**Keywords:** Trigonometry, Teaching Strategies

**Ref: **Ryan1

**Author(s): **Calzada, Maria E; Scariano, Stephen M.

** Date: **2006

**Title: ***A Natural Bridge from Algebra and Geometry to Trigonometry
*

**Journal or Publisher: **Mathematics Teacher

**Volume, Issue, Pages: **Vol. 99, Num. 6, pp 450-453

**Reviewer: **Ryan

**Date of Review: **February 16, 2005

This is a good article on how to help teachers introduce Trigonometry
to students. Because Trig is a totally new mathematical setting, the students
sometimes find themselves overwhelmed by what they are doing and learning.
The article gives a few suggestions as to how a teacher could teach the
transition from Geometry and Algebra into Trigonometry. The authors suggest
the teacher leads a group activity where the students discover the foundational
concept of invariance, which they call the cornerstone of Trigonometry, by
focusing on concepts like the Pythagorean theorem, and the area of a triangle
and rectangle. Although this sounds like a good idea, I'm a bit skeptical
because they don't really say anything about anyone trying this in a class,
they just say do it without providing any evidence. Also there are a little
on the vague side with their descriptions as to what should be done. They
tend to talk more about what to concepts to use and not much about how to
connect it to the Trigonometry the students will be learning, to me that
doesn't make much sense. If they have such a good idea that works, I would
think they would be a bit more specific as to how to go about using their
activity and ideas. Their idea could be a very good one and work very well,
but with such little information, if it was interpreted incorrectly, it could
be very confusing instead of helpful for the student.

**Keywords:** Problem Solving

**Ref: **Ryan2

**Author(s): **Shimizu, Jeanne; Zbiek, Rose Mary.

** Date: **2005

**Title: ***Multiple Solutions: More Pathes to an End or More Opportunities
to Learn Mathematics *

**Journal or Publisher: **Mathematics teacher

**Volume, Issue, Pages: **Vol. 99, Num 4, pg 279-287

**Reviewer: **Ryan

**Date of Review: **February 20, 2006

This article discussed the benefits of choosing problems for problem solving with multiple ways to find a solution. The authors argued that the more solutions there were to the problem and the more mathematically rich it was, the more learning of mathematics that took place among the students. The authors gave examples of two problems that a class had worked on, both problems having a number of possible different solutions. In each case they described a number of ways the students went about the problem and the ways in which they guided the discussion that was taking place.

In each situation they did things very similar, they allowed the students to explore with partners and also gave them some hits as to how to go about the problem. After both problems had been discussed in class, they found that their second problem appeared to be much more mathematically rich and that the students were able to learn much more from the problem. The melon problem, unlike the skin problem, was much more in depth as far as looking at a variety of mathematical concepts. Due to the more variety in the melon problem they felt that more learning took place amongst the students because they were able to ask more enlightening questions and get the students to engage their thought in more mathematical concepts.

The authors go on to conclude that with using problems with different and multiple solutions can be very useful to the students' learning. However, picking a problem that had alternative solutions that differ both procedurally and conceptually is the most beneficial way for the students to learn. When the problems differ procedurally they are able to reflect on similarities and differences in the strategies as well as develop a better sense of when different strategies are applicable. However, when the solutions differ conceptually as well, the students are better able to refine their understandings of definitions and also attend to how different conceptions may produce conflicting or consistent results.

All in all I felt that this was a very insightful article that made a
good point as to how to lead students in exploring problem solving techniques.
I think this would be a good read for anyone who is looking for some help
with problem solving in his or her classroom. Although it may get a little
bogged down with details I would recommend it to anyone who is thinking
of becoming a teacher.

**Keywords:** Algebra, Activities

**Ref: **Ryan3

**Author(s): **Kalchman, Mindy S.

** Date: **2005

**Title: ***Walking through Space: A new approach for teaching functions
*

**Journal or Publisher: **Mathematics Teaching in the Middle School

**Volume, Issue, Pages: **Vol. 11, Num. 1, pgs. 14-17

**Reviewer: **Ryan

**Date of Review: **2/22/06

This is a good article on how to first introduce the concept of functions to students for the first time. Because functions are such a big part of mathematics, both in algebra and courses that follow, it is very important that students are able to understand this concept and the earlier they do so, the better. The article describes an activity a teacher used to introduce the idea of functions and graphing that I think is great; after reading this I wish I was taught functions in this way because it is so much easier to understand.

