My 3rd-grade daughter had to do her math worksheet last night which contained problems such as the following:
“Carly and Dave have 42 CD’s. Carly has 4 more CD’s than Dave. How many CD’s does Carly and Dave have?”
My first response was “OK, here’s how you solve it. Subtract 4 from 42, what does that leave you?”
“38”
“OK, then you divide 38 by 2 (because there are only 2 people involved) and…”
“Dad, we don’t do division!” (apparently they get division about 2 chapters from now)
“Er… how does one solve this problem without division?”
The problem was in a section of her Prentice Hall math book* called “Try, Check, Revise” where the apparent method for solving this problem is to guess at two numbers, see if they add to 42 and then see if the numbers differ by 4.
Huh? What’s that supposed to teach them? Why not, you know, just teach division?
Any educators (or parents) who can explain to me the purpose of such a time-wasting assignment, one which uses concepts that will be discarded in a few weeks? Obviously somebody sees some value in this or it wouldn’t be in the book, but I can’t figure out what it is.
*EnVision Math Textbook, Texas Edition, Authors: Randall I. Childs, Janet H. Caldweel, others.
Maybe they’re just giving her a taste of how math education works: first you learn the hard way how to do something, then, only after mastering that, do they teach you methods/theorems/shortcuts to make life easier.
They do this a lot in math books: make you do it the hard way, then the big reveal comes where they show you the “easy” way. About the time half the kids have decided the whole subject is stupid. Beginning calculus is full of this.
You can’t always trust a kid’s understanding of what the teacher said. My kids have told me too many times that I’m doing the problems the wrong way or “we haven’t studied that yet” for me to fully believe them anymore.
Without looking down the thread - 42 pennies.
Put 4 pennies in Carly’s pile to start. Then 1 in Carly’s, 1 in Dave’s pile until all the pennies are gone. At the end, Carly’s pile has 4 more than Dave’s does.
But that may be too tedious for a 3rd grader. And a pain for the adult who has to go get a jar of pennies (or something else small and countable)
Perhaps it’s meant to introduce the necessity of division before the concept? For example, if you’re introducing multiplication, I think there could be some value in making kids do it the clunky way first, e.g., Bill, Bob, Shirley, Mary, and Ted each have five apples, how many apples do they have together? Kids can add them, of course, but after a few tedious 5+5+5+5+5 calculations they’d probably realize the need for the multiplication operation themselves.
With that said, the exercise in question seems to be an odd way to introduce division, so maybe the book just sucks.
Huh, I’m pretty good at math, but the first thing that popped into my mind was just trial and error. Took a few quick guesses at numbers in the 10-20 range and found the ones that worked, no math needed other than simple adding/subtracting.
And even so, the “guess” method still utilizes division conceptually. The very fact that you “intuitively” understand that each will have less than 42 is, in fact, a form of division. Maybe that’s what the teacher (or the book) was trying to convey. If that’s the case, it’s a useful awareness to underscore, especially for standardized test-taking situations.
Actually, this sounds like a simple algebra problem, and is the kind of word problem one might encounter in a relatively basic algebra class:
Let x = how many CD’s Dave has.
Then Carly has x+4 CD’s (4 more than Dave).
And together they have 42, so x + x+4 = 42.
But, many algebra teachers/books recommend, as an approach to such problems, before you go all x on the problem, guess what they answer might be, then check whether your guess fits the situation described in the problem. It probably won’t, but this helps you understand the problem, and then you can go back and do to x what you just did to the particular number you guessed.
For example, maybe Dave has 15 CD’s. Then Carly has 4 more, so she has 19 (= 15 + 4). So check: is 15 + 19 = 42? No? Then try again, but with x instead of 15:
We need x + x+4 = 42.
If she’s in third grade, she may be too young for algebra, but she isn’t too young to learn to understand what a problem like that is asking for and what the situation it describes means—which is often the part that people who struggle with algebra word problems have trouble with.
My kids had Try, Check, Revise type exercises scattered throughout their elementary school education.
