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LOL, ok so the geek in me had to try after Dudes comment and I got it with the first guess.....(so I pass 2nd grade yay!) (ok so one solution)

Well, that's me told.

I agree that these problems could be harmful. But in most cases when DD11 was doing them it was in the context of an exemplar or problem solving and they would discuss options in class and were not necessarily graded on a correct answer. They would possibly know there was more than one solution and were encouraged to think about it. In some cases making a poster detailing the sections of their work. The criteria for the top mark in the problem solving was noticing the alternate relationships etc vs just solving the problem. Solving and writing down why might get you a 3 or 4 but seeing the alternatives or explaining a "rule" would get you the 5.

They also spent a lot of time coloring and making it neat or creative. (me groaning because mine spent longer on this art to get it just right).

Expectations for the explanations and calculations skills increased as they went from 3rd to 5th. I don't know that they actually counted for a grade until 5th but I was sort of clueless about the 4th grade grading process and DD11 skipped 3rd grade math at the school.

Ma, I have a dysgraphic ds who just finished up 6th grade and who is a math/science type kid. I agree with DeeDee (I think it was DeeDee!) who mentioned how it can be really tough advocating for a child with dysgraphia in early elementary as the teachers often don't understand that the child really is trying. This may not sound very encouraging, but fwiw, my first advice is to prepare yourself that it may take a few years (or most of elementary school) to get to the point where you can step away and not have to face this type of perspective from a teacher. It's also important in early elementary to really focus on getting accommodations in place - sometimes that will mean that you might spend more effort/energy/time on that than on trying to be sure the needs of your ds' gifted intellect are being met, but keep focused on knowing you're putting the necessary foundation in place to allow him to soar when he's further along in school.

Math is one of our ds' greatest intellectual strengths and yet is has been a *huge* stumbling block for our ds. Dysgraphia caused him to be held back in more ways than one - there was misunderstanding on the impact of dysgraphia on math performance on the part of teachers, but another part of it is simply that ds' output is slow relative to other children. This is a quick synopsis of some of our ds' experience and how we've approached it:

Early elementary worksheets: a lot of writing and explaining was required, as well as handwriting in relatively small spaces on paper with no lines - so there was a double-edged gotcha there - the need to produce words as well as the need to control handwriting within a challenging workspace. We had ds scribe to us for all of his homework and we advocated (unsuccessfully at that point) for scribing at school. What happened in reality at school was ds didn't finish most of his work (math or other assignments). His teachers didn't worry too much about it because they could tell he knew his math... but he also wasn't offered any kind of advanced material.

Later elementary, starting in 2nd grade and on through 5th - timed math facts quizzes were a nightmare. My opinion is that you need an accommodation stating that all fluency tests will be given orally - this is especially important for math facts. It took us until 5th grade to get this accommodation in writing for our ds, and once it was in writing it was usually ignored, primarily because his teachers, in spite of our never-ending advocacy, just didn't understand dysgraphia. Sadly this also held him back from advancing in math *at school*.

Also later elementary school - more and more "explain your work" problems - and in our school, explain usually was "explain in 2-3 different ways". We gradually moved ds to keyboarding in math for this type of problem.

Like you, we wanted to give our ds an opportunity to work on math outside of school. We first felt it was important because he'd literally fallen behind thanks to the emphasis on passing math facts timed quizzes, so we sent him to Sylvan for around 6 months. He liked Sylvan but our pocketbook didn't So once we knew he was caught up to grade level, we turned him loose in Aleks, and Aleks was fun for him (but again, it might not be fun for other kids or successful for other kids... all depends on the individual). I liked Aleks (as a parent) because it gave me a record of what he'd accomplished at home that was tied to our state standards. We tried to use the Aleks work to advocate for ds in 4th grade, but were only partially successful at that point. DS picks up math concepts quickly, so he breezed through a lot of the 3rd/4th/5th grade modules quickly. His teacher didn't want to believe he was really understanding the math (since he couldn't do those danged math facts that quickly!) so she requested that we have ds do a certain # of problems in each module to prove mastery, and to save copies of all those problems. So we did that, and then were told ds could be considered to be moved up in math but first had to complete packets put together by the middle school math teacher - these packets were so danged basic ds was beyond frustrated - and he was required to do them in the original packets, using handwriting. We gave up at that point.

Also starting 4th/5th grade - as math calculations became more complex, ds started to have a lot of challenges with accidentally switching numbers while writing them down in the middle of working a problem (writing 23 instead of 32 for instance) - we could tell he was doing this without realizing it because he talks out loud while he works and we'd see it happen. Once you've switched a number like that in the middle of a math calculation and not realized it.... obviously you're going to get the wrong answer. We began encouraging him to double-check all of his work (which was annoying to him since it meant twice as much writing...)... and guess what? He'd make similar mistakes when checking and sometimes a previously correct answer would end up no longer correct!

