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A few weeks ago, I wrote an article for TheAtlantic.com describing some of the problems with how math is currently being taught. Specifically, some math programs strive to teach students to think like "little mathematicians" before giving them the analytic tools they need to actually solve problems.This may sound wildly off topic, but the struggle a third grader has "explaining" why two plus two equals four strikes me as being of a piece with the struggle basic writers have trouble writing a conclusion, especially a conclusion in a 5-paragraph essay, a highly compressed form that leaves you no room to "ask a rhetorical question" or "suggest future lines of inquiry" or "close with a quotation that captures your view" and the like.
Some of us had hoped the situation would improve this school year, as 45 states and the District Columbia adopted the new Common Core Standards. But here are two discouraging emails I received recently. The first was from a parent:
They implemented Common Core this year in our school system in Tennessee. I have a third grader who loved math and got A's in math until this year, where he struggles to get a C. He struggles with "explaining" how he got his answer after using "mental math." In fact, I had no idea how to explain it! It's math 2+2=4. I can't explain it, it just is.The second email came from a teacher in another state:
I am teaching the traditional algorithm this year to my third graders, but was told next year with Common Core I will not be allowed to. They should use mental math, and other strategies, to add. Crazy! I am so outraged that I have decided my child is NOT going to public schools until Common Core falls flat.
With the 5-paragraph essay, when you get to paragraph 5 you've said everything you were going to say (if you're lucky), but the teacher wants you to say something more.
But what?
One thing I always liked about William J. Kerrigan's X-1-2-3 approach is the fact that he didn't bother with introductions and conclusions. The introduction was 1 sentence - Sentence X - and the conclusion was 1 sentence, too. Kerrigan called the final sentence the "rounding off" sentence, as I recall. Really, that's all anyone should do in a very short paper; otherwise your introduction & conclusion - 2 paragraphs out of 5 - take up 40% of the essay.
I've had to abandon Kerrigan's one-sentence policy, though, since I'm pretty sure other instructors don't look kindly upon one-sentence introductions and conclusions, not that I've asked.
So my students, like the 3rd grader trying to explain 2+2, solve the problem they've been set and then struggle to say something else about the something they've just said.
10 comments:
Am I missing something that these teachers are seeing? I've spent any time looking at the CCSS in k-6 math but it seems to me that the Common Core Standards in Math are MINIMUM standards. It's the LEAST a teacher needs to do, and at least they mandate fluency, which is more than some states could say. As the sequence below indicates, students are introduced to the vertical algorithms in grade 2 through place value and manipulatives. In grade 3, students should be able to add & subtract within 100, then grade 4 they should be able to transfer their knowledge to larger numbers. Since the algorithms are based on understanding place value, it seems like a decent sequence.
Will students master the algorithm before grade 4? Sure! The teacher that told the parent this:
I am teaching the traditional algorithm this year to my third graders, but was told next year with Common Core I will not be allowed to.
...is just wrong. Not allowed to? Did the teacher actually READ the standards? What might a strategy based on Place Value be? Teachers need more math content education, not meetings on implementing CCSS-M.
CCSS.Math.2.NBT.B.7 Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
CCSS.Math.3.NBT.A.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.
CCSS.Math.4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.
This means that a teacher can no longer pass a kid up to 5th or 6th grade that does NOT know these algorithms. That would be a change for the positive, wouldn't it? I do wish they had mandated fluency of math facts sooner than 5th grade, or provided some better foundation in Grade 3 (or 2)...
Ok, I'm not sure how to edit the comment, but I messed a zero. This sentence should read: "In grade 3, students should be able to add & subtract within 100"
I'm not sure why it is so hard for parents to explain how they get the answer to a basic problem. I'd agree that many students' problems are an explaining issue, not an understanding issue. Nowhere in the CCSS-M does it say that students can ONLY use mental math strategies.
2 + 2 - "Addition is joining or putting things together. I've been working with this since kindergarten, it's a fact now"
35 + 112 - "In my head I add by hundreds, tens, ones." It's easy because there is no regrouping. Isn't that more efficient than lining numbers up and relying on the vertical algorithm?
