Thursday, 1 December 2011

MPs Debate the Use of Calculators

Yesterday's debate between Elizabeth Truss MP, Justin Thomlinson MP (Chair of the Parliamentary Committee for Financial Education) and Nick Gibb (Minister of State for Schools) is available here:

Here are some of the key insights they lacked:
1. Removing calculators from students will NOT create improvements in mental maths.  There is an intermediate step missing and this is the nurturing of mathematics teaching which uses the core structure and axioms of mathematics and does not teach algorithms and quotable facts without deep structural understanding.  Removing calculators could lead to more teaching of algorithms without structure which would compound current problems.
References: Elementary Mathematics by LiPing Ma. and subsequent posts
In essence overuse of calculators (where it occurs) is a symptom of the problem more than being a cause of it.

2. Developing fluency in tables has very limited benefit unless students do it in the context of developing scaling and other structurally sound semiotic structures for multiplication.

3. Both splitting and chunking are essential primitive structures for division.  It's also of great benefit for students to be aware of the scaling structure for division.
(Think - if you do 264 / 2 you use splitting - pulling 264 apart into two equal parts.  If you do 39 /13 you will probably chunk - thinking how many 13s are there in 39 rather than splitting 39 into 13 equal parts.  Different real life situations prompt different structural uses of division).

4. The issue of questions which should have been on non-calculator SATS papers appearing on calculator SATS papers had been identified by QCDA and they were in the process of addressing it by replacing those questions with appropriate calculator questions.  Appropriate uses of calculators include the use of calculators for questions where the results on the calculator demand interpretations and understanding that students could not access with non-calculator techniques and to assist the subordination of complex calculations in extended multi-stage functions which students could not otherwise access.
(contact me for further justification of this claim and more insight).

5. There are plenty of teenagers who are not experts on an appropriate variety of functions of smartphones and internet searches.  Yes they use smartphone but many of them only use them for a limited range of functions and they need the opportunity to reflect on the results internet searches generate to use them wisely in academic analysis.
(widely understood in education).

6. Yes it is important to focus on core skills but we have been doing this.  Some of the National Strategy has worked extremely well, other aspects have been seriously flawed.  Without insight into what has worked and what has not worked and why you cannot create an effective policy which will improve things.

Post Script:
At the beginning of the discussion Ms Truss asks why the issues of calculators in school has not been debated in parliament for so long.
The answer is because the quality of insight explored was far below that which would have been possible had the ministers involved attended a relevant consultation at ACME or WEF, which is what has been happening instead under the last government.

Wednesday, 9 November 2011


I now devote part of my time to developing the academic discipline of analysing the processes and outputs of mass online discussion (that's conversations in forums and the comments threads to blogs, news articles and so on).

My passion for this subject started in maths education - see for example this article:
And I am delighted to have been asked to manage and moderate the inspirational group:
Math, Math Education, Math Culture on, a role which I love and I would warmly invite any enthusiast of maths education to join this warm and vibrant group.

But my interest in forums has now grown beyond education and so I've started a separate blog specifically for it.  The first post in my new blog (which I shall run in parallel with this one) contains my notes on cyberrhetoric - which is the ways in which participants in forums can post to help conversations which are stuck or abusive become productive and respectful -from the session I ran at the Mozilla Festival in Grenwich on Sunday.  As well as developing this study further, I will also post on the analysis of the ways in which mass online discussion can enhance democracy and generate intellectual capital and I hope to link to other articles about mass online discussion.  Here's the link:

I shall keep this blog for maths education and related issues in education.

Friday, 30 September 2011

Busy, busy, busy....

Just to explain the lack of posts, I've been busy working, writing and sadly positing on political blogs about issues in education.

I wish I didn't feel I had to but I feel that someone should and if not me then who?

Hey ho.

But on a much more positive note one of my jobs has been to research and write this course:

It's given me a context within which to focus my attention on re-analysing the emerging technological infrastructure which can support mathematics education.

What I see being reality in 3-5 years time inspires me. 

In my crystal ball I see the integration of systems which track students progress with both axiomatic and personal skills in mathematics and to which all stakeholders can contribute together.  I see interactive teaching systems which can automatically populate elements of this system.  And I see accreditation based on such a system which can replace SATs with much more worthwhile, robust, diverse and meaningful low-stakes accreditation.

The fundamental elements are all now in place...... now I just have to figure out how to make others see what I see.

Tuesday, 6 September 2011

Ofsted part 9: A tough reality of the job.

When there is substantial failure, as there will sometimes be in public services no matter how hard we try, there is always great personal pain which is most effectively dealt with if somebody carries the can by being prepared to lose their job.  When individual inspectors defend their jobs too actively when there has clearly been substantial failure on their watch a regulator loses credibility.

For all sorts of reasons, schools have become overly deferential to the process and conclusions of inspection to the extent where judgements of doubtful validity with potentially very negative consequences are not properly challenged.  We need processes which expect there may be challenges to potentially contentious judgements and plan to allow such challenges to be constructive processes of  growth in understanding for both the school and the inspector.

In essence I am saying here that the recruitment process for the inspectorate should actively seek out those who would not be afraid to leave their jobs should that appear to be the right thing to do.  There are plenty of such people in education and I have known many who have become inspectors.  But sadly they seem too often to rapidly leave inspection and there is little evidence of their presence among those who define policy direction at Ofsted.

Ofsted should only employ people in core roles who can command the personal and professional respect and trust of our most dedicated and able headteachers.  If that trust breaks down and cannot be properly restored within a reasonable time the inspector be removed, preferable of their own volition.  In other words they should be accountable to the very best in education rather than to themselves.

Schools should only employ headteachers who can command the personal and professional respect and trust of their most dedicated and able teachers.  If that trust breaks down and cannot be properly restored within a reasonable time the headteacher should leave and our inspectors should be there to help in that process if necessary.  At present too often it is the case that a school falls to rock bottom at which point the staff and governors organise the removal of the head.  Then Ofsted turn up and hang around for the next couple of years in order to take the credit for the improvement the staff then bring about.

It's not blooming rocket science.  Is is?  Let's strip out all those processes of inspection which protect inspectors who are not up to the job and which are not needed by those who are and replace them with structures used by credible regulators.

Monday, 5 September 2011

Ofsted part 8: How could a purpose-led inspectorate effectively report on the quality of provision?

A purpose-led inspectorate of the type described here could report on the extent to which schools are reaching the high floor standard specified.

It could also provide detailed qualitative reports on innovative and exemplary practice, which is currently often barely acknowledged. To have access to such published information would be of great benefit to many stakeholders in education.

