Do puzzles and games have a net positive impact on student learning?

Mr Patrick Vieira, Teacher of Maths at WHS, looks at how completing puzzles and games can impact student learning.

One day, while travelling to school as a 12-year-old, I saw somebody solving a Rubik’s cube. This person would scramble the cube and solve it very quickly. He would do this repeatedly, and maybe it was just in my head, but he seemed to get quicker with every solve. Seeing a demonstration of that kind was nothing short of captivating to me at the time. It stayed with me throughout the day and when I got home, I told my mother about it and asked her to buy one for me. Neither of us knew what it was called but we took the trip to Hamleys with the hope that they would know. We were in luck! My mother paid for the Original Rubik’s Cube and I took it home excited to begin trying to solve it.

As does everything after a while, excitement quickly faded. The puzzle was difficult and did not come with any instructions. I had managed to solve one face by what seemed like sheer luck (blue, my favourite colour), but when I tried solving another face, my hard work became undone. It was so frustrating that I left it on the mantelpiece where it collected dust for years. Reflecting now, that must be how some of my students feel now when they are given a problem that seems too hard to solve at first.

Fast forward to 2019 when I first joined Wimbledon High School, where I had the opportunity to join the Rubik’s Cube club as a staff member. Of course, if I needed to help students solve the Rubik’s Cube, I needed to have a good understanding of it myself. This time, I was provided with a set of instructions and I got to work. Solve the white cross, then complete the white face. Finish off the second layer and then begin the top… I repeated the algorithms for each of these over and over again, and eventually I solved my first Rubik’s Cube.

But for me, that was not the part that excited me. As I repeated the moves for each step in isolation, I began to see why these algorithms worked. Every move had a purpose, setting the cube up so that on that final turn, everything comes together. It was as if I were almost tapping into The Matrix of the puzzle and I could feel my perception of 3D space improving with every turn. It was then that it hit me. This could be an amazing educational tool… but has it been researched?

Research related to the Rubik’s Cube is very limited but there are many pieces of anecdotal evidence to suggest that there are huge benefits to learning how to solve the cube. The two which stood out to me were grit and creativity.

Grit

Grit is one of the most mysterious personal traits discussed in education. It is widely regarded as the trait most indicative of whether someone will succeed at a task, no matter if it is in business, in the army, or in school.[1] However, it is difficult to nurture. When we complete a task which requires perseverance, the hormone dopamine gets released in our brain. This is the automatic response of the body which reinforces positive behaviours. The more tasks we complete using grit as our fuel, the more we are comfortable and happy being “grittier” – we create a habit of perseverance.[2]

Solving the Rubik’s cube is one way of helping us reinforce that positive trait of using grit. One Maths teacher writes in her blog that after giving her students an assignment to solve the Rubik’s Cube, they showed increased levels of grit.[3] However, just as Carol Dweck writes in her book Mindset: The New Psychology of Success, as educators, we need to still be encouraging our students to persevere and reward their effort rather than their achievement.[4] These will bring about the best results in development of grit.

Creativity

“Creativity?!” I hear you wonder. “How can you be creative when all you are doing is repeating algorithms?”

I had an interesting experience as I was improving my knowledge on the Rubik’s Cube. After learning the algorithms for the beginner’s method of solving and was able to do it well, I turned to an intermediate stage called the ‘CFOP’ method. There were slightly more algorithms to memorise, but I found my creativity bloom in the process of learning them.

From a fully solved cube, I picked one algorithm and applied it to the cube. Of course, this would mess it up completely. However, just for the fun of it, I kept applying the same algorithm and eventually I got back to a fully solved cube. I wondered why and I tried to see if I could do the same with the other algorithms. It turns out that they do. It takes a different number of repetitions for each algorithm but eventually I end up at a fully solved cube. Just for the fun of it, I also tried to combine algorithms or even reverse them. These made me see different patterns and other ways of solving it. I wasn’t really doing much with the cube but still, I thought to myself, “this is pretty fun.”

Where next?

So pick up your cube. Don’t just leave it on the mantelpiece like I did for years. There is a great opportunity to be had whether you are a teacher or a student. Returning to my opening point, do puzzles really have a positive effect on learning? Nobody really knows yet. But if it helps you develop perseverance and foster your creativity, I think it’s worth a shot to find out for yourself.

References:

[1] See Angela Duckworth – Grit: The Power of Passion and Perseverance

[4] See Dweck – Mindset: The New Psychology of Success

Why is it socially acceptable to say: “I’m bad at Maths”?

Alys Lloyd, a Maths Teacher at Wimbledon High School, looks at society’s attitude towards Maths, what makes a good mathematician, and how you can compare the retaining of mathematics knowledge to that of languages.

Teachers do have social lives, although to our students this might be a shocking idea. A teacher being spotted outside of school, in the supermarket for example, can send some students into a flat spin. So the idea of a teacher being at a party might be difficult to imagine, but I can assure you, it does happen!

