This one comes courtesy of frequent commenter, John Friend. It is an example from Cambridge’s Mathematical Methods 34.
It amazes me at times what does and does not concern some commenters. That’s not intended as a criticism. Well, it is, but it isn’t. And, it is. It’s complicated.
I want to make a point about this example, and it seems to me sufficiently important to make the point hard, and as an update rather than a comment. We can all easily agree that the above example – the question and the solution – is bad. What I’m not sure is appreciated, however, and what I want to make absolutely clear is that the example is poisonous.
One can argue the benefits of a properly solid mathematics education, but one of the undeniably beneficial byproducts, if not the main game, is training and practice in reasoning. Mathematics is so good for this because, by and large, the objects and expressions and words and statements have precise meanings. This permits you to think about and argue and clarify the meaning and the logic in a clear and circumscribed setting. No one’s going to throw in a “maybe Newton’s cooling is just a conspiracy” in the middle of solving a DE.
This logical setting is undermined when a textbook/teacher/examiner uses terms in an unclear manner. Of course some lack of clarity is inevitable, because we have humans conversing. And, to be overly pedantic, particularly when there is no serious danger of confusion – and that is way too common – is to err badly in the other direction. Mathematics is a language for humans to converse with humans, and it should be used as such, balancing precision and clarity.
The example above, however, is not remotely a failed but good faith attempt at this. It’s not an instance of a clumsy error. The example above is consciously teaching students to ignore the clear and plain meaning of the words, and to answer an entirely different question from that asked. That is poison. Moreover, the question easily could have been worded to have asked directly for the symmetric solution. It is entirely gratuitous poison.
Now, I understand why commenters are querying more generally about the example. Honestly. You guys in the Methods World – God rest your souls – you have to figure out how to survive and how to ensure your students survive. It’s your job. But it’s not clear to me that it is sufficiently understood how bad is the above example. It’s not clear to me that your lessons on this will begin, for example,
“OK, so are we all clear on why the person who wrote this is a half-wit, and that in mathematics ‘convention’ means something entirely different? Great! Now, what does the half-wit want us to do …”
I just want to be sure everyone is keeping in mind, always, the Mathematic Oath, and which on this occasion we can reword as
First, don’t poison your students.