Keywords: Activities......
Ref: Elizabeth1
Author(s): Munakata, Mika
Date: 2005
Title: Constructing Cooperative logic problems
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98 (6) p. 386
Reviewer: Elizabeth W
Date of Review: 2/15/05
This article discussed the complexities of writing math problems and uses those skills to create a cooperative learning activity. The activity basically has the students write math logic problems for peers.
The author outlines the important steps for problems that teachers use: 1. Will they be interesting for the student? 2. Are they different from problems in the book? 3. Can student use different strategies to solve the problem? 4. Is the solution of appropriate form? When teachers write these problems, they use math through the problem solving skills. So, Munakata asks why not have the students do it?
The kinds of problems for cooperative learning that Munakata described are to make individual clues that fit together to lead to a common solution. She got the idea from some other activities. What happens is that the groups are to be of 4 or 6 students where each student has at least one clue (6 clues total), but they are not allowed to let anyone else in the group see their clue. Together they must solve the problem. The students can write the problems on their own, or in a group. Then after the problems are written, students can get into groups and can work on the problems.
I like this idea because do design the problem students have to both understand the math that they are writing the problem about, but also they have to use the problem solving skills that Munakata described. Also, once the students get into groups and work on the problems, everyone in the group must participate because they have valuable information that needs to be shared. Also, one person can't take over the problem this way either. And each student will also feel ownership and accoutability for the project and their problem. The article gave several example problems which made it clear exactly how these problems should follow. What I liked is that you could write problems for different areas. She gave examples for both logic problems and geometry problems. And this could work for different age groups as well. I do like the idea of having students write their own problems, and I think that having them be group projects makes it more fun.
Keywords: Activities, Puzzles, Discrete
Ref: Elizabeth2
Author(s): Staples, Susan G.
Date: 2005
Title: patterns jumping out of a simple checker puzzle
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98 (4) p 225
Reviewer: Elizabeth
Date of Review: 2/17/05
The author describes an activity that involves playing with checkers to solve for an equation of minimum moves, that depending on how the game is set up, will form a linear or quadratic equation. Staples presents moving checkers along a single row, where several checkers of black is on one side, and several of white are one the other with a single open space between them. By moving the checkers in slides and jumps, the goal is to find the minimum number of moves.
After playing a little, the teachers gives the students rules: 1. No backtracking 2. avoid forming a block 3. prolong the alternating patterns by using rules 1 and 2. The author showed us the algebraic formula to find the number of least moves when using n black and n white. A s=slide, j=jump. by doing n=2,3,4 we see that the equation s+j=2n+n^2, when n is the number of white and black checkers.
Then she discusses the case when black doesn't equal white. This equation is a linear pattern.
Staples says that students have enjoyed these checker puzzles, that they enjoy the discover process.
I could see this being a a very fun activity to do. I do think that it would require alot of leading questions on my part if I were to expect them to come up with this equation on their own. But part of me thinks that she will have students work with the problem out, but then they could find the equation as a class. I like the activity, it reminds me alot of what we did in our discrete class. I did wonder, however, where this would be added in an algebra class. It deals with linear and quadratic equations, but I'm still not sure exactly how you would tie it in.
Keywords: Statistics......
Ref: Elizabeth4
Author(s): Friedman, Hershey H., Halpern, Noemi, Salb, David
Date: 1999
Title: Teaching Statistics Using Humorous Anecdotes
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92,4 pg 305
Reviewer: Elizabeth
Date of Review: 2/22/05
The thesis of this article is that "humorous vignettes can be used to teach concepts effectively and dispel common misconceptions." The authors believe that humor can 1. instigate discussions by kindling interest and response, 2. demonstrate relevance, and 3. help a wide variety of students.
The article goes on to give humorous and exaggerated stories of Dr. Klutz who is a (really bad) statistician. There was a vignette for almost every part of a high school stats class. Each one is a story is a potential real-life situation where Dr. Klutz comes in and makes some glaring statistical error. Students must then realize what the mistake is and discuss the importance of whatever the statistical idea it is about.
