# Heather's Article Reviews, 2006

Ref: Heather1
Author(s): Olson, Mark
Date: 2001
Title: Two Mathematicians' Perspectives on Standards: Interviews with Judith Roitman and Alfred Manaster.
Journal or Publisher: School Science and Mathematics
Volume, Issue, Pages: Vol. 101, Issue 6, page 305
Reviewer: Heather
Date of Review: 2-15-06

This article summarizes an interview in the spring of 2001 with professional mathematicians Judy Roitman, University of Kansas, and Alfred Manaster, University of California, San Diego. Both mathematicians were members of the Writing Group for the National Council of Teachers of Mathematics Principles and Standards for School Mathematics. They explained what their rationale was and what they thought they were trying to accomplish.

I thought it was an interesting article, because it was from the perspective of the people who had a hand in making the standards instead of perspectives of teachers, districts, and students, which I have seen more articles about. It wasn't a life altering article, because it was pretty straight forward and kind of (common sense?... can't think of the word), but it was interesting and worth reading.

Keywords: Problem Solving
Ref: Heather2
Author(s): Nugent, Christina M.
Date: 2006
Title: "How Many Blades of Grass Are on a Football Field?"
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Volume 12, Issue 6, pages 282-288
Reviewer: Heather
Date of Review: February 20, 2006

This article was about a lesson that Christina Nugent taught her fifth grade class. She asked them, "how many blades of grass are on a football field?" She then guided her students and let them come to their own conclusions.

When she asked the question, the students made some guesses. They then concluded that counting all of the blades of grass would take far too long. Christina asked them to come up with strategies for estimating how many blades of grass there are. They came up with a few ideas and finally decided to count how many blades were in a yard and multiply that number by 100. Christina realized that they were missing the concept of area, but instead of telling them, she brought them outside with the rulers and told them to begin counting. Soon enough the students realized they were only counting a line of grass. After much deliberation they decided to count a square inch. They did this in groups, each group obtaining a different number. They discussed why as a class and decided they'd take the average of their answers and multiply. They concluded that there were about 194 million blades of grass (in case you ever wondered that).

After they found their answer, they discussed who would care about this. They brainstormed some people, and Christina asked them to write a letter to a player or groundskeeper, or whoever they thought would care about this, telling them the problem and the way the went about solving it. They then peer edited each other's letters and she sent them off.

I really liked this article. It was an interesting way to introduce problem solving into a classroom, rather than just doing problems out of a text book. I also like that she had them actually do the problem. I have been in classes before where we have brainstormed how to solve something, but that was as far as it went. We never actually did it. I also liked it because the lesson was good for all learning types and levels. It included guessing, brainstorming, counting, estimation, discovery, group work, and writing. From my experiences, it was rare to have that many concepts in one lesson. I will definitely remember this article when I am teaching, because it was a great discovery lesson and it increased the students abilities to explain their thinking, which I think is difficult.

Keywords: Algebra
Ref: Heather3
Author(s): Armstrong, Alayne C.
Date: Dec 2005/Jan 2006
Title: An "Arithmetic" Thinker Tackles Algebra
Journal or Publisher: Mathematics: Teaching in the Middle School
Volume, Issue, Pages: Vol. 11, No. 5, p. 220-225
Reviewer: Heather
Date of Review: February 26, 2006

This article was about a 13 year old student named Nina who was interviewed and given problems to determine her approach to solving algebraic equations. Alayne Armstrong is an 8th grade teacher and Nina is one of her students.

Armstrong begins by summarizing research done by Kieran about students learning algebra. Kieran separated algebra learners into 2 groups - "algebraic" thinkers and "arithmetic" thinkers. Armstrong wanted to see if being an "arithmetic" thinker would help or hinder further development of algebraic concepts.

Armstrong classified Nina as an arithmetic thinker, because she often used trial and error substitution to solve algebraic equations. She set up three interviews with Nina giving her interactive tasks and observing her thought process.

Armstrong found that Nina's thinking began with estimation, trial and error, and logic. She then progressed to using "step by step" approaches, writing down her steps, thinking through, and eventually produces algorithms for two step equations.

After the interviews, Armstrong concluded that additional repetition and reinforcement, explanations of why one should use certain steps, thinking mathematically, and connections help students better develop algebraic skills. She also noticed that "algebraic" thinkers have more difficulty dealing with equations. She said the most important finding was that if she gave students time to work with new concepts and techniques, they have a much better opportunity of fully understanding them.

