## See below in Green for some corrections.

The math problem listed here has been making the rounds. It’s supposedly from the common core. If you haven’t seen it, it supposedly shows Jack using some number line to subtract 427 minus 316.

A lot of writers have been dumping on it.

I think they’re missing something — there are at least TWO errors in the work of this imaginary Jack.

The idea of trying to figure out where someone else got something wrong isn’t the worst idea in the world. However. what Jack was allegedly doing would need to be done in the head, because this method is so unwieldy if written out — as many people have pointed out. Also, if that was a carefully printed out number line, then I hope the problem is entirely imaginary, because unless we are teaching about logarithmic plots, then mathematicians take care to make sure that scales are linear (meaning that the distance from 100 to 200 equals the distance from 0 to 100, which are each exactly ten times the distance from 90 to 80 or from 57 to 67.)

As a mental exercise, number lines like this are not an entirely useless method.

Nobody seems to have pointed out that Jack made two math errors, not just one.

Since this problem asks 427-316, if you are doing this in your head, you could either count backwards from 4 by 3 units, or ask yourself how far it is from 3 to 4 — obviously 1. Writing the number line out is a lot of work, but saying silently to yourself, “427, 327, 127″ isn’t much work. So far so good.

But it’s not only in the mathematics that the imaginary Jack made an error:

I’m not sure, but it seems like the problem writer wanted Jack to confuse 16 and 60. This is not hard to do when HEARING the number, but more difficult if you are SEEING it written out. So this makes the problem a bit more, well, problematic, because nowhere in the problem is there any hint that Jack tends to hear poorly.

So this imaginary Jack mistakenly counts backwards in the tens place by tens by going 127, 107, 97, 87, 77, 67, 57 — which appears to be his final answer. Maybe. I can’t quite tell on this sheet.

So that means that “Jack” made another error in leaving out the decade 117.

Since it looks like this problem was written by some low-paid contract worker (think of “call centers” in Malaysia or India) with little scrutiny afterwards, we don’t know if the intention of the  problem writer was for the student to realize that Jack is both hard of hearing and mis-counted by skipping the 117? If so, you are really asking a lot of a kid looking at the problem — and notice, if I’m right, then a whole lot of adults missed that point as well.

Did they really intend for the problem to be that difficult?

Sounds like an error on the part of the error-writer, but I could be wrong.

ADDED LATER: If the writer were aware that there were two mistakes in the problem, shouldn’t they have written “Find his error(s)” rather than “find his error”?

#### My mistake was thinking that the numbers the parent wrote on the number line were ones that appeared in the original problem.

So, lots of errors all around.

I agree with my still-working math department colleagues that the Common Core standards in math for the middle school are not too bad, as written – they even include positive ideas about approaching math from many different angles (ahem), which I’ve espoused for a long time. It’s all the stuff that the CCSS are bound up with that is the problem: the constant top-down directives, the idea that every single teacher and every single student in the nation is supposed to be on the exact same page every day (which contradicts much of the verbiage of the standards), filling out umpteen useless data sheets and other paperwork every day, and the fact that a teacher’s job is tied to a random-number generator known as “Value Added”.
—————–
Another critique of this math homework and the parent’s reply can be found here.
Published in: on March 23, 2014 at 10:24 am  Comments (49)

1. When MCAS started in my state I was math dept chair and had to proctor the math portion (given to special needs kids as well). A problem described how “Tim” would estimate the cube root of a number. Some poor kid looked to me and said “What the heck is this?”

I read the problem, and for the only time in my career I cheated on an exam. I told the kid “Ignore Tim!”
The problem was such a mess I couldn’t have helped him make a good answer anyway. There was no hook to the problem to give a kid a clue as to what was expected.

Over the years the test got better, but there were still plenty of clunkers.

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2. As a carpenter interested in understanding traditional methods of work, I occasionally use dividers to step off divisions of a line (it’s surprisingly accurate). Or, I use a ruler to make direct transpositions, to avoid fractional arithmetic. Mostly, I employ analog methods wherever possible, and try to keep away from purely abstracted calculations, because confirmation bias tends to incorporate mistakes instead of revealing them.
Would that children were allowed to begin with geometry: straight edge and compass. The greater part of trigonometry can be taught with the eighth part of a circle.

