learning maths with Built Environment

Study and Exploration of ‘Built space’ for Math Teaching in Schools

Study and Exploration of ‘Built space’ for Math Teaching in Schools

Visual Communication Design Research Seminar Submitted in partial fulfilment for The requirement of the degree, Masters of Design in Visual Communication (2009-2011) at the Industrial Design Centre, Indian Institute of Technology, Bombay

Guide: Prof. A.G Rao

Nimisha Drolia (09625802) Date: 11.04.2011

Approval Sheet: This Visual Communication Design Research Seminar Study and Exploration of ‘Built space’ for Math Teaching in Schools

By Nimisha Drolia, 09625802 is approved in partial fulfilment of the requirements for Master of Design Degree in Visual Communication.

Project Guide: Prof. A G Rao

Date:

Place:

Declaration: I declare that this written submission represents my ideas in my own words and where others ideas and words have been included, I have adequately cited and referenced the original sources.

I also declare that I have adhered to all principles of academic honesty and integrity and have not misrepresented or fabricated or falsified any idea/data/ fact/source in my submission.

I understand that any violation of the above will be cause for disciplinary action by the Institute and can also evoke penal action from the sources which have thus not been properly cited or from whom permission has not been taken when needed.

Signature:

Name: Nimisha Drolia Roll No: 09625802 Date: 4th May, 2011

Acknowledgement:

Thanks to, IDC Prof. A G Rao Poornima Girish Samparc, Bhambarde Friends at IDC Friends everywhere else Family God

Nimisha

Content Synopsis Understanding:

Children Learning

Learning Theories  

Multiple Intelligence - Spatial Intelligence

Study of Mathematics

Learning through Physical Space

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16 17 22 31 36

Reviewing 40

BALA - Building as learning Aid.

NCERT

Maths Content

Samparc School, Bhambarde

Conclusion Bibliography

57 85 86

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Visual Communication, Industrial Design Centre, IIT Bombay.

Synopsis: Education has always been a sector of extensive research and study for people all around. There have been many theories which try to understand the process in which a child learns. In the research which have surfaced in recent years it has shown that environment and personal experience plays a vital role in the learning of any individual.

Thus there have been number of experiments to understand how these parameters affect a child’s growth.

Maths as a subject has always been of great challenge to children and teachers both. Maths is a subject of concepts which has all its application directly and indirectly in day to day life. Its understanding also happens best with real experiences and application instead of abstract concepts and solutions.

Built Environment has good potential to be a part of learning process.

Children spend a major part of their day in Schools. Thus this environment can be a good source of permanent infrastructure which can be explored for teaching. In India, there are numerous government schools, which have very poor infrastructure and teaching facility. Thus we need to come with research which can be helpful in this section.

Thus this research seminar is to Study and Explore the Built Environment which can become Aid to teach Mathematics.

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

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Visual Communication, Industrial Design Centre, IIT Bombay.

Understanding:

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

Children Learning Learning Theories   Multiple Intelligence - Spatial Intelligence

16 17 22

Study of Mathematics

31

Learning through Physical Space

36

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Children Learning: How do children learn? This has been a topic of research for many enthusiasts since many years. For many years people are trying to understand what makes some children fast learners and others slow.

Why are some children good at sports but not music? Why do some children like history more than mathematics? Children learning is a topic of vast references and deep understanding. Lot of researchers have already done a great deal of work and are still doing in this field.

Thus when we look at the education system, we understand that so many of the basic learning methods are not a part of our process. These are natural ways of learning which have not been made a part of current education system. All learning happens in a closed classroom with more of listening and less of doing.

Children learn in number of ways.

Broad divisions of learning on basis of experience with people:

Environment is a great source of learning experience to children. It has great potential to explain and experience many concepts. We are living in this environment and everything that we learn are directly or indirectly related to the real world. Thus to learn it best is to experience it in the real world.

• Self Learning, • Learning with Peers, • Learning with help of Parents/Teachers Directions

Thus there is a need to explore other methods of learning and teaching.

Means which are beyond classrooms and make children experience the lessons, makes the understanding more rewarding.

Learning is a process which is guided by a child’s experience, exposure, surrounding, environment and many other factors. Every child has a unique conditioning and thus has personal understanding of every subject.

Steps in learning: • seeing, listening, exploring, playing • doing, reading, writing, experiencing, observing, • research, reflect, discuss, experiment.

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Visual Communication, Industrial Design Centre, IIT Bombay.

Learning Theories: In psychology and education, learning is commonly defined as a process that brings together cognitive, emotional, and environmental influences and experiences for acquiring, enhancing, or making changes in one’s knowledge, skills, values, and world views (Illeris, 2000; Ormorod, 1995).

There are number of learning theories which have developed from the early 20th century till today. There are three main categories or philosophical frameworks under which learning theories fall:

Behaviorism

Cognitivism

• Behaviorism focuses only on the objectively observable aspects of learning.

• Cognitive theories look beyond behaviour to explain brain-based learning.

• Constructivism views learning as a process in which the learner actively constructs or builds new ideas or concepts.

Constructivism

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

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Behaviorism was the prevalent theory of learning from the 1950’s through the 1970’s. This is a school of psychology that focuses on the observable, measurable aspects of experience and that, education, is stimulus - response based. This attempts to apply the methods of science to the study of human learning. The theory here tells that instruction is very important in the process of learning. In whatever form it comes, it has a role to play in children education.

Individuals like Edward L. Thorndike, B.F Skinner and Robert Gagne have contributed enormously to this perspective. This theory saw a child’s brain as a blank slate where the teacher could write anything.

Robert Gagne emerged in the 1960’s as the spokesperson for Neo - Behaviorism.

Three basic assumptions are held to be true by these researchers.

• First, learning is manifested by a change in behavior.

• Second, the environment shapes behavior.

• Third, the principles of contiguity (how close in time two events must be for a bond to be formed) and reinforcement (any means of increasing the likelihood that an event will be repeated) are central to explaining the learning process.

Points to remember: Though it is not the best explanation and only talks about the scientific part of learning process, it has some application • Instructions play a role in learning. Thus as a classical method of teaching, role of an instructor is primary here.

• This method helps children to acquire some skills which can be useful in further learning. For example when children tables, numbers or are taught to do something in a process that helps in further explorations.

• Gives a scientific explanation to human behaviour.

• Its positive and negative reinforcement techniques can be very effective– such as in treatments for human disorders including autism, anxiety disorders and antisocial behavior.

For behaviorism, learning is the acquisition of new behavior through conditioning.

There are 2 kinds of conditioning: • Classical Conditioning • Operant Conditioning

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Visual Communication, Industrial Design Centre, IIT Bombay.

Cognitive Psychological Perspective majorly promotes active student involvement in the learning process. It stresses cognition and sees learning as occurring within the learner. It focuses on processing rather than behaviour. It places great emphasis on the process dimension of the learning process. The evolution of intelligence involves the continuous organization and reorganization of one’s perception of, and reaction to, the environment.

Two key assumptions underlie this cognitive approach • That the memory system is an active organized processor of information • That prior knowledge plays an important role in learning.

Points to remember: • Previous exposure to a subject gives a context to understanding a subject. Thus explains that if children are familiar of any particular concept with any medium it helps them to learn it better.

• The focus here is process of learning and not the behaviour in which someone learns. Thus the learner is the pivot to the theory and one develops teaching technique around the learner.

• A child learns in stages and thus everything gets organised and reorganised according to one’s perception based on environment.

Thus environment plays an important role to enrich a child’s experience which facilitates his learnings.

Cognitive theories look beyond behavior to explain brain-based learning.

Cognitivists consider how human memory works to promote learning.

For example, the physiological processes of sorting and encoding information and events into short term memory and long term memory are important to educators working under the cognitive theory.

The major difference between gestaltists and behaviorists is the locus of control over the learning activity: the individual learner is more key to gestaltists than the environment that behaviorists emphasize.

Works of Jean Piaget, Jerome Burner and Zoltan Dienes are especially noteworthy in this case. This theory places great emphasis on the process dimension of the learning process.

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

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Constructivism is a theory which is based on cognitive theory.

Constructivism views learning as a process in which the learner actively constructs or builds new ideas or concepts based upon current and past knowledge or experience. In other words, “learning involves constructing one’s own knowledge from one’s own experiences.”

