How many of you have a problem with making correct educated guesses in math?

[Nan in Mass]

I thought it was just Singapore, but Michelle's post has made me wonder if this is a more universal problem. We had a terrible time switching over to actually showing work, even when you don't need to, during NEM 1, and even towards the end of PM it was a problem. Singapore demonstrates new concepts using easy, familiar problems. The idea is that the student will solve the problems using the new concept but will be able to understand what is really happening better because he knows what the answer should be. Well... this obviously has some problems. A Singapore-trained child is good at doing math in his head, good at estimating, and knows full well that there are several ways to do most problems. He'll just use the easy old way, do it in his head, and write down the answer, rather than using the harder new way and writing it all out. This is especially true of the switch to algebra. With Singapore, the children have been solving simple algebra problems for years, either by inspection or using the bar diagrams. Getting them to switch to the new way is hard work. And if you don't do that hard work at the beginning, then later on, when the problems get too difficult to use the old just-by-thinking-about-it method, you'll hit a wall. Both my children, even the not-good-at-math one, can solve those problems where you have one person going at one speed towards another person going a different speed and you want to know where they meet IN THEIR HEADS, just by reasoning it out. We do our math on graph paper, and this means that if they draw the picture fairly accurately for a geometry problem, they don't need to do actually do the algorithm to solve the problem, either. Now don't get me wrong - this is a great thing. It will stand them in good stead the rest of their lives, since probably most of their math will need to be done that way, but... Unfortunately, getting the right answer isn't what math class is really about; it is about learning how to figure out increasingly difficult problems (and, of course, about how to think).

 

I've learned that I have to actually write out the example problems, talking all the time about what I'm doing, and then I usually have to demonstrate a few more problems, too. It helps (but of course I don't always do it - sigh) if I watch them do a few problems and make sure they are solving them correctly. It also helps (sigh) if I make them do one problem, then compare the answer with the SOLUTIONS guide (not just see if they have the right answer) to see if they wrote the problem out correctly, then go on to the next one. This takes a bit of work on my part because there are obviously multiple solutions. I have to figure out whether this is one of those problems where they don't care how you get the answer, or whether it is one of the problems where they are supposed to be practising the new concept and have to get the answer a certain way.

 

I'm in the middle of forcing my youngest to show his work for physics and algebra 1. If anyone has any more good ideas about how to make the switch-over, I'd be very happy for more suggestions. This has been a major worry of mine. I'm happy that my children understand numbers and math enough to do simple problems in their heads, but I am so scared that if I don't do the showing-the-work part well, they won't be able to continue on in math.

 

Just thought anyone else struggling with this might like to know they aren't alone.

-Nan

[Veronica in VA]

I had the same problem with my oldest. He did Singapore through 6 then switched to Lial's for Algebra. He did do a lot in his head, and this became a problem once he went to school and took Algebra II. She started marking the whole problem wrong when he didn't show his work. I wish that had helped, but he just thought she was mean. In a way, I don't think you should penalize someone if they can do the work in their head, but at the same time, in a school setting it can look like they are cheating.

 

No good answers here, just commiserating. I do wonder though if it is a teenage issue. When I do the Algebra II with my daughter, I write out everything. She thinks that is too much writing!

 

Veronica

[Brenda in MA]

My youngest is in a "between" year -- between Singapore PM6 and Algebra. I chose to use Saxon 87 to give him one more year of practice with basic math before tackling Algebra. It has been a constant battle all year trying to get him to write out simple equations and show his work step by step when he can do it all in his head. I don't really know of a good solution, other than just to keep making him write things out. While I know the necessity of it, he just thinks I'm mean, too!!!

 

Maybe someone else will have some suggestions in this area!?!

 

Brenda

[Gwen in VA]

My kids suffer from "don't want to show the steps" - itis as well.

 

I divide the credit for each problem in half --

 

1) I give half-credit for showing the steps (no more than one thing done in your head -- if Mom can't follow it there aren't enoughs teps!)

 

2) I give helf-credit for the right answer. That way even if they get the right answer they still need to show the steps.

