Keywords: Teaching Strategies, Problem
Solving, Algebra
Ref: Lori1
Author(s): Martinez, Joseph G. R.
Date:
April 2001
Title: Thinking and Writing
Mathematically: "Achilles and the Tortoise" as an Algebraic Word
Problem
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 94(4), p. 248-52
Reviewer:
Lori
Date of Review: 2/15/04
This journal article focused on how to actively involve students with algebraic word problems and how to help them move past the basic mechanics and actually understand the real problem that is presented. Martinez feels that in order to help students overcome their negative view of word problems that teachers need to engage their imaginations with “creative, thought-provoking” problems and to involve the students more directly by having them think and write descriptively and critically about their mathematical thinking.
The example that was used to illustrate Martinez’s theory was the “Achilles and the Tortoise” word problem. This problem revolves around a race between the tortoise and Achilles in which Achilles could run 10 times faster than the tortoise and therefore the tortoise was given a 100-yard head start. The problem claims that even though Achilles could run 10 times faster, he will continuously get closer to the tortoise but never quite catch up. Students were then expected to journal their thought process with the problem and make logical sense of the paradox that was presented to them.
The teacher’s role with presenting this word problem is to ask questions that take students’ thinking level beyond where it started. Questions like, “Why did you do this? Why did you not do that? What were you thinking when you...? Is this way better or more effective than that? Would you do anything differently?” were all examples on how to expand their thinking.
Overall, I liked this article and
the idea it presented. It claimed that students were excited
about their active involvement in solving the equation and enjoyed
the process of thinking beyond their normal level of thought. I
hope to use these types of thought-expanding questions and maybe
even the Achilles problem in my own classroom some day.
Keywords: Algebra, Teaching Strategies
Ref: Lori2
Author(s): Choike, James R.
Date: 2000
Title: Teaching Strategies for "Algebra for All"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(7), p. 556-60
Reviewer: Lori
Date of Review: 2/18/04
This article focused on basic teaching strategies that teachers should take into consideration while working with any student, but particularly when they are learning algebra. I will review the key tactics that I found the most helpful. First, teachers should emphasize conceptual understanding or the “big ideas.” There are usually many chapters in algebra books but only a few conceptual themes. Therefore teachers should focus primarily on those big ideas so students will be able to better understand how the topic they are currently learning connects with other materials that they have studied in the past.
Secondly, teachers should eliminate numbers that are distracters in order for students to understand new concepts. In other words, the teacher should replace confusing numbers in new word problems with “friendly” numbers such as 5 or 10. This makes the problem appear to be less intimidating and the student may think he has more of a chance to find the solution.
The next suggestion was for teachers to emphasize multiple representations with their algebra problems. Students should be taught the how to represent mathematics verbally in words, numerically in tables, visually in graphs, and algebraically in symbols. They will then learn and see how each of these different forms of mathematics connects to each other.
A final suggestion given to teachers is to not start out the year by remediating. Students will be coming into your classroom with many different levels of skills. A review of certain materials may either be too confusing or not challenging enough and could leave students disinterested with math. Instead, dive right into the curriculum and then review certain concepts when problems arise. Another way to test students’ knowledge is to post a warm-up problem on the board for the first five minutes of class. Carefully chosen problems can help students review past concepts and also be introduced to new ones.
Overall, I was able to pull out a lot of quality information from this article e that will definitely help me with my student teaching. There were many small details that did not stick out to me as something I should include within my lessons but now I know that if they are added that they will most likely make a difference in the end.
Keywords: Probability, Activities...
Ref: Lori3
Author(s): LeMoine, Shirley
Date:
Title: Probability: The Study of Chance
Journal or Publisher: http://www.col-ed.org/cur/math/math15.txt
Volume, Issue, Pages:
Reviewer: Lori
Date of Review: 2/23/04
The lesson that I found on the Internet was an activity that introduced students to the basic principles of probability. It is based on the game of rock, paper, scissors and by completing the activity the students will be able to conduct an experiment, determine if a game is "fair", collect data and enter it into a table, interpret the data by finding the range, mode, and median, display their information in a line graph, conduct an analysis of the game, and finally state and apply the rule/definition of probability.
I thought this was a good introduction to the topic of probability because it used a game that most students would be familiar with. To begin the lesson with a game of rock, paper, scissors would most likely get them excited about the math topic and grab their attention. I was also able to find applications of each of the five process standards to justify my selection of this lesson. The students would be using problem-solving skills to collect their data, put it in a table, and then graph their results. They would have to apply reasoning skills to understand that not everyone in the class will arrive at the same results and why this happens. Communication would be necessary to share their results with the class and to compile all of the data. The connection of real world applications and math is obviously brought into the lesson by the hand game. The teacher could also have the students brainstorm or discuss other areas in life where probability is found. The use of representation is also found in this lesson because all the students would be graphing and explaining their results in the same way and with the same terms. Therefore, because this rock, paper, scissors lesson fit all the five process standards, I felt it was a great lesson to critique and hopefully use someday in my own classroom.
