Robotics in the Mathematics Classroom

            The student's struggle to master skills, equally shared by instructors who strive to improve pedagogy, is an ongoing problem of schools and universities all over the United States.  Traditional mathematics curriculum, while offering the essential core subject areas, still fails to meet the needs of many students.  For individuals who have difficulty comprehending abstract concepts, math class can become a struggle or an insurmountable obstacle toward high school graduation and/or attending college.  Even if the student succeeds on one level, he or she is soon unable to successfully build on previous skills because of difficulty understanding key concepts.

            Robotics in the mathematics classroom offers hands-on approaches to visualizing otherwise abstract mathematical concepts such as those involved in algebraic expressions, areas, perimeters, fractional notation, ratios, and volume to name a few examples.  Robotics can be one avenue toward removing the mysterious cloud of confusion that looms over traditional classroom mathematics courses. 

Manipulative tools are commonly found in grade school math classes as means of illustrating or clarifying the abstract statements required in understanding mathematics and prove themselves valuable tools, however, many students still do not comprehend the concepts because they have learned to tune out and declare that math is boring.  Robotics education in the modern mathematics classroom becomes a technological manipulative providing students an opportunity to experiment with numbers and their relationships while gaining practical experience with technology and the way mathematics is used in technology.

            Some of the features in favor of including robotics education in the modern mathematics classroom include:

1.  The visualization of abstract concepts

2.  Tangible, concrete, visual examples of written word problems

3.  Construction of memories through aural, visual, and sensory learning

4.  Cognitive enhancement through associative learning

5.  Experience in critical thinking skills and interpretation of data

6.  Contribution toward the students' self-esteem

            The visualization of the abstract is required for the learning of many upper level science and math classes.  There are many concepts for which it is technically impossible to create a viable illustration and where the actual "diagram" must exist within the mind of the student.  The degree to which an individual student will comprehend the unseen and theoretical will depend upon the ease the student has with figuring out exactly what the parameters of such concepts really are.  The use of robotics in the classroom allows students exposure to many experiments that can be planned and discussed in the theoretical and then acted upon by the students, who then "report" or evaluate on the responses they were able to obtain from the robot and under which conditions the behavior or success was observed. The bridge between the abstract and the concrete takes place through the use of the robot in the classroom as a manipulative tool toward building successful, insightful, probative thinking on the part of students.

            Students can study the area and perimeter of robot playing fields, calculate the shortest distance by time or by distance, calculate the changes introduced into equations when gear ratios are changed or directions are given in different orders.  Understanding of sequences and the effects upon a result if the sequence is changed can all be examined.  Models of order of operations can also be introduced through the use of sequential education using the robot's programming algorithms as immediate feedback for the students.  Thinking along a path the robot will take to execute the rules of a given task, writing a sequential program that recognizes and completes the task, and writing evaluations for the given task, (i.e. how well or not the robot achieved the desired objective) are all mental processes necessary for abstract thinking.  The evaluation phase, where the student reports the results, allows the instructor to see how well the student comprehended the experiment.  Both the abstract description and the correct interpretation of the "proof" of what really happened help instructors evaluate the student's competency and progress.

            In consideration of absence of concrete examples of word problems, many robotic experiments can illustrate motion, acceleration, braking, coasting, effects of gravity and other key concepts frequently used as topics of traditional word problems and do so in a real-time environment.  Experiments can easily be rerun to collect data for statistical analysis or comparison.  Students can discuss strategies toward understanding the word problems and relate their discoveries in words.  A "bridge" then forms between the abstractness of the word problem and the observable example that illustrates the concept under consideration.  Given an adequate and correct vocabulary, students can develop conclusions and solve basic word problems involving the use of some very basic mathematics equations using numbers and readings obtained from their robots.  The connections made by the students between the information requested (i.e. the answer) and the process used by the robot to find the answer is powerful and valuable toward raising the student's level of competency in abstract ideologies.  When students program their robots to perform a motion, the same motion mentioned scripted in the word problem presented to them in tandem with the experiment, they soon see the relationship the variables have to one another in an equation.  This empowers the student to change the numbers, hypothesize on expected results, execute the program and report back on success or failure.  While the student is learning through a scientific process, he or she is also learning the relationships between variables that help translate the abstract word problem into a tangible, concrete example with the added bonus that the student can now use the learned formula to extrapolate from and make other predictions or raise other questions.  This allows the mathematics concept to become a dynamic process within the student's active brain.