I was interested about this article because of the title. I like to think I'm funny, I'd like to have humor in my classroom. So I was slightly turned off by the fact that the stories weren't as hilarious as a Will Ferrel movie. There were interesting enough to keep my attention though. Perhaps they were bordering on hoaky too, but hey, if they keep kids attention more than dull case study stories...
One question that it did raise, in my mind, was how often should we teach by giving a story problem that is finished but with a problem in it. But I think that it is probably a good tool to create conversation or to have students explain a concept. This along with problem where they actually do something would be good.
Keywords: Probability
Ref: Elizabeth4
Author(s):
sriraman, Bharath; English, Lyn
Date: 2004
Title: Combinatorial Mathematics: Research into
Practice
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98,3 pg182
Reviewer:
Elizabeth
Date of Review: 3/1/05
The authors illustrate several different types of combinatorial problems. They talk about how for many students theses are normally confusing and difficult. The reason for studying combinatorics is that "when presented effectively, combinatorial problems give students opportunities to reason mathematically, make generalizations, and engage in math proof." They go on to talk about how this fits into the NCTM standards, speak a little about the history and reason for studying it.
The authors talk about educational research and find that there are overlaps with how students learn well and what is involved in combinatorics. 1. foster independent thinking, 2. encourage flexibility 3. encourage a focus on structure 4. encourage sharing solutions, and 5. present problem-posing opportunities. The article goes on to discuss these 5 themes and discuss related research and ways to use it in the classroom.
I've always like combinatorics, but I was introduced to it in college, not really in high school. So while I would like to do in in my classes, I don't really know how it fits in. Therefore, I was interested in this article to see how combinatorics fits into the class. i think that this article has some good ideas on how to engage the students with problems, and how students react.
Keywords: Algebra......
Ref: Elizabeth5
Author(s): Saul, Mark
Date: 2001
Title: The Roles of Representation in School Mathematics
Journal or Publisher: NCTM Yearbook
Volume, Issue, Pages: pg 35
Reviewer: Elizabeth
Date of Review: 2/28/05
This article is about what "we" teach students algrebra, and what they actually learn. Saul reports on two former students that he has had, and talks about what they actually learned (read: understand)...different from what they could be tested on.
Saul goes on to talk about what algebra is, what it isn't, and what students are learning. He believes that algebra is not the use of variables. (Many students can solve these problems by replacement or guess and check). It also in not the study of functions. (Though it sometimes does use algebra to describe them, and algebra is an important application of functions)
Instead algebra is 1. a generalization of arithmetic. 2. Concerns with the solution of polynomial equations. 3. Study of binary operations on sets. He discusses the roots of these and what they mean, and how the two former students could understand some parts, and not others.
Since the teacher is not quite sure how he could have better taught the two students he only makes guesses and suggestions. He suggests "number stories," and story problems that made the students not use the skills they knew but rather work on what the teacher was trying to get them to learn, and likening variables to real number problems.
I think that the real strength that this teacher had was that he was able to realize what his students knew and what they didn't, because it seems that they were very good at knowing what they did to cover up what they didn't know. This teacher seems to also be a special ed teacher, which is probably why he knows so much about these two students and how exactly they learn and what they know. I'm pretty sure I won't have such indepth knowledge about my students. (Not that I don't what to...) I liked what his goal in teaching them math (since they would never become mathematicians) was to enstill them with an understanding of abstractions in general. BR>
Keywords: Representations......
Ref: Elizabeth7
Author(s): Elizabeth George Bremigan
Date: 2004
Title: Note: Figures Not Drawn to Scale
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98 (2) p74
Reviewer: Elizabeth
Date of Review: 3/9/05
Bremigan discusses how often students are given problems in symbolic notation, sometimes with a visual (graph, chart, diagram, figure) to accompany the problem. Although sometimes the visual will provide no new information, she studies how to study the visuals when they do provide new information. She asserts that to succeed students need to be able to interpret visuals as well as symbols and words.
