Monday, March 30, 2020

How on earth are we going to give grades this summer?


This is a premature blog, given that we will know what the plans are inside a week, but I am writing to get my thoughts in order and to raise some of the potential problems ahead of a final decision being made.

There are three important issues here:
  1. There isn’t a way to do this that everyone will be happy with.
  2. Something has to be done, and the organisations charged with doing this have an impossible job.
  3. You can have a simple system, or you can have a fair system, but you can’t have both.
The third of these is important, because if you have a simple system it will be unfair on very many students.  Yet the more complicated the system is, in a bid to make it fairer, the more difficult it will be to understand, so it won’t appear to be fair. 

One reason for writing this is that I can think of only one system that could be vaguely fair, but it is complicated, and I worry that when it is published it will be difficult to understand. 

The statement from the DfE

On the Gov.uk website (1) the DfE’s priorities are set out, shown here in italics.

This year’s summer exam series, including A levels, GCSEs and other qualifications, and all primary assessments, have been cancelled as we fight to stop the spread of coronavirus.

This is unsurprising and I can’t see any way around this.  We do not know how long schools and colleges will be closed for and there is no good way for exams to be taken without breaking the government’s rules on congregating in public.  We are not equipped as a country to sit exams online and there is not the time to set up secure ways of doing this (for example, how would you know that the person sitting at home behind their computer is the candidate and not a sibling or parent?). 

So: no exams, then. 

The Government’s priority is now to ensure affected students can move on as planned to the next stage of their lives, including going into employment, starting university, college or sixth form courses, or an apprenticeship in the autumn.

This means having some way of determining which students have achieved qualifications in particular subjects.  If exam results were only used as a way of proceeding to the next stage of education or training then it would be possible not to award grades at all, but it would seem unfair on the current group of students if they were not to receive their own grades.  Universities have made offers based on grades and will presumably continue to do so.

This means ensuring GCSE, A and AS level students are awarded a grade which fairly reflects the work that they have put in.

This doesn’t say “that fairly reflects the grade they would have got in their exams”. 

There will also be an option to sit an exam early in the next academic year for students who wish to.

There seems to be an expectation that this will happen in September.  This seems to be a fair and reasonable thing to offer students, and the exam papers for the summer exams (which have presumably been written already) can be used for that.  Marking will take several months as usual, so such students will not be able to take up university places in the autumn term. 
It will be interesting to see whether students are required to accept the grade they are ‘awarded’ this summer, or can reject it in favour of the grade they will get in the exam, or whether they will keep the higher grade if they subsequently take the exam and score poorly.  If the exam grade is automatically used, whether higher or lower, then this will be a tough call for some students.

Ofqual will develop and set out a process that will provide a calculated grade to each student which reflects their performance as fairly as possible, and will work with the exam boards to ensure this is consistently applied for all students. The exam boards will be asking teachers, who know their students well, to submit their judgement about the grade that they believe the student would have received if exams had gone ahead.

This is where the difficulties arise.  More on this below.

To produce this, teachers will take into account a range of evidence and data including performance on mock exams and non-exam assessment – clear guidance on how to do this fairly and robustly will be provided to schools and colleges. The exam boards will then combine this information with other relevant data, including prior attainment, and use this information to produce a calculated grade for each student, which will be a best assessment of the work they have put in.

Again – more on this below.

Ofqual and exam boards will be discussing with teachers’ representatives before finalising an approach, to ensure that it is as fair as possible. More information will be provided as soon as possible.

I wonder who they are talking to.  Unions?  Universities?  Subject associations?  Head teachers?  MAT leaders?  I hope and expect it will be a wide range of representatives. 

The aim is to provide these calculated grades to students before the end of July. In terms of a permanent record, the grades will be indistinguishable from those provided in other years.

This means a student will have a ‘grade 7’ and not a ‘coronavirus grade 7’.

We will also aim to ensure that the distribution of grades follows a similar pattern to that in other years, so that this year’s students do not face a systematic disadvantage as a consequence of these extraordinary circumstances.

Rather than not facing disadvantage, I wonder whether this is more about not allowing ‘grade inflation’ for this year.  First off, this will be very hard to achieve, given that a ‘take the exam’ option is being offered.  If an identical distribution to last year is created using the techniques they devise, they cannot know how many students will reject that grade and seek to score higher in the exams in the autumn.
This statement does suggest a way they might create the results (more below).