Constructivist learning, therefore, is a very personal endeavor, whereby internalized concepts, rules, and general principles may consequently be applied in a practical real-world context. It states that learning is an active, contextualized process of constructing knowledge rather than acquiring it. Knowledge is constructed based on personal experiences and hypotheses of the environment. Learners continuously test these hypotheses through social negotiation. Each person has a different interpretation and construction of knowledge process. The learner is not a blank slate (tabula rasa) but brings past experiences and cultural factors to a situation.

There are several guiding principles of constructivism: • Learning is a search for meaning. Therefore, learning must start with the issues around which students are actively trying to construct meaning.

• Meaning requires understanding wholes as well as parts. And parts must be understood in the context of wholes. Therefore, the learning process focuses on primary concepts, not isolated facts.

• In order to teach well, we must understand the mental models that students use to perceive the world and the assumptions they make to support those models.

• The purpose of learning is for an individual to construct his or her own meaning, not just memorize the “right” answers and regurgitate someone else’s meaning. Since education is inherently interdisciplinary, the only valuable way to measure learning is to make the assessment part of the learning process, ensuring it provides students with information on the quality of their learning.

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A constructivist learning intervention is thus an intervention where contextualized activities (tasks) are used to provide learners with an opportunity to discover and corroboratively construct meaning as the intervention unfolds. Learners are respected as unique individuals, and instructors act as facilitators rather than as teachers.

Points to remember: Thus taking the learning experience a step further to environment and experience based, this theory again supports in concept of building an overall environment. It talks about relating theories to real life experiences and giving context to everything. Thus this will make learning more meaningful and relative.

• Constructivism talks about individual customising and learning process on basis of every individual students’ exposure and experience.

• It emphasizes on hands on problem solving.

• In this theory learning happens by always making connections of old and new experiences. Every new lesson is taught with base of the what is already learnt.

• This theory is very activity based and encourages children to experience more and experiment. They learn by self exploration and surrounding environment rather than instructions.

Visual Communication, Industrial Design Centre, IIT Bombay.

Some more: There are number of theories which followed the above mentioned theories. Few in agreement and few in disagreement. Some took the theories into more specific area and others studied them more critically to get better understanding. Motivational and Humanist Theory, Design Theories and Models, Identity Theory, Activity Theory and many more have put their points. There are number of categories and classifications done in these to explain how children learn.

The Fleming VAK/VARK model, one of the most common and widely used categorizations of the various types of learning styles,categorized the various types of learning styles as follows: Visual learners, Auditory learners, Reading/Writing-preference learners, Kinesthetic learners (also known as “tactile learners�)

This is another classification which can be used to understand how learning happens outside the conventional teaching in classrooms.

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

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Multiple Intelligences: “An Intelligence is the ability to solve problems, or to create products, that are valued within one or more cultural settings: - Howard Garner (Frames of Mind) Gardner defines intelligence as “the capacity to solve problems or to fashion products that are valued in one or more cultural setting�

(Gardner & Hatch, 1989). When you hear the word intelligence, the concept of IQ testing may immediately come to mind. Intelligence is often defined as our intellectual potential; something we are born with, something that can be measured and a capacity that is difficult to change.

This theory suggests that traditional psychometric views of intelligence are too limited. Gardner proposed that there are seven intelligences.

In order to capture the full range of abilities and talents that people possess, Gardner suggests that people do not have just one intellectual capacity, but have many different intelligences including musical, interpersonal, spatial-visual and linguistic intelligences. Using biological as well as cultural research, he formulated a list of seven intelligences. This new outlook on intelligence differs greatly from the traditional view which usually recognizes only two intelligences, verbal and computational.

Gardner argues that there is both a biological and cultural basis for the multiple intelligences. Neurobiological research indicates that learning is an outcome of the modifications in the synaptic connections between cells. Primary elements of different types of learning are found in particular areas of the brain where corresponding transformations have occurred. In addition to biology, Gardner (1983) argues that culture also plays a large role in the development of the intelligences. All societies value different types of intelligences. The cultural value placed upon the ability to perform certain tasks provides the motivation to become skilled in those areas.

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Visual Communication, Industrial Design Centre, IIT Bombay.

• Logical-Mathematical Intelligence: consists of the ability to detect patterns, reason deductively and think logically. This intelligence is most often associated with scientific and mathematical thinking.

• Linguistic Intelligence: involves having a mastery of language.

This intelligence includes the ability to effectively manipulate language to express oneself rhetorically or poetically. It also allows one to use language as a means to remember information.

• Spatial Intelligence: gives one the ability to manipulate and create mental images in order to solve problems. This intelligence is not limited to visual domains-- Gardner notes that spatial intelligence is also formed in blind children.

• Musical Intelligence: encompasses the capability to recognize and compose musical pitches, tones, and rhythms. (Auditory functions are required for a person to develop this intelligence in relation to pitch and tone, but it is not needed for the knowledge of rhythm.)

• Bodily-Kinesthetic Intelligence: is the ability to use one’s mental abilities to coordinate one’s own bodily movements. This intelligence challenges the popular belief that mental and physical activity are unrelated.

• Interpersonal Intelligence: the ability to understand and discern the feelings and intentions of others.

• Intrapersonal Intelligence: the ability to understand one’s own feelings and motivations. These two intelligences are separate from each other.

Nevertheless, because of their close association in most cultures, they are often linked together.

• Naturalist Intelligence: ability to recognize and categorize plants, animals and other objects in nature • Existential Intelligence: sensitivity and capacity to tackle deep questions about human existence, such as the meaning of life, why do we die, and how did we get here.

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According to Gardner, All human beings possess all nine intelligences in varying amounts.

• Each person has a different intellectual composition.

• We can improve education by addressing the multiple intelligences of our students.

• These intelligences are located in different areas of the brain and can either work independently or together.

• These intelligences may define the human species. These are the seven different ways in which a person sees around and perceives the world. This theory can be seen as a personality assessment of every child and thus try and teach with that quality.

Multiple intelligence explains that beyond traditional teaching methods which place strong emphasis on development and use of verbal and mathematical intelligence, there are different set of intelligences which needs to be addressed. Thus, • Educator should recognize and teach to a broader range of talents and skills. This will give a wider understanding and kids with different kind of intelligences will also get catered to.

• The teaching should be structured in a way that it engages most or all of the intelligences. Thus one subject can be taught in number of mediums and methods, which addresses various kinds of intelligences.

• Understanding of multiple intelligences also explains why children from similar backgrounds, upbringings, culture and exposure still have differences in learning process.

• Every student has all these 9 intelligences. Each has a different level at which these are found. Thus to teach such that each of these are taken care of is important.

• Use of Multiple Intelligence makes the student more active and involved learner.

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Visual Communication, Industrial Design Centre, IIT Bombay.

Spatial-Visual and Bodily-Kinesthetic Intelligence: In a conventional teaching condition these two methods are not explored completely. We need to explore the spatial and kinesthetic intelligence of children. There are many children who are more comfortable with these understanding and there are many subjects which can be taught better with help of these mediums. (http://www.infed.org/thinkers/gardner.htm)

Spatial-Visual Learning: Visual-spatial intelligence involves the potential to recognize and use the patterns of wide space and more confined areas.

The ability to accurately perceive the visual world and to re-create, manipulate and modify aspects of one’s perceptions (even in the absence of the relevant visual stimuli). Visual-spatial intelligence deals with shapes, patterns, designs and the entire spectrum of colour - and with the placement and relationship of objects in space, including distance and direction. It includes our capacity to visualise, dream and imagine.

Spatial thinking is powerful and pervasive, underpinning everyday life, work, and science. It plays a role in activities ranging from understanding metaphors, becoming good at way finding, and interpreting works of art, to engaging in molecular modelling, generating geometry proofs, and interpreting astronomical data.

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

How does Spatial learning help?

• Children who are good spatial learners understand subjects better with help of spatial means. They develop good visualization and imagination. Thus they can perceive and interpret things fast in visual language.

• Many things around us have visual quality and can be best understood in space. Geometry in maths, light and shadow in physics, botany in biology etc have spatial qualities and needs its references.

Thus by helping children develop their spatial intelligence these subjects can be better taught to children.

• By use different methods the teacher is reinforcing the subject to the student. Students can understand the subject completely in all references and context. The same principle in different references gives complete understanding of the subject.

• Spatial/Visual learning also helps in building up knowledge on experience and exposure. Living in the environment children are all the time coming in contact with environment, space and things around.

We live in this space, interact with the space and talk about space all the time. Thus by developing this quality we can help children know correct references of many subjects.