35 + 89 = " take one from the 35 and add it to 89 to make a simple number." 34 + 90 is easy to do in your head.
35 + 98 = "add one hundred, take away two" or "take two from the 35 and add to the 98 to make 100."
135 - 97 - "Take away one hundred, add back three because.
91 - 18 - "Subtract 20, add two" or compensate..."Add two to both sides to subtract an easier number 93 - 20.
Don't we use these strategies as adults all the time? ...If we know them - my husband uses the vertical algorithm for that last one because he was ONLY taught the algorithm. My senior added three to the 18 then found the difference between 91 and 21. He adds four digit numbers that way as well.
(Not all strategies are created equal, though. I still don't get the "Doubles + 1" that many curricula use. When will that ever come into use again? How is that based on Place Value?)
They should use mental math, and other strategies, to add. Crazy! I am so outraged
I expect my own children to manipulate numbers mentally. To have them sitting in an Algebra class vertically lining up 91 - 18 while solving the quadratic formula is crazy.
I'd be outraged if mental math wasn't expected of all students.
"This means that a teacher can no longer pass a kid up to 5th or 6th grade that does NOT know these algorithms. That would be a change for the positive, wouldn't it?"
There is too much wiggle room. "Fluency" is never really defined and it's used on only a few skills. Also, I don't think the standard says anything about not passing a student along who isn't proficient. THAT would really test their beliefs and force schools to take more responsibility for mastery - the mastery they claim they want for "balance".
We await the PARCC test in our state to define what the stadard really means. Then there is the idea that the state will define a minimum cutoff on a minimum standard. If our state follows its normal path, they will only look at the percent of kids who get over the low cutoff. Then, when our high school gets only a 48% proficient rating on the state (not PARCC yet) test, the school can then claim that it ranks 8th in the state, which it recently did. My wife and I were at a school open house where they put the 48% statistic on the screen. Nobody blinked. By high school, however, many parents know that these standards are not meaningful for their kids. The more knowledgable ones know this in the early grades.
My big beef is that the standards institutionalize low expectations. You CAN do more, but what school is going to do that when they are faced with a 50% proficient rating? Many look at the pseudo-algebra II high school requirememt and think it's tough, but the standard does not solve the problems of mastery in the lower grades. Students will not be much better prepared by the time they get to high school, but they will be required to get to a higher level. And by then, the onus will be completely on them.
My big beef is that the standards institutionalize low expectations.
This is true of most of the old state standards, right? Not just the CCSS. Yet for many states, the CCSS are an improvement over their old standards. Has anyone looked deeply at the new Texas standards?
Fluency is 3 pronged:
Accuracy - you have to know the fact.
Efficiency - you have to know the fact within 3-5 seconds.
Flexibility - You have to know the relationships: 7x8=56, 8x7=56, 56/8=7, 56/7=8
Geez, same typo TWICE. Just not my day..."In grade 3, students should be able to add & subtract within 1000" (not 100)
"This is true of most of the old state standards, right?"
Yes, but it seems that this time (with the national effort, not just state-by-state) it's more official. Some states claim that the standard goes backwards.
I like fluency standards that specify the breadth of problems and the performance, not just a vague fluency for a few things. It appears that Texas does this. However, are students held back if they don't achieve the performace standard?
A number of years ago in our state, they mandated subject certification for anyone teaching 7th grade or higher. After that, we got rid of CMP in the middle school and replaced it with the top-level Glencoe Math series for Pre-Algebra and Algebra. I am now waiting to see a scaling back to a lower rigor CCSS-aligned series.
One of the issues in our small school is that they won't support more than one curriculum. The need for providing a path for the best students to geometry as a freshman was one of the reasons for getting rid of CMP. The other students get the same textbooks, but they cover the material more slowly. I don't know how the school will resolve that issue unless the Glencoe textbooks come out with an edition that correlates units to the standard. One textbook could still cover both levels of math. If only the school would warn parents about the 7th grade math split with enough time to figure out how to fix the problem. When my son went through the process, I remember a number of surprised parents.
I am teaching the traditional algorithm this year to my third graders, but was told next year with Common Core I will not be allowed to.
