Mark Palko passes this one along from high school principal Carol Burris:
My music teacher, Doreen, brought me her second-grade daughter’s math homework. She was already fuming over Education Secretary Arne Duncan’s remark about why “white suburban moms” oppose the Common Core, and the homework added fuel to the fire. The problem that disturbed her the most was the following:
3. Sally did some counting. Look at her work. Explain why you think Sally counted this way.
177, 178, 179, 180, 190, 200, 210, 211, 212, 213, 214.
It was on a homework sheet from the New York State Common Core Mathematics Curriculum for Grade 2, which you can find here.
Doreen’s daughter had no idea how to answer this odd question.
I’m with Doreen’s daughter on this one. Actually, it’s worse than that. I clicked on the link, searched on *Sally* to check that the problem was really there as stated, then I looked at the two previous problems:
I can’t be sure of the answer to either of these! I mean, sure, the first one has 40 stars. So I’d say 40 ones = 4 tens, and 40 stars in all. But I’m not quite sure—is that all they’re asking? It seems a bit tricky to ask me to write “40” twice.
As for #2, I see 14 sticks, so that’s 14 tens, 1.4 hundreds, and 140 sticks in all. But they can’t really be asking me to answer “1.4” to question b, can they? It also just seems weird for them to be asking me to count to 14? I had to count a second time just to make sure I didn’t mess up somewhere. That can’t be the skill they want to be teaching.
But . . . after struggling with all the problems on this page, I figured out the answer they wanted for the “Sally” problem: They want me to say that she is counting from 177 to 214 using tens and ones! Not anything that anyone would ever want to do, but it seems to be part of the curriculum.
This relates to Burris’s other comment:
The teachers in her daughter’s school are also concerned. They are startled to find that the curriculum is often a script. Here is an excerpt to teach students to add using beads from the first-grade module.
T: How many tens do you see?
T: How many ones?
T: Say the number the Say Ten way.
S: Ten 6
Teaching is hard, and developing curricula is hard. So in some way we should be giving the people on these committees a break. But they have power, they waste a lot of people’s time, so when they do things wrong, I do think the right solution is to mock and mock, scream and scream. “Ten 6,” indeed.
If you cut out our poor urban schools, American students perform on par with the rest of the developed world. The outrage over our “general educational decline” and attack on all schools everywhere is a product of our ignorance and deliberate blinkering of the actual causes of social problems and low educational outcomes in poor black and brown urban neighborhoods.
There’s a note: each stick is 10 sticks.
I have no idea about the stars.
Eh. When I was in third grade I brought home math problems for several weeks that my parents had no idea how to help me with. They involved adding and subtracting different color dots that were placed in a two-by-two table. Totally non-sensical if you haven’t been introduced to the rules of the particular activity. But, it turns out, those strange exercises were very effectively teaching me both base 2 and negative numbers. Perhaps the “Say Ten” method works.
Why can’t they just give kids Kumon?
With questions like these I am sure I’d fail high school math.
Not sure that makes me, or the questions, stupid.
Me too. What’s sad is these kind of weird questions aren’t aberrations. They seem the norm. Someone on another blog posted a whole test full of such arithmetical inanities.
Andrew: Have a look at these questions. Some of them are even more ghastly than the two questions you posted.
Looks fine to me – some non-standard terminology (e.g. number sentence) but it does seem to test the kids’ understanding of the concepts as opposed to regurgitating addition and subtraction tables. Loved the multi-ethnic names too!
I tried googling for a cartoon I recall – no luck.
But it depicted a duck who just turned its head and asked what? in response to “duck!” which it mis-interpreted as “hey duck” only to see a cannon ball about to hit him.
Of those who have attended the course, how many will make the same mistake?
(My guess is few, by I don’t really know.)
I agree with KK. Not only do these look right to me, but this is far closer to mathematics than what I remember doing in second grade. As I recall, I spent most of the first five years of math classes learning to replicate a pocket calculator.
These all seem pretty straightforward, but that may be because of the way I always add (in my head) bunches of two-digit numbers. At each step, I take the subtotal, add the tens of the next number, and then add the ones of the next number. I thought this is what most people did, but I’m guessing maybe not since this way of add seems foreign to the rest of you. Or maybe you do add this way, but just never thought about it explicitly enough to see the pattern in these problems, or think that teaching that pattern is constraining the number of methods a person might use to add.
All of these exercises are simply trying to teach place value.
It’s hard for some of us who are deep in the weeds of stats and econometrics to remember not knowing what place value means, but that’s where 7 year olds are. Indeed, a lot of 7 year olds really struggle with it, and probably most of this blog’s readers cannot even fathom struggling with place value.
