Mathematics Instruction: Goals, Tasks and Activities

Yearbook 2018, Association of Mathematics Educators

Pee Choon Toh, Boon Liang Chua

• Publisher: WSPC
• Year: 2018
• ISBN:
  9789813271685
• Topic: Matematica
• Subtopic: Matematica nella didattica

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The book, the tenth volume in the series of yearbooks by the Association of Mathematics Educators in Singapore, comprises 14 chapters written by renowned researchers in mathematics education. The chapters offer mathematics teachers a cache of teaching ideas and resources for classroom instruction. Readers will find various task design principles, examples of mathematical tasks used in classrooms and teaching approaches to implement the tasks. Through these discussions, readers are invited to reflect and rethink their beliefs about mathematics teaching and learning in the 21st century, and reexamine the tasks and activities that they use in the classroom, in order to bring about positive impact on students' learning of mathematics.

This book contributes towards literature in the field of mathematics education, specifically on mathematics instruction and the design of mathematical tasks and activities.

--> Contents:

• Tasks and Activities in the Mathematics Classroom (Boon Liang CHUA and Pee Choon TOH)
• From Task to Activity: Noticing Affordances, Design, and Orchestration (CHOY Ban Heng)
• Affordances of Typical Problems (Jaguthsing DINDYAL)
• Mathematical Tasks Enacted by Two Competent Teachers to Facilitate the Learning of Vectors by Grade Ten Students (Berinderjeet KAUR, Lai Fong WONG and Chong Kiat CHEW)
• Use of Comics and Its Adaptation in the Mathematics Classroom (TOH Tin Lam, CHAN Chun Ming Eric, CHENG Lu Pien, LIM Kam Ming and LIM Lee Hean)
• Designing and Implementing Scientific Calculator Tasks and Activities (Barry KISSANE)
• Engaging the Hearts of Mathematics Learners (Joseph B W YEO)
• Developing Interaction Toward the Goal of the Lesson in a Primary Mathematics Classroom (Keiko HINO)
• Designing and Implementing Activities in the Flipped Classroom in the Singapore Primary Mathematics Classroom (CHENG Lu Pien, NG Swee Fong, TAN Bee Kian Jasmine Susie and NG Ee Noch)
• Designing Mathematical Modelling Activities for the Primary Mathematics Classroom (Chun Ming Eric CHAN, Rashidah VAPUMARICAN and Huanjia Tracy LIU)
• Extending Textbook Exercises into Short Open-Ended Tasks for Primary Mathematics Classroom Instruction (YEO Kai Kow Joseph)
• Integrating Problem Posing into Mathematical Problem Solving: An Experimental Study (JIANG Chunlian and CHUA Boon Liang)
• A Vicennial Walk Through 'A' Level Mathematics in Singapore: Reflecting on the Curriculum Leadership Role of the JC Mathematics Teacher (Weng Kin HO and Christina RATNAM-LIM)
• Probability: Theory and Teaching (YAP Von Bing)

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Readership: Graduate students, researchers, practitioners and teachers in mathematics.
-->Keywords:Mathematics;Instruction;Task Design;Singapore;Teachers;InstructionReview: Key Features:

• Firstly it has a focused theme: Mathematics instruction and task design, which is of prime concern to mathematics educators
• Secondly it is written by university scholars who work closely with classroom mathematics teachers thereby drawing on their research knowledge and classroom experiences
• Lastly, the book is rich resource, of tried and tested practical know-how of approaches that promote mathematics learning, for mathematics educators in Singapore schools and elsewhere

Chapter 1

Tasks and Activities in the Mathematics Classroom

Boon Liang CHUA Pee Choon TOH

This introductory chapter provides an overview of the chapters in the book. The chapters are classified into three parts. The first part considers different sets of design principles and affordances of mathematical tasks. The second part is a showcase of tasks and activities developed by various mathematics educators towards the goal of helping students learn effectively. The third part emphasises the important role that ‘A’ level mathematics teachers play in implementing mathematical tasks.

