Keywords: Assessment, Communication, Teaching Strategies
Ref: Ross19
Author(s): Bracken, L.
Date: 2003
Title: Re-Teaching Algorithms
Journal or Publisher: MCTM Spring Conference
Volume, Issue, Pages:
Reviewer: Ross
Date of Review: 5/2/03
Summary:
Teaching in a state college in Idaho, Laura Bracken has been teaching a basic algebra class for students that have not previously learned many of the mathematical skills required to graduate. In order for her to succeed at teaching math to her students, she has to battle with the wrong methods that they have already been drilled on.
The strategies that she uses are group work and in-class writing activities. Her assessment of the students is done by the group and based on one of the papers, so all students are required to participate in the group and it is everyone's responsibility to make sure that the rest of the group are understanding. The activities are designed such that the students are constructing their knowledge and the teacher acts as a facilitator, checking their work at specific checkpoints. The in-class assignments do two things: (1) build community in a diverse class and (2) requires the students to explain concepts and algorithms in their own words.
Personal Reaction:
Even though she uses this method of instruction with college aged students in a remedial algebra class, many of these strategies can be used in almost any classroom setting. It takes a lot of initial work for the teacher in preparing the in-class assignments and getting the students used to the group work, but the students are spending their time in the class constructing the concepts and algorithms and creating notes in their own words. I think that that strategy is the key to learning math. Often having the teacher just tell the rules and require memorization and practice doesn't make the connections the students need to retain the information and succeed in mathematics.
She included her assessment rubric for group work. It contains a box to enter the names of the group, a column with the amount of points for certain parts (e.g. Descriptions (2 points)), and then the points column that the group gets. It is one sheet that can be carried around the room on a clipboard and the points can be recorded quickly. I think this is setup for assessment is very efficient and quite practical for a classroom that requires you to move around the room and attend to many groups.
Keywords: Connections, History, Activities
Ref: Ross20
Author(s): Kelly, P.
Date: 2003
Title: Loads of Codes: Cryptography Throughout the Ages and in Your Classroom
Journal or Publisher: MCTM Spring Conference
Volume, Issue, Pages:
Reviewer: Ross
Date of Review: 5/2/03
Summary:
Codes are a great way to trick your students into thinking mathematically. Paul Kelly has studied some cryptography and the math involved and has integrated it into his lesson plan. The lesson can be used in many different classes and the students find that creating codes is a fun activity. The lesson also includes connections with history and the Enigma Code, the Jefferson Wheel, and Caesar Shift codes.
The mathematics in codes at the very basic level teaches order of operations. In order to crack a code, you need to work backwards from the way that the code is created. On other levels, encoding messages can involve practicing concepts. Adding the message matrix and a keyword matrix together (with numbers in place of their respective letters) allows students a fun way to practice addition and subtracting to decode the message. There are even codes that use the frequencies of each letter, therefore the letter 'E', which is the most common, is represented by 'A' and 'T', the second most common, as 'B' and so on.
Personal Reaction:
If you need a lesson that is a break from the book, something for a substitute to work with the class on, or additional practice on things like matrix addition of order of operations, cryptography can be a fun avenue to explore. There are many books and internet resources on cryptography and you may find more interesting or personalized codes when you let your students loose. A cryptography lessons also affords you a chance to tie math in with a little history and stories of World War II or Caesar's campaigns. There are many opportunities to integrate this into the classroom, and with many websites dedicated to cryptography more information is easily available.
Keywords: Research , Communication
Ref: Ross18
Author(s): Miller, K. F., Smith, C. M., Zhu, J., & Zhang, H.
Date: 1995
Title: Preschool Origins of Cross-National Differences in Mathematical Competence: The Role of Number-Naming Systems
Journal or Publisher: Psychological Science
Volume, Issue, Pages: 6(1), p.56-60
Reviewer: Ross
Date of Review: 4/27/03
Summary:
English speakers are disadvantaged by their language in ways that make accessing some mathematical relations more difficult. Because of the differences in the naming of numbers between Chinese and English languages, Miller et al found that American children are often delayed in number-naming and comprehension of numbers by the time they are entering preschool.
What are the disadvantages? Miller et al cite where learning the Chinese and English number-naming systems diverge. First, all children need to learn the names of the numbers and their natural order, the difference here is that in English, the numbers between 10 and 20 all have a novel name whereas Chinese follow a consistent base-10 rule (e.g., a literal translation of 12 is 'ten two'). The second difference comes in counting decades. Chinese numbers translate as 'two ten' for 20, 'three ten' for 30 and so on while the English language requires learning words that have a phonetic relationship to the number or the prefix and the rule of adding 'ty' at the end. Numbers beyond 100 are isomorphic.
