Uh, sorry?
Seriously, look at the diagram for 4 x 6 in the OP’s link and read the diagram vertically, as one might, if one had been trained in some language other than English. In that case, there are clearly 4 sixes there, not 6 fours, as claimed by the teacher. (Okay, they are a little wobbly, but still . . . )
I don’t think that hitting the students over the head with order conventions is fruitful, when they are young (when they reach matrix algebra, sure, and it’s useful to know that not all operations are commutative).
@WasatchWriter, I think I understand what you are saying. The goal is not to drill facts into kids heads, but to teach them the relationship between the numbers so they understand what they are doing, not just parroting back a memorized fact. Years ago, my dad (a shop teacher) was given a section of basic algebra to teach. I still remember him at home with a number line asking me to show him how to get an answer for something. I didn’t know how to show him… I just knew. His shop kids weren’t always in the AP classes and he could relate to them not understanding something and wanted to give them a different way of understanding. I think now he was just ahead of his time. 
@WasatchWriter : Assuming your last comment was in response to me, you wrote: “One way of instilling a facility with numbers is to teach kids multiple ways to approach a problem.” So … how is calling 5+5+5 a wrong answer furthering that goal? The only thing you said that made sense to me was that the smart kids are going to be bored and sit in the back of the room.
@AboutTheSame As I said, I apologize if my explanation didn’t help. I’m just a random guy, ya know?
Hopefully, the teacher explained that the answer was right, but the method was not what they were practicing. If the teacher is teaching that one way to solve AxB is to take A groups of B then taking B groups of A is not reflecting what was taught. While many kids can easily get that AxB is the same as BxA, that may confuse others. Of course, the answer of 15 could be obtained in many different ways, but this test was on that method. Smart kids often lose points because they can do the problems in their head or know a short cut way to get the answer and don’t show any work or don’t use the correct method. I don’t know if this is a better way to teach math, but certainly the old ways were not working all that well.
Should kids be allowed to get the answer however they want to? Eventually they will.
Exactly. Normally I explain things as sensibly as mom2and, but I seem to have failed. But that is what I meant
The best way of instilling facility with numbers is to learn how to add, subtract, multiply and divide numbers using single most efficient methods that were used for years and memorize multiplication table.
We should not make kids use 5 different methods to do trivial arithmetic only because everyone is different and if little Johnny did not understand this method we will offer him another 4 artificial ways to do the same thing. We should not celebrate diversity of ideas in basic math. We should not make word problems out of trivial stuff, make kids draw and write poems about adding 1+1 and telling them stories about how important it is to learns arithmetic. You memorize multiplication table because you are told to do so. Every kid without learning disability should be able to do it.
After kids get comfortable with numbers they can spend time on real word problems about pipes that pump water into pools, bicyclists that ride from point A to point B, boats that travel with and against the flow and other good stuff.
But clearly the old methods have not resulted in kids loving math or doing well in it. Why shouldn’t we offer another method to learn math if it gives a few kids an aha moment? When tried and true fails, innovation may succeed.
Kudos to the kid who thought outside the box. I would have seen 3 fives as well. Ambiguous problem because multiplication IS commutative- even if the child hadn’t learned that yet. Different if it had been a story/word problem- then order would matter (3 girls did x 5 times or 5 girls did x 3 times…).
Reminds me of blue book college chemistry exams- some students got the same result with an equally correct but different approach than the professor envisioned- perhaps they deserved extra points.
The equation “5 x 3 = 15” on a placard would have made a good Halloween costume tonight.
We went to the moon on tried and true math. And when computers failed got back home on pencil and paper.
And a slide rule.
Making a chore of simple arithmetic is wrong . And assuming that kids find it hard is wrong too.
I’m trying to understand how inflexibly memorizing multiplication tables has anything to do with “tried and true math.”
My kids would, I think, be classified as the smart, sometimes bored kids. I guess. anyway, we talked multiplication as a concept when we baked cookies. Generally they fit in rows of 3 X 5 on the pan, or 3 X 4. So I showed them how you could count the sides and how multiplying meant 3 groups of 5, or 5 groups of 3, and not something you just memorized. Which is what this exercise is supposed to show, right? And they got it no matter which order we counted --horizontally or vertically. So to then say to the kid, now you’re wrong, because I meant the other set of steps, does indeed sound like social compliance math to me, and likely to lead to “I thought I got it, but maybe I didn’t.”
Concepts of why math works are good; imposing a level of arbitrary goes back to the original problem, I think.
Also, my H follows Christopher Danielson on Twitter (wrote Common Core for Parents for Dummies), and has met and spoken with him several times. Danielson discusses this on Twitter, with multiple mathematicians and physicists, who agreed that CC is not meant to teach that order matters. What they need to do get is the concept of what multiplication is, not the order.
The one under it with the 4 X 6 is even more egregious.
This is more like “in your face, gotcha,” math, than teaching concepts.
If you set a test question, all right answers should be marked correct. There is no sense in which the student’s answer was wrong.
This reminds me of the famous story about Carl Friedrich Gauss summing an arithmetic sequence (often said to be the numbers from 1 to 100) by pairing the numbers at the beginning and the end, and determining the number of pairs. He obtained the answer within a very short time after the problem was announced, and placed his slate with a single number (the correct answer) on the teacher’s desk.
The history of mathematics might have been somewhat different if the teacher had marked the answer wrong, on the grounds that the commutative and associative properties of addition had not yet been covered!
For the aficionados, there is an interesting historical investigation of the origin of this story and its accretions here:
http://www.americanscientist.org/issues/pub/gausss-day-of-reckoning
The author, Brian Hayes, traces the story to comments by Wolfgang Sartorius, Baron von Waltershausen, in a memorial volume published a year after Gauss’s death. Sartorius was a professor of mineralogy at the University of Goettingen, Gauss’s academic home, and wrote that he heard the story from Gauss. The subsequent development of the narrative is interesting, too.
Sorghum. That has not always been true even before common core. Kids would be marked wrong if they didn’t show their work or use the method being tested.
Agree that it shouldn’t matter f it is three groups of 5 or 5 groups of 3, but also not sure what the teacher here was trying to do. We are looking at this with adult eyes so multiplication is easy at this level. May not be for all little kids. Certainly a kid that already understands that 53=35 should be allowed to use that knowledge. But teachers often teach these things n smaller steps, which can be frustrating for the kid that is way beyond.
^which the teacher should realize and deal with accordingly, not lockstep. Isn’t that the whole point?
Well, yeah, a correct answer includes showing the work if required. The student’s solution was equally valid in all ways to the one demanded by the teacher.