Title

The Relevance of Representational Use in the Assessment of Mathematical Understanding and Differentiated Instruction

Location

Harborside Center East and West

Strand #1

Academic Achievement & School Leadership

Relevance

In the current climate of increased attention to differentiated instruction, effective assessment methods take on increased importance. By looking at the types of representational tasks teachers pose for the purpose of assessment through the lens of cognitive skill strengths and skill-based assessment methods, this presentation will provide opportunities for exploring the relevance of these representational tasks to assessment whose goal is the differentiation of instruction. Instructional and evaluative assessments that are geared towards skill development and deeper conceptual understandings, and that engage students in meaningful critical thinking activities have the potential to produce valuable data to inform differentiated instructional decision-making and the development of effective instructional strategies, as well as for closing achievement gaps and promoting learning for all students.

Brief Program Description

Geared towards teacher educators and elementary and middle-school mathematics teachers, the findings of a study that explored elementary and middle-school teachers’ use of representations for the purpose of assessment to inform instruction, and for assessment to determine a grade will be presented. The data will also be reviewed from the perspective of its relevance to differentiated instruction.

Summary

The purpose of this presentation is to engage teacher educators and pre-service/in-service elementary and middle-school teachers in reflecting on effective instructional and evaluative assessment strategies for mathematics and how to use the data collected from those assessments to effectively differentiate instruction. The goal is to open up meaningful dialogue about the types of representational tasks posed by teachers for each assessment type, the importance of alignment between both types of assessments, and how to use the information provided by student performance to develop effective strategies for differentiating instruction. Participants will also spend time developing skill-based assessments in small groups and sharing with the larger group. The findings of a study that looked at the types of representational tasks posed by teachers for the purpose of assessment to inform instruction and assessment to inform a grade will be presented and discussed. The following questions were addressed by the study: What types of representational tasks do teachers pose for these two assessment purposes? a) Describing what students know for the purpose of instruction; and b) Evaluating what students know for the purpose of grading? The findings from this study will be discussed within the framework of using assessment data to inform instructional decision-making, especially as it refers to the potential implications for the differentiation of instruction.

Evidence

The findings of a study conducted by the presenter, which investigated teachers’ use of representations in their mathematics instruction and assessment, will serve as the basis for the discussions and the skill-based assessment activities in which conference participants will engage. The purpose of this study was to examine the types of representational tasks posed by teachers at the various grade levels for assessment purposes. Specifically, the types of representational tasks posed for the purpose of making instructional decisions, and representational tasks posed for the purpose of evaluation to assign a grade were investigated. According to Leinhardt (2001), representations hold the position in education of being powerful and useful supporting mechanisms in explanations; however they can be dangerous tools if they are not properly constructed or used. She states that “…to build good representations during an explanation, the teacher needs the knowledge that will allow him/her to a) select from those representations that are offered in texts, b) build new representations that truly match the context of the explanation, and c) refine representations that are offered by students in the course of their own discussions” (p. 348). She recommended that language be identified as a vital aspect of the use of representations so that explicit interconnections can be built, and either the teacher or the students need to work to build meaning from representations and to build interconnections between them. She also emphasizes the importance of teachers’ understanding of both content and students’ personal lives and experiences in order to build good representations. Attempting to provide representations without discussion is ineffective. For example when used alone, manipulatives do not increase understanding, but when they are well designed, used effectively, and are well connected to core problems, they work well. Further, the importance of problem solving to conceptual development in mathematics and the building of representation in student learning is also emphasized in The National Council for Teachers of Mathematics’ (NCTM) Principles and Standards for School Mathematics (NCTM, 2000). Some evidence has been provided by previous research to indicate that students’ structural representations affect, and are affected by their learning, and that teachers’ beliefs and knowledge about content and pedagogy also affect students’ representations structures (Fennema, et al., 1996). Connections between spatial ability and representations were also detected (Pyke, 2003), as well as connections between real-life problems that are relevant to students’ lives and the representation structures that they build (Irwin, 2001). Studies also suggest that students have the criteria needed to make algebraic representations of real-life situations (Iszak, 2003), and that modeling activities help students to build relationships between and among quantities, and to develop representations of these relationships (Doerr & English, 2003). This study therefore sought to look at the types of representation tasks that teachers pose for the purpose of both instructional and evaluative assessments at various grade levels, to identify any patterns existing between both assessment types, and to uncover evidence of alignment between the two forms of assessment. Three teachers – one Pre-K/Kindergarten, one 3rd/4th, and one 7th/8th grade – were observed over two class sessions. Prior to the observed lessons, each teacher was interviewed. The purpose of these interviews was to uncover each teacher’s definition/understanding of representations, their perceived use of representational tasks in their instruction, to develop working definitions for instructional and evaluative assessments. Follow-up interviews were also held with each teacher (i.e., after the observed lessons) to get answers to questions about any aspect(s) of the lesson that needed clarification. Both interviews were audiotaped. Each lesson was also videotaped and the data collected from the transcripts of both the audio- and video-tapings were analyzed to determine the types of representational tasks that were posed by each of the teachers for both assessment purposes, as well as to identify any patterns existing between both assessment types, and to uncover evidence of alignment between the two forms of assessment. Statistically significant findings indicate that the use of written symbols was the representational form used most frequently for both assessment to inform instruction and assessment for the purpose of evaluation. When looking, specifically, at assessment to inform instruction, the data revealed that spoken word was used most frequently by the Pre-K/Kindergarten teacher, both spoken word and written symbol were used most frequently by the 3rd/4th grade teacher, and written symbol was used most frequently by the 7th/8th grade teacher. The data on the use of assessment for evaluation purposes indicate that both written symbol and written word were used most frequently by the Pre-K.\/Kindergarten teacher, diagrams/pictures by the 3rd/4th grade teacher, and written symbol by the 7th/8th/grade teacher. The data were also analyzed to determine patterns of use for each assessment purpose, based on grade level. Statistically significant findings indicate that for assessment to inform instruction, concrete objects and spoken word were used more frequently by the Pre-K/Kindergarten teacher than by any of the other teachers, and written symbol, written word, and diagrams/pictures were used more frequently by the 7th/8th grade teacher than by any of the other teachers. For assessment for the purpose of evaluation, concrete objects, written symbol, and diagrams/pictures were used more frequently by the 3rd/4th grade teacher, and written word was used more frequently by the Pre-K/Kindergarten teacher than by any of the other teachers. Spoken word was not used by any of the teachers for formal evaluative assessments.

