You know what happened today?
It was 6.30.... We wre done with the class... Students had already left the class and I was wrapping up things around....and suddenly I heard the sound, with increasing volume, heading towards me.... And in no time, the gang zoomed at the door.
"Sir, we saw a pattern on the number plate of your scooter!"
What? I really didn't know how to respond to this... Fortunately, they didnt allow me either :)
"Your number is MH 43 1950 right?"
My smile gave them the answer.
" 4 + 3 = 7 ......... 4 x 3 = 12 ................... And 7 + 12 = 19 "
And while I was still trying to catch up with this observation, embroiled in their wild enthusiasm......
" And 43 + 7 = 50 "
"Wow !! Good one... I am very happy that you have started making observations... This reminded me of Hardy Ramanujan's story of 1729......
"Yes sir,.... you had told us that story......."
"Hmmmm...... Congratulations !! Keep exploring further... But you could have told this to me even tomorrow isn't it? Why climb up Five Floors, for this?"
"We were so excited that we didn't think about this... We just came to tell you.... We were sure, you will be very happy !!"
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Has something similar or more interesting thing happened in your Maths class any day, when your student(s) had blown you off with their Discovery?
If yes, please share that incident, story... I would love to read....
It would be great if you share this story, and your views and comments about this story directly on my blog-post.
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Hope to meet soon this time :)
Indeed ur inspiring n with such company u will b inspired :)
ReplyDeleteGr8 going....
What the students call a 'pattern' in the number 431950 is NOT a pattern in the number, it is simply a property of the representation of that number in the system of Arabic numerals.
ReplyDeleteA pattern of numbers is a general property of a population, not the property of an individual member of the population. Thus, the statements "The sum of three consecutive numbers is divisible by three" and "No side can be longer than the sum of the remaining sides in a polygon" express patterns, but "The sum of 23+24+25 is divisible by three" is not a pattern: it is a statement about a particular triplet.
More importantly, there is an important difference between properties of numbers and properties of the numeral representations of numbers. That 8647 is an odd number is a property of that number, but that the last digit of 8647 is 7, which expresses an odd number, is a statement about the representation of that number in the Arabic numeral system. This staement is not true in the Roman numeral system or in a binary system. (In the Roman numeral system, the number seven is expressed as VII where the last digit is I, and in a binary system, the last digit of the representation of any number is either 0 or 1.)
Ramanujam's remark to Hardy on 1729 was not about the numeral 1729, but the number expressed by that representation.
It is important that students develop the capacity to discover patterns in numbers and figures, express these patterns as conjectures, and prove these conjectures to establish them as theorems. What the students were doing suggests that they are getting interested in the subject, but I don't see what they were doing as the activity of pattern finding, conjecture formulation, or conjecture proving. I would love to see these capacities being developed in your students.
Mohanan
Hello Mohanan...
DeleteThank you very much for reading the blogpost and more importantly, for taking time and effort to nicely elaborate the difference between property and peculiarity.
Incidentally, I am aware of these mathematical terms/ concepts. I understand that what the child saw was not a pattern but a specific property of that particular number. And let me also humbly inform you that even he is aware of this. We have discussed (and keep discussing) about these ideas regularly. It's just that he has not used the correct math term (vocab) in this particular incident. And given the various factors, I can forgive him for this.
We (I mean, 'they') do algebra, geometry, spotting patterns, making and proving conjectures, generalizing patterns towards formulae building, etc... They have solved some interesting challenging problems as well which I have been unable to share as of now... but will probably do so soon... You are meanwhile, welcome to come and interact with them when you come to Mumbai...
Thank you once again, for nicely elaborating with examples...
Dear Rupesh,
DeleteI am glad that your students have developed the capacity to notice patterns, formulating them as conjectures, proving them as theorems, and generalizing the theorems to explore more conjectures. These abilities constitute an important component of mathematical thinking, hardly ever present in mainstream math curricula in India. Once the students have learned how to find patterns, formulate conjectures, and prove conjectures to establish them as theorems within a given mathematical theory, the next step for them to learn how to construct novel theories, by constructing logically possible worlds, setting up definitions of entities and relations that inhabit those worlds, and postulating axioms that govern those entities and relations.
I am very much interested interested in finding out what pedagogical strategies you employ to help students develop these abilities of mathematical thinking. Unfortunately, I am not in a position to come to Mumbai and attend a number of sessions, so the only way I can find out is through the postings you share with us. (I assume that you are sharing these experiences because not everyone can attend your class sessions.) So I am eagerly waiting for your experiences of children going through the experiences of thinking like mathematicians.
Great !
ReplyDelete