We can use Mr Men too!

So, while the Mr Men are enjoying a revival in UK teaching I thought I would jump on the bandwagon with some ideas for using the Mr Men in some Mathematics lessons. I blogged yesterday in support of my colleague Russel Tarr whose 'Mr Men Lesson' came under fire from Michael Gove. Ever since, I have found myself wondering about how we might use Mr Men in some of our mathematics classes! As so often, the more you think about these things, the more possibilities you see and I think there are a few genuine opportunities that I will work on in the future!

Mr Men Quadrilaterals,

I was thinking about a series of 7 books about quadrilaterals, Mr Square, Mr Rhombus and so on.... I thought that the stories could be about the properties of the particular quadrilaterals. Each story might be about 10 paragraphs long, then I thought you could cut out the paragraphs, mix them up and ask students to rebuild the books wih the correct titles. An example sentence might be....

'Mr ........ was feeling a little square today, it was quite rare for him to feel this way, but it could happen from time to time'

The sets and subsets of quadrilaterals and their properties are such a rich topic for investigation, but you do need to present problems well enough to encourage students to investigate them further. There is quite a bit of thought to do to think about the stories, but I think that could be fun and might even get a little cyptic depending the audience I decide to use!

Mr Man Concepts

Quite undeveloped yet, but I would love to have a series of stories about sophisticated mathematical ideas like proof. 'Mr Proof had seen several examples that worked, but wasn't quite satisfied that it would happen everytime' Again, I wonder how well able students might be to write the books themselves. Ultimately that would be fantastic and would depend on the target class. Like the history example, it could be that older students are asked to write the books with a younger target audience in mind. Imagine the adventures that Mr Proof might have on a given day. He could have a whole host of friends too, like 'Little Miss Improbable' who misses out on so many things and is hardly ever included. Every now and then though she turns up and surprises everyone. What about 'Mr Commutative', 'Little Miss Average' or better a reworking of 'Mr Mean' (and  a whole host of friends). We could go on... 'Mr Power', 'Little Miss Infinite', 'Mr Prime' and on and on. The key would really be about how to bring the activity to the students. Like the quadrilaterlas above, I think there might be some mileage in writing the stories and then having students rebuild them. Plenty to think about, but all ideas must start this way.

Anyway, walking down my corridor at school I was reminded that the Mr men have already been in our classooms. The picture below shows Mr Tickle depicing a box and whisker diagram about the spread of armspans in our school population, done during a whole school data week.

Through all of this, I am remided of how foolish it was of Mr Gove to dismiss an idea out of hand without any exploration. So often, the more you explore, the more potential you see. That is the business of teachers! As I write, I am certain that I am not the first to have these ideas and resolve to look for existing ideas before I go any further.

We are Mr Men teachers and proud!

For any of you who are UK based, you will doubtless have followed the now infamous 'Mr Men Speech' by Michael Gove last week. For those of you who are not, follow the links - it is worth a read whatever your base or perspective. The issue hits home on a number of levels for me and us here in Toulouse. Firstly the 'Mr Man Teacher' Russell Tarr is one of my colleagues and I really want to speak out on his behalf, although I will not manage to do so as eloquently as he did himself. Secondly, I want to officially sign up as a 'Mr Man' teacher. I think this could be a cult movement in the making and I want to be a part of it, given what it stands for and its implications for teaching Mathematics.

I have worked with Russell Tarr for nearly 8 years now and he has been a thorn in my side from the start. The way he engages, enthuses and develops critical thinking in his history students has meant, for years, that all our efforts in the mathematics department can only be second best. As such, he has been an inspiration for those of us he works with (and many others around the world) to do the same for our students. Indeed he is in many ways responsible for our decision to develop our own resources to the point where they can be shared on the web. Putting your stuff out there is a big leap with a number of costs and benefits (Russell can now add to that list). Most significantly, it encourages you to think very hard about, and develop your resources fully so that they are as good as they can be. This has a huge benefit for students and our own professional development as teachers. For that inspiration and his willingness to share, help and discuss all things educational, we owe him great thanks. I sincerely hope and believe that ‘Mr Men gate’ will come down on his side in the end.