Education Secretary Gavin Williamson said: […] I have asked exam boards to work closely with the teachers who know their pupils best to ensure their hard work and dedication is rewarded and fairly recognised.

This is not a surprise, but it is worth noting that it forms a fundamental shift in the way students are assessed.  Clearly, this will involve some sort of teacher assessment and teacher input.  Many teachers may well welcome this.  But the biggest shift is that previously it has felt like {me and my students} in the red corner, versus {the examiners} in the blue corner.  In the past we could discuss exam technique (“the examiners like to see…”, “they are looking for…”, “they won’t give you marks unless…”) and if a result wasn’t what a pupil hoped for then they could blame it on the exam.  Now the pupils know that their teacher will have some involvement in determining their grade.  All of a sudden the teachers have changed side and it’s {students} against {teachers, DfE and exam boards}.

We recognise that some students may nevertheless feel disappointed that they haven’t been able to sit their exams. If they do not believe the correct process has been followed in their case they will be able to appeal on that basis.

I am intrigued as to what sort of evidence we might be called on to provide, and how it will be possible to provide that evidence for parents and pupils to see without breaking other pupils’ confidentiality.  It’s going to be interesting!

In addition, if they do not feel their calculated grade reflects their performance, they will have the opportunity to sit an exam at the earliest reasonable opportunity, once schools are open again. Students will also have the option to sit their exams in summer 2021.

This seems absolutely reasonable, with the caveats mentioned above.

There is a very wide range of different vocational and technical qualifications as well as other academic qualifications for which students were expecting to sit exams this summer. These are offered by a large number of awarding organisations, and have differing assessment approaches – in many cases students will already have completed modules or non-exam assessment which could provide evidence to award a grade. We are encouraging these organisations to show the maximum possible flexibility and pragmatism to ensure students are not disadvantaged.

If part of the exam is coursework, or a languages oral exam, then on the one hand it seems reasonable to use that as evidence towards the grade the pupils get.  But it mustn’t be the only thing that is used.  Otherwise, there is little point in students sitting all of the other elements of the course in a ‘normal’ year. 

Ofqual is working urgently with the sector to explore options and we will work with them to provide more details shortly.  The Government will not publish any school or college level educational performance data based on tests, assessments or exams for 2020.

This seems fair on the face of it, and I certainly wouldn’t want schools to be judged on the strength of grades that are not exam grades.  Yet we are saying to pupils “these results are so dodgy we can’t use them to assess schools but we think you should accept them”.  This feels odd.

Some options

Now, let’s look at the meat of this.

Ofqual will develop and set out a process that will provide a calculated grade to each student which reflects their performance as fairly as possible, and will work with the exam boards to ensure this is consistently applied for all students.

A “calculated grade”.  This means there will be some sort of method, formula, or system that will be put in place and which will be applied to all schools and colleges.

The exam boards will be asking teachers, who know their students well, to submit their judgement about the grade that they believe the student would have received if exams had gone ahead.

To lots of people this suggested predicted grades would be used. There are a number of difficulties with just using these grades.

  a)  Predicted grades are notoriously inaccurate, particularly at A-level. This is unsurprising because they are given months ahead of the exam and there is plenty of time for students to change their working practices between the prediction and the exam.
  b)  Sometimes they are given as an ‘aspirational’ grade.
  c)  Sometimes they are depressed so as not to make a student over-confident.
  d)  While teachers will have done their best to produce predicted grades (that are already in the UCAS system), they will not have expected them to be used to determine the pupils’ actual grades.

To produce this, teachers will take into account a range of evidence and data including performance on mock exams and non-exam assessment

Can mock exam grades be used?  Mock exams are carried out in very many different ways and these cannot be compared fairly across schools.

  a)  They happen at different times in different schools. Comparing mocks taken in one school in November with others taken in March is nonsense.
  b)  Some schools set a full exam paper that includes questions on topics that haven’t yet been covered, while others remove such questions.
  c)  Some do all of the papers, while others don’t.
  d)  Some mark the mocks as if they were the real thing, while others take into account that they have been sat months before the actual exam.
  e)  Students know they are mock exams and may not have revised sufficiently, or understood that process, and certainly won’t have expected those exams to create their final grade.
The exam boards will then combine this information with other relevant data, including prior attainment, and use this information to produce a calculated grade for each student, which will be a best assessment of the work they have put in.

