See if this question makes sense to you. It is taken verbatim from my first-grader's math homework worksheet, with a reference to the California curriculum standard it's designed to meet:

Carrie wants to add 2 + 6. What number should she count on from [sic] to find the sum? Explain why.

I suppose it may mean "if you add by starting with one number and counting from it a number of times equal to the number you are adding, which number would you start with, and why?" That still strikes me as nonsensical, because — among other things — it doesn't matter, and she should be adding two and six without counting by now.

The first grader's homework includes numerical problems and word problems, as I would expect. However, it also includes some of these "explain the theory behind arithmetic operations" questions, which are similarly badly worded.

### Last 5 posts by Ken White

- A Response To Marc: Institutions, Agendas, and the "Culture War" - January 13th, 2016
- Lawyering Is About Service, Not Self-Actualization - January 11th, 2016
- Lawsplainer: Was FAU Prof. James Tracy Fired in Violation of His First Amendment Rights? - January 7th, 2016
- Defy, Defy, Defy. - January 7th, 2016
- President Obama And The Rhetoric Of Rights - January 5th, 2016

Roy Haddad says

Assuming she has to count, she should count from 6, because then less of her time will be lost to this bizarre waste of life.

Simon says

The only interpretation that makes sense to me is the one you've already come up with, I'd say. That seems like a really weird way to think about Math.

That said, the answer I think the question wants is something along the lines of "Six, because it takes less effort to count two more to arrive at the answer of eight." I could be totally wrong, though.

Nicholas Weaver says

"Can't do the kid's math homework" is a very old problem. Tom Lehrer wrote a great song about it:

Ken White says

I was tempted to write in "Carrie should count on hundreds of education-bureaucracy wonks to spend millions of dollars to devise thousands of unreadable standards that yield tens of thousands of incomprehensible questions that impede learning."

Simon Gill says

Presumably somewhere in her books is an explanation of the way they want your daughter to think about it.

I'd answer starting from two because you resolve expressions from left to right at the same level of operator precedence (i.e. BODMAS).

It is a very weird phrasing though.

Dan T. says

Start with six, to minimize the counting. Unless you haven't yet learned that addition is commutative, in which case you need to start with 2 because it comes first in the problem.

Nicholas Weaver says

Drat, just realized the video skipped the best part: Base 8…

pedr says

Is it trying to give a grounding to the idea that 2+6 is the same as 6+2 (and, as mentioned, 6+2 is quicker if your process of addition is to start at a number and count forward "six, seven, eight")? This can then be contrasted, perhaps in the lesson, with the fact that you cannot do this with subtraction.

Baz says

Children go through a series of strategies as they learn to add, before they can manage without counting: "counting all", "counting on", "counting all from larger", etc.

http://www.jstor.org/discover/10.2307/749165?uid=3738032&uid=2&uid=4&sid=21102678187061

Presumably they're being taught the last of these.

Baz says

ugh I meant "counting on from larger", obviously.

Ian M. Ireland says

I think the key here is what looks incorrect in the question as posed: "what number should she count on from." She's got two, she should "count on" from 3… Because 3 is the first number after 2.

Rick H. says

That is insane. If I was conspiracy-minded, I'd guess it was intentionally constructed, by an educator who hates/resents mathematics, to systematically bash any affection for the subject out of the students.

AlphaCentauri says

We ended up getting the textbook teachers' editions online used (very cheap!) to deal with some of that nonsense.

You'd have to know the context. Are they trying to teach the kids to use number lines?

Zack says

@Pedr: Sure you can do the same thing with subtraction.

6-2=6+(-2)=(-2)+6. So you start at -2 and count 6 forward. :)

HandOfGod137 says

It's just possible it's a typo-ridden introduction to the number line of the reals. Like others say, you can then move to showing how some operators aren't commutative and introduce minus numbers.

SarahW says

Yes, it's Pedr's suggestion. Even though the process is as stupid as can be, of no use to any person who intends on memorizing sums of any two single digit numbers in base ten math (who would also be anyone who intends to carry out normal activities of everyday life on this planet.) While finger counting is the game, it makes some small sense.

Cheezit crackers, get your kid away from this foolishness.

John says

If I was feeling very, very charitable, I would guess one of the following two explanations:

1. If you're trying to add two large numbers in your head, say 883 and 634, it gets easier if one of them is close to a round number. You can say, for example:

883 + 634 = 900 + (634 – 17) = 900 + 651 = 1551.

"Counting on" sounds like it could be a precursor to teaching this trick.

2. Alternatively, over the next few weeks they're going to be teaching the class why the standard longhand addition algorithm works. This is actually far more important than teaching them to carry it out. (Yes, seriously. I am an actual mathematician and the only arithmetic I do is in my head or on a calculator.) Counting on could be a very simple addition algorithm intended to introduce some ideas (like maybe commutativity) before discussing the usual algorithm.

Probably they're just incompetent, though. Maths education in the UK is pretty awful, and I hear in the US it's a lot worse.

cb says

so simple addition should be done by counting, but students should be able to comprehend/explain the abstraction of addition? The hell?

My guess would be 3(or 7, for simplicity as noted above). A six count would be 3-4-5-6-7-8(or two count 7-8), you don't count what you already have

Michael says

As a math major that spent a lot of time working on out reach programs for kids in the area I went to uni, things like this make me die a little bit inside.

I'm still not sure how everyone could get "4 + [ ] = 7" but lost it at "4 + x = 7".

HandOfGod137 says

I dunno. If it

isthe number line, it introduces geometrical thinking about operators, it makes imaginary numbers seem like a logical extension and it makes it easy to start thinking about stuff like vectors. It's a huge assumption to make from one phrase, but it could be a pretty good introduction to a lot of cool and useful maths.Orpheus says

I took (and benefited from) new math, and generally had good teachers, but aside from a few moments of clarity and insight from my fellow Math League competitors in high school, NO ONE could explain why we started every year from 3rd grade on, with a chapter on sets, groups, properties — certainly none of my teachers! It was only in college (where, having AP'ed out of two years of calculus, I tool abstract algebra as a lark) that I learned the simple, powerful, beautiful and elegant applications of these principles, but one can easily get a Ph.D in a science without ever taking it

eigenperson says

The answer is that she should count on from 3, because 3 is the number of prepositions appearing in a row in the question. For example, if the question had been "Which number should she count on in after from to find the sum?" then the answer would have been 5.

Seriously, any question that can be legitimately answered with "Who cares?" should not have been asked. This is a perfect example — even if you are still adding numbers by counting, it probably takes you longer to decide whether to "count on from" 2 or from 6 than it does to actually count to 6. If you replaced 6 with "935" then maybe the question would be worth asking, although it still assumes the student is using a naive addition algorithm.

Mark - Lord of the Albino Squirrels says

Maybe this is the textbook they are using:

http://newmathdoneright.com/2012/11/07/counting-on-rules-of-arithmetic-cora/

Peano axioms and lucky duck mamas.

John says

cb: No – the idea would be to have a progression very roughly like this:

– It's hard to see how to add two large numbers, but everyone can add and subtract 1 from things.

– Using this (and associativity and commutativity), we can say 66 + 43 = 67 + 42 = … = 109. So now we have a procedure that can add any two numbers together.

– But this is horribly inefficient. Especially since adding 10 or 100 is only about as hard as adding 1. So instead we could say 66 + 43 = … = 69 + 40 = 79 + 30 = … = 109.

– Then replace the counting by single digit addition and add some convenient notation and you've got the standard algorithm. Only now the students hopefully understand why it works.

That's how I'd use the concept anyway, but all my teaching is at university level so I really have no clue about kindergarten teaching.

ketchup says

This is bad on so many levels since Ken claims these types of questions are common. First of all, assuming the worksheet was provided by the textbook company (what teacher is going to write his/her own – they are busy enough as it is), including these questions shows a lack of competence by the authors of the curriculum. Second, it shows that the administrators or committee that chose the curriculum either did not know or did not care that these crazy questions were present. Third, that the teacher either did not review the worksheet for appropriateness, or did so and did think there was a problem.

In other words, the whole system is screwed up.

HandOfGod137 says

@eigenperson

Google tells me the USA's first-grade is for 6-7 year-olds, so a naive addition algorithm sounds a reasonable assumption. And if the answer is "2 or 6: it doesn't matter for addition, but it's different for subtraction", then a profound point has been made.

Apart from the obvious typo, this doesn't have to be as bad as people are making out.

gramps says

Maybe John (above) could explain what he did….

Possibly the parentheses are misplaced leading to a wrong answer. The concept of using a round number has its place in this stuff, but you have to use the correct numbers otherwise.

Nigel Declan says

I'm inclined to agree that the point of the question is to ask that, in the process of doing the addition through counting (which is the method that the teacher or whoever came up with the question seems to believe, rightly or wrongly, that 1st graders use to add), what number should he or she begin counting from. What the answer is depends in exactly what the kids know about whether a+b = b+a as well as whether counting, say, from 6 to 8 is consider starting at 6 (i.e. 6 plus one is 7, plus 2 is 8) or 7 (i.e. we have six, so we count 7 then 8). So the answer is either 2,3,6 or 7. I suspect the answer the teacher is looking for is 6, so the student is picking the larger number, recognizing that it is faster to count from 6 to 8 than from 2 to 8. But the question doesn't reflect a mathematical fact; it reflects a particular style of teaching which, if you are unfamiliar with it, is basically incomprehensible.

