Category Archives: Curriculum

Classroom Maths Dialogue with a Handheld

Now that HP prime is launched and I have a few of them to play with (oh and to run training sessions with …) the implications can be tested of having handheld maths technology that connects easily to the teacher computer in a classroom. Ultimately the connection will be wireless but the dongles will not be available until early next year. So, I bought some 5m micro USB cables and could test it in teaching situations. So far only teacher groups, but I’m getting the hang of it.

Calculators ListThere is free emulator software and the connectivity kit is freely available too. You can try this out yourself by opening a number of instances of the emulator and the connectivity kit on your PC. Just go to http://www.hpgraphingcalc.org/hp-prime-links-and-resources.html for the files. The most impressive thing is how utterly seemless it is. Install the connectivity kit and launch the software. Down the side there are three tabs: ‘Calculators’, ‘Content’ and ‘Class’. Just plug in an HP Prime calculator (or launch an instance of the emulator software and you will see it listed. Here, I’ve plugged in one real machine and am running my emulator. Clicking next to the device brings down a view of all of the content on the machine. Clicking the content tab allows us to author new content and then save it to the machine. You can create tests and polls. (Right click on the thing you want to create and ‘new’ comes up. Click that. When you are done, click the small save icon in the top left of the screen. The dialogues are pretty self-explanatory so I’ll leave you with that. When you are done, go back to the new item you created and right click to ‘send to class’ and it will be on the machine. Imagine being in a classroom in which simply by launching the software and machines being in the room, you can share content with them. No connections no log ins, just get started. But to me it bursts into life when you click the ‘class’ tab. Now you can see the screens of all of the connected machines. (Remember, when wireless is there, connected just means ‘in the same room’.). Class viewThese refresh every couple of seconds (although right click and refresh speeds this up if needs be. Also, right click and ‘project’ creates a resizeable image to show the whole class. So, now I can ask my class to work on a problem and watch what they do. When something interesting happens I can (if I want) bring up that screen to show the whole class. But, there is more, … I can open a messages window. From there I can send a private message to a single machine or a message to the whole class. MessagesThis appears on their screen and they can even respond and we can discuss. Personal in-class responses to individual students work monitoring the outcome of their thinking from the maths appearing on their screen. Now, I always say get students to work one between two, so we’ll have 15 screens up in a full classroom. I tried this with 6 Greenwich PGCE students and it is a bit frantic trying to respond individually. However, really, I should be more teacherly and give the occasional prompt looking our for the ‘Aha’ moments. The possibility to get to the nub of student understanding is tantalising. What do you think? I am dying to get the wireless dongles and try this in a routine classroom with ordinary kids. Then we’ll see what they can do. Exciting times.

Illuminate: Gifted and Talented at Key Stage 3 School Reviews

This is a shameless commercial post because I am really excited that schools who have bought our Illuminate Gifted and Talented Course for key stage 3 have posted on-line reviews on the National STEM centre web site. Obviously I would only be saying this if they like it, but they really like it a lot and that is really exciting.

See here: http://www.stemdirectories.org.uk/scheme/wondermaths-gifted-and-talented-maths/#comments

Our aim was to produce a course in mathematics, so that school students had the opportunity to see what Maths is really all about. It is full of puzzles and games and tricky things to think about. But it takes them to the next level by unpicking fundamental ideas notably proof and isomorphism and giving students an incite. Maths gives a way of definitively saying how we know what we know. We use Pythagoras Theorem to unpick the idea of proof. From the essential structuring idea that sets up the proof to the language needed to be clear and the sequencing of the statements to construct the complete argument. It is thrilling that schools are reporting that students are able and interested to work on this. It is hard, but interesting things are, but students are game to carry on. Then we compare cyclic and Klein groups with isometries and modulo arithmetic. I cannot think there is anything more wonderful for the beginning mathematician to see that we can show that two complete areas of operation, so apparently dissimilar as arithmetic of clocks and transformational geometry have exactly the same underlying structure and hence, if we know something about one, we necessarily know the same thing about the other. That, to me is what maths is really all about. The mechanical processes that students learn for their GCSE and A Levels give no insight into this amazing world.