The study had 3, 4, and 5 year old children from the United States and China perform two tasks. An abstract counting task, which asked the children to count as high as they could, and an object-counting task where children were asked to count the number of stones in three sets: small (3-6 items), middle (7-10 items), and large (14-17 items).
The results showed Chinese children out performing American children on both activities. In the abstract task, there is a considerable drop in the percentage of American children reaching a particular number in the 10-20 range which was anticipated. The percentages of both groups decrease about the same rate after that. In the object-counting task, the Chinese children had a higher percentage of correct responses across all the sets and age groups with a significant difference in the large set where learning the teens names in English would have an effect.
Personal Reaction:
The implications of this study are endless. One of the interesting things that Miller et al state in their discussion is that the differences in mathematical reasoning among different cultures might be a result of this obstacle of language rather then the American educational system.
How we teach children numbers becomes important. The Chinese number-naming system has a clear base-ten system that allows for children to have a better understanding of what the number 1 means when it is in front of 5 in the number 15. Other studies have shown that when given 16 tokens and asked what the six meant, American students would pull out 6 of the tokens. Then when asked what the number 1 meant, students would pull out only one token and would have no response when the experimenter would ask what are the other nine tokens were for.
Keywords: Calculus, Measurement, Geometry
Ref: Ross15
Author(s): Morriss, P.
Date: 1998
Title: Discovering a Geometric Volume Relationship in Calculus
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91(4), p. 334-336
Reviewer: Ross
Date of Review: 4/13/03
Summary:
Patrick Morriss has convinced his advanced calculus students they are capable of original math which is crucial to his lesson plan. The feeling that all mathematics has been done is what he is trying to avoid when he connects geometry and calculus with volumes of solids of revolution. After learning the disc method and shell method, he leads his class on a journey of discovery and the volume of cones. The idea first originated when a teacher from the introductory-calculus class asked her students to use the shell method to find the volume of the region bounded by y=x2, y=0, and x=2 rotated around the y-axis. The discovery occurred when most of the students made the mistake of using the disc method and finding the volume of the region bounded by y=x2, y=4, and x=0. Both of the volumes are equal and it was his class's job to investigate and verify this which led to even more exploration. They soon discovered that the volume was half the volume of the circumscribed cylinder.
They generalized the bounded region using y=ax2, y=ar2, and x=0 and compared it to the volume of the circumscribed cylinder which has radius r and height ar2. They found that the volume of the cylinder was twice the volume of the truncated parabola. The class was on a roll and they examined cases of the function y=xn for n=3, 4, 5, . . .. After a night of working on those cases for homework, the next day they combined their work and developed a theorem. They found that The volume of a solid of revolution formed by revolving about the y-axis the region bounded by the graphs of y=axn, y=arn, and the y-axis, where a and r are positive constants, is n/(n+2) times the volume of the circumscribed cylinder.
Personal Reaction:
I have always enjoyed solids of revolution in calculus; especially using them to obtain the same formula for finding the volume of all the objects that we were told in geometry. This article is yet another activity that students can work on. The generalizations and testing extra cases such as n=3, 4, 5, . . . is great practice in variable manipulation as well as noticing patterns. The fact that there is a general statement for the volume of a cylinder circumscribed by a truncated curve is the icing on the cake.
Keywords: Discrete, Connections, Activities
Ref: Ross16
Author(s): Ruff, J. V.
Date: 2003
Title: A Millennium Prize Problem for Students
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 96(1) p. 26-29
Reviewer: Ross
Date of Review: 4/14/03
Summary:
The problem is P vs. NP. Proof is worth a million dollars. The math is beyond the abilities of a high school math class, but the concepts involved in developing the problem are not, and that is just what Ruff does with his class (possibly in the hopes that one of his students will win the million dollars).
Decision problems can be solved by a computer algorithm with a 'yes' or 'no' response. It takes time to solve decision problems; the time it takes to do one calculation will be multiplied by n when doing multiple calculations. Favorable algorithms have a polynomial-time model which increases slower and allows more calculations to be done faster then other algorithms with exponential-time models. A decision problem that can be solved with a polynomial-time algorithm is considered a P-class algorithm. An NP-class algorithm is the class of decision problems that can be checked using a polynomial-time algorithm. It would be ideal to prove that an NP-class algorithm is an element of a P-class algorithm so that calculations of exponential time models can be converted into the much faster polynomial-time models.
