I HAVE BEEN SUMMONED!
Frankly, I think if the teacher wanted those specific addends, she should have said, "by adding 3's." (Personally, I would have said, "Use addition to show that 5x3 and 3x5 both equal 15," thus having the child write both 3+3+3+3+3 and 5+5+5. This also makes it more clear that, while the result is the same, the order of numbers can be important.) But then, unlike most elementary-school math teachers, I majored in math for undergrad, so I actually have a decent understanding of how numbers work beyond basic fucking arithmetic.
You have to be able to recognize patterns in the way numbers work, or algebra is going to be really fucking hard to learn. The traditional algorithms are useful, but they don't make it easy to see those patterns. I've taught adults who can't look at a number and know whether or not it's divisible by 2. (This is also why I support a lot of "counting by twos, threes, fives, etc." exercises in the primary grades too--put charts on the wall of all the twos, threes, or fours up to 100* and the patterns for multiples and factors get much easier to see.) Students have to learn what happens when you add 2 even numbers, or one odd number and one even number, or 2 odd numbers. They have to see how multiplication works, or the times table is just a bunch of meaningless disconnected facts. This is why arrays and repeated-addition are used so much in common-core math classes--they make the concept of multiplication more intuitive, so that they realize that there are probably patterns to it. If you don't understand what it means to multiply, or that multiples of a number form recognizable patterns, then how are you going to be able to remember that x * x = x2?
Plus, by encouraging exploratory methods, you give students room to discover interesting math facts for themselves. I remember the "Eureka" moment I had when I realized that terminating decimals terminate in the decimal system because if you make them fractions, the denominator is a power of 2, 5, or 10. Other denominators make repeating decimals. This doesn't seem relevant to real life, but the sense of accomplishment from realizing this fact, and that the rule is based on the factors of the "base" of the number system, was huge.
That said, I do disagree with Hemant Mehta on the array thing, but that may also be because I despise matrices with a flaming passion. I failed so-called "Linear Algebra" twice because it was literally nothing but working with matrices. I'd work them out on paper, then check with my TI-83+, and every single problem would be wrong, and I wouldn't be able to find the error.
* Naturally, this would have to only go up to 99 for the threes, since 100 isn't a multiple of 3.