Better mathematics workshop pack spring 2015: secondary

Secondary workshop information pack 1 of 17 Ofsted’s Better mathematics conference 2015
Better mathematics
conference
Spring 2015
Secondary workshops
Information pack
for
participants
This pack contains the information you will need to refer to during the activities.

Questions
Version 1
Version 2
Question 1
Jeans cost £13.95. They are reduced by 1/3 in a sale.
What is their price in the sale?
Dan buys the jeans. He pays with a £10 note.
How much change does he get?
Question 2
Dan buys the jeans. He pays with a £10 note.
How much change does he get?
Question 3
Dan has £10. Does he have enough money to buy the jeans?
Explain why.
1) Calculate x and then  2) Calculate 

Deepening a problem
Problems can be adapted by:
 removing intermediate steps
 reversing the problem
 making the problem more open
 asking for all possible solutions
 asking why, so that pupils explain
 asking directly about a mathematical relationship.
Remember, you can:
 improve routine and repetitive questions by adapting them
 set an open problem or investigation instead.
Area of a rectangle
Approaches to other topics
You might find the following general questions useful in discussions with your colleagues about
the teaching of other topics, perhaps identified through monitoring of teaching or question level
analysis of test results.
1. How well does your introduction develop conceptual understanding?
2. How repetitive are your questions? How soon do you use questions that reflect the
breadth and depth of the topic?
3. At what stage do you set problems? How well do they deepen understanding and
reasoning?
4. Are questions and problems presented in different ways?

Secondary workshop information pack 4 of 17 Ofsted’s Better mathematics conferences 2014
Introduction to the formula for area of a rectangle
Teacher A Teaching Assistant B Teacher E
Ngn,
Teacher D
Rectangle Length Width Squares
A 4 2 8
B 5 3 15
C 6 5 29
D 6 2 10
E 5 4 20
Teacher C
I start by telling pupils the formula L x W. They
work out lots of areas mentally, when I give them
the dimensions or show them a diagram. They write
answers on mini-whiteboards. I check they can
identify the length and width correctly. Then they do
a worksheet with lots of different rectangles; the
first few have their lengths written on and the next
ones pupils have to measure. The high attainers have
questions with larger numbers and decimals.
I try to find something that pupils
can relate to from their experience
– a swimming pool is a good
image because most pupils
understand what ‘a length of the
pool’ means. They then recognise
the ‘length’ as the longest side.
3 x 8
8 x 3
I ask the pupils to count squares inside the
rectangles drawn on the IWB. Then we count
systematically. We add rows, for example 8+8+8,
which they know is the same as 3 lots of 8. We also
add columns, which is 8 lots of 3. The pupils see
why the formula works because the dimensions
match the numbers of rows and columns. They
understand the same area can be built up from rows
or columns.
I give the pupils rectangles on dotty grids so they
don’t waste time drawing them. I mark the length
and width of the first few, and they find the
others. I ask pupils to count the squares inside
the rectangles, complete the table and look for a
connection between the numbers in the table.
They sometimes make mistakes but usually spot
that multiplying gives the number of squares.
Length
5 or 6?
I make a rectangle with 15 tiles and explain its area is
15. We look at it in different orientations using a
visualiser. I describe each image in different ways using
‘rows’ and ‘columns’. They understand it is the same
rectangle in different positions so has the same area.
3 rows of 5
5 columns of 3
5 rows of 3
3 columns of 5
Then I give each pair 12 tiles and ask them to
make a rectangle. I ask if they think they could
make another rectangle with 12 tiles, making sure
they predict its dimensions before they make it. I
mention the word ‘factors’ if they don’t. It’s
important to ask them to check they’ve found them
all and explain why. Then I give them other
numbers of tiles to investigate. In the end, we have
discussed factors, primes, the fact that squares are
special rectangles and the formula for area of a
rectangle being the product of the dimensions.
As well as area, I’ve learned more
about factors, prime numbers and
square numbers.
You can only make one rectangle
with 11 tiles – and I know why!
Rectangle D
width
length

Misconceptions
Several errors are illustrated below. They are caused by:
 underlying misconceptions
 unhelpful rules
 lack of precision with the order of language and symbols.
With your partner, see if you can identify the underlying misconception or cause of each error.
Take 6 away from 11
6 – 11 = 5
In order, smallest first:
3.2 3.6 3.15 3.82 3.140
6 ÷ ½ = 3 2.7 x 10 = 2.70
–
2 x –
8 = +
16
–
1 + –
4 = +
5
10% of 70 = 70 ÷ 10 = 7
20% of 60 = 60 ÷ 20 = 3
The
angle
measures
80°

