Journal of Byzantium, how many schoolgirls do you know with their very own Ministries of Mathematics? or ministries of other kinds?
For my part, I see that John’s daughter’s school has had difficulties teaching handwriting. Capital letters really should be something taught well and truly after students have mastered small letters. If they’re taught at all. They really are superfluous. And the way “yes” was written, that really looks like torture! Handwriting shouldn’t emulate print.
4: That’s kind of my point. The proof is quite simple for those who know how to do those kinds of things, but those people are quite rare. Most such knowledge comes from authority.
On the next page she gets asked the same thing about even + even always equalling even. In response to the ‘why do you believe this?’ question she writes: ‘it works’. Very pragmatic of her.
All odd numbers (y) can be expressed as 2x + 1, where x is some number.
Add even numbers (a) can be expressed as 2x, x is a/2.
Adding two even numbers gets you another even number.
thus:
y + y’
= 2x + 1 + 2x’ + 1
= 2x + 2x’ + 2
= a + a’ + 2
since a, a’ and 2 are all even numbers, thus y+y’ is even.
Speaking as someone with a maths degree, I can give you any number of explanations, but this seems awfully advanced for a girl as little as she is.
That said, I’m sure you could explain to her why: Any odd number is just 1 away from an even number. So adding two odd numbers together is the same as adding two even numbers and adding 1 + 1. She’d get that – but c’mon, asking it in a class test is not so fair.
So adding two odd numbers together is the same as adding two even numbers and adding 1 + 1.
Even that requires exploring whether addition is commutative, of course.
In general, it makes me happy to know these questions are being asked. The question does not ask, “How would you prove this?” or “Justify your answer” or anything that would require algebra or formal logic. (So, Tom has no need to be concerned.) It’s very open-ended, it just encourages students to think about their hypotheses and what makes them reasonable. Hopefully this answer won’t be marked “wrong” — teachers do have a great potential to undermine the thinking this kind of question encourages by demanding certain categories of answers. A follow-up lesson on “trying lots of examples to see if an idea is often true” would be great. “How could you show somebody who didn’t believe you?”
Very pragmatic of her.
:) And really, “It works” is an advanced answer to a “why” question for someone young enough to be first learning about the properties of addition. “I tried enough examples and it works for all of them” (for whatever definition of “enough” makes sense) is implicit in the shorter statement, I think. The idea that a rule must “work” on specific cases is also implicit. The idea of trying examples to check if an idea “works” is exactly what’s at stake here.
The teacher was not asking for a proof, go easy on her, she asked indeed “why”: something like 3+3=6 would have been OK, I guess – or if she was looking for an Einstein: 1+1= 2 and in all the odd numbers there is a “1”.
I was taught that appeals to authority are not fallacious when (a) the issue in question is one that admits of authority, (b) the person appealed to is an authority in that area, and (c) the claim in question is not one about which authorities would disagree.
It seems to me that “basic properties of addition” is an area that admits of authority, that Belle is an authority in that area, along with billions of other people, and that none of these authorities would disagree about the claim in question. Hence the appeal to Belle’s authority, in this case, is perfectly reasonable and not at all fallacious.
Of course, Mom saying so isn’t what it is that makes the claim true. But the question was why the pupils thought the claim true, and Mom saying so seems to be a perfectly good reason to think it true.
Even that requires exploring whether addition is commutative, of course.
If we ignore the difference between abstractly-defined addition and grouping together sets of things (which surely we have to at this level) a small child can understand that commutativity holds in a way which is more than just “it works for the examples I’ve tried” though. It’s is clear to a kid (at least above some low-ish age) that if you empty three bags of marbles into a bucket you get the same number of marbles regardless of the order you empty the bags.
This together what Peter Hollo said would count as knowing odd plus odd is even.
At the risk of taking this discussion too seriously, I wonder: what predicts whether a child will respond in such a way or leave the line blank even if she thinks of such a response? That is, I could see someone thinking: “my MoM said so, but they can’t possibly be looking for that as an answer” or “can’t possibly give credit for that answer” and thus not writing anything. If it turns out that saying “MoM said so” gives you credit then the person who dismissed it as the not-sufficiently-mathematical or relevant response loses out. (I can think of all sorts of related scenarios where a child might decide not to say anything and it wouldn’t be for a lack of a certain response.) Thoughts?
i was asked in infant school what the difference between odd and even numbers was, and i said “odd numbers always have a single spot in the middle of the page” — the teacher passed on till she got an answer more like “1 and 3 and 5 and 7 and so on…”
i was basing my (correct) answer on how the numbers were displayed on the pinned-up frieze round the schoolroom (basically as 1-12, like the dots on dominos)
this happened about 45 years ago — and i went on to get a degree in maths and philosophy — yet i am still burning with fiery resentment that my teenytiny genius went unrecognised (however there were no other instances of unrecognition, if still-burning resentment is a guide)
(also: i was then and remain quite bad at arithmetic)
In high school, we used to answer “Yes” to “Is it A or B?” questions. The first time teachers saw this answer, they were usually surprised enough to give you points, but never if they had seen it before.
In general, a feeling what kind of answer is expected and accepted in a given context is important during school, and also outside of it I guess. But I feel that a teacher should always try to be clear about the intended kind of answer, and keep the be-honest questions apart from normal questions.
Yes, “Belle says that P, so, P,” will do for now — and for some time. But eventually, one wants one’s children to develop the maturity and independence of mind to say, “I don’t know — it just feels right in my gut.”
“being good at arithmetic” is like that though — i was bad at it because i worked everything out longhand from first principles; people who are good at it do it instantly in their heads, and the correctness turned into become a “gut” thing
There’s also that pragmatic distinction, when someone asked you a certain kind of question like “Why do you say that?” (meaning, what is the reasoning that led you to that conclusion?) and you answer “To make you mad.”
Or “So-n-so is a jerk?” “Why?” “Probably because he was born that way.” (Where the question is asking for what he just did to make you conclude he is a jerk, not how he became a jerk.)
So Holbo’s daughter is not committing a fallacy, but answering the question quite literally–just a different question than was asked. “I believe what the textbook says” might be a completely accurate and non-fallacious answer to the question “Why do you think X” on a college exam, if the student believes the question is asking: “What is your source for this information?”