Nigel Declan says

Actually, considering the question is what number Carrie should "count on from", the "correct" answer would seem to be which of the numbers in the problem Carrie should use as the basis for starting her "addition-by-counting". So, 6. The exercise really seems to be to get the student to identify, in an addition context, which number is greater and, in so doing, that the operation is commutative (i.e. works both ways).

ketchup says

This story reminds me of an article I read some time ago where some physics professors did a survey of middle school science textbooks to check them for accuracy. They found hundreds of factual errors. By far the best one was a picture of Linda Ronstadt that was labeled as a silicon crystal.

http://content.time.com/time/magazine/article/0,9171,999123,00.html

cb says

@Mark-LotAS

I tried reading that page, and feel like it was originally written in some other language and translated by some really terrible online tool

@John

I don't quite get how that relates directly to the question. And it doesn't strike me as a particularly good way to reach young children

opendna says

The question assumes that the child get the correct answer so it is asking them to *explain* the process of getting the answer (whatever that process is).

It's very likely that most any of the answers given above will be acceptable. Pretty much the only wrong answer is "2+6 is 8, because it is."

KronWeld says

Ken, do please let us know how well you do on the homework, what your answer is/was and if it was correct. Also, if you can find out, what is the correct answer.

I'd send the question back to the teacher, letting them know that it is grammatically confusing at best and if they couldn't write a question that made sense, how did they get a teaching certificate?

bowlerhattedman says

John, you didn't say what subject you teach at the university level, but I sincerely hope it's not math. Or were you trying to out-Lehrer Lehrer?

You added 883 and 634, and came up with 1551.

(See me after class!)

John says

cb: If this interpretation is right, the question relates pretty well. With the simplest algorithm, you could either write 6 + 2 = 7 + 1 = 8 (presumably "counting on" from 6) or 2 + 6 = 3 + 5 = 4 + 4 = 5 + 3 = 6 + 2 = 7 + 1 = 8 + 0 ("counting on" from 2). Obviously you should count on from the bigger number, and this would probably have been covered in class. So it's just asking for a detail of the algorithm plus the reasoning behind it.

How would you teach children why the standard algorithm works? (This is not a snarky question.)

Stormy Dragon says

There's an interesting book on the history of Algebra out there called

Unknown Quantity. The phenomenon you point out above is a lot less boggling when you know it took human civilization thousands of years to go from the first to the second.lytefoot says

As a mathematician who

* has colleagues that worked on Common Core, and

* works on educating students who got through grade school by memorizing algorithms, hit a wall at algebra, and now need to un-learn everything and start over:

It is NOT obvious to everyone that you can add 2+6 by starting at 2 and counting 6 more, or starting at 6 and counting 2 more. If it's obvious to your kid, GREAT. But make sure it's really obvious, and not just that they're agreeing because you told them. The idea is to teach kids to see math as a coherent, logical structure, not a series of arbitrary rules. (If you never saw math as a series of arbitrary rules… good for you. You're in THAT half of the population.)

Ken, the correct answer to your child's homework (in first grade) is EITHER of:

* Start at 2 and count on 6 more. This gives a number 6 more than 2.

* Start at 6 and count on 2 more. We should start at 6 because it is bigger, and 2+6 is the same as 6+2.

The second one would be more correct in another grade.

If your child is keen to pass future math classes, learn higher math, or communicate with other humans in later life, it is important to be able to explain the reasoning.

ketchup says

Just the fact that there can be a real debate about the meaning of the question among the intelligent adults on this site shows how bad the question is. It is a first grade question for crying out loud!

eddie says

Given Baz' explanation, it seems the question is both correctly worded and appropriate to include in a problem set.

… assuming that they're receiving the corresponding in-class instruction in that particular strategy, of course.

As a parent outside the classroom – and not being a first-grader yourself – you'd probably have no way to know that that's what's going on, which is why the problem seems badly worded and pointless to you.

Mike says

This is all very funny. Look at all of you!

And people wonder why the gap hasn't been closed and test scores have not improved. The people running education don't know anything!

Jacob Schmidt says

Is your kid friends with Joe from Wayside elementary school?

Quantum Mechanic says

Let me guess…this is TERC or Everyday Math, etc.?

Stormy Dragon says

The funny part is that having grown up after the new math came into use, the supposedly confusing part of the song makes perfect sense to me and the supposedly obvious part was what confused me the first time I heard it.

And of course as The Frantics pointed out, old people hating new teaching methods (even when they are better than the old ones) has been around a long time:

eigenperson says

HandOfGod137

When I was in first grade, we were encouraged to think about addition as putting two quantities together. Commutativity is trivial (even for a first-grader) when you think about it that way. And to get 6+2, we didn't "count" — we took 6 and 2 and put them together.

We even had color-coded "Cuisenaire" rods of integer lengths, so we could take a rod of 6 and of length 2 and put them together to get 8 (and of course it didn't matter which way around we did it).

Maybe I'm biased, but I think this is far superior to the "counting" method, which requires the students to treat both sides of the "+" differently before realizing they're actually the same.

Stephen says

Trying to read that abomination of a question has convinced me to home-school when I finally have children.

David K says

I'm with Mark – It looks like a poor attempt to subtly hint at the Peano axioms, which are the underpinnings of arithmetic. But I don't think you just throw first graders into that ocean. They should get at least a little set theory beforehand.

What I think they're really asking is "what does the symbol + mean in terms of the natural (counting) numbers?" And the full answer to that is the Peano axioms, but a good first grade approximation is probably "you start at 2 [or 6] and count out 6 [or 2] places on the number line, and that gives you 2 + 6 = 8."

John says

Mike: None of the people talking have anything to do with the kindergarten curriculum, dude. I'm not even in the same country.

Ken White says

Is it a question with a right answer?

1. The answer could be "it doesn't matter which one you count from because the result will be the same," which teaches one mathematical concept (x+y = y+x).

2. Or the answer could be "she should count from six, because it will require less counting," which teaches a problem-solving strategy.

If the question is fishing for one or the other of these answers, then I think the question is rather vague, particularly for a first-grader. It could be rendered less vague by asking "does it make a difference" or saying "which is the easiest."

If the student ought to know, from first-grade classroom instruction, which one a vaguely worded question means, I'm not sure it teaches good reading strategies.

I have doubts about the pedagogical value in this form.

eigenperson says

Or even:

3. "She should count from 2, because that's the number that comes first in the problem." This doesn't teach a problem-solving strategy or a mathematical concept, but it does teach blind obedience to arbitrary authority, so perhaps this is the most likely intended answer.

Christopher says

So, part of the problem here seems to be the use of a specific vocabulary that folks like Ken and myself weren't taught; I guess "Count on from" has some kind of specific meaning. Judging from the link Mark posted up there, it just seems to be a fancy name for "add".

So, another way to say 2+6 is that you have 2, and you count on by 6 to get the result.

Judging by some more cursory reading, it seems to be an attempt to teach kids math the way they generally already learn it; to add 2+6, you start by saying "2" and then count off six more on your fingers, "three, four, five, six, seven, eight".

Fair enough.

Jargon aside, it doesn't seem like a very well-written question, because like many of you, even making an educated guess about what "count on from" means, I'm not sure what that question is intended to teach or demonstrate.

barry says

I'd start from zero (then you know where you are when you've finished).

0 + 2 + 6

or

0 + 6 + 2

gets to the same place.

Albert Lunde says

Rickard Feynman wrote this article on "New Math" in textbooks in 1965, I'm afraid it still applies today:

http://calteches.library.caltech.edu/2362/1/feynman.pdf

(Feynman also won the Nobel Prize in physics in 1965.)

tomvet475 says

I see what they're doing here. The most common and practical use of "counting on from" is making change from a cash register. They are teaching them to be clerks at WalMart or Mickey D's!

Education Realist says

Hi–David Pinsen sent me here via Twitter.

Ian thinks as I do, that "on" signifies the first number you start to count–that is, the correct answer is 3 or 7. But I teach high school, and only learned a couple years ago that "counting on" is a term used in elementary education. So I'm pretty sure Baz is correct. There was nothing ungrammatical about the directions. I would prefer they teach it differently, since I have a lot of kids still counting on their fingers in high school and it'd be easier if they knew what number to start with–particularly when plotting points. Oh, well.

A secondary purpose to the lesson is something that drives parents and many math experts nuts, which is to encourage children to realize that there's more than one answer, more than one way to do the problem. So the correct answer should be 2 or 6, with either one being correct. Then the student has to explain his or her thinking, a big part of reform math and Common Core.

At this point, people interrupt–oh, my god. We don't want them *thinking* about this! We want FLUENCY! AUTOMATICITY!!! And I get it, I really do. I used to agree with that line of thought and I still think, around the edges, in the middle of the cognitive ability spectrum, teaching more automaticity is a good thing. However, when you have seen several kids with high ability and crap fluency, and a solid fraction of kids with amazing fluency and low ability, you start to realize that fluency is just another form of pixiedust (that's my word this weekend.)

By the way, a huge number of kids will not be math fact fluent in first grade. And for those parents who are worried their little snowflake will be bored, give them enrichment but remember that almost all kids with decent intellects find elementary school easy. Moving them faster in elementary school doesn't guarantee you an advanced high schooler.