Some have worried that this means KS2 levels will be used to give GCSE grades and that GCSE grades will give Year 13 students their A-level grades.  That isn’t what this says, and it would clearly be unfair to say to a student in Yr 11 that everything they have done for the past 5 years will be disregarded and their GCSE grades will instead be based on tests they took at the age of 10 or 11. 

What is to be done?

It is obvious that a grade cannot just “be awarded” by teachers.  In most subjects, the students haven’t been carrying out coursework and there isn’t anything comparable that can be used to moderate grades given by teachers working in different ways in different schools.  Further to this, if there are ‘too many’ grade 7s awarded nationally (remember, there is a desire to keep the overall outcomes this year comparable to those of last year) then how do you decide which students will be given a grade 6 instead (or vice-versa if not enough have been awarded)?

Here is my prediction (with further detail below):

1) Teachers will be asked to rank students in each subject within their school. The highest-attaining student in a particular subject, followed by the second highest … to the student we expect to achieve the lowest grade.
2) The DfE and exam boards will decide, using prior attainment data for these students and prior value-added for the school, how many of each grade in each subject a school ‘ought’ to achieve.
3) The students will receive those grades.

Ranking students

This will not be straightforward to do, but work by Daisy Christodoulou (2) and others involved in ‘comparative judgement’ has shown that deciding whether one piece of work is ‘better’ than another is easier than giving it a grade.  (If two people carry out these two types of grading then they are much more likely to agree on which is better than they are to give identical grades).

Using prior attainment data to give grades for the school

This is not straightforward to explain.  I will start with a naïve way of doing this and will then tweak the model to make it more sophisticated.  This will result in something that is fairer but more difficult to understand. 

We can look at last year’s GCSE results in the school and can see, in a particular subject, how many of each grade there were.  For example, let’s imagine a particular school had 20 students who took GCSE History in 2019 and that these were their grades:

Grade
1
2
3
4
5
6
7
8
9
No. of pupils
1
0
1
4
3
5
4
1
1

We could turn that into percentages
Grade
1
2
3
4
5
6
7
8
9
% of pupils
5%
0%
5%
20%
15%
25%
20%
5%
5%

Then we multiply the number of candidates for 2020 by these percentages.  For example, if there are 37 candidates for History in 2020 then we multiply those percentages by 37 and round them off:
Grade
1
2
3
4
5
6
7
8
9
% of pupils
2
0
2
7
6
9
7
2
2

Then the list put together by the teachers is deployed: the first two students on the list are awarded a grade 9, the next two get a grade 8, the subsequent seven get a grade 7, etc.

This doesn’t take into account that different cohorts of students have different prior attainment (remember, the Gov.uk article mentioned that ‘prior attainment’ would be used.  It doesn’t seem fair to use an individual’s KS2 test result (the only comparable ‘prior attainment’ we have for Yr 11 students), but it could be feasible to use that on a school level.

Here is an increased level of complexity:
This new table shows the KS2 levels achieved by the students who took GCSE History at the school last year, with their GCSE results along the top.  This highlighted number shows that there were two pupils who got a level 4 at KS2 (in English and Maths) and went on to get a grade 6 in GCSE History.

GCSE grades 2019

1
2
3
4
5
6
7
8
9
2
1

1






KS2
3



1
1




level
4



3
1
2



5




1
3
4
1

6








1

At this school there were two pupils who took GCSE History in 2019 who had a level 3 at KS2.  Of them, half got a grade 4 and the other half a grade 5.  So we would look at the KS2 grades of the 37 students who were planning to take GCSE History in 2020, and would say that half of those with a level 3 at KS2 will get a grade 4 and half a grade 5.  We will do this for each of the KS2 levels to produce the grades for the school.

Those grades would then be aggregated, so we will still have a total number of grades at grade 9, grade 8, etc, for the school.  The list devised by the teachers will then be used to assign grades to pupils in the same way as previously.

This is much more nuanced, in that it takes account of differences between cohorts, but is harder to explain.  A further level of complexity could involve using ‘fine KS2 levels’ (eg 4a, 4b, 4c) or the decimal levels used by Progress 8.

What is good about this method (and how can it be tweaked further)?

It is fortunate that the current Year 10s were the first students to take the new version of KS2 SATS, so the Year 11s have the same type of prior attainment as last year’s cohort (KS2 levels and not a scaled score). 