I have to admit that I hate seeing questions like “How do you know this” or “how did you arrive at your answer” in children’s textbooks. I understand what it’s supposed to accomplish, and I think it’s wonderful if the kids actually have a deep understanding of the topic. But for 90% of the kids, a mechanistic understanding is all that can be hoped for for another few grades and it often confuses students who would otherwise understand how to accomplish the real task.
Or maybe I’m scarred from when my 5 year old came to me in tears over his math homework, which he normally enjoyed.
Question 1 was “What is 1 + 3?”
Question 2 was “Explain how you arrived at your answer.”
He was old enough to understand that “I added them” was not an acceptable answer, and probably about 14-15 years away from being able to provide any other answer.
(I was about ready to write my own answer by pulling out my first year textbook with ‘God gives you 1 and the successor function’ and going on from there, but my wife insisted on common sense.)
True, Jonathan, but still in defense of the teacher: it’s not because a question can provoke some such answers that it’s a bad one. This particular question provokes creative answers – even if it’s not credited with points, sounds like a good idea.
But it would be interesting to ask the question to the teacher whether and how she’d quote it.
But seriously, 99.9% of the stuff we “know” is only the basis of authority. I know that the White House is in Washington because I’ve seen it. But I know that President Obama lives there only on the basis of the reports of others. I know Paris is in France because I’ve been there. But I know that Berlin is in Germany only on the basis of the reports of others. Obviously, for Berlin NOT to be in Germany would require an enormous conspiracy involving hundreds of millions of people over a period of centuries – so I don’t have any difficulty believing that Berlin is Germany – but still, I believe it on the basis of authority.
When you get to more difficult stuff – like the nature of electricity or plate tectonics or global warming or how airplanes manage to stay up or Burke’s views on the French Revolution – for me, that’s all authority.
Take evolution. I understand evolution about as well as any other ordinary non-scientist who pays attention to stuff. And I will never understand it well enough to be able to judge its truth other than on the basis of authority. That is, I understand and can articulate the theory, but I have no way of evaluating the theory. That’s beyond me. I can only believe what I’m told by people whose role in society leads me to trust them.
I think anyone who belittles belief on the basis of authority has never really considered how central authority in the formation of their own beliefs.
But seriously, 99.9% of the stuff we â€œknowâ€ is only the basis of authority.
This reminds me: last week I was giving a talk in Qatar. During Q&A, a student stood up to claim that because President Obama had said something, it was true. I started my response as follows: “I reserve the right to disagree with President Obama.” (In fact, I didn’t disagree with Obama here, I just pointed out that his approach needed to be taken a step further. It was a discussion about broadband policy.) The point is that it was fascinating to watch this first-year college student basically argue that something was the way it was simply because the President had said so.
He was old enough to understand that â€œI added themâ€ was not an acceptable answer,
I’d grumble (to the teacher in question), why not? Maybe “I added them in my head” is a marginally better answer, but the answer “I added them” shows he at least recognizes the procedure he used. Further guidance with leading questions is an appropriate teacher response, but not a penalty mark. “Did you add them by counting?”
One of my eternal complaints against the way some math classrooms are run is the emphasis of the right answer over what’s actually important, arriving at a reasonable answer. Of course, an answer other than 4 to 1 + 3 = ? is (in this context) unreasonable. But for answers to questions like “how did you find your answer” there ought to be much greater flexibility in what’s considered reasonable.
I think asking open questions like that, and then criticizing honest responses instead of offering constructive guidance toward more reasonable responses, is every bit as harmful as Tom suggests.
I think anyone who belittles belief on the basis of authority has never really considered how central authority in the formation of their own beliefs.
As a relevant aside, the book Truth: A Guide for the Perplexed is fantastic fun (so long as it’s not taken too seriously), and a good fast read. And it’s on Amazon used for US$2.
It seems to me that if teachers want to ask kids “why” questions in maths they should first be doing it in class, where they can provide rapid feedback on what’s a suitable answer and what isn’t, rather than for homework, where the feedback is necessarily delayed and even educated parents don’t necessarily know what the teacher is looking for, judging by this thread.
It seems to me that if teachers want to ask kids â€œwhyâ€ questions in maths they should first be doing it in class, where they can provide rapid feedback on whatâ€™s a suitable answer and what isnâ€™t,
Agreed, of course — First in class, with rapid (positive and guiding) feedback as you say, and then in homework, and then follow-up guidance in class, rinse/repeat as necessary. On the other hand, it’s not especially damaging to sometimes ask these questions “cold” (no in-class advanced guidance) so long as the student knows that any reasonable, honest answer will be “acceptable.”
I don’t think you ever want a student to ask themselves, “but will this answer be acceptable to Teacher?” You’ve lost ’em, then: they’re no longer engaging the content. They’re playing head games with Teacher.
Sometimes you want students to do self-evaluation after an assignment draft (e.g. when writing an essay), but even then they really should be asking something like, “Will this satisfy the audience?” Thinking about the teacher should be largely absent, thinking about completing the assignment in a reasonable way should be the overall guidance, and teachers generally ought to encourage that kind of focus on the work.
Eszter, had he concluded that X is true “simply” because Obama had said so? Or was he adducing evidence that X is true? Outside a courtroom, I would accept “Obama said X” as evidence in favor of the proposition “X is true,” although it’s clearly not proof that “X is true.”
I dunno about advanced mathematics, but I think that “Why do you think so?” is about as sophisticated a question as you can get. The vast majority of human beings never seem to get round to asking themselves this question in their entire lifetimes. Of anything.
I agree with 40. Belle’s daughter’s answer is awesome, but other answers could have been awesome as well. For instance, how about “I tried many examples and convinced myself”? Sure, not as cute a fallacy, but also revealing.
base case:n=m=0
answer=(2n+1)+(2m+1)=2(n+m+1)=2(0+0+1)=2
now using n=i and m=j we have
answer=(2i+1)+(2j+1)+2(i+j+1)
now for i+1 and j+1 case
answer=(2(i+1)+1)+(2(j+1)+1)=(2i+3)+(2j+3)=2(i+j+3)
which of course is of the form 2n, therefore adding 2 odd numbers always results in an even number.
Though her way was much cuter. Here’s to hoping she aspires to be a math major!
