Developing an assessment probe is different from creating appropriate questions for summative quizzes, tests, or state and national exams. The probes in this book were developed using the process described in Mathematics Curriculum Topic Study: Bridging the Gap Between Standards and Practice (Keeley & Rose, 2006).
The process is summarized as follows:
• Identify the topic you plan to teach, and use national standards to examine concepts and specific ideas related to the topic. The national standards used to develop the probes for this book were NCTM’s (2000) Principles and Standards for School Mathematics and the American Association for the Advancement of Science’s (AAAS, 1993) Benchmarks for Science Literacy.
• Select the specific concepts or ideas you plan to address, and identify the relevant research findings. The source for research findings include NCTM’s (2003) Research Companion to Principles and Standards for School Mathematics, Chapter 15 of AAAS’s (1993) Benchmarks for Science Literacy, and additional supplemental articles related to the topics of the probes.
• Focus on a concept or a specific idea you plan to address with the probe, and identify the related research findings. Choose the type of probe structure that lends itself to the situation (seemore information on probe structure following the Gumballs in a Jar example on page 9). Develop the stem (the prompt), key (correct response), and distracters (incorrect responses derived from research findings) that match the developmental level of your students.
• Share your assessment probes with colleagues for constructive feedback, pilot with students, and modify as needed. Concepts and specific ideas related to the probability of simple events. The information was used as the focus in developing the probe Gumballs in a Jar .
A probe is a cognitively diagnostic paper-and-pencil assessment developed to elicit research-based misunderstandings related to a specific mathematics topic. The individual probes are designed to be (1) easy to use and copy ready for use with students; (2) targeted to one mathematics topic for short-cycle intervention purposes; and (3) practical, with administration time targeted to approximately 5 to 15 minutes.
Each one-page probe consists of selected response items (called Tier 1) and explanation prompts (called Tier 2), which together elicit common understandings and misunderstandings. Each of the tiers is described in
more detail below.
Tier 1: Elicitation
As the elicitation tier is designed to uncover common understandings and misunderstandings, a structured format using a question or series of questions followed by correct answers and incorrect answers (often called distracters) is used to narrow ideas to those found in the related cognitive research. The formats typically fall into one of seven categories.
Tier 2: Elaboration
The second tier of each of the probes is designed for individual elaboration of the reasoning used to respond to the question asked in the first tier. Mathematics teachers gain a wealth of information by delving into the thinking behind students’ answers not just when answers are wrong but also when they are correct (Burns, 2005). Although the Tier 1 answers and distracters are designed around common understandings and misunderstandings, the elaboration tier allows educators to look more deeply at student thinking as sometimes a student chooses a specific response, correct or incorrect, for an atypical reason. Also, there are many different ways to approach a problem correctly; therefore, the elaboration tier allows educators to look for trends in thinking and in methods used. Also important to consider is the idea that in order to address misconceptions, students must be confronted with their own incorrect ideas by participating in instruction that causes cognitive dissonance between existing ideas and new ideas. By having students complete both tiers of a probe and then planning instruction that addresses the identified areas of difficulty, teachers can then use students’ original responses as part of a reflection on what was learned. Without this preassessment commitment of selecting an answer and explaining the choice, new understanding and corrected ideas are not always evident to the student.
In Designing Professional Development for Teachers of Science and Mathematics, Loucks-Horsley, Love, Stiles, Mundry, and Hewson (2003) describe action research as an effective professional development strategy. To use the probes in this manner, it is important to consider the complete implementation process. We refer to an action research quest as working through the full cycle of
• questioning student understanding of a particular concept;
• uncovering understandings and misunderstandings using a probe;
• examining student work;
• seeking links to cognitive research to drive next steps in instruction; and
• teaching implications based on findings and determining impact on learning by asking an additional question.
Grade-span bars are provided to indicate the developmentally appropriate level of mathematics as aligned to the NCTM Standards and cognitive research. The dark band represents the grade levels where the mathematics required of the probe is aligned to the standards, and the lighter band shows grade levels where field testing of the probe has indicated students still have difficulties. The grade spans, although aligned to the standards, should be considered benchmarks as some students at higher grades may have misunderstandings based in understandings from lower grades, while others may be further along the learning progression and need probes designed for older students.
Student answers may reveal misunderstandings regarding methods of addition, including a lack of conceptual
understanding of number properties. Responses also may reveal a common misconception that there is only one correct algorithm for each operation or that, once comfortable with a method, there is no need to understand other methods.
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