Ruff has developed a lesson plan that directs students in a discovery lab that inevitably leads to the statement of the P vs. NP problem. The story is a queen whose daughter, Theta, is having a birthday. As a part of the celebration, the queen employs an artist, a baker, and a bead stringer. Each year the artist is to make a square mosaic with the length and width equal to Theta's age. Similarly, the baker is to make cubecakes and stack them into a larger cake with the number of cubecakes in then length, width, and height equal to Theta's age, and the bead stringer is to make a necklace with two beads for that year and two for every year before, so the number of beads follows a 2n+1 model, where n is Theta's age. It takes each artisan time to complete each task, three minutes per tile in the mosaic, half a minute per cubecake, and five seconds to string a bead. After a few years the baker and artist begin to complain that it is taking them over 2 hours to complete their tasks while the bead stringer is done in minutes. The queen decides that they can switch jobs, but once they have changed they can never switch again. Ruff asks his class to determine if it is wise for the baker and artist to change jobs. After calculating a few values (like Theta's sweet 16) it becomes obvious that the hours it takes the baker and artist pale in comparison to the days of bead stinging that will take place before the princess is old enough to drive.
After Theta's third birthday, the queen decides that she wants to give Theta a tour of n towns in the kingdom and requires that the coach driver only visit each town once and then return to the palace. This map-route problem is the same as the Hamiltonian Circuit Problem and the root of the P vs. NP problem.
Personal Reaction:
The activities in this article can be done with students familiar with exponential functions. The first few exercises are a great way to show varying rates of change and how some functions start as 'greater than' or 'less than' but can change (sometimes rapidly). Travelling into the map activities can allow divergence to road maps of your city or imaginary vacations in other countries.
Keywords: Communications,
Ref: Ross17
Author(s): Rothbart, A.
Date: 1998
Title: Learning to Reason from Lewis Carroll
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 91(1), p. 6-10, 96
Reviewer: Ross
Date of Review: 3/14/03
Summary:
"Ordinary communication is not a science" writes Rothbart who continues by explaining the elements of context: body language and we are able to ask questions to clarify. Written language does not have the same cues and as a result different interpretations arise. "We cannot afford ambiguity when we communicate mathematical ideas", hence our need for inference.
Rothbart begins the article by first describing If-Then forms of sentences, how to rewrite premises into If-Then form, and then building a dictionary and representing the sentence symbolically. She continues much like a lesson plan into the rules of inference including contrapositive, double negation, and transitivity. Using all of these rules of inference you can connect statements and form a logical deductive response and then translate back into English. Finally, she includes many examples of Lewis Carroll puzzles and describes teaching strategies such as using Venn Diagrams.
Personal Reaction:
This is a great way of learning inference based deductive reasoning. It combines many of the zany premises created by Carroll and allows students to play around and develop a logical conclusion from them. It affords students the opportunity to work with Venn Diagrams and further symbolic manipulation while practicing how we communicate mathematics. To encourage creativity, it would be fun to have the students create their own, workable set of Carroll-like premises which could end up in a class book of creative math.
Keywords: Algebra, Representations, Teaching Strategies
Ref: Ross14
Author(s): Forringer, R.
Date: 2000
Title: (A+B+C)3
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(1) p. 6-8
Reviewer: Ross
Date of Review: 4/3/03
Summary:
(A+B+C)3 is a lesson in multinomial expansion. Forringer expects that after this lesson his students will never write (A+B)2=A2+B2 again. His begins by asking, "What is the area of a square whose side measures A+B?" and then has his students think about an answer. Usually students begin by drawing a square and then dividing each side into parts A and B. The result is two smaller squares, one area equal to A2 and the other area equal to B2, and two rectangles with area AB each. Summing together the four parts of the square results in a correct expansion of (A+B)2=A2+2AB+B2.
Forringer then expands on this with (A+B)3 and the concept of the volume of a cube. At this point he asks his students to go further and find the expansion of (A+B+C)2 using the square idea again. He asks the students to take the next step and find the volume of a cube with sides (A+B+C), however, he begins their exploration of the volume activity with different shaped blocks and has the students construct the volume themselves. Concluding the lesson the students work on finding a shortcut which leads them to discover that you can treat the trinomial as a binomial by letting X=(A+B) and Y=C, then substituting back. This shortcut can be proven using the blocks.
Personal Reaction:
I thought this was an interesting way to look at multinomial expansion. There is an engaging exploration for the students and gives them a great visualization for figuring out binomial and trinomial expansion. This process is harder to visualize for higher powers. If you don't limit yourself to squares cubes you can use this method for squaring or cubing other polynomials.
Keywords: Geometry, Connections, Curriculum
Ref: Ross13
Author(s): Kane, J. A.
Date: 1999
Title: Sharing Teaching Ideas: A Book of Creative Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(9) p. 800-801
Reviewer: Ross
Date of Review: 4/2/03
Summary:
The author of this article stresses the importance of writing as a means of communicating, especially in science and mathematics. She writes "creative writing and journals allow students to sort out ideas, reinforce their understanding of concepts, or discover weaknesses in their understanding of concepts." She has introduced an assignment into her curriculum where each student contributes their page into a class Book of Creative Geometry. The students are allowed to be creative with their writings as long as the content includes information about figures and their properties, what makes them special or unique, and fill in details about how the figure thinks and reacts. Submitted material included poetry, comics and cartoons, illustrations of nursery rhymes, greeting cards, and newspaper articles all created by the students.
