods students use to solve fraction problems. Also, the article was intended to show the importance of these different strategies to help students solve problems. Teachers should incorporate asking students to explain why they solve a problem as they did and not just settling for the final answer. The article followed three students through a written fraction test and a oral fraction test, their different methods of solving the problems and how or where they went wrong.

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

This lesson was created to get students familiar with modular numbers and how they work. At the beginning of the lesson, mod 12 was introduced by connecting it to daily life. For example, 10 am + 4 hrs= 2pm ---- 10= 2 mod 4. This lesson was also created to explore the patterns in modulus tables. Attached to the lesson are practice sheets where students can start exploring these patterns found in addition and multiplication modulus tables.

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

The main idea in this book follows the title of the book, accentuate the negative. The book includes different activities that help students understand addition, subtraction, multiplication and division of integers. Visual activies such as number lines and chips are used for students to explore why addition, subtraction, multiplication and division of integers works. The goals of the activities is to help students develop good math habits as well as be able to work through problems effectively. Also this book helps teachers understand how to connect integer operations to big ideas and to prior and future work.

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

This chapter entitled "Manipulative Devices for Elementary School Measurement Activities" provides descriptions and illustrations of manipulative devices to use in teaching measurement. The measurement manipulatives are grouped into categories of length, temperature, area, volume, mass and weight, and metric place-value aids. Along with describing and illustrating uses of the devices, the authors have provided company names that provide the items and prices for each. A few of the manipulatives mentioned are cubes, rulers, protractors, thermometers, fraction circles, etc.

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

This article dealt with the equal sign and students' misconceptions about the meaning of the equal sign. Without this basic knowledge and understanding, it is hard for students to progress into more complicated mathematics. This article looked at four students using different interpretations of the equal sign. Each student was asked to explain what they did and then asked a few other questions from the assessor to further understand what each student did.

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

This article deals with the use of variables in school algebra. Usiskin first states that algebra is not easily defined and that algebra in school is much different from algebra taught in college to math majors. The conceptions of variables have changed over time. Variables can be used in formulas, equations, identities or properties-variables are used as symbols in which things can be substituted. One major question posed in this article is when algebra should be introduced into the curriculum. Usiskin discusses 4 conceptions of algebra; algebra as generalized arithmetic-relationships between numbers using variables, algebra as a study of procedures for solving certain kinds of problems-variables are unknown or constants, algebra as the study of relationships among quantities-variables as arguments-patterns and algebra as the study of structures.

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

This lesson focused on multiplying matrices. The lesson gives about four real world applications for why multiplying matrices might be useful in real life. The examples varied from understanding what the entries of a matrix tell you, to understanding how to get a useful result from multiplying two matrices together. Also, this lesson plan incorporated examples to try on the calculator. It seems as though this lesson is geared towards the student exploring how and why two matrices are multiplied together, instead of just giving them simple examples and answers.

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

I read two chapters in this book: Subtraction With Regrouping: Approaches To Teaching a Topic and Multidigit Number Multiplication: Dealing With Students' Mistakes. These chapters compare and contrast teachers from the United States and China on their procedural knowledge as well as their conceptual understanding of a topic. This is a fairly critical look at how and what teachers are teaching to students in their classrooms.

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.

These chapters were really interesting, but complex, to read. The author's goal of this book was to show how Chinese teachers' methods of teaching contributes to the success of their students. I thought it was interesting to take a look at the way each country's teachers tend to teach their students. However, Ma used quotes from individual teachers which took away a generalizing feel to the book. I would be interested to know how these major differences have developed and how American teachers can get back into teaching conceptual ideas, what we can learn from the Chinese.


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