Or I could turn your remark around: what makes you, the adult, think you are so much smarter than the child that you can correctly judge what lies are OK to tell them? Are all adults really that smart? (Are any of us?)
Is it a "lie" to teach kids just learning chess that queens are worth more than any other piece and you should always protect your queen, even though there are advanced situations when it makes sense to sacrifice your queen for no immediate material gain?
In this specific case, introducing the ideas of "operations" and "counting numbers" into the picture muddies the waters, most kids who are just learning multiplication won't have any idea what you mean by those concepts.
"Multiplication is repeated addition" is not "an abstraction/simplification/shorthand". Doing that for multiplication would be saying something like "multiplication is a distinct primitive operation, but it works like repeated addition for whole numbers, so that's what we'll be learning how to do now." Is that really so hard?
> Is it a "lie" to teach kids just learning chess that queens are worth more than any other piece and you should always protect your queen, even though there are advanced situations when it makes sense to sacrifice your queen for no immediate material gain?
If you tell them everything you just said, no, you're not lying. But if you just tell them "always protect your queen", without explaining anything about why and without saying that there are some advanced situations where you might break this rule, yes, you're lying. It only takes a couple of sentences to add that extra information. Again, is that really so hard?
> In this specific case, introducing the ideas of "operations" and "counting numbers" into the picture muddies the waters, most kids who are just learning multiplication won't have any idea what you mean by those concepts.
Um, what? We're assuming they already know about addition of whole numbers. So it is simple to tell them "this addition thing that you learned, that's an example of an operation", and "those whole number thingies that you learned how to add, those are numbers". Once more, is that really so hard?
I think it's an arbitrary perspective, whether you treat the whole number case as primary or the generalization as primary.
People may prefer to consider the extended definition more "real", but I think the argument for going the other way is that usually the original limited form of something is more likely agreed upon by most people, whereas the generalization can be done in multiple ways which may owe something to history and culture, or context.
I feel like math is fundamentally different than physics, where the more advanced theory is objectively closer to correct. With math, it's more of an arbitrary aesthetic or social judgment. Nothing ever stops you from generalizing anything even more than anyone did yet, right?
Axiomatically, I think it can go either way. But one still has to recognize, as you do, that there are more cases than just the whole number case, and that what works for the whole number case might not work for other cases.
> the generalization can be done in multiple ways
There are certainly cases of this, but I don't think the case under discussion is one of them. There is only one generalization of the whole numbers under discussion here, the one from whole numbers to rationals to reals (and on to complex numbers if you want to take it that far, and still further on to matrices for some people in this discussion). There aren't multiple ways to do that: the rationals, reals, and complex numbers are all unique sets.
And yes, even when I'm in the learner's shoes, I prefer that a teacher start with an approximate model, and then refine it as they go further. Starting with the full thing is barely comprehensible.
It's not a lie to tell them "this is a simplified model that doesn't include everything, but you'll be able to add more complexities to it later". But that's not what "multiplication is repeated addition" says. You would say something like: "repeated addition is a simplified model of multiplication that works for whole numbers, but doesn't work well in more complicated cases that you'll learn about later".
I have never proposed doing the latter, so you are attacking a straw man. Once it's understood that you're teaching a simplified, approximate model, you don't have to repeat in every sentence that you're teaching a simplified, approximate model. You just have to not say it's "the Truth", without approximation and without qualification.
>"multiplication is a separate operation on numbers, but it works like repeated addition for the counting numbers you're familiar with"
>"repeated addition is a simplified model of multiplication that works for whole numbers, but doesn't work well in more complicated cases that you'll learn about later"
If you disagree that that's "long" or would feel that way in having to do it in every sentence, we can have a great discussion about that, but it is not a strawman -- you seem to reject the idea of giving the one caveat at the beginning of the course, and instead want to make each sentence rigorous.
If you recognize that your complicated sentences are probably not ideal for teaching math to second graders, then I think we're in agreement.
I do. Some of the words might be changed, depending on what words have been used to describe the operation of addition and the set of counting numbers. But, as I think I've pointed out elsewhere in this thread, the very fact that the children know about addition and the counting numbers means they know what an operation is ("a thingie like addition") and what a set of numbers is ("a thingie like the counting numbers").
> you seem to reject the idea of giving the one caveat at the beginning of the course
I don't know where you're getting that from. I have already said the contrary--once you've said it, you don't need to repeat in every sentence.
Little kids don’t care about nuance when they’re still having difficulties with carries in addition. Your ideal world where we first explain children that base10 isn’t the only way to represent numbers and whatever other caveats simply doesn’t exist.
