Developing Fractions Knowledge

Amy J. Hackenberg, Anderson Norton, Robert J Wright

• Publisher: SAGE Publications Ltd
• Year: 2016
• Print ISBN: 9781412962209, 9781412962193
• eBook ISBN: 9781473987616
• Edition: 1
• Topic: Education
• Subtopic: Teaching Mathematics

Supporting and understanding your students' fractional knowledge is crucial to their overall grasp of numbers and mathematics. By centralizing around three key stages of development, this effective guide will help you to assess your students' understanding of fractions and modify your teaching accordingly.

These key stages are identified as: 

• Stage 1a: Fair Sharing
• Stage 1b: Part-Whole
• Stage 2a: Disembedding and IteratingStage 2b: Measuring with Unit Fractions
• Stage 2c: Reversing Fractions
• Stage 3a: Fractions as Numbers
• Stage 3b: Operating with Fractions

As the newest addition to the bestselling Maths Recovery Series, this book will be a useful guide for all primary classroom teachers and assistants, including experienced Mathematics Recovery instructors.

1 Professional Learning and Three Grand Organizers for Arithmetic Instruction

Chapter 1 outlines ways in which the book can be used to support professional learning focused on strengthening instruction in arithmetic. In order to demonstrate this, three scenarios of professional learning will be included: (i) a mathematics coach working one-on-one to support a teacher in developing an instructional plan for her class; (ii) a school-based mathematics leader working with a team of teachers to revise their school’s mathematics programme; (iii) a district-wide Math Recovery® Leader meeting with a team of intervention teachers to develop specialist knowledge aimed at advancing low-attaining students’ fractions knowledge. Chapter 1 will also describe links from this text to major generic topics appearing in the current Mathematics Recovery Series, including guiding principles for instruction, domains of arithmetic knowledge and dimensions of mathematizing, such as complexifying, distancing the instructional setting, formalizing, generalizing, notating and unitizing.

Scenario 1: School-Based Mathematics Leader

Mr Phillips is a school-based mathematics leader working with a team of teachers to revise aspects of their school’s mathematics programme. The Professional Learning Team (PLT) plans to meet initially for two hours, and then for one hour per week, for a period of six weeks to focus on the teaching of fractions in 4th and 5th grade. Their goal is to develop an instructional programme that (a) is more focused on arithmetical content than their current programme; and (b) enables teachers to take better account of students’ current levels of fractions knowledge. Two members of the team have already developed a schedule of assessment tasks drawn from Chapters 3–6, and have administered the schedule individually to each of eight students covering a wide range of attainment levels. At their first meeting the team will review the video recordings of the assessment interviews and develop a simple means of coding students’ responses to each of the assessment tasks. Extrapolating from the results of the assessment interviews, they will develop a set of 12 lessons to be taught over a six-week period, based on Chapter 4 (Fragmenting), Chapter 5 (Part–Whole Reasoning), Chapter 6 (Measuring with Unit Fractions) and Chapter 7 (Reversible Reasoning). Instructional plans for each lesson include: (a) a description of the main topic for the lesson; (b) instructional materials; (c) worksheets of relevant exercises incorporating a progression of difficulty; (d) the range of student responses to relevant assessment tasks; and (e) descriptions of common errors, difficulties and misconceptions. For each lesson, the instructional plan includes a 10-minute segment allowing for intensive, targeted intervention with a group of up to six students. This segment focuses on helping students to develop and consolidate their knowledge of the earlier topics of Fragmenting (Chapter 4) and Part–Whole Reasoning (Chapter 5).

Scenario 2: District-Wide Math Recovery® Leader

Ms Gomez is a district-wide Math Recovery® (MR) Leader. In the last two years she has trained two cohorts of 12 MR Intervention Specialists who work in schools across the district. This training has focused on teaching whole number arithmetic across grades K–4 and has involved the following three phases: Phase 1 consists of an initial professional learning programme of three to five days focusing on assessing and profiling students’ arithmetical knowledge. Up to 12 students judged as likely to benefit from intensive intervention are individually administered a schedule of assessment tasks. The schedule of tasks has been developed from the sets of assessment tasks in Chapters 3–9. Each teacher’s pre-assessment interviews are video-recorded for later analysis. Phase 2 consists of selecting up to four students, each of whom is taught individually for up to five 30-minute sessions per week, for teaching cycles of 12–15 weeks. Phase 2 also includes three or four on-going professional learning sessions during the period of the teaching cycle. Phase 3 involves administering one-on-one post-assessments to the 12 students who underwent the pre-assessment. Each teacher routinely video-records all of their teaching sessions as well as their pre- and post-assessment interviews. In all of the professional learning sessions, the participating teachers present case studies highlighting students’ arithmetical strategies and progressions in learning. More detailed descriptions of the professional learning programme described above are available (Wright, 2000, 2003, 2008). This year Ms Gomez will again train two cohorts of Intervention Specialists, but for the first time one cohort will focus on the teaching of fractions at the 5th and 6th grade. In working with this new cohort, Ms Gomez will adopt the year-long professional development model that she has used for several years with other cohorts of teachers in her district.

