An alternative Maths curriculum

Alternative staffing models for Maths

Dave Appleby

24th February 2012.

Dotheboys Hall School faces an impending crisis. Of the 5 Maths teachers, two are leaving and no-one can be replaced. They will only have Glenda Gauss, Nicy Newton and Rajendra Ramanujan.

Fortunately thay have three excellent LSAs called Candice Carnelian, Jade Jet and Peter Pearl.

In each half year group there are 120 pupils in 5 setted classes.

Each class has 5 periods per fortnight.

There is one computer room with 25 machines.


Another little problem facing Dotheboys is that sets split. When the pupils arrive at Dotheboys in year 9 they are tested to see how good they are at Maths and this measure is used to group them. Thus pupil 24 is in set 1 and pupil 25 is in set 2. They then experience mostly whole class teaching aimed at more or less the centre of the class. Thus pupil 24 struggles to keep up with pupil 13 whilst pupil 25 coasts along compared to pupil 38. In general this means that pupil 24 progresses faster in Maths than pupil 25; when they are retested the difference between them is greater than it had been. Generally the setting is confirmed.


The pupils who come from Salem House (whose Maths teacher, Mr Creakle, is less than brilliant) tend to be assigned the lower sets. It soon becomes impossible for them to catch up with the pupils who have been taught Maths better at Dr Blimber's Academy.


This has led to some anomalies. Pupil 60, in set 3, has never been much good at Maths but towards the middle of year 10 something clicks and she suddenly makes progress. By the start of year 11 she has clearly demonstrated her potential for Higher Level Maths. Unfortunately she has been taught Foundation Maths until now. There is no mechanism for her to catch up on the Higher Maths she has missed. Despite having the potential to achieve an A she is entered for a paper where the maximum possible is a grade C.

The headteacher, Wackford Squeers, has a cunning plan.

He has heard of the School Of One in New York City. This school personalises the maths curriculum. It aims to maximise differentiation in ‘Math’ by offering a variety of learning opportunities including:

• A Math computer lab

• A Math investigative and collaborative learning lab
• A Math seminar class of no more than 12 pupils
• One to one online Math tuition
• Proper ‘Math’ lessons taught by a real ‘Math’ teacher
• Individual learning Math sessions in which supervised pupils complete worksheets

The clever bit is that School Of One assesses each pupil daily and then assigns them to the appropriate learning opportunity in next day’s timetable. Wackford doubts that Dotheboys Hall is yet quite ready for such Yankee technology. Nevertheless, he is a fan of giving pupils at different levels of Maths different experiences.

He proposes four types of Maths experience:

• A Maths computer lesson which will offer individualised instruction using a variety of computer systems such as MyMaths, Mangahigh, and Successmaker (for the lowest achievers). This will be supervised by an LSA (Candice)
• A Maths collaborative investigation in which 6 teams of 4 pupils will try to solve a Maths challenge. These sessions will by pedagogically founded on the work by Sugatha Mitra. They will be led by an experienced Maths teacher, Glenda.
• Individualised Maths workshops in which pupils will work through problems hand-picked for them supervised and supported by a pair of LSAs (Jade and Peter).
• There will also be ‘trad Maths’ lessons taught by his other experienced Maths teachers, either Nicky or Rajendra.

Mr Squeers has devised possible schemes based on these experiences. He has one scheme for 5 ppf and one for 6ppf.

5ppf model

	Lesson 1	Lesson 2	Lesson 3	Lesson 4	Lesson 5
Set 1	Computer Maths with CC	Investigation Lab with GG	Maths with NN	Maths with NN	Maths with RR
Set 2	Maths with RR	Computer Maths with CC	Investigation Lab with GG	Maths with RR	Individual Maths with JJ & PP
Set 3	Maths with NN	Individual Maths with JJ & PP	Computer Maths with CC	Investigation Lab with GG	Maths with NN
Set 4	Individual Maths with JJ & PP	Maths with RR	Maths with RR	Computer Maths with CC	Investigation Lab with GG
Set 5	Investigation Lab with GG	Maths with NN	Individual Maths with JJ & PP	Individual Maths with JJ & PP	Computer Maths with CC

• Every set sees a qualified Maths teacher at least twice per fortnight.
• Every set has a computer session which means the IT resources are fully used.
• Every set has an Investigation lab.
• Set 1 has 3 trad Maths sessions; sets 2, 3 and 4 have 2 trad maths sessions; set 5 has a single trad Maths session.
• Sets 2, 3 and 4 have a single individualised maths workshop; set 5 has two such sessions.
• Both RR and NN see 80% of the students; GG sees them all.

