Keywords: Calculus, Curriculum, Assessment
Ref: Paul20
Author(s): Riedesel, M.; Goerdt, S.
Date: 2003
Title: AP Calculus/Calculus Sharing Session
Journal or Publisher: 2003 Minnesota Spring Mathematics Conference
Volume, Issue, Pages:
Reviewer: Paul
Date of Review: 4/28/03
This session was mostly an open discussion among high school and college calculus teachers. The speakers began by distributing the results of a survey that had been completed by 28 local colleges. This was used to elicit discussion from the audience. One of the first things Sonja Goerdt mentioned was that many calculus topics formerly covered on the Calculus AB exam have been shifted to the BC exam. Concepts such as the shell method, work, and L'Hopital's Rule, which I recall covering in my AB course, now belong to the BC curriculum. This change surprises me, because I always envision curriculum changes demanding that students learn information at younger and younger ages.
There was a good deal of discussion regarding calculus exams. As far as AP exams are concerned, the group's general consensus was that students who receive a 4 or 5 should go straight to Calculus II in college, while those who score a 3 or less should enroll in Calculus I. Gateway exams were another assessment topic. These are exams which prohibit calculator use to assure that students can solve a wide range of limit, derivative, and integral problems by hand. After students pass each gateway exam, teachers generally allow more calculator use to save students time in solving problems. From the discussion, it appeared that many college professors give a few gateway exams throughout their courses, while no high school teachers present used such exams.
Alternative calculus assessments were another big issue in the session. Sonja stated that she gives regular written assignments in her classes, with topics such as comparing the integration approximation techniques and discussing the role of the limit as the foundation of the derivative and integral. Another teacher added that he incorporates oral presentations into his classroom activities. One college professor distributed several examples of his exams, lab experiments, and other projects. He invited math teachers to visit his website at www.mnstate.edu/peil for numerous furt her examples of his work. High school AP teachers felt very strongly that the period of time after the AP exam is best for beginning work on a major project or written assignment.
Keywords: Algebra, Activities, Keyword 3, Optional...
Ref: Paul21
Author(s): Larson, R
Date: 2003
Title: "Modeling Real-Life Data: Graphically, Numerically, and Analytically"
Journal or Publisher: Minnesota Spring Mathematics Conference
Volume, Issue, Pages:
Reviewer: Paul
Date of Review: 4/30/03
Ron Larson's session on modeling real-life data was beneficial for teachers seeking a way to explore and connect mathematical models in, say, an Algebra II course. Larson presented a set of six worksheet activities that help students discover linear, quadratic, cubic, square root, exponential, and logarithmic models for themselves. Each worksheet begins with a real data set, such as the carbon dioxide level of Earth's atmosphere since the 1950s.
As the title of the session states, the worksheets have students examine the data using graphical, numerical, and analytical approaches. First, students work analytically, perhaps finding a pattern by investigating differences or ratios. Next, a coordinate plane is provided so that students may plot the data points and examine the approximated curve they create. Finally, students use the regression feature on their graphing calculators to actually write the mathematical model.
Larson strongly emphasized using the graphing calculators after the numerical and analytical approaches. This way, students learn how to identify the type of model that best fits the data on their own and then construct the specific model using technology.
One of the main strengths of these activities is that they use interesting data that are not form-fitted to perfectly match a particular model. Students begin to realize that real-life data are rarely as neat and clean as seen in many lower level textbooks. They also discover that most sets of data possess characteristics of several different math models. The graphical section is well prepared, with pre-labeled coordinate planes that help students connect the algebraic and graphical representations of each model. In general, I agree with Larson's "try it first, calculators later" method. These activities are designed to be workable by hand, so there is no reason to let students have calculators do the work for them without understanding the process. I highly recommend these activities to algebra teachers, and they can be foun d on the web at www.RonLarson.us (E-mail: odx@psu.edu).
Keywords: Communications, Curriculum, Keyword 3, Optional...
Ref: Paul17
Author(s): Williams, N.B.; Wynne, B.D.
Date: 2000
Title: "Journal Writing in the Mathematics Classroom: A Beginner's Approach"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(2), p. 132-135
Reviewer: Paul
Date of Review: 4/17/03
This article describes a joint experiment performed by two high school mathematics teachers. In accordance with the NCTM's Curriculum and Evaluation Standards, the authors felt they needed to add an element of written communication to their respective curricula. Therefore, each teacher selected one class in which to implement weekly journal writing prompts. They alternated between affective and mathematical prompts, so students could communicate both their knowledge of the content and their feelings about the classroom environment. The authors found it most successful to give students ten minutes at the beginning of class to start their writing and have students turn in the entry (approximately one page) at the end of the week. To assure fairness, they distributed a syllabus that described the purpose and grading procedure for the journal writing. The authors affixed half of the grade for each entry to its mathematical content. The other half consisted of proper form, neatness, vocabulary, and length. At first, students complained that written work had no place in a math course, but by the end of the journal experiment, most students voted to continue the journal writing until the end of the grading period.
