**Keywords:** Problem Solving, Assessment

**Ref: **Vanessa1

**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: **Vanessa

**Date of Review: **2/15/06

This article was about a teacher who uses open-ended problems, or problems with no correct answer, as part as his assessments. The students in his math classes have to work in groups to come up with the best answer for each question as 25% of each test. In his article, Goetz gives an example of the type of problems he uses on his tests. He gives data about the percentage of the time different numbers of chimps would have a successful hunt. The students would have look at the data and answer questions like, "give at least 2 reasons why you think Chimpanzees would hunt in groups" or "if you were biologist doing primate research, what might be some questions that you might pursue based on the research presented here?" Goetz feels that he has a great deal of success in using open-ended problems for tests and believes that working in group might ease test anxiety. He also believes that students would be learning even when they are being assessed from doing those types of problems. The open-ended problems are more of what people really do in real life so he feels that his students would have a better feel about how math is used outside of the classroom.

I think open- ended problems are really important in a math class
because
it is challenging to think outside the box and most real life problems
are
open-ended problems where there is no absolute solution. Being able to
solve
real problems and working in groups is a valuable skill to have, but I
am
skeptical about how fair the grades are going to be. There is no real
answer
and the students are working in groups. I feel that he cannot judge one
groups
answer over another group's answer because they both could be right.
Besides
that, I would definitely give open-ended problems to my students in my
future
classroom.

**Keywords:** Problem Solving

**Ref: **Vanessa2

**Author(s): **Sawada, Daiyo

** Date: **1999

**Title: ***Mathematics as Problem Solving: A Japanese Way *

**Journal or Publisher: **Teaching Children Mathematics

**Volume, Issue, Pages: **54-58

**Reviewer: **Vanessa

**Date of Review: **2/20/06

Sawada gives an example of a typical Japanese lesson on population density and compares the Japanese approach with the United States approach. At the beginning of the lesson, the teacher asks the students to figure out which rooms or cities are more crowded. After they figured it out together, they continued to solve problems about which metal is more dense. The teacher purposely links the crowdedness of a room or city to the density of the metals. Sawada notes that it is just as important to emphasize the medium (problem solving) as the message (density). By using problem solving as a medium of teaching, even the errors the students make could be valuable to the lesson because they could figure out together why it doesn't work. Sawada concludes by posing the question, "why has the problem-solving approach as a medium taken hold so pervasively in Japanese elementary schools but remains largely problematic in North American Schools.

I think that this is a great example of how problem solving should be used in a classroom. This article also corresponds with the videos we watched in class comparing the American teacher and the Japanese teacher. I think that problem solving is much more effective than teaching formulas and learning by boring repetitions because it is much more engaging and a lot more fun. Also, students do not have to be afraid of giving the wrong answer; there might be many ways to approach a problem. We should think about why there aren't more schools in America using that the solving problem approach.

**Keywords:** Algebra, Problem Solving

**Ref: **Vanessa3

**Author(s): **Amit, Miriam, Klass-Tsirulnikov, Bella

** Date: **2005

**Title: ***Paving a Way to Algebraic Problems Using a
Non-Algebraic
Route *

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

**Volume, Issue, Pages: **Vol. 10, No. 6, 271-276

**Reviewer: **Vanessa

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

Many students in middle school, especially those in Pre-Algebra or Algebra classes have trouble solving word problems and deciphering the words into something mathematical. Amit and Klass-Tsirulnikov suggests that the students struggle because they cannot connect real life and math. They propose a 3-stage method that could help Pre-algebra and Algebra students do better in problem solving. Stage 1 involves making meaningful problem settings that students could relate to or would like to read about. The second stage suggests that students should tackle the problem in a nonalgebraic way. This is the most important step for Klass-Tsirulnikov and Amit because the students could use logic to solve the problem without being caught up in the mechanical algebra. The third step transforms the logic into algebraic sentences. This way, the students go through a cognitive developmental process to solve problems.

I think the 3-stage method is a wonderful idea to incorporate in middle school classrooms because not only will the students be able to solve word problems easier, they could fully understand what the algebraic formulas and methods mean. This is great for Pre-algebra students who are just starting to learn about variables because the process leads them to algebra. It is also good for Algebra students because they could fully understand the algebra problems by thinking about it logically.

