Most people are familiar with doctors’ “Hippocratic Oath”:
First, do no harm.
Yes, this aphorism does not appear in the Hippocratic Oath, which is also probably not Hippocrates’. And, yes, the meaning of and fealty to this statement are not nearly so straight-forward. Still, the statement gives a beautifully clear and human principle, a guide on how to think about the difficult work of treating a person in one’s care.
Which brings us to mathematics. We feel that mathematics teachers need a similar guiding light, the Mathematic Oath:*
First, tell no lies.
As with the doctors’ oath, the implications of the Mathematic Oath are not obvious, respecting the oath is not always so simple. But it is a light, telling us the way. And it also tells us the wrong way: if you are a mathematics teacher and you are not telling your students the truth, then you are doing wrong.
*) Yes, it is an oath for all teachers.
Marty et al, this argument is going to be quite circular (for reasons explained in detail on many of your previous posts) but one has to first know the truth in order to tell not a lie.
Now, if a person has gone their whole life thinking the only tool in the world is (for want of a better example) a fork, then said person is going to have trouble telling students it is OK to use a knife.
In a world where mathematics teachers seem to know less and less mathematics (and more of a problem, seem to care less that they know less) how does one stop the spread of lies?
There is no such thing as “nerd immunity” unfortunately.
RF, a slightly pedantic point, but I’d argue that if one sincerely and justifiably believes P, and asserts P, even though P is false, one is not lying. Lying has connotations of moral wrongdoing, and I don’t think that’s always a fair description of an assertion of a falsehood.
That aside, I agree with the central point that teachers won’t (unless by cosmic accident) teach correct mathematics if they don’t know it themselves. And I also agree with the point that mathematics teachers have a responsibility to know and learn mathematics well enough to teach it correctly, so ignorance is no excuse.
An unfortunate aspect of this is that many teachers and school leaders are unconcerned about this, so long as students “get results”.
I’d add that teachers also won’t teach correct mathematics if they don’t have the time.
One example: How many times do you hear teachers say to students with total conviction “You can never cross an asymptote.” I’d bet that a lot of those teachers know that you can cross a horizontal asymptote and would not blink an eye at the graph of something like
.
But I rarely see Maths Methods teachers investing time and effort in defining a vertical asymptote (and hence explain why you can’t cross them) and defining a horizontal asymptote (and hence explain why it is possible to cross one), give examples of curves like
etc. This takes a lot of time but is well worth the investment.
Taking the time to do this is important since in Methods the study design only specifies functions that don’t cross their horizontal asymptote and teachers cut corners by giving rote rules for the equations of asymptotes:
has a vertical asymptote x = h and horizontal asymptote y = k. Students are rarely taught why these rules work and to consider asymptotic behaviour.
If the only sheep you ever see are white you will very quickly ‘learn’ that all sheep are white ….
A properly written study design together with salient exam questions would very quickly force teachers to teach this sort of stuff properly.
I am convinced that most teachers could and would teach this stuff if they had more time.
JF, I like your second last sentence, which I think negates your last sentence. I think most maths teachers are dedicated to their students, but less dedicated to mathematics, less convinced that mathematics is something to love. But a good study design means that a devoted (to students) teacher can and must and will teach good mathematics in a good way.
Thanks, RF. In the main, my interpretation of the Oath (which can and hopefully will fend for itself) is in agreement with SRK’s: it is the moral element that is at issue, and which is violated by consciousness of the deception. I understand that many, many teachers are insufficiently trained, or insufficiently confident, to be able to contradict the letter, much less the dispirit, of the curriculum and the VCAA and the textbooks and the PISA and the ACARA and the ACER, and all the other two-bit frauds that are taken as educational authorities. It takes genuine arrogance to shout sanity in an insane world.
On the other hand, a man’s got to know his limitations. Which means, I’m not sure teachers get off that easily. If a teacher knows they’re out of their depth, then it is their moral responsibility to learn to swim or get out of the pool.
You’re assuming the teacher is intelligent enough to know when they’re out of their depth. It takes genuine intelligence (and humility) to know this. In reality most will be lepers without their bell …
I’ve talked to roughly a billion teachers in the last 20 years, although that is a very biased sample. (Many teachers wouldn’t touch me with a barge pole.) My sense is that many teachers in this sample are aware of and are uncomfortable with their weak sense of the mathematical principles underlying at least a few curriculum topics. The difference is how much they consciously address this, how much they blame themselves (wrong), and how much they take responsibility (right).
From a student’s perspective, an effective ‘compromise’ I’ve thought to be useful is for a teacher to state where they are ‘lying’, and to accompany any slides or material with an asterisk and/or footnote to highlight that there is more to the content than what’s been said.
That way, students who are content with learning the bare surface remain blissfully ignorant, and those who are looking for more nuance can pursue it independently or ask their teacher later – but at least they are aware that there is something more beneath the surface. I think a real danger (besides Red Five’s point re incompetence or ignorance) is when the students aren’t aware that they are being lied to.