Like Williams' and Wynne's students, I did not see the value in math journal writing until I learned more about it. Until now, I viewed such writing assignments as a small consolation for students who are convinced that they have strong verbal skills and weak math skills. Given my recent experiences with proof writing and tutoring students who fly through skills exercises without understanding what they are doing, I see that verbal and written explanations of math concepts are essential in truly checking for student learning. It is a pity that written work takes significantly more time for teachers to grade, but I am convinced that the benefit outweighs the additional cost. I was reminded of my Discrete Math class while reading the authors' sample mathematical prompts. They often asked stude
nts to develop several sample problems, say applying the Pythagorean Theorem, and then solve their own problems. As I am finding out this semester, writing good original problems can be difficult. This is another clever way to give students a glimpse of the responsibilities of a teacher.
Keywords: Technology, Probability, Activities
Ref: Paul18
Author(s): Whitney, M.C.
Date: 2001
Title: "Exploring the Birthday Paradox Using a Monte Carlo Simulation and Graphing Calculators"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 94(4), p. 258-262
Reviewer: Paul
Date of Review: 4/21/03
Matthew Whitney discovered a feature of the TI-83 calculator that can be used in an activity that introduces probability in an exciting way. The TI-83 possesses a random-integer generator, which can be used to create random birthdays in order to investigate the Birthday Paradox. This counterintuitive probabilistic event states that more than 50 percent of the time, when groups of random strangers are assembled, only twenty-three people are needed to find a matching pair of birthdays. Whitney consulted the TI-83 because he did not have twenty-three students in his class and feared that students would be unimpressed. He recommends creating three lists using the STAT menu on the calculator. One list generates twenty-three numbers ranging from one to twelve to represent the months. The second list contains random numbers ranging from one to thirty-one to represent days of the month. The third list uses the Sort feature to combine the first two lists and make it easier to find birthday matches (i.e. 116 represents January 16th). Eventually, students can combine their results and divide the number of successful trials by the total number of trials to discover the truth of the paradox.
This article brought me back to one of the first days of my Probability Theory course, when each student reported his birthday and we indeed found two matching days within the class. This method would work well for classes of twenty-three or more students, where the teacher's probability of success is more than 50 percent. However, Whitney's activity is applicable in classes of all sizes, and it is fail-proof, because with enough trials, the desired probability will always approach approximately 50 percent.
As I read this article, I wondered about the availability of graphing calculators in future classrooms. Will the TI-83 still be commonly used, two years from now? How available will such graphing calculators be in public school classrooms? In general, I would say both of these potential obstacles will be surmountab
le. If the TI-83 is outdated by that time, teachers will simply need to adjust their calculator instructions to fit the most prevalent models. As for availability, I hope it does not get to the point where there are not enough calculators for students to work in pairs with one calculator. If this is the case, a teacher could resort to projecting one calculator onto an overhead screen and completing the activity as a large group.
Keywords: Discrete, Problem Solving, Algebra
Ref: Paul19
Author(s): Gannon, G.E.; Martelli, M.U.
Date: 2001
Title: "Discrete Dynamical Systems Meet the Classic Monkey-and-the-Bananas Problem"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 94(4), p. 299-301
Reviewer: Paul
Date of Review: 4/23/03
This article presents an extended solution to a well-known problem using discrete dynamical systems, a relatively new branch of mathematics. The three-sailors-and-the-bananas problem involves a large pile of bananas and three distrustful companions. In the middle of the night, one sailor awakens and divides the pile into three equal shares, hiding his own and giving the leftover banana to a monkey. The second sailor then awakens and goes through the same procedure with the reduced pile, followed by the third sailor later in the night. In the morning, the sailors divide the remaining bananas equally and give the extra banana to the monkey. What was the minimal number of bananas at the beginning? Guessing and checking or using a standard system of equations yields the answer of 79. But what if there are four sailors, five sailors,..., n sailors? The authors take the reader through a method of using point-slope form for three, four, and five sailors, which yields a general formula for the minimal initial number of bananas. This discrete dynamical system relies on a composition of functions, since each sailor does what the previous sailor did.
Initially, I chose this article simply to add a classic math problem to my repertoire. I was not particularly looking forward to trudging my way through the abundance of algebraic equations in the article. In the end, I learned a valuable new technique that could be introduced in an algebra course to incorporate an engaging word problem into the curriculum. The basic algebraic approach to the three-sailor problem could be examined in an introductory algebra course, while this approach could be used as review in an advanced course before moving on to the discrete dynamical systems method.
The more advanced procedure highlights general problem solving strategies, such as starting with a given number of subjects (sailors) and gradually increasing this number until a pattern is found. Then, the general formula can be obtained, offering a chance for students to become more familiar with terminology such as "n sailors," the "ith sailor," and the subscript notation used to label terms and equations.
Keywords: Problem Solving, Discrete, Activities
Ref: Paul16
Author(s): Miller, C.M.
