Someone obviously likes to fuck with their students.

- even showing their work. The point is that the child DID show they understood the relationship between addition and multiplication. At WORST, the teacher could have corrected and NOT TAKEN OFF POINTS.

Thre problem is 5 times 3 not 3 times 5. While the answer to both is the same the way you read and write them is different. Math is done left to right and while that doesn't matter in this particular problem it will make a difference when more variables are thrown in.

variables there are. Not to mention, it is not true that the everything is done left to right - you follow the order of operations.

The kid wrote the equation incorrectly.

- including many of us who went on to get graduate degrees in math.

"Use the repeated addition strategy"

http://www.homeschoolmath.net/teaching/md/multiplication-repeated-addition.php

That lays out pretty simply the concepts being taught. The fact that cumulative law means the answer of 5x3 is the same as 3x5 has no bearing whatsoever on why the kid got the repeated addition portion wrong.

when expressed

5 x 3 = 3+3+3+3+3 = 15
3 x 5 = 5+5+5 = 15

Two completely different things and the kid did the later when the former was asked for. Not so important in that problem but the foundation is being laid for later more complex problems.

If you have a graduate degree in math you should understand that.

obviously before this homeschoolmath.net defined its own version of mathematics.

When this child does go further in mathematics, he or she will learn why 3 x 5 is equal to 5 x 3.

What I can see from your link is that the student is being taught rigid rules that he is told to follow. I assume your point is that it doesn't matter if there is a generally understood law of mathematics that says these two operations result in the same value - they are to follow in robotic fashion - preferably without thinking - the rules to get to the answer.

This teaches the child that math is not to be really understood intuitively but rather that it is something you just follow the rules on. (Then they have the chutzpah to argue that memorizing multiplication tables is just learning by rote.)

He wasn't marked off for the solution of 15 he was marked off for expressing the problem incorrectly.

Or are you trying to argue that you would say in English 5x3 is five times three and you would also say 3x5 is five times three.

It's not the same despite the fact that they are equal and it is an important distinction.

I would say that Five times three is equal to three times five. As to how you solve it via addition, you could decompose it as either 3+3+3+3+3 or 5+5+5. It is this method that has chosen to define the two quantities as different.

the question was how it should be expressed.

its both, memorization and conceptual understanding are BOTH a part of it.

I suppose if people think we should be testing kids on whichever arbitrary order someone decided was best for some particular pedagogical approach it makes sense. I'd prefer we teach kids about mathematical concepts.

beg to differ.

The article explains reasonably well why the focus is on process, rather than results.

The challenge will be the same as it was when set theory was the "new math" when I was in elementary school: Ensuring that teachers who teach it (1) understand it and (2) know why it is extremely powerful in more advanced math. Absent these to factors, they will get it wrong and do more damage that good (both in terms of understanding and in terms of instilling a hatred for meaningless busy-work - as most of us felt set theory was until we hit about the second year of college math).

You are of course, correct, that the commutative property makes the two expressions equivalent. But having taught several levels of math for years, I'm pretty sure the person who answered that question didn't think, "Hmm...5 x 3 is equivalent to 3 x 5 because of the commutative property. It would be easier to write the equivalent addition expression 5 + 5 + 5, than it is to write 3 + 3 + 3 + 3 + 3." It is far more likely that they got the concept of repeated addition, but simply just didn't understand which factor designated how many addends, and which designated the value of each addend.

The entire PURPOSE of teaching kids this is so that they understand what they are doing instead of doing it by rote. Later, it can help them quite a bit.

The child is doing these operations efficiently rather than incorrectly.

Who in their right mind adds five numbers when it is correct to add three? That ability to recast problems efficiently is crucially important for mental arithmetic, for estimating, and for using numbers in daily life.

The matrix just absolutely cracked me up. Recasting an arithmetic problem into a two dimensional structure this way makes it clear that the order makes no difference.

The fact that they have the same answer is not mere coincidence.

Which one is 3+3+3+3+3 and which is 5+5+5?

Not that it matters since, as has been pointed out numerous times, multiplication is commutative. But if you want to go by the most common definition of integer multiplication, it's closer to 5x3 = 5+5+5 than 5x3 = 3+3+3+3+3 (look at a books on real analysis). The fact that the people clamoring about how important the true meaning of 5x3 is or order of operations don't seem to understand these concepts themselves (IE, the numerous claims that 5x3 intrinsically means 3+3+3+3+3) should tell you all you need to know.

3X5 and 5x3 could mean either of those.

It has a clumsy solution and an efficient one.

It is bizarre to tell a child that they have to add 3 five times when it is correct to add 3 fives.

Read the fucking question.

Seriously. We wonder why math education isn't more successful. It's bullshit like this.

At worst, the child's answer should been scored as a correct alternative solution with the preferred one spelled out. Calling the answer incorrect just teaches the child to stop thinking.

Until I took Calculus.

I nearly flunked Algebra and aced Calculus. Why? Because results matter and rigidity produces nothing but more rigidity, not learning.

That kind of thinking is exactly the wrong way of teaching math. You are penalizing kids for not being automatons.

is the same as three added five times.

