“The universe cannot be read until we learn the language in which it is written. It is written in mathematics, and the letters are triangles, circles and other geometrical figures, without which it is humanly impossible to understand a single word.”
– Galileo Galilei
The universe is a source of basic geometric shapes, we discover them through the observation of nature. This understanding of basic shapes and their functions have taught us to mark time and space in a variety of ways which has inspired mathematics, technology, language and ever-evolving civilisation.
As geometry is inspired by nature, children should also understand its elements by discovering the world of shapes around them. LOGO Turtle is one such tool that I experienced, which would help children discover the concepts by exploring on their own.
I remember when I was first introduced to LOGO Turtle, I was so excited as I was trying my hands on it despite lacking the technical background and coding whereabouts. I was more than excited as I am going to learn to code as well as to see a turtle move according to my wish on my computer screen. I could make it move up and down, do right and left and experiment with its movements. Spending time exploring the different commands actually gave me the confidence to experiment. But, initially I got confused in the right and left command as I thought that the turtle would move in that direction but when I understood all the possible angles were explored. I loved exploring reflection and transformation concepts and bringing them alive in the form of turtle art.
Conversations – Math Talk
While exploring the tool, as a teacher I always thought that this tool would give so much of space to explore and learn while talking about mathematics. LOGO Turtle actually provides that space of interaction and talking about the concept which is often missing in mathematical learning. Reuben Hersh in his book “What is mathematics really” talks about “mathematics which is learnt by computing, by solving problems and by conversing more than by reading and listening”. This important element of mathematics can come alive as the teacher can start a discussion on angles, directions, movement and also ask questions like – What happens when you enter 45? What about 180? Some prompts like – Can you try making a shape using what you all have learned or explored till now? Talking about their learning and thinking in a mathematics class would construct many learning dimensions for the children. But some students might struggle putting these pieces together, but combining actual physical movement, concrete experiences or walking like the LOGO Turtle along with verbalising would help them to conceptualise.
A constructivist curriculum focuses on students actively experiencing and building ideas to solve personally meaningful problems along with taking ownership and being self motivated. The traditional geometry curriculum which starts from the concept and then ask the child to solve a problem which may or may not be contextual, while if we teach geometry through LOGO Turtle children would explore on their own and reach to the concept pre-requisite after having a concrete contextual experience. This would mean that children would be inventing basic concepts in mathematics on their own, thereby learning to be a mathematicians. There are numerous reports which revealed that students fail to learn basic geometric concepts especially geometric problem solving due to lack of geometric intuition. The children do not find enough examples to experience for a conceptual and procedural understanding of topics to be studied in higher classes like vectors, coordinates, transformations, and trigonometry.
The whole process for me started with exploring concepts intuitively on the LYNX which helped me to apply knowledge of geometric concepts in making complex patterns. This made me think that LOGO Turtle is a powerful platform for intuitive learning.
As mentioned by Seymour Papert in his book Mindstrom “I take from Jean Piaget ~ a model of children as builders of their own intellectual structures. Children seem to be innately gifted learners, acquiring long before they go to school a vast quantity of knowledge by a process call “Piagetian learning” or “learning without being taught.””
As an educator who believes in the principle that children learn a lot intuitively, I have experienced the same when I observed toddlers playing with loose parts, making shapes or patterns, using things in symmetry while making a pattern or balancing things, and taking decisions intuitively. This process of children experimenting on their own, makes me think about “objects to think with” as mentioned by Seymour Papert which is a powerful concept to reflect upon as this keeps the learner at the centre of the learning process. The best part is that the child does not have to think about creating things but as they learn to use it, they create and discover which can be related to constructive learning as it means that learners construct the mental models to understand the world around them. LOGO Turtle is giving that space for intuitive learning but it is also serving as an object to think which I need to explore myself with children to understand it further.
It can be said that physical actions on concrete objects are necessary to help students construct geometric ideas through concrete materials like geometry rods, blocks, geo-board, isometric papers and many others. Using manipulative facilitates the learning process and it is equally important to see whether the children are able to establish a link between the action of the manipulative to describe the action. Thus students must internalise such physical actions and abstract the corresponding geometric notions.
This learning of geometric ideas can be seen in LOGO Turtle as the children would invent basic concepts which would help them progress to higher levels of thinking in mathematics. Van Hiele has given a theory of geometric thinking levels of students in which the students move from one level of thinking to the next. If a teacher would plan the lesson combining concrete material, experience from the environment as well as combining LOGO Turtle to teach geometry, I strongly believe that students would not only progress into higher levels of thinking but also build conceptual structures about concepts of geometry like shapes and angles which they would be able to use it in other situations to solve other related problems. They would thus be learning geometry relationally.
Reflections and Implementation
Actually this project came at a time when we were exploring revolution as a theme with children in our organisation. The community has been doing embroidery and thread work on things around cloth. So we were exploring ideas as to how cloth had been used as a tool for revolution and in that specifically quilting. The children created quilts and did thread work around revolution. I would write in detail about this project by children in my next blog and also I am excited to give them the exposure of LYNX where they would be exploring more about it and making quilt patterns and then making those quilts physically.
It has been an enriching experience for me to explore LOGO Turtle but at the same time it has been a reflective process as well as I was challenging my own comfort zone and questioning as to how technology can be integrated with hands-on-contextual experiences of the children. One of the aim of our organisation is to give students ownership of their learning which sometimes come with frustration on their part but making them in charge and giving them freedom often push them through roadblocks and they have an eagerness to continue learning. This experience of working on quilt project with other fellows has given me the push to think through as till now concrete hand-on-contextual experiences of making have been the key of my work with children. But this experience has given me many points to ponder over and enhanced my own learning as an educator. This has been a powerful idea for me to explore further.