Date: 2000
Title: "Student-Researched Problem-Solving Strategies"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(2), p. 136-138
Reviewer: Paul
Date of Review: 4/14/03
Catherine M. Miller used to set aside one day of class to describe various problem solving strategies to her class. However, she found that this list of strategies held little meaning for students, because the techniques were not applied to specific problems or situations that students would remember. In response, the author created a project that can help students from the middle school to the college level discover and experience effective problem solving strategies. The first stage involves research, in which each student takes several assigned problems and finds three individuals, such as friends, parents, or siblings, who are willing to solve one or more problems. Students must watch their subjects as they solve the problems, documenting both their strategies and techniques as well as their behaviors and emotions. Then, as a class, students share with one another the strategies they observed. This way, they internalize the strategies and assign their own name to each one. The teacher also creates a chart that students fill in with the helpful/harmful emotions that can arise while solving a problem. With this foundation set, the teacher can inform students that they will be expected to clarify which strategies they use while solving problems later in the course.
This article definitely hit home for me, because I am one of those people who lets himself get bogged down mentally if I am unable to solve a problem reasonably quickly. In addition, I have had teachers who listed out all sorts of problem solving strategies at some point in a lesson, and this list was not something I found valuable enough to use regularly. A general strategy, such as "use algebra," means next to nothing unless it is put into the context of a sample problem. Miller's project takes this one step further, in that students can observe people of varying problem solving skill putting such techniques into action. I agree with the author that the project should be implemented at the beginning of any discrete math course or u
nit, because these strategies can save students a lot of time and frustration. Problem solving is one of the strongest connections between math and "real life," and this project has the potential to prove to students the widespread importance of mathematics.
Keywords: Probability, Activities,
Ref: Paul15
Author(s): Szydlik, J.E.
Date: 2000
Title: "Photographs and Committees: Activities That Help Students Discover Permutations and Combinations"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(2), p. 93-99
Reviewer: Paul
Date of Review: 4/9/03
In this article, Syzdlik presents three dynamic activities designed to teach permutations, combinations, and the connections between the two. These activities are based upon the NCTM's Standards and are highly inquiry-based. In the first activity, students must find the number of ways to arrange four people in a straight line for a photograph. The second activity asks students to calculate the number of committees of various sizes that can be formed from a group of four students. Here, unlike in the first activity, students learn that "order does not matter." The latter stages of these activities have students generalizing their findings to a subset of r students within a group of size n. In this way, students derive the general formulas for a permutation and a combination on their own. The third activity helps students find the connections between permutations and combinations by using physical demonstrations with student volunteers. Whereas nearly all students usually understand the first two activities, Szydlik notes that about half will probably master the third. Perhaps the most challenging portions could be used as extension activities.
Strong evidence of a discovery-based approach is apparent when the teacher does not introduce the formal notation, P(n,r) and C(n,r), until students have derived the two definitions for themselves. Note the nice coincidence that "photograph" and "committee" begin with the same letters as "permutation" and "combination," so students have a personalized way of differentiating the two concepts from one another. The above notation is different than that which we learned in Probability Theory, but one advantage of the author's notation is that it is identical to the one used by students' calculators. Finally, Szydlik concludes her activities with a "problem write-up," in which students describe a problem in such a way that someone who had never seen it would understand it. This written component is a practice too rarely used in math classes and adds nice closur e to the author's modern, carefully coordinated activities.
Keywords: Measurement, Activities,
Ref: Paul14
Author(s): Johnson, A.
Date: 2000
Title: "The Jurassic Classroom"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 93(2), p. 102-113
Reviewer: Paul
Date of Review: 4/7/03
Art Johnson has developed a hands-on activity that uses measurement and analysis to teach proportional reasoning and similarity. Johnson recommends the activity for a typical high school geometry class. He emphasizes its usefulness in fostering an understanding of proportional relationships between solid figures, whereas students usually envision proportional reasoning as an algorithm to solve for a missing term. The activity uses growable dinosaurs, which expand to about six times their original length in twenty-four hours. Using rulers, graph paper, and graduated cylinders, students regularly measure the length, area, and volume of the dinosaurs and can eventually predict their next measurements. By the end of the experiment, students also discover the ratios between corresponding lengths, areas, and volumes. Johnson includes a set of five worksheets with this article, designed to lead students toward an understanding of proportions. He concludes the article by addressing some more general proportion-related questions, such as why a mouse can fall from a height of twenty feet and scurry away unhurt while a human falling from a proportional height would not be so lucky.
When I first skimmed through this article, I had doubts about the activity being challenging enough for high school students. However, given the intricacy of the required area and volume measurements, Johnson may be right on target. Students must do some careful estimating in using graph paper to measure area, as the dinosaurs' shapes are fairly irregular. To measure volume, the activity utilizes a water displacement method, where the amount of water displaced by the dinosaur is equivalent to the dinosaur's volume. At one point, students must do this using a full water container and measuring the overflow when the dinosaur is placed on top. I remember using this technique in high school, myself.
In addition, I liked the practical problems that Johnson addressed at the end of the article. I had not considered that the solution to the mouse problem involved surface area and air resistance. Another problem uses proportions and our knowledge of human bones to explain why a human the size of Paul Bunyan would not be able to stand or function properly. These types of thinking link mathematics to both physics and biology.
