Megan's Article Reviews, 2007

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Keywords: Problem Solving
Ref: Megan1
Author(s): Cohen, Robin
Year of publication : 2006
Title: How Do Students Think?
Journal or Publisher: Mathematics: Teaching in the Middle School
Volume, Issue, Pages: volume 11, number 9, page 434
Reviewer: Megan
Date of Review: February 19, 2007

This article discusses some examples of problems the author has done with her students and all the different ways they went about solving them. The first is a problem involving a basketball game, saying who outscored who by how much in each quarter, but not giving the quarters an order, or specifying a beginning or ending score. Some students used the smallest numbers possible, others branched out with bigger numbers, some solved it putting the quarters in the order they were described, and others mixed things up a bit. Another problem involved spinners and probability, and the last involved adding times. For all the problems, she presented her own method for solving it, then shared a few of the different ways students went about solving them. She encouraged students to teach other students how they had gone about solving it, and to evaluate each other's work (anonymously on both ends).

When I read the title of this article, I have to admit, I thought it would be the answer to all my questions about how kids think. I definitely know the way I think, but often it's very different from how other people see problems, so I was hoping it would be full of helpful hints about how to teach concepts in a multitude of ways so the different styles of thinking would mostly be covered. However, the article was still very helpful. I know I personally am always tempted to have a pencil in my hand, and always be "teaching." This article just reinforced again how important it is for students to find their way on their own as much as they can through exploration and working with each other. I especially liked that the problems they used were so open for exploration and had so many different possible ways to reach a solution. I think it is so important to be flexible when evaluating how students problem solve, since everyone has their own unique way to do it. Human beings are so unique to begin with, there's no way we can possibly expect them to all solve problems the same way. I also liked the criteria she gave for when students evaluate each others' work. It helped them not only to reinforce what needed to be on their homework, but to understand the challenges a teacher faces when evaluating homework that is unclear. Hopefully seeing how difficult it can be to read the chicken scratches they sometimes hand in would help them remember to organize their thoughts on paper more clearly. In the end, the article, while unfortunately not the answer to all my questions, was a very valuable one to read.

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Keywords: Teaching Strategies
Ref: Megan3
Author(s): Reinhart, Steven C.
Year of publication : 2000
Title: Never Say Anything a Kid Can Say!
Journal or Publisher:
Volume, Issue, Pages:
Reviewer: Megan
Date of Review: February 28, 2007

Reinhart's article is about teaching without what people right now sometimes refer to as "traditional" methods. Lecture, demonstration, "telling" students how to do the math, are all thrown out the window. He explains at the beginning of the article that he was very good at explaining how to do math, and that even his superiors acknowledged him to be so, but he was still not happy with his results, so he tried a new approach. He realized that, with his old methods, he was learning, but his students weren't. In order for the students to learn, he says, they must do the discovering. We, as teachers, shouldn't "show" them anything. We must give up our pencils, and start asking instead of telling. Asking good, guided questions leaves it up to the student to figure out what's going on. By them doing the discovering, their understanding of the math is much more deeply rooted and adaptable than for a teacher to tell them how it's done. Encourage them to explore, to make mistakes, to teach each other, and to learn from each other's differences. Make math interactive, push them to teach themselves.

My reaction to this article was, to be perfectly honest, a little bit of fear. I'm a perfectionist, I get down on myself when I don't do things really well the first time, even when I know in my heart that I'm doing something you have to learn over time, such as teaching. Very few teachers are excellent teachers the first time they step up in front of a classroom. I know in my head that I'm not going to be able to successfully teach with these kinds of methods the first couple years of teaching, but I desperately want to, and I know a big part of me is going to be disappointed those first few years. I know that teaching is something you learn from experience. I know that it's going to take a while before I can teach the way I want to. In the meantime, I am going to have to fight the impulse to TELL them, and I really do mean fight. The other day when we were working with the students who came to our class, I found myself working with a girl who was obviously very used to teachers TELLING her how to do a problem. She knew exactly what questions to ask to get me to show her how to do the problem, and I was fighting her and myself the entire time. She almost refused to respond to my questions, she'd always counter them with another question. It was really difficult and more than a little frustrating, to the point where, once the bell rang, I grabbed her pencil and showed her how it was done (bad Megan!). This is what I fear in teaching. I thought I was asking the right questions, tried so many different directions, but I could not get through to her.

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Keywords: Number and Operation
Ref: Megan5
Author(s): Anderson, Dawn L.
Year of publication : 2001
Title: Magic Squares
Journal or Publisher: Mathematics Teaching in the Middle School Journal
Volume, Issue, Pages: http://illuminations.nctm.org/LessonDetail.aspx?id=L263
Reviewer: Megan
Date of Review: March 16, 2007

The lesson plan I chose was about magic squares. A magic square of order n has n2 spaces, with all of the columns, rows, and diagonals adding up to the same number (called the magic constant). The lesson plan gives an in-depth history of magic squares; their origin, how they developed, uses they had, etc. It also shows a couple of different methods for constructing magic squares, complete with illustrations, and provides activity sheets for the students to use.