The vast majority of GCSE subjects have had two years of results under the new 9 to 1 grading system, so it would be possible to combine the school’s results for the past two years for many subjects.  (For English, Maths and English Lit there are three years of grades that could be used.  For some subjects there is only one year of prior data with the 9 to 1 system.)

It is helpful that the teachers are not deciding whether a student should get a particular grade, but instead are only choosing an order for their students.

If the entry cohort to the school changes then that will be reflected by this system.  If the entry cohort for a particular subject changes then that will also be recognised by this system.

Problems with this method

There are lots of potential problems with this system:

1) Schools that have had a significant change (eg they have gone into special measures this year, or have made significant changes, or are improving rapidly) will not see this reflected in the exam results, which will be broadly in line with those of the previous year. I don’t see any way around this.

2) A subject that has undergone a significant change (eg there was a lot of teacher absence in the past but there is now a new and experienced team in place) won’t see this reflected in their results.

3) Previous students with particular issues will affect the current cohort. A school-refuser last year will affect the grade of a student this year.

4) If a subject has a small cohort this could lead to unfair grading.

5) A new subject to the school won’t have prior data that can be used.

6) Schools with a significant number of pupils who join after the start of Yr 7 won’t have as much matched data (KS2 to GCSE) that can be used.

7) Subjects where early entry is reasonable (such as native speakers taking GCSE French in Yr 9) will be difficult to grade.

8) Certain groups of students do not fare well under teacher assessment. For example, a study (3) found that non-white-British pupils achieved lower teacher assessed levels (compared to their test scores) than their WBR counterparts. How do we avoid any unconscious bias on the grounds of race, and also by gender, background, accent, FSM, etc?

9) There are always surprises on results day, where teachers are amazed (in a positive or a negative way) about the grade a certain student has gained.

10) There may be student perception that there will be favouritism (“my teacher doesn’t like me”).

We will need to compare across classes taught by different teachers.  Determining the order of students is where the data will come in.  Going back to the Gov.uk guidance:

To produce this, teachers will take into account a range of evidence and data including performance on mock exams and non-exam assessment – clear guidance on how to do this fairly and robustly will be provided to schools and colleges.

Because we are only ranking students within a single school, the data we have is likely to be similar and comparable for all our students, and it doesn’t matter if it is not comparable to that of other schools.  While we cannot compare easily between schools, we will be able to use teacher assessment and mock exam results within a school to create an order of students.

The exam boards will then combine this information with other relevant data, including prior attainment, and use this information to produce a calculated grade for each student, which will be a best assessment of the work they have put in.

How might A-levels work?

This blog has focused largely on GCSEs, but a similar process could be used for A-levels, making use of GCSE grades as prior attainment.  There will again be problems will small cohorts in certain subjects and most of the other concerns raised above will still apply.

How might KS2 work?

I am outside my area of expertise here.  Could a similar process work for KS2, using KS1 levels as a baseline?  Or, given that teacher assessment is already part of KS2 and there is a system in place for doing this, will that be used instead of this system?  A final thought on this one, is that if this KS2 data won’t be used to publish school level performance data in 2020, presumably that means Progress 8 (which is based on KS2 data) won’t be published in 2025 (when the current Year 6 pupils reach Year 11).

Conclusion

I started with these three issues:

1) There isn’t a way to do this that everyone will be happy with.
2) Something has to be done, and the organisations charged with doing this have an impossible job.
3) You can have a simple system, or you can have a fair system, but you can’t have both.

This blog has explained what I see as being the least-worst way of doing this, and I have tried to acknowledge the problems that are inherent in this, and any other method.

I finish by saying ‘Good luck’ to everyone involved in whatever system is deployed.  Good luck to those who are devising the system, to the teachers who will need to use it, and good luck to those whose grades will depend on it.




Wednesday, March 25, 2020

Some maths with Social Distancing


As part of the advice of what to do to avoid spreading coronavirus, the BBC has some little graphics.  The one about social distancing (where we were advised to keep 2m apart) is this:


Is this to scale?
If we lie the chap down, and assume he is about 6 ft tall, then the circle appears to have a diameter of about 2 metres.


But hold on, we have to be 2m _away_ from other people: I mustn’t just be this distance away from others but should be twice that amount.  The first diagram here is bad, the second is good.


