To gauge their conceptual understanding of fractions, I asked a problem to a bunch of 10th std. private school students --

**"Write a number between 2/5 and 9/14"**

Sadly (or rather not surprisingly) none of them could answer this problem correctly, in fact most of them had not even attempted it. (Why?)

I would however like to share one of the solutions --

What are your views about the way this student has solved this problem? Is it correct or not? Why?

While I was correcting their answer papers, my regular students (sixth standard municipal school ones) were around me working on their assignments. I thought that this problem would be pretty simple for them. Still, I gave it to two of them to study how they see and solve this problem.

Vaishnavi's approach: She converted 2/5 into its equivalent fraction with numerator same as that of 9/14. (Conventional maths teachers might want to read this sentence again :)

2/5 = 2x4.5 / 5x4.5 = 9 / 22.5

So now the game was easy for her.

"Now, we want to find a fraction between 9 / 22.5 and 9 / 14. So it is 9 / 14.5"

What do you feel? Is her answer correct? :-)

While Jeetu was still figuring out his way, I included him in this process and asked him if he agreed with this approach and answer. He studied the solution carefully and gave a thumbs-up.

The trouble-making teacher in me pushed her further. I don't like to see decimal points in fractions.

And she immediately replaced the fraction 9 / 14.5 with 18 / 29.

" Can you find a fraction between these two fractions now?"

She gave me a puzzled look, "Which two fractions?"

**18 / 29 exists between 2 /5 and 9 / 14. Now find a number between 2 / 5 and 18 / 29**

Same as previous one, she converted 2 / 5 into its equivalent 18 / 45.

" 18 / 30 is a number between 18 / 29 and 18 / 45."

While I was still studying her solution - wondering about the next level of challenge, she as if read my mind and --

"Now you will tell me to find a number between 2/5 and 18/30, isn't it?" :-) :-)

"Nope...I know this is easier for you now... So now, I want you to find a number which lies exactly mid-way between 2 / 5 and 18 / 29."

Now I told both of them to work independently, and I observed what both of them were doing.

Contrary to her previous approach, this time Vaishnavi was trying to make both the denominators same rather than numerators. What do you feel, why would have she changed her strategy this time?

2 / 5 = 58 / 145

18 / 29 = 90 / 145

And after doing some scribbling, she said "Sir, the required number is 74 / 145"

This is how she had written finally --

**58 / 45 , 74 / 145 , 90 / 145**

All this research was happening on the floor with chalks and a cloth (duster). Unfortunately, my phone's battery betrayed me and I could not take any of the beautiful snaps.

"How do you know your answer is correct? Did you verify using other way?"

She thought for a moment and then brought the calculator from the drawer. I was happy. (Should a calculator be allowed in the school? :-)

This gave me time to look at Jeetu's work. He too was ready with his answer by then.

"Sir, the required number is 18 / 37"

He had written these 3 numbers one after the other.

**18 / 45 , 18 / 37 , 18 / 29**

Wow! I get super-delighted when my students' answers look different....

Hey wait !! Aren't you wondering what kind of crazy maths teacher I am? Won't a maths teacher want all his students to have the same answer and probably using the same (standard) method? :-)

"Ok. But then Vaishnavi's answer is different than yours. How do I know what is correct?"

(If you observe, I did not ask - 'Who' is correct? I asked 'What' is correct? Do the words we use consciously/ unconsciously in our maths class make any difference towards building the math mindset of our students?)

He got engrossed in studying his solution. Meanwhile, Vaishnavi was doing the number crunching on calculator. After a minute, she screamed --

"Sir, my answer is correct."

She started explaining. "The calculator gave me the decimal representations of the two fractions 2/5 and 18/29. Using these two decimal values I calculated their mid-point which was also in decimal form. This was found to be equal to the decimal form of my answer (i.e. 74 / 145 )"

Are you satisfied with her approach?