Personal Reaction:
What an engaging project for the students to participate in. The final product is distributed among the students as a bound book which allows them to display and take pride in their work. The students are forced to gather information and make connections about the figures they work with. The Creative Book of Geometry project is fun and encourages students to explore the beauty and wonder of geometric shapes. I also think that this project could be expanded upon within each lesson or include other things like Euclid's propositions and postulates. It could also include history and information on geometers. I have seen a "Who Am I?" type poster which includes information on the character and works completed of mathematicians which could be included in the book and make cross-discipline connections as well.
Keywords: Geometry, Problem Solving, Connections
Ref: Ross12
Author(s): Manaster, A. B. & Schlesinger, B. M.
Date: 1999
Title: Geometry Problems Promoting Reasoning and Understanding
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(2), p.114-116
Reviewer: Ross
Date of Review: 03/19/03
Summary:
This article has four problems in Geometry, and their answers, that integrate concepts and techniques from other disciplines. Each question is a continuation of or includes the techniques refined from the previous questions. They are: (A) Find the dimensions of a rectangle with a perimeter of 30 inches and sides of integral length that has the largest possible area; (B) Find the dimensions of a rectangle with a perimeter of 30 inches that has the largest possible area; (C) Consider all rectangles with a perimeter equal to the circumference with radius of one meter. Find the dimensions of the rectangle that has the largest possible area; (D) Is the ratio of the rectangle area to the circle area whose circumference is equal to the perimeter always the same? Why or Why not? Based on the answer of question B, there was an additional question arising from a discussion of the results whose solution was included. It appears that given a perimeter, the rectangle with the largest area is actually a square and it can be proved so.
Personal Reaction:
The order of these problems presents a good outline for a lesson or two. It has the students using the algebra skills that they have acquired making cross connections. The only problem is there is no real world context within which the students would be operating and the problems end up being boiled down exercises. I read an additional article before I read this one (same volume but in the January issue) that discussed "perfect triangles" whose perimeter and area were equal and didn't write that one up because I was worried that it was too algebraic, however, after reviewing this article, there are some other problems that would fit equally well with triangles versus rectangles.
Keywords: Curriculum, Standards, Research
Ref: Ross11
Author(s): Martin, T. S., Hunt, C. A., Lannin, J., Leonard, W., Jr., Marshall, G. L., & Wares, A.
Date: 2001
Title: How Reform Secondary Mathematics Textbooks Stack Up Against NCTM's Principles and Standards
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 94(7), p. 540-545, 589
Reviewer: Ross
Date of Review: March 17, 2003
Summary:
The authors of this article gathered five reform textbooks and compared them to the principles and standards that were created by the National Council of Teachers of Mathematics. The texts they studied included Contemporary Mathematics in Context (Core-Plus), Integrated Mathematics Program (IMP), Math Connections: A Secondary Mathematics Core Curriculum, Mathematics: Modelling Our World (ARISE), and SIMMS Integrated Mathematics: A Modelling Approach Using Technology (SIMMS). In first comparing these texts to the traditional course textbooks, there were a few noticeable differences. The traditional books stressed "teaching by telling" and the exercises were problems stripped of any real-world connection and require specific answers. In contrast, the reform books emphasized problem solving and often require the teacher to have students work in groups. They were more likely to integrate technology and other disciplines into the lessons. While many traditional books focus on a single subject (i.e. Algebra), integrated books covered fewer topics while increasing student understanding.
Comparing the reform textbooks to the NCTM's Principles and Standards was broken down into two categories based on the Process Standards and the Content Standards. The books were rated on the five Process Standards which include problem solving, reasoning and proof, communication, connections, and representation. The raters examined a section of each book and rated them on a three-point scale for each of the processes. Results were positive for all five books but noticed considerably better numbers in the Core-Plus and SIMMS books. Similar results were found when the books were examined for the Content Standards, again with Core-Plus and SIMMS books having the highest marks.
Before concluding, the authors added a section on the distinctive features of each reform textbook and included an example from the text. Most of the books were four year programs with a focus on different concepts of problem solving. The conclusion was that all the books were aligned well with NCTM's Principles and Standards and that teachers need to figure out which book best applies to their individual school's mathematics program.
Personal Reaction:
I felt myself cheering for the Core-Plus and SIMMS books because they had higher numbers and better scores when compared with the standards but after reading the "Distinctive Features" section and the conclusion it became apparent that any of the texts that were rated would no better than the others and depending on the program that a school has would either fit in or not. Each book was funded by the National Science Foundation and approaches mathematics from different values. The IMP program focuses units and problems around one central problem. Core-Plus values learning through investigation and ARISE values learning mathematics by developing models for real-world situations. If your department feels that enriching an understanding of the symbolic representation of mathematics then Mathematics Connections would fit well in your school. Different from all the other books, SIMMS allows you to choose between two tracks, Math and Science or Statistical concepts, organized into six levels. Their book requires more reflection from the students on the limitations of the models they develop.