I'm still not sure how everyone could get "4 + [ ] = 7" but lost it at "4 + x = 7".Stormy Dragon is correct. This task draws on the ability to abstract, and not everyone has it. I mention several students with this problem, including one extended conversation, in the links below.

If you're interested, these posts are directly on point: Math fluencyand Teaching Students with Utilitarian Spectacles

As for the "teachers are stupid" meme, I have terrible news for your tropes. 1) No, they aren't. 2) Considerable evidence, including the TFA study that just came out, shows that smarter teachers have no meaningful impact on student outcomes.

Again, if you're interested, these are on point: Teacher Quality Pseudofacts I and

II

The takeaway from the TFA study

Jacob Schmidt says

Interesting. I typically hear from people that they think school is too technical and specific; it's a change to hear complaints about it being too vague.

Quiet Lurcker says

The answer is 2. But, why oh why doesn't common core require memorization of such basic facts?

tm says

Ken: So, now you know what it's like for us non lawyers to read through an EULA.

Obviously, 1st graders should just learn to use the calculator app on their smartphones. It's what they'll end up doing anyway 30 years hence.

barry says

@Quiet Lurcker You'd lose marks for missing the 'explain why' part of the question.

Bill says

I would say… start with two, because when counting on your fingers (as I'm assuming they are in first grade), holding up six fingers would use two hands, while holding up two fingers would be on a single hand.

Of course, I haven't had to deal with the educational system since back in 1990 so….

eigenperson says

Education Realist

I'm not claiming there's something grammatically wrong with the directions. But how can you expect children who are just learning to read, and whose understanding of addition is still on the level of counting, to understand a jargonistic train wreck like "count on from to"?

And if the answer is "because the teacher uses that term over and over again in class, so they're familiar with it," then surely there is a better use of the class time than making sure the students are familiar with the ed jargon for a algorithm that they ought to have abandoned six months from now?

John Beaty says

When children first start adding, they often start from the first number, instead of the first + 1, i.e. 2+2, they start "2, 3" and get 3 for the answer. "Count on" is a way (an old way, I learned it in the 60's) to talk about which number to start counting WITH, so that 2+2 goes, "2, count on 3, 4". Since many, many first graders have had exactly zero introduction to math, this sounds about right.

Shane says

@Ken

I think it is the phrase "count on" that sux. If you start at 2 you don't count it but you "count on" from it, so you count 3, 4, 5, 6, 7, 8.

Education Realist says

Well, many of them won't have abandoned it. As I mentioned, math fluency is not universal. The kids who learn it easily won't much care what the others use, so it's irrelevant. And yes, I suspect they know the term because the teacher uses it. If not, then of course that's a problem.

But, why oh why doesn't common core require memorization of such basic facts?I think they do, mainly because math profs complained. Requiring it or not won't make much difference, although as I said I'd be happier if more mid-ability kids didn't have to think about them. But that would take hours and hours and hours of math drills that a good third of the class wouldn't get, ever, and the top third would already have down cold.

Jess says

Everywhere. Like, such as.

Jacob Schmidt says

Because I've met children? Because I've been one?

If the kids have a hard time learning a new phrase, we have a bigger problem than math.

eigenperson says

Well, did Ken's daughter understand the phrase? (If so, I doubt Ken would be complaining.)

Besides, this isn't just a matter of learning a new phrase. It's learning a new technical term for an abstract concept. That's a much harder thing to do.

Xenocles says

"Seriously, any question that can be legitimately answered with "Who cares?" should not have been asked. "

My mother told me she once wrote "Who cares?" as a response to a test question. The teacher gave her zero points and wrote back "I do."

Alternatively, I suppose any subject could be legitimately derided as irrelevant by some student.

eigenperson says

Xenocles: That's a fair point; perhaps I was too glib. What I meant was that even if you are fascinated by the intricacies of addition and counting, the answer to this question is probably

stillnot interesting to you. It's a little like asking, in a history class:No matter how much you care about history, this doesn't matter to you. As a result, an answer like "Carrie can start browsing wherever the fuck she wants, because it's a free country" is just as good as some explanation of why starting at the top or bottom shelf is better. And if that non-answer is just as good as an answer, I think it's a stupid question.

Mark - Lord of the Albino Squirrels says

@cb

This is my favorite quote from that text book:

"Zero is a lucky duck. Every lucky duck is a mom duck with one baby, but zero is not a baby, each baby has one mom and a farm with zero and the baby of each mom on the farm has all the lucky ducks."

TobinL says

I will speak up for one, and as someone who majored in math for higher learnin'. For one as stated, it does not matter whether you count from 2 or from 6 because of how addition works. But the why? For first grade? What the heck? Thats a theory level question about reflexiveness of addition and you don't even learn how that works properly in kolleje till you take the modern math (or similar, basic first 300 level course) and you throw out just about everything you learned and start over and do actual proof and build addition, subtraction, etc.

Though we all could be overthinking that part of the question as to what the teacher actually wants to know.

Tracy- Shane's Wife says

Hi all. I'm sure some of the lawyers on here were English majors in their undergrad years. I know some of my professors would have spit out their soup at that sentence. It *is* a grammatical mess. "on from to" are three prepositions strung together. Forces the reader to parse out the sentence in order to make sense of it. Part of the reason this is a mess is because "count on" is being treated as a one-word verb. Something like this might work, if the teacher is looking for 2 or 6 as the answer:

If "what" was used not because the math teacher failed English but because the child is to begin counting at 3 or 7. Then it could be worded like this:

(I am using "begin" and "start" interchangeably here).

This solution does not take into account the teacher wanting to use "count on" as a mathmatical term. If this term is truly important, it could be bolded or underlined to signify to the reader that it is to be treated as one word. You still have "from" and "to" together, which is a common grammatical error as questions are the only sentences (from what I recall) that can begin with a preposition.

:) Now all you intellectuals out there can relax and go back to talking about… uh… intellectual stuff.

Xenocles says

@eigenperson-

I like to dump on the educational system as much as anyone else here, but to me there's a difference between the way a jaded adult would tackle that question and the way a novice having gone through a particular course of study would. With no context, I guessed that it was a question meant to assess if you had learned a particular method of attacking small addition problems. In many ways "It doesn't matter" is a correct answer – owing to the commutative property or to the fact that in a grand enough scale it pretty much costs the same to count to six as it does to two. But at the same time I find myself encouraging my six-year-old to start from the larger number if she feels she has to resort to counting to do a sum because she often loses track and either forgets to stop or has to start over for some reason. It's probably that novice/veteran divide again.

Part of me enjoys finding creative alternative valid answers for questions. Another part of me just says "You know what they're getting at, just show the knowledge they're looking for." I suppose it depends on the scenario.

LauraW says

Feynman also wrote a great essay called "Judging Books by their Covers" about the time he served on the California Curriculum Commission evaluating math textbooks. I first saw it in the book

Surely you're joking, Mr. Feynman!It's reproduced here. His wife referred to him as the volcano in the basement because he would erupt every time he came across this sort of inanity.Matthew Cline says

Why are grade schoolers being taught the axioms used to construct the set of natural numbers?

mud man says

What Xenocles said. Also, when I was a working programmer, this kind of trivial optimization is the sort of thing I spent a lot of time on: count (6, 7, 8) in 2 steps or count (2, 3, 4, 5, 6, 7, 8) in 8 steps? Of course in those days it was ones and zeros ("In the OLD old days we didn't have zeros! Had to do it all with ones!") whereas today it's all magic rabbits and if you can pull one out of any hat at all, it hardly matters how long it takes. So maybe it's clumsy and obscure but it loos to me that what they are trying to teach here is ENGINEERING, long may it wave.

John L says

I'm not necessarily opposed to talking about why arithmetic works the way it does at such an early point in the learning process, but my question is this: if they teachers are going to teach the "why"s, shouldn't they be required to have studied the "why"s? I don't know a single math teacher whose taken discrete math, much less number theory, although that might just be sampling bias on my part.

eigenperson says

Xenocles:

I feel like there's a difference the two scenarios. In one, you're suggesting to a kid who is actually struggling with an addition problem that she might have less trouble if she started with the other number. That seems like a great idea (and a good problem-solving strategy: if you're stuck, try solving the problem a different way). In the other, you're asking the kid to come up with a rule for Carrie to follow, which is not, I think, a good way to learn problem-solving strategies.

I guess I react negatively to this question because it seems to me that our education system causes kids to regard math as a process that has a lot of rules that you have to follow. And, of course, there

arerules that you have to follow if you want to get a correct answer, but "always start adding from the larger number" isn't one of them. Every unnecessary rule you introduce makes the kids more and more anxious, because they always have to worry about accidentally falling afoul of some rule.To be fair, I only interact with students at the opposite end of the educational system, but over and over again I see how things that are supposed to be helpful problem-solving suggestions are transformed, in the minds of the students, into anxiety-producing burdens. The one that comes to mind right now is "FOIL," which is supposed to be a helpful mnemonic for multiplying binomials, but which students treat like a Masonic ritual that can only be performed during certain lunar periods.

rsteinmetz70112 says

Asking lawyers about math is a doomed quest, unless or course it refers to fees.