A number of us found ourselves wondering, which activities of ours do we wish Michael Gove had criticised? A number come to mind! Personally, I think he would like Dancing Vectors the best. Mathematics teaching is a critical field, fighting against years of misconception about the subject, what it is, why we learn it and what it can be for different people. Our job is hugely important and any suggestion that we are making it trivial with our attempts to engage students, indeed misses the point. Any suggestion that our efforts are aimed at anything other than raising confidence, expectation and improving a generation’s understanding of the role mathematics plays in our world is, sadly, very wide of the mark. The point is this; nothing is more inspiring or rewarding that creating opportunities that promote engagement, enquiry, enthusiasm and critical thinking. Anyone with any experience of that will know that, and those without cannot and should not be directing the future of education. As we think about posting resources to this site, this is our mantra and our goal. It may be that it doesn’t always work for all teachers or all classes (ours included) but is our goal all the same.

I am a Mr Man teacher and proud!

Homework - Yuck or Yay

The following blog post is a quick response the question above that has been posed by the #globalmath group on twitter who are having a debate on this topic this evening based on some responses to this  Questionnaire. I love this debate and like many good ones can see lots of different points of view. It often helps to try and come off the fence though and in this case I have written some thoughts about why I would be happy to see the end of homework as we know it. This view is not necessarily shared by my colleagues that help write this website and is really just designed to spark further debate. Few issues are black or white! I should probably add that I have been influenced by  The Homework Myth by Alfie Kohn. The thoughts are about homework in secondary schools in general, but from the perspective of a maths teacher.

Quality of life

• Big cause of tension and conflict in homes.
• Students day is long enough if spent productively.
• Odd to expect longer days from teenagers than of many working adults.
• After school and evening times should be for family, sports, leisure and hobbies - all very valuable pursuits from which people can learn huge amounts.
• There has to be time in the day for students to be doing things they have decided they are going to do. Progress in these fields is likely to be more significant than that which is obliged.
• We should encourage learning to be voluntary and self driven.

The case for homework?

• Where is the evidence supporting the case for homework?
• I would argue that evidence presented is assumed.
• Homework is a long standing feature of education - it is there because it always has been, not because of strong evidence that it should be.

The case for no homework

• Equally, scrapping homework is largely untried. I think it is time to try it for the reasons above. 
• School days must become more productive but students should be more receptive during the day if homework is off the table.
• Too much emphasis is placed on amount of time rather than quality of time. Too much on coverage, rather than nature of time spent.

The nature of homework

This is the crux of the matter for me. Any time spent working at home in the evening is infinitely more valuable when students do it of their own volition rather than out of compulsion. The types of things that students could be doing are many and varied and I think it would be great to be providing possibilities and provoking an interest. Of course, for this to work, the key would be getting home culture to match the schools which is unlikely but we can only really offer possibilities. As its stands, the success of homework as we know it is already highly dependent on the culture of students' homes. To support learning in maths I would be suggesting/encouraging - just a first draft

• Regular playing of logic/strategy games that encourage critical thinking and discussion
• Individual Puzzle solving, giving students practise of working idependently
• A shared stream of interesting and relevant links etc about the subject, encourage an exchange about these via a tool like fb or edmodo or blogs or the like
• Offer various projects that students could volunteer for and offer support with them.
• Offer tasks that involve creativity.
• Have ongoing things like photos of maths in the real world, sudoku competitions etc.
• Keep an eye out for relevant documentaries - give students the chance to respond to these.

If this sort of advice was being offered across all subjects then would be no shortage of things on offer for students to enrich their school based studies.

Anyway, I am looking forward to the debate, both as it unfolds this evening and continues.

Expanding Double Brackets

Generally students enjoy playing games (who doesn't). Whilst we can find many attractive and fun games online I'm often disappointed with the quality of the questions. If it's fun, but the students don't really learn much mathematics what's the point?

I'm keen to create some games and have found a useful platform from http://www.classtools.net

Here are a couple of games to practise expanding double brackets. In the first game the students have to save the world by shooting invaders. In the second they have a choice of which game they prefer including an old classic Manic Miner!

I aim to create a whole actvity on factorising brackets and these games will provide a warm-up.

Enjoy the games and let me know what you think.

Round 1

Click here for larger version

Round 2

Click here for larger version

Building and summing sequences with cubes!

This has come up a couple of times recently and seeing a twitter discussion on 'brilliant activities using manipulatives' I decided to write this down quickly. This is a beautiful and simple activity to teach arithmetic sequences using manipulatives. Of course this could be done in lots of ways and so this blog is really only giving the outline. What is nice about this activity is the possible age ranges it could be used with!

1st - ask students to make a visual representation of the sequence 1, 5, 9, 13 .... and so on depending on howmany cubes you have. you can get all sorts of different shapes. I have been doing this activity for years and always see something new! The picture on the left is one possible answer, the one below another.