He'd also do things like accidentally read part of one problem into a different problem or copy from a previous set of a problem worked into the current problem - if there was more than one problem on a page, he could get them confused and never realize what he'd done.

We moved ds to a different school for 6th grade, showed the school his IQ tests plus his state testing in math and told them which courses he'd completed in Aleks, and they subject-accelerated him to where he needed to be in math without any questions. He's made straight As in math all year, so I think that puts to rest all the doubts his previous school had about his math abilities. OTOH, that doesn't mean there haven't been challenges. He's now using his laptop for all of his math (he uses Efofex)... but even though he can write equations, draw graphs etc on the laptop, it's not as smooth, quick and effortless as it is for kids who can rely on handwriting. I'd guesstimate his math homework takes him 2 times longer than the other kids in his class - and that's just due to mechanics, it's not understanding concepts - he picks up the concepts quickly. And he still makes copy errors, even on the laptop!

I'm sorry I've rambled on so long - and I'm sorry I think it probably sounds very discouraging. It isn't meant to be - I hope it might be a little bit encouraging. In summary, we never did get the accelleration we thought our ds was capable of or needed in elementary and we spent soooo much time focused on fighting for understanding of dysgraphia and appropriate accommodations. It was frustrating and far from ideal - but it turned out ok, actually better than ok. We didn't get any accelleration out of Aleks during elementary school, but once we had ds in middle school at a school that was receptive, he had done the work and was ready to jump ahead in math. All the advocating we did in elementary was difficult and frustrating, but it gave ds the foundation he needed to begin to become his own advocate as he grew into middle school.

Gotta run - I hope some of that made sense!

polarbear

Just be careful, because the accommodations you may think you're going to get aren't always the accommodations you end up with in the end. For example, in many states, dictating answers to a scribe for standardized testing requires a student to spell every single word in an answer and state every punctuation mark. Talk about a tedious process!!! Once that accommodation is described to parents, they often end up refusing. So, while not expecting perfection, I would still work with your son on building his skills independently when it comes to written explanations of his math reasoning.

One thing you can do to help him "practice" is to have him explain his process to you orally while you "take notes" for him -- not in complete sentences, but just copy down key words or phrases that he said. Then, have him take your notes and turn it into an actual written answer. It adds an important step to the process of pulling an answer from his brain and translating it into well-written sentences on a piece of paper. Over time, you can teach him to take his OWN "notes" on a scrap piece of paper before he shifts his brain into "composing sentences" mode. Yes, it takes "longer," but extended time for testing is basically the first accommodation agreed upon, so it won't be a big deal.

I think ALEKS would be okay if your goal is to enrich him over the summer, but I don't think it will result in his school giving him subject acceleration (just a hunch based on what you've described -- they seem to be pushing compacting instead, and his current teacher seems reluctant to recommend that for future years).

After all the discussion here about this problem, I'm intrigued: what was the official answer?

There are 2 solutions. One where S is $1.10 and the other was S is 86 cents.

There was one other S has 76 cents. There are limited solutions because of the number of coins. And because there has to be 4 quarters.

It's just as I feared, then: ignoring the call of the question, encouraging students to think they've got the right answer when they haven't, and even supplying a wrong answer. Note that 76 cents for Stella is not a combination that fits the numeric relationships of the problem:

Stella could not have four quarters and a total less than a dollar. In addition, Stella could not have three quarters and a total of 76 cents, or she wouldn't have five coins. If she were to have two quarters, her maximum possible total would be 80 cents, but Joan would have two quarters as well, already boosting her well over half Stella's total; thus Stella could not have only two quarters. For similar and obvious reasons Stella could not have only one quarter or none.

ETA: (Were you looking at a solution given by the teacher, or just trying to reconstruct it yourself? The latter seems likely.)

The way this question was obviously set up was to give a sense of accomplishment after a bit of easy work. A coin combination that suits the numeric relationships was chosen to be likely to be explored first, as many of the students would want to first try the maximum possible total for Stella, $1.10, to give the most room for a half total for Joan, and because it makes sense to explore the space starting from the end.

Thus in essence it's really another of the many EM problems about adding up coins to a total, except this time it's simply adding up ten coins to 55 cents, and throwing in a false sense of accomplishment for having achieved something greater. It's the same old EM approach, but with extra elements added that will help to insure that many students wind up feeble-minded, happy creatures that can't even read a problem properly, and who quit at the first opportunity.

Its a combinatorics problem. Its actually quite subtle. It can be modeled several different ways.

http://en.wikipedia.org/wiki/Combinatorics

The limiting constraint is that there can be only four quarters.

Given denominations of 1,5,10,25,50, and 100 cents, then there are actually quite a few possible solutions with at least 5 discrete ones. Just playing with a spreadsheet for a few minutes I found several solutions not yet mentioned in this thread.

An edit - I just found a way to generate a lot more solutions. I am up to 20+ now.

Looks like there are around 50 solutions or more. LOL.