I believe good math students will say 5x3=3x5 (they know that long before learning order of operation) and will find the result with the least amount of work and receive extra credit.
Regarding 4x6: I don’t think students are learning linear algebra maxtrices and vector spaces here. It’s just a rudimentary enumeration table. No rank is involved.
Well, if you consider it from the way we all were taught our multiplication tables, ie 1x2=2, 2x2=4, 3x2=6 etc, then 5x3 does indeed mean 5 3’s. And 3x5 means 3 5’s.
Some researchers at the University of Mass, Amherst did a study that included as a typical exercise: “Write an equation using the variables S and P to represent the following statement: ‘There are six times as many students as professors at this university.’ Use S for the number of students and P for the number of professors.”
Error rates ranged from 12% to 67% in groups ranging from calculus students (37%) to high school and university faculty in various disciplines. I have given this exact problem on exams twice to a total of approximately 110 college students who are supposed to have HS algebra skills, with a consistent error rate of 43%.
I agree with WasachWriter. Something was being done that was not working. Now they are trying to do something else. Maybe it will work better, but it won’t if parents and students don’t give it a chance. We’ll just go back to what wasn’t working and everyone will be happy. Except of course those of us who have to try to get your kid through our class when he/she just wants to memorize the answers without grasping the concepts themselves.
Would this still be an outrageous story if the teacher had written, “Of course you are right that 15 is made up of three groups of 5, just as it is made up of five groups of three. Because the 5 was listed first, I expected you to use five groups of three ,but your answer isn’t wrong.”
I think this was a teacher using an answer sheet, a problem that can occur under any curriculum.
I think the teacher marked the problem this way because the state exams under No Child Left Behind have been written to enforce a particular interpretation of 5 x 3. The student would probably be marked wrong by the anonymous scorer on the state exams. So it’s an act of self-protection.
If a student doesn’t know that 5 x 3 and 3 x 5 mean something different, how will the student be amazed when he/she learns that they are actually the same thing? (Written partly in jest, but not 100%)
^Ha - but they are not THE SAME THING. They EQUAL the same thing, but they ARE NOT the same thing. Just like 2+2=4, but 2+2 is not THE SAME AS 4.
And here I thought that questions about identity were mainly confined to literature, like Oedipus Rex! 
I recall when my D was not allowed to read a book in 2nd grade unless she could define for the teacher any random word the teacher pulled from the text (and silly me, I always thought we learn words from the context when we read). That made reading a chore for her, because she had to read “baby books” in school. If they had thrown some of these silly math requirements at her, she would have found math to be a chore, as well. So glad my kids are out of school now.
As someone said above, the kid may have only gotten 1 point off (out of some larger number) due to the methodology. The 15 was NOT Xed out and another answer put in.
Some of my kids had a math curriculum (name escapes me at the moment) that reportedly taught it in a spiral fashion. In the beginning of the year the concept was introduced lightly. It was later revisited in a more complex way. It was sometime hard to help with homework when as a parent you know an easier way to do it, but they haven’t learned that yet.
Our district still teacher algebra in 7th grade to advanced students and algebra 1 is typical in 8th grade for almost all students. Common core has not changed that.
When my kids were in 1st/2nd grades, they learned to visualize with circles and stars, and by using jacks, and dice. That way they could see you could draw 5 circles and put 3 stars inside each circle, or you could draw 3 circles and put 5 stars in each one. When you were done with one picture (say 3 circles - really ovals), you would then draw new circles in a different color to show that 3 groups of 5 are the same as 5 groups of 3.
This was public school. It seems that the concept of “approach is flexible” is the goal, even in this instance, and that the test was testing the ability to break that down along rules not placed on the test itself. It was testing the baby step of 3 X 5. It had to be interpreted in the context of the class. But it should have been presented in a way that did not require context.
In the overall context of the pedagogy, it may be fine; but the test does not stand well in isolation.
I agree that this is probably just a case of a teacher following an answer key too closely and some curriculum company rushing out something that could have a sticker on it that says “Common Core aligned”. (Lots of that going around…)
The Smarter Balanced (Common Core) tests are supposed to be set up to accept multiple correct answers for the math performance tasks.
Maybe it’s just that I went to school in the Dark Ages and at a very “traditional” school even back then, but I learned the multiplication facts pretty much as an abstraction–lots of flashcards and timed tests. I don’t recall any talk about what it was for until well after learning the facts. (Maybe it was explained, and I didn’t pay attention.)