Keywords: Algebra, Connections, History
Ref: Paul13
Author(s): Lesser, L.M.
Date: 2000
Title: "Reunion of Broken Parts: Experiencing Diversity in Algebra"
Journal or Publisher: Mathematics Journal
Volume, Issue, Pages: 93(1), p. 62-67
Reviewer: Paul
Date of Review: 4/1/03
The Arabic root of "algebra" means "the reunion of broken parts." This reminds us of the lack of personal connection with algebra that students often feel. Diversity helps students form both mathematical and personal connections with algebra. The author first examines diversity through history. For instance, algebraic inequalities can be related to gender inequality. Two expressions, representing average women's and men's salaries over past years, can link math to democracy. Diversity through multiple representations is also discussed. For example, the ancient Greeks often solved quadratic equations geometrically rather than our modern algebraic approach. Finally, Lesser touches on diversity through the object concept of function. A useful activity here is to present students with several functions and have them classify the functions as even/odd, strictly increasing/decreasing, continuous, 1-to-1, etc. Rarely do all functions on a list all share a single property, and this can be related to topics such as stereotyping members of an ethnicity or other group as all possessing a certain characteristic. In general, this article can help teachers make algebra more interdisciplinary without taking up significantly more class time.
While reading this article, I wondered where Lesser's goals fit in comparison to those of integrated curricula authors. After all, if algebra is taught using an integrated approach, students should not feel the need to ask, Why do we need to learn this? My conclusion is that exploring diversity in algebra should be a regular component of an integrated curricula. In other words, the applications and constructivist activities that headline integrated teaching should occasionally be based upon math history or multiple representations of a concept. I feel it is important to stress that these explorations can be done without using extra class time. Veteran teachers likely know that time is precious in the classroom, and a class discussion about gender equality or stereotyping is not a top priority. However, by coordinating allusions to such themes through relevant math activities, a quick reference to another discipline is all that it takes.
Keywords: Curriculum
Ref: Paul11
Author(s): Laser, J.K.
Date: 2003
Title: Core-Plus Mathematics Project
Journal or Publisher: Glencoe/McGraw-Hill Publications
Volume, Issue, Pages: http://www.wmich.edu/cpmp/index.html
Reviewer: Paul
Date of Review: 3/17/03
This is the official website of the Core-Plus Mathematics Project. The CPMP is funded by the National Science Foundation and is responsible for the development of a three-year, Standards-based high school mathematics curriculum, entitled Contemporary Mathematics in Context: A Unified Approach. The website includes fairly standard features, such as an overview of the curriculum and descriptions of its four courses, as well as more extensive information, like recent reports of the curriculum's impact upon student test scores and hints for optimal implementation of CPMP materials.
The "Overview" portion of the website is a useful tool for obtaining a brief but accurate understanding of the CPMP curriculum. As with other integrated curricula, these materials support educational ideals such as unified content, active learning, and multi-dimensional assessment. The four strands of math content covered (Algebra & Functions, Statistics & Probability, Geometry & Trigonometry, and Discrete Mathematics) receive significant attention in all four of the curriculum's courses.
The reader should be aware that, as this is the CPMP website, its material is strongly biased in favor of the curriculum at hand. The "FAQ" section contains answers to questions about the effectiveness and applicability of the curriculum in different educational environments. Of course, each of these answers assures the reader that the CPMP's material can be implemented successfully anywhere. In addition, the site displays quotations of praise from teachers and students, while no mention of the curriculum not being warmly welcomed is represented.
This having been said, the CPMP website offers a great deal of research data and visual depiction of how and why its curriculum is apparently succeeding. Links to sample material from each unit within each course are available for anyone interesting in comparing and contrasting this curriculum to others. Units are also broken down quite nicely into individual lessons with objectives that are well defined and easily molded into personalized lesson or unit plans. This is the first curriculum I have thoroughly studied that blends concepts from several math topics over the course of three years. It appears that the CPMP has succeeded thus far in improving students' understanding of mathematics.
Keywords: Geometry, Curriculum,
Ref: Paul12
Author(s): Lornell, R.; Westerberg, J.
Date: 1999
Title: Fractals in High School: Exploring a New Geometry
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(3), p. 260-269
Reviewer: Paul
Date of Review: 3/19/03
In this article, Lornell and Westerberg argue the case for including a unit on fractal geometry in traditional geometry courses. They open with a brief definition of fractals--iterated objects that are self-similar (a part of the whole closely resembles the whole). The majority of the article describes four activities, which have proven to be effective in familiarizing students with a few important fractal types. Activity 1 has students brainstorm 2-D and 3-D shapes that we see in everyday life, and students begin to realize that these shapes are not perfect Euclidean objects. We need a new type of geometry to explain natural objects. Activity 2 introduces students to the Cantor set, a fractal in which the middle third of a line segment is removed, iteratively. Activity 3 allows students to iterate and analyze the Koch Snowflake, a fractal in which the middle third of a line segment is removed, and the inner endpoints of the two remaining segments are connected to create an equilateral triangle with a missing base. In Activity 4, students discover a bit of the counterintuitive nature of fractals; the Cantor set has an infinite number of pieces with a total length of 0, and the Koch Snowflake has an infinite perimeter within a bounded area.
