Keywords: Technology, Connections
Ref: TomD1
Author(s): Picciotto, Henri
Date: 1996
Title: Make The Designs
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 89(5), p.424-427
Reviewer: TomD
Date of Review: 01/29/2000
This article is about using technology, in this article a basic graphing calculator, to make connections between the graphs of functions and that functions parameters. This article, particularly uses the function of the form y = mx + b and connections it has if its parameters (x and b) change. Students will be given a dozen descriptions of functions, all different, in which they must come up with an equation in the form y = mx + b.
After this, the students for each function are to change the parameters, x and b, and record how their new function differs from the original by the changes that they made. For maximum results and student understanding the students must work unguided from the teacher, allowing them to set their own path through the activity and arise questions among themselves on why a function behaved a certain way when the parameters changed. In order for maximal success, the teacher must interact with each group and assist them on pulling out some conclusions about the parameters and relationship with the graphs. For instance, if the students haven't noticed yet that the b in the equation is the y-intercept, then the teacher must pose questions to the students that will help them see this correlation. Another important aspect of this activity is it works for all level of students. There are close to a dozen equations students are asked to come up with and examine, with a couple whose equations could be very difficult to find. This is great for all level of students because there are hard problems for the higher students, and plenty of others for the average level students.
I think the first sentence of the conclusion summed up my feelings, "Technology presents an excellent context for the reversal of standard tasks, which yields powerful educational benefits". I like this activity because it makes use of technology that every high school age student has access to (a graphing calculator), and it would be a very popular activity among students because (a) they aren't doing the every day thing out of the book, and (b) they are allowed to learn at their own pace and on their own path, they are allowed to do their own thinking, evaluations, and questioning of what is going on.
Keywords: Geometry, Curriculum
Ref: TomD2
Author(s): Gregg, Jeff
Date: 1996
Title: The Perils of Conditional Statements and the Notion of
Logical Equivalences
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 90 (7), 48 - 54
Reviewer: TomD
Date of Review: 02/06/2000
In this article, Jeff Gregg, talks about how the traditional two column proofs in no way promotes any mathematical thought processes. He believes that before attempting a proof students need to make conjectures, and justify their arguments through constructing mathematical arguments (in particular conditional statements). For example, the author mentions the use of sketchpad and doing activities very similar to what we did. He suggests that the students should be given a problem and attempt drawing it out. From this they need to make some conjectures about their figure and then use conditional (If ., then .) statements that they are trying to validify or will show are untrue. The students will then use the sketchpad to support or deny their conditions. At this point, the students haven't even started a proof yet. However, by conjecturing, making conditional statements, and supporting or denying those statements these students are going to have a better understandi! ng for this problem now than if they were just doing a two column proof. The author's strong feelings on conjecturing and using conditional statements are summed by this following quote, "Conditional statements and the notion of logical equivalences should not be introduced as individual and isolated topics with no apparent connection to proving but rather should be taught to arise the students efforts to construct valid arguments that will justify their conjectures and communicate their reasoning to others".
Reading this article, I could relate to the author on how he feels that the traditional geometry courses and two column proofs never call for "real" thought to go on by the students. For my practicum this fall, I observed two geometry classes that still taught from the traditional textbooks and were doing two column proofs. Like the author of this author states, the students learned virtually nothing. They were simply regurgitating information, and really had no reasoning or conjecturing about what they were doing. As I would help the students, I would try to pose questions that would make them think about what the problem asked and what would help them. And just about every student said, "We don't need to know that much information, all we need to know is a definition or theorem as a reason for that statement. " It's crazy to think that this is what the students are taking away from their geometry class.
