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Subject Knowledge

Development and

Didactics Journal for initial teacher training in

Mathematics

at Liverpool John Moores University

Name

Year of Study 2019/20

Subject Tutor Paul Killen

Liaison Tutor

Version 1.2 Secondary ITE Programmes Liverpool John Moores University

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Definitions1

ubject

“A branch of knowledge studied or taught in a

school, college, or university.”

nowledge

“Facts, information, and skills acquired through

experience or education; the theoretical or

practical understanding of a subject.”

“The sum of what is known.”

“True, justified belief; certain understanding, as

opposed to opinion.”

“Awareness or familiarity gained by experience of a fact

or situation.”

evelopment

“The process of developing or being developed2.”

“A specified state of growth or advancement.”

“An event constituting a new stage in a changing

situation.”

and

idactics

“The science, art, or practice of teaching.”

From the Greek didaktikós “skilled in teaching”.

Related to Pedagogy, which has its roots in methods and

practice of learning and teaching, with a focus on the

learner (derived from the Greek ἄγω "I lead", and παῖς "a

child")

1 From https://en.oxforddictionaries.com 2 Develop (def.) “Grow or cause to grow and become more mature, advanced, or elaborate.”

S K

D

D

https://en.oxforddictionaries.com/


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Table of Contents

Section 1 Principles for SKDD

Section 2 Guidelines for SKDD Journaling

Section 3 Subject Fundamentals

Section 4 Curricular Frameworks

Section 5 SKDD Activities

Section 6 References and Bibliography

section one

Principles for Subject

Knowledge

Development and

Didactics

Teachers must… “Demonstrate

good subject and curriculum

knowledge… have a secure

knowledge of the relevant

subject(s) and curriculum areas,

foster and maintain pupils’

interest in the subject, and

address misunderstandings.”

(DFE, 2011)


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Teachers’ Standards, and in particular S3, require that trainee teachers demonstrate a

high level of subject knowledge and pedagogy. Subject knowledge is an essential

component of effective teaching, alongside effective pedagogical / didactic skills,

and contextual knowledge of learners, schools and policy.

Subject knowledge is complex and trainee teachers develop it in five key ways:

Trainees’ prior knowledge (degree and industrial experience);

Pre-course subject knowledge enhancement, including self-directed activity and

SKE courses in shortage subjects set as a condition of entry;

In-course subject knowledge development/enhancement and ‘auditing’ in

university;

School-based mentoring and teaching practice, whilst on placement, alongside a

subject specialist teacher;

Trainees’ personal responsibility to undertake self-directed object knowledge

development to expand and deepen knowledge and skill (outside of university

and placement).

Professional Teacher Knowledge

Banks, Leach and Moon (2005) introduced graphic model

Figure 1) to support the development of reflection in Initial Teacher Education (ITE).

Describing teacher professional knowledge as: subject knowledge, pertaining to

disciplinary content knowledge, school knowledge, pertaining to localised policy and

practice (including curriculum, behaviour, etc.), and institutional culture (i.e. what it

means to be a teacher in a specific context). Furthermore pedagogical knowledge,

pertaining to methods for learning, teaching, and assessment. These are not separate

areas, or spheres, of teacher knowledge, but interrelated aspects of teacher

knowledge that converge to develop a ‘personal subject construct’ (PSC); which could

be defined as the individual teacher’s vision for teaching and learning their subject, as

influenced by their knowledge of their subject, the school context and pedagogical

approaches.


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Figure 1

section two

Guidelines for Subject

Knowledge

Development and

Didactics Journaling

SKDD Journaling is an ongoing

process, and disposition of reflection

and self-evaluation, focusing on and

prioritising your subject knowledge

developed during your initial teacher

education and beyond. Curricula

change periodically and knowledge

within subjects can evolve over time,

with emphasis and priorities being

influenced by policy, society and

technology.


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Trainees and mentors should regularly review subject knowledge development and

identify priority areas for development of subject knowledge for teaching. The subject

‘fundamentals’, in section three, provide a broad framework for discussion around

subject knowledge, and aim to provide flexible and inclusive categories that cover

much of the knowledge that you will need to teach your subject.