I thought this article was okay. Armstrong only observed (in depth) one student and drew her conclusions from that. She also said that students can "understand" things better when she practiced certain teaching techniques, however I don't think her definition of "understand" is what we have been learning. She didn't exactly give an explanation of "why and how" one uses and algorithm or certain techniques, and it sounded more like she meant "when and how" one should use them. The article somewhat informative, but I also have a problem with the fact that Armstrong was Nina's teacher, which may have also influenced Nina to make more connections from her interviews to her classroom.

Keywords: Activities, Communication
Ref: Heather4
Author(s): Reynolds, Anne; Cassel, Darlinda; Lillard, Eileen
Date: March 2006
Title: A Mathematical Exploration of Grandpa's Quilt
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: Vol. 12, No. 7, p. 340-345
Reviewer: Heather
Date of Review: February 26, 2006

This article was about students in a class (did not specify grade level) working together to solve a problem from a story book. This lesson took place over a five day period.

On day one, students were to work with partners to construct and image of grandpa's quilt from a book the teacher read to them. They used colored squares to form the quilt. It was red with a yellow star in the middle. When they were finished, students shared with the class their different strategies of how they constructed the quilt. On day two, the teacher read them more of the book, and they needed to find a way to make the quilt longer (with the same amount of squares) so grandpa's feet would not stick out. They brainstormed and discussed strategies as a class. On the third day they worked with different partners and tried out their different ideas of how to lengthen the quilt. Once again, solutions were presented at the end of class. On the fourth day, the teacher read the rest of the book, which showed the solution. The students were puzzled, because the star in the middle was changed to a diamond. The teacher instructed them to cut their quilts up and try to change the shape into a diamond. The students did this and those that were able shared with the class. Finally, on the fifth day they made puzzles of the star/diamond shape to take home and test their parents or friends.

I enjoyed reading this article. Five days seems a little bit long for me to stay on this one topic, but I am not sure if they only did this activity for a certain amount of time each day. I liked how the book showed the solution but not how to get there. I think that for students, it sometimes helps to have an answer for them to work to, because they are more confident in their progress. They demonstrated cooperative learning, because they worked on everything in pairs or as a class, and they figured out different solutions as a class and were able to make sense of all of the ideas together. It also allowed students who were! n't quit e grasping some ideas to see their classmates' rationale. The teacher also remarked that this activity encouraged later discussions among the students about different activities and puzzles done in class. I think that is great, because students can talk through and defend their ideas and solutions, which they may not be used to doing in math class. They may tend to think mathematically and not necessarily know how to express themselves in words. This is important for them to do especially in cooperative learning.

Keywords: Probability
Ref: Heather5
Author(s): Rubel, Laurie
Date: February 2006
Title: Good Things Come in Threes: Three Cards, Three Prisoners, and Three Doors
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 99, No. 6, p. 401-405
Reviewer: Heather
Date of Review: March 1, 2006

This article talked about the famous Monty Hall Door Problem and two other similar problems (Three Card and Three Prisoner). A quick overview of the problem is: There are three doors, two with goats behind them and one with a big prize behind it. A person selects a door and one of the other doors with a goat behind it is raised. The person can then choose to stay with the same door or switch to the other door. It is a conditional probability problem, and the Three Card and Three Prisoner Problems are similar.

The article talked about these problems and how people tend to approach them. Rubel stated that most people assume the probability of the prize being behind either of the two remaining closed doors is 1/2. In fact, the Three Card problem was given to 173 students grades 5, 7, 9, and 11 from the NYC boys school, and over half of them thought the answer was 1/2. Their thought processes varied, but they still arrived at that answer. Rubel advises teachers of these misconceptions and tells us to anticipate them when giving these problems. She suggests diagrams, simulation, and data analysis, with a goal of having students understand conditional probability.

I liked this article, but it wasn't extremely informative. I remember in Probability Theory, my teacher gave us the Three Card Problem. I was one of those students that said 1/2. After he went through and explained the answer and how he had arrived at it, I was kind of amazed. I thought it was a great problem. It did help me understand conditional probability a lot, and it was easier for me to use and apply it to other problems.

This article simply stated the problems and gave us examples of students answering it. Their answers didn't come as a shock to me, since I was one to give the same answer. Obviously I am already aware of the misconceptions people have, but if someone had not already seen the problem and the solution, this article would be more beneficial to them.