 

I do end up doing problems occasionally with the kids to show what I actually mean by "show your work." I think that should be intuitive, but my kids don't!

[Lori D.]

nt

[Myrtle]

I absolutely noticed this. I ended up doing a blog entry on it here after I figured out a way to solve the problem. In short, here is the solution:

 

Teach child to always write the "number sentence." In the primary grades rather than have the child "show his work" vertically, have the child always, always, always show his work horizontally. Now, if he needs to stack up numbers vertically to find the answer, that is done on a separate sheet of paper and we call that scratch work.

 

The hard part, I have found, with younger kids is teaching them to reinterpret the number sentence so that the thing which they are solving for is isolated on one side of the equal sign. For example, in a word problem such as "Jane had 20 stamps and Xiuli gave her some more, now she has 50. How many did Xiuli give her?" All of my kids would begin by writing that as an addition sentence. It took a lot of work to get them to express that as an subtraction sentence. Clearly, they were subtracting in their heads without realizing it and coming up with the correct answer and not knowing how they got to it. To teach them how to convert an addition sentence to subtraction I stopped and added more practice with "number families" which, in effect, is working with the formal definition of subtraction (If a + b = c, then c - a = b) Then you have to repeat this sort of thing when you hit multiplication and division.

 

The other thing you will have to do is explicitly teach the use of parenthesis in equations so that they can express a two step word problem as a single sentence. The way that I got my second grader to do this (and I wish I had a gif uploaded to show you how this works) is to have him write two horizontal sentences which he knew how to do as a result of how I worked with him as I explained in the above paragraph. So he writes the first step and then sees that the answer is used in the second step. I had to explain to him, on a seven year old level, the principle of substitution. I'd circle the expression which yielded the answer in the first step and draw an arrow to where he used the answer in the second step. Then, I had him do that with the arrow as well with his work.

 

When he could do that I said, "I can express those two sentences as a single idea like this.... And I'd write the second step leaving a big gap with empty parenthesis which would be filled with the expression from the first step.

 

For example: Mr. Lin had 112 tomatoes. 8 of them were rotten. He packed the good tomatoes into packets of 8 each. How many packets of tomatoes did he get?

In stage one Child writes:

A) 112 -8 = 104

B) 104 ÷ 8 = 13

 

Stage two: child draws arrow from the 104 in first sentence to 104 in second sentence. Seems mindless, but a seven year old needs to have the fact that you use this number twice pointed out to them.

 

Stage three:" I am going to express these two sentences as a single idea."

(112 - 8 ) ÷ 8 = 13

"Do you see that I replaced the 104 with the expression 112-8?" Child should say yes. "I can replace 104 with that expression because they are equal." In effect, you have to explicitly teach the child this principle of substitution. Because they will be using exactly the same set up each time with two step word problems, even if they don't generalize the principle, they will be able, with enough practice, to catch on to the formula. In other words, rote will do where understanding will not when it comes to a pinch.

 

At any rate, my second grader is now at the stage where he will sometime skip over writing out the A & B steps and immediately write out his equation. Not always. Sometimes he needs the two prior steps to help him think it through, that is fine. No hurry. Later on, when he learns that the fraction bar can also represent division I'll have him start expressing division using the fraction bar. By the end of P6 he'll learn that x can represent the unknown and he won't have to have make sure the answer is to the right of the equal sign. It will just be an x inside an equation or expression. He won't know how to manipulate the equation to get x, that actually manipulation of x is something we save for algebra when he can understand the technical justifications for such manipulation (in our algebra program you aren't allowed to do a thing to x until you can write out a full blown proof of why it's justified)

 

To make myself clear, even though in the sixth grade the child may show his work with an x in the expression, I am not requiring him to solve for x or manipulate these multi-step problems, I'm having him explain the thought process that went behind his thinking. He can solve the problem mentally or by scratch on a seperate sheet of paper if he wishes.

 

The sample you see in the gif to which I posted a link is the last step in this process: Using x for the unknown and formally declaring the variable. You must declare the variable. You must declare the variable.

