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In July 2009 Clare Franklin, a sixth former from Hills Road Sixth Form College in Cambridge, was offered a Nuffield Bursary to work with NRICH and Steve Hewson on the development of the stemNRICH pages. The challenge offered to Clare was to take lots of the problems from stemNRICH for a test drive, write up solutions and give comments for their further development. In short, Clare was asked to describe what the problems felt like from the point of view of a student in the middle of their A-levels. Prior to starting, Clare had relatively little exposure to the sorts of rich tasks offered by NRICH. Steve asked her to log her progress and record her thoughts and feelings about the mathematics she encountered so that she could give advice to students themselves about to embark on a diet of rich mathematics! Here are Clare's thoughts.....
As a student about to enter Year 13, I was excited to be given a Nuffield Bursary Project placement with NRICH in Cambridge. My project, which lasted 4 weeks, involved working alongside five undergraduates. We had to go through the stemNRICH problems and give feedback about about which problems worked best, as well as writing up solutions. Although I was looking forward to the challenge, the first morning when found myself sitting at a table, surrounded by undergraduates, was really daunting. We had three packs of problems (chemNRICH, Core Scientific Maths and Advanced Scientific Maths) to work through which were so unlike standard textbook tasks . When I opened the parcel of chemNRICH problems and began to read through 'Cobalt Decay', I remember the feeling of complete panic; I wasn't used to being completely stuck and I didn't know what to do about it. Four weeks on, I'm now much more relaxed and part of that definitely comes from getting used to a new, much more open, way of thinking. Who knew that stemNRICH would be so much fun?
I have to admit that before I started the problems, I was sceptical about how much they could really improve my maths skills. After 13 years of school, how was working through a few questions going to noticeably develop my competence at problem solving? Well, having spent four weeks working through them, I now acknowledge that I completely underestimated stemNRICH, because the problems really have subtly changed the way I approach questions, forced me to consider my own thinking much more clearly and helped me to develop generic techniques that I could apply to almost any problem. I now feel much more able to tackle new problems with confidence. I know just how intimidating new sorts of problems can seem at first, but I hope that by sharing some of my experiences, you'll be encouraged to crack on with the problems because they are definitely worthwhile.
The most important thing to realise about the NRICH tasks is that they are different from standard A level work, in that they are much more open, more thoughtful and less repetitive. The problems are not designed to help you practice something you've just been taught, but rather to encourage you to be genuinely interested and to allow you to solve unusual problems which are, in some cases, extremely challenging. Partly because of this, getting started can be really scary. I certainly found it so, and on my first morning of doing the problems I tried to stick to the questions that were most similar to the extension work I was familiar with, shying away from any which I found too vague or too open. Questions with which I could obviously use algebra, such as 'Temperature' and 'Peaches Today, Peaches Tomorrow', both from the Core Scientific Maths section, were easily my favourites then, largely because I was so relieved to be able to see how to start a problem. I would still strongly recommend these as starting points, particularly 'Peaches Today, Peaches Tomorrow', which becomes more difficult as the problem progresses and is therefore quite a nice way to gently immerse yourself in the NRICH way of thinking.
After the initial adjustment to the style of the problems, it becomes much easier to relax and to actively enjoy doing the questions. Choosing and completing problems which immediately capture your interest is definitely a good idea and the ones which I think do this best are the questions which are, scientifically, an extension of A level concepts. I would particularly recommend the pH questions in chemNRICH, which define pH mathematically, and then go on to get you to calculate various things which have a clear, real scientific meaning. I also suggest trying at least one more open question, as these can really help to make your thinking more adaptable. Whilst completing 'Approximately Certain' from Core Scientific Maths, for example, I was forced to accept that, as with all estimating questions, there was no 'right' answer, and that clear reasoning was the most important thing. Though initially, I found this lack of one definite answer difficult, there is no doubt that working through questions like this has made me more open to the idea that the process of completing the problem is more important than the answer itself. This gives the problems a wonderful sense of freedom because if you think of a particularly interesting related question, which is not explicitly asked by a problem, you are free to investigate it.
The difficulty rating on questions should not put you off, as different people will find different questions challenging; don't worry if you find a 'one star' problem really difficult, because there may well be a 'three star' problem that you find much easier. The best advice I can give about completing the problems is simply not to panic. The problems are designed to make you think and not only is it therefore okay to be stuck, it is in fact a good thing. This can be really difficult to get your head around, as in A level courses you are rarely encouraged to work through an unfamiliar problem without being taught how to tackle it first, which means that not being able complete a problem immediately can make you feel inferior. When you first tackle the NRICH problems, you will get stuck because the problems are designed to make you stuck. The process of working through this will really help your problem solving skills, which is the ultimate aim of completing the problems. Just remember that these are not A level questions and there is no need to panic when you cannot immediately see the answer. Of course, this is a good thing!