Keywords: Standards, Teaching Strategies, Activities
Ref: Lori4
Author(s): Dengerud-Au, Mary
Date: 2000
Title: Strength of Wood Beams: An Engineering Application
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(7), p. 544-48
Reviewer: Lori
Date of Review: 2/25/04
NCTM's Curriculum and Evaluation Standards for School Mathematics states that allowing students to experience problems "that have multiple solutions, each with different consequences, will better prepare them to solve problems they are likely to encounter in their daily lives." Therefore the article that I evaluated covered an entire unit on the strength of wood beams that gives math students a taste of what it's like to be an engineer and presents them with problems that an engineer might face in the real world.
To begin this unit, the author had her students imagine that they owned a small portion of land with a small stream or pond in the backyard. Their challenge is to design a simple bridge made up of a wood beam that spans the water and allows the person crossing to keep their feet dry. The students are then asked to brainstorm what they will need to know in order to solve this problem. In the end, they should discover that they know very little information in the beginning and will need a considerable amount of information to build their bridge.
The next part of the lesson then is to introduce the students to a few engineering terms. Concepts like a 'simply supported' beam, a 'cantilever' beam, length, span, and load rating are all essential to constructing a bridge. The author also suggested showing a slide show of different bridges from around the area that display the many examples of how a bridge can be built.
A simple experiment of the strength of paper clips can be used next. Students unfold ten paper clips and bend them back and forth until they break. They record this data, share it with the class, create a bar graph, and find the median, mode, and range for the data. It is also important that they discuss or brainstorm about why some paper clips were stronger than others? Why would anyone care about how strong a paper clip is? And Do you care about the strength of bridges that you cross?
The next step is the fun part where the students actually get to test the strength of dif different wood beams. They set up two different types of bridges and hang a bucket in the middle of the beam. Slowly they fill the buckets with sand and continue to do so until the beam fails. The teacher then presents the students with 'bending stress equations' (which can be found in this article) and has them compute the failing force that must be exerted on the beam to make it break.
The assessment involved in this unit is an ongoing process that begins with observing the students while they work in groups to collecting their data sheets to listening to their comments made during group discussions. In the end they are asked to work in groups to build their own bridge that will not fail. They must take cost, safety, and design into consideration and create a presentation for the rest of the class on why they chose to build the bridge they created. The class could even end the unit by determining which group or bridge would get a "bid" to be built.
I was really excited by this unit and thought that it offered a lot of opportunities for class discussion, hands-on learning, problem solving, and real world reasoning. I would love to be a student in this class and therefore would have fun teaching it as well.
Keywords: Teaching Strategies, Activities...
>Ref: Lori5
Author(s): Appelbaum, Elizabeth
Date: 2000
Title: A Simulation to Model Exponential Growth
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(7), p. 614-15
Reviewer: Lori
Date of Review: 3/1/04
This article was a model lesson for how to teach exponential growth. The simulation the author used was with cancer cells and how they reproduce exponentially. One student is named the founder of the cancer population and is given a die to roll. Each roll of the die is representative of a year. If the student rolls a three then the student reproduces and another student is chosen to roll another die. They roll their two dice and if either of them rolls a three then they reproduce again and another student is added to the population. This continues until everyone is chosen and the entire class is part of the cancer cell population. At the same time, the teacher plots the number of years (rolls) and the number of students (cancer cells) on a scatter plot. After the students have finished with the experiment, the teacher can then take their scatter plot of points and lay it over the correct graph for an exponential function. It should be surprisingly close but not precise.
The teacher would then need to talk about the formula for exponential growth of a population (y = Ne^rt) and how r is derived. The students would need to understand the probability of rolling a three on each toss, how that probability will increase each time the die is thrown, and what happens when more dice are added. This could be a discussion for the students to discover on their own or with teacher assistance.
Overall, I thought the idea of rolling the dice and adding more students and dice was a great way to simulate exponential growth. I don't think the example of cancer is necessarily the best since some students could have a personal relationship with someone with cancer and it could make them feel uncomfortable. I also am unsure about the discovery part of the lesson. I feel that more direction is needed to help these students understand how the equation is derived and in the end they could be a little confused about how the growth rate is found. I do not know if I would use this exact lesson but I might take the be ginning simulation into consideration.
Keywords: Planning, Activities...
>Ref: Lori6
Author(s): Kosbob, Sarah; Moyer, Patricia S.
Date: 2004
Title: Picnicking with Fractions
Journal or Publisher: Teaching Children Mathematics
Volume, Issue, Pages: 10(7), p. 375-78
Reviewer: Lori
Date of Review: 3/3/04
I chose this article because I thought it was a great lesson on how to teach younger students fractions and I could imagine students having a lot of fun with the math involved. The lesson is designed to last for about three days and revolves around the idea of going on a picnic with friends or family. Groups of students are given different amounts and types of food (i.e. 25 peanuts, one candy bar, a giant cookie) and told to divide it into equal portions for a certain amount of people to share. They are expected to represent their answers with a drawing, words, and with a fraction. Most likely, the students will come up with different solutions (such as 1/3 and 2/6), which can then turn into a discussion about equivalent fractions. If the teacher wanted to extend the lesson for one more day, then the article suggested having the students create their own picnic lunch. They can choose the food they want to serve, show how it is divided among the people on his/her picnic, and speak of the divisions in fractions.