Before becoming interested in the subject, Bremigan describes how she would use visuals and would make many assumptions about how students would interpret them. But after dealing with frustrated and confused students, she realized that she needed to change how she used visuals.
One thing that she felt she needed to focus on was to interpret/study figures that are not drawn to scale - similar to figures found in standardized tests. The benefit of these are 1. visuals can give answers away if drawn correctly, rather than using logic (which is the goal). 2.Students can have practice for the standardized tests. 3. Figures can be used to represent general cases, and the reasoning used here is good for problem solving.
Bremigan suggests specifically addressing the figures in the classroom by questioning students regarding symbols as well as visual representation. And from this students learn how to distinguish what can be assumed and what can not. In otherwords, it is important to study the figures to determine what is the general case, and what is specific for "this" case.
Overall, I do like the idea of the article. I think that 1. It is important to be able to interpret visual models because it is a very real application of mathematics. 2. It's a good way to understand general cases vs. specific cases. In geometry this past interim we talked a lot about what you can know from a model...I don't think that this idea is just for college math majors, I think that it is applicable for high schoolers. What I didn't like about this article is that she seemed very concerned about standardized tests. Also (and I don't know how bad this is) but she only vaguely mentioned how to use the ideas in the classroom. And maybe she intended to discuss why it's important rather than specific teaching examples. <
Keywords: Connections, History...
Ref: Elizabeth8
Author(s): Will Hansen
Date: 2004
Title: War and Pieces
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98 (2) pg 71
Reviewer: Elizabeth
Date of Review: 3/10/05
The thesis of the article is as follows: "This article describes an interdisciplinary activity in which students see how a famous author, Leo Tolstoy, metaphorically applied the integration steps from calculus to illustrate his ideas about how history should be studied." Apparently, Tolstoy discusses the paradox of Achilles and the tortoise (which is about dividing continuous intervals, leading to incorrect conclusion) and compares it to historians distorting the process of change when focusing on specific events and leaders as causes of movements.
This lesson should be part of an application of interpretation section. Hansen describes two ways to use this idea, depending on time and enthusiasm for the subject. The first is two assign a short chapter from the goob and to discuss it in class the next day. In small groups, then as a class the students would discuss a variety of questions, (ex, In what way is Tolstoy applying calculus in this essay?). The second way is after reading a chapter, to discuss and then have the students write a short paper. From the paper, students get a deeper understanding, and many students are very interested, particularly strong lit. students. He suggests consulting an English teacher, but really in terms of writing, the papers just need to flow well and make sense because it isn't a English class, the point it to discuss mathematical ideas.
I found this idea very interesting and unique. I really had never thought about doing anything like this in a math class. For me, I see some challenges, mostly stemming from the fact that I am not an English person. I think most math teachers may feel a little hesitant leading a discussion on an novel in a math class. However, I think that a positive to this is that it only takes one class period. So it really wouldn't be hard to try it out. If it doesn't work so well, it's only one day lost, not an entire unit. Also, I think that changing things up, and trying to find good connections for calc students would really be appreciated by them.
Keywords: Curriculum, Statistics, Standards
Ref: Elizabeth9
Author(s): NCTM
Date: 1992
Title: Addenda Series, 9-12
Journal or Publisher: NCTM
Volume, Issue, Pages: Data Analysis and Statistics
Reviewer: Elizabeth
Date of Review: 4/5/05
The book/series was written in response to the NCTM standards that had just come out. The series focuses on problem solving, reasoning, communication, and connections. A goal was to be teacher friendly. Since the new NCTM standards included new areas, this series tries new content like emphasis on statistics, which is the book that I'm reading. Another goal of the series in to have more student interaction and problem solving.
There are lots of idea for statistical projects for the students. I really liked how many of them include having students gather their own data. The book also includes grading rubrics for the teachers and gives ideas as to how to grade essays. I think that this would be very helpful because many math teachers are not used to grading reports/essays, but rather homework sets. Many of the activities look like alot of fun.