At parties and in social situations with people who don’t know me, I have found that my job can, unfortunately, put a bit of dampener on things. A typical conversation opener is to ask what someone does for a living. The most common response to my saying that I’m a Maths teacher is “oh, wow” then something along the lines of “I was never any good at Maths in school.” Then the person I was talking to politely excuses themselves. I now tend to dodge that kind of question and stick to safer topics.

Why is it socially acceptable to say you are bad at Maths? I doubt that so many people would be so upfront saying that they can’t read… So why does Maths get such bad press?

My Theory

Mathematics is a very black and white subject, with normally only one right answer, although there may be lots of different ways to get there. Many people have been put off Maths because in the past they have got stuck, had a negative experience and not known how to get to the correct solution.

This may have been because the teaching was poor, or the methods they were taught to use didn’t make sense to them, or they didn’t speak up in class so didn’t get help. I believe that by far the most common reason is that they take getting stuck personally. They believe that they didn’t get the right answer because they themselves are bad at Maths. Unfortunately, I don’t think this is something that just happened in the past; it still happens, and I see it happening with the highly achieving girls at WHS. They are not used to getting things wrong, finding something difficult, having to struggle, and they take it personally – they internalise this as a failure: they are bad at Maths.

Which leads us to the question: what makes someone ‘good’ or ‘bad’ at Maths? Who is someone who is ‘good’ at Maths? A Lecturer or Professor of Mathematics? A Maths teacher? Or someone who simply enjoys doing Maths? Is it about who you are comparing yourself to? As a Maths teacher, my level of mathematics is low compared to a Mathematics Professor. Being good at mental arithmetic is not the same as being good at Maths; possibly conversely in fact – professional Mathematicians are notoriously bad at mental arithmetic, as are some Maths teachers!

So, for people who say: “I’m bad at Maths”, they may think that those people who are ‘good’ at Maths never get stuck; never struggle to get to the answer. But I can assure you, that is not the case. I am a Maths teacher and I get stuck on Maths problems. I definitely don’t always immediately know how to get to the answer.

I believe the difference in how you feel about Maths is about what you do when you get stuck, because we ALL get stuck. Being stuck isn’t bad – it’s part of the process. It is a way of forming new connections in the brain; it’s a part of learning.

When I get stuck on a problem I don’t take it personally; I don’t take it as a reflection of my mathematical ability; I think of it as a challenge, a conundrum to be figured out, a puzzle to be solved. If I can’t find a solution quickly, I stop and try to think about it differently. Could it be thought about in another way? Can I visualise it by drawing a sketch or diagram? Is there an alternative approach or method I haven’t tried? Have I used all the information I have available? These are very important problem-solving skills and have lots of relevance to everyday life.

Use it or lose it

Mathematics in many ways can be considered its own language. When learning languages, you start with basics: hello, please, thank you, and a few important sentences (dos cervezas, por favor); and build up to be able to communicate fluidly. If you have ever tried to learn a language seriously, you will know that it is not a smooth process. You go through phases of thinking you’re doing great, then you feel like you plateau – you realise that there is a whole verb tense you had no idea existed, that you now need to learn.

Maths is similar. You need to know the basics: numbers, patterns, arithmetic, and a few important ideas like algebra; and you build up to some quite complicated Maths like calculus, proof, complex numbers. With Maths numbers and algebra are the words, and rules like BIDMAS are the grammar. They are a means to the same end as languages – to communicate effectively.

One aspect of learning a language (or learning a musical instrument) is that if you don’t practise it regularly, you start to lose the gains you had made; it becomes more difficult, and eventually you forget. I firmly believe – that like a language – if you don’t use Maths, if you don’t practise it regularly, you start to lose it.

For me, this explains why parents can struggle to remember how to do school-level Maths with their children, even if they found it easy when they were young – they haven’t practised it in years. It can seem like an alien language – it’s hard to pick something up again when you have had such a long gap.

Yet even if you, yourself, haven’t used Maths in years, you are constantly using things that have been programmed by someone using Mathematics. Maths underpins everything ‘modern’ around us: the computer at which I am typing this article, the smartphone in your pocket; it keeps planes in the air and stops them crashing into each other; it’s in our buildings, in our clothes; Maths is fundamental to our modern style of living.

We want to encourage our children to feel it is socially unacceptable to be bad at Maths. We want them to be the ones solving the problems of the future, and part of this will certainly require mathematics.

So, what’s the take-home message? I’d like to think it’s this: in Maths, as in life, we all get stuck, but the people who succeed are the ones who don’t give up. And if you are lucky enough to meet a Maths teacher at a party, please be nice!

Mrs Rebecca Brown reviews Craig Barton’s book: How I wish I’d taught Maths

Mrs Rebecca Brown, teacher of Maths at WHS, reviews Craig Barton’s book How I wish I’d taught Maths, focusing on Chapter 11 about formative assessment and diagnostic questions.

“without an effective formative assessment strategy we are in danger of teaching blindly, being completely unresponsive to the needs of our students.”

