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(Project Description)
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Authors: Darlene Kohrman
and Sheila Eisenhauer
from Kalamazoo Valley Community College, Michigan |
Project Description: This project describes a classroom/course assessment project that takes place in a geometry-focused Math for Elementary/Middle School Teachers class.
The goal of the assessment project was to document students' understanding of geometry content knowledge for the purpose of adjusting classroom activities and teaching techniques to improve the learning of future students taking the class.
Click on the assessment cycle steps to the right to see the details of this classroom/course assessment project.
1. DEFINE / REFINE student learning outcomes based on input from stakeholders.
In the winter of 2005, the two authors were asked to redesign the course known as Geometry for Elementary/Middle School Teachers (Math 142). The class meets twice a week for one hour and fifty minutes per session and lasts fifteen weeks. The challenge during the redesign was to follow the recommendations of (Conference Board of the Mathematical Sciences, The Mathematical Education of Teachers, 2001, p.7) which suggests developing a content course for the pre-service teacher that would "make connections between the mathematics being studied and the mathematics prospective teachers will teach".
The redesign began with the rewriting of the course objectives and learning outcomes following the recommendations of Norman E. Gronlund from his book "Writing Instructional Objectives for Teaching and Assessment". The new objectives are defined to reflect deeper understanding of the mathematics content. The former objectives were procedural in nature. The Grade Level Content Expectations (GLCS's) for grades K-8 for Michigan are also guided the rewrite of the course objectives and learning outcomes which are attached. These objectives are shared with students so they are aware of what they must achieve in terms of learning outcomes. Students also see the alignment with the state objectives.
COURSE OBJECTIVES - Students completing Math 142 will be able to:
1. Comprehend the characteristics of geometric shapes.
1.1 Identify the attributes of all types of quadrilaterals.
1.2 Identify the attributes of all types of triangles.
1.3 Identify the characteristics of common three-dimensional shapes.
1.4 Describe an angle in terms of the amount of turn and develop the appropriate unit of measure.
1.5 Determine when it is possible to create triangles and quadrilaterals and determine how many can be created.
2. Comprehend the relationship between geometric shapes.
2.1 Describe the relationship between different types of quadrilaterals.
2.2 Describe the relationship between various types of triangles.
2.3 Describe the relationship between common three-dimensional shapes.
2.4 Illustrate the relationship between common shapes using Venn diagrams.
3. Analyze transformations.
3.1 Perform 900, 1800, and 2700 rotations about the origin.
3.2 Perform reflections over the x-axis and y-axis.
3.3 Perform translations throughout the coordinate plane.
4. Analyze tessellations.
4.1 Identify characteristics of regular tessellations.
4.2 Identify characteristics of semi-regular tessellations.
4.3 Determine whether or not one or more polygons can tessellate a plane.
4.4 Derive a rule to find the measure of the vertex angle of a regular polygon.
5. Know the concepts of similarity and congruence.
5.1 Identify figures that are mathematically similar.
5.2 Identify figures that are mathematically congruent.
5.3 Construct shapes that are similar to a given shape.
5.4 Identify the scale factor between similar figures.
5.5 Explain how to determine whether or not two figures are mathematically similar or congruent.
5.6 Explain what we know about two figures that are mathematically similar or congruent.
5.7 Distinguish between shapes that are similar and shapes that are congruent (and shapes that are both).
5.8 Apply concepts of similarity to solve problems.
5.9 Use similarity concepts in real world situations.
6. Understand the concepts of measurement.
6.1 Measure length using a non-standard unit.
6.2 Measure the length of a line segment on dot paper using more than one strategy.
6.3 Discover the Pythagorean Theorem.
6.4 Describe the meaning of area.
6.5 Develop formulas for calculating the area of parallelograms, trapezoids and triangles.
6.6 Develop formulas for calculating the circumference and area of a circle.
6.7 Determine strategies for finding the area of figures on dot paper.
6.8 Develop formulas for finding the surface area of rectangular prisms and cylinders through the use of flat-shapes.
6.9 Develop formulas for finding the volume of prisms, cylinders, cones and spheres.
6.10 Identify realistic values of measurement for real world situations.
6.11 Solve problems involving area, perimeter, surface area, and volume.
7. Demonstrate a positive attitude towards mathematics.
7.1 Exhibit an open-mind in regard to new mathematical strategies and approaches.
7.2 Engage in collaborative group work and remain on-task.
7.3 Ask group members for help when needed before asking the instructor.
7.4 Provide help to other group members when requested.
7.5 Participate in whole class discussion.
7.6 Exhibit respect for others when others are talking (peers as well as the instructor).
7.7 Submit professional quality work.
8. Use technology to investigate mathematical topics.
8.1 Select appropriate procedures for constructing geometric objects and figures using Geometer's Sketchpad and MicroWorlds EX.
8.2 Investigate geometric concepts using software procedures.
8.3 Verify geometric concepts using Geometer's Sketchpad and MicroWorlds EX.
8.4 Relate software procedures to geometric concepts.
8.5 Examine measurement concepts using Geoboard on the TI-73 graphing calculator.
2. DESIGN assessment tools, criteria and standards directly linked to each outcome.
Permission was obtained from Dr. Zalman Usiskin (University of Chicago) to use the van Hiele Levels Test as a course-level assessment tool. Van Hiele levels indicate where students are in terms of geometric experience and understanding. The first level indicates that students view geometric objects by appearance or holistically. The second level indicates students are beginning to view geometric shapes in terms of their attributes or properties. The third level is called the relationship level where students see that geometric shapes are related. For example, a square is a special rectangle. Level Four is the deduction level where students can write proofs. The test is administered the first day of the course to students as a pre-test in order to determine their current level of geometric understanding. Most of our students enter the course at Level One. Many of our students are products of the "hidden curriculum" referred to by Glenda Lappan (NCTM) in 1989. It is administered again on the last day of the course as a post-test to determine their exiting level of geometric understanding. NOTE - The results of this test are not to be used in grade determination per Dr. Usiskin. Changes in levels are recorded as "no change", "increase in level", "decrease in level" or "has reached Level Four". Level Four is the desired exiting goal for the course.