Where there is a clear argument for their being specific high standards to which schools could or should aspire, these could be provided by other agencies and quality assured and reported on by an education inspectorate.

Pages with lists of numbers would be lost.  Academic and political reviews of practices in inspection and regulation conclude that this would be beneficial to the quality of the services being provided by the organisations being inspected.

Sunday, 28 August 2011

Ofsted part 7: How could a purpose-led inspectorate effectively drive quality improvement?

By mandating, as part of the minimum standards criteria (discussed in parts 5 and 6) that schools participate in a quality assured improvement programs.

You'd neeed someone exceptionally bright (highly experienced in education with at least an MEd and a recent MBA as well) organising the regulation of the frameworks for such programs at the inspectorate.

They'd need a small team of people who worked to quality assure improvement programs (individual schools might be quality assured or bodies which groups of schools worked with may be).

The at last schools would have the freedom to select who they worked with to mentor them on quality improvement.  Wouldn't it be wonderful if we could have PRUs being mentored by bodies with specialist expertise in running PRUs for example?

Thursday, 25 August 2011

Ofsted part 6: How could a purpose-led inspectorate effectively identify and eliminate unacceptable practice? (part 2)

In part 5 of this section of my blog I pointed out that there could be considered to be two contexts in which a purpose-led inspectorate may need to intervene to protect against unacceptable standards.

The first (where specific issues have arisen which do not raise concerns about the leadership at the school) I considered in part 5.   In this section I shall consider:

What happens in cases where there is concern regarding the leadership at a school?

One of the most important ways in which our children can be protected from unacceptable standards in education is by planning to ensure that headteachers who are not performing well are replaced.

Dealing with this issue is extremely difficult and complex, as the same headteacher behaviour may be productive at one school and destructive at another.

It is also natural that the style of leadership exhibited by a headteacher and the decisions they make will not suit all parties at all times.  Such alienated parties may raise concerns which are sincere, but when should they be deemed to require the replacement of the head and when should the head be endorsed despite the concerns raised?

Despite these very difficult issues, the replacement of failing headteachers should be one of the functions of a purpose-led education inspectorate if one of its purposes is to protect against unsatisfactory standards in education.

It is therefore absolutely essentials that inspectors are exceptionally credible to the member of the school community they are inspecting.  They should be credible headteachers with substantial experience which is relevant to the type of school they are inspecting.  Inspection should be a natural options for career progressions for a well reputed headteacher who is seeking to move from full time work to part time work in the later stages of their career.

Consideration of the issues here leads to the conclusion that there will be instances where it is necessary for inspectors to monitor and mentor schools over a period of time.  Questions also arise as to who would be responsible for supporting an incoming replacement head given the current uncertainty regarding the provision of local authorities.

In part 7 of this section of my blog I will consider how a purpose-led education inspectorate body could deliver on the purpose of:
      2.   Ensuring that schools are working in a professional way to improve the quality of their provision
In part 8 I will consider how the purpose of:
3.   Reporting on what is going on in schools / the quality of provision.

Wednesday, 24 August 2011

Ofsted part 5: How could a purpose-led inspectorate effectively identify and eliminate unacceptable practice?

It is right and proper that all stakeholders in education should have the right to be consulted regarding what they consider to be unacceptable practice. 

Although there will be a significant amount of contextual variation in standards set (what is unacceptable in some circumstances is acceptable in others), nonetheless it is right that there should be processes where the boundaries between unacceptable and acceptable practice are defined and any possible exemptions from these boundaries are agreed.

Such a purpose for an educational inspectorate would lead to there being two ways in which a school may raise concerns regarding unacceptable practice.  The first is if it is failing to meet one or more of the standards laid down.  The second is if there is concern regarding the quality of the leadership of the headteacher at the school.  I will discuss the first issue in this post and the second issue in the next.

When schools fail to meet particular standards

Example 1: child protection
Say for example an inspector questions a teacher at a school his is inspecting about child protection issues and find that that teacher doesn't really understand the topic and doesn't know who the child protection officer at their school is.  That inspector might then divert from his original plans in order to take the time to ascertain whether this is a significant issues among staff at the school or not.  If he finds other teachers are properly aware of procedures he is likely to conclude it is not.  However if he find this is a systemic issue at the school he may will raise it with the headteacher.  The headteacher may already have in place a coherent plan for a training update for staff or he may, perhaps, deny the problem or create a defence of there being no need for individual awareness as staff would ask if an issue arose.  The inspector will be well placed to rebut either denial (by taking the headteacher with him while he interviews more staff) or deflection (because he has clear standards by which he is assessing staff awareness).

The inspection report produced will contain a statement of the issue, the proposed plan of action which the inspector should agree with the head and intended consequences (e.g. an unannounced re-inspection on this issue to take place in 3-9 months). There may also be a charge for the school associated with this which might typically be to cover the cost of that re-inspection.   

Of course the inspector would be assessing the headteacher during this interaction.  Do they know their school well?  Are they behaving in an appropriately professional and competent way?  If the behaviour of the head raises concern they will investigate further (to be covered in the next blog post).

Example 2: school security.
Supposing an inspector discovers that a school is failing to comply with particular standards of school security and that the root cause of that failure to comply lies with their being a public right of way through the school grounds, a circumstance not considered when those standards were drawn up.

There is likely to be a lengthy discussion between the inspector and the headteacher about this issue.  Between them they may come up with some improvements which could be made despite this issue.  The headteacher may talk about the enormous cost and possibly impenetrable public issues associated with changing the right of way.  The headteacher may suggest that change is neither necessary nor right for his community. 

In this case the inspectors report may contain a section which discusses this issue.  It may describe changes agreed and mandate the school to carry out a consultation and a cost/funding stream analysis of possible ways forward within the normal inspection cycle.  This would allow time for that inspector to report back to the bodies which set these standards, carefully describing the issues and exploring the possibility of specific exemptions/modifications to the criteria being developed for schools with public rights of way through their grounds.  This issue would then be revisited during the next normal inspection.  There may also be communication between the inspector and the headteacher in the meantime to help that inspector best inform the mandating bodies regarding the issues of public rights of way.

In essence I am saying that there should be professional/human interaction regarding the consequences of failures found.  Inspectors should be competent and properly empowered by their organisational structure to behave in this way.

Wednesday, 17 August 2011

Ofsted Part 4: What principles would underpin the actions of a purpose-led education inspectorate?