Keywords: Representations
Ref: Heather6
Author(s): Walker, Janet M.
Date: Winter 2004-2005
Title: "Going Around in Circles: Connecting the Representations
Journal or Publisher: On-Math
Volume, Issue, Pages: Vol. 3, No. 2
Reviewer: Heather
Date of Review: March 6, 2006

This article was basically the outline and results of an activity that students completed. The activity was done on the Geometer's Sketchpad. Students made a circle and then did different things to it and watched what happened to the radius, circumference, and area. They put their results in a table. After this, they explored and found relationships between the properties of the circle. The author mentioned that most of the students didn't recognize the representations, such as the types of graphs the relationships created, until they made graphs. This activity showed how procedural calculations can be displayed with graphical representation.

In the introduction of this article, the author states that "All math is a circle. Students are the center of the circle, and they should learn to connect every point on the circle to their own knowledge by using various representations and connections." I like this quote, and this activity showed the procedure of how this can be achieved. It helped the students make connections among graphical, symbolic, and tabular representations of a circle. It also used representation to help the students learn about how procedural knowledge relates to other math. The article was interesting and it laid out an effective activity to use in a classroom.

Keywords: Algebra, Activities

Author(s): Alejandre, Suzanne
Date: 1994-2006
Title: Understanding Algebraic Factoring
Journal or Publisher: Drexel School of Education
Volume, Issue, Pages: website
Reviewer: Heather
Date of Review: March 8, 2006

This lesson plan's objective is to show the geometric basis of algebraic factoring. In this lesson, students use algebra tiles(15 1x1 unit squares, 10 1xX unit rectangles, and 3 XxX unit squares) to represent algebraic factoring. For example, they are to show (x+1)(x+3)using one XxX unit square, four 1xX unit rectangle, and 3 unit squares. After they realize they can represent this, they can take apart and re-arrange their shape and see that those 8 shapes also represent x2+4x+3. They are then to practice with other equations. The teacher presents FOIL and shows how it works with the tiles.

After they have mastered using the algebra tiles, they can start another activity using 3 axa unit squares, 4 axb unit rectangles, and 3 bxb unit squares. The first task they must perform is to form a rectangle or square that shows (a+b)(a+b). After this, there are some questions for them to think about, such as "What figure does this make? A square or a rectangle? Why?" or "If you rearrange the squares and rectangles making up the larger square, what do you have?" Stemming from their previous knowledge, they discover that this also represents a2+2ab+b2.

I thought this was a good lesson plan. It worked with the inner ideas of factoring and displaying what exactly is happening when one performs algebraic factoring. It did not simply state FOIL and have the students practice FOILing over and over. I also liked the questions she asked towards the end in order for the students to realize that(a+b)(a+b)=a2+ab+b2 instead of just telling them. I thought the activity was great and very good for visual learners. This lesson caters to visual and mental learners though, because the mental learners can see how the tiles should be and arrange them according to that, while the visual learners can keep re-arranging the tiles until they find their solution.

Keywords: Curriculum, Probability,
Ref: Heather8
Author(s): Zawojewski, Judith S.
Date: 1991
Title: Dealing with Data and Chance
Volume, Issue, Pages:
Reviewer: Heather
Date of Review: April 17, 2006

This Addenda book would be very helpful for teachers, providing them with ideas and activities in order to cover the NCTM standards. It is for grades 5-8, helping students make a transition from elementary to high school level mathematics. This book encourages problem solving, reasoning, communication, connections, etc and steers away from traditional teaching which they deem unsuccessful. Real life situations are also included in this book especially, since it deals with data and chance. This requires students to use technology in order to sort and analyze data, which are important tools to use in society. The book goes in chapter order listing activities, questions, assessments, examples, and illustrations.

I thought this book was good for the type of classroom we are encouraged to form in this class (ed 350). I like the fact that there are little paragraphs in the margins telling the teacher the connections, reasoning, problem solving, etc used in the chapter. I liked the book and how it integrates statistics and literature into a curriculum.

Keywords: Curriculum, ,
Ref: Heather9
Author(s): Bright, George W.; Brewer, Wallece; McClain, Kay; Mooney, Edward S.
Date: 2003
Title: Navigating through Data Analysis in Grades 6-8
Volume, Issue, Pages:
Reviewer: Heather
Date of Review: April 17, 2006

The Navigations book specifically targets the NCTM standards related to the unit. In this case, Data Analysis. The book provides activities where conceptual knowledge and deeper thinking about the mathematics is involved. Each chapter is complete with an introduction including materials, illustrations, and what the students may already know about the chapter. Then each lesson is set up as a lesson plan including the goal, materials, activities, discussion, and extended activities.

