What's 5 x 3?
Recently, the
grading of this answer sheet went viral on the Net. The issue? For the question, what is “5 x 3”, the
kid had solved it by calculating “5 + 5 + 5” but the teacher marked it Wrong
indicating the correct way to solve it was by calculating “3 + 3 + 3 + 3 + 3”.
What’s the difference, the Internet screamed. I agreed. In fact, I had written a blog on a similar topic a year back based on Richard Feynman’s comment in The Pleasure of Finding Things Out:
What’s the difference, the Internet screamed. I agreed. In fact, I had written a blog on a similar topic a year back based on Richard Feynman’s comment in The Pleasure of Finding Things Out:
“The
whole idea was to find out what x was and it didn’t make any difference how you
did it–there’s no such thing as, you know, you do it by arithmetic, you do it
by algebra.”
Of course, this
being the Net, you can find a counter argument (if you choose to look). Sure
enough, I found a good
one by Brett Berry as to why the teacher may be right after all:
“In
the above problem 5 x 3 is equal to 5 + 5 + 5, but they’re not necessarily
equivalent.”
Huh? Another
example from Berry makes the difference between “equal” and “equivalent”
clearer:
“Here’s
another example: 30 ÷ 2 is equal to 15. But does 30 ÷ 2 represent
multiplication? Is it equivalent to repeated addition? No, it represents
division. Therefore, 5 + 5 + 5 is equal to 30 ÷ 2, but they are not equivalent.”
Now apply the
above line of thought to what multiplication means:
“In
the definition of multiplication, the first factor is the number of copies and
the second is the number being repeated.”
Now suddenly the
teacher doesn’t seem wrong. But something still doesn’t feel right, does it?
Why obsess on the way to solve it? Berry has an answer for that: as adults, we
know both ways of solving the problem, but pick the one we know better or the
one that works faster. But with kids, we need to teach them both methods. And the
only way to ensure they learn both methods is to force them to apply the
prescribed one.
As someone who knows software, I am also bought into another point by Berry:
As someone who knows software, I am also bought into another point by Berry:
“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.”
And like it or
not, knowing the basics of computer science is increasingly becoming an
indispensable thing in most fields.
But even if you’re not bought into the computer science connection, perhaps you can still agree with Berry’s closing:
But even if you’re not bought into the computer science connection, perhaps you can still agree with Berry’s closing:
“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 slam and discredit them on the internet.”
There are times when the teacher does something like what this teacher of your blog goes right and sometimes wrong. Should we not look beyond labeling of the answer to the question? Should be not take up the broader issue of "what is teaching?" and "how well it can be done?". Neither specific details not the rights or wrongs of them may be the direction on this matter.
ReplyDeleteA good teacher, as the old saying goes, is one who "draws out" and not one who "puts in". In situations like this, if the teacher knows his/her subject, should be knowing that a x b is equal to b x a. At the level there was no immediate need of discussing, mathematical regulations such as commutative law etc., hence can be set aside. The teacher, if he or she so wishes can gently draw the students to follow a pattern or procedural way of taking up the repeated addition process from the multiplication sum given. A good teacher may actually admit both a x b and b x a happen to be OK in this case. Instead he/she just marked what was not done in the order that he or she preferred and possibly taught as wrong. Maths, the subject, actually states that what is done by the student is valid mathematics and right. Maths never decides issues on the basis of what it likes and what it dislikes, which is a human prerogative!
I can certainly say this: this person was poor teacher because he/she was rigid in attitude. The ability to guide simply vanished with that failing. However, this kind of a thing happens in life to ever so many people in ever so many situations. It is not easy for everyone to view things from a higher plane, so that the perspectives will appear clear enough to avoid firm shut-off mindset. Many of us are of poor ability to view from higher planes about issues, not necessarily classroom lessons.
I recall this example, again in the teacher-student setting. At a college level, one of my relatives said this happened. In a class test, the question "What is entropy?" was on the paper. It was unusual to ask this question because this is not the physics domain that even many lecturers understand in clear terms. To ask this in the test didn't look right, knowing about 3 minutes should be the time to answer that one, after judging by the number and detailed nature of many other questions on the paper. Anyway, my relative chose to answer this question because the subject matter interested him and he had read well beyond what the college text book described or what the lecturer explained on this. He was happy with his answer. When the answer papers were distributed in the class he found the lecturer had given him zero for the answer! After the class the student went to the lecturer's place and confronted him, "What is wrong in my answer?" The lecturer told him bluntly that it was not what is seen in the text book. The book had only one paragraph on the subject and the only thing the lecturer was willing to accept was a line by line reproduction of what was in the book. All he wanted was a mugged up thing vomited out! He was not even competent to check the answer because his knowledge of entropy was nil. What can a student imbibe by way of either spirit of the subject or knowledge in the domain?