Firstly the purposes of inspection would be clearly visible at all times.
These may be:
1.    The identification and elimination of unacceptable practice
2.    Ensuring that schools are working in a professional way to improve the quality of their provision
3.    Reporting on what is going on in schools/the quality of provision.

Secondly, not only the law which explicitly applies to Ofsted:
 regulatory activities should be carried out in a way which is transparent, accountable, proportionate and consistent;”
but the principles behind this law would be embodied in the behaviour of inspectors and in the procedures they follow.

Thirdly, processes which generate spurious results or results which are not robustly valid would be eliminated.

Fourthly, there would be substantial feedback loops whereby conclusions both could and would be challenged according the now visible purposes of inspection.

Fifthly, and perhaps most importantly, where practice of concern was identified interventions would be of the minimum level needed to address them in ways which were appropriate to the institution being inspected.
I understand more detail is needed to bring this to life and I will attempt to provide more of that detail in the posts which will follow.  Please don’t hesitate to raise issues and post questions using the comment facility available here.

Sunday, 14 August 2011

Ofsted Part 3: Cultures of inspection (Process-led v purpose-led)

My journey into understanding why Ofsted is as it is has taken me deeply into the history of inspection in education and has also taken me beyond education into past and present cultures and process of inspection and regulation across many disciplines.
This posts attempts to summarise a key emerging theme which is the difference between process-led and purpose-led inspection

Process-led inspection occurs when regulators establish processes and procedures and the following of these processes becomes the main function of inspection.  
The culture which surrounds this is one where inspectors believe (often correctly) that by following those processes to the letter they can exhonerate themselves from any blame associated with problems arising at the organisations they inspect. 
There is an expectation that inspection processes will identify problems and that the methods for resolving those problems will be as clearly defined at the processes which have been followed have been.
In reality, the processes and consequences which have been laid down quickly age and were probably never appropriate for all circumstances in the first place so there is a poor fit between the outcomes of inspection and the organisations to which those processes are applied.

Purpose-led inspection occurs when the purposes for which regulators exist are clearly defined and the processes employed evolve to fit those purposes.
Questions asked and observation strategies may start from a standard process but will rapidly deviate from this so that the inspector can analyse and understand the situation being inspected. 
Rather judging the behaviour of the organisation being inspected according to abstract criteria, it is judged according to whether or not its behaviour is in accordance with its journey forward given where it currently is and its contexts.
Consequences of negative judgements fit the purpose of the identifying and resolving these issues effectively and efficiently.

It’s a human sign that there will be individual inspectors who will gravitate towards process-led inspection.  Politicians may either implicitly or explicitly demand it.  But it is essential that the directors of a regulatory body work exceptionally hard to ensure that that body is and remains purpose led and that the procedures it employs will support inspectors in delivering on that purpose.

The extent to which Ofsted has failed in this respect is extreme and the consequences of this failure have been devastating for staff in English education.  The duty for measuring the extent of this failure lies with Ofsted who have to justify, on balance, any deviation from its obligation the Legislative and Regulatory Reform Act (2006). I see no evidence that they are even aware of this duty.   

To be continued…. 

Saturday, 13 August 2011

Ofsted Part 2: A journey to the heart of matters.

Why do Ofsted grade the quality of education where a strong floor standard is clearly met when other regulators consider this to be bad practice which damages innovation, diversity and prevents them clearly seeing and understanding practice with which they are not familiar?
I took this question to the Westminster Education Forum Consultation:
Accountability in Education – the future of governance, Ofsted and league tables on the 5th of July 2011.
I put it first to Graham Stuart MP who has recently chaired a review of Ofsted.
He agreed with my observations that Ofsted clearly had a duty to establish and police a high floor standard for educational services and that it had a duty to scrutinise and audit schools detailed self-reports regarding their provision of services.  In response to my question regarding the grading of the quality of services, he said that he felt that there was an opportunity for me to contribute to the Ofsted’s current internal consultation regarding their framework to try and make sure the Ofsted fitted more with what I thought it should do compared to what it does now.
I was satisfied with this response and then addressed the question in a more succinct form to Richard Brooks, the Ofsted Director present at the consultation:
“Where you are inspecting an area of service in an institution and there are no concerns about it, why is better to grade is on a single scale rather than for there to be a qualitative report on it, provided by the school, which is a basis for professional dialogue and on which the inspector comments with any concerns or qualifications?”
His reply was (subject to possible transcript errors):
“I mean of course, there are epistemological attractions from having essentially a single judgment to make about which side of a line do you fall on, satisfactory/unsatisfactory. But you have a real problem about deciding where that line is, and what you taken into account when you think about whether people are
which side of it. So if you think about the overall effectiveness judgment, would we be making essentially a judgment analogous to inadequacy, where all of the other information was lost, where 90% of schools were essentially passing? You would lose a great deal of the richness about the discussion at the boundary between ‘is this practice satisfactory, or is it better than satisfactory? Is it good? If it’s good what can we do to make it excellent? What are the improvements points at a school which actually is not worried about being found inadequate, but really wants to strive to be
outstanding? What are the things that we think institutionally we should focus on?’ All of that would go. It would be a simpler inspection system in some ways, but I think you would lose a lot. Perhaps I have not understood you, but we can talk at the break.”

This was, of course, and answer which attempted to justify the practice of grading the quality of provision without engaging at all with the reasons why this practice is unsatisfactory or to criticise possible alternative practices which serve the same purpose.

We did speak at the break and I was concerned that Mr Brooks did not appear to have any awareness of the arguments make against the status quo by other regulators.  So I suggested that I come and talk to Ofsted as Graham Stuart MP had advised.  That suggestion received a very definite refusal and that categorical statement that the grading of the quality of provision would not change, however Mr Brooks agreed that if I put my question in writing by email he would respond.

Rebecca Hanson to Richard Brooks July 5th

Dear Richard,

Thank you for your time today.

You promised to ponder the question I asked and attempt to answer it more fully.

Just to remind you the question was:
Other regulators interpret the Hampton principle on page 11 of the attached report (regulators are only mandated to intervene when there is a clear case for protection) to mean that they have no duty to attempt to rate the quality of provision where there is not a clear case for protection.  There are considered to be important reasons why rating the quality of provision where there is not a clear case for protection militates against innovation and high quality provision.

Please can you explain to me why Ofsted does rate the quality of provision where there is no clear case for protection?