Keywords: Activities, Problem Solving
Ref: TomD3
Author(s): Gonzales, Nancy; Fernandez, Albert; Knecht, Corine
Date: 1996
Title: Active Participation in the Classroom through Creative
Problem Generation
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 89 (5), 383 - 385
Reviewer: TomD
Date of Review: 02/09/2000
This article focuses on how a mathematical classroom must change the role of students from being passive recipients of mathematical problems to becoming active participants. However, in order for such a task to occur then we as teachers must structure activities within the classroom that will involve all students, provide opportunity for mathematical communication, and link in creative thinking with mathematical content. In this article they describe one such activity, kind of like the game "pass it along". What happens is a designated group makes a mathematical phrase and puts it on the board. The next group must then add an additional phrase to the original, and so on until the final group who then brings closure by summing up the entire phrase. Now, where the inquiry and conjecturing comes in is that while the problem generation is going the other students are commenting on whether or no the phrases fit into the whole overall idea or problem. Then in each of their own groups they follow the four step method of problem solving: (1) Understanding the problem - here the students ask there group "What exactly is the problem?", or "Is this a problem at all?"; (2) Deciding a plan - they inquire with each other some ideas on how the problems could be solved and then the methods that they might use to solve them; (3) Carrying out the plan - at this point they must verify their solution or no solution either by a formal or informal proof, a diagram, a theorem or a postulate; and finally (4) Looking back - each group must evaluate what they did and discuss any changes they would make, and then the entire class will reflect on the problem and discuss maybe how it could be presented differently.
I really like this idea because it incorporates more than one area of mathematics to accomplish. The students are first of all working with word problems, and trying to understand what the problem is. Then each group must decide themselves how they would answer the question, and venture on their own to either proof or disproof their conjectures. Then they must reflect on what they did, listen to what other groups did, and then make an overall evaluation of what they did. Students will learn so much about communication, inquiry, conjecturing, and reasoning that hopefully they will see how useful it is and use it when they work alone on other projects in the future.
Keywords: Connections, Geometry,
Ref: TomD4
Author(s): Dodge, Walter; Goto, Kathleen
Date: 1998
Title: "I would consider the following to be a proof . . . "
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91 (8), Soundoff
Reviewer: TomD
Date of Review: 02/16/2000
This article appeared in the Soundoff section of that Mathematics Teacher in 1998, and it pertains to how important proofs are to geometry but can also but equally important in the other areas of mathematics. It talked about how in prealgebra and algebra classrooms students are first introduced to power series such as 1+2+3+4+....+n=n(n+1)/2. However, how often do you see an algebra or prealgebra teacher proving this. Better yet, how often do you see a teacher asking students to think about this equation and determine if in fact it is a valid statement. Another example from an algebra classroom is the binomial (a + b)2 = a2 + 2ab + b2. How many students in algebra actually know why that is true? Or, how many think very little about it and accept it as fact. This article talks about how geometry (and the point of proofs) attempts to make students look at mathematics and the things they are doing in a new and different manner. Geometry allows students to open their minds, and begin a new reasoning process in mathematics. However, why does this reasoning process be restricted to geometry? We need to have our students reason and rethink about things in all areas of mathematics because that is how our students will learn and be able to apply.
I definitely had some mixed reviews about this article. Primarily I supported the point that too often we as teachers make a statement and without making our students think about it, let our students automatically think it is true. I remember this from my algebra class and never really knew why many laws and rules were true. I was told it, and it worked so I took it as fact. There was no conjecturing, reasoning, or thought process about it. However, one aspect of this article I disagreed with (so I didn't mention above) was the emphasis they put on technology. They talked about how geometry students should be able to use Sketchpad to do proofs. How does that show any thought process or knowledge? They might be able to use a program by fooling around with it, but do they know or understand what justifies what the computer is doing? This is an example where I believe that they are trying to use technology as "the answer", rather than as "a tool".
