Many people have been asking about this viral math problem where a student is asked to use the repeated addition strategy on 5 x 3. The student is marked incorrect for responding 5 + 5 + 5 instead of 3 + 3 + 3 + 3 + 3.
It seems a little absurd, doesn't it?
Equals Versus Equivalency
Just because two things are equal doesn't mean they're equivalent.
Equal means two things are the same in quantity, size, degree or value, whereas equivalent means they are the same in value, function and meaning. In the problem, 5 x 3 is equal to 5 + 5 + 5, but not necessarily equivalent. Equivalence relates to meaning, so it depends on the definition of multiplication.
To be able to apply the commutative property, we must establish a definition of multiplication that applies meaning to the operand order. If operand order isn't important, the commutative property serves no useful distinction.
In Wikipedia's definition of multiplication, the first operand is the number of copies (multiplier) and the second is the number repeated (multiplicand). If this is the same definition the teacher taught, 5 x 3 is equivalent to 5 copies of 3, or 3 + 3 + 3 + 3 + 3. It is equal but not equivalent to 5 + 5 + 5 because 3 copies of 5 represents something different. For example, 3 bundles of 5 bananas is different from 5 bundles of 3 bananas although they total to the same number of bananas because their structures are different.
I get the difference but isn't it a little harsh? Well, it depends.
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 and is awesome that the student realized this! Kudos! What a mathlete!
If the teacher has not covered the commutative property, it might be unwise to let a student continue with this line of thought if they don't understand the reasons. It's common for beginners to get confused as to when it's okay to commute operands in binary operations. This is understandable if one sees that sometimes it's okay to switch the order and other times it's not and hasn't learned when or why yet.
By focusing on the meaning of these operations as they relate to repeated addition, arrays and area, teachers are creating a deeper understanding and trying to prevent students from making mistakes.
But it's the right answer, does it matter if the meanings are different?
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.
In programming, there is a distinction between testing if two things are equal or equivalent (aka identical). Equal means they have the same end value. Equivalent means not only are they equal, they are also of the same data type. In other words, they mean the same thing.
The Right Mindset for Matrix Multiplication
The student was also marked down for drawing 4 x 6 as an array of 6 rows of 4 instead of 4 rows of 6. Why is this important?
Not only does this adhere to the definition, it also teaches students the correct order for diagramming matrices which is rows times columns. Keeping rows and columns straight in matrix multiplication is vital.
Matrices are labeled using a row by column notation, m x n. To multiply matrices together, you multiply the rows of the first matrix by the columns of the second. The number of columns in the first matrix must equal the number of rows in the second, or else you cannot multiply them together. For example, we can multiply a 2 x 3 matrix and a 3 x 4 matrix together. But if we swap the order there will not be sufficient rows and columns and the operation cannot be performed.
Order is essential in the definition of multiplication because not all forms of multiplication are commutative, such as matrix multiplication. This is why commutativity is a separate property.
Respect the Teachers
They are experts on child education. They have the best intentions for the students in mind. This teacher made a decision based off a lot more information about the student and class setting than we can tell from a photo. We don't have to agree with it, but we can respect it. If you are confused, ask them why they did something before you discredit a teacher on the Internet.
This post originally appeared on Math Memoirs.