Overall, I liked this lesson because it covered the process standards that we have been discussing in class. The students are problem solving by collecting data and dividing the data into fractions. They are communicating with each other in groups and to the class when they present their results. There is the instant connection being made by the real-world example of food. Reasoning is being applied by discussing equivalent fractions and representation is found in the students' answers through displaying them as pictures, words, and fractions.
Keywords: Technology, Geometry...
>Ref: Lori7
Author(s): Glass, Brad
Date: 2004
Title: Transformations and Technology: What Path to Follow?
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 9(7), p. 392-97
Reviewer: Lori
Date of Review: 3/9/04
This article describes a technology-based approach for dealing with a misconception that some students hold concerning the equivalency of geometric transformations. Through a series of activities with an interactive geometric software system, middle school students are given the opportunity to conceptualize the transformations in terms of the relationship between preimage and image of the shape rather than the motion or path the shape moves on.
The activities are preset for the students in the files of the geometric software. The first activity shows a transformation of a figure G along a vector (also known as a translation). The students are allowed to play with the vector and derive any relationships between the preimage G, the final image G, and the vector.
The second activity involves a two vectors and the students are to analyze how the preimage and image are related. Most students focused on the "slide" of the object and described the movement along a nonlinear plane. This shows that students are more concerned with the path of the figure rather than the outcome of the transformation. Thus, a third activity is needed.
In the final activity, students are to examine the transformation of a figure under two vectors once again but there is also a third vector present that is equivalent to the first two. Students are asked to describe how the vector chain and the single vector are related and most of them discovered that if they altered the chain vector than the single vector would also adjust. It was also made apparent that a third possibility could be derived to translate the preimage to the final image and therefore showing that there are multiple ways to show equivalent transformations.
Overall, I liked this article because it described a teaching method where students were able to develop their own reasonable conjectures with minimal guidance from a teacher. They were also allowed to explore through the interactive geometric software, which provided them even more freedom to discover dvi different ideas about transformations of shapes on their own. This challenge will better equip the students to think correctly about equivalent transformations for their secondary and post-secondary math in the future.
Keywords: Standards, Teaching Strategies...
>Ref: Lori8
Author(s): Zawojewski, Judith S.
Date: 1991
Title: Dealing with Data and Chance
Journal or Publisher: Curriculum and Evaluation Standards for
School Mathematics
Volume, Issue, Pages:
Reviewer: Lori
Date of Review: 3/10/04
This book is a teacher's manual on how to teach the process standards in mathematics. The first section focuses on data gathering by students and provides activities where the students create their own surveys, conduct experiments with beans and probability games, and use Monte Carlo simulations. All of these activities are aimed at expanding students' knowledge of collecting and analyzing data. I was a little overwhelmed by the survey lesson because it consisted of ten different activities of which some required more than one class hour. It seemed like a lot of work for the teacher to organize the entire project but it would be very beneficial if it worked out in the end.
The next section focuses on communication in mathematics by having students practice writing out survey questions and comparing plots and graphs. Along with these activities are pre-written worksheets that are available for the teacher to copy in the back of the book. I thought this was convenient since there are very few examples in the other lessons and these worksheets will give teachers a place to build off of.
The last three sections focus on problem solving, reasoning, and connections. All of these have great activity ideas for expanding lessons and allowing the students to interact and learn math in a way besides computing number problems. I would definitely use the ideas presented in this book and feel it would be a great resource for any math teacher.
Keywords: Equity/Diversity, Issues, Teaching
Strategies
Ref: Lori9
Author(s): Lee, Hea-Jin; Sikjung, Woo
Date:
2004
Title: Limited-English-Proficient (LEP)
Students: Mathematical Understanding
Journal or
Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 9(5), p. 269-272
Reviewer:
Lori
Date of Review: 3/16/04
Today teachers in the United States are encountering more and more students who are limited English speakers. In fact, during the years 1993-1994 schools across the nation enrolled over 2.1 million LEP students with more than 90 percent of them coming from non-English speaking countries. The problem mathematics teachers are now facing is the reality that for success in the classroom to occur, there must be some proficiency in the English language. Therefore, this article focuses on specific problems and issues which teachers and LEP students may encounter, some effective approaches for teaching, and recommended learning strategies that could bring about a more positive math experience for both in the classroom.
To begin, there are many issues that both teachers and LEP students may struggle with. Students who lack sufficient English skills may have problems communicating in math especially with abstractions, symbols, and written and oral problems. They will most likely have to focus on basic facts, concepts, and algorithms and what they mean in English instead of conceptually thinking about the math in the way the teacher may have intended. These students will have lower self-confidence in themselves and their math knowledge and will want to give up faster than a native speaker of English. Teachers, in the same respect, develop a misunderstanding of students' knowledge because of shorter answers on tests and a poor attitude in class.