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• I’ve gotta see trig done with an eighth of a circle!

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• on a unit circle, all the factors for sine and cosine occur between 0º and 45º. from 45º to 90º, they are inversions. the other quadrants use the same sine/cosine values, assigning + or – according to their relation to number line.
or, we have a right triangle, height<length, rise/run=tangent/1,
if instead the hypotenuse has the value of 1 ( or 10 or 100…) sine and cosine are square-root co-efficients.

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• So that’s what you mean. Well, yes, all of the values of all of the trigonometric functions can be figured out if you know what their values are from 0 to 45 degrees (or 0 to pi/4, or one-eighth of a circle). I thought you were implyiing something else.

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• Right, once you divide rise by run, you’ve cracked the code, the rest is inverse functions on the calculator, or tabulated (remember interpolation).
I had to go back and re-learn trigonometry from a geometric perspective to solve various building problems, rafter length, hips, valleys, etc.
Most builders nowadays just work directly from a book, or plug numbers into a construction calculator. Architects let auto-cad generate dimensions.

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• Carpenters and machinists both use congruent segments and angles a lot, rather than making a measurement in one place and trying to reproduce that measurement somewhere else — which introduces at least three sources of error…

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• Calvin M Woodward created a school for manual training in St. Louis in the 1870’s. That idea eventually gave way to vocational training, under pressure from business. Now, vocational has been mostly abandoned, or moved upscale to the colleges. People still don’t learn how to work with their hands. How do we get back there?

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3. Guy, did you see Chris Danielson’s piece on this yesterday? Pretty much gets it. http://christopherdanielson.wordpress.com/2014/03/22/the-latest-common-core-worksheet/

My comment: As Christopher says, we need both. And it’s just common sense. And pulling apart examples like this one to “prove” that the CCS-Math approach is, frankly, an incredible waste of time for so many reasons that it’s exhausting to contemplate having to go through them once again for the uninitiated, recalcitrant, et al. Having mental facility with arithmetic (not to be generalized into “mathematics,” please) is a very gratifying pursuit in its own right, one that has long been under-taught, under-explained, undervalued, often by teachers who are not particularly facile with it.

I have zero doubt that there are teachers right now and teacher in the making who will botch more flexible, sense-making approaches, too. I’ve said for several decades that lousy teachers can make a hash of anything, while good ones can do much of value with very little in the way of creative problems, materials. My question remains: Why settle for mostly mediocre teachers teaching mathematics in mundane, deadening ways? Why champion that over real mathematics and real teaching and learning?

And then it shows up on Mercedes Schneider’s blogs, too: http://atthechalkface.com/2014/03/23/photo-one-parents-response-to-common-core-math/

I have had a couple of failed attempts at debating her about ‘progressive math teaching’ at the elementary level, to no avail. For a bright person who generally gets important issues surrounding Common Core, etc., she has the most hermetically-sealed mind on anything touching up “Everyday Math = Common Core = Hell in a Handbasket” I’ve seen outside of the land of Mathematically Correct. That we started blogging on At The Chalk Face last fall around the same time, just as I found out about her rigidity on this one issue was a bit disconcerting, but no attempt on my part to engage her in anything might be considered rational discourse has succeeded. She’s adamant. I find it puzzling and troubling, because she speaks for and to a lot of bright folks.

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• Unfortunately, we accept mediocre teachers because that’s all we can get. The tiny salary that teachers get means that many people who are good at math and at explaining things choose to pursue other careers (college professor, accountant, etc.) where they can make a decent living and provide for their own children without having to pick up random outside jobs.

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• Actually these days most folks who go in for college teaching are adjuncts and earn a pittance with virtually no benefits and job security. Kinda like K-12 teachers are becoming.

