Research Based Pedagogic Tasks RBPTs

This has been a long time coming, but I intend to restart sharing thinking about work I’ve been doing in projects around the world! I am part of a Sheffield Hallam, British Council project for the Indian Institute of Science Education and Research in Pune. The idea is to work with teachers of undergraduate science and maths to develop tasks requiring problem solving skills in settings of genuine benefit. The problem with maths, is that the problems are frequently unrealistic or too simplistic. One of our co-tutors wrote to ask my thoughts on this …

The issue is that teachers of mathematics are so fixed in their need to teach some specific mathematics that all they can ever do, is invent some ‘story’ that fits the mathematics they want to teach. In applied mathematics, the solution is led by the application. The mathematics is not fixed and will keep changing. So the vital issue is to start with a real problem that someone really wants to solve. You may have to accept that the mathematics that ends up being used is not necessarily what you wanted to teach.
Probably the best situation is for mathematicians to find a group of scientists who already have a genuine problem and work with them. The vast majority of the RBPTs created by participants at the workshops would have benefited from much better design and analysis of the data. Claims were being made e.g. of the relationship between certain diseases and certain things found in rivers. This needs good statistics. NOT finding a simple correlation and saying that therefore one is caused by the other. On the same subject, flow rates in water courses generate problems needing differential equations as for examples would traffic management problems.
I saw an RBPT on nano technology. The issue of orders of magnitude was very clear. For example we did an experiment with waterproof sand (yes, seriously). This is grains of sand covered in a polymer coating one nanometere thick. What is the relative size of the sand grain to the polymer? Making serious calculations at the nano level of sizes requires some complex handling of numbers. But again – the application must lead.
Finally Space technology requires sophisticated mechanics and calculus and of course India is a world leader. Differential equations are needed to model rocket speed/time trajectories and escape velocities, so a nice RBPT would be to design a satellite flight path to put it onto successful orbit.
So: (i) wean the mathematicians off starting with their bit of maths and onto finding an interesting problems and (ii) force them (!) to work with scientists who already have the problems and need some help with the maths.

What is this maths that we are teaching?

It is with envy that some mathematics educators in England look to our colleagues in the Netherlands where the Freudenthal institute has generated a rich, coherent research debate which has been widely implemented in schools. Realistic Mathematics Education offered the antidote to the formalism of the New Maths based on Hans Freudethal’s view that mathematics was not pre-formed. He said; “… the global structure of mathematics to be taught should be understood: it is not a rigid skeleton, but it rises and perishes with the mathematics that develops in the learning process. Is it not the same with the adult mathematician’s mathematics?” So it is very sad to hear that the Commission for Examinations in the Netherlands is considering banning graphing calculators from public examinations. What is it that a calculator does that could be damaging to mathematics developing in the learning process? A machine can do only what a machine can do. If mathematicians continue to fulfill an important role, then clearly they must be able to things that machines cannot do. In his 2001 novel, Uncle Petros and Goldbach’s Conjecture, Doxiadis’ eponymous mathematician dismisses any process a machine could do as ‘shopping maths’. That of course includes anything a computer algebra system (CAS) could do.

So, learners of Freudenthal’s mathematics should have access to the tools to do the shopping maths, to free up the thinking space to engage with real mathematics; solving problems, generating conjectures, developing proof. These are the art of mathematics, not the mechanical grind. Godfrey Hardy acted as the foil to Ramanujan’s genius, but in the ‘apology’ he makes clear how well he understood that Ramajan’s ability for finding extraordinary new relationships that only he could see, was the real mathematical gift. Getting it into a publishable state was the routine work for afterwards.

The excellent Project Euler takes as it’s premise that mathematicians will have access to a high level programming language (Python, which naturally has a powerful CAS) to engage with problems in number theory. The wonderfully named designed for potential International Maths Olympiad candidates has a whole section of problem solving requiring programming (and hence CAS) available.

Having a machine capable of high level mathematics available in a public examination in mathematics forces examiners to take a considered view of what the maths is that they are examining. It prevents them from asking students to replicate what machines can do and focuses their thinking on the maths that matters. The maths that Hans Freudenthal was so keen to preserve in the Netherlands, against the onslaught of formalism.

This sad situation was brought to my attention through the English translation of a response by Erik Korthof to an advert for the new HP Prime graphing calculator. He suggests that the absence of graphing calculators in the past allowed the construction of ‘proper exams’. The task of mathematics education should not be to make the lives of examiners easy. Clearly, asking a student to complete a mechanical task that would be simply done by a machine is very simple. To construct a question knowing that the student has access to such a machine is hard. Specifically so, because the question must demand genuine mathematical thinking and that puts great demands on examiners. In the UK, the most progressive mathematics education project (MEI) for A Level students (age 18) have just had their first cohort complete an examination module with a CAS calculator available. The result is thoughtful, highly mathematical questions of exactly the type University maths courses are excited to see. The link will take you to their answer to Erik Korthof’s question: “Is secondary education served with a Computer Algebra System?”. Clearly they answer a resounding yes and MEI are major players in the future of maths education in England.