Craig begins this chapter by referencing the 2013 Dylan Wiliam tweet:

Example of a really big mistake: calling formative assessments ‘formative assessment’, rather than something like responsive teaching.

It’s only a too familiar scenario – you mention an assessment and a classroom (or staffroom!) erupts into a panic of more pressure and visions of tests, marking and grades. But how do we understand what our pupils know and where we need to begin or continue teaching them from? Even more crucial now, following a period of prolonged guided home learning. My key quotation from the chapter is when Craig explains that ‘without an effective formative assessment strategy we are in danger of teaching blindly, being completely unresponsive to the needs of our students’.

Formative assessment should be about ‘gathering as much accurate information about students’ understanding as possible in the most efficient way possible and making decisions based on that’. In short, it is about adapting our teaching to meet the needs of our students.

He describes elements of great teaching and cites one of Rosenshine’s (2012) ‘Principles of Instruction’ -to check for student understanding: ‘The more effective teachers frequently checked to see if students were learning the new material. These checks provided some of the processing needed to move new learning into long-term memory. These checks also let teachers know if students were developing misconceptions’.

Teaching is only successful if students have understood and learned something. Successful formative assessment can help us to identify problems and begin to fix things in the here and now much more effectively and efficiently. Asking ourselves, do I need to go over this point one more time or can I move on to the next thing?

Craig suggests the use of diagnostic questions to give quick accurate and useful information about students’ understanding. A good diagnostic question is a multiple choice, four-part question, with three incorrect answers that can help you to identify both mistakes and misconceptions. Each incorrect answer must reveal a specific mistake or misconception. If the question is designed well enough, then you should be able to gain reliable evidence about students’ understanding without having to have further discussions.

Diagnostic questions are designed to help identify, and crucially understand students’ mistakes and misconceptions in an efficient and accurate manner. They can be used at any time in a learning episode and are most effective when used throughout, using follow up questions to test the exact same skill as the first question.

Craig has developed a website of diagnostic questions that can be used in a variety of subjects. This year I will be trying to incorporate these into all of my lessons to ensure I have accurate, timely information on student understanding to enable me to effectively teach the girls that I have before me.

What is the single most important thing for teachers to know?

Cognitive Load Theory – delivering learning experiences that reduce the overload of working memory

Rebecca Brown – GDST Trust Consultant Teacher, Maths and teacher at Wimbledon High School – explores how overload of the working memory can impact pupils’ ability to learn effectively.

Over the summer whilst (attempting to) paint and decorate my house, I was truly inspired listening to Craig Barton’s podcasts[1] and the opinions and theories of the fabulous guests that he has interviewed. In particular, his episode with Greg Ashman[2] where they discuss Cognitive Load Theory. I feel slightly embarrassed that I have managed to get through the last twelve years of my teaching practice and not come across this pivotal theory of how students learn before now!

Delving into this deeper, I have since found out that in 2017, Dylan Wiliam (another of my educational idols) tweeted that he had ‘come to the conclusion Sweller’s Cognitive Load Theory[3] is the single most important thing for teachers to know.’ As a self-confessed pedagogical junkie I immediately wanted to know more – so what is Cognitive Load Theory and what impact could it have on the learning of my students?

What is Cognitive Load Theory and where did it come from?

“If nothing has been changed in long term memory then nothing has been learned” – Sweller

In 1998, in his paper Cognitive architecture and instructional design[4], prominent Educational Psychologist Dr John Sweller helped demonstrate that working memory has a limited capacity. He put forward the idea that our working memory – the part of our mind that processes what we are currently doing – can only deal with a limited amount of information at one time.

In essence, it suggests that human memory can be divided into working memory and long term memory. Long term memory is organised into schemas. If nothing is transferred to long term memory then nothing is learned. Processing new information puts cognitive load on working memory, which has a limited capacity and can, therefore, affect learning outcomes.

If we can design learning experiences that reduce working memory load then this can promote schema acquisition. Sweller’s Cognitive Load Theory suggested that our working memory is only able to hold a limited amount of information (around 4 chunks) at any one time and that our teaching methods should avoid overloading our working memory to maximise learning.

De Jong[5] states that ‘cognitive load theory asserts that learning is hampered when working memory capacity is exceeded in a learning task’.

Put simply, in early knowledge acquisition, if we can simplify how we deliver material to students, to focus on what really needs to be learnt so that they are not using up too much working memory, then we have a much higher chance of being able to help the learning stick in their long term memory.

The theory identifies three different types of cognitive load:

Intrinsic: the inherent difficulty of material being learnt. This can be influenced by prior knowledge that is already stored in the long term memory. For example, if students know that 5×10=50 this can be retrieved without imposing any strain on working memory but if the calculation required as part of a problem was 398 x 34, students would have to begin to retrieve information on how to do long multiplication which would take up working memory required for new material.

Extraneous: the way in which the subject is taught or the manner in which material presented. Extraneous load is a cognitive load that does not aid learning and should be reduced wherever possible.