I like this book better than the Addenda book. This was organized and more of an outline for a teacher to follow. I also liked how there were goals for each lesson and that they were based on the NCTM standards. I also liked that under the "Activity" part of each lesson, there was a description of the activity and how and what it will push students to understand. I also liked the "What Students Might Already Know" section in each introduction, however, it is based on previous Navigation books, so if a student came from a different curriculum it may be different.

All in all, I liked this book. It was very well organized, and the tasks and problem posing were good and worth while. It also ensures coverage of the standards which would help me out a lot in the classroom.

Keywords: Communications, ,
Ref: Heather10
Author(s): Martino, A.M.
Date: 1999
Title: Teacher Questioning to Promote Justification and Generalization in Mathematics: What Research Practice Has Taught Us
Journal or Publisher: Journal of Mathematical Behavior
Volume, Issue, Pages: Vol. 18, No.1, pages 53-78
Reviewer: Heather
Date of Review: April 12, 2006

This article was about the teacher's role in the classroom in which exploration takes place. It was about a study being done about teacher questioning in the classroom. According to the study, the most important role teachers can play is questioner and listener. Also, the teacher's understanding of the mathematics and the children (backgrounds) is important too. When a teacher does these things, it encourages students to express their thinking. Analysis of this study showed a strong relationship between careful monitoring of students' constructions leading to a problem solution, and the posing of a timely question which can challenge learners to advance their understanding.

Keywords: Teaching Strategies, Measurement, Representations
Ref: Heather11
Author(s): Danielson, Christopher
Date:
Title: Perimeter: Three Approaches, Three Definitions
Journal or Publisher: MCTM Conference
Volume, Issue, Pages:
Reviewer: Heather
Date of Review: April 24, 2006

One session I attended was about different ways to teach and learn a concept. In this case, we were "learning" about perimeter. We paired up and were instructed to grab a blue, yellow, or white worksheet and to complete it. We were supposed to pretend that we had no prior knowledge about perimeter and that our worksheet was teaching us through some type of exploration. The blue worksheet used kitchen lay outs and paths to define perimeter, the yellow used measurement with string and rulers to define it, and the white sheet used a bumper car arena in its investigation. After we finished, we came together to discuss our own definition of perimeter based on the worksheet we had. Our speaker gave us a shape on the overhead and told us to find the perimeter of it. We came up with many different answers based on what we had learned.

I thought this session was very interesting. It turns out that each worksheet we used was from a different curriculum, ranging from 100% discovery to traditional teaching. It reminded me a lot of class, and it would have made a good microteaching lesson. I was also the youngest person in the session, and it was fun for me to see actual teachers reason, solve, discuss and defend their way of approaching a problem. I liked the fact that we did an activity instead of being talked at for an hour, and I got a lot out of it. This was well done.

Keywords: Equity/Diversity

Ref: Heather12
Author(s): Neel, Kanwal Singh Date: 2005 Title: Addressing Diversity in the Mathematics Classroom with Cultural Artifacts
Journal or Publisher: Mathematics Teaching in the Middle School Volume, Issue, Pages: Vol. 11, no. 2, p. 54-59 Reviewer: Heather Date of Review: May 7, 2006

The article began by addressing what diversity is. It has many meanings such as a diverse background, ethnicity, culture, gender, language...etc... The article addresses the question, "How should one teach mathematics to all learners regardless of their diversity?" Neel believes that teachers need to vary their practice so each student learns and makes meaning of their learning in a unique way. Teachers also need to create a safe and accepting environment. He also said that classroom practice can embrace and celebrate diversity while ensuring a powerful math program for everyone.

Neel went on to describe a classroom project used in his middle school class that addressed diversity. The project is called the "SCAMP" (Story about a Cultural Artifact from a Mathematical Perspective) project. To complete this project, students were asked to choose any item that they were interested in and outline why they chose it. Then the students researched the math behind their item. For example, a student used pizza, and she found that it related to math through its shape, measurement, probability, and fractions. Other ways items could relate to math would be through number sense, spatial sense, statistics, patterns...etc... After they did their research, they were to write a story, song, or poem about their item and then make a presentation to the class. The assessment for this project was a scoring rubric given to the students at the beginning of the project. The students also completed a self-evaluation and reflection questions.