But that’s why teachers and curriculum guides come up with creative ways to teach what place value means. That may include saying “Ten-6,” which seems like a BRILLIANT way to get struggling kids to realize for the first time that “16” does not mean “1” and “6” together (for a total of 7), but rather refers to a 10 and a 6 together. I mean, what’s your pedagogical technique for helping struggling kids understand that concept?
As for the way Sally counted, of course it’s nothing anyone would want to do: Sally made a mistake. The point is to have kids thinking about common mistakes if you get the ones place and the tens place confused, and start counting by increasing the tens place in each step. I would bet almost anything that “Doreen’s” daughter had gone over this in class and should have known what to look for, but then just forgot. (That is more plausible to me than the suggestion that the second grade teacher was so oblivious or cruel that she sprang a homework question like this without ever discussing it.)
Anyway, I think it is quite ill-advised for anyone to ridicule teaching techniques without even understanding what they are for and why teachers or other experts thought them worthwhile.
I agree with your comments Stuart! Out of context to those who are far moved from this level of learning math it can seem absurd. The “Sally” question does say “Explain why you think Sally counted this way.” After some review it is clear that she started counting by one but when she reached 180 switched to counting by ten, then back to counting by one again at 211. Something a child struggling with these concepts might do.
I think there are good intentions to the changes in the curriculum being imposed on our children. The task put before teachers today is no easy hurdle. I do fear in trying to leave no child behind they do others a disservice in that they hold them back from their potential. I think the key is parents working together with teachers to assure their children learn what they need to the way that server them best.
First, I don’t see why you refer to Doreen as “Doreen” as if she is not a real person. I have no idea but I have no particular reason to doubt that this is her real name.
Second, I don’t think anyone’s saying that the second grade teacher was oblivious or cruel. The issue is that the curriculum appears to be both mysterious and rigid. Follow the links to see the discussion of scripting, for example.
P.S. It would seem most natural to me to teach in innovative ways but then, when evaluating, test to see if the kids got the write answer. In this case we’re not ridiculing the teaching methods, we’re ridiculing the tests. Also, given the importance of testing, once you have these test questions that are tied to very specific teaching methods, there seems to be a big risk of the incentives getting all screwed up. I’d think that tricks like “ten 6” should be the means, not the end. If they really work well as teaching methods (and they well might, at least for some kids), that’s great. But then this success should show up on a more conventional test, I’d think.
This test is absolutely bizarre. I had no idea someone would actually defend these questions!
Innovative teaching methods are great but almost by definition it means there can and will be more than one innovative way to solve an arithmetic problem. Hence standardizing on one method and even worse testing on the methodological detail is quite preposterous.
The only invariant in these sort of problems is the final answer. So that’s what you test on. And leave each kid and each teacher to adopt a method they are most comfortable with.
While I appreciate what you’re saying and agree for higher level students, this is SECOND GRADE, not high school. These students are struggling with the concept of representing a number using place value. I don’t really defend these tests but I don’t really have such a strong reaction to them either. Honestly I just feel basically unqualified to discuss how to teach arithmetic to 2nd graders.
First of all, none of the things anyone has posted are tests. They are worksheets. And they are totally standard. I did worksheets like this in my math classes 25 years ago — in, by the way, a gifted math class in one of the country’s best school systems, as part of a curriculum explicitly designed to get us all into a math magnet high school (it worked). Like others, I would actually say they are more useful for teaching math concepts than endless strings of basic arithmetic (which, of course, we also did, but that’s not nearly as newsworthy.)
I just have to disagree then. What it does remind me was Feynman’s critique of “New Math” back in the 60’s. I guess this is it all over again. I found this anecdote by Feynman hilarious:
Ridiculing the tests? *Nothing* mentioned in the post is a test. It’s all teaching methods and homework sheets.
The problems in the post may be “mysterious” only because we haven’t been in the classroom and heard how the teacher explains place value to kids. Lots of elementary teaching is like that.
My best guess about the hundreds/tens is that they want you to say its 1 hundred and 4 tens, in parallel to seeing that 14 is 1 ten and 4 ones.
It’s not fair to grab a worksheet from a course without knowing what norms they’ve developed for answering. So yeah, I could answer hey THERE ARE MANY ANSWERS like 1.4 hundreds, or 0E in hexadecimal, but that doesn’t make it an outrage.
I would consider it an outrage if Doreen answered 1.4 hundreds and the teacher said “that’s wrong” instead of “that’s clever”.
So the answer is Sally doesn’t know how to count. I can see going over errors as useful in learning what is correct but I am not sure how much value there is in testing for them.
