The Ninja looked smug. He thought that was it, game over. I thought it had been a sneaky trick he’d pulled with the Ninja Bread, but I couldn’t change it now. I finally pulled my mouth apart and took a big swig of tea to eliminate the last of the Ninja Bread.
The Ninja’s smugness had led him into a false sense of security; he reached for his own slice of Ninja Bread and took himself a bite. I thought I would raise some points, to counter his (albeit very strong) argument.
“Ninja”, I said, “The first flaw I feel I should point out is a major contradiction in your argument. You have questioned the necessity of needless constants early on, and then finished with the grandiose statement that “I use two constants instead of one”? Surely it is much better to work with a solitary constant? Leaving less room for silly arithmetical errors?”
He didn’t answer, a victim of his own Ninja Bread ploy.
“To me, this is more a statement than an argument”, I said.
“It may be a special case, but that doesn’t mean it becomes irrelevant. Pupils are introduced to straight lines near the axis and as such the special case gives them a much clearer understanding of the geometrical properties of the line. Given the nature of the A Level content, and the fact that they will always be using the portion of the line close to the axes, then the “grown up formula”, as you call it, gives added chances for confusion and arithmetic errors.”
The ninja hadn’t been too explicit on this one; his argument was based on statements which could quite easily apply to both cases. I did feel I needed to add something though:
“I have found that your “grown up formula” is used purely mechanically in this sense. Pupils see it as a short cut, and follow it like an algorithm, rather than fully understanding what is going on. With $y = mx +c$ they can link it to what they know, and have a better geometrical understanding.”
I wondered if this might be due to my own preference to $y = mx + c$, and made a mental note to try and not prejudice students in the future. The ninja, however, did not need to know.
I could see where the Ninja was coming from with this, but:
"This is all well and good once you get to undergraduate level. It does provide a deeper understanding of what a straight line is, however it is of a level much higher than is needed for A-level students, and I have seen it confuse and perplex many of the brightest. If they are introduced to it later, once their understanding of coordinate geometry is greater, then they are better placed to comprehend it. A-Level needs to build on what they know, which is the geometry of the line around the axes."
This was the one I’d heard before, the one that is often banded around in this debate.
“Your examples make it seem a little more skewed than it is. You have written every step in the y=mx+c version, but have jumped over a few in your own version. A sleight of hand befitting a Ninja!” He glanced at his numchucks.
“I will, however, concede that it is slightly quicker to rearrange with fractional gradients. Although it is being used as a shortcut, rather than to enhance understanding”
I wondered what the ninja would make of my counter. I felt I now had a better understanding of why people preferred $y-y_0 = m(x – x_0)$, but I still feel that there isn’t a need for it, that $y = mx +c$ is better for understanding, particularly at A-Level, and that $y-y_0$ offers more chances for confusion and arithmetic errors.