Best regards,

Rebecca Hanson

Richard Brooks to Rebecca Hanson 6th July

The two short answers are:
-          whilst we act as a regulator in respect of some services, we act as an inspectorate in relation to the education provision within our remit.
 -          we are a creature of statute, and the relevant legislation, including the Education Bill before Parliament now, is explicit that we will report on quality in school inspections. There is of course a valid debate to be had about the merits of taking this approach, and whilst I’m not seeking to dismiss the alternative, I would personally come down on the side of quality ratings. Indeed I think the protection principle can very well be used to justify inspection of quality in relation to many services for children. In the end, however, it is not a decision for us. Kind regards Richard

In November 2009 Ofsted was specifically included under 2006 Legislative and Regulatory Reform Act to which most other regulators had been accountable since 2007.  Therefore it is bound by precisely the same law and principles as other regulators.  Yet although this Act was constructed to militate against poor practices like the grading of the quality of provision of services, Richard Brooks was claiming that Ofsted is forced to grade the quality of the provision of services “it is not a decision for us”.

So I submitted a freedom of information request to Ofsted to find out if there was any justification for Richard Brooks claim.  The response is 3 pages long.  The only possible partial justification for the claim is a reference to a statue which designates that Ofsted must report on the quality of provision.  I cannot find any legal imperative that this must be done by the auditing of quality standards or though qualitative reports with disciplined language.  There is substantial justification for the assertion that the quality of the work done by Ofsted would be much higher if alternative practices were used.

So I ask again:
Why are those involved in education in England subject to inspections which are considered to be bad practice by other British regulators because they militate against diversity, against innovation and against the regulator actually perceiving what is happening?

Richard Brooks claims that Ofsted is forced to do from outside.
By what or by whom is he forced to do that?
His response to me question showed a very serious lack of understanding or either inspection and regulation and a total lack of insight into the realities of the consequences of current inspection practices on education.

Where is Ofsted’s ‘proper conclusion’ that this practice is in the interest of the education of our students?
Where is anyone’s ‘proper conclusion’ that this is in anyone’s interest?
Where is the cost benefit analysis I can look at which properly understands and justifies the costs of this quasi-objective system?

To be continued...

Monday, 8 August 2011

Ofsted part 1: A journey into questioning the status quo.

Time for an interruption for something anglocentric I’m afraid.  I will come back to the yin and yang theme after that and in the meantime please do continue to comment on any old posts.

Like the vast majority of those in English Education, my assumption that there had to be a coherent explanation as to why reason why Ofsted is as it is was deeply ingrained.

I’ve seen such horrors, but I assumed that it was only me – that my experiences were not representative.  I’ve seen the gifted teachers destroyed by ‘unsatisfactory’ ratings.  I watched strong pedagogies disappear because they weren’t ‘Ofsted-proof’.  I've seen great school put into special measures by teams who’ve turned up with scores of ‘fail points’ those schools had to disprove in order to avoid the status and have been deemed to be ‘failing’ despite disproving very nearly all of them.  I’d read the statistics about the correlation between Ofsted gradings and exam outcomes.

Yet still, when I accepted a position as Head of Maths at a school in special measures, I expected the frequent dialogue with inspectors to be a constructive experience.  The reason for this was that our HMI was the top inspector for maths in England.  My personal experiences of such characters had been so positive that I was convinced the interaction would be appropriately challenging and constructive.  With excellent academic and teaching credentials, a credible history of achievements in mathematics education at national and international levels, a clear plan as to how we would harness the substantial technological resources we had to transform results and a strong network of friends and contacts across mathematics education I thought it would be a constructive experience.  I am well known for relishing every challenge and seeing it as being an opportunity to achieve deeper insight.

The reality was…. indescribably awful.  

Yet still I was convinced I must just have been unlucky. I started to listen to others as I attended conferences.   From a distance Ofsted had done a wonderful job at their schools.  Close up the picture was very different.  But I could only get ‘close up’ because teachers trusted me.  They could see I’d been there.  They didn’t feel at all at ease expressing their experiences and had no fluency in doing so.  Most were embarrassed by their experiences.  Few felt that there was any point in talking about it as nothing would change and they would be labelled trouble makers.  

In essence it seemed that the quality of the reports and gradings were often very low.  The constant numerical grading of everything created a pseudo-objectivity to the reports, but it often seemed that those numbers (or at least the average/corollary grading of them) was determined prior to the arrival of inspectors and the reports were cut and paste standard comments which linked those grades with few modifications.  This seemed to be much more the case in schools with challenging cohorts which did not choose to follow strategies where by all staff were ‘singing from the same hymnsheet’ all the time in order to allow teachers to have a degree of professional responsibility and freedom in allowing them to teach to best of their abilities and to adapt rapidly to evolving circumstances.

It seemed to me that the process of grading the quality of provision was counterproductive because it created a false sense of objectivity, it created abstract definitions of what a high quality education was which meant that much great teaching went unrecognised and, most importantly, it seemed to militate against dialogue between schools and weaker inspectors who could hide behind their own negative judgements of the staff who disagreed with their opinions.  The feedback loops were virtually non existent for most staff and the assumption that weak results indicated that ‘weak teachers should be punished’ could only be countered by very strong, eloquent and established heads.    

It seemed to me then that what was needed instead was that Ofsted should be responsible for clearly defining ‘unacceptable’ practice in collaboration with all stakeholders and that it should demand (as part of the process of avoidance of ‘unacceptable practice’) that schools provide detailed qualitative descriptions regarding their current practices and their processes and plans for improvement and that these should be scrutinised, audited and mentored by inspectors.  

Yet still I assumed that Ofsted must have a justification for continuing to use these numerical grades for the quality of provision.  So I started to suggest in both face-to-face and in discussion forums that they shouldn’t in order to try and reveal what this justification was. Coherent answers or justifications for what Ofsted was doing did not arrive*.  Instead I found that international education regulators and other UK regulators considered the grading of the quality of provision above a high floor standard to be poor practice and effectively to be illegal under the Legislative and Regulatory Reform Act (2006) which enshrined the Hampton Principles for regulation for the reasons I suggested.  To define what is high quality provision militates against innovation, diversity and regulators properly tuning in to and seeking to understand what is actually going on.  Improvement is more effectively achieved through detailed scrutiny and mentoring of provision and improvement processes.
(more to follow).

Sunday, 7 August 2011

The yin and yang of maths education - Part 7. The difference in the nature of the red route and the blue route.

File:Yin and Yang.svg

There are two arrows from zone C to zone B.  How are they similar?  How are they different?

When the purple arrows are in operation, the brain is constructing a need and place for the knowledge which it acquires before it receives it.

But what of the red arrow?  Is the opposite true?  Is knowlege being presented before the brain is ready to receive it?