One of the points in the conclusion that I thought was interesting (and something that I saw when my high school switched over) was that the students weren't ready to thrown into that style of learning. The authors said that moving from traditional curricula to the integrated curricula (or vise versa) would be difficult for the students. It seems that making the switch should have been a little more gradual than having students graduate from traditional courses in junior high to the integrated curricula in high school.
Keywords: Algebra, Standards, Technology
Ref: Ross10
Author(s): National Council of Teachers of Mathematics
Date: 1995
Title: Curriculum and Evaluation Standards for School Mathematics: Algebra in a Technological World
Journal or Publisher: National Council of Teachers of Mathematics, Inc.
Volume, Issue, Pages:
Reviewer: Ross
Date of Review: 03/12/03
Summary:
The purpose of this book is to aid in interpreting and implementing curriculum and evaluation standards for Algebra in the high school setting. Based on teaching the standards, the book takes a focus on a function approach to learning/instructing Algebra.
The structure of the book places a new topic of focus in each chapter that ends in some sample activities that can be used in the classroom or developed into a topic that the students may have more interest in. The chapters are broken down into subcategories that outline what a student should understand after the course. For example, the chapter "Extending a Functional Approach" has subcategories on several functions with one or several variables, composite functions, and dynamical systems.
Personal Reaction:
While reading the chapters, I felt that the organization of the chapters was geared to instructing to the activity at the end of the chapter and less about giving examples of activities and more about pushing the use of their activity. They did, however, use very real world situations and stressed the use of technology in calculating and picturing their results. In comparison the Navigating books, I felt they were less organized and more focused on individual topics rather than giving broader topics to borrow for more in depth exploration. One difference that I did appreciate from the Navigating books was code to a simple calculator program that could be integrated into the lesson as a method of finding orbits of functions. Calculator programs are great tools to save time and get to the heart of a topic without the long and sometimes difficult calculations.
Keywords: Geometry, Standards, Curriculum
Ref: Ross9
Author(s): National Council of Teachers of Mathematics
Date: 2001
Title: Navigating through Geometry in Grades 9-12
Journal or Publisher: The National Council of Teachers of Mathematics, Inc.
Volume, Issue, Pages:
Reviewer: Ross
Date of Review: 3/10/03
Summary:
Navigating through Geometry does just that. The book begins with an introduction that restates the standards of Geometry. It continues into well organized chapters that cover topics of transformations, map making, projective geometry, and dimensions.
Each chapter begins by outlining some basic concepts and definitions, moves into a lesson plan with the goals, materials needed, and then some topics to facilitate discussion. Then the chapter continues in the same manner with more concepts and definitions and another lesson plan.
As an appendix to the guide, they include the Black Line Masters worksheets and activity guides (for the students) which are provided in case you want to use their lesson plan (however I got the impression these were places to start to build on). They also include the solutions to each of their projects and the end of the book.
The book also comes with a CD-Rom that includes applets and games that allow students to explore, in a creative and more in depth fashion, the topics covered in each chapter of the book. There are some articles from various periodicals that contribute more information than is covered in each chapter and most of the Black Line Masters worksheets are on the disk as an Acrobat file.
Personal Reaction:
This book, as it is meant to be, is a wonderful resource for designing the curriculum for a course in Geometry. The lesson plans, additional materials, and the solutions all give a teacher a clear and well plotted lesson plan or something to work with and improve upon that promotes a constructivist approach to teaching and learning Geometry.
The CD-Rom that accompanies the book has additional supplemental material. I looked at the Tessellation Exploration applet which allows you to draw your own or open many that are already done, view a tutorial or a slide show, and there are some other activities that promote understanding of transformations such as line reflections, rotations, and glide reflections.
Keywords: Calculus, Activities,
Ref: Ross8
Author(s): Cipra, Barry
Date: 1989
Title: Misteaks ...and how to find them before the teacher does...
Journal or Publisher: Academic Press, Inc.
Volume, Issue, Pages: Second Edition
Reviewer: Ross
Date of Review: 3/5/03
Summary:
Using knowledge of basic calculus and examples of integrals and derivatives, this book is organized into nine chapters with each one focusing on a mistake that is commonly made by students (and heaven forbid teachers) and how to spot it. The first chapter discusses methods of recognizing when a negative is lost, in particular with problems where the answer is in units that shouldn't be negative such as volume, area, and distance. The second chapter suggests plugging in values to double check that an output really is what you would expect, for example, getting a positive output from a derivative of a function that is increasing. Bounding your answer with easily found areas of geometric objects is the topic of the third chapter. This geometric bounding can be done easily, as Cipra notes, by drawing a picture or a rough approximation of the function.