For what it is worth, Lornell and Westerberg convinced me that a fractal unit is a worthwhile component of a high school geometry curriculum. One of their strongest arguments is that examining fractals such as the Cantor set and the Koch Snowflake provides important connections to algebra and calculus through developing formulas for growth in area and perimeter and investigating their limits. Furthermore, the anomalies inherent in fractals, such as infinite perimeter within a bounded area, can foster spirited class discussions and higher level mathematical thinking. Fractals also provide an opportunity to incorporate technology into the classroom. One of the four activity worksheets that supplement this article deals with an exploration in which students can iterate the Koch Snowflake up to ten times with a graphing calculator. Doing so by hand would be far too tedious and time consuming. So, I agree with the authors' conclusion: fractals can easily intrigue students who do not respond to traditional problems, while challenging even the most analytically gifted young minds.
Keywords: Geometry, Activities, Standards
Ref: Paul10
Author(s): Day, R.; Kelley, P.; Krussel, L.; Lott, J.; Hirstein, J.
Date: 2001
Title: Navigating Through Geometry in Grades 9-12
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages:
Reviewer: Paul
Date of Review: 3/12/03
There is no doubt that this book is rooted in the Principles and Standards for School Mathematics. The Geometry Standard is printed in full inside the front cover, and each of the book's four chapters is based upon one of the sets of expectations that comprise the Geometry Standard. The major topics of the chapters are transformations, position and map making, similarity, and visualizations. The book opens with remarks from the authors regarding their lofty expectations for high school students, such as being primed for geometric proof and reasoning by the end of middle school and the importance of familiarizing students with dynamic geometry software. The authors also stress that the activities set forth in this book are examples rather than an entire suggested curriculum. For the most part, each chapter consists of a variety of exploratory activities, complete with listed goals, materials, and a discussion of the activity. In fact, each activity closely resembles a lesson plan. Solutions to the activities are found near the end of the book, and a CD-ROM is included. This disc contains every activity in the book, along with an assortment of suggested geometry readings and four interactive computer applets on topics such as the Koch Snowflake.
Especially compared to the addenda on data analysis and statistics that I examined, this resource is a well-prepared curriculum aid for teachers. The NCTM's presence is clear throughout its Standards-based presentation, so teachers can be confident that the activities provided seek student understanding of "important mathematics." The lesson plan style of the activity descriptions proves that the authors thought long and hard about which components of the Standards each activity teaches. Thus, teachers can select an activity that looks promising, analyze the suggested "lesson plan," and personalize it to their own classroom. This seems to be a much more efficient approach than that of the addenda I examined, which presented several activity ideas but did not formally state activity objectives or use a consistent style to summarize each activity. Like the addenda, this book even includes class handouts and worksheets for many of the activities.
Keywords: Curriculum, Standards, Statistics
Ref: Paul9
Author(s): Burrill, G.; Burrill, J.C.; Coffield, P.; Davis, G.; Lange, J.; Redneck, D.; Sieged, M.
Date: 1992
Title: Curriculum and Evaluation Standards for School Mathematics Addenda Series, Grades 9-12
Journal or Publisher: National Council of Teachers of Mathematics
Volume, Issue, Pages: Data Analysis and Statistics Across the Curriculum
Reviewer: Paul
Date of Review: 3/10/03
This addenda is intended to complement the Curriculum and Evaluation Standards for School Mathematics by providing instructional ideas and materials that can be practically applied in a high school setting. Its eight chapters are divided by subtopic of data analysis and statistics, including "Making Sense of Data," "Exploring Linear Data," and "Chi-Square: A Measure of Difference." Each chapter opens with several paragraphs describing which concepts should be covered. This information is illuminated by graphs and charts as well as hints in the margins, entitled "Teaching Matters" and "Try This." These brief inserts give extremely specific ideas of activities to try. Each chapter concludes with a series of example activity worksheets, which could physically be used in the classroom to implement each of the activities mentioned earlier in the chapter. The addenda also includes sections on "Student Projects," "Assessing Statistical Understanding," and "Solutions and Comments for Activities." Here, the emphasis is on choosing activities that allow students to manipulate real data and assisting teachers in effectively grading and giving feedback on data analysis and statistical projects and assessments.
The first aspect of this work that stood out at me was its decision not to address combinatorial counting problems prior to the study of statistics. I was surprised to hear the former approach referred to as the "usual treatment" in 1992, because my schools' math programs, both in high school and at St. Olaf, dealt with combinatorics and counting before statistics. I would have expected my education to reflect the new approach, but perhaps it was not warmly welcomed by many schools. Furthermore, the addenda includes activities covering such advanced topics as curve-fitting, correlation, and residuals. Perhaps schools are still catching up to these lofty aspirations, because I was not introduced to such concepts until taking an upper level undergraduate statistics course. However, this in no way hinders the exhaustive nature of this addenda. It provides teachers with all of the well-organized tools necessary to assemble an understanding-based data analysis or statistics course. In fact, classroom activities are provided in such quantity and detail that a teacher could likely plan an entire course built around this one document.