I like how the "catchers/launchers" are set up. For example in 'making sense of data', the book discusses how a culture that wants the best and the biggest, like the US, focuses on rankings. But rankings are both objective and subjective. Also, just because someone is a expert in the area of interest does not mean that they are experts in statistics. I think that a teacher could really take this idea and run with it, by connecting statistics to an area that students are interested in, while helping them to view the area of interest with a critical eye.
Also, I'm slightly confused as to how exactly the book is too be used. Is it just a tool for the teacher to refer to and supplement their classroom book with? And how do you use it for different grades? Because clearly a 9th grade class is very different from a 12th grade class.
Keywords: Problem Solving,
Ref: Elizabeth10
Author(s):
Date: August 2004
Title: The extension-reduction strategy: activating Prior knowledge
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 98, No 1, pg. 48
Reviewer: Elizabeth
Date of Review: 4/18/05
Sloyer starts with a problem of finding the volume of a cup with unequal radii at the top and bottom of the cup. His students don't know how to solve this-no ideas even. They do know how to find the volume of a cylinder and a cone.
He then turns to a picture of a circle circumscribed inside a square, where the part inside the box and outside the circle is shaded. Given 5 as the side of the square the students solve for the area of the shaded part, which is 5^2 - 2.5^2*pi.
Then, he leads them to solve the problem of34+35+36...+90+91. The students solve it; 91*92/2 - 33*34/2.
They then went back to the original problem and the students then had ideas of how to solve the problem. They made the cup into a cone, and just subtracted the created part.
I like how Sloyer gives a good example of leading students to the right solution instead of one of the two extremes of giving them the answer or just telling them to solve it. This approach takes more time, but it is an effective way to have students understand a new problem-solving technique.
Keywords: Geometry,
Ref: Elizabeth11
Author(s): Charles Worrall
Date: October 2004
Title: A journey with circumscribable quadrilaterals
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: vol. 98 no3 pg 192
Reviewer: Elizabeth
Date of Review: 4/18/05
Worrall focuses on ways to develop curiosity, he believes it is to encourage them to wrestle with complex objects with deep relationships to be found.
He found the following question, which is to just illustrate one idea, and he expanded it for his students. "A quadrilateral with consecutive sides of length 12, 15, and 17 units circumscribes a circle. Find the 4th side. This problem, he said is to "only teach that 2 tangent segments from a point outside a circle to the circle are congruent, but with prodding it can become an interesting problem." He believes the problem alone is not good because in the solution the variable just disappears like magic.
In class Worrall asked: What are the properties of circumscribable quadrilaterals; and conversely, what are the necessary conditions that make quadrilaterals circumscribable?
First, they looked at the triangle case-unfortunately all triangles work, but interesting nonetheless.
Going back to quadrilaterals, they found a theorem stating that circumscribable quads do have the equal sum property. He then gives proofs to "if the sum of the lengths of a pair of
opposite sides of a quadrilateral is equal to the sum of the lengths of the other pair of sides, then the quadrilateral is circumscribable." There are 3 types of proofs for it.
The rest of the article is proving theorems regarding circumscribable quads.
The article was interesting because the main idea was to study one general problem in-depth, and from that problem, learn a lot of math. I like this approach more than studying many problem that are unrelated because the students will have something to connect their learning too. The article was a bit dry because it was a lot of proof.
Keywords: History,
Ref: Elizabeth12
Author(s): Jeffrey J. Wanko
Date: April 2005
Title: The Legacy of Marin Mersenne: The search for primal order and the
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 98, no8. pg 525
Reviewer: Elizabeth
Date of Review: 4/18/05
The article starts by discussing the importance of primes. Then, Wanko introduces Mersenne as a French monk who was seen as intelligent, but not brilliant. But he had insight to the nature of primes that helped him make conjectures and test numbers for primality. Mersenne was also important because he played a large role in the community of mathematicians and mentored scholars. At the time mathematicians and scientists were isolated from each other, and he tried to bring them together to share ideas. The article then gives a history of primes. Mersenne, unlike contemporaries of Descartes', Fermat, and Desargues liked to read others writings, and he wanted to exchange ideas. He came up with conjectures, and at the time people questioned his methods. He did it my analyzing the series of known primes, and saw potential patterns. Some were wrong, but he gave a new approach for looking at the primes. It was interesting to read about Mersenne, a man I know little about. The article didn't say much else, nor did it offer ideas as to what to do with this information. It did however spark my interest and made me think about how I would want to include some math historical knowledge in my class.
