Keywords: Equity/Diversity
Ref:
Seth1
Author(s): Lattimore, Randy
Date:
January 2001
Title: Gloria Hewitt: Mathematician
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 94, Number 1, pages 9-13
Reviewer: Seth
Date of Review: February 16,
2004
This article detailed the life of Gloria Hewitt, a black, female professor of mathematics, and the obstacles she overcame to achieve her status in the mathematics community. She was born to college educated parents, but had a difficult time in the public elementary and high schools. She was put into remedial math classes in her first two semesters of college, because her high school was so far behind, but after that, she succeeded very well, and went on to grad school to become an expert in the fields of abstract algebra and group theory. She is now a professor emeritus at the University of Montana.
I think this article
is a good example of someone of minority in both race and sex
succeeding in a dominantly white, male profession. She did not
have it easy, growing up in a time of segregation. However, she
was resilient, and has become a role model for many students.
Keywords: Geometry, Proof
Ref: Seth2
Author(s): Dobbs, David E.
Date: January 2001
Title: Analytic Methods in Investigative Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 94, Number 1, pages 28-30
Reviewer: Seth
Date of Review: February 17
Dobbs explains in his article the problems many students have writing proofs, specifically, the elusive first step. It often takes a very creative mind to find that first step without having seen a similar proof performed on a similar problem. Dobbs reasons this is very true in the field of Geometry, and offers some alternative methods to "classic proofs" using analytic methods and trigonometric proofs.
I liked this article, because I have often felt like the students Dobbs mentions in his article. The first step is never easy, and he is offering another way to look at things. He's not abandoning the old ways, but just giving teachers some options for different ways to approach the problem. He explains, "Illuminating the old while introducing the new is quite a balancing act. Creating this balance is part of the art of curriculum design and, ultimately, of teaching itself."
Keywords: Measurement, Communication...
Ref: Seth3
Author(s): Sherin, Miriam Gamoran
Date: 2003
Title: Video Volunteers
Journal or Publisher: ENC Online, http://www.enc.org/features/focus/
Volume, Issue, Pages: Volume 12, Number 6, http://www.enc.org/features/focus/archive/mathvideo/document.shtm?input=FOC-003402-vidvol#IDAMUBEB
Reviewer: Seth
Date of Review: February 22, 2004
This issue discussed a group of middle school math teachers who, tired of monthly department meetings, wanted to do something more to get a better idea of what was going on in their colleagues' classrooms.
They decided to begin showing video of themselves teaching, and students' learning. At first, teachers were nervous that colleagues would be judging and critiquing their teaching, but over time, they realized they weren't watching to videos to understand their colleagues' teaching, but rather, their students' learning.
Through watching these videos, fellow teachers began to get a better understanding of what was going on in their own classrooms. Teachers could communicate about students' behaviors/reactions to certain types of instruction. The teachers gained a greater understanding of student learning, and thus could foster an environment conducive to such an end.
I liked this article, and I feel the video reviews would be a great way for teachers to work together in becoming better educators.
Keywords: Assessment, Standards...
Ref: Seth4
Author(s): Maccini, Paula; Gagnon, Joseph Calvin
Date: Spring 2002
Title: Perceptions and application of NCTM standards by special and general education teachers
Journal or Publisher: Exceptional Children
Volume, Issue, Pages: v68 i3 p325
Reviewer: Seth
Date of Review: February 24, 2004
This article summarizes a study conducted in order to get an idea of teachers' perceptions related to application of, and barriers to implementation of the National Council of Teachers of Mathematics Standards with students labelled learning disabled and emotionally disturbed.
129 secondary education math and special education teachers responded to a survey regarding the NCTM Standards, and a majority of them were not even aware of the standards, although they were mostly special ed. teachers.
Most math teachers responded, saying they were aware of the standards, but encountered difficulties in implementing many activities within the standards. The greatest cause of this problem was insufficient resources.
I was shocked to hear so many teachers are unaware of NCTM Standards. It is disheartening to know there are so many ill-prepared and underresourced teachers out there.
Keywords: Problem Solving, Communication, Representations
Ref: Seth5
Author(s): Stephens, Ana C.
Date: January 2003
Title: Another look at word problems
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 96, Number 1, p. 63-65
Reviewer: Seth
Date of Review: March 30, 2004
This article reviews a study done to interpret first and second year algebra students' ability to describe an arimethic equation in the form of a word problem. Stephens classifies the results into one of seven categories: (1) Correct, (2) Operations error, (3) Direct translation error, (4) Changed unknown, (5) Solved equation only, (6) Set up content only, and (7) No response.