As I’ve said elsewhere the existence of tools like HP Prime which allow access to powerful mathematical visualization and calculation tools in the classroom liberates students from the mechanical processes that prevent them thinking deeply about the mathematics. Certainly there will be many lessons where the calculators are put firmly away and students will learn and practice these mechanical processes, like drawing graphs and manipulating algebra, not only because they need to see how they work, but also to give them a better feel for the outcomes. Happily teachers are sophisticated enough to manage this. They can also find secure ways to use exam modes to ensure devices adhere to local regulations. Schools are expert in this. These logistical issues should not be used as an excuse for not allowing students the tools that professionals have access to and reducing what is called maths in schools to a collection of mechanical processes. Especially not from the birthplace of RME and the beautiful, powerful view of mathematics presented to the world by Hans Freudenthal.

HP Prime The New Future

HP_prime_front_pictureHP Prime will be launched ready for September and the new school year. Have a look at the teaser YouTube HP released to show you what it looks like. Last week I had one in my hands at a launch workshop in Prague led by GT Springer the lead designer. GT has been central to most of the major innovations in graphing calculator design and he has put all of that experience into a genuinely wonderful new device. Read the interview GT gave to the US tech blog Cemetech. First impressions matter to schools who want to show the smart new kit they are buying and to students who want something really flash in an era where new tech does indeed look good. It is interesting that after a stunned response at the NCTM conference in Colorado there has been a lot of buzz around tech sites like Slashgear and Ubergizmo. Well that’s good, because if the tech savvy think it’s worth talking about then bright young teachers and their equally bright students will take a look.

Being an old fogey myself, all I can say is that it looks very smart indeed,with a brushed aluminium front and a smooth bright screen. The colour is bright and very sharp with extremely clear detail and you just have to keep reminding your self that it is a touch screen and that you can drag and move objects and navigate drop down menus. The touch is smooth and very accurate. Younger folk than me will do this instinctively, I’m sure that they will be wondering how it could be done any other way. It is very well made and feels sleek and smooth all round. It is about 300g which feel sufficiently heavy to be solid but easy to hold and it balances really nicely in tho hands with your thumbs over the Home screen and the CAS button. You really feel you are holding a classy piece of kit. So, part one of the battle is won, savvy young people will want one and schools will be proud to show off that they bought them. So, what does it do?

The biggest headline is: wireless connectivity. Files can be transferred via the connectivity software. However, if you plug a small USB dongle (which you purchase seperately) into the top of the PRIME, it will immediately be recognised on the computer, notably the teacher’s computer in class. Files and settings can then be transferred wirelessly. (Only from PRIME to PC not from PRIME to PRIME). More than that, the PRIME screen can be shown on the teacher’s screen. Then their will be class polling functions allowing the teacher to set a question from her computer and students to offer responses from their PRIMES with the results shown in table and chart form. Just like the polling systems many schools are getting which only do this. That will be just the start of what can be done. The critical point is that this a plug-and-play system. No set up, which is a critical factor for classroom use.

The software itself initially looks like an up-rated version of the HP39gII, which it is, so you will find all of the Apps in the HP39gII working exactly the same. So, anyone who has used a HP39gII will get started immediately. However, there are three new Apps which make a big difference. There is a mathematical spreadsheet, a dynamic geometry system and the advanced grapher. Together these represent a major advance in providing an space to explore mathematical ideas. These tie together with the big pause for breath moment. The CAS button.There is no CAS/non-CAS option. A mathematical machine must speak algebra and this one does. There are two home screens; a CAS screen which deals with exact objects and the traditional home screen which deals with approximate objects. The Apps can use the last object from each of these screens and the choice is always there; CAS screen or Home screen. This recognition of the fundamental pure/applied, exact/approximate distinctions is central to an underlying philosophy which has the potential to transform the way we think about exploring mathematics. For me, this is the thing that will determine future research into maths education technology. The spreadsheet, the dynamic geometry and the advanced grapher can all take CAS and non-CAS statements and allow users to explore the results. Just to get a feeling for what this means, have a look at GT’s handouts from the NCTM conference.

Now the sad thing is that exam boards are scared of CAS and we look forward to a future where CAS systems will transform maths exams by getting beyond procedural questions and towards mathematical problem solving. Well done to MEI for getting an A-level module approved allowing CAS and look to Germany and Australia for examples where CAS is embraced. But in the UK CAS is not allowed. Well, no, CAS is not allowed in public maths exams for which any calculator IS allowed. So, it is quite clear that this machine has a CAS system, so could you use it in an exam? To be sure the answer will be yes, the machine includes a comprehensive exam mode. A menu system allows a vast range of features to be turned on or off, CAS is one of the, but suppose a particular exam disallowed solver apps, they can be turned off too. The system is password protected and the user will simply be greeted with a little round exclamation mark if they try to access or disallowed function or suppressed apps will simply be missing from the menu. For school use, the teachers sets the settings they want e.g. turn off the CAS, creates a password and then beams this setting to all of the connected PRIMES, wirelessly. A series of bright LEDS light up in the same sequence while exam mode is engaged. It is immediately clear to the exam secretary that the machine has only those facilities allowed in exams. In discussion with teachers, it became clear that this feature sets up the possibility to allow younger learners to get started with the machine in a simplified mode and actually presented exciting pedagogic possibilities too.

