__Objective:__Assessment

__Why?__-- Not for marks, but to figure out the student's level of understanding and thus to help me plan my lesson.

__Student's details:__Class-6 student (Her maths teacher told me (on her own) that she is among the brilliant ones in the class)

Teacher: What's bigger?

**1/8 or 1/4**(i.e. one-eighth or one-fourth)

Student: 1/4

Teacher (very happy, but emotions concealed) : Why do you feel so?

Student: Because the fraction with smaller denominator is always bigger.

Teacher: (Highly disappointed, but continues without any such communication) Ok. How about this pair now? Which is the bigger of two ? 7/21 or 25/100

Student: (thinking.......for long....)

Teacher (after about half a minute): What happened?

Student: (no answer)

Teacher: (after another 10-15 seconds): Tell me what you are thinking. May be I can help.

Student: Denominator of 7/21 is smaller than that of 25/100..... (Silence)

Teacher: Ok, go ahead....

Student (in a confused tone): But then...... Numerator of 25/100 is bigger than that of 7/21.. So I am confused which of the two fractions is bigger.

Teacher: Okay... But how come you did not face this problem in the previous problem.

Student: I did not look at their numerators then......... But anyways, they were both '1' there.

Teacher (after waiting for another minute): Okay.. How about this comparison now - Which of the two is bigger? 3.5 or 3.28

Student (after about 5-7 sec): 3.5

Teacher: (Excited again!) And how do you know this?

Student: I converted both the decimal numbers in fraction form.

Teacher: (excitement grows!) Plz explain how

Student: 3.5 = 3/5 and 3.28 = 3/28

Now we know the rule that smaller denominator means bigger fraction.

So 3/5 (i.e. 3.5) > 3/28 (i.e. 3.28)

Teacher: (almost wanting to bang his own head, but again - emotions concealed)

Okay.. Let's take the last problem.

Arrange the following decimal numbers in descending order: 3.05 , 3.5 , 3.50

Student (after about a minute): 3.5 is biggest of the three.

Teacher: Can you plz explain how?

Student: Again, I represented these decimal numbers in fraction form.

(this is what she wrote and showed me (pen-work)

So according to her,

**3.5 > 3.50 > 3.05**

**Teacher: Ok, thank you :)**

There is a lot that can be said, discussed and learned from this experience...I learned few things.... But this time, I chose to stay mum and would like to hear from you, if there is anything that you could learn from this...

Please note that I could have helped her discover the 'correct' answers by asking right questions.... However, I chose to not do so in this exercise.... But how would you deal with such a situation, if you happen to teach her?

How about asking these questions to your entire class? -- Not to the class who is being currently taught fractions and decimals..... But to the class who has already been taught these concepts in their previous years.... Worth trying (daring) ? :-)

Thanks and Regards

Rupesh Gesota

I really like your approach! I share the same passion... Hope we get to meet soon... God bless you!

ReplyDeleteThis is the kind of misunderstanding that blindsides me in the College Algebra course I teach at my local community college. When they arise in class, though, I am so pressed for time that I usually am stuck just saying "that's not how that works" and asking the student(s) to come to my office hours to discuss it. I *hate*hate*hate* rules-based math teaching and this is why. Kids will generalize rules incorrectly and create their own rules if they're only given limited exposure (or only do some of their homework), then they move on thinking that they've "got it!" Back in the foundation of this child's education (or homelife even), she/he never used or comprehended the relationships between fractions and decimals (and probably percentages and ratios). I would not have been able to walk away leaving that student happy in her misunderstanding if it was a tutoring situation, but I could envision that the corrective conversation would be long and potentially disheartening to both student and teacher.

ReplyDeleteHi-

ReplyDeleteGreat post - wait time and allowing the student to struggle with their own thoughts is key- also no leading questions- a few follow up ?s:

- what did you do next?

-did other students comment on the problem? Curious if students led students to accurate concept

Thanks again

-Josh

Hi I am a maths student too , I am currently in third year and I guess that what we are taught , even if u ask a college going person this , he/ she have same answer , infact me

ReplyDeleteIt would be asking if you could tell the answer why ��

Hi, I am a maths student too ,and in my final yr of undergrad

ReplyDeleteI would like to say these are the most common answers that even a university kid will endup saying ,may be because thats what we were taught

But please if you have any good explanation , do let us know we would be glad to know more

Really, eye opener, that how we were tought everything through formulazing and generalizing rules rather than concept explanation. Hopefully with your efforts kids will learn how to learn conceptually.

