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Learning About Native Americans Through Math, Art, and Science  
Liz Baca 

As we learn about other cultures, we are often taught about their food, clothing, housing, and maybe some of their celebrations. Although this is often the foundation for “multicultural” education, it does not tell much about the importance of other groups. By studying the contributions to art, math and the sciences, students will come to appreciate the ways other groups attempt to learn about their world, as well as the things they did to try to improve their world. 

            This unit looks at five distinct Native American groups and some of the ways they used math in their culture. We look at the Anasazi of the Southwest. Their contributions include a vast knowledge of astronomy and architecture, and they left some knowledge of that in the actual ruins of their cities. We will study the petroglyphs as records of their lives, and will connect their record keeping with the records we keep. Next, we’ll study the patterns found on Pueblo art, particularly jewelry and pottery, as we learn about the tribes who share our state. We will visit the ancient Maya, who were the first known culture to use zero in their number system, and who also had a vast understanding of the cycles of nature. We will learn about the Pomo Indians, basket makers from California, who played games of chance, similar to games played by many other Native tribes. Finally, we will observe and create patchwork clothing, similar to the Seminole Indians of Florida, who tell a story of resilience and adaptation so common in the Native American tribes. 

            Although these lessons were written for first grade, they can be easily adapted for a variety of grades. The concepts we look at, record keeping, patterns, geometry, probability, and statistics, are math concepts that are taught at any grade, and older students can work with them in more advanced ways. Ideas for adaptation are included in the Context and Background information section of the paper.

  

Covering Geometry with Quilts
Michelle A. Felis

This unit is designed for use in a cross categorical inclusion program at the fourth and fifth grade level.   The students in the program range from those who have been identified as being gifted to those with severe learning disabilities.   Several students are physically challenged or have other diagnosed conditions such as autism or emotional disturbance.

This unit contains five lessons that use quilting techniques to expose students to the basic shapes of geometry.  In addition, students are encouraged to work in teams to compare and contrast the patterns they are able to create. 

              The unit offers an opportunity to make connections with social studies lessons as one can work with quilt patterns from certain regions or historical periods.   In addition, students are able to experience how traditional quilters would have used recycled materials and scraps to make new, beautiful items. 

It is my hope that students who are not typically engaged in “math” will engage in these artistic activities. 

 

“How Do You See It? 
Discovering Mathematical Patterns and Sequences”
Keith Gaudet

For the last three years I have been teaching math and science to middle school children. Teaching these kids have been a pleasure and a challenge. They are born from a variety of ethnical backgrounds, large families to small families, creative and simple. But one concept that surprised me was that a majority of these kids were not as perceptive in mathematics. They choose to take the simple or most obvious approach. Not that there is anything wrong with that approach but these students are not seeing the whole picture. This is crucial when dealing with mathematics and science as well. Scientists need to be able to see if there are other alternatives to a particular science experiment if the original one fails. With all this in mind, I have made looking for patterns my general underlying theme throughout the school year. Thus, this forms the basis for this paper. 

            My topic for this ATI curriculum is on the variety of mathematical patterns and sequences.  Though this curriculum will be designed for the middle school teacher, it includes ways to adapt it for any grade level. I have also been a firm believer in integrating all subject areas. It is also my intention to come up with ways to discover and use patterns and sequences in other subjects. I am planning on this paper to be set up in 4 major sections: Introduction to the Curriculum, Host School Academic Setting, Background Information, and Collection of Lesson Plans.  

            The introduction to the curriculum will inform the reader as to what the unit is about, the target audience of the paper and how the curriculum could be taught. The academic setting section is a general background of my assigned school, Truman Middle School. The section will break down the demographics in a variety of areas including: ethnic background, reduced meals vouchers, staff experience, and community profile. My Background Information is considered the heart of my paper. I shall first attempt to give a historical perspective on a variety of mathematical patterns and sequences. I broke down the sequences section into geometric and arithmetic (similar to what Jacobs' Mathematics: A Human Endeavor textbook does).  I have researched and recounted how the early mathematicians discovered these fascinating patterns. The next section is a collection of lesson plans. These are lessons that could be taught in conjunction with this unit. These lessons will not be in any particular order, for the order will be determined by what subject the teacher is discussing. These lesson plans will include New Mexico's state standards and benchmarks. Finally I will wrap things up in my conclusion. 

 

Beyond Numbers: Why study math?
Where is math apart from math class and the minds of crazy geniuses?
Heather Jenkins

Very often math is presented to students as though it is a system of arbitrary rules and systems thought up by evil geniuses and having no bearing in the world unless one is a mathematician. Particularly, “at risk” students seem to have difficulty or lack of interest in math and very often are skeptical or cynical of the subject by the time they are in middle school. This unit is designed to allow students to see that math does exist in the world beyond the classroom, and that it does not always have to do with numbers or formulas. Through hands-on activities students will experience math as embedded in a social, cultural context. The unit is intended to be a brief survey in which students will investigate math in several “non-numerical” ways. Students will look at and create symmetry patterns in order to view math as an aspect of culture and as molding our perceptions of art and beauty. They will use logic to see that math is a way of thinking about and organizing the world.   They will observe math as a presence in nature by looking at tessellations and honeycombs. Also students will play and investigate games of chance to understand that math is present in many forms of entertainment. The unit is not meant to be a comprehensive examination of how math is used and present in the world but, rather, to let students experience a glimpse of math in ways they may not have thought about previously. If successful, the unit will dispel the notion that math is something only a select few people can master. Math is everywhere, and for everyone.