The exams battle is a big one and many schools still think you cannot use any graphing calculator in a maths exam, so we will need to talk with exam boards and the JCQ to make sure the message is clear enough: you can use this machine in a maths exam and without disabling it as an amazing teaching tool.

I’ve always been a fan of calculators as a learning tool. I’ve said elsewhere that tablets are exciting, but you don’t work and think like that, you need different technological tools for different functions and the resilience of the calculator as a form factor is remarkable  I think for this reason. It’s a highly portable, personal thinking space. I am really excited about PRIME because it has all of the maths you could possibly want with an intuitive touch driven interface, wireless connectivity to support proper classroom dialogue in a package that everyone will want to own.

Please get in touch with me if you see the video and want to be part of early development to get really exciting maths back into our classrooms. I would be delighted to talk to you about the support I can offer.

The New HP39GII

I’ve now had a good time to work with the new HP39GII graphing calculator. It is exactly what you would want if you have been using HP39/40GS or a TI84 or a Casio FX9750 and you are ready for a really fast processor, tons of memory, a grey scale hi-resolution screen and batteries that last forever. The HP39/40GS series is a classic graphing calculator with a very smooth operating system built around apps all of which are controlled by the three ‘multiple representations’ keys … symbolic, plot and table. The HP39GII works in exactly the same way, so you know straight away what to do. But everything works really well. On the home screen calculations can be shown in textbook display with quotients and indices shown correctly. Divisions are shown fraction form where needed and an approx key comes up which converts to a decimal approximation. The graphing screen is superb. Clear smooth lines, clear dark axes and subtle grey grid lines. Pressing the + or – keys zooms in and out. Everything happens really fast. Continue reading

Why Learn Maths

Just before Easter I ran a session for the A Level mathematics groups in the Harris Academy group in South East London. I can tell you it was pretty daunting in the small hall, but with something like 120 sixth form students, who had chosen me over another talk, about options, I think. However, can I publicly thank (a) the teachers at Harris Crystal Palace who invited me and most especially (b) the students who attended for reminding me what fun it is to talk about maths to young people. I’ll be applying for a teaching job again, next … Continue reading

Cape Town Maths

Two weeks ago, I had a very nice trip down to Cape Town. It is a very beautiful city indeed. However, I did a series of sessions mixing HP Graphing Calculators, GeoGebra and Data Streaming to groups of maths teachers, trainee maths teachers and undergraduate engineering students at the Cape Penninsula Technical University and the University of the Western Cape. South Africa, in the post apartheid era has been trying to bring all of its education systems up to the level of the former elite schools. As you can imagine, this is a tough task, although the government’s commitment is clear having one of the highest proportional spends on education anywhere in the world. The universities I visited are excellent examples of that move to change and I was delighted to work with really enthusiastic students and teachers. Continue reading

Is MEI Better for Mathematicians?

We are at the point in our PGCE course where the students get a set of sessions about A level mathematics. It is clear where there is a correspondence between either a strong view of the importance of mathematics as a subject or that A level mathematics can be taught using an open and exploratory pedagogy, then teachers choose the MEI exam for their students. Others are put off by the idea that it may be ‘harder’. What that really means is that it is more mathematical. It requires some thinking, rather than rote practice. The real question is: what will these students be moving on to next? A degree in mathematics is clearly benefitted by working mathematically and a degree in science or engineering is at the heart of the purpose of MEI. It remains a beacon of opportunity for teachers to actually teach their subject. In my experience, entirely in inner city comprehensives, this is always to the benefit of exam results as well as the right thing to do.

Impressive Maths Thinkers

It is easy to be a bit jaundiced about A level maths, when you see the drill and practice text books with the exam board’s logo on the front and student’s in permanent practice, practice mode because of the modular exams. Well, I had an excellent day in a non-selective, state sixth form college, working with two groups of charming A level maths and further maths students. Continue reading

HP Graphing Calculators at A-Level

I am working with a group of A level students who have received an HP39GS graphing calculator. They got the calculators a couple of weeks ago, so hopefully they will at least have opened the package and turned the thing on. I will have about 3 or 4 hours with each of two groups tomorrow to give them ideas of things they can do with the machine. Clearly we live in a gadgety time. A lot of people will get a new phone every year or so, which is basically a computer, running a range of software packages, notably communication systems, but also, pretty much anything you care to mention. Continue reading