6ppf model

	Lesson 1	Lesson 2	Lesson 3	Lesson 4	Lesson 5	Lesson 6
Set 1	Computer Math with CC	Investigation Lab with GG	Maths with NN	Maths with NN	Maths with RR	Investigation Lab with GG
Set 2	Maths with RR	Computer Math with CC	Investigation Lab with GG	Maths with RR	Individual Maths with JJ & PP	Maths with RR
Set 3	Individual Maths with JJ & PP	Maths with NN	Computer Math with CC	Investigation Lab with GG	Maths with NN	Computer Math with CC
Set 4	Maths with NN	Individual Maths with JJ & PP	Individual Maths with JJ & PP	Computer Math with CC	Investigation Lab with GG	Maths with NN
Set 5	Investigation Lab with GG	Maths with RR	Maths with RR	Individual Maths with JJ & PP	Computer Math with CC	Individual Maths with JJ & PP

• Every set sees a qualified Maths teacher at least three times per fortnight.
• Every set has a computer session (set 3 has 2) which means the IT resources are fully used.
• Every set has an Investigation lab (set 1 has 2)
• Set 1 has 3 trad Maths sessions; sets 2, 3 and 4 have 2 trad maths sessions; set 5 has a single trad Maths session.
• Sets 2, 3 and 4 have a single individualised maths workshop; set 5 has two such sessions.
• Both RR and NN see 60% of the students; GG sees them all.

This new system means:

• Every pupil learns Maths using a collaborative investigative method.
• Every pupil uses ICT to aid their learning at least once per fortnight.
• Every pupil experiences a mix of learning styles.
• Every pupil has a qualified Maths teacher at least twice per fortnight.
• The balance of learning styles can be better tailored to the pupil’s prior achievement.
• Where pupils are being given extra one-to-one tuition they can be taken from the individualised workshops without disrupting their learning of content (presuming that pupils in set 1 do not need to come out of lessons).
• GG develops special skills in teaching in this investigative style; CC develops special skills in running computer lessons.
• Given that we normally staff 5 sets with 5 qualified teachers and an LSA and given that qualified teachers earn more than LSAs, we will save on staffing costs. Mr Squeers likes that! But more importantly it solves his problem of the shortage of specialist maths teachers.
• This system builds differentiation in. The Investigation Lab, the Computer lesson and the Individual Workshops are all highly differentiated. Set 1 in the 5ppf model experiences 40% of these types of lessons; every other set experiences more. The average pupil will experience 60% highly differentiated lessons.
• This will reduce the tendency of sets to split. The pupil perceived as the best in set 2 might well be assigned more challenging work than the pupil perceived as the worst in set 1.
• Pupils in the lower sets will experience higher amounts of personalised work. It should be possible for pupil 60 to catch up with a significant amount of Higher Level work; certainly sufficient for her to be entered at this Level.

A Framework for Literacy

A Framework for Literacy

A US framework for literacy (OK, Math, color, sulfur) bases itself on two predicates:

Ø      “Literacy skills are most effectively acquired in contexts that make reading and writing meaningful” (p3)

Ø      “Reading, writing, and critical thinking differ in purpose and emphasis yet draw on a common pool of literacy skills.” (p3)

The report believes that literacy involves three processes: interpretation, expression and deliberation. At a first approximation these appear to correspond to reading, writing and critical thinking but “matters are more complex. Skilled readers write in support of their reading (by taking notes) and employ reflective reading strategies. Skilled writers use reflective strategies to improve writing quality and read every time they revise or in response to material from other texts.” (p4)

There are five discourses which explain literacy skills.

Ø      The social model focuses on “inferences about communicative intent” (p5)

Ø      The conceptual model is most concerned with comprehension and making meaning from text.

Ø      The discourse model looks at how the author works within a genre.

Ø      The verbal model is based upon vocabulary and grammar.

Ø      The “print model represents skills in processing text in formal, phonological, or orthographic terms.” (p6)

It is clearly complicated to propose any coherent model of skill development which (a) takes into account the complexity of interrelation between the three processes of interpretation, expression and deliberation and (b) looks the same when viewed from each of the five perspectives above.

The authors also hypothesise a developmental sequence for the development of the critical thinking inextricably linked to literacy.

Ø      “The minimum prerequisite for rational argument is the recognition of incompatible and conflicting viewpoints.” (p9)

Ø      “Begin to anticipate challenges and accumulate justification strategies that have worked in the past.” (p10)

Ø      “Begin to strategically select justifications and elaborate on arguments where supporting evidence will help to bolster the case.” (p10)

Ø      Learn from the experience of having one’s own arguments refuted and “recognize fallacies, develop rebuttals and reason more generally about the validity of arguments.” (p10)

Ø      Use the knowledge of what arguments should be “as an intellectual tool that helps determine which ideas should be accepted.” (p10)

This then leads to a ladder of literacy skills.