Ways to help colleagues
Help staff to be aware of misconceptions that:
 pupils may bring to the lesson
 might arise in what is being taught.
Encourage staff, when planning a topic, to discuss mistakes that pupils commonly make in that topic
and explore the misconceptions that underpin them. Also help staff to:
 plan lessons to take account of the misconceptions
 look out for misconceptions by circulating in lessons.
Bear in mind, it is more effective to address misconceptions directly than to avoid or describe them.
You could give pupils carefully chosen examples to think about deeply. Pupils then have the
opportunity to reason for themselves why something must be incorrect.
The potential of work scrutiny
To check and improve:
 teaching approaches, including development of conceptual understanding
 depth and breadth of work set and tackled
 levels of challenge
 problem solving
 pupils’ understanding and misconceptions
 assessment and its impact on understanding.
To look back over time and across year groups at:
 progression through concepts for pupils of different abilities
 how well pupils have overcome any earlier misconceptions
 balance and depth of coverage of the scheme of work, including using and applying
mathematics.

An extract from Jacob’s work, which includes his own marking and the teacher’s
comments
An extract from Poppy’s work, which includes the teacher’s marking and comments

Extract of work from a pupil in teacher A’s class shown in the keynote session
What does this extract suggest about how well:
 the pupil understands the topic (collecting like terms)
 the exercise provides breadth and depth of challenge
 the marking identifies misconceptions and develops the pupil’s understanding?
Understanding
 The pupil adds like terms correctly but makes an error with subtraction.
 The exercise has too many straightforward questions on addition before subtractions start.
This makes it too narrow and unchallenging to develop understanding of the whole topic.
 Such narrowness prevents the teacher from checking the extent of the pupil’s understanding.
Teacher’s comments
 The teacher’s comment ‘only 1 wrong’ misses the pupil’s difficulty with subtraction shown by
the wrong answer to Q8 and incomplete Q9.
 It is likely that the pupil left the signs unmoved when rearranging 3j – 4k + 2j + k
to get 3j – 2j + 4k + k
 The comment does not help the pupil to overcome this misconception.
Work scrutiny and lesson observation
 The top priority to bear in mind when scrutinising work is ‘Are the pupils doing the right work
and do they understand it?’
 Thinking back to the extract, the pupil was not doing ‘the right work’. The questions lacked
breadth, did not develop understanding fully, and included no problems. Improving these is
more important than improving marking.
 The skills you have just used in scrutinising the extract of work are equally applicable when
observing lessons.
 A key additional element when observing lessons is seeing how well teachers check and
deepen each pupil’s understanding. Does the teacher move round the class observing and
listening to pupils to check their progress?
Assessment policy
 In accordance with the school’s policy, the teacher has provided a ‘next step’.
 ‘Equations’ is not a ‘next step’ to help the pupil improve in this topic. Understanding
subtraction is needed first.
 Working with brackets, constant terms and powers would give fuller breadth of the topic.

Extract of work from a pupil in teacher B’s class
Note: the pupil’s marking is in pencil and the teacher’s marking is in red.

Extract of work from a pupil in teacher C’s class
Note: the green highlighting shows the pupil’s response to the teacher’s comments.

Work scrutiny record – school W
Teacher name: ____Elspeth Thomas_ Date: 6 Nov 2013
Year group: 5 Sample: 2LAP, 2 MAP, 2 HAP
Aspect Comment
Book is neat, with no
graffiti
   WJ’s book has graffiti on – ask pupil
   to cover book
Work is dated    1 missing date spotted by T for PG
  
Learning objectives
are clear
?   LO copied out at start of lesson
   WJ’s is hard to read
Work is marked
regularly
   Ticks on every page – some pupil marking,
   initialled by T or TA
Marking indicates two
stars
  
  
WJ: no positives but not much work. Others: ‘Neat work’
‘brilliant’ ‘’really good try’, ‘You can times HTU by U now’
Marking includes next
steps (a wish)
   T writes a next step eg ‘factors & multiples’, but it is
   not always a wish e.g. ‘next step: work faster’.
Work is assigned a
level
2b 3c 3b T gave NC levels on every main piece of
3b 3a 3a marking, an improvement since last check
Effort grades are
given
D A A (latest grades - grades given every week)
B C A WJ effort often poor; SB one-off, usually A/B
Evidence of homework    Weekly HW set to practise tables
   WJ not doing HW at all; SB missed latest
DHT comments: Elspeth, you are following the marking policy pretty closely.
I’m glad to see you are now including NC levels. Remember the ‘2 stars and
a wish’ rule. Some of your next steps aren’t really wishes (what could be
done better). NB: I am worried about WJ’s book. It is very scruffy and he
isn’t doing much work in lessons. Can you get him to try a bit harder?
Teacher’s comments: Thank you for noticing the NC levels! It’s hard to think of a wish
sometimes when a pupil gets everything right, but I’ll try to work on this.
As for WJ – he’s been very difficult lately – especially since his LSA was off sick. His
work was much better when she helped him. I made him stay in at play-time a few
times.