Format

Poster Presentation

Biographical Sketch

The presenter has a PhD in Mathematics Education, five years of teacher-education experience as an Asst. Prof. of Mathematics Education in the College of Education at Armstrong State University, and a total of 17+ years of teaching experience at the elementary, middle (mathematics & science), and high school (mathematics) levels.

Grants and awards received by this presenter include:

  • Golden Apple Science Award, 1999 – 2000.
  • GTE/Verizon Growth Initiatives for Teachers (GIFT) Fellow for Mathematics & Science, 2000-2001 (for the development of integrated mathematics & science curriculum).
  • Toyota Tapestry Award for Excellence in Science Education, Diversity of Life: The Study of Ecosystems in Urban Parks and Gardens, Toyota Motor Sales, U.S.A., Inc. & the National Science Teachers Association, $10,000, 2001 – 2002. (This grant was used to extend the curriculum project developed in 2000 for another year.)
  • Co-PI on two teacher quality grants (Summer, 2011) designed to help teachers develop/augment their biology, ecology, and geology curricula, based on their experiences during two week-long workshops. The workshops took teachers on a tour of the Georgia Barrier Islands and on a geological tour of various landforms and mines across the state of Georgia. My responsibilities included developing and presenting activities geared towards helping teachers effectively integrate mathematics into their science curricula.
  • Currently serving as a member of one of the Noyce MASTERS Future Teacher Professional Learning Communities (FTPLC) in Mathematics (Fall, 2013 – Present); responsibilities involve engaging in Math FTPLC research design and other preparation tasks, and supporting the intellectual development of the Noyce student scholars.
  • Recipient of the Georgia Association of Teacher Educators Distinguished Dissertation in Teacher Education Award (October, 2013) for doctoral research – The Contributions of Spatial, Verbal, and Analytical Skills to Problem-Solving Performance.

Previous conference presentations include:

Bennett, K., Fabrikant, K., Foster, A., McGrath, R., Rich, L., Skidmore-Hess, D.,

Williams, J. (April, 2014). Smilla’s Sense Of Synergy? Using Literature To Bridge Professions. 2014 InterProfessional Health Care Summit, Savannah, GA.

Foster, A. (March, 2014). Problem-Solving Across the Curriculum. MatemaTech Conference of the European Territorial Co-Operation: Austria-Czech Republic, Vyssi Brod, Czech Republic.

Foster, A. (March, 2014). The Contributions of Spatial, Verbal, and Analytical Skills to Problem-Solving Performance. National Youth-At-Risk Conference, Savannah, GA.

Foster, A. (March, 2013). The Contributions of Spatial, Verbal, and Analytical Skills to Problem-Solving Performance. Teaching Matters: Making Connections Conference, Gordon College, Barnesville, GA.

Foster, A. (March, 2012). The Value of Math/Science-Integration Workshops to Classroom Practice. Teaching Matters: Ten Years After Conference, Gordon College, Barnesville, GA.

Foster, A. (March, 2007). Representational Use in the Assessment of Mathematical Understanding. Illinois Council of Teachers of Mathematics Western Regional Conference, Western Illinois University, Macomb, IL.

Foster, A. & Zawojewski, J. (April, 2005). Teacher Partners in Action Research. The 2005 National Council of Teachers of Mathematics Conference, Anaheim, CA.

Keyword Descriptors

representation, mathematical understanding, differentiated instruction, instructional strategies, achievement gaps

Presentation Year

2015

Start Date

3-3-2015 4:00 PM

End Date

3-3-2015 5:30 PM

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Mar 3rd, 4:00 PM Mar 3rd, 5:30 PM

The Relevance of Representational Use in the Assessment of Mathematical Understanding and Differentiated Instruction

Harborside Center East and West

Geared towards teacher educators and elementary and middle-school mathematics teachers, the findings of a study that explored elementary and middle-school teachers’ use of representations for the purpose of assessment to inform instruction, and for assessment to determine a grade will be presented. The data will also be reviewed from the perspective of its relevance to differentiated instruction.