OPINION

Is teacher education up to Common Core requirements?

Frank Murray

The new Common Core standards, to take a laudable example in mathematics, ask students to "make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, attend to precision, look for and make use of structure," and so forth. One standard (2.OA.C.3), about odd and even numbers, is made more rigorous, another Common Core goal, by simply placing a third-grade task into the second grade and by asking pupils to determine whether a group of 20 or less objects has an odd or even number of members.

Some years ago Deborah Ball, now dean of education at the University of Michigan in Ann Arbor, was teaching her third grade students about odd and even numbers, when one student, Sean, said that he thought some numbers were both odd and even. Now, the question is how would the teacher adhering to the new Common Core standards respond to this student and, more to the point, would today's teacher education programs provide sufficient training for an appropriate response from the teacher?

As part of its site-visit of teacher education programs seeking accreditation across the country, the Teacher Education Accreditation Council (TEAC), whose system was designed incidentally at the University of Delaware and is now part of the new Council of the Accreditation of Educator Preparation (CAEP) in Washington, D.C., often asked faculty and students at 200 institutions about the degree to which various teacher responses to a student comment exemplified their program's goals and philosophy. In other words, how satisfied or happy would the faculty members be (from "not at all" to "exactly what our program hopes") if a graduate of their program responded as the following three teachers responded when a student said during the teacher's lesson about odd and even numbers that some numbers were both odd and even.

Teacher A responded that there are no numbers that are both odd and even and that you can always tell which are odd and which are even by which are divisible by two and which are not. Those evenly divisible by two are even and the others are odd. She then moved on with her lesson plan.

Teacher B asks the student to explain why he thinks some numbers are both odd and even. He says that because two goes into some even numbers (like 6 or 10) an odd number of times, those numbers are both odd and even. The teacher says that is a very interesting thought, but then goes on to explain the rule to him as had been done by Teacher A and moves on with the lesson.

Teacher C suspends her lesson plan and asks the class to think about the student's conjecture. They talk about it and the teacher provisionally calls these odd/even numbers, Sean numbers, saying that is what mathematicians do when they explore a new idea. The class subsequently discovers or notices that every other even number is a "Sean number" and they then discover what kind of number results (odd, Sean or non-Sean) when "Sean numbers" are added to other "Sean numbers", to "non-Sean even numbers" and to odd numbers. The lesson ends as the class tries to decide whether "Sean numbers" should be added to the list of numbers in their mathematics curriculum, but by then time has run out."

Whose pupils, Teacher A's, B's, or C's, would do better on Common Core standards? Teacher A seems out of step with the Common Core values as she was not even curious about Sean's reasoning, although she did explicitly correct what she took to be his error. Teacher B went further in exploring Sean's reasoning, surely something expected in the Common Core, but Sean was more or less left with the idea he was not doing anything worthwhile mathematically. Teacher C seems to embody the Common Core's values of having Sean "make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, attend to precision, look for and make use of structure." Teacher C moreover encourages the whole class to accept the Common Core goals and see what more Sean and his classmates could make mathematically of his unconventional idea.

A national sample of students and faculty members in recently TEAC accredited teacher education programs gave the lowest marks to Teacher A and the highest to Teacher C with Teacher B falling in middle but closer to Teacher C. Faculty and students were in agreement about how poor and inadequate Teacher A was and how exemplary Teacher C was, but differed on Teacher B with the students being happier than the faculty. Each teacher earned the full range of ratings, however – from "not what we hope for " to "exactly what we hope for." Teacher C earned 76 percent of the top marks for mostly or fully satisfied, while Teacher B received 48 percent of them and Teacher A only 13 percent of them.

But what about those who gave Teacher C low marks, sometimes the lowest marks? Their concerns fell into three categories, all worrisome for the future of the Common Core. Most worrisome were those who felt Sean had made an error that had to be corrected and because of that, it was a mistake to let his error contaminate the whole class. Others felt the teacher's mistake was pedagogical because having the error named after the student inappropriately shamed him. Others felt that Teacher C simply wasted valuable time as this "odd/even number" conjecture was never going to be on a standardized curriculum test and they, and their districts, want their teachers to focus on what the state will test. Clearly, the topic of numbers being both odd and even has no mathematical future and is not going to be raised by anyone ever again.

Sean, of course, has not made a mistake, and as his conjecture shows, and as too many faculty and students overlooked, he fully grasps the principle that Teachers A and B taught. He is in fact doing the kind of thinking that the Common Core standards also hope to see when they ask students to "make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, attend to precision, look for and make use of structure," and so forth. From this perspective three quarters of the faculty and students in accredited teacher education programs have it right. It is furthermore encouraging that the nation's accredited teacher education programs apparently gave their students the intellectual confidence to recognize the value in a novel and provocative mathematical idea they surely never encountered or studied and that seems on first hearing to be of doubtful and unknown value. An uncomfortably large percentage of faculty and students, however, have it wrong with regard to the subject matter and pedagogy embedded in the Common Core. The limitations in their subject matter and pedagogical knowledge unfortunately foster the kind of conventional teaching that would undermine the larger goals of the Common Core and preserve the undesired status quo.

Over and above all this, there is the issue of how we will know if the goals of the Common Core were achieved. To focus only on checking whether pupils can figure out if an array of 20 or so items has an odd or even number of items in it, however, runs the risk that Sean's conjectures will be treated as dismissively as Teacher A and B treated them. Success on this counting task, if it is taken as an exemplary measure of the standard, means that some students will remain as baffled, as were a countable number, but fortunately a relatively small number, of participants in today's teacher education programs by the sense in which some numbers can be both odd and even. Another Common Core standard, the kindergarten mathematics standard (K.CC.4.B) requires that children understand that the number of counted objects is unaffected by their spatial arrangement or the order in which they were counted. We know that most young children can accurately count the arrays of objects the standard requires, but we, but unlike the standard-setters, also know that they will still say the moved array has more objects than it had before even though they have counted the arrays correctly before and after the move. Thus, they also will not in fact fully understand what was required of the standard on which they were credited with success.

So, with regard to teacher education and the Common Core, the glass looks to be about three-quarters full. With regard to how we will know whether we have succeeded in the Common Core experiment, we won't know for sure until the tests we mandate find a way to respect and reward the ingenuity of students like Sean.

Frank B. Murray is the H. Rodney Sharp Professor in the University of Delaware's School of Education.