Keywords: Proof, Standards,
Ref: Noah1
Author(s): Knuth, Eric J.
Date: 2002
Title: Proof as Tool for Learning Mathematics
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 95, Issue 7, 560
Reviewer: Noah
Date of Review: 2/15/2004
This nice little article is about the role of proof in secondary school education. The author, Eric Knuth, feels that not enough time is spent on proof in secondary school math. This is especially not optimal because proof is of primary importance in higher math. Knuth distinguishes between proofs that prove and proofs that explain as well as prove. Knuth is after these later proofs as he feels that these will go to great lengths to promote students’ understanding of the math in question. Knuth gives several examples by what he means by proofs that prove and proofs that explain as well as prove. These later proofs are often more geometrical in form, even for seemingly non-geometrical problems.
Eric makes several important points in this article, and I tend to agree with him on what he is trying to accomplish. It definitely would be nice for students to have more of a background in proofs and always, anything to help promote understanding for students would be a great thing. Explanatory proofs would definitely do just that, though proofs in the more traditional form, strictly non-“explanatory” though they may be, would be very important to add as well. More specifically, in advanced high school classes, I think an introduction to various proof techniques like proof by induction or contradiction may be very helpful for these students. Even if they are not explanatory, they are nevertheless powerful and beauty. And if you have power and beauty, what else do you need?
Keywords: Activities, Probability, Discrete
Ref: Noah2
Author(s): Kent Anderson
Date: 1996
Title: How Many Ways Can A Team Win a 7 Game Series?
Journal or Publisher: SCORE Mathematics
Volume, Issue, Pages: http://score.kings.k12.ca.us/lessons/teamwins.htm
Reviewer: Noah
Date of Review: February 22, 2004
This lesson concerned how many ways there are for a team to win a 7 game series. Honestly I wasn't that impressed with this lesson plan. It was intended for 8-12 grade, but there is not a whole lot of real math in the lesson although it was hinted at under the extensions heading where the lesson plan simply states "Pascal's Triangle Combinations and Permutations". Basically, the lesson plan seems to be geared to simply listing and counting all the different combinations. This seems to me to be too simplistic for a high school lesson. It would be nice to perhaps get at the underlying mathematical principles and be able to generalize for an n game series. It seems like this would be a great lesson to introduce the "n choose r" terminology. Not using it just leaves the students floundering pointlessly to list tedious calculations. The lesson plan indicates that this might be a good extension but gives no indication as to how to develop it. Brahier discusses in chapter 5 about lesson planning that it is really important in the lesson planning to plan out in great detail the guiding questions and plans of developing an understanding of fundamental ideas. This lesson plan really doesn't do this. I think, however, that one could come up with this without too much difficulty. I think that this question of how many ways a 7 game series can end would make a good anticipatory set for a broader lesson dealing with basic combinatorics.
Keywords: Algebra, Activities, Technology
Ref: Noah3
Author(s): Taylor, Sharon E.; Mittag, Kathleen Cage
Date: 2001
Title: Seven Wonders of the ancient and Modern Quadratic World
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 94, Number 5, 349-350, 361
Reviewer: Noah
Date of Review: 2/25/2004
This article is about the quadratic equation and ways of representing and exploring it to best enable students to understand it well. This article mentions briefly the traditional ways for solving and exploring the quadratic equation: factoring, completing the square and using the quadratic formula itself. The main part of the article explores how different calculator (TI-83) functions can be used to further understanding. These calculator methods include graphing the equation, using tables and doing algebra on the calculator. This article proposes using all of these in combination while emphasizing that although these calculator features promote understanding, the other traditional ways are still crucially important. The authors emphasize how important calculators are in allowing a visual aid to the student's understanding of what the quadratic equation is saying. Included is a sample "lab" or worksheet of some sort that is fairly plain and undeveloped. I think more could be done in developing a nice class exploration but I think the ideas are there for a very nice lesson or couple of lessons on the quadratic equation.
Keywords: Geometry, Activities, Representations
>Ref: Noah4
Author(s): Morris, Barbara H.
Date: 2004
Title: The Beauty of Geometry
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: Volume 9, Number 7
Reviewer: Noah
Date of Review: 3/8/2004
This very recent article is about using stained-glass-window designs in a middle school classroom to help students learn geometry. This article states that "geometry is often thought of as being a topic that is dull and that requires analytical thought". This article discusses how in contrast to this, a stained-glass-window project can create an interesting math class, one that students will be enthusiastic about. For this project, students first review 20 important geometric terms from acute triangles to vertical angles. Students define these terms, find an example, draw a picture, and find an application for each term. Then they use all of these geometrical constructions when they make their very own original stained-glass-window. Aside from creating their own stained-glass-windows (which takes up the bulk of the time), students also learn about the history of these designs, from Medieval Times to Frank Lloyd Wright, as well as review the work of one of their classmates. Although the author does not specify a time-frame for this project, it appears that it would take several days with most of the time and energy spent creating stained-glass-windows.
