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Robot Algebra Project: Introduction
Robot Algebra Project

Initial Robotics Education Research
Friction, Ratios and Proportions
Lessons Learned with Teachers
Abstraction Bridges
Design Based Learning Units (DBLs)
Robot Algebra Part 1: Robot Dancing (Example DBL)
Robots In Motion Version One Curriculum

 

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Introduction: What is the Robot Algebra Project?

“This project positions mathematics as a thinking tool.”                                      
Dr. Chris Schunn, University of Pittsburgh, CoPI on the project

It is crucial for teachers to develop students’ algebraic thinking and engineering design skills if we are preparing them to compete in the global economy. Algebraic thinking involves identifying patterns, relationships, and functions between one or more objects and being able to find the interrelationships between the variables that make up the objects; it is the beginning of symbolic reasoning.  Engineering design skills provide students with a systematized methodology for solving complex problems; it is rigorous creativity. The Robot Algebra project uses classroom-friendly technologies to develop students’ algebraic thinking and reasoning skills by placing them in technology-rich problem solving situations where they must find the mathematical rule or principle to unlock the solution to the problem and then apply that rule across multiple contexts.  The goal of the engineering design portion of the project is to teach students a research-based systematized method for solving engineering design problems. The project places mathematics and design engineering in contexts that students understand, encourages teamwork, integrates a systems ways of thinking, and most importantly makes salient, supports, and connects the math with the engineering design activity.

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Robot Algebra Project

The Robot Algebra Project develops of a set of Design Based Learning Units (DBL) that use a combination of the motivational effects of robotics, music, dance, and student success, combined with foregrounded mathematics lessons, engineering design, and competition to promote algebra readiness.  The project is a partnership between Carnegie Mellon (CMU), the University of Pittsburgh’s Learning Research and Development Center (LRDC), and a consortium of industry, government, foundation and education partners committed to improve both the quality and quantity of students pursuing science, technology, engineering, and mathematics (STEM) careers.
Across the region and nation, many schools and community-based organizations are proposing to use robotics to address STEM competencies. Yet, our research has found that most teachers miss the key STEM “teaching moments” that a robotics project places into a real-world context.  Often, teachers working with the robotics will allow students to be haphazard in their design process, avoid mathematics when possible (e.g., using guess-and-check strategies), which leads to weak solutions and reduces student learning.  Building from decades of research on instruction and learning, our project team believes that:

  • Robotics provides unique opportunities for teachers to place engineering design and mathematics in contexts that students find engaging and understand.
  • Learning is a cooperative process between the student, the teacher, and the problem; engagement must be present for optimal learning to take place. The choice of the problem is critical if the goal of the problem is to teach STEM.
  • Design problems are a natural way of teaching design skills and creating a need-to-know for students to learn math and science. DBLs organize extended curriculum units around design challenges.
  • Math is the language of science, engineering and technology. The mathematics in the lesson needs to be thought throughout by the teacher and foregrounded for the student.
  • For STEM education lessons to have a significant impact on a students’ math understanding, the focus of the math instruction must be centered on addressing specific mathematics concepts (not general) and the mathematics in the lesson must be made explicit not implicit.
  • For students to obtain a deep understanding of the focal math concepts, connections need to be made between the applied math problem and everyday math situations.
  • For students to move beyond parroting the teacher’s words, ideas, and solutions, and develop deep understanding, students need the opportunity to struggle with the problem, be able to defend their decisions, and explain their answer in their own words.
  • The ideal STEM curriculum gives students opportunities to solve problems that require them to work cooperatively, to use technology, to address relevant and interesting mathematical ideas, and to experience the power and usefulness of mathematics.
  • Curriculum implementation is important. The ability to vary teaching strategies; connect what the learner already knows to what they need to know; and provide individualized feedback for students needs to be taught to teachers.

The Robot Algebra project measures and iteratively improves curriculum that can be implemented in both traditional and non-traditional educational settings.   

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Initial Robotics Education Research

Over the last two years, CMU and LRDC have been guided by the following questions: “What can we do to improve Robotics’ ability to demonstrate and teach STEM?” and “Can we integrate the successes that LRDC has demonstrated in their science DBLs into CMU’s highly regarded Robotics curricula?”