Kalchman describes how the class used a fundraising example to first learn the idea of functions, she mentions a walk-a-thon in the article, but pretty much any fundraising idea could work. What I particularly liked about this idea was that it was a lot of discovery done by the students. The teacher introduced a few organizational methods, but other than that it was the students who came up with the remainder of the ideas. With the fundraising concept they were able to explore a number of different ways in which functions can behave and look like. However, because walking negatively is a hard concept they primarily kept their x values positive.

This article also did a wonderful job of giving other ideas of what to
do after the initial discover has taken place. It mentioned having the
students give presentations to the class about what they had discovered
and what the relationships between the graph, function and table are. I
was also impressed to read that the students who were taught functions in
this way scored higher than eight graders and equal to tenth graders who
were taught by a more traditional, textbook way. A must read for any middle
school math teacher, I thought it was a brilliant idea, and the article give
so may ideas and options to use.

**Keywords:** Assessment, Activities, Teaching Strategies

**Ref: **Ryan4

**Author(s): **Goetz, Albert

** Date: **2005

**Title: ***Using Open-Ended Problems for Assessment *

**Journal or Publisher: **Mathematics Teacher

**Volume, Issue, Pages: **Vol. 99, No. 1, 12-17

**Reviewer: **Ryan

**Date of Review: **2/27/06

Lately I've found myself thinking a lot about which level I would like to teach. At first I thought that without a doubt I'd want to teach at the high school level. However, after spending some time in the middle school in some of my field experiences, as well as when I was in Hawaii last month I've started to wonder if the middle school would be a good fit for me.

I've always enjoyed working with the middle school age group. Back when I was in high school I was a 4-H camp counsellor and really enjoyed working with the middle school aged children. Another reason I've been wondering what age level I would really prefer to teach.

The main reason I've been asking myself that question lately is because of what I saw in my high school placement while I was in Hawaii. The students who were not in the advanced math classes really struggled with some very basic mathematical operations. It was quite sad to see that these students were struggling with concepts that should have been mastered by the time they were in 7th grade. After seeing this, I thought to myself that maybe middle school would be a better fit for me, in hopes that I could be one that teaches and makes sure the students learn the necessities before they move on to bigger and better things. Then again, high school is the last chance to really reach them and get them to enjoy math. As of right now, I'm exactly sure what I want to do, but I hope I figure it out soon.

**Keywords:** Connections, Activities, Statistics

**Ref: **Ryan5

**Author(s): **Caniglia, Joanne C., Leapard, Barbara B.

** Date: **2005

**Title: ***Conic Sections: Draw it, Write it, Do it *

**Journal or Publisher: **Mathematics Teacher

**Volume, Issue, Pages: **Vol. 99, No. 3, pgs 152-155

**Reviewer: **Ryan

**Date of Review: **3-1-06

This was a really good article on connecting math, to of all things, art. The authors of this article give a description of the activity they have come up with as well as notes for the teacher which prove to be really helpful. However, this activity is not for just any math level, I would think that it would be best at the high school math level and I might even add that it be used in the upper level math classes. I believe this primarily because it deals with conic sections and the graphs that are formed with this method.

I think this is a great activity to for a teacher to use in class. The students who may not be as inclined mathematically are able to show off their artistic design, and those who are less artistically inclined are able to show how mathematics is used in the arts. They also discuss how the students will benefit from such an activity. That is what really got me; anyone can pick an activity and try to make it work in a math class, but it's worthless unless the students benefit from it. They also mentioned a few other activities you could do to branch off of this activity if time permits. I thought this was a great article and a great idea. I hope to use this someday in my own classroom.

**Keywords:** Representations, Problem Solving, Communication

**Ref: **Ryan6

**Author(s): **Garner, Amanda S., Preston, Ronald V.

** Date: **2003

**Title: ***Representation as a Vehicle for Solving and Communicating
*

**Journal or Publisher: **Mathematics Teaching in the Middle School

**Volume, Issue, Pages: **Vol. 9, No. 1, p 38-43

**Reviewer: **Ryan

**Date of Review: **3/7/06

I felt that this was a great article about using representation in a middle school setting. The authors discussed a group activity in which it was up to the students to choose which site would be the best choice to have a party primarily based on price. Each group was allowed to use anyway they liked to figure out the problem, but once they came to a conclusion they had to be able to present their idea to the rest of the class.