Keywords: Proof, Geometry, Problem Solving
Ref: Elizabeth6
Author(s): Pandiscio, Eric
Date: 2004
Title: Using Proportional Reasoning to solve Geometric Problems
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Vol. 98, 1, pg 16
Reviewer: Elizabeth
Date of Review: 4/18/05
This article, whose focus is on reasoning starts by citing the PSSM standard "reasoning mathematically is a habit of mind and like all habits, it must be developed through consistent use of many contexts." He then goes on to discuss the important of the consistency portion, and how it should be used in all(most) lessons.
This article then gives an activity to use reasoning. The 2 goals of the article are to "illuminate the usefulness of mathematical reasoning in the problem solving process" and to illustrate how mathematical reasoning can be an effective instrument for teaching and reinforcing mathematical content."
The task is to take and equilateral triangle, with side of 1, find two points such that when you draw a new segment through those points, the segment is parallel to the third side of the triangle and that the area of the new triangle is half that of the original.
To solve, he suggests that instead of just computing it, you really understand what is going on by finding a general solution. First solve a simpler problem where you are working with a square of length one, and then find the length so that a new square is double the area. Then do the same, but where the original is double the area of the third square. Then, with preparation, students can use reasoning can make a connection that changes by the square change in side lengths can apply to other polygons. So, they will now that the point are sqrt(1/2) units away from the vertex. This approach emphasizes concepts.
The example of using proportional reasoning to understand spatial relationships shows how conceptual reasoning can be a powerful tool.
I think that the idea here is very good. My only concern is that he did not help to explain the connections with multiply polygons. That is the most important step, and will be difficult for student to understand. It seemed that he just glossed over it, advising to work with the square for a while.
Keywords: Technology......
Ref: Elizabeth13
Author(s): Lee, J. Todd
Date: 2005
Title: Fool me twice, Shame on me
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98 (8) p560
Reviewer: Elizabeth
Date of Review: 5/9/05
This article went through a few things in popular technology that tend to give people a hard time. The first tip is for microsoft excel. A problem is that -5^2 will give -25. This can be corrected by typing in 0-5^2. Apparently Excel uses negation and subtration differently. The other problem with excel that was discussed was regarding creating histograms. The article goes through clear instructions as to how to creat histograms. Tip two is for TI-83 Plus; there are 3 tricky parts discussed. The first is linear regression. Apparently the tool to give the coefficents will sometimes turn off-why? no-one knows. DiagnosticOn will do the trick. The other is regarding decimals for exponents. Sometimes, when the integer is 10 digits long, the decimal will not show up, neither will anything to the right of it, implying the answer is an integer. Something to be aware of. The 3rd point is for random digits. If the calc is new, or reset, the randome digits for a whole class will seem awfully similar. This can be corrected by using the calcs, or to reset the seet. The 3rd tip is for geometries sketch pad. IT is a trick for when you have a bunch of lines hidden and just need to unhid one. The article goes through how to do it. Good tips overall, I could see being confused by these things if I used technology (which I will shortly). The info is for teachers, not really a class. This "tech. tips" are in math teacher every month, and are very helpful.
Keywords: Technology......