Keywords: Teaching Strategies, Assessment,
Ref: TomD5
Author(s): Gronseth, Phillip
Date: 1999
Title: Course Diary: A Valuable Information Source
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92 (6), 496 - 497
Reviewer: TomD
Date of Review: 02/16/2000
This article talks about the importance of using a diary, as a teacher, to write down thoughts, feelings and possibly new ideas. The diary is especially useful for new teachers to write down what they did, what worked, and what miserably failed that was thought initially as a brilliant idea. For maximum benefits you should have one journal for each subject that you teach. For example, during my student teaching I know that I have an algebra and a prealgebra for sure. So right now I am planning on having two exclusively separate journals. The diaries should also contain any handouts, quizzes, midterms, chapter tests, or worksheets. Anything that you did with the students; whether they needed turn it in or not you should keep a record of it. This article gave a few good reasons why, as a teacher, this would be an exceptional tool. First, it's an excellent tool for teachers to assess the job they did in the classroom. To many times the only assessment we ever hear about is for the students, well isn't just as important that the teachers are assessed on the job that they do! Secondly, this allows teachers to document on student performances; whether it is used for assessment of those students or to make the course better for the students of the future. Finally, it can be used a useful tool to assess the effectiveness of the course itself or the version of text that the school was using.
Until this class this quarter, I never had to keep a journal on anything. For me, I like the journaling that we are doing because it allows me to reflect on what I'm doing and to make changes or modifications to my approaches in my practicum. This is one reason that this article jumped out at me when I was reading it. I was considering to keep a journal during my student teaching to help me assess my teaching as a tool. Reading this article there is no doubt I will be journaling, in addition putting in anything that I did with the class. It will not only help me as a teacher, but would also be an eye opener if I walked in with that for a job interview !!!
Keywords: Games, Connections, Teaching Strategies
Ref: TomD6
Author(s): Johnson, Carl
Date: 2000
Title: Human Coordinates and Floor Tiles
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93 (1), 13
Reviewer: TomD
Date of Review: 02/16/2000
In this article the author, Carl Johnson, talked about how he used his classroom floor as a coordinate system and the students as points in that system. What he did was assign certain students as coordinates in each of the four quadrants, representing the four vertices of a square. Now, the other students no apart of the square had to answer questions about the length of each side and the length of its diagonals, and the area of the figure. Then they had to verify there work by actually using tape measures and measuring the distances. The students then had to make transformations of the "student figures". For instance, they could do a reflection over any given line. Or, rotate a certain number degrees about a vertice (one of the students representing that point), and even translate the figure in a particular direction. From this he had the students break up into groups (I believe he called them teams) and make up two or three transformations they could do, and then they physically had to do the transformations themselves on the floor (coordinate system).
The main reason I enjoyed this article is that I like to read about how teachers have done different things in the classroom to keep student awareness high. This activity allows students to see the mathematical connections, and inspires the students to have fun and that they can be more creative in an ugly mathematics classroom.
Keywords: Activities
Ref: TomD7
Author(s): Naylor, Michael
Date: 1999
Title: Exploring Fractals in the Classroom
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92 (4), 360 - 364
Reviewer: TomD
Date of Review: 03/04/2000
In this article, Michael Naylor has six investigations that he does/suggests to do with students to further investigate and explore fractals. The first investigation that he talks about in his article is fractal trees. The student starts with a tree trunk, and extends two branches off the trunk, then two more branches off the existing branches, then two more branches off each of the two new branches, and so on. During this activity the students have sheet in which they will determine the number of new branches and then the total number off branches for each row of branches. From their observations, the students must try to come up with the number of new and total branches after n rows. If they counted correctly they should come up formulas for the number of new branches (2n) and for the total number of branches (2n-1 - 1). Another investigation that Naylor discussed doing with his students was what he called the Sierpinski Carpet. In this activity the students will! first start with a square, then divide inside of the square into 9 congruent squares (tic-tac-toe fashion). Then the middle square is considered "removed", and the process is repeated for each of the 8 remaining squares, then the other eight remaining squares, and etc. The key discussions that come out of these activities is how does the area of the congruent squares change as n (the number of squares) increases [area = (1/9)n], and what pieces will never be removed from the square (it will be the edge).
I have very little information and/or experience with fractals and the formulas that come out of investigating different types of fractals. I found this article to be very intriguing because the activities that Naylor was doing in his classroom were both activities I could follow and understand (considering my lacking knowledge of fractals), and activities that the students would follow and be intrigued by. I also enjoyed how Naylor had in each of his investigations what formulas he wants his students to derive, some key discussion questions that would make the students think about what they were doing and some homework tips for the students so they could venture on their own after class.