If I remove the major arc of the circle, suddenly this graphic makes sense:


The artist was thinking of us as Subbuteo figures

(Royalty-free image from Shutterstock)

This then leads to the question of how many people could fit in a room, obeying the requirement of being 2m apart.
My first thought was that this was a ‘packing’ problem, and for a 2m by 5m room I produced this diagram:


But we aren’t actually Subbuteo figures, so it’s fine for our ‘base’ to overlap the edge of the room.  In fact, we can have at least 6 people in the room:


Further questions:

  1. Can we have more than 6 people in this room?
  2. Is it possible for them to get into the room without violating each other’s space?  Does it depend on where the door is?
  3. How many people will fit in rooms of other dimensions?
  4. What is the smallest room that will fit a particular number of people?


Thursday, October 17, 2019

MathsConf21


Here are some brief thoughts and notes about MathsConf21, which took place in Peterborough on Saturday 12 October 2019. 

Background

It was my first time at MathsConf (which is one reason I wanted to blog about it: to tell people who have never been how good it was!).  These happen three times a year in England, on a Saturday, and there was a real buzz.  It was clear that while lots of those who attended were local to Peterborough there were others who had travelled a considerable distance and who were MathsConf regulars!

It’s a conference in a day, with plenary speeches, workshop sessions to choose from, exhibition stalls, the famous tuckshop and other events, such as the TweetUp during lunchtime.

Introductory plenary

After the welcome from Mark McCourt came the big Steve Jobs-style announcement.  The brilliant maths education software Autograph has been bought by La Salle (the organisers of MathsConf) and will be free for everyone to use!  Moreover, there will be (around Easter 2020) a browser-based version, which will presumably allow it to be used on Chromebooks and on PCs/Macs without the need for installation. 


This is an absolute game-changer and is very exciting for those of us who teach maths at secondary and sixth form level.  The only previous barrier to using Autograph has been the need to pay for it.  Over the past few days it has been installed on my laptop; I suspect my half term week will be full of Fun With Autograph!

Session – Manipulatives

Liz Henning (@oxfordedmaths) presented ways that Cuisenaire and Numicon can be used. 
She showed a lovely way to think about factorisation, with this diagram demonstrating two ways to visualise some factors of 15. 


In what ways are these two diagrams the same/different?

This can easily be extended to help explore HCF and LCM.

I often find I learn lots at conferences from informal conversations and from other delegates.  This one was no exception.  Liz asked us to use Cuisenaire rods to work out 1+2+3+4+5+4+3+2+1.


I rearranged in this way:

Someone else on my table did this:

That’s lovely!  Several of us then tried to sort out how this method would work if the middle rod was an even number in length.

Session – Autograph for A-level Statistics

Douglas Butler (@DouglasButler1), Janet Smith (@JanetAdeleSmith) and Leona So (@LWYSO) showed us where to find useful data sets and other information about Autograph (http://www.tsm-resources.com/useful-files.html).  It was, as always, a privilege to work with Douglas, the creator of Autograph.

It was good to see how to draw a histogram, normal dist, etc.  I hadn’t realised before today that the rather weird idea of histograms with unequal width bars is unusual elsewhere in the world!

My initial thoughts are that Autograph:
  • is more powerful than many other pieces of maths software
  • does stats brilliantly
  • is aimed at KS3 – 5
  • has a steeper learning-curve than other software.

Lunch

This was an opportunity to visit the exhibitors (and the charity tuckshop) and to catch up with colleagues from other schools.  There was so much to do that I missed the TweetUp session!

Session – ‘So you think you’ve got problems?’

Alex Bellos (@AlexBellos), author and puzzle-ist (he writes a fortnightly puzzle column for The Guardian) spoke very entertainingly about his forthcoming book.  We had lots of great problems to work on, there was audience participation, and I enjoyed the opportunity to work on problems with those sitting around me.  There was history, there was geography and even the problem Alex described as being the worst he has ever come across turned out to be interesting to hear about!

Session – Exploding dots

Rebecca McAndrew (@MathsMcBec) showed us an introductory video about exploding dots which, at their simplest, are a way to show place-value.  We were then able to work on using ‘exploding dots’ to carry out calculations in different bases (we managed to add in binary and to divide in both binary and in base 8). 

The part that was completely new to me was that exploding dots can be useful in dividing polynomials by other algebraic expressions.

I continued thinking about this session on my drive home – and realised during the journey that we could make use of ‘negative dots’ in numerical calculations (having been shown this as part of algebraic division).  I don’t know how self-explanatory this diagram is…


Any session that keeps me thinking for several hours afterwards must be a good thing!