This thread is reminding me of way too many things. Now I’m thinking about high school math (in Hungary). I dared use Pythagoras’ theorem to prove something at some point only to be told I couldn’t, because we had not yet proven Pythagoras’ theorem in class. Since we already learned the basics back in middle school, it seemed rather artificial to then have to forget about it, because we hadn’t yet gone over it in more detail in high school.
When I was a lad, though probably not quite as young as John’s daughter, the was a fashion for making kids play with arithmetic in bases other than 10 – a fashion beautifully demolished by Tom Lehrer. But this is the first example I can think of where it might actually have been remotely useful:
In binary, all odd numbers end in 1 and all even numbers end in 0;
In binary, 1 + 1 = 10;
So if you add two odd numbers together, the first thing you do is add 1 + 1;
So the answer is going to end in 0.
So there!
That is a horrible question. If I’d have been asked that sort of thing at school I’d have cried like Tom West’s son, and maybe I did And why isn’t “because I added them” an acceptable answer anyway?
Like belle le triste, I recall a question in primary school that I thought I’d got right and was peeved to be told was wrong. The question – why do rivers flow to the sea? My answer – because of the slopes. The correct answer – because of gravity. Bit unfair on an 8 year old that, I think.
I think “I added them” is a perfectly good answer, since the whole point of the addition algorithm is that it’s guaranteed to produce the sum of the inputs if carried out properly. Curry-Howard and all that.
Actually, I would think that”becas i tryed som and it workd” is what the teacher is looking for. Question 3 is supposed to prompt the student to try one or two easy cases – 1 plus 1, 3 plus 5. Question 4 is intended to encourage the student to try a few more – 7 plus 5, 9 plus 7. In my experience, with my own children, math teachers do teach induction fairly early.
In No. 2, Anders Widebrant is impressed by the double-“capitalization” of “MoM”. No such thing. The admittedly very clever girl has had very bad writing instructions, and there is not a single upper–lower case distinction discernable in her handwriting. Her M/m is her (never-ever corrected?) emulation of an M. The first ‘leg’ of an (upper-, or lower-case) M/n should always go down; her’s goes up. Not teaching children to write letters in a very distinct order results in poorly readable handwriting later on.
I am thinking of the usual level of math facility in a typical primary school teacher and imgining what the response would be to a complete proof by mathematical induction, starting with a definition of “odd” as being 2n-1 where n varies through the natural numbers. Probably complete crogglement, without even the recognition Gauss got for his “legget se”.
So, let me congratulate John with the Rithmatic achievements of his Alice, advise him to get her some better Riting teaching, while I apologize for him Reading my typo M/n.
Actually, I would think thatâ€becas i tryed som and it workdâ€ is what the teacher is looking for.
Maybe. Back in the theoretical case, I’d maintain it’s important to not be “looking for” anything specific for these answers, and I hope the teacher wasn’t in this case. It is necessary and sufficient to check that the answer is reasonable and/or cute.
And I mean that at least half-seriously: I’d maintain that cute and creative answers, especially to open-ended questions like these, ought to be entitled to credit. Train of justifications: (1) One goal of a teacher should be, to encourage behavior in students that we would like to see in citizens. (2) We would like for citizens to be creative, clever, honest, and funny within the boundaries of politeness and propriety. (3) A key, essential ingredient of creativity is the ability to hypothesize: to guess, conjecture, and communicate those conjectures clearly. (4) The reasoning supporting a conjecture is often informal or poorly-formed, initially. It is reasonable to expect that students’ initial justifications for their hypotheses will not be as rigorous as proof, and may very well rely on appeals to example or to authority. (5) To encourage creativity, students should initially be asked to suggest support for their conjectures, exactly as DoJ did here,* and young students should be asked for a formal argument only in an interactive setting, as the justifications are discussed in greater detail.
*Daughter of John?**
**If/when she’s acting intolerable in her adolescent years, you’ll have to show her this thread. :)
Do kids still use physical objects to learn math? You can “prove” even+even easily if you’ve been prompted with an intuitive set theoretical definition of “even,” though odd+odd is a little harder (and the proof is no less rigorous than Euclid). I agree with Eszter that questions like these seem likely to be answered intelligently only if the kids were given similar problems in class (though I’ve known teachers who apparently believed otherwise, convinced that someday they would have the true genius in their class who would intuit the technique, never having seen the problem before).
And even if her class didn’t use physical objects, it’s possible the workbook was written with such a procedure in mind, I suppose. Though the “always” is strange. How do you explain to a small child that there is the difference between “when you add an odd number to an odd number” and “always, when you add an odd number to an odd number”?
I always remember being asked the question “Can you prove p …” and being sorely tempted to claim full marks for the answer “No”. Then, of course, there is
Authority is all in the capitalisation. The leader and the Leader; a chairman and the Chairman; the party and the Party; a god and a God. So a mom is not an authority, a Mom is an authority, and a MoM is infallible.
Salient, I didn’t mean “looking for” in the sense of determining the answer for a good grade. I meant that the questions were intended to prompt certain actions and “I tried some examples” would demonstrate that the actions were taken. “My Mom said so” is cute but it shows that what the teacher was trying to accomplish in class didn’t happen, and the teacher needs to try again.
This is thought-provoking on so many levels: infant psychology (my MoM said so), pedagogy (why do you think so?), epistemology (is this always true? Always??), mathematics, you name it… Really wonderful.
What predicts whether a child will respond in such a way or leave the line blank even if she thinks of such a response?
Interestingly, I remember spending a not negligible amount of time during my school years trying to figure out what was teachers expecting of us for open ended questions. In first grade for instance, we were once given a text with the assignment “Circle all the words you don’t understand or that you can’t read”. I was a slightly precocious child, so I could read and understand all the words. I agonized over the exercise, trying to remember if we had ever seen this or that word in class (if not, presumably I should be circling it). In fifth grade, our teacher once wrote an addition, a subtraction, a multiplication and a division on the blackboard, instructing us to compute them. Then she wrote “Devise what is the hardest math problem for you and solve it”, meaning in fact the hardest operation (by the way, I think it is good teaching technique to ask such questions, at least to older students), but we had already been assigned problems (of the John has bought three apples and paid 15 francs, how much is an apple? variety) so I took the assignment literally and tried to think of the hardest math problem I could possibly solve (I eventually failed to solve it, by the way).
Anyway, I think the answers “A number”, “Maybe it won’t be true anymore when human beings have disappeared” and “I am unsure about platonism, perhaps even and odd are external properties, perhaps not” seems perfectly correct.