ReplyDeleteHi, I would ask the child to first 1) use diagrammatic representation of fractions by dividing a rectangular bar for her to understand and see how big each fraction is. That is how i taught my 4th grader and even the school taught him the same way

ReplyDeleteFor example for 1/4, the rectangular bar is divided into 4. I will ask her to shade the number of parts of fraction (1 part) from total 4 parts. Similarly a same size bar should be drawn and divided into 8 and 1 part from total 8 parts should be shaded. So she can see the difference in size of the fractions and say the answer instead of blindly forming a rule without understanding what fraction means. The students should be able to first divide all types of geometrical shapes such as square, rectangle, triangle, circle to understand the line of symmetry and all the possible ways to divide the shapes.

2) Next step is for her to understand the equivalent fractions. Use the same rectangular bar and divide the bar into 4. and now shade 1/4th i.e one part from total 4 parts. Divide the same rectangular bar into 8 parts which can be done by dividing all of the four parts into 2 so there are 8 parts. When we see the number of parts that are shaded. it will be 2. So the earlier fraction is 1/4 and after dividing the same bar into 8 total parts, the fraction is 2/8. The student can try dividing the rectangular bars into different numbers and compare the number of parts that are shaded. Once she understands what equivalent fraction is, she will understand that 7/21 is 1/3 and 25/100 is 1/4.

3) For teaching decimals, i would use 10 X 10 square grid to represent ones, each 10 by 10 square grid will be one whole. Divide each 10 by 10 grid into 100 equal smaller squares (10 rows and 10 columns). each smaller square would represent 0.01, each column with 10 rows would represent 0.1. The student can be asked to shade tenths, by shading each column and see how much they add upto. each column or row would add up to 0.1 (0.01 * 10 times). now student and draw and see what is 3.5 and 3.50. 3.5 is equal to 3 wholes and 5 tenths i.e 5 columns. 3.50 would be 3 wholes and 50 hundredths. 50 hundredth is also same as 5 tenths. so 3.5 = 3.50. Another method to teach decimal is to use number line ( a line that is divided into equal parts similar to a ruler/ scale) . draw a number line dividing it into 10 equal parts. each part would be marked as 0.1, 0.2, 0.3 and so on till 1 whole. Draw another number line exactly below the earlier one and divide the same length line into 100 parts. so each part would be marked as 0.01, 0.02, 0.03 and so on. after 10 marking it would be 0.1. so 0.1 on both lines would align. now student can be asked to mark 0.5 on the first line and 0.50 on the second line. it would be on the same position. student would understand that 0.5 = 0.50.

Yes! As a child I too had difficulty in understanding operations on decimals and fractions! First of all I wud make her understand how faction and decimals are related. And more practical will be the obvious division of rectangle into smaller squares.Also a measuring scale wud come to help for making them understand decimals like 1.2cm. Under things which the children tend to get confused is placing of zeroes!

ReplyDeleteI read a rather dry but insightful book called the Psychology of Learning Mathematics - it had quite a lot to say on the subject of mental models of arithmetic and how it's easy for a student to form an incorrect mental model and for a (poor) teacher to fail to test for whether a student's mental model is correct or not.

ReplyDeleteThe above story clearly says that the student has been taught in a wrong manner, just by following certain set of rules without knowing the reasons behind those rules. The student above doesn't seem to know what is fraction. He / She has to understand the basic meaning for fraction itself and then the various forms of writing fractions.

ReplyDeleteWhen the teacher intends to teach with a purpose of just 'solving' a mathematical exercise then she is likely to teach in a short cut way of following certain rules. But if she teaches with the intention of understanding the concept, she can ask 'correct' questions and guide much better.