I would say sometimes and sometimes not.  I would also say that where this is happening it is not necessarily disastrous.

Sometimes and sometimes not: A wise and experienced teacher will not tell students what they are to learn, they will guide them through experiences and help them develop their own insights which will naturally lead their brains to demand the knowledge which is to be taught.  This is appropriate in many areas of mathematics where knowledge is constructed in an axiomatic way.  In these cases there will be little difference between the red and the purple routes.  However not all mathematical skills lend themselves to axiomatic construction.  This is particularly true of human skills such as resililance and application to contexts, but is also true of other ontological aspects of mathatics such as the definitions of root concepts.

Where this (knowledge being presented) is happening it is not necessarily disastrous. This is particularly true if the student has developed a mind which is used to acting in a constructivist way.  Such a student will look at the knowledge being presented and will work actively to see if they can construct a place and a context for it given what is already authentic to them.  It is useful that students develop the ability to do this.

Tuesday, 12 July 2011

The yin and yang of maths education - Part 6. What are these paradigms?

File:Yin and Yang.svg

Here's a diagram which I offer as a tool for thinking.  All diagrams are partial.  They are only in 2-dimensions after all.

In the traditional paradigm, the maths to be learned in a lesson is selected from the mathematics which has been proved and written down.

In the constructivist paradigm, however, the focus is instead on nurturing the way that students mathematise the world around them. Because the focus is on the thinking processes to be developed, the maths to be learned is not clearly specified.  Hence it becomes irrelevant as to whether it is mathematics which has been previously proven or not.  In my opinion and experience substantial amounts of mathemics is learned during these lessons, but that mathematics is most powerfully learned when it is backed up with some traditional teaching at another time.  There is also a job to be done to fill in the gaps of core mathematical techniques which have not been learned.

But - oh - the magic of invention.  When students have aha moments.  Which a child expresses something incredible that you've never imagined before.  The intellectual journeys you take as a teacher.  Please excuse me while I get a little misty eyed.  I can understand why some teachers who discover the constructivist paradigms for themselves can become rather messianic about it.  I wouldn't want to teach without it being part of my teaching.  It inspires me.  My students inspire me.

Monday, 11 July 2011

The yin and yang of maths education - Part 5. What was new and what was the same?

File:Yin and Yang.svg
I don't think my mathematics department was so different to some of the most inspirational department I had encountered and heard about in my early days of teaching.

Some of the subtle skills of teaching we wanted had been obliterated by our managerialist, ofsted centred culture but because our need and desire for them became obvious, we could spark them back into life with the help of retired teachers who came in and reminded us what to do.  But, as I've said before, they were only part of the story.  There were many types of teaching going on, the majority of which would not have appeared different to mathematics lessons in other schools.

My students matured in a way I liked.  Instead of having to sit down and shut up until they were permitted to express themselves, they were encouraged to express themselves until they learned how to control themselves.  A student passing test at level 6 might suddenly realise that they did not properly understand place value.  If I didn't have the time to work with them one-to-one I would show them the relevant part of mymaths and let them explore it for themselves.  Having done that they might do the same mymaths work again the next lesson, and again until the penny finally completely and irrevocably dropped.  20 years ago a teacher might have done the same with a text book, so my thinking is not so different.

So teachers were freer and students were freer and yet all the disciplines and demands in place previously were still in place.  We quite simply had better tools at our disposal to facilitate what we wanted to achieve.  I have chosen to teach in a school where I had a big, flexible classroom with plenty of space and ICT. It would be in appropriate to claim or recommend that it was just the ICT which mattered.  I had an interactive whiteboard at one end of the room and ordinary whiteboards at the other.  This helped me teach different groups different things.

I don't want to talk too much about exactly what I did.  Each teacher did it differently and that was a big part of the point and I'm sure if another school used the same structure for thinking they would be different again.  I'm not trying to get people to copy the detail of the way I or any of my teachers taught so I won't say any more about it in this part of the blog.  Hopefully I've said enough to stimulate others to think about their situations which is my main aim.

Sunday, 10 July 2011

The yin and yang of maths education - Part 4. Harnessing Technology.

File:Yin and Yang.svg

By the time I was ready to apply to be a head of department, I had a new tool at my disposal.

Mymaths ( provided online interactive lessons on every secondary mathematics topic to tie into the UK National Curriculum.  Although the accompanying homeworks were not yet in place at that stage there were already booster packs which could be used as homeworks.  These gave me detailed information on the progress of each students and, in doing so, empowered me to require that each students work through the curriculum at their own pace.  The quality of the interactive lessons was much higher than previous online tools due both to the high level of mathematical and pedagogical insight of the designers and also to the interactive visual tools they had incorporated.

I had a very clear vision that I wanted use this technology, combined with individual examples of key topics incorporated into lesson starters, to take a great deal of the strain of teaching the National Curriculum, leaving the teachers at my school with substantially more freedom to decide how best to use significant portions of lesson time.

The 'synthesis of paradigms' strategy was probably most evident in students homework, where I set them an online homework each week which was automatically marked and therefore had more energy to spend on the detailed marking of personal investigations, also set as homeworks.  In this second type of homework students were required to think for themselves, independently interpreting and analysing pure and applied mathematical situations.

Classes probably didn't look so different as you might expect.  We did some applied tasks and some rich investigations.  Some lessons the students worked primarily with high quality texts.  Students were explicitly taught to explain their thinking to each other (see or example the previous section of this blog on how the Chinese do it) and their skills in listening to, explaining to and supporting each other were explicitly cherished.  We were dealing with many challenges at the time, such as mixed ability classes (due to the school being small and working towards closure) and having many students with challenging behaviour and specific learning difficulties and it was my perception that the framework I had allowed us more time to focus on using teaching methodologies which responded the needs of our students rather than on our need to deliver a National Curriculum which contained more material than many classes could cover.

By focusing on teaching rather than on the content to be covered a substantial proportion of the time, in our different ways we covered many elements of that National Curriculum in powerfully connected contexts.  Thus they were relatively easy for students to understand during starter questions and during those online homeworks.  Those individual questions and the computer based teaching wouldn't have worked without the rich earth of experience in which they were taking root.

This structure for organising teaching and learning also allowed me and other staff to focus on some of the more challenging pedagogical strategies which are often neglected in the pursuit of curriculum content coverage.  In experimenting in this way I found that I experienced substantial personal growth as a teacher and my student clearly benefitted from this.