I think the point was not whether you could prove this true. The point was how this was a logical fallacy for an argument. Namely the “proof” for the little girl is an appeal to authority. One of the fundamental flaws in logic that from a very early age we teach our children to do even though its wrong. Sadly even people well versed in mathematics fail to understand logic when it is applied to language :P
That’s why its titled Fallacy and not can you prove this with math.
I’m just amazed that there are 65 posts here and not a one realized what it was about.
Even number can be written as 2k where k is any integer 2(1) = 2, 2(2)=4, 2(3) = 6
Odd can then be written as one more then an even so 2k+1, 2+1 = 3, 4+1 = 5, 6+1 = 7
Eszter – in my high school precalculus class, the rule was we could use on an exam (or homework) anything which we’d had in class or shown on the exam. One exam was after we’d learned the definition of a derivative, but not any of the really useful formulae for derivatives of various functions. The exam was a bunch of “find the derivative of this polynomial function”, with the intent of doing it the hard way – going through the polynomial and using the definition of derivative. For problem 1, I worked out the formula for the general case of the derivative of mx^n, and then solved the remaining problems by plugging them into the formula. Teacher was not amused, but did give me full marks.
When first learning even and odd, I was taught that even numbers are numbers that are divisible by two. If she had been told that, then it wouldn’t be out of the ordinary for her to reason that any number added to itself would produce an even number, no matter if it were an odd or even to begin with.
James, yes, that’s actually easier, still Einstein at that age but if you add 2 odd numbers then divide by two you get the add number you started with … It is beyond me why the hell anybody would like to prove this with advanced formalisms.
James @76: That’s only a special case of what she was asked though. The question is adding two (possibly different) odd numbers
Are there any groups where addition is not commutative? I canâ€™t think of any.
This is kind of a question about language rather than mathematics. There are plenty of groups where the operation is non-commutative (the group of symmetries of a equilateral triangle, say) but it wouldn’t be usual to refer to the operation as “addition” in these cases (whereas you might in an arbitrary commutative group). This probably isn’t very relevant to primary school arithmetic, though.
Well, I could easily think of multiplying matrices being non-commutative, but salient’s comment got me wondering just how much of those old math classes I was forgetting.
{ 81 comments }
mpowell 04.02.09 at 9:20 am
Won’t it be a long time before she can give a better answer (if ever)?
Anders Widebrant 04.02.09 at 9:24 am
Dawwww the double-capitalization of “MoM” is adorable beyond description.
JoB 04.02.09 at 9:31 am
Was ‘MoM’ right?
Did she mean Ministry of Mathematics?
Oh, so much confusion …
Felix 04.02.09 at 9:41 am
Journal of Byzantium, how many schoolgirls do you know with their very own Ministries of Mathematics? or ministries of other kinds?
For my part, I see that John’s daughter’s school has had difficulties teaching handwriting. Capital letters really should be something taught well and truly after students have mastered small letters. If they’re taught at all. They really are superfluous. And the way “yes” was written, that really looks like torture! Handwriting shouldn’t emulate print.
Tom Mathews 04.02.09 at 9:50 am
I’m not actually sure how else you would answer that question, other than algebraically, which presumably is not what they want…?
Anyone?
mpowell 04.02.09 at 10:00 am
4: That’s kind of my point. The proof is quite simple for those who know how to do those kinds of things, but those people are quite rare. Most such knowledge comes from authority.
John Holbo 04.02.09 at 10:01 am
I’m wondering the same thing, Tom.
On the next page she gets asked the same thing about even + even always equalling even. In response to the ‘why do you believe this?’ question she writes: ‘it works’. Very pragmatic of her.
Factory 04.02.09 at 10:07 am
All odd numbers (y) can be expressed as 2x + 1, where x is some number.
Add even numbers (a) can be expressed as 2x, x is a/2.
Adding two even numbers gets you another even number.
thus:
y + y’
= 2x + 1 + 2x’ + 1
= 2x + 2x’ + 2
= a + a’ + 2
since a, a’ and 2 are all even numbers, thus y+y’ is even.
Peter Hollo 04.02.09 at 10:11 am
Speaking as someone with a maths degree, I can give you any number of explanations, but this seems awfully advanced for a girl as little as she is.
That said, I’m sure you could explain to her why: Any odd number is just 1 away from an even number. So adding two odd numbers together is the same as adding two even numbers and adding 1 + 1. She’d get that – but c’mon, asking it in a class test is not so fair.
Jamie 04.02.09 at 10:43 am
Anyway,
Belle says that P
So, P.
That looks like a pretty good argument, not a fallacy. Maybe not quite deductively valid, but better than many arguments.
glenn 04.02.09 at 11:05 am
John – how old is your maths wizard? In any event, cheers for her!
salient 04.02.09 at 11:08 am
So adding two odd numbers together is the same as adding two even numbers and adding 1 + 1.
Even that requires exploring whether addition is commutative, of course.
In general, it makes me happy to know these questions are being asked. The question does not ask, “How would you prove this?” or “Justify your answer” or anything that would require algebra or formal logic. (So, Tom has no need to be concerned.) It’s very open-ended, it just encourages students to think about their hypotheses and what makes them reasonable. Hopefully this answer won’t be marked “wrong” — teachers do have a great potential to undermine the thinking this kind of question encourages by demanding certain categories of answers. A follow-up lesson on “trying lots of examples to see if an idea is often true” would be great. “How could you show somebody who didn’t believe you?”
Very pragmatic of her.
:) And really, “It works” is an advanced answer to a “why” question for someone young enough to be first learning about the properties of addition. “I tried enough examples and it works for all of them” (for whatever definition of “enough” makes sense) is implicit in the shorter statement, I think. The idea that a rule must “work” on specific cases is also implicit. The idea of trying examples to check if an idea “works” is exactly what’s at stake here.
Robert the Red 04.02.09 at 11:57 am
Answer 5 is as rigorous as a those made in a recent “scientific” paper I just read in a well-regarded journal, and expressed much more clearly, too.
JoB 04.02.09 at 11:59 am
The teacher was not asking for a proof, go easy on her, she asked indeed “why”: something like 3+3=6 would have been OK, I guess – or if she was looking for an Einstein: 1+1= 2 and in all the odd numbers there is a “1”.