Xenocles says

@eigenperson-

It all depends on the context to me. I could see the course of instruction planned such that when the student faces that question for real it's obvious they're asking about strategy. I could also see that instruction not being clear and resulting in the confusion – that Ken is seeing this so early in the year probably doesn't help.

@mud man-

Optimization is sort of a casual hobby for me, but there are times when you're simply polishing the cannonball before firing it. I absolutely see how small changes like that would make a huge difference over a large number of iterations. I also have a feeling that the extra cost of doing it either way for a human of sufficient skill is negligible. (For a small enough difference might it even cost more than you save to figure out the "right" way to attack it?)

teiglin says

I agree with others that asking kids to think about this sort of thing is not at all nonsensical–traditionally kids learn basic algorithms for arithmetic without any explanation as to why they work, which means they are never taught to understand arithmetic. This is helpful for math of all levels, though admittedly it's only really necessary for college courses like analysis and such. Reasonable answers include "it doesn't matter, addition is commutative" and "adding two to six by counting takes less time"–though the former should be what they are getting at, assuming the course isn't just idiotic.

What actually bothers me about that is not the typo (errors are a part of life, and it's so obvious that it isn't confusing). What bugs me is actually the phrase "add 2 + 6"–the redundancy absolutely drives my anal side up the wall. You can add two and six, or you could evaluate 2+6, which is obviously what they mean. If you tell me you want to "add 2+6" then my question is "what do you want to add to 2+6?"

C. S. P. Schofield says

For most of a century, if not longer, the Professional Educators have been desperately trying to make learning "fun", for certain values of fun. This has led them to attempt to do away with repetitious exercises such as multiplication table, and to introduce complex mathematical concepts much earlier. The latter is complicated by the fact that the vast majority of them are no mathematicians, do not really understand the concepts in any depth, and thus are flailing around.

HAVING and education is fun; it greatly multiplies the power of whatever natural intelligence you have. ACQUIRING and education, however, is frequently tiresome …. especially when dealing with matters such as spelling, grammar, and arithmetic where the answers to "Why is that so" are so complicated that it is often better to simply say "because", at least until the pupil has enough grounding to handle the complex.

IgnatzEsq says

I'm pretty sure that to 'explain why' you first have to define 'addition' from the successor operation a la the Peano axioms.

Todd Knarr says

I get the feeling this is being over-thought. This is 1st grade. They're just done recognizing numbers by how many things that number represents, and starting on the number line and addition/subtraction by counting. It's not unreasonable this early in the term to have questions to test whether the kid recognizes that the starting point for the process is one of the numbers to be added, as opposed to their favorite number or the number the teacher said in a completely unrelated problem a few minutes ago. And yes, I remember from grade school other kids in the class not getting basic concepts like that right off.

John says

Mark: That… thing… is an abomination. The axioms of the natural numbers only make sense in the first place because they match our pre-formed intuition about the natural numbers. When you take that away, even if you replace it with a terrifyingly infinite horde of little quacky ducks, you're left with gibberish. Gibberish which is syntactically equivalent to the Peano axioms, but gibberish nonetheless. And this is being inflicted on people who don't even *have* a pre-formed intuition about the natural numbers.

The author is clearly a harmless(?) lunatic, but whoever humoured them enough to put their work on a syllabus needs firing. From a cannon.

OngChotwI says

Carrie wants to add 2 to 6. Clearly, Carrie has two options in fulfilling this desire. The first option would be to start with 2 ponies, and convince her paternal parent type to pay Farmer Brown a stud fee for his mini Stallion a total of 3 times to service these 2 ponies 3 times. After waiting almost a year, 2 new ponies will be born. 6 months later, the ponies will be old enough for weening, and the second visit from Farmer Brown's mini Stallion could be arranged. In another year and a half, a third visit from Farmer Brown's mini Stallion could be purchased. While this approach will provide some insight into the genetic diversity of the original two mare ponies.. at some point, Carrie's male parental unit might start considering them night mares, and renege on his promise about providing 3 visits from Farmer Brown's mini Stallion. Thus, this has a high chance of being a sub optimal method of adding 2 and 6. A second option would be to start with 6 mares. Carrie's male parental unit would pay Farmer Brown a stud fee to have his mini Stallion service two of the mare ponies. This option has a much higher probability of our being able to add 2 and 6 and getting 8, instead of perhaps only being allowed to add 2 and 2 and thus getting a mere 4 ponies to bring delight to Carietta "Carrie" White.

(Stephen King reference, not picking on Ken.. much.)

(Some of my math teachers maxed out the math tests, but had sub par English scores – and we quickly taught them the K.I.S.S. principle to keep us from arguing over what the correct answer was.)

gramps says

@ tomvet…. If that is the case they are teaching an obsolete skill: those places use cash registers where the clerk punches in the amount of money tendered and computes the change due.

Want to confuse them? give them a $10 bill and the 27 cents for the $6.27 purchase so you can get 4 ones back instead of 3 and a handful of change.

Micky-D even has registers with pictures of the products so the clerk only has to touch the picture… no knowledge of the costs of the items is necessary…. and those came out decades ago.

Not that this has anything to do with teaching number theory to first graders. I am surprised that they expect first graders to understand and/or deal with word problems; those were the bane of my 7th grade math class… but that was quite a while ago.

rmd says

My sister edits math textbooks (though hopefully not the one at issue.) I think from now on, if anyone asks me what she does, I'm going to say she plays piano in a whorehouse.

tm says

eigenperson: From my kids' homework (also in California), there's an emphasis placed on how they got to answer rather than the "this is the one way to get to the answer, so learn this rule!" that I'm sure we're all familiar with. Clearly 2+2 = 4, but how does a child solve 438 + 743, what's their thinking, etc.

John L: Do you mean, when a 1st grader asks "why" addition works, we should break out something from number theory? That we should have our teachers working from "The Higher Arithmetic"? Silly, no?

I think we will be ok as a society if a teacher does not use the formal abstractions and gives a bunch of 1st graders some technically inaccurate, but good enough for non-mathematicians, concreteness. If a precocious kid wants to peek behind the curtain, I hope they're guided to some sort of explanation on the internet.

I read a while back that one of my organic chemistry professors win the Nobel prize. I told my wife, "I fell asleep in his class!" Clearly a great researcher and expert in his field and in chemistry in general. Absolutely terrible at teaching. Good thing he's rewarded for the former and not the latter. And he was by no means an aberration in a school filled with brilliant researchers.

barry says

There are many mathematical terms and concepts which don't have a good english translation, so we get hybrid terms like 'count on from' which the students probably have a better idea of the meaning of from hearing the context than their parents do seeing it for the first time.

To me, not having learnt to add that way, the question looks ambiguous with 6 possible answers (0,1,2,3,6, or 7), and the rightness of the answer depending on how well the 'explain why' matches whichever particular number is given.

Marconi says

This may sound weird, but it is how a nun in a Catholic school taught us, a long time ago.

Hold open a hand, look at the four fingers with three segments each, and the thumb with two.

Count off two. Then count off six from that segment going down the fingers and marking where you end up. Now count all of the segments.

Don't shoot the messenger, but that is one way to interpret the grammatically mangled sentence probably means

AlphaCentauri says

First grade homework isn't going to introduce any new concepts. Whatever it means, it refers to something the kids did in class already.

As I recall, my kids' first grade desks all had number lines taped to them. I suspect the kids have been doing problems together counting on the number lines. That means that to avoid chaos, they have to have a convention of where they will all start counting. It doesn't matter what the answer is (largest number or first term in the equation), only that all the kids doing exercises together in that classroom know what they have agreed to do.

Mike B says

This is the kind of horse excriment that pretty much exemplified why our children are failing miserably in school. Next to "teaching for the test" the worst thing that the dogmatic idiots who run schools and write textbooks do is labor under the illusion that there is a "right" way to do things" The answer to that question is "it doesn't matter what number she counts from because if she can count, she'll get it right either way."

spinetingler says

John L "I don't know a single math teacher whose taken discrete math, much less number theory, although that might just be sampling bias on my part."

Well, now you do. (Discrete was at freaking 7 AM). Of course, mine were graduate level classes, so you may only know teachers with Bachelors degrees.

whheydt says

Having encountered New Math for the first time in High School (because it actually was NEW at that time), it threw me for a loss. It only came in handy about 30 years later when I had the epiphany of realizing that SQL is based on set theory (as was the New Math).

Tom Lehrer's song didn't bother me a bit, because I'd played around with different base systems, like octal years before the New Math came along.

As for the problem at hand…sounds like a lead in the integer number lines. Take a pair of rulers and line them up as a primitive slide rule and send a picture to the teacher labeled "THAT'S why."

p.s. Teach the kid how to use a slide rule and she'll make a lot fewer wildly wrong mistakes because she'll quickly learn how to get the magnitude of the answer right, even if some of the low order digits are off a bit.

Gabe says

Finally, there is a post on Popehat on which I am qualified to comment. I am not an educator, but I am a graduate student in pure mathematics. Ken's daughter's question is badly worded and out of context, pretty much meaningless. However, the comments above that suggested that 6 is the correct answer because 2 is a smaller number to count up are probably right. In order to figure this out, a kid needs to understand that 2 + 6 = 6 + 2 and how one can add numbers using counting, which is pretty much the definition of addition via iterations of the successor function. A lot of posters have made fun of how convoluted this makes a simple idea, but there is a saying in math to "think deeply of simple things" and the ideas underpinning this are a lot deeper and more interesting than it would initially appear. Also, anything that gets children to play with numbers and ideas is good and hopefully these questions (when better worded) inspire a spark of interest when they notice patterns.