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4. I also was chagrined to see this problem show up on Mercedes’ blog. It has been posted several times on the Badass Teachers Association (BATS) page on FB as well. Many of the comments about CC math refer to it as dumbing down the math while others say it is making easy math complicated Neither is the case.

I find it very troubling how many people who oppose the Common Core are conflating the very real issues many of us have with it, with math learning. We do not teach mathematics well in this country (nor do they in most countries) so keeping the status quo makes no sense. I recommend Hung-Hsi Wu’s website for many articles, mostly by him, on why the CC is MUCH better than what we currently have, as he calls it:TSM. Textbook School Mathematics, or the defacto national curriculum is (badly) written by textbook publishers. Here’s the link: http://math.berkeley.edu/~wu/

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• I agree with you for the most part, Sally, but recommend taking Wu’s views on elementary mathematics education with grains of salt. I have serious issues with his rigid insistence, for example, that THE way to introduce rational numbers in elementary grades is as points on the number line. Sorry, but that contradicts my belief that “THE” way, “best way,” etc., is chimerical at best and dangerous at worst. John van de Walle knew a boatload more about teaching math to young kids than Wu and was always cautious about introducing the points-on-the-real-line model overly-early. To suggest it’s the clear winner or that such a winner exists simply reinforces my mistrust of what happens when mathematicians overreach into math education issues on an arbitrary basis.

That said, Wu is correct that we’re not in principle looking at an all-out disaster with CCSS-M either. But I don’t think his analysis is sufficiently nuanced or subtle.

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• I appreciate your perspective as long-time math teacher. I only recently came across Wu’s work so I’ve only read a few of his articles. I agree there is not one best way to teach rational number concepts. Unfortunately, in my experience as a middle school math teacher and principal I have seen far more ineffective methods that teachers themselves often don’t understand. I don’t blame them; they are teaching as they learned and as they were taught to teach.

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• Yes, there’s little doubt that teachers with limited understanding and/or rigid thinking about something as complex (mathematically, semantically, practically) as rational numbers are going to do damage and disservice to kids. I still work with adult students for whom fractions are a complete mystery/nightmare, who can’t connect what they understand about decimal fractions to percents or numerical fractions, who think mixed numbers and improper fractions are inventions from hell, etc.

Wu is, I’m afraid, reacting so heavily in opposition to bad “school mathematics” that he runs the risk of retreating into a corner. We’ve seen this before in the US, in the worst manifestations of the so-called New Math movement (most of which was vastly better than what actually made it into schools on a wide basis) and with NCTM-style reform efforts in the ’90s badly taught by teachers who just didn’t have the chops to handle any sort of conceptual approach to math.

We will always be able to find some elementary math “teachers” who can go through the mechanical steps of K-5 math with kids – arithmetic, rational numbers, signed numbers. But for the majority of kids, that’s simply not going to cut it at some level. Either they’ll be utterly lost, or they’ll be part of the next group who slips through school thinking that being able to do mechanical calculations by following recipes is tantamount to being ‘good at math.’ Then, they get a math course that takes thought and they’re in up to their eyebrows. But try to promote more thoughtful problem-solving, mathematically-challenging work in K-5 and parents and teachers want to string you up. It’s like our repeatedly-failed efforts to “go metric.” 😦

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5. Regarding Michael Goldberg’s comment to the effect that I am “closed minded” to the new math and that I refuse to “engage” in discussion about the issue:

I have tried to tell him that I consider this “new math” to be “the long way home,” and that if teachers want to choose it, that’s fine, but that I don’t like the idea of teachers’ being required to use it– the rigid, no-options signature route that is CCSS.

If Goldberg likes his math this way, fine. However, teachers (and parents) across the US are being forced to ditch traditional calculation in favor of this alternative way. It’s the “forcing” that I find to be the overriding problem.

Goldberg wants to talk me into his position. It isn’t going to happen.

Some believe that requiring young children to “explain
and illustrate” math concepts is fine. I think it is developmentally inappropriate.

Then there is the coup of having companies sell “CCSS-aligned”
curriculum that requires students to complete assignments like
the one in the example in Guy’s post.