To help combat these issues in the classroom, this article suggested some basic instructional strategies. Teachers should use multiple sources when teaching any type of word problem or abstract concept. Manipulatives, pictures, graphs, kinesthetic tools, and body language will help the student pictorially see the question in their mind and gain an overall better understanding. LEP students should also be allowed to work together in their own native language so they can reinforce mathematical concepts in their own tongue. It is beneficial for these students to work with native English speakers as well though so they can build relationships with the other students and spread an understanding of their own culture to the class. Teachers should also encourage LEP students to develop their language skills by questioning them on main ideas, having them present their solutions to the class, asking them to interpret/ read graphs, or giving them oral practice in reading formulas or equations in English. Above all though, the classroom environment has to be supportive and trusting for these students to be successful. Without a caring classroom, these students will fall between the cracks and disappear into the corner.
Keywords: Problem Solving, Activities...
Ref: Lori10
Author(s): Beckmann, Charlene E.; Thompson, Denisse R.;
Austin, Richard A.
Date: 2004
Title: Exploring Proportional Resoning through Movies and
Literature
Journal or Publisher: Mathematics Teaching in the Middle
School
Volume, Issue, Pages: 9(5), p. 256-62
Reviewer: Lori
Date of Review: 3/31/04
This article discussed how teachers could bring the idea of proportional reasoning to life for their students through examples from movies and literature. Certain examples include the size of Barbie if she was a real person, how large the grass and insects had to be made for Honey, I Shrunk the Kids to make the kids look miniature, or how small Hagrid's house and eating utensils were in Harry Potter and the Sorcerer's Stone in order to make him look enormous. Proportional reasoning is an important concept in mathematics and can be found in geometry with a simulation or dilation, algebra when analyzing constant rates of change, measurement with converting units, in numbers when changing the form of a number from a fraction to a decimal to a percent, and statistics when estimating values that are between two known data points. Students should also realize that proportional reasoning is a necessary skill in real life when, for example, using a scale drawing, determining how much gas is needed for a car trip, or even converting a recipe for larger or smaller amounts of people.
The rest of the article offered example questions that a teacher could discuss with and work on with his/her students. It focused on three main movies, which were Harry Potter, Lord of the Rings, and The Perfect Storm. I found these questions to be really helpful and found this article to be a great suggestion on how to relate math to students' interests. I love the idea and hope to use it someday myself in my own classroom.
I also want to list other resources that this article suggested for
this lesson:
Movies – Honey, I Blew up the Baby, Gulliver's Travels, Alice in
Wonderland, King Kong.
T.V. – I Dream of Jeannie, The Littles.
Books – The Littles, Magic School Bus, Harry Potter.
Toys – Model train villages, toy cars, trucks, matchbox cars, GI Joe,
Barbie, furniture/objects that are built for children, battery operated
cars, tea sets.
Ref: Lori11
Author(s): Stutzman, Rodney; Race, Kimberly
Date: 2004
Title: EMRF: Everyday Rubric Grading
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 97(1), p. 34-39
Reviewer: Lori
Date of Review: 4/3/04
The NCTM standards on assessment suggest that the assessment system should reflect the mathematics that all students need to know and should be able to do, enhance mathematics learning, promote equity, be an open process, promote valid inferences, and be a coherent process. The article I read on assessment described a rubric grading system called EMRF that covers all of these standards. The creators of this system were in search of a way to communicate to their students the level of understanding they had achieved in their mathematics. It answers the basic questions for students, such as: "Is my work at an acceptable level? Am I on track to earn the grade that I would like to receive in this course? Do I need to go back and revise or relearn some material?"
There are four levels to this rubric: E signifies excellent, M is meets expectations, R means revisions required, and F is fragmentary. This article suggested that teachers set their standards of each level before setting out to grade their students' work. This will eliminate the tendency to curve the scores towards certain students and will minimize the tough decisions that need to be made in grading.
The standards set by the authors of this article were as follows. E-level of work indicates that the student shows generalization and has clear communication of their ideas. Most of the work is accurate (there is a possibility of trivial errors) and overall the work is an example of excellence. The student has demonstrated solid conceptual understanding and has defended their result with a logical explanation. M-level work meets the basic expectations and does not represent perfection. The student most likely has the procedural understanding mastered but has a little difficulty with the conceptual ideas. This stage is often referred to as 'first-draft quality' and there are minor errors that need to be corrected. At the R-level, there are evident errors that need to be addressed and corrected for complete understanding. The communication i s usually nonexistent or trivial but there is evidence of partial understanding of the procedural process. These students are given the chance to revise their work in order for a higher grade and to ensure complete understanding. The final level is F and this letter if given to the student who has made an attempt but the work is fragmented, completely misdirected, or unsubstantiated. There is no clear evidence of even partial understanding and lacks the structure, foundation, and framework that would be needed for revision. Revisions at this level are considered on a case-by-case basis and depend on the teacher's philosophy on grading.
Overall, I think this rubric would be successful when assignments are based on written work as well as computation and not solely for everyday problems. I like the idea of it though and will take it into consideration when I develop my own grading system.
Keywords: Proof, Representations, Geometry
Ref: Lori12
Author(s): Cox, Rhonda L.