 

When my sixth grader went into algebra he had no problem with it because he'd already been habituated to showing his work in a way in which it would be show in algebra. Having him declare the variable also has made word problems in algebra much easier for him as well as proving theorems. I could go on to explain how the above steps removed a learning burden when it comes to proving theorems (something that I was excited to see "work") but that is getting further off topic.

 

It is much easier to teach a kid who already thinks mathematically to show his work properly than it is to get a kid who can show his work properly to think mathematically, and that is why we stuck with Singapore despite this apparent deficit.

[Nan in Mass]

Where were you when we were doing 2nd grade math?

 

I DID do the PM6 stuff you described (except I didn't make them do the integer part - good idea from a software engineer's perspective GRIN), and we always worked horizontally, since mine did the arithmetic in their heads anyway. That probably saved us. There is a section in NEM that I think is supposed to help the child make this switch. We worked hard at that section, because fortunately I recognized what it was doing. I think in Singapore, a good teacher would be doing what you did all the way along. I've seen hints of that here and there. Mine had no trouble understanding what the algebraic manipulations were doing, once we got to them, and I think it might be because they could see the connection between what they were doing on paper and what they normally did in their heads, once they realized that the manipulations on paper went from last to first compared to the ones in their heads. I've been terribly grateful to Singapore for teaching them to do it in their head. That at least makes the switch possible. As you said, going the other way round involves an awful lot of backing up. When my middle son began homeschooling with me in 5th, he had NO mathematical thinking. Ug. Backing up and doing Singapore fixed that.

 

You know, I think I might take both children and do a review of that chapter in NEM1 where they go from the bar diagrams to the algebraic sentences and review it, using the method you just described. I bet we can do it in a day or two, at their ages, and it might help solidify the procedure and make it easier. The older one has pretty much transitioned over, but he always struggles a bit, and the younger one is still fighting, so it might help. Thanks for writing all that out. It might help some people if you post that procedure on the curriculum board.

 

-Nan

[Mama Lynx]

This is exactly what we're dealing with now.

 

I took my oldest son through Singapore 6B, but I never learned math well in my own education, and did not know to teach him to write out his work properly. My husband, who has taken over and who has far more of a clue, has spent most of the year teaching him to do this. It's been a struggle, and we're still not quite there. The lightbulb has just recently gone of in my head that I need to teach my younger sons to do this *now*, every step of the way.

 

It's interesting that there is a section in NEM to help make this switch. I don't have NEM 1 in my hands, but now I wonder if my dh didn't just spend a year doing from scratch what NEM would have guided him through. (Not that dh needs the guiding, but if my son is already used to the Singapore way, perhaps it would have been more effective to use that instead of switching to an entirely different course.)

 

I appreciate your example, Myrtle. I need all the help I can get. Frankly, I always thought of math as just material to "get through," until you started posting about it.

[LizzyBee]

Hmm, I wonder if this is, as you suggested, more of a problem with a particular type of student rather than the curriculum? I have never forced my kids to show their work for problems that they can do in their head. They use scrap paper to do the work and write their answers on notebook paper. When my oldest moved from PM to NEM, she didn't have any trouble writing out her work. She struggles much more with writing proofs for Foerster than writing out problems for NEM.

[langfam]

When I did Singapore Math in Singapore as a child, we had to write full and complete sentences for all word problems. This was drilled from the time we were very young. Points were taken off by the teacher for not doing so.

 

For example:

Mr. Lin had 112 tomatoes. 8 of them were rotten. He packed the good tomatoes into packets of 8 each. How many packets of tomatoes did he get?

 

Mr. Lin had 112 tomatoes

 

- 8 of them were rotten

 

 

____

 

 

104 good ones were left

He packed 104 good tomatoes into packets of 8

 

There were 104 / 8 = 13 packets of tomatoes

 

I've tried to get my DC to write out their word problems like this, my boys just resist. DD will. My eldest DS always solves everything in his head. It's very frustrating. Sometimes when I'm grading a test for example and his answer is wrong, I can't "see" where he made the mistake. I'll have him do it again to have me show his work, he'll still skip steps and usually gets it right doing it the second time.