If there were no CCSS, we would not have these same frustrating
math examples popping up across the US right now.

Yes. Michael, students, parents, and teachers are frustrated with being forced to complete math the way one is required to complete it in the example above.

“Forced” is the word: The corporate reformers’ goal is to align (and mandate) all: CCSS, curriculum, and tests.

And NGA/CCSSO are now debating how they might “approve”
the CCSS curriculum since they are the CCSS copyright holders. Stay tuned for more on that front.

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• Mercedes: would it be too much to expect you to give me the respect of getting my last name correct? It’s “Goldenberg.”

All that said, I find your attitude towards me unconscionable and deeply offensive. I’ll leave it at that, because your language and tone communicate something deeply entrenched in you on this issue that clearly was there before we ever crossed your path.

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• First off, the error in writing your name was an innocent error.
Second, if you choose to be offended that I do not take your side on this issue, so be it. I’ll write yet again: Your preference for teaching math is your choice. To write that I “have a single right way to teach math” misses the greater point that teachers should have choice in the matter and should not be forced to teach math in an nontraditional manner. And yet, via Common Core, forcing curriculum onto teachers is the order of the day, and the effect of such forcing shows itself in student and parent frustration. And yes, my perspective on the need for teachers to be invested in their curriculum in order for said curriculum to succeed was indeed “entrenched” in me before you “crossed my path.”

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• I don’t believe I saw an apology there.

I would love to know at what point in time history was frozen such that “traditional” math teaching became the correct way to do it, regardless of whether that method worked for students, helped them understand what they were doing, made them love or loathe mathematics, etc.? Could you explain? And what background gave you the hotline to the truth on these questions such that it overturns all the research of the vast majority of K-12 mathematics educators in the entire world over the past century? To me, that smacks of incredible arrogance.

I have no interest in forcing teachers to teach in a single way. But I will not allow frightened, ignorant, incompetent teachers to cripple students because they (the teachers) can barely understand arithmetic (let alone mathematics), and feel compelled to teach it as a mysterious, deadly-dull subject that is about rules, definitions, procedures, and repetitions done mindlessly, AND NOTHING MORE.

So if you are pitting yourself as a defender of a failed system, that’s rather ironic, given your otherwise staunch attacks on corporate idiocy. The central problem with your poor analysis of math education is that it’s both ahistorical and gets just about everything backwards. You’re defending the status quo and think you’re being revolutionary because you’ve missed about 50 years of history.

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6. Hello,
you wrote,
“So this imaginary Jack mistakenly counts backwards in the tens place by tens by going 127, 107, 97, 87, 77, 67, 57 — which appears to be his final answer. Maybe. I can’t quite tell on this sheet.”
In fact, that string of numbers was written by the “Frustrated Parent”, not Jack.
All of the hand writing is from the “Frustrated Parent”.

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7. Discussion above tl;dr. I will cut to the chase.

You expend some energy on the skipped 117. Let’s be extremely clear that this error belongs to the parent, not to the fictional Jack. There is precisely one mathematical error that Jack made. This was leaving the lonely 10 in 316 out of his backwards count.

This accounts for the fact that his answer (121) is 10 units too big.

If you haven’t already, please see my annotations on the original piece from Frustrated Parent

There may be lots of other things to criticize here, but that skipped 117 is not among them. That is the parent. Not the teacher who created the task. Not Common Core. Not Jack.

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• It was very difficult for me to figure out what was written by the parent and what was written by the test maker.

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• OK, so then it was the engineer parent who made the error in counting backwards by tens.

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• But 200 of your 500 words here are expended critiquing an error that was introduced by the supposedly highly educated parent. You ought to keep the critique honest, I think. If you dislike the task, that’s fine. It is imperfect at best.

But nearly half of your argument against it is due to the parent miscounting. Pretending that this is too hard to figure out seems unfair or even disingenuous.

Again, I have no interest in defending the task. But I do have an interest in people being fair when holding instances of classroom practice up for public scrutiny. I don’t think the criticism of this task—either here or in the larger public space—has been fair at all.