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Bodily - Kinesthetic Learning: Bodily-kinesthetic intelligence entails the potential of using one’s whole body or parts of the body to solve problems. It is the ability to use mental abilities to coordinate bodily movements. Howard Gardner sees mental and physical activity as related. (http://www.infed.org/thinkers/gardner.htm) Kinesthetic learning is a learning style in which learning takes place by the student actually carrying out a physical activity, rather than listening to a lecture or merely watching a demonstration. It is also referred to as tactile learning. People with a kinesthetic learning style are also commonly known as do-ers.

Tactile and kinesthetic learners absorb information best by doing, experiencing, touching, moving or being active in some way. Learn by becoming physically involved and actually doing something with what’s being learned.

“Hands-on” activity is needed to get the learning to stick! “Being on the move” helps their memory to work. In the classroom, they benefit from physical activity. Using large diagrams, floor/wall puzzles and large maps on the wall or floor are helpful. Charades, acting, interviewing, pantomiming, skits, and role-playing enhance learning.

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How does Kinesthetic learning help?

• It is a saying that we learn best when we do things. Thus use of Kinesthetic learning is a method to reinforce subjects in more active and participating method. By doing a subject physically rather than seeing or listening is always more rewarding.

• Learning requires a physical response, physical interaction, all of which is natural to children. Thus Kinesthetic learning gives them chance to learn by direct interaction and experience.

• Students need to be provided with structure that allow them to actively process information. Kinesthetic teaching makes subject matter accessible by concretizing the abstract.

• Kinesthetic has element of Time, Space and Shape. Thus subjects which have concepts of Time like estimation, light and shadow etc, Space like mass, volume and Shape like 3-dimentional geometry etc can be best understood by kinesthetic learning.

Visual Communication, Industrial Design Centre, IIT Bombay.

Logical-mathematical learning

Interpersonal intelligence

Logical-mathematical intelligence consists of the capacity to analyse problems logically, carry out mathematical operations, and investigate issues scientifically. In Howard Gardner’s words, it entails the ability to detect patterns, reason deductively and think logically. This intelligence is most often associated with scientific and mathematical thinking.

Interpersonal intelligence is concerned with the capacity to understand the intentions, motivations and desires of other people. It allows people to work effectively with others.

Educators, salespeople, religious and political leaders and counsellors all need a well-developed interpersonal intelligence. (http://www.infed.org/thinkers/gardner.htm)

(http://www.infed.org/thinkers/gardner.htm)

Though people have the physical ability to exist individually and alone, we are also social animals who thrive and grow when involved with others. This ability to interact with others, understand them, and interpret their behavior known as interpersonal intelligence. According to Gardner, interpersonal intelligence is seen in how we “notice distinction among others; in particular, contrasts in their moods, temperaments motivations and intentions.”

The most popularly understood cognitive faculty is that of logical-mathematical intelligence. This intelligence is our ability to mentally process logical problems and equations, the type most often found on multiple choice standardized tests. Logical-mathematical intelligence often does not require verbal articulation, for we can churn a complex problem in our head, only to articulate it out loud once the problem has been solved. Additionally, individuals who have high logical-mathematical abilities are able to process logical questions at an unusually rapid rate.

Before the advent of MI theory, logical-mathematical intelligence was considered the archetypal intelligence, the “raw intellect” on which Western culture has placed a high premium. Though MI theory agrees that logicalmathematical intelligence is indeed a key section of the intellect, it is by no means the only section that must be both explored and cultivated.

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

Why do certain people make excellent politicians, leaders, or clergy? A well-developed interpersonal intelligence plays a substantial role in such success. From a psychological and neurological point of view, the connection between interpersonal intelligence and the brain has been explored for generations; damage the frontal lobe, as was once done in the case of lobotomy patients, and you damage that person’s personality and her ability to interact well with others. Interpersonal intelligence allows us to affect others by understanding others; without it, we lose the ability to exist socially.

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In case of Spatial Learning, built spaces can be used for initial steps of • Familiarisation of concepts, by visuals that can be painted or highlighted in the built space.

• Reinforcement of knowledge, by giving aid which can be used by children and teachers to understand lessons.

• Visualization of subjects, mainly by activity in the class or the playground, which will let children experience the concept.

In case of Kinesthetic Learning, built spaces can be used for • Activity based learning, these can be part of classroom program where the teacher guides and children learn lessons doing the problems practically.

• Interacting with space and experiencing, This can be existing in the environment around which children experience and see the concepts in real situations.

• Hands on experiments of concepts, There are number of concepts which can be learnt with hands on experience.

• Concept of Play and Learn, Many theories can also be explained with method of play and learn. There are number of games which have learning and many are designed with theme of some learning.

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Visual Communication, Industrial Design Centre, IIT Bombay.

Few more Categorization of Learning Research indicates that, human beings tend to represent knowledge in basically three ways (Bruner, 1964; Piaget, 1952)

• Concrete: The key component of the concrete level of representation is the performance of an action on an object or objects. At the concrete level, students use hands-on manipulative to physically arrive at the solution.

• Pictorial: At the pictorial level of representation the child no longer needs to physically manipulate objects to solve problems such as the one given above. In some elementary textbooks, several pictures are already drawn and the student is asked to choose the picture that matches a given problem situation. Often in a disposable textbook, part of a picture is drawn and the student is instructed to complete the drawing and use it to solve the given problem. The student may also solve the problem mentally by visualizing three sets of five and counting.

• Symbolic: At the symbolic level of representation the student has acquired enough experience with the given problem type to represent it symbolically without using concrete operations or pictures or images. ..

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Three Types of Learning Experiences The type of learning experience or activity that you choose to use will depend on the objective of the activity or experience.

Developmental: Developmental activities are used to develop new ideas and introduce new concepts. Developmental activities should be conceptually based; the students should be able to acquire meaning for the concepts and operations that they are learning. These activities should be designed and presented in a way that will make the math understandable to the students.

Reinforcement: These activities are used to reinforce previously introduced skills and concepts. One may use developmental activities or modified developmental activities as reinforcement activities. One may also find or develop new activities to use as reinforcement activities.

Drill and practice: Drill and practice activities are used to practice or memorize previously learned skills, procedures, and concepts. These activities should not be boring. Pick activities that will engage the students. One can use contests, challenges, games, problems where the solution provides the answer to a riddle, and many other fun activities.

To classify an activity, consider main objective or intent.

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Visual Communication, Industrial Design Centre, IIT Bombay.

Mathematics Mathematics is a branch of study that deals with logic, decision-making, deductions, assumptions, precision, clarity of thought and the ability to solve problems in a calculative manner by following a series of steps.

This is an important subject as it is not only useful to learn other subjects but has application in everyday business of every individual.

Every branch of science, be it physics, chemistry, biology, astronomy, engineering, aeronautics, transportation and communication, etc., requires the person studying it to acquire the in-depth knowledge of mathematics so that he or she can understand the theories of the subject. Most scientific theories are based on mathematical concepts.

Apart from being an important part of every branch of science, it is also very important in everyday work of our life. Everytime we go out to buy something, or have to keep an account of our salary, maths becomes helpful. In todays world of fast technology, even if someone is doing a non scientific job, he needs basics to deal with the pace and keep up with the flow of the world.

The mathematics standards are built around five content Standards-numbers and Operations algebra, Geometry, Measurement, Data analysis Probability

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Learning Mathematics: Maths has always been a subject of mystery and fun for kids.

Some are scared of its complications and others love the relations.

But whatever be the case Mathematics is a very important subject which has its application in day to day life of every person. It is a subject of concepts unlike literature or history. Apart from visualization and memory, there is this element of relation and connection involved. 2+2 = 4 is a just a concept which a kid understands well only when he physically adds 2 things to 2 things and get 4 things in total. Also Maths is a subject which keeps building on its basic foundation. Thus if the foundations are not clear in the initial years the later study becomes very disconnected.

Children must create or recreate mathematical relationships in their own minds. To process information, one must interpret it in relation to what is already known or believed. Children should be active in the learning experience in order to fully internalize the experience.

Various mathematics educators have compared two types of learning: • Procedural learning and • Conceptual learning (Hiebert & Lindquist, 1990; Skemp, 1971).

• Procedural learning involves learning processes or algorithms by rote.

• Conceptual learning involves understanding the concepts and meanings underlying the operations as opposed to merely applying rules. The main tenet of conceptually based mathematics is that when students understand the concepts and reasoning underlying a process, they are more likely to be able to correctly apply that process. They are also more likely to be able to apply that process in learning related new skills and procedures.

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Visual Communication, Industrial Design Centre, IIT Bombay.

The four thinking math standards are Problem solving, Communication, Reasoning, and Connections.