Work scrutiny record – school X
Focus on the highlighted cells. The work scrutinised included collecting like terms (CLT).
The scrutiny was by the deputy headteacher, a mathematics specialist, who made this summary
to inform discussions with leaders, including the head of department (HoD). Each teacher
received individual feedback under the same five headings.
Teacher names: KYh, CHi(NQT), SPa(HoD), MKi Date: 19/11/12
Year /sets: 7A 7B 7C 7D (mixed ability) 1 LA, 1 MA, 1 HA Unit: Algebra 2
Curriculum: Links/progression; differentiation & challenge; match to scheme of work; UAM/PS/MR
7A: Topics covered in logical order; links alg ops to
number ops –diff via different exercises. Unnecessary
copying of LO? UAM ‘bolt-on’ activity (no diff)
7B: Appropriate topics but tends to flit so work
doesn’t flow. Limited diff, esp lack of challenge for
HA. Unnecessary copying of LO? No UAM in books
7C: Good links across curriculum. Interesting work-
sheets replace textbook in places. Challenge for HA.
UAM via problem solving, proof , real-life contexts.
7D: Appropriate coverage, heavy use of textbook
exercises. HA do more questions. No UAM evident.
Variation among teachers in differentiation and emphasis on UAM. Variation in use of textbook – 7A & 7D
using textbook a lot while 7C often substitutes more interesting work. 7B coverage is an issue.
Progress: Evidence of learning; gains in knowledge, skills and understanding; progress of groups
7A: Pupils learn via repetition, but only basic cases.
HA wasting time on easy work. Not developing
deeper understanding or independence through PS
7B: Pupils often start OK but get lost as Q get harder.
eg MA pupil OK collecting like terms for + but not -.
Also confusion eg gets 3a+5b but then puts = 8ab
7C: Pupils learn via mix of routine Q and Q that
make them think. Well judged when to move P on
(eg CLT with brackets). Stretch evident for HA.
7D: Some pupils seem to be altering their work when
T reads out answer, so it looks right. T giving big
but not spotting lack of understanding.
Tracking data suggests better progress evident in 7C (HoD) than in others; slowest in 7B (NQT). HA progress
weaker than MA. This is consistent with scrutiny. HA not always stretched enough in 7A,B,D. 7A,D classes
make adequate progress through textbook approach, but not developing understanding or PS skills.
Teaching: Teaching approaches used; focus on understanding, depth, problem solving
7A: Using agreed approach for CLT to develop
understanding. Useful notes but exercises too
similar so not getting enough depth of coverage.
7B: Not using agreed approach of rearranging cards
for CLT. Use of fruit salad algebra risks misconception
– no PS or teaching for understanding.
7C: Agreed app for CLT. Good breadth & depth of Q;
some require PS, deeper thinking or algebra for
proof. Practical resources adapt work for LA.
7D: Uses agreed approach for CLT. P’s comment re T
using ‘2 minuses make a plus’ is a worry; dept agreed
not to use rule . CLT straight into good mix of +/-
All use CLT approach agreed last year except NQT (why? check with mentor) More variation on other topics.
HoD sets good example - needs to support /inspire others to focus more on understanding. Dept needs to
develop more guidance on teaching approaches, so all have similar expectations re depth, use of PS & UAM.
Marking: Regularity; identifying misconceptions, strengths, how to improve; follow-up dialogue
7A: Mostly  marking. Comments re neatness. Next
steps = next topic, not how to improve. Qs constrain
scope for diagnostic mark. Uses NC level (old policy)
7B: Comments made but targets vague, no help for P
to improve. Doesn’t diagnose why P goes wrong with
negative terms in CLT. Uses NC level (old policy)
7C: Good diagnostic marking. Spots source of errors/
misconceptions, also small points eg 2u + 0v = 2u.
Good on how to improve, extra challenge & follow-up.
7D: Mostly  & some guidance. Work provides good
opp for diagnostic marking, but missed. No help for
imp. More comments on quantity than quality.
7A (KYh) still using NC levels from old marking policy and so is her mentee, the NQT! Marking all up to
date, but variable quality. Weak use of targets, next steps, how to improve. Limited diagnostic marking to
root out misconceptions. Only HoD (7C) is marking to good standard but isn’t influencing others enough.
Summary: Imp since last scrutiny, areas for improvement, strengths to share; issues for dept to take forward
7A: KYh improving re links & agreed app to CLT. Now
including UAM, though ‘bolt-on’. Needs to follow
latest policies & ensure best advice for NQT.
7B: CHi no previous scrutiny. Policy/agreed approach
not followed. Were they discussed with mentor?
Suggest focused lesson obs , joint HoD & mentor.
7C: SPa’s own practice is good, but not influencing
dept enough. Could buddy with experienced HoD
(AFo?). Start middle leader course soon.
7D: MKi apart from CLT work, is still relying on
textbook exercises & giving ‘rules’. Marking is not
helpful to P. Discuss further support .
Overall: The department has improved since last year, mainly due to new HoD and NQT replacing supply.
But, pupil progress and teaching still require improvement. Pace of change is not fast enough. Little
differentiation – is mixed ability working? Need to support HoD to develop team, involve AST? Also talk with
mentor. Department CPD needed on developing understanding and problem solving, and marking to
diagnose misconceptions. Also, department meeting time to concentrate on developing agreed approaches
and depth, including UAM and stretching the more able.