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• Christopher, you know, I think, that I agree with you. What I find particularly intriguing is that a lot of folks have a hard time parsing the nature of the problem: it’s error analysis. Now, we can debate at what point in a student’s development it’s productive to try to understand, articulate, correct, praise, etc., another student’s work or one’s own. I have advocated moving towards this sort of thing in stages: using a neutral (unknown to you, possibly fictional) person’s work, then a classmate’s (exchange of papers?), but most importantly one’s own. Self-criticism as part of formative assessment (dare I say “meta-cognition”?) seems like the truly big goal here. How else do we become effective thinkers – mathematical thinkers in particular – if we can’t step back and constructively critique ideas? Of course, every such activity occurs in a context and at a level of general age-appropriateness.

Further, I’m pretty damned sure that, despite some people’s beliefs, that kind of sharp, constructive, critical thinking is NOT what the masters behind the Common Core really are promoting. Far from it. I’m sure most simply want the vast majority of students to “work hard and be nice’: critical discourse of any kind doesn’t enter that scheme in the least.

I personally love error analysis. Without being able to actually interact with the person whose work is under scrutiny, however, it is a bit of a mug’s game: there are loads of questions one should want to ask every step of the way. I know I always do, as Michael Pershan can probably attest to from things I write on his Math Mistakes blog. So often, in the abstract, when it comes to error analysis, “We know nothing, John Snow.” But done with peers, what a potential goldmine of learning.

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8. That’s “before we ever crossed paths.”

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9. Thanks for clarifying things, Christopher. I hope I’ve fixed it well enough.

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10. Another blogger’s take on this:
http://relearningtoteach.blogspot.com/2014/03/ccbs.html?m=1

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11. here is another take on this “problem”… i suspect that the GOAL of the “problem” is NOT to learn how to subtract in some “new” way… the GOAL of the “problem” is just that, it is a puzzle/problem to solve as to how/why jack arrived at the wrong answer! this OBVIOUSLY is no way to TEACH the operation of subtraction… memory and practice of subt. tables is most efficient at that… rather , the source of jack’s problem… he didnt jump in the “ones” enough times: he only jumped 5 times beyond the 10’s jump, so needed to jump once more in the one’s jump… that gets you to 111!! so, the goal here is the puzzle/problem to solve, NOT the learning of subtraction; i would agree tho, IF the goal is to learn subtraction, then this OBVIOUSLY is not the way to do that!

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12. Everybod so darned smart on here.

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13. The goal was indeed to write to Jack,showing him what his mistake is, and that mistake is… omitting a marker for 10’s but counting off the right number of 1’s, and consequently writing 121 instead of 111 at the end. Not hard to figure out. Of course, solving for the right answer is much faster than debugging a problem. 🙂

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14. The problem as written says “Jack used” when the explaination assumes JACK CREATED if the problem were correct the probablity of a correct answer would be much higher, as this number line cannot be “used” to achieve a correct answer through normal logic. The assumption the teacher/instructor uses which is “used” means “create” is poor at best.

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15. Is this real? I saw this floating around the internet and thought it was the dad making fun of the people against Common Core. This problem is very easy. What in the world is all the fuss about this problem? If this is real and this is the state of our understanding of math or number sense, we have been teaching math wrong for years. This problem was probably taught to the student prior to the test, but he either did not pay attention or simply did not care. Just because the father is an engineer means very little when you are a parent trying to rationalize your child getting bad grades. It must be the system, school, No Child Left Behind, Common Core. It would never be my child did not paying attention in class, doing their homework, asking for help, why should I help my child… my parents did not help me what ever the excuse. Number lines are easy and common sense. Each dot is an interval, you count down 9 times from 427 to get 111 (10 dots total, starting at 427). The first couple are done for the student: 427, 327, 227, 127. Notice the distance between dots gets smaller following 127. The next interval or dot is at 117. Notice the next interval is even smaller… Or think from 117 how do I get to 111, by subtracting at 5 intervals or 5 times, 5 dots 117, 115, 114, 113, 112, 111.