John Holbo 04.02.09 at 12:03 pm
“I guess â€“ or if she was looking for an Einstein: 1+1= 2 and in all the odd numbers there is a â€œ1â€.”
I think that’s right. The kids are being taught something like: every odd gets a partner when you add two odds.
philete 04.02.09 at 12:21 pm
What Jamie says at 9.
I was taught that appeals to authority are not fallacious when (a) the issue in question is one that admits of authority, (b) the person appealed to is an authority in that area, and (c) the claim in question is not one about which authorities would disagree.
It seems to me that “basic properties of addition” is an area that admits of authority, that Belle is an authority in that area, along with billions of other people, and that none of these authorities would disagree about the claim in question. Hence the appeal to Belle’s authority, in this case, is perfectly reasonable and not at all fallacious.
Of course, Mom saying so isn’t what it is that makes the claim true. But the question was why the pupils thought the claim true, and Mom saying so seems to be a perfectly good reason to think it true.
engels 04.02.09 at 12:27 pm
I think you’ll find this is a standard proof technique.
Matt Heath 04.02.09 at 12:34 pm
Even that requires exploring whether addition is commutative, of course.
If we ignore the difference between abstractly-defined addition and grouping together sets of things (which surely we have to at this level) a small child can understand that commutativity holds in a way which is more than just “it works for the examples I’ve tried” though. It’s is clear to a kid (at least above some low-ish age) that if you empty three bags of marbles into a bucket you get the same number of marbles regardless of the order you empty the bags.
This together what Peter Hollo said would count as knowing odd plus odd is even.
MarkUp 04.02.09 at 1:01 pm
Would this pass muster with Larry Summers?
Eszter 04.02.09 at 1:22 pm
At the risk of taking this discussion too seriously, I wonder: what predicts whether a child will respond in such a way or leave the line blank even if she thinks of such a response? That is, I could see someone thinking: “my MoM said so, but they can’t possibly be looking for that as an answer” or “can’t possibly give credit for that answer” and thus not writing anything. If it turns out that saying “MoM said so” gives you credit then the person who dismissed it as the not-sufficiently-mathematical or relevant response loses out. (I can think of all sorts of related scenarios where a child might decide not to say anything and it wouldn’t be for a lack of a certain response.) Thoughts?
Matt Heath 04.02.09 at 1:29 pm
Ugh! My comment is full typos. Sorry. At the end it should say “why odd plus odd equals even”.
belle le triste 04.02.09 at 1:30 pm
i was asked in infant school what the difference between odd and even numbers was, and i said “odd numbers always have a single spot in the middle of the page” — the teacher passed on till she got an answer more like “1 and 3 and 5 and 7 and so on…”
i was basing my (correct) answer on how the numbers were displayed on the pinned-up frieze round the schoolroom (basically as 1-12, like the dots on dominos)
this happened about 45 years ago — and i went on to get a degree in maths and philosophy — yet i am still burning with fiery resentment that my teenytiny genius went unrecognised (however there were no other instances of unrecognition, if still-burning resentment is a guide)
(also: i was then and remain quite bad at arithmetic)
belle le triste 04.02.09 at 1:32 pm
also: what jamie said in 9
Zamfir 04.02.09 at 1:37 pm
In high school, we used to answer “Yes” to “Is it A or B?” questions. The first time teachers saw this answer, they were usually surprised enough to give you points, but never if they had seen it before.
In general, a feeling what kind of answer is expected and accepted in a given context is important during school, and also outside of it I guess. But I feel that a teacher should always try to be clear about the intended kind of answer, and keep the be-honest questions apart from normal questions.
Michael BÃ©rubÃ© 04.02.09 at 2:14 pm
Yes, “Belle says that P, so, P,” will do for now — and for some time. But eventually, one wants one’s children to develop the maturity and independence of mind to say, “I don’t know — it just feels right in my gut.”
belle le triste 04.02.09 at 2:26 pm
“being good at arithmetic” is like that though — i was bad at it because i worked everything out longhand from first principles; people who are good at it do it instantly in their heads, and the correctness turned into become a “gut” thing
O.W. 04.02.09 at 2:38 pm
But doesn’t 2+2=5?
Jonathan Mayhew 04.02.09 at 2:40 pm
There’s also that pragmatic distinction, when someone asked you a certain kind of question like “Why do you say that?” (meaning, what is the reasoning that led you to that conclusion?) and you answer “To make you mad.”
Or “So-n-so is a jerk?” “Why?” “Probably because he was born that way.” (Where the question is asking for what he just did to make you conclude he is a jerk, not how he became a jerk.)
So Holbo’s daughter is not committing a fallacy, but answering the question quite literally–just a different question than was asked. “I believe what the textbook says” might be a completely accurate and non-fallacious answer to the question “Why do you think X” on a college exam, if the student believes the question is asking: “What is your source for this information?”
Tom West 04.02.09 at 2:48 pm
I have to admit that I hate seeing questions like “How do you know this” or “how did you arrive at your answer” in children’s textbooks. I understand what it’s supposed to accomplish, and I think it’s wonderful if the kids actually have a deep understanding of the topic. But for 90% of the kids, a mechanistic understanding is all that can be hoped for for another few grades and it often confuses students who would otherwise understand how to accomplish the real task.
Or maybe I’m scarred from when my 5 year old came to me in tears over his math homework, which he normally enjoyed.
Question 1 was “What is 1 + 3?”
Question 2 was “Explain how you arrived at your answer.”
He was old enough to understand that “I added them” was not an acceptable answer, and probably about 14-15 years away from being able to provide any other answer.
(I was about ready to write my own answer by pulling out my first year textbook with ‘God gives you 1 and the successor function’ and going on from there, but my wife insisted on common sense.)
JoB 04.02.09 at 2:48 pm
True, Jonathan, but still in defense of the teacher: it’s not because a question can provoke some such answers that it’s a bad one. This particular question provokes creative answers – even if it’s not credited with points, sounds like a good idea.
But it would be interesting to ask the question to the teacher whether and how she’d quote it.
ajay 04.02.09 at 3:05 pm
The kids are being taught something like: every odd gets a partner when you add two odds.
So don’t worry if you’re odd, kids; somewhere out there is someone equally odd, and the two of you will be very happy together.
Bloix 04.02.09 at 3:07 pm
Berube, stop auditioning for a role on Futurama.