Work scrutiny in your school
Think about your school’s work-scrutiny system and look at any work-scrutiny documents you
brought with you.
 Consider whether your work-scrutiny system is getting to the heart of the matter. How effectively
does it evaluate strengths and weaknesses in teaching, learning and assessment, and contribute
to improvement in them?
 Identify at least one improvement you can make to each of:
 your school’s work-scrutiny form
 the way work scrutiny is carried out
(e.g. frequency, sample, focus, in/out of lessons, by whom)
 the quality of evaluation and of development points recorded on the forms
 the follow-up to work scrutiny.
Observation records – school Y
The emboldened text is the headings on the form.
The italicised text is the comments written by the observer this term.
The plain text is a point for development for the teacher that was written in the previous term.
Extract 1
Progress towards previous points for development:
 Ensure you identify and tackle misconceptions throughout the lesson, for example
through use of mini-whiteboards and questioning during your introduction and
checking during independent work that pupils have overcome them.
You used mini-whiteboards well to spot the pupils who had the misconception that a
rectangle had four lines of symmetry. But your explanations did not ensure they understood
why not – six pupils then went on to show four lines of symmetry on some other rectangles
during their independent work. You still need to work on this point for development.
New points for development:
 Use visual images and practical apparatus to ensure concepts are understood.
Extract 2
Progress towards previous points for development:
 Ensure that key learning is reinforced through the repeated correct modelling of
mathematical vocabulary e.g. ‘What is 1 more than 5?’ rather than ‘What is 1 more?’
You ensured that you and the children used comparative language well, especially ‘than’ e.g.
you checked well that children knew they were taller than the chair but shorter than the
teacher.
New points for development:
 Provide opportunities for children to achieve the differentiated learning outcomes (e.g.
in this lesson, the activities gave no opportunity for the more able children to show
they could meet the criterion ‘recognise numbers to 50’).

Observation record – school Z
School Z uses a form with six aspects to evaluate.
Alongside each aspect are four cells each containing text relating to one of the four grades.
For example, the text in the ‘good’ cell for the aspect ‘planning including differentiation’ is as
shown below.
Aspect Good
Planning including
differentiation
planning is detailed and takes account of
prior attainment of different groups
In the extract below from School Z’s form, the detail of the text in each cell is not shown. It is
represented by the word ‘text’.
When evaluating each of the six aspects during the lesson, the observer has highlighted the text
in one cell (good for the first five aspects and requires improvement for the sixth). The observer
has also written a summary and given a grade for teaching.
Aspect Outstanding Good
Requires
improvement
Inadequate
Planning including
differentiation
Text Text Text Text
Variety of activities Text Text Text Text
Subject knowledge Text Text Text Text
AfL Text Text Text Text
Behaviour Text Text Text Text
Progress Text Text Text Text
Summary
 Interesting range of activities – the SEN pupils enjoyed working on the
computers with the TA. They made good progress.
 Very clear explanations as usual – you have good subject knowledge.
 All the pupils quickly settled down to their work.
 You need to have an extra activity up your sleeve for that able group
when they have finished. I don’t think they learnt anything new, unlike
the middle groups who did well.