I was impressed the majority of both first and second year algebra students correctly interpreted the equations. However, a number of students in both cases fell into the "changed unknown" category, because they see the equals sign as an indication of an "answer" forthcoming, and not a sign of equivalence.
The study suggests an increased focus on language and the
definitions of operations in students' elementary years. I would agree
that language is an integral part of understanding.
Keywords: Algebra, Activities...
Ref: Seth6
Author(s): Burke, Maurice; Erickson, David; Lott, Johnny
W.; Obert, Mindy
Date: 2001
Title: Expanding the notion of function representation
Journal or Publisher: Navigating through algebra in
grades 9-12
Volume, Issue, Pages: chapter 3, pages 26-29
Reviewer: Seth
Date of Review: March 30, 2004
This chapter suggests using recursive or iterative equations to represent relationships. From there, students may approximate and interpret the rates of change based on numerical data, and draw conclusions about the situation being modeled.
It gives some good examples of recursive formulas, and discusses the Babylonian method of finding the square root of two. The lesson also encourages visual representations of the numerical data.
I think the lesson plan ideas are great for students having trouble
visualizing algebraic equations, and it's a nice lead-in to the study
of functions. Students are learning problem solving and
interpretational skills, as well as an understanding of the concepts of
algebra.
Keywords: Activities, Planning, Teaching Strategies
Ref: Seth7
Author(s): Gerver, Robert K., and Sgroi, Richard J.
Date: January 2003
Title: Creating and Using Guided-Discovery Lessons
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 96, Number 1, p.6-13
Reviewer: Seth
Date of Review: April 1, 2004
This article discussed the effectiveness of guided-discovery lessons. These lessons are great, because students come to discover the "punch line" of the lesson, rather than it being served like fast food.
Writing guided-discovery lessons can be quite difficult, and it indeed takes a skilled teacher to develop an effective lesson, however, many new teachers should not shy away from using such lessons. It is recommended that they begin with collegues' guided-discovery lesson plans, or textbook guided-discovery lesson plans. And when they do begin writing their own lessons, it is helpful to "test" them on a relatively naive friend or collegue. In this way, teachers may find problems with the lesson ahead of time, and smooth them out.
I liked this article, because I think guided-discovery is great
method for getting students to retain concepts. I'm sure I'll have to
write some of these lesson plans in the future.
Keywords: Number and Operation, History, Technology
Ref: Seth8
Author(s): Not listed
Date: Not listed
Title: PBS TeacherSource
Journal or Publisher: PBS
Volume, Issue, Pages:
http://www.pbs.org/empires/romans/classroom/lesson3.html
Reviewer: Seth
Date of Review: April 4, 2004
This "article" was actually a series of lessons designed for 3rd to 5th graders, on Roman numerals. Students find examples of Roman numerals in the classroom, in their homes, on TV or the Internet, and they're asked to figure out their corresponding Arabic numerals.
Students then change examples of Arabic numerals (ie, pages in their textbooks) to Roman numerals.
Students try performing some basic operations on Roman numerals (adding, subtracting), and they discuss how life would be different if they had to use Roman numerals instead of Arabic. They also discuss the benefits of our Arabic, base 10 system.
This article points out that although there isn't a lot of use for Roman numerals these days, it is important for students to understand different representations for equal values. An understanding of Roman numerals can help students understand a binary number system, or a base-12 number system.
Keywords: Activities, Management, ...
Ref: Seth9
Author(s): Boyer, Kimberly
Date: September 2002
Title: Using Active Learning Strategies to Motivate Students
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 8, Number 1, p. 48-51
Reviewer: Seth
Date of Review: May 28, 2004
This was a great article about active learning strategies, and how they can motivate students. Ms. Boyer spent years trying to make her lessons more exciting by supplementing them with fun activities, but she realized something was still missing. After reading Merrill Harmin's *Strategies to Inspire Active Learning* she realized she wasn't building a strong classroom community. Building a better community promotes a better learning environment in which students feel comfortable, respected, and motivated to learn. She found five keys to building this better community.
First, the students must find some dignity. They need to have confidence, feel secure, and see themselves as valuable. Second, they need an energetic classroom, in which students are active and engaged. Third, students need to build self-management skills. They need to have self-discipline, make appropriate choices, and want to work. Fourth, students need to cooperate. Sharing of ideas and interdependence support everyone, and help to build community. Fifth, students need to be aware. They need to focus, listen, think, observe, and evaluate.