But seriously, 99.9% of the stuff we “know” is only the basis of authority. I know that the White House is in Washington because I’ve seen it. But I know that President Obama lives there only on the basis of the reports of others. I know Paris is in France because I’ve been there. But I know that Berlin is in Germany only on the basis of the reports of others. Obviously, for Berlin NOT to be in Germany would require an enormous conspiracy involving hundreds of millions of people over a period of centuries – so I don’t have any difficulty believing that Berlin is Germany – but still, I believe it on the basis of authority.
When you get to more difficult stuff – like the nature of electricity or plate tectonics or global warming or how airplanes manage to stay up or Burke’s views on the French Revolution – for me, that’s all authority.
Take evolution. I understand evolution about as well as any other ordinary non-scientist who pays attention to stuff. And I will never understand it well enough to be able to judge its truth other than on the basis of authority. That is, I understand and can articulate the theory, but I have no way of evaluating the theory. That’s beyond me. I can only believe what I’m told by people whose role in society leads me to trust them.
I think anyone who belittles belief on the basis of authority has never really considered how central authority in the formation of their own beliefs.
Eszter Hargittai 04.02.09 at 3:20 pm
But seriously, 99.9% of the stuff we â€œknowâ€ is only the basis of authority.
This reminds me: last week I was giving a talk in Qatar. During Q&A, a student stood up to claim that because President Obama had said something, it was true. I started my response as follows: “I reserve the right to disagree with President Obama.” (In fact, I didn’t disagree with Obama here, I just pointed out that his approach needed to be taken a step further. It was a discussion about broadband policy.) The point is that it was fascinating to watch this first-year college student basically argue that something was the way it was simply because the President had said so.
sara 04.02.09 at 3:25 pm
John: Does your daughter attend a regular Singapore public school, or does she attend an American school or some other type of school?
Jason 04.02.09 at 3:31 pm
“numder” is my new favorite word.
tom s. 04.02.09 at 3:38 pm
Come on people, read the question.
The question is not “Do you think this is true for all pairs of odd numbers?”. It is the completely different “Do you think this is always true?”
It is true today, but will it be true tomorrow?
salient 04.02.09 at 3:41 pm
He was old enough to understand that â€œI added themâ€ was not an acceptable answer,
I’d grumble (to the teacher in question), why not? Maybe “I added them in my head” is a marginally better answer, but the answer “I added them” shows he at least recognizes the procedure he used. Further guidance with leading questions is an appropriate teacher response, but not a penalty mark. “Did you add them by counting?”
One of my eternal complaints against the way some math classrooms are run is the emphasis of the right answer over what’s actually important, arriving at a reasonable answer. Of course, an answer other than 4 to 1 + 3 = ? is (in this context) unreasonable. But for answers to questions like “how did you find your answer” there ought to be much greater flexibility in what’s considered reasonable.
I think asking open questions like that, and then criticizing honest responses instead of offering constructive guidance toward more reasonable responses, is every bit as harmful as Tom suggests.
I think anyone who belittles belief on the basis of authority has never really considered how central authority in the formation of their own beliefs.
As a relevant aside, the book Truth: A Guide for the Perplexed is fantastic fun (so long as it’s not taken too seriously), and a good fast read. And it’s on Amazon used for US$2.
Tracy W 04.02.09 at 3:49 pm
It seems to me that if teachers want to ask kids “why” questions in maths they should first be doing it in class, where they can provide rapid feedback on what’s a suitable answer and what isn’t, rather than for homework, where the feedback is necessarily delayed and even educated parents don’t necessarily know what the teacher is looking for, judging by this thread.
salient 04.02.09 at 3:57 pm
It seems to me that if teachers want to ask kids â€œwhyâ€ questions in maths they should first be doing it in class, where they can provide rapid feedback on whatâ€™s a suitable answer and what isnâ€™t,
Agreed, of course — First in class, with rapid (positive and guiding) feedback as you say, and then in homework, and then follow-up guidance in class, rinse/repeat as necessary. On the other hand, it’s not especially damaging to sometimes ask these questions “cold” (no in-class advanced guidance) so long as the student knows that any reasonable, honest answer will be “acceptable.”
I don’t think you ever want a student to ask themselves, “but will this answer be acceptable to Teacher?” You’ve lost ’em, then: they’re no longer engaging the content. They’re playing head games with Teacher.
Sometimes you want students to do self-evaluation after an assignment draft (e.g. when writing an essay), but even then they really should be asking something like, “Will this satisfy the audience?” Thinking about the teacher should be largely absent, thinking about completing the assignment in a reasonable way should be the overall guidance, and teachers generally ought to encourage that kind of focus on the work.
Bloix 04.02.09 at 4:07 pm
Eszter, had he concluded that X is true “simply” because Obama had said so? Or was he adducing evidence that X is true? Outside a courtroom, I would accept “Obama said X” as evidence in favor of the proposition “X is true,” although it’s clearly not proof that “X is true.”
sharon 04.02.09 at 4:13 pm
I dunno about advanced mathematics, but I think that “Why do you think so?” is about as sophisticated a question as you can get. The vast majority of human beings never seem to get round to asking themselves this question in their entire lifetimes. Of anything.
pedro 04.02.09 at 4:30 pm
I agree with 40. Belle’s daughter’s answer is awesome, but other answers could have been awesome as well. For instance, how about “I tried many examples and convinced myself”? Sure, not as cute a fallacy, but also revealing.
pedro 04.02.09 at 4:31 pm
Oops, I meant to say I agreed with Sharon at 41.
seth 04.02.09 at 4:48 pm
proof by induction:
base case:n=m=0
answer=(2n+1)+(2m+1)=2(n+m+1)=2(0+0+1)=2
now using n=i and m=j we have
answer=(2i+1)+(2j+1)+2(i+j+1)
now for i+1 and j+1 case
answer=(2(i+1)+1)+(2(j+1)+1)=(2i+3)+(2j+3)=2(i+j+3)
which of course is of the form 2n, therefore adding 2 odd numbers always results in an even number.
Though her way was much cuter. Here’s to hoping she aspires to be a math major!
Sean 04.02.09 at 4:57 pm
Doesn’t leave an awful lot of space for anything. I wish I knew what answer they expected. Must’ve been discussed in class.