Germanic: the load imposed on the working memory by the process of learning itself. That is, moving learning from the working memory into the schemas in long term memory.

So, if we can manage intrinsic load, reduce extraneous load, allow more room in the working memory for Germanic load then we have better chance of learning being transferred into long term memory.

Moving forward

In his enlightening and motivational book How I Wish I’d Taught Maths, Craig Barton[6] summarises that the essence of Cognitive Load Theory is getting students to think hard about the right things in order to facilitate the change in the long-term memory necessary for learning to occur.

Whilst I am so far from being an expert in Cognitive Load Theory, from the research that I have already read, I am positive that my teaching practices will be enhanced by continually considering ways of reducing Cognitive Load and ensuring that students working memories are not overloaded with information that is not conducive to learning.

My next steps are to look further into the research from Mayer[7] on Cognitive Theory of Multimedia Learning to develop how I can best present learning opportunities to students.

References

[3] Sweller, J., Van Merriednboer, J. J. G. and Paas F.G. W. C. (1998) ‘Cognitive architecture and instructional design’, Educational Pscycholgy Review 10 (3) pp. 251-296

[4] Sweller, J., Van Merriendboer, J.J.G and Paas, F.G. W. C. (1998( ‘ Cognitive architecture and instructional design’, Educational Psychology Review 10 (3) pp. 251-296

[5] De Jong T (2010) Cognitive Load Theory, educational research, and instructional design: Some food for thought. Instructional Science 38 (2): 105-134.

[6] Barton, Craig 2018 How I wish I’d taught Maths

[7] Mayer, R.E (2008) ‘Applying the science of learning: evidence-based principles for the design of multimedia instruction’, American Psychologist 63 (8) pp. 760-769

What happens when maths goes wrong?

Grace, Year 8, looks at why maths is important in everyday life, and what happens when it goes wrong.

Maths is integrated into our lives. Whether it’s telling the time or looking at our budget for the latest gadget that we want to buy, we all use maths. But sometimes we use it incorrectly.

The Leonard v Pepsi court case

One example of maths going wrong is when in 1995 Pepsi ran an advert where people could collect Pepsi points and trade them in for Pepsi-branded items. Points could be collected through purchasing Pepsi products, or through paying 10 cents per point. For example, a T-shirt was worth a mere 75 points whilst a leather jacket was worth 1,450 points.

To end their campaign, Pepsi stated that the Harrier Jet, which was promoted in their advert, could be bought for 7 million Pepsi points. At the time, each Harrier Jet cost the U.S. Marine Corps around \$20 million. Knowing its worth, a man called John Leonard tried to cash it in. This was an extensive task with particular rules, all of which John followed. His amount totalled \$700,008.50 which he put into an envelope with his attorneys to back the cheque! Pepsi initially refused his claim, but Leonard already had lawyers prepared to take his side and fight. The case involved a lot of discussion, but eventually, the judges sided with Pepsi, even though Leonard v PepsiCo, Inc. is now a part of legal history.

Errors in the news

Sometimes maths goes wrong on a big scale. For example, the Russian shooting team in the 1908 Olympics left with no medals because they turned up nearly two weeks late as the 10th July in Russia, was the 23rd July in the UK. The Russians were using a different calendar.

Lottery complications

Another example is of human confusion with maths. A UK lottery scratch card had to be taken off the market within a week due to players having problems with negative numbers. The card was called Cool Cash, and had a temperature printed on it. If you scratched a temperature lower than the target, you won. But lots of people playing didn’t understand negative numbers… “On one of my cards it said I had to find temperatures lower than -8. The numbers I had uncovered were -6 and -7 so I thought I had won, and so did the woman in the shop. But when she scanned the card, the machine said I hadn’t. I phoned Camelot and they fobbed me off with some story that -6 is higher, not lower, than -8, but I’m not having it.” These players didn’t know how negative numbers worked, so looked for the numbers that were usually lower when they were positive.

Maths is important in everyday lives as we all use it, sometimes without being aware of it. However, it is important that checks are made to ensure the correct figures and calculations are used. After all, our lives may depend on it.

How do you create a whole school academic timetable?

Mr Bob Haythorne, Director of Academic Administration and Data at WHS, looks at the processes involved to craft a whole school academic timetable.

“I don’t know how you do it!”

“I don’t know why I do it!”

If I had £1 for every time I’ve had that exchange about timetabling over the years, I think I’d have been able to retire a few years ago.  If we add the other old chestnuts – “Why don’t you just reuse the same timetable every year?” and “Can’t you just get a computer to do it all?”– I think I’d have been able to retire before I’d even started!  I’ve been asked to write this WimTeach blog about timetabling, but I’m not sure I can do it justice within the allowed wordage:  there are textbooks on it and the standard training course is three days long, so this is just a potted summary of the process.