LEVEL	INTERPRETATION	DELIBERATION	EXPRESSION
Preliminary (oral to sentence)	Can orally restate or identify the reasons someone else has given to support an opinion.	Can distinguish reasons from non-reasons and infer whether reasons would be used to support or oppose a position.	Can give plausible reasons to support an opinion when asked or spontaneously in conversation.
Foundational (sentence to paragraph)	Can restate (list in one’s own words) the supporting reasons provided in a paragraph-length text.	Can self-generate multiple reasons to support an opinion.	Can express lists of reasons in declarative sentence form and embed them in a paragraph-length position statement.
Basic (paragraph to text)	Can recognize and explain the relationship between main and supporting points and keep track of which evidence supports which point.	Can rank and select reasons by how convincing they seem. Can distinguish between reasoning that seems convincing because one agrees with it and reasoning that seems convincing because of the content of the argument.	Can select and arrange reasons and include specific supporting details. Can group reasons with evidence to form (implicit or explicit) paragraph structure.
Intermediate (text to context)	Can track and distinguish multiple positions when they are discussed in the same text. Can evaluate the accuracy of a summary and the credibility of a source text based on strength of arguments and evidence.	Can recognize counterexamples and distinguish facts and details that strengthen a point from those that weaken. Can distinguish valid from invalid arguments and recognize unsupported claims and obvious fallacies.	Can organize reasons/evidence contrastively to compare opposing positions. Can summarize and embed sources as supporting evidence. Can write simple critiques or rebuttals.
Advanced (text and context to discourse)	Can evaluate arguments in light of existing knowledge and discussions, actively verifying, challenging, and corroborating the case presented in terms of other sources of knowledge.	Can identify and question the warrants of arguments, distinguish necessary and sufficient evidence, and synthesize a position from many sources of evidence, using that to identify key evidence and propose new lines of argument.	Can write extended discussions that place arguments in the context of a larger literature or discourse. Can embed critiques and rebuttals effectively into a longer argument.

Deane P, Sabatini J, and  O’Reilly T 2011 English Language Arts Literacy Framework Educational Testing Service Princeton NJ available at http://www.ets.org/s/research/pdf/ela_literacy_framework.pdf accessed 10th February 2012

Household Risks that Undermine Education

Household risks that undermine education

A new study[1] claims that 27% of UK children live with multiple risks to their educational development. The identified risk factors are:

Ø      Mother or her partner often feels depressed

Ø      Mother or partner has a longstanding physical illness that limits daily activities

Ø      Mother smoked during pregnancy

Ø      At least one parent drinks alcohol more than the recommended amounts (14 units weekly for women; 21 for men)

Ø      Mother or partner often gets into a violent rage

Ø      The family finds it difficult to manage financially

Ø      Neither mother nor father in paid employment

Ø      Mother’s first pregnancy was when she was under 20 years old

Ø      Either mother or father lacks basic skills; this limits daily activities

Ø      House is overcrowded (more than 2 people per room not including toilets, hall, kitchen, living room and garage)

Of course some of these risk factors are correlated. If you have no basic skills you are less likely to be in employment; overcrowding is linked with illness and with depression.

“Children living in families with multiple risks are more likely to have long-term disadvantageous cognitive and behavioural consequences.” Already by the age of 5 they have poorer behaviour and cognitive assessments than their peers in no risk households.

Children with 2 or more risk factors are likely to do less well at school.

Number of risk factors	Approximate percentage	Number of risk factors	Approximate cumulative percentage
None	43%	None	43%
1	30%	At least 1	57%
2	15%	At least 2	27%
3	7%	At least 3	13%
4	3%	At least 4	6%
5	1%	At least 5	2%
6	1%	At least 6	1%
7	0.2%	At least 7	0.2%

The authors found that there are ethnic differences in the exposure to risk factors. “48 per cent of Bangladeshi families faced two or more risks, followed in order of prevalence by Pakistani families (34.4 per cent), other mixed (32.9 per cent), black African (31.4 per cent), black Caribbean (29.2 per cent), white (27.8 per cent) and Indian (20.4 per cent) families.”

---

[1]  Ricardo Sabates and Shirley Dex 2012 Multiple Risk Factors in Young Children’s Development Centre for Longitudinal Studies London available at    http://www.cls.ioe.ac.uk/news.aspx?itemid=1661&itemTitle=More+than+one+in+four+UK+children+facing+multiple++risks+to+development%2c+study+finds&sitesectionid=905&sitesectiontitle=Press+Releases accessed 7^th February 2012

Changes to GCSE

Changes to GCSEs

Information from Ofqual.

These changes will affect all candidates starting two-year courses in September 2012.

Candidates will be required to take all GCSE assessments at the end of the course for awards made from summer 2014 onwards.

All assessments will be in the summer except for English, English language and mathematics which will also be available in November.

In GCSE English literature, geography, history and religious studies, additional marks will be awarded for the accuracy of spelling, punctuation and grammar in questions that require extended answers will first be sat by candidates in January 2013

A detailed timeline explaining these changes is available here.

For courses beginning in September 2012 and candidates certificating in summer 2014 there may be revised specifications in GCSE geography together with changes to or fully revised specifications in GCSE English literature, history and religious studies. This is because of the suggestion that some qualifications may permit narrowing of the expected course of study. The regulators are reviewing all GCSE geography specifications and assessment materials. Awarding organisations will be informed of accreditation decisions by Friday, April 20, at the latest. For GCSE English literature, history and religious studies, awarding organisations will be told by mid-February if there is any further action required.