Students at the Year 8 level used LogoWriter software to design computer screens to teach Year 3/4 students Fractions. Students were set the task of doing transformations between words, symbols and pictures using LogoWriter. They recorded their experiences in a journal and identified problems they encountered and solutions to those problems. They helped each other solve problems in Fractions, design and computer programming.

Outcomes from this learning sequence included expressive writing about mathematics, improved scores in a Fraction test, improved fluency in Logo programming, improved self management skills, increased cognitive resilience (overcoming frustration and not giving up), improved time management, and increased faith by the students in their own thinking patterns. Students remained motivated and interested in the Fractions topic for a 7 week block using this approach. The culture of mathematics was perceived by the students to be different and more interesting than traditional textbook maths. Some students dropped in at recess and lunch to work on their projects.

The final combined software product is a useful piece of educational software that can be utilised by other teachers for diagnostic purposes as well as being an exemplar of what can be achieved with LogoWriter when it is used in this way.


A pretest of 41 questions about Fractions (selected from Idit Harel's pretest -- see reference at end for this excellent resource) was administered to the Year 8 class at the beginning of the topic. The test involved a variety of Fraction transformations between words, symbols and pictures with multiple choice answers. Here is a sample of a couple of questions from the pretest:

frac1 (5K)

frac2 (3K)

The same test was then administered to the Year 3/4 students by the Year 8 students. The Year 8 students were asked to explain the questions to the Year 3/4 students if they did not understand them.

Introductory lessons:

As well as the Pretest other introductory lessons were held with the Year 8's to explain the nature of the Project and get the students started on the design project. This included:

Conceptually, the students were being asked to integrate their knowledge and learning about the 3 different areas of Fractions, Logo Programming and Instructional Design. Their brief is diagramatically represented below:

frac4b (2K)

Information for a Logo novice

triProcedure (4K)

To create even a simple Logo screen involves a lot of mathematical learning. For instance, to create an equilateral triangle requires knowledge of the external angle of a triangle (120 degrees). To create a more complex design, such as a title page for the Project, requires more sophisticated manipulation of the turtle, for instance by using cartesian co-ordinates (Logo primitives, show pos and setpos[xvalue yvalue]). Conceptually, this is a fourth transformation of Fraction representations in addition to the word, symbol and picture transformations described above.

Regular lesson format

After the introductory lessons the class then gradually settled into a regular lesson format that went as follows:

The teacher kept his own journal at the same time as the students. During the middle part of the lesson (40 min.) the teacher mainly worked as a facilitator, moving from group to group, answering questions and helping students design and program their screens.

When students completed a Fraction screen then they would go to the Year 3/4 room and ask their partners to return with them to complete the problem on the screen. The teacher would often intervene after this to assist the Year 8 students to evaluate their screens. Did the Year 3/4's find them too easy or too hard? Were there any confusing design aspects of their screens (such as confusing a picture of 3/4 (three-quarters) with 1/4 (one-quarter))? What would be an appropriate question to ask the Year 3/4 students next? The Year 8's were then offered a copy of the pretest to go through it again with the Year 3/4's so as to discover what they understood and did not understand.

This part of the programme carried on for about 5 weeks at 4 lessons a week. In that time each group (1 or 2 students per group) had designed between 1 and 4 Fraction screens. Some groups designed special title pages and special answer pages as well.

FRAC6 (3K)

Here is a design problem that arose in the course of one lesson. The Year 8 Designer intended C to be shaded 3/4 in white and the correct answer to be E. However the Year 3/4 student saw C as shaded 1/4 in the darker colour. After the ambiguity was pointed out by the teacher the Year 8 Designer altered the question to, "What picture shows 1/4 shaded in white?"

The cultural setting

Although this Project used computing technology extensively, it was culturally driven not technology driven. The elements of the cultural setting included the skills and style of the teacher, the background of the students, some important elements of the Paralowie R12 School environment and finally the computing hardware that was available.