 

Linear Transformations in Works of Art
Robert Keeney 

The Western institution of education has evolved starkly from the methods of learning employed by our ancient ancestors.   And though no one can argue the many great successes it has produced over its history, the whittling down of a topic until only a singular, isolated concept remains, can be a difficulty for many students.  This unit attempts to “decorate” the learning of linear transformations in a high school geometry class by relating it to an artistic and cultural environment.

            The unit reviews the concepts of translations, rotations, reflections and dilations and their symmetries.  It then discusses how these transformations are employed in a sample of art mediums.  M.C. Escher and tessellations, frieze and wallpaper groups, and tapestries are covered as two-dimensional representations.  The distortions of curved surfaces are analyzed as Acoma pottery is reviewed. 

Five lesson plans that complement the ideas discussed above are included.  They rely heavily on computer use, particularly Geometer’s Sketchpad software by Key Curriculum Press.  The first covers the definitions of the four transformations by having students transform triangles on graph paper.  The second is a research project where students produce an electronic slide show presentation highlighting several of Escher’s works.  The third uses Geometer’s Sketchpad to create a slide-tessellation where students can then explore its properties of symmetry.  In the fourth, students research a piece or collection of art and the culture from which it came, and produce an electronic brochure.  In the final lesson plan, students use Geometer’s Sketchpad to conduct all four transformations, singularly and as composites, on a polygon they have constructed.  The dynamic features of the program allow for continuous changes of the transformation variables so that unique and creative results can be achieved. 

 

When Will I Ever Use This?
James E. Lewis

Mathematics is a difficult subject for some students.  Usually the students inability to master mathematical concepts can be attributed to “math anxiety” or the fact that the student has given up.   Our function as teachers is to find some way to motivate each and every student.  This unit is designed to find means of reaching those students that are on the verge of giving up or have already left the building.  The mathematics of past generations is of little interest to today’s students.  Often they are convinced  that technology will take care of their mathematical needs and that every thing there is to discover has already been found. 

            This unit provides biographies of many  historical mathematical masters and some individuals alive today.  The biographies should show the students that they have much in common with historical mathematicians.  It looks not only at what they discovered, but at their home situation, education, and the attitude of their era.  I have also included several topics in mathematics to demonstrate to students that what they learn today may not be of immediate use.  It encourages students to look for new uses for concepts they know and extrapolate new concepts from what they have learned.  The work sheets of the appendix are designed to assist the teacher to motivate the students.  This unit is by no means all inclusive of mathematics, but is a simple overview that can be used to answer and motivate those students that have difficulty with mathematics. 

 

Math Game From Around the Word Using Strategy and Chance
Nancy O’Neil

This curriculum unit is intended for fourth grade regular education mathematics classroom.  The purpose of this unit is to assist the teacher in providing students with a better understanding of logical thinking and the understanding of how to use their logical thinking.  Also to assist students with a better understanding of basic math skills, such as adding, subtracting, and multiplying.  The rational for doing so is as follows: those students that need a better understanding of basic math skills will be reinforced, while the students that have that understanding will gain more knowledge of logical thinking and how to use it; students need the opportunities for developing logical thinking; students will be able to acquire and develop logical thinking skills in different situations; students will be better prepared academically.

            This curriculum unit provides a list of the New Mexico State Standards that will be addressed, a detailed rational for the unit, and a brief history of math games.  It also includes the objectives for the lessons, a list of materials needed for implementation of the lessons, and lesson plans.  The unit is intended to be used with your existing math curriculum.   The lessons are intended to be used throughout the year, either in the order given or in an order that best suits your math curriculum.  

Probability and the History of Mathematics
Jason K. Sanchez 

Chance and risk are inherent in every aspect of life.  There is nothing we do that is devoid of risk.  From the time we get out of bed in the morning, until we lay down again at night, we are besieged with risk.  The role that chance, probability, and risk play in our existence does not end while we sleep in the “safety” of our homes; however, for most of us the risk of something bad happening while we sleep is relatively small.  In addition, I would venture to say that the average person would consider a life that is completely safe and devoid of risk as boring.  Chance and risk spice every adventure.  There is something very seductive about risk. 

            Chance is the randomness or unpredictability of events.  Risk is a measure of an event’s tendency to produce an unfavorable outcome.  Since earliest times humans have participated in games that involve chance and risk.  Yet, it was not until relatively modern times that the mathematics of chance were explored or even recognized.  Probability is a measure of the certainty or uncertainty that an event will occur. 