So, well done to those schools for being brave enough to work this way and really well done to the students who are becoming serious young mathematicians. Clearly we would be delighted for you to try it too. Just ask for some trial materials of the Illuminate course.

Also, come to ATM sessions and meet Danny Brown. Danny is the head of maths at the Greenwich Free School and he is getting his kids working on deep mathematical ideas all the time. Danny has presented regularly to ATM London Branch and has a web site of the amazing stuff he does. I persuaded Danny to get this out in book form and the first volume, on Number, is nearly ready, so look out for that.

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 https://brilliant.org/ 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.

The New National Curriulum

Well, mathematical modelling now has a serious place in key stage 4. So, get yourselves ready. Do not look through the document looking for any coherence, though. You won’t find it. This is another pot-pouri. Some very odd things like frequent reference to mechanics as an example of mathematical modeling, even though it is not taught as that at A level. More a collection of known models being applied. The real trick is to get students to develop their own models critically and develop methods for validation. That’s why real engineering projects go through more than one development iteration and A level mechanics problems do not. My Pizza Project article develops the model creating phase here. Also, Venn diagrams are back for probability problems, but set theory is not. Vectors in different format are back, but matrices are not. My favourite is the explicit teaching of Roman numerals (up to 100 in year 4 and then, I kid you not, up to 1000 in year 5).

Mr Gove was on question time yesterday. We cannot learn to be creative until we have a through grounding in the facts and techniques needed, he says. Everyone agrees with this (including our good selves). The trouble is, that no-one questions which facts and techniques and for why. The secretary of state himself quotes long division as an example. But could he tell us which creative mathematics is opened up by being able to do long division. Certainly, at A level we can divide polynomials this way, and for sure, unpicking the process to see how it works, provides deep insights into the power of place value, but, as a technique to be learned, it is just a pandering to an imagined perfect past. The trouble for us, is that everyone … the man from industry, the children’s author and all of the politicians agree with him. I say get kids to memorise Pascal’s/The Chinese triangle and chant their squares, and cubes. That would genuinely help them engage with maths creatively. But 11×12? Why? Is old money making a comeback?

So, there’s a big opportunity. Some interesting if oddly chosen hard maths, that modelling word and even ‘proof’ is in there too. An absence of levels is a major blessing. But, ordinary kids have to be able to do this. Escalante got all of his students to AP calculus with ganas. We must be able to do this too. The stakes have been raised.

Playing Maths Games Makes You Do Better at School!

OK, so I came to this by being responsible for public maths events for maths year 2000. We had 22 shopping centre events, at the end of January 2001, where we set up staffed table stands with maths activities. It was humbling to see ordinary shoppers give up on Sainsburys and spend the day doing maths puzzles and games. So, the I end up being a part time shop keeper selling maths games and puzzles. It is just great to keep being reminded that people love doing this stuff. Continue reading Playing Maths Games Makes You Do Better at School!

Finding Good Maths Resources

The internet for teachers, blessing or curse? In the past, you would have a set of text books or work cards as your basic resource. The department would have bought a small library of additional books and materials from people like the ATM. If you needed a good idea, you would never have to look beyond the maths office or the maths cupboard (do you still have those?) Every department would have a pile of good physical manipulatives like centicubes and logic blocks, cuisenaire rods and probability kits. A set of large compasses and ruler for board work and a good collection of games and puzzles for activity days. There would be copies of those wonderful books by Brian Bolt (which are still available) for practical problem solving and a set of Points of Departure books for maths investigations. Always excellent, always to hand. Continue reading Finding Good Maths Resources

Rubiks Maths for GCSE Revision

We’ve just finished work on updating the Rubik’s maths service so that it acts as a direct, complete GCSE revision tutor. The centrepiece is a neatly arranged assessment and practice system for National Curriculum Maths. A smart space theme is used for the navigation … choose the Planet (e.g. Algebra), then the Continent (e.g. sequences) to find a nice self marking, auto updating, flash based assessment package. If you make mistakes there are web links and video resources to give your practice opportunities (and, if you have a MyMaths licence, links to these resources). Continue reading Rubiks Maths for GCSE Revision