This article discusses the importance of including historical information about mathematics in the curriculum. It emphasizes that mathematics does have strong ties to real life situations, and has been utilized throughout history. The article claims that teaching the history of mathematics increases pride in one’s culture, as well as the student’s understanding of the different practical applications in which mathematics is used.

The article is broken down into sections to discuss various ways mathematics has been used in the past. The first section describes counting and measurement techniques. Most tribes in the Philippines use a base-ten system, with different terms for the first ten numbers and prefixes to represent higher numbers. Many societies in the South Pacific and the Philippines use body parts when counting, most often fingers and toes. Some will go as far as to count using hands, wrists, forearms, elbows, biceps, collarbones, shoulders, eyes, and ears. Measurement is often done by also using body parts. Fingers, thumbs, palms, and paces are all used for measurement. Tallying on stone or animal bone is another popular method of counting. The second section of the article indicates that in ancient times, Filipinos tied knots on a string after every sunset to remember the days and scratched on a stick and stone after every full moon to keep track of the months. In this portion, it also said that the Bagobos (a tribe in the southern Philippines) copied a trap design from the bow design they saw in stars in the Orion constellation using geometry. Apparently, some tribes in Borneo also used geometry by measuring hours and days by shadows cast by poles exposed to the sun. The final two sections of the article, geometry and logic, discuss other applications of mathematics in society such as: getting the proper angles to construct homes, following the path of sea turtles in and out of the sea to find where their eggs are buried (the turtles come out of the sea and enter back into the sea using two different paths, the intersection is where the eggs are located), patterns and designs on clothing, construction of boats and musical instruments, games, music, and logic that applies to procedures used in daily life.

I think that this article brings up many good points about the importance of teaching students about the history of mathematics as well as the practical applications for it. Speaking from a student’s perspective, I am much more likely to be dedicated to learning the material if I know that it can be applied to my life and the situations that I encounter on a day to day basis. I think that most students would share this opinion. Therefore, I agree with the author of this article when she says that students are interested in knowing where the mathematics in their culture as well as other cultures came from.
 
 
 

Keywords – Activities, Algebra, Applications, Problem Solving, Writing
Author(s) – Martinez, Joseph G. R.
Title – Thinking and Writing Mathematically: “Achilles and the Tortoise” as an Algebraic Word Problem
Date – April 2001
Volume:Number – 94:4
Page(s) – 248-252
Reviewed by – Amy Baltmanis

This article discusses that students have a tendency to dislike word problems, and that research has shown that some students lack the ability to understand the meaning of word problems.  Many students view word problems as “too hard, contrived, boring, and irrelevant.”  This article demonstrates how students can become quite interested in word problems and learn a great deal from them when the problems are on topics that are relevant to the students.  These topics include things such as traveling by automobile or airplane, buying groceries, movies, music, etc.  Martinez believes that teachers need to engage the students’ imaginations by using creative problems, and also get the students thinking about problem solving strategies as well as writing descriptively about their mathematical thinking.

Martinez gives an example of a specific problem he used in his class in order to get his students thinking about an algebraic word problem.  The problem is an adaptation of Zeno’s famous paradox, “Achilles and the Tortoise.”  The problem reads:
 Achilles runs a race with the tortoise.  He runs ten times as fast as the tortoise.  The tortoise has 100 yards’ start.  Now, says Zeno, Achilles runs 100 yards and reaches the place where the tortoise started.  Meanwhile the tortoise has gone a tenth as far as Achilles, and is therefore 10 yards ahead of Achilles.  Achilles runs this 10 yards.  Meanwhile the tortoise has run a tneth as far as Achilles, and is therefore 1 yard in front of him.  Achilles runs this 1 yard.  Meanwhile the tortoise has run a tenth of a yard and is therefore a tenth of a yard in front of Achilles.  Achilles runs this tenth of a yard.  Meanwhile the tortoise goes a tenth of a tenth of a yard.  He is now a hundredth of a yard in front of Achilles.  When Achilles has caught up this hundredth of a yard, the tortoise is a thousandth of a yard in front.  So, argued Zeno, Achilles is always getting nearer the tortoise, but can never quite catch up.
Martinez says that he asked probing questions to get the students thinking about the language of the question and why it is confusing.  He even had some of the members of the class act out the problem.  The students worked in groups to try to come up with solutions or ideas.  The students deduced that Achilles had to eventually catch the tortoise, but the question remained at what number of yards.  Students used various algebra techniques, and came up with 11 1/9 yards.

