FWIW, we started Primary Math 6A today. Kicking and screaming ensued.That's where I am -- and I can see exactly how this curriculum gets a student to the point of perfect scores on state math exams.
Then we opened yesterday's mail and got a very pleasant surprise. My daughter's CMT scores (CT's state testing) arrived and were very good. She got a perfect 400 in math. Math is not her favorite, or strongest subject, but it's one we've focused on at home at length. I see the score as reflecting the coherence of the Singapore Math system.
After much oooohhing and ahhhing, I was able to get her to start 6A.
Right now, she doesn't seem to have any real weak areas. She's doing pretty good in decimals and fractions, so we're just going to charge ahead with the whole program.
The Primary Mathematics series is the coherent curriculum (pdf file) par excellence. One thing leads to another, and you have problem solving applications from the get-go.
C. has probably done 50 word problems in his entire math-learning career, if that. He has no idea how anything relates to anything else. He's been getting by on brute memory.
Here's how bad it is.
We've been doing the percent lessons in Singapore Math 5A.
That quickly became so "natural" a learning situation that, when I told C. yesterday morning I wanted him to do an entire page of problems, he looked at the page and said, "OK, that looks easy."
And it was easy. He finished in 5 or 10 minutes & got everything right.
The fact is, going back to 3A will probably be fun, especially since the Premack principle is working so brilliantly that yesterday morning (Saturday), at 10 am, C. came into my office, sat down, and said, "What do I have to do today?"
So I think we've got the think-like-a-behavior-analyst aspect of things down, which will free C. to enjoy being able to comprehend, learn, and do 3rd grade math.
That is already happening, in fact.
Yesterday, after he whipped off his one-page problem set, he said, "That Singapore method is really good."
He was referring to finding the percent equivalent of a fraction by multiplying numerator and denominator by the same number.
This is the state of his math education.
Multiplying 6/20 by 5/5 to get 30% is "that Singapore method."
There is nothing like success to motivate.
ReplyDeleteKnowing how to do a bunch of problems, and then successfully doing those problems, creates a sense that you can do anything. My daughter has just never had to do a problem in SM that she was utterly clueless on. Everything makes sense.
The weird thing is that she is doing much harder problems than in EM, but she knows how to do them and isn't intimidated anymore.
If C. is having fun with SM's Primary Math and you're up for a challenge, try the challenge problems in the Intensive Practice book. AHHHHH!
Knowing how to do a bunch of problems, and then successfully doing those problems, creates a sense that you can do anything. My daughter has just never had to do a problem in SM that she was utterly clueless on. Everything makes sense.
ReplyDeleteThat is miraculous.
We will not be attempting CHALLENGE PROBLEMS until C. is well on his way to being able to do NON-CHALLENGE PROBLEMS.
ReplyDelete(but yeah, those challenge problems are HARD)
ReplyDeleteI did all the challenge problems in the 3rd grade "Challenging Word Problems" book -- and even THOSE were a challenge!
(They were a challenge to do with bar models...)
Here's a question for you... working through the word problems in Singapore 4A/B has been such an epiphany for my 9 year old, that she solves many problems mentally to the point that she only records her answer. I've asked her to explain to me how she got to a particular answer (which is most always correct) just to understand her thinking process.
ReplyDeleteIs this a problem? I worry about the requirement of showing her work (in some other setting). It's almost as if the thinking process she has developed as a result of working with the Primary Math series is so efficient that the answer materializes magically before her eyes.
I agree. I've had a few where I've wanted to use algebra to solve. I had to read the home instructors guide and look at the answers to figure out how we were supposed to do it (since we've had no algebra yet).
ReplyDeleteI am frequently amazed at how much you can do with a bar model.
ConcernedCTParent:
ReplyDeleteI wouldn't worry that much about it at this point. Sometimes the answer does seem really obvious when you get to the point of automaticity; but kids that hate showing their work on daily assignments seem to do okay when they know they have to for school or exams.