**Keywords:** Communications, Teaching Strategies

**Ref: **Vanessa4

**Author(s): **Leikin, Roza, Zaslavsky, Orit

** Date: **1999

**Title: ***Cooperative Learning in Mathematics *

**Journal or Publisher: **Mathematics Teacher

**Volume, Issue, Pages: **Vol. 92, No. 3, 240-246

**Reviewer: **Vanessa

**Date of Review: **2/27

Roza Leikin and Orit Zaslavsky defines the four criteria of a cooperative-learning setting: (1) Students learn in small groups (2-6 members) (2) Students mutually and positively depend on one another. (3) All members of the group has an equal opportunity to interact with one another regarding the learning tasks. (4) Each member of the group has a responsibility to contribute to the group work and is accountable for the learning progress of the group. The authors suggests that the exchange-of-knowledge method is a great way to incorporate cooperative learning in the classroom. The exchange-of-knowledge method gives the students an opportunity to be the teacher or the student. In groups of fours, each student receives a card with a questions and each person in the group has to master their own card. Then, they pair up and explain the card's concepts to their partner. Afterwards, they have to do the problem on the back of their partner's card and they could ask each other questions about the concept. When they are done, they switch partners within their group of fours. Leikin and Zaslavsky also gives guidelines for facilitating cooperative learning in mathematics.

I think that cooperative learning is very important in a classroom especially in a math classroom because students could improve their people skills. In real life, people are always solving problems in teams. I think there are pros and cons to the exchange-of-knowledge method. The exchange-of-knowledge method could be a really good idea for a cooperative-learning activity because every student is engaged in either teaching, learning, or doing the problem individually. When they have problems with something, they can just go to the "expert" on each topic. Although the method is great because the students are teaching each other concepts, I could see it being potentially disastrous because the "expert" could just tell the other members of his groups how to do the problem exactly without the other students fully understanding the concept. This method of learning also contradicts the method of learning by problem solving.

**Keywords:** Proof, Games, Geometry

**Ref: **Vanessa5

**Author(s): **Gernes, Don

** Date: **1999

**Title: *** Sharing Teaching Ideas: The Rules of the Game *

**Journal or Publisher: **Mathematics Teacher

**Volume, Issue, Pages: **Vol. 92, No. 5, 424

**Reviewer: **Vanessa

**Date of Review: **2/28/06

Concept development is very important but challenging especially when students are starting to learn about proofs. It is sometimes hard to teach students the deductive system and how to prove a theorem. Gernes suggests that many games such as monopoly and baseball have a similar concept. Each game has rules, definitions, and plays. After "proving" rules for games they are already familiar with, they are ready to move on to proving theorems in geometry. In one of the games called the "letter game," the students are given undefined terms, definitions and postulates. They are given MI and they have to prove MUIU. One of the rules is that if you have MI, then you may add I to get MII. Therefore, MI turns into MII and so on until students get MUIU. This game is similar to the proofs in geometry and students should be familiar with the concept of proving before they prove geometry theorems.

I think this is a great idea for a geometry class because many students have a hard time proving theorems. I think they have a hard time because they can't get a hold of the concept of the deductive system and how to prove something. Students also don't understand why they're proving something when the answer seems completely obvious. With the "letter game" and other games, students would be able to understand the concept of proving a lot better. Even though this may take more time out of the classroom to prove these games, I think it is worth it.

**Keywords:** Representations, Algebra

**Ref: **Vanessa6

**Author(s): **Piez, Cynthia M., Voxman, Mary H.

** Date: **1997

**Title: ***Multiple Representations- Using Different
Perspectives to
Form a Clearer Picture *

**Journal or Publisher: **Mathematics Teacher

**Volume, Issue, Pages: **Vol. 90, No. 2, Page 164

**Reviewer: **Vanessa

**Date of Review: **3/6/06

To understand any concept, we can study the representations of these concepts and to any concept, there are probably multiple representations of each concept. Piez and Voxman argues that each representation emphasizes or suppresses certain aspects of the concepts and by looking at a concept through multiple representations, students can truly understand the concept. Furthermore, each student has a preference for a different type of representation. One student might like the graphical representations more and another student might like the analytical representation. No matter which one they prefer, understanding multiple representations help students understand a concept more.