Subject knowledge should be discussed with your subject-specialist, school-based ITT

Mentor, in weekly meetings, as part of the wider dialogue around your training and

development. Appropriate areas for development will include:

Gaps in your subject knowledge for content that is new to you;

Content that you have not studied for some time and need to refresh your

knowledge;

Content that you are going to be teaching in the next Phase of your training;

Areas of strength, where you need to break down your high-level/expert

knowledge into component parts to introduce it to learners for the first time;

Challenging your preconceptions and assumptions about what pupils

know/understand and motivations within your subject;

Identifying and exploring misconceptions and complex concepts/principles;

Figure 2 Pyramid model of subject knowledge

During your training, reflect on your developing subject knowledge for teaching and

make notes of where you have done so, setting targets for linked to short, medium and

long term planning for the lessons that are timetabled to teaching. When you are

making notes in relation to the subject ‘fundamentals’ (section three), it is helpful to

include:

The date of the entry (and subsequent updates);

The specific aspect(s) of content knowledge that you focused on within the

‘fundamental’;

The class(es) that you developed the knowledge for and/or why you targeted

this for development;

Indicate how you went about developing the knowledge;

Subject knowledge

High-level subject knowledge: deep

knowledge in a narrower range of content.

Can lead to assumptions about learners’

knowledge and engagement, and need to

be broken down into ‘component’ parts

(concepts, principles, etc.).

Low-level subject knowledge: broad

range of content knowledge across the

range of the subject.


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Note the strategies/approaches you adopted (i.e. pedagogy/didactics) and the

impact they had on pupils’ learning;

Your next steps and targets for further development;

The SKDD Journal is your notebook, recording what subject knowledge you have

developed and how you have gone about it. Notes can include any additional and

relevant information, including concept maps (Figure 3) photographs or scanned

images of examples of your own or pupils’ work (Figure 4).

Figure 3 Example concept map from Geography

Figure 4 Example image from Art and Design


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section three

Subject

Fundamentals

The areas below describe broad,

‘umbrella’ areas of conceptual and

procedural knowledge fundamental

to the subject. You will identify specific

content knowledge and skills for each

category, and write a narrative of your

subject knowledge development

during your initial teacher education,

setting targets appropriate to the

stage of your training and needs (see

section two, above). Specific lists of

content knowledge can be found in

the relevant curricular frameworks (see

section four, below).


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Fundamental 1: Number and Operations for addition & subtraction

Understand the relative place value of numbers, directed numbers and decimals.

the use of a number line and empty number line; estimation & rounding;

The operations of addition and subtraction, including the algorithms for addition and

subtraction of whole numbers.

Notes on my development of subject knowledge for teaching:

Fundamental 2: Operations for division & subtraction

The operations of multiplication and division, including the algorithms for multiplication

and division of whole numbers.

Highest common factors (HCF), multiples and lowest common multiples (LCM);

Arrays; the orders of operation.

of area;

calculation of the mean and problem solving. Notes on my development of subject knowledge for teaching:


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Fundamental 3: Partitioning

Understand how to manipulate and calculate with representations of the partitioning of a

whole, including fractions, decimals, percentages and ratio.

Use of resources to support number acquisition.


Fundamental 4: Number Structures

Understand, classify, manipulate and calculate with number structures, including: even

and odd numbers; prime numbers; powers and roots; surds; rational and irrational

numbers; terminal and recurring decimals and rounding.



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Fundamental 5: Measure

Relationship between metric & imperial measure. Conversion of units. Properties of 2D &

3D shapes. Derivation of formulae for area & perimeter (2D) surface area & volume (3D). Pythagoras’s Theorem.


Fundamental 6: Algebra

Understand how to create and manipulate algebraic expressions; solving equations

(linear & quadratic), including rearranging equations, manipulating algebraic structures

such as brackets, factorising, algebraic fractions, forming and solving equations,

inequalities, functions, the transformation of functions, and algebraic proof.



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Fundamental 7: Trigonometry

Definition of parallel and perpendicular lines, their properties and associated angles, angles

and angle facts, definition and classification of shapes, angle properties of shape, angle

properties of circles, trigonometry for right angled triangles (sine, cosine, tangent) and sine

and cosine rules for non right angles triangles (including proof), the transformation of shapes,

vectors, and other geometric proof.


