What is 76 - 20 --------- 56, 55 or 54 ?

Two days back I was assessing one of the 6th grade students......

(Purpose? -- same as before -- to know her level of conceptual understanding).... So my role at that time was more of an examiner/ assessor than a maths teacher/ guide........but as you will find ahead, the teacher in me couldn't curb his urge to help and hence, I have eventually ended up teaching her, but fortunately, in a way that she could identify her mistakes on her own and then even correct those.. (I did not tell her that she has erred).... 

Also, I ensure that when I do such assessments, me and the student are no longer strangers to each other..... Such assessments are done only after enough contacts.... i.e. after I have conducted few 'Fun with maths' sessions with them and when I am confident that the student jells well with me and trusts me.... Also the purpose of assessment is clearly communicated to the students and hence they too are always eager to go through this diagnostics process honestly and fearlessly because they are now confident that this process is to help their favorite teacher to teach them better....

We had reached the section of 'Mental Math'

Q.1) "What is 50-20?"

"30" (in a flash)

Q.2) "What is 20-7"

No wonder, this time there was a pause. But the sadder part being, I found her using her fingers for this problem. Her lips and tongue moved, but there was no sound... After about 10-12 seconds, she said ---

"It is 11"

"Hmmm... And how did you get this?"

"I counted back from 20"

"Ok.. Is it possible to verify your answer using other method?"

Silent stare....

"Okay... Tell me what is 20-10" ....While I asked this, I also wrote this expression on the table.

​"10"  (and I wrote her answer too)

While writing further, I ask her what will be 20 - 9 then?

She continued to look at the two similar expressions for a while and then looked up with confidence -- "It is 11"

"Hmmm.... How did you do this --- without fingers this time?"

"Sir, 20 - 10 = 10, then if we remove just 9 from 20, then we will have one more left as compared to the previous case i.e. 10+1 = 11"

"Interesting. You have used the logic this time.. So tell me then.....20-9 = 11  and 20-7 will also be equal to 11 ?"

Her smile revealed her embarrassment. It seems she had realized that she had goofed-up in the beginning 20-7.

"Tell me... I am waiting."

"Sir, it is 13."

"How come?"

"20-9 = 11...... So 20-8 = 12...... So 20-7 = 13"

I paused for a while before proceeding further -- to witness her reaction. And no wonder, she was trying to hide her smile.

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Q.3)  "So tell me now, what is 48-10"

Silence! And this time, I even saw her using pen-n-paper for this problem....

Surprised, I peeped into her note-book. This is how it looked.....

​Soon, my Worry was dispelled by the Joy -- this time she was using some logic/ strategy (change in approach.)  But then, a thought pinched me -- I wondered if she will now use this strategy always.....I mean.... even when in cases it is not needed.....  I thought that she being a middle-schooler, must be already fluent with the facts of removing 10.... My Joy was over-ridden back by a Worry -- Has the newly learned strategy influenced with her (mastered) fluency of easy facts??  While I was still lost in this conflict, I saw her write down 41, 42 below 40...... I could see the blunder.... 

"Sir, it is 42....."

I was speechless by now, even cursing myself for a while.... However, I also found myself re-affirming the faith in my belief... I was quite certain, that even if the strategy has disturbed (is disturbing) her fluency/ accuracy, this confusion is transient and can be resolved.... I was in no mood to give up the Logical approach of problem solving.... So I start my investigation...

"I am very happy to see you using Logic to solve this problem..... But can you plz explain me how have you worked out?"

"Sir, I first subtracted 8 from 48..... This gives 40.... But, we need to remove 10 i.e. two more..... So I counted up by two....... 41...42.....similar to what we did in the previous subtraction."

Hold on guys..... Do you realize what has happened??  The strategy used in the previous subtraction problem (20-10 being used to figure out 20-7) required adding to the partial difference (10+3)..... So she has simply done the same (adding) in this case too...without realizing/ understanding the meaning of the new expressions while applying this strategy....  Here, if she wants to use the fact of 48-8 to arrive at the answer of 48-10, then she needs to 'subtract' (and not add) from the partial difference (40-2).... 

{ Of course, one need not / does not need to remember when to add or subtract... if the student is taught in a culture where understanding the meaning of the expression is first and more important than solving the problem, then he/ she 'thinks' while choosing and applying the strategy, and the second step -- addition or subtraction -- happens effortlessly and even fluently... I have personally noticed that students enjoy and love to use strategies and logic to do arithmetic rather than being drilled (and killed) by the carry/ borrow algorithms... (I leave it up to you to figure out the pros/ cons of two approaches).....  }

Coming back to this student's case, when I probed her to think about the meanings of these expressions, she could successfully and easily trace her mistake, and could even correct it.... She realized and even articulated well as to why she needs to remove two more in this case and not add two..... It was a PLEASURE nourishing her different capabilities and even see those blossom !!