It actually starts a long time before September; in fact, we need to have a pretty good idea of what we’re offering to Year 12 over fifteen months earlier, when we hold our ‘Into the Sixth’ events.  This can be affected by the addition of new subjects, the removal of others and changes to the whole structure brought about by new government policies.  Straightaway, we can start to see why it’s not possible to copy the timetable over from one year to the next.  Add in staff moves, random variations in popularity of different optional subjects at GCSE and A-Level, and increases in the size of year groups and it soon becomes obvious why everyone retracts that question after a moment’s thought.  A senior school timetable is critically dependent on the GCSE and A-Level options, so a great deal of time is spent with individual girls in Years 9 and 11, mainly in the Autumn and early Spring Terms, to help them choose and to get the information from them that we need to make a start.

With options in by February Half Term, the fun begins!  The first task is to analyse the numbers to see what the staffing implications are.  Will we have enough Mediæval Tapestry teachers?  Or can we really justify running Industrial Botany for just one girl?  We need to give our part-time teachers a term’s notice of variations in their hours and any recruitment ideally should be sorted before the Easter deadline for giving notice.  Whilst the Head and HR team are resolving those issues, we are crunching these options to fit them into option ‘blocks’ – groups of subjects where classes will be taught simultaneously.  (The number of option blocks is equal to the number of subjects the girls are allowed to opt for).  There is software to help with this, although my experience is that if you ask the program to create a scheme from the raw information, it will give a ridiculous answer, if it can give one at all.  (We don’t want all three Mediæval Tapestry groups in one block if there are only two teachers and only one specialist classroom, for instance;  really we want them spread across three different blocks).  I find that manually allocating about half the groups into blocks based on common sense and experience and letting the software solve the rest works best.  It will then allocate girls to groups and you can see whether some groups are too big or too small, and you can experiment with moving them around until you get a solution you like.  Whether it will timetable is still another matter!

‘Blocking’ is the first stage of actual timetabling and can be a major task in its own right.  This is where we take the 70 periods of our timetable and draw up a table of what could be going on at the same time.  At this stage, its just 70 periods, labelled 1 to 70 – there is no thought about when each period will happen, except for certain fixed periods, like Y11 – 13 Enrichment, which has to be on Thursday afternoon because of its links with outside agencies.  Because Enrichment has so many small groups, it places a strain on staffing, so we have usually scheduled the largest Year group in Years 7 – 10 to have PE at this time.

How the Year 12 and Year 13 option blocks mesh together is critical.  We draw up a table that might look like this:

(In practice, I would show all the actual subjects in each block – usually between 8 and 12 different groups – so that I can see the clashes.)

The ‘No’s are because, say, the planned scheme for Year 12 block A contains three subjects where we only have one teacher, or one necessary specialist room, and these three subjects happen to spread across three different Year 13 (i.e. current Year 12) blocks – W, X and Y.  Something similar will be the case for 12C and 13X clashing, and for 12D and 13W clashing.  In this example, we see that all 12A lessons will have to be timetabled simultaneously with all 13V lessons.  The ‘?’ for 12D and 13Y might mean that it probably could work, but ideally we would keep them apart.  This means there are really only two solutions:

These will then determine the underlying structure of the entire timetable.  Making the decision about which one to go with is nerve-wracking – one might turn out weeks later to have been a poor choice!  These go into the Blocking table:

It gets a lot more fragmented after this.  Ideally, two Year 11 option blocks (7 periods each) would sit nicely under each Year 12/13 pairing, but it never works out that neatly.  For a start, there are only six option blocks, not eight, and Maths and English have more than 7 periods.  Again, there are restrictions caused by limited specialist rooms and/or teachers that determine which Year 11 blocks can and cannot go under which Year 12/13 pairings.  And after playing with this for a week or two, it’s time to insert Year 10, with the same issues, but with complications like Enrichment and PSHE being at the same time for Years 11 – 13, but there’s no Enrichment in Year 10 etc.  Bear in mind that the table above just shows one column for periods 1 to 14:  the reality is one column for each period …and a very wide table!  By the way, it’s probably around Easter by now.

Some timetablers would do all their blocking (down to Year 7) before attempting to schedule the lessons into actual periods on actual days, but either because I’m impatient or because we have so many restrictions on what can happen when, I tend to start scheduling now. (E.g. we have a large proportion of part-time staff, who – funnily enough – don’t think it’s really part-time, if you have to be available for all 70 periods, with a small number of lessons randomly scattered around!)  Now, at last, we turn to the timetabling software.  Again, you could try inputting all the information and hitting the ‘Autoschedule’ button, but when it’s finished trying (a few days later), it will either have produced garbage or – more likely – failed.  The problem is that there are too many arbitrary decisions that you would have to make when inputting all that information, so it’s better to build things up steadily, seeing what’s working and trying to keep as much flexibility as possible for later stages.  An example of an arbitrary decision would be to allocate all the teachers to the incoming Year 7 groups:  there is no point being specific about exactly who is going to teach English to which class, because any of the four Year 7 English teachers could take any group, without needing to worry about continuity from Year 6.  Doing so means creating an unnecessary restriction;  doing so across all subjects is just mad!