Problem solving is key in being able to do all other aspects of mathematics. Through problem solving, children learn that there are many different ways to solve a problem and that more than one answer is possible. It involves the ability to explore, think through an issue, and reason logically to solve routine as well as non routine problems.

In addition to helping with mathematical thinking, this activity builds language and social skills such as working together.

Communication means talking with children and listening to them. It means finding ways to express ideas with words, diagrams, pictures, and symbols. When children talk, either with you or with their friends, it helps them think about what they are doing and makes their own thoughts clearer. As a bonus, talking with children improves their vocabulary and helps develop literacy and early reading skills as well.

Reasoning is used to think through a question and come up with a useful answer. It is a major part of problem solving.

Connections: Mathematics is not isolated skills and procedures.

Mathematics is everywhere and most of what we see is a combination of different concepts. A lot of mathematics relates to other subjects like science, art, and music. Most importantly, math relates to things we do in the real world every day. Connections make mathematics easier for children to understand because they allow children to apply common rules to many different things. What parents can do:

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

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Learning through the Physical Space: Children perceive their world through multiple senses, especially the tactile and visual senses. A three-dimensional space can offer a unique setting for a child to learn because it can introduce a multiple sensory experience to accompany the text book or blackboard.

Spatial Dimensions, texture, shapes, angles, movement and spatial attributes like inside-outside, symmetry, up-down, can be used to communicate some basic concepts of language, science, mathematics and the environment.

A child has experience of different places, people and time. He is all the time participating in the environment around. He is a part of the living world and is experiencing the space and place. He sees, listens and does things. And this is how he learns, with the exposure to the place he is in. Every child has his knowledge base based on the his experiences and exposure to the environment around. Thus a child who has travelled a lot has experienced more and thus many times has potential to learn faster than someone who is stuck to one single place.

Thus environment plays a very important role in learning process of children. When we say environment, there is Natural environment and there is Man made environment. Both have a role to play in learning.

A child spends his time at, Home School Playground Other Public Spaces.

These spaces are Man made spaces and can be well explored to make it a part of learning experience for children. With direct interaction and experience children can learn with help of these built spaces.

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Visual Communication, Industrial Design Centre, IIT Bombay.

In case of Spatial Learning, built spaces can be used for initial steps of • Familiarisation of concepts, by visuals that can be painted or highlighted in the built space.

• Reinforcement of knowledge, by giving aid which can be used by children and teachers to understand lessons.

• Visualization of subjects, mainly by activity in the class or the playground, which will let children experience the concept.

In case of Kinesthetic Learning, built spaces can be used for • Activity based learning, • Interacting with space and experiencing, • Hands on experiments of concepts • Concept of Play and Learn

Built Environment of School: School is a place where a child spends quantity and quality time. He becomes socially and physically aware of his environment. Schools are the foundation to a child’s growth.

Thus the environment of the school is also very important.

Children are constantly interacting with the physical environment of their schools during structured or unstructured time, consciously or unconsciously. Thus we need to pay attention to physical environment for learning.

• Buildings are the most expensive physical assets of a school. Maximum educational value should be derived from them. Creative and practical solutions can be used to maximize this educational value while repairing or upgrading existing schools or making new buildings.

• Playgrounds in school are another very important asset.

Semi open and open spaces in schools can be very well used to incorporate learning for children during their playtime. This space is of great value and will indirectly work to building up the knowledge of the child.

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

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Kind of Spaces In Schools: Classrooms, Laboratories, Activity Space, Semi Open and Spaces like Courtyards, Terraces, Playground, Hostels, Common Area, Mess, Circulation space like corridors, staircase.

Thus Schools can be explored keeping these spaces in mind. Kind of activity that children do in these spaces can be categorised. According to this categorization what kind of learning, spatial or kinesthetic needs to be done can be figured out. On that basis one can try incorporating design ideas in these spaces.

Classrooms (Spatial Learning): Classrooms is place where teachers and students directly interact. There are formal lessons and is a more serious environment. There is good scope of Spatial Learning in this space. Information on the walls, grills, windows, floor, ceiling, doors etc all will help in visualizing concepts. They can be used by teachers to reinforce the lessons they are taking. Furniture in classrooms can be adopted to give visual information.

Laboratory (Spatial Learning): Laboratories are places of experiments. Here also using the building elements to make visuals which can be used to familiarise, and memorize one can bring Spatial learning.

Activity Space (Spatial and Kinesthetic Learning): In many schools there are separate Activity Spaces. These can have both Spatial Learning and Kinesthetic Learning.

Provision for different games can be done in these spaces, which have kinesthetic learning. Also Visual and Spatial learning can be a part of activity Semi Open Spaces (Spatial and Kinesthetic Learning): Courtyards, Terraces, Backyards etc are semi open spaces in schools where Spatial and Kinesthetic learning can take place. Again with help of games, hands on activity and visuals in the space around one can make children learn outside the class.

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Visual Communication, Industrial Design Centre, IIT Bombay.

Playground (Kinesthetic Learning): Playground is an inseparable part of the School campus. Children love to spend time there. During break and also as a part of curriculum at times. This is an open space where a lot of activity takes place. Children play, interact, spend time together in the most informal way. This place is best suited to explore the Kinesthetic way of learning. One can teach many things to children with games and physical activity.

Hostels (Spatial Learning): In many residential schools, hostels can also be explored for some spatial learning. There are furniture, and other building elements like wall, floor, ceiling doors, windows etc which can be used. Also because when children spend time in hostels either they are in relaxed state or busy in their homework. Thus things which can facilitate in the work, and also be of some interest can be done.

Common Area (Kinesthetic and Spatial Learning): With hostels, there are common areas for entertainment, sports and other activity also present in residential schools.

These can have games, activity based learning and visuals included. Some can also have music, dance and other forms of intelligences included in their space.

Mess (Spatial Learning): In many residential and non residential schools there is place to eat. The mess has quality for many spatial activities. These activities can be designed to make children directly experience lessons. Spatial Learning has good scope in this space.

Circulation Spaces (Spatial and Kinesthetic Learning): Circulation space like the staircase, corridors etc are places of constant movement. This can be well used to explore Kinesthetic and Spatial learning possibility. These are used for constant movement thus learning lessons based on movement, time can be explored.

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Schools in India:

Mathematics Content:

Schools In India, are not very fine example of great infrastructure and facilities. Infact government schools in India don’t have even basic required facilities. Thus to be able to get hands on activity material and make separate maths lab for children or install different playing material in the ground is very difficult. Thus in this case, building itself acts as an important infrastructure which can become teaching material. Like said, why make its best to teach a language in its home land, why not teach maths by making school a mathematics land.

The National Council of Teachers of Mathematics (NCTM), the world’s largest organization devoted to improving mathematics education, is developing a set of mathematics concepts, or standards, that are important for teaching and learning mathematics. There are two categories of standards: thinking math standards and content math standards. The thinking standards focus on the nature of mathematical reasoning, while the content standards are specific math topics.

Also when real and obvious examples of principles of science and maths can be seen around in built space, it is important to point it out which will give real sense of things.

When children can experience concepts and phenomenon in real time and space it should be a part of their learning.

Being in Indian scenario, where the population and poverty makes it difficult to invest a lot of money in these kind of venture, it is important to come up with simple and economically cheap solutions so that it can be easily incorporated in schools and can be useful to many schools.

There have not been many examples of such exploration in India. BaLa is one such organization which has worked in this area. Also to study the concept of Maths Lab which is becoming fast popular, where maths is learnt by activity needs to be studied.

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Visual Communication, Industrial Design Centre, IIT Bombay.

The content math standards are Estimation, Number sense, Geometry and Spatial sense, Measurement, Statistics and Probability, Fractions and Decimals, Patterns and Relationships

In The same field of work, there is a good example from an organization called BaLa which has done lot of exploration in the similar subject. It was interesting to go through their work and review it for further explorations.

Going through the NCERT books was also a very important and useful exploration. These books are very well designed and have lot of scope for actual application and implementation.

In the following pages, using the maths content categorization the subjects have been reviewed on basis of these two existing work and what further possibility can be explored. Thus each category of the content with help of NCERT and BaLa has been looked upon with what has been done, and what are other possibility.

Pictures used to explain the work are from the NCERT books and BaLA work. The proposals from BaLA are reviewed and either used directly at appropriate place or modified and further explored as to the context.

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Reviewing the Content with help of Existing Explorations NCERT: The National Council of Educational Research and Training (NCERT) is an apex resource organization set up by the Government of India, with headquarters at New Delhi, to assist and advise the Central and State Governments on academic matters related to school education. It was Established in the year of 1961.

