Wednesday, March 29, 2017

Modeling the Model Diagram

I remember when I was learning to be a teacher, the math professor gave the class of trainee teachers a Primary 4/5 question. She wanted us to solve the question with model drawing. There were about twenty of us, with different backgrounds and age groups. The time given was 15 min. None of us managed to solve it. NONE. Mind you, all of us were university graduates. 

It's been almost 10 years since then and since then, I always believed that it does help kids with their Math. Unfortunately, most people I speak to, don't agree with me. There was a mother whom I met at a school's event recently who actually dissuaded her girl from drawing the model as she thought it was a waste of time. 

Audrey is P3 this year, and it's also this year that her teacher is requiring the class to do their models in the Math sums. I naively thought P3 Math means simple math questions = simple intro to model diagrams = a piece of cake. 
I was wrong. I should have known that the hardest thing to teach are always the basics. Audrey came home with a math worksheet one day, which required her to do her model diagram and solve the questions. It was already painful to see her come up with a diagram because she was quite particular about being neat and later erase all away because she didn't get it. After 30 minutes, she was still at the first (unsolved) question. I tried to do it slow with her, but she was not getting it. After a while, I gave up and basically did her homework with her. (I 'taught' and she wrote) Because we again had an unpleasant learning experience together, I felt lousy and was up the whole night thinking of what I could do to change things. 

We went back to basics  of the model diagram after school one day. So far it has worked for me. And in case you are one of those who were born too early for the model diagram method, see if this works for you. 

1) I did away with the drawing of the diagram
I used strips of construction papers (4 colors would be safe since so far, I haven't used more than 3 colors) of standard varying lengths. I halved a strip for the pink and further halved the lengths for the orange and blue respectively. Whilst the green I divided the paper into threes.
I did this for many reasons. Firstly, it was less time consuming for Audrey to present her model and secondly I'm not sure if it was just her, but she couldn't see at times that certain parts need to be the same (because it represents the same amount) and certain parts need to be longer proportionately to represent a greater amount.

2) She only shows me the diagram
This doesn't mean she doesn't solve the question, but my focus was really the diagram and not her working. The diagram is essentially a working and while she doesn't need to do her working, I still ask her what her steps are.
I got her to do on her white board from school since I really didn't want to keep wasting paper and I thought it was easier to erase any mistakes with the duster than the eraser.

3) Step-by-step intro to model diagram
With all my materials prepared, I had a step-by-step demo. I showed her an example, guided her on the next and let her do the second. She fumbles sometimes, but with practice she does get it.
In short, this is how you show it:
a) Translate line by line of the question to the diagram. (It helps to break down the question. If sometimes the first line does not help much, you can use the second line to help) 
b) Labeling  (It helps to understand the premise of the question)
c) "Layering": When comparing the strips, all similarities must be found in the model (it helps the child to relate to the question)
The above shows an example of how "layering" works. Comparing the first and third strips with the second, the difference is shown by the green and orange strips respectively. However, since the green is longer than the orange strip, it would also mean that the green strip consist of the orange portion inside it. Visually, it helps the child to see which has the most and by how much. 
d) Finally, indicate the question with a question mark (It helps to understand what we are finding out)

Here's an example:
There were 15 more pupils in Class 3A than in Class 3B. 20 pupils from Class 3B moved to Class 3A. How many more pupils were there in Class 3A than Class 3B in the end?
Starting from top left to right.
I always tell Audrey to approach the question line by line since most of the questions are pretty straight-forward. (There are some questions which you do not work on the first line, but because it won't be 'basics' I won't be talking about it here. If you do want to know how to do it, let me know, and I can always share it)

As you can see, she used the green strip to represent 20 pupils from Class 3B in the second picture and immediately did the "layering" step of placing it in the first strip as well. She then "moves" the 20 students to the first strip by adding another green strip to it, while indicating using a dotted line that the students have moved to the first class. (I would tell her to cover the bottom green strip so that she would remember that the 20 students are no more in that class, allowing her to see the 'excess' students  in 3A as compared to 3B)

Obviously, it's not possible to use the strips in a pencil and paper exam mode. Only after she is comfortable moving the colored strips, I would let her attempt drawing it out. 
Yes I know, the question mark for the diagram is missing :P
And I am proud to say, she is slowly becoming a little expert in model drawing. :P

I know it's a little wordy today… Nonetheless, I hope it has helped you to help your child a little. :) Let me know if it did!!!