Class activities are designed to help students overcome their inadequacies in their geometric content knowledge. For example, an activity was designed to help students discover the properties of two-dimensional shapes; follow the van Hiele model of learning. Manipulatives, plastic copies of two-dimensional shapes support the construction of concept knowledge. First shapes are sorted by attributes determined by the students. This free sort is followed by whole class discussion. This is a level two activity as it involves the classification of shapes by properties. The activity then asks students to resort the shapes based on the idea "All quadrilaterals have", "Some quadrilaterals have", "All triangles have", "Some triangles have" and "Others". This type of classification then leads to the idea of inclusion and students draw detailed concept maps to complete the activity on properties of two-dimensional shapes. Further discovery will follow when students use Geometer's Sketchpad to investigate more properties.
Classroom assessment is both formative and summative in nature. Various forms of formative assessment are used including: collaborative group assignments, written reflections on selected articles from NCTM publications and metacognitive written reflections on how students view their own progress in meeting the course objectives. Students also work on developing possible exam questions that align with the course learning outcomes and the class activities they have been experiencing.
Unit exams are written by the instructors to align questions with the course objectives. A blue print for each exam is developed. Click here for sample exam blueprint (doc). Expanded performance assessment or a constructed response format is used on all exams. An example contrasting a traditional question and an expanded performance question is as follows: Click here for example (doc)
NSF funded materials are used to align mathematics content with the mathematics curriculums pre-service teachers will experience. We currently use units from the Connected Math Project as student texts.
3. IMPLEMENT assessment tool(s) to gather evidence of student learning.
Evidence of student learning is collected in many ways. Writings from assigned readings are graded using a rubric. The metacognitive reflections (self-engaged learning, Royal Sadler) are used to design review questions or rewrites of activities to better meet the learning needs of the students. End of unit group questions are practiced whole class, here individual learning outcomes are addressed and aligned with text readings, activities from CMP or GSP to demonstrate that the opportunity for the student to meet the objective did occur in the unit.
4. ANALYZE and evaluate the collected data.
All activities, student reactions and responses are analyzed by the instructors as they happen. Test analysis is completed by the instructors on each question to determine the number of students who responded successfully (using a rubric) and if there is evidence that student learning did occur for each intended objective. Changes are then made on future tests concerning clarification of the question. A sample of a test analysis is attached.
5. IDENTIFY gaps between desired and actual results.
At the end of each class meeting, assignment, project, writing or unit exam, we ask ourselves several questions. These questions include:
- Did each activity meet its intended learning outcome?
- Do we see gaps in the intended outcome and how the students reacted and indicated their understanding?
- Do we need to make changes?
- What else do we see in the data?
Each writing activity is assigned with directions and bullets to which the students respond and must address. If a student answers a bullet with a weak response than our bullet may lack clarity and it will be rewritten. For example, the GSP assignment titled "Practice 9" has been restructured to enable students to discover that as the quadrilateral's properties increase and it becomes a square, the diagonal properties also increase. The restructuring included the addition of several leading questions to help students see the property increase.
Through the use of the metacognitive reflection writings, students tell us which learning outcome(s) is/are their most challenging. For example,
- Describe using complete sentences or thoughts, what you have learned thus far from the classroom activities, readings or text?
- What ideas, concepts, or approaches have been the most challenging? Why do you feel this is the case?
- What strategies have you used to help you learn the concepts, strategies, ideas and connections in this first unit? For example, have you designed 3 by 5 cards with the definitions of your geometric shapes?
Students will often indicate that the most challenging thing has been learning all the properties and terms. We are not surprised by this as many of them pre-test at the lowest van Hiele level of "Visualization". Our suggestions have been to use 3 by 5 note cards where not only a definition can be placed but also pictures both holistically and formally can be added. A new activity was developed to use in groups to help students learn properties and relationships. This activity focuses on their ability to write definitions several in ways and to make them as minimal as possible. For example, a rectangle is a parallelogram with one right angle. Students reflect on why this definition is sufficient.
6. DOCUMENT results and outline needed changes in curriculum, instructional materials or teaching strategies.
From the winter of 2005 until the present, our rather piecemeal course pack has been replaced with the actual student units from CMP. Even CMP has rewritten the unit "Stretching and Shrinking - Understanding Similarity". Similarity is still a weak area for students; therefore an application project will be added to the course next semester to engage students in similarity in a real world context. Students will be asked to design a miniature room thus helping them focus on the idea of scale factor and proportionality.
Our GSP unit has been modified and now includes more leading questions. The new minimal definition activity has helped students become more successful on the true and false portion of the unit exam when asked about the relationship of certain shapes.
Van Hiele results of the pre and post geometric levels test have been tabulated. The results have been promising, for example, this past fall 2006, of the 17 students from the night section taking both pre and post tests, twelve of the seventeen (71%) showed upward changes in levels and ten of the seventeen (59%) exited at Level 4.