Scenario 3: Mathematics Coach

Ms Liang is a mathematics coach working one-on-one to support a 5th grade teacher (Ms Koppel). The focus of the coaching is to use video-recorded, one-on-one assessments to advance Ms Koppel’s knowledge of fractions pedagogy. Ms Liang meets with Ms Koppel to discuss their working together for a two-week period during which they will develop, administer and review a short schedule of assessment tasks adapted from the assessment tasks in Chapters 5 and 6. These tasks will be designed to assess students’ mental actions relating to the fractions topics of partitioning and iterating. Using the assessment schedule, Ms Liang conducts one-on-one, video-recorded assessments with five students from Ms Koppel’s class. The five students are selected as representative of a wide range of attainment levels in the learning of fractions. The purposes of the assessments are (a) to inform both coach and teacher of the range and nature of student responses to the assessment tasks and (b) to induct Ms Koppel into the process of using one-on-one assessments to gauge students’ current levels of fractions knowledge. In their next meeting they review the video records using the assessment schedule to note students’ responses to the tasks. Ms Liang takes the opportunity to highlight the interactive and inquiring nature of the assessment interview. The next day, Ms Koppel conducts one-on-one assessment interviews with a similar group of five students and in their next meeting they use the assessment schedule to review the video records of the second group of students, noting students’ responses. As well, Ms Liang reviews Ms Koppel’s conducting of the assessment interviews, noting the extent to which Ms Koppel successfully elicits valuable information about the fractions thinking of her students.

Three Grand Organizers for Arithmetic Instruction

In professional learning work with teachers focusing on whole number arithmetic, we developed three grand organizers (see Wright et al., 2006a, 2006b, 2012, 2015). These are: Guiding Principles for Instruction; Domains of Arithmetic Learning; and Dimensions of Mathematizing. In the following sections these grand organizers are described and extended to reasoning and arithmetic involving fractions.

1 Guiding Principles for Instruction

The authors of this book have conducted an extensive range of research and development projects focusing on mathematics pedagogy. Many of these projects involved working in close collaboration with teachers, schools and school systems to advance mathematics instruction. Below we set out nine guiding principles of mathematics instruction which aptly summarize the approach to fractions pedagogy that we advocate. In our collaborative research and development work, these principles have been applied extensively to guide the teaching of mathematics.

1. The teaching approach is inquiry-based, that is, problem-based. Students routinely are engaged in thinking hard to solve fractions problems that for them are quite challenging.
2. Teaching is informed by an initial, comprehensive assessment and on-going assessment through teaching. The latter refers to the teacher’s informed understanding of students’ current knowledge and problem-solving strategies, and continual revision of this understanding.
3. Teaching is focused just beyond the ‘cutting-edge’ of students’ current knowledge.
4. Teachers exercise their professional...

Table of contents

• Cover
• Half Title
• Publisher Note
• Title Page
• Copyright Page
• Acknowledgements
• Brief Contents
• Contents
• Illustration List
• Table List
• About the Authors
• Acknowledgments
• Series Page
• Introduction
• 1 Professional Learning and Three Grand Organizers for Arithmetic Instruction
• 2 From Whole Numbers to Fractions
• 3 Assessing Stages of Units Coordination
• 4 Teaching Students at Stage 1: Fragmenting
• 5 Transitioning to Stage 2: Part–Whole Reasoning
• 6 Teaching Students at Stage 2: Measuring with Unit Fractions
• 7 Teaching Students at Stage 2: Reversible Reasoning
• 8 Teaching Students at Stage 3: Fractions as Numbers
• 9 Teaching Students at Stages 2 and 3: Equal Sharing of Multiple Items
• 10 Teaching Students at Stages 2 and 3: Multiplying Fractions
• 11 Teaching Students at Stages 2 and 3: Adding and Subtracting Fractions
• 12 Teaching Students at Stages 2 and 3: Dividing Fractions
• 13 From Fractions to Algebra
• Glossary
• Appendix: Templates Marked and Unmarked Fraction Strips
• References
• Index