I agree with Piez and Voxman's article because I think that if
students
could look at one concept through different perspectives, they could
really
understand the deep meaning of each concept. Even though each person
might
have a different preference for the types of representations, it is
also
important to look at the other ones because each of them emphasizes
different points. Proper representations of a problem can make it much
easier to solve.

**Keywords:** Algebra, Problem Solving,

**Ref: **Vanessa7

**Author(s): **Engel, Bill, Schmidt, Diane L.

** Date: **2004

**Title: ***How Big is a 16-Penny Nail? A Measurement Lesson for
Middle
Grades *

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

**Volume, Issue, Pages: **Vol. 9, Issue 8, Page 422

**Reviewer: **Vanessa

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

In the lesson "How Big is a 16-Penny Nail?" the work in groups to predict the size of a 16-penny, or 16d, nail by observing nails of other sizes. Groups are either given Set A data which consists of the 4d, 6d, 8d, and 10d size nails, or Set B data which consists of 20d, 30d, and 60d size nails. They are also given a bag of pennies and they are allowed to use rulers, calculators, and a balance. Each group has to use their materials they are given to predict how big a 16d nail is. Most students would start out by finding the relationship between pennies and the nails. They measure the nails with pennies but will soon find out that it is not very accurate. They will then try other methods for measuring the nails with rulers. After experimenting with the length, students are advised to look for a pattern by using a chart and graphing the results on graph paper. Then, students should make an derive an equation from their data and graph. Finally, students can predict how big a 16d nail is by using their equation. However, when groups with set A and set B compare their predictions, they realize that they have different predictions. When the teacher reveals the real answer, the students would be surprised to find out that both teams did not get the right answer. At this point, students are confused about why they did not get the right answer and how might they do their measurements so that they would get a more correct answer. Students would be perplexed at the situation and come up with other theories of why their answer is not correct. Some students might even suggest that the name does not have anything to do with their size, but how much the nails use to cost. Also, students would be wondering how pennies fit in to this situation.

I think this is a wonderful lesson plan because students would be
actively
engaged in experimenting, thinking about possible solutions, and
finding
patterns. Students can also make connections between formulas and real
world
situations. This shows the students how mathematics and algebra is
useful
in real life. Also, students get to communicate with each other about
possible theories about the penny nail-labelling system. Through this
lesson, students
can also see how real life situations do not always have obvious
"right" answers.
This activity mimics the tasks of mathematicians and scientists in
real-world situations. I also love how this lesson could be extended by
having students weigh the nails to see if there are any correlation
between the nail label and its weight and the weight of the pennies.
Students also get the opportunity to learn about the history of nails
and wonder about how nails were named back in the old days. Lessons
like this one helps motivate students because it is more interesting to
learn about something that occurs in real life than
to do problems out of their textbooks. Also, there is also a
competitive factor
involved when groups "compete" to see who gets the most accurate
answer.
Without realizing it, the students are practicing measuring, graphing,
deriving equations, and solving equations. This is a lesson that would
be perfect for
a algebra class or even a pre-algebra class.

**Keywords:** Equity/Diversity, Representations

**Ref: **Vanessa8

**Author(s): ** Boognl, Mary A.

** Date: **2006

**Title: ***Hand-On Approach to Teaching Composition of Functions
to
a Diverse Population *

**Journal or Publisher: **Mathematics Teacher

**Volume, Issue, Pages: **Volume 99, Issue 7, Page 516

**Reviewer: **Vanessa

**Date of Review: **3/20/2006

NCTM's standard of equity in the classroom encourages that "reasonable and appropriate accommodations be made as needed to promote access and attainment for all students" and Boognl has designed a lesson that can accommodate all ethnic groups and learning styles, particularly her target group, Native American students. Her lesson fits all ethnic groups because it provides for cooperative learning groups, visual learners, self-paced learning, and the natural emergence of language. Native Americans, such as the Apaches and the Navajos, view the world differently from other cultures. Therefore, she feels that her lesson of cooperative learning is perfect for them. In the lesson, students are learning about functions, composite functions, and inverse of functions by playing a "game" and guessing what is the "rule" of each function. By doing this, the students develop what composite functions and inverses mean, and also develop the appropriate language they need to discuss functions.

