Using Logo In Mathematics

USING LOGO IN MATHEMATICS

John Turner

PLC

Use of Computers for Enhancing Classroom Outcomes - Using Logo in Mathematics

In line with the school's "Computing Across the Curriculum" approach to using Computers, the PLC Mathematics Department has developed many computer related programs designed to enhance each student's learning. The use of Logo highlights the potential for using the computer to promote and develop a student based approach to learning.

For the unaware, Logo is a programming language developed in the late 1960s at the Massachusetts Institute of Technology by Seymour Papert and associates. While the consideration of Logo can take many forms, from educational philosophy to computer science language, this example is built on the use of turtle graphics within Logo as a learning environment to promote the geometry learning. There are many books available on Logo, two of which deserve special mention are Papert's Mindstorms (1980) and The Children's Machine (1993)

Teachers decided that if Logo was of value then every student should benefit. Therefore Logo is used by every mathematics teacher within his or her Mathematics classes. The basis of the approach is built around material that enables students to work at their own pace, work with the teacher in discovering 'new" mathernarics, and which promotes the experimental and debugging opportunities that are built into Logo.

The process in using the material is critical. For this programme a model has been built around the material, and not the teacher, providing the initial guide to the student. The student is asked to:

READ This is designed to wean the student off expecting the teacher to be the source of all that is required The material is designed to encourage the student to generate their own tasks and interpretations, but making sure they are within parameters that will advance their mathematical understanding

PLAN The student needs to realize that thinking can be both independent of the computer and in conjunction with it. By planning away from the direct input of the computer the student can put their thoughts into a coherent structure with which to then enhance through interaction with the computer.

EXPERIMENT This is designed to allow students to take full advantage of the interactive potential of the Logo environment.

DEBUG Students need to appreciate that a "problem" or bug is not necessarily a negative occurrence, but can be an opportunity for them to address their own thinking shortcomings, work through to a personally satisfying outcome, and learn to appreciate that they can solve their own problems

SOLVE Traditional Mathematics is built around the need to solve problems and we do not believe that students should lose sight of such requirements

COMMUNICATE Too often in computer work the final product is left to the students. Our approach to computers in the school is that students need to be aware of the need to work in a wider context. In this case the student is called on not only to describe their thinking as part of the process, but also the mathematics contained in their solutions

There is no clear stated demarcation between each of these requirements as students are encouraged to reflect, develop and improve their process thinking. Interaction between students, and with the teacher is encouraged within PEEL guidelines at all stages.

As an example of this Logo approach, the following material is provide. While it is specifically designed to complement the Year 9 Mathematics curriculum, it is in fact included as part of a "Logo" workbook that students are provided with in Year 7. This is to allow students who may have developed Logo and/or Mathematics skills in their primary schooling to start off at a level applicable to their ability and potential. Hence, we see no reason why some Year 7 students cannot complete this material.

The research of Anne McDougall at Monash University testifies to the potential for students to understand concepts thought previously beyond them at a certain age if given the opportunity to learn through using a computer in a personally constructivist way.

For those mathematically inclined, I hope you can see how the tasks "guide" the student towards discovering trigonometry.

For the Logo inclined, you might note that the tasks do not require any Logo knowledge beyond the basic level of moving the turtle (FD, BK), turning the turtle (RT, LT) and writing simple procedures. It is an area of mathematical thinking development, as guided by the tasks, that the real power in learning lies.

Task One Complete the following procedures to make the turtle draw complete triangles.

Question 1.1: What are the internal angles for the three triangles you have created?

Question 1.2: In what way are the internal angles different from the turning angles used in the Logo procedures?

Task Two Complete the following procedures to make the turtle draw complete triangles.

Question 2.1: What are the similarities between the three triangles in task two?

Task Three Complete the following procedures to get the turtle home by adding two extra commands.

Question 3.1: What was common to the three triangles in this task?

Question 3.2: How would this knowledge help you with future problems?

Task Four Complete the following procedures to get the turtle home by adding two extra commands.

Question 4.1: Explain how you obtained your solution for problems 1 and 2. Design your own movement problems and try them out on a friend.

READ	This is designed to wean the student off expecting the teacher to be the source of all that is required The material is designed to encourage the student to generate their own tasks and interpretations, but making sure they are within parameters that will advance their mathematical understanding
PLAN	The student needs to realize that thinking can be both independent of the computer and in conjunction with it. By planning away from the direct input of the computer the student can put their thoughts into a coherent structure with which to then enhance through interaction with the computer.
EXPERIMENT	This is designed to allow students to take full advantage of the interactive potential of the Logo environment.
DEBUG	Students need to appreciate that a "problem" or bug is not necessarily a negative occurrence, but can be an opportunity for them to address their own thinking shortcomings, work through to a personally satisfying outcome, and learn to appreciate that they can solve their own problems
SOLVE	Traditional Mathematics is built around the need to solve problems and we do not believe that students should lose sight of such requirements
COMMUNICATE	Too often in computer work the final product is left to the students. Our approach to computers in the school is that students need to be aware of the need to work in a wider context. In this case the student is called on not only to describe their thinking as part of the process, but also the mathematics contained in their solutions