When “a x b” is not equals “b x a”…

When I first saw this article on how some apparently correct answers were marked wrong, my first thought was “what rubbish is this?”.

More recently though, my wife talked about the incident in which a colleague of hers will be switching her kids to private school because her kids was deducted points for not showing the steps, even though their answers were correct. Is this justified?

If this is what is resulting from “Common Core”, I will dare say that the next generation of US-educated kids may just end up being the laughing stock of the world. And this is my take on it.

Math the Universal Language

Math defines operators that work using a certain set of rules and properties. For simple arithmetic, we know these rules instinctively. Take addition for example, we know that “1 + 2 = 2 + 1”. It is commutative. Multiplication is a little bit more involved. It is commutative on numbers but is not on matrices. “3 x 5 = 5 x 3”. Order does not matter.

Regardless what language Math is taught in, you get the same answer, hence Math is also known as the universal language. We expect aliens in space to know this too, because Math describes the underpinnings of nature.

Now in “Common Core”, “3 x 5” is taught as 3 sets of 5, and “5 x 3” is 5 sets of 3. This is a reasonable way of understanding the question. To teach the math, we impose a context on the operation by expressing in English. Now herein lies the issue; The rules changed… 3 sets of 5 is not the same as 5 sets of 3, even though the underlying Math is commutative.

By penalizing a child on it, we are now teaching kids the wrong basic mathematical rules which can be detrimental when solving equations which rely on these properties. To me, “preparing them for matrices in High School” does not fly as a reason. It is easier to teach something new than to correct a misconception. Here, we are simply teaching the wrong fundamental.

And it matters to me as a parent, because I have to go in and correct all the bad knowledge for my own kids.

Hence this is what I think teachers should do. In the absence of a clear context, a student should not be penalized for invoking the commutative rule. In other words, if a question is given as simply: 5 x 3. Then solving the question as 3 x 5 is correct. However if we ask a question along the lines of: “A grocer has packed 5 bags of 3 apples…How many are there in all?”. Then solving for 3 x 5 in this case should be penalized.

In the viral image, a question is given as “Draw an array to show and solve: 4 x 6”. The student is penalized when he drew a 6 by 4 array. This is retarded, because an array can look 4×6 or 6×4 when we simply turn the paper 90 degrees. In my definition, this is a “vague context” and so the student should not be penalized.

Getting to the final answer…

Moving on to penalizing students who do not show steps and who just writes down the final answer. I do not believe this is the way we should be teaching the kids. The way I was taught when I was a kid feels so much better. Full credit will be given on the correct answer but if the answer is wrong, there will obviously be no partial credit. Showing workings is just a means to get that partial credit and is of course, optional.

The rationale behind this type of grading is that not everyone thinks on the same level or at the same speed. “A step” that someone finds natural, might equate to “2 steps” for another person. Are we to penalize the person that can think faster and better? Does it really matter if they skip a step or 2?

There should be a balance between process and result. By penalizing a student for not showing workings, we are teaching him that the process is more important than the result. By giving full/partial credit, we are teaching that the result is important, and that showing the workings is a means of getting the process right. And that the process is there to let them get the correct answer faster, should they get it wrong.

There are real world implications as well. The best processes count for naught if it doesn’t yield desirable results. Conversely, the worst processes can work as long as the result is good. A company that overly focuses on process may find itself slowed down significantly, relative to more nimble, result driven competitors.



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