I got a strange request few weeks back...

Two 10th class students were brought to me.. Their parents wanted me to "clear their concepts" of maths that they had learned (I think, they actually meant rote-learned) in their lower classes.... Why? because both of them were now entering into the most important (!) year... Yes, you guessed it right -- Tenth standard !!

My maths class is generally full of debates, arguments, confusions, conjectures, etc. And hence

**More is the Merrier in our kitchen**. So working with only two students, and that too with a time-bound and syllabus-bound objective was no longer my interest. However for some reasons, I accepted this odd assignment, perhaps because I saw the yearning for understanding more than the greed for score in their requirement....
I must confess, its a different experience working this way after a long time. After my couple of sessions, I thought I would write and share the lessons, trajectories, experiences and reflections of each of these sessions. But sadly, this could not happen..

In fact, I soon realized that I have gradually recoiled back to my natural style - Inquiry leading to Discovery..

I did their diagnosis in the beginning so as to gauge their conceptual clarity. No wonder, it was a dismal performance (their mistake??)

It was our 8th session now and I wanted to unfold the tons of rules they had (rote-)learned in fraction arithmetic...

I told them to show me

I then asked for

So far so good.

I pray for the moment to see the students answer this question correctly. But my prayer is hardly ever heard. And this time was no different. After thinking for almost a minute, one of them reluctantly pointed out the remaining portion of the bar above as 1/3. When I probed her for the explanation, she said --

While pointing at half, she described it as one-upon-two. The last small piece is one-upon-four. And the portion 'half plus quarter is one-third'.

I noted that she used the terms 'one fourth', 'quarter' and 'one-upon-four' interchangeably.

"But what makes you feel that this portion is 1/3?", I ask her while pointing at her 1/3.

"We have divided the whole into 3 parts..and 1/3 should be bigger than 1/2.. So that's why."

I asked her peer for confirmation. She simply conformed.

I was amused and even amazed by the fact that they wanted 1/3 to be bigger than 1/2. But at the same time, they did not see that their 1/4 was smaller than 1/3 :-)

I could have drawn her attention to this contradiction, or may be I could have even asked her the meaning of one-third and one-fourth, but rather I dragged her into another zone.

I asked her to show me three-fourths. And she showed this...

In fact, I soon realized that I have gradually recoiled back to my natural style - Inquiry leading to Discovery..

I did their diagnosis in the beginning so as to gauge their conceptual clarity. No wonder, it was a dismal performance (their mistake??)

It was our 8th session now and I wanted to unfold the tons of rules they had (rote-)learned in fraction arithmetic...

I told them to show me

**half**with a diagram. Surprisingly both of them drew a horizontal bar rather than circles to show half (generally I get only pizzas from the private school students :)I then asked for

**one-fourth**which was shown this way.**Then I asked them to show me my favorite fraction: One-third**I pray for the moment to see the students answer this question correctly. But my prayer is hardly ever heard. And this time was no different. After thinking for almost a minute, one of them reluctantly pointed out the remaining portion of the bar above as 1/3. When I probed her for the explanation, she said --

fig - 1 |

I noted that she used the terms 'one fourth', 'quarter' and 'one-upon-four' interchangeably.

"But what makes you feel that this portion is 1/3?", I ask her while pointing at her 1/3.

"We have divided the whole into 3 parts..and 1/3 should be bigger than 1/2.. So that's why."

I asked her peer for confirmation. She simply conformed.

I was amused and even amazed by the fact that they wanted 1/3 to be bigger than 1/2. But at the same time, they did not see that their 1/4 was smaller than 1/3 :-)

I could have drawn her attention to this contradiction, or may be I could have even asked her the meaning of one-third and one-fourth, but rather I dragged her into another zone.

I asked her to show me three-fourths. And she showed this...

fig - 2 |

When asked for the explanation,

"I have made 4 parts and taken 3 out of it... So 3 upon 4"

I thought the bulb might glow now. But it did not. So I took her deeper. I drew this on the slate --

fig - 3 |

"Okay... So going by your definition of 3 upon 4, what I have shown now is 2 upon 3, right?"

She stared at it mum for almost half a minute... I could see her eyes even roll down to her first figure at times...

"What happened?"

"No sir... this is not 2 upon 3."

"Why? I have made 3 parts... and then took 2 out of 3... So shouldn't it be called as 2 upon 3? That's how you argued for 3 upon 4.."