On a separate note, much of the criticism of math pedagogy, especially of new math, strikes me much as being akin to complaints that legal documents require too much specialized knowledge and so cannot be read by a layperson. I'll grant that new math was a disaster, but math research resembles "new math" more than anything else that is not exclusively taught to math majors in college (e.g. analysis and abstract algebra). New math didn't work for many reasons, including that it ignored the natural mathematical development of most children and required teachers who were experts in mathematics to teach the material, but it was the exact opposite of lowering standards and teaching down. It was an attempt to teach math the way that professionals mathematicians think about it. In modern mathematics, the definitions initially seem abstract and ridiculous, but you need precision and clarity in order to prove theorems and you want to pinpoint exactly what the essential ideas are. That was the goal of new math, and the problem it takes years of mathematical experience to understand why the way it presented ideas give the right way to tackle problems.

It is a shame that new math scared so many students away from mathematics and I suppose it is the equivalent of throwing children into an Olympic swimming pool and telling them to swim IMs, but I can appreciate the idea behind it and it did a fair amount of good as well. We are now in an age where computers can do many of the difficult computations and so conceptual understanding has reached paramount importance, and new math shifted grade school math from rote calculations. Plus, that sort of brutal math at a young age produced Grigori Perelman.

Finally to tm, The Higher Arithmetic is far too elementary for 1st graders. If you want to really challenge the kids, I recommend "A Course in Arithmetic" by Serre.

rxc says

As someone who learned math the old way, just barely excaped most of the "New Math", became an engineer and then lost most of the advanced math that I learned because I didn't use most of it, this discussion is fascinating. I don't understand the language at all, which I attribute to the need of the education-academic industry to continuously invent new methods to show that they are performing real "research", and discovering new paradigms that need to be applied to the education field. I.e., experimenting on children.

I think that the creation of a new language for talking about subjects like this which is taught to children is a symptom of the modern educational system. I think that it is intended to get children to think differently from their parents, and inculcate them in new social paradigms. The concept of sets, which can be very useful later in life, in many ways, is introduced early so that children learn how to aggregate things (and people) by their characteristics, and then identify commonalities among groups of things (and people). Sound familiar? BTW, this idea just occurred to me, from reading the comments about sets.

I know that I currently use all sorts of strategies in doing simple arithmetic, from using multiples of 2 and 3 and 5, to rote memorization of table, to simple counting on my fingers, when I don't have a machine to do it. I have no idea when I started to develop any of these techniques. I don't think it is a good technique to teach that there is only one "good way" to do it.

V says

Why is it even a word problem. Where are the apples or other inane object to count?

Carrie wants something and suddenly it's your child's problem? If Carrie needs help with her homework, she can ask politely.

Alan W. says

Yo, the answer is 42 bitches!

(Sorry…I've been watching too much Breaking Bad lately, so my phraseology is janked.)

Dan says

The answer is 6. Carrie then needs to count two more numbers to reach the sum.

Grant Gould says

My first grade daughter says: "Start counting at 8. Because it's always shortest to start from the right answer."

So there is a data point of some sort.

Robert White says

If you're interested in correct results you always start counting from zero.

Depending on the presuppositions of your school you may need to start counting from Xenu, jehovah, Yahweh, or Allah instead.

Advanced placement schools may have you start counting from "1" because the first one is so special that it doesn't actually count.

A computer class doesn't start counting _from_ zero but instead starts you counting the first one _as_ zero, because computers want to know how many you have to "skip over" to get to something and you don't skip over any to get to the first one. (e.g. the first apple is zero apples away from the first apple.)

In really higher computer maths there is no such thing as two (2) let alone _six_.

So it all depends really on how well your child can reverse engineer their school and teacher.

HandOfGod137 says

@rxc

A comment like that really call for a stylish tinfoil fedora. Sets are taught because they are fundamental to all maths. More fundamental, in fact, than numbers (you can derive numbers from set theory). And it's probably a good thing that children are being shown a different way of looking at maths to their parents, given that we have a large section of the adult population who can discount some of the best, most accurate collaborative science currently being done on the basis that it's "just models".

Basic logic and Boolean algebra (and hence computer science) rely heavily on set theory. Introducing it, the number line and basic concepts of number theory at an early age are essential if we're not to end up with a generation as dim as some of their parents apparently are.

CJK Fossman says

About computer arithmetic:

Computer Arithmetic-Logic Unit (ALU)

And Two's complement number representation.

CJK Fossman says

I wish it were as simple as teaching set theory and number theory. I fear it is not.

We have so many who cling to their beliefs tighter when faced with contrary evidence.

Skip Intro says

My immediate reaction was "teaches basic work on the real number line". This thread does a good job of illustrating basic set theory by demonstrating that the intersection of the set of people who are numerate and the set of people who understand pedagogy is a very small subset of the set of all internet commentators.

John Beaty says

I hope that, when Ken has asked the teacher about this, he will post the answer.

In the meantime the 4 or 5 people who have elementary school children and have encountered this, or who have taught it, will have to bide their time amid the conspiracies and fallacies.

Duncan Byers says

Ok…..yes, I'm a lawyer and no I'm not a mathematician. But I do have a BS and MS in Chemistry, and did some gawd-awful mathematical type stuff. So here's my two cents.

It's terribly worded (as numerous people have already pointed out; I won't belabor the point) but I *think* the correct answer is…. zero. It's a fundamental theoretical point that the numbers should be considered as a continuum on the number line, including the negative numbers. So, we start at zero, count out two (add two) and then count out six more (add six). Every problem is "anchored" to the zero on the number line. That way, visually, the problem "Add -2 + 6" is counting backwards (negative) -2, then forwards (positive) +6 giving 4 (positive) as the answer. Prepares a child for algebra and trig (think x, y, z coordinate system).

That's my two cents worth. But you get what you pay for here.

Can't do math says

Monty Python Math. Start at 1:29. http://www.youtube.com/watch?v=xOrgLj9lOwk

Dan Weber says

How should I try to teach the Peano axioms to an eighth grader who doesn't particularly excel at math?

Gabe says

Please don't teach the Peano axioms to any 8th grader. Show them pictures of the Mandelbrot set or explain the statement of Fermat's Last Theorem instead. If you can spark interest in the subject, they will eventually understand that they need axioms and be able to appreciate Peano axioms then.

ysth says

It's not elegant language, but not technically incorrect.

"on" there is used in its sense as an adverb, though "onward" would have been a better choice.

With a discrete set of choices, "which" would be preferable to "what", but "what" certainly is not *wrong*.

I personally try to avoid dangling prepositions; here, perhaps unusually, the non-dangling "From which number should she count onward" would actually sound less contrived, but again, it is not an error, just inelegance.

CJK Fossman says

@Dan Weber

There is really only one way to teach anything.

1. Describe it and verify the student understands the description.

2. Demonstrate it and verify the student follows your demonstration.

3. Have the student do it until he gets it right.

Everything else comes under the heading of tools.

Are you sure the eighth grader isn't math averse?

Of my three kids, the most math averse is a successful salesman who has no trouble figuring his commissions.

A Person (Probably?) says

The answer is Three or Seven. Because you go 3(1), 4(2), 5(3), 6(4), 7(5), 8(6). A lot of kids would start counting with '2' which would leave them short by a number.

It's basically a weird and badly phrased way of teaching kids inclusive/exclusive rules for numbering.

mud man says

@xencles, if you're still around…. also @eigenperson

If you are firing for accuracy at extreme range, a well-polished cannonball would be a good thing. But never mind. It's just HOMEWORK. It isn't the whole admissions test.

This is a school problem; the correct answer has ONLY social value, no real-world consequences at all. And it doesn't ASK for a RULE: The point (I assume) is to get at the bizarre constraints of this particular problem and merge it with a good grip on the domain of small number arithmetic. Us programmers hate on customers/managers that want to tell us HOW things should be done; stick to WHAT, plz.

Granted this is clumsy, but I'll bet it's a lot more like what Little Suzie will see in a few years on the Math AP test than reciting the addition table. Which she absolutely should be able to do also.

CJK Fossman says

In my world, confusing = incorrect, even if one can parse it.

Xenocles says

mud man-

Sorry. "Polishing the cannonball" is an expression in my field (and perhaps others, but I don't know) that directly implies that the benefit derived from exerting the effort is too small to justify the effort. You're right about the extrema, of course, this is just an example of the limitations of the idiom. You can read it as trying to predict where the cannonball will land down to the precise molecule or something similar – interesting in theory but unimportant in battle.

And I certainly agree that the curriculum could be inappropriately focused here. But I hesitate to infer that from a single sample question.

CJK Fossman says

Totally correct.

What I don't get is how this dose of number theory, even generously assuming the teacher understands it, is going to help little Suzie master the real-world situation where she has to add 6 + 2 in her head.

Do we somehow assume a magical process whereby understanding the theoretical underpinnings leads to mastery of performance? In my experience quite the opposite is true.