They don’t have the knowledge required yet to even understand the scenarios when “exceptions to the rule” apply.
Kids actually understand the point, and they can decide if they care enough to ask more questions or if it’s good enough for them.
We faced that when teaching divisions. Saying upfront we’d explain falsehoods for the sake of simplicity helped set aside the more difficult questions (infinity, etc.) that came right after. We just say it’s the complicated parts and move on.
The best aspect is they are more receptive to have their mental model broken afterwards, instead of clinging to what you explained as fully true.
There's also the flip side response which is always asking a lot of questions about "well then what are the other sorts of numbers" and eventually getting shut down "we're not talking about that now" which comes back to the "who decides what you're smart enough to hear about now" question in its own way. Or "oh the teacher doesn't actually know what the difference is, or why this isn't 'true' 100%."
Despite the cutesy name, I don't think omission of detail is the same as lying. It's often impossible to tell 100% the truth. You probably don't even know it yourself!
If confidence gets priority over truth in explanations, then society will churn out people who are confidently wrong. This is a bad idea even if everybody does that and even if it is the traditional approach.
If people were honest that they don't know something then the world at large would be a lot nicer to live in.
I really don't see people being confidently wrong about abstract mathematics as an issue. I certainly don't know the rigorous definitions of an integral, but I can apply the concepts of calculus to everyday life perfectly fine. People who care about rigorous math can do rigorous math, and I'm glad if they teach me an intuitive understanding that lets me live a happy, productive life.
Personally, I find layers of abstraction necessary for learning. Maybe there are people who don't, I suspect they would have to be prodigies though. Tell me how to add fractions practically, then teach me the principles when I need to know them. Framing that as a lie seems wrong to me, I'd call it "bounded knowledge".
The qualifications are there; that's just a fact. Being told about them, or at least about their existence if not every detail of them, up front seems better to me than finding out about them later on when your mental model is solidified around the simplified version that you then find out doesn't always work.
> I don't think omission of detail is the same as lying.
Saying "multiplication is repeated addition", without qualification and without any caveats, is not "omission of detail". It's a false, categorical statement, i.e., lying.
As for where the line is where you stop giving details, obviously that will depend on the circumstances. A teacher who says "we don't have time to talk about that during class today, but yes, there is much more detail here that you can look into on your own" is not lying and is not saying the child is "not smart enough" to take in all the detail now. (Bonus points if the teacher says "see me after class and I'll give you some pointers on where to go for more information".) A parent who says something similar because they have to get dinner ready and the child needs to do the rest of their homework before bed is also not lying and not saying the child is "not smart enough". Limitations of time are a fact of life, and children need to deal with it just like the rest of us.
A teacher who just says "we're not talking about that", or who doesn't even know about the qualifications, or who gets snippy when a child asks a natural question, is obviously not doing the child any good; but that is because of the teacher fixating on a simplified model and treating it as "the Truth", so doing more of that won't fix it.
I'm not the one that used that word. If people are going to reference a concept that's been published in a book and uses that word, they should own it and be prepared to justify it. If they can't do that, they should not use the word in the first place.
To be clear, I'm not saying you are one of those people. But the poster I was responding to is.
They also don't need the "information" that multiplication is repeated addition, period, full stop. But that's what others in this discussion appear to be trying to argue for.
Arriving at an understanding of one particular special case of multiplication might indeed be a necessary step towards arriving at an understanding of multiplication in general, but that doesn't mean the two are the same.
Do you have a specific example of where you think you need to lie to teach physics?
My high school chemistry teacher had no problem explaining this to me, when teaching the periodic table of the elements, without telling any lies and without going into the details of the quantum mechanics involved. The Pauli exclusion principle and a general statement that the details of the quantum mechanics were out of scope for that class was enough.
> Explaining classical mechanics without including a bunch of caveats about relativistic speeds.
My high school physics teacher had no problem explaining classical mechanics including the caveats. The caveats took only a few minutes early in the semester. What's the problem?
We started with the JJ Thomson "plumb pudding model"
Then we learnt about Rutherford and the gold foil experiment which invalidated Thomson's model.
Then we learnt about emission spectrum and the Bohr model which invalidated Rutherford's model.
The whole time it was clear that nothing was "settled" the models were useful at explaining the observed phenomena but were incomplete.
Same thing in physics. We started with debate about nature of light and examined things like lumniferous aether and Michaelson Morley experiment, double slit experiment etc. Then Maxwell and electromagnetism leading to Heinrich Hertz and the photoelectric effect, which led to Planck and Einstein.
I certainly never thought I was being lied to. We were treading along the same path as those who had came before.