I strongly agree with Martinez that if teachers want to get students interested in the subject matter, they have to make it apply to something in “real life.”  Students are much more likely to become involved and interested in what they are doing if mathematical problems are interesting and something that is relevant to their life.  I also believe it is important to get students thinking about their own thinking.  It is not only important to be able to solve problems, but also understand the methods used when coming up with those solutions.
 
 
 

Keywords – Algebra, Applications, Geometry, History, Problem Solving
Author(s) – Allaire, Patricia R. and Robert E. Bradley
Title – Geometric Approaches to Quadratic Equations from Other Times and Places
Date – April 2001
Volume:Number – 94:4
Page(s) – 308-313, 318-319
Reviewed by – Amy Baltmanis

This article focuses on studying geometric solutions of the quadratic.  The authors think that there are many good reasons to use geometry when trying to solve algebraic equations.  One of the main reasons is for some people who are visually oriented (as opposed to symbolically oriented), it might be easier for them to make sense out of algebra problems by viewing what is happening instead of just looking at symbols.  Also, for those that are symbolically oriented, it can provide another method for solving problems if a person gets “stuck” on something.  The article gives a collection of geometric techniques from ancient Babylonia, classical Greece, medieval Arabia, and early modern Europe that can enhance the quadratic-equation portion of an algebra course.

In geometrical algebra, simple quantities are represented by physical objects, typically a line segment whose length is what is to be determined.  A product of two quantities is then looked at as the area of a rectangle, and a product of three quantities is viewed as the volume of a right triangular prism.  For example if we look at the quantity (a + b) 2:

This allows people to see what (a + b) 2 looks like.  In the upper left box, the result is ab.  In the upper right box, the result is b 2.  In the bottom left box, the result is a 2.  In the bottom right box, the result is ab.  This allows a person to then see that:
 (a + b) 2  =  a 2 + 2ab +  b 2.

This portion of the article seemed to be the most applicable to what is taught in middle and high schools.  The rest of the article deals with the classification of quadratic equations, the construction of the square root of three as seen in Euclid’s The Elements, application of areas (a Greek method), the Babylonian solution of a quadratic equation, Al-Khwarizmi’s method of “completing the square”, Descartes’ solution to the quadratic equation x2 + bx = c, Carlyle’s method of solving the equation x2 + bx + c = , and proof of the Varignon parallelogram theorem.

I believe that the portion of the article that is most beneficial to future math teachers is the idea of representing algebra geometrically.  It can be a great technique to use in the classroom, and there are many activities that can be done with it.  Every student has a way that they learn best.  If you can present the algebra material in various ways, it will give students who cannot follow the concepts when the teacher uses symbols a chance to visualize what is happening geometrically.
 
 
 

Keywords – Technology, Calculus, Geometry
Author(s) – Cantrell, Martha and Riddle, Ira Lee
Title – Technology
Date – April 2001
Volume:Number – 94:4
Page(s) – 326
Reviewed by – Amy Baltmanis

This article discusses two different computer programs that can be helpful to students and teachers.  The first program is the APCD Calculus AB software reviewed by Cantrell.  This software is designed to help students prepare for the Advanced Placement Calculus AP examination.  The orientation workshops explain the examination, calculator use, and test-taking strategies.  There is also a multiple choice practice session, which allows students to obtain immediate feedback on the correctness of an answer and to look at the explanations of correct solutions.  The difficulty level of each question is included, and the session is not timed.  There is also a section in which students can take a timed test.  Cantrell feels that the activity will maximize student learning and prepare the students for the Advanced Placement calculus AB examination.   This software can be purchased for an individual or as a network version.