It starts with the fixed items:  Enrichment, Year 12/13 PE, PSHE, HoYs meeting, SMT meeting and lock these in place.  Then the Year 12/13 pairings, bearing in mind we want double periods etc..  Then Year 11.  Then realise this isn’t going to work, because of some staffing issue.  Rip out Years 11 – 13 and start again.  Repeat as necessary until there’s a satisfactory solution.  Add in Year 10.  Take out Year 10 and adjust the option scheme.  Retry Year 10….

This constant back and forth continues for a few weeks.  We now have some limited options in Year 9, so it’s a similar story and that takes about a week to sort out.  Finally, Years 8 and then 7, where there is much more flexibility, as one class can have Geography whilst another is studying English and another Art etc.. Even so, that’s another week and it’s May Half Term already  – or later!

So, why do I do this?  Because it’s a huge puzzle, but it’s a puzzle where the answer isn’t in the back of the book or in tomorrow’s edition;  it’s a puzzle where you have to keep changing the problem when it can’t be solved to one you think you might be able to solve… or might not, so you try turning it into yet another problem.

But when that final piece of the jigsaw slots into place, that multi-way swap that you think will do the trick does do the trick, that opportunity to move something that you thought you remembered turns out to be right, then I know why I do this!

Can the Harkness approach to delivering Maths lead to a deeper understanding?

Mrs Clare Duncan, Director of Studies at WHS @MATHS_WHS, describes the Harkness approach she observed at Wellington College and the impact that this collaborative approach has in the understanding of A Level Maths.

Named after its founder, Edward Harkness, Harkness it is a pedagogical approach that promotes collaborative thinking. Edward Harkness’ view was that learning should not be a solitary activity instead it would benefit from groups of minds joining forces to take on a challenging question or issue. What Harkness wanted was a method of schooling that would train young people not only to confer with one another to solve problems but that would give them the necessary skills for effective discussion. Harkness teaching is a philosophy that began at Philips Exeter Academy in New Hampshire in the 1930s.

Edward Harkness stated:

“What I have in mind is [a classroom] where [students] could sit around a table with a teacher who would talk with them and instruct them by a sort of tutorial or conference method, where [each student] would feel encouraged to speak up. This would be a real revolution in methods.”

This was very much what the classroom looked like when I was lucky enough to observe Maths teaching at Wellington College last term. Their newly refurbished Maths rooms had floor to ceiling whiteboards on all the walls. On entering the classroom, the students were already writing their solutions to problems that were set at preparatory work for the lesson. Whether the solution was correct or not was irrelevant, it was a focal point which allowed students to engage in discussion and offer their own views, problems and suggestions. The discussion was student led with the teacher only interjecting to reinforce a significant Maths principle or concept.  The key learning point is giving the students their own time before the lesson to get to grips with something before listening to the views of others.

The Maths teachers at Wellington College have developed their own sets of worksheets which the students complete prior to the lesson. Unlike conventional schemes of work, the worksheets follow an ‘interleaving’ approach whereby multiple topics are studied at once. Time is set aside at the start of the lesson for students to put their solutions on whiteboards, they then walk around the room comparing their solutions to those of others. Discussion follows in which students would discuss how they got to their answers and why they selected the approach they are trying to use. In convincing others that their method was correct, there was a need for them to justify mathematical concepts in a clear and articulate manner. The students sit at tables in an oval formation, they can see one another and no-one is left out of the discussion. The teacher would develop the idea further by asking questions such as ‘why did this work?’ or ‘where else could this come up?’.

The aim of Harkness teaching is to cultivate independence and allows student individual time to consume a new idea before being expected to understand it in a high-pressured classroom environment. This approach can help students of all abilities. Students who find topics hard have more time than they would have in class to think about and engage with new material and students who can move on and progress are allowed to do so too. In class, the teacher can direct questioning in such a way that all students feel valued and all are progressing towards the end objectives.  It involves interaction throughout the whole class instead of the teacher simply delivering a lecture with students listening. It was clear that the quality of the teachers questioning and ability to lead the discussion was key to the success of the lesson.

Figure 1: WHS pupils in a Maths lesson solving problems using the Harkness approach

This was certainly confirmed by my observations. The level of Maths discussed was impressive, students could not only articulate why a concept worked but suggested how it could be developed further. I was also struck by how students were openly discussing where they went wrong and what they couldn’t understand; a clear case of learning from your mistakes. Whenever possible the teaching was student led. Even when teachers were writing up the ‘exemplar’ solutions, one teacher was saying ‘Talk me through what you want me to do next’. Technology was used to support the learning with it all captured on OneNote for students to refer to later. In one lesson, a student was selected as a scribe for notes. He typed them up directly to OneNote; a great way of the majority focusing on learning yet still having notes as an aide memoir.