That's because you weren't. You said:
> The whole time it was clear that nothing was "settled" the models were useful at explaining the observed phenomena but were incomplete.
Which is exactly what I think should be told to students when a simplified model is being taught. But the people I've been responding to are advocating for telling students the simplified model as if it were exactly correct and covered all cases. That is what I am saying is lying.
Someone who thinks they understand physics without considering experimental accuracy doesn't understand physics.
People seem to be interpreting "lie-to-children" as meaning "[you must] lie[ ]to[ ]children". I don't think there's anyone who thinks that children should be actively deceived into not knowing about quantum mechanics, just that it's OK to use simplified explanations as one step in the process of learning.
It's much more productive pedagogically to get an intuition for slope and area than it is to get an intuition for compactness and the infinite intersection of open sets, but slope and area are A LIE.
If we are defining "lie" to mean "simplifying the model to allow students to understand what's going on" (which is what we're talking about) then basically every field of physics required some simplifications in order to teach.
For instance, it would be "lying" to teach Newtonian dynamics and Newton's theory of gravity at all without explaining relativity -- Newton's theory of gravity is "wrong" after all. It would be "lying" to teach electromagnetism without first explaining quantum mechanics. However it would be impossible to teach anything to a high-school or undergraduate student if you first had to explain everything else in order to teach F=ma or Gauss's law.
I think there's also a limit to how many times you can say "this is simplified" to students before they feel like they're being pandered to or that they're being told they're "not smart enough to understand the full theory". Physics was developed over many thousands of years, it's not really reasonable to expect a student to be able to learn it in one shot. Physics courses are often structured so that you learn concepts in historical order so that you learn how concepts were developed and what motivated future discoveries -- that is also a very useful thing to learn in addition to the actual physics.
I believe the same applies to mathematics education.
Source: I majored in physics. Every year we had an electromagnetism course and we learned that the model we had learned last year was too simplified to work with. Same goes for thermodynamics.
No, it isn't what we're talking about. We're talking about making categorical statements that are false. Telling students that you're teaching them a simplified model and there are complications that will be added back later is not lying. But telling them categorically that the simplified model is true and not even talking about the fact that it's a simplified model and there are complications being left out, is lying. And the latter is what the posters I've been responding to are attempting to defend.
> I think there's also a limit to how many times you can say "this is simplified" to students before they feel like they're being pandered to
I have never said that is necessary. Of course it's not. Once you've explained at the outset that you're teaching them a simplified model and there are complications being left out, obviously you're not going to repeat it with every single sentence. Any more than the people who are advocating telling students categorically "multiplication is repeated addition" with no mention of all the complexities lurking underneath are advocating repeating that with every single sentence.
> or that they're being told they're "not smart enough to understand the full theory"
I have never said they should be told that. Teaching the simplified model could be due to nothing more than limited teacher time and knowledge. There's nothing wrong with the teacher honestly telling the students that. Bonus points for encouraging them to investigate further on their own if they're interested.
That's not what we're talking about. Describing multiplication as repeated addition as a way of teaching a new concept to elementary school children is simply not lying.
Your issue is that the fact it is a simplified model is not being explicitly told to students, which is being called lying. While it would be false to say that a simplified model is the complete picture, teachers aren't standing up in class and saying "this is all there is to this subject, no need to learn any more!".
My main issue with your suggestion is that young students simply aren't going to remember a side comment at the beginning of their classes that this isn't the full picture -- so you will either have to repeat this regularly (which will be demoralising and confusing) or there really is no strong benefit to doing so.
You say that the "categorical statement" is not being repeated with each sentence -- but the thing is that when students use or discuss the tools they've learned, they're reinforcing their mental model of how the tool works. So in a way, the simplified model is being repeatedly reinforced in their mind. If you want to counteract it, saying once at the beginning of each class that introduces a new concept "this isn't the full picture" won't help most students overcome the issues they hit when they learn their old model doesn't work for more complicated problems.
> I have never said they should be told that.
You misunderstood what I said -- I said that constantly being told that what they're learning is "not the real theory" (or however you want to phrase) is demoralising in of itself -- it needlessly gives the impression to students that they're not smart enough to understand "the real theory". I never claimed that you said that teachers should say that explicitly.
It seems elsewhere in the thread that you don't consider it lying if one gives a general disclaimer that models aren't perfect, but I wonder whether or not that general disclaimer wouldn't inoculate the idea that multiplication is repeated addition? It's an imperfect model that super useful for all the numbers available to students when they learn what multiplication is. And I wonder further whether there's not an implicit disclaimer that what you're learning isn't the final word inherent in the structure of the educational system?