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Most recent publication:

*DGA Item Characteristics*
(Technical Report, August 2010)

Masters, J. (2010). *Diagnostic Geometry Assessment Project: Item Characteristics.* Chestnut Hill, MA: Technology and Assessment Study Collaborative, Lynch School of Education, Boston College

Diagnostic Geometry Assessment

The Diagnostic Geometry Assessment (DGA) project is a federally-funded research project. The DGA aims to develop a formative assessment that middle-school teachers can use to identify the misconceptions and degrees of understanding that their students hold in geometry. The DGA will be an online assessment that can be administered from any Internet-connected computer, and that provides instant feedback to teachers. The DGA will enable teachers to identify underdeveloped reasoning in specific areas of geometry, while also providing information about why students struggle with concepts. Teachers can use this feedback, along with a package of instructional resource materials, to target instruction to each student's individual need. The DGA project is a collaboration between researchers at inTASC and the Center for Leadership and Learning Communities at the Education Development Center, Inc.

The DGA project is based on findings from cognitive research related to student geometric conceptions. The initial focus of the DGA is on three topics:

- Shape Properties,
- Measurement, and,
- Transformations.

The difficulties that students experience related to Shape Properties often stem from students having developed a concept image (a mental image of a shape) without a concept definition (a specified definition of the shape or its properties). Students who have this deficit of understanding can often identify examples of shapes, but will also fail to identify examples of shapes that are not identical to their own mental image of the shape or the shape prototype, i.e., the figure does not "look like" the shape. Characteristics such as orientation and proportions affect whether students recognize certain shapes, despite the fact that these characteristics are irrelevant to the defining properties of a shape. Rather than rely solely on a mental or visual image, students should have both a set of visual examples and a description of the necessary and sufficient properties that define a shape. Instruction can help students move towards this understanding.

Transformations refer to the mapping of *every* point in a plane, or, in other words, transforming the entire plane. Students often think of "the plane" as the empty space behind the figure, rather than thinking of the figure as a subset of points within the plane. This belief leads them to make transformations on the set of points that make up the figure, rather than the entire plane. The distinction between rotating the whole plane or the object becomes especially critical with distant points of rotation or distant lines of reflection.

Students often have a variety of misconceptions and underdeveloped reasoning related to two-dimensional measurement. Two common sources of misconceptions lie in students' ability to mentally structure space and their ability to connect structured space to mathematical formulas. Students frequently do not intuitively understand the concept of area as covering space. When mathematical formulas are introduced, students who cannot mentally structure space will memorize the formulas and apply them blindly, without understanding what the formulas mean or what the calculations represent. It is critical for teachers to know how their students are reasoning, and thus to help them develop more sophisticated reasoning and link that reasoning to mathematical formulas.

The National Council of Teachers of Mathematics defines geometry as one of the five content strands critical for the middle grades, and states the importance of geometry both for later studies of mathematics as well as for life outside of the mathematics classroom. Existing research has linked students' geometric skills to competence with higher-order mathematics processes, including logical reasoning, application of knowledge in arithmetic and geometry, management of data and procedures, and proportional reasoning. Most high school geometry classes attempt to develop student's ability to reason at higher levels, and these classes are often the only subject in which students are asked to construct proofs based on logic, higher order thinking skills, scientific thinking, and rigor. Because of this, geometry class is often the sole opportunity for students to experience these types of sophisticated thinking and reasoning, both as they apply to mathematics as well as to other domains. It is therefore critical for students to have the prerequisite knowledge required to attempt these tasks, and this prerequisite knowledge is initially developed in the middle school classroom. The research linking geometry to other mathematical and logistic processes suggests that teaching geometric thinking in the middle grades is at least as important as teaching algebraic thinking.

Despite the importance of learning geometry in the middle grades, many assessments have found that students in the U.S. perform poorly in geometry. In the National Council of Education Statistics' 2003 and 2005 National Assessment of Educational Progress results, the geometry and measurement strands had the lowest average score, compared with numbers and operations, algebra, and data analysis. In 1999 and 2003, the Trends in International Mathematics and Science Study reported that eighth grade students showed:

- U.S. students got only 44–45% of geometry items correct;
- For the within-country analysis, geometry was the weakest strand for U.S. students (compared with number sense, algebra, measurement, and data);
- 42–43% of countries performed significantly better than the U.S. on geometry items.

While geometry in the middle grades is a critical content area, both as it relates to other areas of mathematics and of its own merit, student achievement is falling short, compared with students in other countries, as well as with other areas of mathematics within the U.S. It is our assertion that one cause of the current disconnect between standards and performance is that teachers lack the information and resources required to diagnose and understand their students' geometric conceptions. The visual and spatial nature of geometry, more so than other areas of mathematics, allows students to develop their own ideas about geometry before they enter the classroom. It is only when teachers can access, identify, and explore these ideas that they can help students develop, refine, and build upon their understandings. The DGA will be a tool that teachers can use for this purpose.

The DGA is an extension of the Diagnostic Algebra Assessment (DAA) project, also conducted by inTASC researchers. Click here to read about the DAA project.