On Wed, Aug 10, 2016 at 11:30 AM, caroljudith@yahoo.com [alt-ed-india] wrote:

ReplyDeleteHi, I would ask the child to first 1) use diagrammatic representation of fractions by dividing a rectangular bar for her to understand and see how big each fraction is. That is how i taught my 4th grader and even the school taught him the same way

For example for 1/4, the rectangular bar is divided into 4. I will ask her to shade the number of parts of fraction (1 part) from total 4 parts. Similarly a same size bar should be drawn and divided into 8 and 1 part from total 8 parts should be shaded. So she can see the difference in size of the fractions and say the answer instead of blindly forming a rule without understanding what fraction means. The students should be able to first divide all types of geometrical shapes such as square, rectangle, triangle, circle to understand the line of symmetry and all the possible ways to divide the shapes.

2) Next step is for her to understand the equivalent fractions. Use the same rectangular bar and divide the bar into 4. and now shade 1/4th i.e one part from total 4 parts. Divide the same rectangular bar into 8 parts which can be done by dividing all of the four parts into 2 so there are 8 parts. When we see the number of parts that are shaded. it will be 2. So the earlier fraction is 1/4 and after dividing the same bar into 8 total parts, the fraction is 2/8. The student can try dividing the rectangular bars into different numbers and compare the number of parts that are shaded. Once she understands what equivalent fraction is, she will understand that 7/21 is 1/3 and 25/100 is 1/4.

3) For teaching decimals, i would use 10 X 10 square grid to represent ones, each 10 by 10 square grid will be one whole. Divide each 10 by 10 grid into 100 equal smaller squares (10 rows and 10 columns). each smaller square would represent 0.01, each column with 10 rows would represent 0.1. The student can be asked to shade tenths, by shading each column and see how much they add upto. each column or row would add up to 0.1 (0.01 * 10 times). now student and draw and see what is 3.5 and 3.50. 3.5 is equal to 3 wholes and 5 tenths i.e 5 columns. 3.50 would be 3 wholes and 50 hundredths. 50 hundredth is also same as 5 tenths. so 3.5 = 3.50. Another method to teach decimal is to use number line ( a line that is divided into equal parts similar to a ruler/ scale) . draw a number line dividing it into 10 equal parts. each part would be marked as 0.1, 0.2, 0.3 and so on till 1 whole. Draw another number line exactly below the earlier one and divide the same length line into 100 parts. so each part would be marked as 0.01, 0.02, 0.03 and so on. after 10 marking it would be 0.1. so 0.1 on both lines would align. now student can be asked to mark 0.5 on the first line and 0.50 on the second line. it would be on the same position. student would understand that 0.5 = 0.50.

------------- On Wed, Aug 10, 2016 at 9:16 AM, Sachin Jadhav wrote: -----------------

ReplyDeleteInteresting Rupesh

I am travelling and wont be able to access blog....but here is what I am thinking.

I am not a teacher so wont be able to teach although but surely this will hepl will able to handle it when my kids would reach that stage.

I will look into answer of these solving them like follows

I may be wrong here although..... :)

1/8 is equal to 1 x 8 times =8 parts

1/ 4 is equal to 1 x 4 times =4 parts

------------

so 4 having large share of pie it will be bigger

7/21 will be 3

25/100 will be 4

so here 7/21 in a pie will have bigger share

----------

3.5 is certainly bigger than 3.28

well why ??? hummmmm

of the pie it will have bigger share, simple.....right na ?

also when decimal moved two places from both it will be 350 and 328 hence 350 is big.

-------

3.5,3.50 are same as here after decimal 0 does not have any value....

or does ir Rupesh.... that will be shocker if it does

and 3.28 is cerainly small than that as said above.

---------

keep me in the loop Rupesh to solve this.

Warmly,

Sachin

--------------------------------

----------- On Wed, Aug 10, 2016 at 8:42 AM, Tara Kini wrote: ---------------

Dear Rupesh,

Amazing!

So many misconceptions in physics understanding come to mind when I read this. How differently can a learner perceive a concept from what you think you have communicated!

This is also so true for any form of communication. In fact, for perception itself. Is not the entire world for each one of us, a perception that is entirely unique to each of us? So is not the world what is in our heads and not out there?

Tara

Dear Rupesh,

ReplyDeleteAs usual you have put up a wonderful situation. One which each one of us as teachers may have encountered and ignored. The mistakes that have been made are wonderful talk points and should be used to spearhead a discussion to get to a solution that is accepted by each and every child. The ones who get the answer should be asked to convince the rest of the class, which could make use of diagrams too as suggested by someone. Teachers should absolutely desist telling the right answers.

Anjali