Not only was this a great activity but it got the students thinking about
something in a mathematical way that they typically wouldn't think about
in such a way. They mentioned a number of ways the students went about solving
the problem as well as ways that the teacher may guide students towards and
away from. I particularly liked the table they had of the representations
and giving the advantages, disadvantages and in which situations each is
typically used. All in all I felt it was a great article and a great idea
for a future activity in the middle school classroom, one might even use
it in high school, but some changes would be necessary.

**Keywords:** Algebra

**Ref: **Ryan7

**Author(s): **Rizzo, Erica.

** Date: **N/A

**Title: ***Multiplying Polynomials *

**Journal or Publisher: **LessonPlansPage.com

**Volume, Issue, Pages: **N/A

**Reviewer: **Ryan

**Date of Review: **3-8-06

This lesson plan is written for an 8th or 9th grade Algebra class learning
to multiply polynomials. The lesson plan itself is very thorough and detailed.
It contains pretty much everything I have been taught a lesson plan should
contain and then some. I am very impressed with the content of Ms. Rizzo's
lesson plan, she has everything from the obvious like goals and objectives
to accommodations for students with specials needs and reflective notes so
she can assess herself after the lesson has been completed. She also goes
into great detail with what she is going to use for the lesson and what her
objectives are. All in all I think it is a very well written lesson plan.

**Keywords:** Algebra

**Ref: **Ryan8

**Author(s): **Heid, M. Kathleen; Choate Jonathan; Sheets, Charlene; Zbiek,
Rose Mary

** Date: **1995

**Title: ***Curriculum and Evaluation Standards for School Mathematics
*

**Journal or Publisher: **National Council of Teachers of Mathematics

**Volume, Issue, Pages: **Algebra in a Technological World

**Reviewer: **Ryan

**Date of Review: **4-5-06

I feel as though the Algebra in a Technological World addenda book can be very useful and practical in certain situations. The book does a very good job of relating algebra to real world situations that would be appropriate to a high school age student. They also use a number of technological graphing tools to represent the data. Also each section has a few activities where the students can use the technology along with the skills they learn to interpret, represent and analyze data.

However, I feel that this book is geared toward the more advanced math
student. Also I don't think it would be very usefully to my particular unit
because my unit consists of linear equations and their graphs where as the
book's primary focus is on polynomial functions. It could be useful in a
unit yet to come.

**Keywords:** Algebra

**Ref: **Ryan9

**Author(s): **Friel, Susan; Rachlin, Sid; Doyle, Dot

** Date: **2001

**Title: ***Principles and Standards for School Mathematics: Navigations
Series *

**Journal or Publisher: **NCTM

**Volume, Issue, Pages: **Navigating through Algebra, Grades 6-8

**Reviewer: **Ryan

**Date of Review: **4-10-06

I thought this book was much more helpful than the Addenda series. Overall I felt that it was put together in a manor that was very easy to follow and understand. On the back of the front cover, I particularly liked how NCTM Algebra Standard expectations were printed just as they are in the PSSM for grade 6-8. This makes it very clear for the teacher what is expected of the student in those grade levels. Also there is a very thorough introduction to the book which seemed to have some very helpful information.

I personally thought that this book would be a great tool for the teacher.
Each chapter has a number of activities and each activity has a list of goals,
materials needed and it gives a section on discussing it. I particularly
liked how each activity showed examples of how students actually went about
doing the activity. They actually took the work of a few students and scanned
it into the book, which I think is a great idea for teachers to see how an
actual student may do it rather than how some professional thinks a student
might do something. There was also more than one example for each activity,
this gives the teacher an idea of how their students may react. Also in
the beginning of each chapter it discusses the important mathematical ideas
that are covered in that particular chapter. I think this is something that
can be very helpful if the teacher knows of another activity that could fit
in the chapter to further enhance what they are already doing or have done.