The NCERT syllabus is well designed by experts in the field of education.

The books have good references to help children relate the theories and concepts. The books for younger children are more colourful with more visuals, as that gives them more clues.

One very important feature of NCERT course is the addition of activity in the books. These activities are based on real life experiences and can be used by teachers to give correct context. There are number of activities and experiments which children can do during and after class.

In maths books, there are number of activities which can in real space become a part of the school. Thus while teaching teachers can use them as direct references and after class children can again use them.

These activities are very well designed and can be used as reference to design spaces.

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Visual Communication, Industrial Design Centre, IIT Bombay.

BALA, Building as Learning Aid BaLA is an inspirational effort by Kabir Vajpayi and his wife Preeti, both architects leading an outfit in Delhi called Vanyas.

BaLA is based on improving the physical parameters of the learning environment. It relies on open spaces, greenery, surfaces on which to doodle and scribble,contraptions to learn from.

BaLA is about developing the school’s entire physical environment as a learning aid – the inside, the outside, the semi-open spaces –every where. It is based on ‘how children learn’.

BaLA is about being child-friendly, fun-oriented, using local resources and so on. Within this frame the administrator gets the freedom to innovate with management and monitoring systems. The engineer gets the freedom to improvise and adapt or develop a new idea. An artisan who is making a BaLA element with a new refined process of construction is given credit for that improvement. A teacher has the space to come up with a new use for a learning aid.

BaLA aims to use the built elements like the floor, walls, pillars, staircases, windows, doors, ceilings, fans, trees, flowers, or even rainwater falling on the building as learning resource.

There are number of examples of work done by BaLA. These can be adapted and interpreted according to the new built space. These first needs to be understood and categorised.

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Patterns and Relationships: Patterns are things that repeat; relationships are things that are connected by some kind of reason. They are important because they help children understand the underlying structure of things; they help children feel confident and capable of knowing what will come next, even when we they can’t see it yet. Patterns and relationships are found in music, art, and clothing, as well as in other aspects of math such as counting and geometry. Understanding patterns and relationships means understanding rhythm and repetition as well as ordering from shortest to longest, smallest to largest, sorting, and categorizing.

NCERT • Dot board on the wall.

• Recognizing the patterns in the environments around.

• Square box to make pattern.

• Space on the board to let children write and learn number pattern.

• Pattern on the floor.

• Making rangoli, or other things on the floor to learn pattern as an activity.

Exploring the Built Space.

• Patterns can be made for visual learning. They can be painted on the wall, floor, with help of pavement in the circulation space, in window grills. Pattern can be with numbers, art or natural things around.

• Pattern can also be meant to bring kinesthetic learning, where base can be made and then children in games or class activity make patterns and learn relationships.

• Pattern finding can become some activity which can be a part of building. Also one can show naturally occurring pattern in nature and space to explain.

• Relationship can also be easily found in environment and be created in space around. These can be enhanced in design which can be seen and experienced by children.

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Visual Communication, Industrial Design Centre, IIT Bombay.

BaLA

Things done: a. Pattern painted on Wall: Patterns painted on wall gives visual exposure to children. It makes children get familiar to pattern and relationship which can become base to later learning.

Because it is static, after a while children may not notice. The learning is dependent on the child to interpret and he might not do so.

1.

Pictorial 2.

Reinforcement

b. Board provided to draw: It is activity based learning and thus will help children learn better. Again if not used during class and not guided with teacher might not get noticed and used by children.

1.

Concrete 2.

Developmental

What else can be done: • Should be changed regularly. • Teachers should make it a part of lessons and use it as references during class. • Not be used as decoration only but also have some interpretation,and puzzle so that children get curious and learn better.

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Number sense and Numeration: Number sense is much more than merely counting, it involves the ability to think and work with numbers easily and to understand their uses and relationships. Number sense is about understanding the different uses for numbers (describe quantities and relationships, informational tools). Number sense is the ability to count accurately and competently, to be able to continue counting—or count on—from a specific number as well as to count backwards, to see relationships between numbers, and to be able to take a specific number apart and put it back together again.

It is about counting, adding, and subtracting. Counting and becoming familiar with numbers will help your children understand all other aspects of math.

NCERT Making square box on the wall Number line in the outdoor space.

The kerbs and planters can have number line, number pattern and numeration.

Exploring the Built Space?

• Number line can be incorporated in the walkways in the ground.

• Staircases, corridors can be used for numbering of things.

• Blank walls, floor can be used to paint with numbers and pictures.

• Abacus can be used on some window grill, or made a part of some wall thus giving kids learn counting.

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Visual Communication, Industrial Design Centre, IIT Bombay.

BaLA

Things done: a. Numbers in the staircase: By regular use it helps children to memorize and get familiarised to numbers and counting.

This is first level of learning where one gets initial experience of numeration.

1.

2.

Pictorial Reinforcement

b. Numbers are painted on wall, with corresponding pictures. This will help them to get pictorial reference of concepts.

This can be used by teacher during class for activity and reference learning.

1.

2.

Pictorial Reinforcement

What else can be done: • Its only visual right now. Thus children can tend to ignore. Needs to be incorporated in class curriculum. • If some activity is involved, where children need to draw, or play with help of board, or games on steps they will learn in a more concrete learning.

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Geometry and Spatial sense: Geometry is the area of mathematics that involves shape, size, space, position, direction, and movement, and describes and classifies the physical world in which we live. Young children can learn about angles, shapes, and solids by looking at the physical world. Spatial sense gives children an awareness of themselves in relation to the people and objects around them.

NCERT Magnet board where kids can use different shapes to draw compositions.

Black or green board which can be used again by children to draw.

Shapes in the windows and doors.

Shapes in the grills of the windows.

Shapes in 3 dimensional form by use of space.

Exploring the Built Space.

• Shapes can be shown with highlights, like the window corners can be painted to show square, rectangle. Grills can have shapes. Circles in the planters, triangles in the trusses can be highlighted.

• Space can be perceived with built environment. Teachers can use some space to explain volume, mass, gravity etc.

• Direction can be shown with real directions in the building which can be part of everyday usage. With help of milestones in corridors and playground one can talk about left, right etc.

• Angles in geometry can come in the swing of the doors, windows. Also in the truss of the roof.

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Visual Communication, Industrial Design Centre, IIT Bombay.

BaLA Things done: 1. Angles are painted around the swing of the door on the floor. The door when opens, makes angles, which can be painted on the floor. Thus this gives direct reference to children of angles.

2. Angles which are seen in the staircase, are highlighted.

3. Planter, roof truss etc have angles, which are highlighted to show different angles.

1.

2.

Pictorial Reinforcement

What else can be done: • These are mainly methods for references, and don’t have self experiencing and inferring quality. Thus teachers can use as references during class.

•

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Measurement: Measurement is finding the length, height, and weight of an object using units like inches, feet, and pounds.

Time is measured using hours, seconds, and minutes.

Measurement is an important way for young children to look for relationships in the real world. By practicing measurement children will learn how big or little things are and how to figure that out.

NCERT Scale in the class furniture like benches, tables, chairs.

Scale around the blackboard.

Scale with help of things on floor.

Measuring of different things with help of scaled furniture Mile stone on ground telling about different ways of route.

Exploration of Built Space: • Measurement from small unit like millimetre to large units like kilometre can be experienced in real space very easily.

The furniture of the class can be marked with measurement in different units. The corridors can be marked with scale and finally the milestones in the playground can tell how much have you walked.

• Weight of different things can also be written and also be a part of some activity during and after class. The real life using stuff like buckets, or bottles etc can tell their capacity.

• Height chart can be painted on the wall which can be used by children to measure their own height and then compare compile when in group.

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Visual Communication, Industrial Design Centre, IIT Bombay.

BaLA Things done:

What else can be done:

1. Scale with different units is painted on the benches.

Children can check for themselves the measurement, which is not understood because it is experiences in real.

• There is need to establish relationship that exist in measurement units. • When children can see direct application, by means of activity or some work of these measuring units, they can understand it completely.

2. Height chart is painted vertically on the wall.

Children can use the chart to measure their heights and understand relationship in units.

Other exercises of estimation, statistics can also be done with help of these.

3. Weight of the benches etc are written.

By real experience children get real references of units and measurements. Once a reference is set, they can always base their further understanding on that.

4. Milestones are used to tell the distances from one place to other.

In classroom, one can never understand the real scale of measurement. Blackboards can be used to explain till meter, but units beyond that like kilometre can be shown with help of space.