I am a big fan of "mathology" (recall Halmos' term for pure math), especially as it relates to mathematical beauty. When I saw the cover of this issue in the library which said "The Beauty of Geometry", I was ecstatic as I had not yet seen an article on mathematical beauty in an NCTM periodical. It turns out that I still haven't. This article really was not about mathematical beauty but rather the beauty of art. I was surprised that at the beginning of the article, Barbara Morris almost seems to agree with the statement that analytic geometry is "dull" and that this project is one way around it. Besides the fact that analytic geometry (aka coordinate geometry) is not (in my experience) covered in middle school math, analytic geometry is far from dull, for in this analytic mathematics is universal truth and pure mathematical beauty. This is being ignored from the very beginning and it is disappointing.
I think that a much simplified version of this project (one that would fit in a day), would be a great way to review all the important geometric terms and would be a great opportunity for students to combine geometric concepts and see connections. The emphasis in this lesson unfortunately is on the art and crafts aspects. In the grading rubric provided, only 40% of the student's grade is based on actual "math". I would tweak this project, condense it into a day, and emphasize the math parts more, maybe even talk about mathematical beauty. It is math class after all.
Keywords: Communications, Teaching Strategies,
Research
Ref: Noah5
Author(s): Kitchen, Richard S.
Date:
2003
Title: Challenges Associated with
Developing Discursive Classrooms in High-Poverty, Rural Schools
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 97, Number 1, pages 28-31
Reviewer: Noah
Date of Review: 3/8/2004
The ideas of communication in math classes and student-centered approaches to teaching math are important issues to study. This article is about challenges to the creation of student-centered discussions in the math classroom and how prospective teachers can be shown how to overcome these challenges. The author discusses three specific challenges: domination of class discussion by students with greater mathematical knowledge, mathematical discourse that really isn't deep or significant, and finally the challenge of helping/encouraging some students to become involved in discussions. Kitchen suggests that prospective teachers ought to be taught strategies to involve all students such as grouping less-verbal students together or making sure that all students are given a turn to speak. In regards to the second challenge of lower-level disourse, the author suggests the use of "high-quality, cognitively demanding curriculum materials". To face the third challenge, prospective teachers must be given strategies to engage, motivate and reach out to these children who may resist mathematical dialogue in class.
Although the title suggested this would be a high-poverty, rural school issue, the challenges faced in creating a classroom where good discussions take place is broader than this and the issues addressed in this article are more universal. I wish the author had gone into greater depth on ways to address all three of these challenges. His responses left me without a really good sense of how to facilitate better discussions. Perhaps detailed situations or a deeper look in to the research would have helped. A bibliography is provided which perhaps will provide good reference for those wanting to learn more.
Keywords: Geometry, Research , Planning
Ref: Noah6
Author(s):
Hollebrands, Karen F.
Date: 2004
Title:
High School Students' Intuitive Understandings of Geometric
Transformations
Journal or Publisher: Mathematics
Teacher
Volume, Issue, Pages: Volume 97, Number 3,
pages 207-214
Reviewer: Noah
Date of Review:
3/12/2004
This wonderful article examines the research concerning 8th and 9th graders intuitive understandings of geometric transformations before they have seen these topics in class. She examines students conceptions and misconceptions of all the basic geometric transformations (reflections, rotations, and translations). This is an invaluable resource for teachers about to start a unit on the geometric transformations because after reading this article teachers will have a good idea of common student misconceptions and will have a chance to prepare to help students understand what is going on. The teacher can start where the students are at and build on their knowledge. This will help to reduce wasting time on things that students already know and enable the teacher to help students with things they don't or even to explore further topics or applications of these basic transformations. She also gives suggestions on how the Geometer's Sketchpad (a marvellous program and almost as good as Cabri) can be used in the teaching of transformations.
Keywords: Connections, Standards,
Representations
Ref: Noah8
Author(s): Froelich, Gary W.
Date: 1991
Title: Curriculum and Evaluation Standards for
School Mathematics: Connecting Mathematics
Journal or
Publisher: NCTM
Volume, Issue, Pages: Addenda
Series: Grades 9-12
Reviewer: Noah
Date of
Review: 3/10/2004
The purpose of this publication by the NCTM is to explore in depth the idea of "Connecting Mathematics" in mathematics education at the high school level. More specifically it was intended to "clarify the recommendations of selected standards and to illustrate how the standards could be realistically implemented" (iv). In both of these regards, this book does a fine job.