1 Mathematics Instruction: Goals, Tasks and Activities

This yearbook of the Association of Mathematics Educators in Singapore invites mathematics teachers to reflect and rethink their beliefs about mathematics teaching and learning in the 21^st century to improve mathematics instruction in the classroom. Like the previous yearbooks such as Developing 21^st Century Competencies in the Mathematics Classroom (Toh & Kaur, 2016), and Empowering Mathematics Learners (Kaur & Lee, 2017), the theme of this present yearbook is shaped not only by the school mathematics curriculum developed by the Ministry of Education and the needs of mathematics teachers in Singapore schools, but also the current international trend in mathematics education.

In recent years, there is a growing interest amongst mathematics educators in task design. The growth of attention to task design is not surprising at all when viewed through the lenses of practical, cognitive and cultural perspectives. From the practical perspective, mathematical tasks are the bedrock of mathematics lessons in many countries (Watson & Ohtani, 2015). Mathematics classroom instruction is normally organised and delivered through the activities in mathematical tasks developed by mathematics teachers or found in curriculum materials. In other words, mathematics instructions are information guiding students how to carry out the activities in the mathematical task. They may take different forms, but typically as a series of easy-to-follow steps and guiding questions that prompt students to explore and discover the mathematical concepts embodied in the task. To examine classroom instructions, it is common to look at and analyse the amount of lesson time that students spend on doing the task (Shimizu, Kaur, Huang, & Clarke, 2010). For instance, the Year 8 students in all seven participating countries in the TIMSS 1999 Video Study were found to have spent at least 80% of their time in mathematics lessons working on mathematics tasks (Hiebert et al., 2003). Watson and Ohtani (2015) also point out, from the cognitive viewpoint, that the goal and content of a mathematical task are important and can have significant effect on students’ learning. In a similar way, they highlight the cultural aspect of mathematical tasks when they remark that mathematical tasks can shape students’ learning experiences of mathematics and their understanding of the nature of mathematical activity.

2 Mathematical Tasks vs Mathematical Activities

Mathematical tasks are crucial vehicles in the classroom for enhancing students’ mathematical thinking and reasoning (Stein, Grover, & Henningsen, 1996; Watson & Ohtani, 2015). The phrase “mathematical task” is often used alongside “mathematical activity” by many authors in the mathematics education literature, but the distinction between them is rarely noticed and discussed by the authors.

According to Watson and Sullivan (2008), mathematical tasks refer to the questions, situations and instructions that are accessible to students in the mathematics lessons. So a task could range from a mathematics textbook exercise, or an examination question, to an exploration using the guided-discovery approach. Tasks such as these, regardless of their cognitive demands, embody mathematical ideas and trigger activities which then offer students opportunities to encounter mathematical concepts, skills and processes (Mason & Johnston-Wilder, 2006; Watson & Ohtani, 2015). Consider the factorisation task involving the Multiplication Frame method in Chua (2017). The learning goal of the task for students is to master the technique of factorising quadratic expressions of the form ax² + bx + c using the Multiplication Frame method. However, the activities within the task that students have to engage with include examining how the terms are positioned in the frame and establishing the relationship between the pair of quadratic term and the constant, and the pair of linear terms in the frame. Hence, mathematical tasks are different from mathematical activities. In Chapter 2 of this book, From Task to Activity: Noticing Affordances, Design, and Orchestration, Choy draws the same distinction between mathematical tasks and mathematical activities, and argues why such a distinction is crucial.

In the following sections, 13 peer-reviewed chapters resulting from the keynote lectures and workshops from the Mathematics Teachers Conference 2017 will be introduced. These chapters are classified broadly into three parts. In the first part are two chapters who offer sets of principles for designing mathematical tasks so as to provide meaningful learning experiences to students.

The second part of the book comprises nine chapters that showcase mathematical tasks and activities developed by educators to facilitate effective mathematics learning in the classrooms. The content of these chapters varies widely in the choice of teaching approaches, ranging from flipped classroom and the use of comics to mathematical modelling and problem posing.

