Tuesday, August 30, 2016

"There seems to be some problem in Table of 15...."

"Estimate the length of this table"

He thought for a while observing the table and said, “More than 45 inches and less than 75 inches.”

“How did you guess?”

“Our small scale is of 15 inch. I feel around 3 and half such scales will fit along the table.”

(Can you figure out his two errors? What will YOU do in this case? Take some time to think about your strategy before your read ahead)

“Yes, I agree with you that around 3 and half scales would fit along the table. However, can you explain how you got these two figures: 45 and 75?”

“Sir, 15 x 3 = 45 and 15 x 4 = 75.”

After some thinking – “What if it were exactly 5 scales?”

"Then it would be 90 inches."

“What if it were 10 scales?

"15 x 10 = 150 inches."

“Okay…  So you mean 10 scales would mean 150 inches and 5 scales would imply 90 inches….”

While he was about to nod in agreement, he paused…. I could see him being puzzled at something….

After 3-4 seconds: “What happened?”

“Sir, wait… I think there is some problem.”

“What problem?”

“Some problem in the table of 15.”

“Oh, is it? How do you know?”   

(I also succeeded in NOT reacting to his response with my laughter/ shock/ anger)

“Because 15 x 10 = 150, then how can 15 x 5 = 90?”

“Hmm…. So then, what is the contradiction according to you?”

“If 15 x 5 = 90, then 15 x 10 should be double of 90 i.e. 180.”

“True… So then…….?  Is 15 x 10 = 150 or 180?”

“180 does not seem to be correct……150 is correct.”

“Hmmm…. So?”

“But then 15 x 5 = 90 is also correct…..Table of 15 has something special it seems…. This is surprising, I never saw this…”

 I just ensured that he found an authentic enquirer along with him in this process/ problem.

“Can we check other tables also then?”

“Yes… 12 x 10 = 120……. 12 x 5 = 60…. Here it is working properly….”

“Working means?”

“Means 60 is half of 120…”

“Okay….. So?”

“Wait Sir…. Let me check some more……   11 x 10 = 110   and 11 x 5 = 55…… Here also, it’s working well…”

“Hmm…… Should we check some more tables…?”

“No sir… I feel it should work in the table of 15 also then…. 15 x 5 should be half of 15 x 10…….? I am not getting why we are not getting so in 15?”

“Okay… Let’s recite the table of 15 together….”   (Why would I have done so? Was there any other way?)

“15x 1 = ……”


“15x 2 = ……”


“15 x 3 = ……”


“15 x 4 = …….”


“15 x 5 = ……..”


“15 x 6 = …….”


I paused, waiting for him to pick up the clue….. And Yes! He did it!

“Sir, Wait… I think I have done some mistake…   How can 15 x 5 and 15 x 6 be both 90?”

“Hmmm….. So?”

He started telling the tables again but slowly and more thoughtfully this time. I could see him making the table of 15 via repeated addition (adding 15 to the previous multiple).

“Sir, I got the mistake!!  I had missed out 60.”

The JOY of Discovery on his face was unmatchable!!

“Hmmm….  So…?  What about the mistake in the table of 15….?”  (This time, I couldn’t hide my teasing smile :-)

And even he continued to laugh -- with little shades of embarrassment  :-)

Rupesh Gesota


  1. I myself cannot remember tables more than 12. These kids are learning how to learn.

  2. It's so amazing to read your posts....and I can't help but wonder what's the best way to impart this 'questioning technique' to other teachers. Pedagogical content knowledge is a must,practice and patience.... Or may be something much more.

    1. Thank you for appreciation... Its nice to know you understand the imp.of questioning as an imp pedagogical tool... while practice and patience are important, but whats more imp. is the availability of space for such practice and patience... sadly, teachers often don't get the "regular " learning culture within their own schools .. what's seen at some places is just " occasional " training programs...

    2. Amit, I was at a training recently that was all about teaching in this way. Two things I came away with that may help you. 1) Arbitrary vs. essential information. Arbitrary information is information the child could not get on his own. Things like we call this '=' the equal sign, and the word for this symbol '2' is two. Someone must provide a child with this information. Essential information is the stuff they can figure out on their own. We don't give them this informtion, we let them get it for themselves. Our job as teachers is exactly what Rupesh did above.

      The other thing I took away is a quote from Dr. Arthur Powell. Someone said he must have a lot of patience. His response, "I'm not patient. I'm curious to know how the child is thinking."

      An excellent treatment of arbitrary and necessary is found here: http://flm-journal.org/index.php?do=search&search=Dave%20Hewitt&operator=and

      And Rupesh, there is no place for this kind of work inside American classrooms. This was an excellent post.

    3. Mrs. Post, thanks for the kind words of appreciation.. I am glad you can so much value in my conversation with the student... Also Thanks for sharing the idea of Arbitrary v/s Essential information... It's beautiful... I have checked the link given by you and enjoying the article...Thanks again :)

    4. Thanks Mrs post and Rupesh for the reply.
      I'll go through the link shared above.
      Mrs Post,Im curious to know why do you say that there is no place for this kind of work inside American classroom. Not very familiar with the American education system and the typical classroom culture, I would like to believe that the average American classroom would be more open to such progressive pedagogies and would have a more liberal timetable to allow for such experimentation.

    5. Amit, you will find pockets of dedicated teachers who are looking to change the system. BUt most are so busy with testing and other requirements that they are not free to take up this kind of thing. There are some, indeed, who are bucking the system. I know some very good ones but it is not most, and I doubt even 1/4 of teachers. I don't blame them, I blame the system.

  3. This is a fantastic line of questioning. Made my day!

  4. Thank you for sharing. I love reading your posts.

  5. I really liked that you shared the whole conversation step by step. This example will really help me with my boys as I would normally have just pointed out the mistake or worse- given them the answer. I'm learning not to interrupt their discoveries.

  6. We an adults by now are so much well
    TRAINED and STAMPED with conventional
    method of learning. This example is a relection of my education as a child. Many basic and important topics in any subject are not naturally understood and goes unattend