Overall, I think that this lesson plan was much too long-winded. The "Brief History of Magic Squares" is eight paragraphs long. There are details in the intro and history that really aren't necessary for appreciating magic squares. Especially for a middle school audience, this lesson plan is definitely a little too boring. I know I, personally, would start to zone out after the first paragraph if a teacher used this for a lesson. I actually started spacing out as I was reading it! The descriptions of the construction methods were also very verbose, and could have been simplified for the sake of the teacher and students alike. I think it would be to the benefit of both to not be so detailed; it would leave room for exploration and questioning. As it is, you would just be following instructions. Question 2 on the activity sheet is better about this, but it would be important to have the students do this activity before you described the construction methods so that they would have to try to find their own method first.

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Keywords: Connections, Number and Operation, Activities
Ref: Megan7
Author(s): Ernie, Katherine T.
Year of publication : 1995
Title: Article: Mathematics and Quilting. Book: Connecting Mathematics Across the Curriculum
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: P. 170-176
Reviewer: Megan
Date of Review: April 4, 2007

The article that I read describes a rather clever way of introducing and applying modular arithmetic. She starts by giving a brief history of quilting; names of patterns, uses, etc. She then introduces a well-known Amish pattern, called "Sunshine and Shadow", which is essentially a diamond pattern radiating out from the center in bands of color, which appear in a repeating order. She takes the top left quarter of the pattern (as shown in an illustration in the book), and assigns the bands of color numbers 0-9 beginning in the top left corner, and makes a chart for the color pattern. She then uses this chart as an example for mod-10 counting, shows how to use a "clock" with 0-9 on it for counting in mod-10, and does the same for mod-5.

I really liked how she used multiple methods for teaching the students how to count using modular arithmetic. The chart for the pattern in the Sunshine and Shadow pattern turned into an addition table for mod-10, and the clock method facilitated using modular arithmetic in subtraction and multiplication. She also suggested having the students come up with their own quilting patterns, which would give the students the hands-on experience that most middle schoolers enjoy. By having this activity cover so many different levels, I really think she came up with a very successful lesson.

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Keywords: Number and Operation
Ref: Megan8
Author(s): Falkner, Karen P; Levi, Linda; Carpenter, Thomas P.
Year of publication : 1999
Title: Children's Understanding of Equality: A Foundation for Algebra
Journal or Publisher:
Volume, Issue, Pages: Teaching Children Mathematics
Reviewer: Megan
Date of Review: April 14, 2007

In this article, the authors write about the different conceptions students have of the equals sign and how to work with them to help them understand its true meaning. They start by giving a classic example of a problem where students often confuse the equals sign to mean "do something", instead of representing a relationship. It was a fill in the blank problem, 8 + 4 = _ + 5, that most of the students got wrong, with answers ranging from 7, to 12 and 17. They give lots of examples of dialogues between students and teachers when approaching problems like these as a way of illustrating how the students are thinking, and how the teacher is responding to them. They also give a lot of examples for how to help students take their thinking to the next level. I really liked this article because it was so detailed and descriptive. It had a lot of examples of how to teach what they were talking about, which to me makes a good article for teachers. It's one thing to write an article about what's going on, but sometimes it's not as easy to go from what to how, and this article really did that. I thought that the conversations between teacher and student were good to read, it gave a really good idea of what we as teachers have to look out for and good examples of how to respond to it. However, I unfortunately could not read the Teacher Commentaries, the printer did not translate them well. The examples they used were fantastic, really well thought out in their sequence, and how they fit into the students' frame of reference. The benchmarks were also very valuable, since it gives the teacher something tangible to look for when evaluating how well their students are developing their understanding of the equals sign. The whole thing was somehow detailed, yet concise, so it was easy to follow yet extremely helpful at the same time.

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Keywords: Algebra, Representations
Ref: Megan9
Author(s): Usiskin, Zalman
Year of publication :
Title: Conceptions of School Algebra and Uses of Variables
Journal or Publisher:
Volume, Issue, Pages:
Reviewer: Megan
Date of Review: April 24, 2007

This article focuses mainly on the many different ways a variable can be conceptualized. It goes through each of them in detail, basing the discussion off of the two major problems in teaching algebra today. The first of these is knowing how much of algebra students should be able to manipulate by hand, what with all the technology we have available. The second is knowing what role functions should play in the teaching of algebra, and when they should be introduced. The first conceptualizing of algebra is as a generalizing of arithmetic, and uses the variable to generalize patterns. The second concept sees it as a study of procedures for certain kinds of problems, where the variable is used to generalize mathematical relationships. Thirdly, it is seen as a relationship among quantities, where variables don't necessarily take on a definite value. Fourth, it represents the study of structures, where the variables are simply arbitrary marks on the paper that we use to manipulate and find a solution.

While I found this article interesting, I found it lacking a lot of the "so what" quality that I find helpful in these sort of things. Ok, we've found some different conceptualizations for algebra, now how do we go about applying them to teaching? What do I do with this information? He talks a little bit about how students might get confused with the different concepts of algebra, but gives no examples of how we're supposed to teach them, how to distinguish them, or even how to watch out that we don't screw them up. There isn't really a good discussion about how we should integrate these ideas into how we teach our students algebra, just a description.

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