Eszter Hargittai 04.02.09 at 5:34 pm
Bloix, it sounded like he meant it as proof.
Sharon, good point!
This thread is reminding me of way too many things. Now I’m thinking about high school math (in Hungary). I dared use Pythagoras’ theorem to prove something at some point only to be told I couldn’t, because we had not yet proven Pythagoras’ theorem in class. Since we already learned the basics back in middle school, it seemed rather artificial to then have to forget about it, because we hadn’t yet gone over it in more detail in high school.
chris y 04.02.09 at 5:59 pm
When I was a lad, though probably not quite as young as John’s daughter, the was a fashion for making kids play with arithmetic in bases other than 10 – a fashion beautifully demolished by Tom Lehrer. But this is the first example I can think of where it might actually have been remotely useful:
In binary, all odd numbers end in 1 and all even numbers end in 0;
In binary, 1 + 1 = 10;
So if you add two odd numbers together, the first thing you do is add 1 + 1;
So the answer is going to end in 0.
So there!
Wade 04.02.09 at 6:07 pm
If you add two odds together it makes an even number because you can cut an even number perfectly in half
Katherine 04.02.09 at 6:11 pm
That is a horrible question. If I’d have been asked that sort of thing at school I’d have cried like Tom West’s son, and maybe I did And why isn’t “because I added them” an acceptable answer anyway?
Like belle le triste, I recall a question in primary school that I thought I’d got right and was peeved to be told was wrong. The question – why do rivers flow to the sea? My answer – because of the slopes. The correct answer – because of gravity. Bit unfair on an 8 year old that, I think.
recursivelyenumerable 04.02.09 at 6:28 pm
I think “I added them” is a perfectly good answer, since the whole point of the addition algorithm is that it’s guaranteed to produce the sum of the inputs if carried out properly. Curry-Howard and all that.
recursivelyenumerable 04.02.09 at 6:30 pm
should’ve typed “an” addition algorithm.
Bloix 04.02.09 at 6:33 pm
Actually, I would think that”becas i tryed som and it workd” is what the teacher is looking for. Question 3 is supposed to prompt the student to try one or two easy cases – 1 plus 1, 3 plus 5. Question 4 is intended to encourage the student to try a few more – 7 plus 5, 9 plus 7. In my experience, with my own children, math teachers do teach induction fairly early.
RenÃ© 04.02.09 at 6:51 pm
In No. 2, Anders Widebrant is impressed by the double-“capitalization” of “MoM”. No such thing. The admittedly very clever girl has had very bad writing instructions, and there is not a single upper–lower case distinction discernable in her handwriting. Her M/m is her (never-ever corrected?) emulation of an M. The first ‘leg’ of an (upper-, or lower-case) M/n should always go down; her’s goes up. Not teaching children to write letters in a very distinct order results in poorly readable handwriting later on.
James 04.02.09 at 6:52 pm
I am thinking of the usual level of math facility in a typical primary school teacher and imgining what the response would be to a complete proof by mathematical induction, starting with a definition of “odd” as being 2n-1 where n varies through the natural numbers. Probably complete crogglement, without even the recognition Gauss got for his “legget se”.
RenÃ© 04.02.09 at 7:11 pm
So, let me congratulate John with the Rithmatic achievements of his Alice, advise him to get her some better Riting teaching, while I apologize for him Reading my typo M/n.
salient 04.02.09 at 7:40 pm
Actually, I would think thatâ€becas i tryed som and it workdâ€ is what the teacher is looking for.
Maybe. Back in the theoretical case, I’d maintain it’s important to not be “looking for” anything specific for these answers, and I hope the teacher wasn’t in this case. It is necessary and sufficient to check that the answer is reasonable and/or cute.
And I mean that at least half-seriously: I’d maintain that cute and creative answers, especially to open-ended questions like these, ought to be entitled to credit. Train of justifications: (1) One goal of a teacher should be, to encourage behavior in students that we would like to see in citizens. (2) We would like for citizens to be creative, clever, honest, and funny within the boundaries of politeness and propriety. (3) A key, essential ingredient of creativity is the ability to hypothesize: to guess, conjecture, and communicate those conjectures clearly. (4) The reasoning supporting a conjecture is often informal or poorly-formed, initially. It is reasonable to expect that students’ initial justifications for their hypotheses will not be as rigorous as proof, and may very well rely on appeals to example or to authority. (5) To encourage creativity, students should initially be asked to suggest support for their conjectures, exactly as DoJ did here,* and young students should be asked for a formal argument only in an interactive setting, as the justifications are discussed in greater detail.
*Daughter of John?**
**If/when she’s acting intolerable in her adolescent years, you’ll have to show her this thread. :)
bianca steele 04.02.09 at 8:14 pm
Do kids still use physical objects to learn math? You can “prove” even+even easily if you’ve been prompted with an intuitive set theoretical definition of “even,” though odd+odd is a little harder (and the proof is no less rigorous than Euclid). I agree with Eszter that questions like these seem likely to be answered intelligently only if the kids were given similar problems in class (though I’ve known teachers who apparently believed otherwise, convinced that someday they would have the true genius in their class who would intuit the technique, never having seen the problem before).
bianca steele 04.02.09 at 8:42 pm
And even if her class didn’t use physical objects, it’s possible the workbook was written with such a procedure in mind, I suppose. Though the “always” is strange. How do you explain to a small child that there is the difference between “when you add an odd number to an odd number” and “always, when you add an odd number to an odd number”?
John Quiggin 04.02.09 at 9:04 pm
I always remember being asked the question “Can you prove p …” and being sorely tempted to claim full marks for the answer “No”. Then, of course, there is
http://www.justtellmehowtomanage.com/2008/10/feedback-outgoing.html
James Wimberley 04.02.09 at 9:45 pm
Authority is all in the capitalisation. The leader and the Leader; a chairman and the Chairman; the party and the Party; a god and a God. So a mom is not an authority, a Mom is an authority, and a MoM is infallible.
Bloix 04.02.09 at 10:02 pm
Salient, I didn’t mean “looking for” in the sense of determining the answer for a good grade. I meant that the questions were intended to prompt certain actions and “I tried some examples” would demonstrate that the actions were taken. “My Mom said so” is cute but it shows that what the teacher was trying to accomplish in class didn’t happen, and the teacher needs to try again.