Sometimes just a little talking can help a kid see how she got an answer. I might start off by saying something like, "Wow I can't believe you got that, I was sure you were going to ask for help on that one. How did you do it?" She might give me a short answer and I might respond, Did you find the common denominator? Did you divide first or did you subtract first?
Asking leading questions can help them realize what they are doing. This is all just kind of practice for when she has to explain herself at school. I guess I'm giving her a procedure for thinking about what she might have done.
"She got a perfect 400 in math."
ReplyDeleteCongratulate your daughter from the gang at KTM. Great job!
"Here's a question for you... working through the word problems in Singapore 4A/B has been such an epiphany for my 9 year old, that she solves many problems mentally to the point that she only records her answer."
ReplyDeleteThat I wish I could have done with the PSLE!
The PSLE markers always wanted you to show your working, and after two terms' worth of exams of getting stupid deductions for not showing working, I ended up explicitly writing out every step, even steps that I suspected were unnecessary.
But I was kiasu and didn't want to lose any marks, so I generally went for the overkill.
So even if you did your problem by trial and error, etc. and you got yours on the first or second try (got good instinct, right?), you generally had to make some fake steps up to make it look like a real guess and check problem.
A 5 year old rant I wanted to express.
"We will not be attempting CHALLENGE PROBLEMS until C. is well on his way to being able to do NON-CHALLENGE PROBLEMS."
ReplyDeleteAh, could you reprint some of those? I haven't done those problems in years, and yes, they are difficult even for adults! That's why I found them fun!
The rest could get tedious after a while. Challenge problems were like a good Sudoku puzzle, even better, because sometimes the methodology isn't obvious to even adults at first.
But I think it's something your son could do. Those were the problems I enjoyed diving in head-first. I would try out random strategies, analyse and think, not even knowing where I was going at first. But each completed problem was like a mental conquest.
And yes, success encourages.
ReplyDeleteWhen I first came back from the US to find myself in the Singaporean primary mathematics syllabus, my confidence started dropping after a while after seeing all those red X's on my mathematics paper. I started fearing them.
Something turned for me on my third term's exam. Suddenly there were a lot less red X's, and most of the point deductions had arisen from careless mistakes rather than total cluelessness.
It's the sense that once your ability is revealed, and once you're that high up, you don't want to disappoint, so you are determined to keep up the good work.
After a while, having done miserably before, you begin to feel like a warrior in mathematics. The problems may outnumber you, and their tactics may be diverse, but you know you are skilled. You know you can deftly deal with anything they throw at you!
And after you do something like challenge problems, confidence really soars. You feel you can do any problem. Anything!
"I've asked her to explain to me how she got to a particular answer (which is most always correct) just to understand her thinking process. Is this a problem?"
ReplyDeleteIf she can explain how she got the answer, then everything is fine. I tell my son that I want him to write everything down carefully, line by line. I don't care if he can do it in his head. What I see happening is that it's easy for him to figure out problems with nice numbers, but change the numbers and he can draw a blank.
Just today, we had the following problem:
75% of what number is 15?
It's almost as if he is doing a bar model in his head - three parts or boxes for the 15, so the fourth quarter (25%) or box is 5 to make 20 the number you're looking for.
I've mentioned before that bar models fall apart when the numbers or percentages are not nice, and that's what seems to happen with his mental process.
Next, I gave him something like:
7/34 of what number is 17?
I could have used a percent, but I figured that having a fraction is closer to his thought process of seeing 75% as 3/4 in the first problem. I told him that I just wanted him to set up the equation. He didn't have to come up with the final number.
He still struggled. There is clearly a mental way to "see" the answer to problems with nice numbers that doesn't translate to other numbers.
To help, I told him to:
1. Translate percentages to decimal or fractional form, whichever is easier.
2. Translate "of" to times
3. Translate "is" to equals
4. Translate "what number to 'X'
This would give him:
7/34 * X = 17
I told him that if he can write down the equation, he doesn't have to think anymore, and the difficulty of writing an equation does not depend on he numbers.
"I ended up explicitly writing out every step, even steps that I suspected were unnecessary."