Ms. Boyer noticed almost immediately the positive change in her classroom. Students commented on the positive changes, and after two years of implementing these five keys, she has yet to hear "This class is boring" or "When are we going to use this in real life?" Ms. Boyer's lesson is that becoming a great teacher involves a lot more than just fun activities. Teachers should set goals for themselves and their students, and one of these goals should be a positive classroom community.
Keywords: Algebra, Planning...
Ref: Seth10
Author(s): Johnsen, Connie and Wilkerson, Trena L.
Date: October 2003
Title: My Jouney Toward a New Slant on Slope
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 96, Number 7, p. 504-508
Reviewer: Seth
Date of Review: April 20, 2004
This article is about a woman's continually evolving lesson/unit plan on the concept of slope. She discusses the trouble she had with a direct lecture approach, and students repeating her class, or passing but with a poor understanding.
She has eventually come to a very hands-on approach to the concept. She begins with describing the concept through pictures. Even before the students start working with numbers, they're interpreting graphs and writing stories to go along with them, such as "The Tortoise and the Hare." Then they begin graphing data themselves, and developing the slope formula, as well as the slope-intercept equation. Finally they come to an understanding of the variables, and have a basis for the concept of functions.
This particular woman has spent 14 years refining her unit on slope, working hard to use current, real-world examples to motivate and interest students. She has also moved towards more modern approaches to evaluating her students' progress.
Keywords: Geometry, Proof...
Ref: Seth11
Author(s): Gole, Andy M.
Date: November 2003
Title: Sherlock Holmes, Geometry Proofs, and Backward Reasoning
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 96, Number 8, pages 544-547
Reviewer: Seth
Date of Review: April 20, 2004
This was a great article about backward reasoning and it's application to proofs, especially geometric proofs. Often times, students' toughest task in proof writing is starting the proof. They have a terrible time looking at a blank sheet of paper and finding that critical first step.
This article suggests working backwards--figure out the step that finishes your proof first, and then figure out what step is necessary to complete that last step, and so on and so forth.
The article also discusses the value of backward reasoning in real-life situations. Suppose you're expected to develop a new product and this product must meet certain criteria; you must work backward to figure out if this hypothetical product is even possible to produce.
I really liked this article because backward reasoning relates to creating a lesson plan, or a unit plan, or a curriculum as well. In all cases, we start out with certain criteria that need to be met--usually state standards, or district standards, as well as concepts we find to be the most essential--and we construct our lessons based on these criteria. Backward reasoning, although it shouldn't always be depended upon, is a very effective means to finding an end.
Keywords: Probability, Statistics...
Ref: Seth12
Author(s): Teppo, Anne R, and Hodgson, Ted
Date: 2001
Title: Dinosaurs, Dinosaur Eggs, and Probability
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 94, Number 2, pages 86-92
Reviewer: Seth
Date of Review: 5-3-04
This was a great article on the use of real-world problems in the classroom. It took the same problem, and broke it into three difficulty levels, each one applicable for different grade levels.
On the surface of the problem, the measurements of the dinosaur eggs provided plenty of interesting material for junior high aged students, to give them an introduction to the idea of probability and statistics. On a deeper level, sophomores and juniors could ponder some other questions, including if the nests found were truly random samples of nests of a singular species. Finally, seniors and college level students could really delve into the problem, and program technological re- creations of the nest, and the placing of the eggs, to get a much better idea of the likelihood, and significance of the evidence.
This article is great because it looks at a very interesting real-world problem from many different perspectives, yet each one gives a slightly better understanding of the central concepts behind probability and statistics.
Keywords: History, Probability...
Ref: Seth13
Author(s): Kiernan, James F.
Date: March 2001
Title: Points on the Path to Probability
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 94, Number 3, pages 180-183
Reviewer: Seth
Date of Review: 5-4-04
This article discusses how many probability textbooks open with a story of "how the Chevalier de Mere approached Pascal with the problem of points." The article explains that it is important for students to understand that mathematical reasoning is not always the result of a single brilliant idea, but often a long, cumulative process of successes and failures.
This problem, apparently first associated with the Arabs, was tackled by many great thinkers of the 15th, 16th, and 17th centuries. It involved a game in which 60 points are required to win, but supposing the game must end before someone has scored 60 points, and the opposing teams have scored 50 and 30 points, respectively, what share of the prize money belongs to each side?