These types of lessons are not only great for Native American students, they are great for all students because I think most students would benefit from this. Although this is a cooperative group activity or game, all students have to think for themselves and come up with an answer themselves because students are not allowed to tell their teammates the answer during the activity. I really liked the Native American philosophy that Boognl mentions about how the elders are a source of knowledge and wisdom in the community but "learning flows not only from the old to the young, but from the young to the old." I totally agree that teacher could learn tons just by observing their students.

**Keywords:** Statistics

**Ref: **Vanessa9

**Author(s): **

** Date: **1992

**Title: ***Curriculum and Evaluation Standards for School
Mathematics Addenda Series, Grades 9-12 *

**Journal or Publisher: **NCTM

**Volume, Issue, Pages: **Data Analysis and Statistics Across the
Curriculum

**Reviewer: **Vanessa

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

The Addenda book for data analysis and statistics wants to shift our perspective on how statistics is treated in schools. It wants to emphasize the importance of developing statistical thinking, instead of just learning statistical procedures. It integrates statistics into algebra, functions, and geometry so that it could be understood in many ways. The book gives suggestions of how statistics should be integrated in the algebra or geometry classroom and give examples of how to do it. There are also worksheets included in the book that teachers could use in their class.

I think this is a good tool for teachers to use in the classroom to
get ideas about what to teach students and how to teach them about
statistics.
The NCTM approach to statistics is very similar to the approach in the
Arise textbook in that they both focus on exploring the relationships
between data and scatter plot graph and how to find a linear equation
that fits. I also enjoy that the examples of activities teachers could
do with their class have
interesting data that students could relate to or would be interesting
to
them to work with.

**Keywords:** Probability, Statistics

**Ref: **Vanessa10

**Author(s): **Burrill, Gail, Franklin, Christine A., Godbold
Landy,
Young, Linda J.

** Date: **2003

**Title: ***Navigating through Data Analysis *

**Journal or Publisher: **NCTM

**Volume, Issue, Pages: **

**Reviewer: **Vanessa

**Date of Review: **4/11/06

The Navigating through Data Analysis book believes that statistical and probabilistic thinking is based on formulating questions that can be addressed with data, selecting and using appropriate statistical methods to analyze data, developing and evaluating inferences and predictions that are based on data, and understanding and applying basic concepts of probability. Before each chapter, there is an explanation of each chapter talking about its main focus of each chapter and activity. There are a few activities in each chapter. For example, in chapter 1, there are three activities: Sampling Rectangles, Sample Size and Sample Methods. I think that each activity is presented in a very clean and organized manner. Goals are always on top so that it is clear what the activity is about. There is also a Discussion section in each activity. I think the discussion is more about instructions than discussions. I also think that there is a good balance of "real life" stuff and mathematical stuff in the book.

Return to Index

**Keywords:** Technology, Algebra,

**Ref: **Vanessa11

**Author(s): **Lassak, Marshall, Heller, Brad

** Date: **2006

**Title: ***Delving Deeper: The Where, Why, and How of Solving ax
=
Log(ax) *

**Journal or Publisher: **Mathematics Teacher

**Volume, Issue, Pages: **Volume 99, Issue 9, Page 650

**Reviewer: **Vanessa

**Date of Review: **4/26

In this article, Marshall Lassak writes about how technology is a good way to help students discover concepts and ideas, it could also help teachers discover deeper meaning to a problem. In his class, he wanted his students to explore the functions f(x)= a^x and g(x)=log a x when 0<1 and a>1. They are supposed to explore and come to a conclusion that the graphs are inverses, the functions are equal when their graphs intersect, a cannot equal 1 or 0, and a calculator can directly graph only two logarithms (e and 10). In the process of exploring, some students found that for 0<1, some values of a produced 3 intersections of the functions that the teacher has never discovered himself. This discovery led the class to dive deeper into the subject and have an even better understanding of the subject.

**Keywords:** Communications

**Ref: **Vanessa12

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

** Date: **2003

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

**Journal or Publisher: **Mathematics Teacher

**Volume, Issue, Pages: **Volume 96, Issue 8, Page 262

**Reviewer: **Vanessa

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

This article starts out by giving an example of a teacher who asks questions that do not have any significance. The teacher was only asking questions that required one word answers or procedural answers and it doesn't allow the students to demonstrate what they really understand. Instead, the article says that teachers should ask questions that give the students an opportunity to express their reasoning processes. There are three things that teachers should be considering when asking questions: form, content, and purpose of the questions. There are closed form questions and open form questions. Open form questions or questions that begin with how or why are the questions that usually the ones that really help teachers understand if students understand the concepts.