Fundamental 8: Probability

Understand how to represent outcomes and calculate probabilities, including theoretical and

experimental probability, listing outcomes and sample spaces, conditional probability, Venn

diagrams, intersection, union and set notation, combined events, and tree diagrams.

Independent and mutually exclusive events.



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Fundamental 9: Collection and Analysis of Data

Understand how to collect and analyse mathematical data, including populations and

samples, collection and organisation of data, meaningful representation of data, analysis

of data including comparison of data sets, correlation, and time series. Histograms.


Fundamental 10: Problem Solving

Understand how to problem solve in the context of any of the above fundamentals.



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section four

Curricular

Frameworks

Content knowledge taught in subjects

changes over time, through curriculum

development and as new knowledge

is generated in the field. An effective

teacher must be able to manage

change and have a mindset that

enables them to develop and

construct their subject knowledge. This

is an important disposition for the

resilient and effective teacher.

This section identifies the current

curricular frameworks for your subject,

and other supporting information and

guidance. These provide the specific

content knowledge that is taught in

schools. Use the subject

‘fundamentals’ (section three, above)

to aid your reflection regarding

related ideas and activity in your

subject.


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Curricular Frameworks Document URL

National Curriculum Programme

of Study for key stages 1 to 3

https://www.gov.uk/government/collections/nati

onal-curriculum

GCSE Subject Content https://www.gov.uk/government/publications/g

cse-mathematics-subject-content-and-

assessment-objectives

GCSE AS and A Level Subject

Content

https://www.gov.uk/government/publications/g

ce-as-and-a-level-mathematics

Subject Associations and Groups Name URL

Association of Teachers of

Mathematics (ATM)

https://www.atm.org.uk/

Mathematical Association http://www.m-a.org.uk/

Awarding organisations Organisation URL

Assessment and Qualifications

Alliance (AQA)

http://www.aqa.org.uk/

Oxford, Cambridge and RSA

(OCR)

http://www.ocr.org.uk/

Pearson Qualifications (Edexcel) https://qualifications.pearson.com/en/home.html

Welsh Joint Examination Council

(WJEC)

http://www.wjec.co.uk/

https://www.gov.uk/government/collections/national-curriculum

https://www.gov.uk/government/collections/national-curriculum

https://www.gov.uk/government/publications/gcse-mathematics-subject-content-and-assessment-objectives



https://www.gov.uk/government/publications/gce-as-and-a-level-mathematics

https://www.gov.uk/government/publications/gce-as-and-a-level-mathematics

https://www.atm.org.uk/

http://www.m-a.org.uk/

http://www.aqa.org.uk/

http://www.ocr.org.uk/

https://qualifications.pearson.com/en/home.html

http://www.wjec.co.uk/


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section five

SKDD activities

The SKDD activities, below, are

designed to support trainees in their

self-directed subject knowledge

development. The structured activities

are mandatory and linked to SKDD

sessions (with subject tutors), with

recommended activities for

personalised learning. The

recommended activities should be

discussed in weekly meetings (trainees

and mentors) and used to personalise

training and encourage deeper

reflection on content and

pedagogical knowledge.


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Activity 1 Microteaching and Sharing Practice During the introductory SKDD week, in September, you will plan and deliver a 10-minute lesson with your

peers. The topic and teaching style for the session will be agreed in advance with your subject tutor, and

you will have time to prepare before delivering. As part of the preparation you should read the following

chapter on teaching styles: Carpenter, C and Bryan, H. 5.3 Teaching Styles. In S. Capel, M. Leask and S. Younie (2016). Learning to teach in the

Secondary School: a companion to school experience (seventh edition). Abingdon, UK: Routledge. pp 368-384.

As part of your planning, you should produce an appropriate learning resource (e.g. visual aid,

worksheet, etc.). ICT (i.e. a computer and project) will not be available for the microteach session. As a

follow up to this activity, you will share what you have learnt with teachers in your Home School

placement, in a Mentor Meeting with your ITT Mentor, a Subject Department Meeting or by producing a

resource to summarise a range of teaching styles and how they might apply in your subject (e.g.

Mosston’s Spectrum).