In year one, we evaluated the mathematic concepts taught in a middle school level robotics curriculum.  CMU and LRDC began by performing a content analysis on CMU’s “Introduction to Robotics Engineering, Volume 1” LEGO curriculum, which is currently being used in thousands of classrooms. In this analysis, researchers identified many types of mathematical topics that students would have an opportunity to learn, and investigated the extent to which those topics were aligned with national mathematics standards. They found that the robotics lessons aligned well at the category level, but not as well at the individual math topic level. Simultaneously, the group conducted a case study analysis of an implementation of the robotics curriculum in an eighth grade technology classroom to assess whether mathematics ideas were salient as students were engaged with the tasks. Indeed, when prompted by the teacher during whole-class discussion, students brought in a wide range of formal mathematics ideas. However, because of the multitude and diversity of those ideas (e.g., measurement, algebra, geometry, statistics), we were not able to develop significant gains in math understanding with so little exposure to any one mathematics concept.  Similar findings were uncovered during a second test in a ninth grade technology classroom.  Our initial research suggests that robotics provides a promising engineering context to teach STEM concepts and that we can use this organizer to teach meaningfully relevant algebraic mathematic concepts in ways that make sense to students but this will take a redesign of the curriculum. 

After the end of the first year, the CMU/LRDC team decided that to improve the curriculum’s ability to teach math, that we would need to focus on a narrower set of mathematical principles. The mathematical focus of the Robot Algebra curriculum is to teach ratio and proportion; the ultimate goals of the curriculum are to develop algebraic reasoning skills, support the development of technological literacy, and to teach engineering design to students.  We chose ratio and proportion because research supports that these foundational mathematical concepts are not understood by students and these concepts are found in every math class that a student will study from middle school through college. We chose to teach engineering design to provide students with a systematized methodology to solve complex problems. We chose LEGO because of its ease of implementation into the conventional classroom as well as its ability to allow our project to scale. This project will focus, support, and align a set of robotic engineering activities with algebraic concepts that will effectively enable students’ mastery of algebra thinking skills.

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Fractions, Ratios, and Proportions

Traditionally, fractions, ratios and proportions have been considered middle school topics, but testing shows that high school and college students struggle with these foundational mathematic concepts. Fractions, ratios, and proportions are arguably the most mathematically complex and cognitively challenging concepts for students to understand.  In addition, ratio and proportion problems can be solved multiple ways, which often leads to student confusion.  The Robot Algebra DBLs give students opportunities to work on, struggle with, and eventually solve contextually rich applied ratio and proportion problems. For example, the Dancing Robots unit asks students to create programs that allow a set of physically different robots to dance in synchrony to music. In these lessons, students will learn that there is a linear proportional relationship between:

  • Speed and power,
  • Speed and wheel diameter,
  • Wheel diameter and distance traveled,
  • And, the center to center distance across the robots wheels and the angle of turn that the robot makes.

These proportional relationships, once discovered, give the student programmers the control that they will need to synchronize their robot dances. This DBL presents many teaching moments where the teacher can demonstrate proportional relationships that lead to student understanding.

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Lessons Learned with Teachers

Carnegie Mellon presented a scaled down version of the Robot Algebra project to a group of thirty mathematics teachers at a professional development seminar.  Teachers were posed with the following challenge: Given a robot with 5.6 cm diameter wheels and wheel encoders accurate to 1 degree of wheel rotation, program your robots to travel exactly 31 centimeters (the length of a ruler, it is easier to find rulers than it is to find meter sticks The teachers were broken into teams and tasked to calculate the mathematics to solve this problem, they were shown how to enter the values they calculated into their robot to test their results.  Within 15 minutes, all teacher teams solved the problem.  During the debriefing session teachers were asked to explain how they calculated the number of degrees the robot traveled. Below are the results. There are slight variations due to rounding, but the answers are basically the same.

Strategy 1 (direct proportion):

360 degrees / 17.59cm = (x)degrees / 31cm =
17.59 * x = 360 * 31
x= 634 degrees

Strategy 2 (unit ratio):

31cm / (.0488cm/degree) =
31/.0488 = 635 degrees

Strategy 3 (unit ratio):

31cm * 20.408 degrees/cm =
31 * 20.408 = 632 degrees

Strategy 4 (scale factor strategy)

31 cm is the distance to travel
17.59 is the distance traveled per rotation
31 / 17.59 = scale factor

360 degrees * scale factor = new number of degrees;
360 degrees * 31 /17.59 = new number of degrees ;
360 degrees * 1.76  = 634 degrees

Strategy 5 (iterative testing - guess and check)

This sample group of math teachers identified five different ways to solve this simple robot math problem.  And each group of teachers might argue that their method was the best one to solve this problem. At the very least, it was the method that they selected.