"Ok.. But what if your method to find out the mid-point itself might be incorrect? After all, you have used the same method for vulgar fractions as well as for decimal fractions."

She thought for a while and then replied with a confident voice -- "No sir, my method for calculating mid point is correct. We have used it many times in our class."

"Oh.. is it? Can you explain?"

"I first found the difference between 58 and 90. Then added half of this difference to 58 to get their mid-point 74."

Did you get what she said? Is it the same way we have been 'taught' to get the mid-point? or were we taught some 'standard' method?

I asked Jeetu for his opinion on her method. He agreed and even confessed that he too has used the same approach to find the mid-point of 29 and 45 to get 37.

"I know a little easier & faster method to find the number half way between the two numbers. Can you think of that?"

After a while, Vaishnavi exclaimed --

**"Do the half of both the numbers and then add them."**

I was like Wow... She got it...!!

Jeetu wore my cap.. He questioned her... "How do you know?"

"I just guessed... and it works in couple of cases I tried..."

Jeetu looked at me for my views....

"Why are you looking at me? Why don't you verify?"

He tried few cases and found it working....

Vaishnavi started flying with her discovery.... but soon I brought her back -

"Hey,,,wait... your job is not yet done.... You also need to "prove" your method now....."

So both of them started thinking about the proof now.... When I saw that they were unable to take ahead, I intervened.

"Can you express your method in mathematical form for any two numbers a and b?"

They wrote it immediately --

(b-a)/2 + a .... assuming b > a

"Is it possible to simplify this expression?"

Vaishnavi wrote b/2 - a/2 + a but both of them were surprisingly unable to take this ahead. So she just wiped it away (and my heart sank !! :-(

Jeetu then directly wrote (b - a + 2a) / 2

While he was continuing further, I stopped him -- "What have you done?"

He explained,

**"Sir, we have to add 'a' after dividing (b-a) by 2. So if we want to take 'a' inside the bracket with (b-a) then we will have to first multiply it by 2 and then put it inside the bracket."**
Did you get this explanation? :-) Tell me honestly, what was your method?

Hmmm....Let me guess -- It was either cross multiplying or making both the denominators common, isn't it? :-)

"Ok.. go ahead..."

"we are subtracting 'a' and adding '2a' so this becomes

(b+a)/2

"Done.. we cannot simplify this further.."

"Hmm.... Actually you knew this method. But you did not apply it here."

They were puzzled .... After a while Jeetu sparkled --

"yes sir... If I have some pebbles and you have some pebbles.... and if both of us need to have same number of pebbles, then we will first put them all together and then just halve it.."

I looked at Vaishnavi for her nod --- "It was so simple.... In fact, I can even see my method existing here....." and while saying this, she simplified (a+b)/2 to a/2 + b/2 :-)

"Why is this method easier and faster than your previous method?"

Jeetu said -- "It requires just two operations rather than three operations in our former method.. So even lesser chances of error..."

And we now shifted our focus back to the original problem now ---

Which answer is correct -- 74 / 145 or 18 / 37

Vaishnavi was confident about the accuracy of her answer because she had verified it using calculator, and hence I could see her not taking the responsibility of solving this 'new' problem of which answer is correct...

However it didn't take much time and effort for me to motivate her to get engaged in this problem solving / fault-finding process...

Jeetu thought for 5 minutes and then said -- "Sir, I feel my answer is not correct... Vaishnavi's answer is correct... 2/5 is less than half and 18/29 is more than half..... Her answer i.e. 74/145 is close to half......"

Vaishnavi responded -- "Even your answer 18/37 is close to half... even that can be correct.."

I was glad to find that Vaishnavi had started analyzing Jeetu's answer and could see some sense even in her peer's work...