This publication helped to clarify in my mind what is meant by "connecting mathematics". There are two major forms of connections that are discussed here: (1) connections from the mathematical to the real world, and (2) connections between different areas of mathematics. Through the in-depth examination of multiple lessons in several different units, I was able to see very clearly both types of connections that were being made.
For example, chapter 2 explores "Connecting with Matrices". Various activities involving matrices are presented and discussed. Matrices are a way to represent information or sort knowledge about the world. In this way, students can see how a very mathematical idea can be connected to the world they see. These activities also demonstrate how a matrix can express algebraic or geometric knowledge. These are the two kinds of connections that the NCTM is talking about, and in this publication they do a fine job of explaining what these are and how they can be seen in the classroom.
Keywords: Curriculum, Standards, Algebra
Ref: Noah10
Author(s): Askey, Richard
Date: 1999
Title: The Third Mathematics Education Revolution
Journal or Publisher: Cambridge University Press
Volume, Issue, Pages: Contemporary Issues in Mathematics
Education, Pages 95-107
Reviewer: Noah
Date of Review: 4/4/2004
This article is a critique of the current mathematics education revolution, in his opinion best illustrated by the NCTM standards. The author advocates a "balanced view" of mathematics education with good problems, technical skill, and a "broader view, which contains the abstract nature of mathematics and proofs". He emphasizes that traditional math is not enough, but does not think that traditional math is obsolete either. He considers the classical curriculum of arithmetic, algebra and geometry leading up to calculus to still be of great importance. He acknowledges the importance of good problems and teaching for understanding, but doesn't always see this happening in textbooks following NCTM standards.
The bulk of this article is a case study examining and critiquing a new textbook, Addison-Wesley's "Focus on Algebra" in detail. The author has a whole series of complaints about the book that he expands on and illustrates using examples from this text. His main criticisms of this text are a lack of reasoning and a lack of algebra. As regards the lack of reasoning, he cites several examples in which "teaching students is passed over in favor of having them do calculations which do not lead anywhere." The examples cited fail to do a sufficient job in being mathematically vigorous and in having concrete goals. The book often asks students to explain things, but doesn't offer sufficient explanations itself. As regards a lack of algebra, the author is dismayed that although this is algebra, equations are not introduced until page 165, and the first solution to one doesn't occur until page 218. He also calls into question the extensive use of algebra tiles in this text. Besides the fact that algebra tiles have a quite restricted usage and are inappropriate for some students, the author is concerned that "they become a crutch which makes it hard to progress to a higher, more abstract level".
This is clearly a well-researched and thought-out article. In his
three-pronged vision of mathematics education (good problems, technical
skill, and a "broader view which contains the abstract nature of
mathematics and proofs"), he has synthesized the important aspects of
different visions of education to form a well-balanced ideal that is
hard to argue with and one which, at least in theory, many NCTM minded
people might agree with. The problem as always is to bring the
theoretical goals into concrete reality. NCTM textbooks strive to
provide understanding, but sometimes they don't live up to that. I
suppose the moral is that guided by research, we should continue to
strive towards a curriculum and textbooks that best accomplishes these
goals.
Keywords: Number and Operation, Research ...
Ref: Noah11
Author(s): Dehaene, Stanislas
Date: 1997
Title: The Number Sense: How the Mind Creates Mathematics
Journal or Publisher: Oxford University Press
Volume, Issue, Pages: Chapter 5, pages 118-133
Reviewer: Noah
Date of Review: 4/5/2004
This chapter of the book explores answers to the question "Why is mental calculation so difficult?". Dehaene begins with considering how the idea of counting is developed in the minds of children. Everybody hears children counting from a very early age "onetwothreefourfive", but it isn't until later (age 4 or so), that this idea of counting is related to the world; until then it is only a verbal string of sounds. This connection is eventually developed though. Another fact Dehaene addressed is the trouble that children have learning basic multiplication facts. Multiplication facts like 9x6 and 7x8 pose great difficulties for many students. Yet these very students before adulthood "will have learned at least 20,000 words and their pronunciation". Dehaene explores various answers to why this might be, especially concerning the reliance on the verbal and interconnectedness of numbers and the problems that the combination of these two things can result in.
Especially in the later half of the chapter, Dehaene talks about "Number Sense" and its importance since humans in general have such a tough time with mental calculations. He commends the NCTM for emphasizing this. Dehaene feels that the emphasis on algorithms combined with this problem with mental calculations that people have, help to destroy whatever ideas of number sense people have. The development of ideas of relationships and operations can go a long way in furthering mathematical understandings. Unfortunately this is simply not done.
