According to Wolfram MathWorld, which I would personally trust more than Wikipedia, it says: “In simple algebra, multiplication is the process of calculating the result when a number a is taken b times.” (http://mathworld.wolfram.com/Multiplication.html) Hence, on this basis, I would argue that the student is right if we’re going to be so technical about the order.
In any case, whether or not a x b is a groups of b or b groups of a, penalizing the student for using the commutative property is comical. Clearly this student understood the main concept of grouping correctly. Being forced into a box that the teacher may incorrectly be penalizing the student for (and people commenting on here buying into) just further proves to me the mass level of math ignorance in society. Don’t get me wrong, I don’t think students should disrespect teachers, especially since I’m a math teacher myself, but with plenty of “math” educators in America and Canada not having a proper math background themselves, I think we must be wary of simply just believing someone who teaches math knows what they are talking about. I know many “math” teachers who really have a limited background in math and will even go as far as saying they despise teaching math but have to, especially in elementary schools.
There is so much else I can argue with about in detail, such as no context being provided in the question on the test (such as 3 groups of 5 bananas) or the matrix multiplication example since that also has a context such as a 2 x 3 matrix being related to something like 2 equations comprised of 3 variables so obviously it can’t be randomly turned into 3 equations with 2 variables. However, I’ll stop here for now.