Keywords: Technology, Issues, Assessment
Ref: Paul8
Author(s): Podlesni, J.
Date: 1999
Title: A New Breed of Calculators: Do They Change the Way We Teach?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(2), p. 88-89
Reviewer: Paul
Date of Review: 3/5/03
The "new breed" of calculators discussed in this article includes the TI-89 and the Casio 9970. These are among the first calculators to use computer algebra systems (CAS), which allow for the manipulation of symbols. Thus, today's students can look no further than their calculators to factor polynomial expressions and calculate derivatives and integrals. While the author asserts that "calculators are the single most positive development in teaching mathematics within my lifetime," he emphasizes that such technology should be used to remove unnecessary tediousness and provide unique graphical representations. He goes on to suggest that the use of these newest calculators not be allowed during any standardize tests until the NCTM develops a clear position on the issue. Podlesni also touches on the role of the calculator industry in this affair. Obviously, these corporations are pushing for widespread use of their products, because they seek to make a profit. In the author's words, "teachers need to distinguish between what is 'neat' and what is good." He adds, "What is technologically possible may not always be educationally desirable."
I share Podlesni's concern with the improper use of calculators in modern math classrooms, but I strongly believe that each teacher can eliminate this danger by making wise curriculum choices. For instance, students must learn how to use the quadratic equation--how it works and what its function is. However, after a significant period of instruction and assessment, in which the teacher requires students to list their steps and give reasoning, the teacher should introduce the class to a calculator function that calculates quadratic formula values in seconds. This way, students understand both the important math concepts and how to use modern technology to their advantage. In my mind, all available means of technology should be implemented if they can enhance one's teaching objectives. Now, as far as standardized testing is concerned, I believe that exam questions should be written so that students should not need the use of a calculator. Given the broad range of family incomes and student familiarity with calculators, I think it is most equitable to use standardized tests to measure what students can achieve using only their minds, pencils, and paper.
Keywords: Games, Proof, Activities
Ref: Paul7
Author(s): Gernes, D.
Date: 1999
Title: The Rules of the Game
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(5), p. 424-429
Reviewer: Paul
Date of Review: 3/3/03
Don Gernes was imaginative enough to combine two math entities that usually are not thought of in the same sentence: games and proofs. In his article, Gernes describes his frequent use of fun activities and everyday examples to introduce students to deductive systems. His reasoning is that games are made up of materials, rules, and plays, which are completely analogous to the undefined terms, postulates/axioms, and theorems of deductive systems. In particular, the author gives extensive information about two of his games, entitled the "Letter Game" and "Euclid's Game." The former involves constructing strings of letters given specific rules, while the latter serves as a very basic introduction of proving facts about parallel lines in Euclidean geometry. Also, Gernes suggests topics such as Monopoly and sports for which students could easily generate undefined terms, postulates, and so forth. In general, the author argues that students will enjoy and quickly understand deductive systems if the topic is first introduced using a game. Based upon his experience with this method, Gernes reports that "usually students are successfully writing proofs within ten to fifteen minutes."
The first thing that impresses me about this idea is that it allows teachers to introduce students to deductive reasoning and proof writing at a much younger age. One would be tempted to hold off on presenting this topic using a straight-forward method because of intimidating terms such as "postulate" and "theorem," and younger students may not be able to grasp the technical proof writing style. Using the author's procedure, however, the teacher can start by using one of many game ideas and then take the learning activity as far as is deemed appropriate. For instance, with a game such as soccer, students could generate undefined terms (equipment) and postulates (rules), and proofs could simply be touched on through a verbal discussion. In general, it was refreshing to learn how secondary school students can associate proof writing with games. I would wager that few undergraduate students think in such a way.
Keywords: Issues, Teaching Strategies, Management
Ref: Paul6
Author(s): Fiore, G.
Date: 1999
Title: Math-Abused Students: Are We Prepared to Teach Them?
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(5), p. 403-406
Reviewer: Paul
Date of Review: 2/25/03
Greg Fiore teaches developmental-algebra at Dundalk Community College. The average student enrolled at DCC is thirty-three years old, so the author was quite surprised when he discovered that two students in one of his classes suffered from extreme math anxiety. "Terry" appeared to be very tense and disturbed during Fiore's first in-class test. "Lenore" froze and panicked when the author called on her to answer an easy question during class. Fiore's concern drove him to assign a "Math and Me" paper, in which students were instructed to describe their previous math experiences. In this way, he discovered that Terry's third grade teacher frequently ordered Terry to stand in front of the chalkboard until she could solve an assigned problem. On at least one occasion, Terry was forced to stand there from 9:15 A.M. until 3:30 P.M. As a result, Terry became very anxious when expected to solve math problems in a given amount of time. Lenore's experience with abuse was more physical than Terry's. When Lenore asked her father for math help in the third grade, he slapped her if she did not understand his explanation. As a result, she dreaded being called on in class. Knowing these histories, the author made every effort to accommodate Terry and Lenore through encouragement and careful procedural decisions. As a result, both students passed his class and went on to achieve positions in their desired career fields.