            By receiving digital instruction in binary logic from which education into bases and change of base can branch from, such concepts are immediately seen as valuable and necessary by the student in a robotic math environment.  Each step in the concept of changing bases, for example, takes on a degree of immediate concern and interest.  Students are challenged to "decode" what messages have been sent by their robot and how to communicate with their robot by writing "binary telegrams" to send commands to a robot to do a particular task.  Students can learn how to send the message first in base two, and then are empowered to learn other bases using skills learned in communicating with their robots in binary.  Hexadecimal is also an easily and completely legitimate expansion toward understanding different bases for communication in our standard decimal system.  Once the basic "map" for conversion is introduced, students are able to convert from decimal to binary, binary to decimal, decimal to any base, any base to any other base with ease.  While the robot merely introduces the concept, students are motivated through their desire to communicate with their robot and to understand the commands they send to the robots that they pay keen attention in the learning phase.

            A typical mathematics problem requires not only an understanding of the mechanics required to solve the problem, but also skills in decoding the actual scenario through which those mathematical concepts are revealed.  For example, a word problem advising the reader that "Susan's robot is always 6 seconds faster than Pete's" requires some abstract thinking to decode.  First, a student must realize that from the statement above, neither Susan's or Pete's true speed is given.  The student must decide what information is required to give as complete an answer as possible.  Since he/she cannot calculate either person's speed, the concept of an algebraic expression is the only concept that will allow the student to present a viable answer to the statement above.  Without understanding how language decodes into mathematical concepts, a student cannot understand the type of answer prompted by the statement above, nor can he/she fully understand the rational expectation of the instructor for such a response.  The student must understand the idea of using variables.  Additionally, the concept of using variables to represent desired, unknown, definite values must also be understood. 

            Through the use of robots as manipulative tools, students not only think about the words of the problem, but also have robots to manipulate on a racetrack, for instance, to understand the answer.  In the statement "Susan's robot is always 6 seconds faster than Pete's," no matter what number is used for Susan's age, a student can clearly see the concept of "6 seconds faster than" and can see visually that the manipulative tools do not change in number for "6 seconds faster than."  The concept of variables (Susan's speed, which can be any speed) and of constants ("6 seconds faster than", which never changes while manipulating), are clearly illustrated in front of the student.  Then "s = p + 6" can emerge as an understandable, defendable answer (s = Susan's speed, p = Pete's speed).  Understanding HOW a solution works is the key for many to retaining memory of the concept for later retrieval and application.  Such education allows the student to OWN the actual methods that allow the solution of otherwise abstract concepts.  Such students no longer have to rely on solely memorizing formulas without understanding WHY the formula works, because visually, he/she has seen that the formula works and how. 

            While it is true that the human mind learns in many different ways utilizing many capabilities both in isolation and in tandem, a known mnemonic is that of association.  Robotic education, specifically the programming of the robots, mimics this concept for the student.  The robot must be programmed to "understand" certain commands and does this by looking up meanings in a library of robot commands.  This process is parallel to the student's own learning process as the student quickly discovers that a robot uses the libraries of functions much like he or she would use a dictionary.

            Understanding how the robot interprets fractional rotations, for example, requires that the robot understand what certain commands mean, AND how to execute them.  By using robots, students are able to see associations between programmed code for example, and the resulting behavior.  Changes can then be hypothesized and developed which require the student to manipulate the mathematics that allowed the robot to achieve its goal previously, but is now inadequate to achieve a new goal.  Students can plan, program, execute and interpret the behavior of their robots and determine which variables are dependent and which are independent in the process.  Data can be sorted, charted and analyzed based upon associations made evident by the robot's behavior.  A student can experience that their ability to make an association increases when given the immediate confirmation of their hypothesis by the robot.  The relevant associations between commands and resulting behavior take on immediate impact and the virtually immediate and repeatable for observations obtainable by using the robots allows students to study the math concepts in more detail and to actually make their own associations and "tinker" around or experiment.  The value of free experimentation allows an otherwise bored math student the freedom to challenge him or herself and find new ways in which to make mathematics important and personal.