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(If I were taking their class, I would have made her master these facts using  patterns like :  48-10=38,   78-10= 68,    56 - 10 = 46.... etc. However, I just made a note of this point now and continued with the current task at hand ---- Assessment)

Q.4)   How much is 76-20

She again resorted to pen-n-paper, while I observed her patiently....This is how it looked after few seconds.

​Yes, as you must have observed that this time, she has not erred.... She is on the right track...... But soon, I saw her leaving this work incomplete and switching to some other approach.... This is what she did...

​"Sir, it is 56."

"Hmmm... I see you have done something interesting. Can you please explain me?"

"Sir, I kept the 6 of 76 aside..Then I removed 20 from 70 which is 50.. then I added that 6 back to get 56."

"Okay...You have devised a new strategy.... But I saw you solving this problem in a different way at first? " I drew her attention to her first approach (incomplete work).

"Sir, I could not work that way... It was difficult."

"Oh.. is it? Can you tell me what's the difficulty you faced?"

"I first subtracted 6 from 76 to get 70.... then I subtracted 4 more..... so that I removed 10 in all from 76......I got 66..... Now I needed to remove 10 more....  That's again too much of work... I felt that I will do some mistake there....so I thought of another strategy...."

"Ohkk.... But is it possible for us to complete that approach too, so that we can verify if our answer '56' that we got using second approach, is correct or not?"

She thought for a while...."Sir, 56 is correct!"

"Oh dear, I am not saying that 56 is incorrect.... But wouldn't it be great if we get the same answer using TWO approaches?"

I could finally persuade her :-)  I galloped her from 76 till 66..... And then I threw the ball back in her court... "What next?"

After few seconds -- 

"We need to remove 10 more from 66.....and we can do this in steps of 5....... 66 - 5 =.........60..........And 60 - 5 =........... 55........."

And this did shock her.... she was now wondering because this 55 did not match with her former 56... I don't know how she had arrived at these answers... But she had not used fingers for sure....

"Kyaa huaaa?"

Silence...

"Sir, 56 is correct."

"Ok.... But yahaan pe 55 kaise aayaa?"

"I must have done some mistake there..."

"Lets find out then...."

After some murmur, she said, "Sir, I have done mistake in subtraction..."

And this delighted me, but with no clue that this happiness would be --- Short-lived !

"66 - 5 = ......... 59 and 59 - 5 =......... 56"

Uffff  !! A REAL Test of my patience !!  By now I was so anxious to 'directly' point out her mistakes...... I also felt like scolding her for the 'silly' subtraction errors she makes in Grade-6.. .....  But, I didn't react.... I responded.... I asked her, "Can you just check these subtractions on paper?"

I don't know why, but she worked out only the 2nd part on paper i.e. 59 - 5 

​A closer look at the image will reveal some overwriting at 4.... Yes, she had erred here as well in the beginning.... her first answer was 55 and not 54.....  So you might then wonder -- why did she err, how did she realize that she has erred and then, finally how did she arrive at the correct answer i.e. 54

Why had she erred?
I was alert about the method of her computation for 59 - 5.... She had used fingers to arrive at 55.... She began with all 5 fingers closed.... A finger was opened out for every number that she uttered.... . 59 (one finger out)..... 58 (2nd finger out)....... 57.... 56........55.... Realizing that her job was done i.e. 5 was removed (5 open fingers), she now simply placed the last number that was uttered out as the solution of the problem....  i.e. 59 - 5 = 55

How did I help her find out her mistake? 
I asked her -- what's 20 - 5?  And to this, she promptly replied --  5.

"Ok....Can you do this the way you did 59 - 5 now?"

She was surprised at this strange request/ command,.... however, luckily she obeyed..... the video as above got played again... one finger out for every number spoken.....

20.....19.....18....17.....16.....   

And now, I seized this opportunity instantly.... "So 20 - 5 is 16, right?"

"No..no... it is 15."

After thinking for a while, she understood that she had goofed up....

She explained -- " The answer is not the last number we speak, but the number before the last number...."

I pondered at this statement/ rule for a while.... I don't know if someone has taught this way to her....or whether she has figured out this pattern/strategy to arrive at the correct answer... I could have probed her for this,,,,, But I did not... (we were already deviated a lot from the original question).... But I definitely made a note of this point in my pad so that I can raise this point during my session.....