2nd - ask how many cubes it will take to build the next, 10th, 100th pattern etc which is really asking students to think about the structure of the sequences and work towards a formula that is built on intuition.

3rd - ask how many terms of the sequence you can build with the cubes you have - you get some wild answers! Test it by building them

4th - Once they are built arrange them like the picture below...

5th - Carefully get students to demonstrate how this can be made in to a rectangle like the the picture below...

and that the area of this rectangle is the sum of the sequence and that the formula for the sum is easily derived from the way this was put together.

As I said, this is just a quick blog to sow the seeds of an idea. There is so much going on here, so many alternate paths that it has become for me an essential activity!

It's always nice to know that other mathematics teachers appreciate the site so it was a proud moment for the site to get a glowing testimony from Craig Barton AKA MrBartonmaths and the main man behind mathematics resources in the TES.

Here's Craig's short video:

Here are the three activities and the training resources that Craig mentioned in his video

 Angry Birds

Age: 15+ Time 1 hr. This set of games asks students to find the correct equation of the parabola in order to hit the pig!  Three set of coordinates are given and students are required to calculate the equation of the parabola.  They will be required to understand the equation of a quadratic, in particular the form y=a(x - p)(x - q) would be helpful.  Great fun!

 3D Perception

Age: 12+ Time 1h  The aim of this resource is to develop student’s association of nets, hence surface area, with 3D solids, hence volume. The activity starts with a matching activity, nets and solids, some of which work, some don’t, students can cut and fold to check. Two virtual manipulative websites are then used, one aimed at inspiring them with a wide, and unusual range of 3D shapes.

 Roll 'em

Roll, roll, roll… This carefully structured activity aims to get students to discover that experimental probability approaches theoretical probability as we increase the number of trials.  We often overlook the importance of carrying out games of chance to build up an intuition for probability.  In this case we roll a dice then use a lifelike simulator on Excel to produce up to 2000 rolls. Age: 11+ Time: 1h

TSM - July 2012 - Introductory Geogebra

Objectives - To support workshop participants to get to grips with geogebra software and demonstrate its uses in the mathematics classroom.

Aims - Participants should observe potential, get practice, be inspired to use it in the classroom!

Content - Learn the basics of creating geometrical figures, measurement and calculation; coordinate geometry and functions for elementary and advance topics.

The power of visualising

The abstraction of numbers is a fascinating topic for debate and discussion and the root of much confusion in secondary school matehmatics students, not to mention many of the worlds adults! As teachers, our challenge is to keep developing successful activities that help people to develop their powers of abstraction and understand ideas in different ways. Visualisation is a key tool in this process and, as such, I am always on the look out for new, and successful old ways of building activities around the idea. In the digital age, it is easy to forget the value of physical manipulatives and practical activity. Here are five number based activities that invite students to create their own visualisations using manipulatives and practical activity!

	Multiple Factors Age 9+ Time: 1h. Students form interesting groups using their bodies to get a physical appreciation for factors and multiples. They then create a wide range of imaginative designs using counters, beads, collage etc. to represent numbers in terms of their factors for other groups to decipher. Finish with the multiple factors game. Be prepared for a lot of fun!
	Recreating Ratios This lesson requires students to produce a range of images for a given ratio. The aim is to draw out the equivalence of different ratios and how, and why, they can be simplified. The more creative and imaginative students are in creating different images to fit a given ratio, the clearer the true concept of ratio becomes (great opportunity for display work). It also provides a good lead in to sequences and graphs. Age: 11+  Time: 1h+
	The Art of Fractions Create some great art and display work whilst practising calculating fractions of different quantities. This activity involves the repeated splitting of rectangles into carefully chosen proportions defined by a single fraction and can lead to some lovely 'Mondrian inspired' pictures! Age: 11+ Time: 1 hr
	Visualising Indices 'Squared' and 'Cubed' can be explained by using 2 and 3 dimensions. The area of a square with length 5 is 52, volume explains cubed, so how can we represent 54? This activity explores visual representations of indices and draws on a little creativity! Age: 14+ Time: 1 hr
	The Rice Show This acivity is inspired by 'Of All The People in All the World' from 'Stan's Cafe'. Use grains of rice to represent different numbers of people! How can we make a pile of rice with 1,000,000 grains in it? Do the estimation then use some statistics to make a powerful display! Age: 10+   Time: 1 - 3hrs

Stop animation video!