I think this article breaks down and analyzes the different types of questions and what kinds of questions might be better when teachers are checking for understanding. This article is also helpful because it talks about what form to put questions in to make them more efficient. For example, instead of asking "what is the next step?" you can ask "what could you do next and why? How could you proceed from here?"

**Keywords:** Equity/Diversity, Probability

**Ref: **Vanessa14

**Author(s): **George, Jenni, MacCormack, Dana

** Date: **2006

**Title: ***What is Racial Profiling
*

**Journal or Publisher: **MCTM Conference

**Volume, Issue, Pages: **

**Reviewer: **Vanessa

**Date of Review: **5/1/2006

In this session called “What is Racial Profiling,” two students from Metro State University presented an activity that teachers could do in their classroom to promote inequality within their own neighborhood while learning probability. Before the activity, the class looks at percentages of people of each race in Minnesota or the United States. After looking at the statistics, the class has to guess how many percentages of people pulled over by the cops are of each race. Then in the activity, each group of threes have a bag containing color blocks. They are not allowed to see what’s inside the bag, but they are allowed to pick a block out of the bag one by one and record them on a chart. They repeat this process 100 times and count up how many they picked of each color. The class will see that the number they picked out of each bag would be close to the number of blocks the bag contains of each color. Then, they look at the real percentages of people pulled over and the discrepancy between the percentages of people in the population of each color and the percentages of people who get pulled over. The teacher could have a discussion with the class about what could be causing this difference and have the class think about these issues.

I think that this activity is a very good idea for students to learn
about racial profiling and racism while doing probability. I would
definitely do this in my classroom if I ever had a chance to teach
probability. I believe that students should learn about this even if it
could be a sensitive subject.

**Keywords:** Activities, Games

**Ref: **Vanessa13

**Author(s): **Westegaard, Sue

** Date: **2006

**Title: ***MTCM Convention
*

**Journal or Publisher: **

**Volume, Issue, Pages: **

**Reviewer: **Vanessa

**Date of Review: **4/30/2006

I felt that the session called “Oldies but Goodies” had a lot of “goodies” that I could definitely use in my future middle or high school classroom. The whole session was just example of interesting things she did in her classroom before. Two of my favorites is the Algebra Walk and Mathemania.

In Algebra Walk, have student stand in front of the where they stand in a line representing the number line from -7 to 7, arm lengths apart. The teacher then gives direction of what each student should do such as “look at the number between your feet and multiply that number by 2.” Each person in the line would have to do that to their number and they would form the line y=2x together. I think this is a great way for students to visualize what each graph looks like. Also, there are many possibilities with this activity. The students could learn absolute values and parabolas with this activity.

Mathemania can be done as a class activity or in groups. The class or group should be split up into two teams and they are given a deck of cards. The teacher gives the class or group exercises to do such as C+C or C-C. When the teacher pulls out two cards from the deck, the students from each team have to apply the numbers to the exercise and solve. You could also make the exercises harder by making any exercise you like, such as finding the roots of a quadratic formula or finding the slope of two points.

I think these activities would be very interesting for the students and could help them understand math concept with a lot more fun.

**Keywords:** Assessment

**Ref: **Vanessa15

**Author(s): **Kennedy, Dan ** Date: **1999 **Title: ***
Assessing True Academic Success: The Next Frontier of Reform *

**Journal or Publisher: **Mathematics Teacher **Volume, Issue,
Pages: **Volume 92, Issue 6, Page 462 **Reviewer: **Vanessa **Date
of Review: **5/8/2006

Kennedy's taught math for many years and is also involved with writing problems for the Advanced Placement (AP) calculus examinations. It was when he was making up problems that he started looking at issues of assessment. He saw that math classes was like a "game" where the teacher shows the students how to do it, the students practice it for a while, then the students are tested on how closely they can follow what the teacher did. He reasons that when a student masters a test, he or she is not thinking about it at all because when a student can do something very well, they can do it without really thinking about it. Thus, he believes that students should be given problems on a test that they are not fully prepared for, so that they could think about it. This is because Kennedy believes that we learn "about" everything else, but we learn how to "do" math. Therefore, he started to give problems for his students to solve each day so that they would be familiar with solving their own problems, not copying what the teacher's doing. He also devised a curving system that would make his grades "fair" since the students are solving problems instead of reciting the answer.