            A “winner” in the chancy game of life is either “lucky,” favored by God, or able to make good decisions based on proper assessment of probability, chance, and risk.  Perhaps, all three factors are involved.  This curriculum unit aims to help students learn how to calculate probability and, thereby, to assess risk.   It strives to do so while also providing an opportunity for students to experience probability and the history of mathematics in a holistic manner.  That is, students will be encouraged to investigate how mathematics and specifically, probability has been shaped by, and has shaped history, philosophy, religion, and society.

 

Learning One Thing Well:  Perimeter
Ann Stevenson

This unit was designed after I taught for one year at a relatively small middle school of 600 students in Albuquerque’s north valley.  The population is approximately sixty percent Hispanic and forty percent Anglo, with a small minority of Asian, American Indian, and African-American students.  Just fewer than 40% of the students receive free or reduced lunch.  The diversity of income levels is high, as the neighborhood is very diverse.   Many of my students had very low math skills and performed poorly on the Terra Nova and in the previous year’s math class.   As most teachers do, I needed to design lessons to meet the needs of a wide variety of abilities. 

It became clear that my students had difficulty solving problems on perimeter.  For instance, a figure was drawn with perpendicular angles and the length of some of the sides given, and they were expected to find the length of the missing sides.  The majority of the class had difficulty reasoning the answer from the information given.  I decided they had not had enough experience with actual measuring of objects and that this would help.   However, they also needed to be exposed to the kind of activity that would promote abstract thinking.  This unit was designed with these goals in mind.  

The purpose of this curriculum unit is to identify what must occur before geometrical thinking can advance in middle school-aged children.  The unit must proceed from identifying the geometric thought level of each student, introducing lessons that are appropriate for each level and that challenge the student to advance to the next level, then validate that indeed the next level has been reached.   So there are three parts to this unit:   assessing entry level, designing appropriate experiences, and assessing exit level. 

Math and Games of Ancient Civilizations
Tamara Werner 

This unit will be based on mathematicians and mathematics from their time (ex. Egyptian writing).  It will be more meaningful to the students if they can find out and know where the things that they have to learn came from.  Plus, each lesson will include a mathematical game from their civilization that will help bring home that math was everywhere.  Most of the games will be board games, because of all the mathematical thinking that goes into trying to win.  

            The last part of my unit will involve a research project that my students will have to complete.  They will be researching one mathematician.  They will be looking for who these people were, what they did in their life, and more.  I want my students to really get a good idea about one mathematician, one that they can relate to!  They can present this research project in a couple different ways.  The students can write a one page typed paper about their mathematician. Or, they can do a report on the mathematician.  This will include a poster board with pictures or drawings, and information about that mathematician.  The last selection that the students can pick from is to become that mathematician for the day.  They will dress up like they were that person and come into the class to introduce themselves.  They have to be prepared to answer all questions that the other students will have for that mathematician.   So they can not slack on the research of their person. 

This unit will be given over the length of the year.  Each new mathematician and mathematics from their time will be introduced when the students are studying each civilization that the mathematicians came from.  I want this unit to also be cross-curricular for the social studies class.  This will bring home that all subjects in school relate to each other. 

Heron, Pythagoras, Euler, Ptolemy and Ceva
Some Contributions to Euclidean Geometry
Stanley Wong

When students learn mathematics in school, they may be fascinated by what it offers.  They may be able to understand a theorem that leads to a formula.  They may be able to use that formula to solve a practical problem.  But who proposed that formula?  When was it first used?  Why was it discovered at that time?  Would the world be different if some mathematical formula was discovered say 1000 years before or 1000 years after when it was discovered?  Were formulas discovered as of necessity or did they happen randomly?  I do not propose to answer these questions.  I just want mathematics to be appreciated not only in the use of the theorems themselves but in the history of their development. 

            I will look at five mathematicians and their discoveries.  This will give students a greater appreciation and enrichment of the geometry that they are studying.  Heron’s Formula uses just the sides of a triangle to find the area.  If a, b and c are the lengths of the three sides of a triangle and s is the so called semiperimeter of the triangle (s= ½(a+b+c)    ), then the area of the triangle A =.  The Pythagorean Theorem states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.  Euler’s Line states that the circumcenter, orthocenter and centroid of a triangle are collinear.  If we take the midpoint of the segment whose endpoints are the circumcenter and the orthocenter of a triangle, that point is the center of a circle that passes through the following six points:   the midpoints of the three sides of the triangle and the point where each of the three altitudes intersect the sides of the triangle.  This is Euler’s Nine-Point Circle.  Ptolemy’s Theorem states that in an quadrilateral inscribed in a circle, the sum of the products of its opposite sides is equal to the product of its diagonals.  Ceva’s Theorem   states that if three concurrent lines, one from each vertex of a triangle, are drawn, they divide the opposite sides into six segments such that the product of three segments having no common endpoint is equal to the product of the other three segments. Heron’s Formula and Ptolemy’s Theorem can also be used to prove the Pythagorean Theorem.  It is hoped that students will also see that the geometry that has been studied throughout the ages is intertwined.