**Keywords:** Teaching Strategies, Communication, Keyword 3, Optional...

**Ref: **Ryan10

**Author(s): **Lapp, Douglas A; Manouchehri, Azita.

** Date: **2003

**Title: ***Unveiling Student Understanding: The Role of Questioning
in Instruction *

**Journal or Publisher: **Mathematics Teacher

**Volume, Issue, Pages: **Vol. 96, No. 8 p 562-566

**Reviewer: **Ryan

**Date of Review: **4-12-06

I felt that this particular article was a great article and should be read by all teachers, not just math teachers. The article has a lot of good advice and information on posing questions to check for student understanding. More often than not, the question of the top of our heads may not always be the best choice. Choosing what information we want to receive from the students takes time and planning.

I liked how the article breaks down questions in to three parts, form, content and purpose. They then describe each part of the question and refer to examples to help illustrate their point. In the end we see how each part of the question is very important and that changing each part at different times and situations can be very useful for the teacher. Of the three parts of questions content sticks out as the most important to the teaching process. Making sure the content is correct is a must, and with time and planning the other two parts will come. However, there is some questions we may want to stay away from at specific times, it all goes back to what we want to know and planning. Again I thought that this was a very good article and very help, a copy on hand would not be a bad idea.

**Keywords:** Probability

**Ref: **Ryan11

**Author(s): **Christopher Smith

** Date: **2006

**Title: ***Is the Price Right? *

**Journal or Publisher: **

**Volume, Issue, Pages: **

**Reviewer: **Ryan

**Date of Review: **4-24-06

During this session we looked at probability and statistics problems that can be formed from the TV show the Price is Right. I thought this was a great idea; almost everyone has heard of the Price Is Right and has probably seen it. I never really thought about all of the probability in the games until it was pointed out, and it would be a great way to demonstrate some probabilities for the class. You could even make it more fun by taping it and showing the class and pausing to think about and compute the probabilities to determine what choice the contestant should have made.

Although the book for the convention said the session was for grades 6-college, I felt that high school to college would be more appropriate. I felt that most if not all of the computations would be much too difficult and confusing for anyone who has not had much of a math background. I am in probability now and some of what he was saying was even a bit confusing to me. I thought it was very interesting and very practical.

**Keywords:** Technology, Algebra, Geometry

**Ref: **Ryan12

**Author(s): **Erbas, A. Kursat; Ledford, Sarah D.; Orrill, Chandra Hawley;
Polly, Drew

** Date: **2005

**Title: ***Promoting Problem Solving across Geometry and Algebra by
Using Technology *

**Journal or Publisher: **Mathematics Teacher

**Volume, Issue, Pages: **Vol. 98, No. 9, pgs 599-603

**Reviewer: **Ryan

**Date of Review: **4-26-06

This article discusses a few ways how teachers can use technology in their classroom lessons to help students with problem solving, particularly in Geometry and Algebra. The authors discuss in depth one example and how the students can use two different computer software programs to explore the problem as well as look for any patterns. With the computer programs students are able to manipulate and look at different situations very quickly; this allows the students to look at a number of different possible answers in one class period.

Near the end of the article the authors make a few points that I think are very important. One that I liked was that even if students may not have the algorithms memorized, they are still developing their mathematical thinking. Also they made sure to mention just because a teacher uses technology in their lesson does not relieve teachers from being active in their classrooms, something that I have noticed in a few of my past experiences.

Although the article only looks at one particular problem and goes quite in depth on it, I still feel that it is a very good, informative article. However, I feel that the article could be better if it were to talk about more than one problem and a number of different technologies rather than just computer software. I know that there is a lot of technology out there to be used in the classroom; I'm just not exactly sure how it all can be incorporated.

**Keywords:** Equity/Diversity, ,

**Ref: **Ryan13

**Author(s): **Gilbert, Melissa C.

** Date: **2001

**Title: ***Applying the Equity Principle *

**Journal or Publisher: **Mathematics Teaching in the Middle School

**Volume, Issue, Pages: **Vol. 7, No. 1, 18-9-36

**Reviewer: **Ryan

**Date of Review: **4-28-06

This particular article seemed to hit on everything we talked about in Ed 330 when it came to equity. I felt that it was a nice review to keep the ideas fresh in my mind. The very first thing mentioned in the article was the idea of students needing a safe and secure classroom in order to succeed. This was something that was stressed a lot in Ed 330. The article went on to discuss a number of ways to accomplish this issue of security.

