kitchen table math, the sequel: Singapore Math, day one

Wednesday, August 29, 2007

Singapore Math, day one

I've just walked C. through his first day of Primary Mathematics 3A, including the 3A workbook.*

Then I had him do one "Challenge Problem" from Intensive Practice 3A:

1. I am a three-digit number. All the three digits add up to 9. My tens digit is twice my hundreds digit and my ones digit is three times my tens digit. I have no zeros. Who am I?
(page 10)

He took a look at it and said, "That's hard!"

I thought so, too. As a matter of fact, I am currently doing word problems only slightly more advanced than this in Saxon Algebra 2, at the end of the book. [update: RJ and Cheng noted that the sum of the digits of their two-digit counting number was 9. If the digits were reversed, they found that the new number was 27 less than the original number. What was the original number? Saxon Algebra 2, Second Edition, p. 432, no. 1.]

I handed him the workbook to look at while I wrote the problem on separate sheets of graph paper for both of us to do. I figured it might be a good idea to get some practice drawing bar models.

A couple of seconds later C. said, "I have the answer."

He did. answer: 126

I pushed him to tell me how he got it, and finally he said (paraphrasing), "I asked what is 2 times, and then I realized it was 2, and then that was 6."

I believe this is called "making children's thinking processes visible," right? And, yes, this is the way kids talk when they're describing a mental process. Mental processes are slippery. When you get an answer fast, you don't necessarily remember -- or even know -- how you did it

What I take Christopher's report to mean is that he intuitively -- unconsciously -- understood immediately that the hundreds digit would have to be 1, then consciously asked himself what two times 1 was, had a eureka moment (I could tell this from his tone of voice) when he realized that 2 was put him in the running to end up with a grand total of 9, and from there semi-consciously realized that the ones digit was 6.

At least, that's the way I piece it together. Obviously I don't know how he came up with the answer five seconds after he read the problem.

This gives me hope. I was saying yesterday that Ed and I were hoping the fact that C. appears to be correctly guessing answers to fraction word problems means he's developed some number (& fraction) sense; today's Challenge problem seems to be evidence that he has.

I think I've mentioned before that "Math Dad," the parent here who is a veteran NY state math teacher, has told me, and I'm pretty close to quoting here, "It's too late for these kids. They are lost." He was speaking of Christopher's group, the kids who need effective instruction and a sound curriculum in order to succeed in accelerated math.

When I heard that I thought: over my dead body.

Maybe, just maybe, these past 3 years of slave labor and protracted Math War have produced some "growth" after all.


These are the three questions for which C. correctly guessed the answer. I'd forgotten that one of them is extremely simple algebra. Sigh.

Robin and Jim took cherries from a basket. Robin took 1/3 of the cherries and Jim took 1/6 of the cherries. What fraction of the cherries remain in the basket?
answer: 1/2

___ represents the number of magazines that Lina reads each week. Which of these represents the total number of magazines that Lina reads in 6 weeks?

6 + ____

6 x ____

____ + 6

( ____ + ____ ) x 6

[yes, I am horrified that I have a "rising" 8th grader who had to guess the answer to this question - though this does serve the purpose of illustrating why "Math Dad" goes around saying things like "It's too late for these kids."]

Penny had a bag of marbles. She gave one-third of them to Rebecca, and then left one-fourth of the remaining marbles to Jim. Penny then had 24 marbles left in the bag. How many marbles were in the bag to start with?
answer: 48

* For newbies, C. is entering 8th grade and began taking Math A -- integrated algebra and geometry -- in the second semester of 7th.


Me said...

That's terrific your son could do this problem so quickly! I think this shows a good number sense plus an ability to decode word problems.

However, it's a strange (trick?) problem in that there is so much extraneous information ["I have no zeros"; the answers are positive integers; seems to have something to do with place value but doesn't] and is essentially two (unrelated) problems in one.

The hard part can be rewritten as follows.

Three numbers add up to 9. The second number is twice the first. The third number is three times the second. What are the numbers?
x+y+z = 9
y = 2x
z = 3y = 6x
x+2x+6x = 9
9x = 9
x = 1

The second part is to state the number which has 1 as its hundreds digit, 2 as its tens digit, and 6 as its ones digit.

Instructivist said...

If you convert the problem to an equation, the answer sorta suggests itself:

x + 2x + 6x = 9

Anonymous said...

if "that's hard" is an appropriate
response (and maybe it is), i'd like
to suggest that it's because
*reading* is hard (not because
math, or let's say, reasoning,
is hard ... which maybe it is).

once you've got the nerve
to actually try something,
it's pretty much *bound* to work.
but you've got to be able to
believe you've understood
what you're trying to achieve.
failures in this area are common.
math teachers get more than
their share of the blame.

le radical galoisien said...

Hmm, number theory. Not a bad place to start.

The Math Olympiad has similar problems, only that it's not as easy.

Well, "similar" might be a stretch. I suppose a (Singapore) Math Olympiad type problem would ask you, "how many 3 digit numbers add up to 9?"

Catherine Johnson said...

plus an ability to decode word problems

I was THRILLED about this.

Interestingly (well, interesting to me...), I didn't read the problem as well as he did; I didn't notice that the solution to the problem went "left to right" -- that is, that you start on the left, with the hundreds digit.

I was semi-focused on the ones digit.

I wish now I had read the problem carefully. I'd like to know whether my problem was skimming or breaking set (not being able to stop thinking a number "starts" with the ones & then gets bigger and to "start" with the hundreds digit in order to solve the problem).

Do you remember that study saying adolescence is a better time to learn algebra than other ages??

In any case, this isn't an easy word problem to read (IMO) and he obviously read it quickly and accurately.


Catherine Johnson said...

it's a strange (trick?) problem in that there is so much extraneous information ["I have no zeros"; the answers are positive integers; seems to have something to do with place value but doesn't] and is essentially two (unrelated) problems in one.

Just read this -- THANK YOU!

I feel better.

As I say, I didn't instantly see the answer. Not at all.

Catherine Johnson said...

yes, it's the reading that's hard - although....I'm not completely sure I can separate the "reading" from the "solving".....

Probably someone else can say this better than I can.

Catherine Johnson said...

Definitely "hard" in terms of working memory.

Catherine Johnson said...

I'm thinking I'll show him the algebraic formulation tomorrow .... and draw the bar model, too.

I think.

Tex said...

This sounds like progress and I think you’re right to be hopeful. Does Christopher express satisfaction about his progress? Maybe every once in a while when he’s not complaining about all the work you make him do? ;- )

I’ve noticed my daughter “guessing” correct answers to word problems. She astounds herself sometimes when she knows the answer so quickly and she says her lips just know what to say!

When I ask her to explain how she got the answer, she’ll reflect for a few seconds and then she’ll usually provide a coherent explanation.

I partly attribute this to her very good mental math abilities and I tend to believe she’s developing a good number sense. We’ve been doing Singapore bar modeling problems this summer, and I think (hope, really) that’s helping her develop a better sense of how to solve word problems.

All this progress comes from lots of practice, “procedural fluency preceding conceptual understanding” IMHO.

Catherine Johnson said...

When I ask her to explain how she got the answer, she’ll reflect for a few seconds and then she’ll usually provide a coherent explanation.

That's exactly what C. did.

He was happy to have gotten it right -- and to have gotten it so quickly.

This is exactly the right thing to be doing, going back to the beginning.

He needs to do interesting math HE CAN DO.

That's Singapore.