Although new to me, at Wimbledon we have been teaching using the Harkness approach to the Sixth Form Further Maths students for the past couple of years. Having used this approach since September it has been a delight to see how much the Year 12 Further Maths pupils have progressed. Being able to their articulate mathematical thinking in a clear and concise way is an invaluable skill and, although hesitant at first, is now demonstrated ably by all the students. The questions posed and the discussions that ensue take the students beyond the confinements of the specifications.

Does the Harkness Method improve our understanding of Maths?

Elena and Amelia, Y12 Further Mathematicians, explore how the Harkness Method has opened up a new way of thinking about Pure Maths and how it allows them to enhance their mathematical abilities.

For Further Maths A Level, the Maths department has picked a new style of teaching: the Harkness Method. It involves learning by working through problem sets. The problems give clues as to how to get to the answer and this is better than stating the rules and giving examples; we have to work them out ourselves. These problem sets are given for homework, and then we discuss them together during the next lesson by writing the answers on the board and comparing our results with each other.

Elena:

At the beginning of term, I found it quite challenging to complete exercises without knowing what rules I was expected to apply to the problems, as each question seemed to be completely different to the one preceding it. The tasks also require us to use our previous GCSE knowledge and try to extend it ourselves through trial and error and by applying it to different situations and problems. I found it difficult to understand how to apply a method to solve different problems as previously each problem came with a defined method.

As the lessons progressed, I started enjoying this method of teaching as it allowed me to understand not only how each formula and rule had come to be, but also how to derive them and prove them myself – something which I find incredibly satisfying. I also particularly like the fact that a specific problem set will test me on many topics. This means that I am constantly practising every topic and so am less likely to forget it. Also, if I get stuck, I can easily move on to the next question.

Furthermore, not only do I improve my problem-solving skills with every problem sheet I complete, I also see how the other girls in my class think about each problem and so see how each question can be approached in more than one way to get the same answer – there is no set way of thinking for a problem.

This is what I love about maths: that there are many ways of solving a problem. Overall, I have grown to like and understand how the Harkness Method aims to challenge and extend my maths skills, and how it has made me improve the way I think of maths problems.

Amelia:

When I first started the Harkness approach for Pure Maths in September, I remember feeling rather sceptical about it as it was unlike any method of learning I had encountered before. To begin with, I found it slightly challenging to answer the questions without knowing what topic they were leading to and found confusing how each sheet contained a mixture of topics.

However, I gradually began to like this as it meant I could easily move on and still complete most of the homework, something which you cannot do with the normal method of teaching. Moreover, I found it extremely beneficial to learn the different topics gradually over many lessons as I think that this improved my understanding, for example for differentiation we learnt it from first principles which gave me the opportunity to comprehend how it actually works instead of merely just remembering how to do it.

Furthermore, I think that the best part of the Harkness Method is that you are learning many topics at a time which means that you cannot forget them as compared to in the normal method which I remember finding difficult when it came to revision for GCSEs as I had forgotten the topics I learnt at the beginning of Year 10. I also began to enjoy the sheets more and more because the majority of the questions are more like problem-solving which I have always found very enjoyable and helpful as it means you have to think of what you need to use instead of the question just simply telling you.

Moreover, I very much enjoyed seeing how other people completed the questions as they would often have other methods, which I found far easier than the way I had used. The other benefit of the lesson being in more like a discussion is that it has often felt like having multiple teachers as my fellow class member have all been able to explain the topics to me. I have found this very useful as I am in a small class of only five however, I certainly think that the method would not work as well in larger classes.

Although I have found the Harkness method very good for Pure Maths, I definitely think that it would work far less well for other parts of maths such as statistics. This is because I think that statistics is more about learning rules many of which you cannot learn gradually.

To what extent can Bitcoin replace Sterling?

Phoebe, Year 13, explores the use of Bitcoin as a currency and its potential to replace Sterling by discussing the limitations of the new currency.

Bitcoin is a cryptocurrency, which is a digital currency that uses cryptography for security, and a worldwide payment system. It is the first decentralized digital currency, meaning the system works without a central bank or single administrator. It is based on a special field of maths called cryptography which is the study of how to secure communications, this being one of the main issues with not having transactions being overseen by a central administrator. Bitcoins are created through the process of mining; where miners use special software to solve mathematical problems and are issued in exchange with bitcoins. So, to what extent does this new unregulated technology have the ability to replace sterling?

Despite the fact that Bitcoin supports the attractive libertarian utopia of a society free from government intervention, where welfare is cheaper and wealth more distributed, in reality Bitcoin currently does not pose a threat to the sterling. One of the major reasons that I will be focusing on is the unsustainable scale of computer computational power that is required in order for miners to verify transactions within the block chain system due to the increasing marginal costs for them. Miners are being imposed with a direct cost as they are forced to require more bandwidth to enable them to solve the increasingly difficult puzzles in the same time frame.

Distributed systems such as Bitcoin’s involve a negative externality that causes over investment in computer hardware as the expected marginal revenue from the individual miners is increasing with the amount of computing power that they individually have. Not only does this increase their own marginal cost but it increases the competition within the system and thus the cost is also increasing across the entire network. “Cetirus paribus” economic theory would suggest that in equilibrium all miners are inefficiently investing in hardware while receiving the same revenue that they would have had they not invested in the extra computing power. This behaviour is irrational as it is increasing the computing power across the entire network making it harder for them to succeed individually.