Then all he had to do was write a letter explaining what Jack did wrong, not counting by 10, 2, 1 , 1, 1, 1. This shows that he comprehends the problem and can verbally explain it. I was taught how to do this in public school 20 years ago. I think this was the dad trying to get under the skin of those against Common Core.

If a math teacher cannot understand this problem, they should not be teaching math.

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• what happened to 116?….five dots from 117 is 112

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16. As in life, if a problem is not presented properly then the conclusion can be misinterpreted. It would have helped both student and parent if the increments were presented.

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17. I don’t know whose writing that is between the 121 and the 127, but it’s not the parent’s. The 7s are different.

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18. I don’t think children have the math maturity to learn math this way.

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19. Problem is they are trying to repackage the instruction of math to our children. Being a father who has sat at the kitchen table to help my son do his math homework I discovered that his school wasn’t teaching him the basics. This has been going on for quite sometime. They have dumbed it down and were not reinforcing good math practices such as proofing, multiplication tables, etc. This parent’s comments on the subtraction line problem are valid and in my opinion highlights the substandard math instruction our kids are getting. I work in the semiconductor business and I see Indians, Korean, Chinese engineers gaining in numbers because they can do the work not Americans. In fact many schools just use a calculator instead of teaching math. The comment I got was “well at least they are using it”. Remember, someone needs to know the math to program the calculator!! Disgraceful!!

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• We actually continue to churn out way more engineers and scientists than there are jobs for them, by a factor if two or three. The visa program to bring in foreign engineers is simply a way for American firms to reduce wages and benefits even further.

When I taught in DCPS I had kids at all points in the math-ability-and-interest spectrum. Some were truly outstanding and competed successfully in local and national MathCounts competitions. Every so often I run into those kids. Most of the time I find that their tremendous math talents are woefully utilized….

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• Bill said in part, “Being a father who has sat at the kitchen table to help my son do his math homework I discovered that his school wasn’t teaching him the basics. This has been going on for quite sometime. They have dumbed it down and were not reinforcing good math practices such as proofing, multiplication tables, etc.”

1) Why are you helping your son with his math homework?

2) Would “the basics” be, say, the arithmetic facts associated with addition and multiplication (with subtraction and division being related to those as inverse operations)? And would what “they” didn’t teach your son be a set of tables that you could at any point have handed to him or had him construct or put on flash cards or gotten one of those cute little boards “Push this button down to see what 3 + 4 equals” deals (non-electronic), or any of a host of other readily available low tech resources that have been around for about a century? Are you suggesting that this knowledge is only available to elementary school teachers and can only be studied, practiced, or otherwise learned in a classroom?

But let’s get to your third sentence in which you claim that “they” have “dumbed it down.” Please explain. How can you “dumb down” tables of facts? If anything, math teacher educators for the last 25-30 years have been trying to get teachers to “smarten up” the teaching of basic arithmetic by focusing not only on facts associated with operations, but on why the operations work as they do, and what sorts of patterns can be seen in the tables. That requires thinking. How is thinking “dumbing down,” pray tell?

I’m not sure what you mean by “proofing.” I presume it’s not “proofing” as I’ve used it when I worked as a legal proofreader in the ’80s. So it must be “proving.” At which point, I must ask what grade your son is in and what sort of proofs you were doing at that age. Other than 10th grade geometry, I never had to prove anything in K-12 math, and graduated h.s. in 1968 utterly unaware that any math subject outside of Euclidean geometry involved proofs (which I imagined were all of the “classic” two-column variety.

That said, NCTM and NCSM have long been calling for more proving in K-12 math than has been there for the last half-century or so. I personally have argued in favor of age-appropriate “proving” in just about every grade, which for primary-grade kids means making informal but reasonable arguments about why the sum of, say, any two odd numbers is always an even number. I’ve seen a fabulous conversation about odd and even numbers in a third grade classroom from 1989-90 in East Lansing, MI, led by Deborah Ball, a leading mathematics teacher educator (and current Dean of the University of Michigan School of Education). But Ball, who has served on GWB’s Presidential Math Panel and other influential committees and groups, would general be thought of as someone who supports the general drift of the Common Core Standards for Math Content and Math Practice. I don’t know if she’s made any public pronouncements on those, but my sense from what I know of her work is that she would view enough of what’s in them favorably that she would qualify as a proponent of what you appear to despise.

