Research

That Matters

Pressing for Mathematical Understanding

The problem, on its surface, is simple:
A toy is hidden in one of two cakes. One cake is a circle, cut into fourths.
The other is a rectangle, cut into sixths. Students must choose the cake that
gives them the best chance of finding the toy.
Some choose the rectangle. Why? Because “most toys come in square boxes.”

Logical enough, if you apply a child’s real-life experience to
a theoretical situation. Teachers who mark the rectangle answer incorrect, who move onto the next problem without asking “Why?” may miss the thinking behind the child’s solution to the problem. They may believe that their students did not “get” probability, and the answer is just to reteach says Elham Kazemi, associate professor at the UW‘s College of Education.

Boy peeking “The question is what we do with errors,” says Kazemi. “Teachers can use them to reconceptualize a problem, explore contradictions in student understanding, and try out alternative instructional strategies.”

In the real world, Kazemi points out, people study errors to avoid the same mistakes. Football coaches study bad plays; diving coaches study splash patterns; analysts study the processes that led to flaws in a financial report.
Kazemi has spent countless hours listening to student-teacher exchanges in classrooms, recording dialogues and analyzing them to see how successful teachers develop children’s mathematical thinking. Her research shows that teachers who press students with strategic questions and carefully monitor their answers can move pupils to genuine mathematical argument and reasoning, even within elementary school classrooms. However, Kazemi notes that such instruction is not yet the norm.

“I wouldn’t say that the quality of mathematics instruction is universally high. We still have a long way to go,” says Kazemi. “The good news is that there now exists an array of professional resources to help.”

With a team of colleagues at the University of Washington collectively known as the Mathematics Education Project, Kazemi is working to build capacity to support excellent mathematics instruction in elementary and middle schools. Partially funded by a grant from the National Science Foundation, the project’s goal is to help systems support the professional education of teachers, teacher educators, and administrators and help them effectively engage with families.
Research by Kazemi and others has shown that well-organized, long-term professional development is needed to support teachers in creating the ambitious instructional practices that will allow all students to learn. The goals for the Mathematics Education Project grew out of the team’s understanding of the challenges schools and districts face in creating coherent plans for elementary mathematics professional development. The project has identified resources in elementary mathematics education to deepen teachers’ content knowledge, help them elicit and interpret student thinking, and advance children’s thinking through instruction. Resources are also identified to help administrators and parents understand their key roles in supporting children’s mathematics learning. Because using these new materials is complex, members of the project work with districts and schools to develop coherent implementation plans.

Elham Kazemi teahing “We still see too many districts adopting a one-shot approach to professional education,” says Kazemi. “In the Mathematics Education Project, we’re committed to helping schools and districts learn what these resources offer to create a long-term plan to engage teachers and the broader system in substantive work on their own teaching.”

At the core of the project’s work with teachers, teacher educators, administrators and families is the view that teachers should use a deep understanding of students’ mathematical thinking as well as a clear understanding of mathematical content to guide instruction. School leaders learn how particular resources can support knowledge and skill building. Educators, leaders, and families come to appreciate how students’ thinking develops as they explore students’ understandings and, especially, their misunderstandings. The resources recommended by the Mathematics Education Project reflect the goals to deepen mathematical understanding and bring to the surface the significant work that teachers do when they anticipate, elicit, and advance students’ mathematical knowledge.

“When you get kids to show you what they’re capable of, you are amazed,” says Kazemi.

One student may add 28 + 34 with traditional column carry-over. Another adds 2 to 28 and subtracts 2 from 34 before adding the two results. A third student adds 8 and 4 to make 12, then 12 and 30 to make 42, and 20 more to make 62. In an effective classroom, all those solutions are studied, the links between them established, and the connection made to larger mathematical concepts (such as place value, the properties of addition, and developing generalized strategies).
“As teachers talk with students, they need to press to find out what, exactly, they know. What students know should determine what the teacher does next, what kind of conversations the teacher prompts, and what comes next in the learning trajectory,” says Kazemi. “It’s important to understand the kinds of connections that students make between ideas. Mathematics is a body of knowledge that makes sense because of the relationships.”

Eliciting children’s mathematical thinking is a skill Kazemi also emphasizes in her math methods courses for teachers-in-training. To provide her student teachers with first-hand experience exploring children’s ideas, Kazemi has taken her methods classes out into elementary schools partnering with UW’s Teacher Education Program.

In one first-grade classroom, Kazemi had her UW students interview first-grade students in conjunction with a district-mandated assessment of counting and computational skills. After the interviews were completed, she and her students gathered to interpret and score the assessment and interview data, using the very framework the classroom teacher was using through the school’s professional development, and then collated the results across the whole class to share with the teacher.

“It became apparent, through our analysis of the whole class, that the first-graders were ready to be challenged to move their problem solving approaches beyond their fundamental counting strategies,” says Kazemi. “We also noted by compiling our interview data that students were very comfortable with a particular kind of word problem and struggled with others.”

The classroom teacher benefited from this opportunity for each of her first graders to be interviewed by an adult, and she left the conference with Kazemi and her students strategizing the next set of problems she would pose to her class. The student teachers experienced the direct link between what they were learning in their methods class and the work they would be engaged in as teachers.
For Kazemi, these are crucial steps in preventing these first graders from sharing the fate of more than 40 percent of the state’s tenth graders in 2007 who failed the math portion of the Washington Assessment of Student Learning (WASL). “When we see such high failure, it’s not as simple as ‘The kids don’t get it,’ ” says Kazemi. “When kids are getting wrong answers on a whole slew of problems, we should be asking, ‘What is it that they aren’t understanding? What skills do they have? What do we need to build? and What does that say about our instruction?’”

For more information about the resources used in the Mathematics Education Project see:
Kazemi, E. (2007). Supporting elementary mathematics through long-term professional education. Curriculum in Context, 34, 10-12.
depts.washington.edu/matheduc
For more information on Dr. Kazemi’s work on student thinking see:
Kazemi, E. (2002). Exploring test performance in mathematics:
The questions children’s answers raise. Journal of Mathematical Behavior, 21, 203-224.
Kazemi, E. & Franke, M.L. (2004). Teacher learning in mathematics: Using student work to promote collective inquiry. Journal of Mathematics Teacher Education, 7, 203-235.