Forgive me if there’s a thread on this already but people must have seen the test marked wrong because 5 x 3 was written as three 5’s instead of five 3’s. And the second problem is marked wrong for the same reason, the student made an array of six 4’s instead of four 6’s. [url=<a href=“https://medium.com/i-math/why-5-x-3-5-5-5-was-marked-wrong-b34607a5b74c#.jcjcu4yc2%5DHere%5B/url”>https://medium.com/i-math/why-5-x-3-5-5-5-was-marked-wrong-b34607a5b74c#.jcjcu4yc2]Here[/url] is the most reasoned defense I’ve found of why the problems are marked wrong and I find it infuriating.
To me, the defense is idiotic and it even says that: “If the teacher has already taught the commutative property of multiplication (the law that says a x b = b x a), then this is a fine substitution to make.” Or, in simple words, the answer is correct but it’s marked wrong, which means this isn’t math but a reading quiz which ignores the basic rules of math; commutation is the most basic law of addition and multiplication and is what separates those operations from subtraction and division. That means, to me, this is an example of poorly written crap testing: if you don’t want people to use commutation, then say that because the entire field of mathematics depends on using what is known UNLESS it is specifically excluded. This fundamental of math (and physics) is what makes it interesting and hard; to prove things, you have to show this is the only possible answer or you need to exclude all the other cases, even the absurd ones.
Mathematics works by picking away at the cases. A famous example is Fermat’s Last Theorem: people first excluded n=3 and then a bunch and then an actual solution is derived using a bunch of weird techniques creatively applied because no one said the problem had to be limited to basic algebra but rather was open to creative solution. A more important example is the entire field of set theory: the entire axiomatic structure is designed to define the limits of what nature excludes, meaning a naive universal set. If they didn’t state the specific limiting requirements of a set, then it could be anything even though that “anything” would contain paradoxical contradictions. In physics, a famous example is renormalization: if you don’t think about the denominator going all the way toward 0, meaning distances become infinitely small, then you’re just excluding all the hard or even impossible cases without a bleeping reason. Or in the news this past week, a possibly complete Bell’s Theorem test: back in the 1960’s John Bell showed that if you add up the probabilities from entanglement experiments then classical physics says the result would be this and quantum says it would be this. For decades people have been coming up with better and better tests even though each test shows the same result, that the probabilities always match quantum (and thus exceed what classical physics allows). Why bother? Because just like in this simple math problem, if it’s not excluded, then you have to consider that it’s not absolutely proven.
To me, marking commutation wrong is a sin and, in this case, a sin that covers up the original sin of not stating mathematically what was excluded. The importance of clearly excluding or including operations and characteristics is a far more important math lesson, one the teachers and writers of this test don’t grasp. Sadly, the defense I link to also gets this wrong by citing Javascript: the reason “4” is not 4 is defined in the language rules that you learn, meaning it’s specifically stated as an exclusion that while the value of “4” equals 4 the quote marks surrounding 4 define it as a string which may take other values and that is not the same as the value 4. You don’t learn Javascript without learning that. In fact, you can’t learn any computer language without learning that. It’s as basic as knowing you can’t switch order in subtraction.
The problem with this math test isn’t the answer but the wording of the test itself. It tests what the teacher says the words mean: that “using the repeated addition strategy” somehow in this case excludes the basic operation of commutation. That isn’t math.
