I haven't written much on the Common Core school standards mess (just this), but since Florida give us the opportunity to take sample tests, I couldn't resist checking out what was expected of third graders in mathematics.  I was a math major in college and usually enjoy taking standardized tests, so it should have been a piece of cookie, as we say in our family in honor of one of Heather's college math instructors, who was, Ziva-like, idiom-challenged in English.

I'm strongly in favor of holding students, teachers, and schools accountable for what is learned in school.  What's more, I have always had little sympathy for those who whine about the standardized testing that comes with a welcome concern for such accountability.  For endless years schools have failed to work with parents, to open their doors and records to parents, and to provide parents any reasonable assurance that the massive amount of their children's time spent at school is not being wasted.  They brought it all on themselves with their high-handed, "we know best, you just have to trust us" attitude.

And to those who complain that too much time is being wasted in school with teaching to and practicing for the tests, I always say the fault is not in the test, but in teaching to it and practicing for it.  Any generalized testing system worth its salt should be able to count on the fact that test results are a representative sample of a student's knowledge; teaching to the sample undermines its reliability.

All that said, this is a test that requires practice, and specific, test-related teaching.  First, doing math by mouse clicks instead of paper and pencil is a non-trivial exercise.  In this I was aided by my hours of Khan Academy math work.  But certainly students need time and practice to learn the specific testing interface.

Second, and most important, even with a bachelor's degree in math I found questions that made me stare blankly at the screen.  I don't just mean i didn't know the answer:  I hadn't a clue how to begin answering the question.

I'd like to tell you how I did on the test, but when I finished I was presented with this annoying screen (for all images, click to enlarge):


Now that's just lazy website design.  Being lazy myself, and fully confident of most of my answers, I didn't bother to confirm them.  However, I did pull up the answers to the two questions I couldn't figure out for love nor money.  Here are the problems, for your own amusement, edification, or frustration.  (Scroll down after the second problem to see the answers.)


If I wanted to graph the data, I would simply set Blue to 3, Green to 6, Brown to 12, and Hazel to 9.  But what does "Use a scale of 3" mean?  That on the Number of Students axis, the number 1 actually represents 3 students?  If so, I need to make the Eye Color bars 1, 2, 4, and 3, which is not only ridiculous but can't be done:  the bar heights snap to multiples of three.  Does it mean that I should scale each entry up by a factor of 3?  No, because then I'd need one entry to be 36, which is beyond the range of the graph.


Okay, this seems reasonable at first, with the little trick thrown in that one X on the graph means 2 students.  I'm glad to see that third graders are expected to notice such subleties.  So I plot "Two students like camping" with a single X.  But wait a minute.  "At least 5 students like playing basketball."  What am I supposed to do with that?  I can plot 2 1/2 X's, but there's no obvious way to indicate that the number might or might not be higher than 5.  And what about "Twice as many students like hiking more than playing basketball"?  Who cares if they like hiking better than basketball?  Maybe they like camping better than either one.  And twice as many as what?  Even if you rewrite it as "twice as many students like hiking best as like playing basketball best," you're still left with the problem of how to graph "at least."

Scroll down for the answers:












It seems that all they mean by "use a scale of 3" is that the vertical axis labels should be multiples of three, which is totally unnecessary, and thus confusing, since the axis is already labelled.


Except for the poor wording, this turns out to be a good puzzle, and I like to think that if I hadn't been rushing through the test—because I was in a hurry, and it was for third graders, after all!—I would have caught it.  Assuming that "twice as many students like hiking best as like playing basketball best" was meant, embedded within the graphing question is a simple algebra problem:  if B = the number of students who prefer basketball, then the number of students who prefer hiking is 2B, and thus 3B + 2 = 32, or B = 10 and 2B = 20.  But for third grade?

My limited experience more than once made me question the language abilities of whoever wrote the questions, but I have to say that a problem of this complexity on a third grade test is impressive.  It's not a hard problem, but it requires realizing that it is not a simple graphing question.  That's a subtlety I'm not accustomed to on standardized tests at this grade level, and might be indicative of an exam that actually measures students at the higher levels of skill and knowledge, instead of levelling them all out at "99%."  Unless, of course, you teach to the specific questions.

I came out of the exercise with no strong feelings one way or the other about the new Common Core approach to math.  All in all, it's very like the Khan Academy approach, so there's plenty of practice available for those who want it.  But I'm not sure what I think of the Khan approach.  It's not unreasonable to demonstrate multiplication of multidigit numbers via diagrams like the one below, but is it useful as a primary teaching method?  Does it really increase mathematical insight?  That's more than I know.  But I'm pretty sure that taking someone who already knows how to multiply and making him spend much time on this kind of exercise is less helpful than educators would like to think.  Especially if it's just to pass a test.


Posted by sursumcorda on Thursday, October 2, 2014 at 1:24 pm | Edit
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Although I noticed the weird wording for question 9, I decided to just ignore it based upon how the graph was set up and would have gotten the correct answer.

On question 10, it did not even occur to me that they were trying to have the results add to 32. I knew he asked 32 kids, but I guess I was thinking that maybe not all were included in the results. Not sure why I thought that. You are right though, that it seems like a complex question to ask a 3rd grader.