The second program is a CD-ROM called Visual Plane Geometry.  According to Riddle, the CD has impressive graphics, and it is part of a series of CD-ROMs that include Algebra I and II, Solid Geometry, and Trigonometry.  The CD-ROM has a table of contents so that students can select the topic that they would like to learn about.  There are questions and answers given throughout the program.  Possible uses for this program include a stand-alone teaching device for a student who is trying to get ahead, an aid for students who need review on a topic, or a way for a student who is stuck at home to learn.  It is not an expensive piece of software, and teaches on a wide variety of mathematical topics.

I think both of these programs could be extremely useful in the classroom.  It is important to give students variety in how they go about learning.  Allowing students to work with the Visual Plane Geometry can let them work through things at their own pace.  It also allows the student to choose which topic he/she wants or needs to focus on.  With as much as our society depends on computers, I think it is important for students to be able to become familiar with them, and how they can be used in all subject areas.  I think the APCD Calculus AB software is a great idea because of the fact that it allows students to immediately find out if they are performing problems correctly.  When students receive immediate feedback, it allows them to see what it is they are doing wrong and correct it before students incorrectly learn how to do problems.  I also think it could be very useful because students can take a timed version of the test.  This is excellent for students who have problems budgeting their time on tests.  This will allow the student ahead of time to plan on how much time they can allot themselves for each question.  Making use of test taking strategies can improve scores, and prevent students from getting hung up on tough questions.
 
 
 
 

Keywords – Activities, Applications, Geometry, Measurement, Problem Solving
Author(s) – Moyer, Patricia S. and Wei Shen Hsia
Title – The Archaeological Dig Site: Using Geometry to Reconstruct the Past
Date – March 2001
Volume:Number – 94:3
Page(s) – 193-197
Reviewed by – Amy Baltmanis

This article discusses the fact that students often see little connection between geometry and real-life mathematical situations around them.  It is important for teachers to help students make these connections.  Many secondary school students enter into geometry classes without any informal geometry experiences.  It is the teachers’ job to allow students to engage in activities/investigations that will fill in the gap between their preconceived ideas of geometry and the formal relationships that they need to comprehend.  The authors mention that even though students may lack the vocabulary and conceptual understanding to verbalize or write geometric relationships in formal terms, teachers can help students obtain these skills by starting off with basic geometry and then move on to more complex analytical activities.

This article specifically describes an investigation in which students apply basic understandings of geometric properties to work with polygons and their properties, and later on demonstrates a method for using chords and arcs to find the measurements of circular objects when only a portion of the circular object is given.  In part one of the activity, students transfer scale drawings of a given polygon to a life-sized dig site by using string, stakes, and measuring devices.  In part two, students are given broken pieces of circular artifacts and asked to use measurements from the broken pieces to determine the original circumferences of the artifacts.  Specific details, instructions, and suggestions of what to do with the students are given in the article for each portion of the activity.

I think this is a great activity for modeling a real-life mathematical situation.  Archaeologists have to use geometry in order to help map out dig sites as well as reconstruct artifacts that they find.  This is something hands-on that would really appeal to the students that need to “do” something in order to understand it.  It also provides an alternative to lecturing, reading, or doing worksheets in order to get teach a mathematical concept.  This activity also deals with a specific standard in Principles and Standards for School Mathematics.  It focuses on the expectation that, “students should…explore relationships among classes of two- and three-dimensional objects, make and test conjectures about them, and solve problems involving them; establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others;…use geometric ideas to solve problems in, and gain insight into, other disciplines and other areas of interest such as art and architecture.”  I think that this activity can be used for many different ability levels and could even be differentiated for students with varying skills within a classroom.