I thought that this chapter, as well as many other parts of the book
are crucial for math educators to read. This book, though entirely
about mathematics, is classified as psychology since it looks at the
mind and how the mind understands and interprets mathematics. This is
essential knowledge to have to be an effective teacher. It is
important to clarify what ought to be taught, but it is equally crucial
to know how to human minds works to best impart that knowledge.
Keywords: Research , Curriculum, Standards
Ref: Noah12
Author(s): Roitman, Judith
Date: 1999
Title: Beyond the Math Wars
Journal or Publisher: Cambridge University Press
Volume, Issue, Pages: Contemporary Issues in Mathematics
Education, pages 123-134
Reviewer: Noah
Date of Review: 4/10/2004
As the title indicates, this essay is an attempt to move beyond certain disagreements math educators and policymakers have over how and what mathematics should be taught. The author's main point is that the math "war" is in the minds of the combatants and that the sides agree on a lot more than they acknowledge. To illustrate this later point, she has selected ten quotes from those for reform (mainly NCTM) and those generally regarded as opposing reform. These direct quotes are chosen to illustrate how reform minded individuals or groups still regard skills as important, and how more pro-traditional individuals or groups also recognize the importance of understanding and exploration. In short, the two sides recognize many of the same values and taken out of context it is hard to recognize who is saying what.
The author laments the politics of this disagreement over math education as well as the gross generalizations and simplifications of the opposing viewpoint that dominate the scene. She would like to see both sides sit down together and discuss what she thanks are the 10 key issues in mathematics education: relative performance, equity, technology, demography, subject matter, pedagogy, teacher preparation and certification, assessment, high performers, new curricula. She thinks that this is the best option, but also contends that there are at least four delusions that make this difficult, especially "the delusion of assessment" and "the delusion that curriculum can be judged on the page". The underlying problem is that it is really tough to know what works and what does not. Setting up adequate scientific models to ascertain what works and what doesn't is nearly impossible, due to the incredible number of different variables. Therefore it is all the more important for the sides to sit down together, start with all the things they have in common and work together towards improving and working on the ten key issues listed above.
I thought that this was a very good and timely article that more people
should read. There are disagreements among and between math educators
and
policymakers but they are not nearly so big as people imagine them to
be.
Clearly both sides value both computational abilities and mathematical
thought processes, but emphasize one or the other a bit more. Both are
necessary. We should stop focusing on the difference, and work together
to
improve education. It's not quite as easy as Judith Roitman makes it
sound, because there are times when direct disagreements will occur,
but
the two sides still have a lot in common and can benefit by working
together.
Keywords: Planning, Problem Solving, Teaching Strategies
Ref: Noah13
Author(s): Gerver, Robert; Sgroi, Richard
Date: 2003
Title: Creating and Using Guided-Discovery Lessons
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 96, Number 1, pages 6-13
Reviewer: Noah
Date of Review: 4/11/2004
This article is a reference and how-to guide for teachers wanting to create student-involved discovery lessons. The authors first clarify what a "guided-discovery lesson" is and what it involves through an example of one such lesson. The authors emphasize that a good guided- discovery lesson will have a story-line that will interest and engage the students. Specifically, some sort of dilemma or counterintuitive fact should be presented to facilitate this. The students should understand what the problem is asking and know what they want to find out.
The authors then move on to discuss 8 steps teachers can follow to help in the creation of a guided-discovery lesson. There are eight specific steps that they outline: 1. "Selecting the content", 2. "Stating the aim", 3. "Identifying the prerequisites", 4. "Setting up a graphic organizer", 5. "Writing the lesson", 6. "Using a naïve proofreader", 7. "Writing a follow-up activity to check for accountability", and finally, 8. "Field-testing and revising". They emphasize several of these parts, especially part 5 and the idea of "reflection points" and how these are important in keeping the class together and on track. They also emphasize the idea of review in parts 6 and 8. It seems like a very good idea to have a colleague review the discovery activity by working through it themselves. This will to ensure that the directions and intent are clear. Through these 8 steps a teacher can arrive at a well-thought-out, clear lesson plan designed to engage students and help them to explore some part of mathematics.
This week I need to develop in detail a lesson-plan for my micro- lesson, and of course I want to do a discovery-based lesson. I am using these guidelines to assist me and I think that in the end I will arrive at a very good lesson plan. Their 8 steps to develop a good lesson-plan are clear, straightforward and logical. I think that this could very well be a guide I keep by my side as I plan future lessons as well.