Date: 2004
Title: Using Conjectures to Teach Students the Role of Proof
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 97(1), p. 48-52
Reviewer: Lori
Date of Review: 4/13/04
This article focused on one teacher's attempt to teach her students how to write proofs through her own unique unit plan. She wanted to see her students become mathematicians and make conjectures, test them, and work to justify them through proofs. Mathematicians are also required to critique their fellow mathematicians and accepting their conjectures as truths, so her students have practice with this as well.
Before she begins her unit, there are some preliminary ideas that must be addressed. First, students need to be mentally prepared before they can learn how to write proofs. This teacher has her students study the basic parts of a proof, learn about different styles of proofs, teaches them how to write 'given' and 'prove' statements from conditional statements, and has them work on fill-in-the-blank proofs and simple proofs with some assistance. She collects all of the students' textbooks because she wants them to think and conjecture on their own and without assistance from a book. Students are placed in groups and continue to work together throughout the unit. It is a huge benefit to talk about conjectures and hear other students' ideas, which could eventually lead to other ideas. Finally the teacher must change his/her role and act as a facilitator, discussion leader, and sometimes the devil's advocate. This unit was created for students to work on their own and the teacher must give up all control and let the student find their own way.
The unit itself is divided into seven parts: parallelograms, proving quadrilaterals are parallelograms, rhombi, rectangles, squares, isosceles trapezoids, and kites. The students perform the same steps for each part of the unit, which include making conjectures, writing a proof for the conjecture, critiquing the proofs as a class, and accepting or rejecting the proof as a truth. To begin, the students are given a conjecture sheet of parallelograms and must work together by measuring sides and angles to create a definition. Computer software is also hel pful with this process. For the first two parts of the unit, the teacher and students work together to figure out the proof and after that, the students are on their own. In order to review the proofs, the teacher puts an overhead up of each proof and everyone reads through the proof together. They decide what could be changed and discuss what is different from other students' ideas. After they come up with a satisfactory product, the class decides on a name for the proof and they each add it to their personal list of proofs.
I really liked this article. Even though I'm not a huge fan of proof writing myself, I can tell that this type of unit where students discover on their own would be very beneficial. The students end up finishing the unit with a feeling of accomplishment that they improved from the beginning. It is also neat to see that students have the ability to communicate mathematically with each other. Overall, I enjoyed this reading and would suggest this unit idea to anyone who is thinking about teaching geometry.
Keywords: Geometry, Technology...
Ref: Lori13
Author(s): Edwards, Michael Todd
Date: 2004
Title: Fostering Mathematical Inquiry with Explorations of
Facial Symmetry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 97(4), p. 234-41
Reviewer: Lori
Date of Review: 4/19/04
This article describes two technology-based activities that introduce entry-level students to symmetry in geometry. Michael Edwards wanted to create a lesson that involved real-world applications and symmetry so that he could grab his students' interests and get them excited about math. In his research, he found a Newsweek article called "The Biology of Beauty" which suggests that reflection symmetry can influence the health, sexual activity, and longevity of various organisms, including humans. After reading through this article with his class, his students became interested in how facial symmetry could play a role in the lives of actors, entertainers, and politicians. Therefore, Edwards created activities that involve photo- processing software to manipulate and test facial symmetry.
For the first activity, Edwards collects 20 or more headshots of famous celebrities/personalities for his students to analyze. The students select a face and using computer software select half of the face for manipulation. By then performing the correct transformation, a symmetric face is relatively similar to the original while a nonsymmetrical face is very different. The students experiment with different faces and come to conclusions over which face is the most symmetrical and which one is most asymmetrical.
The next activity introduces students to building their own algorithm in order to describe their symmetrical findings from the previous computer activity. First the students use protractors, rulers, and calculators to connect points and measure angles from different parts of the face. Most students discover that by connecting the outer points of the eyes and nose and also from the nose to the mouth creates inverse trapezoids (which preserve symmetry in angle measure). After deriving their formula, they find that a face that yields a value closer to 1 is most symmetrical while those with a value closer to 0 is the least. Edwards also creates cartoon-like faces using geometry software and has his students an alyze these faces with precise measurements.
Overall, I liked the real-world application and the use of technology in this lesson. There is definitely a lot of material to prepare but it would be well worth the effort if the students walk away understanding and appreciating geometry a little more.
Keywords: Statistics, Activities...
Ref: Lori15
Author(s): Goldsby, Dianne S.
Date: 2003
Title: "Lollipop" Statistics
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 9 (1), p. 12-15
Reviewer: Lori
Date of Review: 5/4/04
The NCTM Principles and Standards say that students should work directly with data to understand the fundamentals of statistical ideas. It is the teacher's responsibility then to present these statistics in a way that can appeal to all students' abilities and interests. The way this article suggests to present the idea of statistics is through music since most students in middle and high school can relate to this topic. The particular song they recommended was the 1950s hit "Lollipop" which can be found in the movie Stand by Me (1986).