"No... because it is 1 upon 3... It is the same figure that we saw at first.."

"Oh !! Is it? But then I just followed the way you had defined 3 upon 4.." and while saying this I smartly drew her attention to 3/4 now :-)

I knew this would baffle her completely....and yes, she did !

I made it more spicy !!

**"Oh.. What is this?? Three identical portions having Three different names...? I am more confused now... What's correct? Is it 'one third' or 'three fourths' or 'two thirds" ?"**

Pin drop silence for almost 2-3 minutes... She was just staring at all the three diagrams... I was about to intervene, but just then ...

"Sir.. the last one is not 2/3 ... It is same as 3 upon 4... "

"Really?"

"yes... "

"But then.. your definition......of taking 3 out of 4 parts......?"

"We have to make 4 EQUAL parts first.... and then take 3 parts....."

Yes !! I was so happy.... One milestone achieved :)

"Okay.... Would you like to rephrase the meaning of "3 upon 4" again?"

"Yes... Make 4 equal parts... and taking 3 out of those..."

This time, I could see her emphasizing on the word 'equal', along with a shy smile :-)

But still our problem is not yet solved.....

"If both the portions in the last two diagrams show '3 upon 4', then what does the portion in the first figure show?"

She was quick this time... "Sir, that's also 3 upon 4.."

"Sure?"

"yes"

"Okay... then show me one-third now..."

I thought she would be able to pick up now... But no.... After few seconds, I asked her -

"what does 4 represent in one-fourth?", while pointing at the quarter in the diagram.

"It means we have made 4 equal parts."

"So then..... for one-third......?"

I paused....waiting for her to take it ahead......

".... three equal parts....?"

She uttered this reluctantly, looking at me for confirmation....

I raised my thumb and she instantly heaved a sigh of relief :)

"Wait, wait dear... Draw and show me one-third now..."

fig - 4 |

Just to implant this idea deeper in her memory, I asked her to show few more unit fractions like, 1/6 , 1/8 etc... and she showed each of these correctly....

I noted that she had also showed and reasoned for '3 upon 4' some time back...

So now I asked her to show me 3/8....

fig 5 |

"Can you explain?"

"I made 8 equal parts.... and then took 3 out of it..."

Did you notice the presence of word 'equal' in her explanation now? :-)

**So are you satisfied with her responses and explanations now?**

Did you say 'Yes'?

Hmmm... But I am not yet..... And so, I asked her another question....

**Can you guess what could be the next question ??**

Waiting for your 'question'... I mean, response......

You need to be quick....as I have already started working on the Part-2 of this post this time :)

Regards

Rupesh Gesota

Rupesh this is the basic problem many students have. Even I found the same. Rather using rectangles and circles to devide i use roti to be devided equally..... eg. 2/3 two rotis shared equally between three kids then what will be share of each kid?

ReplyDeleteThis help them estimate the size of the fraction.

The way you probe is really nice, it helps atudents to reflect on their own thought ..... metacognition is most imp part of learning.

Rupesh, you are really doing a great job! Fractions are indeed tough for students. Rotis, paper cutting activities may help.

ReplyDeleteVery interesting Rupesh.

ReplyDeleteVery interesting observation.

ReplyDeleteThe paper folding activity and the chapati activity are surely worthwhile.

ReplyDeleteWe at Fountainhead school are surely following what you taught us at your workshop.. " Let the students construct the meaning."

I personally feel that rotis or circles are even more difficult to divide equally and therefore children opt for rectangles and that is quite alright. I think that the way you make them talk about their solutions is great. That is the need of the hour. To engage with a problem and be able to reason

ReplyDeleteI work with students with these kinds of misconceptions and this will probably need to be re-built many times before it really sticks. Thirds really are trickier for some reason!

ReplyDeleteFrom couple of Parents ---

ReplyDelete1) Fantastic...it was so simple to understand...fraction for me was always googly...as I kept reading ahead...clarity was more transparent... actually learning with squares and rectangle is much more easier..good learnings.

.it matters how it's projected and made understood...perfect thumbrules..thanx

This is nice.

ReplyDeleteA fraction has different connotations i.e. (1)3/5 means take one whole, divide it into 5 equal parts and take 3 parts out of that AND (2)Take 3 wholes, divide them into 5 parts and take 1 part.

But both unify to give the same outcome. Is it possible that 3/5 was created as a notation first and later it was realized that it unifies with division?

I find it unlikely that both were discovered at the same time.