Tuesday, June 2, 2015

"We can divide the product only with the first factor"

Hello friends, 

Our First "Maths Teachers Study Group" meet on 18th May was fantastic. I will soon share its proceedings. Thanks to all the teachers who participated in this initiative :)

Yesterday night, I saw the video of Deborah Ball where she shares some interesting insights about the competency required by the maths teachers to spot the error in their students’ work. I don’t want to spoil your excitement by narrating her video and hence would recommend you to first watch this video and then resume your reading. https://www.youtube.com/watch?v=nrwDM4ejNqs

Who is Deborah Ball?

Deborah shows three multiplication problems to the committee members and asks them if they can figure out the errors made by the three students. You may call me crazy but, something drove me next day to pose the same challenge to my class-7 students J I wrote these 3 problems as it is on the board. Believe me, it was a real pleasure and a rich learning experience to watch and hear them analyze these problems - hunt for others’ mistakes and base reason on their arguments. I strongly felt the need to video record these conversations. We will discuss about this process at length during our next ‘Maths Teachers Study Group’ scheduled on Sunday, 31st May.

But in this email, I want to share with you some equally (if not more) interesting observations that came out as a by-product of the discussions on the above problem.

The students were very happy (and even surprised) to come across one of the new ways of doing multiplication.  I leave this up to you now to figure out (and tell me) the reason for their excitement. You will have to watch the video for this. 

In fact Poonam was so thrilled by this experience that she wanted to test/ apply this method (‘new’ method - according to her) on a different problem now. This is how she did it.

She asked me to check if it’s correct. (Hold on - What do you feel about her way of doing multiplication this way?) 

Now, they have heard enough from me that it is not the teacher’s business to check if the student’s work is correct. But my experiences have taught me that the Undoing and Unlearning effect takes some time J

She got the message. “I can cross-check this answer by using division. Shall I?”

“Go ahead.”

And this is how she did.

I noticed that she has erred.

“Sir, how come the quotient is 154? Shouldn’t I get 198?”

“How do I know dear?” I see to it that my pretended innocence is not caught J

Perplexed, she thought for a while, and decides to solve the same problem using the ‘old’ multiplication method now.

She errs again and gets the product (3770) which is different than the one she had got in the former one (4950)

“Oh sir, it means I have some done some mistake in the first multiplication process. Or, is it that the ‘new’ method doesn’t work?”

“What do you feel?”

“But we had seen earlier that we get the same and correct answers using both the methods....because they are actually similar methods.”

(We had already discussed about the use of commutative property in these two methods.)

“Hmm....So what do we do now?”

She thinks for a while.... And I was so happy to hear her say, “Shall I cross-check the product that I have got using division?”

(Of course, there were other smarter ways for verifying the correctness of product, but talking about all that at this juncture had the potential danger of disturbing her present flow. Hence I choose to just let her go - her way(struggle)).

So this is how she did it.

“Oh my god! Sir, I am getting a remainder of 20 now!”

I also noticed that her process of division was not complete. Look at the quotient. (In fact, it’s one of the most common mistakes made by most of the students (why so?) However, I choose to ignore this error as of now. And I will surely bring this to her notice, but later, and with the help of this snap :-)  ...To continue to read, click on 'Read More' below...

“So what’s next?” I join her with the same curiosity.

She was completely confused by now. As her errors were compounded by now (cumulative effect :).

But I was extremely impressed by her perseverance. “Sir, I will do the multiplication again, but carefully this time.”

I observe that she had naturally switched to her ‘old’ method by now and she could also arrive at the correct answer. But at first, she was puzzled, because her previous multiplication work had produced an answer (3770) which was different than the present one (4950). However, probably the figure 4950 seemed a little familiar to her, as if she had seen it or worked with it. And probably this intuition might have encouraged her to drive her eyeballs at the product of her very first multiplication process, which was still (intentionally) kept unerased on the board.. (compare image-1 and image-5)

“Oh!!!   Sir, I have got the same answer that I had got in the very first try. Also, I notice that the answers are same using both the methods – the old one and the new one.”

“I am glad that you could notice this. So what does this mean now?”

She paused for a while. “It means, I had goofed up in the division process somewhere. I will perform the division again.”

Thankfully, she did not err this time and could arrive at the quotient that she had expected. But there are two imp. things to note:

a) Notice the beautiful strategy that she has used to calculate 198 x 5 = 990 (do spend some time thinking about this)
b) Notice the divisor she has used this time. It is 198. However, she has used 25 as the divisor everywhere till now.