In the third part that consists of two chapters, the role of ‘A’ level mathematics teachers in task design and implementation is discussed. The authors call for mathematics teachers to deepen their understanding of curriculum policy and mathematics content so that they can enrich the learning experiences of their Grades 11 and 12 students. The following three sections provide summaries of each chapter, specifically highlighting how the chapter addresses the theme of the book, Mathematics Instruction: Goals, Tasks and Activities.

3 Task Design Principles and Affordances of Mathematical Tasks

In the second chapter From Task to Activity: Noticing Affordances, Design, and Orchestration, Choy observes that very few teachers perceive task and activity as totally different things and illustrates the difference between the two using an example of a task on investigating the graph of a trigonometric function using graphing software. He also points out how important it is for mathematics teachers to notice the mathematics embedded in a task and to think about how to use the task to orchestrate meaningful learning experiences for their students. He then introduces the MAD (Mathematics, Activity and Documentation) framework and follows by demonstrating how to orchestrate discussions in the mathematics lessons using the five practices developed by Stein, Engle, Smith, and Hughes (2008). Additionally, Choy draws a vignette to show how a typical examination question can be used beyond the purpose of honing procedural skills to engage students in mathematical thinking about the concept.

In Chapter 3, Affordances of Typical Problems, Dindyal offers a description of typical problems as mathematical tasks that are readily available and are often used to develop procedural skills. He holds the view that the affordances of typical problems for developing conceptual fluency can only be realised when mathematics teachers are able to modify them in various ways. He then suggests eight strategies, with examples, to illustrate how typical problems can be modified and then used productively in mathematics lessons.

4 Mathematical Tasks and Activities for Effective Learning

In Mathematical Tasks Enacted by Two Competent Teachers to Facilitate the Learning of Vectors by Grade Ten Students (Chapter 4), Kaur, Wong, and Chew describe how two competent teachers, each with 20 years of teaching experience, enacted mathematical tasks to facilitate the learning of vectors. The teachers used similar types of tasks to achieve the goals of their lessons. But their tasks differed in context and cognitive demands due to different students’ interest and ability. The teachers also hold different beliefs in how students should learn. One believes in mastery by practice after the mathematical concept is properly understood whereas the other believes in both developing understanding and building skills for the topic. Kaur et al. emphasise the important role that mathematics teachers play when they enact the mathematical tasks so as to support meaningful connection amongst concepts, procedures and contexts, and to provide opportunities for students to engage in mathematical reasoning and problem solving.

Chapter 5, Use of Comics and Its Adaptation in the Mathematics Classroom, by Toh, Chan, Cheng, Lim, and Lim describes the case of two mathematics teachers from a secondary school co-teaching the topic of Percentages using comics. The teachers were mindful of the instructional goals and adapted the tasks in the comics teaching package developed by Toh et al. accordingly to enhance students’ conceptual fluency as well as procedural fluency.

In Designing and Implementing Scientific Calculator Tasks and Activities (Chapter 6), Kissane considers the educational potential of scientific calculators in mathematics learning, including offering different representations of the same result, allowing computation, encouraging exploration, and seeking affirmation. He points out that for calculator tasks to achieve maximum educational value, providing teachers with adequate guidance on the implementation of the tasks is crucial. The teacher guide can provide information such as the answers to the tasks, the nature and purpose of the task, and suggested classroom organisation and time needed to complete the task. Examples of calculator tasks, with analysis of some of them, are provided to show mathematics teachers how worthwhile tasks can be designed.

In Engaging the Hearts of Mathematics Learners (Chapter 7), Joseph Yeo B. W. introduces the LOVE Mathematics framework (Linking Opportunities in a Variety of Experiences to the learning of mathematics) to illustrate how students can be engaged in mathematics lessons through three principles: namely, variety, opportunity, and linkage. Mathematics activities incorporating the use of amusing mathematics videos, catchy mathematics song, witty mathematics comics, puzzles and games are described to demonstrate how mathematics lessons can become more engaging.

