Reminds me of the poem Foundations Of Arithmatic by J.A. Lindon.
http://komplexify.com/math/harmony/FoundationsOfArithmetic.html
The original problem said “Use the repeated addition strategy to solve 5x3” (bolding in the original) and every example of repeated addition that I have found writes out the second digit. 1x3 would be 3, 3x1 would be 1 1 1. I don’t know why, but that’s how it is and I assume that’s what they teach. I don’t know of repeated addition is useful or not, but I have no problem with it being graded wrong. They weren’t learning addition or multiplication, they were learning repeated addition and the student did it wrong.
Sorry, morals don’t come into play in math. In math, most everything is black or white, right or wrong.
Again, if a student just happens to figure out that 5x3=3x5 before the teacher gets around to that part of the lesson, it can become a problem when they try to shoehorn it into other areas of math because the don’t understand why it just happened to work in that specific instance.
Can’t you think of some non-math area of your life where you happened upon a shortcut that worked, but wouldn’t apply to other, seemingly similar things in the same field? If you were teaching someone, wouldn’t you rather they learned the basics before the tricks?
The article I read on this basically said that the curriculum is trying to prevent habits that students will have to unlearn later.
Back when New Math was a fad some wag referred to it as “teaching basic arithmetic in terms of set theory”. 
Yes, because it makes no difference mathematically.
Arithmetic is an important craft which is being lost, partly due to the prevalence of calculaters. I see CompSci students who cannot do simple math, which in turn can start with laziness or apathy about arithmetic. Pedantry about a 3x5,5x3 distinction can only interfere, at least for many students, with the need to get basic arithmetic skills down.
If the Core Curriculum designers think that boring 2nd-graders with a 3x5,5x3 distinction will help them 12 years later when they’re working with matrices, then I agree with all the bad things anyone wants to say about those designers.
Wow–This is exactly this thread.
Not sure I buy it entirely, but here’s Friendly Atheist blogger Hemant Mehta’s justification.
No one has implied that.
No one has disputed that “five times three is equal to three times five” or “5 X 3 = 3 X 5”
They are equal or equivalent or of equal value, but they’re clearly not the same or identical statement. Try putting a 3x5 shed on a 5x3 lot and you’ll see that.
One of the commentors on that post found at least one dictionary with a definition of multiplication by repeated addition that would make the student’s answer right. I found definitions both ways so it seems wrong to me to dock the student points for the ‘wrong’ answer.
The student wasn’t docked for getting the wrong answer. The answer was correct. They got points off for not using the “correct method” that the teacher was looking for. Sometimes they’re trying to teach a process, and doing it the wrong way will get points taken off even if the answer is right. Has something like this never happened to anybody else here during our collective centuries of schooling?
I think basic single-digit multiplication is the wrong time to be picky about the method, but it wasn’t my classroom, and obviously the teacher thought differently.
I know what you mean but I’m going to phrase it differently. The answer was not correct. The question didn’t say show one of the ways to solve 3x5 using addition. It specified a particular method for which there was only one answer.
Now maybe the teacher didn’t do their job and explain it properly to the students, but there’s nothing wrong with the question and what the correct answer is supposed to be. You are getting at the core of the problem though, it’s not about teaching multiplication, it should be about teaching the nature of numbers and mathematics, and more importantly than that, the use of processes. If those kids can learn any one valuable life skill it is how to follow a set of steps to complete a process.
Do you have a shred of evidence for this claim? How do you know the student didn’t know that multiplication wasn’t commutative?
As RTFirefly points out, the rectangle visualization makes it completely obvious that three groups of five vs. five groups of three are utterly equivalent. One has to be deliberately obtuse to not make this connection immediately.
My entire pre-college mathematical education consisted of not only knowing how to solve the given problems already, but knowing them well enough that I could solve them in any required fashion. Tests consisted of writing down the answer first and then figuring out which steps the teacher wanted to see. It was a dumb waste of time, but at least I never had to guess on anything as stupid as the example here.
My point is interpreting the expression 5x3 as meaning 3+3+3+3+3 and not meaning 5+5+5 is simply creating an arbitrary convention. And, so far as I can tell, that convention** is not universal**. The dictionary definitions I found (not the best source for rigorous mathematical information I know, but this is gradeschool math) are split on which one is ‘correct’.
Arbitrary conventions are fine, if there’s a good reason for insisting that everyone do it the same way; like driving on the right in the US, or the right hand rule for vectors. But if there’s isn’t (at least local) unanimity on which arbitrary convention is correct, should you really be insisting on one. Especially if both lead to the exact same answer.
This is all doubly true when the kids are going to take it home and get help from their parents. Unless you’re teaching the convention to the parents, too, you’ve got nothing but a bunch of confusion.
Nope, none. How about you, got any to back up your side? It’s not the point, it’s not about a single student, it’s about learning math. Teachers can’t just say “okay kids, here’s your homework, I don’t care how you do it, as long as you get the right answer”.
Do you not feel that kids should apply what was taught that day to their homework? At some point over the coming weeks or months they’ll be taught the commutative property and how and when to use it, but that’s not in the lesson plan for today. Telling Johnny that since he was smart enough to sidestep today’s math lesson he doesn’t have to learn it is kind of silly. Honestly, why not just give them calculators and to hell with arithmetic. Most kids use them anyways. Next time you go to the store and your change is $4.21, say "hold on, I have 80 cents and watch them say ‘uhhhhh’ and pull their phone out of their pocket…and probably still not know what to add to what to figure out how to reconcile what you just did.