In general, I believe that
assigning a "Math and Me" paper is a great teaching strategy. With a simple one-page paper, a
teacher can bring himself or herself up to speed with the most vital information of each student's
unique math background. This could also save the teacher hours of frustration with students
who refuse or are too afraid to participate in class. It has become increasingly evident that
mathematics teaching is a delicate field, in that teachers are single-handedly responsible for
getting even "non-math people" interested. Simple adjustments, such as allowing a student to
complete a test at an alternative time or location, can help troubled students achieve their
mathematical potential.
Keywords: Issues, Teaching Strategies, Management
Ref: Paul4
Author(s): Murdock, T.B.
Date: 1999
Title: Discouraging Cheating in Your Classroom
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(7), p. 587-591
Reviewer: Paul
Date of Review: 2/19/03
When asked how teachers can discourage cheating, students recommend that teachers increase surveillance and punishment. Tamera Murdock believes that this course of action attempts to treat the symptom rather than the disease. The disease in question is that students do not have confidence in their teacher or in their own understanding of mathematical concepts. In a recent study, 25-30% of middle school students and 55% of high school students admitted to cheating in the previous academic year. Also, cheating is more frequent in math and science classes than in other academic areas. Students' reasons for cheating include fear of failure, desire for a better grade, and pressure from parents to do well in school. Teachers can decrease cheating by implementing task-focused curricula rather than performance-focused ones. Obviously, task-focused classes encourage understanding, while performance-focused ones look for high assessment scores. Teachers can also encourage "fair play" by aligning assessments with course objectives, giving frequent feedback on student performance, using multiple forms of assessment, and emphasizing problem solving rather than computation.
When I read the title of this article, I wasn't so sure that it had much to say
about mathematics specifically. However, I ended up learning several tips on how to construct
math assessments which optimize student motivation. Teachers who don't sufficiently prepare
students for assessments or suddenly forbid calculator use on an exam seem to give students the
justification needed to cheat, because their teacher has done something that is "unfair." I was
glad that the article addressed grading on a curve as a practice that fosters unnecessary
competition between students and increases the chances of cheating. To me, grading on a curve
seems like a way of balancing out student grades in case a teacher's test is too easy or difficult.
Finally, I thought Murdock had a great point that assessments emphasizing problem solving not
only promote higher-level thinking but also make it more difficult for students to copy off of one
another. Strategies like these allow teachers to discourage cheating without decreasing the
quality of their teaching objectives.
Keywords: Algebra, Activities, Teaching Strategies
Ref: Paul5
Author(s): Lewkowicz, Marjorie
Date: 2003
Title: The Use of "Intrigue" to Enhance Motivation in Beginning Algebra
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 96(2), p. 92-95
Reviewer: Paul
Date of Review: 2/23/03
This article introduces the reader to the intrigue model, a new attempt to help students begin to enjoy mathematics at the early secondary level. Lewkowicz's model is built around a particular type of problem that she uses to get her beginning algebra classes thinking mathematically. Generally, the problems begin with each student thinking of a number. Then, Lewkowicz gives students a series of arithmetic procedures to apply to their numbers. At the end, she asks several students to tell her the number they ended up with, and based upon the arithmetic procedures, Lewkowicz can quickly respond with each student's original number. Rather than immediately explaining the "trick," the author takes this opportunity to let students try to figure out what she did, and eventually, each problem breaks down into a very understandable algebraic analysis. Lewkowicz presents six examples of these problems along with their underlying algebraic analyses and several suggestions of which concepts each problem can best introduce.
I would definitely consider using the intrigue model in a beginning algebra class. Along with introducing such algebraic concepts as variables, factoring, and the division of polynomials, Lewkowicz's method does a great job of providing students with a glimpse of proof writing. The algebraic analysis of each problem shows students that the use of numerical examples is insufficient to verify a given statement or procedure. Instead, they learn that by using variables, they can account for any given number with which one may start each problem. I also liked how the author links her model back to "Principles and Standards for School Mathematics." She presents a quote from "Principles," and then gives evidence of how the intrigue model coincides with these guidelines in that it utilizes "well-chosen and worthwhile tasks to engage, challenge, and excite students."
Keywords: Standards, Curriculum, Research
Ref: Paul3
Author(s): Martin, T.S.; Hunt, C.A
Date: 2001
Title: How Reform Textbooks Stack Up Against NCTM's "Principles and
Standards"
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 94(7), p. 540-545, 589
Reviewer: Paul
Date of Review: 2/17/03
Mathematics teachers are not alone in the struggle to assure that the NCTM's "Principles and Standards" are satisfied in the classroom. Many textbooks, which cater to the national Standards, have already been funded by the National Science Foundation. The authors of this article analyzed five such textbooks by comparing their content to that desired by the NCTM's "Principles and Standards." The Standards involved are separated into Process Standards and Content Standards. The Process Standards require emphases on problem solving, connections within math and with other disciplines, and reasoning. There are ten Content Standards, and they include such areas as: Algebra, Trigonometry, and Probability. The authors rated the five textbooks on a scale of 1-3 based upon evidence of the Process Standards and with a "+" or "-" to signify the inclusion of each Content Standard. To provide a more descriptive evaluation, the authors also listed several distinctive features of each text. In the end, they recommended the consideration of all five texts to any district looking to align its curriculum with "Principles and Standards." In fact, two of the five textbooks ("Contemporary Mathematics in Context" and "Interactive Mathematics Program" have already been identified by the U.S. Department of Education's Mathematics and Science Expert Panel as exemplary math programs.