Students arrive in class with an intuitive understanding of proportional reasoning. Imagine a less than confident math student sitting in a classroom being taught a strategy to solve the problem that was incompatible with the student’s cognitive model.  If the intuitive understanding that the student has is incompatible with the way that the teacher presents the solution to the problem, and if the teacher insists that their way is the best or only way to solve the problem, then the student may experience uncomfortable levels of cognitive dissonance which can lead to low self esteem and a less math confident student. It is important that teachers recognize and position mathematics as a thinking tool and that the tool can be used many different ways to the same problem similar to the way that a person driving car can go from point “A” to point “B” using many different routes, but still getting to the same destination.

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Abstraction Bridges

We have found that teachers see robotics as offering great opportunities to teach STEM.  We also found that many teachers miss these opportunities because solving the robotics project becomes the focus of the class and the STEM concepts in the lesson are either assumed or implied instead of foregrounded, scaffolded and made explicit.  In order to foreground the academic STEM content, our team has developed a concept that we call an abstraction bridge. Abstraction bridges are easy for teachers to implement and are designed to:

  • Refocus the teacher and student’s attention to the academic component of the problem.
  • Provide a set of everyday problems designed to develop generalized set of problem solving strategies across multiple contexts for the student.
  • Provide formative assessment tools for the teacher enabling individualized remediation.
  • Tie the lesson to outcomes measured by NCLB standardized tests.

An added benefit to the development of the abstraction bridge concept is that it can be used by all STEM teachers who are using project-based learning and authentic assessment to teach. 

Robot math demonstrates specific mathematical principles in a focused-applied setting.  Students apply ratio, proportion, conversion of units, and measurement when they program their robots; the robot math context is much different than what is being taught in the mathematics classroom and assessed on NCLB-required state standardized tests.  The abstraction bridge concept is designed to enable students to form a cognitive bridge between what they learn in a focused-applied robotics setting and the types of mathematics that students encounter everyday. The Robot Algebra abstraction bridge model is a tool that the teacher will use everyday. Students will be required to solve at least one non-robotic math problem per day for the duration of the project.  Initially, the problems will be solved in class as a group; eventually the problems can serve as warm-up activities checking student understanding or will become homework assignments.  Students will be required to both solve the problem and also explain how they derived their answer. Documentation will be kept in the student engineering journal.

Example Problems are below:

______________________________

Robot Problem:
Faster Robot

Generalized Problem:
Better Deal

Directions: Show all work, describe how you got the answer using mathematics and words, and circle your final answer.

The Problem: Dontay has the choice of placing two different diameter wheels on his robot; 5.6 cm or 8.15 cm. Which robot wheel will go faster? Explain your answer using math and words.

Directions: Show all work, describe how you got the answer using mathematics and words, and circle your final answer.

The Problem: Two girls got into the theater on State Street for $3. Five boys got into the theater on Main Street for $6. Which group, the girls or the boys, got the better deal? Explain your answer using math and words.

So for the above robot problem example, in the physics problem you can ask the exact same question, and then follow up with a question on how much more torque is needed. This teaches conservation laws and a deeper lesson that you don’t get something for free.  In fact, the latter, the deeper life lessons, is the final thing i like to get out of the robotics courses. So for me, robots 1) captivate the students, 2) provide a vehicle to learn math and science and 3) can teach deeper lessons, even if they are design ones

Books on a Shelf
Directions: Show all work, describe how you got the answer using mathematics and words, and circle your final answer.
The Problem: If 6 books are 2/3 of all the books on Robert’s shelf, figure out how many books are 5/9 of the books on his shelf. Explain your answer using math and words.

Halloween
Directions: Show all work, describe how you got the answer using mathematics and words, and circle your final answer.
The problem: Suppose you are making treats to hand out on Halloween. Each treat is a small bag that contains that contains 5 Jolly Ranchers and 13 Jaw Breakers.  If you have 50 Jolly Ranchers and 125 Jaw Breakers, how many complete small bags can you make?

Better Deal
Directions: Show all work, describe how you got the answer using mathematics and words, and circle your final answer.
The Problem: Who gets more pizza, a girl or a boy? Explain your answer using math and words.

Lamon2007-MorePizza.pdf
How Far
Directions: Show all work, describe how you got the answer using mathematics and words, and circle your final answer.
The Problem: Car A and Car B are leaving the same place and going in the same direction. If it takes Car A 6 hours to get to the destination driving 20 miles per hour, how long will it take Car B to get to the same destination driving 50 miles per hour? Explain your answer using math and words.

Proportional Equation Example
Directions: Show all work, describe how you got the answer using mathematics and words, and circle your final answer.
The Problem: Write an equation for the following statement: There are six times as many students at this school as there are teachers at this school.  Use “S” for the number of students and “T” for the number of teachers. Explain your answer using math and words.