The article goes on to discuss many incorrect solutions to the problem, and how
these incorrect solutions helped lead to the correct solution, devised by both
Pascal and Fermat. The story is a shining example of how much mathematical
reasoning is derived.
Keywords: Proof, Activities
Ref: Seth14
Author(s): Craine, Timothy V., and Rubenstein, Rheta N.
Date: April 2000
Title: Traveling Toward Proof
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 93, Number 4, pages 289-291
Reviewer: Seth
Date of Review: June 23, 2004
The authors of this article discussed the problems they had encountered when trying to teach proof-writing to students. It's a very important skill, but students had trouble bridging the gap between their logic and their writing. However, the authors devised an activity which they've discovered works as a great metaphor for proof-writing.
The authors created Aristotle Airlines, which flies all over the country. Travel must originate at the starting city (the given information) and arrive at the ultimate destination (what is to be proved). And a city (a statement) that links two points cannot be skipped. For example, a flight from New York to Los Angeles must go through Chicago. The authors note that there are some drawbacks (all flights are one way, thus, the traveller must be going in the right direction). However, the metaphor may also encompass if and only if statements, using two-way flights.
The activity overall has been a very positive experience, supporting students' writing of two column proofs, and acting as a stepping stone towards paragraph proofs. Students make more sense of proof-writing with this analogy, and tackle the problems with much more enthusiasm.
Keywords: Puzzles, Communication, Planning
Ref: Seth15
Author(s): Copes, Larry
Date: April 2000
Title: Messy Monk Mathematics: An NCTM Standards-Inspired Class
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 93, Number 4, pages 292-298
Reviewer: Seth
Date of Review: June 23, 2004
This was a great narration of a teacher's thoughts during a lesson. The article itself is just an account of a (probably fictionalized) classroom, where students are asked an interesting question regarding a monk with an odd habit. On the last day of every month, he walks to the top of his small mountain, being sure to leave exactly at sunrise, and arriving at the top exactly at sunset. Then the next morning, on the first day of every month, he walks back down the mountain, being sure to leave exactly at sunrise, and arriving home again exactly at sunset. The students were asked if there was necessarily a point on the path up and down the mountain that the monk crossed at the same time on both days. The article narrates all the quick decisions the teacher must make in posing such a question, and trying to evoke some critical thinking out of the students, especially when there's a jokester in the class, and a student that knows the answer right away, and many students that seem disinterested in the problem. The teacher is able to get students to listen to one another, and support their arguments, and eventually come to a logical conclusion, but the teacher doesn't end the thinking. He asks more questions and relates what they've done to mathematical research.
The class period described in this article is a great example of putting a lot of teaching theory into practice. It gives me an idea of what to prepare for when teaching an inquiry-based lesson.
Reflecting after a week: I actually read this article last week, but it's still pretty vivid in my memory. The 'article' is actually a story told about a guest teacher in a high school mathematics classroom, who poses the students a problem about monk with a very interesting habit.
On the last day of every month, this monk walks a path from his house, which is at the base of a small mountain, to the summit of the mountain. He is very careful to leave precisely at sunrise, and arrive precisely at sunset. Then on the next day (the first of every new month), the monk walks down the mountain, back home. Again, he is very careful to leave precisely at sunrise, and arrive home precisely at sunset. The question the students were asked, was "Is there necessarily a point on the path that the monk crosses at the exact same time on both days?"
I really liked this article, because it accounts all the students' conversation, but it also accounts everything the teacher is thinking. We get to see all the quick decisions he's making, and understand his motives for just about every action he takes. The teacher is very careful not to dominate the class, but he is also mindful of what's going on. He gently leads the students in a direction, but they must figure out their own arguments; he won't give them a "right" answer.
In this article, the teacher must deal with a class he's never taught before, including a class clown, some lazy/sleepy students, and one that knows the answer to his problem immediately. However, he gracefully meets these challenges, and the students have a productive, interesting discussion, and as a group, move toward the correct solution. The article is a great example of teaching methods put into play in a classroom. It's given me a different perspective from an observer's point of view. I've got a little better idea of what to watch for.
Keywords: Connections, History, ...
Ref: Seth16
Author(s): Natsoulas, Anthula
Date: May 2000
Title: Group Symmetries Connect Art and History with Mathematics
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 93, Number 5, pages 364-370
Reviewer: Seth
Date of Review: June 28, 2004
This article presented many pictures of ancient drawings, paintings, and carvings of the people of Cyprus and Ethiopia. The art was amazing, and it also demonstrated rotational and reflective symmetry. The article discussed lesson ideas working with the artwork as a joint history and mathematics lesson.
