Primary mathematics : extending knowledge in practice

141 Pages·2008·2.174 MB·English

by  Alice Hansen, Dawsonera

Extending
Knowledge in Practice
Primary
Mathematics

Extending
Knowledge in Practice
Primary
Mathematics
Alice Hansen
Firstpublishedin2008byLearningMattersLtd.
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Contents
The author vi
Introduction 1
1
Using and applying mathematics 5
2
Counting and understanding number 18
3
Knowing and using number facts 32
4
Calculating 48
5
Understanding shape 67
6
Measuring 83
7
Handling data 99
Objectives Index 115
References 123
Index 131

v

The Author
Alice Hansen is a Principal Lecturer in Primary Mathematics Education at the
University of Cumbria. She has taught extensively at primary level in England
and abroad. Alice has particular interests in how children construct geometrical
definitions,thedesignofmathematicaltasksandtheuseofICTtoenhancemathe-
matics teaching andlearning.

vii

Introduction
This book hasbeen designedto help youidentifyhowyourmathematicssubject
knowledge and pedagogical knowledge support each other in order for you to
become a better teacher of mathematics. Through case studies, a number of
mathematical aspects will be considered. These cover some of the Early
Learning Goals (DfES, 2007c) and a range of Programmes of Study in Key
Stages 1 and 2 of the National Curriculum for Mathematics (DfEE, 1999).
However, they are arranged in chapters according to the strands of the Primary
FrameworkforTeachingMathematics(DfES,2006).Thesecasestudiesaredrawn
from a wide range of year groups, from working with young children in nursery
settings to working with more able children in Year 6. Other case studies focus
specifically on relevant and interesting research that covers an important part of
children’smathematicaldevelopment.Accompanyingeachcasestudyisadiscus-
sion aimed at developing your pedagogical and subject knowledge while it
highlights the subject knowledge that the teacher requires in order to teach the
objectives. Where appropriate, consideration is also given to how the teacher’s
subject knowledgedevelopedwhile working withthechildren.
Althoughthebookissetoutinthisway,itisessentialtoconsidercasestudiesfrom
yeargroupsotherthantheone(s)youaredirectlyplanningfor.Inmostcasesthe
discussionthatensuesrelatestoawideraudiencethantheteachersofthespecific
yeargroupinthecasestudyandthereforetheideasthatyoufindthroughoutthe
chapterscanbeadaptedformostattainmentlevels.
Throughout the book you will see a number of cross references, shown by this
symbolinthemargin.Thesewillhelpyoutofindotherplacesinthebookwhere
similaraspectsarebeingdiscussed.Thehighnumberofcrossreferencesdemon-
strateshowtheareasoftheEarlyYearsFoundationStage,NationalCurriculumand
the Primary Framework for Teaching Mathematics are interrelated, and how you
cannot effectively teach one objective in isolation. One way to be able to make
meaningful links between thes e areas of mathematics is to have good subject
knowledge.
Inadditiontothecasestudiesandthediscussion,youwillfindstickynoteremin-
ders scattered throughout the chapter. These are useful prompts for your own
subject knowledge, links to curriculum content or related reading/research for
you to follow up if you require more information. There are also links to further
readingtosupportyourpedagogicalknowledgeandyoursubjectknowledgeofthe
aspectof mathematics being discussed.
This book aims to help you develop your mathematical subject knowledge and
pedagogicalknowledgesothatyourconfidenceinteachingmathematicsdevelops,
thusimprovingthelearningandmathematicaldevelopmentofthechildreninyour
care.
1
Introduction
What is the difference between mathematical
subject knowledge and mathematical pedagogical
knowledge?
Shulman (1986, 1987) defined three categories of teacher knowledge on which
manyhavebasedtheirresearchintomathematicalknowledge.Theseare:subject
matter content knowledge (the amount and organisation of knowledge per se in
themindoftheteacher),pedagogicalcontentknowledge(thatgoesbeyondknowl-
edge of subject matter per se to the dimension of subject matter knowledge for
teaching) and curricular knowledge (the full range of programs and materials
designed for the teaching of particular subjects and topics at a given level)
(Shulman, 1986,p9).
Many others have also considered the knowledge that teachers require to teach
mathematics (Morris, 2001). For example, Ball and Bass (2000) identified the
connections between mathematics content knowledgeand teaching in what they
termed ‘mathematics knowledge for teaching’. They explained how this involved
two elements, both ‘common’ knowledge (which any educated adult should
possess)and‘specialised’knowledge(theknowledgeofteachers)(Balletal.,2005).
Whatis clearfrom theresearchisthatthereare a numberof elements thatinter-
relate to enable teachers to effectively develop children’s mathematical
understanding. Whatever the labels different researchers assign to these, within
this book the edges of the elements have been intentionally blurred in order to
discussthemostappropriateaspectswithinthechaptersmoreeffectively,without
being constrainedby discrete(and sometimesartificial) groupings.
Do I really need good subject knowledge to teach
mathematics well?
Forthelast25yearsthesubjectknowledgeofteachersandstudentteachershas
been a focus of government and initial teacher educators and it has recently
becomeembedded within the Professional Standards for Teachers (TDA, 2007a).
Indeed,ifteachinginvolveshelpingotherstolearn,thenunderstandingthesubject
contenttobetaughtisafundamentalrequirementofteaching(Aubrey,1997,p3).

Within initial teacher education the current focus on developing students’ subject
knowledgeisevident.Forexample,itisarequirementforQualifiedTeacherStatus
that student teachers pass their numeracy skills test (TDA, 2007a) and the TDA
currently funds subject knowledge booster courses for students entering PGCE
teacher training (TDA, 2008).
ResearchbyRowlandetal.(2001)demonstratedthatsecureknowledgeofmathe-
matics (including and extending beyond the areas of the primary curriculum) is
stronglylinkedwithmorecompetentmathematicsteachinginstudentteachers.In
addition to this, they found that poor subject knowledge was connected to less
competentteaching.
Ball et al. (2005, p16) remind us that although many studies demonstrate that
teachers’mathematicalknowledgehelpssupportincreasedstudentachievement,
theactualnatureandextentofthatknowledge–whetheritissimplybasicskillsat
2