I felt I was delivering on Barcelona.  The two sides of maths education were empowering each other:

Personal investigation of the world is much more effective if  the personal investigation has mastered the use of many key tools and the universal vocabulary which will help them analyse and interpret what they see.

It is easier to learn and more deeply understand can personally make connections in a curriculum if you have spent time considering the situations where the maths it involves is generated and you have experienced the need for that maths.

Verification of my claim for the success of this strategy is limited.  Our results were very good despite the considerable challenges we faced but there were only four externally examined year groups and they were small.  But I think I can validate my claim in the theory of education and in the intuition of experienced teachers.  Much of the content based curriculum was being discovered in context and then taught in abstract.  Students needed core techniques in order to be able to think mathematically and mathematise situation.  To me it just all made perfect sense.  And I think that, as we move from Web2.0 and 3.0 to Web4.0 it will make easy and fluent sense to many others too. I'll continue to work on this validation in my future posts in this section.

I wrote up my efforts to synthesise the paradigms of teaching when the school where I was working closed in 2008 and these can be found here, under my previous name (Rebecca Teasdale).  Sadly there is a £3 charge to access this article.

Saturday, 9 July 2011

The yin and yang of maths education - Part 3. How to make enemies of natural befellows.

File:Yin and Yang.svg

I wondered about the tendency of some to attack the paradigm of maths education which is not their favourite.  Why do it?

I think there is a natural human tendency to universalise conclusions.  When you find something which is good it is natural for you to want everyone to share your experience.  It's easy not to notice that others may also have found something else which is equally exciting and fulfilling.  I think this is more true for the young and less true for those who have experienced the limitations of thinking in this way many times before.

But there is another force which has made enemies of these natural bedfellows.  It is that the organisational forces which are required to make maths education function in a school have needed to be configured predominantly one way or the other.  Schools have either had programs of study, testing and teaching which are organised according to the mathematical techniques and vocabulary to be learned (with perhaps specific times allowed for constructivist activities) or the curriculum has been child thinking centred (with perhaps specific times allowed for exam preparation).  Those teachers who have been in departments configured for teaching methodologies which have not suited them have suffered and so they have complained.  If they have not been heard then they have come to resent a status quo which prevents them being the teacher they are capable of being.

Let's imagine some parallels:
Say schools had to choose either to teach French or Spanish.  There would be great debate as to which was better and such debate inevitably includes consideration of which is worse.  Now suppose schools were required to teach both French and Spanish.  After the initial strains of reorganisation do you not think that concerns associated with which language is worse would simply disappear?  Do you not think that teachers would instead look at the synergies between the two languages and at ways of teaching which would enhance students' progress with both?

There is another parallel in society as a whole.  I have long appreciated the benefits of the philosophy of liberal freedom.  I have also understood socialist values.  And in my appreciation of both positions I have come to understand that society has had to configure itself either one way or the other.  Either you have centrally organised socialism or you have liberal freedoms with much less central organisation.  But that has changed, hasn't it?  For me the most obvious joy of the 21st century is that ICT has given us the power to develop much more complex and versatile administrative infrastructures which can more easily and effectively integrate social and liberal ideals.  Isn't socialism much more powerful when it is powered by the financial contributions to society of those who free to generate substantial wealth?  Aren't we actually more free if we know that should we become vulnerable in an aspect of our life there will be support available to us?

Broadband is a magic bullet we can use to eliminate that jar between many pairs of traditionally opposing paradigms.  By 2006, when I was preparing to become a head of department, I knew that I had new and very powerful tool at my disposal in my question to cherish both the teaching and learning of traditional mathematics vocabulary and techniques and the development my students as creators and appliers of mathematics.

Thursday, 30 June 2011

The yin and yang of maths education - part 2: My journey into this thinking.

File:Yin and Yang.svg
I took my GCSE in maths in the very early days of that exam - when it was supposed to be a revolutionary change from O-level.  And what a revolution it was!  Previously my set would have done O-level in a year (at 15) and then AO-level in the next year (age 16) so that the most able could go on and do A-level the next year and Further Maths the year after that.

All of a sudden we were doing GCSE.  Over 2 years so we had time to grow.  With barely any defined content apart from some odd stuff on shears I seem to remember.  Instead we had to do 6 extended projects which we designed ourselves.

It was complete and utter chaos.  Our poor teacher had no experience in running classes in this way and suddenly he had to supervise 34 students all doing their own independent research but being marked to a core scheme.  The intensity of the strain on him meant that those of us who had completed two pieces of work to a decent standard (only two were graded) we left to our own devices while he struggled to help the others through.  We played a lot of cards.  It was fun.  It made me think, a lot, about what the heck was going on?

After 2 years of that the school thought we simply couldn't do A-level in a year.  We hadn't done AO-level- how on earth would we cope with a whole A-level in 9 months (Sept to May)?  They didn't know what to do with us.  We told them to let us try and, for reasons no-one could fathom, we coped.  Instead of one or two completing further maths, 7 of us did and instead of 1 or 2 going to Oxbridge, 6 of us got in that year.  Weird!

What we'd been through wasn't sensible.  It wasn't sustainable.  It wasn't fair to teachers!  But it was interesting.  I can still so clearly remember the contents of one of my independent investigations.  I developed a great deal through doing it.  But how much time did we waste which we could have spent studying stuff?

It was no wonder the traditional teachers complained and there was a swing back to knowledge and skills based learning.

When I entered teaching I came to understand the frustrations of the teacher who was great at teaching kids mathematical knowledge and skills but who felt overwhelmed and frustrated by the idea of teaching them how mathematise the real world or to construct mathematics for themselves.  Until you've seen people doing the latter well and have experimented with it for yourself, it's really difficult to imagine, especially if you're working in a teaching environment which is structured with the assumption that traditional teaching will be taking place. I've watched teachers try and fail to make the shift in their teaching.  I've watched some come to demonise constructivist teaching because it hasn't worked for them.

On the other hand I've seen the most incredible and inspiration teacher from the constructivist paradigm lock themselves in a world of beliefs which denigrate traditional modes of instruction.  "Exams are bad", "you should never test kids", "you shouldn't grade kids and so on."  For me these arguments have some meaning and validity.  I have always tried to be aware of the contexts and particular students where such actions can be counterproductive.  But on the other hand I've seen how most kids love to collect badges, to tick boxes, to achieve closure and completeness.  How if you ask them to climb a ladder they will say 'how high?'  And I don't see that this is a bad thing. 

Most teachers clearly belonged either in one paradigm or the other and I deeply admired teachers from both, provided that they were at ease talking about why they were as they were, their motives were sound, they were sensitive to childrens' needs and they were at ease with and, when they had the energy, interested in teachers who were different to them and ideally keen to see if there was anything worthwhile for them they could learn from them. 