I thought this article was significant, because it connects two very different areas of study. The lesson options offer new, intriguing alternatives to fuel students' enthusiasm, and the concept of looking at artifacts of ancient history to investigate mathematical concepts has great potential as an entire unit.
Keywords: Activities, Connections...
Ref: Seth17
Author(s): Fernandez, Maria L.
Date: February 1999
Title: Making Music with Mathematics
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 92, number 2, pages 90-95
Reviewer: Seth
Date of Review: July 16, 2004
This article was about an activity that could be jointly held in mathematics and a physics classroom. Students were asked to play "Mary Had a Little Lamb" or "La Cucaracha" with 20 ounce plastic bottles. In order to find the correct pitch, students needed to be able to measure the frequency of the musical note they were playing. By setting up a CBL (Calculator-Based Laboratory), a CBL-microphone, and a TI-80-something calculator, students can record data right into the calculator.
Based on their data, students may find their frequency, and thus alter the water amount in the bottle to find the correct frequency of the note they're trying to play.
The lesson may experiment with different bottle sizes and lengths, as well. And students will gain a much greater understanding of sine functions, and their relation to the physics of sound production.
Keywords: Geometry, Activities...
Ref: Seth18
Author(s): Naylor, Michael
Date: February 1999
Title: The Amazing Octacube
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 92, number 2, pages 102-104
Reviewer: Seth
Date of Review: July 16, 2004
This article was written by a teacher who sort of accidentally stumbled upon a great activity for exploring polyhedral duals. He was teaching about properties and relationships between the Platonic polyhedra, and he decided to join along in creating something for the final project. The teacher constructed a model of an octahedron inside a cube, and by fastening a hinge it could be "turned inside out". The resulting shape was a cuboctahedron.
The images in this article do an excellent job of helping explain the relationships between these polyhedral duals, as does a table listing the number of faces, corners, and edges of the regular polyhedra. The article also gives explicit instructions on how to build your own "octacube".
This article provides ideas for a very interesting lesson investigating the properties of the regular polygons, and the new properties of the shapes created when you "combine" the polyhedral pairs.
Keywords: Proof, Connections, Algebra
Ref: Seth19
Author(s): Manaster, Alfred B., and Schlesinger, Beth M.
Date: February 1999
Title: Geometry Problems Promoting Reasoning and Understanding
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 92, number 2, pages 114-116
Reviewer: Seth
Date of Review: July 16, 2004
This article focuses on the observation that students do not work with much proof writing until Geometry. In order to enhance students' reasoning skills, Manaster and Schlesinger offer these four problems, which may be offered in an Algebra 2 class, to familiarize students with the concept of a chain of reason.
The first two problems deal with students finding the dimensions of a rectangle with a perimeter of 30 inches that has the largest possible area. Students may guess the shape must be a square, but they must algebraically prove any theories. The third and fourth problems deal with the relationship between the perimeter of a square and the circumference of a circle. Again, all solutions require proofs, and the problems are staggered throughout the term, so as to reflect students' knowledge gained.
I think this is a good article, because it stresses the many relationships and connections between different fields of mathematics. Many skills are useful, and many concepts are found in multiple areas of mathematics, and that is what this article is getting at. Students should not simply associate proofs with geometry; it is a skill useful all over.
Keywords: Activities, Calculus, Management
Ref: Seth20
Author(s):
Date: February 1999
Title: The Life Expectancy of a Jawbreaker: An Application of the Composition of Functions
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 92, number 2, pages 125-127
Reviewer: Seth
Date of Review: July 16, 2004
I really liked this article, because the author (I misplaced his name) started it by posing questions he had had trouble with his first year of teaching Algebra 2. "How can I teach practical applications for abstract ideas?" and "How can I make students understand that applied problems often require techniques from several branches of mathematics?" I'm sure these are questions I would be asking myself during my first year.
The teacher was trying to teach the concept of composition of functions to a class of upperclassmen, but for some reason students were not motivated by the rational that this concept was essential for differential calculus.
So, the teacher came up with an activity involving jawbreakers. Students would record the surface area and volume of the jawbreaker at time intervals, monitoring its changes, and graph their results. This also required students to construct volume and surface area functions for their jawbreakers, to interpret their data. Students overcame some small obstacles in figuring out how to actually measure the jawbreaker, but in the end their work paid off immensely.
I'm always thankful to run across these specific little articles that deal with some difficult questions, but have some great activities to counter them.