This was an interesting article about Kennedy's philosophy about assessment. I agree with him that students will not have a lot of problem solving skills when they are just expected to do exactly what the teacher shows them. This idea is also present in the NTCM standards of problem solving. I also liked his idea that assessment should be an instrument of teaching. Although I like his philosophy about what assessments should be about, I'm not sure how his idea of curving works for grades. I feel that when a student can get a 70% after a curve with a 20% raw score, the student would think that he or she could do bad on all the tests and still receive a decent grade. This article raises a lot of questions about assessments, such as, how could we give grades to students when they are expected to make mistakes when they are problem solving on a test?

**Keywords:** Equity/Diversity

**Ref: **Vanessa16

**Author(s): **Gilbert. Melissa C. ** Date: **2002 **Title: ***Challenges
in Implementing Strategies for Gender-Aware Teaching *

**Journal or Publisher: **Mathematics Teaching in the Middle School
**Volume, Issue, Pages: **Volume 7, Issue 9, Page 522 **Reviewer:
**Vanessa **Date of Review: **5/8/2006

Melissa Gilbert gives a lot of tips about how not to be gender stereotyping in classroom, even when you don't notice it. She starts out by giving advice on creating a safe learning environment for all students. In a large classroom, she suggests that you should control students who want to dominate class discussions so that everyone has a chance to contribute. Also, she talks about how you should stop the class when you feel that they are being disrespectful. For example, when a student is giving her opinion or answer and another student is going to sharpen her pencil, you should tell the other student to wait until later to do that because every student's answer is valuable in the classroom. Then, she goes on talking about not how girls can feel misplaced in a mathematical classroom and it is the teacher's role to help girls see that both girls and guys can do math. You can do this by showing them that women have contributed a lot in the field of math and science and encouraging girls to be more confident in math. One of the most interesting thing in this article is that sometimes we use "guys" to address both girls and boys. This is one of the ways that Gilbert suggest that we should not do because it implies that guys are more important in the classroom.

Although this article is meant for middle school teachers, I think that teachers of every grade level could benefit from this article because gender stereotyping goes on in every grade level. I agree with Gilbert in that all students should be encouraged to speak out in a math class and be confident, not only boys. There are many things in this article that we have already learned about but this is a great reminder. This article is not only has great advice on gender stereotypes but it also has good suggestions for making everyone in the classroom feel comfortable.

**Keywords:** Measurement, Geometry, Activities

**Ref: **Vanessa17

**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: **Volume 99, Issue 9, Page 622

**Reviewer: **Vanessa

**Date of Review: **5/14/2006

This article presented a new and interesting way teachers can have their students apply what they’ve learned in geometry in real life. After doing this activity, students would really understand similar triangles angles and how it could be used in real life. The “thumb method” is an indirect measurement of height. Students have to use their body parts and sightlines to find the distance of an object. This activity is similar to the classical similar triangles activity of finding height lengths by using shadows. First, Wong explains the steps of how to measure using your thumb and then he explains the why it works by using geometry. This is a flexible lesson in that it can adapt to many classrooms and conditions.

I think this is a very good idea to use in a geometry classroom when you get bored of using the same old shadow and height problems. In this activity, you can tell the students to go outside and find the distance to certain objects. Wong suggests that you should tell the students the first few steps but have them figure out how to come up with a formula or method to find the answer. Then the students should figure out how the geometry works on their own. I think this activity is great because you can discuss with the students about what kind of assumptions they are making when they’re measuring with their thumbs and how accurate the method is. I will definitely try this with my future classes because it is a fun activity and it’s also a skill that students can take with them when they leave my class. It is also a good lesson to practice spatial reasoning and drawing diagrams. I think students would be interested in using the thumbs to measure far away objects because most people have seen artists or photographers do it, but most students probably don’t know how to do it.