Ref: Noah7
Author(s): Rubenstein, Rheta and Thompson, Denisse
Date: 2001
Title: Learning Mathematical Symbolism
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: Volume 94, Number 4, pages 265-271
Reviewer: Noah
Date of Review: 3/26/2004
This article is about mathematical symbolism, problems students have with it, and thoughts about how to teach it more effectively. Mathematical symbolism is obviously of great importance. It enables us to express complex ideas very simply and also to manipulate these ideas simply and effectively. Mathematics is expressed through a whole separate language, and just as it is impossible for me to understand a sentence in swahili if I don't know what the words are, it is impossible to understand a mathematical statement if you can't read the language and don't know what is being said. The authors emphasize the multitude of different ways that mathematical symbolism can be used. For example, symbols can name concepts, state relationships, indicate an operation etc. Mathematical symbolism is such a central feature of mathematics that clearly time should be spent to make sure it is understood.
The authors discuss three specific challenges related to mathematical symbolism: verbalization, reading and writing difficulties. They discuss how these difficulties can be addressed. Basically what is most important is for a teacher to be aware of these difficulties, spot them, and be prepared to help students overcome them. They emphasize in general that meaning should precede symbolism and in this way troubles with symbolism can be overcome. The authors also discusses how certain projects and visual strategies can help students as well.
The understanding of mathematical symbolism is obviously of incredible
importance for one's students. I think that incorporating the history
of mathematical symbolism in class might be interesting and a fun way
to appreciate its value as well as review it.
Ref: Noah9
Author(s): Uhl, Jerry and Davis, William
Date: 1999
Title: Is the Mathematics We Do the Mathematics We Teach?
Journal or Publisher: Cambridge University PRess
Volume, Issue, Pages: Contemporary Issues in Mathematics
Education, pages 67-74
Reviewer: Noah
Date of Review: 3/27/2004
This article is written as an answer to the question of whether the mathematics that is taught even qualifies as mathematics, that is, as mathematicians view mathematics. The article then examines in depth an alternative program for mathematics instruction.
The authors note that for a variety of reasons, the general sequence in many math classes ends up being "lecture - memorization - test". This is clearly not what mathematicians do. What mathematicians do is more in the "intuition - trial - error - speculation - conjecture - proof" sequence. So how can we as math educators make math education more like actual mathematics and help our students to think like mathematicians?
This article examines a new laboratory calculus course called "Calculus&Mathematica" as a way of doing this. This course approaches mathematics more like a science, as an investigative endeavor. It is only after investigation and exploration occurs that more detailed explanation is attempted. There is no textbook for this course; instead, students are given a series of electronic notebooks which facilitate the exploration of different ideas. Students spend a good deal of time observing and investigating these ideas and through this personal investigation hopefully gain a better understanding of calculus. This of course sounds very similar to what Core-Plus and other integrative textbooks are striving to do, though there are some slight differences in the manner in which this is attempted. Calculus&Mathematica especially tries to emphasize visual understandings before moving on to a more formal approach. Calculus&Mathematica also sounds quite different from integrative textbooks as Calculus&Mathematica does not emphasize real world connections too much, but instead focuses on the pure mathematical ideas of calculus.
Overall this sounds like a great program and something I would really
like to teach with at some point. This program I think will truly
facilitate a good understanding of calculus but it also doesn't seem to
be a watered down form of calculus like some integrated texts. This
program gets at the core ideas and in a way that moves through the
sequence of "visualization - trial - error - speculation -
explanation" and helps students to think like mathematicians and not
robots.
Keywords: Proof, Representations, Geometry
Ref: Noah14
Author(s): Nelson, Roger
Date: 1993
Title: Proofs Without Words: Exercises in Visual Thinking
Journal or Publisher: Mathematical Association of America
Volume, Issue, Pages:
Reviewer: Noah
Date of Review: 4/25/2004
This is one of the greatest math books I have ever seen.
Really. This book belongs on the shelf of every person interested in math (from junior-high student to PhD) and especially every math educator. The identities and theorems "proved" in this book are fairly complex and are definitely not self-evident. Yet these proofs, purely geometric in form, are simple and often stunningly beautiful. It is incredible that such seemingly complex ideas can be drawn without a word of explanation and in a way that makes is simple and straightforward. I spent the last half-hour "reading" through large parts of this book with a fellow math freak oohing and aahing the proofs (NB: you don't have to be a math freak to ooh and aah these proofs). A lot of the identities we were already intimately familiar with; nevertheless, to see them in this new, very visual way (as opposed to purely abstractly), is incredible. The author has done an incredible job of gathering these geometric "proofs" from all over the world into this one little book. These proofs range from junior-high topics like the Pythagorean Theorem to E.R.A. topics like the Cauchy-Schwartz inequality and integration by parts. All are beautiful.
As a teacher, this book is an important book to have for several reasons. First of all, these proofs are a way for students to visualize what they are studying. Secondly, students will gain in understanding of what they are studying by seeing the geometric representation put forward so clearly and simply. Thirdly, students will appreciate the variety of ways in which something can be proved; not only can things be proved geometrically as well as algebraically, but there often multiple ways of doing both (for example, this text offers six graphical proofs of the Pythagorean Theorem). Finally, this book can help students to see the beauty of mathematics and the that way that everything fits together so beautifully and in such incredible order.