This activity comes around year after year for me and I love that each time you do an activity you come up with new ideas about how to develop it. It is even better when those new ideas come from students! The activity is  Visualising Indices. This year, one of my students decided to create a stop motion video to represent a particular power. I think this is a brilliant development of the idea and have added the video to the activity page and embedded it below as well with some more images of this years efforts and a link to a  school blog entry about it! The whole thing also got me interested in the use of stop motion video for this type of thing and so I got straight in to  iStopmotion for iPad and iPhone which is fabulous and highly recommended. The iPad runs the show and works with the iPhone as a remote camera so that the camera can stay in the same place.

Stopmotion video

Some more photos

Are computers a natural medium for mathematics?

One of the reasons I both love and hate twitter! I am casually flicking through some pages over breakfast and I happen on this blog post from Dan Meyer. In fairness the blog post seemed mostly to point out how helping mathematics education has not ever risen to the top of silicon valley's priority list. Whilst this is an interesting question, it was the question phrased in the title above that caught my attention. I love this because it is great when some one else's writing makes you stop and think - I hate it when the question pre-occupies your mind when you are trying to do other things. The result is that I am writing this long after I should be asleep, getting ready for tomorrow. Anyway, I think the below can stand alone, but can be put in to context by reading the blog post linked above. This was the response I left on the blog post.

As #57 says, who is still reading! I find though that putting these thoughts and reactions in writing is mostly only for my own benefit! In this case, it is beacuse, whilst I understand and sympathise with the general view being expressed, I think I actually disagree with the statement about 'natural medium'! I read most of the responses and scanned the rest but the response from David Wees came closest to my reaction when he said '

'There are some tasks for which computers are perfectly suited in terms of mathematics'

and

'What you have suggested is that they are less than ideal for the quick communication of mathematics, and for deeper assessment of what mathematics students understand.'

Regarding the first point....

My relatively short teaching career (13 years) has spanned 'almost no access to computers' to 'working with a one to one program at my current school'. There is no doubt in my mind that computers have had a hugely significant effect on the way mathematics can be taught and, more importantly, discovered, beacuse they provide a considerably more natural, able and versatile medium. A lengthy description of cases could follow, but I will limt myself to just a few...

Dynamic geometry, as has been mentioned by some already. This tool has done amazing things for helping teachers to create opportunitites for students to make discoveries on their own and thus enage with mathematics. It can go beyond the teaching of geometry as well. Examples of activities  Indestructible Quadrilaterals,  Discovering circle theorems,  Making a trig Calculator allof these activities involve students creating mathematical objects in the medium of dynamic geometry.

Graphing software - largely by labour saving, but also through dynamic functionality - these tools as well have created new opportunities for exploring relationships. Examples of activties  Olympic Records,  Straight line Graphs

Data Handing - This has come to life through computers with access to real, live data, the functionality to collect it and the ability to process it. All this means that the nature of data handling tasks can now vary in new ways. (I will not say 'more mathematical ways' although that it is what I think.) Examples of activities  Predict the future,  Dynamic Scattergraphs

As suggested, I could go on and will in my head!

Regarding the second point....

Yes I agree that progress is slow on more able and intuitive user interfaces for communicating mathematics. I think that this has worked in our favour as teachers though. For example, taking the fractions, modern calculators now make it much easier to input and work with fractions than it used to be and this may have resulted in a poorer understanding of what fractions actually mean. The fact that computers dont find it easy to accept fractions means that users have to think about what the fraction actually means in order to input it. A fraction is easily written on a piece of paper with no understanding of its meaning.

Likewise, when programming with dynamic geometry (and I do consider constructions a type of programming), there is no 'rectangle tool', in order to construct one you have to know that a rectangle is made by two pairs of parallel sides intersecting at right angles. When you program it correctly it will always be a rectangle regardless of which points are moved. The process of drawing a rectangle on a piece of paper is not at all the same.

In summary, dont get me wrong, I estimate that computers are used for about 50% of our lesson time and I am a committed believer in variety of tasks that range from the pencil and paper, to the practical, to the virtual. That said, I am a passionate supporter of what computers have done for mathematics education. I am also a relatively new blogger and always have a sense of fear when 'submitting' such responses. I think most bloggers understand that expressing your views and reactions is the best way to develop them, so thanks Dan for making me think! Apologies if I have missed the point somewhere along the line, I feel better for writing this down either way.

