Keywords: Activities, Connections,
Teaching Strategies
Ref: Mark1
Author(s): Davidson, Doris; Keller, Rod
Date: 2001
Title: The Math Poem: Incorporating
Mathematical Terms in Poetry
Journal or Publisher:
Mathematics Teacher
Volume, Issue, Pages: 94(5), p.
342-347
Reviewer: Mark
Date of Review: 15
February 2004
"The Math Poem" is about two teachers, one English and one mathematics, and their joint project. They had their students write poetry, incorporating mathematical terms in the poems. There were different requirements for students in advanced classes, but the general idea was the same. This project came at the end of the school year, so students had a large word bank of mathematics terms to use. Topics were mostly left open to the students; to prevent 'I hate math' poems, even from students who enjoy mathematics, they could not write about mathematics or English class. This stipulation also meant that students would have to think about the terminology in a different way.
The end result was surprising. Some of the poems showed minimal effort, but many were thoughtful and enjoyable. Use of mathematics words was sometimes forced, but there was some obvious consideration too. The topics varied widely, as did style. Some rhymed, most didn't. There were poems about basketball and love. One example printed in the article dealt with lofty ideas of living fully in the face of death. The majority of the students enjoyed the activity. Having some type of constraint (use mathematics words) was helpful for some, a hindrance for others. Nearly all talked about how surprised they were at their poems.
This is something I would have enjoyed in high school. I would like to befriend an English teacher when I am working and try this idea out. Thinking mathematically about non-math subjects is a valuable skill, I believe, and this project is a step in the right direction. The value of interdisciplinary work is well documented, but, as the article points out, 'interdisciplinary' mostly means mathematics and science or English and history. Throwing English with mathematics is always a good idea. Just look at Professor Zorn.
Keywords: Proof, Teaching Strategies...
Ref: Mark2
Author(s): Knuth, Eric J.
Date: 2002
Title: Proof as a Tool for Learning Mathematics
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 95(7), p.486-490
Reviewer: Mark
Date of Review: 23 February 2004
Article discusses the role of proof in mathematics education at the secondary level. While we all likely agree that using proof in the classroom is a useful teaching tool, Knuth elucidates the reasons why this is important. Interestingly, Knuth argues that this is one of the major reasons to teach using proof – it helps students understand the 'why' something is correct, rather than simple right or wrong. Another point Knuth makes is that mathematics and proof go hand in hand (think peas and carrots). Without proof in the classroom, mathematics education is incomplete.
There are often many methods of proving the same conjecture, some of which are more enlightening. In order for an educational use of proof to work, these must be used. Not only will these proofs be easier to understand for student, but they will likely also help grasp the technique and mindset that accompany proof. Knuth warns against using excessive, pedantic, uninspiring and untrue proofs, which seems like a good idea to me. Many examples are presented, showing what a 'better' proof might look like.
Using proof in the classroom appeals to me. There were many excellent points arguing why its use should be more widespread. However, nothing was said about student attitude when working with proof. I didn't have much experience with proof in High School, but what I did have wasn't positive. Proof was a chore, and difficult to do well. I dreaded it. And I believe students still have similar opinions of proof. However, this could be remedied – starting with how proof is taught. In fact, if taught well, I could see student attitudes turning 180 degrees. Proof could be fun.
Keywords: Geometry......
Ref: Mark3
Author(s): Basden, Jon
Date: ???
Title: The Derivation of Pi
Journal or Publisher: Highland Middle School, Madison, IL
Volume, Issue, Pages: http://www.highland.madison.k12.il.us/jbasden/lessons/pi_3_14159265358.html - via mathforum.org
Reviewer: Mark
Date of Review: 24 February 2004
This lesson plan is geared for a seventh grade classroom and supposedly takes two forty minute class periods. The main objective is for students to 'discover' the value of pi through measurement of circular objects and calculation of proportions (circumference/diameter). Listed under "Goals" are standards for the state of Illinois (eg: Select and apply instruments including rulers... and units of measure to the degree of accuracy required.). After students make their measurements and calculations, as a class they plug their numbers into an Excel spreadsheet to show the class data. Finally, the teacher is to guide students through derivations of formulas using the class' assumption that circumference/diameter=pi (C=pi*D, C=2*pi*R, etc.)
I am not sure how I feel about this lesson – mixed feelings, I guess. There are some very positive aspects in this lesson. First of all, the focus is on discovery learning. Time and again, we have seen this constructivist approach to learning is often best. Also, the students are active during the lesson. They are to measure objects, calculate proportions of their measurements, and then compare with their classmates'. This comparison is accomplished through the use of technology, another positive aspect. The teacher is to show the class wide data in an Excel spreadsheet, calculating proportions and displaying outputs.
In spite of the positive characteristics this lesson possesses, there are some negative points that seem nearly irreconcilable. According to the plan, this lesson takes two class periods. I have a hard time seeing how this lesson will last for one of the periods, let alone two. This lesson doesn't seem deep enough to fill that amount of time. Also, there doesn't' seem to be very much room for enrichment of the lesson. There are some attached websites that were interesting, but that would involve the students leaving the room and going to a computer lab. Mostly, this lesson isn't my favorite because it does not mention any deeper mathematical skills.
I am sure there's more to say, but that's all for now.
Keywords: Standards, Representations...
>Ref: Mark4
Author(s): Schultz, James E.; Waters, Michael S.