Ref: Elizabeth14
Author(s): Erbas, A. Kursat; Ledford, Sarah D.; Orill, Chandra Howley; Polly, Drew
Date: 2005
Title: Promoting Problem Solving by using technology
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 98 (9) p599
Reviewer: Elizabeth
Date of Review: 5/11/05
The article starts out by describing how technology can enhance a classroom and a lesson. It leads to creativity in problem solving and techniques, it allows for using data in many ways, can lead to better understanding. This article has an example with algebra and geometry so that connections are made between the two.(note: they cite the intermath web site-www.intermathuga.gatech.edu). The problem they explore is essentially taking a rectangle ABCD, and want to see when the distance AB, BC is 40 units more than AC. Here, technology can help students visualize what is going on, what is fixed, and what varies. They use sketchpad to make the rectangle, so they can move points around and see the units, and they can compare different rectangles that solve the problem. The authors describe how essential it is for teachers to help out the process of seeing patterns and shaping the understanding. The article then shows how to graphically make the dependent relationship independant. Another peice of technology to use is a spreadsheet. Here students can show the ABC's distance vs. AC's distance and find the difference...AB+BC-AC. Then they can show how many dimensions work. Teachers should then challenge students to represent solutions with an equation by finding a pattern in the spreadsheet. Algebraically they should find y=40(20-x)/(40-x). The pros to the spreadsheet is that they will help test and numerically tabulate their conjectures. Also they can help to explore why a pattern is happening. The authors discuss how teachers are very important when using technology because the right questioning will encourage students to make connections. Overall, I think that this could be an interesting lesson. It certainly is the explore then learn the "math" or equations. I think that this is certainly something for teachers with a experience (or maybe for those who are just really smart)who could lead the discussion well.
Keywords: Equity/Diversity......
Ref: Elizabeth15
Author(s): Tim Sauer
Date: MCTM spring conference
Title: Tommy, Pay Attention! Multiple Intellegence Math
Journal or Publisher:
Volume, Issue, Pages:
Reviewer: Elizabeth
Date of Review: 5/11/05
Using Multiple Intellegences (MI) is an effective teaching tool, but should not be used as the only tool. It should be one power source. Sauer discussed 2 NCTM standards which using MI satisfies 1. Equity-Excellent mathematics education should be accessible to all students regardless of personal characteristics or backgrounds. 2. Teaching-teachers must be able to analyze what they and their students are doing and consider were those actions are affecting students' learning. Use a variety of strategies. Sauer then went on to discuss how intellegence has to do with problem solving and working in a natural setting. For this reason, MI is good because math is just one intellegence, so it is important to explore math in other arenas. The 8 intellegences are: linguistic (oral and written words), mathematical (all of us), body kinesthetic (using body to express ideas and feelings), spatial (perception of the the visual world), musical, naturalistic (expertise in recognition and classification of flora and fauna, sensitivity to natural phenomena, capacity to discriminate among nonliving forms), Interpersonal (working with others), and intrapersonal (self-knowledge). In the talk, we got into pairs and were assigned one of 6 intellegences (excluding math and interpersonal). The groups were assigned the mathematical problem: 20 increased by 40%. Then we presented ideas...many of them were very interesting and new ways of seeing the problem. Some examples include: Spatial-taking a ladder with 20 wrungs against the wall and then adding 40% more wrungs, how does this look. Musical-taking a beat of 20 beats per minute, hearing what it sounds like and then increasing the beat by 40%. Naturalistic-talking about how increasing something (people, radius of a city, ect) by 40% and discuss how that would affect nature. After this, we discussed how different intellegences are used in different ways than they used to be (eg. explorers used to be way more important than today). We also talked about how each person has all 8 MI, only some are more prominant than others. Overall, I really liked this talk, it was good to get us to think about how we ought to use MI, not in every class, but every once and a while. And, while I don't think it was a point of his, from this talk, I learned that its important to talk to others and get their ideas, because some of these I would not have ever thought of them.
Keywords: Activities, Games, Discrete
Ref: Elizabeth16
Author(s): Sue Haller
Date: MCTM spring conference
Title: Fair? Well...