I can't recommend this book highly enough. Go buy yourself a copy. Now.
Keywords: History, Manipulatives...
Ref: Noah15
Author(s): Arlton, Stanley L.
Date: 2004
Title: Fascinating Mathematical Relations
Journal or Publisher: MCTM Conference
Volume, Issue, Pages: Session #60
Reviewer: Noah
Date of Review: 5/3/2004
Stanley Arlton gave an extremely interesting talk in which he talked about all kinds of fun and interesting mathematical facts. The topics varied considerably from historical anecdotes to the relationship between music and math, to the mathematics of the Miss America Contest, and to Fibonacci squares and other purely mathematical relationships and ideas. What all of these things had in common was that these mathematical stories and ideas he brought up were all extremely interesting and intriguing, often getting at the beauty and wonder of mathematics. He handed out a 44 page booklet filled with these "Interesting Mathematical Relationships", all written by hand. His love and dedication for math is obvious.
Stanley Arlton seemed to have an underlying message for teachers in his talk as well. He continually asked if the mathematical things he was bringing up were useful. Often, the answer was "no". But this is okay; what matters is that the students enjoy and learn from these things. He was especially insistent on developing the "human face of mathematics", especially through the use of historical anecdotes, to interest the students and to facilitate understanding. His teaching philosophy included a wonderful combination of wanting to make mathematics real by putting a human face on it, while not being obsessed with making it "real" by applying it. His goal was to show how interesting math is and share the stories that lie behind it, and in this way engaging them and helping them to develop a sense of wonder about math and the world.
Stanley Arlton is clearly very passionate about and truly enjoys mathematics. Unfortunately, he seemed a fairly bitter about the fact that he never had a teacher to inspire him in mathematics as he would have liked. Nevertheless, he is clearly a knowledgeable, passionate man; somebody who I wish had taught me in high school.
Keywords: Problem Solving, Number and Operation...
Ref: Noah16
Author(s): Mrachek, Len; Duranczyk, Irene
Date: 2004
Title: Math in the Movies and Other Math Applications
Journal or Publisher: MCTM Conference
Volume, Issue, Pages: Session #29
Reviewer: Noah
Date of Review: 5/4/2004
There were two parts to this talk: first, Irene Duranczyk talked about "Math in the Movies", and then Len Mrachek talked about some other applications.
Irene Duranczyk showed several clips from movies in which math and mathematical abilities (or the lack thereof) played a role. The first and most important clip she played was an Abbot and Costello clip in which the character bakes 28 donuts for 7 officers so that each of the officers can get 13 donuts. This of course makes no sense, but the character goes through and justifies his answer for the number of donuts in three different ways by misusing basic algorithms for addition, multiplication and division. She also showed clips from Little Big League and Die Hard.
Although she didn't explicitly specify a goal she had in mind, it seems that a teacher should be able to benefit from the views of mathematics that are presented in some of these clips. A couple times, especially in the first Abbot and Costello clip, the view of math as a bunch of algorithms was presented. A teacher should be able to recognize this and work even harder to present algorithms in a way that is sense-making and clear. Also, understanding what is being asked in a word problem is another important theme that came up a couple times in these clips, and it is clear how hilariously necessary it is to have an understanding of the goal of a problem to be able to solve it. This is something that a lot of people struggle with and thus something that teachers should help their students to develop.
The second half of the talk, given by Len Mrachek, was about other math applications. This part of the talk was cut short by time, but I wasn't overly impressed by it. Basically his talk consisted of showing and going through a number of word problems that he has written. These word problems were good and really required students to understand the problems and to know what they are doing. Unfortunately, a lot of the problems were very similar and a lot of them simply focused on unit conversions, which although an important skill to have, gets tedious after a while. I think there are some excellent problems in here, but beyond that I didn't get too much out of the second part of the talk.
Keywords: Teaching Strategies, Algebra...
Ref: Noah17
Author(s): Bracken, Laura
Date: 2004
Title: Re-teaching Algorithms
Journal or Publisher: MCTM Conference
Volume, Issue, Pages: Session #104
Reviewer: Noah
Date of Review: 5/5/2004
The speaker, Laura Bracken, teaches at a state college. Every year, she has many many students who come into her class with years of elementary and high school math, yet these students struggle profoundly because they simply don't understand the basic algorithms and theorems of mathematics. She realizes that students who take her class can once again get by, simply by memorizing the algorithms again. What Laura Breacken strives to do, however, is to finally get these students to understand the reasoning in mathematics behind these concepts, algorithms and theorems. This talk is about how this understanding can best be accomplished.