It is important, as part of the problem solving process, to develop techniques to use when you're stuck. With so many questions to choose from, it is easy to give up when things get particularly difficult and move on to another problem, but try not to. Having initially taken this approach, I found myself flicking between questions, rarely completing one and making excuses to myself about why I wasn't sticking at it. I would definitely say now that it's not only much more satisfying but also much more beneficial to spend the time thinking about and exploring a question which you find particularly hard.
Of course, it is important to have some idea of what to do when you're stuck. The best advice anyone gave me about this was simply to clearly think through each of the following questions:
A Journey Into stemNRICH
In July 2009 Clare Franklin, a sixth former from Hills Road Sixth Form College in Cambridge, was offered a Nuffield Bursary to work with NRICH and Steve Hewson on the development of the stemNRICH pages. The challenge offered to Clare was to take lots of the problems from stemNRICH for a test drive, write up solutions and give comments for their further development. In short, Clare was asked to describe what the problems felt like from the point of view of a student in the middle of their A-levels. Prior to starting, Clare had relatively little exposure to the sorts of rich tasks offered by NRICH. Steve asked her to log her progress and record her thoughts and feelings about the mathematics she encountered so that she could give advice to students themselves about to embark on a diet of rich mathematics! Here are Clare's thoughts.....
As a student about to enter Year 13, I was excited to be given a Nuffield Bursary Project placement with NRICH in Cambridge. My project, which lasted 4 weeks, involved working alongside five undergraduates. We had to go through the stemNRICH problems and give feedback about about which problems worked best, as well as writing up solutions. Although I was looking forward to the challenge, the first morning when found myself sitting at a table, surrounded by undergraduates, was really daunting. We had three packs of problems (chemNRICH, Core Scientific Maths and Advanced Scientific Maths) to work through which were so unlike standard textbook tasks . When I opened the parcel of chemNRICH problems and began to read through 'Cobalt Decay', I remember the feeling of complete panic; I wasn't used to being completely stuck and I didn't know what to do about it. Four weeks on, I'm now much more relaxed and part of that definitely comes from getting used to a new, much more open, way of thinking. Who knew that stemNRICH would be so much fun?
I have to admit that before I started the problems, I was sceptical about how much they could really improve my maths skills. After 13 years of school, how was working through a few questions going to noticeably develop my competence at problem solving? Well, having spent four weeks working through them, I now acknowledge that I completely underestimated stemNRICH, because the problems really have subtly changed the way I approach questions, forced me to consider my own thinking much more clearly and helped me to develop generic techniques that I could apply to almost any problem. I now feel much more able to tackle new problems with confidence. I know just how intimidating new sorts of problems can seem at first, but I hope that by sharing some of my experiences, you'll be encouraged to crack on with the problems because they are definitely worthwhile.
The most important thing to realise about the NRICH tasks is that they are different from standard A level work, in that they are much more open, more thoughtful and less repetitive. The problems are not designed to help you practice something you've just been taught, but rather to encourage you to be genuinely interested and to allow you to solve unusual problems which are, in some cases, extremely challenging. Partly because of this, getting started can be really scary. I certainly found it so, and on my first morning of doing the problems I tried to stick to the questions that were most similar to the extension work I was familiar with, shying away from any which I found too vague or too open. Questions with which I could obviously use algebra, such as 'Temperature' and 'Peaches Today, Peaches Tomorrow', both from the Core Scientific Maths section, were easily my favourites then, largely because I was so relieved to be able to see how to start a problem. I would still strongly recommend these as starting points, particularly 'Peaches Today, Peaches Tomorrow', which becomes more difficult as the problem progresses and is therefore quite a nice way to gently immerse yourself in the NRICH way of thinking.
After the initial adjustment to the style of the problems, it becomes much easier to relax and to actively enjoy doing the questions. Choosing and completing problems which immediately capture your interest is definitely a good idea and the ones which I think do this best are the questions which are, scientifically, an extension of A level concepts. I would particularly recommend the pH questions in chemNRICH, which define pH mathematically, and then go on to get you to calculate various things which have a clear, real scientific meaning. I also suggest trying at least one more open question, as these can really help to make your thinking more adaptable. Whilst completing 'Approximately Certain' from Core Scientific Maths, for example, I was forced to accept that, as with all estimating questions, there was no 'right' answer, and that clear reasoning was the most important thing. Though initially, I found this lack of one definite answer difficult, there is no doubt that working through questions like this has made me more open to the idea that the process of completing the problem is more important than the answer itself. This gives the problems a wonderful sense of freedom because if you think of a particularly interesting related question, which is not explicitly asked by a problem, you are free to investigate it.