But again, what proving are you talking about, at what grade band? And when was such work the norm for those grades?

Finally, the “Asians are kicking our asses in math” argument doesn’t hold up to close scrutiny. It is well-known that our top kids (mostly, of course, from affluent or at least middle-class schools and districts), fare quite well in math. Could we do better by then? Of course. But that’s not new. When I graduated a good high school in 1968 with somewhere between 700 and 800 classmates, 80% of whom were college-bound, what percentage of them had taken calculus? my best guess is that at most about 50 students had (enough for two sections) and it may have been less than half that number. That’s somewhere between 3 and 7 percent. Not exactly earth-shattering, and that was in a good district nearly half a century ago. We still had graduates who went to MIT, Yale, Chicago, etc. However did the country survive the next 50 years with those sorts of numbers?

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20. Again, an over-complicated response to a simple math question. The directive of the the simple math method proved the point, and your response to the “Common Core” math problem proves over thinking a simple problem.

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21. Did anyone suggest that perhaps this particular model was used in a class that the kid attended but the parent didn’t?

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22. Paul, I wonder why you have such harsh words for teachers,”I will not allow frightened, ignorant, incompetent teachers to cripple students…” (3/24). Most US teachers learned math for rote memorization as opposed conceptual understanding so how can you put the blame on them? The research shows that we teach the way we learned. One methods course can not undo 16 years of rote memorization. We need to support teachers and help them to learn mathematics deeply, which will take much time and effort. Please do not make teachers out to be ignorant fools because they are not. Most are dedicated individuals who work exceptionally hard and are attacked in ways they do not deserve. I was surprised to read those words in your post because as a former teacher I would hope you had more respect for your own colleagues.

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• It’s Michael, Mary. And you quote one sentence out of dozens I wrote here, do so out of context, and suggest my comment speaks to all or most teachers, rather than to specific ones to whom those adjectives do in fact apply. If you need to see it otherwise, I can’t help you. If you want to read what was actually written, interpret it reasonably, and talk about what can be done to help those teachers who really don’t get it and particuarly those who don’t WANT to get it, then there’s a basis for discussion here.

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23. Just say no to common core it is only for retarded libtards

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• Now there’s an insightful, well-argued comment if ever I’ve read one. Guaranteed to enlighten us all: not about mathematics or the Common Core or this problem, but about people who think simplistic epithets are the same as facts and reasoning.

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24. I am a little late to this post, but you just “followed” my blog, so I came to check out yours.

Studies show us that advance problem solvers always have a number of different ways to solve a problem and tend to automatically gravitate to the easiest one. I think Jack’s problem here is he is using a sledgehammer to crack walnuts.

Is this a reasonable way to solve this problem? It certainly is one way, but it reduces math to counting on the tips of one’s fingers, so I wouldn’t count it as being subtle or efficient or imaginative or …

There is a classic story (probably apocryphal) about a student taking a physics test which asked the question: “state a procedure to determine the height of a very tall building using a mercury barometer.” The student responded with “hold the barometer up until its length appears to be as tall as the building and use “similar triangles” to calculate the height of the building (diagrams included). The professor, expecting an answer that used the barometer as a barometer gave him a score of zero on the question. The student complained that the question did not require the barometer to be so used. This lead to a rather large dispute ending up with the combatents in the Dean’s office. The Dean was not amused, and put the student at a table with paper and pencil and told him to answer the question (again). What followed was a list of ten different ways to determine the height of the building, correct ways, none of which involved using the barometer as a barometer. #10 was “Go to the Building Supervisor and say “I will give you this barometer if you will tell me how tall the building is.”