Making crude estimates (based of course on educated knowledge of derivatives and integrals) is the topic of chapter four. It is reasonable to simplify problems to check your work and work well except on occasion when they neglect reality of practical answers. Chapter five highlights a very effective method for determining accurate answers. Taking problems that involve specific numbers can be converted into general formulas which simplify your thinking. Using your general formula you can check "special cases" such as zero or one to see if the results coincide with a realistic expectation. Another convenient application of this simplification is remembering derivative facts by replacing a large equation for instance: x4-3x3+7x2+1 with a simpler one like: x2. One other advantage that works in appropriate situations (you'll have to read the chapter to figure those out) is to associate a dimension like feet to the variable. This dimension allows a person to check that every "foot" is conserved through a derivative or integral.
Symmetry is a shortcut suggested by Cipra in chapter seven. The benefits of symmetry are in applications of repetition of similar computations such as partial derivatives, expanding or factoring polynomials, deriving formulas, and quickly solving integrals of even functions or odd functions. Expectations for a result are key to understanding where a mistake is made. Chapter eight is all about expectation. Cipra suggests a general rule when differentiating or integrating functions: exponentials and square roots don't disappear or morph into new polynomials when differentiated or integrated and there is certain permanence to trigonometric functions. The final chapter outlines hints for taking a test. There are six common mistakes made, loss of a minus sign, dropping parentheses, coefficients disappearing, missing or damaged exponents, multiplying when you want to divide and vise versa, and uncontrollable computations (remember you only have an hour to complete a test and the teacher isn't going to give you a question that requires difficult computations to waste your time).
Personal Reaction:
The entire time I was thinking about how this book could be used in a class room and thankfully in the spirit of any mathematics textbook, each chapter ends with a few exercises to practice weeding out the error. Using those exercises in conjunction with your lessons promotes an in depth understanding of how derivatives and integrals work. This book would be a great companion to any calculus. The book is simple and geared to the student who has an introductory understanding of derivatives and integrals, however, some chapters can be used before derivatives are covered, others can be read and used while working with derivatives, and the rest of them after an introduction to integrals, making this book an excellent addition to the span of the course curriculum.
Cipra uses many appropriately placed jokes to humor the reader and convey his point. He uses his own methods (and promotes combing them) in later chapters to fill out their use and show how each can be used as a powerful tool to thin out the field of wrong answers. And in the end Cipra leaves you wondering "what kind of fool am I?"
Keywords: Standards, Technology, Teaching Strategies
Ref: Ross7
Author(s): Hudnutt, Bethany S., & Panoff, Robert M.
Date: 2002
Title: Mathematically Appropriate Uses of Technology
Journal or Publisher: On-Math: Online Journal of School Mathematics
Volume, Issue, Pages: Winter(2002)
Reviewer: Ross
Date of Review: 2/28/03
Summary:
There is very little controversy over whether or not mathematics instruction should integrate technology into the classroom. The authors of this article agree but also see the difficulties in appropriate uses of these technologies. Often teachers feel that they could better teach topics without the aid of technology and thereby circumvent technical problems, errors, or time issues. The authors understand this, but they also suggest a website that can solve almost all your teaching with technology anxieties. They suggest the free web service dedicated to providing robust interactive applets for teaching math concepts. The website is run by Shodor Education Foundation called Project Interactive which has the goal to produce courseware that simultaneously presenting material and promoting exploration. The rest of the article has some sample applets that can be found at the website linked above. Some of the courseware examples include topics of probability, coordinate plotting, and some coefficient manipulation in functions. The Project Interactive website also has links for further exploration and information, areas for students and teachers, and lesson plans created to use the technology and meet the standards or textbook that is being used in your classroom.
Personal Reaction:
This article reinforced the importance of technology in classrooms and suggests effective uses of technology as a device to enhance a lesson. I spent some time on the Project Interactive website which is geared towards the early and middle school levels of math instruction but can be integrated into a high school math lesson for classes that need a refresher or are experiencing it for the first time. Part of Project Interactive is taking teacher suggestions and creating courseware that requires few directions and requires no support. The website will always be changing and a great resource at this point for almost any middle and elementary school math teacher.
Keywords: History, Planning,
Ref: Ross6
Author(s): Lee-Chua, Queena N.