1.

2.

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Concrete Developmental

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Fractions: Fractions represent parts of a whole. A very young child will see something cut into three pieces and will believe that there is more after cutting it than before it was cut.

Exploration of Built Spaces: • With help of visuals on the window panes and grills where real fraction takes place, one can explain the concept.

• Also paint on the wall, or the floor.

• There are number of games on fraction, which can a part of built space and give children experience by games.

• Some activity based on fraction for which some infrastructure is needed can be a part of the built environment.

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NCERT Fraction in the grill of the window.

On the floor with help of tiles.

On furniture with paint Some activity of fraction with some games and movable mechanism.

Visual Communication, Industrial Design Centre, IIT Bombay.

BaLA What else can be done: Things done: 1. Window grills are used to show the concept of fraction.

Gives an idea, that when whole is divided it makes divisions which is used as fractions.

1.

2.

Pictorial Reinforcement

• There should be some concrete learning.

Fraction is a concept of whole and part, which needs to understood. First this concept that fraction is a part of whole, needs to be understood. • More games can help to understand the concept. Also further working with fractions can be taught with these games.

2. Dots made on the blackboard which can be used to play games based on fractions.

Children learn the rules by playing these games.

1.

2.

Concrete Developmental

3. Directly painted on the wall.

Can be used to familiarise the symbolism of the concept. Thus when children move to symbols in fraction from pictorial representation they are already aware of it.

1.

2.

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Pictorial Reinforcement

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Estimation: To estimate is to make an educated guess as to the amount or size of something. To estimate accurately, numbers and size have to have meaning. When children use estimation, they learn to make appropriate predictions, to obtain reasonable results, and they learn math vocabulary such as “about,” “more than,” and “less than.”

NCERT: Different options given to estimate.

Estimation of distances with help of mile stones.

Infrastructure for games that can help in estimation.

Estimation of weight and volume with help of everyday tasks.

Exploration of Built Space: Can be part of many things, like distance, quantity etc. Thus things done for measurement and Geometry Spatial sense can be used for this as well.

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Visual Communication, Industrial Design Centre, IIT Bombay.

Statistics and Probability: Using graphs and charts, people organize and interpret information and see relationships. Graphing is another way to show and see information mathematically. Charts, including calendars, can be used to organize everyone’s weekly activities. Even older children in elementary school may find it hard to keep track of calendars, but, when adults use them with children, calendars can be helpful tools to learning and understanding how we organize information.

NCERT: Some Charts on the walls, which can be updated and modified.

Activity statistics with help of material.

Black board and green boards in the semi open spaces, where different records can be taken and marked. This becomes part of daily activity and children learn theory of statistics and probability.

Exploration Built Space: • Calender or timetable, and charts etc can be painted on wall. There can a blackboard in the back of the class, which can be used by children with their teacher to regularly update their calender and charts etc.

• It can also be an activity with facility like green board, blackboard etc in the back of the class, or the corridor walls, where children can regularly participate and do activities based on probability and Statistics.

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Maths Lab:

Geoboard:

A mathematics laboratory is a place where we find a collection of games, puzzles, teaching aids and other materials for carrying out activities. These are meant to be used both by the students by their own and together with their teacher to explore the world of mathematics, to discover, to learn and to develop an interest in mathematics. Although mathematics is not an experimental science in the way in which physics, chemistry and biology are, a mathematics laboratory can contribute greatly to the learning of mathematical concepts and skills.

A geoboard is a mathematical manipulative often used to explore basic concepts in plane geometry such as perimeter, area or the characteristics of triangles and other polygons. Consisting of a physical board with a certain number of nails half driven in, in a symmetrical square five-by-five array, students are encouraged to place rubber bands around the pegs to model various geometric concepts or to solve other mathematical puzzles. Twodimensional representations of the geoboard may be applied to ordinary paper using rubber stamps or special “geoboard paper” with diagrams of geoboards may be used to help capture a student’s explanations of the concept they have discovered or illustrated on the geoboard.

Geoboards were invented and popularized by Egyptian mathematician Caleb Gattegno in the 1950s

Some ways in which a mathematics laboratory could contribute to learning mathematics: •A mathematics laboratory provides an opportunity for children to discover through doing. In many of the activities, students learn to deal with problems while doing concrete activity, which lays down a base for more abstract thinking.

•It gives more scope for individual participation. It encourages children to become autonomous learners and allows a student to learn at his or her own space.

•It widens the experiential base, and prepares the ground for later learning of new areas in mathematics and of making appropriate connections.

•In various puzzles and games, children learn the use of rules and constraints and have an opportunity to change these rules and constraints. In this process they become aware of the role that rules and constraints play in mathematical problems.

•Because of the larger time available individually to the student and opportunity to repeat an activity several times, students can revise, re see and rethink the problem and solution. This helps to develop metacognitive abilities.

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Visual Communication, Industrial Design Centre, IIT Bombay.

Inferences: Thus Bala has many different proposals which are easily adaptable and applicable. Some of these are at basic level of pictorial learning. They are more visual and few are activity based. These proposals can be easily adapted to other learning environment. These can be categorised into sections and seen as to how it can be further modified to look at it at a further learning level.

There are many good, and experimented models that the organization has incorporated. These can be read, understood and applied as to requirement and context of the environment.

Thus looking at the NCERT text book and their syllabus, lot of references and inspirations can be taken to explore mathematics in the built space. There are number of examples of activity and drawings which of provided in the school can be helpful. Children apart from reading in the book, if can do these activities will learn them better.

Also teachers can then easily relate the environment and give real references to children during class. She can make them do activities, and show pictorial references which becomes reinforcement to the lessons in the book.

Schools which don’t have NCERT syllabus can also take advantage from these books. These can be a part of their curriculum and be helpful in learning.

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Visual Communication, Industrial Design Centre, IIT Bombay.

Samparc School, Bhambarde

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Samparc, Bhambarde

General Information:

Way back in 1998, SAMPARC started a School & Hostel in the remote terrain village of Bhambarde in Pune District, at a distance of 70 km from Pune & 130 km from Mumbai. The sole purpose behind this difficult task was to educate the poor tribal children in Mulshi Taluka up to Std. X

Locations: Located in Bhambarde in Pune District, at a distance of 70 km from Pune and 130 km from Mumbai. The terrain is hilly with mountains on around the school. It gives a perfect setup for focussed learning for children. Staying in contact with nature here, children stay in a healthy and quite environment here.

In this hilly terrain where the importance of education plays least significance, SAMPARC is proud that children regularly attend classes since the last 11 years. Today 246 students study in the school.

Counselling and creating awareness among the villagers, enabled us to win their confidence and trust. This school is first of its kind in Mulshi Taluka. The school provides a good platform for various extracurricular and cultural activities.

How to reach: One can reach Samparc either from Mumbai or from Pune. There is a direct bus which brings you from pune. From Mumbai, one needs to first come to Lonavala and then take a state bus which will bring you to Bhambarde.

Climate: Temperature - Summer (39 to 22) degree and Winter (34 to 12) degree.

Rainfall - Maximum rainfall here is upto 910 mm/month.

Students: Students from 7th to 10th get educated here.

There are 246 students. The Schools is completely residential, with teachers, students and other staff living together.

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Visual Communication, Industrial Design Centre, IIT Bombay.

1st Trip to Samparc Bhambarde: Purpose of Visit: In January we made our first trip to the School, to get a basic idea of the setting and the infrastructure. We spent one day there and looked around, met children and understood the environment of the place.

In the first trip, I mainly looked around the school and tried to understand the infrastructure of the School. I went through the place and took pictures of the classrooms, circulation spaces, outdoor spaces and understood the kind of space that exist.

My first reflection: The setting of the school is naturally very beautiful, with mountains all around and everything green to see. Against the expectations, the school was well maintained and had good basic infrastructure. Being so remote from urban facility they had basic requirements in place. It was better than most of municipality schools and had a healthy environment for children to study and live.

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Visual Communication, Industrial Design Centre, IIT Bombay.

Built Environment of Samparc, Bhambarde.

Plan of the School

Hostel and Mess

Classroom

Faculty Residence

Faculty Residence

Playground

• The school is in the valley with mountains all around.

It has 5 acre of agricultural land around, and 2.5 acre of playground.

• There are 4 classroom in two buildings. The buildings are all ground floor with sloping roofs.

• A New building for agriculture teaching has come up in the campus.

• There are labs for science, maths and computer learning.

