Teachers Need Help, Too
Professor Heather Hill sees much commonality among teachers teaching math and those instructing students in other disciplines. They have to manage their students, deal with the flow of classroom life, and engage students in discussion about the topic at hand.
But math is also a discipline with very specific practices needed to develop an aptitude for quantitative reasoning. Over the past decade, Hill has developed an assessment tool called the Mathematical Quality of Instruction (MQI), which has shown promise in evaluating teaching performance and creating pathways for improvement.
Hill, who began developing the evaluation tool a decade ago while teaching at the University of Michigan, has continued to refine it at the Ed School. She's currently conducting a study sponsored by the National Science Foundation that's giving teachers in grades four and five in a suburban Boston school the tools to conduct self-evaluations through online videos that model effective math instruction.
"It's a way to provide feedback on math teaching in ways that make sense," says Hill, a political scientist by training. "And now we are training teachers to rate their own instruction, and through that lens, teachers say it has changed the way they teach."
Hill says the quality of math instruction varies widely, especially in elementary schools, where generalists typically teach all subjects, up through fourth or fifth grade.
"In elementary school we often find that the teachers aren't teaching math," says Hill. "They might have their students coloring or be involved in the logistics of the class, but there is very little math going on. I was just in a fourth-grade math class where there wasn't any mathematical content for 20 minutes. Students were asked to write about what excited them about first-grade math."
To rate a teacher, two MQI observers independently rate classroom segments on five elements and provide an overall score on the lesson. The five elements of mathematical pedagogy are: 1. how teachers work with students, 2. student participation in meaning-making and reasoning, 3. the richness of the mathematical material, 4. what errors or imprecision are conveyed by the teacher, and 5. how classroom work is connected to mathematics.
The material's richness is shown through the attention that a teacher gives to the meaning of math facts and how a teacher engages students by using math practices and the language of math. To judge whether teachers are making meaning of the math, observers look at how teachers explain math ideas and draw connections among different mathematical concepts, such as fractions and ratios. Mathematical practices include such concepts as the presence of multiple solutions to a problem, and the use of specific problems to develop broader generalizations.
MQI focuses on whether teachers understand what their students are saying about math. Teachers get rated on how they respond to student errors.
"When they hear a kid talking and then see that the kid is wrong, the teacher needs to ask a series of questions to understand why they are wrong," Hill says. "They need to get familiar with their kids, know how they talk about math, and help them find strategies to get them out of their misconception."
One common misconception happens as children learn multiplication, Hill says. At first, they come to understand that multiplying numbers results in larger numbers. But then they multiply fractions, and suddenly the numbers get smaller.
"Teachers need to draw them into a space where they can get this understanding," she says. "They need to understand that multiplying doesn't always make numbers larger."
Raters also look for major errors, which can reveal a lack of math knowledge by the teacher, imprecise language and notation, or a lack of clarity in the presentation of tasks the teacher wants students to perform.
In another recent observation, the teacher had taught the students that 0.5 percent equals one-half (not what it really equals: one-half of one percent).
"That's what the teacher said," says Hill. "That's very common."
Student engagement with mathematical content provides insight into whether a teacher's lessons are getting through. Raters discern whether students provide mathematical explanations, either spontaneously or at their teacher's urging. They also look at the cognitive requirements needed to carry out a certain mathematical task — finding patterns, drawing conclusions, or explaining how they got their answers.
"Our assessment tool gives teachers specific ideas about how to improve," she says. "It helps them develop a better understanding of best practices."