Another one of the main points was how teachers shouldn't just tell students how to do something, but rather guide them to the solution. This is again something that we have discussed a number of times. They also briefly mentioned ignoring the stereotype of girls and mathematics. As I said before this was a good review article to keep the issues that many math teachers face fresh in their minds so they may create an equitable classroom to foster their students' learning.

**Keywords:** Geometry, Activities, Keyword 3, Optional...
**Ref: **Ryan14
**Author(s): **Kelley, Paul.
** Date: **2006
**Title: ***Geometry Projects: Escher, Sierpinski and Snowflakes in your classroom
***Journal or Publisher: **MCTM Conference
**Volume, Issue, Pages: **
**Reviewer: **Ryan
**Date of Review: **5-1-06

I thought this session was very practical and a great opportunity for a teacher to get more ideas to use as activities in a Geometry class. The two main ideas that were talked about during the hour were tessellations and fractals. For each of the units/lessons he did on each topic he gave a few different activities that he has done with his classes in the past.

For tessellations he used the example of M.C. Escher and his paintings. He has done many paintings that are tessellations and are great examples to show the students. After showing them examples of tessellations he had 2 activities for them, one was hand drawing tessellations and the other was creating snowflakes with tessellations. He showed a number of examples his past students have done and they were amazing. You don’t really realize how creative and artistic students can be until you really allow themselves to express them selves.

Much like the tessellations, he also had two great activities for students to explore fractals. Again he had one activity where the students were hand drawing fractals and could do whatever they wished, and the other was building a Sierpinski pyramid, which I actually did when I was back in high school. Again he showed many examples that his past students have done, including a 19 foot Sierpinski pyramid, which was incredible.

I felt as though this session was very worth my time. He had a lot of good ideas, and showed examples to back them up. The ideas were great ways students could further explore math in a more exciting way.

**Keywords:** Geometry

**Ref: **Ryan15
**Author(s): **Kochaver, Betty. ** Date: **2006 **Title: ***The Math of Machu Picchu
***Journal or Publisher: **MCTM Conference **Volume, Issue, Pages: ** **Reviewer: **Ryan **Date of Review: **5-3-06

When I first saw the name of this session, I was very intrigued as to how they were talking about the math of Machu Picchu. When I first got there, it was very interesting, this lady told us about her trip to Machu Picchu and personally I always like hearing about trips and seeing pictures of new places. It was very apparent how the ancient Incas used a great deal of mathematics when constructing and maintaining their great city that was once lost. She pointed out many ways in which mathematics was used to create such advanced and rare architecture for the time.

However, it quickly got very boring when she started getting it to great detail of her and her husbands trip with little side stories as well; I actually had a hard time keeping my eyes open. When she finally got to how she incorporated the math from Machu Picchu in her classroom, I personally thought the exercise she presented was a bit too difficult for 9-12 graders, maybe an upper level senior class could have handled it but nothing less I don't think. I was having a hard time following along; maybe it was because I haven't seen much Geometry since 9th grade. I think this could be a very useful session, however, I think there could be more than just mathematical idea pulled from it, I think there was more ideas she could have taken from the lost city and used as activities in her class. A little less trip detail and a little more exploring with math wouldn't hurt either.

**Keywords:** Problem Solving, Activities

**Ref: **Ryan16
**Author(s): **Britton, Barbara. ** Date: **2006 **Title: ***Solutions to the Candy Conundrum Problem
***Journal or Publisher: **Teaching Children Mathematics **Volume, Issue, Pages: **Vol. 12, No. 9, 468-469 **Reviewer: **Ryan **Date of Review: **5-10-06

This article primarily discussed one way a particular class went about solving a problem that was presented in an earlier edition of Teaching Children Mathematics. I particularly liked hearing how the class went about solving this problem, one that could be solved in a number of ways and could be used at a number of different grade levels. It was also good to hear what some of the students had to say about the project, what was fun about it, what was difficult and how they got through the challenging parts. Also the author made a great point at the end of the article saying that if the activity doesn't interest you, there's no way that it would interest your students. Personally I would have liked to have read about how other classes may have gone about the problem rather than just one.

**Keywords:** Teaching Strategies

**Ref: **Ryan17
**Author(s): **Cohen, Robin. ** Date: **2006 **Title: ***How Do Students Think?
***Journal or Publisher: **Mathematics Teaching in the Middle School **Volume, Issue, Pages: **Vol. 11, No. 9, 434-436 **Reviewer: **Ryan **Date of Review: **5-10-06

To me this article is a little misleading. When I saw the title I thought there would be more examples of how different students approach the same problem and come up with a correct answer. However, it was basically just all about how each student is in fact unique and how each of them look at problems in their own ways coming up with their own answers.

Although they did offer a few examples, that's all it was, a few. Typically one example for each situation they described and personally I would think for an article with the title, "How Do Students Think?" there would be a ton of examples, but there wasn't. This was basically common sense article about something that you learn in your early education classes.

**Keywords:** Geometry

**Ref: **Ryan18
**Author(s): **Jones, MaryClara; Soto-Johnson, Hortensia. ** Date: **2006 **Title: ***Rotations of the Regular Polyhedra
***Journal or Publisher: **Mathematics teacher **Volume, Issue, Pages: **Vol. 99, No. 9, pg 606-609 **Reviewer: **Ryan **Date of Review: **5-10-06

This article discusses looking at the rotational symmetries of three-dimensional objects in a high school geometry class. Although the third dimension transformations are not really covered in a high school math class, the authors discuss how the third-dimension can be useful for the students understanding of the second-dimension. They believe this can be done through exploring rotational symmetry of three-dimensional objects which is a transformation which rotates the solid about an axis such that the solid will occupy the same space as before. They offer a number of examples and explain what to do very well in the article. I personally think this would be great for anyone teaching Geometry to read. Personally I wish I would have covered something similar to this in high school, I think it would have better prepared me for what I would see in college and allow me to become familiar with more mathematical vocabulary.

**Keywords:** Measurement

**Ref: **Ryan19
**Author(s): **Wong, Michael. ** Date: **2006 **Title: ***The Human Body's built-in Range Finder: The Thumb Method of indirect distance measurement
***Journal or Publisher: **Mathematics Teacher **Volume, Issue, Pages: **Vol. 99, No. 9, p622-626 **Reviewer: **Ryan **Date of Review: **5/10/06

When I saw the title to this article I had to read it. When I spent my interim in Hawaii, we sat in on a talk and listened to a guy discuss how the ancient Polynesians used to navigate the sea with only their body parts, no compass and poor maps. Seeing this article I had to see what was inside.

I felt that this was a very good article, not only did it give you a number of interesting activities to use in class, but they even explained how you should lead up the activates. Personally, I thought that the activities would be very appealing to the students, demonstrating yourself how to use your thumb to tell a distance could really fascinate the students. Also by not telling them everything they need to know they could use the activity as a great opportunity to explore similar triangles in Geometry. I thought it was a great article and activities like these should be used in a classroom and they would have great results.

**Keywords:** Geometry, Research

**Ref: **Ryan20
**Author(s): **Hickman, Aaron; Walmsley, Angela L.E. ** Date: **2006 **Title: ***A Study of Note Taking and its Impact on Student Perception of use in a Geometry Classroom
***Journal or Publisher: **Mathematics Teacher **Volume, Issue, Pages: **Vol. 99, No. 9, p614-619 **Reviewer: **Ryan **Date of Review: **5-10-06

Over all I thought this was a good article. In all honesty I was expecting a little more, but what I got out of it I think is very useful. The two authors conducted an informal study to see which of three types of note taking was better for student recollection and understanding. They examined traditional note taking, a columnar format and what they called a "mini-textbook." In the article they mention how students become turned off in a class by a lack of power, and when students have more control over what they are doing they take pride in their work and are more likely to participate. Because of this they initially thought that at first the students would do better with the "mini-textbook" because they could personalize it more and it would be more effective for them to study from, going off of the whole more control for the students.

However, they actually found out through surveys that the columnar format seemed to be the most preferred way for students to take notes. Also it is not just the style of notes the students are taking, but getting the students actively engaged in the lesson is crucial. Although I was expecting a bigger, formal study to be presented in the article I felt that what they said would something very important for teachers, especially math teachers to know. I say especially math teacher because they mention how many students don't even take notes in their math classes.