Assessment in the mathematics classroom : yearbook 2011 Association of Mathematics Educators

ASSESSMENT IN THE MATHEMATICS CLASSROOM Yearbook 2011 Association of Mathematics Educators 8216.9789814360982-tp.indd 2 4/1/11 9:29 AM TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk ASSESSMENT IN THE MATHEMATICS CLASSROOM Yearbook 2011 Association of Mathematics Educators editors Berinderjeet Kaur Wong Khoon Yoong National Institute of Education, Singapore 3 x + 1 2 = 5 x 1 0 x + Y World Scientific 8216.9789814360982-tp.indd 1 4/1/11 9:29 AM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ASSESSMENT IN THE MATHEMATICS CLASSROOM Yearbook 2011, Association of Mathematics Educators Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. ISBN-13 978-981-4360-97-5 ISBN-10 981-4360-97-X Printed in Singapore. ZhangJi - Assessment in the Maths.pmd 1 4/4/2011, 4:38 PM Contents Chapter 1 Introduction: Assessment Matters 1 WONG Khoon Yoong Berinderjeet KAUR Chapter 2 Using a Multi-Dimensional Approach to 17 Understanding to Assess Students’ Mathematical Knowledge Denisse R. THOMPSON Berinderjeet KAUR Chapter 3 Assessing Problem Solving in the Mathematics 33 Curriculum: A New Approach TOH Tin Lam QUEK Khiok Seng LEONG Yew Hoong Jaguthsing DINDYAL TAY Eng Guan Chapter 4 Assessing Conceptual Understanding in 67 Mathematics with Concept Mapping JIN Haiyue WONG Khoon Yoong Chapter 5 Using Journal Writing to Empower Learning 91 Berinderjeet KAUR CHAN Chun Ming Eric Chapter 6 Implementing Alternative Assessment in the 113 Lower Primary Mathematics Classroom YEO Kai Kow Joseph v vi Assessment in the Mathematics Classroom Chapter 7 Open-Ended Tasks and Assessment: 131 The Nettle or the Rose David J. CLARKE Chapter 8 Using ICT to Improve Assessment 165 Marja van den HEUVEL-PANHUIZEN Angeliki KOLOVOU Marjolijn PELTENBURG Chapter 9 The Assessment for, of and as Learning 187 in Mathematics: The Application of SLOA Magdalena Mo Ching MOK Chapter 10 Building Bridges Between Large-Scale External 217 Assessment and Mathematics Classrooms: A Japanese Perspective Yoshinori SHIMIZU Chapter 11 Errors in Mathematics Assessment Items 237 Written by Pre-Service Teachers Jaguthsing DINDYAL Chapter 12 Affective Assessment in the Mathematics 257 Classroom: A Quick Start TAY Eng Guan QUEK Khiok Seng TOH Tin Lam Chapter 13 Implementing Self-Assessment to Develop 275 Reflective Teaching and Learning in Mathematics Lianghuo FAN Contributing Authors Chapter 1 Introduction: Assessment Matters WONG Khoon Yoong Berinderjeet KAUR This introductory chapter provides an overview of the subsequent chapters and relates them to significant issues about assessment in mathematics education relevant to Singapore and other countries. It ends with a set of reflective questions that teachers might think about while reading the chapters and thinking about making changes to their practices and beliefs about assessment. 1 Why a Yearbook on Assessment? Ordinary yearbooks, also called annuals, normally report significant events and statistics about the past year, but academic or scholarly yearbooks in education tend to be issue-based. These issues may or may not reflect discussion that has taken over the past year by members of that scholarly community, yet the issue is important enough for the community to produce a timely collection of articles to address the issue in depth. One of the longest running education yearbooks is the Yearbook of the National Society for the Study of Education (NSSE)1, which began publishing in 1901 and is now published under Teachers College, Columbia University. Its 2007 yearbook is entitled “Evidence and Decision Making” and it focuses on the roles that education professionals, including researchers, teachers, and administrators, can play “in constructing, interpreting, and using evidence to make decisions that support learning” (Moss and Piety, 2007, p.2), referring also to the 1 http://nsse-chicago.org/Yearbooks.asp 1 2 Assessment in the Mathematics Classroom “data-driven” movement. In mathematics education, the National Council of Teachers of Mathematics (NCTM) has published yearbooks since 1926, and the purpose is to “focus concerted attention on timely issues by viewing them in depth, from multiple perspectives, through interdisciplinary lenses, and across different grade bands”2. Its latest, the 72nd yearbook published in 2010, is about mathematics curriculum, but its chapter 6, by Peter Kloosterman and Crystal Walcott, is an example of using national assessment in the United States to investigate the link between what we teach and what the students learn. The NCTM’s 1993 yearbook was on assessment in the mathematics classroom, and this might have laid the groundwork for the assessment standards announced two years later (National Council of Teachers of Mathematics, 1995). The Association of Mathematics Educators has followed this trend of producing yearbooks based on major issues of interest to the Singapore teachers and educators. The first two yearbooks were about Mathematical Problem Solving (2009) and Mathematical Applications and Modelling (2010). This yearbook, the third one in the series, focuses on assessment. This theme follows naturally from the earlier two volumes because, having implemented problem solving, applications, and modelling in the classrooms, the teachers should then focus on assessing how well these attempts have been successful in improving mathematics learning for the students. Furthermore, there is a need to encourage teachers to include assessment of non-cognitive attributes and to use techniques in addition to standard paper-and-pencil tests that focus on typical problems. In fact, the first Mathematics Teachers’ Conference organised by the Association in 2005 was on assessment, and this yearbook can be considered an update on the same theme. In his keynote address delivered at that conference, Professor David Clarke discussed how international educators could implement assessment in many different ways to serve common visions. Readers will be pleased to read his chapter (7) to find out his latest view about the advantages and disadvantages of using open-ended tasks in mathematics assessment, six years after his earlier address. 