Optimising understanding with 3D shapes - A quick idea for playing with cubes and cuboids

This blog post is just a quick way of sharing an idea that is developing. I love it when ideas pop in to your head as you are teaching and you just go with them. Its a risk, but some times brilliant things happen and great ideas are born. I tried this in class this week and it got me thinking about a series of questions and challenges that could be really engaging and help students to get to grips with 3D shapes.

Objectives

• For students to play with different nets for a cube.
• For students to explore the nets and thus the surface area of cuboids.
• For students to consider what is an appropriate measure of 'bigness' and thus consider the idea of volume.
• Students think about 'Optimisation'.

The Task

• Students are given a piece of A4 card from which they must do the following;
• Draw the net, cut out and make a cube 5cm by 5cm by 5cm
• From the card that remains, students must draw the net, cut out and make the 'biggest' cuboid that they can!

The thinking!

The following are some of the thoughts and observations related to this activity that came out as we did it. They are in no particular order!

• Students straight away wanted to know if their nets had to be a single piece - i answered yes so as to stick with the definition of a net and make it more of a challenge. I was pleased that students seemed to recognise a key point early on.
• Students had to think about the different nets for a cube so they could choose one that left a maximum area of card for the cuboid. 
• What does 'biggest' mean? And so surface area and volume are born as ideas!
• There is no substitute for building 3D shapes for understanding how the nets work and which sides have to correspond.
• Students are thinking about optimisation at an early age! A super concept to introduce.
• Most importantly, students were engaged from start to finish with solving the problem and all of the objectives listed above.
• On a technology note - I had just had a new document camera delivered to my room and it was perfect to be able to use it to show the class all the cuboids up close so we could decide which one was the biggest.
• The ensuing debate was terrific.

The more I think about this, the more possibilities I see and I want to go away and devise a series of questions involving more complex shapes! Watch this space - I plan to post a resourced activity on this idea in the future! Thoughts and suggestions welcome!

Just some thoughts on the topic!

In this entry I am writing down some of the thoughts I have following two things that I have paid attention to this week. The first is the TEDxLondon event on the theme ‘Education Revolution’. The second is Carol Vorderman’s report to the UK government on the state of mathematics education in the UK (BTW this is interesting reading wherever you live and work). 

As in most cases with blogs, I suspect that the primary beneficiary of this exercise will be me! Articulating thoughts, reactions and emotions in to coherent statements takes me far too long, but can be satisfying. Importantly, I reserve the right to change my mind in the future based on subsequent thoughts and reactions!

The current State of Education

I have a great fear that those who speak so clearly, well and influentially on the current state of education are not familiar enough with it and thus not qualified enough to do so. Whilst this does not invalidate their arguments it does begin to undermine them. Far too many sweeping statements are made about the terrible things that happen in current education. Most are very careful not to blame teachers, but rather government and micro management, but all tend to imply that teachers follow enforced strategies blindly.... most teachers, from my experience, are educators and capable of taking directives, standards and tests etc in their stride, whilst remembering that their primary role is to provide an education for their students. As such, what happens in classrooms is seldom the blind delivery of someone else’s plan. Maybe I am lucky, but that has been my experience of teachers to date. For interest you can read here about our philosophy on creating 'mathematical experiences'.

Sage on the stage

Much is spoken of how the ‘Sage on the Stage’ idea is outmoded and it is time for change. This relates to what I have said above. I ask, who is teaching like that? There is no way I could ‘lecture’ for the 19 hours a week I spend with my classes. Apart from being pretty dull for all of us, I would not have the energy. I just don’t think this is happening. One of my colleagues, @russelltarr, pointed out the irony of the format of TED events in this context and I was reminded of this from Jeff Jarvis on the same topic. It is not rocket science, but worth remembering that variety is a huge tool in sustaining engagement and interest. This is as true of a group of adults as it is students. Sometimes I enjoy listening to the sage on the stage – sometimes I enjoy trying to be it, but this makes up a small proportion of what happens.

Revolution/Evolution

I am increasingly leaning towards evolution in this debate. Again, we could easily get caught up in semantics here, but I think I have achieved some clarity on this point. Education – that which happens in schools – has a constant need to ‘evolve’. From my experience it does so all the time. If it didn’t, my job would be easy but dull. Constant reflection, openness and willingness to engage with new ideas and the views of others are key ingredients. How individuals are judged by the wider world as a result of their ‘education’ is quite possibly in need of a revolution. This is perhaps best illustrated by the fact that what happens in schools evolves despite the stranglehold exam boards have on the notion of ‘terminal assessment’. Many courses offer a very sound philosophical basis and then use a horribly blunt assessment tool that does match that philosophy. For example, while schools are embracing technology, we still seem light years away from technology being used in assessment. Mathematics education and technology are deeply interwoven, but students still sit terminal exams without a computer. Revolution is required at that end to allow the natural evolution to happen in schools.