Date: 2001
Title: Mathematics in Tribal Philippines and Other Societies in the South
Pacific
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 94(1), p. 50-55
Reviewer: Ross
Date of Review: 2/25/03
Summary:
This article is written on something near to Lee-Chua who is of Filipino heritage. Often mathematics courses who survey the history of mathematics focus primarily on Egypt, Babylon, China, Greece, Renaissance Europe, and modern day technology, but this article examines the development of mathematics on the islands of the South Pacific. The article discusses 4 topics in which there is evidence of mathematical reasoning; counting and measuring, time, geometry, and logic. Examples of mathematics show in the games played, farming calendars, art, and hunting strategies. In the Philippines counting was heavily based on body parts and was not limited to fingers and toes. Lengths and sizes were described in units based on paces, hand lengths, finger lengths, and arm spans that provide evidence for the body part counting strategies. Tallying with knots on rope or notches in bone and stone were used to track phases in the moon and their understanding of the duration of a month. Similarly, the inclination of the midday sun tracked the seasons and organized planting and harvesting. Geometry is used in home construction and in the patterns in cloth, weapons, and tattoos. The notion that two lines intersect at a point was understood by turtle egg hunters who would use the mother's tracks in the sand to find their point of intersection and location of the eggs. Finally, mathematical logic is apparent in many of the games played by children and adults of the islands.
Personal Reaction:
I really think this article does an excellent job of organizing the development of mathematics in the Philippines. I had forgotten that many important everyday things, like keeping time, stemmed from mathematics. This article incorporates history with mathematics as well as suggesting possible alternates for countries of origins for mathematical thought. For Lee-Chua this is an opportunity to feel pride in their ethnic and ancestral achievements. This could be a possible avenue for getting diverse students, particularly Filipino students but could easily be altered for others, interested in math.
Keywords: Standards, Assessment,
Ref: Ross5
Author(s): Romagnano, Lew
Date: 2001
Title: the Assessment Standards for School Mathematics: The Myth of
Objectivity in Mathematics Assessment
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 94(1), p. 31-37
Reviewer: Ross
Date of Review: 2/21/03
Summary:
How do we maintain objectivity when writing an assessment exam for our class? Romagnano begins by stating that objectivity can never be achieved and that anything created by a person is subjective. He writes about three examples of tests, a quadratic question written by a teacher, a calculus questions on the AP Calculus test, and the mathematical reasoning section on the SAT-I test, and how each of these tests compare on objectivity.
One main point this article makes about assessment, is not objectivity but, validity. Does the exam test what you are trying to assess? Are the answers to the questions provided by the students representative of a knowledge and understanding of the material? These are important questions to ask when preparing your own exam.
Romagnano uses the AP and SAT tests to demonstrate methods of improving objectivity. The grading process for the AP test is done by two trained graders (working independently) who are assigning points based on a defined rubric for the problem. When a rubric is used grading exams, it removes some grader bias and adds equity to the assessment. The SAT tries to increase objectivity by standardizing the test. Using the same instructions, amount of time given, environment, and the norm-referenced scoring make the test objective in those variables, however, the testers assume that a taker possesses a certain amount of knowledge when they are designing the test and this according to Romagnano is a subjective method of test design.
Personal Reaction:
This article demonstrated to me that writing an "objective" exam can be a difficult task. First, I need to make sure that my exam questions are representative of what I'm trying to assess. There are helpful hints to increase the likelihood of this, for example, asking understanding questions rather than ones with simple "solve" directions. Second, after administering the test I need to score it. Again Romagnano suggests that sharing your rubric with the students before the tasks clarifies what your expectations are and makes meeting them more likely. I felt that his conclusion, that while no objectivity can be obtained, teachers should try to make the method of assessment consistent and useful in determining understanding, was important to remember.
In the appendix of the article there is a calculation of the variance and reliability for the
SAT test. I never made the connection between the norm-referenced scoring and statistics, but
like the example in the article pointed out; a student scoring 470 and one scoring 530 on the
math portion may not possess any difference in mathematical understanding due of the
unreported 95% confidence intervals for each score.
Keywords: History, Puzzles, Technology
Ref: Ross4
Author(s): Mahoney, John F.
Date: 2003
Title: Benjamin Banneker's Mathematical Puzzles
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(2), p. 86-91
Reviewer: Ross
Date of Review: February 19, 2003
Summary:
In honor of African American History Month, Mathematics Teacher published this article on Benjamin Banneker. Banneker was a mathematician, astronomer, scientist, and scholar in a time of American History when African-Americans were regarded as inferior and still enslaved all over the country. Born a free man, Banneker taught himself the tools of algebra, geometry, and other formulas used in his study of planetary motion. He published a Farmer's Almanac for 6 years predicting the phases of the moon, eclipses, and weather. Along with his almanac, Banneker also kept a journal of math puzzles for which he had solved and others that he had no solution to. Four of these puzzles are discussed in this article and how modern technology can be used to solve them. The first and third puzzles deal with solving systems of equations; the second, an application of the Pythagorean Theorem; and the fourth puzzle is written as a poem with rhyming couplets.