I thought this article was interesting. I think it is difficult to address diversity in a math classroom. The SCAMP project is a good idea, because it allows students to learn math in different ways and to educate their peers about it. However, there are some items that would be a lot more challenging to find mathematical connections to than others. I think the story/song/poem part of the project is good, because it gives the student a chance to be creative and also explain why their item is important to them. From the teacher's and students reactions, the project seemed worthwhile and it demonstrated understanding of math in diverse contexts.

Keywords: Technology

Ref: Heather13
Author(s): Sinclaire, Nathalie; Crespo, Sandra Date: 2006 Title: Learning Mathematics in Dynamic Computer Environments
Journal or Publisher: Teaching Children Mathematics Volume, Issue, Pages: Vol. 12, no. 9, p. 436-443 Reviewer: Heather Date of Review: May 7, 2006

This article was informative for me, because I am not familiar with the Geometer's Sketchpad. It gave a good review of the tasks the Sketchpad could accomplish and convinced me that it is an effective technological tool for the classroom. I think technology should play a role in education, because it allows students to visually see what is happening so they can better understand the mathematics behind it. Effective technology makes it possible for students to learn important mathematical content and processes that are challenging to learn without it.

Keywords: Measurement, Activities

Ref: Heather14
Author(s): Horak, Virginia M. Date: 2006 Title: A Science Application of Area and Ratio Concepts
Journal or Publisher: Mathematics Teaching in the Middle School Volume, Issue, Pages: Vol. 11, no. 8, p. 360-364 Reviewer: Heather Date of Review: May 7, 2006

According to this article, when math and science are integrated, students can investigate questions like "Why do I sink into fresh snow and a rabbit does not?" or "Are 'human' giants of fables and fiction possible?" It goes on to describe an activity used in an integrated math and science course where students explored the first question. They used area and ratio concepts, which are important cornerstones of the middle school math curriculum. Students investigated the impact of an object's weight and the area of its "footprint." They used different sized weights and set them on/in soft surfaces such as mud, sand, sugar, and whipped topping. They then did trials where the surface area changed, but not the weight. They described what they saw using observation skills from science class and made a ratio of weight to area and found a sinking value or SV. They then went on and generalized their findings to paw and footprints and discussed the results.

I thought this was an interesting activity. I feel like math was a tool that the students used in a science classroom. For example, they found and examined the SV ratio in science, but they used the ratio in math. This activity made good use of mathematical and scientific resources. I guess the more I think about it, this could be an activity in a math classroom as well, but it fit perfectly into the integrated course. I agree that it is definitely an advantage to integrate math and science, because it allows math to explain concepts and occurrences in science.

Keywords: Technology, Statistics, Probability

Ref: Heather15
Author(s): Franklin, Christine A,; Mullker, Madhuri S. Date: 2006 Title: Is Central Park Warming?
Journal or Publisher: Mathematics Teacher Volume, Issue, Pages: Vol. 99, no. 9, p. 600-605 Reviewer: Heather Date of Review: May 7, 2006

The article began by addressing the fact that students find probability a difficult topic when presented in a formal mathematical way. It suggested simulation as an alternate and effective approach in teaching things like p-value and distributions of variables or statistics. One activity that uses simulation is about the temperatures in Central Park. This simulation is useful in demonstrating concepts of hypothesis testing, p-value, and sampling distribution. It is to be used in a high school or college classroom. Students are instructed to collect data (provided on a web page) of mean annual temperatures in Central Park from 1901-2000. They then must make a time series plot and look for fluctuations and patterns. In order to broaden their information and see if fluctuations were by chance or pattern, they can simulate temperatures from year to year. Next they analyze their data and see how close the simulated probability is to the mathematically derived probability. The students can then think of questions to further investigate with histograms or charts. After this, they come to a conclusion that Central Park is or is not warming.

While I was reading this article, it reminded me of my Statistics 212 course that I took last semester. We used a lot of simulations and applets to look for trends and make predictions. Simulation is obviously an efficient way to randomly represent things like 1000 coin tosses. I found it very helpful and efficient in the classroom. Another positive aspect was that we each did our own simulation, so we could compare ours with our classmates' results. I thought simulation was effective and easy and a good tool for the classroom.