Session Focus:

Teaching Style:

Session Outline

Resources

Self-evaluation (including tutor & peer feedback) Areas for development (targets and actions)


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Activity 2 Level 2 Specification Audit

Identify a relevant Level 2 (e.g. GCSE, BTEC, etc.) qualification taught in your Home School that you will

be solo or team teaching. Read the content knowledge section in the specification document, auditing

your knowledge and identifying key areas of strength. Next, identify areas for development in knowledge

and/or skills that you will be required to teach; and set targets for how you will address them.

Examination paper/Unit

Awarding

Organisation Qualification

Areas of strength (highlight ‘new’ knowledge for teaching)

How/where were they developed? (e.g. on your degree, personal study, etc.)

Areas for development (highlight when achieved)

Key action point (to be developed in the Weekly Meeting Record)

Date completed:

Mentor comment


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Activity 3 Level 2 Examination Paper Audit Consider the most recent examination (e.g. GCSE, BTEC, etc.) for the Level 2 specification used in the

specification audit.

Read last years Past Paper materials on the awarding organisation (AO).

Discuss with the ITT Mentor and identify key areas for development.

Sit the paper, or appropriate elements (linked to individual needs), in exam conditions.

Read the Mark Scheme provided by the AO.

Read the Examiners Report for the specification.

Mark and discuss your results with your ITT Mentor.

Examination paper/Unit

Awarding

Organisation Qualification

Areas of strength (highlight ‘new’ knowledge for teaching)

How/where were they developed? (e.g. on your degree, personal study, etc.)

Areas for development (highlight when achieved)

Key action point (to be developed in the Weekly Meeting Record)

Date completed:

Mentor comment


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Activity 4 Evaluating and Adapting Learning Resources

Trainees should spend time selecting, evaluating and adapting a range of current resources that are

used at their Home School, as part of their units and schemes of work, or obtained from external sources

(e.g. free or paid online teacher resources). As part of your development, select a resource (e.g.

homework, worksheet, etc.) to critically review and adapt for one of your lessons taking into

consideration standard S5 (Differentiation).

Description of resource3 (including the source, age range and expected outcomes)

Critique of the original resource (including the benefits and limitations)

Summary of adaptions (including differentiation of learning and links to assessment)

Evaluation of the adapted resource4 (considering the impact on learning)

3 Include a rough sketch, screenshot, photo or scanned image. 4 Ibid.


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Activity 5 Transition (key stage 2 to 3)

Read the statutory requirements for key stage 2, including formal assessment arrangements and how

your subject is taught in this phase of education. Discuss the Year 6/7 transition arrangements in your

Home School, for pupils from local ‘feeder’ primaries, with the Head of Department / Subject Lead.

Consider the questions below:

What does the department know about the primary curriculum experiences of their current Y7

students? (curriculum content, teaching and learning activities)

How do teachers gain insights about these curriculum experiences? What helps or hinders this

process?

Does the school/department have assessment data about their students’ performance in the

subject at the end of Key Stage 2? Why/ why not? Do they find this data useful? Why / why not?

How is the subject taught in key stage 2? What are the similarities and difference to key stage 3?

Write a reflection, below, about how effectively you have catered for Year 7 learners; based on your

knowledge about prior learning and/or attainment in key stage 2. What are the emerging issues and

actions you may wish to take?

Reflection (including how your perspective has been renewed)


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Implications for future practice

Activity 6 Transition (key stage 4 to post 16)

Set up a focus group with a group of year 11 students in your subject, and discuss their aspirations for their

education and career following GCSEs. Then discuss the findings with your ITT Mentor and/or the Head of

Department. Consider the questions below:

What are the key differences between key stage 4 and post 16? (These may include: the

physical, social and cognitive development of teenagers, curriculum content, teaching and

learning activities, etc.)

How does the key stage 4 curriculum and the teachers prepare students for the transition to post

16? What helps or hinders this process?

How is the subject taught post 16? What are the similarities and differences to key stage 4?

Write a reflection below about your readiness to support transition from key stage 4 to post 16. This may

be based on your knowledge about the curriculum, student aspirations and development. What are the

emerging issues and actions you may wish to take?