In Chapter 8, Developing Interaction Toward the Goal of the Lesson in a Primary Mathematics Classroom, Hino maintains that the goal of the lesson that a mathematics teacher has in mind shapes the mathematics instruction in the classroom. Drawing evidence from classroom interactions in two Grade 5 mathematics lessons on comparing fractions taught by an experienced teacher, she illustrates how being clear about the goal of the lesson helped the teacher not only engage his students in meaningful interactions but also manage various students’ responses. For instance, the mathematics teacher, other than guiding his students to develop conceptual understanding through questioning, also paid particular attention to their choice of methods as well as use of mathematical language and notations.

Chapter 9 by Cheng, Ng, Tan, and Ng, Designing and Implementing Activities in the Flipped Classroom in the Singapore Primary Mathematics Classroom, describes a teaching model involving the flipped classroom approach that was used by a group of Primary 5 mathematics teachers to teach the topic of triangles. The teachers’ involvement in the development of mathematical tasks for the teaching package has benefitted them greatly. Cheng et al. describe both the benefits gained and challenges encountered by the teachers in the chapter.

The next chapter, Designing Mathematical Modelling Activities for the Primary Mathematics Classroom (Chapter 10), by Chan, Vapumarican, and Liu, offers those who are new to mathematical modelling a brief description of the Model-Eliciting Activities perspective, which involves complex, open, and non-routine modelling tasks situated in a real-world context to allow students to exercise both informal and formal mathematical knowledge interactively. Chan et al. present the design principles and exemplify the principles through two examples, one on recommending a suitable phone plan and the other on modelling the spread of mosquito borne diseases.

Yeo K. K. Joseph offers two strategies to modify and extend closed textbook exercises typically used in primary schools into short open-ended tasks in the chapter Extending Textbook Exercises into Short Open-Ended Tasks for Primary Mathematics Classroom Instruction (Chapter 11). He also discusses the benefits of implementing open-ended tasks and the implications for teaching and learning from using such tasks.

The last chapter in the second part of the book, Integrating Problem Posing into Mathematical Problem Solving: An Experimental Study (Chapter 12) by Jiang and Chua, explores the use of the what-if-not strategy to encourage students to pose problems in an experimental study involving 56 Grade 7 students in Macao. The study, conducted over four lessons on the topic of solving simultaneous linear equations in two variables, involved a pre-test and a post-test. The results show that students exposed to the what-if-not strategy performed better in both problem solving and problem posing items.

5 Role of ‘A’ Level Mathematics Teachers

The two chapters in this third part of the book underscore the role of teachers in selecting and designing mathematics tasks for use in classroom instruction. In Chapter 13, A Vicennial Walk Through ‘A’ Level Mathematics in Singapore: Reflecting on the Curriculum Leadership Role of the JC Mathematics Teacher, Ho and Ratnam-Lim call for teachers to be curriculum leaders. By curriculum leaders, they expect teachers and heads of department to take on the responsibility of understanding the purpose for changes in the syllabus, the shifts in educational orientations, and how these are then translated into the scope and sequence of teaching and learning experiences and assessments. By assuming the role of an active curriculum leader, they believe that teachers are then able to formulate goals for the lessons, design mathematical tasks and implement classroom activities to bring about a more enriching learning experience for ‘A’ level students.

Chapter 14 by Yap, Probability: Theory and Teaching, is the final chapter of this yearbook. Yap, a statistician, was invited to lecture on a mathematical topic that many teachers find difficult to teach. This chapter explains briefly Kolmogorov’s axioms and explores three different meanings of probability, paying particular attention to the frequency interpretation. Yap provides a list of ten problems and their solutions, with the aim of strengthening the appreciation of the frequency approach in probability.

6 Concluding Remarks

Mathematical tasks are important vehicles for classroom instruction to bring about positive impact on student learning. The relationship between mathematical tasks, teaching and learning needs to be managed carefully by the teachers. Otherwise, gaps may appear between what the teacher on one side intends and what students on the other side perceive.

The chapters in this yearbook provide readers and specifically classroom teachers with a cache of resources to help them implement ma...