            Students are able to manipulate the sequence in which the robot does various tasks within its routine and observe the results.  Very quickly, as in the rotation of the wheels, students can collect data on distance traveled, time traveled, how change of speed, change of gear ratios and other variables affect the outcome of their experiments.  Students quickly discover both direct and indirect proportionality as it relates to the function of their robots.  Charts, Venn diagrams, statistical spreadsheets and other such mathematically valid forms of analysis can be used by the students to record and report their results.  As a result of experimentation with the immediate feedback (success or failure) of a given set of data, robotics provides fast confirmation of an individual's hypothesis, rational logical connections between concepts, and compelling insight into the correct meaning suggested by the data received.  A student is better equipped to imagine future fractional relationships because the concepts of association and relationship are concretely seen, believed, and proven.

            The value to a student's self esteem that arises from successes in the math classroom cannot be understated.  The robot becomes a focus of motivation for the student and removes the immediate panic usually associated with mathematics.  Many students relate well with the robots and will take many failures of the robot more easily than they will take a math paper with many wrong answers in a row.  The robots are not judgmental; they are inorganic interpreters of the student's commands.  Students are less threatened by the mathematics involved as they aid their friend, the robot, to do many fun tasks that just happen to require the use of math to accomplish.  Students are able to understand that the robot may not always do what they want it to do, but it always does what they have told it to do.  It will faithfully execute the program the student places in its memory.  If the robot fails, students usually display keen interest in understanding what went wrong and how to help the robot get to the goal.  For many students, the satisfaction that comes from getting their robot to perform is very empowering, and for many others, it is the only time when they feel completely in control of their own learning process.

Cross-Curriculum Education and Robotics

Today's science classroom is increasingly becoming limited through budget cuts in struggling school districts and the lack of qualified teachers to fill open positions.  Vernon J. Ehlers, Chairman of the US Committee on Environment, Technology and Standards stated that “Hudson Institute estimate[s] that 60% of all jobs in the early 21st century will require skills possessed by only 20% of the current workforce.[1]”  In short, the nation is not educating enough scientists to handle the expected needs of the future.  Given that the United States is a world leader in the area of technology as evidenced by the contributions from Silicon Valley, major biotechnology concerns, and successful NASA space missions, ways must be developed to incorporate more science education into contemporary classrooms.

This paper will attempt to illustrate how science and technology in the form of robotics education can be introduced into the science classroom through the selection of astronomy as the core curriculum addressed.  The rationale for attempting such inclusive education, a description of the method of delivery, suggested cross-curriculum links, and a suggested lesson will also be discussed.  The goal of this paper is to show that scientific technology can be implemented effectively in today's existing classrooms, and that such topics as robotics can be discussed in the classroom whether or not the teacher has access to hands-on equipment.

Currently, NASA has an extensive Robotics Education Project that allows teachers to have access to cutting edge curriculum at the click of a mouse button.[2]  Information is available from NASA on all the major missions, planets, celestial phenomena and mathematics useful in understanding key astronomy concepts.  Since a large amount of information is already available and robotic missions are either currently underway or planned in the near future, introducing students to scientific technology through the avenue of robots in space is one that is both timely, informative, and well within the reach of any motivated teacher.

The current statistics on educational procedure in today's classrooms reveal that little, if any, time is spent on science and technology.  A study by Horizon Research, Inc. conducted in 2000 attempted to quantify the quality of education provided in the K-12 grade range.[3] The data below is a sample of the statistics obtained from the Horizon study.  Percents are meant to include the percentage of time each category or type of science education listed took up from the total science curriculum presented to students.

Table A:  Percent of Science Education by Type

The Horizon study was conducted by observing classroom time spent on various types of scientific education.  Researchers also stated, "The prevalence of life and physical science lessons at the high school level mirrors patterns of course offerings reported in the 2000 National Survey of Science and Mathematics Education, where three-quarters of courses are classified as either life or physical science.  The percentage of lessons with a focus on science inquiry (typically in combination with another topic) varies from 2 percent of lessons in grades 9-12 to 15 percent of lessons in elementary school.[4]"  Statistics like these illustrate that school children in the United States are not being offered complete, compelling or competent educational opportunities in the areas of science and technology despite the fact that these areas are at the forefront of many major fields of study needing qualified workers. 