The difficulty rating on questions should not put you off, as different people will find different questions challenging; don't worry if you find a 'one star' problem really difficult, because there may well be a 'three star' problem that you find much easier. The best advice I can give about completing the problems is simply not to panic. The problems are designed to make you think and not only is it therefore okay to be stuck, it is in fact a good thing. This can be really difficult to get your head around, as in A level courses you are rarely encouraged to work through an unfamiliar problem without being taught how to tackle it first, which means that not being able complete a problem immediately can make you feel inferior. When you first tackle the NRICH problems, you will get stuck because the problems are designed to make you stuck. The process of working through this will really help your problem solving skills, which is the ultimate aim of completing the problems. Just remember that these are not A level questions and there is no need to panic when you cannot immediately see the answer. Of course, this is a good thing!
It is important, as part of the problem solving process, to develop techniques to use when you're stuck. With so many questions to choose from, it is easy to give up when things get particularly difficult and move on to another problem, but try not to. Having initially taken this approach, I found myself flicking between questions, rarely completing one and making excuses to myself about why I wasn't sticking at it. I would definitely say now that it's not only much more satisfying but also much more beneficial to spend the time thinking about and exploring a question which you find particularly hard.
Of course, it is important to have some idea of what to do when you're stuck. The best advice anyone gave me about this was simply to clearly think through each of the following questions:
- Why am I stuck?
- What am I stuck on?
- What do I know?
- What do I need to know?
By taking the questions one at a time and really thinking
about the answers, the next stage in the calculation, or at least
something reasonable to explore in order to get to the next stage
usually becomes obvious. I've particularly found that I'm stuck
more often because I have failed to understand the question fully
than because the problem itself is overly difficult. Having
recognised this, the first thing I now check when I'm stuck is
whether I know what the question is asking me to find (the last
question above). The questions will also help you to realise if you
need to look anything up, as you do for many of the problems. this
is fine. Above all, don't be afraid to experiment: trying a few
unsuccessful methods might bring you closer to a solution or help
you to discover something else interesting.
Occasionally I found that, even after asking myself the above
questions, I was still stuck. Here I would advise looking at the
online hint (which is linked to at the top of the individual
problem page) as this should help to channel your thoughts in the
right direction and, if you are still unsure, have a look at the
teachers' notes (again, linked to at the top of the problem page),
as these give key questions which will often lead you in the
direction of a solution. Though to begin with using these resources
felt like cheating or admitting defeat, I realised as I worked
through the problems that the hints are there to help move your
thoughts in the right direction (rather than to spoon-feed you the
answer). They can, and do, raise interesting issues which you may
not have considered.
Talking through a problem with someone was also something that
I found extremely helpful. Whether someone of a similar
mathematical level to you or someone more advanced, just talking
through your thought process and comparing it with theirs is a
really nice way to lead each other closer to a solution. It is also
really interesting, even if neither of you are stuck, to compare
methods, especially for the questions which have no definitive
answers, as these allow you to not only explain your method but
also to discuss what assumptions you've both made and therefore
which answer is most valid.
If I could, having completed the project, go back in time to
prepare myself for the challenges provided by the stemNRICH
problems, I would emphasise how important it is to enjoy them. The
best way to have fun with the tasks is to relax and explore the
problems, because finding an interesting method or investigating a
special, specific case can be much more satisfying than just
finding a solution and moving on. I certainly feel now that working
with the NRICH team has made me think a lot more carefully about
the way I've been taught and the way I learn things. I'm now much
more conscious of the need to have a go at problems which are not
just a harder version of what's been taught but which are genuinely
different. The problems themselves have given me a huge number of
invaluable skills and I'm also now much more confident in my own
ability to problem solve but, in some ways more importantly, my
interest in and enjoyment of maths has also been extended. In
particular the questions combine science with maths, and are
challenging in both; this is something I've not really found before
and find truly fascinating. In finding the stemNRICH tasks like
this, which have the ability to capture and hold my attention, I've
found a new, and subtly different, side of maths to love.
Thanks to Clare -- It was a pleasure to work
with you!