Success and Failure

Related to the above is a need to revise perceptions of success and failure. It is true that success in most schools is still measured mostly by academic achievement and this really does need to change. I believe that lots of schools do a fabulous job of providing a broad range of opportunities for students to succeed but still there are lots of students who leave schools as very able, broad, caring and considerate people with little to show in the way of ‘Official success’. Sure exam results open doors, but being a successful person is about so much more and I would like to see teacher references for students carry a lot more weight than they do at present. I could tell you more about my students than any set of exam results.

Play Vs Work

A colleague tweeted during the TedxLondon event that ‘School leaders need to learn not to see playing and learning as mutually exclusive’ and I could not agree more. I do subscribe to the point of view that ‘play’ is a fantastic way to learn but want to be careful not to imagine it as the only or the best way of learning. It is important here not to get caught up in semantics and I think the word play can be defined very broadly, but on its own can easily be misinterpreted. I much prefer engagement as a term and I base this on my own experiences as a learner. On the one hand we can see that students will be more likely to engage when what they are doing is not perceived as work. On the other hand would it not be better to change the perception of ‘work’?

Technology

There is far too much to discuss here to even think about adding a ‘paragraph’ that sums it up, so I will try and do it in a sentence. Technology it seems is generally considered, toylike, frivolous, flashy, dangerous and unnecessary etc until proven otherwise. This needs to be reversed!

For example - If you are a player in the ‘twittersphere’ then you will not get this impression because of the obvious bias of those most likely to engage with each other about education through social media, but Facebook and Twitter are still dirty words in most educational establishments. I am not unaware of the risks, but it is unbelievable to me that we take the ‘communication tools of choice’ for most of our students and brand them too dangerous and frivolous to use for education.

As suggested already, there is so much to discuss here regarding hardware, software, access and philosophy, but the world around us will change and move on and schools and education simply cannot afford to be left behind.

Cultural Importance….

The Vorderman report does talk about the differing cultural importance of mathematics in different countries and concedes that we can’t just take the methods used in other places and expect them to work. It would be a lot to expect that the report offers suggestions for how we can change the cultural perception of mathematics, but I feel the point is rather glossed over. I can’t speak for other subjects but I feel strongly that the general understanding and perception of mathematics as a subject is often misguided and its cultural value is very low in the UK. These two things are of great significance to the future of mathematics education.

Talking and Doing…

I am in danger of finishing this piece ironically. I have been prompted to write this by both the TEDxLondon event and the the Vorderman report on the state of mathematics education. Whilst this has been very good in helping me reflect on some of these issues, I can't help but feel that so much is spoken and written about what needs to change in education, whilst most teachers are in the practice of simply doing it! On that note I am going to stop writing and do some planning.....

Pick it up and move it around!

This is a brief reflection on my experiences this week in thinking about the merits of physical Vs virtual manipulatives. Working in a school with a one to one laptop programme always invites You to think about what a computer can add to an experience. The number of virtual manipulatives available is staggering and some of them have really helped the evolution of teaching methods in mathematics. Having that programme really allows us to take advantage of them. With that in mind, please don't consider this blog post an 'anti technology' entry.

Task design happens in a number of different ways. It is probably fair to say that most of us start by thinking about what it is we want to teach. At that point we either go looking for existing resources or start thinking of new ways to do it. Being that it is the start of a new term, I am prone to the latter given the energy I have after a summer break. I would like to think that I am disciplined enough not to overlook some fabulous existing ones either. The trouble with coming up with new ideas is that they usually involve the creation of new materials! As an optimist, I will always go and look on the internet to see if what I am looking for is already out there, but this is really the wrong way round. The internet is like a garage sale - go in search of something in particular and you are likely to be disappointed, go with some money in your pocket and you will probably find something useful!

Having done some work on introducing the concept of tree diagrams, I decided that what I wanted was some online, interactive tree diagrams where the probabilities were listed but not put in the right place so that the task was to move them into the right places! This would just remove one area for possible error and in the early stages of an idea, I find it a very useful checking mechanism to know that if you have one left over that doesn't make sense then you may well have made a mistake. Anyway, I looked and I looked and I couldn't find it anywhere - I wondered about how long it would take me to program something like this using flash, but resolved that this was not the best use of my time at this time of year. I then decided that I really believed that in this case the 'physical manipulative' would be better. (I am not sure it saved me any time). A consequence of having technology at our disposal is that the benefits of the physical manipulative can be overlooked.

