# Experts at UC study how students learn math

- March 06, 2004
- Nanette Asimov, Chronicle Staff Writer
- SAN FRANCISCO CHRONICLE
- http://www.sfgate.com/cgi-bin/article.cgi?file=/chronicle/archive/2004/03/06/BAGR45FQRU1.DTL

The world's bravest sixth-grader will step onto a Berkeley stage Monday morning to answer questions about fractions before an audience of 130 of the nation's leading math experts.

If he is like nearly two-thirds of California sixth-graders -- that's 65.9 percent, or 32,368 kids -- he'll get many answers wrong. But the question-answer session is meant to help the nation's experts solve the trickiest math problem of all: how to measure students' true understanding of math -- not just their ability to pick the right answer.

By agreeing to the public grilling, the sixth-grader shows not just courage, but savvy: He is withholding his name from those attending the four- day workshop called "Assessing Students' Mathematics Learning: Issues, Costs and Benefits" at UC Berkeley's Mathematical Sciences Research Institute.

"The main problem with math testing is that we don't test the qualities or skills that matter," said Deborah Loewenberg Ball, a math education professor at the University of Michigan who will interview the 11-year-old. "We test a rather narrow range of abilities. You want to test understanding and application. But it's easier to measure people getting the right answer."

Most state exams, including the California Standards Test, rely on multiple-choice questions that measure a narrow set of skills, Ball said. Another common problem is that "we don't yet really know how to make sure that assessments are fair to the diverse populations in the country, which can give us really biased results," she said.

These issues will be debated Sunday through Wednesday. Educators care about testing not only to learn how well students are doing, but because exams influence instruction.

For years, disagreements on teaching and testing have led to the so- called Math Wars. At their most slanderous, some experts accuse others of practicing "fuzzy math," while the return volley involves charges of "drill and kill."

By interviewing the sixth-grader, Ball said, she hopes to show the academics, think-tankers, psychometricians and policy wonks at the workshop that they agree on more than they thought.

"No mathematician is going to argue that it's fine for kids to get the right answers but not understand what they're doing," Ball said. "When they look at this student, they won't have the typical arguments they usually have."

Call it reducing fractiousness to simplest terms: What a child understands, and how to determine that.

Ball will ask the young man such questions as whether he can divide 5 by 2/3.

To do it, California students who have completed the first semester of sixth grade are supposed know that "5" is really "5 over 1," and that they should invert the second fraction and complete this operation: 5/1 x 3/2. To do that, they multiply the numerators, then the denominators, and get 15/2 (which is 7 1/2).

"Most people will remember the procedure," Ball said. "The technique is like a magic trick we learn."

But though a typical test would consider the question done, Ball would not quit there. To learn how well students understood what they did, she would ask: Why do you think the result is bigger than the numbers you started with?

Hmm ... Logically, quotients are less than dividends. (Right?)

"No one ever stops to look at that," Ball said. "Sometimes people say that's weird. Some people say, 'Who cares, as long as you get the right answer?' Or they say, this is a stupid school thing. Who ever divides fractions anyway?"

If asked the follow-up question, students might try to visualize what it means to divide 5 by 2/3.

Ball, a former elementary math teacher, makes it easy:

"Suppose I tell you I have 5 cups flour," she said. "And some sauce requires 2/3 cup flour. That's a division problem. I'll want to know how many batches I can make -- and the answer is that I can make 7.5 batches."

Voila -- a quotient bigger than the dividend.

Here's another puzzle, but one that suggests dessert metaphors don't always sweeten the problem: Suppose Janice buys half a cake. She eats a quarter of the cake she bought. Then she cuts the remainder into two equal pieces. What fraction of a whole cake was each piece?

"This problem is not so easy," Ball said.

That's an understatement.

The point of Janice and her cake is to learn how far a student can go before stumbling -- and what the stumbling tells testers about testing. "This is really good for measuring problem-solving skills, which everyone will agree matters," Ball said.

If Ball were to pose the Janice question Monday, she would first ask the student to explain what is being asked. And she would ask him to draw a picture of what Janice did.

"He'll know what half a cake is. But the question is staged so that each question gets harder." (See graphic for answer.)

Ultimately, a good test should reveal how students reach their answers and whether they can explain what they're doing, Ball said. Sometimes, that can matter more than a precise answer, as in this example: Approximately how much is 19/21 + 52/55?

Forget calculators, least-common denominators, fingers or Ask Jeeves. If you understand the idea of fractions, this question is, well, a piece of cake.

The answer is "nearly 2."

For more information about the conference, visit: