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!!!

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!

When a Picture says More than a Thousand words

The school term has started and so has my revision time with the kids.
Looking at last term's work, I wanted to do addition word problems with Audrey.
This week, instead of eggs, let me introduce our new character - Gingerbread Men!

This seems easy right? Ha! If only...

This (thank God) is not a question from Audrey's test, it's one of the questions we went through today.
To be fair, this is not a difficult question. I am sure suspect when teachers teach such questions, they would basically highlight the words "at first"and explain to them that it means finding out the number of items at the beginning. To do so, it would mean to add what is left and and is missing.

But here's what my dear girl did…

I'm not sure if you can see, but she struck 2 gingerbread men from the picture, and wrote as 6-2=4. (I don't even know to mark right or wrong)
After the egg incident, I have learnt it is more important (especially for kids) to understand rather than to be understood.
I asked her what made her do this way and here's her explanation:

There were 6 gingerbread men (Like why did they even draw that out?)
4 gingerbread were left.
So… She thought her working was supposed to be 6 - 2 = 4.
(I checked with D and he gave the same explanation as to why she could have done this way. Brilliant! My daughter is going to be a dentist too!)
I thanked her for teaching me something new and told her to take a break, since we would be continuing after dinner.

I tried to ask some friends whose kids are in Primary 1 to let their child try the above question. Basically I wanted to understand if their kids got it right or wrong, and if they did was it because they read the question or looked at the picture.

So one of my girlfriends got her daughter to try and she got it right. I asked did she solve it by reading the question or looking at the picture. She told me the former. She didn't even bother with the pictures.
Another two did the same thing as my girl… (okay now we got 2 more dentists!)

My concern, at this point, wasn't Audrey's train of thought. I was worried. Worried that she would not be able to proceed on questions when it didn't have any pictures (misleading or otherwise) and more importantly, that she will not be able to fit into what the system wants of her.

After dinner, I got her to redo the question again, this time, I covered the picture. I asked her to solve the question for me.

Just to showcase her artistic talent...

So I covered the pictures and she got the right answer. (did you know how loud my sigh of relief was????) I know why schools needed to provide pictures for word problems - for the consideration of other students who can't read.
However, through this one incident, I am not sure if the use of the pictures here has made matters worse. Or rather, the use of pictures has indirectly led to a failure of showing her concept understanding here.

SO if the kids generally didn't get it right because of the picture and if they get it wrong because of the picture… I'm trying to understand… does it mean:

1) Word problems should henceforth not have pictures?
2) If the child can't read, they shouldn't be given word problems?

It's sad that a good intention (picture in this case) didn't produce a good result.
Honestly, if I didn't have this incident with Audrey, I won't have known better as well. (Because I read the question instead of looking at the picture)
And if i were in the school management team, I won't have known that kids may be confused by the pictures and/or whether pictures should be drawn for word problems.
But instead of being critical of what has been done, I hope this post entry would create an opportunity to discuss on what could be done… after all, at some point, chances are your child will be going through the same education as mine.
So I'm posing to you readers, IF you were the setter, what can you do differently?

I do hope you share your thoughts on this… if not, I hope, this will help you further understand how a normal 7 year old think. :)


added note: my friend told me that it's probably easier to let audrey learn how to give answers people want.. Maybe.. but I would also hope that from this post, people would also learn to accept what answers my child (and many other children give) because of the little things we as adults overlook. Will it change overnight, no. But will that make a difference to what you do with a child like this? Maybe… :)

When a Math question teaches you humility… Part 3

So… for all you Audrey's supporters… I got this email from her teacher today..

Dear Michelle,

Thanks for your patience.  I have shared the alternative solution provided by Audrey and her answer can be accepted in this case.  I have informed Audrey about it before she was dismissed.

However, as mentioned, it is also important for Audrey to be familiar with the part-whole concept and be able to use both addition and subtraction equations appropriately.

I am glad we also had the opportunity to discuss Audrey's progress.  I hope she gets a good break and enjoy herself during the one-week term break.

Warmest Regards,

Mrs Lee

Through the whole episode, I am very sure she didn't know she was causing quite a sensation on the internet… To her, the Math paper was already history… Her being on time for school and having sushi during recess with her girlfriends are more important than whether the teacher appreciated or accepted her way of counting. (Boy I love her way of thinking…)

Yes, I heard some of you, the education system has its flaws but it's not unreasonable… So have a little bit more patience and faith it will be better. :)

That said, I do hope that through this sharing, it has reminded you to appreciate the little one(s) who are sent along your way… They are truly a blessing (I mean why would I have 4 if i don't think so..tsk tsk..) 

For those who have dropped me a message to tell me how special Audrey is, and how much you appreciated reading the blog…. Thank you… I really appreciated it. :)