Laura Bracken provided, and then went through a number of activities that illustrated how she would reintroduce these old concepts to students for understanding. The way she set these examples up was by beginning with basic concepts and definitions and then slowly building upon these. Students do these in groups with teacher checks at various points. She discussed practical advice for how these activities can best be structured as well as advice for writing activities like this. She really emphasized the importance of having students explain things in their own words.
I thought this was a very good talk. Although geared to a college
audience, what she was talking about can be applied to high school and
grade school as well. Understanding mathematics knows no bounds. The
investigation lesson plans she has developed to reteach concepts and
algorithms I think would work to introduce them for the first time as
well. In both cases one wants to facilitate understanding by letting the
students work though examples and justify the concept in question
themselves.
Return to Index
Keywords: Technology......
Ref: Noah18
Author(s): Brahier, Daniel J.
Date: 2000
Title: Integrating Technology in Mathematics Instruction
Journal or Publisher: Teaching Secondary and Middle School
Mathematics
Volume, Issue, Pages: Chapter 7, pages 171-208
Reviewer: Noah
Date of Review: May 16, 2004
This chapter in Brahier's textbook discusses integrating technology in math classes. The chapter discusses various kinds of technology including computers, graphing calculators, audio/video devices, and the internet. Within the sections on each of these technologies, Brahier discusses why and how each might be used in the classroom to further education.
There are many benefits to these various technologies. For example, computers can be used for spreadsheet and data gathering purposes, as well as for programs such as Geometer's Sketchpad. Both of these facilitate personal exploration in mathematics. Calculators can also be used to a smaller extent for spreadsheet and data gathering purposes. Probably the biggest thing that calculators are used for these days is for graphing. The ability to graph a number of different graphs quickly and efficiently is of incredible utility and importance. Back in the days before calculators, students would spend most of their time carefully graphing a few selected graphs. Graphing calculators, however, allow students to change parameters and options with ease, allowing them to investigate topics in mathematics in ways not possible before. "Computers and calculators... allow students to visualize parameter changes on their resulting calculations and graphs, again, making it possible to explore many problems in a relatively short period of time." Math is the study of patterns and relationships. If students can see a plethora of information quickly, clearly, and easily, they will be all the more able to look at patterns and relationships and ultimately further their understanding of mathematics. Using technology can enable students to investigate mathematics in ways not possible before.
This chapter highlights well the importance of technology and illustrates through examples how the use of technology can most aptly be implemented. As a teacher, I will definitely make use of technology, especially in investigative activities. As discussed in the previous paragraph, graphing calculators and computers allow students to view many examples quickly. In my classes, I will have students use technology to help them investigate mathematics, and to discover and see patterns. At this point, students will be more able and better equipped to make conjectures. They can then explore why their conjectures are true. Since they are making the conjectures themselves from what they have observed, what is being investigated is already "theirs" in a sense, they have taken ownership. From this point, investigating why and proving are much easier. Technology not only provides students with a sense of ownership of the material since they are able to make conjectures for themselves, but also technology allows students in many cases to have a better conceptual understanding of the material.
Finally, though, I think it is important to note that one needs to think carefully about when and how technology should be used. Technology can be very beneficial, but can also be a waste of time if used in an improper fashion. Technology should be used appropriately. One can't just go into class and hand all the students a calculator and then leave the room. When technology is used, their needs to be a purpose and using it ought to increase student understanding. Lessons using technology especially need to be guided and structured. A teacher ought to give much thought before hand to whether and precisely how technology would be beneficial in each specific case. Before I use technology in any lesson, I will ask myself why technology will further student understanding. I will also keep in mind the downsides of technology, especially student reliance on technology.
Overall, a nice chapter providing an in-depth and broad understanding of
technology and its uses in the math classroom.
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Keywords: Activities, Puzzles...
Ref: Noah19
Author(s): Maganzini, Christy
Date: 1997
Title: Cool Math: Math Tricks, Amazing Math Activities, Cool
Calculations, Awesome Math Factoids, and More!
Journal or Publisher: Price Stern Sloan
Volume, Issue, Pages:
Reviewer: Noah
Date of Review: May 17, 2004
This little book has a noble purpose - to present math to kids in a fun and exciting way; to show students how "cool" it is. The book is geared to students 9 and older. The whole book (about a hundred pages) is filled with "math tricks, amazing math activities, cool calculations, awesome math factoids, and more!" Indeed, the book is filled with these things, especially the amazing math activities for which there are sometimes solutions in the back of the book. Also spread around in this book is some math history especially brief biographies of some of the great mathematicians. This book covers briefly a huge number of interesting and important mathematical anecdotes and ideas like Pi, the Konigsberg Bridge problem, perfect numbers, triangular numbers, codes and ciphers, and much much more.
