Mathematics Ali Institute of Education
A Teacher and Material Developer Should Think How A Child Thinks
A few days ago my youngest daughter, Pipi who is completing her pre-schooling, was solving a worksheet. I was observing her from a distance. Initially, I was seeing excitement and confidence from her facial expressions, but after a while these vanished. Instead I saw anxiety in her face. I realised that she was struggling with something. I silently approached her and looked at the work she had produced; I got a shock. I was sure her teacher would cross-out her work if she presented it, because the counting she had done was not in sequence. It showed a child who was not having a command of numbers and who was unable to write numbers. I realize, however, that the situation was not as easy as that. We, math teachers, try to guide toddlers to write numbers horizontally or vertically. Usually we provide them grid-notebooks having squares neatly aligned. Looking at the worksheet my daughter had, I could see that the layout was confusing her.
Take a look at the picture of my daughter’s work in this blog. From the number 20, Pipi had difficulty connecting the next figure (21) because she was distracted by the layout, and this pushed her to write the number 21 in the square below. She continued to write numbers up to 24 following the line she had started, and then write 25 on the previous line which suggested she didn’t know how to count.
The purpose of sharing the picture and this experience is that maths teachers who develop resources for kids should think how a child thinks. Developers can make some layouts which look very stylish, colourful and attractive, but jeopardise the process of learning. So, if you are teacher always think how a user, a toddler, child or young-one, could think and interpret the resource or instructions you are using.
Yesterday was a really pleasant afternoon. I thought to leave the institute a little early and avail the chance to pay my utility bills and school fees of my two kids. I found my kids, Mimi and Pipi, at home. I took them for a ride to the bank so that it could be a little outing for them and I could pay the bills.
We reached the bank well before bank closing. We had time and I thought instead of simply paying the bills myself why not turn this into a learning experience for my kids. I asked my elder daughter, Mimi who is recently promoted to grade four, to take a paper and pen, make a list of all the bills and make two fee slips. Initially she didn’t have any idea how to complete the task. With a little help she drew a table and filled it with the information. I asked her how much we need to pay to clear our dues. She started adding five four-digit amounts. Then I extended the situation and asked her, if we pay fifteen thousand rupees to the person at the bill collection window, how much will he return to us. She confirmed the sum she needed to do with me and then started subtracting amounts. She told me that we will get thirty rupees in change. I handed over the bills to Pipi (recently promoted to preparatory grade) and the money to Mimi, and asked them to go to the window and submit the dues. I sat comfortably to watch what happened. Both were a bit anxious but very excited as it was their first experience at submitting bills at the bank. Both went to the window. Pipi found it a little difficult to reach her hand up to the counter but she managed it by standing on her toes. It was funny to see the cashier stand up to see who was beneath the bills and fee slips! Both sisters submitted their first ever payment slips in the bank and came back with the body language of queens! This gave me immeasurable pleasure. Almost everyone - including some mothers - in the bank had noticed us, and there were some passing smiles of appreciation. I am quite sure, if not all then at least some of them will certainly use this idea with their kids.
This situation took ten to twelve minutes. I can do it in a few seconds but then I would have missed the sight of both sisters walking towards me. Those additional ten minutes had changed them; more confident and closer to the real world.
Importance of rough-work to solve maths sums
I remember my childhood experience of doing mathematics. My maths teachers created a habit of maintaining rough-work along with the actual working of a maths sum. My classmates and I used to draw a vertical line about one third down the page and wrote the heading prominently: ‘rough-work’. This provided evidence for our thinking process as we used it to solve questions. I remember my teacher was very concerned about the rough-work along with the actual solution of the sum. That style of working, which was rigorously maintained, helped the teacher to address our way of approaching maths problems.
Last Friday my daughter – a grade 4 student - came to me to seek help regarding a maths sum. Rather than telling her straightforward steps to solve the sum, I used a student-centred approach. I asked, “Mimi where you will do rough-work?” In response to this question she didn’t say anything. She was quiet, her body language and facial expression made it seem like I was speaking some alien language. I asked her about the routine she was guided to use by her maths teacher when solving a maths question. She described the process but there was no role of rough-work.
I thought about that - Why didn’t the maths teacher value the use of rough-work along with the solution to the question. It may be because of time; teachers have more work to do and guiding students about rough-work and ‘the right answer’ is problematic for them. It could because of students; they have more to do and maintaining both portions side-by-side is difficult for them. It could be because of school management who are focused on a ‘traditional-style of Mathematics teaching’ or they figure that rough-work makes the exercise book look messy! May be there is no need to doing both types of work?
I pose both questions to you, to my readers (maths teachers or parents):
‘Is it beneficial to maintain rough-work along with the solution of a maths sum?’ and
‘What could be the reasons students are not maintaining rough-work along with solving questions?’