Paralowie R12 School

Paralowie School is located in one of the lowest socio-economic regions of Australia. Absenteeism and lateness to lesson by students are chronic problems in the School and a variety of programmes already exist to meet special student needs. Students in the school are under some pressure NOT to embrace the traditional culture of maths and science since they are likely to be labelled "squares" by their peers. However, it was noticeable that some of the students from different cultures (eg. Serbian, Vietnamese) overtly rejected this cultural stereotype. The School Administration supports innovative teaching practice and so I have been encouraged to pursue my investigations into the effectiveness of transforming a maths learning culture into something more relevant and meaningful to students by using the LogoWriter medium. However, Logo is NOT an established part of the whole school culture at this stage. The Year 8 class is part of the new Paralowie Middle School (Years 6-9). As such I taught the class for 10 lessons a week (4 Maths, 4 Science and 2 Personal Development). This enabled me to establish closer personal relationships with many of the students than is normally possible for High School teachers.

Teacher input into the class culture

The central element of my teaching style can be described by the metaphor of relationship. I believe that learning occurs best when students develop a positive relationship with the teacher, their classmates and the subject matter, in this case maths. I select teaching materials with the idea of building a positive relationship at the forefront. This is a central reason for using Logo, for Logo is closely associated with an educational philosophy of making Maths personally meaningful or appropriable. My students would see me as an evangelical promoter of Logo and someone who can answer any question they have about it. Other maths teaching materials that I use extensively are Australian developed "hands on" products called RIME (Reality in Maths Education) and MCTP (Maths Curriculum and Teaching Program).


This Year 8 class had a high proportion of English as a Second Language students of a variety of backgrounds. 5 students had Khmer background, 2 were Australian Aboriginal, 2 Latin American, 1 Vietnamese, 1 Vietnamese / Maltese, 1 Serbian and the remaining 13 were Anglo-Saxon Australian.

Each student brought into the classroom certain cultural attitudes -- attitudes to mathematics and Fractions that have developed over 8 years of Schooling, attitudes to computers ranging along a continuum from extreme reticence (initially) to extreme interest, attitudes about being put into the role of being expected to teach younger kids, attitudes about how to be "cool" in the classroom. I would loosely and simplistically group my students as follows:

Although I believe that this Project could succeed in many classes it is worth stressing that it did succeed in this class with its high proportion of Socials and Strugglers (15 out of the 25 students)


At the time of this project there were 17 computers in the room shared between 24 students. Hence some students had to double up on the computers. The computers are mainly ageing XT's (5 years old) with a variety of monitor formats. All of the computers were old and some were unreliable. Time and work was sometimes lost because of mechanical failure. Only 8 out of 17 computers had colour screens which was a big drawback because the students love to use colour.

Background knowledge

In Logo: Students had very little knowledge (if any) of Logo at the beginning of the school year. During 1994 they had been exposed to it in a fairly intensive way over 3 terms (10 weeks per term) as part of the Maths course prior to commencing this project. A closed book test held during Term 3 indicated that students knew between 12 and 68 LogoWriter primitives each, with a mean score of 38 primitives.

In Fractions: Students came from a variety of feeder schools with diverse curricula and teacher expertise in maths. Initially knowledge in Fractions was ascertained by a Fraction pretest (taken from Idit Harel's thesis). Scores in the pretest varied between 13 and 36 out of 41 with a mean of 25 out of 41.


Students were assessed for this unit of work as follows:

  1. Quality of their written journals, marked about every 1.5 weeks.
  2. The number of problems identified in their journals and the number of solutions to the identified problems
  3. How many times they helped other students as recorded in the journals
  4. Quality of the Logo Fractions screens that students designed
  5. How many screen that were designed (ie. how many times that Year 3/4 students were invited to the room).
  6. Post test of Fractions (same as the pretest)
  7. Open book test at end with this question:
    Place Logo primitives into groups or categories of your own choosing.