The activity begins with students recalling what they did in their first mathematics lesson. Most students replied that they did something with counting. The teacher then continued to ask them why counting was so important and how we can use counting in our daily lives. After discussing their real-life experiences with counting (planning for a party or determining the number of students in the classroom), the teacher leads into the song "Lollipop". It is played one time for the class and they are asked what phrases occurred more than once in the song. This brings up the topic of frequency and the students brainstorm on the white board which phrases or words they heard the most often. The students are then broken into groups and each member of the group is assigned one phrase to count. After the song is over, the students who are assigned the same phrase are asked to compare their numbers. Not all of them will be the same. Should they take the average of the two numbers? Not in this case. The students decide it would be better to listen and recount the frequency of their phrase again. These counts are then displayed on the overhead or whiteboard and the students must decide which form of representation would be the best for their data.
The author of this article had a lot of success with this lesson and found that students were very excited and involved. She had students interacting with each other and even had the attention of the students who have difficult y concentrating in mathematics. At the end of the article, there were recommendations on how to extend this activity with other music ideas such as conduction surveys on what types of music is preferred by parents, teachers, or students in other grades; number of CDs owned by students; or number of new songs released in a month in various categories.
Keywords: Connections, Problem Solving, Algebra
Ref: Lori16
Author(s): Kelley, Paul
Date:
Title: Loads of Codes - Cryptography Through the Ages and in
Your
Classroom
Journal or Publisher: MCTM conference
Volume, Issue, Pages:
Reviewer: Lori
Date of Review: 5/4/04
At the MCTM conference, I attended the session on cryptography and really enjoyed the information I learned from Paul Kelley. He began by inquiring what the definition of mathematics would be if there was only one word to describe it. We came up with patterns and therefore since cryptology is the study of many different patterns, then it was math! He then introduced the three main types of ciphers that are found in cryptology. The first one was the transition cipher and this pattern doesn't change the letters but simply scrambles them in a different order. The next was an anagram where you use the letters of one word/phrase and form a different word/phrase. He said that many examples of these anagrams could be found in the popular book, The Da Vinci Code. The third type was a substitution cipher where you replace the numbers with letters, symbols, or other letters. He told us that Caesar was the most famous for using this cipher and created the cipher wheel as his coding/decoding device.
The second part of Kelley's lecture focused on the history behind cryptology. Ciphers have been used for thousands and thousands of years. Some of the most famous users of ciphers were Mary, Queen of Scots who used codes and ciphers to try and escape from jail. Thomas Jefferson was another famous cryptographer and he created the Jefferson wheel. Kelley showed us how to make our own Jefferson wheel from Styrofoam cups and strips of paper alphabets. Also, during WWII, the Germans created an almost unbreakable code called Enigma. By luck though, the code was broken and the war was able to come to an end.
The final section of the session revolved around ways to involve cryptography in the classroom. It could be used in practicing matrix addition or multiplication by assigning each letter in the alphabet a number, writing out a message in the number code and then adding/multiplying that matrix by a keyword matrix (in Kelley's example, he used the word math). Another way ciphers could be used is in algebra. The students could begin with a base equation (such as c=3p+1) where c stands for the cipher's letter position and p stands for the plaintext's letter position. The students could practice coding a message, giving it to a friend, and then have them try to solve it.
Overall, I really liked this session and hope to use the information I learned in the talk in my future classroom. I had done some previous study on cryptography and it was good to hear that I could use this knowledge in my lessons at some point.
Keywords: Issues......
Ref: Lori17
Author(s): Luck, Larry; Schoeder, Becky; Blaisdel, Fred;
Barwick,
Gertrude Flowers
Date: 2004
Title: Go to the Principal's Office!
Journal or Publisher: Future Teacher's Conference (March 6,
2004)
Volume, Issue, Pages:
Reviewer: Lori
Date of Review: 5/5/04
I attended the Future Teachers Conference in March and learned some valuable information from the sessions I went to and therefore I wanted to share some helpful hints with the rest of you. This particular session that I attended focused on the interview process and what you should expect to happen during and after the interview. I have never actually had to participate in a real professional interview before so I found a lot of the tips these principals gave to be very helpful.
To begin, they suggested researching about a little information on the school you are interviewing for. Find out little details about the school population, colors, mascot, and what the needs are of the school. That way you can come into the interview showing that you have knowledge and interest in what that school stands for, who they represent, and what they are looking for in a teacher. On the day of the interview, they recommend that you show up early so you aren't pressed for time and wear appropriate clothing that shows that you are a neat, responsible, and orderly person. They described it as "dressing for success" so guys should at least wear a tie and jacket and ladies should be in a nice pants/skirt outfit. During the interview, always answer directly and be brief in your responses. Make sure you show enthusiasm for your job and your subject. A few questions they are likely to ask are for you to describe your strengths in teaching and also your weaknesses (or areas that you could grow in). They could also ask you to describe yourself in three adjectives. They might want to know why you want to teach in their school and what you would bring to your subject area that makes you stand out from the rest. Therefore it is important to have some examples prepared about what you would teach and how you would go about teaching it. It was also suggested that you should have a few questions prepared for them. A few examples include, "As a new teacher in this district, what kind of support would you be able to give me?" "What are the courses available that I would be teaching?" "What are the class sizes?"
