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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

This article was written by two middle school math teachers from Rice Lake, Wisconsin, who expressed a desire to “make education more meaningful to our middle school students and their parents.” To do this, Kristine and Jennifer, the two teachers, decided to try switching the focus from grades to learning and opening up communication between schools and home. They used standards-based assessment, along with portfolios, to work for the change they want to see.

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

In this article, Steven Reinhart begins by briefly describes some of his struggles with how to teach, which eventually lead him to search for better ways to engage students. When he first began teaching, Reinhart felt he was good at explaining math in ways that kids could understand, and he know his subject matter very well. However, despite these things he found that kids were not engaged and learning math up to the potential he knew they had. Reinhart decided to do some research and gradually started changing his teaching style to incorporate other strategies.

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

This article looked at the different ways children understand fractions. It discusses issues surrounding the results of a test on fractions, which was given to fifth graders in a particular district. It focuses in depth on 3 of 26 fifth graders who were selected for more extensive interviews.

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

This introduction to the Accentuate the Negative book begins by explaining how students need to be introduced to integers, beginning with games, then number lines, and then colored chips. Adding comes first, then subtraction, and then multiplication and division.

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

This article begins by defining subtraction with regrouping. It explains that a deep understanding of mathematics is necessary for teaching this topic, and a teacher’s subject matter knowledge contributes to students’ learning.

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

This article begins by showing research that reveals many elementary students do not understand the true meaning of the equal sign. By asking students to make an open number sentence, which showed addition of two numbers on both sides with the value of one number missing, a true number sentence, they found that many students see the equal sign as a symbol to do something and not to show equality between two things.

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

This article begins by explaining the various ways that variables are used in algebra: in formulas, in equations, in properties, in identities, and in expressions not to be solved. Our idea of variable, it argues, has changed over time and it is not appropriate to limit that idea to one conception.

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

As the title suggests, the book Motivation Counts gives many strategies for motivating students in the secondary mathematics classroom.

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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