"So then 59 - 5 is .......? "

Without saying a word, she instantly overwrote the 5 with 4...thus correcting 55 to 54......It was now imp. to re-draw her attention back to the original problem.....

"So now, we have three different answers to the problem 76 - 20 -------- 54, 55 and 56"

No doubt, she was in a sheer confusion now... not being aware that she was nested in her own loop of errors....I was wondering as to how will she come out of this....when after a while, there came a confident tone --

"Sir, 56 is correct."

"Hmm...ok... May be its correct,... but then how come we got different answers using other approaches... Weren't those approaches correct?"

Silence...(she was thinking....)

I was unable to understand as to why this student who had some time back had worked out 76 - 20 and even 48 - 10 using beautiful strategies was unable to apply any of those in this problem.... What was stopping her from using any of these strategies to work out 66 - 10? Why was she struggling in this second part of the problem? I wanted her to leave with confidence and not being puzzled/ confused... So I had to intervene now... 

I could have asked her to re-check her previous subtraction problems -- her errors ( 66 - 5 = 60 and then 66 - 5 = 59), but I don't know why I didn't do this.... I rather guided her this way -- while pointing at her first incomplete approach --- "You see, you have worked out 76 - 10 = 66 correctly over here.... We are stuck up now at 66 - 10... You tried removing 10 in steps of 5.... Can we do 66 - 10 in any other way? I mean, you have Rs.66 and you have to give me Rs.10 right?"

This luckily clicked with her....and this is how she responded....

"Sir, I will keep one rupee of 66 aside... So I now have 65..... I give you Rs.5... so I am left with 60.... I now remove only 4 from this 60 to be left with 56... We got 56! "

What??  What did she do?? I was completely puzzled.... unable to fathom her reasoning.... It was only after she explained this to me again, I could construct my feedback for her approach....

"Hey...wait !! I understood that you have kept one aside from 66... Also 65 - 5 = 60... But how come you have removed only 4 from 60,..you still need to remove 5 right? And what about that Re.1 that you had kept aside? "

"Sir............, that one rupee is added in that four rupees that I gave you...."

What?? I could still not understand this...

"Plz explain this to me slowly... I am really unable to understand something... How can you add that one in the four?"

{ And believe me, she went all the way to really make me understand.... Of course, I too was not joking/ fooling her... Perhaps, she could sense that her teacher was genuinely trying to listen to (help) her.....And probably, that's why she even continued to be with him, and persist with this one problem for last 30 minutes!  }

And this time, while she was explaining/ reasoning, I tried to record her mental steps on the desk, so that we can see and track the path... This is how it looked...

​Can you see how beautifully she has removed 10 from 66?  :-)

I appreciated her effort but was also quite overwhelmed by this (complicated) approach..... And hence, I couldn't resist myself and wanted her to figure out the easier approach...which was not only devised but even used by her some time back...

"Just check, how you have removed 20 from 76 in your one of your approaches..." She started studying her previous work...

​

I broke the silence after 10-15 seconds...  "What have you done here?"

When she was done with her explanation, I asked her if she could do something similar for 66 - 10...

And this as if electrified her.... 

"Yes...Yes..... We keep the 6 of 66 aside... remove 10 from 60 to give 50 and then add the 6 back to get 56... Sir, this was so easy.... !!"

And while she was saying this, I again recorded this mental sequence on the desk, thus allowing her to monitor her thought process and eve relieve her of the cognitive load (eliminate chances of error due to memory)....

​I could see the Joy in her eyes, face, smile and even in her body movements !! Perhaps, she might also be amused by the thought of mistakes, struggle,, etc...that she went through during the course of other approaches ...and how they could have been easily avoided in this strategy.....

Perhaps, she might realize this after few years, that this Struggle was Planned.... by her Teacher ! The teacher knows that if 'you steal the struggle from the child's life, then you are stealing away her learning..."  

The teacher wanted her to persevere, analyze, reason, justify, communicate, strategize.... The teacher wanted her to do Real Math ! ...... to experience and appreciate the beauty and learning hidden in the process of problem solving, she needs to realize that the process is more crucial than the mere answer of 76 - 20 :)

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Thanks & Regards

Rupesh Gesota

www.about.me/rupesh.gesota

www.rupeshgesota.blogspot.com

"The true value of a teacher is determined not by what he knows, not by his ability to impart what he knows, but by his ability to stimulate in others a desire to know."