I currently am on a second career after retirement—I teach math in middle school. During my last few years of work, I started taking courses in ed school at night. The first course I took was taught by a professor who had what seemed to me to be a unique gift. He managed to agree with whatever anyone said about teaching. I learned very quickly that this was pretty much the norm, and that ed school was the place where there are no wrong answers—just the “greater truth,” which will eventually prevail. It is the place where future teachers see the light and embrace the principles of student-centered, inquiry-based, discovery-based teaching, and answering students’ questions is “handing it to the student” (aka the “struggle is good” philosophy).
I am seeing something similar with respect to the Common Core math standards. Peter Greene, on his blog Curmudgucation, puts it this way: “If the Common Core were to collapse and everyone in the country came to see it as a disaster and a Huge Mistake, exactly whose head would roll? Who would be held responsible?” And he answers it as follows: “To use the language of the ed revolution, nobody is accountable for Common Core.”
And another perspective is offered by Katharine Beals at her blog Out in Left Field. She points out a constant refrain heard about Common Core:
‘The Common Core is pedagogically neutral’; ‘The Standards are guidelines, not a curriculum’; ‘Teachers can use whatever tools they want to help students meet the standards’. … And yet, whenever we look up close at an assignment or activity or classroom showcased as having been inspired by the Common Core Standards, we see the same old Constructivist imprints: student-centered; discovery-driven; group-based; real-life-relevant; and ‘critical-thinking’-fostering.
This raises an interesting question. If no one is in charge, who’s to say that the interpretation and implementation of Common Core’s math standards by a teacher, or school or school district or state is wrong? Currently, the “instructional shifts” that the Common Core literature says are called for manifest themselves in the manner that Beals describes above. But it can just as easily be read another way. Specifically, math education—particularly in K-8—has been dominated by “reform” ideas for the past 20-plus years. That is, student-centered approaches, delaying of teaching the standard algorithms and procedures until alternative “strategies” are mastered (in the name of “deeper understanding”), and on and on. Thus, the “instructional shifts” required by Common Core could just as easily be interpreted to require a standard, conventional (dare I say “traditional”) approach, teachers teaching and answering students’ questions rather than facilitating, reliance on whole-class and direct instruction in class rather than on videos at night, textbooks that contain worked examples, students working individually, with occasional—not continual—group work. After all, the Common Core literature does say that the standards do not dictate pedagogy.
Go Ahead—Do Better than Common Core
As far as the alternative strategies for simple computational procedures that have been garnering much attention on the Internet and in the news, their use is also a matter of interpretation. In fact, I have written about this in a four-part series on a common-sense approach to Common Core math standards, in which I interpret selected standards and offer recommendations for their implementation. (See parts one, two, three, and four.) One particular recommendation I make is to teach the standard procedures or algorithms for common computation methods earlier than the grades in which they appear. The Common Core grade-level standards are minimum levels of expectation and goals, and are to be met no later than that particular grade level. Thus, the standards do not prohibit teaching a particular standard earlier than the grade level in which it appears.
Specifically, the standard algorithm—which is the classic and most efficient procedure—for multi-digit addition and subtraction is developmentally appropriate for second graders. The algorithm appears in the fourth grade standards (4.NBT.B.4) and states: “Fluently add and subtract multi-digit whole numbers using the standard algorithm.” I recommended that it should be the prevailing method for adding and subtracting numbers, starting in second grade. In that way, the standard algorithm becomes the “main dish” and the alternative methods are side dishes, that offer students a different way of doing a computation, should they desire.
For example, the news and Internet have been full of articles on the various alternate ways schools are teaching simple addition and subtraction “the Common Core way.” NBC News featured a story that showed the “counting up” method for subtraction. And this online news story showed how to add by “making tens.” Both techniques derive from examples suggested in the following first-grade standards:
Apply properties of operations as strategies to add and subtract.2 Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)
Understand subtraction as an unknown-addend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8.
Note that, first of all, these techniques are examples and, second, that they are suggested, not required. I have recommended that instead of holding all first graders (and subsequent grades) accountable for mastering a method of adding known as “making tens,” that teachers “differentiate instruction.” First graders who are ready can go ahead and master it; those who are not can do so in later grades.
I also make the point that these alternative strategies are nothing new. They have been in textbooks for years, but are offered to students as alternatives after mastery of the standard procedures. In fact, I believe it is possible to meet the Common Core math standards—and achieve the sought after “understanding” that many feel is missing from traditionally taught math—by using many of the techniques and explanations contained in textbooks from the 1950s and ’60s, with some slight modifications.
From Arithmetic We Need, Grade 3 (Brownell et al; 1955).
From Growth in Arithmetic, Grade 4 (Clark, et. al.; 1952): shows “counting up” method for making change.
After publication of this four-part series, I have yet to hear anyone tell me that I’m wrong in my interpretation of and recommendations for implementing the Common Core math standards. (This includes some of the writers of the math standards who have read the articles and with whom I have had some discussions.) If anyone out there thinks I am wrong about anything I have discussed here or in my articles, please tell me. Give me the reasons why, citing the sections of the standards that say otherwise. I would particularly like to hear from state departments of education, school districts, publishers, and professional development vendors—these last two seeming to hold a monopoly on how Common Core is to be interpreted.
Of course, there is one fly in the ointment to all this: the test-makers. They seem to take the view that students will be held to not understand unless they can explain their reasoning via pictures, alternate strategies or written explanations. Well, at least there’s one entity who is accountable. Right?
Brownell, William A., G. T. Buswell, I. Saubel. “Arithmetic We Need; Grade 3”; Ginn and Company. 1955.
Clark, John R., C.W. Junge, H.E. Moser. “Growth in Arithmetic; Grade Four”; World Book Company. 1952.
Image by Liz.