...is just wrong. Not allowed to? Did the teacher actually READ the standards? What might a strategy based on Place Value be? Teachers need more math content education, not meetings on implementing CCSS-M.
She was told this. So someone is wrong, I agree. There is a lot of misinterpretation of the CC standards, per the comment that William McCallum (lead writer of the CC standards) left at The Atlantic article. CC standards also do not dictate that inquiry-based, collaboarative, student-centered learning is required, yet PD courses are saying that Common Core requires this via the Standards of Mathematical Practice.
As for mental math, fine, but why the big emphasis on it in second and third grades. Let students get fluency with written methods before they start to do mental math techniques. You say you use mental math all the time. You're an adult, not a 7 year old.
The problem with the Common Core as I see it is the problem with any vague standards dictated from on high: they're used by those in power to further justify, and further entrench, current practices-- in this case, Comstructivism. If that's not what the CC want, they need to say so loudly and clearly, with specifics. As far as our children are concerned, what matters is what actually happens as a result of the CC, not what was intended.
I was about to say exactly what Katharine has just said.
Writers -- and the authors of the Common Core standards are functioning as writers when they publish a final draft -- have to take responsibility not just for what they say but also for what people think they've said.
Especially in a political context (I've spent years writing political posts in my town) you have to be EXTREMELY careful to make sure that your text can't be misinterpreted or misconstrued. Your text needs to say what you mean it to say, no more, and no less.
Obviously, people can think what they want, and reading can always go awry.
BUT your job as a writer is to do your level best to make sure that the meaning you intend to impart is the meaning that actually does get imparted.
In the context of constructivism and the math wars, it should be clear to any purveyor of academic standards that constructivists will read constructivism into any text that fails to explicitly reject constructivism.
The Common Core standards allow a constructivist reading, and we now have a great deal of evidence that the Common Core standards are indeed being given a constructivist reading (to the point that one of the local candidates for school board last spring told voters that Common Core requires a "different kind of teaching").
Now that the authors of Common Core know that the standards are being misinterpreted, they need to revise the standards for clarity.
The fact that they aren't doing so is significant.
"If that's not what the CC want, they need to say so loudly and clearly, with specifics."
Exactly. The standard tries to be neutral in terms of pedagogy. In doing so, it ignores the biggest cause of math failure in K-6. It throws in a few requirements of "fluency" without definition and we are still waiting for the PARCC test to see what that means. The writers of the standard are crazy if they don't see what will happen,... or not happen. As I've said before, you can't change what's in the hearts and minds of K-6 educators, especially if the standard is neutral. Maybe the CCSS writers believe the stories that schools just need a little more time to get differentiated instruction to work. Maybe that means getting more parents to do the work at home.
Standards like this always deal with the lower end cutoff. They would like to think that it ensures the chance for a STEM career, but it doesn't. They can't even guarantee that you won't need to take remedial math in college. They could define a STEM level for the standard, but they don't. Do they honestly think that a proficient on this standard will ensure a STEM career? It can't even ensure it in K-6.
Most parents figure out that these state standards are meaningless for their kids. My son took our state's junior year test and I couldn't care less about his scores. I care about his PSAT scores that he will get before the holiday break. I care about his grades in his honors and AP classes. I care a lot about his preparation for the SAT. Who needs the CCSS? It's a statistical tool, not one designed to help individual students.
Michigan requires that all high school kids take the ACT. How are they going to calibrate that with CCSS? There is a lot of data for colleges and the ACT. Why didn't the people who came up with the standards start with the ACT or SAT and work backwards? Working backwards, we have the PSAT and the PLAN, and they could define a consistent progression of tests that go back to the lower grades and use the same grading scale. We would then have data not just for a low cutoff, but data on what scores lead to the ACT score for particular colleges and/or STEM programs. If you have a separate CCSS test (philosophy-wise too), how will parents be able to tell whether their kids are on a path to get any particular ACT or SAT score?
Life will go on the same for most parents. We will still have to teach at home. We will have to set higher standards. We will ignore the PARCC test. For those kids who cannot get help at home, the onus for mastery will still be on their shoulders. The finish line might be moved slightly further away, but they will get little extra help at school.
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