How did I Lose My First Ten Chess Games?
There are any number of activities we can do in our free time. Different activities require different skills. Personally, I value those activities which require knowledge, abilities and skills…and chess is at the top. My personal experience of losing my first ten chess games will help you understand the complexity of this game:
I lost my first game because I could not distinguish between the shapes of the different pieces. At that time, I suppose I used my visual-spatial intelligence poorly. I lost my second game because I could not understand the rules of the game. The third I lost because I lost concentration. The fourth - I was just looking at my own pieces and I neglected to watch the pieces of my opponent, and I lost my fifth match because I paid more attention to my opponent’s pieces rather than my own. The sixth was lost due to lack of planning. I lost my seventh game because I was playing aggressively and the eighth because I was playing defensively. The ninth game was lost because I was not thinking what my opponent was thinking, and the tenth was lost because my opponent guessed what I was thinking.
Actually, the list does not end here. In every game I play now, I try to improve – to build on my deficiencies. Every game makes me learn. With every game you grow more composed, focused and logical.
Is there any game in the world which requires various intelligences, skills and abilities in the same way? Even if you feel you don’t have time or you are burdened with lots of work, if you are tense, one game of chess will be time well spent.
MATH IS IN THE AIR: MAKING OF A GIANT-SIZE CHESS SET
I have a strong belief ‘THERE IS NO MEANINGFUL SURVIVAL WITHOUT MATHEMATICS’. Every maths teachers, every mathematician, is doing, applying and taking help from a very important field of knowledge. If anyone disagrees with this argument, just recall today’s routine and consider when you didn’t use maths!
Every field, occupation, task and activity is incomplete without maths. To reinforce my point I’ll share my recent experience. Ali Institute of Education is arranging a chess tournament between school kids for the first time ever. It is planned that the finals will be played with giant size pieces on a huge chess board, 8 metres square, on the Ali Institute front lawn. The task was to make a huge chess piece to understand the mechanics of producing a stable and long-lasting chess piece. The problem solving process was started with observation. I looked at regular chess pieces with a mathematical eye. They are three-dimensional objects which needed to be converted into a two dimensional drawing, disintegrated into small segments; the measurements were taken for each part, a scale is used and a scaling factor was identified. Proportionality between the actual piece and the large-size piece was maintained by cutting and using circular disks. Contours were made through a series of sequential disks of various sizes. The amount of material and time required to complete the task was estimated. A table containing a list of material, the unit price and the cumulative prices were entered and a request was made to the purchase office.
Finding the Right Balance
I am standing at the front of the classroom, beside the whiteboard, carrying a few equations in my arms, standing still like a witch has cast a spell on me. I feel like my feet are stuck in concrete and my joints and muscles are frozen. I am looking at twenty five faces. My eyes making contact with fifty eyes, all conveying different messages. Their features, their expressions and their body language are conveying different messages. The words that describe the feelings here are: accepting, accomplishing, aggravating, amusing, angry, annoying, anxious, awaking, blank, bored, calm, cheerful, cold, complacent, confused, cranky, curious, depressed, determined, disappointed, drained, ecstatic, energetic, excited, exhausted, happy, impressed, irritated, lethargic, naughty, pessimistic, pleased, shocked, sleepy, stressed, tired and many more. My God, twenty five in the audience and more than twenty five feelings. More than that, my lesson plan is telling me to go on – I can’t repeat the session as it will affect the whole course, and I can’t offer off-course time as I have lots of tasks to complete. I can’t refer to any other teacher as no-one is there.
Suddenly I get some flashbacks, a chat between me and a student teacher:
“Sir, what should I do for my students? I have a diverse group of students. Sometimes I run out of ideas, my plans get fouled up and I face difficulty making myself understood?”
In response I lecture her about the area under the normal curve and distribution of various types of students in a natural setting. The last thing I said to her, proudly and confidently, was “DEAR, FIND THE RIGHT BALANCE”.
I am hearing the echo ‘FIND THE RIGHT BALANCE’ and realising perhaps this is the easiest proverb to say but certainly the most difficult action to do.
What is the purpose of maths education?
I think about this question quite frequently and my thoughts swing between two themes; maths education as a means to develop inquisitive humans, aware of the world, or maths education as a means to support the rigours of day-to-day living.
Undoubtedly one’s understanding of language is hugely enhanced by knowledge of basic grammar. Apart from that, it is an exciting phenomenon that human’s learn to speak different languages especially without being aware of grammar. Children and adults can easily make sense of ungrammatical sentences!
The case is exactly the same with the language of mathematics. Up to a certain level, one can do and speak mathematics without knowing how to classify the different sorts of words one is using. Daily conversation or human involvement in daily life activities sensitises the brain to understand patterns and structure, called the grammar of mathematics.