I am not joining in with pseudo-context bashing

I'm not. I mean, I get it, there are some lame and damaging attempts at putting mathematics in to context. I understand the problem and am both equally amused and frustrated by some of the things that come up, but sweeping pseudo-context bashing is too easy. As I see the thread streaming past I am reminded of 2 things..

1. Einstein's wonderful quote about mathematics and reality - As far as the laws of mathematics refer to reality, they are not certain, and as far as they are certain, they do not refer to reality.Ok so there is a lot there, but the opposite of easy pseudo context bashing might be to claim that - in the light of this quote - all context is questionable. That position would be equally flawed.

2. In part of my brief research in to context I explored Ole Skovsmose's 'An invitation to critical mathematics education' and his 6 milieu for activity that can take place in the 'abstract', the 'real' or the 'semi real'.The semi real has a role. 

The only problem with pseudo context is when there is an attempt to convince students its not a pseudo context. We should always think about it, but being sweepingly dismissive is usually a mistake. The definition and role of context in mathematics is a rich area for investigation. For that reason, it is important to be clear that I am not dismissing pseudo-context bashing, I am just not joining in!

What can I safely assume......

Here are some thoughts I have been having about running sessions with teachers. Conference sessions, training etc...

'What can I safely assume about my audience?'

1. That they will have something to offer in return and will appreciate the opportunity to exchange.
2. There will be people there with wildly different experiences from my own and each other.
3. That some, many, maybe all will have at least seen/crossed paths with the key ideas the session is focussing on ...
4. Someone will be frustrated with something I said or did. They will appreciate at least the acknowledgment of that.
5. The list of things I can't assume is much longer......

And now, to think about what assumptions I can safely make about each of the classes of students that I will see on any given day.......

Creativity and Mathematics

Something I found my self saying.....

'It is less a question of identifying when one can be creative with mathematics than it is recognising that mathematics, and doing mathematics, is inherently creative.  For a given mathematical scenario, success is less about 'Knowing what we are supposed to do' and more about speculating on things that we could do. Doing mathematics is about taking bits of mathematical information, and putting them together in various ways to create new bits of mathematical information.' The very essence of constructing knowledge in mathematics is a creative process.

Algorithms.....

Gut Reaction - Just saw it and wanted to write something down!

Just picked this up from TED ED. I enjoyed it and thought it would be useful! It made we wonder though about using simple examples to explain complex ideas. Will not most people watch this and say - 'I don't need an algorithm for that?' An example that showed how an algorithm worked to do something more complex might be more convincing. Don't we do this all the time in Maths classes? Take things people understand, then explain them in a different, more complex way, so as to demonstrate a method that will be more useful for more complex problems, even though the method is cumbersome and inefficient for the example we use.

I am sure there must be research on this so i will look for it now, but I have a feeling that I want, more often to show the power of a tool during its initial demonstration so that it answers a really difficult question straight away, rather than one we had already answered much more efficiently......

At least I know what I am thinking about!

Qama Calculators

The calculator that only thinks if you do!

So we have just invested in a few of these Qama calculators to play with. I think I just needed to see one for myself and see how it worked. The basic logic is really quite simple. You enter a calculation and press 'equals'. Then the cursor moves to a new line where it expects you to make an 'estimation' for the answer. If the calculator is happy with your estimation,then it will give you the actual answer. If it is not, then you must try again. What a beautiful idea! It has clearly been a terrific effort from conception and design to manufacture and distribution and these are now really easy to get hold of a relatively inexpensive. The question is to think really carefully about how we might choose to use them in mathematics education. There are a few questions that people might think about...

On investment

1. Would we consider a wholesale swap and ask all our students to get one of these?
2. Would we like a situation where students had one of these alongside anoher traditional calculator?
3. Would we go for something like a class set that can be brought out for particular activities?
4. Is it just something we might recommend for some students?

Well, my usual take on something like this would be option 2, because this leaves all options for use open. You would need to be pretty certain that they were going to get used before you asked parents or schools to make an investment.

On activity

If this was the default calculator, then we imagine that there would be long term improvement in estimating ability and the asociated number sense, but it is worth thinking about the implications. Calculators are often a bridge that allow an activity to focus on particular skills without letting calculating skills hinder progress.

1. Would permanent use of these calculators slow down activity? Does that matter?
2. How would we adjust our teaching style to accomodate for the extra stage of involvement?
3. What would be the implications fo increasing the emphasis on estimation from an early stage?
4. Is there potential for thinking about specific activities that might make the best use of this calculator?

I think there is a need for experimentation here and possibly picking a class to trial them with one way or another. The first question here helps me answer the first one above, in that I dont think I would want it to be a permanent replacement. I do think that it would interupt some teaching and learning activity. I also think there will be a significant implication for teaching style.
Really importantly and potentially one of the most exciting implications of using these calculators is likely to be the increased emphasis on estimation as a skill. There are so many reasons to encourage this, but my favourite is because of the potential impact on understanding.

Specific activities

I think that these calculators could introduce a whole new level to activity design. Here are a few ideas that have popped in to mind in the first few days I have had this calculator....
Percentage error - I am really curious about the margin of error that the calculator is accepting in different contexts. My colleague tried log120 and was denied with an estimation of 2.1 Quite demanding I think, but there is a lot of potential to investgiate different types of operation and the percentage error allowed.
Least guesses - I like the idea of some activity where students have to try and estimate difficult operations in as few guesses as possible.
Trigonometry - I had fun estimating some trig ratios. I found myself thinking about the ratio between different sides of a traingle and how it would change as the angle increased. What a great way to encourage students to think about what trig ratios actually mean.
Again - this is just a start of lots of ideas that are bound to come from playing with these calculators.
Thanks Qama for these - a really exciting development.

