As a young teacher in an East Lansing, Mich., elementary school, Deborah Loewenberg Ball realized teaching mathematics required a special kind of knowledge. Unfortunately, she didn鈥檛 have it.
Take long division, for example.
鈥淭here was no way of explaining to students what the procedure means, and what they鈥檙e really doing,鈥� said Ms. Ball, now an education professor at the University of Michigan in Ann Arbor. 鈥淜ids forget that stuff all the time.鈥�
Without the right explanations, she added, 鈥渁ll you can do is keep saying, 鈥楤ring down the zero.鈥� 鈥�
Now, though, after 20 years of systematic study, Ms. Ball is beginning to get a handle on just what that special kind of knowledge is. With her university colleagues, she has defined what teachers need to know to teach math effectively and devised new ways of measuring whether they know it.
This is no mere exercise in academic hairsplitting. Ms. Ball鈥檚 work suggests that having what she calls 鈥渕athematical knowledge for teaching鈥� matters for students鈥� learning. Her research shows that students whose teachers score high on the measures developed by Ms. Ball and her research partners learn more math over the school year than do students of low-scoring teachers.
鈥淒eborah鈥檚 work really is the model that we look to to set the standard,鈥� said Lee S. Shulman, the president of the Carnegie Foundation for the Advancement of Teaching, based in Stanford, Calif.
Having coined the term 鈥減edagogical content knowledge鈥� in the mid-1980s, Mr. Shulman is the godfather of the idea that teachers need to know their subject matter in special ways to teach well. But he said Ms. Ball has found concrete ways to elaborate, describe, and measure it in her own field of mathematics and to show that it matters.
鈥淚t鈥檚 what we should be doing in all the disciplines,鈥� he said.
As Ms. Ball sees it, the knowledge good math teachers need consists of more than knowing math well or understanding how children think at particular developmental stages. It comes from knowing how to apply mathematical knowledge, quickly, in ways that make sense to students.
Reacting to Students
In universities that prepare teachers, she said, 鈥渋t鈥檚 almost as though this kind of thing fell in the crack between the math department and the school of education.鈥�
For example, when a student uses an unorthodox method for multiplying 25 times 35 and still gets the right answer, an effective teacher must determine in a split second whether that student鈥檚 approach is a genuine method that generalizes to other multidigit multiplication problems or pure dumb luck.
When the teacher, flummoxed, instead instructs the student to stick to the standard method, that can send the wrong message to the class, said Hyman Bass, a research mathematician who has been working with Ms. Ball since the late 1980s. 鈥淭he message to the student in that circumstance is that mathematics is a subject where you鈥檙e not supposed to think,鈥� he said.
Or suppose a teacher is teaching about polygons, the researchers write. A polygon is defined as 鈥渁 closed plane figure consisting of straight lines,鈥� according to Webster鈥檚 New World College Dictionary. To begin with, the teacher has to recognize whether students could even begin to understand the textbook definition of a polygon鈥攁nd, if necessary, come up with another definition that is more user-friendly, but just as precise.
A teacher would also have to be able to pinpoint the sources of the errors their students make, to come up with word problems on the spot that illustrate key concepts, and carefully choose assessments that show whether students are 鈥済etting it,鈥� not just making lucky guesses.
鈥淭his is a subset of knowledge that teachers need to know that mathematicians don鈥檛 even know,鈥� Mr. Bass said.
In fact, the Michigan researchers have given some of their tests of mathematical knowledge for teaching to mathematicians. While the mathematicians answer most questions correctly, Ms. Ball said, they also get some wrong鈥攁nd they often take much longer than experienced teachers do to puzzle out a correct response.
Deborah Ball鈥檚 work in defining teachers鈥� knowledge in mathematics grew out of earlier work showing that most teachers, like Ms. Ball herself at one point, didn鈥檛 know enough math. To better devise strategies for filling in the gaps, she decided to go back to the source: classrooms.
She and her colleagues studied a 3rd grade classroom intensively for a year, collecting videotape footage, student work, teacher journals, lesson plans, textbooks, and other 鈥渞ecords of practice.鈥� They collected videotapes from dozens of other classrooms as well and invited mathematicians and educators to sit with them to pore over all the material and make observations.
鈥淵ou start realizing that no matter how you teach, kids make mistakes and teachers who can鈥檛 figure out what kids are doing are really ill-prepared to do the work,鈥� Ms. Ball said.
From those real-life observations, the researchers have developed a bank of 200 test items, which can be used to evaluate professional-development, preservice, and other kinds of preparation programs for math teachers. However, the tests are not designed to evaluate the skills of individual teachers, said Heather C. Hill, an assistant research scientist who works closely with Ms. Ball on the assessments.
In addition, Ms. Ball, Ms. Hill, and colleague Brian Rowan gave a 30-item assessment to a total of 699 1st and 3rd grade teachers at 115 predominantly low-income schools across the country. The schools were drawn from the massive Study of Instructional Improvement, in which Ms. Ball is also participating.
The researchers also tested the students of those teachers twice鈥攐nce in the fall and, later, in the spring of the same school year. The results showed that, while most students learned math over the course of the year, the learning gains were greatest for pupils whose teachers had also scored high on the 30-item 鈥渕athematical knowledge for teaching鈥� test. In other words, the researchers figured, students got an extra one-third to one-half of a month鈥檚 worth of learning growth for every standard-deviation rise on their teachers鈥� test scores.
In addition, the researchers said that teachers鈥� scores on the mathematical-knowledge assessment seemed to matter more than how much time they spend teaching math during an average school day, whether they were certified, or whether they had taken extensive mathematics or math teaching courses.
Classroom Observations
To check against the possibility that the high-scoring teachers were simply smarter than their other colleagues, the researchers compared teachers鈥� scores on reading and language-skills tests with their scores on mathematical knowledge for teaching.
鈥淚f teachers scored high on both of them, then you begin to think maybe what you鈥檙e measuring has more to do with general intelligence,鈥� Ms. Ball said.
But that did not turn out to be the case.
As an added check, the researchers also visited the classrooms of the highest- and lowest-scoring teachers.
鈥淲hat if somebody scores great and you visit that teacher, and they鈥檙e really clumsy and you can鈥檛 understand what they鈥檙e saying,鈥� Ms. Ball said. 鈥淥r what if someone scores badly, and they鈥檙e really skillful?鈥�
鈥淪o far,鈥� she added, 鈥渨e鈥檙e seeing that there is some relationship between their scores and the way they teach.鈥�
Ms. Ball said her findings are important because so few studies link what teachers know to their students鈥� math achievement.
The results also suggest that efforts afoot now to require teachers to take particular math courses鈥攐r to open up the field to individuals who might have degrees in math but no education training鈥攃ould be on the wrong track.
鈥淲hile those may be nice ideas,鈥� Ms. Ball said, 鈥渢hese results suggest that might not be the most promising way to think about improving math instruction.鈥�
Interesting ideas?
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