Z 04.02.09 at 10:50 pm
This is thought-provoking on so many levels: infant psychology (my MoM said so), pedagogy (why do you think so?), epistemology (is this always true? Always??), mathematics, you name it… Really wonderful.
What predicts whether a child will respond in such a way or leave the line blank even if she thinks of such a response?
Interestingly, I remember spending a not negligible amount of time during my school years trying to figure out what was teachers expecting of us for open ended questions. In first grade for instance, we were once given a text with the assignment “Circle all the words you don’t understand or that you can’t read”. I was a slightly precocious child, so I could read and understand all the words. I agonized over the exercise, trying to remember if we had ever seen this or that word in class (if not, presumably I should be circling it). In fifth grade, our teacher once wrote an addition, a subtraction, a multiplication and a division on the blackboard, instructing us to compute them. Then she wrote “Devise what is the hardest math problem for you and solve it”, meaning in fact the hardest operation (by the way, I think it is good teaching technique to ask such questions, at least to older students), but we had already been assigned problems (of the John has bought three apples and paid 15 francs, how much is an apple? variety) so I took the assignment literally and tried to think of the hardest math problem I could possibly solve (I eventually failed to solve it, by the way).
Anyway, I think the answers “A number”, “Maybe it won’t be true anymore when human beings have disappeared” and “I am unsure about platonism, perhaps even and odd are external properties, perhaps not” seems perfectly correct.
Ferris 04.03.09 at 10:22 am
You mean you don’t teach modular arithmetic in schools? Not even mod 2?
Smokinace 04.03.09 at 11:12 am
It’s actually pretty easy to explain arithmetically:
take 2 odd numbers… 5 and 7
7= 5+2
thus:
5+7=5+5+2
which can be expressed as:
7 + 5 =2 x 5 + 2
which is
7 + 5 = 2 x ( 5+ 1)
observe the 2. an even number by definition is a multiple of 2, thus 2 odd numbers added together will always result in an even number.
this works with any 2 odd numbers.
it even works with integers
-3 + 5
-3 = 5 – 8
5 – 8 + 5
which is 2 x 5 – 8
which, again is a multiple of 2: 2 x (5 – 4)
Maths > All
JoB 04.03.09 at 1:33 pm
Smokinace,
Shorter is:
1+1= 2 and in all the odd numbers there is a 1
I still believe that an answer like this would be plausible from a 7 or 8-year old. It wouldn’t be cute but it would be damned good.
Kyle 04.03.09 at 1:52 pm
You’re all wrong. Here’s what I’d say.
1) What do you get when you add two odd numbers together?
A: You get their sum.
2) Do you think this is always true?
A: Yes.
3) Why do you think so?
A: I doubt they’ll change the meaning of the word ‘sum’.
armand 04.03.09 at 2:13 pm
No, it’s definitely a Miata. Or maybe a Del Sol… not sure.
Sock Puppet of the Great Satan 04.03.09 at 3:19 pm
“Belle says that P
So, P.”
I hope that Belle’s Husband uses the same logical analytic framework.
go4broke 04.03.09 at 5:31 pm
I think the point was not whether you could prove this true. The point was how this was a logical fallacy for an argument. Namely the “proof” for the little girl is an appeal to authority. One of the fundamental flaws in logic that from a very early age we teach our children to do even though its wrong. Sadly even people well versed in mathematics fail to understand logic when it is applied to language :P
That’s why its titled Fallacy and not can you prove this with math.
I’m just amazed that there are 65 posts here and not a one realized what it was about.
Calum 04.03.09 at 6:43 pm
Can also be proven by modular arithmetic or induction. But c’mon that’s asking a lot from a six-year old or however old she is.
Curtis 04.03.09 at 7:15 pm
Even number can be written as 2k where k is any integer 2(1) = 2, 2(2)=4, 2(3) = 6
Odd can then be written as one more then an even so 2k+1, 2+1 = 3, 4+1 = 5, 6+1 = 7
(2k+1) + (2k+1) = 4k+2 = 2(2k+1)
2k+1 is an integer so 2(2k+1) must be even
Anthony 04.03.09 at 10:47 pm
Eszter – in my high school precalculus class, the rule was we could use on an exam (or homework) anything which we’d had in class or shown on the exam. One exam was after we’d learned the definition of a derivative, but not any of the really useful formulae for derivatives of various functions. The exam was a bunch of “find the derivative of this polynomial function”, with the intent of doing it the hard way – going through the polynomial and using the definition of derivative. For problem 1, I worked out the formula for the general case of the derivative of mx^n, and then solved the remaining problems by plugging them into the formula. Teacher was not amused, but did give me full marks.
Mom 04.03.09 at 11:46 pm
Looks right to me.
Michael Drake 04.04.09 at 1:46 am
Well, you have to know these things when you’re a Mom, you know.
Michael Drake 04.04.09 at 1:46 am
(BTW, this is why we need socialized epistemology.)
James 04.04.09 at 3:57 am
When first learning even and odd, I was taught that even numbers are numbers that are divisible by two. If she had been told that, then it wouldn’t be out of the ordinary for her to reason that any number added to itself would produce an even number, no matter if it were an odd or even to begin with.
JoB 04.04.09 at 10:54 am
James, yes, that’s actually easier, still Einstein at that age but if you add 2 odd numbers then divide by two you get the add number you started with … It is beyond me why the hell anybody would like to prove this with advanced formalisms.
Tangurena 04.05.09 at 4:52 am
Even that requires exploring whether addition is commutative, of course.
Are there any groups where addition is not commutative? I can’t think of any.
Matt Heath 04.05.09 at 11:01 am
James @76: That’s only a special case of what she was asked though. The question is adding two (possibly different) odd numbers
This is kind of a question about language rather than mathematics. There are plenty of groups where the operation is non-commutative (the group of symmetries of a equilateral triangle, say) but it wouldn’t be usual to refer to the operation as “addition” in these cases (whereas you might in an arbitrary commutative group). This probably isn’t very relevant to primary school arithmetic, though.
Tangurena 04.05.09 at 2:06 pm
Well, I could easily think of multiplying matrices being non-commutative, but salient’s comment got me wondering just how much of those old math classes I was forgetting.
Frank 04.05.09 at 8:36 pm
I think you have all missed the point: a very intellegent young lady was taking the piss.
Bless her.
Comments on this entry are closed.