I get they are trying to teach place value but I agree with Rahul we ought to separate the teaching method from the evaluation. Where you see a BRILLIANT solution I see a SILLY one but that is just my view.
As for your point: “Anyway, I think it is quite ill-advised for anyone to ridicule teaching techniques without even understanding what they are for and why teachers or other experts thought them worthwhile.” I’m afraid I disagree with your “leave it to the experts” view on the strongest terms.
For starters, I’ll soon have to pay taxes to finance, among other things, a public school system that is mediocre at best. I believe that gives me the right to question. Second, do you have any hard evidence (e.g. from a RCT or whatever) that this BRILLIANT method is better at teaching place value, or addition, or whatever it is trying to teach compared to whatever standard practice it is trying to replace? (Note that in the implicit comparative effectiveness trials, outcomes cannot be measured with questions, like the ones above, that favor one method over another.) Third, even is the method is best on average, surely no method is best for all. Hence all the more reason from separating the test from the method.
PS Just to give you an example, I have a graphical mind. When I look at the number 124 I see one big blob, 2 small blobs, and 4 tiny blobs, and I know each is ten times the other. I don’t like to SAY Ten Six for 16. I like to DRAW a small blob and 6 tiny blobs. Now suppose the test asked “Draw the number 465”, without any context. Those who like to say things might be a little confused.
My kids are doing 3rd grade “common core,” and I would say it’s going well. Many of these problems require the parent to know quite a bit of context. For example, on the Sally counting problem, I would suspect that they are preparing the children for doing a subtraction problem like 214-177, for which the answer 3 tens and 7 ones is a nice one. The emphasis in the program is on various ways of understanding. But there is the real risk that this turns into various algorithms (“tricks”, “mysteries”) for getting the right answer. For a parent or a blogger looking at a single problem, it often looks like a “trick”.
I suspect they are preparing them for more tests like this. The longer tests are used, the more common they become, the less effective them become at separating students. To get around this they have to come up with more obtuse questions to see how fast they can pick up the sources intent and internalize it to give an answer the source deems valid. They aren’t really testing math but the ability to discern the test makers intention, and they aren’t concerned with them acquiring a set of knowledge but with being able to separate those fastest at internalizing the makers mindset. At the limit, this becomes a rorschach with what the maker sees as the answer, an intelligence test to identify those most capable of reading the maker. Expect ever more obtuse tests to circumvent students learning what is expected and thwarting the ability to rank them. The problem is while this tests intelligence, it isn’t that useful of a skill.
In the “New York State Common Core Mathematics Curriculum for Grade 2” document, lesson 2 homework, which is the subject of this post, follows several pages of lesson 2. Presumably the second-graders were taught the contents of lesson 2 before being given the homework. If you read the content of lesson 2 you will probably aquire the mindset required to do the homework.
As a PhD in Statistics who spent a working life in research in Industry I have always mixed with highly numerate people. Now as a retired volunteer I work with poor people who get into debt. There are poor people out there who can’t add up, which makes it tricky to keep out of dept. It is very difficult to balance the financial needs of every day living when you do not have a surplus of cash. Perhaps the curriculum is designed to help all second-graders learn about numbers, to add and subtract and so forth. I doubt that the curriculum is designed to test inmtelligence or rank students, as Lord suggests. Yo should want all students to leave school with the ability to use basic arithmetic.
My reaction to Sally’s method was that she was counting up a hand full of change. Using dimes and pennies as markers might be a very real life practical way to demonstrate place value. Of course, I do remember reading Murray Gell-Mann’s ( a distant relative maybe?) book “The Quark and the Jaguar” where several atypical and ingenious solutions to problems were given by children. An infinite number of points can be drawn through any finite numbers of points. The full context of these problems might make us think that the program is good/bad/indifferent.
The American educational establishment is pretty much betting the country on David Coleman, a charismatic former McKinsey consultant, who is the driving force behind the Common Core (K-12) and now heads the College Board, where he’s revamping the SAT. Here’s a profile of Coleman from the Forward:
Only half off topic: my dad used to like to stump math majors with this series of questions (probably works best if said out loud):
What do you get?
Hint: it works (to stump people, that is)!
Was a PhD candidate in mathematics. The number sequence question makes no sense to me.
But it has nothing to do about sequences. It’s about teaching addition and subtraction to 2nd graders, and from this perspective it makes a lot of sense – count by ones to the nearest tens and then count by tens (and then by ones). Reminds me of the good old method for making change.
“How many ones in all?”
Answer: there are no ones in “all”
The Sally question seems odd, but the #1 and #2 were standard place-value type questions when I was learning that stuff in Australia in the early eighties.