Post-test results for Fractions test for Year 8 class: out of 41


This improvement occurred over 7 weeks without any organised formal instruction from the teacher to the whole class about how to solve Fraction problems. Twelve students improved their score substantially (between 5 and 18 extra), 8 marginally (between 1 and 4 extra) while 4 obtained the same score or less.

Samples of students work

From the journal of Ngoc Tran 9/11/94

"I have brought 3 girls up from Ms Munro's class, and show them my animation on computer of Fraction, and they all got incorrect answers by guessing. One of my year 8 friend who not bad at maths but couldn't even get it right, the problem is that they cannot recognize the equal shapes or areas."
frac7 (4K)

By the design of her question, Ngoc is clearly identifying a common problem students have about Fractions, that the parts have to be divided into equal areas.

From the journal of Daniel Curnow Monday 21/11/94

"Today I am going to make a harder procedure maybe one that the younger kids found hard in the test they had. The last procedure we did the younger kids found it easy but it took a while before they got the answer. they said that they did not know that one fourth is the same as one quarter."

Daniel's screen:


Daniel is reflecting on as aspect of language in maths. Students sometimes become confused when different words are used to represent the same value, in this case one fourth and one quarter.

From the journal of Sarah Scott Monday 14/11/94

"I showed them (the Year 3/4 students) my screen and they found it easy. I showed them the fractions test and pointed out the hard ones and they knew the answers to all of them. I don't know what screen to do that they wont find easy. I will design that screen in planning tomorrow."

Tuesday 15/11/94

"Today I will ask Mr. Kerr what type of screen I can do now since I am not sure. I just thought of one."

frac8 (3K)

Sarah's was paired with a talented student in the Year 3/4 class who had found her previous screens easy to solve. Sarah thought up this more difficult screen without teacher help so as to offer the Year 3/4 student a real challenge. Her journal entry clearly documents the problem and the moment of creation.


Rich Learning Outcomes

As well as the learning about Fractions my strong impression was that significant amounts of learning were also occuring in such diverse areas as:

It's hard to prove this and unfortunately you, the reader, were not there. Also the merits of the whole approach rests or falls on this claim. The best I can do is to refer you to Idit Harel's thesis for a far more comprehensive documentation of these claims.

What follows is a discussion of some of the claims and connected issues.

Improved Fraction Knowledge

How come students improved their Fraction knowledge (shown by the Pre and Post test results) without being formally instructed in Fractions?

The environmental framework was constructed by the teacher by setting the students a teaching task, a design task and a medium to work in. These were non negotiables but beyond that the students had the freedom to do their own thing. Students were put into the role of a teacher and all teachers know that having to teach a topic is a very good way to learn it. Students were set the task of doing transformations between words, symbols and pictures using LogoWriter. The LogoWriter procedures written by the students became a fourth type of transformation that kept students focused on the manipulation of Fractions. They were learning in constructionist fashion using Logo as a medium over an extended period of time. By constructionist I mean active, self directed exploration providing the opportunity for internal representations of fractions to evolve.

Dealing with complexity

A complex learning sequence where students designed computer screens to teach other students Fractions was completed successfully by the class. The students did not find it particularly difficult or confusing to be learning different skills at the same time. The teacher did not have to nag the class to get on with their work, apart from the occasional individual exception. By and large students self managed their own progress with the teacher (or another student) acting as a helper or facilitator when they became "stuck" with a particular problem.

Inclusive learning activity

All of the students, except two latecomers to the class, contributed to the final instructional software design product. Most of the students designed and made their own screens. The quality of the final screens varied considerably but the final collective class product is a useful piece of instructional software. Some students copied designs from the pretest, which was made readily available throughout. The teacher did not interfere if students chose to do this interpreting it as a lack of confidence that would be overcome in time.

