Distributing money : Two methods -- Mentally v/s Long division

The other day I was assessing a middle school student one-on-one.

I asked her 'If you wish to distribute Rs.4,000 equally among 5 people, how much will each of them get?"

She thought for almost 1-2 minutes. I guess, she was working out mentally. I generally don't open my mouth (interfere) when students are thinking. Then I noticed that she resorted to pen and paper. I could see her work and she was trying out this problem using long division method. 

"Sir, each one will get Rs.800."

"Ok. I saw that you gave enough time to think before using pen and paper. Can you please share what were you thinking?"

She replied with a smile -- "Sir, I was trying mentally but I messed up somewhere."

"Oh... and Where?"

"I first did half of 4000, gave 2000 to two of them, further halved this 2000 and now 4 people got 10000. I then took 200 from each of the 4 people and gave the collected amount to the 5th person. So each got 800."

"Wow !! You have worked out correctly. What made you feel that you have messed up?"

"Yes, now I did correctly. But in the beginning, I had, by mistake, taken 300 from one of the 4 persons, and hence, had given 900 to the last person. So four of them got 800 and the 5th person got 900."

I paused for a second. 

"Would each of the 4 guys get 800, if the 5th one has got 900?"

She thinks for a while. " No..No... I took away 200 from the 4 guys but I did a mistake in adding four 200s... I calculated the sum as 900 instead of 800... So I thought I have done a mistake as each of them have not got the same amount. So I switched to pen and paper (log division method.)"

"Okay.."

"Shall we try one more similar problem?"

"Yes Sir", she happily agreed. :)

"How will you distribute Rs.2,500 equally among 4 people?"

I wanted her to do this mentally, but she directly jumped to pen/paper. I did not interrupt her (will push her for mental work after this).

"Sir, each person gets Rs.435." 

I was happy with her confidence, but was also disappointed to note that she was victimized!  This is how she did.

​I would request you to hold on for a while and..... Reflect...... Think what will be your reaction to such a situation -- i.e. if possible, write down sequence of steps -- statements/ questions/ actions -- you would take when your student (s) err out this way. 

I hope, the 'learners' will honestly work (pause and think), before reading on further !!   :-)

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I looked at her written work for about 5 seconds (actually I was thinking as to how do I respond now) ... and then I finally asked her if she can also solve this problem mentally, the way she had solved the previous problem. I tried my best that my words/ body language would not hint her about the mistake. She agreed to my request and started scribbling something on pen/paper after 5 seconds of mental work.

She continued to work while I was patiently observing her as well as thinking as to what must be going on in her head :)

After about a minute, she said, " Sir, I am unable to do this way."

"Why? I think you were going correct way."

"I am unable to do the half of 2500." 

I looked at her work. I noticed that she had worked with 2000 in the beginning and then, had overwritten 2000 with 2500.

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"I see that you have already done some work well. Can you explain?"

"I first halved 2000 and gave 1000 to two people as before. I then halved this 1000 again and so all 4 people got 500 now."

Silence.

"Yes, then?"

"We are still left with 500. So I cancelled 2000 and then again started working with 2500. But I could not halve this 2500 properly"

I thought for a while and said, " But you have  already distributed 2000, haven't you? Its just the business of 500 that's left."

I was happy that she got the hint and she again started working on paper, but this time only with 500. (Check her work (image above))

"I will give half of 500 i.e. 250 to two people... So two people got 750 now..."

Silence.

I realized that she was again stuck. She had added the newly distributed 250 to the previous 500. I had to scaffold again. 

"How much money are we trying to distribute in this second round?"

"500 to four people."

"Yes. Because one 500 is already given to each of them, isn't it?"

I was glad that she could take it well from here.

After thinking for a while, she continued -- "We will now half the 250 and give 125, 125 to all the four. So each of them will now get...................  500 + 125....... 600.......625."

"Hmmm... See, you could do it !! "

She looked at me with a smile of confidence, not knowing that the ball will be back in her court again :)

"Just see if you have got the same answer even by long division method."

And no wonder, she was stoned... She again looked at me with a smile, but this time with a pinch of embarrassment. 

"Why? What happened?"

"Sir, I have done some mistake.", she confessed almost immediately. 

"Where?"

I wanted her to think.... but she had meanwhile started working on the long division method again. I thought she will be able to find out her mistake this time, but..........surprisingly, the exactly same errors and hence the same (incorrect) figure - 435.

And now she was more puzzled. I broke the silence again. 

"So what is your opinion?"

"I again messed up in the mental work.", with a stroke a sheer confidence.

And this came as a Big Blow to me this time !! 

"How can you be so sure, that you have messed up  in mental work? You might have erred up in the long division algorithm."

"No sir. I am sure, This written work is correct. The method we did mentally is incorrect. We have done some mistake there."

I looked at her for a while, with the hope that she will change her mind and might do some more investigation. But I was wrong. She had given up. 

Though I wanted to and even could have cajoled her for investigation, but I choose not to at that time for some reasons. 

I want to share about what I felt at that instant and even about my views about this experience. But I would rather like to leave this space open for you all folks now as to what are your comments and views on this experience. I hope you will appreciate the effort I take to share these stories with you and hence will continue to motivate me by replying back to this email with your reflections so that we can have a healthy discussion/ learning among the peers in this group. 

Waiting for you....  :)

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Regards

Rupesh Gesota