More Popular Math

More Popular Maths

I have just watched my first episode of 'School of Hard Sums' and really enjoyed it. I dont live in the UK and so am playing catch up. It is another great example of what I put under the heading of Popular Maths and I am delighted that so many people are making an effort to get more people watching, thinking and talking about maths. Terrific. I want to watch more for sure and certainly before I make any general comments about the show. As ever though, my first thought is about how this sort of stuff can be used in schools. As I have said before, I always think the challenge is how to take this sort of media and focus on the maths involved and making this in to some sort of activity. Here is the you youtube clip I watched.

My focus is on the dancing problem. In short, if you are stood still in a dance hall and everyone kisses the person they are closest to, how can you position people so that you get the most number of kisses and what is the most number of kisses you can get? I liked this problem because I thought the context was fun - I can see playing musical statues with a class a few times. Secondly, I was interested in the three aproaches used to solve the problem, all of which were apparently different to mine. This is always the sign of a rich problem for me. As ever, it makes it more difficult to control which 'learning objective' can be taught in a given lesson, but I think that is a sacrifice worth making for the potential to get students using mathematical reasoning, testing, evidence gathering and so on. Perhaps also, there could be an activity that offers the different explanations as a starting point and asks students to discern between them. Either way there is a lot of potential and now I have more things I want to watch, do and plan! It is frustrating (lack of time) but exciting that there seems to be a never ending stream of possibilities in this profession! Thanks to the School of Hard Sums team - I am looking forward to more.

Ideas to activities

'An essential teacher skill'

So how many mathematics teachers have had the experiencing of sharing a really cool bit of mathematics with their students only to be disappointed that their students didn't get caught up in the magic like they did? I know it has happened to me and these are some of the more salient moments of teaching. Mathematical ideas will grab different peoples attention in different ways because we are all different and this is just another factor that makes mathematics teaching such a challenge. In this blog I am trying to describe the process of taking a fascinating mathematical idea and turning it in to an activity so that it a) gives students an opportunity to experience the same fascination that we did, b) engages students in mathematical thinking and c) leads to some useful conclusions. As is so often true, I am going to find it easier to explain with an example. I am going to start by looking at the prime number visualisations shown here by the  Ptolemy Primitaves application. This is just one of many awesome tools and ideas out there related to prime number visualisation. The video below gives a quick overview of the application.

So, if you haven't seen the application yet then you are probably thinking just how cool it is. (I am making an assumption about people who might be reading this blog post!). In the comments below, you can see people who are saying things like 'I am using this with my students' and this really sums up the issue I am trying to present in this blog post. Just how exactly are teachers 'Using' this with their students? I am sure there are loads of good answers to that question, I am just making the point that once we have found things like this then, our next and often most challenging job is to create activities that allow students to experience the mathematics. I have shown this to students before and they do seem interested and will play with it for a while, but without any associated activity there have been no real moments of mathematical thinking and discovery. It is almost like telling a joke in wrong order with the punchline first. This year I decided to do something I have wanted to for a long time with this wonderful app. I started by asking myself what things I would want my students to get out of playing with this idea Here is a list. I wanted students to.....

• Above all, be engaged in mathematical activity.
• Discover the 'coding' of the diagrams for themselves and not tell them it was anything to do with prime numbers.
• To see for themselves how numbers can be expressed as products of their prime numbers.
• To reason with each other about why the above is true.
• To understand how and why there is more than one way to show the same numbers in this way.
• To understnd how they could create a visualisation for any number this way.

For any of the above things to happen there needed to be an activity with an objective as an entry point for students and, not being able to settle on one, I created three that I thought could be used together or could be used to differentiate.

1. I captured an screen shot of each of the first 30 intergers, jumbled them up and asked students to work out which image should go with which number.
2. I made a similar sheet of 30 images, but made it clear that there were not 30 different numbers. Numbers would appear either 2, 3 or 4 times in different ways. The objective here was to work out which pictures represented the numbers and then what the numbers were. The order in which those things happened was for students to decide.
3. A third sheet was made of 30 different numbers (but not the first 30 integers) and the objective was to work out what the numbers were.

Each task was prepared with some prompts to help students get started. In the end , I decided to go with the 3rd sheet for everybody and gibve out the 1st and 2nd as and when I felt it might help a group or individual. I was completely delighted with the result which achieved, in different measures for different students, all of the objectives set out above. What I find most important is the reasoning that students use with each other when arguing a particular conjecture. The conversations were rich, the students were engaged and my role was to step in as and when required to make sure these things were true for as long as possible.

There is a rich irony in that I had a great fear that students would crack the code by working out that they could simple count the black dots. It is ironic because, for the activity to go well, I needed students to search for something more complex than a relatively simple solution that is just in front of them. Although it alone would not completely crack the code, it would have taken the sting out of the activity somewhat.

Thinking about and creating what seems like a simple activity took a long time, but I find this to be my favourite kind of work. The resourecs I made can be found here 'Prime Pictures'. Returning to the title of the blog, this, for me, is one of the very key elements of mathematics teaching - turning great ideas and tools into real activities that help us to engage students in mathematics. There are so many other examples of this and the one below is what is on my mind at the moment.

I have always loved this problem I recently come to know as  Curry's paradox. This is nicely explained through the link. This is such a lovely bit of maths and I had never considered the range of cases to which it could apply. I read the link with relish and enjoyment and am left thinking 'mmmmm how can make this in to a great activity for students?'. It must be possible! (image below taken from 'Cut-the-knot')

Happy creating!