Individuality was expressed

Some students displayed their individuality, initiative and skill by designing special features, such as:

  1. Attractive title pages, designed using LogoWriter
  2. An elaborate answer screen where a truck backed up to pickup a "YOUR RIGHT" shape and towed it across the screen
  3. Flashing colour screens. One group discovered this by accident and it quickly spread throughout the class.

The teacher did not ask students to do any of this but did approve and encourage it when it happened.


Motivation and interest in the Project by both students and the teacher remained high throughout the whole 7 week block. Usually, the teacher could work intensively with a small group of students with his back turned to most of the class. Ihave taught the same class using other more teacher directed methods and found this method the most effective for maintaining motivation and interest over an extended time period.

Problem Finding and Solving

Nearly all of the students systematically identified and recorded problems that occurred in the course of their work and solutions to many of those problems. According to my records, in the course of the Project 123 problems were identified by the students and solutions to 47 of those problems were recorded. The sort of problems that were identified included programming problems, technical problems, design problems, maths problems and personal problems.


Students appraised the suitability of the product they made for the target audience (Year 3/4 students) and in many cases made plans to improve their subsequent designs to better fit the target audience. eg. in some cases the first design was too easy for the particular Year 3/4 partner and so a more complex question was designed next time. This is a similar process that real life teachers go through in learning how to teach effectively.

Improved Fluency and Confidence in Technological Competence

Students became more fluent in their use of Logo primitives so that certain strings became second nature to them. For example, I have seen one particular programming sequence that involves trialling something on the front of the LogoWriter page in the command centre and then selecting, copying and pasting it to the flip side, which involves about 12 different keystrokes in correct order, gradually become second nature to a large proportion of the class. This is just one illustration of the improvement in programming fluency and increasing confidence of students in working with complex technology that could be readily observed in the classroom. Students, to varying degrees, developed a positive relationship with the computer and a sense of self as a technically competent person.

Expressive Writing about Maths and Technology

Students wrote systematically about mathematical and technical questions and in many cases included how they felt about these events. They wrote with feeling about technical questions and their collaboration with other students.

Genuinely Useful End Product

The "final" product is educationally valuable. The student software designs have been compiled and edited by the teacher and some of the more enthusiastic students. It is envisaged that the end product will be a useful diagnostic tool for maths teachers as well as an exemplar of what can be done with LogoWriter.

The "final" product could be developed and refined further in the future, simulating within the School the process that commercial software developers have to go through. It might even be possible to work on the product over an extended time with a select group of students to improve the software to commercial standard and then market it.


Methodology: Objects to think with

Teachers face the task everyday of how to make their subjects relevant and interesting to their students and this is seen to be a particular problem with maths. One way to look at this is from the point of view of objects to think with. The teacher and students co-construct a learning environment that is replete with "objects to think with". These "objects" include:

Taken together these objects represent the ISDP (Instructional Software Design Project)

Harel and Papert (1990) argue that some materials are better with regard to the following criteria:

When used in the way described above LogoWriter is a most effective learning medium to think about Fractions and Design according to these criteria.


The approach adopted in this learning sequence was inspired from Idit Harel's PhD thesis titled: Software Design for Learning: Children's Construction of Meaning for Fractions in Logo Programming (MIT, June 1988). I obtained a copy of the thesis for US$20 by writing to:

Epistemology and Learning
MIT Media Lab
20 Ames Street
Cambridge, MA 02139

Idit Harel's thesis was subsequently published as a book called Children Designers (1991), published by Norwood: Ablex.

Harel, I. & Papert, S. (1990) Software Design as a Learning Environment. Interactive Learning Environment, 1, 1-32

Kafai, Yasmin B., Minds in Play: Computer Game Design as a Context for Children's Learning (1993). This thesis is available from the same 'Epistemology and Learning' address given above for the Idit Harel thesis.


Helen Munro, teacher of the 3/4 class at Paralowie R12 School in 1994, for her flexibility and collaboration