-----

Well, I can see that this post has already grown quite a lot and hence I would share the further story with you all in the next -- Part-2 -- of this post.... :-)

Meanwhile, I will be happy to ---

a) know your views/ opinions about the way maths is done with students in this post..

b) read your answers to the questions asked in blue in the post...

c) How would you solve this problem?

d) What would have you done if your child or students would have solved this problem differently than the way you solve or expect?

e) What do you feel what would happen in the 2nd part of this story i.e. how will we/ they decide which answer is correct ?

Waiting for your reply :-)

Rupesh,

ReplyDeleteI really loved the way you made it safe and challenging for the students to explore without the pressure of a desirable outcome. Also your step by step approach to nudge them to go deeper is truly fascinating. Thank you I learnt something important about fractions and how to fund numbers in between..

Hi,

ReplyDeleteThanks for the post. It is always fascinating to see your posts. The narration you give and how the children can find many different ways of solving a problem is enlightening.

I don't know how Vaishnavi switched from numerator equating in the first method (we just need to compare the denominator to find the answer) to making denominator the same in the second method. The first one can have many (infinite number) answers. The second on has exactly one answer. I hope they understand to solve the second question the denominators need to be same.

Thanks a lot once again.

Best Regards,

Suresh.

Ive been fortunate to have got the chance to observe some of your sessions... What strikes me most about how your class operates, Rupesh, is the complete absence of any readymade methods and techniques… It’s filled with the joy of exploration and discovery….

ReplyDeleteVaishnavi first started with making the numerators equal… My conditioned mind first dismissed it and said, ‘oh that’s wrong. You have to make the denominators equal.’ And then I thought… but why? All you need is some common factor to be able to compare…. Making the numerators equal may work just fine…. Voila.. there crashes one ‘condition’ of my mind… Thank you…

Another surprise… In order to make the numerators equal, she did not do the LCM (:)) of 2 and 9…. She multiplied both the numerator and denominator of 2/5 by 4.5…. To see a fraction with decimal points is odd to any conventional student… But when you understand, what’s odd about it??

Finding the number that lies exactly halfway between two fractions.. I just can’t guess why Vaishnavi changed her strategy… maybe I will in the middle of the night as I sometimes do…. :)

This part reminded me of the time that they had found the need during one class to find the average of a set of numbers… The word average was not even mentioned… But they said that they needed to find one number that would best represent the whole set of numbers…. And found a method for it…

Maybe this learning would have helped them find the midpoint on this day…

Coming to the use of the calculator…. It’s great to see that instead of demonizing technology, students are just being guided to use it judiciously to their advantage…

Rupesh, it’s an important life skill they are learning from you, when you say ‘Which answer is correct’ rather than ‘Who is correct’… Did you notice, that the students too have responded likewise… Jeetu said, ‘I think Vaishnavi’s answer is correct’… He didn’t say, ‘Vaishnavi is correct’.

If I were asked to find the midway between two numbers I would unthinkingly add them both up and halve them without batting an eyelid… But the two other methods they used…. Finding the difference, halving it and adding it to the smaller number… (you can also substract half the difference from the bigger number, right?) Alternatively, halving both numbers and adding them… Really amazing…

Finding out the standard method on their own and seeing how it is just a simplification of their own methods must have again been a ‘high’ producing discovery for them….

Again the connection I see is Jeetu’s observation that it is a better method because it involves 2 operations instead of 3, hence lesser prone to errors… This concept of minimizing steps was discussed when they learnt flowcharts, isn’t it? Beautifully applied here….

What always amazes me is that finding the right answer (destination) is never the focus of the class…. It’s always about the process (journey), finding out the different ways of getting there… finding the most efficient way based on many (self discovered) parameters… Sometimes adding twists and wondering how things would be if this or that constraint/change were there in the problem… So gratifying…..

The difference between a conventional method/formula based class and this way of learning through exploration/ inquiry/ discovery is like that between grabbing a sandwich on your way to office, because you just have to eat something versus enjoying a 7 course meal in the sweetest of company and atmosphere till you are fully satiated…..

Ekta Khakhar