The result was the creation of this resource  Probability trees in which students work in groups with cut out bits of paper to solve problems where they have the answers and just need to put them in the right order. The physical manipulative really helped the group work aspect because more than one person can be involved in the arranging, and the absence of the computer screen allowed both more space and encouraged conversation and reasoning between the group members. In my search I had hoped that I would find something that was 'self-checking' so that students would get instant feedback on their efforts. This is a principle that can be very helpful, but is not without its faults. When no answer is instantly available, students need to reason with each other and reach some kind of consensus before settling on an answer. None of this is to say that the physical manipulative was 'better' in this context, but rather to say that there are lots of benefits to experiences based around physical manipulatives. Below are some pictures of the bits of paper!

On this note, the following are just a few examples of similar activities where physical manipulatives are used.

 Meeting Functions

A classification exercise with different functions, domains and ranges.

 OXO

Using Multilink cubes to relate algebraic sequences to physical situations.

 Quadratic Links

Linking the different representations of a quadratic function together.

Using multilink cubes to look at linear sequences!

This blog entry was written as part of a reflection on the ICTMT 10 conference held in July in Portsmouth, UK.

CAS for schools is tricky decision at the moment, but I must confess that this conference has helped me to narrow down some of those decisions.  Mathematica,  Maple and  TI-Nspire are the names that seem to rise to the top and I am interested in all three. The first two are probably more capable then we actually need for secondary school. At our school, we still use and enjoy an old version of Derive that we have on our system at school and have been looking at ways to get TI Nspire for all our computers. We have been playing with single user licenses and like what it does, but we don’t want to invest in the handheld technology. We have the luxury of 1 to 1 computers and I am still of the view that advances in smartphone technology will take over handheld calculators as soon as exam boards catch up. This is no small point however, and the exam boards are the ‘joker in the pack’ as one teacher put it, for CAS enabled calculators as the leap into allowing smartphones or effectively computers into exams is huge and one that will cost exam boards a large amount of time and money and so they are likely to hold out as long as they can. More frustrating than this is that TI appear, not surprisingly, much more interested in selling the handhelds than they do the emulator software. As a result, I feel priced out of all three of those top runners, TI, Mathematica and Maple.

I was then, fascinated to learn more about the development of the freely available  Maxima from Chris Sangwin. Even better than this is the collaboration with Geogebra to integrate maxima thus creating one great tool that will combine so many of our needs in secondary schools. CAS, graphing and dynamic geometry all integrated and linked! A beta version of this software is available  GeogebraCAS and the official launch is due this month. I for one am looking forward to it. Of course, the next discussion is to think in more detail about how to get the maximum benefit from this software in the classroom. See Investigating Quadratic Factors and  Finding Factors for examples of investigations, using CAS (with "how to" video help for Geogebra 4.0 CAS, WolframAlpha, TiNspire Derive), that offer the opportunity for students to 'discover' for themselves the concept of factorising. Watch this space for further applications ....
 

This blog is a short reflection on my experience of the ICTMT10 conference in Portsmouth 5 – 8 July 2011. I will not attempt a complete review of the conference for two reasons. Firstly, it would be impossible because of the number of parallel sessions on offer and secondly because I would not do it justice. As a secondary school mathematics teacher, the most useful thing for me to do is record some of the highlights and, perhaps most importantly, the resulting points of action that I have. Rather than publish this all in one go, I will publish a series of blogs over the next few weeks on some of the main themes. The following serves as an overview of what may follow.

It is worth noting from the start that the conference is aimed at a mixture of educational researchers and practitioners from secondary and tertiary education and the sessions are a mixture of keynotes, presenting research and workshops. As such it is a rich mixture of possibilities for all kinds of people. I attended the previous conference, ICTMT9 in Metz, 2009 and I would be lying if I said I wasn’t a little disappointed that there were quite a few less secondary school teachers in Portsmouth and a considerably smaller North American presence. That said, there was a truly international feel to the conference and enough of my peers for some really rich exchange, not to mention some good company and good times.