• Classrooms have metal furniture and adequate light and ventilation. The labs have all basic facilities.

• There are three buildings for the faculty.

• The hostel for the student is situated behind the school playground. It has 3 wings. One for boys accommodation, one for girls accommodation and one for mess. A huge courtyard is there in the centre.

• The furniture in the rooms of the hostel are metal bunker beds. Storage is created in the walls.

• Mess has metal furniture. The space is spacious and well lighted.

• There are corridors all around the buildings.

• Many brick platers and landscaping can be seen in the ground.

• A good portion of the land is used for agriculture.

• There is solar heater installed in the backyard of the kitchen.

Classroom

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1

4

7

2

5

8

3

6

9

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Visual Communication, Industrial Design Centre, IIT Bombay.

First set of Proposals:

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• The window edges can be highlighted to show the shape of the window.

• Planters have a semicircle. The edge can be painted to show the shape. Metal rim can be fixed to show the shape.

• The window panes can be painted in different colours, to show fractions, shape. The grill can be painted to show fraction.

• Furniture in the hostel and class can be painted to show measurement and conversion.

• The truss of the roof, can have the measurement on the runners.

The angles can be highlighted to show the angle.

• The storage space in the rooms, can show vertical measurement and the volume can be explored.

• The corridors can have bigger measurement painted along the skirting. The half wall can be painted with many more things.

• The playground can be given some infrastructure so that number of games can be played.

• The wall is blank and long. Can be explored to make more useful.

• The kerb along the planters can be used to explain numbering and pattern.

• The stage can be used to explain the concept of perimeter, numbering system. Measurement.

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2nd Trip to Samparc Bhambarde: In the last week of February, I made a second trip to the school. While others were there to conduct a workshop on Learning Maths with Bamboo, I went to explore the place more and understand its scope to my project. The workshop was for 4 days, and I stayed for two and a half days.

One can best understand the project, its scope and limitations by direct interaction and seeing the place. When I stayed there I tried to understand the school, place, children, teachers which became very important for me to completely understand the school and its environment.

The workshop was being done for students of class seventh, and thus I also focused my work to class seventh. Thus I tried to understand the syllabus and the age group of these people.

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Visual Communication, Industrial Design Centre, IIT Bombay.

Observations: Time

Space

People

Activity

Remarks

Early Morning

Hostel, Courtyard, Mess

Self, Peer, Warden, Cook, Gardner, Helper.

Physical Activity, Study, Morning activities

Self learning and exploration, Learning with peers.

Day in School

Classrooms, Labs, Circulation Space, Mess.

Teachers, Peer, Self.

Study.

Learning with teachers thus as reinforcement to what they teach, Self learning. Learning with peer as a group activity, discussion.

Evening

Playground

Self, Peer

Play (physical games)

Learn while playing, which are mostly physical games. Thus infrastructure to help children in their games with friends.

Night

Hostel, Mess

Self, Peer, Warden, Cook, Helper.

Play, Self study, Rest

Self Learning and Exploration. Learning with peer.

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Focus for the Proposals: Based on the Day of these Students Life I observed how they interact with their space. And keeping this Space as the focus came up with few design proposals to incorporate maths learning.

With these proposals I will talk about how they will help children in learning.

What subjects can be covered with each proposal will be talked about.

Hostel and Mess

Spaces recognized:

Classroom

Faculty Residence

Faculty Residence

• Interacting with students of 7th standard, and seeing availability of the classroom in the new building I decided to use that for exploring the classroom design activities.

• Second place was the playground which made me explore more and more games which can be used to learn maths.

Playground

• A good amount of their early morning is spent in the courtyard and corridors around the hostel. Thus things which can help them with self learning can be proposed here.

• Children use the mess three times a day, which can also be used.

• Circulation space specially from classrooms to the hostel/mess block is frequently and repeatedly used. Thus this can be place to explore.

Classroom

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Visual Communication, Industrial Design Centre, IIT Bombay.

Some categories to be able to become base to the study:

Design Proposals:

On the basis of Day:

Based on my visit to the school, and experience of the place, I came up with few Design Proposals. These proposals are categorised on basis of the Spaces. Then they are analysed to their use.

Student have 3 major chinks in the day • Early Morning: Students spend their early hours of the day in some physical activity and studying.

• Day in School: The most important part of the day is spent in school and classrooms.

• Evening: Children spend their evenings in the playground.

• Night: Children spend time studying in their hostel.

Later these proposals are put into table where it is further analysed in relation to Multiple Intelligence theory.

On the basis of Interaction with People: • Self learning and Exploration • Learning with Peer • Learning with Teachers • Learning with other People Around like Warden, Cook, Gardner.

On the basis of Space: • School with classrooms, labs and activity spaces • Hostel Block with mess, courtyard, semi open space.

• Playground • Circulation Space

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Classroom: Children spend maximum time of their day in the classrooms. They have classes from 10 in the morning to 4 in the evening. Also classroom is where they have their lessons with teachers. Thus this place a very good potential for exploration.

Location of the Classroom

Time

Space

People

Activity

Remarks

Early Morning

Hostel, Courtyard, Mess

Self, Peer, Warden, Cook, Gardner, Helper.

Physical Activity, Study, Morning activities

Self learning and exploration, Learning with peers.

Day in School

Classrooms, Labs, Circulation Space, Mess.

Teachers, Peer, Self.

Study.

Learning with teachers thus as reinforcement to what they teach, Self learning. Learning with peer as a group activity, discussion.

Evening

Playground

Self, Peer

Play (physical games)

Learn while playing, which are mostly physical games. Thus infrastructure to help children in their games with friends.

Night

Hostel, Mess

Self, Peer, Warden, Cook, Helper.

Play, Self study, Rest

Self Learning and Exploration. Learning with peer.

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Visual Communication, Industrial Design Centre, IIT Bombay.

1. Scale on the furniture - The metal furniture can be painted with scale both horizontal and vertical.

Back Wall with writing board

2. Scale around the blackboard - The blackboard can have scale painted around.

Subject it will teach • Measurement • Estimation • Conversion of units • Scale of real things.

Useful as Aid to teachers to show the real situation when teaching the subject in class Used by students during and after class to experience and experiment the lessons.

Plan

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

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3. Angle on the floor of the Door - Angles can be painted around the swing of the door.

4. Angles with the truss - Angles formed in the trusses of the roof can be highlighted with paint or metal rims

Subject it will teach • Angles in Geometry • Concept of kind of triangle based on angle Useful when Aid to teachers to show the real situation when teaching the subject in class Used by students during and after class to experience and experiment the lessons.

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Visual Communication, Industrial Design Centre, IIT Bombay.

5. A writing wall on the back wall of the class: A black board or green writing board can be installed on the back wall of the class.

6. Pegs provided in the classroom to make space for display.

Useful as Used by children for group work Used when self exploring.

Infrastructure for many everyday activity for children.

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

7. Geoboard - Geoboard can be painted on any of the wall.

Subjects it will teach: Geometry Area, Perimeter etc Pattern Useful as Aid to teachers to gives hands on experience of concepts to children.

Used by students during and after class to experience and experiment the lessons.

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Residential Block: This school is a residential school with occupancy for both students and teachers. There is a residential block which has three building wings, one for girls, one for boys and third is the mess with kitchen. A part of the boys hostel wing has a small medical room, guest room and wardens room. This has a huge courtyard. The blocks have corridors, and a blank wall is there covering the entire area.

Time

Space

People

Activity

Remarks

Early Morning

Hostel, Courtyard, Mess

Self, Peer, Warden, Cook, Gardner, Helper.

Physical Activity, Study, Morning activities

Self learning and exploration, Learning with peers.

Day in School

Classrooms, Labs, Circulation Space, Mess.

Teachers, Peer, Self.

Study.

Learning with teachers thus as reinforcement to what they teach, Self learning. Learning with peer as a group activity, discussion.

Evening

Playground

Self, Peer

Play (physical games)

Learn while playing, which are mostly physical games. Thus infrastructure to help children in their games with friends.

Night

Hostel, Mess

Self, Peer, Warden, Cook, Helper.

Play, Self study, Rest

Self Learning and Exploration. Learning with peer.

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Visual Communication, Industrial Design Centre, IIT Bombay.

8. Measurement scale on the bunker beds.

9. Fraction on the circular seats of the mess

Subject it will teach • Measurement • Estimation • Conversion of units • Scale of real things.

Subject it will teach • Fractions

Useful as Used by students after class to experience and experiment the lessons.

Useful asThe students look at different fractions everyday and this help in visual understanding and memory.