Journal or Publisher:
Volume, Issue, Pages:
Reviewer: Elizabeth
Date of Review: 5/11/05
This talk was mostly learn about different games, discuss if we think they are fair, play the game, compare results, and then through the math to see why they are fair or not. If not fair we'd discuss how we could make them fair. I was not present for the first two situations on the handout. We played the game diceracer which can be played with up to 6 players. The two die are rolled and their results are added. One player gets to move ahead if a 6 or 8 is rolled, another moves if 5 or 9 is rolled, another for 4 or 10, another for 7, another for 3 or 11, and the last for 2 or 12. The goal is to move 6 spots. We played a couple rounds, and as we guessed this game is completely unfair. Because a 6 or 8 has 10/36 chance of being rolled. 5 or 9 has a chance of 8/36 .... So we changed the game to the number of rolls that each player has to go. We did this inversly proportionally to the chance of rolling you number. Thus, 2 or 12 moves only 2, 3 or 11 needs to go 4, 7-6 spaces, 4 or 11-also 6, 5 or 9 needs 8, and 6 or 8 needs to move 10. This seems fair to us because it works with the probability of being rolled. So, we play and find that 2/12 is going to win most of the time. We then realize that in addition to the probabilites, the succession matters. So we consider the chance of rolling two 6/8 in a row is (10/36)^2. Haller explains how this works by showing a third game that has only two outcomes. Throw two die again, this time multiply their results. And odd product has 1/4 probability of being rolled, and a even product has a 3/4 probability. It would not work to play the game where the even must move 3 and the odd must move 1. To make it really fair, we decide that at least 3 rolls must be rolled. For the even person to win, their probability is (3/4)^3=about 42%. Therefore, the odd person has a 58% chance of winning. Overall it was a really great talk. I think that students will get really into the games and will want to figure out why games are not fair because we play before we do the math behind the game. It also offers alot of possibilities to explore and can also be good for classes where there are many different levels of ability. It'll also be good for projects because of the many possibilities.
Keywords: Geometry......
Ref: Elizabeth17
Author(s): Richgels, Amber; Lofgren, Joanie
Date: MCTM spring conference
Title: Taxicab Geometry and How it can be used in the classroom
Journal or Publisher:
Volume, Issue, Pages:
Reviewer: Elizabeth
Date of Review: 5/11/05
Taxicab is a noneuclidean geometry that, essentially is done on graph paper where you can only move on the lines (as though you are driving in a city and can only drive on roads). As a group, we went through definitions of circle, midpoint, perpendicular bisector. Everyone in the group was familiar with these in euclidean geometry. So, we took graph paper and one at a time, tried to draw these for a specific case, then looked at the generalization for taxicab geometry. For example, a circle in taxicab geometry looks like a square rotated 45 degrees. THis is because, if your radius is 4, you start at your center and go 4 "blocks". So, you can go up 4, up 3 over 1, up 2 over 2, up 1 over 3, ect. The midpoint is a bit different from taxicab and euclid because there are multiple midpoints between your two points. Basically, you'll form a rectangle around your two points and can take multiple paths to get from one point to the other, thus there are multiple midpoints. The perpendicular bisector is different also because when two comparsion points form a rectangle, the bisector will be along the midpoints, but once outside the rectangle, it will change it's angle and head (depending on which two vertices of the rectangle you bisecting) either staight up/down, or straight left/right. We found that the distance formula is |x1-x2|=|y1-y2| and that pi is 4. The presentation is hard to describe because it was so much of our own work. I did like it quite a bit though because it really tests your knowledge of definitions. So instead of thinking of a circle as circular object, you realize it is the points that are equal distance from the center. I think that it is also a good introduction and understanding of non-euclidean geometry that might be easier for younger students to understand.
Keywords: Equity/Diversity......