The rest of the day

There were then the closing remarks, an announcement of how much had been raised for Macmillan during the day and a chance for more catching up and more conversations.

It was a brilliant day of CPD, with lots of interesting things going on and people to talk to (and, of course, the announcement about Autograph!).

I am very keen to go to the next one!
MathsConf 22 is in Manchester on 14 March 2020 (the day that shouldn’t be Pi Day!)
MathsConf 23 is in Oxford on 27 June 2020.

Many thanks to all who led workshops and to the organisers.  Much appreciated!

Saturday, May 25, 2019

Exam nerves


For the first time in 30 years I am sitting a GCSE exam.

I started learning Spanish a couple of years ago and have been working very hard at it.  I attend evening classes, have conversation lessons via Skype, have made visits to Spain, watch TV programmes with the Spanish audio and with subtitles in English (Netflix does that for lots of its shows) and, my favourite, listen to Spanish podcasts while I go running.

I have clearly been having lots of fun with Spanish, so what’s the point in doing the GCSE?

I really enjoy the speaking, listening and reading elements of Spanish, but have very little reason to do any writing and I don’t want to have a big gap in my knowledge.  Listening and reading come close to being problem solving.  It’s often possible to find the root of a well-known word in something unfamiliar, or to get some meaning from the context or from the sound of a word.  Speaking can be difficult because it all about thinking on your feet.  [I frequently think about cognitive load theory when I am trying to speak Spanish; as I am floundering around for the particular word I know I will need to finish off a sentence I then find myself making silly mistakes elsewhere in what I am saying!]  Writing in Spanish is something I just don’t need to do very often.

One reason, then, for signing up to take the exam was that it would force me to work on the things I don’t find easy and the things I don’t often need to do.

At the time of writing this I have done the speaking, listening and reading exams and am now focusing on preparing for the written exam.

What have I learned by doing GCSE Spanish (apart from ‘more Spanish’)?

   1)  I was surprised at how nervous I was before the exams.

Despite having nothing riding on this at all (I don’t need a particular grade, my life isn’t going to change depending on whether I get a grade 1 or a grade 9) I was still nervous.

I know I can speak/understand/read Spanish.  I have been a successful learner and a successful taker-of-exams in the past.  Yet I was still nervous!

   2) Exam technique is important.

It really has been good for me to prepare and to take the exam.  I now have a good knowledge of GCSE Spanish vocabulary and am much more confident at writing, for example.  I wasn’t expecting to need to spend so much time thinking about exam technique and rubric though. 

For example, in the speaking exam you have to ask the examiner a question at some point.   I checked several times at the end of the reading paper that I had answered the question in the correct language (some stipulated Spanish and others wanted answers in English).  None of this really impacts on how well you understand Spanish!

   3)      Revision plans are difficult to keep to.

I had a 46-page booklet of vocabulary to learn, set out in two columns on each page.  Three months before the exam I decided I would learn a column each day.  Seven weeks before the exam, having not done much, I decided it was fine because I could learn a page a day... .  Guess what I was doing the weekend before the exam?  Learning several dozen pages of vocab each day!

What are the implications for my teaching?
  • I need to recognise that pupils will get anxious about their tests and exams, even if they really shouldn’t.  
  • It has worked well for me to learn lots of Spanish and then to prepare for the exam shortly beforehand.  I need to think about what point to start mentioning exam technique and other exam-related things with my students.
  • Setting regular deadlines for revision is a good thing.  I would have benefitted from having regular vocab checks!

So hats off to Year 6, Year 11 and Year 13 students for the huge numbers of public tests and exams they do during the summer term.  And “¡buena suerte!” to those who still have more exams to do.


Writing blogs

I have the enormous privilege of working with excellent colleagues as part of the Cambridge Maths Hub.

This has given me regular input from people I wouldn't usually have an opportunity to spend so much time with, to be able to reflect on my own teaching in different ways and to reflect on teaching in general.

I also have the privilege of being able to write, alongside others, regularly for the Cambridge Maths Hub blog.  This has given me more opportunities to sit down and think about an aspect of teaching and learning mathematics (and explains why I haven't posted here recently!).

One of the interesting things in what I have written here is that everything I do still revolves around reflecting on, and attempting to improve, my own classroom teaching.  Long may that continue!

Thursday, July 12, 2018

ATM #beingmathematical. Cows eat grass

This is my 'homework' in advance of the ATM #beingmathematical session on 12 July 2018.

