I love this concept. On your immediate first impression, do you think of “equal” and “equivalent” being the same thing? Think again. Understanding the difference between equivalency and equality could be a really valuable concept for students to understand as soon as they are old enough to comprehend it. It would help their learning process in school (and in the real world) considerably.
Equal is defined as, “being the same in quantity, size, degree, or value.” Whereas 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.
Take a look at this case in Medium. I think the school made a bonehead move here because if it is not covered in class there is no way for a student to answer correctly on their standardized test. But it demonstrates by it should be covered in class.
The question is also misleading because the colloquial meaning of “equal” is different than the precise mathematical definition. I understand the difference very well, but if there were a generic question like does 5×3 = 5+5+5, I would get it wrong too because I would assume they meant the colloquial definition. I wonder just how many standardized test questions make that mistake . . .
On the other hand, think of how often in the messy reality of the real world that this distinction is important. There is a direct parallel between this concept and the “Separate but Equal” issue in Brown v Board of Education. Equal is not equivalent if there are differences on other relevant variables that are also important but not part of the dimension that is equal.
If you know your computer science, you will also see the parallel with “==” and “===”. “4” == 4 is true, but “4” === 4 is false. Having the quotes is critical to differentiate equal and equivalent for the computer.
If you know matrix multiplication, you will see the fundamental difference between multiplying a 3×5 matrix versus multiplying the same matrix rotated into 5×3. In this case, we get a totally different answer. Not just a subtle one.
- Is this making a mountain out of a molehill? Or is it an important distinction for a student to learn by the time they get through high school?
- Are there real world applications? Or is it just a technical issue for computer system and matrix math?
- Does my Brown v Board example resonate? Or am I stretching it?
Image Credit: Cloakenn