Thursday, March 18, 2010


1. The final division practice session in the "Multiplication and Division" chapter of the 3rd grade Everyday Math Student Math Journal Volume I, p. 84:

2. The final division practice session in the "Multiplication and Division" chapter in the 3rd grade Singapore Math Primary Mathematics 3A, p. 102:

3. Extra Credit:

Should Singapore math students, like Everyday Math students, be solving 3rd grade division problems using counters?


John said...

5 people sharing 12 objects "evenly" hurts my brain.

If they meant "5 children have 12 markers to choose from and they wish to share them evenly, how many markers can each person have? How many do they ignore?" That might make more sense.

But as phrased, this question is malformed - if they are all black markers, people will just each take one, and its done. If they are all different colored markers, and conceivably each kid would want access to each at some point, the question can really only answer "at time n(initial), what is the maximum number of markers each student can have without their being a fight" Which is not exactly math, and probably is closer to "1" so the number of markers they can trade theirs for is maximized so that when they do wish to use a new marker the chance that their first or second choice is available is much higher.

If they each started with 2, then you either have to make sure they are really good at sharing (not math) or impose some sort of "only use those two markers ruke" (again, not math.)

I've known kids smart enough to look at questions like this. How can they intelligently answer a malformed question?

If for a moment I accept this form of teaching division (Lets make it more complicated and build a system around it instead of doing some remainder problems, then moving onto long division) - the questions need to be a lot better. They need to demonstrate situations where remainder is an important consideration, not just mathematical laziness.

"You are making 12 hotdogs, but you have to buy buns in packages of 8, how many buns will you have left over? Is there a way to buy the same number of hotdogs and buns? What might you do with the extra hotdogs and buns so that they don't go bad?"

The last question isn't math, but its useful - it demonstrates situations where thoughts of remainders make sense.

As far as long division? The idea of a remainder is built into each step, and its a good opportunity to demonstrate important properties of addition and multiplication when checking the answers "kids, this works BECAUSE of it is valid to break multiplication into a set of additions (or a set of additions of multiplications), also this is how base-10 works! etc...)

I think the long division route is a lot better for this.

Alex Francis said...

This idea of presenting division in terms of the metaphor of "assigning items to groups" seems intuitive, but it actually does children a disservice when it comes to (eventually) learining to divide by anything other than whole numbers. How does a kid who has internalized division as simply assinging items to groups begin to tackle the problem of dividing 12 by one quarter? What can it even mean to assign twelve items to one-quarter person? What is a quarter person? At least with the long division drills the kids have not overly reinforced an inappropriately over-simplified metaphor for division...

LexAequitas said...

This is the first comparison set where the Singapore math question makes me roll my eyes.

The answer is undoubtedly "Computers" -- and thus you could just transfer the numeric answers to the right boxes above.

Kennic said...

The Primary Math explains two ways of thinking of division. Sharing: making groups and finding how many in each group, and Grouping: putting that much into each group and finding how many groups. It also teaches that the answer is the same whichever way you do it: 12 divided by 3: make 3 groups and find that there are 4 in each group, or put 3 in each group and find that there are 4 groups. So 12 divided by 1/4 could be thought of as 1/4 in each group, how many groups?

The Everyday Math is showing both ways too. The first problem is sharing and the second is grouping, but the terminology is the same, which I guess makes it confusing, but kids may have done the same thing in class with counters or pennies and maybe won't find it confusing. This is practice, it is not teaching the concept.

I think the main issue is the level this is being taught at.
Long division is not even taught until grade 5 in most US texts. If it is taught at all. Most state standards do not include long division until grade 5. Even those newly released national core standards do not have long division until grade 5. If it is so hard for US students that it must remain untaught until grade 5, then I suppose grade 3 and 4 can only have simple problems like this example from Everyday Math.

In the Primary Math, they use counters to share and group numbers within 40 in grade 1. By grade 2 they do not use counters, but rather are supposed to think of the inverse multiplication problem to find the answer for division by 2, 3, 4, and 5. So in grade 3 they concentrate on remainders and briefly divide within 100 to introduce the algorithm and very soon dividing 3-digit numbers. They do start out doing this concretely, with place-value discs (discs with 100, 10 or 1 written on them) and making groups of hundreds, tens, and ones and renaming remainder. Then by half way through first semester they can divide 3-digit numbers by 2, 3, 4 and 5 without manipulatives. Then the rest of the first semester they work on the rest of the facts and at the same time multiply and divide 3-digit numbers by 6, 7, 8, and 9. 4th grade they keep practicing and do 4-digit numbers. Plus they divide decimal numbers by 1-digit whole numbers, something that is not done at all in US 4th grade (even in these new core standards). 5th grade it is assumed they know the division algorithm for dividing by a single digit and they go right into dividing by 2-digits.

They do have to show their work, so they can't just transfer the answers. It is a silly puzzle since the answer is so obvious, but it allows for self-checking.