Overall, the principals in this panel said that they are looking for teachers who have experience with and enjoy being around kids. They want to see what you have learned over the years and how you have been able to implement it into your teaching philosophy. They want to know how you manage your classroom. Not simply the rules you enforce but how you are able to interact with students and form relationships with them. These principals stressed that the interview process should not be a scary or intimidating thing but just a way for you to show them how much you love your job and that you are the best one that they could hire.
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Keywords: Equity/Diversity, Geometry, Activities
Ref: Lori14
Author(s): Neumann, Maureen D.
Date: 2003
Title: The Mathematics of Native American Star Quilts
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 9(4), p. 230-36
Reviewer: Lori
Date of Review: 5/15/04
I always like to read about different ideas on how to involve real world activities and culture into mathematics lessons and this article was a great example on how Native American quilts can be applied to such topics as algebra, geometry, and arithmetic. In this activity students first examine the mathematics that are needed to create the star quilt, then they make their own star quilt, and the final section extends the students' knowledge of the eight pointed star to different shaped stars and focuses on the numerical patterns found within the star quilt.
It is important that students learn about the history of quilting in Native American tribes. Students can explore the Internet to learn about these quilts and then report back to the class the information they discovered. Some interesting facts about the history include the fact that missionaries introduced quilting to Native American girls at boarding schools in the late 1800s. These girls applied their traditional geometric patterns to quilting and one of their most famous creations was the eight-pointed star, which is also known as the Northern Star on northern plains reservations. These quilts have many personal meaning to different tribes. For example, the Lakota Sioux use quilts in honor of births, deaths, marriages, and graduations. On a Montana reservation, the star quilts are given to athletes on basketball teams as a symbol of respect, honor, and admiration for that person. The colors that are chosen for the quilt are also symbolic to certain tribes. For example, the Lakota use the colors red, black, white, and yellow for representing the four cardinal directions and what they bring to them.
After researching about the Native American quilts, students are given a worksheet where they investigate the mathematics behind making the quilt. They discuss the geometric properties of an isosceles triangle and how the triangular templates relate. They also share ideas on how an error in measurement or cutting their templates will effect and influence their results. By the end of this class session, the students should be able to explain how triangles and rhombi are related and why they are essential to creating the eight-pointed star.
The next step is to have the students apply what they learned about the mathematics and create their own star quilt. Have the students select three colors that are meaningful to them and to be ready to explain why they chose those colors in the end of the lesson. The students should cut out their own rhombi and create their own eight-pointed star. Some important questions that could be asked at this time could be, "Where do you see octagons in the quilt?" "Why is that an octagon?" Where do you see lines of symmetry?" "Why did you choose these colors?"
When the
students are finished creating their quilts, they should journal and reflect on
what mathematical properties they learned during this activity. Where did they
use mathematics in their quilt design? What did they learn about Native
American culture? What did they learn that was unexpected? In the end,
students should be able to relate to the Native American culture through their
joint understanding of mathematics and should therefore have an increased
appreciation for them and their art.
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Keywords: Management......
Ref: Lori18
Author(s): Anderson, Megan; Johnson, James
Date: 2004
Title: Help! I'm Only 6 Years Older Than My Students
Journal or Publisher: 2004 Future Teachers Conference
Volume, Issue, Pages:
Reviewer: Lori
Date of Review: 5/15/04
This was another session that I went to at the Future Teachers Conference in March and I thought it was pretty helpful and informative (and will most likely relate to all of us in the future). Two new teachers who had only been teaching for a year gave this session and they wanted to share their advice about starting out in the teaching field when you are only 22 years old.
They began by stating that there are some advantages to being young. Young teachers are enthusiastic which students and employers love to see. They are creative and willing to try new things. They said that it's easier to level with students since you are more likely to remember what it was like to be in their shoes. Also, being young means that you have a much better idea of what your students are interested in. They told us not to be afraid of using those common ties and pulling the students in. However, there are a few disadvantages that we should be aware of about being young. Students can smell inexperience and will call you on it if you aren't prepared to support what you are trying to teach. The best thing to do when you are first starting out is to create an illusion of experience for your students. Dress maturely and have your students address you by your title of Mr. or Ms. and not by some nickname. Act confidently and remember that you should be the definitive expert of your topic so make sure you know your subject. They suggested not to fall into the "friend" vs. "teacher" trap and to remember that you are in the classroom to be the adult role model. The students aren't looking for another friend but a teacher who is competent enough to teach them the material. It's okay to show interest in the students' lives but don't cross the line that separates teacher and student.
They then went on to discuss how they were able to manage the classroom (or in other terms: how to be the ringmaster in a circus). The first thing is to establish clear and simple expectations such as respecting my right to teach, respecting others right to learn, and to expect that your subject matter will be worked on for the entire class period. It is important that you remain consistent with these expectations but also to somewhat flexible. Be selective on the battles you want to get into and be firm in your decisions and actions. A good way to avoid problems is to simply keep your class engaged in what they are learning. Plan more than one activity, tell the occasional story, and pick stuff that you know your students will be interested in. A final way to avoid problems is to know all of your students' names and make sure to learn them within the first two days of school. Use seating charts to your advantage and let them help you with attendance, names, and management problems.