Here is the question:


The instructions were to think about how we thought about the problem.  Below are my ideas, including things I did wrong!  My thinking is in red and the maths is in blue.




Was there a way to do this without so much algebra?

Thursday, April 12, 2018

Ebbinghaus


I have some questions about the Ebbinghaus Curve.  

One example of the curve is linked to here:

While I am comfortable with the idea of needing to revisit material and that this can help with long-term retention (one of the authors of a 'Learn Spanish' podcast that I listen to says that you have to use a word in 6 different contexts before it becomes embedded), I find the Ebbinghaus Curve to be somewhat dubious. 

It looks rather specific, it has multiple copies of the same 'shape' of curve (subject to a stretch-factor) and it also still involves 'forgetting', so all we are doing is staving off the inevitable forgetting that will occur with everything.

I had a brief look at Ebbinghaus' work and was surprised by a couple of things:

1) Ebbinghaus had a single subject.  One person!  This seems to be a ripe and obvious field for research to be carried out with large numbers of people.  The idea that such an idealised curve can be produced after testing a single individual doesn't seem sensible.

2) The person was an adult.  The curve is now being used to predict how children might behave.  There are lots of differences between adults and children, but of particular relevance is that the adult knew they were part of a research study and presumably wanted to learn and recall things as well as they could.  This contrasts with some children in the classroom.

3) The adult learned strings of nonsense-words.  Again, this is vastly different to what happens in the classroom.  Is memorising strings of unconnected information related to learning connected information?  Is it linked to learning maths?  What about the role of understanding?

In summary: I don't have a problem with the idea of needing to revisit material to help with its retention.  I do think the 'Ebbinghaus Curve' is overly-specific and is an example of applying an idealised graph created from research carried out in a particular way to a completely different scenario.


Edit:
David Wees is exploring the data behind this in detail.  Worth looking at this thread:

Wednesday, November 22, 2017

£600 for post-16 mathematicians

The government initially seemed keen to make a success of Core Maths.  They put some very good people onto promoting and supporting it (David Burghes, Mick Blaylock, Paul Glaister, the CMSP group) and seemed to have nurtured this new course well.  

Then it became clear that many sixth forms would not be able to run Core Maths because it isn't funded in the same way as A-levels and they would not be able to afford to teach it.

In the budget today (22 Nov 2017) the Chancellor announced the following (taken from gov website):


Let's take the case of Core Maths.  One period per week across a whole year costs a school about £2000.  If a Core Maths class has 3 lessons per week over two years then that will be 3 x 2 x £2000 = £12,000.

Now let's divide that by the £600 per extra pupil taking the qualification:
£12,000 / £600 = 20.

So, a class of 20 extra students doing Core Maths will fund itself.

In the case of Core Maths then, this essentially sorts out the funding required to run the qualification.

And working this out could form the basis of a Core Maths lesson!

Friday, October 20, 2017

A Geographer Does Core Maths

A couple of years ago I made a presentation to geography teachers about Core Maths.  (One motivation for doing Core Maths is to support the statistical work that A-level geography students need to do.)

My colleague Scott, head of the geography department has admitted that he now “does core maths” in real life!  When watching Scotland fail to qualify for the World Cup finals he heard the crowd singing “I’d walk a million miles for one of your goals” .  As any self-respecting Core Mathematician should, he wondered how long it would take to walk a million miles.

Scott reckoned it would be sensible to walk 30 miles a day.  He then told me it would take 92 years to walk a million miles.  What’s nice is that I can use this with a core maths class.  I could give them these figures and ask whether they are reasonable.  Or I could play them the chant and ask for their thoughts: they might create a different question.  The students could also decide on their own ‘daily-distance’ to walk.  They might decide that it’s reasonable for people to walk long distances from the age of, say, 15 up to the age of 65 and could work out how far you would need to walk each day to cover the million miles.  (About 55 miles per day – that feels like a lot!)

Over at the Quibans website there are lots of ‘Questions Inspired By A News Story’.  They work in a similar way to this, with some sort of stimulus (usually from a newspaper or news website) and then some related questions to answer.  This is ideal for Core Maths and can be used to work on problem-solving as well as to practise mathematical skills (such as percentages) in different contexts. 

The tasks on the Quibans site can usually be projected in class.  This question has been set up as a Quibans and can be found here.  Do explore the rest of the site too.