If the cost of verification for the miners is constantly increasing, then eventually the incentive to secure the network will disappear and lead to the collapse of the system.

Therefore, due to this increasing cost of mining, Bitcoin, in its current state, does not have the potential to replace sterling.

As teachers, do we need to know about big data?

Clare Roper, the Director of Science, Technology and Engineering at WHS explores the world of big data.  As teachers should we be aware of big data? Why, and what data is being collected on our students every day… but equally relevant questions about how we could increase awareness of the almost unimaginable possibilities that big data might expose our students to in the future.

The term ‘big data’ was first included in the Oxford English Dictionary in 2013 where it was defined as “extremely large data sets that may be analysed computationally to reveal patterns, trends, and associations.”[1] In the same year it was listed by the UK government as one of the eight great technologies that now receives significant investment with the aim of ensuring the country is a world leader in innovation and development.[2]

‘Large data sets’ with approximately 10000 data points in a spreadsheet have recently introduced into the A Level Mathematics curriculum, but ‘big data’ is on a different scale entirely with the amount of data expanding at such speed, that it cannot be stored or analysed using traditional methods. In fact, it is predicted that between 2012 and 2020 the global volume of data will increase exponentially from 4.4 zettabytes to 44 zettabytes (ie. 44 x1021 bytes)[3] and data scientists now talk of ‘data lakes’ and ‘dark data’ (data that you do not know about).

But should we be collecting every piece of data imaginable in the hope it might be useful one day, and is that even sustainable or might we be sinking in these so-called lakes of data? Many data scientists argue that data on its own actually has no value at all and that it is only when it is analysed in context that it becomes valuable. With the introduction of GDPR in the EU, there has been a lot of focus on data protection, data ethics and the ownership and security of personal data.

At a recent talk at the Royal Institute, my attention was drawn to the bias that exists in some big data sets. Even our astute Key Stage 3 scientists will be aware that if the data you collect is biased, then inevitably any conclusions drawn from it will at best be misleading, but more likely, be meaningless. The same premise applies to big data. The example given by Maja Pantic from the Samsung AI Lab in Cambridge, referred to facial recognition, and the cultural and gender bias that currently exist within some of the big data behind the related software – but this is only one of countless examples of bias within the big data on humans. With more than half the world’s population online, digital data on humans makes up the majority of a phenomenal volume of big data that is generated every second. Needless to say, those people who are not online are not included in this big data, and therein lies the bias.

There are many examples in science where the approach to big data collection has been different to that collected on humans (unlike us, chemical molecules do not generate an online footprint by themselves) and new fields in many sciences are advancing because of big data. Weather forecasting and satellite navigation rely on big data and new technologies have emerged including astroinformatics, bioinformatics (boosted even further recently thanks to an ambitious goal to sequence the DNA of all life – Earth Biogenome project ), geoinformatics and pharmogenomics to name just a few. Despite the fact that the term ‘big data’ is too new to be found in any school syllabi as yet, here at WHS we are already dabbling in big data (eg. MELT project, IRIS with Ark Putney Academy, Twinkle Orbyts, UCL with Tolcross Girls’ and Tiffin Girls’ and the Missing Maps project).

To grapple with the idea of the value of big data collections and what we should or should not be storing and analysing, I turned to CERN (European Organisation of Nuclear Research). They generate millions of collisions every second from the Large Hadron Collider and therefore will have carefully considered big data collection. It was thanks to the forward thinking of the British scientist, Tim Berners-Lee at CERN that the world wide web exists as a public entity today and it seems scientists at CERN are also pioneering in their outlook on big data. Rather than store all the information from every one of the 600 million collisions per second (and create a data lake), they discard 99.99% of this data as it is produced and only store data for approximately 100 collisions per second. Their approach is born from the idea that although they might not know what they are looking for, they do know what they have already seen [4]. Although CERN is not using DNA molecules for the long-term storage of their data yet, it seems not so far-fetched that one of a number of new start-up companies may well make this a possibility soon. [5]

None of us know what challenges lie ahead for ourselves as teachers, nor our students as we prepare them for careers we have not even heard of, but it does seem that big data will influence more of what we do and invariably how we do it. Smart data, i.e. filtered big data that is actionable, seems a more attractive prospect as we work out how balance intuition and experience over newer technologies reliant on big data where there is a potential for us to unwittingly drown in the “data lakes” we are now capable of generating. Big data is an exciting, rapidly evolving entity and it is our responsibility to decide how we engage with it.

[1] Oxford Dictionaries: www.oxforddictionaries.com/definition//big-data, 2015.

[3] The Digital Universe of Opportunities: Rich Data and the Increasing Value of the Internet of Things, 2014, https://www.emc.com/leadership/digital-universe/