Date: 2000
Title: Discuss with you Colleagues: Why Representations?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(6),p.448-453
Reviewer: Mark
Date of Review: 29 February 2004
This article, rather than answering the title's question, discusses the idea of using multiple representations of a single problem to elicit different types and degrees of understanding. At first, I wasn't entirely sure what the term 'representation' meant, but after a few examples, I think I have a pretty good idea. A representation is any depiction of a problem or mathematical concept. Through a typical example problem (solving two linear equations in two unknowns), the article offers 5 different representations that all lead to the problem's solution but in unique ways. The analysis about this specific question gives way to a discussion about how to choose which representation is best. Authors put forward 6 criteria to help the teacher decide: 1: Which representation best promotes conceptual understanding? 2: Best generalizes to higher-level mathematics? 3: Best applies to finding approximate solutions? 4: Best applies to finding exact solutions? 5: Best for a given type of technology? 6: Best suits the learning style and comfort of the student? This last question is obviously the most important one to consider when planning lessons and units.
Really a nice article. The information was generally straightforward, but it was helpful to see it spelled out. This topic is one that we all easily understand, and it is easy to agree with the authors, but we seldom think about (or I seldom think about, at least). One question that was raised in the article was 'which skills are more important and which are less important, given your decision about representations?' These types of questions are especially helpful for me to remember - I have trouble looking for different ways to solve problems after I have already solved them. Something to keep in mind.
Keywords: Technology, Geometry...
>Ref: Mark5
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.392-397
Reviewer: Mark
Date of Review: 29 February 2004
This article is broadly concerned with the use of technology in the middle school classroom and specifically about the use of The Geometer's Sketchpad to help students understand the concept of transformations in geometry. The author had eighth-graders work on three activities. The first two involved two figures, a preimage and its translated image along with sketched vectors indicating the translation. Students could change the direction and size of the vectors and observe the corresponding change in the image. The author noted that these activities had the students focused on the paths of the objects. The goal of the third activity was to help students think about translations as "more than the path of a shape," and instead focus on the outcome and thereby understanding translations as a relationship between vectors independent of shapes. To that end, the third activity held the preimage and the image fixed in place. The students would then manipulate the vectors of the set - one change to this vector changes this other vector, so that the image is always the same. I am not explaining this very well, but the article uses many illustrations, which make understanding these activities much easier.
This was a very interesting article. First, this distinction between outcome and path of translations was one that I would not have made - mathematically enlightening. Second, the use of technology to help students avoid this misunderstanding, I thought, was original and creative. I would only hope to be so innovative when I am a teacher (I'll try hard). Finally, it was good for me to read about middle school mathematics. I haven't had much experience lately in the middle grades, and the little experience I've had was not that great. This was a refreshing refresher on the sixth-eighth perspective.
Keywords: Teaching Strategies, Statistics, Planning
>Ref: Mark6
Author(s): Rumsey, Deborah J.
Date: 1999
Title: Cooperative Teaching Opportunities for Introductory Statistics Teachers
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(8), 734-737
Reviewer: Mark
Date of Review: 4 March 2004
The article discusses the author's colleagues (college level) efforts to restructure an introductory statistics curriculum in light of mathematics reform. The courses were changed so that all teachers would cover roughly the same material, with the same basic philosophy of the importance of statistics and the need for statistical knowledge. There were 6 overarching themes: data awareness, variation, sampling, decision making, scientific investigation, and relationship between variables. Cooperation was a major theme, with the creation of a curriculum to weekly meetings to a common resource file.
While the statistics information is quite interesting, the main point of the article is this: reform requires support. Quotation that sums this up well: "As...teachers undertake changes toward educational reform, they need a strong, supportive environment in which they can learn more about [their subject], pool their efforts, share their resources, and try out new ideas."
Related story: I remember when I was observing at Rosemount High School, January 2003, there was a staff development meeting one day after parent-teacher conferences when students had the day off. I attended that meeting - very good idea. During the morning, the principal presented and discussed data from a survey the faculty had taken at the last staff development day. There were a few key issues, one of which was improving collegiality. They were going to implement a system where each department would meet more often. Collegiality is key to quality education, said the principal.
Keywords: Problem Solving, Probability,
Teaching Strategies
Ref: Mark7
Author(s): Kahan, Jeremy A.; Wyberg, Terry R.
Date: 2003
Title: Problem Solving Can
Generate New Approaches to Mathematics: The Case of Probability
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 96(5),p.328-332
Reviewer:
Mark
Date of Review: 14 March 2003
This article focuses on "the World Series problem," a variation of the Problem of Points. Here's the situation: the Yankees and the Mets are playing the world series (best of 7, hypothetical $1,000,000 prize money). Yankees are up 2 games to 1 when an umpire's union strike cancels the rest of the series. How do they split up the pot? Specifically, this article discusses three methods of solving the problem: the Monte Carlo method, using trees, and generating functions. The roots of probability theory are sunk into this type of problem, exploring binomial distribution, Pascal's triangle and more. Because of the problem's richness and importance, students connect the different representations for the situation (Monte Carlo, trees, etc.) while repeating the "historical development of the mathematics of probability."
Ahh…probability-it's stupendous! I really like this problem. In fact, Professor Roback used the same type of problem in the beginning of last semester in Probability Theory to introduce probability concepts. It was very effective, I thought. Using the history of mathematics to engage students (at least college mathematics students) puts things into a helpful perspective. For this reason, I didn't like the article so much. They used a Problem of Points- type problem, but not the actual Problem. Pascal's triangle coincides so well with this type of problem because Pascal used it to grapple with this very problem. Why not have students do the same? Authors discuss ways to make the problem more appealing to students (they suggest some strange rewording of the problem that has to do with a frog that hops only south and east on its way to a river), but I contend that the actual problem could be the most broadly interesting.
At any rate, I believe this is the best way to learn mathematics. A problem like this, with so many opportunities for extension and generalization, gets students thinking about how to solve problems, about using mathematics to achieve a goal. This both demonstrates the power of mathematics, and gives students power over mathematics.