I think for me, the most awkward part is not writing the problem out with pencil and paper.

I think one of the problems with Common Core (from stuff I have heard) is that, like Everyday Math, it is being taught with no flexibility. They may expect you to do division problems a certain way, even if you already know how to do them another way.


Posted by dstb on Thursday, October 02, 2014 at 8:09 pm

"I guess I was thinking that maybe not all were included in the results." That's the problem: thinking. Keep thinking and you'll find lots of reasons why a test question is poorly worded. Half the battle is figuring out what the real question is.

Posted by SursumCorda on Friday, October 03, 2014 at 6:57 am

The strange thing is that they somehow think doing it their way prooves understanding, but a child could memorize the "Everyday Math way" just as easily as any other method. Proving understanding is actually rather difficult. You might as well just stick to the facts and assume if they get the questions right, they understand it enough and their understanding will deepen as needed.

Posted by Janet on Friday, October 03, 2014 at 2:29 pm

"assume if they get the questions right, they understand it enough and their understanding will deepen as needed."

This reminds me of how I react when I'm told (and if you read much in the education field, you'll hear it a lot) that it's not sufficient for a child to merely read a book—he must prove that he understands it. This is used to justify endlessly dull "reading comprehension" questions—and to sneer at the mother who announces that her eight-year-old has just finished reading Little Women. Does anyone fully understand a great book on first (or second, or third) reading? I say that if a child understands a book well enough for it to hold his interest, that's all that is necessary. Fuller comprehension will come with time (and no need for questions at the end of each chapter).

Posted by SursumCorda on Friday, October 03, 2014 at 3:39 pm

I scored like Sarah: ignoring the weirdness of the multiples of three, I got that one right; overlooking the number of students and focusing on the red herring of giving a minimum number of basketball lovers, I just figured that with 6 basketball lovers and 12 hikers the problem was sufficiently solved, though undetermined. (I can rationalize that nowhere does it say that all students answered one of those three options, but really, I just didn't pay attention to the "32" up there.)

Posted by Stephan on Friday, October 03, 2014 at 3:53 pm

Sarah just referenced this post in a conversation, so I looked it up.

You've probably heard me say this before, but I don't think you referenced it, so maybe you haven't heard me argue on Facebook about the silly anti-common-core arguments that are routinely made (I think there are rational arguments against it, but showing the "new, required" methods of addition are not one of them).

If you read the actual standards (Common Core Math Standards) you will see that these methods that the Florida testing company is using have nothing to do with Common Core.

Same for Khan Academy's method (I was confused in Khan Academy when they wanted me to fill out the area chart).

The Core standards hardly specify anything regarding the means in which to teach.

I just re-read the third grade standards regarding multiplication again, and I don't see anything objectionable in the standard at all. "Multiply two digit numbers within 100", "understand the relationship between multiplication and division", "represent and solve problems involving multiplication and division. (For example, describe a context in which a total number of objects can be expressed as 5×7)"

Similarly for Anthony Esolen's arguemnts - I haven't looked at the English standards as much, but a quick review of them doesn't show anything that is all that interesting, and appears that my English education would fit within the standards.

"Cite strong and thorough textual evidence to support analysis of what the text says explicitly as well as inferences drawn from the text, including determining where the text leaves matters uncertain."

"Analyze how an author's choices concerning how to structure specific parts of a text (e.g., the choice of where to begin or end a story, the choice to provide a comedic or tragic resolution) contribute to its overall structure and meaning as well as its aesthetic impact."

"Analyze multiple interpretations of a story, drama, or poem (e.g., recorded or live production of a play or recorded novel or poetry), evaluating how each version interprets the source text. (Include at least one play by Shakespeare and one play by an American dramatist.)"

"The Common Core requires certain critical content for all students, including classic myths and stories from around the world, America’s founding documents, foundational American literature, and Shakespeare. Appropriately, the remaining crucial decisions about what content should be taught are made at the state and local levels."

In some cases, people are objecting to how their local administrators or teachers are testing or teaching, but it really has nothing to do with Common Core.

Posted by Jon Daley on Wednesday, November 26, 2014 at 1:03 pm

Thanks for the analysis, Jon. (Sorry for the late moderation.) It seems the confusion between Common Core Standards and the various implementations is everywhere, not just among the complainers but in the media and even among the supporters. In the end, I suppose it doesn't matter if the standards are fine, as long as everyone is going to go all weird with the implementation, but we should at least know what we're really dealing with.

There is such a big difference between "know now to multiply two-digit numbers" and "this is the one and only way you multiply two-digit numbers." I've lived through any number of changes to the "one and only way to teach" math, reading, grammar, biology, foreign languages, you kname it. The only thing they have in common is that all work for some students and none works for all.

Posted by SursumCorda on Sunday, November 30, 2014 at 12:52 pm

Now that was funny. I had to moderate my own comment. I guess I've never commented from my phone's hotspot before.

Posted by SursumCorda on Sunday, November 30, 2014 at 12:55 pm
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