How Maths is made difficult
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| Children should use tools |
This progression is often designated by ELPS:
- E for Experience
- L for Language
- P for Pictures
- S for Symbols
Typically the maths books start from the Picture (P) stage and, then immediately move to the Symbol stage (S).
Without proper exposure of Experience stage (E) and development of the necessary Language stage (L), we naively expect that a student would directly comprehend from pictures (#3) and would effortlessly graduate to the symbols stage (#4).
We start talking about a room being 20'x30' without realizing that a student has no experience of what is meant by measuring a room. He had probably never used a measuring tape to size up the room. He has not used a ruler to measure things around him. His estimation of 1" or 1' or something being 5" or 5' does not build a mental image of the relative sizes. 1", 1', 5" or 5' are symbols without meaning. Similarly, his estimation of the passage of time is weak. Word problems therefore create the biggest hurdle. First of all they lack the experiential feel and an intutitive understanding of what is being mentioned, secondly their understanding of the language and the phrases being used is imprecise. .
For e.g., the operation "minus" is often phrased in world problem as "subtracted from", "decrease by", "lesser than", "difference of", and so on. However, the synonymous use of these phrases in some contexts makes sense and in others do not. This creates a huge word/language barrier.
We thus make maths harder when we ignore the importance of E and L. We must in order to make maths easy for the students give them a feel and intutive understanding of all the measure through physical experience and then working on the explicit development of language.
This may not have been a big problem about 50 years ago, when people were used to doing so many things by their hands. By age eight most children had obtained experience of using tools like hammer, pliers, nose pliers, saw, hand drills, tester, manual weighing machines. They had already observed people repairing watches with their precision tools, clothes being sewed on sewing machines, carpenters making furniture, plumbers repairing and threading GI pipes and using pipe wrenches for tightening and opening pipes, digging with hand showels, welding and soldering, overhauling of cycles by taking apart all the parts and then reassembing them, pulling out of water from the well, filling the buckets and taking them around the house, watering the plants with hand sprinklers, etc. Normal maintenance in the house was done manually by people from the household or was done by workers coming to the house and doing the repair and maintenance. Making of cheeze, ghee, washing soap, dahi (curd), lassi was done at home by hands. Juices were extracted through manual machines, qeema was grounded by hand, spices were grounded by hand, and chutneys and so many items were processed at home.
All these exposures gave a child of a feel and experience of forces at work, friction, resistence, circular motion, transfer of motion from one plane to another, push, pull, passage of time, estimation, calculation, conversions, translations, transformations and other physics, maths and chemistry concepts. Armed with these experiences, they had a head start, and for them to move from sr #2 to #4 was straight forward.
However, in today's city life and even in the village life this exposure has become minimal with the availability of automatic devices, packaged food, wash n wear shirts, ready-made garments, bottled water, disposable furniture, disposable watches and disposable and pre-packaged other stuff. I was surprised to note that even in villages the only "dahi" and "doodh" available is the packaged one. Hence, E and L must now be introduced first, explicitly and conciously.
Another problem is in communication and in understanding the intent of mathematical questions. One of the classic examples is that of a teacher asking a student how many halves are there in three and a half ( 3 1/2). The student may give an answer which can be one or seven depending upon how he understands the question. However, the teacher may overlook the ambiguity in the question and may mark the answer of the student wrong, because only one answer can be right. A student going through several such mistakes may errorneously conclude that using his mind is a dangerous activity. He then starts trying to foresee what the teacher wants, or looks for clues or simply starts guessing. He loses his confidence and thinks that maths is unpredictable.
This loss of confidence in a student's ability to use his mind is the fundamental reason for maths anxiety.
Without proper exposure of Experience stage (E) and development of the necessary Language stage (L), we naively expect that a student would directly comprehend from pictures (#3) and would effortlessly graduate to the symbols stage (#4).
We start talking about a room being 20'x30' without realizing that a student has no experience of what is meant by measuring a room. He had probably never used a measuring tape to size up the room. He has not used a ruler to measure things around him. His estimation of 1" or 1' or something being 5" or 5' does not build a mental image of the relative sizes. 1", 1', 5" or 5' are symbols without meaning. Similarly, his estimation of the passage of time is weak. Word problems therefore create the biggest hurdle. First of all they lack the experiential feel and an intutitive understanding of what is being mentioned, secondly their understanding of the language and the phrases being used is imprecise. .