Reflection (including how your perspective has been renewed)


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Activity 7 Sharing Personal Subject Knowledge Development During the SKDD sessions in the February training block, you will deliver a 5-minute presentation to your

peers on an aspect of subject knowledge for teaching that you have developed in Phase 1 or 2

Rationale (Why did you develop the subject knowledge for teaching?) (Preparation before)

Process (How did you develop the subject knowledge for teaching?) (Preparation before)

Reflection (What are the implications for your future subject knowledge development including peer

feedback?) Following the session


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Activity 8 Learning Outside The Classroom (LOTC)

The ‘classroom’ can be described as the ‘normal’ environment for learning in your subject in timetabled

lesson. What are the alternative environments when your subject can be taught to develop and enrich

children’s knowledge and understanding?

Write a reflection, below, on an experience of learning outside of the classroom in your subject. This could

include activities that you have supported or planned in school, or as part of your university-based SKDD.

What are the emerging issues and actions you may wish to take?

Reflection (including the practical aspects of planning, risk assessment, communication, etc. and

how your perspective has been renewed as a result of the activity)



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Recommended activities

The activities below may be used by trainees, in discussion with their ITT Mentor, Personal/Liaison Tutor

and/or Subject Tutor to personalise subject knowledge for teaching. Any relevant documentation

produced should be printed and stored in the relevant section of the QTS Training and Development File

and/or Placement Experience File; and recorded in your LJMU ITT Tracker audit page for the appropriate

standard.

1. Engage with your subject association (see www.subjectassociations.org.uk).

2. Undertake a formal risk assessment, including the hazards, risks and controls, for a

relevant activity in your subject. Store it in your Placement Experience File.

3. Review a relevant Level 3 qualification and examination material, and set targets for

subject knowledge development. Record targets and actions in your Weekly Meeting

Record(s).

4. Review and critique an off-the-shelf learning resource with your ITT Mentor. Record in a

lesson plan and evaluation.

5. Design an original learning resource for one of the classes you teach. Record in a lesson

plan and evaluation, and store it in your Placement Experience File.

http://www.subjectassociations.org.uk/


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section six

References and

Bibliography


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References

Banks, F. Leach, J. and Moon, R. (2005). Extract from new understandings of teachers'

pedagogic knowledge, The Curriculum Journal, 16(3), pp.331-340, DOI:

10.1080/09585170500256446

DFE (2011). Teachers’ Standards. London: Department for Education. Available at

http://www.education.gov.uk/publications

Department for Education (DFE) www.education.gov.uk

National Foundation for Educational Research (NfER): Research publications

www.nfer.ac.uk/home

National Whiteboard Network. www.nwnet.org.uk

nRich Mathematical enrichment (a great source for investigation tasks) nrich.maths.org

National Centre for Excellence in the Teaching of Mathematics www.ncetm.org.uk

Bibliography

The list below provides a wide range of sources of relevant subject knowledge, which may

support you in consolidating existing and developing new knowledge; as well as supporting

lesson planning and preparation, and resource design.

BOALER, J. (2013) (2nd Edition) The elephant in the classroom: Helping children learn and love

maths, London: Souvenir Press Ltd

CHAMBERS, P & TIMLIN, R. (2013) Teaching mathematics in the secondary school, London: Sage

COTTON T (2013) Understanding and teaching Primary Mathematics Pearson

HAYLOCK, D. (2014) Mathematics explained for primary teachers. 5th ed. Paul Chapman Press /

Sage

HANSEN A (2014) Children’s Errors in Mathematics 3rd Ed Learning Matters

JOHNSTON-WILDER, S et al (2017) Learning to Teach Mathematics in the Secondary School 4th

edition Routledge

KILLEN P & HINDHAUGH S (2018) How Big is a Big Number Sage: Learning Matters

MOONEY, C., BRIGGS, M., FLETCHER, M. and McCULLOCH, J (2018). Primary mathematics:

teaching, theory and practice. (8th Edition) Learning Matters

Newell R (2017) Big Ideas in Primary Mathematics Sage

POUND L & LEE T (2011) Teaching Mathematics Creatively Routledge

SOUTHALL, E (2017) Yes, but why? Teaching for understanding in mathematics Sage: Learning

Matters

http://www.education.gov.uk/publications

http://www.education.gov.uk/

http://www.nwnet.org.uk/

http://nrich.maths.org/

http://www.ncetm.org.uk/


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Developed by Paul Killen (2019)

Department of Teacher Education

School of Education

Liverpool John Moores University

IM Marsh Campus,

Barkhill Road, Liverpool, L17 6BD

Email: [email protected]

mailto:[email protected]

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