This book has a noble goal. In the end though, I think it doubtful whether this end has been obtained. The book's main failure lies in its lack of explanations that severely limit student understanding. For example, several of the amazing math activities consist solely of one student asking another student to pick a number and then do something with it. The student who doesn't know the number is then able to tell the student who chose the number what that number is, or something else cool as a result. This is presented a couple times as "Math Magic". The problem is that there is no explanation for why the "magic" works. I think that students will enjoy this, but chances are they really won't learn about math - it will just be a trick to show their friends once or twice. Even more disturbing, classic problems like the Konigsberg Bridge problem were answered, but not really answered. The book asks the student whether crossing all the bridges without crossing any twice is possible. The back of the book says that in fact it is impossible to do so. The back of the book does not give one word of explanation as to why it is impossible. I think this might be frustrating to students who want to know why. One cool calculation asks students to determine if the 4th perfect number (8,128) is also a triangular number (the first three perfect numbers are triangular as well). I was able to determine very quickly that it was in fact triangular (the answer wasn't in the back), but I want to know if all perfect numbers are triangular as well - I think this is the meaningful question here. What bothers me is that students who read this will only have the brute force method of listing triangular numbers up to 8128 to determine if 8128 is indeed triangular. A lot of the cool calculations like this still involve much tedious calculation on the part of the student. The only difference is that the problem is now dressed up a bit and is called a "cool" calculation instead of just a calculation.
If I were a student reading this book, I would be frustrated. Many things I
would be interested in, but would be extraordinarily frustrated by a lack of
explanations and by my own lack of understanding of the topics that I would have
after reading the section. My suggestion to the author would be to slow down,
not to cover so many topics in the book, but instead to focus on how one might
better explain (or explain at all) these things to students so that they can
understand the "cool" mathematics as well as have knowledge of why it works. It
is through this understanding that students will progress mathematically.
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Keywords: Gifted, Issues, Number and Operation
Ref: Noah20
Author(s): Dehaene
Date: 1997
Title: The Number Sense: How the Mind Creates Mathematics
Journal or Publisher: Oxford University Press
Volume, Issue, Pages: Chapter 6 - Geniuses and Prodigies - pages 144-
172
Reviewer: Noah
Date of Review: May 18, 2004
This chapter in Stanislas Dehaene's marvelous book is entitled "Geniuses and Prodigies". In this chapter, Dehaene discusses mathematical prodigies and geniuses and explores reasons for their exceptional abilities. He begins by looking at Ramanujan and the amazing formulas which he was able to come up with. There is, of course, the famous story about Ramanujan and Hardy in taxi # 1729. Dehaene then explores the PERSONAL relationship that mathematical prodigies have with numbers. People with this kind of relationships with numbers see the interconnectedness of numbers. Dehaene quotes Wim Klein as follows: "Numbers are friends to me, more or less. It doesn't mean the same for you, does it, 3,844? For you its just a three and an eight and a four and a four. But I say, 'Hi 62 squared'!". Math prodigies through experience with these numbers, develop a good sense of numbers and their relationships and the nature of those relationships with other numbers. Dehaene thinks that mathematical prodigies don't think so much about calculations, rather, the greatness of their ability lies in their "direct perception of significant relations". For Ramanujan, many of the incredible equations and identities appeared in his mind when he woke up - he did not spend hours calculating.
Dehaene spends much of the rest of the chapter investigating whether mathematical talent is a biological gift. Dehaene concludes that biological variables do play at least some role in mathematical ability. There is some evidence that mathematical talent is linked to certain parts of the brain and that the development of these certain parts is partially determined by genetics. Furthermore, there is some evidence that certain activities that use specific parts of the brain (especially at an early age), help to significantly develop that little corner of the brain. He also spends time exploring in depth how practice and time can help to develop these mathematical gifts. In the end he concludes that "...biological factors, however, do not weigh much when compared to the power of learning, fueled by a passion for numbers. Great calculators are so passionate about arithmetic that many prefer the company of numbers to that of fellow humans".
This was a very interesting chapter if nothing else. As a teacher though, it is important to have some kind of idea of current neuroscience research about how the mind works especially as regards to the subject matter you teach. In that sense this chapter is crucially important. Although it is unlikely that you will have a Ramanujan in your class ever, you will have some brilliant students, perhaps even geniuses, in your classes over the years. It is important to know something about geniuses and how they view mathematics so that you can better understand and help these notable students to push their abilities to the fullest extent. Even with regular students, if you can help them to explore relationships among numbers (especially in areas like Number Theory), you will go a long way in developing their ability to really understand math and perform at a higher level.