Keywords: Communications, Teaching Strategies

Ref: Heather16
Author(s): Manouchehri, Azita; St. John, Dennis Date: 2006 Title: From Classroom Discussions to Group Discourse
Journal or Publisher: Mathematics Teacher Volume, Issue, Pages: Vol. 99, no. 8, p. 544-551 Reviewer: Heather Date of Review: May 8, 2006

This article talks about moving classroom discussion to a learning community with discussions, participation, agreement, disagreement, thinking, etc... The authors believe that simply sharing ideas is not enough and not what the standard of classroom discourse envisions. The standard envisions the generation of productive math dialogue among learners. Worthwhile discourse also includes conversation with reflection and action. Students gain insight and try to influence others. It also requires participation, commitment, and reciprocity.

In a traditional classroom, discourse is predictable and polished. Students often arrive at answers, but they may or may not make connections. In addition, the quality and quantity of student participation is left up to the teacher by asking questions, hearing an answer, and giving feedback. On the other hand, within a learning community, discourse means seeking connections, distinguishing valid from invalid arguments, and constructing knowledge through dialogue that is intellectually challenging. Discourse within a learning community is not predictable and the teacher builds on the students' ideas to design a curriculum and instruction.

The authors also believe that the purposes of discourse in traditional classrooms and learning communities differ. According to them, the purpose of discourse in a traditional classroom is to transfer information, monitor progress, listen to provided information, and standardize student thinking. Its purpose in a learning community is assist the students and teachers in learning about the subject, explore and explain ideas and connections, establish knowledge, and seek new understanding and inquire. The outcome of discourse also differs from the traditional classroom to the learning community. In a traditional classroom, discourse develops techniques while the teacher designs and delivers lectures and communicates "important information" to students. Discourse in this setting may or may not change the students ! or teach er's overall thinking about math. However, in a learning community, discourse constructs, negotiates, and verifies mathematical ideas while everyone is involved. It also brings about a new development of understanding, new insights, and deeper analysis for both the teacher and the student. In this setting, the teacher participates but ensures that students reach a shared understanding of math.

The article continues with two classroom narrations - one traditional and one a learning community. The classrooms were geometry classes within the same school. The narrations gave good examples of discourse in each of the environments described above.

Keywords: Activities, Games, Measurement

Ref: Heather17
Author(s): Tzur, Ron; Clark, Matthew R. Date: 2006 Title: Riding the Mathematical Merry-Go-Round to Foster Conceptual Understanding of Angle
Journal or Publisher: Teaching Children Mathematics Volume, Issue, Pages: vol. 12, no. 8, p. 388-393 Reviewer: Heather Date of Review: May 9, 2006

This article was about worthwhile activities that can be used in a classroom to foster understanding. The authors believe that one should gain understanding from all activities done in the classroom. and that learning is an active process. Their thinking also corresponds to the NCTM Principles and Standards. The authors stated that typical angle activities are confusing and boring and the definition of an angle is not a good conceptual basis for distinguishing an angle. Therefore, activities that develop better understanding should be used in this case.

The activity that this article goes in depth about is called the "Mathematical Merry-Go-Round." It is appropriate for the fourth grade level. The teacher, Kay, did little to introduce this activity to her class. She did not inform them before-hand of the concept she was going to be teaching either. She simply explained to the class and had one student volunteer to demonstrate what they were going to be doing.

Player a puts a blank piece of paper on a piece of cardboard with a merry-go-round transparency on top. They then insert a push pin into the center of the merry-go-round circle and through the cardboard. On the transparency there is a ray extending from the center point to the edge of the circle with four labeled points along it (A,B,C, & D). The player then places a push pin through one of the points on the ray so it leaves a hole in the cardboard. They then pull the pin out of the cardboard but not out of the transparency. Next, they use the pin to slowly rotate the transparency clockwise about the center. Player B then closes their eyes and tells Player A when to stop rotating, and player A pushes the pin into the cardboard at that point (not more than 1 full circle). Next the pins and transparency are removed from the cardboard, and Player A draws segments going from the center through each hole made by the push pin. Now player B must use a new point on the ray and try to make the same angle. Points are awarded for! which a ttempt Player B matches the angle. After this, the players switch roles and repeat.

