Why 5 x 3 = 5 and 5 and 5 Was Marked Wrong

The student got a grade of 4/6. There is no evidence from what can be seen on the photo if there are more than two questions or not - however it appears that after #2 there are instructions about showing work, etc. So it's entirely within reason to think that there were only two questions, yet many many people seem to accuse the teacher of being awful for not assigning partial credit. The evidence present on the page is not enough to conclude either way - there are solid arguments and inferences in each direction, yet everyone seems to argue from a "the answer was all right" or "the answer was all wrong" perspective.

Similarly - everyone is focusing just on the "learning to multiply whole numbers" aspect of this. But I presume this assignment was given in the course of a broader teaching curriculum. Perhaps the goal is to get the kids to arrange things in a certain way, because it provides a bit of foundational knowledge for next steps. Some next steps where pushing the "3+3+3+3+3" version of this is a "better" representation:

* algebraic concepts: 2x becomes x+x, etc. I don't know a way to write 2x in terms of 2+...

* fractions: (this is basically the same as above) 4 * 1/2 is 1/2 + 1/2 + 1/2 + 1/2, but again I don't know how to write it in terms of 4+ without requiring a bigger transform of first doing the multiplication then switching the sign.

* Matricies - here is a case of multiplication that isn't commutative

Point being - when teaching sometimes things are left out at first, for the sake of a simple consistent framework to built more concepts upon. Later - those concepts can help to understand additional properties or adjustments to the original facts. It's not required to teach the rules of commutative and associative and so on immediately.

Another funny thing about this is everyone just assumes that the paper as shown represents the totality of output from the teacher. It's entirely possible that after the quiz or assignment, the teacher gave a lesson on why this (perhaps common) 'mistake' is wrong.

I guess I'm rambling off my original point - but I don't really understand how this entire thing is causing so much vitriol and hate and unfounded speculation - other than a bunch of people projecting their own frustrations with some shitty teachers they had in the past.

It’s more important than ever for students to understand the difference between equal as a result and equivalence in meaning from a young age because it is a fundamental computer science concept (...) Equivalent means not only are they equal, they are also of the same data type. In other words, they mean the same thing.

Except that this point is totally misguided because 5+5+5 and 3+3+3+3+3 are, in fact, the same thing. A member of the set of natural numbers, commonly known as 15, and that you can write as S(S(S(S(S(S(S(S(S(S(S(S(S(S(S(0))))))))))))))) if you have the patience to do so.

In fact, the author uses the == and === operators in JavaScript to illustrate his point, but of course, 5+5+5 === 3+3+3+3+3 resolves to true in Javascript, and in any language under the sun that compares stuff by value.

And then, if you compare by reference, the result of == or similar operators doesn't depend at all on whether you made your integer by adding up three fives or five threes: in Java,

  Integer i1 = new Integer(5+5+5);
  Integer i2 = new Integer(3+3+3+3+3);
  Integer i3 = new Integer(3+3+3+3+3);
  System.out.println(i1==i2 + " and " + i2==i3);

So I don't see how this nonsense would teach kids anything useful about computer science. The only thing it can do is confuse them.

equivalent is defined as, “equal in value, amount, function, or meaning.” In the above problem 5 x 3 is equal to 5 + 5 + 5, but they’re not necessarily equivalent. Equivalence relates to meaning, so it depends on the meaning of multiplication, as the directions indicate.

First of all, the sign in the exam statement is an equal sign, not an equivalence sign. So if 15 is equal to 5x3, what the student wrote is perfectly fine. Also, "solving" a multiplication means finding out its value, which is what he did.

Secondly, you know what also is a fundamental computer science concept? The logical operation "or". The definition of "equivalent" in the blog is reported as “equal in value, amount, function, or meaning.” "Or", not "and". So the definition doesn't say anything about equality in meaning (however you define it) being a necessary condition for equivalence.

Maybe teachers are "experts on child education" but that doesn't exempt them from knowing something about maths if they are supposed to teach them.

1. The teacher understands (and is expecting 9 year olds to learn) the pedantic difference between equivalence and equality. Keep in mind, this being elementary school, the teacher likely is a generalist and also teaches reading, science, and social studies.

OR

2. The teacher has a very rigid grading guide that specifies how much credit is given for any given answer/technique, and is simply blowing through 50+ tests at 1AM, applying this standard grading. Ironically, in this scenario, the teacher is just as pointlessly constrained to "following the rules" as the students. He/she may even agree that it's ridiculous to deduct a point.