Finally, they offered some
advice about how to maintain your sanity while student teaching. First and
foremost, minimize the amount of work that you take on. Don't let grading over
take your life. Save your time to interact with students and to plan valuable
lessons. This will pay off in the end with management problems and your
emotional sanity. Save the 'good stuff' like little notes and cards from
teachers and students. When you're having a bad day, it's always nice to pull
out your goodie bag and remind yourself why you love your job and how much it
means to you. Also, organize your materials so that it is easier the next time
and have special 'holidays' like work days, reading days, fake math holidays so
that you can take a breath and catch up with your work and life. In the end
though, always appreciate your staff around you. They also know what you are
going through and will be there to support and offer advice as much as possible.
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Keywords: Statistics, Activities...
Ref: Lori19
Author(s): Attia, Tamar Lisa
Date: 2003
Title: Using School Lunches to Study Proportion
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 9(1), p. 17-21
Reviewer: Lori
Date of Review: 5/15/04
This lesson idea was designed for middle school students to apply mathematics in a way that they could analyze and write about a practical and relevant issue to them: the nutritional value of their school lunches. Kids are definitely interested in food (even though it is mostly junk food) so this lesson was a good way to grab kids' attention and inform them of the food choices they were making on a day-to-day basis.
Before starting the project, the students wrote down everything they ate for lunch for one week. When they had this information, they discussed the implications of what they had eaten and found they knew little about nutrition. The school nurse was brought in to discuss some of the nutrients that people need in large quantities like carbohydrates, fat, protein, and fiber. The students were then required to do some Internet research to find out two specific kinds of information for their study. They needed to find the nutritional content of the foods they were eating and the amount of various nutrients that students require.
The next stage of this lesson required students to transfer their lunch choices to a spreadsheet along with the days' corresponding grams of protein, carbohydrates, fat, and fiber contained in the food. Then the students used the average function on the spreadsheet to find the mean amount of nutrients they consumed in lunch during that week. With this information, students could now compare their nutrition data to the RDA daily-recommended number of grams of each nutrient and determine if they were eating healthfully or not. The idea of ratios is brought into use here since students must consider how much of their daily food intake happens during lunch. Things like not liking the school lunch for that day and other circumstances could affect this data.
The final stage of this lesson had the students create two circle graphs. One was to illustrate their weekly average amounts of nutrients taken in at their lunchtime and the other graph was to show the RDA of th e nutrients. This was a good way to have the students visually see what nutrients might be lacking or taking over their diets. The students were then encouraged to analyze and discuss their data and think about how they could make better choices in the future.
I really enjoyed reading about this lesson
because I have always been interested in nutrition and I think it is so
important that students understand how important the food choices they make are.
This is a lesson that involves proportion and ratios to learn about math but it
also covers real life issues that could influence a child for the rest of
his/her life.
Keywords: Teaching Strategies, Keyword 2, Optional..., Keyword 3, Optional...
Ref: Lori20
Author(s): Saltar; Chuck
Date: 2001
Title: 16 Ways to Be a Smarter Teacher
Journal or Publisher: http://www.fastcompany.com/magazine/53/teaching.html
Volume, Issue, Pages: (website)
Reviewer: Lori
Date of Review: 5/22/04
I read an article on the Internet called, “Attention Class!!! 16 Ways to be a smarter Teacher” and found it to be a little list to live by on a day-to-day basis. It was a great list of ideas that I think everyone should take into consideration throughout life whether they become a teacher or not. I will explain the five that I thought were the most important and then list the rest of them.
To begin, the article stressed that “It’s not about you, it’s about them.” Teachers are not there to stand up in front of the class and preach all day long. The best teachers are there to be guides for their students and to listen to what they have to say. The students are the most important things in the classroom and the teacher needs to recognize them and their needs.
The next step to being a great teacher is to study your students. You need to know whom you are teaching to and should be aware of your students’ interests, hobbies, and past experiences. It will help you to relate more with the kids in the classroom and will make you a better teacher by being able to gear lessons towards their likes.
Another way for you to win your students over as a teacher is to teach from the heart. If the students can tell that you love your subject and are passionate about what you are teaching, then they will be more responsive and take more of an interest to what you have to say. No one wants to watch someone teach who looks bored themselves. As long as you put your heart into whatever you are doing, people will respond positively and be excited themselves.
Great teachers ask great questions. How can you go wrong with a teacher who is able to ask you just the right question to make you think that much further? I appreciate it when a teacher is intelligent enough to be up to date on their subject and understands how to push me to deeper into a subject. It only propels my interest and makes me enjoy the subject that much more.
A final point that I thought I would include that makes a teacher great is to learn what to listen for. A lot of teachers will only listen for the right answer (especially in math) and won’t look deeper into a student’s thought process. However, I feel that even though the student didn’t get the right answer, he/she might have an alternative process or way of looking at things. Therefore I feel that students should be expected to explain themselves thoroughly and teachers should take on the responsibility to listen to all that they have to say.
These were my top five points out of the article. I really enjoyed reading this because it offered some great advice for inside and outside the classroom. I hope to take these with me as I enter into my student teaching as well.