From what I understood they served a variety of purposes, from familiarizing the kids with the process of parsing the word problem for key points, to showing them the “hard” way and then the “easy” way, to emphasizing the point that in algebra if you’re stuck plugging different numbers in and seeing how it changes what comes out can often help you understand the problem better.
The purpose of these types of exercises is to teach you skills in situations where you don’t know the correct problem-solving methodology. As you said, one method of solving this problem is to subtract out 4 CDs, divide by 2, and then give those 4 CDs to Carly. Something similar, which is what first popped into my head, is to first divide by 2, and then transfer 2 CDs from Dave to Carly for a net difference of 4 CDs. But what if you didn’t immediate “see” this methodology? Then you’d have to use trial and error.
Trial and error techniques are used in other disciplines as well. I remember in my beginning physics class, we were taught the equations of motion. Some students intuitively grasped how distance, velocity, acceleration, and time all related to one another right off the bat. For those students, the equations were no problem; the formulas had underlying meaning. Other students initially saw the equations of motion as a series of math problems with no particular physical meaning. For those folks, they were taught to fall back on a procedure of asking:
What do we know?
What equations do we know that relate to the information in #1?
What additional information can we derive given #1 and #2? Derive the information, and add it to the list of things we know.
Repeat #2 and #3 until we obtain the information we’re looking for.
This works fine (if painfully) until either the student develops a real-world interpretation of the math or pattern recognition kicks and they start intuitively grasping the problem solving methodologies.
This in and of itself is an application of division, because you deliberately didn’t choose a number over 42, but a fraction of it.
The OP wants to know how to do it–whether by “trial-and-error” or not–without division. For any approach which uses “Let’s-just-try-this-number-and-see-what-happens” to be at all useful, you implicitly must apply division, or you could be guessing forever.
Not really. True, I chose a number significantly less than 42 (which didn’t require division), but it wouldn’t invalidate anything I said if I had used the number 50 rather than 15.
Of course, if I had gone on to solve the algebraic equation I came up with, that would have involved division. In fact, I would have been doing the same operations the OP talked about doing (subtract 4, divide by 2) in solving that equation.
Well, you could do division itself by a “Let’s-just-try-this-number-and-see-what-happens” approach rather than by “applying division,” when you get right down to it: What’s 76 divided by 4? Pick an artibtrary number and multiply it by 4. If you get 76, your pick was the answer. If not, increase or decrease your guess, depending on whether the product was too small or too large, until you do get 76.
If it makes any difference, I’m coming at this from the point of view of someone who regularly teaches algebra to college students (of varying degrees of mathematical ability) but who has had no direct experience of elementary-level mathematics education since my own childhood. I’ve given problems similar to the OP’s example to algebra students (hoping they’ll solve them with the help of an algebraic equation). Sometimes they’ll take an approach like the OP did, and sometimes they’ll get it wrong (e.g. dividing by 2 and then subtracting 4).
The main advantage of this I can see would be to teach them how much easier division can make things? Ie rather than trial and error you can do it systematically.
Edit: As in I think the methods used to do it by trial and error may make it easier to learn division. Its not simply a ‘reward’ but getting them to get more of a ‘feel’ for how numbers work as well.
So they develop critical thinking skills and the ability to look at a misworked problem down the road and go “Woah! That can’t be right!”
Might sound useless now, but in another 15 years when she’s in nursing or med school, you don’t want her being the student who thinks that giving a patient a 200 Liter IV bolus is the right answer. And yes, I’ve had classmates who couldn’t immediately look at that and go, “Woah! That can’t be right!” :smack:
It’s developing an internal sense of mathematics and judgment, rather that relying on rote memorization of functions. (Not that there shouldn’t be some rote memorization in math, too. Good programs include both.)
I still use Try, Check, Revise in my business and personal life, even though they didn’t call it that when I was in school. I’ve seen situations where people who understood formulas came up with numbers that made utterly no sense, but they didn’t notice because they were using formulas rather than common sense.