I believe humans reason better from the concrete to the abstract, rather than the reverse.

James says

Everyone assumes "Count on from (sic)" is a typo. It's not.

I don't remember who (above) pointed out that you could either start "counting on" with either 3 or 7, because you "don't start with what you already have."

Yes, eventually, the child will have to use rote skills to memorize simple addition and multiplication facts. Modern education theory teaches that understanding that 2+6 is actually 2, counting on to 3, 4, 5, 6, 7, then 8 is essential to understanding, where simply reciting the addition tables scars little minds and prevents understanding of higher concepts.

What's hysterical to me is that a bunch of folks who parse complex sentences and understand the importance of specific phrasing for clarity discount – or fail to recognize – the jargon of elementary education – and immediately assume the educator(s) must be idiots for failing to correct an obvious typo.

And really – do any of you truly think that "commutative" is a concept for the second or third month of first grade?

GeoffreyK says

Another math classic: Abbott and Costello, 7×13=28

http://www.youtube.com/watch?v=xkbQDEXJy2k

Jokes aside, I'm sympathetic to the idea that there is value in teaching this kind of "how math works" reasoning rather than "math is magic". As a counterpoint, as a student, I was endlessly frustrated by the need to work out problems in the style being taught, rather than just doing it the way it made sense to me (and realizing after the fact, sometimes, that they were trying to teach a method which was more robust). What is frustrating to a student and what is worthwhile to a student are not mutually exclusive

sets.As to the grammar fascists, relying on the assumption that "count on" is a technical term, the only grammatical error in the sentence is the ending of the phrase in a preposition. I'll let Churchill (or the anonymous contributor it really came from) handle that one: "This is the sort of English up with which I will not put."

Alternatively, if you want to re-arrange to avoid most of the hard-to-parse grammatical phrasings: "To find the sum, from what number should she count on?" I don't find that much improved, to be honest, but if you want to harp on the lack of adherence to a strict, un-justified ideological set of rules, enjoy.

And finally, if "count on" is jargon, and you're unhappy about the need/value of jargon for communicating concepts, I'll just refer you here: http://xkcd.com/1133/

Ron Larson says

I read it as start with 6, and add 2. That way you have less change of getting lost on your way to the total. It all stays on one hand.

The point it is trying to make is that you can switch the operands in addition and multiplication. They don't have to be done in the order presented.

I agree, very poorly worded.

Hoare says

I diddn't read 118 comments so pardon me if this has been said…

Kids learn to start counting/adding using a number line.

In this case one would start at 2 on the number line and count 6 places to the right. Hopefully ending on 8.

This form of teaching is why some kids count on their fingers for too long IMHO

Paul Baxter says

Two things:

first, this is an excellent study on exactly why China outperforms the US in elementary math education, despite Chinese teachers having far less education: http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-Understanding/dp/0415873843

Second, on a related note, I would highly recommend just using the Singapore Math curriculum at home and more or less ignoring whatever the school does with math. My oldest just finished the full Singapore elementary math set the day before he turned 9. He's ready to move into Algebra now and has a fantastic understanding of all of the basic math and geometry concepts.

HandOfGod137 says

@CJK Fossman

But you're not explicitly teaching number theory. You're introducing addition as a simple translation along a number line that will make introducing more complex concepts loads easier later.

whheydt says

From my experience, there was nothing particularly *wrong* with the New Math, it's just that–when first encountering it in High School–it presumed that one had been getting it fed to you for the previous ten years…an invalid assumption when the program swept through school districts.

I managed to scrape by in Integral Calculus, but my Mathematical downfall was matrices, particularly, matrix inversions.

A major part of the calculus issue is that many engineering problems don't actually have solutions, so they have to be dealt with by other methods, like numerical solutions and successive approximation. As is sometimes explained in engineering courses, in order to solve a problem in integral calculus, you have to already know what the solution is (hence, the section in the math tables of the CRC Handbook of such solutions).

Ashera says

My husband is a mathematician, and even he sometimes has problems figuring out what the heck is being asked in my kid's math homework–sometimes leading to two (or more) correct answers for the problem. The manner in which a lot of the word problems are phrased is extremely ambiguous.

JWH says

When trying to reach a settlement between 2 and 6, one should start at 2.1 (assuming representation of 2), but should be prepared for something in the 3.5-4.5 range.

Grant Gould says

The problem with this problem as I see it — and I would also pick on Hoare's comment above — is that it is assumes that there is one way to do, or to learn, addition.

Children learn and do addition in a variety of ways and, particularly if you wait until first grade when many students will have already had to do practical addition for other purposes (like helping Daddy make pancakes, in my family's case), all that a question like this does is make the students who already learned "wrong" ways to add feel very, very confused.

There is no reason that a first grader should be obliged to count in order to add. Even if they learned to add by counting (which is hardly universal), that could be a couple of years in the past by then, and all that asking them to count does is force them backward and alienate them from their own mathematical intuition.

This problem screams of a desire to know or, failing that, to enforce a single, universal, correct, and uniform progression of childhood skills. Education professionals may believe that sort of nonsense, but no half-decent teacher does.

Devil's Advocate says

I have to admit that my first reaction was something to the effect of: "Since this is the California curriculum, you first have to figure out which number is more disadvantaged before you can decide which number to start with. On the other hand, if this was the Texas curriculum, the correct answer would be that she needs to ask the nearest man what the answer is."

JeffM says

I tutor math at a free math help website. I also was in a class that were guinea pigs for the "new math" way back when.

The "new math" and its watered down modern version are UNBELIEVABLY stupid. They are of no use to grade school kids. Furthermore, few public school teachers are competent at it. I took an abstract algebra course, which was one of the hardest math courses I took, and it was a requirement for education students in math. They were almost all totally lost. So why do we try to teach concepts that are not required for learning arithmetic and elementary algebra (meaning the algebra applicable to real numbers) and that most teachers understand at only the most superficial level?

6 * 2 = 12 = 2 * 6. Kids learn that naturally. You can explain it to them by drawing two rows of six dots and turning the drawing on its side to show six rows of two dots. Either way, the number of dots is 12. Over and done. The kid gets it. Calling it the commutative law does not add anything to that. Why do true mathematicians bother? Because there are advanced forms of mathematics where a * b does not equal b * a, such as matrix multiplication. So if you know a lot of mathematics, you have to specify if what you are talking about follows a * b = b * a or not. But it is useless baggage for a first grader. Usually, a student first encounters a non-commutative algebra when studying matrices, and that is usually late in high school or early in college, and even then only for students who have a flair for math.

Rowan says

The school system now is just a bit crazy. Like my son is learning to read and the school says he just needs to know the words rather than sound them out. Hello how is he going to learn new words if he doesn't sound them out?!

sorrykb says

I think part of the problem is the challenge of translating mathematical concepts into English, particularly first-grade English. Perhaps in the language of mathematics, is "What number should she count on from to find the sum?" positively sings. In English, however, the sentence is a bit of a train wreck.

It doesn't help that prepositions, especially in the vernacular, are ambiguous little monsters. (I hate them.) Example: "My alarm went off at 6:45 this morning. It was annoying, so I shut it off." (Since it "went off", one might assume that it was already off, but noooo.)

I

thinkthat this homework problem is asking students to "count on" from a particular starting number.The homework problem might be confusing to most of the adults here (myself include) because the way we were taught math (while "good enough for most of us to get by") might actually be lacking. I think it's a good thing if kids are being taught to reason through problems, rather than just memorizing arithmetic or multiplication tables. If they give this curriculum a chance to succeed (rather than switching to a completely different method in two years' time, as so often happens with educational reform), it might reap real benefits.

Still, we're left with the problem that a generation of kids will have parents (and other adults around them) who struggle to help them with their first-grade homework.

Maybe the homework assignment was really meant for the parents. Look at this thread, where we have a group of adults discussing "2+6". Or is it "6+2"?

Keep calm and count on.

mud man says

@Xenocles

Actually I don't think the curriculum is badly focused, just maybe a little clumsy. I like that the question is open-ended … the very confusion that it engenders is an opportunity. Look at all the people, starting with Ken, who are frustrated because they don't know what rule to apply. Having appropriately confused the tykes, there's an opportunity for an edifyin' (if probably inconclusive) class discussion.

Give it up, Clark! Humans are not advanced enough for Anarchy.

I do get the point of your idiom, but it's a lazy man's excuse, begging your pardon. In my business we used to say "They think we're making toasters down here." I claim educating kids is more like polishing cannonballs than like making toasters. (Good topic for class discussion.)

@JeffM, bad teacher prep is a whole different problem. You could argue that the problem those M.Ed. students have is, they were badly taught in grade school. Chickens and eggs. Since they don't get it, all they can do is pass on the same old simplistic generalizations. AKA "rules". Which has also the advantage of not confusing the School Board or the parents who vote for them.

BTW, a simple intuitive example of non-commutivity is 90 degree rotations of a sphere, like a classroom globe. Or Google Earth. Of course you have to be thinking of such as legitimate math, and what does THAT have to do with small-number arithmetic, I ask you???

Todd Knarr says

@sorrykb: "The homework problem might be confusing to most of the adults here (myself include) because the way we were taught math (while "good enough for most of us to get by") might actually be lacking."