For e.g., the operation "minus" is often phrased in world problem as "subtracted from", "decrease by", "lesser than", "difference of", and so on. However, the synonymous use of these phrases in some contexts makes sense and in others do not. This creates a huge word/language barrier.
We thus make maths harder when we ignore the importance of E and L. We must in order to make maths easy for the students give them a feel and intutive understanding of all the measure through physical experience and then working on the explicit development of language.
This may not have been a big problem about 50 years ago, when people were used to doing so many things by their hands. By age eight most children had obtained experience of using tools like hammer, pliers, nose pliers, saw, hand drills, tester, manual weighing machines. They had already observed people repairing watches with their precision tools, clothes being sewed on sewing machines, carpenters making furniture, plumbers repairing and threading GI pipes and using pipe wrenches for tightening and opening pipes, digging with hand showels, welding and soldering, overhauling of cycles by taking apart all the parts and then reassembing them, pulling out of water from the well, filling the buckets and taking them around the house, watering the plants with hand sprinklers, etc. Normal maintenance in the house was done manually by people from the household or was done by workers coming to the house and doing the repair and maintenance. Making of cheeze, ghee, washing soap, dahi (curd), lassi was done at home by hands. Juices were extracted through manual machines, qeema was grounded by hand, spices were grounded by hand, and chutneys and so many items were processed at home.
All these exposures gave a child of a feel and experience of forces at work, friction, resistence, circular motion, transfer of motion from one plane to another, push, pull, passage of time, estimation, calculation, conversions, translations, transformations and other physics, maths and chemistry concepts. Armed with these experiences, they had a head start, and for them to move from sr #2 to #4 was straight forward.
However, in today's city life and even in the village life this exposure has become minimal with the availability of automatic devices, packaged food, wash n wear shirts, ready-made garments, bottled water, disposable furniture, disposable watches and disposable and pre-packaged other stuff. I was surprised to note that even in villages the only "dahi" and "doodh" available is the packaged one. Hence, E and L must now be introduced first, explicitly and conciously.
Another problem is in communication and in understanding the intent of mathematical questions. One of the classic examples is that of a teacher asking a student how many halves are there in three and a half ( 3 1/2). The student may give an answer which can be one or seven depending upon how he understands the question. However, the teacher may overlook the ambiguity in the question and may mark the answer of the student wrong, because only one answer can be right. A student going through several such mistakes may errorneously conclude that using his mind is a dangerous activity. He then starts trying to foresee what the teacher wants, or looks for clues or simply starts guessing. He loses his confidence and thinks that maths is unpredictable.
This loss of confidence in a student's ability to use his mind is the fundamental reason for maths anxiety.
See also:
- How to define success of a school or a student
- Iqbal's view on What is Meant to be Educated
- Bell-curve assumption about the distribution of intelligence of students
- Charter of Children's Recognition
- How Maths is Made More Difficult
- Beauty is our Business: Dijkstra and Mathematics
- Holistic Learning and Whole Life Orientation
- Education as Tazkia: Is a child like a clean slate?
Dr.irfan, i have been thinking of how to go about math with my homeschooled 6 year old. I've been hesitant to begin for the very reasons you stated above. They were sort of ambiguous in my mind but alhamdolilah i see with clarity now.
ReplyDeleteI've bought a second hand book on the abacus....i thought that might have more E in it. Do you think that would be the right approach?
I would recommend to you the book "How CHildren Learn Mathematics". You can get a copy of the book from ERDC. www.erdconline.org
ReplyDeleteAssalam-o-alaikum Dr. Sahib!
ReplyDeleteFirst of all I want to congratulate you for such a useful and thought-provoking blog. It is no doubt excellent of its kind. Specially the second to last paragraph addresses one of the major obstacles to developing mathematics skills in students.
I can remember the outstanding opinion that you gave at a meeting on the topic of "How to teach better" at Jamia Qurtuba, Clifton Block 1.
We are thankful to you for your sincere remarks and advices. Looking forward to seeing you again.
Thanks.
Regards,
Muhammad Asif,
Jamia Qurtuba Secondary School.