The theory under which this paper is conceived is that by combining subject matter, it is possible to even out the time spent on these three major science subcategories, thus enhancing the quality of scientific education received by today's students.  It is the belief by this author that science and technology are within the reach of motivated teachers, and that such incorporation of that educational opportunity can be achieved without the need of expensive equipment.  As stated earlier in the Horizon Research, Inc. report, 0% of students in grades 6-8, the middle school years, were receiving any science or technology lessons at all in their typical science classes.  This translates into students not receiving the message that such careers are worthy of their consideration.  Students are also not pursuing enough mathematics or higher sciences when they reach high school, and when they do, as the Horizon report shows, only about 8% of them receive any Earth and space science.

The Third International Mathematics and Science Study revealed that 4^th graders were found to be above average in math, but that by the 8^th grade, US students had fallen an entire grade level behind, and by 12^th grade, US students were only able to score higher than students from the countries of Cyprus and South Africa from a list of 41 countries surveyed.[5]  The National Center for Educational Statistics in the statement from the Commissioner states that “post secondary institutions provided remedial coursework for 28 percent of entering freshmen in fall 2000....Public 2-year colleges provided such coursework for 42 percent of their entering

students.”[6]  The fall-off in mathematics education parallels the Horizon study on Science and Technology stated earlier and reinforces the fact that once the system fails to provide an adequate education for its students, the effects spiral downward.  The numbers of students attending college who require remedial classes is high, and is a further symptom of the current state of math and science education.

It is clear that one of the reasons that America is not fulfilling its need to teach future scientists is that it is not providing the necessary introductory education.  The failure to inspire students early on is further compounded at the secondary and college levels.  Failure to capture the collective imaginations of students when they are young and building the appropriate educational bases for their future collegiate career bears a direct and detrimental effect upon their ability to choose technological careers for their future.

Time is pressing in many contemporary classrooms, however, through the use of cross-curriculum techniques many lessons can be taught while at the same time addressing several educational standards at once.  Cross-curriculum education is not new to the contemporary classroom.  Many of today's current textbooks offer supplemental materials and resources to bridge the cultural, subject matter, or gender boundaries.  In the scenario projected in this paper, we are attempting to show, as an example, how scientific technology can be introduced into a classroom while studying the necessary concepts provided in astronomy.  It is possible to include history of major astronomical discoveries such as Johannes Kepler's discoveries regarding the elliptical orbits of the planets, or Newton's Universal Law of Gravity.  Such concepts apply to Earth, but students can also learn that they also apply to the other planets as well.  Reading about robots and astronomy, even in the form of science fiction, can encourage students to read for pleasure and knowledge.  If teachers carefully choose the supplemental reading they allow students to access, important information can be offered to students.  Even art can play a role in the science and technology classroom as students design robots to travel to distant planets.  Speculation on what the outer realm of our universe might look like, what laws are obeyed by the objects that orbit there, and how robots might aid in the collection of future data are all topics easily discussed in the classroom setting using contemporary, inexpensive, and publicly available information.

Provided below is a suggestion for a lesson incorporating the subject areas of mathematics, science history, astronomy, Earth science, robotics and technology into one project.  Since the age group most neglected in science and technology as identified by the Horizon study is the 6-8 grade level, this lesson is aimed for this age group.

Mission to Planet Z

Assumptions:

·        Students will work in small groups of 4-5 students.

·        This project has several components and is done over an extended period of time.

·        Students will have studied Earth as a planet member of our solar system.  Teachers may review the essential facts to guarantee availability of essential knowledge to entire class.

Scenario:

·        Students will be comparing the Earth to one of the other inner planets.

·        Teachers can assign a planet based upon student interest.

·        Students are informed that they are going to plan a mission to visit an inner planet.  They are to compare this planet to what they know about Earth.  The mission will be to deliver a robot to study the planet.  The following information will be required:

Phase One:

·        History of who discovered the planet.