Some heads of department lay in the traditional paradigm but tried to make as much time as they could (without compromising the teaching of core skills) in curriculum planning for activities which focused on developing students' investigational and thinking skills.  I understood exactly where these teachers were coming from.

Other departments put the constructivist approach first (in a wide variety of ways) and signficantly subordinated the teaching of core skills and exam preparation.  Now where this worked well it worked brilliantly and I think this was because teachers who constantly teach in this way are constantly educated and developed through their own teaching.  They therefore become exceptionally skilled teachers.  If they can combine this level of skill with secure exam preparation you get an incredible maths department that students and educated or aware parents love.  But if less skilled teachers try to imitate this structure of teaching without proper support it can all go rather wrong.  So wrong, it seems, that we needed an inspection regime which abolished all departments of this type, good or bad (my apologies to those departments which struggle on against the odds and are, in my opinion, the exceptions which proves the rule).

So that was the background to my thinking.  The interactive teaching technology with personalised tracking came along..... and I'll write about how that changed things in my next post.

Saturday, 25 June 2011

The yin and yang of maths education - Part 1.

File:Yin and Yang.svg
Many authors have written about the need to move from a curriculum which teaches an established curriculum to one which encourges students to mathematise their experiences of the world for themselves.

I see things a little differently.

I believe that both forms of mathematics education are justified and of great importance.

Modern technology offers students a wide variety of ways by which they can efficiently and effecitvely acquire orthodox knowledge.  Teachers who utilise this technology can create more time in the classrooms on the types of tasks which effectively develop students' abilities to interpret real life situations, struggle with connected and extended task and to learn to express and develop their own thinking. 

Modern technology can also enhance the quality and relevance of the experience a students has when they explore mathematics for themselves because they can rapidly interact with established thinking, notation and invididuals who are interested in their area of enquiry.

So it is natural both that more constructivist teaching can and will occur and also that such teaching will lead to a greater coverage of core concepts.  But I prefer not to think of the outcome as being a shade of grey although that's how my lessons might appear to others. I prefer the visualise black and white, yin and yang.

Here is an analogy.
Which is better - pop music?  classical music?  something which is halfway between the two?  or something which is unashamedly both, black and white, yin and yang with no compromise to either?  Here's a video to illustrate my preference.

More orthodox justifications for and desriptions of this perspective will follow as and when I have time.

Sunday, 19 June 2011

How do the Chinese do it? Chapter 8 - A vision of much more

Last night I attended an e-seminar with Alexandre Borovik.  I struggled to follow much of it as the conversation was heavily accented and the sound quality poor.  But then, in the last few minutes of the hour, something tremendous.  I opened a draft of his book:
Shadows of the Truth: Metamathematics of Elementary Mathematics
and in it, muddled among many other paths of thought, I could see that he has this incredible insight into the deeper concepts of mathematics that only children and highly eduated pure mathematicians can see.  Insights of which I have only begun to scratch the surface in my teaching.

Thank you Alexandre.

How do the Chinese do it?  Links to my other blogs in this series.
4. Part 4
5. Part 5
6. Part 6

Sunday, 5 June 2011

How do the Chinese do it? Part 7: The Last Chapter?

After the initial blast (maybe 6 lessons) of starter questions of this type, I simply come back to this starter activity whenever I feel like it for one-off questions.  It's worked well for me right up throught topics like surds and complex numbers.

There's nothing whizz bang or whistles and bells about it.  Students aren't inspired as they walk into the room, there's usually a bit of grumbling (oh no not this again) but their engagement builds gradually as they have to think for themselves mathematically.

It's effortless to set up (just 4 letters and 2 numbers on the board). I like the fact that I can spend my time listening to students and helping them give voice to their personal journeys into understanding primitive structures for mathematics rather than on class management. 

This task rests easily with there being a little off task chat and it taking some students a minute or two to properly engage.  As I've said before I've not taught in schools where there have been high standards of behaviour and I have never had the luxury of expecting all my students to come to my classes bright-eyed, bushy-tailed and well behaved.  I've had to learn to take my students from being badly behaved to being fully engaged and therefore on task through the way I teach and the type of task I use rather than through demanding that they behave well before I start to teach.

It's important to understand that this task will become more powerful for you as you use it with more classes.  You will gradually become aware of a wider and wider variety of structures that students use to support their thinking and as you become aware of them you become better at spotting what it is that a students is struggling to explain and at helping them to express it clearly to their peers. I think you will be surprised how your own thinking expands and takes you in the direction of confidently using flexibly the wide range of structures the teachers in Liping Ma's book used for calculation.

Is this the last post on this topic?  Maybe not.  Maybe you or I will come back with new ideas stimulated by this task.  I hope so.  Thanks to all who have asked questions.  Please keep them coming!

How do the Chinese do it?  Links to my other blogs in this series.
4. Part 4
5. Part 5
6. Part 6

Saturday, 4 June 2011

A little light relief

Many thanks to the teachers at our local school for putting the spirit I know so well onto Youtube.

This was there way of livening up a year 11 leavers' assembly which would have been otherwise weighed down too much by thoughts of the loss of a student.

Cockermouth School Teachers do Glee 2011!!!!

The news article

How do the Chinese do it? Part 6

I always ask students to describe their thinking on these problems.  To compare and contrast the structures they are using. 

Sometimes they work individually and I pick people at random to talk to the class, somtimes they work in pairs or groups. 

Then I'll 'go at little crazy'.  We'll do an SDPQ with two 1cm lines.   Or two lines with obviously different but unmarked lengths.  It sounds bizarre but the conversations are wonderful.  Structural insights are everywhere.  The students are really thinking hard.  By now they know I'm serious when I say this is all about the journey and that I'm only a tiny bit interested in the answers.

When they're ready we'll move on to algebra.  We can use letters, constants, linear term, quadratic terms and so on.  Remember learned tricks only get half marks.  They need to be able to explain structurally what's going on.  They will need to take those structures they've developed with numbers and transfer them to the algebra.  It's powerful, it's challenging, it's rewarding and its deep.  You should expect the unexpected, namely that when students describe their thinking you will be hearing things you have never heard before and will need to go away and think about.  Don't worry - if you get stuck you'll probably find the student who came up with it has had deeper thoughts themselves after a day or two.  Those who say something innovative often think about it a lot after the lesson.