There were keynote speeches from Richard Noss, Paul Drijvers, Colin White and Collette Laborde all of which were thought provoking. One theme that seemed to run through all of these was that there seems to be a general disappointment that progress with ICT in Mathematics teaching has not achieved the potential that was thought to exist 20 - 30 years ago. Having only taught for 13 years I am not able to comment on this, but do think I can reflect on some significant changes during those 13 years!

There was research presented on the benefits of using technology as a modeling tool and motivator, particularly in tertiary education. One example was given in the study of sports science where students did not necessarily have a strong mathematical background, but could make progress with modeling tools such as 'Matlab'.

There was a particularly interesting workshop on students making videos of themselves solving problems and using these videos of getting students to reflect on their 'working out' and the stages they went through.

There was much discussion of various electronic assessment tools and, in particular, their ability to give relevant feedback where mistakes were made. This technology is clearly advancing and does have a place in secondary schools.

Handheld technology, mobile apps, GDCs and screen sharing software were all on show and this has prompted me to think about how to move forward with this in my school.

Which CAS technology should we be embracing? This is a tough call but I became aware of some very interesting developments with Geogebra and Maxima, that may make this choice a little easier.

With London 2012 on the horizon, sport and mathematics was a running theme through the conference and without too many concrete ideas, I am determined to think about strengthening this link in my school.

We were treated to an excellent talk from Richard Noble about the 'Bloodhound project' that made me think about the whole land speed thing in a new way. As well as this we had dinner on the HMS Warrior - a 150 year old battle ship restored in all of its glory to round off an excellent week.

As I said, I could not do the whole conference justice in one blog entry and so only aimed to give a brief summary. Many of the issues deserve to be returned to in future weeks in more depth.

We, from the International School of Toulouse, offered 2 sessions during the conference, the details of which can be found here;

 Future curriculum 'Use of technology to significantly enhance the development, engagement and skillset of students in the third industrial revolution.

 Animated Questions 'New types of question afforded by developments in technology'

I recently attended the ICTMT10 conference in Portsmouth. Amongst the many presentations was a thought provoking keynote from Richard Noss in which one of the themes was considering the merits of different representations for complex and abstract mathematical ideas. He discussed a particular example of a ‘typical’ sequences question in which students were expected to draw the next two patterns of a given tiling sequence and deduce the algebraic generalisation that gave the number of tiles required for any given term of that sequence. The discussion went on to look at ‘typical’ responses where students will spot a term to term rule and mistakenly try to express this algebraically rather than considering a position to term relationship. It was then noted that teachers will try to help students move on by asking them to predict the number of tiles for the 100th term. This may achieve some success, but will draw, from some students, the reaction of counting the number of tiles in the 10th problem and multiplying it by 10.

The example was one of many that alluded to the notion that numbers have different meanings in different contexts with different representations that is both testimony to the richness of numbers and algebra and the ease with which a student can be confused. This was amusingly demonstrated with a picture of two buses, one the number 9 and the other the number 18, which was twice as long, but half as high!

Sticking with the first example and being aware of the problems based on my own classroom experience, I wanted to add that another factor that contributes to general confusion is the lack of ‘incentive to generalise’. For my money a good activity will stop students from asking ‘Why do we need to do or learn this’ not by answering the question directly, but by engaging students with activity and incentives that pre-occupy them before they think to ask. A simple example of this is ‘Sudoku’. Why does anyone need to do that? I could come up with lots of good answers but millions of people worldwide never ask! With this in mind I wanted to share some examples of activities that I think provide that engagement and incentive to generalise.

These activities involve real physical situations in which students are actively involved in solving problems that can be modeled algebraically. The puzzle provides engagement and incentive by appealing to a natural puzzle solving instinct and better yet, it provides a physical, structural link from which students can generalise, thus taking the edge off the otherwise abstract nature of the process. Each of the links below is to a teaching activity ready to go for classrooms or ready for twisting and turning by teachers and students alike. The activities come with descriptions, resources and teacher notes to help think about how to get the most out of them.

 The Tower of Hanoi

The classic puzzle that can be modeled by an exponential sequence which means that only a small increase in the number of discs would make the problem unsolvable within our own lifetimes! Get a real one and let students physically do it themselves.

 Frogs

Truly one of my favourite lessons year after year. There is so much going on with this puzzle, not least of which is a nice example of a quadratic sequence. Get students hopping and sliding around the room to get under the skin of this one.

 The Sliding bus puzzle

Another good one for physical activity and packed with a couple of nice surprises. It’s a linear sequence that can be broken down into smaller ones. It is really accessible and good fun!