Math Word Problems

Do you mean that she is supposed to get that there are 15 red and 5 green? If so, that seems hard to me for a second grader. I wonder how they were teaching her to solve that?

Janis (who doesn’t know the answer to your question!)

I can’t believe they are asking this in second grade. My son had a few questions in his math text just like this in third grade and I thought it was ridiculous then. This is the sort of problem I would naturally be inclined to solve through algebra. Alternatively, you could go through a number of steps to solve it, but setting them up and explaining why you chose the ones you did to a seven year old so he can understand, let alone replicate, is near impossible.

The teacher (like my son’s third grade teacher) must know very little about teaching math—which sadly is pretty much the norm for elementary teachers. Kids at this age should be doing one-step word problems that involve either subtraction or addition, possibly multiplication if they have advanced that far. They need to learn and overlearn (to use one of Victoria’s useful terms) what word problems call for which operation. Believe me there are a lot of fourth and fifth graders that still have difficulty choosing the right operation in a one-step word problem—probably because this wasn’t done properly in second and third grade. Ramping up the expectation to a multistep problem this complicated is truly counterproductive. A teacher who knows anything about teaching math would not have assigned this to a second grader.

Sorry for the rant—could go on and on about this subject, but allow me one more. I am betting that the math program doesn’t even call for solving many word problems. Chances are they have one unit on it sometime during the year (or maybe not even that) and then a stray word problem every two weeks or so. Word problems really need to be a daily thing—not just an occasional add-on—particularly when it this complex and seems to be given simply for the purpose of dviding the sheep from the goats. I’ll let Victoria take up from here.

Thanks for the vote of confidence!

The problem being discussed was that two kids have a total of twenty marbles, and one has ten more than the other, so how many does each have?
This was assigned in Grade 2

This particular problem is a typical introductory algebra problem, and as such would be given in a good Grade 8 or 9 algebra class, along with a lot of models of how to set up such a problem.
One point on that - **of course** you can figure this out without algebra. It is a typical example of an easy problem which is given, not because it has any intrinsic interest, but as a *practice* in setting up problems with algebra. If you concentrate solely on answers, you’ll mis the entire subject of the lesson, which is how to use variables to set up a problem in algebra.

That said, again algebra is not the only way to solve this, just the most efficient. This problem could be solved by estimate-and-check (a phrase I prefer to guess and check, as it shouldn’t be a wild guess). An advanced student in upper elementary school could solve this symbolically with numbers this way.

This problem also lends itself to a concrete modelling solution, for example playing with twenty marbles or twenty beads on an abacus and dividing them into two sets and then comparing the sets. A bright student age 8 or up could do this, if they had the backup — either some hints and a handful of marbles, or lots of experience in similar problems.

HOWEVER besides the computational difficulty, this problem also includes a classic Piaget word game and a developmental problem. In his researches, Piaget tried various questions of comparing part to whole; one of the classics was to show a picture of a family with two parents and three children and ask if there were more children or more people. Of course most of us, including adults, want to divide the family into two distinct sets, children and adults, and then we say more children. But the question actually is meant to ask us to take a subset (the three children) and compare it to the entire set (the five-person family). This leads also to the philosophical question of whether children count as people. Children age eight and up (into Piaget’s “concrete operational” stage) can get around the issue and understand what they are being asked and after a few examples give the correct answer as defined by the examiner. Children age seven and below (in Piaget’s “pre-operational” stage) cannot re-set their point of view like this, insist on two distinct sets, and insist more children, because this is the only way they can make sense of the question.

OK, back to the marbles. There are actually three different sets and three different numbers being compared here, as in the family photo. There is the entire set of twenty marbles, the subset of fifteen held by kid #1 and the sunset of five held by kid #2. The two subsets have to add up to twenty, and the difference between the two *subsets* has to be ten. A lot more complex than it looked on first reading, isn’t it?

In general, a six-seven-eight year old in Grade 2 is just not going to see the interrelations being discussed here. He can deal with two parts adding up to a whole, or a large whole minus one part goves the other part; but the subtlety of comparing part to part is just going to cause confusion and frustration in almost all cases. (There are the few gifted exceptions, but if you have one of those, you aren’t asking this question, you’re asking how to find enough challenges for him.) I would question the applicability of this problem being given as a regular assignment below Grade 6 or 7, when logic should be starting to come to the fore.

That said, this problem might usefully be presented as a “challenge”, *optional*, to keep those gifted kids thinking, and it would be reasonable in that manner - again with those marbles or an abacus at hand to demonstrate.

I ran into a similar issue on challenges when tutoring a boy in Grade 4. His teacher was a plodder. There were three hundred pages in the book, and about 150 days for teaching without other activities and tests, so the teacher faithfully did two pages a day, every day, without reference to start or stop of topics. The students were expected to do every problem on those two pages, every day. This was actually a great improvement on many other teachers in the area, who didn’t teach much at all. Anyway, the teacher ignored the very clear instructions in the text that certain problems presented in boxes were challenges or puzzles for advanced students; he figured if it was on the page they had to do it. My student was very gifted but had never actually been taught to read (amazing how a bright kid can pass multiple choice if nobody ever asks him to read for them), so he had rather a lot to catch up. He wanted to do all the work and get good grades, but he wasn’t ready yet for these challenge problems.

Your situation might be similar; the teacher wants to teach thinking skills and higher-order thinking skills and problem-solving, an excellent goal and one she should be encouraged to continue. But she is using either a badly-designed text, and there are lots of those out there, or a general-purpose text that is designed for a wide range of grades, or challenging problems that are meant for the gifted and advanced.

If that was the only problem assigned and the teacher told ‘em to get out marbles and play with ‘em until they found the answer (though I think I’d pick something less likely to roll around or be choked on by a little sibling!) then it could be a good exercise in working with numbers.
As a “get a pencil and paper and solve this and the next ten problems” exercise… I’ve got junior college students who struggle with this kind of problem.

We are using Singapore Math in our school and our third graders are doing similar problems. If you consider making bars(rectangles) for each unit (color) and making their total 20, then add a small bar which represents 10 to the first color, you have a great representation of the problem. The children can then figure out that they can subtract 10 from 20, which they can tell from the picture, which leaves 10 and this has to go into the other 2 bars evenly. That would make each bar equal to 5. Then 10 +5 = 15 and 5 are the answers.

This agrees exactly with my theoretical outline above, that this kind of problem is doable after age eight-nine (Grade 3 and up) using lots of concrete methods (bars, abacus beads) and lots of modelling and support from the teacher.

Problem-solving is THE thing to do in math; judgement is necessary on appropriate level and a planned and organized introduction.

Sue—Okay, maybe. But I doubt this was the context this problem came home in. When my son had it in third grade, it came home on a practice sheet for a unit test. The practice sheet was from the publisher, and the questions on the publisher-provided test were just like those on the practice sheet, but different numbers were used. It wasn’t an extra credit problem either—it was just presented as a test problem like any of the other 24 or so problems. But clearly the kids weren’t being tested on anything that they had actually been taught. The series was in general very light on word problems. Again, I go back to my theory that the problem was there to tell the teachers who was really gifted in math. Everyone else would just get it wrong, including kids who are actually very good (if not truly gifted) in math. I just don’t see anything positive, and a lot negative, about purposely baffling even very good students with a (non extra credit) test question on a second or third grade math test.