Personal Reaction:
This article has the potential to integrate history into a fun lesson of puzzle solving. The history of Benjamin Banneker's life is fraught fun trivia and amazing accomplishments. There is good documentation (including a letter from Thomas Jefferson) on the sentiment of slavery in that era and the struggles that Banneker had to deal with to achieve what he did.
The problems presented in this article use systems of equations and geometry which would make a fun activity for students who have learned the Pythagorean Theorem, volumes, or Algebra students that just learned solving systems of equations. The clever wording of the problems disguises the math and makes solving the puzzles fun.
Keywords: Standards, Curriculum, Teaching Strategies
Ref: Ross3
Author(s): Bay-Williams, Jennifer M. and Martinie, Sherri L.
Date: 2003
Title: Spotlight on Standards: Thinking Rationally about Numbers and
Operations in Middle School
Journal or Publisher: Mathematics Teaching in the Middle School
Volume, Issue, Pages: 8(6), p. 282-87
Reviewer: Ross
Date of Review: 02/17/03
Summary:
"Thinking Rationally About Numbers and Operations in Middle School" addresses the standards of developing an understanding rational numbers "through experience with many problems revolving a range of topics" (NCTM 2000; p.212).
The topics that the authors discussed were Understanding Rational Numbers, The Meaning of Operations, and Proportionality. Their main focus was on how each of those topics could be taught using various contexts and real-world problem situations to allow the students to develop the algorithms for themselves. The goal aimed for in Understanding Rational Numbers was for students to visualize where fractions and decimals are in relation to each other on the number line. Finding the algorithms for multiplying and dividing fractions was the focus of Meaning of Operations. Similarly, discovering cross-multiplication or a process of their own as a way of finding relationships or comparisons is the goal in Proportionality.
Personal Reaction:
This article is my first experience with any discussion on how teaching the standards of
mathematics are done. I felt the authors laid out a clear discussion of the standard and
interpretation on the important topics to cover. They gave many suggestions on how to teach
each topic and outlined how further questions of situations could be developed for use in the
classroom. They also provided many examples of how the instruction method is received by
students and that they do in fact develop the algorithms that are aimed for in the lesson.
Keywords: Algebra, Problem Solving, Connections
Ref: Ross1
Author(s): Szetela, Walter
Date: 1999
Title: Triangular Numbers in Problem Solving
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 92(9), p. 820-824
Reviewer: Ross
Date of Review: February 10, 2003
Summary:
Szetela shows throughout this article the many applications triangular numbers have in solving problems and the benefits of a classroom lesson involving these numbers. A triangular number is the partial sum of the sequence of natural numbers or n+1 objects taken two at a time (n+1C2). Therefore the nth triangular number is n(n+1)/2. He shows that triangular numbers are not limited to triangles but can be used in solving many other problems involving other geometric figures such as circles and squares and he suggests further questions for exploration.
Personal Reaction:
This article is an excellent resource for activities that can be used as a way of reinforcing or practicing the topic of arithmetic sums. Triangular number problems are a fun physical way to practice finding patterns in sums and converting the sums into something recognizable and easy to calculate an nth term in the sequence. The physical problems like the cannonball pile or rectangles in a checkerboard also help visualize where sums like these occur in the "real world."
Keywords: Calculus, Curriculum,
Ref: Ross2
Author(s): Greive, Cedric E.
Date: 1999
Title: Sum of i-Squared and the Volume of a Cone
Journal or Publisher: The Mathematics Teacher
Volume, Issue, Pages: 92(9), p. 825
Reviewer: Ross
Date of Review: February 12, 2003
Summary:
This article has two sections. The first is the derivation of the formula for finding the volume of a right circular cone, and then incorporating the activity into the classroom.
The derivation is quite simple based on splitting the cone into a tower of n-disks of equal height and radius changing at a rate or r/n. The formula ends up being equal to pi r-squared h divided by n-cubed times the sum of i-squared from 1 to n.
The suggestions for the classroom are to give the students the information about the n-disks of equal height and depending on how much time you want to spend, allowing them to derive the change in radius from similar triangles and then deriving the formula for themselves. Or there is a worksheet suggestion that has the student find the radius and volume for increasing i-values.
Personal Reaction:
This article offers a great introduction to the study of Calculus. Understanding Integrals and Derivatives relies on the concept of steps and how their changing affects the outcome. The activity of finding the volume of a right circular cone by splitting it up into n equal parts will start the students thinking of how changing steps changes outcomes and how an infinite number of steps produces the exact value.
This same demonstration can be returned to the classroom at a later time when students begin considering the volume of 3-D objects rotated around an axis. The volume of the discs used in this activity can be related to a similar cone generated from rotating the line y=mx about the x-axis.
Finally, there are helpful suggestions for teaching this activity that will improve the
facilitation and overall comprehension of the activity.