The method here was actually how I was taught. First, recognizing numbers by "How many is this?". Then addition and subtraction using the number line, learning that we had to start from one of the numbers and count right or left by the other number. That included teaching the kids that they couldn't just start with their favorite number. Then on to multiplication by repetitive addition, which led to the single-digit multiplication tables and how to do multiple-digit multiplication as a sum of single-digit multiplications (which showed us

whymemorizing the multiplication tables was a Good Idea). Are there really that few people who don't remember their first few years of elementary school?tom says

Start with 8. Then count to ten, and take a deep breath. Why? I can add 2 and 6, but I do need to manage my anger to stay out of jail. Any more stupid questions?

Start with 4, the average between the two numbers. Then, count 4 more, because twice the average is the sum.

Factor 2 = 2 and 6 = 3*2. Divide by the gcd (2), and solve the simpler counting problem: start at 3 (6/2) and count up once (2/2) to find 3+1=4. Finally, multiply by the gcd, 4*2 = 8, our final answer.

Start at zero. Count up 2. Then, count up 6. Because if counting is a good way for a child to spend her time, then more counting is better.

Xenocles says

"I do get the point of your idiom, but it's a lazy man's excuse, begging your pardon."

You say that as if it were a bad thing. As far as I can tell, all progress depends on lazy people putting in effort on the front end to avoid putting forth more effort later. It's also a question of resource and requirements management. There has to be a "good enough" for any real system attribute, or you blow up the project. I could give you something "better," but would you pay for it (or wait for it, or be able to move the extra weight, etc)?

But I digress.

Aaron S. says

It's more than a little concerning that the "educators" who designed the math homework are trying to ingrain a correct method of addition – "start with a number and then count up." I think the correct answer is "whichever way gets to the correct answer" (i.e. "26" :P)

sorrykb says

Todd Knarr wrote:

It's entirely possible I simply wasn't paying attention in grade school. I have a fairly clear memory of the view out the classroom windows.

(Come to think of it, that's probably why I ended up thinking pi was a variable. Math becomes more challenging when you think pi is just a funny-looking n.)

mud man says

Sure, and sometimes you ARE making toasters and all you want is cheap, but there is to be said for solving the

wholeproblem and not just cutting a corner off it because that's all you need today. Don't just run off the alligators, go ahead and drain the swamp. "For every hard problem there is an answer that is elegant, efficient, and wrong."… but basically we are in violent agreement.

HandOfGod137 says

I come from a distant and strange land, but I distinctly remember the moment of revelation when I "got" sets and the number line. By a non-spooky totally causally connected event, that's also when maths became interesting

En Passant says

sorrykb wrote Sep 30, 2013 @12:55 pm:

You were obviously an advanced student. I read the entire comment thread to reach that conclusion.

If Pi is defined as the ratio of a circle's circumference C to its diameter D, it is indeed a variable. In Euclidean geometries that ratio C/D, and thus Pi, is always the same, thus a constant. But in non-Euclidean geometries that ratio is not necessarily a constant 3.14159… So, Pi is a variable, depending on the particulars of the geometry.

It's a pity that our all wise and knowing educrats have not tried to smuggle that into elementary school classrooms disguised in bad English syntax just as they are trying to inculcate foundations of mathematics by stealth and bad English.

If parents taught their kids to walk like the educrats want to teach them basic arithmetic operations, infants would know everything about Newtonian mechanics and still be unable to walk across the room.

CJK Fossman says

"Whud you larn in school today, boy?"

"Um, I, uh,"

"Don't stutter. Whud you larn?"

"I uh, learned some geometry."

"Do tell. Say some of that geometer to me."

Pausing, then finally and quietly, "Pi r squared."

Boxing the boy's ears, "Don't be a fool boy. Evvybody knows pie are round."

HandOfGod137 says

@En Passant

I'm not so sure it is bad English. If "count on" means "translate one unit to the right on the number line", it makes total sense. And while teaching the under-10s Riemannian geometry might be a bit of a stretch, showing kids the pi relationship doesn't hold on the surface of a sphere would be useful and informative (and doable on a class globe with string and a ruler).

Blimey chaps, why are you all so against stuff that could make maths interesting and fun?

David C says

@HandofGod: "on from to" is 3 prepositions in a row.

It doesn't seem like a well constructed prepositional phrase. Certainly, if you believe that you can't end a sentence with a preposition, then you can't say that this is OK.

HandOfGod137 says

@David C

If you accept "count on" as a phrasal verb, it's perfectly fine to end with a proposition, so I don't really have a problem with it.

David C says

But then what about "from"?

En Passant says

HandOfGod137 wrote Sep 30, 2013 @1:58 pm:

So why should teachers not say that instead of the syntactically clumsy and ambiguous?

Because knowing how to perform elementary arithmetic operations is not "maths", it is arithmetic. Just as knowing how to walk is not knowing how to perform gymnastics or knowing how to calculate and predict Newtonian dynamics.

First a child must learn to walk. Later, if they're interested, they will learn gymnastics or Newtonian physics.

Smuggling advanced theory by stealth into teaching elementary arithmetic operations could be characterized as

upaya, or skilful and expeditious means employed to convey advanced knowledge by teaching elementary subjects. But exercisingupayarequires both skill by the teacher, as well as interest and capacity to comprehend the advanced subject by the student. Attempted by the clumsy uninvited, it can be disastrous.Every student must learn elementary arithmetic operations to get along in the world, even to count change in the lunchroom. But every student does not need to know number theory or foundations of mathematics. Attempting to smuggle in the latter by clumsy means will discourage all students from learning what they must know, and is unlikely to inspire students who might eventually wish to know it.

CJK Fossman says

Amen to that.

Every student should know the fundamentals of statistics, since they are so often abused to mislead the unknowing.

But one can be reasonably fluent in statistics without knowing number theory or the foundations of math.

AlphaCentauri says

All I know is that my kids' math courses in junior and senior high school were two years ahead of what I took in school – and that I had been unable to comprehend even that until two years later than it was presented. So my 7th graders were understanding algebraic concepts that suddenly became clear to me in 11th grade. If new math is responsible for them attaining abstract reasoning skills earlier by teaching them to understand what the concept of "addition" actually means before rote-memorizing "math facts," I'm not going to argue with results.

Education Realist says

Blimey chaps, why are you all so against stuff that could make maths interesting and fun?Because the stuff that makes math interesting and fun is outside the IQ range of at least half the kids.

All I know is that my kids' math courses in junior and senior high school were two years ahead of what I took in schoolThat's not "new math". "New math" is what was taught from 1960 to around 1975 or so, that began with set theory, did a lot of work in bases, and so on. We haven't done that in a while. Elementary school is teaching reform math, mostly because elementary school likes it.

The reason your kids math courses are two years ahead is because it's more normal for kids to take algebra in 8th grade than it was 30 or more years ago, not because we've gotten substantially better at teaching higher order math.

I took an abstract algebra course, which was one of the hardest math courses I took, and it was a requirement for education students in math.Very few education students end up teaching high school math. Most high school teachers major in something else and then get a credential. So the predictive value of your experience with undergrad ed school students, half of whom become elementary school teachers, a quarter of whom never even teach, as a guide to high school math teacher ability is very close to 0.

Xenocles says

"… but basically we are in violent agreement."

My favorite kind!

Asher says

I'm still not sure how everyone could get "4 + [ ] = 7" but lost it at "4 + x = 7".A cognitive deficit in the ability to engage in abstract reasoning. No amount of "education" can overcome it. That people don't realize this is evidence for the power of propaganda.

Dragonmum says

@Grant Gould

Dragonmum says

SRY. Blockquote fail.

I Was Anonymous says

@Nicholas

Don't worry about forgetting base 8. Base 8 is just like base 10…. if you're missing two fingers!

I Was Anonymous says

@ketchup,

In "What Do You Care What Other People Think?", Richard Feynman told a story from when he served on the California board that approved texts. The board was going to approve a completely blank textbook until he pointed out it was blank.

RandomDBA says

Hmm,

"count on from" almost sounds like some sort of esoteric SQL window function syntax.

Never to early for set theory I suppose…

JeffM says

"Very few education students end up teaching high school math. Most high school teachers major in something else and then get a credential. So the predictive value of your experience with undergrad ed school students, half of whom become elementary school teachers, a quarter of whom never even teach, as a guide to high school math teacher ability is very close to 0."

In case you did not notice, the topic of this post was elementary school, first grade in fact. So how many grade school teachers are there in the public school system that were not ed students. A very small percentage, I suspect. That is the relevant statistic, not what percentage of ed students end up teaching or what percentage of ed students teach high school math in the public schools. So my experience that those ed students required to study abstract algebra did not have a clue about varieties of algebra, mathematical rigor, non-commutative systems, etc. is directly pertinent. Once such students become teachers, they may be able to parrot some rules and definitions, but they do not in fact manage to teach most kids basic arithmetic or make them comfortable with math and certainly cannot challenge the Gausses of the future.

Education Realist says

I understand that this entry was on elementary school. However, it's entirely irrelevant whether or not ed school students understand abstract algebra, since they won't have to teach it or anything close to it.

And available evidence–a lot of it–shows that increasing elementary teacher's math understanding doesn't do much for their students' test scores.

jh says

If it was the result of the Texas Board of Education's curriculum standards, the correct answer is "God did it."