Of course in an ideal world there would be no ‘lies’ and everything would be taught in exacting detail and correctness, but I can also understand that teachers (wrongly or rightly) will, for the sake of expedience or convenience, or in sheer ignorance, take liberties with the truth.
Agreed; not lying can sometimes result in teaching far more material than lying. It’s probably a lot easier to say “for now, dy/dx is just notation” or “for now, dy/dx should be treated as a ratio of very small changes” and acknowledge/emphasize that it’s a lie you’re giving to simplify things, than to actually explain what dy and dx are (which, the way I learned it at least, relies heavily on linear algebra). Physics classes “lie” all the time by ignoring friction and air resistance, but they outright state that they ignore friction and air resistance since not ignoring these would require too much calculation for students, so clearly this is an OK thing to do.
Well, in physics we call these ‘lies’ approximations (or simplifying assumptions). And I completely agree that they are both OK (and indeed necessary: https://en.wikipedia.org/wiki/Spherical_cow).
(And of course theoretical physicists use Lie Algebra all the time)
And I completely agree with ‘lies’ in mathematics such as “for now, dy/dx is just notation” or “for now, dy/dx should be treated as a ratio of very small changes” that are declared as ‘lies’ up front.
For me a key element that defines a lie is deliberate dishonesty or deception, which is what VCAA is famous for. But then VCAA typically uses the Mathematical Oaf.
JF,
The simplifying assumptions in Newtonian mechanics have made the calculations so must easier for generations of physics students provided you state them …
eg small oscillations of a simple pendulum bob with string of length l of negligible mass ,ignoring air resistance,variations in g and the rotation of the earth …might demonstrate SHM with the period dependent only on l and g once experimental error has been adjusted for.
… but if you are studying topology you might not be able to differentiate your coffee cup from a doughnut
https://en.wikipedia.org/wiki/Homeomorphism.
Steve R
So I guess Red Five could also say “it doesn’t matter how much sugar you put on a turd, it will never be a [coffee cup].” (https://mathematicalcrap.com/2020/07/14/letter-from-a-concerned-student/#comment-3481) (Nuh, still prefer mine).
eddieofer, I’m not quite sure of the point you’re making.
“for now, dy/dx is just notation” is not a lie.
“for now, dy/dx should be treated as a ratio of very small changes” is a lie.
The fact that later on sense can be made of differentials doesn’t make the first claim false or the second claim true.
There is a truth spectrum and everything we teach sits somewhere on that spectrum.
So I see three issues:
1) Where we think we sit on the spectrum versus we we actually sit (this will depend on our understanding).
2) Where we decide to sit on the spectrum (this will depend on our honesty).
3) The combination of 1) and 2).
Regarding 2), I totally agree with Tau that when a teacher decides to sit at the ‘lying end’ of the truth spectrum they should “… accompany any slides or material with an asterisk and/or footnote to highlight that there is more to the content than what’s been said.”
Well, sort of …
Thanks, Tau. I think your first point is a great one. Even if it’s just a five-second flag that “there’s more here than I’m telling you”, I think that makes a huge difference.
I disagree with your second point. Exacting detail and correctness is usually the enemy of good teaching. You don’t use equivalence classes when teaching fractions to Year 2 kids, you don’t introduce limits in Year 11 with epsilons and deltas, and so on. That’s one of the reasons why the Oath is tricky; the notion of mathematical truth and falsehood, and the manner in which one might try to present it, is very much dependent upon the age and mathematical maturity of the participants.
Agreed on SRK’s point, although will concede JF’s point as well as potentially problematic.
As an example, today I was asked by a student why the graph of
is the graph of 
translated right, not left. I don’t think I lied to the student, but did suggest that a lot more detail on this was coming in future years (maybe it is, depending on their future teacher…)
I honestly try to tell the truth, but at the same time knowing that the whole truth is simply beyond many students at the point of their learning at which they ask the question.
Genuinely curious to hear how other teachers (and students) handle this conundrum.
RF,
My 2 cents
In the same that y = m( x – x0) is a straight line slope m cutting the x axis at x0 is
Translated x0 units to the right of y =mx for x0>= 0
Alternatively what x value gives a minimum for the parabola in each case?
As to potential white lies on future teaching that is no different from election promises perhaps
Steve R
Thanks RF. Of course one truth to tell about your example is that x-based transformations confuse the hell out of everyone. That’s the inverse-y nature of them: you’re basically asking what value of x gives you a certain output y, and after the –h is chucked in the formula, you need h more (which means you gotta move h to the right). Inverses are always confusing.
I agree with your general point, I think there’s often a trade-off teachers have to make, because of time pressure or simply the difficulty of the task: how much effort to put in to a genuine explanation of something difficult instead of going straight for the method. In VCE, sometimes, many times, you just gotta go for the method. But then Tau’s point kicks in: at minimum, you must raise the warning flag, high so it can be seen.