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.

"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.... :)

Regards

Rupesh Gesota

Student is more inclined towards reliability of written algorithmic processes. The confidence that 'I' am correct is associated more with written stuff than mental calculations.

ReplyDeleteThis is really thought provoking!! Thanks!!

ReplyDeleteRupesh this is an excellent minute level sharing of responses from the child comparing mental work to procedural work. useful insights about what made her think eventually that her procedure was fine and not her logical thinking process? what is at play here? confidence on the procedure more than on her own distributive logic and why?

ReplyDeleteThis is really inspiring. Keep up the good work.

ReplyDeleteOn Thu, Aug 27, 2015 at 5:58 AM, Arun Kumar Mahajan wrote:

ReplyDeleteInspiring.. What I feel after going through it that a quality time can definitely help kids to grow better.

All the best..

Arun Mahajan

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On Mon, Aug 24, 2015 at 10:59 AM, Vivekmonteiro wrote:

Rupesh,

I think adding a third dimension i.e. Doing it with actual money or cards representing money would also be invaluable scaffolding.

Dr. Vivek

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On Mon, Aug 24, 2015 at 11:16 AM, Anjali wrote:

Dear all,

Though actual distribution helps when forming the concept, it is more for the primary stages. At the middle school level, I love the way Rupeshji encourages mental calculations and manipulation. Much better way to understand division than an algorithm.

Anjali Gupte

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On Mon, Aug 24, 2015 at 11:25 AM, vivek monteiro wrote:

The problem is that most children are put into a position of doing mental manipulations or estimations without a sound and confident experiental foundation. So in my opinion there is no short cut. Middle school muddles have their foundation in weak primary understandings.

On Tue, Aug 25, 2015 at 10:30 AM, Subodh Kembhavi wrote:

ReplyDeleteHello all,

A nice discussion. We should have such discussions frequently and regularly.

I also appreciate the efforts taken by Rupesh for such a detailed documentation. This would be quite useful as a sample when 'how to' of documentation gets discussed in the teacher training workshops.

(I think there are a few typos : e.g. where it says 'distribute Rs.4000 equally among 4 people' , it should be 'among 5 people'. Am I right ? )

Here is my view :

In their everyday life, students in middle school do logical thinking, use abstract concepts and visualization.

Hence, even an activity like drawing a picture could be 'concrete enough' for many students. Using actual objects might not be required in middle schools.

Anyhow, we need to make sure that the journey from concrete to abstract is made by each student with confidence and clarity. Handling objects, drawing pictures, listening to a story should be used as required to facilitate this journey.

( How could we know this for sure ? What should be the evaluation method? These are the important and interesting questions but let's keep them aside for some other thread.)

So, in this case, maybe we could try drawing pictures.

Four persons on the left and 3 notes on the right (Rs.1000, Rs.1000 and Rs.500) could be the first picture.

Then the second picture could be four persons and 'a lot of ' Rs 100 notes on the right.

Questions like - How many Rs.100 notes will we have to draw? What do we do next? need to be discussed here.

Also, the journey from picture 1 to picture 2 needs to be facilitated through discussion.

Hope all this is pertinent and useful.

Regards,

Subodh

On Wed, Aug 26, 2015 at 11:58 AM, Jyoti Francis wrote:

ReplyDeleteThanks Rupesh for sharing your learning experiences with children.

Agree with Jayasree - Children and most of us have been taught to naively believe in the ways and methods of algorithm taught in schools as the only possibility correctness. More over, lack of multiple opportunities to bank upon and develop alternative thought building options, most often, we believe what ever being taught in the school as the only absolute form of ‘right answer'.

The blind rote learning of the rules and procedures and following of these methods win over the simple human logic gained from experiences of daily life. It is tragic that children are learning to doubt their own logic deduced from their own life experiences. They are learning how not to believe in their own thinking. How not to critically evaluate.

Thus logic and its outcomes fails and illogical ways prevails. This doesn't stop just at math but as we grow we can see the impact in every walk of our lives.

On Sun, Aug 23, 2015 at 11:02 AM, Jayasree Subramanian wrote:

ReplyDeleteThanks for sharing Rupesh. Quite an interesting read.

The phenomenon you have described is unfortunate, but common - they way students rely on some procedures and less on their own sense. Probably the way we teach, takes that confidence away! I have seen this happen with kids who use maths in their daily life in a particular way - they are able to answer in that context but struggle with the formal procedures.

This study on street Mathematics echoes the same thing.

http://ci512-summer2011.wikispaces.com/file/view/Carraher+(1985)+Street+Math.pdf

The answer 435 for dvision seems to have been arrived at by some weird logic - di you try to probe her on how she came to that?