2 http://www.nctm.org/catalog/productsview.aspx?id=98 Introduction: Assessment Matters 3 The authors of the 12 peer-reviewed chapters have been asked to focus on evidence-based practices that school teachers can experiment in their lessons to bring about meaningful learning outcomes. These chapters are organised into two major sections covering assessment in the cognitive and affective domains. Most of the authors explain practical ideas that the teachers can implement in their classrooms, but they also point to existing research of their own or in the literature to support these practices. Many examples are instructive, some are promising, while others hold potentials that need to be further developed and investigated. Our task here is to provide a roadmap to the chapters by briefly summarising the main points of the chapters and using them to raise related assessment issues not covered by the authors. This will widen the scope of coverage and analysis, hopefully to stimulate the readers to ponder and to form discussion groups to explore important issues based on the authors’ work. Of course, the readers can choose to start with any chapter they are particularly interested in. A recurring theme in most chapters is the widely circulated notions of formative assessment and assessment for learning. Michael Scriven (1967) was credited as the first educator to distinguish between formative evaluation and summative evaluation (some writers use assessment and evaluation interchangeably, while others distinguish between them), when he was focusing on evaluation of curriculum materials. The theoretical conception of formative assessment is further developed by Sadler (1989, 1998) and others. The evidence for formative assessment becomes well established through the comprehensive and widely cited review by Black and Wiliam (1998), and their Inside the Black Box series. Wiliam (2007) provides an updated review specially for mathematics assessment. According to Popham’s levels of implementing formative assessment (2009), teachers who make instructional adjustments after examining assessment-elicited evidence are at level 1, and teachers who train students to adjust their learning tactic after obtaining assessment results are at level 2. These and similar writings have led to the approach called assessment for learning (shortened to AfL or A4L in some writings). An internal review report about assessment by the Singapore Ministry of Education (MOE) in 2009 calls for a balance between assessment of learning with assessment for learning and points out the need to raise 4 Assessment in the Mathematics Classroom assessment literacy of its teachers and education officers. In the following sections, we will indicate how the chapters may contribute toward this important assessment goal. 2 Assessment of Mathematics Cognitive Domain The mathematics cognitive domain, as for all other school subjects, is multi-dimensional. Mathematics understanding is a term widely used in a vague and generic sense to refer to what students know about a piece of mathematics, including the skill or procedure component. When students make some mistakes in carrying out a procedure, teachers often claim that they have not understood the mathematics, failing to explicate what this means as well as confounding the distinctions between concepts and skills. This vagueness poses some problems for valid assessment of mathematical understanding. This has been a controversial issue at least since the publication of the original taxonomy of cognitive domain by Bloom (1956), but in mathematics education, the nature of mathematical understanding has been subject to extensive analysis, for example, it has been classified as instrumental and relational understanding, know what, know how, and know why, and other possibilities (e.g., Schoenfeld, 2008; Sierpinska, 1994; Skemp, 1979; Wong, 1985). In chapter 2, Thompson and Kaur propose the SPUR approach to unpack the multi-dimensions of understanding by focusing on Skills, Properties, Uses, and Representations. They provide examples of SPUR at the primary and secondary level, and teachers may be able to design similar tasks “to help students develop a well-rounded and balanced view of a given topic” (p. 25). The authors then report the results of a comparative study called the International Project on Mathematical Attainment (IPMA) of a sample of Singapore and US primary school pupils in these four dimensions by topics (Number, Algebra, Measurement, Geometry, Data/Chance). The relatively few items in IPMA covering Properties and Uses suggest that it is harder to design such items compared to Skills and Representations. This highlights an important area for teacher professional development. Thus, their chapter contributes to the ongoing discourse about the nature of understanding, an ubiquitous and all-embracing construct.