·        Statistics on the planet as compared to Earth.

·        Students will assemble a two-column list of attributes labeled “Same” and “Different”.  Under these headings they will sort data about Earth and their planet into the appropriate column.  For instance, if they discover that both planets have seasons, they would indicate this under the “Same” category, while an attribute like "atmospheric composition" would be recorded under the “Different” category.

·        Description of the planet's atmosphere, composition, orbital period, temperature, etc. will be written on cards to be placed upon a group poster.

·        Students will prepare a group poster about their planet and share with the entire class. Artwork, bibliography and rough drafts will be turned in by each student to document their participation in Phase One.

Phase Two:

·        Discussion of robots and what attributes a robot must have to make a successful mission to another planet.  NASA's Robotics Education Project can be accessed here for ideas.

·        Students list the tasks they want their robot to complete during its mission.

·        Students compare the attributes of the assigned planet and describe any limitations expected by the robot while on its mission.  Questions to be considered might include:

·        Is the planet so hot the robot might melt?  So cold it might freeze and refuse to operate?

·        Should the robot remain in orbit to do its mission?  Should it contact the surface?  How?

·        What are the dangers to a robot?  What will it have to sense in order to remain safe?

·        What are the advantages to sending a robot to the planet?  Why not humans?

·        Has NASA sent a mission to this planet already? What was discovered?

·        Does NASA plan on visiting this planet again?  When?  What do they plan to study?

·        Students draw up a final mission and present to the class for comments or suggestions.

Phase Three:

·        Using common household discards like bottles, straws, paper towel tubes, newspapers, cardboard, paper, wire, meat trays, and other materials, students build a model of their robot for presentation to the class.

·        Students prepare a final mission report as though their robot has already been to the planet in question. 

·        Students prepare a list of questions that they feel have not been adequately addressed by past missions.

·        NOTE:  This project can be expanded in the following ways:

·        Essential scientists who made discoveries in supporting mathematics or physics can be included in teacher discussions with an emphasis upon the need for others to take an interest in studying these topics and the importance of mathematics to space travel and discovery.

·        Historical coincidences such as literary evidences of comet sightings or discovery of planet information as recorded by Galileo, Tycho Brahe, Copernicus, Newton, Ptolemy and others can be included by teachers to introduce a history strand into the curriculum.

·        Current event logs from newspaper readings regarding space phenomena, astronomy, mathematics, science, robotics, and technology can be introduced by the classroom teacher as a reading component that is directly related to the science strand.  Such a log would encourage students in the practical knowledge that such careers are possible and which skills are useful in acquiring these occupations.

·        If the school has a computer lab with Internet access, introduce the students to the NASA Robotics Education Project website and assign readings and explorations from the website to be shared with the class.

·        Mathematics formulas using simple algebraic principles and related to planet orbits, distances from the sun, density, gravity, temperature conversion, etc. can be introduced to the students and calculated for all the planets in the terrestrial group.

·        This project can be repeated for the outer Jovian planets for double the fun and double the consideration of the material covered.

It is very apparent that the current state of education in the US does not meet the needs of students in mathematics, science and technology.  It is also known that there is a projected deficit between the number of available careers and the number of qualified personnel to fill those positions in the very near future.  America must change its educational methods to provide a more robust and comprehensive format for her children to be effective and competitive in the world marketplace.  Through the use of cross-curriculum educational methods teachers can provide a more comprehensive and connected education for their students; and through the use of already existing government assets, such as the NASA Robotics Education Project, teachers can implement a cross-curriculum program with relatively little additional cost.  By introducing students to the combined subject matter, as in the case of the Mission to Planet Z project, the foundation is firmly established as to the links between mathematics, science, technology, robotics, astronomy, and Earth science.  Students have the opportunity to see the link between history, current events and future explorations.  Today's students, through the use of cross-curriculum educational methods, come away from their primary and secondary education ready for challenging collegiate curriculum that will finish educating them on pathways set down and planned for in their earlier educational endeavors.  Waiting until college to address remedial skills wastes valuable educational resources and time.  Preparation must begin in the earlier grades by engaging students to prepare themselves for the mission of a lifetime:  employment in the fields of science and technology.[7]