When I probed her for the rationale behind replacing 198 by 25, she argued as follows (while drawing my attention to her first division work (4950 ÷ 25))   (look at image-2) --

“You see when we divided 4950 by 25, we did not get the desired quotient i.e. 198. We got 154 as the quotient.”

“Oh!”  And I was almost shocked at this observation of hers and the conclusion she has arrived at from all these trials. “So what do you want to say?”

“It means that when the problem is given as 198 x 25, we can cross check the result by dividing the product only by the first factor (198) and not by the second factor (25).”

For a while, I was just speechless. She has truly (and beautifully) seen something and made some connections (though erroneous) between the way the problem was written and the way it should be solved, just on the basis of her analysis of her failure and success. I was now thinking as to how I make her discover the actual truth.

“Well, you have made some interesting conjecture. Now, how about verifying it?”

“Sir, we have already verified it. Check the quotient of 4950 ÷ 25. We have got it as 154. It is not 198. But we have got the quotient of 4950 ÷ 198 as 25 correctly.”

So, my first attempt had failed. As a teacher, I was completely aware about the error she had made while solving her very first division problem 4950 ÷ 25. (image-2)  But I was now unable to figure out any approach that would ‘stimulate’ her to challenge (check) her work and spot the error. Probably because we come too far, some interesting observations too were made and hence there was no need perceived by her to go back to that first division problem. But with some thought, I could strategize this way –

“You have got 4950 as the answer using two multiplication methods. But one method has also given you 3770.”  (image-3)

“Sir, that must be incorrect then, isn’t it? I must have done some mistake over there.”

“Well then, you haven’t spotted the error in your solution yet. Some time back, the same 3770 was correct for you, isn’t it?”

She quickly picked up the marker and figured out the error. And while she was just about to settle down, I stopped her.

“Hey wait! And what about this division work then?” I draw her attention to the place where I wanted to. (4950÷25)

Probably she might think for a strong counter-argument this time. “Sir, my division work is correct over there. We did not obtain the desired quotient because we had divided using the wrong divisor (25)”

“Yes, I remember you had told this to me. But is it possible for us to give it a one more look before just signing it off?”

And by the time I was done with my appeal, she was already on the job (checking her work)

Silence! And now ---- I was just anxiously waiting for a scream. Will she able to spot her error? If yes, then how is she going to react to this? Will she be able to relate the new ‘actual’ truth (we can divide with any factor) to the old ‘her’ truth (we can divide with only the first factor)? After all, this realization had full potential to challenge the strong opinions that she had confidently formed over her repeated failures and success. It was the moment when the Truth was going to Destroy her Discovery! J J

And yes! She could spot the error. She could relate. And she could even realize.

But you know, how did she react?

Well, not with the scream, but with a Staring look!

“Sir, now I know the reason why you insisted so much, to re-look at the division work.”

“Why?” And this time I couldn’t resist my smile J

“It is because you wanted to prove me wrong! I have realized that we can divide with ‘any’ factor.”    :-)

1 comment:

  1. On Fri, May 22, 2015 at 9:13 AM, Meera mvmeera@yahoo.com [ActiveMaths] wrote:

    Really interesting interaction with Poonam. Working out just one problem with her clears so much for her and tells the teacher so much about her understanding/ misconceptions. More importantly it is wonderful that you let her work and think this problem through.
    Her working out 198 * 25 the first time, tells that she has intuitively got the idea of number - by first multiplying 25 with 8, then 25 with 90 and then 25 with 100. But then, on checking by long division method and getting the wrong answer she reverted back to procedure and persevered till she got both her division and multiplication answers to match and in the process cleared her misconception that the answer that you get on division depends on which factor you take first. Kudos for this.
    Now, you could go on to asking what was different in her approach in her first method of multiplication and second method and why intermediate steps were different but the product was the same. This would bring out the clarity that she took 25 as a whole and multiplied this with 198 and in the second she split 25 to 20 and 5 and multiplied this with 198 .
    Then, what exactly was happening when she divided by 25 etc. this is so important for students. Teachers need to bring in 'place value' along with the four operations .
    You could look up Jo Boaler's 'you cubed' and she calls this 'number talk' .
    Thanks for taking the time to write about this interesting 'number talk'.
    Meera Raghavan.

    Sent from my iPad