Functional understanding
Directions: Show all work, describe how you got the answer using mathematics and words, and circle your final answer.
The Problem: Angela makes and sell special-occasion greeting cards.  The table below shows the relationship between the number of cards sold and her profit.  Based on the data in the table, which of the following equations shows how the number of cards sold and the profit are related.  Explain your answer using math and words.

 

Mon.

Tues.

Wed.

Thurs.

Fri.

Sat.

Number sold, n

4

0

5

2

3

6

Profit, p

$2.00

$0.00

$2.50

$1.00

$1.50

$3.00

  1. p = 2n
  2. p = 0.5n
  3. p = n – 2
  4. p = 6 – n
  5. p = n + 1

Scale Factor Problem Example
Directions: Show all work, describe how you got the answer using mathematics and words, and circle your final answer.
The Problem: Roxanne plans to enlarge her photograph, which is 4 inches by 6 inches. Which of the following enlargements maintains the same proportions as the original photograph? Explain your answer using math and words.

5 inches by 7 inches             5 inches by 7 ½ inches

Proportional Reasoning Example
Directions: Show all work, describe how you got the answer using mathematics and words, and circle your final answer.
The Problem: A giraffe moves forward 10 meters every step that she takes.  A lion moves forward 2 meters every step that she takes.  If the giraffe takes 80 steps, how many steps must the lion take to cover the same distance? Explain your answer using math and words.

There are limitless numbers of ratio and proportion problems that can be developed as part of this project.  Below are the strategies that we will use to create a user friendly database for teachers to access:

  • Continue to build, sort, and qualify the database of ratio and proportion problems that currently reside at the Robot Algebra site. (This database will have many potential users as it is can be used by all teachers incorporating authentic assessment STEM challenges ensuring that math is covered in their lessons.)
  • Develop a structure in the database that helps teachers to quickly identify different types of problems in the database i.e. ratio word problems, graphs, tables, proportional algebra problems, fractional relationship problems…
  • Rate the problems from basic to sophisticated
  • Provide ongoing teacher training in the form of webinars, seminars, and multiday classes that will enable teachers to become expert math teachers.
  • Continual upgrade of the database based on teacher usage and research.

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Design Based Learning Units (DBLs)

According to the Standards for Technological Literacy published by the International Technology Education of Association, “All students need to develop an understanding of Engineering Design.”  Design projects are being used to motivate and teach science and technology in elementary, middle, and high-school classrooms across America: they can serve to open doors to possible science or engineering careers. Over the last six years, the University of Pittsburgh’s Learning Research and Development Center (LRDC) has spent a considerable amount of effort researching and refining how to teach “engineering design”. They have developed a particular methodology called the Design-Based Learning Unit (DBL). The DBL is a well thought out design-project where the student develops a technological solution using limited resources. The DBL is carefully designed and orchestrated so that students feel that they have a lot of choice and freedom in their design, but the DBL is designed so that the student can only successfully solve the problem by applying the mathematical concept that the teacher intended to highlight. The DBL, a highly scripted but open-ended engineering design problem, has the dual benefit of being able to teach academic concepts in rich interactive environments as it develops the students’ problem solving, scientific inquiry skills, and engineering design competencies.

For the DBL to motivate students and teach core concepts effectively and efficiently across a variety of settings, the units must be developed in very particular ways. Just giving students a design challenge produces chaos. Telling students exactly what to do at every step produces boredom and little learning. A well designed DBL begins with an engineering design problem with clearly defined rubrics that define what a successful solution looks like, coupled with seemingly unlimited student choice about how they might solve the problem. The engineering design process systematizes the thinking and the learning issues that must be supported. Thus, students practice a real design process but also receive carefully designed lessons that foreground the STEM concepts the teacher has pre-determined important to teach. The figure below shows the typical macrostructure of a DBL unit. The DBL teaches systematic design process while motivating and supporting STEM learning.



The DBL provides a working, tested framework upon which Robot Algebra design problems are built. They provide a natural mode for teaching the topics which are inherent to robotics and technology and have proven success in the classroom over several years of research and refinement.
The fundamental elements of the engineering design process include:

  • Defining the problem, including thorough research enabling the formulation of well developed potential solutions
  • Establishing clear objectives and criteria enabling the development of a requirements document
  • Systems decomposition - break the problem down into granular modules
  • The use of time management tools like PERT and Gantt Charts
  • Ideation, brainstorming, and design reviews
  • The development of working prototypes
  • Iterative testing, evaluation, and improvement of the prototype
  • Selecting the best solution based on established criteria
  • Iterative improvement based on research and testing

 This methodology is valued in the workplace and needs to be taught to students.

Robot Algebra Part 1: Robot Dancing (Example DBL)

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