The problem with the question is it provides an inefficient solution to a problem with a mistake built in and asks the student to troubleshoot. Specifically what ability is being tested here? Should the test taker “fix” a basically poor choice of solutions or should he suggest that Jack’s mistake was to not select a more straightforward approach?

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• I will reiterate a point others have made: there’s no “Jack.” The scenario is almost certainly hypothetical. And I’m guessing (with the little we have to go on) the following: the method “Jack” uses isn’t intended to be either instructional or exemplary. It’s just a method. If this is early in the instructional process with primary (K-2) grade kids, I have little doubt that there are many who are going to do arithmetic of any kind with some amount (possibly quite a lot) of counting – whether on fingers, other concrete objects, items in a picture, dots on some sort of number line, etc. And that is quite normal.

Adults who already know and have mastered the “standard algorithm” for subtraction are aghast at the very idea that any time might be spent on ANY other approach, regardless of developmental psych., math education research, or anything else. There’s one right way to do math: the way they learned. Period. End of conversation.

But what if this were something I’ve witnessed both live and in videos, including from Japanese elementary classrooms, in which kids are asked to add fractions and at least one student proposes that 1/2 + 1/3 = 2/5? Would people object to the teacher, who is introducting the subject for the first time, just putting that answer on the board along with other proposed answers, and without comment, and then asking students to speak about the suggested sums and/or methods for arriving at them? Or is it incumbent on all teachers to: a) immediately point out mistakes; and b) immediately give the standard method?

I think the answer to both of those questions is a resounding “NO!” Kids benefit in many ways from hearing from peers and learning to think critically about the answers others come up with, along with the answers they come up with themselves. That’s how one develops as a mathematical thinker. So this exercise creates a (probably) artificial scenario and merely asks the student to look at the work and critique it. It’s not asking for a judgment of “Jack” but rather to find one (or perhaps more) error(s) in the proposed solution to a question. Period.

The adult who wrote the note to “Jack” couldn’t even manage to do that, likely because it was SO important for that person to make snide comments about a method of which s/he didn’t approve. What a glowing accomplishment on his/her part! Something to be truly proud of. Of course, I have to wonder if this person has some reading difficulties. But more likely, it’s a matter of being blinded by bile. And that’s commonplace in conversations about math teaching and learning in this country. Someone citing allegedly high-end math credentials weighs in snidely and almost assuredly misses the point.

If some child who already had learned the standard algorithm weighed in with the argument that it’s “wrong” because it’s not the standard algorithm, that would be very interesting. But most kids just learning multidigit subtraction would probably try to puzzle out what “Jack” was thinking. Some (many, I hope) would see the trouble and explain the error. Some might also comment about “efficiency,” but I think that’s an adult concern. Kids become aware (if allowed to do so by adults) that there are advantages and disadvantages to various strategies for lots of things (including sports & games), and make choices based on what suits them best, if not pressured to instantly conform. Seems like far too many adults are only concerned with conforming. As if any modestly bright child would be incapable of switching horses if s/he found a good reason to do so later on down the road. So many of us seem to have so little respect for or confidence in kids, even though we were kids at some point and at least some of us did learn arithmetic.

Give kids a little breathing room and they’ll make good choices. Stifle their opportunity to think and they won’t think. And therein lies the problem with so many American kids when it comes to mathematics. They are taught to believe that it’s beyond their capacity to handle via their own thinking and ingenuity. By 3rd or 4th grade, they’re passive, waiting to be spoon-fed the next magic method to be memorized but (likely) never digested.

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25. I just found this and while I know it is an old thread I cannot help but wonder …what exactly does this “problem” aim to teach? It is not critical thinking , and it is not problem solving, it is definitely not math or arithmetic(unless we teach kids to count after they teach calculations) …. so what is it? Couldn’t this be put in a normal thinking form i.e. how many hundreds tens and ones are in 316 (if that is the intent)? As you said there is more than one way to solve a problem and none is wrong…unless common core aims to create fortune tellers, whomever comes up with these needs to stop converting math in riddles.

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