Overview

The book that I am proposing is a comprehensive series of Logo-Math geometry lessons that requires the use of computer programming with Terrapin Logo to learn mathematics. Immediate feedback from the Logo program provides “just in time” information students can use to test their thinking. Over time, students learn that it is more efficient to develop a plan than to use trial and error. The use of programming in Logo or “turtle math” is one of the most often researched topics. Programming is currently referred to as computational thinking. Robotics provides a good example of how programming is relevant to everyday life.


Wikipedia accurately describes computational thinking as the thought processes involved in formulating a problem and expressing its solution(s) in such a way that a computer—human or machine—can effectively carry out.[1]
Computational Thinking is an iterative process based on three stages (graphic omitted):
  1. Problem formulation (abstraction);
  2. Solution expression (automation);
  3. Solution execution and evaluation (analyses).

The history of computational thinking dates back at least to the 1950s but most ideas are much older.[2] The term computational thinking was first used by Seymour Papert in 1980[3] and again in 1996.[4] Computational thinking can be used to algorithmically solve complicated problems of scale, and is often used to realize large improvements in efficiency.[5]


  • The Logo-Math Wiki contains a series of 15 introductory Logo-Math geometry lessons written for use with students in grades 4-10. The lessons are specifically scaffolded to introduce the Logo programming language and the relevant mathematics in a “just in time” teaching model. The purpose of the lessons is to facilitate an understanding of the why behind the mathematics through guided exploration and practice. Each lesson is correlated with Logo commands and the mathematics vocabulary (see the Topic Tool Matrix). Each lesson contains completed examples, tasks and a mini-project.
  • Currently, excerpts from each Logo-Math geometry lesson appears on the Logo-Math Wiki. Statistics indicate that the Wiki generates, on average, 30 hits per day from users across the world. Lessons can be downloaded and printed individually or in two parts: Lessons 1-8 and Lessons 9-15.
  • The logo lessons were originally included in a course pack that was developed for an undergraduate geometry course. As a result, additional mathematical information that could be used to augment the focus on learning mathematics through coding is available.
  • The following comments about the Logo-Math Geometry Lessons were written by a pre-service teacher in my spring 2010 technology course.

I think the way Dr. Shafer set up this Logo module was a great way of demonstrating the Well-Order Problems principle (Gee, 2005). We were to start with some simple polygons—triangles, and quadrilaterals, then n-gons—and gradually move our way up until we could combine shapes to create more complex figures. We could make generalizations about the simpler figures (squares easily comes to mind, as they were the simplest for me to make), which could lead us to try some things on more difficult drawings (which may or may not have worked, but at least it was tried).

My Logo experience definitely left me feeling that my mind had been stretched in various ways. I have never really written computer procedures before this semester, and this experience definitely opened me up to it. It was fun to try and loop procedures and notice connections between similar but different results (for instance, the difference between the commands repeat 360 [fd 1 rt 1] and repeat 180 [fd 1 rt 2]). It truly seemed that I could manipulate whatever I was doing in Logo to fit whatever I wanted.

Reference
Gee, J. P. (2005). Learning by design: Good video games as learning machines. ELearning 2(1), 5-16.



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Primary Audience: Mathematics Teachers and Home-School Parents

  • The documents and videos used in the 15 introductory Logo-Math lessons were created for teacher to use with their students.
  • Lessons can be duplicated and used by classroom teachers or parents. Teachers making substantial improvements or additions to the lessons are encouraged to become a member of the Wiki. Teachers wanting to contribute additional lessons or projects are also encouraged to join this Wiki. Note that the lessons copyrighted under the Creative Commons License cannot be used in commercial ventures.


About Me

Kathryn G. Shafer is an associate professor of mathematics education at Ball State University. She holds a PH.D. in Mathematics Education from Western Michigan University. Kathryn joined Ball State University in 2008 and received tenure in 2014. Prior to that, she was a faculty member for seven years at Bethel College (Mishawaka, Indiana). Prior middle school and high school mathematics teaching experience includes both a suburban (two years) and a rural setting (three years) in Illinois.


Relevant Publication

The "Digital Duck" publication showcases Gina and Zach's use of Logo to create the net of their shape.

Shafer, K. G., Severt, G., & Olson, Z. A. (2011). Sketching up the digital duck. Mathematics Teacher, 105(4), 262-268. National Council of Teachers of Mathematics: Reston, VA.