Linnea's Article Reviews, 2007

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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

This article offers an idea for a routine project that a middle school math class can be working on throughout the year. Students work in small groups to collect, organize, and describe data; summarize the data in charts and graphs; make predictions using the data; and testing the accuracy of these predictions. The project has the potential to incorporate many forms of technology, many math topics, and cooperative learning strategies.

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

This article offered good suggestions about how to be a more effective questioner and teacher in general. For example, it reiterated what we have been told many times now about not ever taking a student’s pencil or even carrying your own around when answering questions, and it focused on the importance of using questioning to get students to discover solutions on their own. Other suggestions include using the think-pair-share strategy; always requiring several answers to a question so that students can elaborate, agree or disagree, offer other ways to think about the same process, and thus learn something more concretely; and avoid answering your own questions.

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

This is an interesting article about the different ways in which students think about fractions while they are learning about them. As they are developing fraction sense and comprehension skills, students can think about doing tasks such as ordering fractions (determining which is smallest or greatest) or adding fractions (or finding an estimate for the sum) in several ways. Some students might use direct modeling with fraction circles, paper strips, or manipulatives such as pie pieces. Others make mental images and have different strategies that they might employ, depending on the particular problem they are solving. Others use formal strategies. This article had many concrete, specific examples of students who made use of all of these strategies.

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

This lesson is aimed at algebra skills for high schoolers, relating to finding linear equations, solving systems of equations, and measuring and graphing data. Students have a physical model in front of them in which there are two bowls hanging from springs at different heights and they have tokens and marbles to put in them and they measure how far the bowls drop with the addition of one object. After making a table of data, they make a graph (and hopefully see that the lines are parallel when the bowls start at different heights but both have marbles or both have tokens, but they intersect if the higher bowl has marbles and the lower one has tokens. They determine equations for these lines and also solve the system algebraically to find the amount of marbles and tokens needed respectively to have the bowls hang at the same height.

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

Reading reaction 6 Overview to curriculum “Accentuating the Negative” It seems like this will provide students with a very thorough array of activities and lessons in which they can learn and develop their understanding of negative numbers (integers in particular) and operations such as addition, subtraction, multiplication, and division with negative integers. The activities and lessons presented in the overview and lessons seem appropriate for middle schoolers – there are relevant connections made, such as the fact that a thermometer is essentially a number line with both positive and negative numbers. Examples are relevant to middle schoolers’ lives and experiences. So it seems like these ideas would be good ones to steal (since our job as teachers is to steal ideas)!

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.

Reviewer: Linnea

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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

This chapter highlights four points of view as to how a secondary math curriculum should be organized and what its focuses should be. The four options are to use a Basic Skills approach, a Conceptual Mathematics approach, an Applied Mathematics approach, or a Pure Mathematics approach. I found it interesting to have the distinctions between these laid out, because even in college I hear and use the terms but I have never had them defined explicitly. With a Basic Skills approach, the level of mathematical content is low, the teacher's role is "show and tell" and to fix errors and the student's role is to replicate what the teacher has shown and told. In the Conceptual Mathematics approach, top priority goes to comprehension instead of computation and students are expected to assume more responsibility for their own learning. In the Applied Mathematics approach, the goal is to be able to understand and describe real-world situations mathematically and apply math to problems in physics, engineering, etc. The Pure Mathematics approach focuses on discovering and developing new mathematics and responsibility is almost completely on the student to independtly produce solutions and proofs.

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 enjoyed reading this article after having talked about many of the conceptions kids have and issues teachers run into with the idea of the equals sign, what it means, and how it is used. I thought it was interesting to think about the topic, because when asked what number goes in the box for 8+4=?+5, I immediately solved it using the “Benchmark 4” technique. I like the four benchmark categories, because I can see the definite distinction between comprehension skills at each of the four levels – not that one answer should diagnose a student’s abilities, but if they have consistent thinking patterns in one of the stages, this could be a good tool for a teacher to know how to help them progress in their understanding of solving equalities.

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

This article brought out interesting points regarding the understandings of what a variable is. Over the years, the definitions that teachers and curriculum authors have chosen to use in explaining variables to their students have changed and evolved, and because of this, students in different generations probably had different understandings of variables. Some nuances or focuses within definitions have been the degree of importance put on understanding exactly what a variable stands for (to know what it is that you’re even dealing with) or how important it is to be able to manipulate the algebraic equations regardless of what the variable stands for (because the techniques are independent of the variable’s value).

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

I really liked this lesson about multiplying matrices. I like how it guides the students and class as a whole through the lesson in a natural, logical way that leads them to further develop their thoughts and understanding of matrices and matrix multiplication. I think it is necessary that the authors used real-world situation examples so that the matrices mean something to students. If any given entry had no relevance to a student, there would be no way to explain the inevitable “why do you do it that way?” when students ask. If there is a real-life situation giving the exercise some sort of context, we teachers will be able to go back and forth between the arithmetic and the situation in our explanation so that the students understand the background.

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

This book is one of the sequels to Every Minute Counts, which we read earlier in the semester (and which I really liked!). It is written in the same light-hearted, easy-to-read style, and the author puts his humor and sarcasm into the writing again, so it is fun to read. He also makes many good points. I think this book (like the other one he wrote) would be a good back-to-school book for teachers to read each fall at the start of a new school year and re-focus their teaching plans and strategies.

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.



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