Rgds,

JSree

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On Sun, Aug 23, 2015 at 5:27 PM, Puvidham wrote:

Wonderful honest narration! I would move to material thousands, hundreds, tens, units to help the child realize that she was right doing it mentally and made a mistake in her long decision.

Meenakshi

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On Sun, Aug 23, 2015 at 7:21 AM, Sonali Durgam wrote:

Sir

The story u shared is one which most of us experience in our class. But the problem is , in a class the intelligent ones don't allow the weak ones to think, I.e they give out the answer before the slow learners can even start thinking of. How to cater to such group. How to engage the intelligent ones so that the weak ones can think independently.

Please suggest some method with example.

Regards

Sonali Durgam.

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On Sun, Aug 23, 2015 at 9:07 AM, niral gada wrote:

I too have used this method with my learners.. Itz interesting that you too thought in the same line.

On Sun, Aug 23, 2015 at 8:31 PM, Rushikesh Kirtikar wrote:

ReplyDeleteThanks Rupesh again for finding time to share your experiences. Sometimes I feel a lot respect for your selfless work and efforts.

I have some thoughts on your story.

If I were in your place, maybe I would have pushed her towards the answer much sooner than you usually do. It’s a good practice to allow children to find out their own mistakes. But after a point if they aren’t, then the teacher can step in. May be I would have asked her to write down the table of ‘Four’ which would have helped her to find her mistake, or anything else. There shouldn’t be much problem with scaffolding in any way. I don’t know why you didn’t cajole her any more. At least we have given enough time for her to try. And like you said that she gave up after that. That is okay. Children can feel drained after some time, especially when it is only a mental activity with less practical implication. All may not have that much motivation.

And that’s one of the reasons why I prefer linking such maths or any subject with real activities in a way that you need it somewhere in practice. You cannot escape of finding an answer. And the activity itself provides a lot of enthusiasm and clues.

Do share what were your thoughts on the situation.

Thanks again

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On Mon, Aug 24, 2015 at 1:39 PM, Sangeeta Puri wrote:

Hi,

I seriously appreciate your efforts of sharing these experiences and giving us some opportunities to introspect.

I feel we as teachers never give such an opportunity to the child to reason out and make an attempt to correct his/her mistake.

If I were in the same situation, I would also ask her to try mentally.

After all the interventions if she was successful with her mental calculations, I would ask her to correct her long division method by informing her that the answer from mental calculation is the correct one. That would help her cross check each step of long division method.

I will check your blog as well. Thanks for sharing this story.

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On Sun, Aug 23, 2015 at 10:31 AM, Vasanti Padte wrote:

Dear Rupesh

You have tremendous patience which is really needed for any teacher to allow a child to think and reason independently. All the best.

Vasanti Padte .

On Tue, Aug 25, 2015 at 1:35 PM, Aarti Mulani wrote:

ReplyDeleteThanks for the share Rupesh!

These errors could be remediated and rectified using adequate practice with manipulatives and simultaneous representations using pictures /drawings.

Alongside a step wise discussion of the division algorithm with paper pencil, for each of the practice examples should help in the concept formation and clarity.

Warm Regards

Aarti

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On Sun, Aug 23, 2015 at 1:17 PM, Ruchi wrote:

Beautiful demonstration, Sir, of how we must teach. Not by giving algorithms, but by asking questions and making them think, and allowing them to think wrong, then help correct the understanding. It's a lesson in itself!!!

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On Sun, Aug 23, 2015 at 12:42 PM, Hemangi Joshi wrote:

Dear Rupesh,

It is wonderful to see your mail. A few days back I was wondering as to why I'm not getting mails from you... Thought of writing to you and here I found your mail.

It's so good that you write in detail the process of learning and facilitating the session. I wish I could be a trainer f teachers and have discussion with them on this.

There is a problem on the blog, I am unable to post the comment.

Why don't you make this group open so that everyone knows what other people are commenting. Since I got a bcc mail, I can respond only to you and not to others in your group.

Best wishes,

Hemangi

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On Mon, Aug 24, 2015 at 2:11 PM, frincy pulikan wrote:

Hello sir!

For the money distribution experience is a good eye opener on how kids have become like robots - mechanical.

If I would have faced such a situation I would have at the end of the class told the child her mental method answer was right but now she has to get the same using the other method. So she gets a boost to check her long division method. Think and recheck the problem which she has done using the long method. Maybe rechecking twice or thrice may make her find the correct answer and also prove to her that she may not be always dead sure her mental method is wrong.

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On Sun, Aug 23, 2015 at 8:26 PM, Shubha Sharma wrote:

Nice conversation, i guess the happy moment while working logically and mentally she arrived at right results.

Division method can be easy to learn or even by analysing later on herself, she can easily find errors(like this) on methods.