Skip counting by 5s is a Common Core standard for 2nd grade. Skip counting by 3s isn’t specifically on the list for any grade. So, the answer the student provided would be expected based on which numbers are easier to add multiple times. A general 2nd grade standard is “Work with equal groups of objects to gain foundations for multiplication.” The standards do call for more “using models” to understand the math, so that’s where drawing the arrays comes in.
3rd grade standards treat multiplication more abstractly after the model-based foundation in 2nd grade:
– Represent and solve problems involving multiplication and division.
– Understand properties of multiplication and the relationship between multiplication and division.
– Multiply and divide within 100.
– Solve problems involving the four operations, and identify and explain patterns in arithmetic.
@sylvan8798 That is surprising regarding setting up the equation even for calculus students. My daughter is currently doing more complicated word problems than that (add inequalities and multi-step to the mix) in 7th grade Common Core compacted math. (But, my husband tells the story of teaching a discussion section for freshman physics for premeds. He was trying to break down a problem students were having, and asked “How do you find the slope of a straight line?” and a student said “Oh, it’s been so long, I don’t remember.”)
^Your H’s experience not at all unusual, @Ynotgo. Researchers have also found that introductory physics students have trouble with the operational definitions of some concepts that we are utterly taking for granted as instructors, such as “vertical” and “horizontal”. Differences in language between the way words are used in an everyday vernacular and the way they are defined in science are also a big problem.
Things like that are why early instruction is so important and why it’s a pity we do this badly. To do modern physics, you need imaginary number and to understand those you really only need the concepts of a coordinate system with axes and vectors. This kind of stuff isn’t hard for kids with their flexible minds and I think one of the biggest issues is that materials are written by adults who know the stuff too well to put it into learning words or by people who don’t grasp the end meaning, etc. I hated every math book I had. Figured the game was this: take a hard concept, give the most extremely basic intro - like 2 + 2 level - and then skip 27 steps and say “thus” or “from this we see”. Never found one that actually explained.
Feynman used to say we should teach quantum mechanics to 10 year olds because it would make intuitive sense to them. He was right. When you’re older, the concepts of probability amplitudes - going back to vectors and imaginary numbers, etc. - are how the infinite states cancel, etc. become too hard.
I have difficulty interpreting this specific type of statement since my intuition is 6S = P. Checking this one is easy since substituting 1 for P yields S = 1/6. Teaching better problem solving and critical thinking skills would help more people write the correct statement.
Since there are six times as many students as teachers, the value of P has to be less than the value of S. That should be a flag that your intuition is off. The check you did to see if your intuition was correct is exactly the type of critical thinking that can assist a student in setting up formulas. Playing with numbers we know can lead to being able to write a formula to solve problems with any set of numbers. First, though, we have to master things like basic math facts. I think this is why the questions that were marked wrong bother me so much. To me, they seem to intimate that there is only one way to think when it comes to solving math problems. The truth is, there are many ways to think … and whether I think of the problem in terms of three fives or five threes, I am thinking correctly. Don’t dis my thought process. 
If the teacher feels that she has no control on how the standardized test will be graded, isn’t she better off marking the answer wrong, and then telling the student, “yes, I see exactly what you’re doing,and I have nothing against it, but the fact of the matter is you have to take a test where we don’t make the rules, so that’s why you need to change…”
Off-topic, but during the Jurassic era when I grew up, sans calculators (log tables and slide rules only), we often needed to do things like 2 digit times 2- or 3-digit multiplications, such 75 * 18 or 63 19 and we spent a lot of time working out techniques of doing them in our heads, or with minimal writing. eg. a slightly more complicated version of 53 like 5 * 358 may have been done in our heads as 10* 358/2 or perhaps 10 * (350/2 + 8/2) or 10 * (360/2 - 2/2) - If you checked with ten of us in class, the problem would have been done in a half dozen ways.
It’s funny how different the effects of technology have been, even among my own kids - one of them used it as a tool to get more time to attack more complex problems than trying to outdo a calculator, while for the others, the calculator just gave them a way out and didn’t bother use the advantage elsewhere…
Yes, the “Number Talks” our elementary district uses are just like your multiple ways of mentally solving 5 * 358 example.
But it’s not really a matter of “you need to change” it’s more a matter of "you need to learn these various methods of doing something.’ It’s not the 15 that matters. Why is it so hard for people to understand that?
Probably because people who have experience with math know that 15 matters very much and that you do not need to learn these various methods of doing trivial calculations.
http://www.wsj.com/articles/marina-ratner-making-math-education-even-worse-1407283282