Observation records
Evidence on progress in observation records
Evidence on progress should include evaluation and mathematical detail of gains in knowledge,
skills or understanding for groups of pupils.
Common weaknesses in evaluating pupils’ progress are that comments in the progress section
consist of:
 ticks only with no written evaluation
 information about what was taught e.g. ‘harder problems’
 description of what pupils did e.g. ‘all measured angles’
 evaluation of pupils’ effort, being on task or enthusiasm
 amount of work completed e.g. ‘finished the worksheet’
 unquantified evaluation of progress e.g. ‘they made progress’
 no detail of progress of groups, such as higher attainers
 evaluation of work pupils did e.g. ‘added the fractions correctly’, without evaluation of the
gains in learning for pupils or groups (could they do this before the lesson?)
 evaluation of gain in knowledge, skills or understanding but with no mathematical detail
e.g. ‘they learnt a lot’.
Evaluating progress and teaching
Pupils’ gains in knowledge, skills and understanding need to be evaluated in terms of how well
pupils have developed conceptual understanding, problem-solving and reasoning skills, including
accurate use of language and symbols, and overcome any misconceptions.
The quality of teaching is evaluated through its impact on pupils’ progress by gathering a wide
range of evidence, not just observation of a single lesson. However, features of the observed
teaching, including how well the teacher monitors and enhances pupils’ progress in lessons, can
give useful insight into how well pupils are making gains in their mathematical knowledge, skills
and understanding over time. Sharp evaluation also allows timely support and challenge to help
teachers improve their practice.
Learning and progress are the most important areas of focus during lesson observations, with
the following areas contributing:
 monitoring to enhance progress
 conceptual understanding
 problem solving
 misconceptions
 reasoning, including correct language and symbols.
The examples overleaf show how a school might organise recording of teaching input and its
impact on pupils’ learning in each of these areas.

Lesson observation records
Example of blank form for recording evidence of input and impact for five key aspects
contributing to progress
aspect teacher: input pupils: impact (individuals & groups)
progress
monitoring to
enhance progress
conceptual
understanding
problem solving
misconceptions
reasoning, language
and symbols
Example with input & impact for five key aspects contributing to progress – cells contain prompts
for evidence to be recorded
aspect teacher: input pupils: impact (individuals & groups)
progress quality of teaching
mathematical detail of gains in
understanding, knowledge and skills
monitoring to
enhance progress
observe, question, listen,
circulate to check and improve
pupils’ progress
details of how this increments
learning or misses opportunities/fails
to enhance it
conceptual
understanding
approach: structure, images,
reasoning, links
depth of conceptual understanding
problem solving
real thinking required, for all
pupils; used to introduce a
concept or early on
confidence to tackle and persistence;
depth of thinking; detail of pupils’
chosen methods and mathematics
misconceptions
identify and deal with; design
activities that reveal them
detail of misconception and degree to
which overcome
reasoning, language
and symbols
model, check and correct
correct reasoning and use of
language/symbols; detail of missed or
unresolved inaccuracy

Lesson observation in your school
Think about your school’s lesson-observation system and look at the lesson-observation forms
you brought with you.
Consider whether your lesson-observation system is getting to the heart of the matter.
In particular, how effectively does it:
 evaluate aspects of teaching in terms of their impact on learning and progress
 give due weight to learning and progress from a range of evidence in informing an overall
judgement on teaching
 contribute to improvement in teaching, learning and progress?
Identify at least one improvement you can make to each of:
 your school’s lesson-observation form to:
 focus on learning and progress, and take account of:
 monitoring to enhance progress
 conceptual understanding
 problem solving
 misconceptions
 accuracy of language and symbols
 the way lesson observation is carried out (e.g. frequency, focus, range of classes and
teachers, by whom)
 the quality of evaluation and of development points recorded on the forms
 the range of additional evidence gathered on teaching and its impact (such as work
scrutiny, assessment/homework, interventions)
 the weight given to learning and progress in evaluating teaching
 the follow-up to lesson observation.
Think for a moment …
How could you support colleagues in deepening problems for the next topic
they will be teaching?
How is the formula for area of a rectangle taught in your school?
Identify a topic you would find it helpful to discuss with a group of colleagues in this way:
1. How well does your introduction develop conceptual understanding?
2. How repetitive are the questions you set?
3. At what stage do you set problems? How well do they deepen understanding and
reasoning?
4. Are questions and problems presented in different ways?
Take 6 away from 11
6 – 11 = 5
What future learning might be impeded?
How might you find out about misconceptions across the school?
How might you help colleagues use misconceptions well in their teaching?
How might you work with colleagues and senior leaders to bring about improvement?

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