 

I remember having to write full sentences for all work all the way up to Additional Maths. We weren't allowed to skip steps. It was like part of the requirement for your grade. Teachers were very strict and adhered to this rule. School was very rigorous in Singapore........it's like a cake-walk for my DC here. I always tell my DC how easy they have it, but they don't pay attention.

 

I'm no help....sorry....

[langfam]

Thinking of this brings back a memory of Mrs. Brockett flinging my workbook across the classroom and out the door for not writing neatly in full sentences. Poor 8 year old me....:(

I do not suggest this method!!!

[Myrtle]

How many word problems would you do in a set?

[Storm Bay]

I've always made my children write out their work, but I've recently upgraded some of this based on Myrtle's post, which I printed out (but not for resale:001_smile:)

 

I would like to address this a bit, though, from another POV. I do think children need to learn to show their work, and was taught to do that myself. However, even I, who always was taught math showing work starting in Gr 1, and was never taught to estimate (at least not in Gr.'s 1-4), started knowing Algebraic answers in my head. I did not solve them the way the textbook said, but it wasn't a guess as for the most part the only error I would make would be to always revert to x, y, z for my letters instead of the ones in the problems. So when teachers showed how I solved problems, I inwardly screamed, NO, I did not solve it that way. My point being that it's important to recognize that some people are able to solve without guessing or doing all the steps and I never ended up in a place where I couldn't solve the work (but also didn't make it to Calculus or college level non-engineering math.) So when I teach my kids to show their work, I recognize that they might not solve it that way. And I know how to do it the correct way, it just takes so much longer that when I was in high school it seemed like a complete waste of time.

[langfam]

Word problems..... maybe 5 a day in 2nd grade and up to 10 in 6th. But that's just homework for the teacher to grade. Most kids in Singapore also have a tutor, so there are extra workbooks to complete. I did *alot* of extra practice type workbooks.

 

Preparing for the big exams like the PSLE , "O" level, and "A" level would mean more revision type workbooks......It was not uncommon for me to be working 14 hours a day studying all subjects for exams. It's very competitive.

 

Personally, I think this is the reason Singapore kids do well in the international competitions....not because they are smarter, but because they have put alot of time and effort in preparing.

[Nan in Mass]

I suspected all the way along that Singpore would have been teaching it differently than I taught it. I was lazy and didn't make my children show their work until they had to, about PM5, but I knew they should have been doing it. It is helpful to have that confirmed because I did always wonder if I was wrong about that.

 

It also is helpful to see it written out in sentences like that. I've never seen it done that way before! How clever! I wish I'd know this all the way along. I would have made my children do it. I'm doing a bit of backup work with the info Myrtle gave me, and I'm going to show my children how to do it this way, too, and make sure they can do it if they need to.

 

Thank you!

-Nan

[Myrtle]

Word problems..... maybe 5 a day in 2nd grade and up to 10 in 6th. But that's just homework for the teacher to grade. Most kids in Singapore also have a tutor, so there are extra workbooks to complete. I did *alot* of extra practice type workbooks.

 

Preparing for the big exams like the PSLE , "O" level, and "A" level would mean more revision type workbooks......It was not uncommon for me to be working 14 hours a day studying all subjects for exams. It's very competitive.

 

Personally, I think this is the reason Singapore kids do well in the international competitions....not because they are smarter, but because they have put alot of time and effort in preparing.

 

 

Reminds me of a post I recently read....

 

"You can measure the amount of mathematics in any number of ways. I use pounds and ounces.

 

When asked why my son is so much farther ahead than his peers, I explain that if you take out all the pages of problems and exercises that students have done, and weigh them, my son will have done many more pounds of mathematics than his peers."

[Mandy in TN]

Thank you, Myrtle. I also have printed out your post. I am finishing up PM 1B with my youngest ds and this will be very helpful. I also agree with the post about the amount of math work completed. My little guy began Kumon math last summer and this is the same response I would have for why he is ahead.

 

I wanted to add that after Singapore 6 my middle ds did Saxon 87. I did not have any knowledge of this, but I did feel like he needed a transition before high school math.

 

Mandy