**Keywords:** Technology, Algebra, Keyword 3, Optional...

**Ref: **Vanessa18

**Author(s): **Wade, William R.

** Date: **2006

**Title: *** Sound Off! A Dialogue between Calculator and Algebra
*

**Journal or Publisher: **Mathematics Teacher

**Volume, Issue, Pages: **Volume 99, Issue 6, Page 391

**Reviewer: **Vanessa

**Date of Review: **5/14/2006

Wade tries to persuade us that algebra and calculators are a good pair. Algebra is great on its own and so is calculator, but when they are used together, they can create greater results. In this article, algebra is hanging out with calculator one day and they began showing off their talents. They were each trying to show the other that they can do more. Calculator showed off his skills in graphing complex functions and calculating big numbers while algebra showed off his thinking skills. In the end, they realized that they there are things algebra cannot do and there are things that calculator cannot do. So they decided that it is best for them to work together to create the best results.

I really enjoyed reading this article because Wade presented his ideas in a very fun manner. The whole article was just a dialogue between calculator and algebra. I agree with him that calculators can really aid students in learning algebra even though some might think that when students use calculators, they’ll forget how to add 2+2. Wade really demonstrated the different ways calculators can help algebra, but also showing the limits of a calculator at the same time. That’s why calculators and algebra are such a good match.

**Keywords:** Technology, Number and Operation,

**Ref: **Vanessa19

**Author(s): **Mahoney, John F.

** Date: **2006

**Title: ***Fill 'n Pour
*

**Journal or Publisher: **ON Math

**Volume, Issue, Pages: **Volume 4, Number 1

**Reviewer: **Vanessa

**Date of Review: **5/15/2006

In the applet “Pour and Fill,” students have two containers where they can only fill, empty, or pour one container into another container. The goal of the activity is to get a certain amount of water in one container. This could be an activity that introduces your students to the idea of combinatorics and number theory, or it could just a fun activity the class can do in groups to get the students thinking. At first, the students would probably use trial and error to get the answer. After a while, the teacher can encourage the students to figure out a strategy to get the correct answer. This activity could also be a “hands-on” activity where the students can use cups of different sizes and water to do the same activity. The cups and water method could be a bit messy in the classroom because of the water, but this could also be fun for the students. The applet would get things moving a lot faster and I think students would probably learn more from the applet.

This is a good activity for a lot of grade levels, ranging from middle school to high school. I’m not sure when the students would do this though because it seems different from a lot of things students are usually doing in the classroom. I wouldn’t know what to relate it to, but I would still do this activity because it gets the students thinking and experimenting. I also like this activity because it caters to different student levels. For students who figure it out really quickly, there is a button for a “challenge” on the applet. These are questions where it is impossible to get a certain number in a cup. The students would have to figure out why it doesn’t work. Although I think that all students would get a chance to do the challenge because I don’t think it is too difficult to figure out. Overall, I like this applet and would use it in my classroom.

**Keywords:** Calculus, Technology

**Ref: **Vanessa20

**Author(s): **Hodgson,Ted

** Date: **2004

**Title: ***An Interactive Approach to Projectile Motion
*

**Journal or Publisher: **ON- Math

**Volume, Issue, Pages: **Volume 3, Number 1

**Reviewer: **Vanessa

**Date of Review: **5/15/2006

This applet allows students to explore projectile motion and make conjectures about them. Hodgson believes that to completely understand projectiles motions, you have to understand it intuitively and be able to use mathematics to represent the motion. With the applet, students could easily use it to build their intuition about projectile motions. In the applet, students can change the mass, initial velocity, and angle. There is also a button the students can press to add air resistance into the equation. First, students can play around with it and see how the projectile changes when you change the mass, initial velocity and the angle. Then, students can come up with conjectures about what happens when you change the settings. For example, students should conjecture that vertical distance, maximum height, final velocity, and time of flight are not affected by the mass of the projectile.

This applet is definitely important when students are learning about projectile motion in high school math classes. I remember having a lot of trouble figuring out how to do them and how to think about them when I was in high school. The concept of projectile motion is a hard concept to grasp for a lot of high school students. This applet would help students understand the concept more intuitively. Also, students are making up their own conjectures so hopefully they would remember the information better that way. Overall, I think these applets can be very helpful in a classroom. It can really help students to understand math intuitively.