Better mathematics workshop pack spring 2015: primary

Primary workshop information pack 1 of 17 Ofsted’s Better mathematics conference 2015

Better mathematics

conference

Spring 2015

Primary 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

Jeans cost £13.95. They are reduced by 1/3 in a sale.

Dan buys the jeans. He pays with a £10 note. How much change does he get?

Question 3

Jeans cost £13.95. They are reduced by 1/3 in a sale.

Dan has £10. Does he have enough money to buy the jeans? Explain why.


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?

Primary 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

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



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


observe, question, listen, circulate to check and improve pupils’ progress

details of how this increments learning or misses opportunities/fails to enhance it


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:



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?

Better mathematics workshop pack spring 2015: primary

Education

Transcript of Better mathematics workshop pack spring 2015: primary