How do the Chinese do it?  Links to my other blogs in this series.
4. Part 4
5. Part 5
6. Part 6

How do the Chinese do it? Part 5

If I was teaching a lesson on long multiplication or long division, I would set an SDPQ starter which contained one of the toughest questions they would face.  It was fascinating to see them trying to wrestle out an answer.  The method I was teaching that lesson would then become closely contextualised into their existing thinking as this existing thinking was now active, rather than being developed somewhere separate in their mind. 

They didn't object to doing a lesson on long multiplication or division even if they could do the initial example, because by now they understood the validity and power of having multiple methods and of revisiting structural methods.  But, of course, in general I found they needed far less teaching on basic operations, because we were covering it very effectively through these starters.

One structural lesson I would never miss would be counting the squares in rectancles which leads to grouping them in hundreds tens and unit in a visual representation of the grid method for multiplication, because this so beautifully scaffolds the expansion of quadratic brackets and more.

How do the Chinese do it?  Links to my other blogs in this series.

Wednesday, 25 May 2011

No I don't want to write about this.

Big sigh.  All I want to do is be a maths teacher and to contribute to the maths teaching profession.  I just want to carry on with my blog about maths education.

But there is a need to interject.

He's really doing it isn't he? described, in December, his intention to launch a cultural revolution on teaching in England.

And, with his plausible lies about 'increasing teacher freedom' and Red Ofsted Guard already in place he's off and running. Every day there are more edicts, each more disconnected and ludicrous than the first.

Here are today's:  It takes a while to sort out failing teacheres because, er, we sort them out.....failing situations can happen to anyone during their career.  Problems at home, a personality clash at school, changes from outside, stressful and bullying inspections.  How many great teachers have each of us seen go down?

Failing, contentious and problematic teaching situations occur for all sorts of reasons, usually there are multiple contributory sources which it takes a while to unravel and put right.  There aren't thousands and thousands of teachers who clearly are failing teachers Mr Gove, this is a fantasy of your imagination.  There weren't millions of enemies of the state in China either, but the cultural revolution demanded they exist so people had to invent reasons to blame others to avoid the beatings themselves. 

At consultations groups Conservative MPs happily tell us that the days of policy in education being generated by those in education are over.  Mathematics education policy will now be gerated in accordance with the voting preferences of average members of the middle class public.


I sometimes wonder, do they really believe that Maggie Thatcher pulled her policies out of the air?  Do they not understand that she liaised with a substantial professional and business community of experts who understood what she was doing much better than she did?  Who had analysed the consequences and saw how horrific they would be but had also analysed the consquences of doing nothing.....  Who wrote and consulted through both the main press and the IOD and the journals and so on.

This government is nothing like that.  Honestly for those of you who are not involved in the upper echelons of education please believe me that there is no link between what those who are respected in education are advising and what Gove is actually doing.  There are just people floating round desparately trying to limit the damage and make good from bad despite the horrors of it all.

This is surely just a nightmare.  I'll wake up soon.  Please.  This blog post wasn't needed.  We can go back to undertanding the possibilities broadband brings for enhancing education in the 21st century rather than suddenly having to ditch all that and focus on sacking teachers of subjects students find relevant and forcing reluctant 14-16 years old from tough backgrounds through a 1950s grammar school education. 

Please reassure me this isn't real.


Thursday, 19 May 2011

How do the Chinese do it? part 4

S,D,P,Q  7,7012

Try it - it's hard!  Students will rapidly write:
Sum 7019
Difference 7005, -7005   (I like to ask them to fully justify that -7005, remember that the trick:  it's the negative of the other difference, which they will have discovered by now, only gets them half a mark)

The multiplication will keep some busy a bit longer.  Many will naturally split the 7000 and the 12.  It's nice to get them to explain and notice what they are doing here too.

But the divisions?  Nasty!  The obvious division is hard enough but the reciprocal?  They will wrestle with it.

I often give full credit for a numerical answer which is very nearly right and has a correct method and a lot of effort behind it.  It helps to emphasise the aspects of their work which I value and to keep them engaged.
They will more deeply consider their thinking about reciprocals because it is challenged. 

The answers aren't the point yet. It's all about the method and their thinking.  If they get that right then the answers will come and the methods won't be forgotten so rapidly as they are when we teach them abstracted algorithms.  Because they have discovered their method for themselves they will have the confidence to believe they can do so again as and when they need to.

One nice aspect of this pair of numbers is that they encourage students to estimate.  One of the answers is going to be about 1000 isn't it.  Does that mean the other will be close to 1/1000?  Can we be more accurate than that?  When students get the hang of writing divisions as fractions (which is a really useful skill), change the rules - allow only mixed numbers or allow only decimals to at least 3dp.  It's your classroom - you can have different rules for different students at different times provided the underlying theme is fair - that all students are being stretched.

Choose another couple of examples like this yourself.  That takes us to about an hour of classroom time doesn't it?  What have the students learned in this time?  What have you the teacher discovered?

Thursday, 5 May 2011

How do the Chinese do it? 30 minutes in. Reciprocals in Calculation.

SDPQ 3,9
SDPQ 2,5
SDPQ 1,7
Each is a 5-10 minute starter.
Watch as you and your class discover and explore the fundamental structures of mathematical calculation.
One of the most surprising things which reliably emerged with my classes was their insight into reciprocals.

They're always looking for short cuts so they rapidly spot the links in the results for division:
4 and 1/4
3 and 1/3
2/5 and 5/2
7 and 1/7
They're reciprocal pairs.  Even though they only get half a mark for a second quotient calculated in this way, they still spot the pattern.  So my students have a strong familiarity with reciprocal pairs and an insight into their relevance for multiplication and division from early in year 7 (age 11) when I first teach them.  I'm sure that younger students could cope with this too.

It's so easy and natural but I've never seen this understanding in other classrooms. Usually in the UK reciprocals are taught as a disconnected entity at GCSE level or are found in algorithms, again without their connected context.  Yet a structural understanding of them is clearly there in the Chinese classrooms described by Liping Ma.  Chinese teachers use them fluently and flexibly for calculation.

Of course I'm not saying that the Chinese teachers taught this in the way I did.  I'm just trying to convince teachers that it is possible to replicate significant aspects of the Chinese teaching strategies with classes of students and teachers who have never thought structurally about calculations before.  Oh and I always taught in tough schools.  No docile classes for me.  And I found that students were more settled if I taught them in this way.  It suited them to focus on structures rather than to learn recipes.  They felt more in control of what they were doing and their education became more relevant to them.  They weren't good at 'learning recipes'.

There's more to come....