Wednesday, March 1, 2017

The Eureka Moment of teaching the Concept of "How Many More/Fewer"

And so, today's entry revisits my 'love-hate' relationship with P1 Math.
For the past month, I have been revising Math with Isaac and I found that one of the concepts that he seems to be struggling with is this concept of "How many more/fewer?"

A typical question would look like that:
Taken from one past year paper
Isaac's answer would be 2. In case you are wondering, the correct answer is 3. 
Well, if your child is gifted, your child is gifted. And this is of little or no use to you.
If your child is like mine, then you may want to read on. 

You see, throughout the last month, Isaac has been having problems getting it right. Actually, I looked back when Audrey was in P1 and she too had the same problem. It may well be a usual developmental 'kink' that most kids need to overcome, but that would not bring any form of consolation when the parent teaches this to the kid.

We sat through many other practices and he will not be able to get these type of questions right.
Trust me, I actually been through 3 different methods.
Method 1: Crossing the items from each group and asking to count what was not crossed out. 
Result: He crossed them out (though I think he didn't understand why he needed to) and still didn't help. In fact the next time when he saw a similar question, he didn't cross anything out.

Method 2 (similar to the first): Matching and partnering each object from the two groups to form an unit. 
Result: Same as the above.

Method 3: Just subtract one from the other. (Obviously, I'm getting desperate and wanted him to just get the answer) 
Result: He was not able to carry it out when he saw such a question again.

As an adult, you may not have any problems getting this. But a seven year old may. If most 7 year olds go through this like what Kathy Richardson's "How Children Learn Number Concepts" is saying, that what children is understanding from this type of question is how much is the number that has fewer, rather than how many fewer.

On a side note, I have to say that Isaac has no problem with simple subtraction. So 5-2 is manageable for him.
Now, based on Kathy Richardson, such a question could really be because of language rather than concept. 

So this is what I did. I took 5 sweets in one group and put 3 in the other. 
Step 1: Check whether the child even knows which has more. (Isaac knew… phew)
Step 2: Ask the child how many more sweets must you put in the other group to make both the same. (EUREKA… ok at least for Isaac)

Now of course if the group asks for fewer, then you may have to change the questioning a little. So…
Step 1: Check whether the child knows which has less.
Step 2: Ask the child how many sweets you must remove from the other group to make both the same.

Try it for a few more examples. But try not to give one example which has the same answer as the number of items in the group. 
So for example, Group A has 8 girls. Group B has 4 boys. How many fewer boys are there? The problem with this is you really cannot catch whether they got the concept or they were just giving you the number in the group that has the fewer items.

With that, now I explained that Math had a special language and in order to find those answers, the lingo in Math would be to ask "How many more/fewer?" (Which obviously was BS, but it doesn't matter

And it really doesn't because Isaac now was able to use that knowledge and answer these type of questions correctly. I found some other practice worksheets online and printed out to let him try… and he had NO problem. In fact, he was very confident and he managed to solve all within minutes.  

You can't believe how wide my smile was when I saw his answers. (To be fair, I'm not sure if I was prouder of him or myself :P) But HE was also beaming with pride. 

Regardless, let me know if you have tried this and your child had his Eureka moment too. Or better still, if you have any ways or methods you have tried and it worked, I would be really happy to learn from you! :) Otherwise, onward to P1 Math!