Can be used by teachers for reinforcement of class chapters.

This is Spatial learning.

In Spatial learning it is reinforcing the previous knowledge

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

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10. Writing wall on the low height walls in both the wings of the hostels.

Useful as Used by children for group work Used when self exploring.

Infrastructure for many everyday activity for children.

11. Geoboard on the wall.

12. Magic Square on the wall 13. Dot square on the wall Subjects it will teach: Geometry Area, Perimeter etc Pattern Useful as Used by students after class to experience and experiment the lessons.

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Visual Communication, Industrial Design Centre, IIT Bombay.

Playground The school has a huge playground which children use twice a day. Early morning for physical training and evenings after school for play activities. During evening there can be games related to maths that can be played on the ground. For this need of basic infrastructure, which can be commonly used for number of games can be is there.

Time

Space

People

Activity

Remarks

Early Morning

Hostel, Courtyard, Mess

Self, Peer, Warden, Cook, Gardner, Helper.

Physical Activity, Study, Morning activities

Self learning and exploration, Learning with peers.

Day in School

Classrooms, Labs, Circulation Space, Mess.

Teachers, Peer, Self.

Study.

Learning with teachers thus as reinforcement to what they teach, Self learning. Learning with peer as a group activity, discussion.

Evening

Playground

Self, Peer

Play (physical games)

Learn while playing, which are mostly physical games. Thus infrastructure to help children in their games with friends.

Night

Hostel, Mess

Self, Peer, Warden, Cook, Helper.

Play, Self study, Rest

Self Learning and Exploration. Learning with peer.

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Circulation Space: Circulation of all buildings are interesting spaces which are not very well explored and has maximum use and value.

Here the circulation area mostly used by children are the corridors around the hostel block, courtyard of the hostel block, way connecting the hostel block and the classrooms and corridors of the classroom buildings.

Time

Space

People

Activity

Remarks

Early Morning

Hostel, Courtyard, Mess, Circulation space

Self, Peer, Warden, Cook, Gardner, Helper.

Physical Activity, Study, Morning activities

Self learning and exploration, Learning with peers.

Teachers, Peer, Self.

Study.

Learning with teachers thus as reinforcement to what they teach, Self learning. Learning with peer as a group activity, discussion.

Self, Peer

Play (physical games)

Learn while playing, which are mostly physical games. Thus infrastructure to help children in their games with friends.

Self, Peer, Warden, Cook, Helper.

Play, Self study, Rest

Self Learning and Exploration. Learning with peer.

Day in School Classrooms, Labs, Circulation Space, Mess.

Evening Playground, Circulation Space Night

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Hostel, Mess, Circulation Space

Visual Communication, Industrial Design Centre, IIT Bombay.

14. Measuring Scale on the skirting of the corridor along the classroom.

15. Milestone in the corridor of the classroom telling the distances from one place to other.

Useful as Aid to teachers to show the real situation when teaching the subject in class Used by students during and after class to experience and experiment the lessons.

16. Corridor around the hostel with milestone telling about the distance covered.

17. The wall of the boys hostel wing, with a vertical measurement scale and a chart painted.

Subject it will teach • Measurement • Estimation • Conversion of units • Scale of real things.

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

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18. Circulation space connecting the classroom and the hostel block with a number line.

19. Kerbs along the path of circulation with number pattern and picture pattern.

Useful as By regularly use will help children memorize numbers.

Games with numbers can be played, thus different concepts of numbers can be taught.

Subjects it will teach: Numbering and Numeration.

Pattern

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Visual Communication, Industrial Design Centre, IIT Bombay.

Categorizing the proposals according to the Multiple Intelligence Theory. This helps to see and analyse the proposal according the theory set by Howard Garner.

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

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Proposals

Spatial Intelligence Familiarization of Concept

Reinforcement Of knowledge

Interpersonal Intelligence Visualization Of subjects

1. Scale on the furniture 2. Scale around the blackboard 3. Angle on the floor of the Door 4. Angles with the truss 5. A writing wall on the back wall of the class: 6. Pegs provided in the classroom to make space for display.

7. Geoboard 8. Measurement scale on the bunker beds.

9. Fraction on the circular seats of the mess 10. Writing wall on the low height walls in both the wings of the hostels.

12. Magic Square on the wall 13. Dot square on the wall 80

Visual Communication, Industrial Design Centre, IIT Bombay.

Kinesthetic Intelligence Activity based learning

Interacting With space and experience

Hands on experiments of concepts

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

Logical-mathematical Intelligence Concept of play and learn

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Proposals

Spatial Intelligence Familiarization of Concept

Reinforcement Of knowledge

Visualization Of subjects

Interpersonal Intelligence

14. Measuring Scale on the skirting of the corridor along the classroom.

15. Milestone in the corridor of the classroom telling the distances from one place to other.

16. Corridor around the hostel with milestone telling about the distance covered.

17. The wall of the boys hostel wing, with a vertical measurement scale and a chart painted.

18. Circulation space connecting the classroom and the hostel block with a number line.

19. Kerbs along the path of circulation with number pattern and picture pattern.

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Visual Communication, Industrial Design Centre, IIT Bombay.

Kinesthetic Intelligence Activity based learning

Interacting With space and experience

Hands on experiments of concepts

Study and Exploration of Built Space to Teach Maths, Design Research Seminar

Logical-mathematical Intelligence Concept of play and learn

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Visual Communication, Industrial Design Centre, IIT Bombay.

Conclusion: The project gave me first exposure towards design for education of children. There is a lot research already done and a lot more is still needed to be done.

Here I have tried to look into relationship between space, maths and children learning. This is an area which needs to be well explored. Built space has lot of potential and scope to adapt and incorporate learning.

In the process of this project, I interacted with a children from Samparc Bhambarde. Here I spent some time, and made some proposals as to the project. This helped me understand the problem and solution better and more contextually.

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Bibliography: 1. National focus group on Teaching Mathematics, NCERT, 2006 2. <http://www.ncert.nic.in> 3. Building as Learning Aid, Kabir Vajpayi 3. Guidelines for Vidyalaya Kalyan Samiti, Department of Education, Government of NCT of Delhi 4. Gardner.H, Frames of Mind, Basic Books, New York, 2004 5. http://www.sciencebuddies.org/science-fair-projects/project_apa_format_examples.shtml 6. http://www.learning-theories.com/ 7. http://www.harappa.com/wall/1930.html 8. http://www.springhurst.org/articles/MItheory.htm 9. http://www.thirteen.org/edonline/concept2class/mi/index_sub6.html 10. http://www.learningdoorway.com/multiple-intelligences.html 11. http://www.education.com/topic/how-children-learn/ 12. http://ezinemark.com/a/kinesthetics-learning/ 13. http://www2.ed.gov/pubs/EarlyMath/whatis.html 1 14. http://salsa.missioncollege.org/mss/stories/storyReader$33 15. http://homepage.mac.com/efithian/Geometry/Activity-03.html 86

Visual Communication, Industrial Design Centre, IIT Bombay.

16. http://www.ncert.nic.in/rightside/links/pdf/framework/School%20and%20Classroom%20environment.pdf 17. http://www.infed.org/thinkers/bruner.htm 18. http://www.psych.nyu.edu/bruner 19. Significant Influences on Children’s Learning of Mathematics, Science and Technology Education, UNESCO 20. Kristin. E Reedal, Jean Piagets Cognitive Development Theory in Mathematics Education, summation 2010 (pp- 16-20) 21. Brenda Mergel, Instructional Design and Learning Theory, Educational Communications and Technology University of Saskatchewan, May 1998 22. Vicky Lara, Professional Development Module: Learning Theory and the Adult Learner, El Paso Community College, May 2011 23. John Munro, Mathematics underachievers learning spatial knowledge 24. Stella Vosniadou, How Children Learn, UNESCO, Educational Practices Series 7.

25. Ormrod, J.E. (1999). Human learning (3rd ed.). Upper Saddle River, NJ: Prentice-Hall.]

26. Piaget J (1952) The Child’s Conception of Number. London: Routledge & Kegan Paul.

27. Marie Sjölinder, Spatial cognition and environmental descriptions.

28. Post, T. (1988). Some notes on the nature of mathematics learning. In T. Post (Ed.), Teaching Mathematics in Grades K-8: Research Based Methods (pp. 1-19). Boston: Allyn & Bacon.

29. Ojose, B. (2008). Applying Piagets’ Theory of Cognitive Development to Mathematics Instructions.

Pictures from NCERT books, BaLa book and Samparc Images.

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