Ref: Elizabeth18
Author(s): Berry, Robert
Date: 2004
Title: The Eqity Principle through the voices of African American Males
Journal or Publisher: Mathematics teaching in the middle school
Volume, Issue, Pages: 10 (2) p100
Reviewer: Elizabeth
Date of Review: 5/11/05
This article starts out with a story of a gifted African American boy who was not placed into the higher math class because of his behavior, and the school pretty much ignored the fact that he was very good at math. The story illustrates how many black boys are not getting to take the challenging math courses they probably ought to be taking. NCTM has been concerned about this prblem for a while now. In fact, in 2000 they made one of their standards the equity standard. Schools should have high expectations of all their students. This means demanding alot of them. It also means teachers are motivating their students and believing in their students. It also means that teachers help thier students see math as fun and important. Students that are gifted need additional, more challenging work. And students who may not be at the same level as the class needs extra help to catch/stay up with their classmates. Another important thing for students is that teachers realize and embrace the fact that students do have different backgrounds. This is particularly true for black boys-their cultural background is likely quite different from their teachers. These students learn well in contexts that are part of their everday. Going back to the student who was discussed in the beginning to the article, Berry states that many gifted but high energy students do not respond well to classrooms. he says for these students, teachers need to accomodate by having activities, moving around, and working in groups. I think that the author is right about alot of these things. If teachers are to do what he asks, all students will benifit-they are just good teaching techniques. But as there is inequity in classes around the country, it is good for teachers to realize this when dealing with their black students (particularly the males).
Keywords: Connections, Activities...
Ref: Elizabeth19
Author(s): Bay-Williams, Jennifer
Date: 2005
Title: Poetry in Motion: Using Shel Silverstein's works to engage students in mathematics
Journal or Publisher: Mathematics teaching in the middle school
Volume, Issue, Pages: 10 (8) p 386
Reviewer: Elizabeth
Date of Review: 5/11/05
This article goes through poems written by Shel Silverstein and describes how math teachers could use them. The first poem is about a ice cream cone with 18 scoops that ultimately falls over because it was too big. Bay-williams uses this to discuss with her class how they might find out how big the ice cream cone is. Student estimate scoops of ice cream, the "smush factor" the bottom and top scoop, and the cone. Then they find an equation, verbally and mathematically to describe the heigth. The next poem is about being one inch tall. Here, we can use the poem to discuss proportions. If we were one inch tall, how tall would everything around have to be? Closet full of shoes is a poem about a closet that has alot of shoes and you are trying to find a pair. This leads easily to probability. You can even collect students shoes and draw shoes and learn about conditional probability. The missing peice is a book about a circle whose peice is missing that tries to find it's missing part. It finds peices that don't fit, but finally finds it's match. With this story, students write a want ad, or a "missing pet" type sign where they take a circle, make a missing peice, then need to describe the missing part. This article was really fun to read. I think that students would really enjoy it. Not only are the activities really cool, but tying it to Shel Silverstein, the greatest poet ever, makes it that much better.
Keywords: Connections......
Ref: Elizabeth20
Author(s): McShea, Betsy; Vogel, Judith; Yarnevich, Maureen
Date: 2005
Title: Harry Potter and the magic of mathematics
Journal or Publisher: Mathematics teacching in the middle school
Volume, Issue, Pages: 10 (8) p408
Reviewer: Elizabeth
Date of Review: 5/11/05
The article starts by describing how literature, particularly popular and current literature can engage and excite students in areas like mathematics. Harry Potter, full creativity and problem solving really is a good example of literature to use. The first quotation used is when Harry is introduced to wizard money, which clearly is different from human money. The book gives the conversions between the different types of wizard money. This clearly, lends to measurments and conversion problems in both wizard examples and human examples. The next quotation is when Harry gains freedom and can buy different types of candy. Problems regarding functions and linear equations can be made from this. After giving students the price of candy and how much wizard money Harry has, the teachers ask how much of what he can buy. After playing aroudn with it for a bit, the teachers then introduce linear modeling. The last quotation given is when three characters are placed in their respective houses, and luckily the three are in the same house. This quotation leads to a discussion on probabilities. I think that using Harry Potter in the classroom is a really fun idea. I think that the one caution that one must take is not just to use the old idea of story problems and just insert the Harry Potter language into it. Instead, really engaging in the book would make the problems more interesting thus giving students more motivation. <