In my college Physics 202 class, if we started to run short on time during our labs, we figured out the “correct” answer for the experiments, decided on an appropriate deviation, then made up experiment results to match the answer we wanted.
But I’m not sure what that has to do with 3rd and 4th grade math.
Nope. But in the case of ambiguity, I prefer giving people the benefit of the doubt. This includes children.
This isn’t like the student that simplifies 64/16 to 4/1 by cancelling the sixes. The steps were correct, but possibly made an assumption that hadn’t been taught yet. Students should get credit if the steps are correct.
The idea that the entire class has to be taught a single way of doing things is bogus. It actively harms students that think ahead at all, and passively harms the remainder by implying that math is just about following a sequence of steps instead of a creative process.
I recognize the need for “lies for children” when it comes to education–sometimes you need to oversimplify things just to move forward, and correct the misconceptions later. But I do not think that LfC should be held as an end over and above actually knowing how things work properly. If a student demonstrates correct use of a process, even if that process is not quite what was taught, then I don’t think they should be penalized.
Calculators are even worse in this regard, because they reduce math to a particular sequence of button presses instead of a creative process, and impede understanding of what’s actually going on.
Take the percent button on most calculators. God, I hate that button. It confounds so many people, because it works so unreliably between calculator models. No student can go from training in how to use that button to the principle that percent is just a modifier for a number that means the real value is 1/100 of the number. It’s a simple principle that becomes horribly confusing when used with a calculator.
While I fundamentally agree with you I have three question/comments.
1)First off, math is rarely about being creative, elegant, yes, but math is very concrete, it’s not an art, it’s a science, but I fully understand what you’re saying
2)“possibly made an assumption” that’s my issue here. The student (possibly) made an assumption. You don’t know, I don’t know, maybe we should move past it, but if it was marked wrong, it means it wasn’t being taught, wasn’t part of the homework and wasn’t meant to be done that way.
3)That is a fantastic example of what I’m trying to explain. the student got the right answer for the wrong reason. Should he get it marked right or wrong (assuming they were being taught division and not reducing fractions)? He found a shortcut, it yielded the right answer but it, very clearly, works for an extremely limited set. Should the teacher say “Well, Johnny, you found a great way divide 64 by 16, so you don’t put your head down while I teach the rest of the class how to do it”?. (IMO) No, he needs to learn the correct way to do it.
Similarly, a student that just falls assbackwards on to the commutative property isn’t learning anything. He just found a trick and doesn’t know how or why it works or when it can and can’t be applied (and as I said earlier, it’s going to cause headaches if he applies it in the wrong place (hint, he’ll learn division before being taught this, what happens when he tries to apply it to that, or subtraction)) He did the problem incorrectly and it just happened to work, he still needs to learn the correct way to do it.
Math is a passion of mine and I firmly believe that students should learn each and every step as they go. There’s a few (new) ones that I think are stupid (I’m looking at you, tape diagrams), but I think they should learn how to do 5x3 AND 3x5 before they learn that they can just rearrange them if they can do one faster than the other.
This is such a tough one because the commutative property rarely doesn’t apply to multiplication, but there’s other things that you can’t gloss over, even if you happen to find the shortcut.
I don’t know if I’d call them misconceptions, but I do agree with you. IIRC, I was just handed the formula for a circle. (even if I wasn’t) I remember looking at the board in wide eyed wonder when we derived it in calc.
There are a lot of things in math you sort of have to accept, but being this isn’t one of them.
Also, don’t forget, at least in part, this is about learning multiplication tables. Very soon they’ll learn that they can rearrage the numbers so instead of doing 8x7, they can do 7x8. We’ll get there, but lets just go in order. It’s math, it’s important to go in order.
And all were advocating by letting a kid skip ahead to the commutative property is not understanding what’s going on and letting him use, what is essentially, a mnemonic device he happened to find. It’s like a kid that can memorize and spout out M-I-double-S-I-Double-S-I-Double-P-I, but wouldn’t know what it spelled if you didn’t tell them and couldn’t spell it without saying it like that.
Ask an 3rd grader WHY 5x3 = 3x5, he probably can’t tell you any more than 6th grader can explain to you what the weird looking check mark button on his calculator does.
Let’s wait until the teacher gets to that part of the lesson before they start use it.
Or (and I’d be for this, to hell with CC), if the kid wants to use the commutative property, he needs to be able to explain, at least kinda, why it works. Frankly, I’m not sure how they’d put it into words, maybe they could show that that rectangle is the same rectangle on it’s side, I’m not a teacher, that’s not for me to figure out.
To be fair, I think this is just something that anti-common core people have latched on to. They see it and scream ‘CC is stupid, get rid of it, look at this meme’. Don’t get me wrong, I’ve railed against CC in the past, I just don’t think this is an example of CC gone wrong. IMO, this is a somewhat cromulant lesson.
Once again, this has nothing to do with the Common Core. This test (assuming that it’s real, which isn’t clear) was created and graded by a single teacher. The teacher may not have understood the lesson that they were teaching.