I found this
article to be an excellent resource for anyone searching for a text that satisfies the requirements
of "Principles and Standards." The results of the three evaluations performed are listed in tables
within the article, along with a brief summary of the breadth of each text. Sample pages from
each book are also included along the outside of the article, so the reader can get a glimpse of
how each text presents its information. In general, I was optimistic to see the emergence of these
focused, coordinated texts, which seek to unify our nation's math education programs. I was
pleasantly surprised to see attention paid to Standards such as Statistics, Probability, and proofs.
These were three areas that my high school education paid little attention to, and I believe that
today's math students deserve to be equipped with such skills prior to entering college.
Keywords: Proof, Teaching Strategies, Assessment
Ref: Paul1
Author(s): Fidler, Mark
Date: 1999
Title: Chipping Away at Proofs: A Cooperative Approach
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(7), p. 565-567
Reviewer: Paul
Date of Review: 2/9/03
This article takes the reader through Fidler's trial-and-error process of finding the best method for teaching proof writing. The author, a secondary-level honors geometry teacher, originally assigned all proofs to be completed individually by students. He soon found that on long, difficult proofs, most students either gave up or bridged the gaps in their proofs with statements they knew to be logically incorrect. Offering additional credit for incomplete proofs where students admitted dead-ends that they ran into provided marginally improved results for Fidler, as did having students work in groups on a one-week take-home exam. It was evident during this group work, however, that students needed more time to learn how to build off of one another's ideas in constructing high-quality proofs. So, Fidler issued a group quiz at the beginning of the following school year to see how his new students would respond. As expected, their grades were low, because they had never written proofs cooperatively. But without having built a foundation of individual-style proof writing this time, these students improved consistently on subsequent quizzes and exams when Fidler continued using a small group format.
Having recently completed a proof writing geometry course, I agree with
Fidler that working cooperatively on proofs is very effective in making the learning process less
frustrating and more enjoyable. I was still surprised, however, to hear Fidler's students refer to
proofs as "exciting" and "fun," since I didn't come to appreciate proof writing until the end of my
sophomore year in college. My junior high geometry teacher's approach to proof writing was
far less innovative than the author's, of course, and we did all work individually. Another of
Fidler's techniques that I concur with but had not thought of was his idea of assigning groups
according to previous individual performance. It makes good sense to put students in groups of
two, three, or four, with A students usually in pairs, B students in trios, and other students in
foursomes. In general, I would be optimistic about implementing Fidler's instructional method
into my own classroom. Perhaps the results wouldn't be quite as radical for some students as
for Fidler's honors geometry classes, but the basics of his cooperative technique are a worthy
approach to teaching any proof writing course.
Keywords: Issues, Communication, Management
Ref: Paul2
Author(s): Jackson, C.D.; Leffingwell, R.J.
Date: 1999
Title: The Role of Instructors in Creating Math Anxiety
Journal or Publisher: Mathematics Teacher
Volume, Issue, Pages: 92(7), p. 583
Reviewer: Paul
Date of Review: 2/12/03
Why do some students grow to dislike mathematics? How do teachers catalyze or add to this aversion? Jackson and Leffingwell set out to answer these very questions. They conducted a study of 157 college seniors who were pursuing certification in elementary education. In the study, subjects were asked to describe their worst or most stressful math classroom experiences from kindergarten through college. The authors found that a mere 7% of students had only positive experiences in their math classes. 16% of students experienced their first traumatic encounter in grades 3 or 4, while 26% did in grades 9-11, and 27% did during their first year in college. Leading contributors to math anxiety at all levels included gender bias and insensitive and uncaring attitude of the instructor. The article closes with tips for teachers regarding the minimization of math anxiety. Examples of these tips include projecting one's own enjoyment of math to the class and offering alternative times for testing.
I've often wondered why so many students have that "I just don't like math" attitude. Until now, I hadn't really considered the origins of such feelings. I was shocked, to say the least, to see the consistency with which students reported hostile instructor behavior and gender bias. Even in grades 3 and 4, students said that teachers grew angry when a student asked for additional help. Also, this was the first time I'd heard of math teachers favoring male students. I'd heard of the old studies about males being more prone to succeed in math and science, but I had no idea that many math teachers act condescendingly toward female students. All in all, I viewed the article as a wake-up call. If one truly believes in "math for all," then he or she must be mindful of any subconscious components of his or her teaching style that could produce math anxiety. The essence of mathematics is one of careful analysis, not stressful intimidation.