Introducing a 'strategy' which is not an aritmetic or algebraic rule actually breaks the law of commutation in algebra and will just confuse students when they move from basic arithmetic to algebra

The law of commutation:

a+b=b+a

aXb=bXa

There is no violation of order of operations in the kid's answer.

teaching simple rules fixation in modern math education

that has ever calculated physics wasn't quite so rigid. Word problems, calculating acceleration and volume was as easy as pie for me. Hell, when I was driving somewhere I used to do it in my head to calculate how many extra mph I could go and how much time it would save me. I kind of still do it.

"Orders of operation" and rigid crap like that drove me nuts. I can do tons useful things with math, geometry, calculus - I can't do a damn bit of anything with "orders of operation".

It's like defining that unless you arrived at the same answer with a different method means that the answer is invalid. Can you build a house without "orders of operation" but by knowing fundamental principles involved in the quadratic equation? Fundamental principles to cut the wood to achieve a certain angle?

Yes. You can.

How do you dig half of a hole?

With half a shovel. <--- That is orders of operation ideas.

It's meaningless.

An Abelian grape. (There were lots of those jokes, and elephant jokes too, in the 60s.)

and add it to itself the left-hand number of times, not vice versa.

Similarly, in the array, the left-hand number is supposed to be on the vertical axis and the right-hand number on the horizontal.

Yes, that is plain stupid.

It was always my understanding that new math was about novel ways to approach old problems. This approach seems as dogmatic as the old way.

Agreed, it seems pretty stupid.

s/he might have gotten extra credit!

Man, am I ever glad I'm old. This is exactly the kind of shit that I hated about school - where process trumps results.

Now that kid is just going to be confused. Great job teach! Let's make math even MORE byzantine for kids.

WTF teach?

If i have 5 sets of three apples each and I want to distribute those three apples to five different people, then I show that distribution as follows: 3+3+3+3+3. It's not the same thing if I gave 5 apples to three people, if my objective was to sort those apples evenly amongst 5 people. There is a difference. 3 multiplied by whatever means that you are multiplying the first number by the second number. This is important in teaching kids how to calculate distribution equations that can be very important in certain fields. It may seem trivial to you, but it's not. Multiplying 3 by 5 is not the same thing as multiplying 5 by 3, if your goal is to distribute a certain set or constant number of items evenly over a certain number of times.

3x5 and 5x3 both equal 15, but those are naked numbers while numbers in the real world represent actual things.

I'm getting old, but apparently not so old that that I don't see this as an exciting way to teach the fundamentals behind mathematical concepts.

3x5 can be seen as EITHER 3 sets of five, or five sts of three. Without being given an explicit context, it can be EITHER. The problem with this method is that it is confusing and obscures one of the most important multiplicative properties: the communative property. This method implies that the order of the multiplication matters mathmatically, but the communative property explicitly says it DOES NOT, and it will be extremely important for students to understand that as they progress.

This stark approach to teaching multiplication is a complete and utter fail. Without context, EITHER ANSWER should be accepted as correct. I'd eat that teacher for lunch over that bullshit.

while you and others are technically right about the answer, none of you have explicit context of this quiz. What was the exact question in the context of that class?

The entire point appears to be about deciding which sets are to be used, and how they are to be used. As others here have said, once the student understands the basic concepts of sets and the matrix, all sorts of new paths are open to them.

Assuming the student is bright enough to be in that class in the first place, the point deduction should not have been punishment, but a guidepost to get back on the right path. He, or she, was simply being a bit intellectually lazy, or confused, that day.

Any teaching method which directs a student AWAY from the actual underlying math is just WRONG. I can see wanting to use a summation method to teach multiplication. But if done so, the method should emphasize that either 3+3+3+3+3=15 OR 5+5+5=15 is correct because they both are. The grading of the quiz seems to imply one is correct and one is not, which is WRONG. WE should be emphasizing the math involved with helper methods as a mere way to get to the math. We shouldn't be testing kids on BS helper methods.

I am an engineer. But math didn't really come easily for me, and one reason is stuff like this. I now tutor kids in math and science and I'd say at the 3-6th grade level more than half their problems they have are trying to make sense of helper methods that are just not making any sense to them. Dear god, my daughter was able to do complex algebra problem using a method that made sense to her (and was mathematically sound), but struggled to get a B because she couldn't fill out the stupid little table the teacher insisted they use to solve the problem. Heck, with my degree and years of experience, I found the damn table confusing.

I don't give a crap about "operation of order" just results. Calculating volume or acceleration is just as good my way, and it produces the same result.

Fortunately, I had both physics professors and a calculus professor that didn't think it mattered either.

I nearly failed Algebra because I had one that if you couldn't read his damn mind on how to calculate the equation (which I couldn't) you'd still get points docked off.

Should be fired. Not kidding. Its teachers like this that lead to math problems in this country. Order is IRRELEVANT in multiplication.

Those exprssions are mathmatically equivalent.

Is our goal to teach kids math, mor some bullshit "helper method." This kid CLEARLY understood the forces at work here. But instead of a 6/6, he got a 4/6. And themkid is now likely embittered and confused.

We need to keep our eye on the ball. Our goal should be to educate, not force conformity.

and that is, surely, five plus five plus five, using repeated addition?

Your comment makes no sense.

If the problem was to divide 15 five times, your comment would make sense.

That seriously matters when you move on to higher level math. Which is why they're trying to prep them to think about it the correct way now.