Keywords: Representations, Connections

Ref: Heather18
Author(s): Moore-Russo, Deborah; Golzy, John B. Date: 2005 Title: Helping Students Connect Functions and Their Representations
Journal or Publisher: Mathematics Teacher Volume, Issue, Pages: vol. 99, no. 3, p. 156-160 Reviewer: Heather Date of Review: May 10, 2006

The article began by addressing the issue that students view functions and their graphs as two separate things, rather than different representations of each other. Students should be using graphs as visual insight to the behavior of a function. The authors of this article believe that there should be a change in instruction to encourage student exploration and understanding of functions and their graphs. They also believe that when solving a problem, algebraic work can lead to graphical solutions and vice versa. However, when the student works from algebra to graphs, they tend to use algebraic symbols without completely understanding the meaning behind them. Some positive aspects of working with graphical data are: students can make predictions about other graphs from these initial graphs and they can see and understand the algebraic solution better from analyzing the graphs. The article goes on to describe a quadratic function activity to use in the classroom. This activity helps link graphs and functions and helps the students realize the benefit of functional representation. It also helps students build on previous experience in a way that challenges them.

I thought this was an important article and an important subject. When I was in grade school and high school, one thing I never fully understood was the connection between functions and their graphs. We would randomly have homework questions where we were supposed to match a function to its graph, and unless it was a straight line, no one would know how to do it. For example, we would need to circle a graph of a function, then the graph of its derivative, and then the graph of the derivative's derivative. Without being taught good connections between the functions and their graphs, how is a student supposed to recognize them? This article addressed this topic and gave some ideas of how to incorporate these connections into the classroom. I think it is valuable and worthwhile.

Keywords: Technology

Ref: Heather19
Author(s): Watson, John W.; Ciesla, Barbara A. Date: 2005 Title: Finding Complex Roots: Can you Trust Your Calculator?
Journal or Publisher: Mathematics Teacher Volume, Issue, Pages: vol. 99, no. 5 Reviewer: Heather Date of Review: May 10, 2006

This article began by talking about technology, such as calculators, that allow a student to go more in depth with investigations. However, the authors were quick to say that technology can give incorrect or misleading information. It can also take away the meaning for students. For example, a student could type in 3^(root 2) and get an answer of about 4.73, but when asked what that means... students don't know.

The article continued on with an example from a pre-Calculus class where the students found a different answer with their TI-83 calculators then the one in the back of their text. The problem involved finding the root of a complex number. The students and teacher discovered that in order to get the answer in the back of the text, they must enter the problem into the calculator a certain way. The article goes into detail about the problem and a rule that the calculator violates when the problem is entered in a certain way.

I thought the whole concept of this article was important. Technology is a powerful tool, but people must be careful not to rely completely on it. And although it can do great things, people must be careful not to be misled by its information. In the example used in the article, the students entered two problems in two different ways (but equivalent) into their calculators and got two different answers. Students must find other ways of solving these types of problems, because guessing which way to enter a problem is stupid and a waste of time. I am not saying technology is bad, but I just don't think people should rely on it completely. At the end of the article, the author said, "Students shouldn't take an answer at face value no matter what its source." I thought that was a very good point, and I agree.

Keywords: Algebra, Geometry, Keyword 3, Optional...
Ref: Heather20
Author(s): Grandau, Laura; Stephens, Ana
Date: 2006
Title: Algebra: Thinking and Geometry
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: vol. 11, no. 7
Reviewer: Heather
Date of Review: May 14, 2006

This article was about incorporating algebraic thinking into all math. The authors are doing a project for the University of Wisconsin, Madison and are involved with professional development activities that focus on developing and helping teachers recognize the potential offered by tasks to engage students in algebraic thinking. Through research on learning and teaching, algebra is identified as a priority by the math education research community. They believe algebra should be a "strand that weaves throughout other areas of math in the k-12 curriculum. They have also stated that slgebraic ideas can be engaged in as early as 1st grade.

The article goes on to address the CMP curriculum and how this curriculum is more difficult to incorporate algebra into. The authors wanted a design or activity that would encourage the development of algebra use within the context of the curriculum. They found two teachers to try to incoroporate algebra into their geometry lessons from the CMP curriculum. The article went on to describe their lessons.

I had a good time reading this article, especially the lesson plans of the two teachers. They both thought of creative ways to incorporate algebra into their geometry lessons. I agree with the authors that algebra is a very essential part of mathematics, and I think that people get the wrong idea when algebra is talked about. One may think that algebra is just equations with x's and y's, but not realize the importance or applications that it can make. So I think algebra should and can be integrated into other subjects, however, integrating takes strong mathematical knowledge and time from the teacher to help their students develop the necessay skills for this task.

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