ReplyDeleteI remember the arguments we students had over exactly how much change we were allowed to make in each line of algebra. It was always too little, but if the rules were explicit, there were no surprises.
For example, given
4(x-5) = 2x + 4
I would like to write:
2x = 24
or even solve the problem in my head and write:
x = 12
You might get away with this in algebra II, but not in algebra. Part of the reasoning is that they want you to minimize dumb mistakes. I used to tell students that it was for their own benefit. If they make a dumb mistake, then more steps will give them more partial credit. I was not able to read their minds if there was nothing on the page.
If they make a dumb mistake, then more steps will give them more partial credit. I was not able to read their minds if there was nothing on the page.
ReplyDeleteThis is what finally convinced my 15 year old to write down his steps in geometry. He started getting wrong answers and couldn't easily solve many of the problems in his head.
When you are unsure of your answer, you're a lot more interested in partial credit for showing your work.
Sometimes you just have to learn this the hard way.
Ah, could you reprint some of those? I haven't done those problems in years, and yes, they are difficult even for adults! That's why I found them fun!
ReplyDeleteWell, you asked. So here's one from Intensive Practice 5B.
Tap A alone can fill a tank in 3 minutes. Tap B alone can fill the same tank in 4 minutes. When the 2 taps are turned on at the same time, how long will it take the two taps to fill the tank completely?
I did not ask my post-5th grader to solve this one. This came from the "take the challenge" section of the book, which typically is much harder.
Earlier in the unit, kids had to solve a problem where a tank is filled with 3 pipes (of different size) and the per minute fill rate is given for each. Also the total volume was given.
The Singapore Method, if there is such a thing, is to give similar problems to kids over and over, each time leaving out a different piece of known information. By the time you get to the hard problems, the child should have a pretty solid understanding of how time, rate, and volume are interconnected and how to find each one when given the others.
"Tap A alone can fill a tank in 3 minutes. Tap B alone can fill the same tank in 4 minutes. When the 2 taps are turned on at the same time, how long will it take the two taps to fill the tank completely?"
ReplyDelete1 and 5/7 minutes, right?
Heh, happened that a few days I was reading this post (the blogger of which happens to tutor both secondary and primary students, e.g. my friend). It contained roughly the same type of problem, except with a slightly different type of information missing. I remember this sort of question stumped me, because no "concrete" rate was given. Ah, it pains me I still can't derive the methodology myself, after all these years. I remember the other problem was that I kept on confusing inverses. (But this blogger's students doesn't seem to have that problem!)
"Earlier in the unit, kids had to solve a problem where a tank is filled with 3 pipes (of different size) and the per minute fill rate is given for each. Also the total volume was given."
Ah, and to think, when they hit calculus, they will get almost the same problem, only now the fill rate is variable (yay for dV/dt!).
I remember the PSLE had some pretty ingenious volume problems too, especially ones related to pools. Now that I think about it, I know why some of those AP Calc problems were so deja vu now; they reminded me of PSLE problems, only involving derivatives and integrals.
Also, the tutor I mentioned above is only 15 years old. But already he teaches the PSLE syllabus with such dazzling proficiency and seems to have a strong sense of ownership of his classroom.
ReplyDeleteBut yet, from the way he talks about his students and he seems to command his class, it seems like he really is wise and sagely old teacher. Then, there are the students. Did I mention I feel inadequate?
I am frequently amazed at how much you can do with a bar model.I am frequently amazed at how much you can do with a bar model.
ReplyDeleteIt really is amazing -- though not at all easy.
Who left the comment about hard problems in arithmetic being easy problems in algebra? (Was it "price"?)
Eat your Maths!
ReplyDeleteI love it!
You know, maybe I should give him a "challenge" problem AT HIS LEVEL -- i.e. a 3rd grade challenge problem.
ReplyDeleteLynng,
ReplyDeleteCongratulations to your daughter on her perfect score! My son made a 592 in math on the Virginia SOL test. The cutoff score was 600.
PaulaV
Paula - unbelievable!
ReplyDelete(Which reminds me; I have another comment from you I want to pull up front...)