We need better standards. Can't California step up and take at least part of that away from Texas (I know it's based on number of students – aka "textbook sales" – but hey, a guy can dream, can't he?)

m-/

To my view, though, the question (at least without the greater context of the math unit involved) appears to have been written by someone who was the product of a poor education system. It certainly could be written more clearly in any event.

Assuming it's a number line, counting would start on the number 2, and then continue through 6, and the intervals would be counted (2-3, 3-4, 4-5, 5-6 = 4 intervals) to demonstrate that 2+4=6.

jh says

Or rather, the 6 intervals counted (2-3, 3-4, …, 5-6) with a starting point at 2 to get 8.

Dock me a point for bad memory – there were a lot of comments. :)

whheydt says

re jh…

But at least you understand the concepts!

Doug says

So they're teaching number theory to elementary school kids? Awesome. I remember way back in grade school, one of the things that really hacked me off was a total lack of explanation as to why one was one and not two or twelve or eight million, five thousand and thirty-three. "It just is, now PLEASE do the worksheet," never actually satisfies the kid who _has_ to understand things to be able to do them. It wasn't until I got to college that I encountered number theory and my VERY FIRST THOUGHT about it was, "My god, I wish Ms. N. had been able to explain this to me back in the first grade."

Bill Watterson had a really good Calvin & Hobbes strip that pretty much summed up my feelings about math until college.

Sami says

I would say the correct answer is: "Neither. She should know that 2+6 is 8."

Encouraging the "count on your fingers" mindset is the kind of thing that actively impedes mathematical competence.

Taemyr says

@Sami Are you sure? I am fairly sure that my didactics courses indicated that mastery of backup solution strategies is a prerequisite for use of more sophisticated solution stratigies. Granted, I failed that course so I might be misremembering.

Dan T. says

Pie are round… unless they're Sicilian pizza pies.

anonymous says

For a first grader with a developing mind, the question "Explain why?" is the more important of the two. For example, you start a new job where you're hired to do X. Your predecessor did X in a certain manner. Do you: (1) continue to do X in the same manner; (2) improve the manner in which you do X; or (3) question why X is even necessary. The "explain why" part of the math question is designed to teach children to think and, hopefully, do (3) instead of (1).

En Passant says

Sami wrote Oct 1, 2013 @1:47 am:

I think that teaching young children how to count 0-1023 on their fingers (and thumbs) would increase their mathematical competence, not just arithmetical computation competence. And kids would like it because they'd get to show off wacky hand jive signs. It would also make any educrats' complaints about unapproved pedagogical methods look pretty silly.

CJK Fossman says

The number line doesn't explain the why of simple arithmetic any more than the addition table explains it.

Peano's axioms don't explain it, either. Axioms are just the set of assumptions that form the basis for further reasoning.

JeffM says

@ educationrealist

"I understand that this entry was on elementary school."

Then why babble about high school teachers in your previous post.

"However, it's entirely irrelevant whether or not ed school students understand abstract algebra, since they won't have to teach it or anything close to it."

Now you are getting close to my point. Elementary and middle school teachers are generally not competent to teach number theory or abstract algebra so they are forced to teach dumbed-down versions that are irrelevant to kids. I am tutoring a bright kid in middle school right now, and she has no clue why it is important to memorize a name for the fact that 6 * 8 = 8 * 6 when what is important in learning arithmetic and elementary algebra is the fact, of which she is fully cognizant. I can explain commutivity of addition and multiplication of numbers to a first grader with a simple diagram in a way that is unforgettable without burdening the child's mind with a distinction that will not be pertinent for ten or eleven years at best. It is simply a way for teachers who know little math to show off their vocabulary. I have told her that there is no reason to learn this vocabulary now except that it will improve her grade.

"And available evidence–a lot of it–shows that increasing elementary teacher's math understanding doesn't do much for their students' test scores."

I do not doubt it.

Education Realist says

Then why babble about high school teachers in your previous post.Because you said, "Furthermore, few public school teachers are competent at it."

Had you said "few elementary school teachers are competent at it" I wouldn't have posted.

Middle school math teachers have to meet the same criteria has high school teachers these days, unless they are grandfathered in and have proved they've taken the necessary coursework. That's a smaller and smaller number these days.

It is simply a way for teachers who know little math to show off their vocabulary.Aaaaand you're back at being foolish, even assuming you mean elementary school teachers, which you neglect, again, to specify.

Education Realist says

We need better standards. Can't California step up and take at least part of that away from Texas (I know it's based on number of students – aka "textbook sales" – but hey, a guy can dream, can't he?)1) California does control as big a share of the market as texas does, or close to it.

2) There's this thing called Common Core. Perhaps you've heard of it. If you have, your comment is odd. If you haven't, then I recommend you read up.

CJK Fossman says

Option 3 is likely to lead to loss of that new job. How many people are going to do that? In my vast and regrettable experience of corporate life, that's a vanishingly small number.

tm says

@Gabe: Aha, a mathematician appears! I'm glad someone knows that book, which I think is long out of print (or has it reappeared like a number theory zombie?)

Your comment about "thinking deeply of simple things" is at the core of what actual math is. Fermat's Last Theorem is deceptively simple, and I suspect many of you could easily explain it to a middle schooler. But the proof? My number theory professor said of FLT (before Wiles published his proof): "It is a great theorem. It has destroyed so many great mathematicians."

I made a snark earlier about how we all rely on calculators for arithmetic in our old age, but as Gabe points out, even more complicated calculations are well within easy reach (e.g. Wolfram Alpha). I imagine that there are a fair number of you who still don't understand the notion of "adding letters", but are still presumably productive members of society. So whither math education?

JeffM: There is always jargon in technical fields. If your bright student asks "why do I need to know the name?", the short answer is "you don't need to", which will satisfy your snark. If they are truly as bright as you say, then they will soon find out more advanced texts assume they know the jargon and proceed from there, and even more interesting is to know where it breaks down (e.g. is commutation always true and why?) and what has happened when people question the underlying assumptions of the field (e.g. classical physics to relativity and quantum mechanics).

Grigori Perelman, I suspect, would've still declined the Fields Medal regardless of how he was taught.

PonyMaster2k says

It's not obvious that 6+2 is the same as 2+6.

It's very non obvious that 5*4 = 4*5. I remember having a hard time with this in grade school. When I realized why, it was a very elucidating moment.

JeffM says

David Dyer-Bennet says

Kids who wonder if 6 + 2 = 2 + 6 should be encouraged! Because once they start wondering, they won't be nearly so shocked when they discover that for some other operations it's not true. And it's easier to talk about it if it has a name.

My father was involved in writing textbooks for new math back at the beginning (spent the summer of 1963 at SMSG at Stanford doing that). I have no idea what they've got as "new math" now, but that new math made sense to me (and was 3 years ahead of where I was in school at the time; then again I'm the son of a math professor and took a math major in college).

Nobody says

Count from six, because it's larger and therefore 6,7,8 takes less time than 2,3,4,5,6,7,8. But yes, knowing what they want is definitely a challenge.

Robert says

Math person here! My undergraduate degree is in Math (but my graduate degree is in Computer Science because Math got too hard.)

I have NO IDEA what this question is asking

DaveK says

I think a lot of people have been unnecessarily thrown by the wording. "count on from" = "count onward from"; it seems pretty clear to me, but I can see how the "on from" bigram could make your brain stumble slightly over the parsing.

Rich Rostrom says

PonyMaster2k • Oct 1, 2013 @11:23 am:

It's not obvious that 6+2 is the same as 2+6.Steve says

Speaking as a college-educated literate mathematician, my only conclusion as to the proper (if not desired or intended) answer to the question actually asked is: "Carrie should count 8 on from whatever number to which she wants to add '2+6', as the assumption of an unstated third constant is the only construction satisfying both the mathematics and the semantics of the problem as stated."

If this is the intended content of the question, however, I will eat my hat. (Note, the verbiage "count 8 on from 12" for example is, as far as I know, a correct and proper, if somewhat archaic and outdated, english construction.)

NickM says

A few years ago, I was asked by a friend to help her daughter, who was in 7th grade and was, according to the mother, completely lost in a math chapter that the mom said she didn't understand either.

About 5 minutes with the textbook showed me what the problem was – of the 10 examples given in the chapter on combinatorials (permutations and combinations), 4 were incorrectly assigned (2 of each group).

Unsurprisingly, the homework assignment was an utter disaster – the daughter was obviously guessing wildly with no understanding, but had as many wrong answers marked right as right ones marked wrong.

On my recommendation, the mother sent a letter to the teacher informing her of the serious problems with the textbook and suggesting that she speak with a particular local professor involved with Mathematically Correct about how to undo the damage. I don't know what the teacher did, but I was told that the class didn't have a test on that chapter and that the homework assignment wasn't counted in the final grade.

King's Rook says

@Gabe, @tm : It's no longer Fermat's Last Theorem — it's Wiles' Theorem now. He proved it.

Shaun says

I agree with some of the other posts. I'd probably say to start counting from zero, and then it doesn't matter the order that you're adding the other numbers, and you still end up with the same result. However, I also agree that this question seems absolutely ridiculous. What is so hard about asking "what is 2 + 6" if that is ultimately what you're trying to teach? Why bother teaching where to start counting from, as if that makes a difference in arriving at the answer.